diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzecyc" "b/data_all_eng_slimpj/shuffled/split2/finalzzecyc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzecyc" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLet $K$ be an algebraic number field, and let $\\fok$ be its ring of integers. Let ${\\rm N}: K \\to \\bq$ be\nthe absolute value of the norm map. The number field $K$ is said to be {\\it Euclidean} (with respect to the norm) if for every $a,b \\in \\fok$ with $b \\not = 0$ there exist\n$c, d \\in \\fok$ such that $a = bc + d$ and ${\\rm N}(d) < {\\rm N}(b)$.\nIt is easy to check that $K$ is Euclidean if and only if for every $x \\in K$ there exists $c \\in \\fok$ such that ${\\rm N}(x-c) < 1$. This suggests to look at\n$$\nM(K) = {\\rm sup}_{x \\in K} {\\rm inf}_{c \\in \\fok} {\\rm N}(x-c),\n$$\ncalled the {\\it Euclidean minimum} of $K$.\n\n\nThe study of Euclidean number fields and Euclidean minima is a classical one,\nsee for instance~\\cite{lemmermeyer} for a survey. The present paper is\nconcerned with {\\it upper bounds} for $M(K)$ in the case where $K$ is\nan abelian field of odd prime power conductor. Let us recall some previous results.\nLet $n$ be the degree of $K$ and $D_K$ the absolute value of its discriminant.\nIt is shown in~\\cite{bayer} that for any number field $K$, we have $M(K) \\le 2^{-n}D_K$.\nThe case of {\\it totally real} fields is especially interesting, and has been the\nsubject matter of several papers. In particular, a conjecture attributed to Minkowski\nstates that if $K$ is totally real, then $$M(K) \\le 2^{-n} \\sqrt {D_K}.$$ This conjecture\nis proved for $n \\le 8$, cf.~\\cite{grs7, grs8, mcmullen}; see also McMullen's paper~\\cite{mcmullen} for a proof of the\ncase $n \\le 6$ and a survey of the topic.\n\n\nThe point of view taken in the present paper is to study this conjecture for totally\nreal abelian fields, and more generally give upper bounds for Euclidean minima\nof abelian fields. A starting point of this investigation is~\\cite{bayer} where it is\nproved that we have $M(K) \\le 2^{-n} \\sqrt {D_K}$ if $K$ is a cyclotomic field\nof prime power conductor or the maximal totally real subfield of such a field.\nThe present paper contains some results concerning abelian fields\nof odd prime power conductor. In particular, we show that if $K$ is such a field, then there exist constants $C=C(K) \\leq \\frac{1}{3}$ and\n$\\varepsilon = \\varepsilon(K) \\leq 2$ such that\n\\[\n M(K) \\leq C^n \\, (\\sqrt{D_K})^{\\varepsilon}.\n\\]\nIf $[K : \\bq] > 2$, then one may choose $\\varepsilon(K) < 2$. Moreover, we show that $\\varepsilon$ is asymptotically equal to $1$\nand that under certain assumptions $C$ is asymptotically equal to $\\frac{1}{2\\sqrt{3}}$; see Theorem~(\\ref{T:bound1}) for the precise statement.\nIn Theorem~(\\ref{T:bound2}) we obtain the bound\n\\[\n M(K) \\leq \\omega^n \\sqrt{D_K},\n\\]\nwhere $\\omega = \\omega(K)$ is a constant which under certain assumptions is asymptotically equal to $\\frac{1}{2\\sqrt{3}}$.\nIn particular, using these bounds we show\n\\begin{thm}\n Suppose that K is a totally real field of conductor $p^r$, where $p$ is an odd prime and $r \\ge 2$. Let $n$ be the degree of $K$ and let\n$D_K$ be its discriminant. Then $$M(K) \\le 2^{-n} \\sqrt{D_K}.$$\n\\end{thm}\n\\noindent In other words, Minkowski's conjecture holds for this family of fields.\n\n\\bigskip\nThe strategy of the proofs is the following. If $K$ is an algebraic number field, we consider\n lattices defined on the ring of integers $\\fok$ in the sense of \\cite{ideal}, \\S 1. This leads to a\n Hermite--like invariant of $\\fok$, denoted by $\\tau_{\\rm min}(\\fok)$, cf. \\cite{ideal}, Definition 9, and\n \\S 4 of the present paper. By\n \\cite{bayer}, Corollary (5.2), we have\n $$M(K) \\le \\left( {\\tau_{\\rm\nmin}(\\fok) \\over n } \\right) ^{n\/2} \\sqrt{D_K},$$\nwhere $n$ is the degree of $K$ and $D_K$ the absolute value of its discriminant.\nIn order to apply this result, we have to estimate $\\tau_{\\rm min}(\\fok)$. The main technical\ntask of this paper is to do this in the case of abelian fields of odd prime power conductor.\n\n\\bigskip\nThe paper is structured as follows. After a brief section containing the notation used\nthroughout the paper, \\S 3 describes the main results (theorems (\\ref{T:bound1}) and (\\ref{T:bound2})).\nThe rest of the paper is devoted\nto their proofs, starting in \\S 4 with a summary of some notions and results\nconcerning lattices and number fields, and their relation to Euclidean minima.\nSuppose now that $K$ is an abelian field of prime power conductor. In \\S 5,\nwe construct integral bases of $\\fok$.\nIn \\S 6, these bases are used to describe the lattice obtained by the canonical embedding\nof $\\fok$ (equivalently, the lattice given by the trace form). It turns out that this lattice is\nisomorphic to the\northogonal sum of lattices similar to the dual of a root lattice of type A and of a\nlattice invariant by a symmetric group which already appears in [2]. Using this information, we obtain\nan estimate of the Hermite--like thickness of the lattice $\\fok$, leading to an upper bound of $\\tau_{\\rm min}(\\fok)$ that we apply in \\S 7 to prove theorems~(\\ref{T:bound1}) and (\\ref{T:bound2}).\nFinally, \\S 8 contains some partial results and open questions concerning abelian fields\nof odd prime conductor.\n\n\\section{Notation and a definition}\nThe following notation will be used throughout this paper. The set of all abelian extensions of $\\bq$ of odd prime power conductor will be denoted by $\\mathcal{A}$.\nFor $K \\in \\mathcal{A}$ we set:\n\\begin{align*}\n n &- \\textnormal{ the degree of } K \/ \\bq,\\\\\n D &- \\textnormal{ the absolute value of the discriminant of } K,\\\\\n p &- \\textnormal{ the unique prime dividing the conductor of } K, \\\\\n r &- \\textnormal{ the } \\textnormal{$p$-adic additive valuation of the conductor of } K,\\\\\n \\zeta &- \\textnormal{ a primitive root of unity of order } p^r,\\\\\n e &- \\textnormal{ the degree $[\\cyclo : K]$}.\n\\end{align*}\nIf the dependence on the field $K$ needs to be emphasized, we shall add the index $K$ to the above symbols. For example, we shall write $n_K$ instead of $n$.\n\\bigskip\nWe also need the following definition\n\n\\begin{definition}\nLet $\\psi : \\mathcal{D} \\to \\mathbb{R}$ be a function, where $\\mathcal{D} \\subset \\mathcal{A}$. We shall say that\n$\\psi_o \\in \\mathbb{R}$ is the limit of $\\psi$ as $n_K$\ngoes to infinity and write\n\\[\n \\lim_{n_K \\to \\infty} \\psi(K) = \\psi_0\n\\]\nif for every $\\epsilon > 0$ there exists $N >0$ such that for every field $K \\in \\mathcal{D}$\n\\[\n n_K > N \\implies |\\psi(K) - \\psi_0| < \\epsilon.\n\\]\nWe shall also write\n\\[\n \\lim_{p_K \\to \\infty} \\psi(K) = \\psi_0\n\\]\nif for every $\\epsilon > 0$ there exists $N >0$ such that for every field $K \\in \\mathcal{D}$\n\\[\np_K > N \\implies |\\psi(K) - \\psi_0| < \\epsilon.\n\\]\n\\end{definition}\n\n\n\n\n\\section{Euclidean minima -- statement of the main results}\nIn this section we present the main results of the paper; the proofs will be given in \\S 7.\nWe keep the notation and definitions of the previous sections.\n\n\\begin{theorem}\\label{T:bound1}\nLet $K \\in \\mathcal{A}$. Then there exist constants $\\varepsilon=\\varepsilon(K) \\leq 2$ and $C=C(K) \\leq \\frac{1}{3}$ such that\n\\[\n M(K) \\leq C^n \\, (\\sqrt{D_K})^{\\varepsilon}.\n\\]\nIf $[K : \\bq] > 2$, then one may choose $\\varepsilon(K) < 2$. Moreover,\n\\[\n \\lim_{n_K \\to \\infty} \\varepsilon(K) = 1.\n\\]\nIf $r_K \\geq 2$, or $r_K=1$ and $[\\cyclo:K]$ is constant, then we also have\n\\[\n \\lim_{p_K \\to \\infty} C(K) = \\frac{1}{2\\sqrt{3}}.\n\\]\n\\end{theorem}\n\n\\bigskip\n\n\\begin{theorem}\\label{T:bound2}\nLet $K \\in \\mathcal{A}$. Then there is a constant $\\omega=\\omega(K)$ such that\n\\[\n M(K) \\leq \\omega^n \\sqrt{D_K}.\n\\]\nIf $r_K \\geq 2$, or $r_K=1$ and $[\\cyclo:K]$ is constant, then\n\\[\n \\lim_{p_K \\to \\infty} \\omega(K) = \\frac{1}{2\\sqrt{3}}.\n\\]\nMoreover, if $r_K \\geq 2$, then $\\omega(K) \\leq 3^{-2\/3}$.\n\\end{theorem}\n\n\\bigskip\n\nNote that this implies that Minkowski's conjecture holds for all totally real fields\n $K \\in \\mathcal{A}$ with composite conductor\n\n \\begin{corollary}\\label{T:Minkowski}\n\n Let $K \\in \\mathcal{A}$, and suppose that the conductor of $K$ is of the form $p^r$ with\n $r > 1$. Then\n\\[\n M(K) \\leq 2^{-n} \\sqrt{D_K}.\n\\]\n\n\\end{corollary}\n\n\\bigskip\nThis follows from Theorem~(\\ref{T:bound2}), since $3^{-2\/3} < 1\/2$, and for $K$ totally real this is precisely\nMinkowski's conjecture.\n\n\n\n\\section{Lattices and number fields}\n\nWe start by recalling some standard notion concerning Euclidean lattices (see for instance\n\\cite {conway} and \\cite {martinet}. A {\\it lattice} is a pair $(L,q)$, where $L$ is a free $\\bz$--module of finite rank, and $q : L_{\\br} \\times L_{\\br} \\to \\br$\nis a positive definite symmetric bilinear form, where $L_{\\br} = L \\otimes_\\bz \\br$. \nIf $(L,q)$ is a lattice and $a \\in \\br$, then we denote by $a(L,q)$ the lattice\n$(L,aq)$. Two lattices $(L,q)$ and $(L',q')$ are said to be\n{\\it similar} if and only if there exists $a \\in \\br$ such that $(L',q')$ and $a(L,q)$ are isomorphic,\nin other words if there exists an isomorphism of $\\bz$-modules $f : L \\to L'$ such that $q'(f(x),f(y)) = a q(x,y)$.\n\n\\medskip\n\nLet $(L,q)$ be a lattice, and set $q(x) = q(x,x)$. The {\\it maximum }of $(L,q)$ is defined by\n\\[\n {\\rm max}(L,q) = \\sup_{x \\in L_{\\br}} \\inf_{c \\in L} q(x-c).\n\\]\nNote that ${\\rm max}(L,q)$ is the square of the covering radius of the associated sphere covering.\nThe {\\it determinant} of $(L,q)$ is denoted by ${\\rm det}(L,q)$. It is by definition the determinant of the matrix of $q$ in a $\\bz$--basis of $L$.\nThe {\\it Hermite--like thickness} of $(L,q)$ is\n$$\n\\tau(L,q) = {{\\rm max}(L,q) \\over {\\rm det}(L,q)^{1\/m}},\n$$\nwhere $m$ is the rank of $L$. Note that $\\tau(L,q)$ only depends on the similarity class of the lattice $(L,q)$.\n\n\\bigskip\n{\\it A family of lattices}\n\\medskip\n\nLet $m\\in {\\mathbb {N}}$, and $b\\in {\\br}$ with $b>m$.\nLet $L=L_{b,m}$ be a lattice in ${\\br} ^m$ with Gram matrix\n$$b I_m - J_m = \\left( \\begin{array}{cccc}\nb-1 & -1 & \\ldots & -1 \\\\\n-1 & \\ddots & \\ddots & \\vdots \\\\\n\\vdots & \\ddots & \\ddots & -1 \\\\\n-1 & \\ldots & -1 & b-1\n\\end{array} \\right),$$\nwhere $I_m$ is the $m\\times m$-identity matrix and $J_m \\in \\{1 \\} ^{m\\times m}$ is the all-ones matrix.\nThen $L$ is a lattice of determinant $(b-m)b^{m-1}$. Moreover the automorphism group of $L$ contains $\\langle -I_m \\rangle \\times S_m$, where\nthe symmetric group $S_m$ acts by permuting the coordinates. These lattices were defined in \\cite{bayer-nebe}, (4.1).\nNote that the lattice $L_{m+1,m} $ is similar to the dual lattice $A_m^{\\#}$ of the\nroot lattice $A_m$ (see for instance \\cite{conway}, Chapter 4, \\S 6, or \\cite{martinet} for the definition of the root lattice $A_m$).\n\n\n\\bigskip\n{\\it Lattices defined over number fields}\n\n\\medskip In the sequel, we will be concerned with lattices defined on rings of integers\nof abelian number fields.\nLet $K$ be an number field of degree $n$, and suppose that $K$ is either totally\nreal or totally complex. Let us denote by\n$^{\\overline {\\ }} : K \\to K$ the identity in the first case and the\ncomplex conjugation in the second one, and let $P$ be the set of totally\npositive elements of the fixed field of this involution. Let us denote by ${\\rm Tr} : K \\to \\bq$ the trace\nmap. For any $\\alpha \\in P$, set $q_{\\alpha}(x,y) = {\\rm Tr}(\\alpha x \\overline y)$ for all $x, y \\in K$.\nThen $(\\fok,q_{\\alpha})$ is a lattice. Set\n$$\n\\tau_{\\rm min}(\\fok) = {\\rm inf} \\{ \\tau(\\fok,q_{\\alpha}) \\ | \\ \\alpha \\in P \\}.\n$$\n\nIf $D_K$ is the absolute value of the discriminant of $K$, then, by \\cite{bayer}, Corollary (5.2), we have\n\\begin{equation}\\label{E:gen-est}\n M(K) \\leq \\left(\\frac{\\tau_{\\min}(\\fok)}{n}\\right)^{\\frac{n}{2}} \\sqrt{D_K},\n\\end{equation}\nThis upper bound will be used in \\S 7 to prove theorems~(\\ref{T:bound1}) and (\\ref{T:bound2}).\n\\section{Gaussian periods and integral bases}\n\nLet $K \\in \\mathcal{A}$. In order to exploit the upper bound of \\S 4, we need some information\nconcerning the lattices defined on the ring of integers $\\fok$, and these will be described\nusing integral bases of $\\fok$. The aim of this section is to find such bases. This will be done\nin the spirit of the work of Leopold~\\cite{leopold}, see also Lettl~\\cite{lettl}.\n\n\\medskip\nRecall that the $p^r$ is the conductor of $K$ and that $e=[\\cyclo:K]$. Then\n$e$ divides $p-1$. This implies that \nthe extension $\\cyclo \/ K$ is tamely ramified, and hence \nthe trace map ${\\rm Tr} : \\bz[\\zeta] \\to \\fo_K$ is surjective.\n\n\\medskip\nSet $R=\\bz \/ p^{r} \\bz$ and \nlet us denote by $H$ the unique subgroup of order $e$ of $R^{\\ast}$. Then $H$ acts\non $R$ by left multiplication. The orbit $H 0$ will be denoted by $\\mathbf{0}$ and called the zero orbit.\n\n\n\\begin{definition}\nFor $\\mathbf{x} \\in R \/H \\setminus \\set{\\mathbf{0}}$, we define a \\textit{Gaussian period}\n\\[\n f_{\\mathbf{x}} = \\sum_{x \\in \\mathbf{x}} \\zeta^x.\n\\]\nIn addition, we set $f_{\\mathbf{0}} = e$.\n\\end{definition}\n\nIf $\\mathbf{x}=H x$, then $f_{\\mathbf{x}} = \\tr_{\\cyclo \/ K}(\\zeta^x)$. As the trace map ${\\rm Tr} : \\bz[\\zeta] \\to \\fo_K$ is surjective., Gaussian periods are generators of $\\fok$ over $\\bz$. The next\nproposition will be used to show that $\\fok$ has actually an integral basis consisting of Gaussian periods. \n\n\\medskip\nSet $S=\\bz \/ p^{r-1} \\bz$, and let\n$\\pi : R \\to S$ be the canonical projection. The group $H$ acts on $S$ by $h \\cdot s = \\pi(h) s$.\nClearly, $\\pi$ is a morphism of $H$-sets and hence it induces the unique map between the orbit sets $\\rho : R \/ H \\to S \/ H$ such that $\\rho(H x) = H \\pi(x)$ for all $x \\in R$.\nIn other words, if $\\mu_R : R \\to R \/ H$ and $\\mu_S : S \\to S \/ H$ are the canonical projections, then $\\rho \\mu_R = \\mu_S \\pi$. In particular, $\\rho$ is surjective.\nFor any subset $A$ of $R$, set $\\zeta^A = \\{\\zeta^a \\ | \\ a \\in A \\}$, and let us denote by $A^c$ the\ncomplement of $A$ in $R$. For a finite set $X$, we denote by $|X|$ the number of elements of $X$.\n\n\n\\medskip\n We thank H.W. Lenstra, Jr, for sending us the part (1) $\\Leftrightarrow$ (2) of\nthe following proposition.\n\n\n\\begin{proposition}\\label{P:set-A}\nLet $A \\subset R \\setminus \\set{0}$ be an $H$-invariant set. The following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\zeta^A$ is a basis of $\\bz[\\zeta]$.\n \\item The restriction $\\pi :A^c \\to S$ is a bijection.\n \\item The restriction $\\rho : A^c \/ H \\to S \/ H$ is a bijection.\n \\item\\label{P:set-A-card} For every $\\mathbf{y} \\in S \/ H$ we have\n \\[\n |\\rho^{-1}(\\mathbf{y}) \\cap A \/ H| = |\\rho^{-1}(\\mathbf{y})| -1.\n \\]\n \\item $\\trck(\\zeta^{A}) = \\set{f_{\\mathbf{x}} \\mid \\mathbf{x} \\in A\/ H}$ is a basis of $\\fok$.\n\\end{enumerate}\n\n\n\\end{proposition}\n\n\\noindent\n{\\it Proof}.\n (1) $\\Leftrightarrow$ (2) Note that the sum of powers of $\\zeta$ over any coset of $\\ker \\pi$ in $R$ equals zero. Thus, if $\\zeta^A$ is a $\\bz$-basis, then $A$ must miss at least one element of each coset.\nIt cannot miss more than one since then the cardinality of $A$ would be too small. Conversely, if $A$ misses exactly one element from\neach coset, then the sum relation mentioned shows that the $\\bz$-span of $\\zeta^A$ contains all roots of unity of order $p^r$. Hence $\\zeta^A$ forms an integral basis. \n\n\\medskip\n\n\\noindent (2) $\\Rightarrow$ (3) This follows immediately from the fact that $\\pi$ is a morphism of $H$-sets and $A^c$ is $H$-invariant.\n\n\\medskip\n\n\\noindent (3) $\\Rightarrow$ (2) First, we shall show that $\\pi_{|A^c}$ is onto. Suppose that this is not true. Then $\\pi$ must miss at least one full orbit since it is an $H$-map\nbut in such a case the restriction of $\\rho: A^c \/ H \\to S \/ H$ would not be surjective. Thus $\\pi_{|A^c}$ is onto. We claim that $|A^c| = |S|$, which implies that $\\pi_{|A^c}$\nis a bijection. Indeed, the set $A^c \/H$ maps bijectively onto $S\/H$, which implies that they have the same cardinality. Since both $A^c \/H$ and $S \/H$ contain the respective zero orbits, it is easy\nto check that\n\\[\n \\left|A^c \/ H \\right| = 1 + \\frac{|A^c|-1}{e} \\quad \\textnormal{ and } \\quad \\left|S \/ H \\right| = 1 + \\frac{|S|-1}{e},\n\\]\nwhich readily implies $|A^c| = |S|$.\n\n\\medskip\n\n\\noindent (3) $\\Leftrightarrow$ (4) Since $A$ is $H$-invariant, it follows that the sets $A\/H$, $A^c \/H$ form a partition of the orbit space $R \/H$. Consequently, for every $\\mathbf{y} \\in S \/ H$ we have\n\\[\n |\\rho^{-1}(\\mathbf{y}) \\cap A \/ H| = |\\rho^{-1}(\\mathbf{y})| - |\\rho^{-1}(\\mathbf{y}) \\cap A^c \/ H|.\n\\]\nThe restriction $\\rho : A^c \/ H \\to S \/ H$ is a bijection \\ifft $|\\rho^{-1}(\\mathbf{y}) \\cap A^c \/ H| = 1$ for every $\\mathbf{y} \\in S \/ H$.\n\n\\medskip\n\n\\noindent (1) $\\Rightarrow$ (5) Assume now that $\\zeta^A$ is a basis of $\\bz[\\zeta]$. We shall show that $\\trck(\\zeta^{A})$ is an integral basis of $K$. Since $0 \\notin A$, it follows that $A$ is a union of $n$ orbits, each of cardinality $e$.\nConsequently,\n\\[\n |\\trck(\\zeta^{A})| \\leq n = \\rank \\, \\fok.\n\\]\nSince $\\cyclo \/ K$ is tamely ramified, $\\trck(\\zeta^A)$ generates $\\fok$, which in turn implies that we have in fact $|\\trck(\\zeta^{A})| = n$\nand that $\\trck(\\zeta^{A})$ is an integral basis of $K$. \n\n\\medskip\n\n\n\\noindent (5) $\\Rightarrow$ (1) Since $\\trck(\\zeta^A)$ is a basis, we have $|\\trck(\\zeta^{A})| = n$. It follows that $|A| \\geq n e = p^{r-1} (p-1)$. Suppose by contradiction that $\\zeta^A$ is not a basis.\nThen $\\pi : A^c \\to S$ is not a bijection. Since $$|A^c| = |R| - |A| \\leq p^r - p^{r-1} (p-1) = p^{r-1} = |S|,$$ it follows that $\\pi : A^c \\to S$ is not a surjection. Consequently,\nthere exists $x_0 \\notin \\ker \\pi$ such that the coset $C=x_0 + \\ker \\pi$ is contained in $A$. Note that the intersection of $C$ with any $H$-orbit is either empty or a singleton. Indeed, if\n$x_0 + z_1 = h (x_0 + z_2)$ for some $z_1, z_2 \\in \\ker \\pi$ and $h \\in H \\setminus \\set{1}$, then $(1-h) x_0 \\in \\ker \\pi$. Note that $1-h$ is invertible. Hence $x_0 \\in \\ker \\pi$,\nwhich is a contradiction. Therefore the set $HC = \\set{hc \\mid h \\in H, c \\in C}$ is contained in $A$ and it has $|H| \\cdot |C| $ elements. Furthermore,\n$HC \/H$ is contained in $A \/ H$ and\n\\[\n \\sum_{\\mathbf{x} \\in HC \/H} f_{\\mathbf{x}} = \\sum_{hc \\in HC} \\zeta^{hc} = \\sum_{h \\in H} \\sum_{c \\in C} \\zeta^{hc} = 0,\n\\]\nwhich contradicts the linear independence of elements of $\\trck(\\zeta^A)$. Thus $\\zeta^A$ is a basis of $\\bz[\\zeta]$.\n\n\\bigskip\nThis proposition implies that $\\fok$ has an integral basis consisting of Gaussian periods.\nIndeed, we have\n\n\\begin{corollary}\\label{C:basis}\n There exists an $H$--invariant set $A \\subset R \\setminus \\set{0}$\n such that\n$$\n\\trck(\\zeta^{A}) = \\set{f_{\\mathbf{x}} \\mid \\mathbf{x} \\in A\/ H}\n$$\n is a basis of $\\fok$.\n \\end{corollary}\n\n\\noindent\n{\\it Proof.} For all $\\mathbf{y} \\in S\/ H$ with $\\mathbf{y} \\not = \\mathbf{0}$, let\nus choose $\\mathbf{x}_{\\mathbf{y} } \\in R\/ H$ such that $\\rho(\\mathbf{x}_{\\mathbf{y} } ) =\n\\mathbf{y}$.\nSet $\\mathbf{x}_{\\mathbf{0}} = \\mathbf{0}$, and let $B = \\cup_{\\mathbf{y} \\in S\/ H}\\mathbf{x}_{\\mathbf{y} }$. Then $B$ is an $H$--invariant subset of $R$ containing $0$, and\nthe restriction $\\rho : B \/ H \\to S \/ H$ is a bijection. Set $A = B^c$; then $A$\nis an $H$--invariant subset of $R \\setminus \\set{0}$, and the restriction\n$\\rho : A^c \/ H \\to S \/ H$ is a bijection. By Proposition~(\\ref{P:set-A}), this implies that\n$\\trck(\\zeta^{A}) = \\set{f_{\\mathbf{x}} \\mid \\mathbf{x} \\in A\/ H}$ is a basis of $\\fok$.\n\n\\bigskip\n\\bigskip\n\n\n\n\\section{Geometry of the ring of integers}\n\nWe keep the notation of the previous section; in particular, $K \\in \\mathcal{A}$ and\n$p = p_K$. Recall that $\\fok$ is the ring of integers of $K$, and let us consider the\nlattice $(\\fok,q)$, where $q$ is defined by $q(x,y) = \\trkq(x \\overline y)$. As we have\nseen in \\S 4, the Hermite--like thickness of this lattice can be used to give an upper bound of\nthe Euclidean minimum of $K$. The purpose\nof this section is to describe the lattice $(\\fok,q)$ using the results of \\S 5, so that we can\ncompute its Hermite--like thickness.\n\n\n\n\\bigskip\nWe will see that $(\\fok,q)$ decomposes in a natural way into the\northogonal sum of a lattice $\\Gamma_K$, which is similar to the orthogonal sum of copies of the dual lattice of\nthe root lattice $A_{p-1}$, and of a lattice $\\Lambda_K$, which is similar to a certain lattice of type $L_{b,m}$ defined in \\S 4.\nThe Hermite--like thickness of these lattices can be estimated, cf.~\\cite{bayer-nebe}, Theorem (4.1).\nThis allows us to give good upper bounds for the Euclidean minima of fields $K \\in \\mathcal{A}$,\nfollowing the strategy outlined in the introduction and in \\S 4.\n\n\n\n\\bigskip Let $\\Gamma_K$ be the orthogonal sum of\n$\\frac{p^{r-1}-1}{e}$ copies of the lattice $p^{r-1}A_{p-1}^{\\#}$.\nSet $d=\\frac{p-1}{e}$, and let $\\Lambda_K = e p^{r-1} L_{\\frac{p}{e},d}$ (note that\nthe scaling is taken in the sense of \\S 4, that is it refers to multiplying the quadratic form by\nthe scaling factor). \n\n\\begin{theorem}\\label{T:ortho-sum}\nThe lattice $(\\fok,q)$ is isometric to the orthogonal sum of $\\Gamma_K$ and of $\\Lambda_K$.\n\\end{theorem}\n\n\n\n\n\nBefore proving this theorem, we need a few lemmas.\nRecall that $R$ denotes\nthe ring $\\bz \/ p^r \\bz$, and let $\\gtm = p \\bz \/p^r \\bz$ be the maximal ideal of $R$. Note that if $r=1$, then $\\gtm$ is the zero ideal.\nFor $\\mathbf{x} \\in R \/ H$ we set $\\ord_p(\\mathbf{x}) = {\\rm max} \\{ k \\in {\\bf N} \\ | \\ \\mathbf{x} \\subset \\gtm^k \\}$. Let us denote by $\\mu$ the M\\\"obius function.\n\n\n\\begin{lemma}\\label{L:trace-of-gp}\nLet $\\mathbf{x} \\in R \/ H$. Then,\n \\[\n \\trkq(f_{\\mathbf{x}}) = \\frac{\\phi(p^r)}{\\phi(p^{r-s})} \\cdot \\mu(p^{r-s}),\n \\]\nwhere $s = \\ord_p(\\mathbf{x})$\n\\end{lemma}\n\n\\noindent\n{\\it Proof.}\nLet $x_0 \\in \\mathbf{x}$. We have\n\\[\n \\trkq(f_{\\mathbf{x}}) = \\trkq(\\trck(\\zeta^{x_0})) = \\trcq(\\zeta^{x_0})\n\\]\n Assume first that $s=0$. Then, $\\mathbf{x} \\subset R^{\\ast}$ and hence $x_0 \\in R^{\\ast}$. Consequently,\n\\[\n \\trkq(f_{\\mathbf{x}}) = \\sum_{x \\in R^{\\ast}} \\zeta^{x_0 x} = \\sum_{x \\in R^{\\ast}} \\zeta^{x} = \\mu(p^r) = \\frac{\\phi(p^r)}{\\phi(p^{r-s})} \\cdot \\mu(p^{r-s}).\n\\]\nNow, assume that $1 \\leq s < r$. Then, $x_0 = p^s x_1$ with $x_1 \\in R^{\\ast}$. Set $\\xi = \\zeta^{p^s}$ and $T=\\bz \/ p^{r-s} \\bz$. Then $\\xi$ is a primitive root of unity of order $p^{r-s}$.\nIf $\\tau : R^{\\ast} \\to T^{\\ast}$ is the natural map with kernel $G$, and the set $Y \\subset R^{\\ast}$ is mapped by $\\tau$ bijectively onto $(\\bz \/ p^{r-s} \\bz)^{\\ast}$, then\n\\begin{align*}\n \\trkq(f_{\\mathbf{x}}) &= \\sum_{x \\in R^{\\ast}} \\xi^{x_1 x} = \\sum_{x \\in R^{\\ast}} \\xi^x = \\sum_{g \\in G} \\sum_{y \\in Y} \\xi^{gy} = \\sum_{g \\in G} \\sum_{y \\in Y} \\xi^{y}\\\\\n\t\t &= |G| \\cdot \\sum_{t \\in T^{\\ast}} \\xi^{t} = \\frac{\\phi(p^r)}{\\phi(p^{r-s})} \\cdot \\mu(p^{r-s}).\n\\end{align*}\nFinally, if $s=r$, then $\\mathbf{x}=\\mathbf{0}$ and $x_0=0$ and hence\n\\[\n \\trkq(f_{\\mathbf{x}}) = \\trcq(1) = \\phi(p^r) = \\frac{\\phi(p^r)}{\\phi(p^{r-s})} \\cdot \\mu(p^{r-s}).\n\\]\n\n\\bigskip\n\n\n\n\\begin{proposition}\\label{P:Gram-prep}\n Let $\\mathbf{x_1}, \\mathbf{x_2} \\in R \/ H \\setminus \\set{\\mathbf{0}}$. Then,\n\\[\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) =\n\\begin{cases}\np^r-p^{r-1} \\quad &\\text{ if }\\, \\mathbf{x_1} = \\mathbf{x_2} \\text{ and } \\rho(\\mathbf{x_1})=\\rho(\\mathbf{x_2}) \\neq \\mathbf{0},\\\\\np^r - e p^{r-1} \\quad &\\text{ if }\\, \\mathbf{x_1} = \\mathbf{x_2} \\text{ and } \\rho(\\mathbf{x_1})=\\rho(\\mathbf{x_2}) = \\mathbf{0},\\\\\n-p^{r-1} \\quad &\\text{ if }\\, \\mathbf{x_1} \\neq \\mathbf{x_2} \\text{ and } \\rho(\\mathbf{x_1})=\\rho(\\mathbf{x_2}) \\neq \\mathbf{0},\\\\\n-e p^{r-1} \\quad &\\text{ if }\\, \\mathbf{x_1} \\neq \\mathbf{x_2} \\text{ and } \\rho(\\mathbf{x_1})=\\rho(\\mathbf{x_2}) = \\mathbf{0},\\\\\n0 \\quad &\\text{ if }\\, \\mathbf{x_1} \\neq \\mathbf{x_2} \\text{ and } \\rho(\\mathbf{x_1}) \\neq \\rho(\\mathbf{x_2}).\n\\end{cases}\n\\]\n\\end{proposition}\n\n\\noindent\n{\\it Proof.}\n Let $x_1 \\in \\mathbf{x_1}$ and $x_2 \\in \\mathbf{x_2}$. For $h \\in H$ we set $\\mathbf{x}(h) = H (x_1-x_2h)$ and $s(h) = \\ord_p \\mathbf{x}(h)$.\nThen,\n \\begin{align*}\n f_{\\mathbf{x_1}} \\overline{f_{\\mathbf{x_2}}} &= (\\sum_{h_1 \\in \\hbar} \\zeta^{x_1h_1}) (\\sum_{h_2 \\in \\hbar} \\zeta^{-x_2h_2}) = \\sum_{h_1 \\in \\hbar} \\sum_{h_2 \\in \\hbar} \\zeta^{x_1h_1-x_2h_2}\n\t = \\sum_{h_1 \\in \\hbar} \\sum_{h \\in \\hbar} \\zeta^{x_1h_1-x_2hh_1}\\\\ &= \\sum_{h_1 \\in \\hbar} \\sum_{h \\in \\hbar} \\zeta^{(x_1-x_2h)h_1}\n\t = \\sum_{h \\in \\hbar} \\sum_{h_1 \\in \\hbar} \\zeta^{(x_1-x_2h)h_1} = \\sum_{h \\in \\hbar} f_{\\mathbf{x}(h)}.\n \\end{align*}\nBy Lemma~(\\ref{L:trace-of-gp}), we have\n\\begin{equation}\\label{E:tr-form}\n\\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) = \\sum_{h \\in H} \\trkq(f_{\\mathbf{x}(h)}) = \\sum_{h \\in H} \\frac{\\phi(p^r)}{\\phi(p^{r-s(h)})} \\cdot \\mu(p^{r-s(h)}).\n\\end{equation}\nIf $\\mathbf{x_1} = \\mathbf{x_2}$, we can take $x_1 = x_2$ and then $\\mathbf{x}(h) = H x_1(1-h)$. Clearly, $s(1) = r$. If $h\\neq 1$, then we have\n$s(h) = \\ord_p\\mathbf{x_1}$. Thus, if $\\ord_p\\mathbf{x_1} < r-1$, then $\\rho(\\mathbf{x_1}) \\neq \\mathbf{0}$ and the only non-zero term of the sum~(\\ref{E:tr-form})\nis the one corresponding to $h=1$. Thus, we have\n\\[\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) = \\frac{\\phi(p^r)}{\\phi(p^{r-s(1)})} \\cdot \\mu(p^{r-s(1)}) = \\phi(p^r) = p^r-p^{r-1}.\n\\]\nIf $\\ord_p(\\mathbf{x_1}) = r-1$, then $\\rho(\\mathbf{x_1}) = \\mathbf{0}$ and the sum~(\\ref{E:tr-form}) becomes\n\\begin{align*}\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) &= \\sum_{h \\in H} \\frac{\\phi(p^r)}{\\phi(p^{r-s(h)})} \\cdot \\mu(p^{r-s(h)})\\\\\n & = \\phi(p^r) + (e-1) \\cdot \\frac{\\phi(p^r)}{\\phi(p^{r-1})} \\cdot \\mu(p)\\\\\n &= p^r - e p^{r-1}.\n\\end{align*}\nSuppose now that $\\mathbf{x_1} \\neq \\mathbf{x_2}$. Observe that $\\rho(\\mathbf{x_1}) = \\rho(\\mathbf{x_2})$ \\ifft there is an $h \\in H$ such that $s(h) =r-1$. Moreover,\n in such a case an element $h$ with this property is unique unless $\\rho(\\mathbf{x_1}) = \\mathbf{0}$,\nin which case we have $s(h) = r-1$ for\nall $h \\in H$. Thus, assuming that $\\rho(\\mathbf{x_1}) = \\rho(\\mathbf{x_2})$ and $\\rho(\\mathbf{x_1}) \\neq \\mathbf{0}$, we have\n\\begin{align*}\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) &= \\sum_{h \\in H} \\frac{\\phi(p^r)}{\\phi(p^{r-s(h)})} \\cdot \\mu(p^{r-s(h)})\\\\\n\t\t\t\t\t\t &= \\frac{\\phi(p^r)}{\\phi(p)} \\cdot \\mu(p) = -p^{r-1}.\n\\end{align*}\nIf $\\rho(\\mathbf{x_1}) = \\mathbf{0}$, then\n\\begin{align*}\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) &= \\sum_{h \\in H} \\frac{\\phi(p^r)}{\\phi(p^{r-s(h)})} \\cdot \\mu(p^{r-s(h)})\\\\\n\t\t\t\t\t\t &= \\sum_{h \\in H} \\frac{\\phi(p^r)}{\\phi(p)} \\cdot \\mu(p) = -e p^{r-1}.\n\\end{align*}\nFinally, if $\\rho(\\mathbf{x_1}) \\neq \\rho(\\mathbf{x_2})$, then $s(h) \\leq r-2$ for all $h \\in H$, which gives\n\\[\n \\trkq(f_{\\mathbf{x_1}} \\overline{f_{ \\mathbf{x_2}}}) = 0.\n\\]\n\\bigskip\n\n\\begin{lemma}\\label{L:rho-inv}\n Let $\\mathbf{y} \\in S \/ H$. Then,\n\\[\n |\\rho^{-1}(\\mathbf{y})| =\n\\begin{cases}\n 1+\\frac{p-1}{e} \\quad &\\text{ if } \\, \\mathbf{y}=\\mathbf{0},\\\\\n p \\quad &\\text{ if } \\, \\mathbf{y} \\neq \\mathbf{0}.\n\\end{cases}\n\\]\n\\end{lemma}\n\n\\noindent\n{\\it Proof.}\nLet $X = \\rho^{-1}(\\mathbf{y})$. Note that $\\mathbf{0} \\in X$ if and only if $\\mathbf{y}=\\mathbf{0}$,\nhence\n\\begin{equation*}\n |\\mu_R^{-1}(X)| =\n\\begin{cases}\n 1 + e \\cdot(|X|-1) \\quad &\\text{ if } \\, \\mathbf{y}=\\mathbf{0},\\\\\n |X| \\cdot e \\quad &\\text{ if } \\, \\mathbf{y} \\neq \\mathbf{0}.\n\\end{cases}\n\\end{equation*}\nOn the other hand,\n$\n\\mu_R^{-1}(X) = \\mu_R^{-1}(\\rho^{-1}(\\mathbf{y})) = (\\rho \\mu_R)^{-1}(\\mathbf{y}) = (\\mu_S \\pi)^{-1}(\\mathbf{y}).\n$\n\nIf $\\mathbf{y}=\\mathbf{0}$, then\n\\[\n \\mu_R^{-1}(X) = (\\mu_S \\pi)^{-1}(\\mathbf{0}) = \\set{x \\in R \\mid H\\pi(x) = \\mathbf{0}} = \\gtm^{r-1}\n\\]\nand hence $|(\\mu_S \\pi)^{-1}(\\mathbf{0})| = p$, which implies that $|X| = 1+\\frac{p-1}{e}$.\n\nIf $\\mathbf{y} \\neq \\mathbf{0}$, then there is an element $x_0 \\in R \\setminus \\gtm^{r-1}$ such that $\\mathbf{y} = H \\pi(x_0)$. Consequently,\n\\begin{align*}\n \\mu_R^{-1}(X) &= (\\mu_S \\pi)^{-1}(\\mathbf{y}) = \\set{x \\in R \\mid H\\pi(x) = H \\pi(x_0)}\\\\\n\t\t &= \\set{x \\in R \\mid x = k + hx_0 \\text{ for some } k \\in \\gtm^{r-1} \\text{ and } h \\in H }.\n\\end{align*}\nLet $k_1, k_2 \\in \\gtm^{r-1}$ and $h_1, h_2 \\in H$. If $k_1 + h_1 x_0 = k_2 + h_2 x_0$, then $h_1 (1 - h_1^{-1}h_2 ) x_0 \\in \\gtm^{r-1}$. Since $x_0 \\notin \\gtm^{r-1}$, it follows that\n$1 - h_1^{-1}h_2$ is not invertible. Hence $h_1=h_2$, which in turn implies that $k_1=k_2$. \nTherefore\n$|(\\mu_S \\pi)^{-1}(\\mathbf{y})| = |\\gtm^{r-1}| \\cdot |H| = p e$, which gives $|X| = p$.\n\\bigskip\n\n\\noindent\n{\\it Proof of Theorem~(\\ref{T:ortho-sum}).}\nLet $A \\subset R \\setminus \\set{0}$ be an $H$-invariant set such that $\\trck(\\zeta^{A})$ is a basis of $\\fok$. For $\\mathbf{y} \\in S \/ H$ we set\n\\[\n B_{\\mathbf{y}} = \\set{f_{\\mathbf{x}} \\in \\trck(\\zeta^{A}) \\mid \\rho({\\mathbf{x}}) = \\mathbf{y}} \\quad \\textnormal{ and } \\quad L_{\\mathbf{y}} = \\spanp_{\\bz} B_{\\mathbf{y}}.\n\\]\nBy Proposition~(\\ref{P:Gram-prep}), we have\n\\begin{equation}\\label{E:OK-decomp}\n \\fok = \\perp_{\\mathbf{y} \\in S \/H} L_{\\mathbf{y}},\n\\end{equation}\nin other words the lattice $(\\fok, q)$ is the orthogonal sum of the lattices obtained by the restriction of $q$ to $L_{\\mathbf{y}}$ for $\\mathbf{y} \\in S\/H$.\nCombining Lemma~(\\ref{L:rho-inv}) and the condition (\\ref{P:set-A-card}) of Proposition~(\\ref{P:set-A}), we obtain that\n$$\n|B_{\\mathbf{y}}|=\n\\begin{cases}\n \\frac{p-1}{e} \\quad &\\text{ if } \\, \\mathbf{y}=\\mathbf{0},\\\\\n p-1 \\quad &\\text{ if } \\, \\mathbf{y} \\neq \\mathbf{0}.\n\\end{cases}\n$$\nFurthermore, using Proposition~(\\ref{P:Gram-prep}) again, we conclude that the Gram matrix of the lattice $L_{\\mathbf{y}}$ with respect to $B_{\\mathbf{y}}$ is $p^{r-1} (p I_{p-1} - J_{p-1})$ unless\n$\\mathbf{y} = \\mathbf{0}$ in which case it equals $e p^{r-1} (\\frac{p}{e} I_d - J_d)$, where $d=\\frac{p-1}{e}$. Consequently, we have $L_{\\mathbf{0}} = \\Lambda_K$. Moreover, $S\/H$ has $\\frac{p^{r-1}-1}{e}$ nonzero orbits. As a result,\n\\[\n \\perp_{\\mathbf{y} \\neq \\mathbf{0}} L_{\\mathbf{y}} \\simeq \\Gamma_K.\n\\]\nThus the equality~(\\ref{E:OK-decomp}) implies that $(\\fok,q)$ is isometric to the orthogonal sum of $\\Gamma_K$ and $\\Lambda_K$.\n\\bigskip\n\nWe now apply Theorem~(\\ref{T:ortho-sum}) to give an upper bound of the Hermite--like thickness of the lattice $(\\fok,q)$. The following is well--known\n\n\\begin{lemma}\\label{L:det-fok}\n We have\n\\[\n \\det (\\fok,q) = p^{\\upsilon},\n\\]\nwhere\n\\[\n \\upsilon = rn -\\frac{(p^{r-1}-1)}{e} -1.\n\\]\n\\end{lemma}\n\n\\noindent {\\it Proof.} Note that $\\det (\\fok,q)$ is the absolute value of the discriminant of $K$. \nThe result follows from Theorem (4.1) in~\\cite{disc}. Alternatively, one can compute $\\det (\\fok,q)$ directly using Theorem~(\\ref{T:ortho-sum}).\n\n\\begin{lemma}\\label{L:max-fok}\n We have\n\\[\n \\max(\\fok,q) \\leq n \\cdot \\frac{p^{r+1}+p^r+1-e^2}{12 p}.\n\\]\n\\end{lemma}\n\n\\noindent {\\it Proof.}\n By Theorem (4.1) in~\\cite{bayer-nebe}, we have\n\\[\n \\max (L_{\\frac{p}{e},d}) \\leq \\frac{d(p^2+p+1-e^2)}{12ep}.\n\\]\nFurthermore, $\\max (L_{p, p-1}) = \\frac{p^2-1}{12}$. Consequently,\n\\begin{align*}\n \\max (\\fok,q) = \\sum_{\\mathbf{y} \\in S \/ H} \\max(L_{\\mathbf{y}}) &\\leq \\frac{p^{r-1}-1}{e} \\cdot p^{r-1} \\cdot \\frac{p^2-1}{12} + p^{r-1} \\cdot \\frac{d(p^2+p+1-e^2)}{12p}\\\\\n\t &= d p^{r-1} \\left[ \\left(\\frac{p^{r-1}-1}{p-1}\\right) \\cdot \\left(\\frac{p^2-1}{12}\\right) + \\frac{p^2+p+1-e^2}{12 p} \\right]\\\\\n\t &= n \\cdot \\frac{(p^{r-1}-1)(p+1)p+(p^2+p+1-e^2)}{12p}\\\\\n\t &= n \\cdot \\frac{p^{r+1}+p^r+1-e^2}{12 p}.\n\\end{align*}\n\n\nAs a direct consequence of the above lemmas, we obtain the following \nupper bound\nof $\\tau_{\\rm min}(\\fok)$\n\n\\begin{corollary}\\label{c:tau-min}\nWe have\n\\[\n \\tau_{\\rm min}(\\fok) \\leq \\tau(\\fok,q) \\leq n \\cdot p^{r-\\frac{\\upsilon}{n}} \\cdot \\frac{p^{r+1}+p^r+1-e^2}{12 p^{r+1}}.\n\\]\n\n\\end{corollary}\n\n\n\n\\bigskip\nThis bound will be used in the next section to prove theorems~(\\ref{T:bound1}) and (\\ref{T:bound2}).\n\n\n\\section{Euclidean minima -- proof of the main results}\nIn this section we prove the main results of the paper, namely the upper bounds\nfor Euclidean minima stated in \\S 3. Recall that for any number field $K$ of degree $n$,\nwe have $$ M(K) \\leq \\left(\\frac{\\tau_{\\min}(\\fok}{n}\\right)^{\\frac{n}{2}} \\sqrt{D_K},$$\nwhere $D_K$ is the absolute value of the discriminant of $K$. For $K \\in \\mathcal{A}$, we\nnow have an upper bound (see Corollary (\\ref{c:tau-min})) and this will be used in the proofs.\n\n\\medskip\n\\noindent\n{\\it Proof of th. 3.1}\nSet\n\\[\n f = \\frac{p^{r+1}+p^r+1-e^2}{12 p^{r+1}}, \\quad C = \\sqrt{f}, \\quad \\varepsilon = \\frac{rn}{\\upsilon}.\n\\]\nThen, by the inequality~(\\ref{E:gen-est}) and Corollary (\\ref{c:tau-min}), we get\n\\[\n M(K) \\leq \\left(\\frac{\\tau(\\fok)}{n}\\right)^{\\frac{n}{2}} \\sqrt{D_K} \\leq C^n \\cdot (\\sqrt{D_K})^{\\varepsilon}.\t\n\\]\n\nFirst we shall prove that $\\varepsilon(K)$ has the stated properties. If $r=1$, then $\\upsilon = n-1$ and $\\varepsilon = \\frac{n}{n-1}$,\nwhich implies that $\\varepsilon \\leq 2$ with the equality only for $n=2$. It is also clear that $\\varepsilon \\to 1$ if $n \\to \\infty$.\nAssume that $r \\geq 2$. A simple calculation shows that\n\\[\n 2 \\upsilon - rn = r d p^{r-1}- 2 d\\left( \\frac{p^{r-1}-1}{p-1} \\right) - 2.\n\\]\nClearly,\n\\[\n \\frac{p^{r-1}-1}{p-1} \\leq (r-1) p^{r-2},\n\\]\nwhich gives\n\\[\n 2 \\upsilon - rn \\geq r d p^{r-1}- 2d (r-1) p^{r-2} - 2 = rdp^{r-2}(p-1) + p^{r-2}d-2 > 0,\n\\]\nwhich implies that $\\varepsilon < 2$. Another simple calculation shows that\n\\begin{equation}\\label{E:rna}\n \\varepsilon = \\frac{r\\left(1+\\frac{1}{\\upsilon}\\right)}{\\left( r - \\delta \\right)},\n\\end{equation}\nwhere\n\\[\n \\delta = \\frac{p^{r-1}-1}{p^{r-1}(p-1)}.\n\\]\nObserve that\n$$\n\\ln n < r \\ln p < r (p-1) \\leq rp -1.\n$$\nSince $0 \\leq \\delta < \\frac{1}{p}$, it follows that for $n \\geq 3$ we have\n$$\n0 \\leq \\frac{r}{r-\\delta} - 1< \\frac{1}{rp -1} < \\frac{1}{\\ln n}.\n$$\nThus we get\n\\begin{equation}\\label{E:gt1}\n\\lim_{n \\to \\infty} \\frac{r}{r - \\delta} = 1.\n\\end{equation}\nFinally, it follows from~(\\ref{E:rna}) that $\\upsilon + 1 = (r - \\delta) n$.\nSince $r - \\delta > r - 1 \\geq 1$, we get $\\upsilon > n-1$. Consequently\n\\begin{equation}\\label{E:gt2}\n \\lim_{n \\to \\infty} \\left( 1 + \\frac{1}{\\upsilon} \\right) = 1.\n\\end{equation}\nCombining the equalities~(\\ref{E:gt1}), (\\ref{E:gt2}), we obtain\n\\[\n \\lim_{n_K \\to \\infty} \\varepsilon(K) = 1.\n\\]\n\nNow we prove the properties of $C(K)$. We have\n \\begin{equation*}\n f = \\frac{p^{r+1}+p^r+1-e^2}{12 p^{r+1}} \\leq \\frac{p^{r+1}+p^r}{12 p^{r+1}} = \\frac{p+1}{12p} \\leq \\frac{1}{9}\n \\end{equation*}\nand hence $C \\leq \\frac{1}{3}$. We can also write\n\\[\n f = \\frac{p^{r+1}+p^r+1-e^2}{12 p^{r+1}} = \\frac{1}{12} \\left( 1+ \\frac{1}{p} + \\frac{1}{p^{r+1}} - \\frac{e^2}{p^{r+1}} \\right).\n\\]\nIf $r=1$ and $e$ is constant, then $f$ clearly approaches $\\frac{1}{12}$ as $p \\to \\infty$ and hence $C(K) \\to \\frac{1}{2\\sqrt{3}}$. Assume now that $r \\geq 2$. Since $1 \\leq e \\leq p-1$,\nit follows that\n\\[\n 0 < f -\\frac{1}{12} \\leq \\frac{1}{12p}\n\\]\nand thus\n\\[\n 0 < C(K) - \\frac{1}{2\\sqrt{3}} \\leq \\frac{\\sqrt{3}}{12p}.\n\\]\nConsequently,\n\\[\n \\lim_{p_K \\to \\infty} C(K) = \\frac{1}{2\\sqrt{3}}\n\\]\nand this concludes the proof of the theorem.\n\n\n\n\\bigskip\n\\noindent\n{\\it Proof of th. 3.2.}\nWe shall use the same notation as in the proof of Theorem~(\\ref{T:bound1}). In addition, we set\n\\[\n \\omega(K)= C(K) \\cdot (\\sqrt{p})^{\\delta + \\frac{1}{n}}.\n\\]\nA simple calculation using Corollary (\\ref{c:tau-min}) and formulas derived in the proof of Theorem~(\\ref{T:bound1}) gives\n\\[\n \\left( \\frac{\\tau(\\fok)}{n} \\right)^{\\frac{1}{2}} \\leq \\omega(K)\n\\]\nThen, by the inequality~(\\ref{E:gen-est}), we have\n\\[\n M(K) \\leq \\omega^n \\sqrt{D_K}.\n\\]\nIf $r \\geq 2$, then $\\delta + \\frac{1}{n} \\leq \\frac{2}{p}$ and hence\n\\[\n1 < (\\sqrt{p})^{\\delta + \\frac{1}{n}} \\leq \\sqrt[p]{p}.\n\\]\nConsequently, using Theorem~(\\ref{T:bound1}), we obtain\n\\begin{equation}\\label{E:omega-lim}\n\\lim_{p \\to \\infty} \\omega(K) = \\lim_{p \\to \\infty} C(K) \\cdot \\lim_{p \\to \\infty} (\\sqrt{p})^{\\delta + \\frac{1}{n}} = \\frac{1}{2\\sqrt{3}}.