diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfqyk" "b/data_all_eng_slimpj/shuffled/split2/finalzzfqyk"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfqyk"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\nThe continuity of the spectra for families of self-adjoint operators in a Hilbert space\nhas been considered for several decades, but many natural questions have only received partial answers yet.\nIn this paper we consider a fairly general family of\nmagnetic Schr\\\"odinger operators acting on $\\mathbb Z^d$ and exhibit some continuity properties of\nthe spectra under suitable modifications of the magnetic fields and of the symbols defining the operators.\nIn rough terms, the continuity we are dealing with corresponds to the stability of the spectral gaps\nas well as the stability of the spectral compounds. In a more precise terminology\nwe shall prove inner and outer continuity for the family of spectra, as defined below.\n\nIn the discrete setting, the Harper operator is certainly the preeminent example and much efforts\nhave been dedicated to its study and to generalizations of this model.\nIt is certainly impossible to mention all papers dealing with continuity properties of families of such operators, but let us\ncite a few of them which are relevant for our investigations.\nFirst of all, let us mention the seminal paper \\cite{Bel} in which the author proves the Lipschitz continuity of gap boundaries with respect to the variation\nof a constant magnetic field for a family of pseudodifferential operators acting on $\\mathbb Z^2$.\nIn \\cite{Kot} and based on the framework introduced in \\cite{Sun}, similar Lipschitz continuity is proved for\nself-adjoint operators acting on a crystal lattice, a natural generalization of $\\mathbb Z^d$.\nNote that in these two references a $C^*$-algebraic framework is used, as we shall do it later on.\nOn the other hand, papers \\cite{Nen} and \\cite{Cor} deal with families of magnetic pseudodifferential operators on $\\mathbb Z^2$ only but\ncontinuity results are shown for more general symbols and magnetic fields.\n\nBefore introducing the precise framework of our investigations, let us still mention two additional papers\nwhich are at the root of our work: \\cite{MPR} in which a general framework for magnetic systems,\ninvolving twisted crossed product $C^*$-algebras, is introduced and \\cite{AMP} which contains results similar to ours but in a continuous setting.\n\nIn the Hilbert space $\\mathcal H:=l^2(\\mathbb Z^d)$ and for some fixed parameter $\\epsilon$ let us consider operators of the form\n\\begin{equation}\\label{eq_def_H}\n[H^\\epsilon u](x):=\\sum_{y\\in \\mathbb Z^d} h^\\epsilon(x;y-x)\\mathop{\\mathrm{e}}\\nolimits^{i\\phi^\\epsilon(x,y)}u(y)\n\\end{equation}\nwith $u\\in \\mathcal H$ of finite support, $x\\in \\mathbb Z^d$ and where $h^\\epsilon: \\mathbb Z^d\\times \\mathbb Z^d\\to \\mathbb C$ and $\\phi^\\epsilon:\\mathbb Z^d\\times \\mathbb Z^d \\to \\mathbb R$ satisfy\n\\begin{enumerate}\n\\item[(i)] $\\sum_{x\\in \\mathbb Z^d}\\sup_{q\\in \\mathbb Z^d}|h^\\epsilon(q;x)|<\\infty$,\n\\item[(ii)] $\\overline{h^\\epsilon(q+x;-x)}=h^\\epsilon(q;x)$ for any $q,x\\in