\n\\end{equation}\nMoreover, using Theorem~(\\ref{T:bound1}) and the fact that the sequence $\\{\\sqrt[p]{p}\\}_{p \\geq 3}$ is decreasing, we also get\n\\[\n\\omega(K) \\leq C(K) \\cdot \\sqrt[p]{p} \\leq \\frac{1}{3} \\sqrt[3]{3} = 3^{-2\/3}.\n\\]\nIf $r=1$, then $\\delta = 0$ and\n\\[\n (\\sqrt{p})^{\\delta + \\frac{1}{n}} = (\\sqrt{p^e})^{\\frac{1}{p-1}}.\n\\]\nThus assuming that $e$ is constant, we see that~(\\ref{E:omega-lim}) holds as well.\nThis concludes the proof of the theorem.\n\\bigskip\n\n\\section{Abelian fields of prime conductor}\n\nIf $K$ is an abelian field of conductor $p^r$ with $r \\ge 2$, then we have seen that\n $M(K) \\leq 2^{-n} \\sqrt{D_K}$, cf. (\\ref{T:Minkowski}).\nIn particular, if $K$ is totally real, then Minkowski's conjecture holds for $K$. If $r = 1$, that\nis if the conductor of $K$ is prime, then our results are less complete. The aim of this\nsection is to have a closer look at this case. As we will see, one can prove Minkowski's\nconjecture in a number of special cases when $K$ is totally real.\n\\subsection{Totally real fields}\nLet us consider the set $\\mathcal{S}_e$ of all totally real abelian fields of prime conductor such that $[\\cyclo : K]=e$, where $e$ is an even positive integer.\nThe Dirichlet prime number theorem implies that the set $\\mathcal{S}_e$ is infinite. By Theorem~(\\ref{T:bound2}), we have\n\\[\n \\lim_{p_K \\to \\infty} \\omega(K) = \\frac{1}{2\\sqrt{3}}.\n\\]\nIn particular, for each $e$ there exists $N=N(e)$ such that for every field $K \\in \\mathcal{S}_e$ with $p_K > N$ we have\n\\[\n \\omega(K) \\leq \\frac{1}{2}.\n\\]\nand hence Minkowski's conjecture holds for these fields. The next result shows that we can take $N(e)=2e^2$.\n\\begin{proposition}\\label{P:fixed-codim}\n Let $e$ be an even positive integer and $K \\in \\mathcal{S}_e$. If $p_K > 2e^2$, then\n\\[\n M(K) \\leq 2^{-n} \\sqrt{D_K}\n\\]\n\\end{proposition}\n\n\\noindent\n{\\it Proof.}\nWe shall use the same notation as in the proofs of theorems~(\\ref{T:bound1}) and (\\ref{T:bound2}). Additionally, let\n\\[\nT = \\set{(x,y) \\in \\br^2 \\mid x \\geq 2 \\textnormal{ and } y \\geq 2x+1}\n\\]\nand $h: T \\to \\br$ be a function given by\n\\[\n h(x,y) = \\frac{y^2+y+1-x^2}{3 y^2} \\cdot y^{\\frac{x}{y-1}}.\n\\]\nThen we have\n\\begin{equation*}\n \\omega(K) = C(K) \\cdot p^{1\/2n} = \\frac{1}{2} \\cdot \\sqrt{h(e,p)}.\n\\end{equation*}\nThus it is enough to show that $h(e,p) \\leq 1$ for $p > 2e^2$. We set\n\\begin{align*}\n h_1(x,y)&=\\frac{y^2+y+1-x^2}{3 y^2}\\\\\n h_2(x,y) &= y^{\\frac{x}{y-1}}.\n\\end{align*}\nFor every $x \\geq 2$ and every $y > 2(x^2-1)$ we have\n\\[\n \\frac{\\partial h_1}{\\partial y}(x,y)= \\frac{1}{3} \\left( \\frac{2(x^2-1)}{y^3} - \\frac{1}{y^2} \\right) < 0.\n\\]\nFurthermore, for every $x \\geq 2$ and every $y \\geq 2 x + 1$ we have\n\\[\n \\frac{\\partial h_2}{\\partial y}(x,y) = \\frac{x h_2(x,y)}{(y-1)^2} \\left( \\frac{y-1}{y} - \\ln y \\right) < 0.\n\\]\nConsequently, for every $x \\geq 2$ and every $i=1,2$ the function $y \\mapsto h_i(x,y)$ is positive and decreasing on the interval $[2x^2, \\infty)$. Hence the function\n$y \\mapsto h(x,y)$ is decreasing on the interval $[2x^2, \\infty)$. Moreover,\n\\[\n h(x,2x^2) = (2x^2)^{\\frac{x}{2x^2-1}} \\cdot \\frac{4x^4+x^2+1}{12x^4} < (2x^2)^{\\frac{1}{2x-1}} \\frac{1}{2} \\leq 1.\n\\]\nConsequently, $h(x,y) < 1$ for all $x \\geq 2$ and $y \\geq 2 x^2$. The result follows.\n\\bigskip\n\nWe can have $h(e,p) \\leq 1$ even if $p < 2e^2$, which in many cases allows us to show that Minkowski's conjecture holds for every $K \\in \\mathcal{S}_e$.\n\n\\begin{example}\n If $e \\leq 1202$ is an even integer, then Minkowski's conjecture holds for every $K \\in \\mathcal{S}_e$.\n\\end{example}\n\n\\noindent\n{\\it Proof.}\n If $p > 2 e^2$, then the result follows from Proposition~(\\ref{P:fixed-codim}). For $p < 2 e^2$, it has been verified using Magma Computational Algebra System~\\cite{magma} that either $h(e,p) \\leq 1$ or $n_K \\leq 8$.\nIn the first case, the result follows from the proof of Proposition~(\\ref{P:fixed-codim}). In the second case, it follows from the fact that Minkowski's conjecture is known to hold for fields of degree not exceeding $8$.\n\n\\bigskip The previous results are based on upper bounds of $\\tau_{\\rm min}(\\fok)$ obtained\nthrough the lattice $(\\fok,q)$.\nAnother approach is to estimate $\\tau_{\\rm min}(\\fok)$ using a scaling factor $\\alpha$,\ngiving rise to the lattice $(\\fok,q_{\\alpha})$, see\n\\S4. A computation shows that if $p < 400$, then for an appropriate $\\alpha \\in \\mathcal P$\nthe lattice $(\\fok,q_{\\alpha})$ is isomorphic to the unit lattice. Then \\cite{bayer}, Corollary (5.5)\nimplies that Minkowski's conjecture holds.\n\\bigskip\n\\subsection{Totally imaginary fields}\nIf $\\przystaje{p}{3}{4}$, then $K=\\bq(\\sqrt{-p}) \\in \\mathcal{A}$. Using formulas derived in the proof of Theorem~(\\ref{T:bound1}), we get\n\\[\n M(K) \\leq \\frac{(p+1)^2}{16 p}.\n\\]\nNote that this bound is known to be the exact\nvalue of the Euclidean minimum of $K$ (see for instance Proposition (4.2) in~\\cite{lemmermeyer}). In particular, the inequality\n$$\n M(K) \\leq 2^{-n} \\sqrt{D_K}\n$$\ndoes not hold in general for number fields that are not totally real. Just as in the totally real case, we have\n\\begin{equation}\\label{E:tot-im-1}\n M(K) \\leq 3^{-2n\/3} \\sqrt{D_K} < 2^{-n} \\sqrt{D_K}\n\\end{equation}\nfor all totally imaginary fields $K \\in \\mathcal{A}$ with composite conductors. If the conductor of $K$ is prime and \n$n_K > 2$, then by Theorem~(\\ref{T:bound1}) we have\n\\begin{equation}\\label{E:tot-im-2}\n M(K) \\leq 3^{-n} (\\sqrt{D_K})^{\\varepsilon}\n\\end{equation}\nwith $\\varepsilon < 2$. Note that given the asymptotic behavior of the expressions $\\varepsilon(K)$, $C(K)$, $\\omega(K)$,\nusing the formulas derived in the proofs of Theorems~(\\ref{T:bound1}) and (\\ref{T:bound2}) directly will often lead to better bounds.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently, the virtual element method (VEM) was proposed in \\cite{beirao2013basic} as a generalization of the finite element method (FEM) to general polygonal and polyhedral meshes. In VEMs, the local discrete spaces on the mesh polygons\/polyhedrons, called local virtual element spaces, consist of polynomials of certain degrees and some other non-polynomial functions that are solutions of specific partial differential equations. Although such functions are not defined explicitly, they are characterized by degrees of freedom, such as values at mesh vertices, the moments on mesh edges\/faces, and the moments on mesh polygons\/polyhedrons. On each (polygonal or polyhedral) element, the discrete bilinear form can be computed using only the degrees of freedom, and satisfies two properties, called consistency and stability. The consistency means that the discrete bilinear form is equal to the continuous bilinear form when one of the arguments is a polynomial, and the stability means that the discrete bilinear form is coercive for general virtual elements. Moreover, the virtual element spaces can be extended to arbitrary order in straightforward way. Because of such advantages, VEMs have been developed for many different types of equations, and successfully applied to various problems. For more thorough survey, we refer to \\cite{beirao2013basic,brezzi2014basic,ahmad2013equivalent,beirao2014hitchhiker,de2016nonconforming,cangiani2017conforming,beirao2016virtual,da2013virtual,zhang2019nonconforming,antonietti2018fully} and references therein. \n\nThere have appeared some results concering the VEMs for the Stokes problem as well. In \\cite{antonietti2014stream}, a stream formulation of the VEM for the Stokes problem was presented. In \\cite{cangiani2016nonconforming,liu2017nonconforming}, the nonconforming VEM of arbitrary order for the Stokes problem on polygonal and polyhedral meshes was first introduced. Therein, each component of the velocity is approximated by the nonconforming virtual element space presented in \\cite{de2016nonconforming}. However, the velocity approximation in \\cite{cangiani2016nonconforming,liu2017nonconforming} is not pointwise divergence-free, and it is merely divergence-free in a relaxed (projected) sense. \n\nIn the two-dimensional case, some researchers have developed VEMs for the Stokes problem in which the velocity approximation is pointwise divergence-free. In \\cite{da2017divergence}, the divergence-free velocity approximation is presented in the conforming virtual element space of order $k\\geq 2$. On each polygon, the virtual element space consists of velocity solutions of the local Stokes problem with Dirichlet boundary condition. On the other hand, the nonconforming virtual element space of arbitrary order was constructed by enriching a $\\vH(\\div)$-conforming virtual element space in \\cite{zhao2019divergence}. However, the proposed methods in \\cite{da2017divergence,zhao2019divergence} only showed that the computed velocity approximation is pointwise divergence-free. They do not discuss the construction of divergence-free basis functions. To the best of our knowledge, a formal construction of divergence-free bases in these VEMs has never been considered and developed. \n\nThe main goal of this paper is to present a formal construction of a divergence-free basis in the two-dimensional nonconforming VEMs for the Stokes problem introduced in \\cite{zhao2019divergence}. We first compute the dimension of the divergence-free subspace of the nonconforming virtual element space, using Euler's formula. We then construct basis functions of the subspace, in a similar fashion to the divergence-free basis functions proposed in \\cite{thomassetimplementation,brenner1990nonconforming} but we generalize to polygonal meshes and higher-order virtual elements. Using the construction of a divergence-free basis, we can eliminate the pressure variable from the coupled system and reduce the saddle point problem to a symmetric positive definite system having fewer unknowns in velocity variable only. Although we only consider the Stokes problem in this paper, we expect that our construction can be applied to more complicated problems, such as the incompressible Navier-Stokes problem.\n\nThe rest of this paper is organized as follows. In section 2, we state the stationary Stokes problem and its variational formulation. In section 3, we review the divergence-free nonconforming VEM for the Stokes problem introduced in \\cite{zhao2019divergence}. In section 4, we discuss a formal construction of divergence-free basis of the nonconforming virtual element space. In section 5, we discuss implementations including nonhomogeneous Dirichlet boundary conditions. In section 6, we offer some numerical experiments that verify the efficiency and the accuracy of our construction. Finally, conclusions are given in section 7.\n\n\\section{Model Problem}\n\nLet $\\Omega\\sus\\R^2$ be a bounded, convex polygonal domain with boundary $\\pd\\Omega$. We consider the Stokes problem on $\\Omega$: Given $\\vf:\\Omega\\to\\R^2$ and $\\vg:\\pd\\Omega\\to\\R^2$, find $\\vu:\\Omega\\to\\R^2$ and $p:\\Omega\\to\\R$ such that\n\\begin{eqn}\\label{eqn:Stokes123}\n\\left\\{\\begin{array}{rcl}\n-\\Delta\\vu + \\nabla p = \\vf & \\tr{in} & \\Omega, \\\\\n\\div\\vu = 0 & \\tr{in} & \\Omega, \\\\\n\\vu = \\vg & \\tr{on} & \\pd\\Omega.\n\\end{array}\\right.\n\\end{eqn}\n\nIn order to obtain the variational formulation of \\eqref{eqn:Stokes123}, we introduce the usual notation for Sobolev spaces, norms, seminorms, and inner products. Let $D$ be a bounded domain in $\\R^2$. We then define $\\vL^2(D) = [L^2(D)]^2$ and $\\vH^s(D) = [H^s(D)]^2$ for $s > 0$. The $L^2$-inner product of $L^2(D)$ and $\\vL^2(D)$ is denoted by $(\\cdot,\\cdot)_{0,D}$. Next, for $s \\geq 0$, the $H^s$-norm of $H^s(D)$ and $\\vH^s(D)$ is denoted by $\\|\\cdot\\|_{s,D}$. Similarly, for $s > 0$, the $H^s$-seminorm of $H^s(D)$ and $\\vH^s(D)$ is denoted by $|\\cdot|_{s,D}$. The subspace $L_0^2(D)$ of $L^2(D)$ is defined by\n\\begin{dis}\nL_0^2(D) = \\left\\{q\\in L^2(D) : \\int_Dq\\diff\\vx = 0\\right\\}.\n\\end{dis}\nLet us define\n\\begin{eqnarray*}\n\\vH_0^1(\\Omega) & = & \\left\\{\\vv\\in \\vH^1(\\Omega) : \\vv = \\zz \\ \\tr{on} \\ \\pd\\Omega\\right\\}, \\\\\n\\vH_{\\vg}^1(\\Omega) & = & \\left\\{\\vv\\in\\vH^1(\\Omega) : \\vv = \\vg \\ \\tr{on} \\ \\pd\\Omega\\right\\}.\n\\end{eqnarray*}\nThen the variational form of the Stokes problem \\eqref{eqn:Stokes123} is written as follows: For a given $\\vf\\in \\vL^2(\\Omega)$ and a given $\\vg\\in \\vH^{1\/2}(\\pd\\Omega)$ satisfying\n\\begin{eqn}\\label{eqn:StokesBdry}\n\\int_{\\pd\\Omega}\\vg\\cdot\\vn_{\\Omega}\\diff s = 0\n\\end{eqn}\nwhere $\\vn_{\\Omega}$ is the unit normal vector on $\\pd\\Omega$ in the outward direction with respect to $\\Omega$, find $\\vu\\in\\vH_{\\vg}^1(\\Omega)$ and $p\\in L_0^2(\\Omega)$ such that\n\\begin{eqn}\\label{eqn:StokesVarCon}\n\\left\\{\\begin{array}{rcll}\na(\\vu,\\vv) + b(\\vv,p) & = & (\\vf,\\vv)_{0,\\Omega} & \\forall \\vv\\in \\vH_0^1(\\Omega), \\\\\nb(\\vu,q) & = & 0 & \\forall q\\in L_0^2(\\Omega),\n\\end{array}\\right.\n\\end{eqn}\nwhere \n\\begin{eqn}\\label{eqn:StokesVarConBilinear}\na(\\vu,\\vv) = \\int_{\\Omega}\\nabla\\vv:\\nabla\\vu\\diff\\vx, \\quad b(\\vv,q) = -\\int_{\\Omega}q\\div\\vv\\diff\\vx.\n\\end{eqn}\nThe functions $\\vu$ and $p$ are called velocity and pressure, respectively. \n\n\n\\section{Divergence-Free Nonconforming VEM for the Stokes Problem}\\label{sec:DivFreeNCVEM}\n\nIn this section, we summarize some preliminaries and review the divergence-free nonconforming VEM for the Stokes problem introduced in \\cite{zhao2019divergence}.\n\n\\subsection{Notations and preliminaries}\n\nLet $\\{\\mc{P}_h\\}_h$ be a family of decompositions (meshes) of the domain $\\Omega$ into polygonal elements $K$ with maximum diameter $h$. We assume that the decompositions satisfy the following regularity properties \\cite{beirao2013basic,de2016nonconforming,da2017divergence,zhao2019divergence}.\n\\begin{assum}\nThere exists $\\rho > 0$ independent of $h$ such that\n\\begin{itemize\n\\item the decomposition $\\mc{P}_h$ consists of a finite number of nonoverlapping convex polygonal elements;\n\\item if $K\\in\\mc{P}_h$ and $e$ is an edge of $K$ then $h_e \\geq \\rho h_K$, where $h_e$ and $h_K$ denote the diameter of $e$ and $K$, respectively;\n\\item every element $K$ of $\\mc{P}_h$ is star-shaped with respect to the ball of radius $\\rho h_K$.\n\\end{itemize} \n\\end{assum}\nWe next define some notations for sets of mesh items. We denote by $\\mc{V}_{h}$ and $\\mc{E}_h$ the set of all mesh vertices and mesh edges in $\\mc{P}_h$, respectively. We also denote by $\\mc{V}^i_h$ and $\\mc{V}^\\pd_h$ the set of all mesh vertices in the internal and the boundary of $\\mc{P}_h$, respectively. Similarly $\\mc{E}_h^i$ is the set of all mesh edges in the internal of $\\mc{P}_h$, and $\\mc{E}_h^\\pd$ the set of all mesh edges in the boundary of $\\mc{P}_h$. We also define\n\\begin{eqnarray*}\nN_P & = & \\tr{the number of polygons in $\\mc{P}_h$}, \\\\\nN_E & = & \\tr{the number of edges in $\\mc{E}_h$}, \\\\\nN_{E,i} & = & \\tr{the number of edges in $\\mc{E}_h^i$}, \\\\\nN_{E,\\pd} & = & \\tr{the number of edges in $\\mc{E}_h^{\\pd}$}, \\\\\nN_V & = & \\tr{the number of vertices in $\\mc{V}_h$}, \\\\\nN_{V,i} & = & \\tr{the number of vertices in $\\mc{V}_h^i$}, \\\\\nN_{V,\\pd} & = & \\tr{the number of vertices in $\\mc{V}_h^{\\pd}$}.\n\\end{eqnarray*}\nFor each $K\\in\\mc{P}_h$, let $\\vn_K$ and $\\vt_K$ denote its exterior unit normal vector and counterclockwise tangential vector, respectively. Let $e\\in\\mc{E}_h^i$. We then define respectively $\\vn_e$ and $\\vt_e$ as the unit normal and tangential vector of $e$ with orientation fixed once and for all. Next let $e\\in\\mc{E}_h^{\\pd}$, we define respectively $\\vn_e$ and $\\vt_e$ as the unit normal and tangential vector on $e$ in the outward and counterclockwise direction with respect to $\\Omega$.\n\nLet $e\\in\\mc{E}_h^i$ and let $K^-$ and $K^+$ be the polygons in $\\mc{P}_h$ that have $e$ as a common edge, and satisfy $\\vn_e = \\vn_{K_+}$ on $e$ (i.e., $\\vn_e$ points from $K^+$ to $K^-$). If $e\\in\\mc{E}_h^{\\pd}$, we define $\\vn_e$ by the unit normal vector in the outward direction with respect to $\\Omega$.\n\nAgain let $e\\in\\mc{E}_h^i$ and let $K^-$ and $K^+$ are the polygons in $\\mc{P}_h$ having $e$ as a common edge. For $\\vv:\\Omega\\to\\R^2$ satisfying $\\vv|_{K^+}\\in \\vH^1(K^+)$ and $\\vv|_{K^-}\\in \\vH^1(K^-)$, we define the jump of $\\vv$ on the edge $e$ by\n\\begin{dis}\n[\\vv]_e = \\vv|_{K^+}(\\vn_e\\cdot\\vn_{K^+}) + \\vv|_{K^-}(\\vn_e\\cdot\\vn_{K^-}).\n\\end{dis}\nIf $e\\in\\mc{E}_h^{\\pd}$, we define $[\\vv]_e = \\vv|_e$. \n\nWe define the broken Sobolev space $\\vH^1(\\Omega;\\mc{P}_h)$ by\n\\begin{dis}\n\\vH^1(\\Omega;\\mc{P}_h) = \\left\\{\\vv\\in\\vL^2(\\Omega) : \\vv|_K\\in \\vH^1(K) \\ \\forall K\\in\\mc{P}_h\\right\\}\n\\end{dis}\nand define its norm and seminorm by\n\\begin{dis}\n\\|\\vv\\|_{1,h} = \\left(\\sum_{K\\in\\mc{P}_h}\\|\\vv\\|_{1,K}^2\\right)^{1\/2}, \\quad |\\vv|_{1,h} = \\left(\\sum_{K\\in\\mc{P}_h}|\\vv|_{1,K}^2\\right)^{1\/2}.\n\\end{dis}\nWe also define\n\\begin{dis}\n\\vH^{1,nc}(\\Omega;\\mc{P}_h) = \\left\\{\\vv\\in\\vH^1(\\Omega;\\mc{P}_h) : \\int_e[\\vv]_e\\cdot\\vq\\diff\\vx = 0 \\ \\forall \\vq\\in \\vP_{k-1}(e), \\ \\forall e\\in\\mc{E}_h^i\\right\\}.\n\\end{dis}\n\nLet $O$ be an $1$ or $2$ dimensional geometrical object (edge or polygon). For an integer $k\\geq 0$, $P_k(O)$ denotes the space of polynomials of degree $\\leq k$ on $O$. $M_k(O)$ denotes the set of scaled monomials of degree $\\leq k$ on $O$, that is,\n\\begin{dis}\nM_k(O) = \\left\\{\\left(\\frac{\\vx - \\vx_O}{h_O}\\right)^{\\valpha}: |\\valpha| \\leq k \\right\\},\n\\end{dis}\nwhere $\\vx$ is a local coordinate system on $O$, $\\vx_O$ is the barycenter of $O$ in the local coordinate system, $\\valpha$ is a multi-index, and $h_O = \\diam(O)$. \n\nConventionally we define $P_{-1}(O) = \\{0\\}$. We also define $\\vP_k(O) = (P_k(O))^2$ for $k\\geq -1$ and $\\vM_k(O) = (M_k(O))^2$ for any nonnegative integer $k$. \n\nLet $K\\in\\mc{P}_h$ and let $k$ be a nonnegative integer. We define $(\\nabla P_{k+1}(K))^\\oplus$ as the subspace of $\\vP_k(K)$ satisfying \n\\begin{dis}\n\\vP_k(K) = \\nabla P_{k+1}(K) \\oplus (\\nabla P_{k+1}(K))^\\oplus,\n\\end{dis}\nand denote by $\\vM_k^\\oplus$ a basis of the space $(\\nabla P_{k+1}(K))^\\oplus$.\nFor example, one can choose\n\\begin{dis}\n(\\nabla P_{k+1}(K))^\\oplus = \\vx^\\perp P_{k-1}(K), \\quad \\vM_k^\\oplus = \\left\\{m(\\vx)\\vx^{\\perp} : m\\in M_{k-1}(K)\\right\\},\n\\end{dis}\nwhere $\\vx^{\\perp} = (x_2,-x_1)$ with $\\vx = (x_1,x_2)$.\n\n\\subsection{Virtual element space}\n\nWe first define a local virtual element space on each element $K\\in\\mc{P}_h$. Let $k$ be a fixed positive integer. Let \n\\begin{dis}\n\\vW_h^1(K) := \\left\\{\\vv\\in \\vH^1(K) : \\div\\vv\\in P_{k-1}(K), \\ \\rot\\vv = 0, \\ \\vv\\cdot\\vn_K|_e\\in P_k(e), \\ \\forall e\\sus\\pd K\\right\\},\n\\end{dis}\nwhere $\\rot\\vv = \\frac{\\pd v_1}{\\pd x_2} - \\frac{\\pd v_2}{\\pd x_1}$ for $\\vv = (v_1,v_2)\\in\\vH^1(K)$. Also, let\n\\begin{dis}\n\\Phi_h(K) := \\left\\{\\phi\\in H^2(K) : \\Delta^2\\phi\\in P_{k-3}(K), \\ \\phi|_e = 0, \\ \\Delta\\phi|_e\\in P_{k-1}(e), \\ \\forall e\\sus\\pd K\\right\\}\n\\end{dis}\nwith the convention that $P_{-1}(K) = P_{-2}(K) = \\{0\\}$. In \\cite[Lemma 2]{zhao2019divergence}, it was shown that $\\vW_h^1(K) \\cap \\curl\\Phi_h(K) = \\{0\\}$, where $\\curl q = (-\\frac{\\pd q}{\\pd x_2},\\frac{\\pd q}{\\pd x_1})$ for $q\\in H^1(K)$. It was also shown in \\cite[Lemma 3]{zhao2019divergence} that if the local space $\\tilde{\\vV}_h(K)$ is defined by\n\\begin{dis}\n\\tilde{\\vV}_h(K) = \\vW_h^1(K) \\oplus \\curl\\Phi_h(K),\n\\end{dis}\nthen the following degrees of freedom (DOFs) are unisolvent for $\\tilde{\\vV}_h(K)$:\n\\begin{eqnarray*}\n\\tr{the moments} \\ \\frac{1}{|e|}\\int_e\\vv\\cdot\\vn_eq\\diff s, & \\quad & q \\in M_{k}(e), \\\\\n\\tr{the moments} \\ \\frac{1}{|e|}\\int_e\\vv\\cdot\\vt_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\\\\n\\tr{the moments} \\ \\frac{1}{|K|}\\int_K\\vv\\cdot\\vq\\diff\\vx, & \\quad & \\vq \\in \\vM_{k-2}(K).\n\\end{eqnarray*}\nWe define a local projection $\\Pi^{\\nabla}_K:\\vH^1(K)\\to \\vP_k(K)$ on each polygon $K$ in $\\mc{P}_h$. It is defined by\n\\begin{eqnarray*}\n\\int_K\\nabla(\\Pi^{\\nabla}_K\\vv):\\nabla\\vq\\diff\\vx & = & \\int_K\\nabla\\vv:\\nabla\\vq\\diff\\vx, \\quad \\forall \\vv\\in\\vH^1(K), \\ \\forall \\vq\\in\\vP_k(K), \\\\\n\\int_{\\pd K}\\Pi^{\\nabla}_K\\vv\\diff s & = & \\int_{\\pd K}\\vv\\diff s,\n\\end{eqnarray*}\nfor $\\vv\\in \\vH^1(K)$. Note that $\\Pi_K^\\nabla\\vq = \\vq$ for any $\\vq\\in \\vP_k(K)$ and the local projection $\\Pi_K^\\nabla$ is computable using only the moments of $\\vv$ up to order $(k-1)$ on each edge $e\\sus \\pd K$ and the moments of $\\vv$ up to order $(k-2)$ on $K$. \n\nNow the local nonconforming virtual element space $\\vV_h(K)$ on $K$ is defined by\n\\begin{dis}\n\\vV_h(K) = \\left\\{\\vv\\in\\tilde{\\vV}_h(K) : \\int_e(\\vv - \\Pi^{\\nabla}_K\\vv)\\cdot\\vn_eq\\diff s = 0, \\ \\forall q\\in P_k(e)\/P_{k-1}(e), \\ \\forall e\\sus \\pd K\\right\\},\n\\end{dis}\nwhere $P_k(e)\/P_{k-1}(e)$ is the subspace of polynomials in $P_k(e)$ that are $L^2(e)$-orthogonal to $P_{k-1}(e)$. It was shown in \\cite{zhao2019divergence} that the following DOFs are unisolvent for $\\vV_h(K)$:\n\\begin{eqnarray*}\n\\frac{1}{|e|}\\int_e\\vv\\cdot\\vn_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\\\\n\\frac{1}{|e|}\\int_e\\vv\\cdot\\vt_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\\\\n\\frac{1}{|K|}\\int_K\\vv\\cdot\\vq\\diff\\vx, & \\quad & \\vq \\in \\vM_{k-2}(K).\n\\end{eqnarray*}\nFor each $i = 1,2,\\cdots,N_K := \\dim\\vV_h(K)$, let $\\chi_i$ be the operator associated to the $i$-th local DOF. Then for any $\\vv\\in\\vH^1(K)$ there exists a unique element $I_h^K\\vv\\in\\vV_h(K)$ such that \n\\begin{dis}\n\\chi_i(\\vv - I_h^K\\vv) = 0 \\quad \\forall i = 1,2,\\cdots,N_K.\n\\end{dis}\nThe operator $\\vv\\mapsto I_h^K\\vv$ is called a local interpolation operator for $\\vV_h(K)$. It was shown in \\cite{zhao2019divergence} that we can obtain the following interpolation error estimates.\n\n\\begin{spro}[see {\\cite[Lemma 6]{zhao2019divergence}}]\nThere exists a positive constant $C$ independent of $h$ such that for every $K\\in\\mc{P}_h$ and every $\\vv\\in\\vH^s(K)$ with $1\\leq s\\leq k+1$,\n\\begin{dis}\n\\|\\vv - I_h^K\\vv\\|_{0,K} + h|\\vv - I_h^K\\vv|_{1,K} \\leq Ch^{s}|\\vv|_{s,K}.\n\\end{dis}\n\\end{spro}\n\nThe global nonconforming virtual element spaces are defined as follows:\n\\begin{eqnarray*}\n\\vV_h & = & \\left\\{\\vv_h\\in \\vL^2(\\Omega) : \\vv_h|_K\\in \\vV_h(K) \\quad \\forall K\\in\\mc{P}_h, \\ \\int_e[\\vv_h]_e\\cdot\\vq\\diff s = 0 \\quad \\forall \\vq\\in \\vP_{k-1}(e), \\ \\forall e\\in\\mc{E}_h^i\\right\\}, \\\\\n\\vV_{h,0} & = & \\left\\{\\vv_h\\in \\vL^2(\\Omega) : \\vv_h|_K\\in \\vV_h(K) \\quad \\forall K\\in\\mc{P}_h, \\ \\int_e[\\vv_h]_e\\cdot\\vq\\diff s = 0 \\quad \\forall \\vq\\in \\vP_{k-1}(e), \\ \\forall e\\in\\mc{E}_h\\right\\}.\n\\end{eqnarray*}\nThe global DOFs for $\\vV_h$ can be chosen as, for any edge $e$ and polygon $K$ in $\\mc{P}_h$,\n\\begin{eqnarray}\n\\chi_{e,q}^n(\\vv_h) := \\frac{1}{|e|}\\int_e\\vv_h\\cdot\\vn_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\label{eqn:EdgeNDOFG} \\\\\n\\chi_{e,q}^t(\\vv_h) := \\frac{1}{|e|}\\int_e\\vv_h\\cdot\\vt_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\label{eqn:EdgeTDOFG} \\\\\n\\chi_{K,\\vq}(\\vv_h) := \\frac{1}{|K|}\\int_K\\vv_h\\cdot\\vq\\diff\\vx, & \\quad & \\vq \\in \\nabla M_{k-1}(K) + \\vM_{k-2}^{\\oplus}(K). \\label{eqn:CellDOFG}\n\\end{eqnarray}\nSimilarly, the global DOFs for $\\vV_{h,0}$ can be chosen. We also define the global interpolation operator $I_h:\\vH^{1,nc}(\\Omega;\\mc{P}_h)\\to \\vV_h$ by $(I_h\\vv)|_K = I_h^K(\\vv|_K)$ for each $K\\in\\mc{P}_h$ and $\\vv\\in\\vH^{1,nc}(\\Omega;\\mc{P}_h)$.\n\nThe discrete pressure space $Q_h$ is defined by\n\\begin{dis}\nQ_h = \\{q_h\\in L^2_0(\\Omega) : q_h|_K\\in P_{k-1}(K) \\ \\forall K\\in\\mc{P}_h\\}.\n\\end{dis}\nThe global DOFs for the space $Q_h$ can be chosen as\n\\begin{dis}\n\\frac{1}{|K|}\\int_Kq_h\\phi \\diff\\vx, \\quad \\phi\\in M_{k-1}(K), \\ K\\in\\mc{P}_h.\n\\end{dis}\nIt was shown in \\cite{zhao2019divergence} that $\\div\\vV_h(K)\\sus P_{k-1}(K)$ for each $K\\in\\mc{P}_h$, and $\\div_h\\vV_{h,0}\\sus Q_h$, where $\\div_h$ denotes the discrete divergence operator defined by $(\\div_h\\vv_h)|_K = \\div(\\vv_h|_K)$ for each $K\\in\\mc{P}_h$ and $\\vv_h\\in\\vV_h$. Therefore, the nonconforming virtual element space $\\vV_h$ is divergence-free. \n\n\\subsection{The discrete problem}\n\nWe define a local discrete bilinear form $a_h^K$ for each polygon $K$ in $\\mc{P}_h$, as follows.\n\\begin{dis}\na_h^K(\\vv_h,\\vw_h) = a^K(\\Pi_K^\\nabla(\\vv_h),\\Pi_K^\\nabla(\\vw_h)) + S^K((I - \\Pi_K^\\nabla)\\vv_h,(I - \\Pi_K^\\nabla)\\vw_h), \\quad \\vv_h,\\vw_h\\in \\vV_h(K),\n\\end{dis}\nwhere $a^K$ is the bilinear form defined by\n\\begin{dis}\na^K(\\vv,\\vw) = \\int_K\\nabla\\vv:\\nabla\\vw\\diff\\vx, \\quad \\vv,\\vw\\in\\vH^1(K),\n\\end{dis}\nand $S^K$ is a symmetric positive definite bilinear form defined as\n\\begin{dis}\nS^K(\\vv_h,\\vw_h) = \\sum_{i=1}^{N_K}\\chi_i(\\vv_h)\\chi_i(\\vw_h), \\quad \\vv_h,\\vw_h\\in \\vV_h(K),\n\\end{dis}\nwhere $N_K = \\dim(\\vV_h(K))$ and $\\chi_i$ denotes the operator associated to the $i$-th local DOF for $i = 1,2,\\cdots,N_K$. As described in \\cite{beirao2013basic,zhao2019divergence}, we obtain the $k$-consistency and stability of $a_h^K$:\n\\begin{itemize}\n\\item ($k$-consistency) $a_h^K(\\vq,\\vv_h) = a^K(\\vq,\\vv_h)$ for any $\\vq\\in \\vP_k(K)$, $\\vv_h\\in\\vV_h(K)$;\n\\item (stability) there exist constants $c_*,c^*>0$ independent of $h$ such that\n\\begin{dis}\nc_*a^K(\\vv_h,\\vv_h) \\leq a_h^K(\\vv_h,\\vv_h) \\leq c^*a^K(\\vv_h,\\vv_h) \\quad \\forall \\vv_h\\in \\vV_h(K).\n\\end{dis}\n\\end{itemize}\n\nThe global bilinear form $a_h$ is defined by\n\\begin{dis}\na_h(\\vv_h,\\vw_h) = \\sum_{K\\in\\mc{P}_h}a_h^K(\\vv_h,\\vw_h), \\quad \\vv_h,\\vw_h\\in \\vV_h.\n\\end{dis}\nOn the other hand, the discrete bilinear form $b_h$ is simply defined by\n\\begin{dis}\nb_h(\\vv_h,q_h) = \\sum_{K\\in\\mc{P}_h}b^K(\\vv_h,q_h), \\quad \\vv_h\\in\\vV_h,\\ q_h\\in Q_h,\n\\end{dis}\nwhere\n\\begin{dis}\nb^K(\\vv,q) = -\\int_Kq\\div\\vv\\diff\\vx,\n\\end{dis}\nfor $\\vv\\in\\vH^1(K)$, $q\\in P_{k-1}(K)$, and $K\\in\\mc{P}_h$. Note that $b_h(\\vv_h,q_h)$ is also computable using only the DOFs \\eqref{eqn:EdgeNDOFG}-\\eqref{eqn:CellDOFG} and we do not rely on the discrete version of it, indeed we omit the subscript $h$ on such bilinear form.\n\nWe next discretize the right-hand side $(\\vf,\\cdot)_{0,\\Omega}$ as follows:\n\\begin{dis}\n\\inn{\\vf_h,\\vv_h} = \\left\\{\\begin{array}{ll}\n(\\vf_h,\\ol{\\vv}_h)_{0,\\Omega} & \\tr{if $k = 1$} \\\\\n(\\vf_h,\\vv_h)_{0,\\Omega} & \\tr{if $k > 1$}\n\\end{array}\\right., \\quad \\vv_h\\in\\vV_h,\n\\end{dis}\nwhere $\\vf_h,\\ol{\\vv}_h\\in \\vL^2(\\Omega)$ are defined by\n\\begin{dis}\n\\vf_h|_K = \\left\\{\\begin{array}{ll}\n\\Pi_0^K\\vf & \\tr{if $k = 1$} \\\\\n\\Pi_{k-2}^K\\vf & \\tr{if $k > 1$}\n\\end{array}\\right., \\quad \\ol{\\vv}_h|_K = \\frac{1}{|\\pd K|}\\int_{\\pd K}\\vv_h\\diff s, \\quad K\\in\\mc{P}_h.\n\\end{dis}\nHere, $\\Pi_{\\ell}^K$ denotes the $L^2$-projection operator onto $\\vP_{\\ell}(K)$ for each $K\\in\\mc{P}_h$.\n\nIn order to consider the nonhomogeneous Dirichlet boundary condition, let\n\\begin{dis}\n\\vV_{h,\\vg} = \\left\\{\\vv_h\\in\\vV_h : \\int_e\\vg\\cdot\\vq\\diff s = \\int_e\\vv_h\\cdot\\vq\\diff s, \\ \\forall \\vq\\in \\vP_{k-1}(e), \\ \\forall e\\in\\mc{E}_h^{\\pd}\\right\\}.\n\\end{dis}\nWe formulate the nonconforming VEM for the Stokes problem \\eqref{eqn:StokesVarCon} as follows: Find $\\vu_h\\in \\vV_{h,\\vg}$ and $p_h\\in Q_h$ such that\n\\begin{eqn}\\label{eqn:StokesDis}\n\\left\\{\\begin{array}{rcll}\na_h(\\vu_h,\\vv_h) + b_h(\\vv_h,p_h) & = & \\inn{\\vf_h,\\vv_h} & \\forall \\vv_h\\in\\vV_{h,0}, \\\\\nb_h(\\vu_h,q_h) & = & 0 & \\forall q_h\\in Q_h.\n\\end{array}\\right.\n\\end{eqn}\nHere $\\vu_h$ and $p_h$ will be called discrete velocity and discrete pressure, respectively. It was shown in \\cite{zhao2019divergence} that the discrete problem \\eqref{eqn:StokesDis} is well-posed. Moreover, for the case $\\vg = \\zz$, we can obtain the following error estimate.\n\n\\begin{theorem}[see {\\cite[Theorem 13]{zhao2019divergence}}]\\label{thm:Error}\nSuppose that $\\vf\\in \\vH^{k-1}(\\Omega)$ and $\\vg = \\zz$. Let $(\\vu,p)\\in (\\vH_{0}^1(\\Omega)\\cap \\vH^{k+1}(\\Omega))\\times(L_0^2(\\Omega)\\cap H^k(\\Omega))$ be the solution of the continuous problem \\eqref{eqn:StokesVarCon}. Let $(\\vu_h,p_h)\\in\\vV_{h,0}\\times Q_h$ be the solution of the discrete problem \\eqref{eqn:StokesDis}. Then \n\\begin{dis}\n\\abs{\\vu - \\vu_h}_{1,h} + \\|p - p_h\\|_{0,\\Omega} \\leq Ch^k\\left(|\\vu|_{k+1,\\Omega} + |p|_{k,\\Omega} + |\\vf|_{k-1,\\Omega}\\right),\n\\end{dis}\nwhere $C$ is a positive constant independent on $h$. \n\\end{theorem}\n\n\\section{A Formal Construction of Divergence-Free Basis}\\label{sec:NCVEM1}\n\nIn this section, we present a formal construction of a divergence-free basis for the virtual element space $\\vV_{h,0}$. \n\nWe first define the canonical basis associated with the DOFs \\eqref{eqn:EdgeNDOFG}-\\eqref{eqn:CellDOFG} of the space $\\vV_h$. Recall that the global DOFs of $\\vV_h$ are given by\n\\begin{eqnarray*}\n\\chi_{e,q}^n(\\vv_h) = \\frac{1}{|e|}\\int_e\\vv_h\\cdot\\vn_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\ e\\in\\mc{E}_h, \\\\\n\\chi_{e,q}^t(\\vv_h) = \\frac{1}{|e|}\\int_e\\vv_h\\cdot\\vt_eq\\diff s, & \\quad & q \\in M_{k-1}(e), \\ e\\in\\mc{E}_h, \\\\\n\\chi_{K,\\vq}(\\vv_h) = \\frac{1}{|K|}\\int_K\\vv_h\\cdot\\vq\\diff\\vx, & \\quad & \\vq \\in \\nabla M_{k-1}(K) + \\vM_{k-2}^{\\oplus}(K), \\ K\\in\\mc{P}_h.\n\\end{eqnarray*}\nWe sometimes write $\\chi$ to denote $\\chi_{e,q}^n$, $\\chi_{e,q}^t$, or $\\chi_{K,\\vq}$ when it is clear from the context. Using these notations, we define the canonical basis functions of $\\vV_h$ associated to the DOFs \\eqref{eqn:EdgeNDOFG}-\\eqref{eqn:CellDOFG} as follows:\n\\begin{itemize}\n\\item For $e\\in\\mc{E}_h$ and $q\\in M_{k-1}(e)$, let $\\vvph_{e,q}^n$ be the function in $\\vV_h$ such that $\\chi_{e,q}^n(\\vvph_{e,q}^n) = 1$ and $\\chi(\\vvph_{e,q}^n) = 0$ for all other DOFs.\n\\item For $e\\in\\mc{E}_h$ and $q\\in M_{k-1}(e)$, let $\\vvph_{e,q}^t$ be the function in $\\vV_h$ such that $\\chi_{e,q}^t(\\vvph_{e,q}^t) = 1$ and $\\chi(\\vvph_{e,q}^t) = 0$ for all other DOFs.\n\\item For $K\\in\\mc{P}_h$ and $\\vq\\in(\\nabla M_{k-1}(K)) + \\vM_{k-2}^{\\oplus}(K)$, let $\\vvph_{K,\\vq}$ be the function in $\\vV_h$ such that $\\chi_{K,\\vq}(\\vvph_{K,\\vq}) = 1$ and $\\chi = 0$ for all other DOFs. \n\\end{itemize}\n\nLet us define\n\\begin{dis}\n\\vZ_{h} = \\left\\{\\vv_h\\in\\vV_{h} : \\div_h\\vv_h = 0\\right\\}, \\quad \\vZ_{h,0} = \\left\\{\\vv_h\\in\\vV_{h,0} : \\div_h\\vv_h = 0\\right\\}.\n\\end{dis}\nWe first compute the dimension of $\\vZ_{h,0}$. \n\n\\begin{spro}\\label{prop:DimZh}\nThe dimension of $\\vZ_{h,0}$ is \n\\begin{dis}\nN_{V,i} + kN_{E,i} + (k-1)N_{E,i} + \\frac{(k-1)(k-2)}{2}N_{P}.\n\\end{dis}\n\\end{spro}\n\n\\begin{proof}\nSince $\\div_h\\vV_{h,0} = Q_h$ and since $\\div_h\\vV_{h,0} \\cong \\vV_{h,0}\/\\vZ_{h,0}$, we obtain\n\\begin{dis}\n\\dim\\vZ_{h,0} = \\dim\\vV_{h,0} - \\dim\\left(\\div_h\\vV_{h,0}\\right) = \\dim\\vV_{h,0} - \\dim Q_h.\n\\end{dis}\nNote that\n\\begin{dis}\n\\dim\\vV_{h,0} = 2\\left(\\frac{k(k-1)}{2}N_{P} + kN_{E,i}\\right), \\quad \\dim Q_{h} = \\frac{k(k+1)}{2}N_P - 1,\n\\end{dis}\nSince $N_{P} - N_{E,i} + N_{V,i} = 1$ from Euler's formula, we have\n\\begin{eqnarray*}\n\\dim\\vZ_{h,0} & = & \\dim\\vV_{h,0} - \\dim Q_{h} = k(k-1)N_{P} + 2kN_{E,i} - \\frac{k(k+1)}{2}N_{P} + 1 \\\\\n& = & k(k-1)N_{P} + 2kN_{E,i} - \\frac{k(k+1)}{2}N_{P} + N_{P} - N_{E,i} + N_{V,i} \\\\\n& = & N_{V,i} + kN_{E,i} + (k-1)N_{E,i} + \\frac{(k-1)(k-2)}{2}N_{P}.\n\\end{eqnarray*}\nThis concludes the proof of the proposition.\n\\end{proof}\n\nIn order to construct a basis of $\\vZ_{h,0}$, we first define some functions in $\\vV_{h}$. \n\\begin{enumerate}[\\bfseries\\sffamily (D1)]\n\\item For each vertex $v\\in\\mc{V}_{h}$, let $e_1,\\cdots,e_l$ be the edges in $\\mc{E}_h$ having $v$ as an end point, and let $K_1,\\cdots,K_l$ be the elements in $\\mc{P}_h$ having $v$ as a vertex. For each $i = 1,\\cdots,l$, let $\\vn_{e_i,v}$ be a unit vector normal to $e_i$ pointing in the counterclockwise direction with respect to the vertex $v$ (see \\Cref{fig:ConstructD1}). Define $\\vpsi_v\\in\\vV_h$ by\n\\begin{dis}\n\\vpsi_v := h\\sum_{i=1}^l\\frac{\\inn{\\vn_{e_i},\\vn_{e_i,v}}}{|e_i|}\\vvph_{e_i,1}^n + \\sum_{j=1}^l\\sum_{q\\in M_{k-1}(K_j)\\setminus\\{1\\}}c_{j,q}\\vvph_{K_j,\\nabla q},\n\\end{dis}\nwhere\n\\begin{dis}\nc_{j,q} = \\frac{h}{|K_j|}\\sum_{i=1}^l\\frac{\\inn{\\vn_{e_i},\\vn_{e_i,v}}}{|e_i|}\\int_{e_i\\cap \\pd K_j}\\vvph_{e_i,1}^n\\cdot\\vn_{K_j}q\\diff s\n\\end{dis}\nfor $q\\in M_{k-1}(K_j)\\setminus\\{1\\}$ and $j = 1,\\cdots,l$.\n\\item For each edge $e\\in\\mc{E}_{h}$ and each $q\\in M_{k-1}(e)$, define $\\vpsi_{e,q}^t$ by\n\\begin{dis}\n\\vpsi_{e,q}^t = \\vvph_{e,q}^t.\n\\end{dis}\n\\item Assume $k \\geq 2$. For each edge $e\\in\\mc{E}_{h}$ and each $q\\in M_{k-1}(e)\\setminus\\{1\\}$, define $\\vpsi_{e,q}^n$ by\n\\begin{dis}\n\\vpsi_{e,q}^n = \\vvph_{e,q}^n + \\sum_{K\\in\\mc{P}_h}\\sum_{r\\in M_{k-1}(K)\\setminus\\{1\\}}c_{K,r}\\vvph_{K,\\nabla r},\n\\end{dis}\nwhere\n\\begin{dis}\nc_{K,r} = \\frac{1}{|K|}\\int_{e\\cap \\pd K}\\vvph_{e,q}^n\\cdot\\vn_Kr\\diff s, \\quad r\\in M_{k-1}(K)\\setminus\\{1\\}, \\ K\\in\\mc{P}_h.\n\\end{dis}\n(See \\Cref{fig:ConstructD3}.)\n\\item Assume $k \\geq 3$. For each $K\\in\\mc{P}_h$ and each $\\vq \\in \\vM_k^\\oplus$, define $\\vpsi_{K,\\vq}$ by\n\\begin{dis}\n\\vpsi_{K,\\vq} = \\vvph_{K,\\vq}.\n\\end{dis}\n\\end{enumerate}\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{minipage}[b]{.4\\linewidth}\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick] (0,0) -- (0.4,-0.8) -- (1,-1) -- (2,-0.5) -- (1.5,0.5) -- (0.9,0.8) -- (0,0);\n\\draw[thick] (0.9,0.8) -- (0.5,1.4) -- (-0.5,1.4) -- (-1,0.8) -- (-1.2,0.2) -- (0,0);\n\\draw[thick] (-1.2,0.2) -- (-1.5,-0.1) -- (-1.5,-0.8) -- (-1.1,-1.2) -- (-0.6,-1.2) -- (0.4,-0.8);\n\n\\draw[thick, ->](0.2,-0.4) -- (0.6,-0.2);\n\\draw[thick, ->](-0.6,0.1) -- (-0.675,-0.35);\n\\draw[thick, ->](0.45,0.4) -- (0.05,0.85);\n\n\\fill (0,0) circle [radius = 0.075];\n\n\\node at (-0.1,0.2) {$v$};\n\n\\draw (0.966,-0.166) circle (1.5pt);\n\\draw (-0.15,0.633) circle (1.5pt);\n\\draw (-0.6,-0.55) circle (1.5pt);\n\n\\node at (0.1,-0.6){\\footnotesize{$e_1$}};\n\\node at (0.6,0.25){\\footnotesize{$e_2$}};\n\\node at (-0.5,0.25){\\footnotesize{$e_3$}};\n\n\\node at (0.75,-0.55){\\footnotesize{$\\vn_{e_1,v}$}};\n\\node at (0.05,1){\\footnotesize{$\\vn_{e_2,v}$}};\n\\node at (-1.07,-0.3){\\footnotesize{$\\vn_{e_3,v}$}};\n\\end{tikzpicture}\n\\end{center}\n\\subcaption{}\n\\label{fig:ConstructD1}\n\\end{minipage}\n\\begin{minipage}[b]{.4\\linewidth}\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick] (0,0) -- (0.4,-0.8) -- (1.5,-1) -- (2,0.3) -- (0.9,0.8) -- (0,0);\n\\draw[thick] (0,0) -- (-1,0.2) -- (-1.4,-0.6) -- (-1.2,-1.2) -- (-0.5,-1.5) -- (0.4,-0.8) -- (0,0);\n\n\\draw[thick, ->] (0.2,-0.4) -- (0.6,-0.2);\n\n\\node at (0.0,-0.5) {$e$};\n\\node at (0.7,-0.5) {\\footnotesize{$\\vn_{e}$}};\n\n\\draw (0.96,-0.14) circle (1.5pt);\n\\draw (-0.62,-0.65) circle (1.5pt);\n\\end{tikzpicture}\n\\end{center}\n\\subcaption{}\n\\label{fig:ConstructD3}\n\\end{minipage}\n\\caption{Examples of functions defined in (D1) (left) and (D3) (right).}\n\\label{fig:ConstructD1D3}\n\\end{figure}\n\n\\begin{remark}\nThe coefficients $c_{j,q}$ and $c_{K,r}$ defined in (D1) and (D3) are exactly computable using the DOFs \\eqref{eqn:EdgeNDOFG}-\\eqref{eqn:CellDOFG}. Moreover, since it is computed elementwise, the cost of computing the coefficients is negligible.\n\\end{remark}\n\nWe first show that the functions defined in (D1)-(D4) are indeed contained in $\\vZ_h$. \n\n\\begin{lemma}\\label{lem:D1D4DivFree}\nThe functions defined in (D1)-(D4) are contained in $\\vZ_h$. \n\\end{lemma}\n\n\\begin{proof}\nSince $\\div\\vV_h(K)\\sus P_{k-1}(K)$ for each $K\\in\\mc{P}_h$, if $\\vv_h\\in\\vV_h$ then\n\\begin{eqn}\\label{eqn:DivFreeIFF}\n\\vv_h\\in\\vZ_h \\quad \\tr{if and only if} \\quad \\int_Kq\\div\\vv_h\\diff\\vx = 0 \\ \\forall q\\in M_{k-1}(K), \\ \\forall K\\in\\mc{P}_h.\n\\end{eqn}\nFrom \\eqref{eqn:DivFreeIFF}, the functions in (D2) and (D4) are obviously contained in $\\vZ_h$. We first show that the functions in (D1) belong to $\\vZ_h$. Note that\n\\begin{dis}\n\\int_Kq\\div\\vpsi_v\\diff\\vx = 0 \\quad \\forall q\\in M_{k-1}(K), \\ \\forall K\\in\\mc{P}_h \\ \\tr{with} \\ K\\neq K_1,\\cdots,K_l.\n\\end{dis}\nLet $j = 1,\\cdots,l$. Since $K_j$ is a polygon having $v$ as a vertex, there are exactly two edges $e_{i_1,v}$ and $e_{i_2,v}$ with $1\\leq i_1,i_2 \\leq l$ such that $e_{i_1,v},e_{i_2,v}\\sus \\pd K_j$. Moreover, one of the normal vectors $\\vn_{e_1,v}$ and $\\vn_{e_2,v}$ coincides with $\\vn_{K_j}$, and the other has the opposite direction of $\\vn_{K_j}$. We may assume that $\\vn_{e_{i_1,v}} = \\vn_{K_j}|_{e_{i_1}}$ and $\\vn_{e_{i_2,v}} = - \\vn_{K_j}|_{e_{i_2}}$. Then\n\\begin{eqnarray*}\n&& \\int_{K_j}\\div\\vpsi_v\\diff\\vx = \\int_{\\pd K_j}\\vpsi_v\\cdot\\vn_{K_j}\\diff s \\\\\n& = & h\\frac{\\inn{\\vn_{e_{i_1}},\\vn_{e_{i_1,v}}}}{|e_{i_1}|}\\int_{e_{i_1}}\\vvph_{e_{i_1},1}^n\\cdot\\vn_{K_j}|_{e_{i_1}}\\diff s + h\\frac{\\inn{\\vn_{e_{i_2}},\\vn_{e_{i_2,v}}}}{|e_{i_2}|}\\int_{e_{i_2}}\\vvph_{e_{i_2},1}^n\\cdot\\vn_{K_j}|_{e_{i_2}}\\diff s \\\\\n& = & h\\inn{\\vn_{e_{i_1}},\\vn_{e_{i_1,v}}}\\inn{\\vn_{K_j}|_{e_{i_1}},\\vn_{e_{i_1}}} + h\\inn{\\vn_{e_{i_2}},\\vn_{e_{i_2,v}}}\\inn{\\vn_{K_j}|_{e_{i_2}},\\vn_{e_{i_2}}} \\\\\n& = & h\\inn{\\vn_{e_{i_1}},\\vn_{e_{i_1,v}}}\\inn{\\vn_{e_{i_1,v}},\\vn_{e_{i_1}}} - h\\inn{\\vn_{e_{i_2}},\\vn_{e_{i_2,v}}}\\inn{\\vn_{e_{i_2,v}},\\vn_{e_{i_2}}} \\\\\n& = & 0.\n\\end{eqnarray*}\nSuppose $q\\in M_{k-1}(K_j)\\setminus\\{1\\}$. Then\n\\begin{eqnarray*}\n&& \\int_{K_j}q\\div\\vpsi_v\\diff\\vx \\\\\n& = & \\int_{\\pd K_j}q\\vpsi_v\\cdot\\vn_K\\diff s - \\int_{K_j}\\vpsi_v\\cdot\\nabla q\\diff\\vx \\\\\n& = & h\\sum_{i=1}^l\\frac{\\inn{\\vn_{e_i},\\vn_{e_i,v}}}{|e_i|}\\int_{e_i\\cap \\pd K_j}q\\vvph_{e_i,1}^n\\cdot\\vn_{K_j}\\diff s - \\sum_{q'\\in M_{k-1}(K_j)\\setminus\\{1\\}}c_{j,q'}\\int_{K_j}\\vvph_{K_j,\\nabla q'}\\cdot\\nabla q\\diff\\vx \\\\\n& = & h\\sum_{i=1}^l\\frac{\\inn{\\vn_{e_i},\\vn_{e_i,v}}}{|e_i|}\\int_{e_i\\cap \\pd K_j}q\\vvph_{e_i,1}^n\\cdot\\vn_{K_j}\\diff s - c_{j,q}\\int_{K_j}\\vvph_{K_j,\\nabla q}\\cdot\\nabla q\\diff\\vx \\\\\n& = & h\\sum_{i=1}^l\\frac{\\inn{\\vn_{e_i},\\vn_{e_i,v}}}{|e_i|}\\int_{e_i\\cap \\pd K_j}q\\vvph_{e_i,1}^n\\cdot\\vn_{K_j}\\diff s - |K_j|c_{j,q} \\\\\n& = & 0.\n\\end{eqnarray*}\nHere we used the relations\n\\begin{dis}\n\\int_{K_j}\\vvph_{K_j,\\nabla q'}\\cdot\\nabla q\\diff\\vx = \\left\\{\\begin{array}{ll}\n|K_j| & \\tr{if $q = q'$} \\\\\n0 & \\tr{if $q \\neq q'$}.\n\\end{array}\\right.\n\\end{dis}\nThus $\\vpsi_v\\in \\vZ_{h}$. We next show that the functions $\\vpsi_{e,q}^n$ in (D3) belong to $\\vZ_h$. Note that $c_{K,r} = 0$ for any $r\\in M_{k-1}(K)\\setminus\\{1\\}$ and any $K\\in\\mc{P}_h$ with $e\\not\\sus\\pd K$. Then\n\\begin{eqnarray*}\n\\int_Kr\\div\\vpsi_{e,q}^n\\diff\\vx & = & \\int_{\\pd K}r\\vpsi_{e,q}^n\\cdot\\vn_K\\diff s - \\int_K\\vpsi_{e,q}^n\\cdot\\nabla r\\diff\\vx \\\\\n& = & \\int_{\\pd K}r\\vpsi_{e,q}^n\\cdot\\vn_K\\diff s = 0\n\\end{eqnarray*}\nfor any $r\\in M_{k-1}(K)\\setminus\\{1\\}$ and $K\\in\\mc{P}_h$ with $e\\not\\sus\\pd K$. We next suppose that $K\\in\\mc{P}_h$ satisfies $e\\sus\\pd K$. Since $q\\neq 1$,\n\\begin{dis}\n\\int_K\\div\\vpsi_{e,q}^n\\diff\\vx = \\int_{\\pd K}\\vpsi_{e,q}^n\\cdot\\vn_K\\diff s = 0,\n\\end{dis}\nand\n\\begin{eqnarray*}\n\\int_Kr\\div\\vpsi_{e,q}^n\\diff\\vx & = & \\int_{\\pd K}r\\vpsi_{e,q}^n\\cdot\\vn_K\\diff s - \\int_K\\vpsi_{e,q}^n\\cdot\\nabla r\\diff\\vx \\\\\n& = & \\int_{e\\cap\\pd K}r\\vvph_{e,q}^n\\cdot\\vn_K\\diff s - \\sum_{r'\\in M_{k-1}(K)\\setminus\\{1\\}}c_{K,r'}\\int_K\\vvph_{K,\\nabla r'}\\cdot\\nabla r\\diff\\vx \\\\\n& = & \\int_{e\\cap\\pd K}r\\vvph_{e,q}^n\\cdot\\vn_K\\diff s - c_{K,r}\\int_K\\vvph_{K,\\nabla r}\\cdot\\nabla r\\diff\\vx \\\\\n& = & \\int_{e\\cap\\pd K}r\\vvph_{e,q}^n\\cdot\\vn_K\\diff s - |K|c_{K,r} \\\\\n& = & 0\n\\end{eqnarray*}\nfor any $r\\in M_{k-1}(K)\\setminus \\{1\\}$. Here, as before, we used the relations\n\\begin{dis}\n\\int_K\\vvph_{K,\\nabla r'}\\cdot\\nabla r\\diff\\vx = \\left\\{\\begin{array}{ll}\n|K| & \\tr{if $r = r'$} \\\\\n0 & \\tr{if $r \\neq r'$}.\n\\end{array}\\right.\n\\end{dis}\nThus $\\vpsi_{e,q}^n\\in\\vZ_{h}$. This concludes the proof of the lemma.\n\\end{proof}\n\nThe next theorem shows that some of these functions generate a basis for $\\vZ_{h,0}$. \n\n\\begin{theorem}\\label{thm:ConstructDivFree}\nLet $\\vZ_1,\\vZ_2,\\vZ_3,\\vZ_4$ be the subspaces of $\\vV_{h}$ defined by\n\\begin{eqnarray*}\n\\vZ_1 & = & \\spn\\left(\\left\\{\\vpsi_v : v\\in\\mc{V}_h^i\\right\\}\\right), \\\\\n\\vZ_2 & = & \\spn\\left(\\left\\{\\vpsi_{e,q}^t : q\\in M_{k-1}(e), e\\in\\mc{E}_h^i\\right\\}\\right), \\\\\n\\vZ_3 & = & \\left\\{\\begin{array}{ll}\n\\spn\\left(\\left\\{\\vpsi_{e,q}^n : q\\in M_{k-1}(e)\\setminus\\{1\\}, e\\in\\mc{E}_h^i\\right\\}\\right) & \\tr{if $k \\geq 2$} \\\\\n\\{0\\} & \\tr{otherwise}\n\\end{array}\\right.,\\\\\n\\vZ_4 & = & \\left\\{\\begin{array}{ll}\n\\spn\\left(\\left\\{\\vpsi_{K,\\vq} : \\vq\\in \\vM_k^\\oplus, K\\in\\mc{P}_h\\right\\}\\right) & \\tr{if $k \\geq 3$} \\\\\n\\{0\\} & \\tr{otherwise}\n\\end{array}\\right.,\n\\end{eqnarray*}\nwhere $\\vpsi_v$, $\\vpsi_{e,q}^t$, $\\vpsi_{e,q}^n$, and $\\vpsi_{K,\\vq}$ are the functions given in (D1)-(D4), respevtively. Then the following hold.\n\\begin{enumerate}[(i)]\n\\item $\\vZ_1,\\vZ_2,\\vZ_3,\\vZ_4\\sus\\vZ_{h,0}$.\n\\item $\\vZ_i\\cap \\vZ_j = \\{0\\}$ for any pair $(i,j)$ with $i\\neq j$.\n\\item The dimensions of the subspaces $\\vZ_1$, $\\vZ_2$, $\\vZ_3$, and $\\vZ_4$ satisfy\n\\begin{dis}\n\\dim \\vZ_1 = N_{V,i}, \\ \\dim \\vZ_2 = kN_{E,i}, \\ \\dim \\vZ_3 = (k-1)N_{E,i}, \\ \\dim \\vZ_4 = \\frac{(k-1)(k-2)}{2}N_P.\n\\end{dis}\n\\end{enumerate}\nConsequently, $\\vZ_{h,0} = \\vZ_1 \\oplus \\vZ_2 \\oplus \\vZ_3 \\oplus \\vZ_4$. \n\\end{theorem}\n\n\\begin{proof}\nSince $\\vZ_{h,0} = \\vZ_h\\cap\\vV_{h,0}$, and from \\Cref{lem:D1D4DivFree}, it suffices to show that the functions in (D1)-(D4) are contained in $\\vV_{h,0}$. Clearly $\\vpsi_{K,\\vq}\\in\\vV_{h,0}$ for any $K\\in\\mc{P}_h$ and any $\\vq\\in \\vM_k^\\oplus$. If $e\\in\\mc{E}_h^i$, then $\\vpsi_{e,q}^t\\in\\vV_{h,0}$ for any $q\\in M_{k-1}(e)$ and $\\vpsi_{e,q}^n\\in\\vV_{h,0}$ for any $q\\in M_{k-1}(e)\\setminus\\{1\\}$. If $v\\in\\mc{V}_h^i$, then the edges in $\\mc{E}_h$ that have $v$ as an end point are contained in $\\mc{E}_h^i$. Thus $\\vpsi_v\\in\\vV_{h,0}$ for any $v\\in\\mc{V}_h^i$. Hence $\\vZ_1$, $\\vZ_2$, $\\vZ_3$, and $\\vZ_4$ are subspaces of $\\vZ_{h,0}$.\n\nOn the other hand, it is easy to show that $\\vZ_i\\cap \\vZ_j = \\{0\\}$ for any pair $(i,j)$ with $i\\neq j$ and\n\\begin{dis}\n\\dim \\vZ_1 = N_{V,i}, \\ \\dim \\vZ_2 = kN_{E,i}, \\ \\dim \\vZ_3 = (k-1)N_{E,i}, \\ \\dim \\vZ_4 = \\frac{(k-1)(k-2)}{2}N_P.\n\\end{dis}\nThen, since $\\vZ_1 \\oplus \\vZ_2 \\oplus \\vZ_3 \\oplus \\vZ_4 \\sus \\vZ_{h,0}$ and since $\\dim\\vZ_{h,0} = \\dim\\vZ_1 + \\dim\\vZ_2 + \\dim\\vZ_3 + \\dim\\vZ_4$ by \\Cref{prop:DimZh}, we obtain\n\\begin{dis}\n\\vZ_{h,0} = \\vZ_1 \\oplus \\vZ_2 \\oplus \\vZ_3 \\oplus \\vZ_4.\n\\end{dis}\nThis concludes the proof of the theorem.\n\\end{proof}\n\n\\begin{remark}\nIf $k = 1$ and the mesh $\\mc{P}_h$ is a triangular mesh, then the construction of the basis of $\\vZ_{h,0}$ described in \\Cref{thm:ConstructDivFree} is exactly the same with the divergence-free basis in the Crouzeix-Raviart finite element space \\cite{thomassetimplementation,brenner1990nonconforming}.\n\\end{remark}\n\n\\section{Implementation Details}\n\nIn this section, we present how to compute the solution $(\\vu_h,p_h)$ of the discrete problem \\eqref{eqn:StokesDis} by using the construction of $\\vZ_{h,0}$ presented in \\Cref{sec:NCVEM1}. \n\n\\subsection{Computing the discrete velocity $\\vu_h$}\n\nWe first consider the case $\\vg = \\zz$. Note that the discrete velocity $\\vu_h$ is the solution of the following discrete problem \\cite{crouzeix1973conforming}: Find $\\vu_h\\in\\vZ_{h,0}$ such that\n\\begin{eqn}\\label{eqn:StokesDisVel}\na_h(\\vu_h,\\vv_h) = \\inn{\\vf_h,\\vv_h} \\quad \\forall \\vv_h\\in \\vZ_{h,0}.\n\\end{eqn}\nSince $\\dim\\vZ_{h,0} = \\dim\\vV_{h,0} - \\dim Q_h$, the system \\eqref{eqn:StokesDisVel} has a smaller number of unknowns than system \\eqref{eqn:StokesDis}. Moreover, system \\eqref{eqn:StokesDisVel} is symmetric positive definite, while problem \\eqref{eqn:StokesDis} is a saddle point problem. Thus, it is more efficient to compute $\\vu_h$ from \\eqref{eqn:StokesDisVel} than from the problem \\eqref{eqn:StokesDis}.\n\nWe next consider the case $\\vg \\neq \\zz$. Let us decompose $\\vu_h\\in\\vV_{h,\\vg}$ into\n\\begin{dis}\n\\vu_h = \\vu_{h,0} + \\wt{\\vu}_{h},\n\\end{dis}\nwhere $\\wt{\\vu}_h\\in \\vV_{h,\\vg} \\cap \\vZ_h$ and $\\vu_{h,0}\\in \\vZ_{h,0}$ is the solution of the problem\n\\begin{dis}\na_h(\\vu_{h,0},\\vv_h) = \\inn{\\vf_h,\\vv_h} - a_h(\\wt{\\vu}_h,\\vv_h) \\quad \\forall \\vv_h\\in \\vZ_{h,0}.\n\\end{dis}\nUsing the construction of $\\vZ_{h,0}$ presented in \\Cref{thm:ConstructDivFree}, we can compute $\\vu_{h,0}$ by solving a symmetric positive definite system of linear equations, as explained in the case $\\vg = \\zz$. It remains to find a function $\\wt{\\vu}_h\\in \\vV_{h,\\vg} \\cap \\vZ_h$. The following theorem shows that we can easily find such a function.\n\n\\begin{theorem}\\label{thm:DivFreeNonzeroBdry}\nLet $N = N_V^\\pd$ and label the vertices in $\\mc{V}_h^{\\pd}$ by $1,2,\\cdots,N$ such that $v_1,\\cdots,v_N$ are in counterclockwise order with respect to $\\Omega$. We also label the edges in $\\mc{E}_h^{\\pd}$ by $1,2,\\cdots,N$, such that the endpoints of the edge $e_i$ are $v_i$ and $v_{i+1}$ for $i = 1,2,\\cdots,N-1$, and the endpoints of the edge $e_{N}$ are $v_{N}$ and $v_1$ (since $\\Omega$ is a simply connected polygon, $N_V^{\\pd} = N_E^{\\pd}$). Let $\\wt{\\vu}_h$ be the function in $\\vV_h$ defined by\n\\begin{dis}\n\\wt{\\vu}_h = \\sum_{v\\in\\mc{V}_h^{\\pd}}C_{1,v}\\vpsi_{v} + \\sum_{e\\in\\mc{E}_h^{\\pd}}\\sum_{q\\in M_{k-1}(e)}C_{2,e,q}\\vpsi_{e,q}^t + \\sum_{e\\in\\mc{E}_h^{\\pd}}\\sum_{q\\in M_{k-1}(e)\\setminus\\{1\\}}C_{3,e,q}\\vpsi_{e,q}^n,\n\\end{dis}\nwhere the coefficients $(C_{1,v})_v = (C_{1,v_1},\\cdots,C_{1,v_N})$ are given by\n\\begin{dis}\nC_{1,v_k} = -\\sum_{i=k}^{N}\\int_{e_i}\\vg\\cdot\\vn_{e_i}\\diff s, \\quad k = 1,2,\\cdots,N,\n\\end{dis}\nand the coefficients $(C_{2,e,q})_{e,q}$, and $(C_{3,e,q})_{e,q}$ are given by\n\\begin{eqnarray*}\nC_{2,e,q} & = & \\frac{1}{|e|}\\int_e\\vg\\cdot\\vt_eq\\diff s, \\quad q\\in M_{k-1}(e), \\ e\\in\\mc{E}_h^{\\pd}, \\\\\nC_{3,e,q} & = & \\frac{1}{|e|}\\int_e\\vg\\cdot\\vn_eq\\diff s, \\quad q\\in M_{k-1}(e)\\setminus\\{1\\}, \\ e\\in\\mc{E}_h^{\\pd}.\n\\end{eqnarray*}\nThen $\\wt{\\vu}_h\\in \\vV_{h,\\vg}\\cap \\vZ_h$. \n\\end{theorem}\n\n\\begin{proof}\nFrom the construction of $\\wt{\\vu}_h$, it is obvious that $\\div_h\\wt{\\vu}_h = 0$. Thus it remains to show that $\\wt{\\vu}_h\\in \\vV_{h,\\vg}$. From the definition of the coefficients $(C_{2,e,q})_{e,q}$, and $(C_{3,e,q})_{e,q}$, we obtain\n\\begin{eqnarray*}\n\\int_e\\wt{\\vu}_h\\cdot\\vt_eq\\diff s & = & \\int_e\\vg\\cdot\\vt_eq\\diff s, \\quad q\\in M_{k-1}(e), \\ e\\in\\mc{E}_h^{\\pd}, \\\\\n\\int_e\\wt{\\vu}_h\\cdot\\vn_eq\\diff s & = & \\int_e\\vg\\cdot\\vn_eq\\diff s, \\quad q\\in M_{k-1}(e)\\setminus\\{1\\}, \\ e\\in\\mc{E}_h^{\\pd}.\n\\end{eqnarray*}\nSince the boundary edge $e_i$ with $1\\leq i \\leq N-1$ has endpoints $v_i$ and $v_{i+1}$, and since the vertices $v_1,\\cdots,v_N$ are labeled in counterclockwise order with respect to $\\Omega$, we obtain\n\\begin{dis}\n\\vn_{e_i,v_{i+1}} = \\vn_{e_i} = -\\vn_{e_i,v_{i}},\n\\end{dis}\nwhere $\\vn_{e_i}$ is a unit normal vector in the outward direction with respect to $\\Omega$, and $\\vn_{e_i,v_i}$ and $\\vn_{e_i,v_{i+1}}$ are unit vectors normal to $e_i$ pointing in the counterclockwise direction with respect to $v_i$ and $v_{i+1}$, respectively (see \\Cref{fig:NormalBdry}). Similarly, we obtain\n\\begin{dis}\n\\vn_{e_N,v_1} = \\vn_{e_N} = -\\vn_{e_N,v_N}.\n\\end{dis}\nThus we have\n\\begin{eqnarray*}\n\\int_{e_i}\\wt{\\vu}_h\\cdot\\vn_{e_i}\\diff s & = & -C_{1,v_i} + C_{1,v_{i+1}} \\quad \\forall i = 1,2,\\cdots,N-1, \\\\\n\\int_{e_N}\\wt{\\vu}_h\\cdot\\vn_{e_N}\\diff s & = & -C_{1,v_{N}} + C_{1,v_1}.\n\\end{eqnarray*}\nUsing the definition of the coefficients $(C_{1,v})$, \n\\begin{dis}\n-C_{1,v_i} + C_{1,v_{i+1}} = \\int_{e_i}\\vg\\cdot\\vn_{e_i}\\diff s \\quad \\forall i = 1,2,\\cdots,N-1. \n\\end{dis}\nFrom \\eqref{eqn:StokesBdry} we obtain $C_{1,v_1} = -\\int_{\\pd\\Omega}\\vg\\cdot\\vn_{\\Omega}\\diff s = 0$ and thus\n\\begin{eqnarray*}\n-C_{1,v_{N}} + C_{1,v_{1}} = \\int_{e_{N}}\\vg\\cdot\\vn_{e_{N}}\\diff s.\n\\end{eqnarray*}\nTherefore $\\wt{\\vu}_h\\in \\vV_{h,\\vg}$. \n\\end{proof}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\\fill (-1,-1.5) circle [radius = 0.075];\n\\fill (1.5,0) circle [radius = 0.075];\n\\draw [thick] (-3,-1.5) -- (-1,-1.5);\n\\draw [thick] (-1,-1.5) -- (1.5,0);\n\\draw [thick, dashed] (1.5,0) -- (0.5,1.5);\n\\draw [thick] (1.5,0) -- (3,1.75);\n\\draw [thick, dashed] (-1,-1.5) -- (-1.5,0.5);\n\\draw [thick, dashed] (0.5,1.5) -- (0.5,2.5);\n\\draw [thick, dashed] (-1.5,0.5) -- (-3,1);\n\\draw [thick, dashed] (-1.5,0.5) -- (-1,1.5);\n\\draw [thick, dashed] (-1,1.5) -- (0.5,1.5);\n\\draw [thick, ->] (0.25,-0.75) -- (1,-2);\n\\draw [thick, ->] (0.25,-0.75) -- (-0.5,0.5);\n\\node at (-2.75,2.25) {$\\Omega$};\n\\node at (2.75,-2.25) {$\\Omega^c$};\n\\node at (-1,-1.75){$v_{i}$};\n\\node[right] at (1.5,-0.25){$v_{i+1}$};\n\\node[right] at (0.8,-1.25){$\\vn_{e_i} = \\vn_{e_i,v_{i+1}}$};\n\\node[right] at (-0.25,0.25){$\\vn_{e_i,v_{i}}$};\n\\node at (0.375,-0.5){$e_i$};\n\\end{tikzpicture}\n\\caption{}\n\\label{fig:NormalBdry}\n\\end{center}\n\\end{figure}\n\n\\subsection{Recovery of the discrete pressure $p_h$}\n\nOnce we have the discrete velocity $\\vu_h$, the discrete pressure $p_h$ can be obtained by solving the overdetermined system\n\\begin{eqn}\\label{eqn:StokesDisPre}\nb_h(\\vv_h,p_h) = \\inn{\\vf_h,\\vv_h} - a_h(\\vu_h,\\vv_h) \\quad \\forall \\vv_h\\in\\vV_{h,0}.\n\\end{eqn}\n\n\\section{Numerical Experiments}\n\nIn this section, we present several numerical experiments for the symmetric positive definite linear system \\eqref{eqn:StokesDisVel} and the overdetermined linear system \\eqref{eqn:StokesDisPre}. Consider the Stokes problem \\eqref{eqn:Stokes123} on the unit square domain $\\Omega = [0,1]^2$, where the exact solution is given by\n\\begin{eqnarray*}\n\\vu(x,y) & = & ((1 - \\cos(2\\pi x))\\sin(2\\pi y), -(1-\\cos(2\\pi y))\\sin(2\\pi x)), \\\\\np(x,y) & = & e^x - e^y.\n\\end{eqnarray*}\nWe solve both \\eqref{eqn:StokesDisVel} and \\eqref{eqn:StokesDisPre} for $k = 1,2,3$, and we compute the velocity error in the discrete energy norm\n\\begin{dis}\nE_v := a_h(\\vu_h - I_h\\vu,\\vu_h - I_h\\vu)^{1\/2}\n\\end{dis}\nand the pressure error in the $L^2$-norm\n\\begin{dis}\nE_p := \\|p_h - \\Pi_hp\\|_{0,\\Omega},\n\\end{dis}\nwhere $\\Pi_hp$ is the piecewise polynomial function such that for each $K\\in\\mc{P}_h$ the restriction $\\Pi_hp|_K$ is the $L^2$-projection of $p$ onto $P_{k-1}(K)$. \n\n\nWe decompose $\\Omega$ into the following sequences of convex polygonal meshes:\n\\begin{enumerate}[(i)]\n\\item uniform square meshes $\\mc{P}_h^1$ with $h = 1\/4$, $1\/8$, $1\/16$, $1\/32$, $1\/64$, $1\/128$,\n\\item unstructured polygonal meshes $\\mc{P}_h^2$ with $h = 1\/4$, $1\/8$, $1\/16$, $1\/32$, $1\/64$, $1\/128$.\n\\end{enumerate}\nSome examples of the meshes are shown in \\Cref{fig:mesh}. The unstructured polygonal meshes $\\{\\mc{P}_h^2\\}_h$ are generated from PolyMesher \\cite{talischi2012polymesher}. Mesh data (the number of polygons, interior edges, and interior vertices) for each $h$ are given in \\Cref{tab:MeshInfo}. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.33\\textwidth]{squarem_level_1.eps}\n\\includegraphics[width = 0.33\\textwidth]{polym_level_1.eps}\n\\includegraphics[width = 0.33\\textwidth]{squarem_level_2.eps}\n\\includegraphics[width = 0.33\\textwidth]{polym_level_2.eps}\n\\caption{The meshes $\\mc{P}_h^1$ (left), and $\\mc{P}_h^2$ (right).}\n\\label{fig:mesh}\n\\end{figure}\n\n\\begin{table}[!ht]\n\\footnotesize\n\\centering\n\\caption{Mesh information}\n\\label{tab:MeshInfo}\n\\begin{tabular}{|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$h$} & \\multicolumn{3}{c||}{$\\mc{P}_h^1$}& \\multicolumn{3}{c|}{$\\mc{P}_h^2$} \\\\ \\cline{2-7} \n & $N_P$ & $N_{E,i}$ & $N_{V,i}$ & $N_P$ & $N_{E,i}$ & $N_{V,i}$ \\\\ \\hline\\hline\n1\/4 & 16 & 24 & 9 & 16 & 33 & 18 \\\\ \\hline\n1\/8 & 64 & 112 & 49 & 64 & 162 & 99 \\\\ \\hline\n1\/16 & 256 & 480 & 225 & 256 & 707 & 452 \\\\ \\hline\n1\/32 & 1024 & 1984 & 961 & 1024 & 2953 & 1930 \\\\ \\hline\n1\/64 & 4096 & 8064 & 3969 & 4096 & 12043 & 7948 \\\\ \\hline\n1\/128 & 16384 & 32512 & 16129 & 16384 & 48655 & 32272 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nIn \\Crefrange{tab:DimDiscreteSpace1}{tab:DimDiscreteSpace3}, we present the dimensions of the spaces $\\vV_{h,0}$, $Q_h$, and $\\vZ_{h,0}$, for each mesh $\\mc{P}_h^1$, $\\mc{P}_h^2$ and each $k = 1,2,3$. Since the number of unknowns of the system \\eqref{eqn:StokesDis} is $\\dim\\vV_{h,0} + \\dim Q_h$ and the number of unknowns of the system \\eqref{eqn:StokesDisVel} is $\\dim\\vZ_{h,0}$, we can see that the system \\eqref{eqn:StokesDisVel} has fewer unknowns than the system \\eqref{eqn:StokesDis}.\n\n\\begin{table}[!ht]\n\\footnotesize\n\\centering\n\\caption{Dimensions of the discrete spaces ($k = 1$)}\n\\label{tab:DimDiscreteSpace1}\n\\begin{tabular}{|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$h$} & \\multicolumn{3}{c||}{$\\mc{P}_h^1$} & \\multicolumn{3}{c|}{$\\mc{P}_h^2$} \\\\ \\cline{2-7} \n & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ \\\\ \\hline\\hline\n1\/4 & 48 & 15 & 33 & 66 & 15 & 51 \\\\ \\hline\n1\/8 & 224 & 63 & 161 & 324 & 63 & 261 \\\\ \\hline\n1\/16 & 960 & 255 & 705 & 1414 & 255 & 1159 \\\\ \\hline\n1\/32 & 3968 & 1023 & 2945 & 5906 & 1023 & 4883 \\\\ \\hline\n1\/64 & 16128 & 4095 & 12033 & 24086 & 4095 & 19991 \\\\ \\hline\n1\/128 & 65024 & 16383 & 48641 & 97310 & 16383 & 80927 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!ht]\n\\footnotesize\n\\centering\n\\caption{Dimensions of the discrete spaces ($k = 2$)}\n\\label{tab:DimDiscreteSpace2}\n\\begin{tabular}{|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$h$} & \\multicolumn{3}{c||}{$\\mc{P}_h^1$} & \\multicolumn{3}{c|}{$\\mc{P}_h^2$} \\\\ \\cline{2-7} \n & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ \\\\ \\hline\\hline\n1\/4 & 128 & 47 & 81 & 164 & 47 & 117 \\\\ \\hline\n1\/8 & 576 & 191 & 385 & 776 & 191 & 585 \\\\ \\hline\n1\/16 & 2432 & 767 & 1665 & 3340 & 767 & 2573 \\\\ \\hline\n1\/32 & 9984 & 3071 & 6913 & 13860 & 3071 & 10789 \\\\ \\hline\n1\/64 & 40448 & 12287 & 28161 & 56364 & 12287 & 44077 \\\\ \\hline\n1\/128 & 162816 & 49151 & 113665 & 227388 & 49151\t\t& 178237 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!ht]\n\\footnotesize\n\\centering\n\\caption{Dimensions of the discrete spaces ($k = 3$)}\n\\label{tab:DimDiscreteSpace3}\n\\begin{tabular}{|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$h$} & \\multicolumn{3}{c||}{$\\mc{P}_h^1$} & \\multicolumn{3}{c|}{$\\mc{P}_h^2$} \\\\ \\cline{2-7} \n & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ & $\\dim\\vV_{h,0}$ & $\\dim Q_h$ & $\\dim\\vZ_{h,0}$ \\\\ \\hline\\hline\n1\/4 & 240 & 95 & 145 & 294 & 95 & 199 \\\\ \\hline\n1\/8 & 1056 & 383 & 673 & 1356 & 383 & 973 \\\\ \\hline\n1\/16 & 4416 & 1535 & 2881 & 5778 & 1535 & 4243 \\\\ \\hline\n1\/32 & 18048 & 6143 & 11905 & 23862 & 6143 & 17719 \\\\ \\hline\n1\/64 & 72960 & 24575 & 48385 & 96834 & 24575 & 72259 \\\\ \\hline\n1\/128 & 293376 & 98303 & 195073 & 390234 & 98303\t\t& 291931 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nThe errors $E_v$ and $E_p$ and their orders on the sequences of the meshes for $k = 1,2,3$ are given in \\Crefrange{fig:error1}{fig:error3}. In these figures, we see that the convergence order of the errors $E_v$ and $E_p$ are\n$O(h^k)$ for $k = 1,2,3$. Thus the numerical results confirm the theoretical analysis in \\Cref{thm:Error}. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{mesh_square_k1.eps}\n\\includegraphics[width = 0.4\\textwidth]{mesh_poly_k1.eps}\n\\caption{Error curves with respect to $h$ for the velocity and pressure on the sequences of meshes $\\mc{P}_h^1$ (left) and $\\mc{P}_h^2$ (right) with $k = 1$.}\n\\label{fig:error1}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{mesh_square_k2.eps}\n\\includegraphics[width = 0.4\\textwidth]{mesh_poly_k2.eps}\n\\caption{Error curves with respect to $h$ for the velocity and pressure on the sequences of meshes $\\mc{P}_h^1$ (left) and $\\mc{P}_h^2$ (right) with $k = 2$.}\n\\label{fig:error2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{mesh_square_k3.eps}\n\\includegraphics[width = 0.4\\textwidth]{mesh_poly_k3.eps}\n\\caption{Error curves with respect to $h$ for the velocity and pressure on the sequences of meshes $\\mc{P}_h^1$ (left) and $\\mc{P}_h^2$ (right) with $k = 3$.}\n\\label{fig:error3}\n\\end{figure}\n\nIn \\Cref{tab:CPUTimeTotal}, we compare the CPU running times (on a PC with an Intel Core i5 processor and 8GB RAM) required to solve the reduced system \\eqref{eqn:StokesDisVel} and the original saddle-point system \\eqref{eqn:StokesDis}, for the uniform square meshes $\\{\\mc{P}_h^1\\}_h$ and $k = 1,2,3$. For a fair comparison, we use unpreconditioned conjugate gradient method (CG) to solve \\eqref{eqn:StokesDisVel} and the standard Uzawa method to solve \\eqref{eqn:StokesDis}. The cost for computing the discrete pressure (by solving \\eqref{eqn:StokesDisPre}) is a fraction of CG, hence it is not included. For each experiment, we write ``$*$'' if the CPU time is more than 518,400 seconds (6 days). For all cases, the CPU time of solving the reduced system is much smaller than that of solving the saddle-point system. \n\n\\begin{table}[!ht]\n\\footnotesize\n\\caption{CPU running times}\n\\label{tab:CPUTimeTotal}\n\\begin{tabular}{|c||c|c||c|c||c|c|}\n\\hline\n\\multirow{3}{*}{$h$} & \\multicolumn{6}{c|}{CPU time (secs)} \\\\ \\cline{2-7}\n & \\multicolumn{2}{c||}{$k=1$} & \\multicolumn{2}{c||}{$k=2$} & \\multicolumn{2}{c|}{$k=3$} \\\\ \\cline{2-7} \n & CG & Uzawa & CG & Uzawa & CG & Uzawa \\\\ \\hline\\hline\n$1\/4$ & 0.001 & 0.095 & 0.001 & 3.274 & 0.004 & 77.824 \\\\ \\hline\n$1\/8$ & 0.005 & 2.202 & 0.011 & 78.528 & 0.034 & 1713.996 \\\\ \\hline\n$1\/16$ & 0.034 & 61.202 & 0.066 & 1756.959 & 0.389 & 29664.469 \\\\ \\hline\n$1\/32$ & 0.275 & 1822.032 & 0.776 & 36869.009 & 4.255 & $*$ \\\\ \\hline\n$1\/64$ & 4.700 & 53004.713 & 9.581 & $*$ & 53.712 & $*$ \\\\ \\hline\n$1\/128$ & 70.250 & $*$ & 128.755 & $*$ & 721.477 & $*$ \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.4\\textwidth]{CPU_time_k1_new.eps}\n\\includegraphics[width = 0.4\\textwidth]{CPU_time_k2_new.eps}\n\\caption{CPU time curves of CG and Uzawa with respect to $1\/h$ with $k = 1$ (left) and $k = 2$ (right).}\n\\label{fig:CPUtime}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe presented a formal construction of divergence-free bases in the nonconforming VEM for solving the stationary Stokes problem on arbitrary polygonal meshes introduced in \\cite{zhao2019divergence}. If $k = 1$ and the mesh is triangular, then the proposed construction of the basis is exactly the same as the divergence-free basis in the Crouzeix-Raviart finite element space \\cite{thomassetimplementation,brenner1990nonconforming}. Using our construction, we are able to eliminate the pressure variable from the discrete saddle point formulation, and reduce it to a symmetric positive definite linear system in the velocity variable only. Thus, we can apply many efficient solvers available for symmetric positive definite systems. Finally, we provided some numerical experiments confirming the theoretical results and the efficiency of our construction of divergence-free bases in the nonconforming VEM for the Stokes problem. \n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nLogistic regression \\cite{agresti2010analysis} is one of the most popular tools for describing item functioning in multi-item measurements. This method might be used in various contexts, including educational measurements, admission tests, cognitive assessments, and other health-related inventories.\nHowever, its broad applications are not limited to behavioural disciplines. In the context of psychometrics, logistic regression can be seen as a score-based counterpart to the 2 \\gls{pl} \\gls{irt} model \\cite{birnbaum1968statistical}, since in contrast to this class of generalised linear mixed effect models, this method uses an observed estimate of the underlying latent trait. \n\nFurthermore, logistic regression has become a widely used method for identifying between-group differences on item level when responding to multi-item measurements\n\\cite{swaminathan1990detecting}. The phenomenon which is known as \\gls{dif} indicates whether responses to an item vary for respondents with the same level of an underlying latent trait but from different social groups (e.g., defined by gender, age, or socio-economic status).\nIn this vein, \\gls{dif} detection is essential for a deeper understanding of group differences, for assessing effectiveness of various treatments, or for uncovering potential unfairness in educational tests. Beyond the logistic regression, various psychometrics and statistical methods have been proposed for an important task of \\gls{dif} identification which are still being studied extensively \\cite{schneider2021r, schauberger2016detection, peak2015investigation, suh2011nested}. \n\nA natural extension of the logistic regression model to describe item functioning is a generalised logistic regression model, which may account for~the~possibility that an~item can be correctly answered or endorsed without the~necessary knowledge or trait. In this case, the model is extended by including a parameter defining a~lower asymptote of the probability curve, which may be larger than zero. Similarly, the model can consider the possibility that an~item is incorrectly answered or opposed by a~respondent with a~high level of a certain trait due to issues such as inattention or lack of~time. That is, the model includes an~upper asymptote of~the probability curve which may be lower than one.\nAnalogous to the logistic regression model's being a counterpart to the 2\\gls{pl} \\gls{irt} model, generalised logistic regression models can be seen as score-based counterparts to 3--4\\gls{pl} \\gls{irt} models \\cite{birnbaum1968statistical, barton1981upper}. \n\nThe estimation in the logistic regression model is a straightforward procedure, but including additional parameters in this model makes it more statistically and computationally challenging and demanding. Therefore, this article examines innovations in the item parameter estimation for the~generalised logistic regression model in the context of \\gls{dif} detection. This work proposes novel iterative algorithms and compares the newly proposed methods to existing ones in a simulation study. \n\nTo begin, Section \\ref{sec:methodology} introduces generalised logistic regression models, examining the estimation techniques. This section provides a detailed description of two existing methods for parameter estimation (\\gls{nls} and the \\gls{ml} method) and their application to fitting a generalised logistic regression model. Furthermore, this study proposes a novel implementation of the \\gls{em} algorithm and a new approach based on a \\gls{plf}. Additionally, this section provides asymptotic properties of the estimates, an estimation of standard errors, and a software implementation including a specification of starting values in iterative algorithms. Subsequently, Section \\ref{sec:simulation} describes the design and results of the simulation study. To illustrate differences and challenges between the existing and newly proposed methods in practice, this work provides a real data analysis of anxiety measure in Section \\ref{sec:real-data}. Finally, Section \\ref{sec:discussion} contains the discussion and conclusion remarks. \n\n\n\\section{Methodology}\\label{sec:methodology}\n\n\n\\subsection{Generalised logistic\nmodel for\nitem functioning}\n\nGeneralised logistic regression models are extensions of~the~logistic regression model which may account for~the~possibility of guessing or inattention when answering an item. The \\highlight{simple 4\\gls{pl} model} describes functioning of the item $i$, meaning the probability of endorsing item $i$ by respondent $p$, by introducing four parameters: \n\\begin{align}\\label{eq:4PL:simple}\n \\pi_{pi} = \\mathrm{P}(Y_{pi} = 1|X_p) = c_i + (d_i - c_i)\\ \\frac{\\exp(b_{i0} + b_{i1} X_p)}{1 + \\exp(b_{i0} + b_{i1} X_p)},\n\\end{align}\nwith $X_p$ being an observed trait of respondent $p$. \n\n\\paragraph{Parameter interpretation. }\nAll four parameters have an intuitive interpretation: The parameters $c_i$ and $d_i$ are the upper and lower asymptotes of the probability sigmoid function $\\pi_{pi}(x)$ since\n\\begin{align*}\n \\lim_{x \\to -\\infty}\\pi_{pi}(x) = c_i \\quad \\text{and} \\quad\n \\lim_{x \\to \\infty}\\pi_{pi}(x) = d_i,\n\\end{align*}\nwhere $c_i \\in [0, 1], d_i \\in [0, 1]$ and $c_i < d_i$ if $b_{i1} > 0$ and $c_i > d_i$ otherwise. Evidently, with $c_i = 0$ and $d_i = 1$, this model recovers a standard logistic regression for item $i$.\n\nIn the context of psychological and health-related assessments, the asymptotes $c_i$ and $d_i$ may represent reluctance to admit difficulties due to social norms. In educational testing, parameter $c_i$ can be interpreted as the probability that the respondents guessed the correct answer without possessing the necessary knowledge $X_p$, also known as a pseudo-guessing parameter. On the other hand, $1 - d_i$ can be viewed as the probability that respondents were inattentive while their knowledge $X_p$ was sufficient \\cite{hladka2020difnlr} or as a lapse-rate \\cite{kingdom2016psychophysics}. Next, parameter $b_{i0}$ is a shift parameter, a midpoint between two asymptotes $c_i$ and $d_i$ related to the difficulty or easiness of item $i$. Finally, parameter $b_{i1}$ is linked to a slope of the sigmoid curve $\\pi_{pi}(x)$, which is also called a discrimination of the respective item.\n\n\\paragraph{Adding covariates, group-specific 4\\gls{pl} model. }\nThe simple model \\eqref{eq:4PL:simple} can be further extended by incorporating additional respondents' characteristics. That is, instead of using a single variable $X_p$ to describe item functioning, a vector of covariates $\\boldsymbol{X}_p = (1, X_{p1}, \\dots, X_{pk})$, $p = 1, \\dots, n$, is involved, including the original observed trait and an intercept term. This process produces extra parameters $\\boldsymbol{b}_i = (b_{i0}, \\dots, b_{ik})$. Beyond this, even asymptotes may depend on respondents' characteristics $\\boldsymbol{Z}_p = (1, Z_{p1}, \\dots, Z_{pj})$, $p = 1, \\dots, n$, which are not necessarily the same as $\\boldsymbol{X}_p$. This, the general \\highlight{covariate-specific 4\\gls{pl} model} is of form \n\\begin{align}\\label{eq:4PL:gen}\n \\pi_{pi} = \\mathrm{P}(Y_{pi} = 1|\\boldsymbol{X}_p, \\boldsymbol{Z}_p) = \\boldsymbol{Z}_p\\boldsymbol{c}_i + (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\ \\frac{\\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}{1 + \\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)},\n\\end{align}\nwhere $\\boldsymbol{c}_i = (c_{i0}, \\dots, c_{ij})$ and $\\boldsymbol{d}_i = (d_{i0}, \\dots, d_{ij})$ are related asymptote parameters for item $i$. \n\nAs a special case of the covariate-specific 4\\gls{pl} model \\eqref{eq:4PL:gen},\nan additional single binary covariate $G_p$ might be considered. This grouping variable describes a respondent's membership to a~social group ($G_p = 0$ for~the~reference group and $G_p = 1$ for~the~focal group). In other words, this special case assumes $\\boldsymbol{X}_p = (1, X_p, G_p)$ and $\\boldsymbol{Z}_p = (1, G_p)$, which reduces the covariate-specific 4\\gls{pl} model~\\eqref{eq:4PL:gen} to a group-specific form:\n\\begin{align}\\label{eq:4PL:dif}\n \\begin{split}\n \\pi_{pi} =\\ &\\mathrm{P}(Y_{pi} =\\ 1|X_p, G_p) = c_i + c_{i\\text{DIF}}G_p\\\\\n & + (d_i - d_{i\\text{DIF}}G_p - c_i - c_{i\\text{DIF}}G_p)\\ \\frac{\\exp(b_{i0} + b_{i1} X_p + b_{i2}G_p + b_{i3}G_pX_p)}{1 + \\exp(b_{i0} + b_{i1} X_p + b_{i2}G_p + b_{i3}G_pX_p)}.\n \\end{split}\n\\end{align}\nThe \\highlight{group-specific 4\\gls{pl} model}~\\eqref{eq:4PL:dif} can be used for testing between-group differences on the item level with a \\gls{dif} analysis \\cite{hladka2020difnlr}. \n\nIn this model, $X_p$ is an observed variable describing the measured trait of the respondent, such as anxiety, fatigue, quality of life, or math ability, here called the~\\highlight{matching criterion}. In the context of the logistic regression method for \\gls{dif} detection, the total test score is typically used as the matching criterion \\cite{swaminathan1990detecting}.\nOther options for the matching criterion include a pre-test score,\na score on another test measuring the same construct, or an estimate of the latent trait provided by an \\gls{irt} model. \n\n\n\\subsection{Estimation of item parameters}\n\nNumerous algorithms are available to estimate item parameters in the covariate-specific 4\\gls{pl} model~\\eqref{eq:4PL:gen}. First, this section describes two methods, which may be directly implemented in the existing software: The \\gls{nls} method and the \\gls{ml} method. This study discusses the asymptotic properties of the estimates. Next, the study introduces two newly proposed iterative algorithms, which might improve implementation of the computationally demanding \\gls{ml} method: The \\gls{em} algorithm inspired by the work of \\citeA{dinse2011algorithm} and an iterative algorithm based on \\gls{plf}. \n\n\n\\subsubsection{Nonlinear least squares}\n\nThe parameter estimates of the covariate-specific 4\\gls{pl} model~\\eqref{eq:4PL:gen} can be determined using the \\gls{nls} method \\cite{dennis1981adaptive}, which is based on minimisation of the \\gls{rss} of item $i$ with respect to item parameters $(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$:\n\\begin{align}\\label{eq:nls:rss}\n\t\\text{RSS}_i(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i) = \\sum_{p = 1}^n \\left[Y_{pi} - \\pi_{pi}\\right]^2 = \\sum_{p = 1}^n \\left[Y_{pi} - \\boldsymbol{Z}_p\\boldsymbol{c}_i - (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\frac{\\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}{1 + \\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}\\right]^2,\n\\end{align}\nwhere $n$ is a number of respondents. Since the criterion function $\\text{RSS}_i(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$ is continuously differentiable with respect to item parameters $(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$, the minimiser can be obtained when the gradient is zero. Thus, the minimisation process involves a calculation of the first partial derivatives with respect to item parameters $(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$ and finding a solution of relevant nonlinear estimating equations \\cite[Chapter 5]{van2000asymptotic}. Since $\\boldsymbol{Z}_p\\boldsymbol{c}_i$ and $\\boldsymbol{Z}_p\\boldsymbol{d}_i$ asymptotes represent probabilities, it is necessary to ensure that these expressions are kept in the interval of $[0, 1]$ which is accomplished using numerical approaches. \n\nThe asymptotic properties of the \\gls{nls} estimator, such as consistency and asymptotic distribution, can be derived under the classical set of regularity conditions \\cite[Theorems 5.41 and 5.42; see also Appendix \\ref{app:asymptotics:nls}]{van2000asymptotic}. This study proposes a sandwich estimator \\eqref{app:eq:nls:sandwich} which can be used as a natural estimate of the asymptotic variance of the \\gls{nls} estimate. \n\n\n\\subsubsection{Maximum likelihood}\n\nThe second option for estimating item parameters in the covariate-specific 4\\gls{pl} model \\eqref{eq:4PL:gen} is the~\\gls{ml} method \\cite{ren2019algorithm}. Using a notation $\\phi_{pi} = \\frac{\\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}{1 + \\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}$, the corresponding likelihood function for item $i$ has the~following form:\n\\begin{align*}\n L_i(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i) = \\prod_{p = 1}^n & \\left[\\boldsymbol{Z}_p\\boldsymbol{c}_i + (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\phi_{pi}\\right]^{Y_{pi}} \\left[1 - \\boldsymbol{Z}_p\\boldsymbol{c}_i - (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\phi_{pi}\\right]^{1 - Y_{pi}},\n\\end{align*}\nand the log-likelihood function is then given by \n\\begin{align*}\n l_{i}(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i) = \\sum_{p = 1}^n &\\left\\{Y_{pi} \\log(\\boldsymbol{Z}_p\\boldsymbol{c}_i + (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\phi_{pi})\\right. \\\\\n &\\ \\left. +\\ (1 - Y_{pi})\\log(1 - \\boldsymbol{Z}_p\\boldsymbol{c}_i - (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\phi_{pi})\\right\\}.\n\\end{align*}\n\nThe parameter estimates are obtained by a maximisation of the log-likelihood function. Thus this study proceeds similarly to the logistic regression model, except for a larger dimension of the parametric space. To find the maximiser of the log-likelihood function $l_{i}(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$, the first partial derivatives are set to zero and these so-called likelihood equations must be solved. However, the solution of a system of the nonlinear equations cannot be derived algebraically and needs to be numerically estimated using a suitable iterative process. \n\nUsing \\citeauthor{van2000asymptotic}'s \\citeyear{van2000asymptotic} Theorems 5.41 and 5.42, consistency and asymptotic normality can be shown for the \\gls{ml} estimator, see Appendix \\ref{app:asymptotics:ml}. Additionally, the estimate of the asymptotic variance of the~item parameters is an inverse of the~observed information matrix \\eqref{app:eq:mle:variance}. \n\n\n\\subsubsection{EM algorithm}\n\nThe \\gls{ml} method may be computationally demanding and iterative algorithms might help in those situations. Inspired by the work of \\citeA{dinse2011algorithm}, this study adopts a version of the \\gls{em} algorithm \\cite{dempster1977maximum} for parameter estimation in the covariate-specific 4\\gls{pl} model \\eqref{eq:4PL:gen}.\n\nNext, the original problem can be reformulated using latent variables which describe hypothetical responses status of test-takers \\cite{dinse2011algorithm}. In this study's setting, the work considers four mutually exclusive latent variables ($W_{pi1}$, $W_{pi2}$, $W_{pi3}$, $W_{pi4}$), where variable $W_{pij} = 1$ indicates that respondent $p$ belongs in the~category $j = 1, \\dots, 4$ for~an~item $i$, whereas $W_{pij} = 0$ indicates that respondent does not belong in this category.\n\nIn the context of educational, psychological, health-related, or other types of multi-item measurement, the four categories can be interpreted as follows: Categories 1 and 2 indicate whether a respondent who responded correctly to item $i$ or endorsed it (i.e., $Y_{pi} = 1$) was determined to do so ($W_{pi1} = 1$, e.g., the respondent guessed correct answer while their knowledge or ability was insufficient) or not ($W_{pi2} = 1$, e.g., had a~sufficient knowledge or ability to answer correctly and did not guessed). On the other hand, Categories 3 and 4 indicate whether the respondent who did not respond correctly or did not endorse the item (i.e., $Y_{pi} = 0$) was prone to do so ($W_{pi3} = 1$, e.g., did not have sufficient knowledge or ability) or not ($W_{pi4} = 1$, e.g., incorrectly answered due to another reason such as inattention or lack of time). Thus, the observed indicator $Y_{pi}$ and its complement $1 - Y_{pi}$ could be rewritten as $Y_{pi} = W_{pi1} + W_{pi2}$ and $1 - Y_{pi} = W_{pi3} + W_{pi4}$ (Figure \\ref{fig:EM_latent_variables}). \n\n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}\n \\node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm] (answer) {\\textbf{Observed answer} \\\\ $Y_{pi} \\in \\left\\{0, 1\\right\\}$}\n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below left=1cm and 0.25cm of answer](guessed) {\\textbf{Guessing}} \n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below left=0.75cm and -0.75cm of guessed](cat1) {\\textbf{Category 1} \\\\ $W_{pi1} = 1$} }\n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below right=0.75cm and -0.75cm of guessed](cat2) {\\textbf{Category 2} \\\\ $W_{pi2} = 1$} } }\n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below right=1cm and 0.25cm of answer](inattentive) {\\textbf{Inattention}} \n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below left=0.75cm and -0.75cm of inattentive](cat3) {\\textbf{Category 3} \\\\ $W_{pi3} = 1$} }\n child { node [shape=rectangle, rounded corners, draw, align=center, minimum width=25mm, minimum height=8mm, below right=0.75cm and -0.75cm of inattentive](cat4) {\\textbf{Category 4} \\\\ $W_{pi4} = 1$} } };\n \\draw (answer) -- (guessed) node [shape=rectangle, rounded corners, align=center, minimum width=25mm, minimum height=8mm, midway, left, draw=none] {Correct \\, \\\\ $Y_{pi} = 1$ \\,};\n \\draw (answer) -- (inattentive) node [shape=rectangle, rounded corners, align=center, minimum width=25mm, minimum height=8mm, midway, right, draw=none] {\\, Incorrect \\\\ \\, $Y_{pi} = 0$};\n \\draw (guessed) -- (cat1) node [midway, left, draw=none] {Yes \\,};\n \\draw (guessed) -- (cat2) node [midway, right, draw=none] {\\, No};\n \\draw (inattentive) -- (cat3) node [midway, left, draw=none] {No \\,};\n \\draw (inattentive) -- (cat4) node [midway, right, draw=none] {\\, Yes};\n \\end{tikzpicture}\n \\caption{Latent variables for \\gls{em} algorithm}\n \\label{fig:EM_latent_variables}\n\\end{figure}\n\nLet $\\boldsymbol{Z}_p\\boldsymbol{c}_i$ be the regressor-based probability that the respondent was determined to respond to item $i$ correctly or endorse it (Category 1), and let $\\boldsymbol{Z}_p\\boldsymbol{d}_i$ be the regressor-based probability of the respondent not prone to respond correctly or endorse item $i$ (Categories 1--3). Then $\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i$ gives the regressor-based probability that the respondent was not determined but prone to (Categories 2 and 3). Further, we denote $\\phi_{pi}$ and $1 - \\phi_{pi}$ -- the probabilities to answer given item correctly (Category 2) and incorrectly (Category 3), respectively, depending on~the~regressors $\\boldsymbol{X}_p$. Finally, the probability that the respondent did not respond correctly and was not prone to do so is given by $1 - (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i) - \\boldsymbol{Z}_p\\boldsymbol{c}_i = 1 - \\boldsymbol{Z}_p\\boldsymbol{d}_i$ (Category 4). In~summary, the~expected values of~the~latent variables are then given by the following terms \n\\begin{align*}\n \\boldsymbol{Z}_p\\boldsymbol{c}_i, \\ \\ (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i) \\phi_{pi}, \\ \\ (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)(1 - \\phi_{pi}), \\ \\ 1 - \\boldsymbol{Z}_p\\boldsymbol{d}_i,\n\\end{align*}\nand the probability of correct response or endorsement is given by\n\\begin{align*}\n \\mathrm{P}(Y_{pi} = 1|\\boldsymbol{X}_p) =& \\ \\mathrm{P}(W_{pi1} + W_{pi2} = 1|\\boldsymbol{X}_p)\n = \\mathrm{P}(W_{pi1} = 1|\\boldsymbol{X}_p) + \\mathrm{P}(W_{pi2} = 1|\\boldsymbol{X}_p) \\\\\n =&\\ \\boldsymbol{Z}_p\\boldsymbol{c}_i + (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\phi_{pi},\n\\end{align*}\nwhich under the logistic model $\\phi_{pi} = \\frac{\\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}{1 + \\exp(\\boldsymbol{X}_p\\boldsymbol{b}_i)}$ produces the covariate-specific 4\\gls{pl} model \\eqref{eq:4PL:gen}.\n\nUsing the setting of the latent variables, the~corresponding log-likelihood function for~item $i$ takes the~following form:\n\\begin{align*}\n l_i^{\\text{EM}}\n =& \\sum_{p = 1}^n \\left[W_{pi2}\\log\\left(\\phi_{pi}\\right) + W_{pi3}\\log\\left(1 - \\phi_{pi}\\right)\\right] \\\\\n & + \\sum_{p = 1}^n \\left[W_{pi1} \\log\\left(\\boldsymbol{Z}_p\\boldsymbol{c}_i\\right) + W_{pi4}\\log\\left(1 - \\boldsymbol{Z}_p\\boldsymbol{d}_i\\right) + \\ \\left(W_{pi2} + W_{pi3}\\right) \\log\\left(\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i\\right)\\right] \\\\\n =& \\ l_{i1}^{\\text{EM}} + l_{i2}^{\\text{EM}}.\n\\end{align*}\nThe log-likelihood function $l_{i1}^{\\text{EM}}$ includes only parameters $\\boldsymbol{b}_i$ and regressors $\\boldsymbol{X}_p$, whereas the~log-likelihood function $l_{i2}^{\\text{EM}}$ incorporates only parameters related to the~asymptotes of~the~sigmoid function and includes only regressors $\\boldsymbol{Z}_p$. Notably, the log-likelihood function $l_{i1}^{\\text{EM}}$ has a~form of the~log-likelihood function for~the~logistic regression. However, in~contrast to the logistic regression model, in this setting it does not necessary hold that $W_{pi2} + W_{pi3} = 1$ since the~correct answer could be guessed or the respondent could be inattentive, producing $W_{pi2} + W_{pi3} = 0$. The log-likelihood function $l_{i2}^{\\text{EM}}$ takes the form of~the~log-likelihood for~multinomial data with one trial and with~the regressor-based probabilities $\\boldsymbol{Z}_p\\boldsymbol{c}_i$, $\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i$, and $1 - \\boldsymbol{Z}_p\\boldsymbol{d}_i$.\n\nThe \\gls{em} algorithm estimates item parameters in two steps -- expectation and maximisation. These two steps are repeated until the convergence criterion is met, such as until the change in log-likelihood is lower than\na predefined value. \n\n\\paragraph{Expectation. }\nAt the E-step, conditionally on the item responses $Y_{pi}$ and the current parameter estimate $(\\widehat{\\boldsymbol{b}}_i, \\widehat{\\boldsymbol{c}}_i, \\widehat{\\boldsymbol{d}}_i)$, the estimates of latent variables are calculated as their expected values:\n\\begin{align}\\label{eq:em:expectation}\n \\begin{aligned}\n \\widehat{W}_{pi1} =& \\ \\frac{\\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i Y_{pi}}{\\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i + (\\boldsymbol{Z}_p\\widehat{\\boldsymbol{d}}_i - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i) \\widehat{\\phi}_{pi}}, \n \\ \\ \\ &\\widehat{W}_{pi2} =& \\ Y_{pi} - \\widehat{W}_{pi1}, \\\\\n %\n \\widehat{W}_{pi4} =& \\ \\frac{\\left(1 - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{d}}_i\\right) \\left(1 - Y_{pi}\\right)}{1 - \\widehat{\\boldsymbol{Z}_p\\boldsymbol{c}}_i - (\\boldsymbol{Z}_p\\widehat{\\boldsymbol{d}}_i - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i) \\widehat{\\phi}_{pi}}, \n %\n \\ \\ \\ &\\widehat{W}_{pi3} =& \\ 1 - Y_{pi} - \\widehat{W}_{pi4}.\n \\end{aligned}\n\\end{align}\n\n\\paragraph{Maximisation. }\nAt the~M-step, conditionally on the~current estimates of~the~latent variables $\\widehat{W}_{pi2}$ and $\\widehat{W}_{pi3}$, the estimates of parameters $\\boldsymbol{b}_i$ maximise the~log-likelihood function $l_{i1}^{\\text{EM}}$. The~estimates $\\widehat{\\boldsymbol{c}}_i$ and $\\widehat{\\boldsymbol{d}}_i$ are given by a maximisation of~the~log-likelihood function $l_{i2}^{\\text{EM}}$ conditionally on~current estimates of~the~latent variables $\\widehat{W}_{pi1}$, $\\widehat{W}_{pi2}$, $\\widehat{W}_{pi3}$, and $\\widehat{W}_{pi4}$.\n\nThe \\gls{em} algorithm is designed to gain the \\gls{ml} estimates of the item parameters, so estimates have the same asymptotic properties as described above. \n\n\n\\subsubsection{Parametric link function}\n\nIn~this study's setting, the covariate-specific 4\\gls{pl} model \\eqref{eq:4PL:gen} can be viewed as a generalised linear model with a known \\gls{plf} \n\\begin{align}\\label{eq:plf}\n g(\\mu_{pi}; \\boldsymbol{c}_i, \\boldsymbol{d}_i) = \\log\\left(\\frac{\\mu_{pi} - \\boldsymbol{Z}_p\\boldsymbol{c}_i}{\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\mu_{pi}}\\right),\n\\end{align}\nwhere the~parameters $\\boldsymbol{c}_i$ and $\\boldsymbol{d}_i$ are unknown and may depend on regressors $\\boldsymbol{Z}_p$. Subsequently, the~mean function is determined by $\\mu_{pi} = \\pi_{pi}$ as given by \\eqref{eq:4PL:gen} with a~linear predictor $\\boldsymbol{X}_p\\boldsymbol{b}_i$. \n\nKeeping this setting in mind, this study proposes a new two-stage algorithm to estimate item parameters using the \\gls{plf} \\eqref{eq:plf}, which involves repeating two steps until the convergence criterion is fulfilled. \n\n\\paragraph{Step one. }\nFirst, conditionally on~current estimates $\\widehat{\\boldsymbol{c}}_i$ and $\\widehat{\\boldsymbol{d}}_i$ of~the~\\gls{plf}, the~estimates of~parameters $\\boldsymbol{b}_i$ maximise the following log-likelihood function:\n\\begin{align*}\n l_{i1}^{\\text{PL}}(\\boldsymbol{b}_i|\\widehat{\\boldsymbol{c}}_i, \\widehat{\\boldsymbol{d}}_i) = \\sum_{p = 1}^n &\\left\\{Y_{pi} \\log(\\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i + (\\boldsymbol{Z}_p\\widehat{\\boldsymbol{d}}_i - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i)\\phi_{pi}) \\right. \\\\\n & \\ \\ \\left. +\\ (1 - Y_{pi})\\log(1 - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i - (\\boldsymbol{Z}_p\\widehat{\\boldsymbol{d}}_i - \\boldsymbol{Z}_p\\widehat{\\boldsymbol{c}}_i)\\phi_{pi})\\right\\}.\n\\end{align*}\nThe log-likelihood function $l_{i1}^{\\text{PL}}(\\boldsymbol{b}_i|\\widehat{\\boldsymbol{c}}_i, \\widehat{\\boldsymbol{d}}_i)$ has a~similar form to the log-likelihood function $l_i(\\boldsymbol{b}_i, \\boldsymbol{c}_i, \\boldsymbol{d}_i)$ using the~\\gls{ml} method. However, the parameters $\\boldsymbol{c}_i$ and $\\boldsymbol{d}_i$ are here replaced by their current estimates, $\\widehat{\\boldsymbol{c}}_i$ and $\\widehat{\\boldsymbol{d}}_i$. \n\n\\paragraph{Step two. }\nNext, estimates $\\widehat{\\boldsymbol{c}}_i$ and $\\widehat{\\boldsymbol{d}}_i$ of~the~\\gls{plf} \\eqref{eq:plf} are calculated conditionally on~the~current estimates $\\widehat{\\boldsymbol{b}}_i$ as the~arguments of~the~maxima of~the~following log-likelihood function\n\\begin{align*}\n l_{i2}^{\\text{PL}}(\\boldsymbol{c}_i, \\boldsymbol{d}_i|\\widehat{\\boldsymbol{b}}_i) = \\sum_{p = 1}^n &\\left\\{Y_{pi} \\log(\\boldsymbol{Z}_p\\boldsymbol{c}_i + (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\widehat{\\phi}_{pi})\\right. \\\\\n &\\left. \\ \\ +\\ (1 - Y_{pi})\\log(1 - \\boldsymbol{Z}_p\\boldsymbol{c}_i - (\\boldsymbol{Z}_p\\boldsymbol{d}_i - \\boldsymbol{Z}_p\\boldsymbol{c}_i)\\widehat{\\phi}_{pi})\\right\\}.\n\\end{align*}\nAgain, the~parameters $\\boldsymbol{b}_i$ are replaced by their estimates $\\widehat{\\boldsymbol{b}}_i$, and $\\phi_{pi}$ is thus replaced by $\\widehat{\\phi}_{pi}$.\n\nIn summary, the division into the two sets of parameters makes the algorithm based on \\gls{plf} easy to implement in the \\pkg{R} software and can take an advantage of its existing functions. Because the algorithm is designed to produce the \\gls{ml} estimates, their asymptotic properties are the same as described above. \n\n\n\\subsection{Implementation and software}\n\nFor~all analyses, software \\pkg{R}, version 4.1 \\cite{r2022} was used. The \\gls{nls} method was implemented using the base \\Rcode{nls()} function and the \\Rcode{\"port\"} algorithm \\cite{port}. The sandwich estimator \\eqref{app:eq:nls:sandwich} of the~asymptotic covariance matrix was computed using the \\pkg{calculus} package \\cite{guidotti2022calculus}. The \\gls{ml} estimation was performed with the base \\Rcode{optim()} function and the \\Rcode{\"L-BFGS-B\"} algorithm \\cite{byrd1995limited}. The \\gls{em} algorithm implements directly \\eqref{eq:em:expectation} in the expectation step using the base \\Rcode{glm()} function and the \\Rcode{multinom()} function from the \\pkg{nnet} package \\cite{venables2002modern} in the maximisation step. Next, the step one of the newly proposed algorithm based on \\gls{plf} is implemented with the base \\Rcode{glm()} function with the modified logit link, which includes asymptote parameters. The estimation of the asymptote parameters in the step two is conducted using the base \\Rcode{optim()} function. The maximum number of iterations was set to 2,000 for all four methods, and the convergence criterion was set to $10^{-6}$ when possible. \n\n\\paragraph{Initial values. }\nStarting values for item parameters were calculated as follows: The respondents were divided into three groups based upon tertiles of the matching criterion $X_p$. Next, the asymptote parameters were estimated: $c$ was computed as an empirical probability for those whose matching criterion was smaller than its average value in the first group defined by tertiles. The asymptote $d$ was calculated as an empirical probability of those whose matching criterion was greater than its average value in the last group defined by tertiles. The slope parameter $b_1$ was estimated as a difference between mean empirical probabilities of the last and the first group multiplied by 4. This difference is sometimes called upper-lower index.\nFinally, the intercept $b_0$ was calculated as follows: First, a centre point between the asymptotes was computed, and then we looked for the level of the matching criterion which would have corresponded to this empirical probability. Additionally, smoothing and corrections for variability of the matching criterion were applied.\n\n\n\\section{Simulation study}\\label{sec:simulation}\n\nA simulation study was performed to compare various procedures to estimate parameters in~the generalised logistic regression model, including the \\gls{nls}, the \\gls{ml} method, the \\gls{em} algorithm, and the newly proposed algorithm based on \\gls{plf}. Two models were considered -- the simple 4\\gls{pl} model~\\eqref{eq:4PL:simple} and the group-specific 4\\gls{pl} model~\\eqref{eq:4PL:dif}. \n\n\n\\subsection{Simulation design}\n\n\\paragraph{Data generation. }\nTo generate data with the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}, the following parameters were used: $b_0 = 0$, $b_1 = 1.5$, $c = 0.25$, and $d = 0.9$. In the case of the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}, additionally $b_2 = -1$, $b_3 = 0.5$, $c_{\\text{DIF}} = -0.15$, and $d_{\\text{DIF}} = 0.1$ were considered. Next, the~matching criterion $X_p$ was generated from the standard normal distribution for all respondents. Binary responses were generated from the Bernoulli distribution with the calculated probabilities based upon the chosen 4\\gls{pl} model, true parameters, and the matching criterion variable. The sample size was set to $n = 500; 1,000; 2,500$; and $5,000$, i.e., $250$; $500$; $1,250$; and $2,500$ per group in the case of the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}. Each scenario was replicated $1,000$ times. \n\n\\paragraph{Simulation evaluation. }\nTo compare estimation methods, we first computed mean and median numbers of iteration runs and the convergence status of the methods, meaning the percentage of converged simulation runs; the percentage of runs which crashed (caused error when fitting, e.g., due to singularities); and the percentage of those which reached the~maximum number of iterations without convergence. Next, we selected only those simulation runs for which all four estimation methods converged successfully and computed the mean parameter estimates, together with parametric confidence intervals. When confidence intervals for asymptote parameters exceeded their boundaries of 0 or 1, confidence intervals were truncated at the boundary value.\n\n\n\\subsection{Simulation results\n\n\\paragraph{Convergence status. }\nAll four methods had low percentages of iterations that crashed for all sample sizes in the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}, but the rate was mildly increased in the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif} for the~\\gls{nls} method (4.3\\%) and for the algorithm based on \\gls{plf} (3.6\\%) when $n = 500$. With the increasing sample size, convergence issues disappeared. The \\gls{em} algorithm struggled to converge in a predefined number of iterations, especially for small sample sizes in both models. Additionally, the method based on \\gls{plf} reached the~maximum limit of 2,000 iterations only in a small percentage of simulation runs when smaller sample sizes were considered in the group-specific 4\\gls{pl} model (Table \\ref{tab:convergence}). \n\n\\paragraph{Number of iterations. }\nFurthermore, the methods differed in a~number of iterations needed until the~estimation process successfully ended. The \\gls{em} algorithm yielded the~largest mean and median numbers of iterations, which were somehow overestimated by simulation runs which did not finish without convergence (i.e., the maximum limit of 2,000 iterations was reached). The fewest iterations were needed for the \\gls{nls} method. As expected, all the methods required fewer simulation runs when the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} was considered than in the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}. Beyond this, the number of iterations was decreasing with the increasing sample size in both models for all the methods except the \\gls{em} algorithm, where the number of iterations was not monotone (Table \\ref{tab:convergence}).\n\n\\vspace{1em}\n\nIn the group-based model \\eqref{eq:4PL:dif} with a sample size of $n = 500$, some of the estimation procedures produced non-meaningful estimates of parameters $b_0$--$b_3$ (absolute value over 100) despite successful convergence. Those 11 simulations affected mean values significantly, so they were removed from a computation of the mean estimates and their confidence intervals for all four estimation methods. In those 11 simulations, non-meaningful estimates were received twice for the \\gls{nls}, three times for the \\gls{ml}, eight times for the \\gls{em} algorithm, and once for the \\gls{plf}-based method. \n\n\\begin{table}[ht]\n\\centering\n\\caption{Convergence status and a number of iterations for the four estimation methods} \n\\label{tab:convergence}\n\\begin{threeparttable}\n\\begin{tabular}{@{}p{\\textwidth}@{}}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l rrr rr rrr rr}\n \\toprule\n & \\multicolumn{5}{c}{Simple 4\\gls{pl} model \\eqref{eq:4PL:simple}} & \\multicolumn{5}{c}{Group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}} \\\\ \\cmidrule(lr){2-6} \\cmidrule(lr){7-11}\n \\multirow{2}{*}{Method} & \\multicolumn{3}{c}{Convergence status [\\%]} & \\multicolumn{2}{c}{Number of iterations} & \\multicolumn{3}{c}{Convergence status [\\%]} & \\multicolumn{2}{c}{Number of iterations} \\\\ \\cmidrule(lr){2-4} \\cmidrule(lr){5-6} \\cmidrule(lr){7-9} \\cmidrule(lr){10-11}\n & \\mc{Converged} & \\mc{Crashed} & \\mc{DNF} & Mean & Median & \\mc{Converged} & \\mc{Crashed} & \\mc{DNF} & Mean & Median \\\\ \n \\midrule\n \\multicolumn{11}{c}{$n = 500$}\\\\ \\midrule\n NLS & 99.60 & 0.40 & 0.00 & 10.11 & 9.00 & 95.70 & 4.30 & 0.00 & 14.36 & 12.00 \\\\ \n MLE & 99.60 & 0.40 & 0.00 & 22.99 & 22.00 & 99.90 & 0.10 & 0.00 & 90.63 & 80.00 \\\\ \n EM & 84.70 & 0.00 & 15.30 & 684.83 & 361.00 & 89.30 & 0.10 & 10.60 & 627.13 & 354.00 \\\\ \n PLF & 99.80 & 0.20 & 0.00 & 43.73 & 20.00 & 95.90 & 3.60 & 0.50 & 131.48 & 35.50 \\\\ \\midrule\n \\multicolumn{11}{c}{$n = 1,000$}\\\\ \\midrule\n NLS & 100.00 & 0.00 & 0.00 & 7.98 & 7.00 & 99.60 & 0.40 & 0.00 & 11.07 & 10.00 \\\\ \n MLE & 100.00 & 0.00 & 0.00 & 21.40 & 21.00 & 100.00 & 0.00 & 0.00 & 80.25 & 75.00 \\\\ \n EM & 83.70 & 0.00 & 16.30 & 744.01 & 437.00 & 88.20 & 0.00 & 11.80 & 769.27 & 542.00 \\\\ \n PLF & 100.00 & 0.00 & 0.00 & 31.13 & 18.00 & 99.30 & 0.60 & 0.10 & 76.85 & 27.00 \\\\ \\midrule\n \\multicolumn{11}{c}{$n = 2,500$}\\\\ \\midrule\n NLS & 99.90 & 0.10 & 0.00 & 5.90 & 6.00 & 99.90 & 0.10 & 0.00 & 7.58 & 7.00 \\\\ \n MLE & 99.90 & 0.10 & 0.00 & 19.98 & 19.00 & 100.00 & 0.00 & 0.00 & 73.01 & 72.00 \\\\ \n EM & 92.60 & 0.20 & 7.20 & 634.03 & 475.50 & 90.90 & 0.00 & 9.10 & 695.99 & 499.00 \\\\ \n PLF & 100.00 & 0.00 & 0.00 & 17.77 & 16.00 & 100.00 & 0.00 & 0.00 & 38.08 & 19.50 \\\\ \\midrule\n \\multicolumn{11}{c}{$n = 5,000$}\\\\ \\midrule\n NLS & 99.90 & 0.10 & 0.00 & 5.07 & 5.00 & 99.80 & 0.20 & 0.00 & 6.02 & 6.00 \\\\ \n MLE & 100.00 & 0.00 & 0.00 & 19.21 & 19.00 & 100.00 & 0.00 & 0.00 & 69.81 & 69.00 \\\\ \n EM & 95.60 & 0.00 & 4.40 & 588.82 & 477.50 & 92.60 & 0.00 & 7.40 & 808.35 & 647.50 \\\\ \n PLF & 100.00 & 0.00 & 0.00 & 15.40 & 15.00 & 100.00 & 0.00 & 0.00 & 26.07 & 16.00 \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{tabular}\n\\tablenotet{DNF = did not finish, NLS = nonlinear least squares, MLE = maximum likelihood estimation, EM = expectation-maximisation algorithm, PLF = algorithm based on parametric link function. }\n\\end{threeparttable}\n\\end{table}\n\n\\paragraph{Parameter estimates. }\nIn the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}, the smallest biases in estimates of parameters $b_0$ and $b_1$ were gained by the \\gls{plf}-based algorithm with the narrowest confidence intervals when smaller sample sizes were considered ($n = 500$ or $n = 1,000$). Additionally, in these scenarios, the \\gls{nls} method yielded slightly more biased estimates with wider confidence intervals. The precision of the estimation improved for both parameters when the sample size increased in all four methods, whereas differences between estimation procedures narrowed. The precision of the estimates of the asymptote parameters $c$ and $d$ was similar for all four methods, while the differences between estimation approaches were small. The \\gls{nls} and the \\gls{em} algorithm provided slightly wider confidence intervals for a small sample size of $n = 500$ (Figure~\\ref{fig:estimates_simple_parametric}, Table~\\ref{tab:simulation:pars_simple}). \n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width = \\textwidth]{Figures\/estimates_simple_parametric_alt.png}\n \\caption{Mean estimated parameters in the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} with confidence intervals with respect to sample size; horizontal lines represent true values of parameters}\n \\label{fig:estimates_simple_parametric}\n\\end{figure}\n\nIn the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}, the \\gls{plf}-based algorithm yielded the least biased estimates of parameters $b_0$--$b_3$, especially for the smaller sample sizes. On the other hand, the \\gls{nls} method produced the most biased estimates with somewhat wider confidence intervals in such scenarios. The \\gls{ml} method provided less biased estimates than the \\gls{nls}, but accompanied with wider confidence intervals, even for a sample size of $n = 1,000$. Similar to the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}, the differences in the precision of the parameter estimates were narrowing with the increasing sample size, and all four estimation approaches gave estimates close to the true values of the item parameters (Figure~\\ref{fig:estimates_dif_parametric}, Table~\\ref{tab:simulation:pars_DIF}). The estimates of the asymptote parameters $c$, $c_{\\text{DIF}}$, $d$, and $d_{\\text{DIF}}$ were similar for all four methods. The \\gls{em} algorithm provided slightly less biased mean estimates of the asymptote parameters, but with slightly wider confidence intervals, especially for $n = 1,000$. \n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width = \\textwidth]{Figures\/estimates_DIF_parametric.png}\n \\caption{Mean estimated parameters in the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif} with confidence intervals with respect to sample size; horizontal lines represent true values of parameters }\n \\label{fig:estimates_dif_parametric}\n\\end{figure}\n\n\n\\section{Real data example}\\label{sec:real-data}\n\n\n\\subsection{Data description}\n\nThis study demonstrated the estimation on a real-data example of the PROMIS Anxiety scale\\footnote{http:\/\/www.nihpromis.org} dataset. The dataset consisted of responses to 29 Likert-type questions (1 = Never, 2 = Rarely, 3 = Sometimes, 4 = Often, and 5 = Always) from 766 respondents. Additionally, the dataset included information on the respondents' age (0 = Younger than 65 and 1 = 65 and older), gender (0 = Male and 1 = Female), and education (0 = Some college or higher and 1 = High school or lower). \n\nFor this work, item responses were dichotomised as follows: 0 = Never (i.e., response $= 1$ on original scale) or 1 = At least rarely (i.e., response $\\geq 2$ on original scale). The overall level of anxiety was calculated as a standardised sum of non-dichotomized item responses. This work considered the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} and the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif} using all four estimation methods: \\gls{nls}, \\gls{ml}, the \\gls{em} algorithm, and the algorithm based on \\gls{plf}. In both models, the computed overall level of anxiety was used as the matching criterion $X_p$. In the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}, respondents' genders were included as the grouping variable $G_p$. Overall, there were 369 male participants and 397 female participants. \n\n\n\\subsection{Analysis design}\n\nThe same approach used in the simulation study for computing starting values was used for the analysis of the Anxiety dataset. In the case of convergence issues, the initial values were re-calculated based on successfully converged estimates using other methods.\n\nIn this study, item parameter estimates were computed and reported with their confidence intervals. Confidence intervals of the asymptote parameters were truncated at boundary values when necessary. Next, this work compared the estimation methods by calculating the differences in fitted item characteristic curves (i.e., estimated probabilities of endorsing the item) on the matching criterion for both models. Finally, likelihood ratio tests were performed to compare the two nested models (simple and group-specific) to identify the \\gls{dif} for all items and all four estimation methods. Significance level of 0.05 was used for all the tests. \n\n\n\\subsection{Results}\n\n\\paragraph{Simple 4PL model. }\nThe smallest differences between the four estimation methods and fitted item characteristic curves in the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} were observed for item R8 (\\highlight{\"I had a racing or pounding heart\"}; Figure~\\ref{fig:icc_simple:R8}). The greatest differences were observed for item R29 (\\highlight{\"I had difficulty calming down\"}; Figure~\\ref{fig:icc_simple:R29}). The smallest overall differences were found between the \\gls{em} algorithm and the algorithm based on \\gls{plf}, whereas the greatest overall differences were noted between the \\gls{nls} and the algorithm based on \\gls{plf}. Beyond this, similar patterns appeared in the estimated item parameters (Figure~\\ref{fig:estimates_simple}, Table~\\ref{tab:anxiety:pars_simple}). Although the lower asymptotes were mostly estimated at 0, the upper asymptotes were often estimated below 1, suggesting a reluctance of the respondents to admit certain difficulties, such as those due to social norms. \n\n\\begin{figure}[h]\n \\centering\n \\begin{subfigure}[t]{0.495\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_simple_R8.png}\n \\caption{Anxiety item R8}\n \\label{fig:icc_simple:R8}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.495\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_simple_R29.png}\n \\caption{Anxiety item R29}\n \\label{fig:icc_simple:R29}\n \\end{subfigure}\n \\caption{Estimated item characteristic curves for the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}}\\label{fig:icc_simple}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_simple_estimates.png}\n \\caption{Item parameter estimates of the Anxiety items for the simple 4\\gls{pl} model \\eqref{eq:4PL:simple}. }\n \\label{fig:estimates_simple}\n\\end{figure}\n\n\\paragraph{Group-specific 4PL model. }\nThe smallest differences between the four estimation algorithms and the fitted item characteristic curves in the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif} were once more observed in item R8 (\\highlight{\"I had a racing or pounding heart\"}; Figure~\\ref{fig:icc_DIF:R8}). The greatest differences were noticed in item R24 (\\highlight{\"Many situations made me worry\"}; Figure~\\ref{fig:icc_DIF:R24}). The smallest overall differences were found between the \\gls{em} algorithm and the algorithm based on \\gls{plf}. The greatest overall differences were observed between the \\gls{nls} and the algorithm based on \\gls{plf}, analogous to the simple 4\\gls{pl} model. Additionally, similar patterns were seen in the estimated item parameters (Figure~\\ref{fig:estimates_DIF}, Table~\\ref{tab:anxiety:pars_dif}). \n\n\\begin{figure}[h!]\n \\centering\n \\begin{subfigure}[t]{0.495\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_DIF_R8.png}\n \\caption{Anxiety item R8}\n \\label{fig:icc_DIF:R8}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.495\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_DIF_R24.png}\n \\caption{Anxiety item R24}\n \\label{fig:icc_DIF:R24}\n \\end{subfigure}\n \\caption{Estimated item characteristic curves for the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}}\\label{fig:icc_DIF}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/anxiety_DIF_estimates.png}\n \\caption{Item parameter estimates of the Anxiety items for the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}}\n \\label{fig:estimates_DIF}\n\\end{figure}\n\n\\paragraph{DIF detection. }\nUsing the likelihood ratio test, the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} was rejected for item R6 (\\highlight{\"I was concerned about my mental health\"}), item R10 (\\highlight{\"I had sudden feelings of panic\"}), and item R12 (\\highlight{\"I had trouble paying attention\"}) when considering at least one estimation method (i.e., these items functioned differently). While item R6 was identified as a \\gls{dif} item by all four of the estimation methods (all $p$-values $< 0.004$), items R10 and R12 were only identified as functioning differently with the \\gls{nls} ($p$-value = 0.042) and with the algorithm based on \\gls{plf} ($p$-value = 0.047), respectively. \n\nIn item R6, there were no significant differences between the estimated asymptotes, and \\gls{dif} was caused by various intercepts and slopes of the two genders (Figure~\\ref{fig:estimates_DIF}). Male participants seemed to have a higher probability of being concerned about their mental health at least rarely (original response $\\geq 2$) than female participants of the same overall anxiety level. This difference was especially apparent for those with lower levels of anxiety, whereas the differences between these two genders narrowed as the overall anxiety level increased (Figure~\\ref{fig:icc_DIF_R6}). \n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.495\\textwidth]{Figures\/anxiety_DIF_R6.png}\n \\caption{Estimated item characteristic curves for item R6 of the Anxiety dataset for the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}}\n \\label{fig:icc_DIF_R6}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nThis work explored novel approaches for estimating item parameters in the generalised logistic regression model. We described in detail two existing procedures (\\gls{nls} and \\gls{ml}) and their applications for fitting the covariate-specific 4\\gls{pl} model~\\eqref{eq:4PL:gen}. Additionally, the study proposed two iterative procedures (a procedure using the \\gls{em} algorithm and a method based on \\gls{plf}). With a simulation study, we demonstrated satisfactory precision of the newly proposed \\gls{plf}-based procedure even for small sample sizes and when additional covariates were considered. However, these pleasant properties were not observed for the \\gls{nls} and \\gls{ml} methods, which produced either biased estimates or wide confidence intervals. On the other hand, the \\gls{em} algorithm performed satisfactorily, but it sometimes failed to converge in a predefined number of iterations, so its fitting was inefficient. As the sample size increased, differences between the estimation methods vanished, and all estimates were near the true values of the item parameters. \n\nUsing a real data example for the anxiety measure, we illustrated practical challenges in estimation procedures, including specification of initial values. The smallest differences between the estimation procedures were observed for the \\gls{em} algorithm and the procedure based on \\gls{plf}. Additionally the largest dissimilarities were found for the \\gls{nls} and the \\gls{plf}-based method, supporting the findings of the simulation study, especially when considering a smaller sample size in the Anxiety dataset. \n\nIn recent decades, the topic of the~parametric link function has been extensively discussed in the literature by many authors, including \\citeA{basu2005estimating}, \\citeA{flach2014generalized}, and \\citeA{scallan1984fitting}. For example, \\citeA{pregibon1980goodness} proposed the~\\gls{ml} estimation of the~link parameters using a~weighted least squares algorithm. In the same vein, \\citeA{mccullagh1989generalized} adapted this approach and presented an~algorithm in which several models with the~fixed link functions were fitted. Furthermore, \\citeA{kaiser1997maximum} proposed a~modified scoring algorithm to perform simultaneous \\gls{ml} estimation of all parameters. \\citeA{scallan1984fitting} proposed an iterative two-stage algorithm, building on the work of \\citeA{richards1961method}. In this study's approach, we examined generalised logistic regression, accounting for the possibility of guessing and inattention or lapse rate, whereas these features may depend upon the respondents' characteristics. \n\nThe crucial part of the estimation process is specifying starting values for item parameters because these values may significantly impact the speed of the estimation process and its precision. For instance, initial values which are far from the true item parameters may lead to situations in which the estimation algorithm returns only a local extreme or even it does not converge. In this work we used an approach based on upper-lower index which resulted in a low rate of convergence issues with the satisfactory estimation precision. However, other possible naive estimates of discrimination (and other parameters), such as a correlation between an item score and the total test score without given item, could be considered. \n\nThere were several limitations to this study, and several possible further directions for study exist. First, the simulation study was limited to two models -- the simple 4\\gls{pl} model \\eqref{eq:4PL:simple} and the group-specific 4\\gls{pl} model \\eqref{eq:4PL:dif}, both of which included only one or two covariates. The simulation study suggested requiring a larger sample size with the increasing number of covariates. Second, the work considered only one set of item parameters, noting that various values of asymptotes were especially prone to producing computational challenges. Third, this article described the \\gls{nls} method as a simple approach, not accounting for the heteroscedasticity of binary data. For such data, the~Pearson's residuals might be more appropriate to use. This weighted form \\cite{ritz2015dose} takes the original squares of residuals and divides them by the variance $\\pi_{pi}(1 - \\pi_{pi})$. Next, the~\\gls{rss} of item $i$ \\eqref{eq:nls:rss} would take the following form:\n\\begin{align*}\n \\text{RSS}_i(\\boldsymbol{\\gamma}_i) = \\sum_{p = 1}^n \\frac{\\left(Y_{pi} - \\pi_{pi}\\right)^2}{\\pi_{pi}\\left(1 - \\pi_{pi}\\right)}.\n\\end{align*}\nHowever, the number of observations on tails of the matching criterion is typically small and provides only small variability at most. These heavy weights would require a nearly exact fit for cases with few observations. Nevertheless, the computation of the \\gls{nls} estimates demonstrated in this work was straightforward and efficient, providing sufficient precision. Thus, this method could be useful in certain cases, such as producing an initial idea about parameter values and using these estimates as starting values for other approaches. \n\nAdditionally, this study's real data example explored item functioning in the multi-item measurement related to anxiety. However, the task of the parameter estimation in the presented models would also be relevant for several other situations. The later could include data from educational measurement, where the lower asymptote may represent item guessing, and the upper asymptote may represent the lapse rate (item slipping or inattention). Moreover, the generalised logistic regression model is not limited to multi-item measurements since the class determined by Equation \\eqref{eq:4PL:gen} represents a wide family of the covariate-specific 4\\gls{pl} models. This model might be used and further extended in various study fields, including but not limited to quantitative pharmacology \\cite{dinse2011algorithm}; applied microbiology \\cite{brands2020method}; modelling patterns of urban electricity usage \\cite{to2012growth}; and plant growth modelling \\cite{zub2012late}. Therefore, estimating parameters and understanding limitations of used methods are crucial for a wide range of researchers and practitioners. \n\nTo conclude, this study illustrated differences and challenges in fitting generalised logistic regression models using various estimation techniques. This work demonstrated the superiority of the novel implementation of the \\gls{em} algorithm and the newly proposed method based on \\gls{plf} over the existing \\gls{nls} and \\gls{ml} methods. Thus, improving the estimation algorithms is critical since it could increase precision while maintaining a user-friendly implementation. \n\n\n\\section*{Acknowledgement}\n\nThe study was funded by the Czech Science Foundation grant number 21-03658S.\n\\section*{Supplementary Material}\n\nAdditional tables and figures, and accompanying \\texttt{R} scripts are available at \\href{https:\/\/osf.io\/eu5zm\/}{https:\/\/osf.io\/bk8a7\/}.\n\n\n\\bibliographystyle{apacite}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec1}\n\n\\subsection{Infinity-Laplacian and games}\n\nFor an integer $m\\ge2$,\\vspace*{1pt} let a bounded $\\mathcaligr{C}^2$ domain\n$G\\subset{\\mathbb{R}}^m$, functions $g\\in\\mathcaligr{C}(\\partial\nG,{\\mathbb{R}})$ and\n$h\\in\\mathcaligr{C}(\\overline G,{\\mathbb{R}}\\setminus\\{0\\})$ be given. We\nstudy a two-player\nzero-sum stochastic differential game (SDG), defined in terms of an\n$m$-dimensional state process\nthat is driven by a one-dimensional Brownian motion, played until\nthe state exits the domain. The functions $g$ and $h$ serve as\nterminal, and, respectively, running payoffs. The players' controls\nenter in a diffusion coefficient and in an unbounded drift\ncoefficient of the state process. The dynamics are degenerate in\nthat it is possible for the players to completely switch off the\nBrownian motion.\nWe show that the game has value, and characterize the value function\nas the unique viscosity solution $u$ (uniqueness of solutions is\nknown from \\cite{pssw}) of the equation\n\\begin{equation}\\label{07}\n\\cases{-2\\Delta_\\infty u=h, &\\quad in $G$,\\cr\nu=g, &\\quad on $\\partial G$.}\n\\end{equation}\nHere, $\\Delta_\\infty$ is the infinity-Laplacian defined as $\\Delta\n_{\\infty}f =\n(Df)'(D^2f) (Df)\/ |Df|^2$, provided $Df \\neq0$, where for a\n$\\mathcaligr{C}^2$ function $f$ we denote by $Df$ the gradient and by $D^2f$\nthe Hessian matrix.\nOur work is motivated by a representation for $u$ of Peres et al.\n\\cite{pssw} (established in fact in a far greater generality), as\nthe limit, as $\\varepsilon\\to0$, of the value function $V^\\varepsilon\n$ of a\ndiscrete time random turn game, referred to as \\textit{Tug-of-War}, in\nwhich $\\varepsilon$ is a parameter. The contribution of the current\nwork is\nthe identification of a game for which the value function is\nprecisely equal to $u$.\n\nThe infinity-Laplacian was first considered by Aronsson \\cite{Aron}\nin the study of absolutely minimal (AM) extensions of Lipschitz\nfunctions. Given a Lipschitz function $u$ defined on the boundary\n$\\partial G$ of a domain $G$, a Lipschitz function $\\widehat u$ extending\n$u$ to $\\overline G$ is called an AM extension of $u$ if, for every open\n$U \\subset G$, $\\operatorname{Lip}_{\\overline U}\\widehat u = \\operatorname\n{Lip}_{\\partial U}u$, where for\na real\nfunction $f$ defined on $F \\subset{\\mathbb{R}}^m$,\n$\\operatorname{Lip}_F f = {\\sup_{x,y \\in F, x\\neq y}} |f(x) - f(y)|\/|x-y|$.\nIt was shown in \\cite{Aron} that a Lipschitz function $\\widehat u$ on\n$\\overline G$\nthat is $\\mathcaligr{C}^2$\non $G$ is an AM extension of $\\widehat u|_{\\partial G}$ if and only if\n$\\widehat u$ is infinity-harmonic, namely satisfies $\\Delta_{\\infty\n}\\widehat u = 0$\nin $G$.\nThis connection enables in some cases to prove uniqueness of AM\nextensions via PDE tools.\nHowever, due to the degeneracy of this elliptic equation, classical\nPDE approach in general is not applicable.\nJensen \\cite{Jen} showed that an appropriate framework\nis through the theory of viscosity solutions,\nby establishing existence and uniqueness of viscosity solutions to\nthe homogeneous version ($h=0$) of (\\ref{07}), and showing that if\n$g$ is Lipschitz then the solution is an AM extension of $g$. In\naddition to the relation to AM extensions, the infinity-Laplacian\narises in a variety of other situations \\cite{BEJ}. Some examples\ninclude models for sand-pile evolution \\cite{Aron2}, motion by mean\ncurvature and stochastic target problems \\cite{KoSe,SoTo}.\n\nWe do not treat the homogenous equation for reasons mentioned later\nin this section. The inhomogeneous equation may admit multiple\nsolutions when $h$ assumes both signs \\cite{pssw}. Our assumption on\n$h$ implies that either $h > 0 $ or $h < 0$. Uniqueness for the\ncase where these strict inequalities are replaced with weak inequalities\nis unknown \\cite{pssw}. Thus, the assumption we make on $h$ is the\nminimal one under which\nuniqueness is known to hold in general (except\nthe case $h=0$).\n\nLet us describe the Tug-of-War game introduced in \\cite{pssw}. Fix\n$\\varepsilon>0$.\nLet a token be placed at $x\\in G$, and set $X_0=x$. At the $k$th\nstep of the game ($k\\ge1$), an independent toss of a fair coin\ndetermines which player takes the turn. The selected player is\nallowed to move the token from its current position $X_{k-1} \\in G$\nto a new position $X_k$ in $\\overline G$, in such a way that $|X_k -\nX_{k-1}| \\le\\varepsilon$ (\\cite{pssw} requires $|X_k - X_{k-1}| <\n\\varepsilon$\nbut this is an equivalent formulation in the setting described\nhere).\nThe game ends at the first time $K$ when $X_K\\in\\partial G$. The\nassociated payoff is given by\n\\begin{equation}\n\\label{PF1051}\n\\mathbf{E} \\Biggl[g(X_K) + \\frac{\\varepsilon^2}{4} \\sum_{k=0}^{K-1}\nh(X_k) \\Biggr].\n\\end{equation}\nPlayer I attempts to maximize the payoff and player II's goal is to\nminimize it. It is shown in \\cite{pssw} that\nthe value of the game, defined in a\nstandard way and denoted $V^\\varepsilon(x)$, exists, that\n$V^{\\varepsilon}$\nconverges uniformly to a function $V$ referred to as the ``continuum\nvalue function'' and that $V$ is the unique viscosity solution of\n(\\ref{07}) (these results are in fact also proved for the\nhomogeneous case, and in generality greater than the scope of the\ncurrent paper).\nThe question of associating a game directly with the continuum value\nwas posed and some basic technical challenges associated with it\nwere discussed in\n\\cite{pssw}.\n\nOur approach to the question above is via a SDG formulation. To\nmotivate the form of the SDG, we start with the Tug-of-War game and\npresent some formal calculations (a precise definition of the SDG\nwill appear later). Let $\\{\\xi_k, k \\in\\mathbb{N}\\}$ be a sequence\nof i.i.d. random variables on some probability space $(\\Omega,\n\\mathcaligr{F}, \\mathbf{P})$ with $\\mathbf{P}(\\xi_k = 1) = \\mathbf\n{P}(\\xi_k = -1)=1\/2$,\ninterpreted as the sequence of coin tosses.\nLet\n$\\{\\mathcaligr{F}_k\\}_{k\\ge0}$ be a filtration of $\\mathcaligr{F}$ to which\n$\\{\\xi_k\\}$ is adapted and such that $\\{\\xi_{k+1},\\xi_{k+2},\\ldots\\}$\nis independent of $\\mathcaligr{F}_k$ for every $k\\ge0$. Let $\\{a_k\\}$,\n$\\{b_k\\}$ be $\\{\\mathcaligr{F}_k\\}$-predictable\nsequences of random variables with values in\n$\\overline{\\mathbb{B}_{\\varepsilon}(0)} = \\{x \\in{\\mathbb{R}}^m\\dvtx\n|x| \\le\\varepsilon\\}$. These\nsequences correspond to control actions of players I and II;\nthat is, $a_k$ (resp., $b_k$) is the displacement exercised by\nplayer I (resp., player II) if it wins the $k$th coin toss.\nAssociating the event $\\{\\xi_k = 1\\}$ with player I winning the\n$k$th toss, one can write the following representation for the\nposition of the token, starting from initial state $x$. For $j \\in\n\\mathbb{N}$,\n\\[\nX_j = x + \\sum_{k=1}^j \\biggl[\na_k\\frac{1+\\xi_k}{2}+b_k\\frac{1-\\xi_k}{2} \\biggr]\n=\\sum_{k=1}^j\\frac{a_k-b_k}{2}\\xi_k+\\sum_{k=1}^j\\frac{a_k+b_k}{2}.\n\\]\nWe shall refer to $\\{X_j\\}$ as the ``state process.'' This\nrepresentation, in which turns are not taken at random\nbut both players select an action at each step, and the noise enters\nin the dynamics, is more convenient for the development that\nfollows. Let $\\varepsilon= 1\/\\sqrt{n}$ and rescale the control processes\nby defining, for $t \\ge0$, $A^n_t=\\sqrt n a_{[nt]}$, $B^n_t=\\sqrt\nn b_{[nt]}$. Consider the continuous time state process $X^n_t =\nX_{[nt]}$, and define\n$\\{W^n_t\\}_{t\\ge0}$ by setting $W^n_0=0$ and using the relation\n\\[\nW^n_t = W^n_{(k-1)\/n}+ \\biggl(t-\\frac{k-1}{n} \\biggr)\\sqrt{n}\n\\xi_k,\\qquad\nt\\in \\biggl(\\frac{k-1}{n}, \\frac{k}{n} \\biggr], k \\in\\mathbb{N}.\n\\]\nThen we have\n\\begin{equation}\\label{eq330} X^n_t = x + \\frac{1}{2} \\int_0^t\n(A^n_s - B^n_s) \\,dW^n_s + \\frac{1}{2} \\int_0^t \\sqrt{n}(A^n_s +\nB^n_s) \\,ds.\n\\end{equation}\nNote that $W^n$ converges weakly to a standard Brownian motion, and\nsince $|A^n_t| \\vee|B^n_t| \\le1$, the second term on the\nright-hand side of (\\ref{eq330}) forms a tight sequence. Thus, it is easy\nto guess a substitute for it in the continuous game.\nInterpretation of the asymptotics of the third term is more subtle,\nand is a key element of the formulation.\nOne possible approach is to\nreplace the factor $\\sqrt{n}$ by a large quantity that is\ndynamically controlled by the two players. This point of view\nmotivates one to consider the identity (that we prove in Proposition \\ref{prop2})\n\\begin{eqnarray}\\label{44}\n&&-2\\Delta_\\infty f = \\sup_{|b|=1, d \\ge0 } \\inf_{|a|=1, c\\ge0}\n\\biggl\\{-\\frac{1}{2} (a-b)' (D^2f) (a-b)\\nonumber\\\\\n&&\\hspace*{148.4pt}{} - (c+d)(a+b)\\cdot Df\n\\biggr\\},\\\\\n\\eqntext{f \\in\\mathcaligr{C}^2, Df \\neq0,}\n\\end{eqnarray}\nfor the following reason.\nLet $\\mathcaligr{H} = \\mathcaligr{S}^{m-1} \\times[0, \\infty)$\nwhere $\\mathcaligr{S}^{m-1}$ is the unit sphere in ${\\mathbb{R}}^m$. The\nexpression in curly brackets is equal to $\\mathcaligr{L}^{a,b,c,d}f(x)$,\nwhere for $(a,c),(b,d)\\in\\mathcaligr{H}$, $\\mathcaligr{L}^{a,b,c,d}$ is\nthe controlled\ngenerator associated with the process\n\\begin{equation}\\label{star1050}\nX_t = x+\\int_0^t(A_s-B_s)\\,dW_s+\\int_0^t(C_s+D_s)(A_s+B_s)\\,ds,\n\\qquad\nt\\in[0,\\infty),\\hspace*{-33pt}\n\\end{equation}\nand $(A, C)$ and $(B, D)$ are control processes taking values in\n$\\mathcaligr{H}$.\nSince $\\Delta_\\infty$ is related to (\\ref{eq330}) via the\nTug-of-War, and\n$\\mathcaligr{L}^{a,b,c,d}$ to (\\ref{star1050}), identity (\\ref{44}) suggests\nto regard (\\ref{star1050}) as a formal limit of (\\ref{eq330}).\nConsequently the SDG will have (\\ref{star1050}) as a state process,\nwhere the controls $(A,C)$ and $(B,D)$ are chosen by the two players.\nFinally, the payoff functional, as a formal limit of (\\ref{PF1051}), and\naccounting for the extra factor of $1\/2$ in (\\ref{eq330}), will be\ngiven by\n$\\mathbf{E}[\\int_0^{\\tau} h(X_s) \\,ds + g(X_{\\tau})]$, where\n$\\tau=\\inf\\{t\\dvtx X_t \\notin G \\}$\n(with an appropriate convention regarding $\\tau= \\infty$).\n\nA precise formulation of this game is given in Section \\ref{sec2},\nalong with a statement of the main result.\nSection \\ref{sec1.3} discusses the technique and some open problems.\n\nThroughout, we will denote by $\\mathscr{S}(m)$ the space of symmetric $m\n\\times m$\nmatrices, and by $I_m\\in\\mathscr{S}(m)$ the identity matrix.\nA function $\\vartheta\\dvtx[0,\\infty)\\to[0,\\infty)$ will be said to be a\n\\textit{modulus} if it is continuous, nondecreasing, and satisfies\n$\\vartheta(0)=0$.\n\n\\subsection{SDG formulation and main result}\\label{sec2}\n\nRecall that $G$ is a bounded $\\mathcaligr{C}^2$ domain in ${\\mathbb\n{R}}^m$, and that\n$g\\dvtx\\partial G\\to{\\mathbb{R}}$ and $h\\dvtx\\overline G\\to{\\mathbb\n{R}}\\setminus\\{0\\}$ are given continuous\nfunctions. In particular we have that either $h > 0$ or $h < 0$.\nSince the two cases are similar, we will only consider $h>0$, and\nuse the notation $\\underline h:=\\inf_{\\overline G} h>0$. Let\n$(\\Omega,\\mathcaligr{F},\\{\\mathcaligr{F}_t\\},\\mathbf{P})$ be a complete\nfiltered probability\nspace with right-continuous filtration, supporting an\n$(m+1)$-dimensional $\\{\\mathcaligr{F}_t\\}$-Brownian motion $\\overline\nW=(W,\\widetilde\nW)$, where $W$ and $\\widetilde W$ are one- and $m$-dimensional Brownian\nmotions, respectively. Let $\\mathbf{E}$ denote expectation with\nrespect to\n$\\mathbf{P}$. Let $X_t$ be a process taking values in ${\\mathbb\n{R}}^m$, given by\n\\begin{equation}\\label{01}\nX_t = x+\\int_0^t(A_s-B_s)\\,dW_s+\\int_0^t(C_s+D_s)(A_s+B_s)\\,ds,\\qquad\nt\\in[0,\\infty),\\hspace*{-33pt}\n\\end{equation}\nwhere $x\\in\\overline G$, $A_t$ and $B_t$ take values in the unit sphere\n$\\mathcaligr{S}^{m-1}\\subset{\\mathbb{R}}^m$, and $C_t$ and $D_t$ take\nvalues in\n$[0,\\infty)$. Denote\n\\begin{equation}\\label{23}\nY^0=(A,C),\\qquad Z^0=(B,D).\n\\end{equation}\nThe processes $Y^0$ and $Z^0$ take values in\n$\\mathcaligr{H}=\\mathcaligr{S}^{m-1}\\times[0,\\infty)$. These processes\nwill correspond to\ncontrol actions of the maximizing and minimizing player,\nrespectively. We remark that,\\vspace*{1pt} although $\\widetilde W$ does not appear\nexplicitly in the dynamics~(\\ref{01}), the control processes $Y^0,\nZ^0$ will be required to be $\\{\\mathcaligr{F}_t\\}$-adapted, and thus may\ndepend on\nit. In Section \\ref{sec1.3}, we comment on the need for including\nthis auxiliary Brownian motion in our formulation. Let\n\\[\n\\tau=\\inf\\{t\\dvtx X_t\\in\\partial G\\}.\n\\]\nThroughout, we will follow the convention that the infimum over an\nempty set is~$\\infty$. We write\n\\begin{equation}\n\\label{30}\nX(x,Y^0,Z^0) \\qquad [\\mbox{resp., } \\tau(x,Y^0,Z^0)]\n\\end{equation}\nfor the process $X$ (resp., the random time $\\tau$) when it is\nimportant to specify the explicit dependence on $(x,Y^0,Z^0)$. If\n$\\tau<\\infty$ a.s., then the payoff $J(x, Y^0, Z^0)$ is well defined\nwith values in $(-\\infty,\\infty]$, where\n\\begin{equation}\\label{02}\nJ(x,Y^0,Z^0)=\\mathbf{E}\\biggl[\\int_0^\\tau h(X_s)\\,ds+g(X_\\tau)\\biggr]\n\\end{equation}\nand $X$ is given by (\\ref{01}). When $\\mathbf{P}(\\tau\n(x,Y^0,Z^0)=\\infty)>0$,\nwe set $J(x,Y^0,Z^0)=\\infty$, in agreement with the expectation of the\nfirst term in (\\ref{02}).\n\nWe turn to the precise definition of the SDG.\nFor a process $H^0=(A,C)$ taking values in $\\mathcaligr{H}$, we let\n$S(H^0)=\\operatorname{ess}\\sup \\sup_{t\\in[0,\\infty)}C_t$. In the\nformulation below,\neach player initially declares a bound $S$, and then plays so as to\nkeep $S(H^0)\\le S$.\n\\begin{definition}\\label{def1}\n(i) A pair $H=(\\{H^0_t\\},S)$, where $S\\in{\\mathbb{N}}$ and $\\{H^0_t\\}\n$ is a\nprocess taking values in $\\mathcaligr{H}$, is said to be an admissible\ncontrol if $\\{H^0_t\\}$ is $\\{\\mathcaligr{F}_t\\}$-progressively measurable,\nand\n$S(H^0)\\le S$.\nThe set of all admissible controls is denoted by\n$M$.\nFor $H=(\\{H^0_t\\},S)\\in M$, denote $\\mathbf{S}(H)=S$.\n\n(ii) A mapping $\\varrho\\dvtx M\\to M$ is said to be a strategy if, for\nevery $t$,\n\\[\n\\mathbf{P}(H^0_s=\\widetilde H^0_s \\mbox{ for a.e. } s\\in[0,t])=1\n\\quad\\mbox{and}\\quad\nS=\\widetilde S\n\\]\nimplies\n\\[\n\\mathbf{P}(I^0_s=\\widetilde I^0_s \\mbox{ for a.e. } s\\in[0,t])=1\n\\quad\\mbox{and}\\quad\nT=\\widetilde T,\n\\]\nwhere $(I^0,T)=\\varrho[(H^0,S)]$ and $(\\widetilde I^0,\\widetilde\nT)=\\varrho[(\\widetilde\nH^0,\\widetilde\nS)]$.\nThe set of all strategies is denoted by $\\widetilde\\Gamma$. For\n$\\varrho\n\\in\\widetilde\\Gamma$, let $\\mathbf{S}(\\varrho) = \\sup_{H \\in\nM}\\mathbf{S}(\\varrho[H])$.\nLet\n\\[\n\\Gamma= \\{\\varrho\\in\\widetilde\\Gamma\\dvtx \\mathbf{S}(\\varrho)\n<\\infty\\}.\n\\]\n\\end{definition}\n\nWe will use the symbols $Y$ and $\\alpha$ for generic control and\nstrategy\\break for the maximizing player, and $Z$ and $\\beta$ for the\nminimizing player. If\\break\n$Y = (Y^0,K), Z =(Z^0,L)\\in M$, we sometimes write\n$J(x,Y,Z) = J(x,(Y^0,\\break K),(Z^0,L))$ for $J(x$,$Y^0,Z^0)$. Similar\nconventions will be used for $X(x, Y, Z)$ and $\\tau(x, Y, Z)$. Let\n\\begin{eqnarray*}\nJ^x(Y,\\beta) &=& J(x,Y,\\beta[Y]), \\qquad x\\in\\overline G, Y\\in M, \\beta\\in\n\\Gamma,\n\\\\\nJ^x(\\alpha,Z) &=& J(x,\\alpha[Z],Z),\\qquad x\\in\\overline G, \\alpha\\in\\Gamma\n, Z\\in M.\n\\end{eqnarray*}\nDefine analogously $X^x(Y,\\beta)$, $X^x(\\alpha,Z)$, $\\tau^x(Y,\\beta)$\nand $\\tau^x(\\alpha,Z)$ via (\\ref{30}). Define the lower value of the\nSDG by\n\\begin{equation}\\label{05}\nV(x)=\\inf_{\\beta\\in\\Gamma}\\sup_{Y\\in M}J^x(Y,\\beta)\n\\end{equation}\nand the upper value by\n\\begin{equation}\\label{06}\nU(x)=\\sup_{\\alpha\\in\\Gamma}\\inf_{Z\\in M}J^x(\\alpha,Z).\n\\end{equation}\nThe game is said to have a value if $U=V$.\n\nRecall that the infinity-Laplacian is defined by $\\Delta_\\infty\nf=p'\\Sigma\np\/|p|^2$, where $f$ is a $\\mathcaligr{C}^2$ function, $p=Df$ and $\\Sigma=D^2f$,\nprovided that $p\\ne0$. Thus, $\\Delta_\\infty f$ is equal to the second\nderivative in the direction of the gradient. In the special case\nwhere $D^2f(x)$ is of the form $\\lambda I_m$ for some real $\\lambda$,\nit is\ntherefore natural to define $\\Delta_\\infty f(x)=\\lambda$ even if $Df(x)=0$\n\\cite{pssw}. This will be reflected in the definition of viscosity\nsolutions of (\\ref{07}), that we state below.\nLet\n\\begin{eqnarray*}\n\\mathcaligr{D}_0 &=& \\{(0,\\lambda I_m)\\in{\\mathbb{R}}^m\\times\\mathscr\n{S}(m)\\dvtx \\lambda\\in{\\mathbb{R}}\\},\n\\\\\n\\mathcaligr{D}_1 &=& ({\\mathbb{R}}^m\\setminus\\{0\\})\\times\\mathscr{S}(m),\n\\\\\n\\mathcaligr{D} &=& \\mathcaligr{D}_0\\cup\\mathcaligr{D}_1\n\\end{eqnarray*}\nand\n\\[\n\\Lambda(p,\\Sigma)= \\cases{-2\\lambda, &\\quad $(p,\\Sigma)=(0,\\lambda\nI_m)\\in\n\\mathcaligr{D}_0$, \\vspace*{2pt}\\cr\n-2\\dfrac{p'\\Sigma p}{|p|^2}, &\\quad $(p,\\Sigma)\\in\\mathcaligr{D}_1$.}\n\\]\n\\begin{definition}\\label{def3}\nA continuous function $u\\dvtx \\overline G\\to{\\mathbb{R}}$ is said to be a viscosity\nsupersolution (resp., subsolution) of (\\ref{07}), if:\n\n\\begin{longlist}\n\\item for every $x \\in G$ and $\\varphi\\in\\mathcaligr{C}^2(G)$ for which\n$(p,\\Sigma):=(D\\varphi(x),D^2\\varphi(x))\\in\\mathcaligr{D}$, and\n$u - \\varphi$ has a global minimum [maximum] on $G$ at $x$, one has\n\\begin{equation}\\label{24}\n\\Lambda(p,\\Sigma)-h(x)\\ge0 \\qquad [\\le0];\n\\end{equation}\nand\n\\item $u=g$ on $\\partial G$.\n\nA viscosity solution is a function which is both a super- and a\nsubsolution.\n\\end{longlist}\n\\end{definition}\n\nThe result below has been established in \\cite{pssw}.\n\\begin{theorem}\\label{thpssw}\nThere exists a unique viscosity solution to (\\ref{07}).\n\\end{theorem}\n\nThe following is our main result.\n\\begin{theorem}\\label{th1}\nThe functions $U$ and $V$ are both viscosity solutions to\n(\\ref{07}). Consequently, the SDG has a value.\n\\end{theorem}\n\nIn what follows, we use the terms subsolution, supersolution and\nsolution as shorthand for viscosity subsolution, etc.\n\n\\subsection{Discussion}\n\\label{sec1.3}\n\nWe describe here our approach to proving the main result, and\nmention some obstacles in extending it.\n\nA common approach to showing solvability of Bellman--Isaacs (BI)\nequations [(\\ref{07}) can be viewed as such an equation due to\n(\\ref{44})] by the associated value function, is by proving that the\nvalue function satisfies a dynamic programming principle (DPP).\nRoughly speaking, this is an equation expressing the fact that,\nrather than attempting to maximize their profit by considering\ndirectly the payoff functional, the players may consider the payoff\nincurred up to a time $t$ plus the value function evaluated at the\nposition $X_t$ that the state reaches at that time. Although in a\nsingle player setting (i.e., in pure control problems) DPP are well\nunderstood, game theoretic settings as in this paper are\nsignificantly harder. In particular, as we shall shortly point out,\nthere are some basic open problems related to such DPP. In a setting\nwith a finite time horizon, Fleming and Souganidis \\cite{FS}\nestablished a DPP\nbased on careful discretization and approximation arguments. We have\nbeen unable to carry out a similar proof in the current setting,\nwhich includes a payoff given in terms of an exit time, degenerate\ndiffusion and unbounded controls.\n\nSwiech \\cite{swi} has developed an alternative approach to the above\nproblem that relies on existence of solutions. Instead of\nestablishing a DPP for the\nvalue function, the idea of \\cite{swi} is to show that\nany \\textit{solution} must satisfy a DPP. To see what is meant by such\na DPP and how it is used, consider the equation, $-2\\Delta_\\infty\nu+\\lambda\nu=h$ in $G$, $u=g$ on $\\partial G$, where $\\lambda\\ge0$ is a constant,\nassociated with the payoff in (\\ref{02}) modified by a discount\nfactor. Assume that one can show that whenever $u$ and $v$ are sub-\nand supersolutions, respectively, then\n\\begin{eqnarray}\\label{45}\nu(x)&\\le&\\sup_{\\alpha\\in\\Gamma}\\inf_{Z\\in M}\\mathbf{E} \\biggl[\\int\n_0^\\sigma\ne^{-\\lambda s}h(X_s)\\,ds+e^{-\\lambda\\sigma}u(X_\\sigma) \\biggr],\n\\\\\n\\label{46}\nv(x)&\\ge&\\sup_{\\alpha\\in\\Gamma}\\inf_{Z\\in M}\\mathbf{E} \\biggl[\\int\n_0^\\sigma\ne^{-\\lambda s}h(X_s)\\,ds+e^{-\\lambda\\sigma}v(X_\\sigma) \\biggr],\n\\end{eqnarray}\nfor $X=X[x,\\alpha[Z],Z]$, $\\tau=\\tau[x,\\alpha[Z],Z]$ and $\\sigma\n=\\sigma(t)=\\tau\n\\wedge\nt$. Sending $t \\to\\infty$ in the above equations, one would\nformally obtain\n\\begin{equation}\\label{ins1155}u(x) \\le\n\\sup_{\\alpha\\in\\Gamma}\\inf_{Z\\in M}\\mathbf{E} \\biggl[\\int_0^{\\tau}\ne^{-\\lambda s}h(X_s)\\,ds+e^{-\\lambda\\tau}g(X_{\\tau}) \\biggr] \\le v(x),\n\\end{equation}\nin particular yielding that if $u=v$ is a solution to the equation\nthen it must equal the upper value function. This would establish\nunique solvability of the equation by the upper value function,\nprovided there exists a solution. In\nthe case $\\lambda> 0$, justifying the above formal limit is\nstraightforward (see \\cite{swi})\nbut the case $\\lambda= 0$, as in our setting, requires a more careful\nargument.\nOur proofs exploit the uniform positivity of $h$ due to which the\nminimizing player will not allow $\\tau$ to be too large. This leads\nto uniform estimates on the decay of $\\mathbf{P}(\\tau> t)$ as $t\\to\n\\infty$,\nfrom which an inequality as in (\\ref{ins1155}) follows readily. This\ndiscussion also explains why we are unable to treat the case $h=0$.\n\nEstablishing DPP as in (\\ref{45}), (\\ref{46}) is thus a key\ningredient in this approach. For a class of BI equations, defined\non all of ${\\mathbb{R}}^m$, for which the associated game has a\nbounded action\nset and a fixed, finite time horizon, such a DPP was proved in\nSwiech \\cite{swi}. In the current paper, although we do not\nestablish (\\ref{45}), (\\ref{46}) in the above form, we derive\nsimilar inequalities (for $\\lambda= 0$) for a related bounded action\ngame, defined on $G$. The characterization of the value function\nfor the original unbounded action game is then treated by taking\nsuitable limits.\n\nBoth \\cite{FS} and \\cite{swi} require some assumptions on the\nsample space and underlying filtration. In \\cite{FS}, the underlying\nfiltration is the one generated by the driving Brownian motion. The\napproach taken in \\cite{swi}, which the current paper follows,\nallows for a general filtration as long as it is rich enough to\nsupport an $m$-dimensional Brownian motion, independent of the\nBrownian motion driving the state process [for example, it could be\nthe filtration generated by an $(m+1)$-dimensional Brownian motion].\nThe reason for imposing this requirement in \\cite{swi} is that\ninequalities similar to (\\ref{45}) and (\\ref{46}) are proved by\nfirst establishing them for a game associated with a nondegenerate\nelliptic equation, and then taking a vanishing viscosity limit. This\ntechnical issue is the reason for including the auxiliary process\n$\\widetilde W$ in our formulation as well. As pointed out in \\cite{swi},\nthe question of validity of the DPP and the characterization of the\nvalue as the unique solution to the PDE, under an arbitrary\nfiltration, remains a basic open problem on SDGs.\n\n The unboundedness of the action space, on one hand, and the combination of\ndegeneracy of the dynamics and an exit time criterion on the other hand, make it hard\nto adapt the results of \\cite{swi} to our setting. In order to\novercome the first difficulty, we approximate the original SDG by a\nsequence of games with bounded action spaces, that are more readily\nanalyzed.\nFor the bounded action game, existence of solutions to the upper and\nlower BI equations follow from \\cite{CKLS}. We show that the\nsolutions to these equations satisfy a DPP similar to\n(\\ref{45}) and (\\ref{46}) (Proposition \\ref{prop1}). As discussed\nabove, existence of solutions along with the DPP yields the\ncharacterization of these solutions as the corresponding value\nfunctions. Next, as we show in Lemma \\ref{lem5}, the upper and lower\nvalue functions for the bounded action games approach the\ncorresponding value functions of the original game, pointwise, as\nthe bounds approach~$\\infty$. Moreover, in Lemma \\ref{lem4}, we\nshow that any uniform subsequential limit, as the bounds\napproach $\\infty$, of\nsolutions to the BI equation for bounded action games is a viscosity\nsolution of (\\ref{07}). The last piece in the proof of the main\nresult is then showing existence of uniform (subsequential) limits.\nThis is established in Theorem \\ref{th3} by proving equicontinuity,\nin the parameters governing the bounds,\nof the value functions for bounded action games. The proof of\nequicontinuity is the most technical part of this paper and the main\nplace where the $\\mathcaligr{C}^2$ assumption on the domain is used.\nThis is also the place where the possibility of degenerate dynamics\nclose to the exit time needs to be carefully analyzed.\n\nThe rest of this paper is organized as follows. In Section\n\\ref{sec3}, we prove Theorem~\\ref{th1} based on results on BI\nequations for bounded action SDG. These results are established in\nSections \\ref{sec4} (equicontinuity of the value functions) and\n\\ref{sec5} (relating the value function to the PDE). Finally, it is\nnatural to ask whether the state process, obtained under\n$\\delta$-optimal play by both players, converges in law as $\\delta$\ntends to zero. Section \\ref{sec6} describes a recently obtained\nresult \\cite{AtBu2} that addresses this issue.\n\n\\section{Relation to Bellman--Isaacs equation}\\label{sec3}\n\nIn this section, we prove Theorem \\ref{th1} by relating the value\nfunctions $U$ and $V$ to value functions of SDG with bounded action\nsets, and similarly, the solution to (\\ref{07}) to that of the\ncorresponding Bellman--Isaacs equations.\n\nLet $p\\in{\\mathbb{R}}^m$, $p\\ne0$ and $S\\in\\mathscr{S}(m)$ be\ngiven, and, for\n$n\\in{\\mathbb{N}}$, fix $p_n\\in{\\mathbb{R}}^m$, $p_n\\ne0$ and\n$S_n\\in\\mathscr{S}(m)$, such\nthat $p_n\\to p$, $S_n\\to S$. Denote $\\overline p=p\/|p|$ and $\\overline\np_n=p_n\/|p_n|$. Let $\\{k_n\\}$ and $\\{l_n\\}$ be positive, increasing\nsequences such that $k_n\\to\\infty$, $l_n\\to\\infty$.\n\nDenote\n\\begin{equation}\\label{09}\n\\Phi(a,b,c,d;p,S)= -\\tfrac12 (a-b)'S(a-b)-(c+d)(a+b)\\cdot p,\n\\end{equation}\nand let\n\\begin{eqnarray}\\label{18}\n\\Lambda_{kl}^+(p,S)&=&\n\\max_{|b|=1, 0\\le d\\le l} \\min_{|a|=1, 0\\le c\\le k}\n\\Phi(a,b,c,d;p,S),\n\\\\\n\\label{10}\n\\Lambda_{kl}^-(p,S)&=&\\min_{|a|=1, 0\\le c\\le k} \\max_{|b|=1, 0\\le\nd\\le\nl} \\Phi(a,b,c,d;p,S).\n\\end{eqnarray}\nSet\n\\[\n\\Lambda_n^+(p,S)=\\Lambda^+_{k_nl_n}(p,S),\\qquad\n\\Lambda_n^-(p,S)=\\Lambda^-_{k_nl_n}(p,S).\n\\]\n\\begin{lemma}\\label{lem1}\nOne has $\\Lambda^+_n(p_n,S_n)\\to\\Lambda(p,S)$, and\n$\\Lambda^-_n(p_n,S_n)\\to\\Lambda(p,S)$, as $n\\to\\infty$.\n\\end{lemma}\n\\begin{pf}\nWe prove only the statement regarding $\\Lambda^-_n$, since the\nother statement can be proved analogously. We omit the superscript\n``$-$'' from the notation. Denote\n$\\Phi_n(a,b,c,d) = \\Phi(a,b,c,d;p_n,S_n)$. Let\n\\[\n\\overline\\Lambda_n(a,c)=\\max_{|b|=1, 0\\le d\\le l_n}\\Phi_n(a,b,c,d).\n\\]\nLet $(a^*_n,c^*_n)$ be such that\n$\\Lambda_n^*:=\\Lambda_n(p_n,S_n)=\\overline\\Lambda_n(a^*_n,c^*_n)$.\nNote that $\\Lambda_n^*\\le\\overline\\Lambda_n(\\overline p_n,0)$, which is\nbounded from\nabove as $n\\to\\infty$, since $(b + \\overline p_n)\\cdot\\overline p_n \\ge0$ for\nall $b \\in\\mathcaligr{S}^{m-1}$, $n \\ge1$.\nOn the other hand, if for some fixed $\\varepsilon>0$,\n$a^*_n\\cdot p_n<|p_n|-\\varepsilon$ holds for infinitely many $n$, then\n$\\limsup\\overline\\Lambda_n(a^*_n,c_n)=\\infty$ for any choice of $c_n$\ncontradicting the statement that $\\Lambda_n^*$ is bounded from above.\nThis shows, for every $\\varepsilon> 0$,\n\\[\n|p_n|-\\varepsilon\\le a^*_n\\cdot p_n\\le|p_n|\n\\]\nfor all large $n$. In particular, $a^*_n\\to\\overline p$. Next note that\n\\[\n\\Lambda_n^*=\\overline\\Lambda_n(a^*_n,c^*_n)\\ge\\Phi_n(-\\overline p_n,a^*_n,l_n,c^*_n)\n\\ge-\\tfrac12 (\\overline p_n+a^*_n)'S_n(\\overline p_n+a^*_n)\n\\]\nhence,\n\\[\n\\liminf\\Lambda_n^*\\ge-2\\overline p'S\\overline p=\\Lambda(p,S).\n\\]\nAlso, with $(\\widetilde b_n,\\widetilde d_n)\\in\\arg\\max_{(b,d)}\\Phi\n_n(b,\\overline\np_n,d,k_n)$,\n\\begin{eqnarray}\\label{star1106}\n\\Lambda_n^* &=& \\overline\\Lambda_n(a^*_n,c^*_n)\\le\\overline\\Lambda_n(\\overline\np_n,k_n) \\nonumber\\\\\n&=&-\\tfrac12\n(\\widetilde b_n-\\overline p_n)'S_n(\\widetilde b_n-\\overline p_n) - (\\widetilde\nd_n + k_n)(\\widetilde b_n\n+ \\overline p_n) \\cdot p_n\n\\\\\n&\\le& -\\tfrac12 (\\widetilde b_n-\\overline p_n)'S_n(\\widetilde b_n-\\overline\np_n).\\nonumber\n\\end{eqnarray}\nIf $\\widetilde b_n\\to-\\overline p$ does not hold, then $\\liminf\\Lambda\n_n^*=-\\infty$\nby the first line of (\\ref{star1106}) which\ncontradicts the previous display. This shows $\\widetilde\nb_n\\to-\\overline p$. Hence, from the second line of (\\ref{star1106})\n\\[\n\\limsup\\Lambda_n^*\\le-2\\overline p'S\\overline p=\\Lambda(p,S).\n\\]\n\\upqed\\end{pf}\n\nWe now consider two formulations of SDG with bounded controls, the\nfirst being based on Definition \\ref{def1} whereas the second is more\nstandard.\nFor $k,l\\in{\\mathbb{N}}$,\nlet\n\\begin{eqnarray*}\nM_k &=& \\{Y\\in M\\dvtx\\mathbf{S}(Y)\\le k\\},\n\\\\\n\\Gamma_l &=& \\{\\beta\\in\\Gamma\\dvtx\\mathbf{S}(\\beta)\\le l\\}.\n\\end{eqnarray*}\nDefine accordingly the lower value\n\\begin{equation}\\label{20}\nV_{kl}(x)=\\inf_{\\beta\\in\\Gamma_l}\\sup_{Y\\in M_k}J^x(Y,\\beta),\n\\end{equation}\nand the upper value\n\\begin{equation}\\label{21}\nU_{kl}(x)=\\sup_{\\alpha\\in\\Gamma_k}\\inf_{Z\\in M_l}J^x(\\alpha,Z).\n\\end{equation}\n\\begin{definition}\\label{def2}\n(i) A process $\\{H_t\\}$ taking values in $\\mathcaligr{H}$ is said to be a\nsimple admissible control if it is $\\{\\mathcaligr{F}_t\\}$-progressively\nmeasurable.\nWe denote by $M^0$ the set of all simple admissible controls, and\nlet $M^0_k=\\{H\\in M^0\\dvtx S(H)\\le k\\}$.\n\n(ii) Given $k,l\\in{\\mathbb{N}}$, we say that a mapping $\\varrho\n\\dvtx M^0_k\\to M^0_l$\nis a simple strategy, and write $\\varrho\\in\\Gamma^0_{kl}$ if, for every\n$t$,\n\\[\n\\mathbf{P}(H_s=\\widetilde H_s \\mbox{ for a.e. } s\\in[0,t])=1\n\\]\nimplies\n\\[\n\\mathbf{P}(\\varrho[H]_s=\\varrho[\\widetilde H]_s \\mbox{ for a.e. }\ns\\in[0,t])=1.\n\\]\n\\end{definition}\n\nFor $\\beta\\in\\Gamma^0_{kl}, Y \\in M^0_k$, we write\n$J^x(Y, \\beta(Y))$ as $J^x(Y, \\beta)$. For $\\alpha\\in\\Gamma\n^0_{lk}, Z \\in M^0_l$,\n$J^x(\\alpha, Z)$ is defined similarly.\n\nFor $k,l\\in{\\mathbb{N}}$, let\n\\begin{eqnarray}\\label{50}\nV^0_{kl}(x)&=&\\inf_{\\beta\\in\\Gamma^0_{kl}}\\sup_{Y\\in\nM^0_k}J^x(Y,\\beta),\n\\\\\n\\label{51}\nU^0_{kl}(x)&=&\\sup_{\\alpha\\in\\Gamma^0_{lk}}\\inf_{Z\\in\nM^0_l}J^x(\\alpha,Z).\n\\end{eqnarray}\nThe following shows that the two formulations are equivalent.\n\\begin{lemma}\n\\label{lem7}\nFor every $k,l$, $V^0_{kl}=V_{kl}$ and $U^0_{kl}=U_{kl}$.\n\\end{lemma}\n\\begin{pf}\nWe only show the claim regarding $V_{kl}$.\nLet $\\beta\\in\\Gamma_l$. Define $\\beta^0\\in\\Gamma^0_{kl}$ by\nletting, for\nevery $Y\\in M^0_k$, $\\beta^0[Y]$ be the process component of the\npair $\\beta[(Y,k)]$. Clearly, for every $Y\\in M^0_{k}$,\n$J^x((Y,k),\\beta)=J^x(Y,\\beta^0)$, whence $\\sup_{Y\\in\nM_k}J^x(Y,\\beta)\\ge\\sup_{Y\\in M^0_k}J^x(Y,\\beta^0)$, and\n$V_{kl}(x)\\ge V^0_{kl}(x)$.\n\nNext, let $\\beta^0\\in\\Gamma^0_{kl}$. Define $\\beta\\dvtx M\\to M_l$ as\nfollows. Given $Y\\equiv(Y^0,K)\\equiv(A,C,K)\\in M$, let\n$Y^k=(A,C\\wedge\nk)$, and set $\\beta[Y]=(\\beta^0[Y^k],l)$. Note that if, for some\n$K$, $Y^0$ and $\\widetilde Y^0$ are elements of $M^0_K$ and\n$Y^0(s)=\\widetilde\nY^0(s)$ on $[0,t]$ then $Y^k(s)=\\widetilde Y^k(s)$ on $[0,t]$ and so\n$\\beta^0[Y^k]_s=\\beta^0[\\widetilde Y^k]_s$ on $[0,t]$. By definition of\n$\\beta$, it follows that $\\beta\\in\\Gamma_l$. Also, if $(Y^0,K)\\in M_k$\nthen $K\\le k$ and thus $J^x((Y^0,K),\\beta)=J^x(Y^0,\\beta^0)$. This\nshows that $\\sup_{Y\\in M_k}J^x(Y,\\beta)\\le\\sup_{Y^0\\in\nM^0_k}J^x(Y^0,\\beta^0)$. Consequently, $V_{kl}(x)\\le V^0_{kl}(x)$.\n\\end{pf}\n\nDenote\n$V_n=V_{k_nl_n}$ and $U_n=U_{k_nl_n}$.\nThe following result is proved in Section \\ref{sec4}.\n\\begin{theorem}\\label{th3}\nFor some $n_0 \\in\\mathbb{N}$,\nthe family $\\{V_n; n \\ge n_0\\}$ is equicontinuous, and so is the family\n$\\{U_n; n\\ge n_0\\}$.\n\\end{theorem}\n\nConsider the Bellman--Isaacs equations for the upper and,\nrespectively, lower values of the game with bounded controls, namely\n\\begin{eqnarray}\\label{08}\n&&\\cases{ \\Lambda_n^+(Du,D^2u)-h=0, &\\quad in $G$,\\cr\nu=g, &\\quad on $\\partial G$,}\n\\\\\n\\label{19}\n&&\\cases{\\Lambda_n^-(Du,D^2u)-h=0, &\\quad in $G$,\\cr\nu=g, &\\quad on $\\partial G$.}\n\\end{eqnarray}\nSolutions to these equations are defined analogously to Definition\n\\ref{def3}, with $\\Lambda^\\pm_n$ replacing $\\Lambda$, and where\nthere is no\nrestriction on the derivatives of the test function, that is,\n$\\mathcaligr{D}$ is replaced with\n${\\mathbb{R}}^m\\times\\mathscr{S}(m)$.\n\\begin{lemma}\\label{lem3}\nThere exists $n_1 \\in{\\mathbb{N}}$ such that for each $n \\ge n_1$,\n$U_n$ is\nthe unique solution to (\\ref{08}), and $V_n$ is the unique solution\nto (\\ref{19}).\n\\end{lemma}\n\\begin{pf}\nThis follows from a more general result, Theorem \\ref{th2} in\nSection \\ref{sec5}.\n\\end{pf}\n\\begin{lemma}\\label{lem4}\nAny subsequential uniform limit of $U_n$ or $V_n$ is a solution of\n(\\ref{07}).\n\\end{lemma}\n\\begin{pf}\nDenote by $U_0$ (resp., $V_0$) a subsequential limit of $U_n$\n[$V_n$]. By relabeling, we assume without loss that\n$U_n$ (resp., $V_n$) converges to $U_0$ [$V_0$]. We will show that\n$U_0$ and $V_0$ are\nsubsolutions of (\\ref{07}). The\nproof that these are supersolutions is parallel.\n\nWe start with the proof that $U_0$ is a subsolution. Fix $x_0\\in G$.\nLet $\\varphi\\in\\mathcaligr{C}^2(G)$ be such that $U_0-\\varphi$ is strictly\nmaximized at $x_0$. Assume first that $D\\varphi(x_0)\\ne0$. Since\n$U_n\\to U_0$ uniformly, we can find $\\{x_n\\}\\subset G$, $x_n\\to\nx_0$, where $x_n$ is a local maximum of $U_n-\\varphi$ for $n\\ge N$. We take\n$N$ to be larger than $n_1$ of Lemma \\ref{lem3}.\nSince by Lemma \\ref{lem3} $U_n$ is a subsolution of (\\ref{08}), we\nhave that for $n \\ge N$\n\\[\n\\Lambda_n^+(D\\varphi(x_n),D^2\\varphi(x_n))-h(x_n)\\le0.\n\\]\nThus, by Lemma \\ref{lem1},\n\\[\n\\Lambda(D\\varphi(x_0),D^2\\varphi(x_0))-h(x_0)\\le0\n\\]\nas required.\n\nNext, assume that $D\\varphi(x_0)=0$ and $D^2\\varphi(x_0)=\\lambda I_m$\nfor some\n$\\lambda\\in{\\mathbb{R}}$. In particular,\n$\\varphi(x)=\\varphi(x_0)+\\frac\\lambda2|x-x_0|^2+o(|x-x_0|^2)$. We\nneed to show\nthat\n\\begin{equation}\n\\label{1.1}\n-2\\lambda-h(x_0)\\le0.\n\\end{equation}\nConsider the case $\\lambda\\ge0$. Fix $\\delta>0$ and let\n$\\psi_\\delta(x)=\\frac{\\lambda+\\delta}{2}|x-x_0|^2$. Then $U_0-\\psi\n_\\delta$ has\na strict maximum at $x_0$. Since $U_n\\to U_0$ uniformly, we can find\n$\\{x_n\\}\\subset G$, $x_n\\to x_0$, where $x_n$ is a local maximum of\n$U_n-\\psi_\\delta$. To prove (\\ref{1.1}), it suffices to show that for\neach $\\varepsilon>0$,\n\\begin{equation}\n\\label{1.2}\n-2(\\lambda+\\delta)-\\sup_{x\\in\\mathbb{B}_\\varepsilon(x_0)}h(x)\\le0.\n\\end{equation}\nTo prove (\\ref{1.2}), argue by contradiction and assume that it\nfails. Then there exists $\\varepsilon>0$ such that\n\\begin{equation}\n\\label{1.25}\n-2(\\lambda+\\delta)-\\sup_{x\\in\\mathbb{B}_\\varepsilon(x_0)}h(x)>0.\n\\end{equation}\nLet $N \\ge n_1$ be such that $|x_n-x_0|<\\varepsilon$ for all $n\\ge N$.\nSince $U_n$\nis a subsolution of (\\ref{08}),\n\\begin{equation}\n\\label{1.3}\n\\mu_n:=\\Lambda^+_n(D\\psi_\\delta(x_n),D^2\\psi_\\delta(x_n))\\le h(x_n).\n\\end{equation}\nAlso,\n\\begin{eqnarray}\\label{1.325}\\qquad\n\\mu_n &=& \\max_{|b|=1,0\\le d\\le l_n}\\min_{|a|=1,0\\le c\\le\nk_n} \\biggl[-\\frac12(\\lambda+\\delta)|a-b|^2\\nonumber\\\\\n&&\\hspace*{107pt}{} -(\\lambda+\\delta)(c+d)(a+b)\\cdot(x_n-x_0) \\biggr]\n\\\\\n&\\ge&\n\\min_{|a|=1,0\\le c\\le k_n} \\biggl[-\\frac12(\\lambda+\\delta)\n|a-b_n|^2 \\biggr] =\n-2(\\lambda+\\delta),\\nonumber\n\\end{eqnarray}\nwhere $b_n=-(x_n-x_0)\/|x_n-x_0|$ if $x_n\\ne x_0$ and arbitrary\notherwise. Thus by (\\ref{1.25}),\n\\begin{equation}\\label{1.34}\n\\mu_n > h(x_n).\n\\end{equation}\nHowever, this contradicts (\\ref{1.3}). Hence, (\\ref{1.2}) holds and\nso (\\ref{1.1}) follows.\n\nConsider now the case $\\lambda<0$. Let $\\delta>0$ be such that\n$\\lambda+\\delta<0$. Let $\\psi_\\delta$ be as above. Then $U_0-\\psi\n_\\delta$ has\na strict maximum at $x_0$. Fix $\\varepsilon>0$. Then one can find\n$\\gamma<\\varepsilon$ such that\n\\begin{equation}\n\\label{1.35}\\qquad\nU_0(x_0)=U_0(x_0)-\\psi_\\delta(x_0)>U_0(x)-\\psi_\\delta(x)\\qquad\n\\forall 0<|x-x_0|\\le\\gamma.\n\\end{equation}\nThus, one can find $\\eta\\in{\\mathbb{R}}^m$ such that $0<|\\eta\n|<\\gamma$ and\n\\begin{equation}\n\\label{1.4}\nU_0(x_0)>U_0(x)-\\psi_\\delta(x+\\eta)\\qquad \\forall x\\in\\partial\n\\mathbb{B}_\\gamma(x_0).\n\\end{equation}\nLet $\\psi_{\\delta,\\eta}(x)=\\psi_\\delta(x+\\eta)$. Let\n$x_\\eta\\in\\overline{\\mathbb{B}_\\gamma(x_0)}$ be a maximum point for\n$U_0-\\psi_{\\delta,\\eta}$ over $\\overline{\\mathbb{B}_\\gamma\n(x_0)}$. We claim that\n\\begin{equation}\n\\label{1.8}\nx_\\eta\\notin\\partial\\mathbb{B}_\\gamma(x_0) \\quad\\mbox{and}\\quad\nx_\\eta\\ne\nx_0-\\eta.\n\\end{equation}\nSuppose the claim holds. Then $D\\psi_{\\delta,\\eta}(x_\\eta)\\ne0$, and\nso from the first part of the proof\n\\[\n-2(\\lambda+\\delta)-h(x_\\eta)=\\Lambda(D\\psi_{\\delta,\\eta}(x_\\eta\n),D^2\\psi_{\\delta,\\eta}(x_\\eta))-h(x_\\eta)\\le0.\n\\]\nSince $|x_\\eta-x_0|\\le\\gamma<\\varepsilon$, sending $\\varepsilon\\to\n0$ and then\n$\\delta\\to0$ yields (\\ref{1.1}).\n\nWe now prove (\\ref{1.8}). From (\\ref{1.4}) and the fact that\n$\\lambda+\\delta<0$,\n\\[\n\\sup_{x \\in\\partial\\mathbb{B}_\\gamma(x_0)}[U_0(x)-\\psi_{\\delta\n,\\eta\n}(x)] m_0$. By~(\\ref{01}), with\n$\\alpha_t=a^0\\cdot B_t$, on the event $\\tau>T$ one has\n\\[\n\\int_0^T(1-\\alpha_s)\\,dW_s+\\int_0^T(1+\\alpha_s)\\,ds\\le a^0\\cdot\n(X_T-X_0) T\\}$ we have $|M_T|\n\\ge\nT - m_0$. Letting $\\sigma= \\inf\\{s\\dvtx\\langle M \\rangle_s \\ge T\\}$,\n\\begin{eqnarray}\\label{ab442}\n\\mathbf{P}(\\tau> T; \\langle M \\rangle_T < T) &\\le&\n\\mathbf{P}(|M_{T\\wedge\\sigma}| \\ge T - m_0) \\nonumber\\\\[-8pt]\\\\[-8pt]\n&\\le&\\frac{m_1\\mathbf{E}\n\\langle\nM \\rangle^2_{T\\wedge\\sigma}}{(T-m_0)^4} \\le\\frac{m_1\nT^2}{(T-m_0)^4}.\\nonumber\n\\end{eqnarray}\nWe now consider the event $\\{\\tau> T ; \\langle M \\rangle_T \\ge T\\}$. One\ncan find $m_2, m_3 \\in(0, \\infty)$ such that for all\nnondecreasing, nonnegative processes $\\{\\widehat\\gamma_t\\}$,\n\\begin{equation}\\label{ab455}\n\\mathbf{P}\\bigl(H_s + \\widehat\\gamma_s \\in(-m_0, m_0); 0\n\\le s \\le T \\bigr) \\le m_2e^{-m_3 T},\n\\end{equation}\nwhere $H$ is a one-dimensional Brownian motion. Letting $\\gamma_t =\n\\int_0^t (1+D_s) (1 + \\alpha_s) \\,ds$, where $Z = (B,D)$, we see that\n\\[\n\\{\\tau> T; \\langle M \\rangle_T \\ge T\\} \\subset\n\\{M_s + \\gamma_s \\in(-m_0, m_0), 0 \\le s \\le T; \\langle M \\rangle\n_T \\ge T\\}.\n\\]\nFor $u\\ge0$, let $S_u = \\inf\\{s\\dvtx\\langle M \\rangle_s > u\\}$. Then, with\n$\\widehat\\gamma_s = \\gamma_{S_s}$,\n\\[\n\\mathbf{P}(\\tau> T; \\langle M \\rangle_T \\ge T) \\le\n\\mathbf{P}\\bigl(H_s + \\widehat\\gamma_s \\in(-m_0, m_0); 0 \\le s \\le T\\bigr) \\le\nm_2e^{-m_3 T},\n\\]\nwhere the last inequality follows from (\\ref{ab455}). The result now\nfollows on combining the above display with (\\ref{ab442})\n\\end{pf}\n\nThe inequality $J(x, Y^0, Z) \\le|h|_{\\infty} \\mathbf{E}(\\tau(x,\nY^0, Z)) +\n|g|_{\\infty}$,\nwhere $|h|_{\\infty} = {\\sup_x} |h(x)|$ and $|g|_{\\infty} = {\\sup_x} |g(x)|$,\nimmediately implies the following.\n\\begin{corollary}\n\\label{cor01}\nThere exists a constant $c_2<\\infty$ such that $|V_n^{\\gamma}(x)|\\vee\n|U_n^{\\gamma}(x)|\\le c_2$, for\nall $x\\in\\overline G$, $\\gamma\\in[0, 1)$ and $n\\in{\\mathbb{N}}$.\n\\end{corollary}\n\nThe idea of the proof of equicontinuity, explained in a heuristic\nmanner, is as follows. Let $x_1$ and $x_2$ be in $G$, let\n$\\varepsilon=|x_1-x_2|$, and let $\\delta>0$. Consider the game with bounded\ncontrols for which $V_n$ is the lower value function, for some\n$n\\in{\\mathbb{N}}$. Let the minimizing player select a strategy\n$\\beta^n$ that\nis $\\delta$-optimal for the initial position $x_1$; namely $\\sup_{Y\n\\in M^0_{k_n}} J^{x_1}(Y, \\beta^n) \\le V_n(x_1) + \\delta$. Denote\nthe exit time by $\\tau_1=\\tau^{x_1}(Y,\\beta^n)$ and the exit\nposition by $\\xi_1=X^{x_1}(\\tau_1)$. Now, modify the strategy is\nsuch a way that the resulting control $Z=\\beta^n[Y]$ is only\naffected for times $t\\ge\\tau_1$. This way, the payoff incurred\nremains unchanged. Thus, denoting the modified strategy by\n$\\widetilde\\beta^n$, we have, for every $Y\\in M^0_{k_n}$,\n\\[\nJ^{x_1}(Y,\\widetilde\\beta^n)\\le V_n(x_1)+\\delta.\n\\]\nGiven a\npoint $\\xi_2$ located inside $G$, $\\varepsilon$ away from $\\xi_1$,\nand a\nnew state process which, at time $\\tau_1$ is located at $\\xi_2$, the\nmodified strategy attempts to force this process to exit the domain\nsoon after $\\tau_1$ and with a small displacement from $\\xi_2$\n(provided that $\\varepsilon$ is small).\n\nLet now the maximizing player select a control $Y^n$ that is\n$\\delta$-optimal for playing against $\\widetilde\\beta^n$, when\nstarting from\n$x_2$. This control is modified after the exit time $\\tau^{x_2}(Y^n,\n\\widetilde\\beta^n)$\nin a similar\nmanner to the above. Denoting the modified control by~$\\widetilde Y^n$, we\nhave\n\\[\nV_n(x_2)\\le J^{x_2}(\\widetilde Y^n,\\widetilde\\beta^n)+\\delta.\n\\]\nHence, $V_n(x_2)-V_n(x_1)\\le J^{x_2}(\\widetilde\nY^n,\\widetilde\\beta^n)-J^{x_1}(\\widetilde Y^n,\\widetilde\\beta\n^n)+2\\delta$. One can thus\nestimate the modulus of continuity of $V_n$ by analyzing the payoff\nincurred when $(\\widetilde Y^n,\\widetilde\\beta^n)$ is played, considering\nsimultaneously two state processes, starting from $x_1$ and~$x_2$.\nThe form (\\ref{01}) of the dynamics ensures that the processes\nremain at relative position $x_1-x_2$ until, at time $\\sigma$, one of\nthem leaves the domain. The difference between the running payoffs\nincurred up to that time can be estimated in terms of~$\\varepsilon$, the\nmodulus of continuity of $h$, and the expectation of $\\sigma$. It is not\nhard to see that the latter is uniformly bounded, owing to Corollary\n\\ref{cor01} and the boundedness of $h$ away from zero. By\nconstruction, one of the players will now attempt to force the state\nprocess that is still in $G$ to exit. If one can ensure that exit\noccurs soon after $\\sigma$ and with a small displacement (uniformly in\n$n$), then the running payoff incurred between time $\\sigma$ and the\nexit time is small, and the difference between the terminal payoffs\nis bounded in terms of $\\varepsilon$ and the modulus of continuity of $g$,\nresulting in an estimate that is uniform in $n$.\n\nThis argument is made precise in the proof of the theorem. Lemmas\n\\ref{lem02} and \\ref{lem03} provide the main tools for showing that\nstarting at a state near the boundary, each player may force exit\nwithin a short time and with a small displacement. To state these\nlemmas, we first need to introduce some notation.\n\nWe have assumed that $G$ is a bounded $C^2$ domain in ${\\mathbb\n{R}}^m$. Thus,\nthere exist $\\overline\\rho\\in(0,\\frac18)$, $k\\in{\\mathbb{N}}$, $z_j\\in\n\\partial G$, $E_j\\in\\mathcaligr{O}(m)$, $\\xi_j\\in C^2({\\mathbb{R}}^{m-1})$,\n$j=1,\\ldots,k$, such\nthat, with $\\mathbb{B}_j=\\mathbb{B}_{\\overline\\rho}(z_j)$, $j=1,\\ldots\n,k$, one has\n$\\partial G\\subset\\bigcup_{j=1}^k\\mathbb{B}_j$, and\n\\[\nG\\cap\\mathbb{B}_j=\\{E_jy\\dvtx y_1>\\xi_j(y_2,\\ldots,y_m)\\}\\cap\n\\mathbb{B}_j,\\qquad j=1,\\ldots,k.\n\\]\nHere, $\\mathcaligr{O}(m)$ is the space of $m\\times m$ orthonormal matrices.\nDefine for $j=1,\\ldots,k$, $\\widetilde\\varphi_j\\dvtx{\\mathbb{R}}^m\\to\n{\\mathbb{R}}$ as\n\\[\n\\widetilde\\varphi_j(y)=y_1-\\xi_j(y_2,\\ldots,y_m),\\qquad y\\in\n{\\mathbb{R}}^m.\n\\]\nLet $\\varphi_j(x)=\\widetilde\\varphi_j(E_j^{-1}x)$, $x\\in{\\mathbb\n{R}}^m$. Then\n$|D\\varphi_j(x)|\\ge1$, $x\\in{\\mathbb{R}}^m$, $j=1,\\ldots,k$. Furthermore,\n\\[\nG\\cap\\mathbb{B}_j=\\{x\\dvtx\\varphi_j(x)>0\\}\\cap\\mathbb{B}_j,\\qquad\nj=1,\\ldots,k.\n\\]\nLet $0<\\rho_0<\\overline\\rho$ be such that $\\partial\nG\\subset\\bigcup_{j=1}^k\\mathbb{B}_{\\rho_0}(z_j)$. For $\\varepsilon\n>0$, denote\n\\[\n\\mathbf{X}_\\varepsilon=\\{(x_1,x_2)\\dvtx x_1\\in\\partial G, x_2\\in G,\n|x_1-x_2|\\le\\varepsilon\n\\}.\n\\]\nLet $\\underline j\\dvtx \\partial G\\to\\{1,\\ldots,k\\}$ be a measurable map\nwith the property\n\\[\nx\\in\\mathbb{B}_{\\rho_0}\\bigl(z_{\\underline j(x)}\\bigr)\\qquad \\mbox{for all }\nx\\in\\partial G.\n\\]\nFor existence of such a map see, for example, Theorem 10.1 of\n\\cite{EK}. Then, for every\n$\\varepsilon\\le\\rho_1:=\\frac{\\overline\\rho-\\rho_0}{4}$,\n\\begin{equation}\\label{34}\n(x_1,x_2)\\in\\mathbf{X}_\\varepsilon\\qquad\\mbox{implies }\n\\overline{\\mathbb{B}_{\\rho_1}(x_i)}\\subset\\mathbb{B}_{\\underline\nj(x_1)},\\qquad i=1,2.\n\\end{equation}\nFor $j=1,\\ldots,k$ and $x_0\\in\\mathbb{B}_j$, define\n$\\psi^{x_0}_j\\dvtx{\\mathbb{R}}^m\\to{\\mathbb{R}}$ as\n\\[\n\\psi^{x_0}_j(x)=\\varphi_j(x)+|x-x_0|^2.\n\\]\nAlso, note that $|D\\psi^{x_0}_j|\\ge\\frac12$ in $\\mathbb{B}_j$. Define\n$\\pi_j^{x_0}\\dvtx{\\mathbb{R}}^m\\to\\mathcaligr{S}^{m-1}$ such that it is\nLipschitz, and\n\\begin{equation}\\label{31}\n\\pi_j^{x_0}(x)=-\\frac{D\\psi^{x_0}_j(x)}{|D\\psi^{x_0}_j(x)|},\\qquad\nx\\in\\mathbb{B}_j.\n\\end{equation}\n\nGiven a strategy $\\beta$, and a point $x_2$, we seek a control\n$Y=(A,C)$ that forces a state process starting from $x_2$ to exit in\na short time and with a small displacement from $x_2$ (provided that\n$x_2$ is close to the boundary). We would like to determine $Y$ via\nthe functions $\\pi_j$ just constructed, in such a way that the\nfollowing relation holds:\n\\begin{equation}\\label{39}\nA(t)=\\pi(X(t)), \\qquad C(t)=c^0,\n\\end{equation}\nwhere $\\pi=\\pi^{x_2}_{\\underline j(x_1)}$ and $c^0>0$ is some constant.\nMaking $A$ be oriented in the negative direction of the gradient of\n$\\varphi_j$ allows us to show that the state is ``pushed'' toward the\nboundary. The inclusion of a quadratic term in $\\psi_j^{x_0}$\nensures in addition that the sublevel sets $\\{\\psi_j\\eta_i.\\nonumber\n\\end{eqnarray}\nIt is easy to check that $\\eta_0<\\eta_1<\\cdots$ and $\\eta_i\\to\n\\infty$\na.s. Define $X_t=X^{(i)}_t$, $Y_t=Y^{(i)}_t$ if $t\\le\\eta_i$. Let\n$\\rho=\\rho_1^2$ and\n\\begin{eqnarray}\\label{4.1}\n\\tau_{\\rho} &=& \\inf\\{t\\ge\\sigma\\dvtx\\Psi(X_t)\\ge\\rho\\},\\nonumber\\\\\n\\tau_G &=& \\inf\\{t\\ge\\sigma\\dvtx X_t\\in G^c\\},\\\\\n\\tau &=& \\tau_{\\rho}\\wedge\\tau_G.\\nonumber\n\\end{eqnarray}\nDefine\n\\begin{eqnarray*}\n\\overline Y_t &\\equiv& (\\overline A_t,\\overline C_t)=\n\\cases{Y_t, &\\quad $t<\\tau$, \\cr\n(a^0,c^0), &\\quad $t\\ge\\tau$,}\n\\\\\n(\\overline B,\\overline D)&=&\\beta(\\overline Y),\n\\end{eqnarray*}\nwhere $a^0$ is as fixed at the beginning of the section. Let\n\\[\n\\overline X_t=\\xi_2+\\int\\mathbf{1}_{[\\sigma,t]}(s) ([\\overline A_s-\\overline\nB_s]\\,dW_s+ \\gamma \\,d\\widetilde W_s ) +\\int_\\sigma^t[\\overline C_s+\\overline\nD_s][\\overline\nA_s+\\overline B_s]\\,ds.\n\\]\nWe write\n\\[\n\\overline X=\\overline X[\\sigma,\\xi_1,\\xi_2,\\beta,Y],\\qquad\n\\overline Y=\\overline Y[\\sigma,\\xi_1,\\xi_2,\\beta,Y].\n\\]\nNote that if $\\overline\\tau_{\\rho}$ and $\\overline\\tau_G$ are defined by\n(\\ref{4.1}) upon replacing $X$ by $\\overline X$ then\n$\\overline\\tau:=\\overline\\tau_{\\rho}\\wedge\\overline\\tau_G=\\tau_{\\rho}\\wedge\n\\tau\n_G=\\tau$,\nbecause $\\overline X$ differs from $X$ only after time $\\tau$. We write\n$\\overline\\tau=\\overline\\tau[\\sigma,\\xi_1,\\xi_2,\\beta,Y]$. Similar\nnotation will\nbe used for $\\overline\\tau_{\\rho}$ and $\\overline\\tau_G$.\n\\begin{lemma}\n\\label{lem02}\nThere exists a $c^0 \\in(0, \\infty)$ and\na modulus $\\vartheta$ such that for every $\\varepsilon\\in(0, \\rho\n_1)$, and\n$\\gamma\\in[0, 1)$, if $\\bolds{\\sigma}=(\\sigma,\\xi_1,\\xi_2,\\beta\n,Y)\\in\\bolds{\\Sigma}_\\varepsilon$\nand $\\overline\\tau_G=\\overline\\tau_G[\\bolds{\\sigma}]$, one has:\n\n\\begin{longlist}\n\\item\n$\\mathbf{E}\\{\\overline\\tau_G-\\sigma |\\mathcaligr{F}_\\sigma\\}\\le\n\\vartheta(\\varepsilon)$,\n\\item\n$\\mathbf{E}\\{|\\overline X-\\xi_2|^2_{*,\\overline\\tau_G} |\\mathcaligr\n{F}_\\sigma\\}\\le\\vartheta(\\varepsilon)$,\nwhere $|\\overline X-\\xi_2|_{*,\\overline\\tau_G} = {\\sup_{t \\in[\\sigma, \\overline\n\\tau_G]}}|\\overline\nX (t) - \\xi_2|$.\n\\end{longlist}\n\\end{lemma}\n\nProof of the lemma is provided after the proof of Theorem \\ref{th3plus}.\n\nNext, we construct a strategy $\\beta^*\\in\\Gamma^0$ with analogous\nproperties. Here, existence of solutions is not an issue, and\ndiscretization is not needed.\n\nFix $(x_1,x_2)\\in\\mathbf{X}_{\\rho_1}$. Let $j=\\underline j(x_1)$,\n$\\widetilde\\Psi=\\psi_j^{x_2}$, and $\\widetilde\\Pi=\\pi_j^{x_2}$. Given\n$Y=(A,C)\\in\nM^0$, let $\\widetilde X$ solve\n\\[\n\\widetilde X_t=x_2+\\int_0^t \\bigl([A_s-\\widetilde\\Pi(\\widetilde\nX_s)]\\,dW_s + \\gamma \\,d\\widetilde W_s \\bigr)+\\int\n_0^t[C_s+d^0][A_s+\\widetilde\\Pi\n(\\widetilde X_s)]\\,ds,\n\\]\nwhere $d^0$ is a constant to be determined later. Let\n\\begin{eqnarray}\\label{5.1}\n\\widetilde\\tau_{\\rho}\n&=&\\inf\\{t\\dvtx\\widetilde\\Psi(\\widetilde X_t)\\ge\\rho\\},\\nonumber\\\\\n\\widetilde\\tau_G\n&=&\\inf\\{t\\dvtx\\widetilde X_t\\in G^c\\},\n\\\\\n\\widetilde\\tau\n&=&\\widetilde\\tau_{\\rho}\\wedge\\widetilde\\tau_G.\\nonumber\n\\end{eqnarray}\nDefine $Z^*\\in M^0$ as\n\\[\nZ^*_s\\equiv(B^*_s,D^*_s)=\n\\cases{(\\widetilde\\Pi(\\widetilde X_s),d^0), &\\quad $s<\\widetilde\\tau\n$,\\vspace*{2pt}\\cr\n(a^0, d^0), &\\quad $s\\ge\\widetilde\\tau$.}\n\\]\nNote that $\\beta^*[Y](s):=\\widetilde Z_s$, $s\\ge0$ defines a\nstrategy. Let\n\\[\nX^*_t=x_2+\\int_0^t ([A_s-B^*_s]\\,dW_s + \\gamma \\,d\\widetilde W_s\n)+\\int\n_0^t[C_s+d^0][A_s+B^*_s]\\,ds.\n\\]\nDefine $\\tau_{\\rho}^*$, $\\tau_{G}^*$ and $\\tau^*$\nby replacing $\\widetilde X$ with $X^*$ in (\\ref{5.1}), and note that\n$\\tau^*=\\widetilde\\tau$. To make the dependence explicit, we write\n\\[\nX^*=X^*[x_1,x_2,Y,\\overline W], \\qquad Z^*=Z^*[x_1,x_2,Y,\\overline\nW],\\qquad\n\\tau^*=\\tau^*[x_1,x_2,Y,W].\n\\]\n\\begin{lemma}\n\\label{lem03}\nThere exists $d^0 \\in(0, \\infty)$ and a modulus $\\widetilde\\vartheta\n$ such\nthat for all $\\varepsilon\\in(0,\n\\rho_1)$, $Y\\in M^0$, $(x_1,x_2)\\in\\mathbf{X}_\\varepsilon$, if\n$\\tau^*_G=\\tau^*_G[x_1,x_2,Y,W]$, one has:\n\n\\begin{longlist}\n\\item\n$\\mathbf{E}[\\tau^*_G]\\le\\widetilde\\vartheta(\\varepsilon)$,\n\\item $\\mathbf{E}[|X^*-x_2|^2_{*,\\tau^*_G} ]\\le\\widetilde\\vartheta\n(\\varepsilon)$,\nwhere $|X^*-x_2|_{*,\\tau^*_G} = {\\sup_{t \\in[0, \\tau^*_G]}}|X^*(t) - x_2|$.\n\\end{longlist}\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem03} is very similar to (in fact somewhat\nsimpler than) the proof of Lemma \\ref{lem02}, and therefore will be\nomitted.\n\n\\begin{figure}[b]\n\n\\includegraphics{494f01.eps}\n\n\\caption{}\n\\label{figure20}\n\\end{figure}\n\nIf $\\sigma$ is an a.s. finite $\\{\\mathcaligr{F}_t\\}$-stopping time and\n$(\\xi_1,\\xi_2)$ are $\\mathcaligr{F}_\\sigma$-measurable random\nvariables such that\n$(\\xi_1,\\xi_2)\\in\\mathbf{X}_{\\rho_1}$ a.s., then we define the\n$\\mathcaligr{G}_t=\\mathcaligr{F}_{t+\\sigma}$ adapted processes\n\\begin{eqnarray*}\n\\overline X{}^*_t &=& X^*[\\xi_1,\\xi_2,\\widehat Y_\\sigma,\\widehat W_\\sigma](t),\n\\\\\n\\overline Z{}^*_t &=& Z^*[\\xi_1,\\xi_2,\\widehat Y_\\sigma,\\widehat W_\\sigma](t),\n\\end{eqnarray*}\nwhere $\\widehat Y_\\sigma(t)=Y(t+\\sigma)$ and $\\widehat W_\\sigma\n(t)=\\overline W(t+\\sigma)-\\overline W(\\sigma)$,\n$t\\ge0$. To make the dependence explicit, write\n\\[\n\\overline X{}^*=\\overline X{}^*[\\sigma,\\xi_1,\\xi_2,Y],\\qquad\n\\overline Z{}^*=\\overline Z{}^*[\\sigma,\\xi_1,\\xi_2,Y].\n\\]\n\\begin{pf*}{Proof of Theorem \\protect\\ref{th3}}\nFix $x_1,x_2\\in G$ and $\\gamma\\in[0,1)$. We will suppress $\\gamma$\nfrom the notation. Assume that $|x_1-x_2|=\\varepsilon<\\rho_1$, so\nthat Lemmas\n\\ref{lem02} and \\ref{lem03} are in force (see Figure \\ref{figure20}). Let $n_0$ be large enough so\nthat $l_n, k_n \\ge\\max\\{c^0, d^0\\}$ for\nall $n \\ge n_0$.\nGiven\n$\\delta\\in(0,1)$ and $n\\ge n_0$, let $\\beta^n\\in\\Gamma^0_{k_nl_n}$ be\nsuch that\n\\[\n\\sup_{Y\\in M^0_{k_n}}J^{x_1}(Y,\\beta^n)-\\delta\\le V_n(x_1)\\le c_1.\n\\]\nFor $Y\\in M^0$ write $\\tau^{1,n}(Y):=\\tau^{x_1}(Y,\\beta^n)$ and\n$X^{1,n}(Y):=X^{x_1}(Y,\\beta^n)$. Note that\n\\[\n\\underline h \\mathbf{E}[\\tau^{1,n}(Y)]-|g|_\\infty\\le c_1+1,\n\\]\nhence for every $n$ and every $Y\\in M^0_{k_n}$,\n\\begin{equation}\\label{35}\n\\mathbf{E}[\\tau^{1,n}(Y)]\\le m_1,\n\\end{equation}\nwhere $m_1<\\infty$ is a constant that does not depend on $n$.\n\nDefine $\\widetilde\\beta^n\\in\\Gamma^0_{k_nl_n}$ as follows.\nFor $Y\\in M^0$, let\n\\begin{eqnarray*}\n\\xi_1^{1,n}(Y)&=&X^{1,n}_{\\tau^{1,n}(Y)}(Y),\\qquad\n\\xi_2^{1,n}(Y)=\\xi_1^{1,n}(Y)+x_2-x_1,\n\\\\\n\\widetilde\\beta^n[Y]_t &=&\n\\cases{\\beta^n[Y]_t, &\\quad $t<\\tau^{1,n}(Y)$,\\vspace*{2pt}\\cr\n\\overline Z{}^*[\\tau^{1,n}(Y),\\xi_1^{1,n}(Y),\\xi_2^{1,n}(Y),Y],\n&\\quad $t\\ge\\tau^{1,n}(Y), \\xi_2^{1,n}(Y)\\in\\overline G$,\\cr\n\\mbox{arbitrarily defined}, &\\quad $t\\ge\\tau^{1,n}(Y), \\xi\n_2^{1,n}(Y)\\in\\overline\nG^c$.}\n\\end{eqnarray*}\nNote that for every $Y\\in M^0_{k_n}$,\n$J^{x_1}(Y,\\beta^n)=J^{x_1}(Y,\\widetilde\\beta^n)$. Next, choose\n$Y^n\\in\nM^0_{k_n}$ such that\n\\[\nV_n(x_2)\\le\\sup_{Y\\in M^0_{k_n}}J^{x_2}(Y,\\widetilde\\beta^n)\n\\le J^{x_2}(Y^n,\\widetilde\\beta^n)+\\delta.\n\\]\nLet $\\tau^{2,n}=\\tau^{x_2}(Y^n,\\widetilde\\beta^n)$, and\n$X^{2,n}=X^{x_2}(Y^n,\\widetilde\\beta^n)$. Let\n\\[\n\\xi_2^{2,n}=X^{2,n}_{\\tau^{2,n}},\\qquad\n\\xi_1^{2,n}=\\xi_2^{2,n}+x_1-x_2.\n\\]\nDefine $\\widetilde Y^n\\in M^0_{k_n}$ as\n\\[\n\\widetilde Y^n_t=\\cases{Y^n_t, &\\quad $t<\\tau^{2,n}$,\\vspace*{2pt}\\cr\n\\overline Y[\\tau^{2,n},\\xi_2^{2,n},\\xi_1^{2,n},\\widetilde\\beta\n^n,Y^n](t), &\\quad\n$t\\ge\\tau^{2,n}, \\xi_1^{2,n}\\in\\overline G$,\\vspace*{2pt}\\cr\n\\mbox{arbitrarily defined}, &\\quad $t\\ge\\tau^{2,n}, \\xi_1^{2,n}\\in\n\\overline\nG^c$.}\n\\]\nNote that $J^{x_2}(Y^n,\\widetilde\\beta^n)=J^{x_2}(\\widetilde\nY^n,\\widetilde\\beta^n)$.\nThus\n\\begin{equation}\n\\label{8.1}\nV_n(x_2)-V_n(x_1)-2\\delta\\le\nJ^{x_2}(\\widetilde Y^n,\\widetilde\\beta^n)- J^{x_1}(\\widetilde\nY^n,\\widetilde\\beta^n).\n\\end{equation}\nFor $k=1,2$, let\n\\[\n\\sigma^{k,n}=\\tau^{x_k}(\\widetilde Y^n,\\widetilde\\beta^n),\\qquad\n\\widetilde X^{k,n}=X^{x_k}(\\widetilde Y^n,\\widetilde\\beta^n),\\qquad\n\\Xi^{k,n}=\\widetilde X^{k,n}_{\\sigma^{k,n}}.\n\\]\nFor $m_0\\ge0$, let $\\vartheta_g(m_0)=\\sup\\{|g(x)-g(y)|\\dvtx x,y\\in\n\\partial G,|x-y|\\le\nm_0\\}$ and $\\vartheta_h(m_0)=\\sup\\{|h(x)-h(y)|\\dvtx x,y\\in G,|x-y|\\le\nm_0\\}$. Using\n(\\ref{35}), the right-hand side of (\\ref{8.1}) can be bounded by\n\\begin{equation}\\hspace*{33pt}\n\\label{8.2}\n\\mathbf{E}\\vartheta_g(|\\Xi^{1,n}-\\Xi^{2,n}|)+c_3\\vartheta\n_h(\\varepsilon)+ |h|_{\\infty}\n\\mathbf{E}[(\\sigma^{1,n}\\vee\\sigma^{2,n})-(\\sigma^{1,n}\\wedge\n\\sigma^{2,n})].\n\\end{equation}\nOn the set $\\sigma^{1,n}\\le\\sigma^{2,n}$, we have $|\\Xi^{1,n}-\\Xi^{2,n}|\n\\le\\varepsilon+|\\widetilde X_{\\sigma^{1,n}}^{2,n}-\\widetilde\nX_{\\sigma^{2,n}}^{2,n}|$.\nHence, by Lem\\-ma \\ref{lem03}(ii),\n\\[\n\\mathbf{E} \\bigl[|\\Xi^{1,n}-\\Xi^{2,n}|^2 \\mathbf{1}_{\\{\\sigma\n^{1,n}\\le\\sigma^{2,n}\\}\n} \\bigr]\\le\\vartheta_1(\\varepsilon)\n\\]\nfor some modulus $\\vartheta_1$. Using Lemma \\ref{lem02}(ii), a similar\nestimate holds on the complement set, and consequently, the first term\nof (\\ref{8.2}) is bounded by\n$\\vartheta_2(\\varepsilon)$, for some modulus $\\vartheta_2$. By\nLemmas \\ref{lem02}(i)\nand \\ref{lem03}(i), the last term of (\\ref{8.2}) is bounded by\n$|h|_{\\infty}(\\vartheta(\\varepsilon)+\\widetilde\\vartheta\n(\\varepsilon))$. Hence,\n$V_n(x_2)-V_n(x_1)\\le2\\delta+\\vartheta_3(|x_1-x_2|)$ for some modulus\n$\\vartheta_3$, and the equicontinuity of $\\{V_n^{\\gamma}; n,\\gamma\\}$\nfollows on sending\n$\\delta\\to0$. The proof of equicontinuity of $\\{U_n^{\\gamma};\nn,\\gamma\\}$ is similar, and\ntherefore omitted.\n\\end{pf*}\n\\begin{pf*}{Proof of Lemma \\protect\\ref{lem02}}\nWe will only present the proof for the case\n$\\gamma= 0$. The general case follows upon minor modifications.\nDenote\n\\[\n\\psi_{1,\\infty}={\\sup_{x_0,x\\in\\overline\nG}\\sup_j}|D\\psi^{x_0}_j(x)|,\\qquad\n\\psi_{2,\\infty}={\\sup_{x_0,x\\in\\overline G}\\sup_j}|D^2\\psi^{x_0}_j(x)|,\n\\]\nand let $\\varphi_{1,\\infty}$, $\\varphi_{2,\\infty}$ be defined analogously.\nLet $\\varepsilon>0$ and $\\bolds{\\sigma}\\equiv(\\sigma,\\xi_1,\\xi\n_2,\\beta\n,Y)\\in\\bolds{\\Sigma}_\\varepsilon$ be\ngiven, let $\\overline X=\\overline X[\\bolds{\\sigma}]$, $\\overline Y=\\overline\nY[\\bolds{\\sigma}]$,\n$\\overline\\tau_\\rho=\\overline\\tau_\\rho[\\bolds{\\sigma}]$, $\\overline\\tau\n_G=\\overline\n\\tau_G[\\bolds{\\sigma}]$,\nand $\\overline\\tau=\\overline\\tau[\\bolds{\\sigma}]$.\nLet\n\\[\n\\overline\\tau_0=\\inf\\{t\\ge\\sigma\\dvtx\\Psi(\\overline X_t)\\le0\\},\\qquad\n\\overline\\tau_B=\\inf\\{t\\ge\\sigma\\dvtx\\overline X_t\\notin\\mathbb{B}_{j^*}\\},\n\\]\nwhere we recall that $j^* = \\underline{j}(\\xi_1)$.\nWe have $\\overline\\tau\\equiv\\overline\\tau_{\\rho}\\wedge\\overline\\tau_G\\le\\overline\n\\tau_{\\rho}\\wedge\\overline\\tau_0$,\nbecause $\\Psi\\ge\\Phi$. Also, by (\\ref{34}), $\\overline\\tau\\le\\overline\n\\tau_B$. By It\\^{o}'s\nformula, for $t>\\sigma$,\n\\begin{eqnarray*}\n\\Psi(\\overline X_t)\n&=&\\Phi(\\xi_2)+\\int\\mathbf{1}_{[\\sigma,t]}(s)D\\Psi(\\overline\nX_s)[\\overline\nA_s-\\overline B_s]\\,dW_s\n\\\\\n&&{}\n+\\int_\\sigma^tD\\Psi(\\overline X_s)[\\overline A_s+\\overline B_s][\\overline C_s+\\overline D_s]\\,ds\n\\\\\n&&{}\n+\\frac12\\int_\\sigma^t[\\overline A_s-\\overline B_s]'D^2\\Psi(\\overline X_s)[\\overline\nA_s-\\overline B_s]\\,ds.\n\\end{eqnarray*}\nFor $t\\le\\overline\\tau$, using (\\ref{31}) and (\\ref{32}),\n\\begin{eqnarray*}\nD\\Psi(\\overline X_s)[\\overline A_s-\\overline B_s] &=&\nD\\Psi(\\overline X_s)[\\Pi(\\overline X_s)-\\overline B_s]+D\\Psi(\\overline X_s)[\\overline A_s-\\Pi\n(\\overline X_s)]\n\\\\\n&=&-|D\\Psi(\\overline X_s)| (1-\\alpha_s-\\delta_s),\n\\end{eqnarray*}\nwhere $\\alpha_s=-|D\\Psi(\\overline X_s)|^{-1}D\\Psi(\\overline X_s)\\cdot\\overline\nB_s$, and\n$\\delta_s=-\\Pi(\\overline X_s)\\cdot[\\overline A_s-\\Pi(\\overline X_s)]$. Note that\n$|\\alpha_s|\\le1$. Moreover, using the inequality\n$|\\frac{v}{|v|}\\cdot(\\frac{u}{|u|}-\\frac{v}{|v|})|\\le\n2|v|^{-1}|u-v|$ along with\n(\\ref{33}), recalling the definition of $\\overline A$ and the fact $|D\\Psi\n(\\overline X_s)|\\ge1\/2$, we see that\n\\[\n|\\delta_s|\\le4\\varepsilon\\psi_{2,\\infty}.\n\\]\nFurthermore,\n\\begin{eqnarray*}\n&&D\\Psi(\\overline X_s)[\\overline A_s+\\overline B_s][\\overline C_s+\\overline D_s]\n\\\\\n&&\\qquad=\nD\\Psi(\\overline X_s)[\\Pi(\\overline X_s)+\\overline B_s][\\overline C_s+\\overline D_s]+D\\Psi\n(\\overline\nX_s)[\\overline A_s-\\Pi(\\overline X_s)][\\overline C_s+\\overline D_s]\n\\\\\n&&\\qquad=\n-|D\\Psi(\\overline X_s)| (1+\\alpha_s)(c^0+\\overline D_s)+e_s,\n\\end{eqnarray*}\nwhere, by (\\ref{33}), for $\\sigma\\le t_1 \\le t_2 \\le\\overline\\tau$\n\\[\n\\int_{t_1}^{t_2}e_s\\,ds\\le\\varepsilon+t_2-t_1.\n\\]\nFinally, we can estimate\n\\[\np_s:=\\tfrac12[\\overline A_s-\\overline B_s]'D^2\\Psi(\\overline X_s)[\\overline A_s-\\overline B_s]\n\\]\nby $|p_s|\\le2\\psi_{2,\\infty}$.\nShifting time by $\\sigma$, we denote $\\mathcaligr{G}_t=\\mathcaligr\n{F}_{t+\\sigma}$, $\\check\nW_t=W_{t+\\sigma}-W_\\sigma$, and\n\\[\n(\\check X_t,\\check D_t,\\check\\alpha_t,\\check\\delta_t,\\check\ne_t,\\check p_t)\n=(\\overline X_{t+\\sigma},\\overline D_{t+\\sigma},\\alpha_{t+\\sigma},\\delta\n_{t+\\sigma}, e_{t+\\sigma\n},p_{t+\\sigma}).\n\\]\nDenote also $m_t=|D\\Psi(\\check X_s)|$, let $M$ be the\n$\\mathcaligr{G}_t$-martingale\n\\[\nM_t=-\\int_0^tm_s(1-\\check\\alpha_s-\\check\\delta_s)\\,d\\check W_s\n\\]\nand set\n\\begin{eqnarray}\\label{36}\\quad\n\\mu_t:\\!&=&\\langle M\\rangle_t=\\int_0^t m_s^2(1-\\check\\alpha_s-\\check\n\\delta_s)^2\\,ds,\n\\nonumber\\\\\nP_t&=&-\\int_0^tm_s(1+\\check\\alpha_s)(c^0+\\check D_s)\\,ds,\\qquad\nQ_t=\\int_0^t(\\check e_s+\\check p_s)\\,ds,\\\\\n\\Psi_t &=&\\Psi(\\check X_t).\\nonumber\n\\end{eqnarray}\nCombining the above estimates, we have for $0\\le s \\le t\\le\\overline\\tau\n-\\sigma$,\n\\begin{eqnarray}\n\\label{11.1}\n\\Psi_t\n&=& \\Psi_0+M_t+P_t+Q_t,\n\\\\\n\\label{38}\nQ_t-Q_s&\\le&\\varepsilon+r(t-s),\n\\end{eqnarray}\nwhere $r=2\\psi_{2,\\infty}+1$.\nNote that $m_t\\ge1\/2$ for $s\\le\\overline\\tau_B-\\sigma$, and recall that\n$\\overline\\tau_B\\ge\\overline\\tau\\equiv\\overline\\tau_{\\rho}\\wedge\\overline\\tau\n_G$. We have for $t\\le\n\\overline\\tau-\\sigma$,\nassuming without loss of generality $4\\varepsilon\\psi_{2,\\infty}<1\/32$,\n\\begin{eqnarray*}\nr &=&\n\\frac{r}{4} (1-\\check\\alpha_t-\\check\\delta_t+1+\\check\\alpha\n_t+\\check\\delta_t)^2\n\\le\n\\frac{r}{2}(1-\\check\\alpha_t-\\check\\delta_t)^2+2r(1+\\check\\alpha\n_t+\\check\\delta\n_t)\\\\\n&\\le&\n2rm_t^2(1-\\check\\alpha_t-\\check\\delta_t)^2+4rm_t(1+\\check\\alpha\n_t)+2r\\check\\delta_t\n\\end{eqnarray*}\nand\n\\[\n\\check\\delta_t \\le(1+\\check\\alpha_t)+\\tfrac18(1-\\check\\alpha\n_t-\\check\\delta_t)^2\n\\le2m_t(1+\\check\\alpha_t)+\\tfrac12m_t^2(1-\\check\\alpha_t-\\check\n\\delta_t)^2.\n\\]\nHence, for $t\\le\\overline\\tau-\\sigma$, we have\n\\(\nr\\le3rm_t^2(1-\\check\\alpha_t-\\check\\delta_t)^2+8rm_t(1+\\check\n\\alpha_t).\n\\)\nThus, by~(\\ref{11.1}), if $c^0$ is chosen larger than $8r$, we have\n\\begin{equation}\n\\label{15.0}\n\\Psi_t=\\Psi_0+M_t+3r\\mu_t+\\widetilde P_t+\\widetilde Q_t,\n\\end{equation}\nwhere\n\\[\n\\widetilde P_t=-\\int_0^tm_s(1+\\check\\alpha_s)(c^0+\\check D_s-8r)\\,ds,\n\\]\n$\\widetilde Q_0=0$, and\n\\begin{equation}\\label{37}\n\\widetilde P_t-\\widetilde P_s\\le0,\\qquad\n\\widetilde Q_t-\\widetilde Q_s\\le\\varepsilon,\\qquad 0 \\le s\\le t \\le\n\\overline\\tau- \\sigma.\n\\end{equation}\n\nWe will write $\\widehat{\\mathbf{P}}$ for $\\mathbf{P}[ \\cdot\n|\\mathcaligr{G}_0]$, and $\\widehat{\\mathbf{E}}$\nfor the respective conditional expectation.\n\nThe proof will proceed in several steps.\n\\begin{Step}\\label{Step1}\nFor some $\\nu_1 \\in(0, \\infty)$,\n\\[\n\\sup_{\\bolds{\\sigma}\\in\\bolds{\\Sigma}_{\\rho_1}} \\widehat{\\mathbf\n{E}}[(\\overline\\tau-\\sigma\n)^2] \\le\n\\nu_1,\\qquad \\mbox{a.s.}\n\\]\n\\end{Step}\n\\begin{Step}\\label{Step2}\nFor some $\\nu_2 \\in(0, \\infty)$,\n\\[\n\\sup_{\\bolds{\\sigma}\\in\\bolds{\\Sigma}_{\\rho_1}} \\widehat{\\mathbf\n{E}}[(\\overline\\tau\n_G-\\sigma)^2] \\le\n\\nu_2,\\qquad \\mbox{a.s.}\n\\]\n\\end{Step}\n\nNote that Step \\ref{Step2} is immediate from Step \\ref{Step1} and Lemma \\ref{lem01}\nbecause by construction, a constant control is used after time\n$\\overline\\tau$.\n\\begin{Step}\\label{Step3}\nThere exists a modulus $\\vartheta_1$ such that\n\\[\n\\sup_{\\bolds{\\sigma}\\in\\bolds{\\Sigma}_\\varepsilon}\\widehat\n{\\mathbf{P}}[\\overline\\tau_G-\\sigma\n>\\vartheta_1(\\varepsilon), \\overline\\tau_{\\rho}>\\overline\\tau_G]\\le\n\\vartheta_1(\\varepsilon),\\qquad \\varepsilon>0.\n\\]\n\\end{Step}\n\\begin{Step}\\label{Step4}\nThere exists a modulus $\\vartheta_2$ such that\n\\[\n\\sup_{\\bolds{\\sigma}\\in\\bolds{\\Sigma}_\\varepsilon}\\widehat\n{\\mathbf{P}}[\\overline\\tau_{\\rho\n}\\le\\overline\\tau_G]\\le\\vartheta_2(\\varepsilon),\\qquad \\varepsilon>0.\n\\]\n\\end{Step}\n\nBased on these steps, part (i) of the lemma is established as\nfollows. Writing $E_\\varepsilon$ for the event $\\overline\\tau_G-\\sigma\n>\\vartheta_1(\\varepsilon)$,\n\\begin{eqnarray*}\n\\widehat{\\mathbf{E}}[\\overline\\tau_G-\\sigma]\n&=&\n\\widehat{\\mathbf{E}}[(\\overline\\tau_G-\\sigma)\\mathbf{1}_{E_\\varepsilon\n}]+\\widehat{\\mathbf{E}}[(\\overline\\tau_G-\\sigma)\\mathbf{1}\n_{E_\\varepsilon^c}]\\\\\n&\\le&\n [\\widehat{\\mathbf{E}}[(\\overline\\tau_G-\\sigma)^2]\\widehat{\\mathbf\n{P}}(E_\\varepsilon)\n ]^{1\/2}+\\vartheta_1(\\varepsilon)\\\\\n&\\le&\\nu_2^{1\/2}[\\vartheta_1(\\varepsilon)+\\vartheta_2(\\varepsilon\n)]^{1\/2}+\\vartheta_1(\\varepsilon),\n\\end{eqnarray*}\nwhere the first inequality uses Cauchy--Schwarz, and the second uses\nSteps \\ref{Step2}, \\ref{Step3} and \\ref{Step4}.\n\nTo show part (ii) of the lemma, use Steps \\ref{Step3} and \\ref{Step4} to write\n\\begin{eqnarray}\\label{13.1}\n\\widehat{\\mathbf{E}}[|\\overline X-\\xi_2|_{*,\\overline\\tau_G}^2]\n&\\le&\\widehat{\\mathbf{E}}\\bigl[|\\overline\nX-\\xi_2|_{*,\\overline\\tau_G}^2\\mathbf{1}_{E_\\varepsilon^c\\cap\\{\\overline\n\\tau_{\\rho}>\\overline\\tau_G\\}}\\bigr]\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}+[\\vartheta_1(\\varepsilon)+\\vartheta_2(\\varepsilon)]\n\\operatorname{diam}(G)^2.\\nonumber\n\\end{eqnarray}\nBy (\\ref{11.1}) and (\\ref{38}),\nwe can estimate\n\\begin{equation}\n\\label{13.2}\\qquad\\quad\n\\widehat{\\mathbf{E}}\\Bigl[\\sup_{t \\in[\\sigma, \\overline\\tau_G]}\\Psi(\\overline\nX_{t})\\mathbf{1}_{E_\\varepsilon\n^c\\cap\\{\\overline\\tau_{\\rho}>\\overline\\tau_G\\}}\\Bigr]\n\\le\\varphi_{1,\\infty}\\varepsilon+3\\vartheta_1(\\varepsilon\n)^{1\/2}\\psi_{1,\\varepsilon}+r\\vartheta\n_1(\\varepsilon)+\\varepsilon.\n\\end{equation}\nThus, noting that $\\Phi(\\overline X_{t})\\ge0$ on $\\{\\overline\\tau\n_{\\rho}>\\overline\\tau_G; t \\in[\\sigma, \\overline\\tau_G]\\}$,\nwe have on this set,\n\\[\n\\Psi(\\overline X_{t})\\ge|\\overline X_{t}-\\xi_2|^2.\n\\]\nPart (ii) of the lemma now follows on using the above inequality and\n(\\ref{13.2}) in (\\ref{13.1}).\n\nIn order to complete the proof, we need to establish the statements\nin Steps \\ref{Step1}, \\ref{Step3} and \\ref{Step4}.\n\\begin{pf*}{Proof of Step \\ref{Step1}}\nLet $t$ be given. Let $F_t$ denote the\nevent $\\{\\Psi_s\\in(0,\\rho), 0\\le s\\le t\\}$. We have\n\\begin{eqnarray}\n\\label{15.1}\n\\widehat{\\mathbf{P}}(\\overline\\tau-\\sigma>t)&=&\\widehat{\\mathbf{P}}(\\overline\n\\tau-\\sigma>t, \\overline\\tau_0 - \\sigma> t, \\overline\\tau\n_B-\\sigma>t)\\nonumber\\\\[-8pt]\\\\[-8pt]\n&\\le&\n\\widehat{\\mathbf{P}}(F_t,\\overline\\tau_B-\\sigma>t).\\nonumber\n\\end{eqnarray}\nDenote $S_u=\\inf\\{s\\dvtx\\mu_s>u\\}$, where we recall that the infimum\nover an empty set is taken to be $\\infty$. Let $\\kappa\\in(0,1\/16)$.\nThen\n\\begin{eqnarray}\\label{16.15}\n&&\\widehat{\\mathbf{P}}(F_t,\\overline\\tau_B-\\sigma>t,\\mu_t>\\kappa t)\n\\nonumber\\\\\n&&\\qquad\n\\le\\widehat{\\mathbf{P}}\\bigl(\\mu_t>\\kappa t,\n\\Psi_{S_s}\\in(0,\\rho), \\overline\\tau_B - \\sigma> t, 0\\le s\\le\n\\mu_t\\bigr)\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\le\\widehat{\\mathbf{P}}\\bigl(\\Psi_0+H_s+3rs+\\widehat P_s\\in(0,\\rho),\n0\\le\ns\\le\\kappa t\\bigr)\\nonumber\\\\\n&&\\qquad=\\widehat{\\mathbf{P}}\\bigl(\\widehat H_s+\\widehat P_s\\in(0,\\rho), 0\\le\ns\\le\\kappa t\\bigr),\\nonumber\n\\end{eqnarray}\nwhere $H$ is a standard Brownian motion (in particular, $H_s=M_{S_s}$\nfor $s < \\mu_t$), $\\widehat\nP_t$~is a process that satisfies $\\widehat P_s-\\widehat P_u\\le\n\\varepsilon$, $u\\le\ns$, and $\\widehat H_s=\\Psi_0+H_s+3rs$. On the event indicated in the last\nline of (\\ref{16.15}), one has, for every integer $k<\\kappa t$, that\n$\\widehat H_k-\\widehat H_{k-1}\\ge-2$. Hence, the right-hand side of (\\ref\n{16.15}) can be\nestimated by $m_1e^{-m_2\\kappa t}$, for some positive constants\n$m_1$\nand $m_2$, independent of $t$ and $\\kappa$, $\\varepsilon$, and as a result,\n\\begin{equation}\n\\label{16.2}\n\\widehat{\\mathbf{P}}(F_t,\\overline\\tau_B-\\sigma>t,\\mu_t>\\kappa t)\\le\nm_1e^{-m_2\\kappa t}.\n\\end{equation}\n\nNext, on the event $F_t\\cap\\{\\overline\\tau_B-\\sigma>t,\\mu_t\\le\\kappa\nt\\}$ we have\n$\\int_0^tm_s^2(1-\\check\\alpha_s-\\check\\delta_s)^2\\,ds\\le\\kappa t$, thus\n\\[\n\\int_0^t m_s^2(1-\\check\\alpha_s)^2\\,ds\\le2\\kappa t\n+2\\int_0^tm_s^2\\check\\delta_s^2\\,ds\\le(2\\kappa+16\\psi_{1,\\infty\n}^2\\psi\n_{2,\\infty}^2\\varepsilon^2)t\\le\\frac t4,\n\\]\nwhere we assumed without loss that\n$2\\kappa+16\\psi_{1,\\infty}^2\\psi_{2,\\infty}^2\\varepsilon^2\\le\n1\/4$. Consequently,\n$\\int_0^tm_s(1-\\check\\alpha_s)\\,ds\\le t\/2$. Using $m_s\\ge1\/2$, we have\n$\\int_0^t(1+\\check\\alpha_s)\\,ds\\ge2t-t=t$, whence, letting $c^0$ be so\nlarge that $c^0-8 r>8r$,\n\\[\n3r\\mu_t+\\widetilde P_t\n\\le3rt-4rt=-rt,\n\\]\nwhere we used $\\mu_t\\le\\kappa t\\le t$. Using (\\ref{15.0}), on this\nevent we have $0\\le\\Psi_t\\le\\Psi_0+M_t-rt+\\widetilde Q_t$. Hence, recalling\nthat $\\widetilde Q_t\\le\\varepsilon$ and $\\Psi_0\\le\\psi_{1,\\infty\n}\\varepsilon$, denoting\n$m_0=\\psi_{1,\\infty}+1$, and letting $\\gamma$ be the ($t$-dependent)\nstopping time $\\gamma=\\inf\\{s\\dvtx\\mu_s>\\kappa t\\}$, we have, for $t\\ge\nr^{-1}m_0\\varepsilon$,\n\\begin{eqnarray}\n\\label{17.1}\n\\widehat{\\mathbf{P}}(F_t,\\overline\\tau_B-\\sigma>t,\\mu_t\\le\\kappa t)\n&\\le&\n\\widehat{\\mathbf{P}}(M_t\\ge rt-m_0\\varepsilon, \\mu_t\\le\\kappa t)\\nonumber\\\\\n&\\le&\n\\widehat{\\mathbf{P}}(M_{t\\wedge\n\\gamma}\\ge rt-m_0\\varepsilon)\\\\\n&\\le&\\frac{m_3\\widehat{\\mathbf{E}}[\\langle M\\rangle_{t\\wedge\\gamma\n}^2]}{(rt-m\n_0\\varepsilon)^{4}}\\le\\frac{m_3\n(\\kappa t)^2}{(rt-m_0\\varepsilon)^4}.\\nonumber\n\\end{eqnarray}\nIn the second inequality above, we used the fact that $\\mu_t\\le\\kappa\nt$ implies $\\gamma\\ge t$, and in the third we used Burkholder's\ninequality. In particular, $m_3$ does not depend on $t$ or $\\kappa$\n(which will allow us to use this estimate more efficiently in Step \\ref{Step3}\nbelow).\nCombining (\\ref{15.1}), (\\ref{16.15}) and (\\ref{17.1}) we obtain the\nstatement in Step \\ref{Step1}.\n\\end{pf*}\n\\begin{pf*}{Proof of Step \\protect\\ref{Step3}}\nWe begin by observing that, from (\\ref{16.15}),\n\\begin{equation}\n\\label{18.1}\n\\widehat{\\mathbf{P}}(F_t,\\overline\\tau_B-\\sigma>t,\\mu_t>\\kappa\nt)\\le\\mathbf{P}(\\widehat H_s+\\widehat P_s>0, 0\\le s\\le\\kappa t),\n\\end{equation}\nand since $\\Psi_0+\\widehat P_s\\le m_0\\varepsilon$, this probability is\nbounded by\n\\[\np(\\varepsilon,\\kappa t):=\\mathbf{P}(m_0\\varepsilon+H_s+3rs>0, 0\\le\ns\\le\\kappa t).\n\\]\nThe latter converges to zero as $\\varepsilon\\to0$ (for fixed $\\kappa\n$ and\n$t$). Let $\\overline\\vartheta$ be a modulus such that\n$p(\\varepsilon,\\overline\\vartheta(\\varepsilon))\\le\\overline\\vartheta\n(\\varepsilon)$,\nand $\\frac12\\overline\\vartheta(\\varepsilon)^{1\/4}\\ge m_0\\varepsilon$. Taking\n$t=r^{-1}(\\overline\\vartheta(\\varepsilon))^{1\/4}$ and $\\kappa=r\\overline\n\\vartheta(\\varepsilon)^{3\/4}$,\ncombining (\\ref{17.1}) and (\\ref{18.1}),\n\\[\n\\widehat{\\mathbf{P}}\\bigl(F_t,\\overline\\tau_B-\\sigma>r^{-1}\\overline\\vartheta\n(\\varepsilon)^{1\/4}\\bigr)\n\\le\\overline\\vartheta(\\varepsilon)+\\frac{m_3\\overline\\vartheta(\\varepsilon\n)^2}{(1\/2\\overline\n\\vartheta(\\varepsilon)^{1\/4})^4}\n=(1+16m_3)\\overline\\vartheta(\\varepsilon).\n\\]\nUsing the above estimate in (\\ref{15.1}), Step \\ref{Step3} follows.\n\\end{pf*}\n\\begin{pf*}{Proof of Step \\protect\\ref{Step4}}\nFor $a>0$, let $\\tau_a$ and $\\tau_0$\ndenote the first time $[a,\\infty)$, and, respectively, $(-\\infty,0]$, is\nhit by $\\widehat H$. Since $\\widehat H$ is a Brownian motion (with\ndrift $3r$)\nstarting from $\\widehat H(0)\\le\\psi_{1,\\infty}\\varepsilon$, we have that\n$\\mathbf{P}(\\tau_{\\rho-\\varepsilon}\\le\\tau_0)$ converges to zero\nas $\\varepsilon\\to0$.\nThe proof is completed on noting that\n\\[\n\\widehat{\\mathbf{P}}(\\overline\\tau_{\\rho}\\le\\overline\\tau_G)\\le\\mathbf\n{P}(\\tau_{\\rho-\\varepsilon}\\le\\tau_0),\n\\]\nwhich follows from (\\ref{15.0}), (\\ref{37}), the relation $\\widehat\nH_s=\\Psi_0+M_{S_s}+3r\\mu_{S_s}$ for all $s < \\mu_{\\infty} \\equiv\n\\sup_{t \\ge0} \\mu_t$ and observing that on the set where $\\sigma_0 =\n\\sup\\{S_s\\dvtx s < \\mu_{\\infty}\\} < \\infty$ we have that $M_t + 3r\\mu_t\n= M_{\\sigma_0} + 3r\\mu_{\\sigma_0}$, for $t \\ge\\sigma_0$.\\qed\n\\noqed\\end{pf*}\n\\noqed\\end{pf*}\n\n\\section{Analysis of the game with bounded controls}\\label{sec5}\n\nThe main result of this section, Theorem \\ref{th2}, implies Lemma\n\\ref{lem3}. Fix $k,l$ such that $\\min\\{k, l\\} \\ge\\max\\{c^0, d^0$,\n$k_{n_2},l_{n_2}\\}$, where $n_2$ is as in Theorem \\ref{th3plus}.\nThroughout this section, $(k,l)$ will be omitted from the notation.\nAs in the previous section, only simple controls and strategies will be used.\nRecall that\n\\[\n\\Phi(a,b,c,d;p,S)= -\\tfrac12 (a-b)'S(a-b)-(c+d)(a+b)\\cdot p.\n\\]\nFix $\\gamma\\in[0,1)$ and write\n\\begin{eqnarray*}\n\\Lambda^+_{\\gamma}(p,S)&=&\n\\max_{|a|=1, 0\\le c\\le k} \\min_{|b|=1, 0\\le d\\le l}\n\\Phi(a,b,c,d;p,S) -\\frac{\\gamma^2}{2} \\operatorname{Tr}(S),\n\\\\\n\\Lambda^-_{\\gamma}(p,S)&=&\\min_{|b|=1, 0\\le d\\le l} \\max_{|a|=1,\n0\\le\nc\\le k}\n\\Phi(a,b,c,d;p,S)-\\frac{\\gamma^2}{2} \\operatorname{Tr}(S),\n\\end{eqnarray*}\nand consider the equations\n\\begin{eqnarray}\\label{11}\n&&\\cases{\\Lambda^+_{\\gamma}(Du,D^2u)-h=0, &\\quad in $G$,\\cr\nu=g, &\\quad on $\\partial G$,}\n\\\\\n\\label{17}\n&&\\cases{\\Lambda^-_{\\gamma}(Du,D^2u)-h=0, &\\quad in $G$,\\cr\nu=g &\\quad on $\\partial G$.}\n\\end{eqnarray}\nWe will write $V^{\\gamma}$ and, respectively, $U^{\\gamma}$ for the\nfunctions $V^{\\gamma}_{kl}$\nand $U^{\\gamma}_{kl}$ introduced at the beginning of Section \\ref{sec4}.\n\\begin{theorem}\\label{th2} For each $\\gamma\\in[0,1)$, one has the\nfollowing:\n\n\\begin{longlist}\n\\item\nThe function $U^{\\gamma}$ uniquely solves (\\ref{11}).\n\\item The function $V^{\\gamma}$ uniquely solves (\\ref{17}).\n\\end{longlist}\n\\end{theorem}\n\nThe proof of the theorem is based on a result on a finite time\nhorizon, Proposition \\ref{prop1}, in which we adopt a technique of\n\\cite{swi}. Given a function $u\\in\\mathcaligr{C}(\\overline G)$, $x_0\\in\n\\overline G$,\n$T\\ge0$, and $Y\\in M^0$, $Z\\in M^0$, let\n\\begin{equation}\\label{12}\nJ^{\\gamma}(x_0,T,u,Y,Z)=\\mathbf{E} \\biggl[\\int_0^{T\\wedge\\tau\n^{\\gamma\n}}h(X_s^{\\gamma})\n\\,ds+u(X^{\\gamma}_{T\\wedge\\tau^{\\gamma}}) \\biggr],\n\\end{equation}\nwhere $X^{\\gamma}$ and $\\tau^{\\gamma} =\\tau(x_0,Y,Z)$ are as\nintroduced in Section \\ref{sec4}\nwith $X_0=x_0$, $Y=(A,C)$ and\n$Z=(B,D)$.\n\\begin{proposition}\\label{prop1}\nLet $x_0\\in\\overline G$, $T\\in[0,\\infty)$ and $\\gamma\\in[0, 1)$.\nLet $u\\in\n\\mathcaligr{C}(\\overline G)$.\n\n\\begin{longlist}\n\\item\nIf $u$ is a subsolution of (\\ref{11}), then\n\\begin{equation}\\label{13}\nu(x_0)\\le\\sup_{\\alpha\\in\\Gamma^0_{lk}}\\inf_{Z\\in\nM^0_l}J^{\\gamma}(x_0,T,u,\\alpha[Z],Z).\n\\end{equation}\n\\item If $u$ is a supersolution of (\\ref{11}), then\n\\begin{equation}\\label{14}\nu(x_0)\\ge\\sup_{\\alpha\\in\\Gamma^0_{lk}}\\inf_{Z\\in\nM^0_l}J^{\\gamma}(x_0,T,u,\\alpha[Z],Z).\n\\end{equation}\n\\item If $u$ is a subsolution of (\\ref{17}), then\n\\begin{equation}\\label{15}\nu(x_0)\\le\\inf_{\\beta\\in\\Gamma^0_{kl}}\\sup_{Y\\in\nM^0_k}J^{\\gamma}(x_0,Y,T,u,\\beta[Y]).\n\\end{equation}\n\\item If $u$ is a supersolution of (\\ref{17}), then\n\\begin{equation}\\label{16}\nu(x_0)\\ge\\inf_{\\beta\\in\\Gamma^0_{kl}}\\sup_{Y\\in\nM^0_k}J^{\\gamma}(x_0,T,u,Y,\\beta[Y]).\n\\end{equation}\n\\end{longlist}\n\\end{proposition}\n\nBefore proving Proposition \\ref{prop1}, we show how it implies the\ntheorem.\n\\begin{pf*}{Proof of Theorem \\protect\\ref{th2}}\nWe only prove (i) since the\nproof of (ii) is similar. We first argue that any solution of\n(\\ref{11}) must equal $U^{\\gamma}$, and then show that a solution\nexists. Let a solution $u$ of (\\ref{11}) be given.\nFix $x_0\\in\\overline G$ and $\\varepsilon>0$. Fix\n$\\alpha\\in\\Gamma^0_{lk}$ such that\n\\begin{equation}\n\\label{27}\nU^{\\gamma}(x_0)\\le\\inf_{Z\\in M^0_l}J^{x_0}_{\\gamma}(\\alpha\n,Z)+\\varepsilon.\n\\end{equation}\nBy Proposition \\ref{prop1}(ii),\n\\begin{equation}\n\\label{28}\nu(x_0)\\ge\\inf_{Z\\in M^0_l}j^{\\gamma}(T,Z),\n\\end{equation}\nwhere we denote\n\\[\nj^{\\gamma}(T,Z)=J^{\\gamma}(x_0,T,u,\\alpha[Z],Z).\n\\]\nFor the rest of the proof, we suppress $\\gamma$ from the notation.\nLemma \\ref{lem01} shows that there is $m_1<\\infty$ such that, for\nevery $T\\in[0,\\infty)$, $\\inf_{Z\\in M^0_l}j(T,Z)\\le m_1$.\\vspace*{-2pt} Letting\n$M(T)=\\{Z\\in M^0_l\\dvtx j(T,Z)\\le c_1\\}$, it follows from the lower bound\non $h$ that for some $T<\\infty$ that does not depend on $Z$, one has\n$\\mathbf{P}(\\tau>T)<\\varepsilon$ for all $Z\\in M(T)$, where\n$\\tau=\\tau^{x_0}(\\alpha,Z)$. Fix such a $T$. Given $Z\\in M(T)$, let\n$\\widehat Z\\in M^0_l$ be equal to $Z$ on $[0,T)$, and let it assume the\nconstant value $(a^0,1)$ on $[T,\\infty)$. Clearly $j(T,\\widehat\nZ)=j(T,Z)$. Also, by Lemma \\ref{lem01}, denoting\n$\\widehat\\tau=\\tau^{x_0}(\\alpha,\\widehat Z)$, we have\n\\[\n\\mathbf{E}[(\\widehat\\tau-T)^+| \\widehat\\tau>T]\\le m_2\n\\]\nfor some constant $m_2$ independent of $\\varepsilon$ and $T$. By\n(\\ref{12}), (\\ref{27}), the definition of the payoff, and using the\nboundary condition $u|_{\\partial G}=g$, we have for some $m_3 \\in(0,\n\\infty)$\n\\begin{eqnarray*}\nU(x_0)&\\le& J^{x_0}(\\alpha,\\widehat Z)+\\varepsilon\\\\\n&\\le&\nJ(x_0,T,u,\\alpha[\\widehat Z],\\widehat Z)+m_3\\{\\mathbf{E}[(\\widehat\\tau\n-T)^+]+\\mathbf{P}\n(\\widehat\\tau>T)\\}+\\varepsilon.\n\\end{eqnarray*}\nUsing $\\mathbf{P}(\\widehat\\tau>T)=\\mathbf{P}(\\tau>T)$ yields\n$U(x_0)\\le j(T,\\widehat\nZ)+m_4\\varepsilon=j(T,Z)+m_4\\varepsilon$. Note that the infimum of\n$j(T,Z)$ over $M^0_l$ is equal to that over $M(T)$. Thus, using\n(\\ref{28}) and sending $\\varepsilon\\to0$ proves that $U(x_0)\\le u(x_0)$.\n\nTo obtain the reverse inequality, fix $x_0\\in\\overline G$. From Lemma\n\\ref{lem01}, there exists $m_5<\\infty$ and $Z_1\\in M^0_l$ such that,\nfor every $\\alpha$,\n\\[\nJ^{x_0}(\\alpha,Z_1)\\le m_5.\n\\]\nDenote $N(\\alpha)=\\{Z\\dvtx J^{x_0}(\\alpha,Z)\\le m_5\\}$. Clearly, for each\n$\\alpha$, the infimum of $J^{x_0}(\\alpha,Z)$ over all $Z\\in M^0_l$ is\nequal to that over $Z\\in N(\\alpha)$. Hence,\n\\[\nU(x_0)\\ge\\inf_{Z\\in N(\\alpha)}J^{x_0}(\\alpha,Z),\\qquad \\alpha\\in\n\\Gamma^0_{lk}.\n\\]\nUsing the positive lower bound on $h$ as before, it follows that\nthere exists a function $r\\dvtx [0, \\infty) \\to[0, \\infty)$ with $\\lim\n_{T\\to\\infty}r(T)=0$, such that\nfor every $\\alpha$ and $Z\\in N(\\alpha)$ we have\n$\\mathbf{P}(\\tau^{x_0}(\\alpha,Z)>T)\\le r(T)$. Therefore, for some $m_6\n\\in(0, \\infty)$\n\\[\nJ^{x_0}(\\alpha,Z)\\ge J(x_0,T,u,\\alpha[Z],Z)-m_6 r(T),\\qquad \\alpha\n\\in\n\\Gamma^0_{lk}, Z\\in N(\\alpha).\n\\]\nIn conjunction with Proposition \\ref{prop1}(i), this shows that\n$U(x_0)\\ge u(x_0)- m_6 r(T)$. Since $T$ is arbitrary, we obtain\n$U(x_0)\\ge u(x_0)$.\n\nFinally, we argue existence of solutions to (\\ref{11}). Let\nus write (\\ref{11})${}_\\gamma$ for (\\ref{11}) with a\nspecific $\\gamma$. For $\\gamma\\in(0,1)$, existence of solutions to\n(\\ref{11})${}_\\gamma$ follows from Theorem 1.1 of \\cite{CKLS}. To\nhandle the case $\\gamma=0$, we will use the fact that any uniform\nlimit, as $\\gamma\\to0$, of solutions to (\\ref{11})${}_\\gamma$ is a\nsolution to (\\ref{11})${}_0$. This fact follows by a standard\nargument, that we omit. Now, since for $\\gamma\\in(0,1)$ we have\nexistence, the uniqueness statement established above shows that\n$U^{\\gamma}$ solves (\\ref{11})${}_\\gamma$. From Theorem\n\\ref{th3plus}, we have that the family $\\{U^{\\gamma}, \\gamma\\in\n(0,1)\\}$ is equicontinuous, and thus a uniform limit of solutions,\nand in turn a solution to (\\ref{11})${}_0$, exists.\n\\end{pf*}\n\nIn the rest of this section, we prove Proposition \\ref{prop1}.\n\nLet $G_n$ be a sequence of domains compactly contained in $G$ and\nincreasing to~$G$. Let $J_n^{\\gamma}$ be defined as $J^{\\gamma}$ of\n(\\ref{12}), with\n$\\tau^{\\gamma}=\\tau^{\\gamma}(x_0,Y,Z)$ replaced by $\\tau_n^{\\gamma\n}=\\tau_n^{\\gamma}(x_0,Y,Z)$, where\n\\[\n\\tau_n^{\\gamma}=\\inf\\{t\\dvtx X_t^{\\gamma}\\in\\partial G_n\\}.\n\\]\n\\begin{lemma}\\label{lem6}\nFor every $n$, and $\\gamma\\in[0, 1)$ Proposition \\ref{prop1} holds\nwith $J^{\\gamma}$ replaced by\n$J_n^{\\gamma}$.\n\\end{lemma}\n\\begin{pf}\nWe follow the proof of \\cite{swi}, Lemma 2.3 and Theorem 2.1.\nAssume without loss that $G_0\\subset\\subset G_1\\subset\\subset\nG_2\\subset\\subset G$. We will prove the lemma for $n=0$. Since the\nclaim is trivial, if $x_0\\notin G_0$, assume $x_0\\in G_0$. In this\nproof only, write $\\tau$ for $\\tau_0^{\\gamma}$, the exit time of\n$X^{\\gamma}$ from\n$G_0$. Fix $\\widetilde\\gamma>\\gamma$, let $\\widetilde\\tau= \\tau_1^{\\widetilde\n\\gamma\n}$ and\n$\\sigma=\\tau\\wedge\\widetilde\\tau$. For $\\varepsilon>0$, consider the\nsup convolution\n\\[\nu_\\varepsilon(x)=\\sup_{\\xi\\in{\\mathbb{R}}^m} \\biggl\\{u(\\xi)-\\frac\n{|\\xi-x|^2}{2\\varepsilon\n} \\biggr\\},\\qquad\nx\\in G_2,\n\\]\nwhere, in the above equation only, $u$ is extended to ${\\mathbb{R}}^m$ by\nsetting $u=0$ outside $G$. It is easy to see that there exists\n$\\varepsilon_0$ such that the supremum is attained inside $G$ for all\n$(x,\\varepsilon)\\in G_2\\times(0,\\varepsilon_0)$. The standard mollification\n$u_\\varepsilon^\\delta\\dvtx\\overline G_1\\to{\\mathbb{R}}$ of\n$u_\\varepsilon\\dvtx G_2\\to{\\mathbb{R}}$ is well defined,\nprovided that $\\delta$ is sufficiently small. The result \\cite{swi},\nLemma 2.3, for the smooth function $u_\\varepsilon^\\delta$ and the\nargument in\nthe proof of \\cite{swi}, Theorem 2.1, show\n\\[\nu_\\varepsilon^\\delta(x_0)\\le\\sup_{\\alpha\\in\\Gamma^0}\\inf_{Z\\in\nM^0}\\mathbf{E}\n\\biggl[\\int_0^{T\\wedge\\sigma}h(X^{\\widetilde\\gamma}_s\n)\\,ds+u_\\varepsilon^\\delta(X^{\\widetilde\\gamma}_{T\\wedge\\sigma})\n\\biggr]+\\rho(\\varepsilon,\\delta\n,\\gamma, \\widetilde\\gamma),\n\\]\nwhere\n$\\lim_{\\varepsilon\\to0}\\lim_{\\widetilde\\gamma\\to\\gamma}\\lim\n_{\\delta\\to\n0}\\rho(\\varepsilon,\\delta,\\gamma, \\widetilde\\gamma)=0$.\nWe remark here that Lemma 2.3 of \\cite{swi} is written for the case\nwhere $u$ is a subsolution of a PDE of the form\n(\\ref{11}) on all of $\\mathbb{R}^m$ and $T\\wedge\\sigma$ is replaced\nby $T$,\nhowever the\nproof with $u$ and $T\\wedge\\sigma$\nas in the current setting can be carried out in exactly the same\nway. Since $G_0$ is compactly contained in $G_1$, we have that for\nevery $\\theta> 0$\n\\[\n\\sup_{\\alpha\\in\\Gamma^0} \\sup_{Z \\in M^0} \\Bigl\\{ \\mathbf\n{P} \\Bigl( |T\\wedge\n\\sigma\n- T\\wedge\\tau| + {\\sup_{0 \\le s \\le T}} |X^{\\widetilde\\gamma}_s -\nX^{\\gamma\n}_s| >\n\\theta \\Bigr) \\Bigr\\}\n\\]\nconverges to $0$ as $\\widetilde\\gamma\\to\n\\gamma$.\nMoreover, $u_\\varepsilon^\\delta\\to u_\\varepsilon$ as\n$\\delta\\to0$ and $u_\\varepsilon\\to u$ as $\\varepsilon\\to0$, where\nin both cases, the\nconvergence is uniform on $\\overline G_0$ (see ibid.). Hence, the result\nfollows on taking $\\delta\\to0$, then $\\widetilde\\gamma\\to\\gamma$ and finally\n$\\varepsilon\\to0$.\n\\end{pf}\n\\begin{pf*}{Proof of Proposition \\protect\\ref{prop1}}\nThe main argument is similar to that of Theorem \\ref{th3}, and so we\nomit some of the details. We will prove only item (iv) of the\nproposition, since the other items can be proved in a similar way.\n\nFix $x$ and $T$. Let $u$ be a supersolution of (\\ref{17}).\nLet $n$ be large enough so that $\\operatorname{dist}(\\partial G_n,\n\\partial G) < \\rho_1$.\nWrite\n$j_n^{\\gamma}(Y,\\beta)$ for $J_n^{\\gamma}(x,T,u,Y,\\beta[Y])$ and\n$j^{\\gamma}(Y,\\beta)$ for\n$J^{\\gamma}(x,T,u,Y,\\beta[Y])$. Below we will keep $\\gamma$ in the\nnotation only if there\nis scope for confusion. By Lemma \\ref{lem6}, $u(x)\\ge\nv_n:=\\inf_\\beta\\sup_Yj_n(Y,\\beta)$, for every $n$. We need to show\n$u(x)\\ge v:=\\inf_\\beta\\sup_Yj(Y,\\beta)$.\n\nFix $\\varepsilon>0$. Let $\\beta_n$ be such that\n\\begin{equation}\\label{29}\n\\sup_Y j_n(Y,\\beta_n)\\le v_n+\\varepsilon\n\\end{equation}\nand let $\\tau^n_1(Y)=\\tau_n^{\\gamma}(x,Y,\\beta_n[Y])$, $Y\\in\nM^0_l$. Let\n$\\widetilde\\beta_n$ be constructed from $\\beta_n$ as in the proof of\nTheorem \\ref{th3}, where in particular, $\\beta_n[Y]$ and\n$\\widetilde\\beta_n[Y]$ differ only on $[\\tau^n_1,\\infty)$, by which\n$j_n(Y,\\widetilde\\beta_n)=j_n(Y,\\beta_n)$. Choose $Y_n$ such that\n\\[\nv\\le\\sup_Yj(Y,\\widetilde\\beta_n)\\le j(Y_n,\\widetilde\\beta\n_n)+\\varepsilon\n\\]\nand set\n$\\tau^n_2(Y)=\\tau^{\\gamma}(x,Y,\\widetilde\\beta_n[Y])$.\nThen\n\\[\nv-v_n-2\\varepsilon\\le j(Y_n,\\widetilde\\beta_n)-j_n(Y_n,\\widetilde\n\\beta_n)=:\\delta_n.\n\\]\nDenote $X_n=X^x(Y_n,\\widetilde\\beta_n)$. Using Lemma \\ref{lem02},\n\\[\n0\\le\\tau^n_2-\\tau^n_1<\\varepsilon \\quad\\mbox{and}\\quad\n|X_n(\\tau^n_1\\wedge T)-X_n(\\tau^n_2\\wedge T)|<\\varepsilon\n\\]\nwith probability tending to 1 as $n\\to\\infty$. It now follows from the\ndefinition of $J_n$ and $J$ [cf. (\\ref{12})] that\n$\\limsup_n\\delta_n\\le\\rho(\\varepsilon)$ for some modulus $\\rho$. Since\n$\\varepsilon$ is arbitrary, this proves the result.\n\\end{pf*}\n\n\\section{Concluding remarks}\\label{sec6}\n\n\\subsection[Identity (1.4)]{Identity (\\protect\\ref{44})}\n\nRecall from (\\ref{09}) that\n\\begin{equation}\\label{48}\n\\Phi(a,b,c,d;p,S)= -\\tfrac12 (a-b)'S(a-b)-(c+d)(a+b)\\cdot p\n\\end{equation}\nand denote\n\\begin{eqnarray}\\label{40}\n\\Lambda^+(p,S)&=&\n\\sup_{|b|=1, 0\\le d<\\infty} \\inf_{|a|=1, 0\\le c<\\infty}\n\\Phi(a,b,c,d;p,S),\n\\\\\n\\label{41}\n\\Lambda^-(p,S)&=&\\inf_{|a|=1, 0\\le c<\\infty} \\sup_{|b|=1, 0\\le\nd<\\infty}\n\\Phi(a,b,c,d;p,S)\n\\end{eqnarray}\n[compare with (\\ref{18}) and (\\ref{10})].\nThe following proposition establishes identity (\\ref{44}) that, as\ndiscussed in the introduction, allows one to view the\ninfinity-Laplacian equation as a Bellman--Issacs type equation.\nThe result states that for the SDG of Section \\ref{sec2}, the\nassociated Isaacs condition, $\\Lambda^+=\\Lambda^-$, holds. Although\nwe do\nnot make use of it in our proofs, such a condition is often invoked\nin showing that the game has value (cf. \\cite{FS,swi}).\n\\begin{proposition}\\label{prop2}\nFor $p\\in{\\mathbb{R}}^m$, $p\\ne0$ and $S\\in\\mathscr{S}(m)$,\n$\\Lambda^+(p,S)=\\Lambda(p,S)$\nand $\\Lambda^-(p,S)=\\Lambda(p,S)$. In particular, identity (\\ref\n{44}) holds.\n\\end{proposition}\n\\begin{pf}\nWe will only show $\\Lambda^-=\\Lambda$ (the proof of $\\Lambda\n^+=\\Lambda$ being\nsimilar). Fix $p$, $S$, and omit them from the notation.\nWrite $\\mathcaligr{H}_k$ for $\\{(a,c)\\in\\mathcaligr{H}\\dvtx c\\le k\\}$ and\n$\\phi(y,z)$ for $\\Phi(a,b,c,d)$, where $y=(a,c)$, $z=(b,d)$. Given\n$\\delta>0$ let\n$k$ be such that\n$\\Lambda^-\\ge\\inf_{y\\in\\mathcaligr{H}_k}\\sup_{z\\in\\mathcaligr{H}}\\phi\n(y,z)-\\delta$.\nThen\n\\[\n\\Lambda^-\\ge\\inf_{y\\in\\mathcaligr{H}_k}\\sup_{z\\in\\mathcaligr\n{H}_l}\\phi(y,z)-\\delta\n=\\Lambda^-_{kl}-\\delta.\n\\]\nThus, by Lemma \\ref{lem1}, $\\Lambda^-\\ge\\Lambda$.\n\nNext, let $\\overline\\phi(y) = \\sup_{z \\in\\mathcaligr{H}} \\phi(y,z)$. Fix\n$\\delta\n\\in(0, \\infty)$, let $y_{\\delta} = (\\overline p, \\delta^{-1})$, where\n$\\overline p=p\/|p|$, and let\n$z_{\\delta} = (b_{\\delta}, d_{\\delta}) \\in\\mathcaligr{H}$ be such that\n$\\overline\n\\phi(y_{\\delta}) \\le\\phi(y_{\\delta}, z_{\\delta}) + \\delta$. Then\n\\begin{eqnarray*}\n\\Lambda^{-} &\\le& \\overline\\phi(y_{\\delta}) \\le- \\tfrac{1}{2}(\\overline p -\nb_{\\delta})'S(\\overline p - b_{\\delta}) - (\\delta^{-1} + d_{\\delta})\n(\\overline p +\nb_{\\delta}) \\cdot p + \\delta\\\\\n&\\le&- \\tfrac{1}{2}(\\overline p -\nb_{\\delta})'S(\\overline p - b_{\\delta}) + \\delta.\n\\end{eqnarray*}\nNote that $b_{\\delta}$ must converge to $-\\overline p$ or else the middle\ninequality above will say $\\Lambda^{-} = - \\infty$, contradicting the\nbound $\\Lambda^{-} \\ge\\Lambda$. Letting $\\delta\\to0$, we now have from\nthe third inequality that $\\Lambda^{-} \\le\\Lambda$. The result follows.\n\\end{pf}\n\n\\subsection{Limit trajectory under a nearly optimal play}\n\nIn \\cite{pssw}, the authors\nraise questions\nabout the form of the limit trajectory under optimal play of the\nTug-of-War game, as the\nstep size approaches zero (see Section 7 therein).\nIt is natural to ask, similarly, whether one can characterize\n(near) optimal trajectories for the SDG studied in the current\npaper. Let $V$ be as given in (\\ref{05}). Let $x\\in\\overline G$ and\n$\\delta>0$ be given. We say that a policy\n$\\beta\\in\\Gamma$ is $\\delta$-optimal for the lower game and initial\ncondition $x$ if $\\sup_{Y\\in M}J^x(Y,\\beta)\\le V(x)+\\delta$. When a\nstrategy $\\beta\\in\\Gamma$ is given, we say that a control $Y\\in M$ is\n$\\delta$-optimal for play against $\\beta$ with initial condition $x$,\nif $J^x(Y,\\beta)\\ge\\sup_{Y'\\in M}J^x(Y',\\beta)-\\delta$. A pair\n$(Y,\\beta)$ is said to be a $\\delta$-optimal play for the lower game\nwith initial condition $x$, if $\\beta$ is $\\delta$-optimal for the\nlower game and $Y$ is $\\delta$-optimal for play against $\\beta$ (both\nconsidered with initial condition $x$).\nOne may ask\nwhether the law of the process $X^{\\delta}$, under an arbitrary\n$\\delta$-optimal play $(\\beta^{\\delta},Y^{\\delta})$, converges to a limit\nlaw as $\\delta\\to0$; whether this limit law is the same for any choice\nof such $(\\beta^{\\delta},Y^{\\delta})$ pairs; and finally, whether an\nexplicit characterization of this limit law can be provided.\nA somewhat less ambitious goal, that is the subject of a forthcoming\nwork \\cite{AtBu2}\nis the characterization of the limit law of $X^\\delta$ under\n\\textit{some} choice of a $\\delta$-optimal play.\nThe result from \\cite{AtBu2} states the following.\n\\begin{theorem}\n\\label{th1new}\nSuppose that $V$ is a $C^2(\\overline{G})$ function and $DV \\neq0$ on\n$\\overline{G}$. Assume there exist uniformly continuous bounded extensions,\n$p$ and $q$ of $\\frac{Du}{|Du|}$ and\n$\\frac{1}{|Du|^2}(D^2u Du-\\Delta_\\infty u Du)$, respectively, to\n${\\mathbb{R}}^m$\nsuch that, for every $x\\in{\\mathbb{R}}^m$, weak uniqueness holds for\nthe SDE\n\\[\ndX_t=2 p(X_t)\\,dW_t+2q(X_t)\\,dt,\\qquad X_0=x.\n\\]\nFix $x \\in\\overline G$ and let $X$ and $\\tau$ denote such a solution\nand, respectively, the corresponding exit time from $G$.\nThen, given any sequence\n$\\{\\delta_n\\}_{n\\ge1}$, $\\delta_n \\downarrow0$, there exists a\nsequence of strategy-control\npairs $(\\beta^{n}, Y^{n}) \\in M\\times\\Gamma$,\n$n \\ge1$, with the following properties:\n\\begin{longlist}\n\\item\nFor every $n$, the pair $(\\beta^{n},Y^{n})$ forms a\n$\\delta_n$-optimal play for the lower game with initial condition $x$.\n\\item Denoting $X^{n} = X(x, Y^{n}, \\beta^{n})$ and $\\tau^n\n=\\tau(x, Y^{n}, \\beta^{n})$,\none has that $(X^{n}(\\cdot\\wedge\\tau^n), \\tau^n)$ converges in distribution\nto $(X(\\cdot\\wedge\\tau), \\tau)$, as a sequence of random\nvariables with values in $C([0, \\infty)\\dvtx\\overline G) \\times[0,\\infty]$.\n\\end{longlist}\nAn analogous result holds for the upper game.\n\\end{theorem}\n\nA sufficient condition for the uniqueness to hold is that $D^2u$ is\nLipschitz on $\\overline G$, since then both $ p$ and $q$ are Lipschitz,\nand thus admit bounded Lipschitz extensions to ${\\mathbb{R}}^m$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Abstract}\nWe present analytical and numerical results on localized fluidization within a granular layer subjected to a local injection of fluid. As the injection rate increases the three different regimes previously reported in the literature are recovered: homogeneous expansion of the bed, fluidized cavity in which fluidization starts developing above the injection area, and finally the chimney of fluidized grains when the fluidization zone reaches the free surface. The analytical approach is at the continuum scale, based on Darcy's law and Therzaghi's effective stress principle. It provides a good description of the phenomenon as long as the porosity of the granular assembly remains relatively homogeneous, i.e. for small injection rates.\nThe numerical approach is at the particle scale based on the coupled DEM-PFV method. It tackles the more heterogeneous situations which occur at larger injection rates. The results from both methods are in qualitative agreement with data published independently. A more quantitative agreement is achieved by the numerical model. A direct link is evidenced between the occurrence of the different regimes of fluidization and the injection aperture. While narrow apertures let the three different regimes be distinguished clearly, larger apertures tend to produce a single homogeneous fluidization regime. \nIn the former case, it is found that the transition between the cavity regime and the chimney regime for an increasing injection rate coincides with a peak in the evolution of inlet pressure.\nFinally, the occurrence of the different regimes is defined in terms of the normalized flux and aperture.\n\n\n\n\\section{Introduction}\n\n\nIn a broad sense, fluidization refers to the fluid-induced mobility of solid grains in a granular material subjected to upward seepage flow \\cite{payne2008remediation}. Fluidization is employed in a wide variety of industrial processes such as heat transfer, petroleum refining, coal conversion and water treatment \\cite{peng1997hydrodynamic,weisman1994design}. It may also occur as a result of seepage flow in a soil, in which case it can be the cause of internal soil erosion that can lead to serious failures of hydraulic works (dykes, levees, dams, etc) \\cite{bonelli2013erosion,foster2000statistics,fry1997erosion}. A particular case is when there is very localized influx of fluid, leading to a spatial heterogeneity of the phenomenon, this situation is generally termed \\textit{localized fluidization} in the literature. Such a configuration appears in tapered fluidized bed reactors found in many industrial process (drying, coating crystallization, mixing, etc) \\cite{schaafsma2006investigation,\nsutar2012mixing}, spouted beds, or in some natural geological formations \\cite{sutkar2013spout}. Eventually, fluidized zones induced by underground pipe leakage are also a major concern \nas ground surface may collapse due to the leak, causing important accidents \\cite{soderlund2007evaluating}. In addition, channelling can be observed in some applications of fluid bed reactors. Channelling is a condition wherein the fluid passes through the bed along localized paths \\cite{briens1997characterization}. This phenomenon should be avoided due to its adverse effects. Hydro-mechanical instabilities have been observed experimentally and simulated numerically in the case of a saturated granular medium when a localized flux is injected through a small orifice \\cite{gallo2004steady,kohl2014magnetic,philippe2013localized,zoueshtiagh2007effect}. \n\n\nDespite the large number of works dedicated to fluidized beds \\cite{anderson1967fluid, gidaspow1991hydrodynamics,mickley1955mechanism}, only a few have focused on the initial and developing phases of a localized fluidized zone inside a granular medium \\cite{cui2014coupled, ngomainteraction,philippe2013localized,zoueshtiagh2007effect}. The present study is devoted to this specific aspect.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=10cm]{case_of_study.png} \n \\tiny\n \\caption{Sample geometry and boundary conditions.}\n \\label{new_figures:studycase}\n\\end{figure}\n\n\nA typical configuration for laboratory experiments on localized fluidization is shown in figure \\ref{new_figures:studycase}. The previous works on such configurations evidenced three successive regimes during a gradual increase of the injection rate \\cite{ngomainteraction,philippe2013localized,zoueshtiagh2007effect}. At very low rates, the bed is stable. Larger rates cause bed expansion even before any fluidization zone can be observed (expansion regime). For a yet larger rate, the hydrodynamic forces exerted on some particles are sufficient to counterbalance their weight, triggering movements above the injection point in the so called \"fluidized\" zone (cavity regime). Eventually, the height of the fluidized zone increases with the injection rate, until it reaches the top of the granular layer (chimney regime). Experimental results on the progressive development of a fluidized zone in a saturated bed of grains under the effect of a localized upward flow \\cite{philippe2013localized} enabled to gain insight into the fluidized regimes. However, information at the grain scale and details of the pressure field were not accessible by this way. In order to overcome these limitations and analyze more deeply the local mechanisms responsible for fluidization and grain destabilization a numerical model, embedding fluid-solid coupling on the micro-scale, is used in this work.\n\nOverall, the process is highly non-linear, which makes the analytical study impossible without crude simplifications. Nevertheless, it will be seen that a continuum scale model can give some insight into some governing mechanisms. In this work, we provide closed form solutions for the field of initial pore pressure in the problem of figure \\ref{new_figures:studycase}. It enables the determination of the effective stress at every point of the problem. Then the effective stress field will be used to determine the extents of the fluidized zone.\n\nA more realistic simulation of the process is possible using a numerical model of grain-fluid systems. To this aim, we use a coupling between the discrete element method and a pore-scale finite volume method (DEM-PFV) \\cite{catalano2014pore,chareyre2012pore} to simulate the complete process from the expansion regime to the chimney regime. \n\nThis work is devoted to two objectives. Firstly, analytical and numerical simulations performed in this study enable a better understanding of the local mechanisms responsible for fluidization and which variables govern the phenomenon describing the initial and developing phases of fluidization from punctual to a source of infinite size and completing the experimental previous work \\cite{philippe2013localized}. Secondly, by tracking the effective stress within the medium it is shown that the effective stress constitutes a relevant parameter assessing the occurrence of fluidization. Thus, the effective stress field, as well as the porosity field, gives a good approach to describe the internal configuration and define the fluidized zones inside the granular assembly. \n\n\nThe paper is organized as follows: First, we define the physical model and the relevant dimensionless variables. Secondly, the theoretical and numerical models are introduced in section 2. The results are presented in section 3 and compared to data available in the literature. Finally, numerical results are used to highlight the role of flow rate, particle sizes, aperture of the injection zone, and viscosity of the pore fluid. The domains corresponding to the different fluidization regimes are defined in terms of dimensionless injection rate and dimensionless aperture. \n\n\\section{Methodology}\n\\subsection{Problem statement and experimental set-up}\\label{data_analytical}\n\nAs a model system we consider a layer of mono-disperse spheres immersed in a viscous fluid inside a rigid box. The system is subjected to gravity and the density of the solid particles is larger than that of the fluid (i.e. the particles sink). The granular layer is initially at static equilibrium then subjected to a local injection of fluid through the bottom face of the box (figure \\ref{new_figures:studycase}). At the free surface of the granular layer the total stress and the fluid pressure are both null. The lateral and bottom faces are fixed (i.e. zero-displacement condition) and impermeable - the injection orifice excepted. The injection occurs through a rectangular area that covers the whole depth of the box. In this injection area the boundary condition for the solid particles is the same as for the rest of the bottom face: no displacement, as if a rigid grid was stopping the particles while letting the fluid pass through.\n\nHereafter, the theoretical analysis of this problem is two-dimensional (2D), thus exploiting the invariance along depth when the distance between the front and back plates is much smaller than the other extents (as found in the published data). On the other hand, the numerical model considers three-dimensional (3D) sphere assemblies. Granular systems are indeed 3D at the micro-scale even when the average displacement field is 2D. It was thus considered a requirement to simulate three-dimensional grains to approach realistic responses. Periodic boundary conditions are assumed along the horizontal directions, consistently with the two-dimensionality at the macroscale. They are preferred over rigid faces in order to not introduce heterogeneities of the microstructure near the boundaries.\n\nThe setup described here is inspired by the physical experiments of \\cite{philippe2013localized}. Spherical beads were poured into a box before the box was slowly filled with oil via the bottom injection hole. The experiments were carried out by imposing different injection rates. Flow through the porous medium remained in the Stokesian regime as Reynolds number was kept low during the experiments. Stokesian regime is considered in the theoretical and numerical models as well. Visualization of the granular structure was made possible by the combined use of two optical techniques: refractive index-matching between the liquid and the beads and planar laser-induced fluorescence. Fluidization - in fact particles mobility - was then evaluated by image processing. The injection was done through a circular aperture of diameter $D$ = 14 mm at the center of the bottom face. We will not consider a circular aperture in our model system in general since it breaks the invariance along depth, thus introducing additional and superfluous complexity. However, one numerical simulation with a circular aperture has been carried out and will be reported for direct comparisons with the experimental result. \n\n\nThe physical variables of the problem are summarized in table \\ref{tab:Solid_fluid_properties}.\n\n\n\n\\begin{table}[H]\n \\centering\n \\small{\\begin{tabular}{|p{40mm}|p{16mm}|p{16mm}|}\n \\hline\n \\textbf{Variable} & \\textbf{Dimension} & \\textbf{SI units}\\\\\n \\hline\n Discharge per unit depth ($q$) & $L^{2}\\cdot T^{-1}$ & $[m^2\/s]$\\\\\n \\hline\n Pressure ($P$) & $FL^{-2}$ & $[Pa]$\\\\\n \\hline\n Height ($H$) & $L$& $[m]$\\\\\n \\hline\n Length ($l$) & $L$& $[m]$\\\\\n \\hline \n Grain diameter ($D$) & $L$& $[m]$\\\\\n \\hline\n Dynamic viscosity ($\\mu$) & $FL^{-2}T$& $[Pa\\cdot s]$ \\\\\n \\hline\n Aperture($a$) & $L$& $[m]$ \\\\\n \\hline\n Solid eight density ($\\gamma_s$) & $FL^{-3}$ & $[N\\cdot m^{-3}] $\\\\\n \\hline\n Fluid weight density ($\\gamma_w$) & $FL^{-3}$ & $[N\\cdot m^{-3}] $\\\\\n \\hline \n Porosity ($n$) & $-$ & $[-]$ \\\\\n \\hline\n \\end{tabular}\n }\n \\caption{Physical and geometrical variables of the problem. The dimensions are defined in the [FLT] (force, length, time) system.}\n \\label{tab:Solid_fluid_properties}\n\\end{table}\n\n\n\nThe porosity of the porous medium appearing in table \\ref{tab:Solid_fluid_properties} is defined by the ratio $n=\\dfrac{V_v}{V_t}$, where $V_v$ is the volume of void-space and $V_t$ the total volume of material (note that $n=1-\\phi$ if $\\phi$ denotes the solid fraction).\n\n\nThe physical properties of the materials used in the experiments are specified in table \\ref{tab:sampleCharacteritics}:\n\n\\begin{table}[H]\n \\centering\n \\small{\\begin{tabular}{|p{42mm}|p{18mm}|}\n \\hline\n \\textbf{Characteristic} & \\textbf{Experiment} \\\\\n \\hline\n Width ($L$) & 0.20 $m$ \\\\\n \\hline\n Initial height ($H_o$) & 0.12 $m$ \\\\\n \\hline\n Depth ($s$) & 0.08 $m$\\\\\n \\hline \n Mean radius ($r_m$) & 0.00250 $m$\\\\\n \\hline\n Density of the solid phase ($\\rho_s$) & 2230 $kg\\cdot m^{-3} $ \\\\\n \\hline\n Density of the fluid phase ($\\rho_f$) & 850 $kg\\cdot m^{-3} $ \\\\\n \\hline \n Dynamic viscosity ($\\mu_s$) & 0.0183 $Pa\\cdot s $ \\\\\n \\hline\n \\end{tabular}\n }\n \\caption{Solid and fluid properties of the experiments.}\n \\label{tab:sampleCharacteritics}\n\\end{table}\n\n\\subsection{Dimensionless variables}\\label{Dimensionless_analysis}\n\n\n\nBased on the above variables we introduce the so-called submerged (or apparent) density of the solid phase $\\gamma'=(1-n)(\\gamma_s-\\gamma_w)$, and the reference vertical effective stress $\\sigma'_0$ corresponding to the intergranular stress at the bottom of the layer (see next section), i.e.\n\\begin{equation} \\label{eq:refSigma} \\sigma'_0=(1-n)(\\gamma_s-\\gamma_w)H= \\gamma'H. \\end{equation}\nNormalization by this reference pressure leads to the following dimensionless group, where the normalized form of each variable is denoted by the ``$*$''. The relevance of this set of dimensionless variables will be demonstrated in section 4. Note that the dimensionless fluid pressure is a normalized \\textit{excess} pore pressure, i.e. the difference between absolute pressure and hydrostatic pressure.\n\n\\begin{itemize}\n \\item Normalized fluid pressure \n \\begin{equation} \\label{eq:Ec6} \\qquad p^*=\\dfrac{P+\\gamma_w(y-H)}{\\sigma'_0} \\end{equation}\n \\item Normalized flux\n \\begin{equation} \\label{eq:Ec4} \\qquad q^*= \\dfrac{q\\mu}{D^{2}\\sigma'_0} \\end{equation}\n \\item Normalized coordinates\n \\begin{equation} \\label{eq:Ec7} \\qquad x^*=\\dfrac{x}{l} \\end{equation}\n and \n \\begin{equation} \\label{eq:Ec8} \\qquad y^*=\\dfrac{y}{H} \\end{equation}\n\n \\item Normalized aperture\n \\begin{equation} \\label{eq:Ec8.5} \\qquad a^*=\\dfrac{a}{l} \\end{equation}\n\n\\end{itemize}\n\n\n\n \nAs it will be shown later, the response of the system for a given injection rate strongly depends on the macro-scale hydraulic conductivity $K$ of the granular material (ratio between seepage velocity and pressure gradient). $K$ is proportional to the squared particle size as in $K=\\kappa_0 \\dfrac{D^{2}}{\\mu}$, where $\\kappa_0$ is dimensionless and depends on porosity only (see e.g. the Kozeny-Carman form of this relationship). Through $K$ there is an effect of particle size in the continuum scale modeling. An alternative definition of the normalized flux is thus, instead of Eq.\\ref{eq:Ec4}, \n\\begin{equation}\\label{eq:Ec5} q^*_k= \\dfrac{q}{K \\sigma'_0}. \\phantom{\\hspace{3.4cm}} \\end{equation}\nBoth $q^*$ and $q^*_k$ will be used in the analysis. The definition of $q^*$ is simpler and, since $K$ has not been measured in the experiments of \\cite{philippe2013localized}, it is the only form we can use for comparing data and simulations. $q^*_k$ has the advantage of reflecting the change of porosity at any step of fluidization and will be used for the interpretation of some numerical results. \n\n\n\n\n\\subsection{Theoretical model}\\label{analytical_solution}\n\n\nWe suggest that fluidization can be seen as a special case of the so called \\textit{liquefaction}. The latest refers to situations in which the total stress tensor $\\sigma$ in a saturated material and the pressure of the pore fluid $P$ are such that the \\textit{effective stress} tensor\n\\begin{equation}\n\\label{effective}\n \\sigma'=\\sigma+P\\boldsymbol I\n\\end{equation}\nvanishes or has at least one vanishing eigen value. In this context, fluidization is simply the liquefaction produced by a particular combination of gravitational acceleration and upward seepage flow (while liquefaction in general can occur in non-gravitational systems and regardless of seepage flow).\n\nMomentum balance for the solid-fluid mixture in the Stokesian regime lets one deduce the component $\\sigma_{yy}$ of the total stress in the problem \n\\begin{equation}\n \\sigma_{yy}=\\gamma_{sat}(y-H),\n\\end{equation}\nwhere $\\gamma_{sat}$ is the average weight density of the saturated material, defined by $\\gamma_{sat}=n\\gamma_w+(1-n)\\gamma_s=\\gamma_w+\\gamma'$.\n\nAs long as the free surface is approximately flat and the porosity (hence $\\gamma_{sat}$) is approximately uniform, $\\sigma_{yy}$ is constant. Consequently, \nthe changes in the effective stress in Eq. \\ref{effective} can only result from a change of the excess pore pressure, itself controlled by the injection rate.\nThe key part of the theoretical modeling is thus to determine the spatial distribution of pore pressure within the specimen to identify the zones in which it reaches (or exceeds) the total stress.\n\nThe granular material will then be considered fluidized at a particular location $(x,y)$ if\n\\begin{equation}\n \\gamma_{sat} \\cdot (y-H)+P(x,y) \\geq 0,\n\\end{equation}\nor in an equivalent dimensionless form, introducing the normalized effective stress $\\sigma'^*$:\n\n\\begin{equation}\n\\label{fluidizationThreshold2}\n \\sigma'^* := \\dfrac{y-H}{H}+p^*(x,y) \\geq 0,\n\\end{equation}\n\n\nIn order to find closed form solutions for $P$, the following assumptions have been considered:\n\\begin{itemize}\n \\item Darcy's law applies at the bulk scale, i.e. the seepage flow is driven by the gradient of excess pore pressure with a velocity $\\boldsymbol v=-K\\nabla(P+\\gamma_wy)$.\n \\item the porous medium is homogeneous and the deformation is null or negligible, hence conductivity $K$ is uniform in space and time.\n\\end{itemize}\n\nUnder these assumptions, the final expression of the pressure at any point of the specimen can be obtained as a sum of the pressures induced at this point by an infinite set of punctual sources\/sinks which symmetries replicate the actual boundary conditions (see appendix for further details):\n\n \\thickmuskip=0mu\n \\makeatletter\n \\def\\@eqnnum{{\\normalsize \\normalcolor (\\theequation)}}\n \\makeatother\n { \\small \\begin{equation}\\label{eq:Ec3} P+\\gamma_wy=\\dfrac{q}{2 \\pi K} \\sum\\limits_{j=-\\infty}^\\infty \\sum\\limits_{i=-\\infty}^\\infty -1^{|j|} \\left[ln(\\sqrt{(x-i\\ l)^{2}+(y-jc)^{2}}) \\right]\n \\end{equation} }\nwhere $c=2H$ as sources and sinks are spaced by a distance equal to twice the sample height.\nIf a finite injection area is considered rather than an injection point, the above expression needs to be integrated on the aperture width:\n\n \\thickmuskip=0mu\n \\makeatletter\n \\def\\@eqnnum{{\\normalsize \\normalcolor (\\theequation)}}\n \\makeatother\n { \\small\\begin{equation}\\label{eq:Ec3.2} P+\\gamma_wy=\\dfrac{q}{2 \\pi K} \\int_{-a\/2}^{a\/2} \\sum\\limits_{j=-\\infty}^\\infty \\sum\\limits_{i=-\\infty}^\\infty -1^{|j|} \\left[ln(\\sqrt{(x-i\\ l-s)^{2}+(y-jc)^{2}}) \\right] ds\n \\end{equation} }\n \nNote that the pressure defined by Eq. \\ref{eq:Ec3} is singular at the injection point, where the pore pressure is infinite. In the limit $a^*=0$ there is therefore a finite-sized fluidized zone for any value of the injection rate. Conversely, Eq. \\ref{eq:Ec3.2} takes finite values in the injection area, thus defining a clear threshold in terms of injection rate below which the effective stress is strictly compressive everywhere.\n \n \nAccording to Eq. \\ref{eq:Ec3.2}, $K$ must be known in order to find the pore pressure. In the DEM-PFV numerical model, $K$ is a result, and it depends on the particle size and the porosity of the assembly. Consequently, the analytical and the numerical methods can be compared directly by plugging $K$ from the numerical model into Eq. \\ref{eq:Ec3.2}.\n\nIt is important to recall, however, that Eq. \\ref{eq:Ec3.2} is only valid for a homogeneous medium, which is not necessarily the case in experiments or numerical simulations. As soon as localized fluidization occurs, significant differences are expected between this equation and the actual or simulated fluid pressure.\n\n\n\n\\subsection{Numerical model}\\label{Numerical_method}\n\nThe behavior of a granular bed subjected to localized upward flux can be investigated using different coupled DEM-fluid models. Some previous works were based on couplings between the DEM and the Lattice Boltzmann \\cite{chen1998lattice} (LBM) method in two dimensions \\cite{cui2014coupled, ngomainteraction,cui20122d,cui2013numerical}. \n\n2D granular systems are peculiar from a mechanical point of view and their hydraulic properties are unclear - strictly speaking they are impermeable since they don't offer any free path to the fluid.\nA quantitative approach of the problem thus requires 3D models. To this aim, the so-called DEM-PFV coupling was used for the present study \\cite{catalano2011pore}. DEM-PFV refers to a micro-hydromechanical model combining the DEM and a pore scale finite volume formulation of the viscous flow of an incompressible pore fluid \\cite{chareyre2012pore,catalano2014pore}. It enabled 3D simulations at a reduced cost compared to 3D DEM-LBM simulations.\n\nThe solid particles in the model are spherical and slightly poly-disperse (uniform distribution deviating by 2\\% from the mean diameter). The interaction between them are elastic-plastic, with normal and tangential stiffness $k_{n}$ and $k_{s}$, and Coulomb friction angle $\\phi$. Newton's second law of motion is integrated explicitly through iterative time-stepping (implementation details can be found in \\cite{yade:doc2}). The fluid flow model is based on a pore scale discretization of Stokes equations, where the pores are defined by the tetrahedra of a \\textit{regular triangulation} \\cite{chareyre2012pore}. At each time step, the geometry and rate of deformation of each pore is updated on the basis of particles motions. In turn, the fluxes are determined and the fluid forces on the particles are obtained. They are integrated in the law of motion for each particle. This work is carried out using the PFV implementation provided by the open source code YADE-DEM\\cite{yade:doc2}. \n\nThe initial granular layer is obtained by simulating the gravitational deposition of a cloud of particles in a periodic box. The deposition stage stops when the particles reach static equilibrium. The layer is then subjected to an influx of fluid at the bottom, as shown in figure \\ref{new_figures:studycase}. The simulated granular layer is made up of 5000 spheres, the mean diameter of the grain is $D = 0.0166m$ and the height of the sample is $H \\approx 19D$.\n\nThe effective stress in simulated granular systems can be computed directly based on the contact network. Following \\cite{catalano2014pore}, the average effective stress tensor associated to one particle of a saturated material is defined by\n\\begin{equation}\n\\label{eq:effectiveS}\n \\sigma'=\\dfrac{1}{V_p}\\sum_{k=1}^{N_c}\\boldsymbol f_k \\otimes \\boldsymbol x_k,\n\\end{equation}\nwhere $N_c$ is the number of contact with other particles, $\\boldsymbol f_k$ is a contact force, $\\boldsymbol x_k$ the position vector of the contact point, and $V_p$ the volume of the Voronoi cell enclosing the particle. This definition results in rather scattered values when plotted per particle but meaningful results can be obtained when they are averaged locally (see section \\ref{numerical_results}).\n\n\n\n\n\n\n\\section{Results and discussion}\n\\subsection{Analytical solution}\\label{Analytical_solution}\n\nFigure \\ref{new_figures:pressure_field} shows the evolution of the pressure field after Eq. \\ref{eq:Ec3.2} for increasing flux values and an aperture of 10$\\%$ of the width of the specimen ($a^*=0.1$ (Eq.\\ref{eq:Ec8.5})). Near the injection area, the pressure contours tend to concentric half-circular shapes due to the quasi-radial flow distribution. On the other hand, the isolines are horizontal near the side walls, consistently with the no-flux condition.\nFollowing section \\ref{analytical_solution}, the fluidized zone can be identified with respect to the sign of the effective stress. In order to show the evolution of the fluidized zone and to identify the flux values triggering different steps of the fluidization phenomenon, the normalized effective stress is plotted in figure \\ref{new_figures:effective_field} for apertures $a^*$ = 0.1 (plots $(a)$, $(b)$ and $(c)$) and $a^*$ = 0.8 (plots $(d)$, $(e)$ and $(f)$). Therein the extent of the fluidized zone can be defined through the shape of the null-pressure isoline.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=8cm]{pressure_field.png} \n \\caption{Evolution of the dimensionless pressure field $p^*$ for an injection aperture $a^*$ = 0.1 and two injection rates $q^*$ = 0.00019 (a) and $q^*$ = 0.00067 (b).}\n \\label{new_figures:pressure_field}\n\\end{figure}\n\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=17cm]{effective_field_title.png} \n \\caption{Evolution of the dimensionless effective stress field when the injected flux increases. Narrow aperture ($a^*=0.1$) on the first line and wide aperture ($a^*$ = 0.8) on the second line. The normalized fluxes are $q^*$ = 0.00019 (a,d), $q^*$ = 0.00105 (b,e), $q^*$ = 0.00114 (c) and $q^*$ = 0.00126 (f).}\n \\label{new_figures:effective_field}\n\\end{figure*}\n\nFrom the first series of plots (narrow aperture) we can distinguish the regimes described in \\cite{philippe2013localized}. Low flux values correspond to the expansion regime in which no fluidization zone is detected (plot $(a)$ in figure \\ref{new_figures:effective_field}).\n\nAs the flux increases, pore pressure keeps building up until it balances the total stress. At this point a fluidized zone starts developing above the injection area, corresponding to the cavity regime (plot $(b)$ in figure \\ref{new_figures:effective_field}). Eventually, the fluidized zone reaches the top of the specimen, leading to a chimney of fluidized grains (plot $(c)$ in figure \\ref{new_figures:effective_field}).\n\nThe solution with a wider injection area (bottom series, plots $(d)$, $(e)$ and $(f)$ in figure \\ref{new_figures:effective_field}) shows that a slightly larger flow rate is required to initiate the fluidization above a wide aperture.\n\nDistinguishing the cavity regime from the chimney regime in this situation is made uneasy by the fact that large aperture tends to fluidize the granular layer simultaneously at every point in space, thus merging the cavity and chimney regimes into one single regime as $a^*\\rightarrow 1$.\n\n\n\n\n\\subsection{Numerical simulations}\\label{numerical_results}\n\nFigure \\ref{new_figures:porosity_expansion} and \\ref{new_figures:porosity_cavity} show the evolution of porosity within the simulated layer. Porosity is defined for each particle as the ratio of the volume of the void to the total volume of the Voronoi cell enclosing the particle \\cite{catalano2014pore}. The fields of figures \\ref{new_figures:porosity_expansion} and \\ref{new_figures:porosity_cavity} are in fact a moving average of the per-particle porosity for reducing the noise due to the scattered local values. The large porosity values at the top of the layer are artifacts and should be disregarded: the particles of the free surface have large porosity values by definition as their Voronoi cells enclose some void space above the free surface. This artifact gives a fringe of high porosity through the moving average procedure. We notice that, with the exception of this top boundary layer, the deposition process results in a rather uniform porosity throughout the sample. To some extent a high porosity layer is also noticeable near the bottom plate, in this case not an artifact but a perturbation of the microstructure by the rigid wall. There is no perturbation near the vertical boundaries thanks to the periodicity.\n\nThe porosity changes inside the layer are shown in figure \\ref{new_figures:porosity_expansion} and \\ref{new_figures:porosity_cavity} for $a^*=0.1$. A narrow color scale is used in the former to emphasize the expansion regime, which appears as a rather homogeneous process. During the expansion regime the porosity increases over all the points within the sample (from $n=0.360$ to $n=0.368$ in average, excluding the free surface) and no significant heterogeneities are detected.\nThe height of the granular layer increases almost uniformly during this expansion, reaching values over $y^*=1$ (see figure \\ref{new_figures:experimental_DEM_curves}).\n\nFigure \\ref{new_figures:porosity_cavity} highlights the heterogeneous changes of porosity as localized fluidization starts developing: a cavity appears for $q^*=1.16\\times10^{-3}$, followed by a chimney for $q^* \\geq 1.45\\times10^{-3}$.\nIn both cases the particles located in the regions of low porosity are moving and have only transient contacts with each other, while the particles of the dense regions are static and contribute to a permanent contact network. \n\n \n \n \n \n \n \n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=16cm]{Porosity_map_expansion.png} \n\\caption{Evolution of the porosity for narrow aperture ($a^*$ = 0.1) in a very detailed color scale. (a) Static regime, $q^* $= 0. (b) Expansion regime, $q^*$ = 0.00095.} \\label{new_figures:porosity_expansion}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=16cm]{Porosity_map_chimney.png} \n\\caption{Evolution of the porosity for narrow aperture ($a^*$ = 0.1). (a) Static regime, $q^*$= 0. (b) Cavity regime, $q^*$ = 0.00116. (c) Chimney regime, $q^*$ = 0.00145. (d) Irreversible changes in porosity after reducing the flux back to $q^*$ = 0.} \\label{new_figures:porosity_cavity}\n\\end{figure}\n\n\n\nPlot $(c)$ in figure \\ref{new_figures:porosity_cavity} ($q^* = 1.45\\times10^{-3}$) shows inside the chimney a region of even higher porosity ($n\\approx 0.5$ near $y^*=0.7$). This pattern is typical and was found in most simulations. In fact, this bubble of high porosity is not fixed in time and space: the figure only shows a snapshot at one particular time. The bubble actually tends to move up until it reaches the free surface, then another bubble appears at the bottom in a cyclic manner - very much like air bubbles produced in a water tank. This is in clear contrast with the cavity regime in plot $(b)$ ($q^*=1.16\\times10^{-3}$) in which the porosity field is stationary.\n\nAfter setting the injection rate back to zero and reaching a final equilibrium state, a region of high porosity remains ($n\\approx 0.42$ locally) above the injection point and throughout the layer (plot $(d)$ in figure \\ref{new_figures:porosity_cavity}). It denotes an irreversible change of porosity in the layer after the injection steps.\n\nCavities and chimneys may also be analyzed by means of the effective stress (Eq. \\ref{eq:effectiveS}). However, null effective stress was never clearly found in the simulations since particle collisions occur in the fluidized zone and the corresponding contacts are reflected in the effective stress (arguably, zero-contact states hardly exist in agitated granular suspensions - see e.g. \\cite{marzougui2015microscopic}). Conventionally, the fluidized zone in the simulations is defined by the points where the effective stress is less than 10$\\%$ of the initial effective stress. This threshold is such that the cavity shape in plot $(b)$ in figure \\ref{new_figures:porosity_cavity} matches the cavity in the top map in figure \\ref{new_figures:effective_stress_map_narrow}. After some iterations we have found 10$\\%$ of the initial effective stress is a good criterion to predict fluidized zones located inside the specimen. More restrictive thresholds (i.e. 5$\\%$ of the initial effective stress) would delay the actual formation of the cavity and introduce some noise as effective stresses are never zero due to internal collisions. A coarse-grained function has been used as well when plotting the dimensionless effective stress maps in order to obtain accurate and consistent data. Therefore, the averaged effective stress zones leading to fluidization never reach, in appearance, the top of the layer. Furthermore, the height of the cavity can be found by means of the effective stress criterion as the highest fluidized point inside the sample. Cavity (plot $(a)$ in figure \\ref{new_figures:effective_stress_map_narrow}) and chimney (plot $(b)$ in figure \\ref{new_figures:effective_stress_map_narrow}) of fluidization can be easily identified when the injection area is small ($a^* = 0.1$). On the contrary, concerning large apertures (figure \\ref{new_figures:effective_stress_map_wide}), fluidized zone does not look as a chimney due to the fact that an important part of the sample has liquefied.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=10cm]{effective_stress_map_narrow.png} \n \\caption{Evolution of the fluidized zone for narrow aperture ($a^*$ = 0.1). Blue zone represents non-fluidized zone ($\\dfrac{\\sigma}{\\sigma_{o}} \\geq 0.1$). Red zone corresponds to fluidized zone ($\\dfrac{\\sigma}{\\sigma_{o}} < 0.1$). (a) Cavity regime, $q^*$ = 0.00116. (b) Chimney regime, $q^*$ = 0.00145.}\n \\label{new_figures:effective_stress_map_narrow}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=10cm]{effective_stress_map_wide.png} \n \\caption{Fluidized zone for a wide aperture (a* = 0.9) and $q^*$ = 0.00156, an important part of the sample has liquefied. Blue zone represents non-fluidized zone ($\\dfrac{\\sigma}{\\sigma_{o}} \\geq 0.1$). Red zone corresponds to fluidized zone ($\\dfrac{\\sigma}{\\sigma_{o}} < 0.1$).}\n \\label{new_figures:effective_stress_map_wide}\n\\end{figure}\n\n\n\n\n\n\\subsection{Comparison with experiments}\\label{comparison2}\n\nThe results from the previous sections let the height of the cavity be defined by considering in both models the maximum height where the fluidization criterion is met.\nIt is thus possible to compare the models with the data from \\cite{philippe2013localized}, given in terms of height of cavity versus flow rate.\n\nPractically the fluidized zone in the tests was identified using time-averaged images of the granular layer, from which the regions with moving grains could be distinguished from the static ones according to a particular threshold. It was not possible to apply the very same definition of fluidization in the models, mainly because the details of the experimental steps and the thresholds were not reported. This possible cause of discrepancy has to be kept in mind.\n\n\n\nIn the models, the injection area has been defined so far as a band covering the whole depth of the sample while the laboratory tests were done with a smaller and circular orifice. For this comparison, the injection area has been modified in the DEM-PFV model to match that of the experiment. It is obviously not possible to reflect this particular geometry in the theoretical model, which is strictly two-dimensional.\nBesides, the physical properties of the materials used in the test (table \\ref{tab:sampleCharacteritics}) have been used directly as input parameters of both the theoretical and the numerical model. The only exception is the particle size: in the numerical model the grains are larger than the real ones. We do not expect significant bias from this upscaling, as demonstrated in section \\ref{Sensitivity_analysis}.\n\n\n\nFigure \\ref{new_figures:experimental_DEM_curves} shows the ratio of cavity height $H_{c}$ to initial height of the specimen $H_{o}$. A qualitative agreement is found between the tests and the DEM-PFV simulations. The experimental increasing path is slightly steeper than the numerical one. Regarding the decreasing path, both experimental and numerical curves have the same tendency when $H_{c}\/H_{o}$ >0.5. On the contrary, the gap between the decreasing curves widens when $H_{c}\/H_{o}$ < 0.5, as the numerical curves is much steeper than the experimental one. We can clearly observe similar slopes when $H_{c}\/H_{o}$ >0.5 for all the curves except for the experimental increasing path. This is related to the unknown initial packing from the experiments, which is presumably looser than the numerical samples and ease the propagation of the cavity. Besides, the numerical model overestimates the flux for cavity and chimney development (a shift is evidenced between numerical and experimental curves). This difference is mainly attributed to the fact that the permeability resulting from the numerical model is not exactly the same as the one of the real packing of glass beads. However, it is worth noting that permeability is not an input of the model but results from the description of solid fluid interactions at particle scales. Such interactions are described in the numerical model without any fitting parameter. In these conditions, numerical predictions may be considered satisfying from a quantitative point of view (it would be easy to fit very closely the experimental data by adding a single fitting parameter to the computation of the permeability in the model). The gap between the curves in the decreasing path when $H_{c}\/H_{o}$ < 0.5 could be attributed to diameter differences. Experimental specimen height is $H \\approx 24D$, where $D$ is the mean diameter. On the other hand, numerical sample height is $H \\approx 10D$. Furthermore, experimental and DEM results may differ as a consequence of the different criteria used to obtain the height of the fluidized zone. \n\nThe analytical solution has been included in figure \\ref{new_figures:experimental_DEM_curves} as well. We can observe a smooth transition between the origin of the cavity and the point it reaches the top of the sample forming a chimney. Despite the fact analytical curve is near the experimental and numerical ones, analytical solution is based on a 2D media. On the contrary, numerical and laboratory simulations have a three-dimensional effect within the specimen due to the upward flow was injected through a circular orifice at the bottom of the bed rather than a rectangle covering the depth of the sample (see figure \\ref{new_figures:studycase}).\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=10cm]{experimental_DEM_curves.png} \n \\caption{Normalized height of the cavity in the theoretical and numerical models, compared with the experimental data from \\cite{philippe2013localized}.}\n \\label{new_figures:experimental_DEM_curves}\n\\end{figure}\n\n\n\n\\subsection{Flux-pressure relationship and fluidization regimes}\\label{comparison}\n\n\nIn figure \\ref{new_figures:pressure_inc_dec} the fluid pressure at the inlet in the DEM-PFV simulation is plotted as a function of the imposed flow rate. The imposed rate is increased over time up to a maximum, then decreased back to zero, with increments $\\Delta q^*=0.000057$. Each flux value is kept constant over enough simulated time to exhibit the stationary solutions.\n\nVery low flux values correspond to the situation when the particles do not move significantly and the pore pressure increases linearly with the flow rate, as expected from Eq. \\ref{eq:Ec3.2}. At larger discharges the expansion regime leads to a noticeable increase of the porosity and hydraulic conductivity, such that the pressure is no longer proportional to the flux. Pore pressure keeps increasing until it reaches a peak ($p^* \\approx 1.10$). As the peak value is reached, the fluidized cavity starts developing. Further increase of the flux results in a decreasing pressure as the cavity progressively approach the free surface. When the chimney is fully developed the pressure tends to a residual value in average, although the bubbling trend commented in previous section produces some fluctuation around that value (see figure \\ref{new_figures:pressure_inc_dec}).\n\nIn the flux-decreasing phase, pressure values are always below the first flux-increasing curve. It is easily explained by the increased porosity leading to greater conductivity. The irreversible increment of porosity in the granular layer is evidenced in figure \\ref{new_figures:porosity_cavity}$(d)$ . Starting from an initially dense material ($n \\approx$ 0.365, close to the random close packing RCP), fluidization results in making looser the granular bed (at least locally, reaching porosity $n \\approx$ 0.41 not so far from the random loose packing RLP). Then if the flux is increased again (not shown in the present work), no peak is observed in the $p^*$-$q^*$ plot, following closely the flux decreasing curve. \n\n\n\nHereafter, we focus on the increasing phase since one the main focus of this work is on the initiation and development of the fluidized zone.\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=10cm]{Pressure_inc_dec.png} \n \\caption{Dimensionless pressure-flux, $p^*$-$q^*$, curve for an increasing-decreasing cycle of flux at the bottom of the sample. DEM-PFV simulation with an injection aperture $a^*$ = 0.1}\n \\label{new_figures:pressure_inc_dec}\n\\end{figure}\n\nThe non-linearity of the flux-pressure relation before the peak can be explained by the increase of the hydraulic conductivity as porosity increases in the expansion regime. In order to isolate this effect the results can be plotted considering the second definition of dimensionless flux $q^*_{k}$ (see Eq.~\\ref{eq:Ec5}), where the updated conductivity is used. The hydraulic conductivity at one particular time is obtained based on Eq.\\ref{eq:Ec5} and considering the average porosity above the injection area. In figure \\ref{new_figures:pressure_comparison} three curves are considered: the analytical solution (red curve with circle symbols), the numerical results interpreted as if conductivity was constant in time (blue line with square symbols) and the interpretation including the updated conductivity (green line with triangle symbols). \n\n\nFigure \\ref{new_figures:pressure_comparison} shows a perfect fit of the analytical solution by the numerical results at low fluxes (\"0\"-\"1\" path) when plotted with the updated conductivity. After point \"1\" the granular assembly starts expanding significantly and some non-linearity appears but it is less significant in the $p^*-q^*_{k}$ plot. \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=10cm]{Pressure_comparison.png} \n \\caption{$p^*$-$q^*_k$ curves comparison at the center of the injection orifice with an injection aperture $a^*$ = 0.1.}\n \\label{new_figures:pressure_comparison}\n\\end{figure}\n\n\nThe cavity regime begins immediately after the pressure peak is reached (point \"2\"). As can be seen in figure \\ref{new_figures:pressure_comparison}, the blue-squared $p^*-q^*_{k}$ curve clearly deviates from the analytical solution after this point, reaching lower values of $q^*_k$. This is to be understood as a deviation from the homogeneous porosity field assumed for the theoretical derivation: the conductivity increases much more above and around the inlet than in the rest of the simulated layer. Finally, the chimney regime is reached (point \"3\" in figure \\ref{new_figures:pressure_comparison}), triggering bubbling events through the layer. The non-steady nature of the chimney is visible in the scattering of the results after point \"3\" in terms of both $p^*$ and $q_k^*$. \n\nIn order to compare the numerical and the analytical results in more details the difference between the pressure fields is shown in figure \\ref{new_figures:pressure_map}. The difference is normalized by the analytical pressure at the inlet.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=10cm]{pressure_map.png} \n \\caption{Relative error between numerical and analytical pressure within the entire sample. $(a)$ Expansion regime, $q^*$ = 0.00095. $(a)$ Cavity regime, $q^*$ = 0.00116.}\n \\label{new_figures:pressure_map}\n\\end{figure}\n\nFigure \\ref{new_figures:pressure_map} evidences that the error between the pressure obtained numerically and the analytical solution is especially high in the vicinity of the inlet, where error reaches values up to 30$\\%$. However, error rapidly decreases far from the injection point where the average error is usually lower than 10$\\%$ as permeabilities do not significantly change. Moreover, error increases within the entire specimen when fluidization begins, as we can see in plot $(b)$ of figure \\ref{new_figures:pressure_map}, though the error is still low and acceptable.\n\nThe shape of the fluidized zone resulting from the analytical and the numerical simulations can be compared through the map of effective stress. The analytical fluidized zone is defined by the null-pressure isoline, in the simulation we retain the $\\sigma'^{*} = 0.1$ isoline as discussed previously. Figure \\ref{new_figures:effective_comparison} shows that the isolines are relatively similar in the undisturbed regions on both sides of the sample. On the contrary, they differ significantly above the inlet. As a matter of fact, a chimney-shaped fluidized zone appears in the analytical solution in the middle image ($q^* = 1.14\\times10^{-3}$) while at the same flow rate the cavity only starts developing in the simulation. The chimney regime is attained for a value of $q^* = 1.20 \\times10^{-3}$ in the numerical simulation, in this situation, $\\sigma'^{*} < 0.1$ is found in every point located vertically between the free surface and the inlet.\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=17cm]{effective_comparison.png} \n \\caption{Evolution of the analytical and numerical effective stress fields for different flow rates (injection aperture $a^*$ = 0.1). $(a)$ $q^*$ = 0.00105: cavity regime; $(b)$ $q^*$ = 0.00114: chimney regime in the theoretical model; cavity regime in the numerical model. $(c)$ $q^*$ = 0.00120: chimney regime.}\n \\label{new_figures:effective_comparison}\n\\end{figure*}\n\n\n\n\n\n\\section{Sensitivity analysis}\\label{Sensitivity_analysis}\n\nProvided that the normalized pressure ($p^*$) and flux ($q^*$) given in Eq. \\ref{eq:Ec6} and \\ref{eq:Ec4} are relevant dimensionless variables, there should be a unique relationship between them independently of the mean particle size and fluid viscosity. This is verified in this section. Figure \\ref{new_figures:sensitivity_curves} shows the pressure-flux relations for a monotonously increasing flux and $a^*$ = 0.1, in terms of both the physical units ($P-Q$) and the dimensionless quantities ($p^*$-$q^*$). For fluid viscosity ranging from $\\mu = 10^{-3}$ to $2 \\cdot 10^{-2} Pa \\cdot s$, and mean particle size ranging from $d$ = 1.66 to 2.48 $cm$. The pressure-flux results are collapsed in one single curve when expressed with $p^*$ and $q^*$, which confirms the relevance of the chosen variables and validates the proposed dimensionless numbers. Besides, the dimensional plots (Figure \\ref{new_figures:sensitivity_curves}, diagrams on the right) all show a marked peak at approximately $P=3400Pa$ ($p^*=1.05$) regardless of the flow rate. This value corresponds to pressure gradients balancing gravitational and frictional forces. This suggests that fluid pressure, instead of flow rate, is the most natural parameter for defining fluidization criteria.\n\nNevertheless, the $p^*$-$q^*$ relation is not absolutely unique as it depends on the remaining dimensionless variable: normalized aperture of the injection area. This dependency is shown in Figure \\ref{new_figures:dimensionless_aperture_curves} for $a^*$ ranging from 0.02 to 1 ($a^*=1$ means uniform influx through the surface of an infinite half-space).\n\n\\begin{figure*}[ht]\n \\centering\n \\includegraphics[width=16cm]{sensitivity_curves.png} \n \\caption{Sensitivity analysis on pressure-flux curves with injection aperture $a^*$ = 0.1. Viscosity dependency with normalized $p^*$-$q^*$ variables $(a)$ and physical units $(b)$; diameter dependency with normalized $p^*$-$q^*$ variables $(c)$ and physical units $(d)$.}\n \\label{new_figures:sensitivity_curves}\n\\end{figure*}\n\n\nA clear pattern can be identified in the figure \\ref{new_figures:dimensionless_aperture_curves}. Large apertures induce liquefaction over all the granular medium when the pressure peak attains a value close to $p^* = 1$, the expected value for homogeneous fluidization.\n\nOn the other hand, the excess of pore pressure required to reach fluidization is larger for small apertures ($p^* \\approx 1.1$). The reason why $p^*$ exceeds unity with small apertures is that the mobility of the column above the injection zone is constrained not only by gravity (gravity alone would lead to $p^*=1$ as an upper bound) but also by interactions with particles on both sides of the column. The pore pressure required to develop a cavity and a chimney of fluidization in the specimen must then exceed the sum of weight and a downward force coming from contact interactions between the stable mass and the mobile particles. As aperture becomes larger the mass of stable particles is progressively reduced and it eventually disappears in homogeneous fluidization ($a^*\\rightarrow 1$), hence the additional downward force vanishes and no peak is observed. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=10cm]{dimensionless_aperture_curves.png} \n \\caption{Aperture dependency. $p^*$-$q^*$ curves at the bottom of the sample with different injection apertures.}\n \\label{new_figures:dimensionless_aperture_curves}\n\\end{figure}\n\nThe aperture dependency is summarized in the diagram of figure \\ref{new_figures:DEM_diagram} where the different fluidization regimes are identified for particular combinations of simulated flow rate and aperture. As mentioned before and confirmed in figure \\ref{new_figures:DEM_diagram}, large apertures ($a^*$ close to 1) lead to the entire liquefaction of the granular assembly rather than forming a cavity and a chimney of fluidization (cavity regime gets narrower near $a^*=1$.).\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=9cm]{DEM_diagram.png} \n \\caption{Occurence of the different fluidization regimes (static\/expansion, cavity or chimney) depending on aperture $a^*$.}\n \\label{new_figures:DEM_diagram}\n\\end{figure}\n\nQualitatively, similar tendencies are predicted by Eq. \\ref{eq:Ec3.2} of the theoretical model (see also Figure \\ref{new_figures:effective_field}). The regimes deduced in this way are presented in figure \\ref{new_figures:analytical_diagram}. Nevertheless, three significant differences are found. Firstly, the transitions between the different regimes are shifted to lower fluxes compared to the simulations since the increase of hydraulic conductivity induced by the injection is not accounted for in Eq. \\ref{eq:Ec3.2}, which thus overestimates the actual pore pressure for a given flux and anticipate the fluidization.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=9cm]{analytical_diagram.png} \n \\caption{Theoretical prediction of the different regimes (static\/expansion, cavity or chimney regimes) in the dimensionless aperture-flux, $a^*$-$q^*$, plane.}\n \\label{new_figures:analytical_diagram}\n\\end{figure}\n\nSecondly, the line separating the expansion and the cavity regimes for narrow apertures is nearly vertical in the former diagram while it is getting horizontal when the aperture decreases. This is due to the divergence of the theoretical inlet pressure when $a^*\\rightarrow 0$, which results in a finite sized cavity for every non-zero influx in this limit. It does not occur in the numerical simulations because the spatial discretization of the flow problem introduces a lower bound for the range of effective aperture: when $a^*$ is of the order of the average distance between two solid particles further decrease in $a^*$ means no change in the solution. Likewise, we would not expect any difference if the injection was done with syringes using needles of different sizes, as soon as the needles are smaller than the pores of the material.\n\nFinally, the transition between the cavity and the chimney regimes has a \"D\" shape in the simulation (maximum $q^*$ values for $a^* = 0.5$ - $0.6$, and $q^*$ values decrease when narrower or wider apertures are considered) whereas it is approximately a straight line theoretically. The shift toward lower $q^*$ values for wide apertures ($a^*$ near 1) is expected because no interaction remains between fluidized and non-fluidized zones, then fluidization is reached for lower flux. As a matter of fact, this phenomenon can be observed in figure \\ref{new_figures:dimensionless_aperture_curves}. In $a^*=0.80$, $a^*=0.90 $ and $a^*=1.00 $ cases, liquefaction starts when the plateau is reached for $q^* \\approx 1.5 \\times10^{-3}$. In $a^*=0.50$ and $a^*=0.60$ cases, chimney of fluidization is attained for slightly larger $q^* \\approx 1.65 \\times10^{-3}$ (after this point pressure values oscillate as a consequence of the bubbling effect).\n\n\\section{Conclusions}\nIn this work, fluidization of a granular bed through an injection orifice has been investigated numerically using DEM-PFV simulations and analytically using a simplified continuum model.\n\nQualitatively the DEM simulations were found to compare well with available data in terms of the different fluidization regimes. To some extent a reasonable quantitative agreement was also found, without introducing any fitting parameter, although some differences remain. The origin of these differences is unclear as long as no published data include pore pressure measurements. \n\nThe theoretical model was assuming a constant hydraulic conductivity in space and time and the effective stress was used for defining a fluidization criterion. It could approximate the DEM solution quite well at low fluxes. It deviates from the DEM solution at larger fluxes when expansion of the bed and localized fluidization leads to significant changes of the conductivity in space and time. Nevertheless, the analytical model predicts the transitions between the different regimes and it provides a simple framework to explain the main trends. This theoretical approach may also shed light on the interactions between two or more adjacent chimneys in the case of multiple injection points \\cite{ngomainteraction,philippe2013localized}. \n\nThe numerical model allowed to describe the internal configuration by means of the effective stress and the porosity fields within the granular medium. This approach enabled to define accurately the cavity and chimney shape. Besides, compression-decompression cycles were evidenced once the chimney regime was reached.\n\nThe size of the injection area was found to determine whether or not a chimney regime exists for certain injection rates. Small injection areas lead to an early regime of fluidization with respect to the injected flux. In this case, gradual increase of the injection rate results in a peak of the inlet pressure at the transition between the cavity regime and the chimney regime. On the contrary, large apertures produce liquefaction for larger injection rates, the chimney is wider and difficult to identify due to liquefaction occurring over most of the sample instead of a localized zone. Consistently, the peak in the pressure-flux curves tend to disappear and the evolution of pressure increases monotonically as a function of the injection rate until it reaches a plateau. The domains of occurrence of the different regimes have been defined in terms of dimensionless aperture and dimensionless flux.\n\nOur analysis suggests that the inlet pressure is the primary variable controlling fluidization more directly than the injection rate. The published data reports the injection rate only, hence the difficulty in comparing experiments and simulations.\n\nOur suggestion for future experiments is to measure the inlet pressure, ideally. Alternatively, an independent measurement of the permeability would enable the calculation of inlet pressure at least for the initial stages before any significant change of the local porosity. \n\n\\section{Acknowledgement}\nThis work has been partially supported by the LabEx Tec 21 (Investissements d'Avenir - grant agreement $n^o$ ANR-11-LABX-0030) \n\n\n\\newpage\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}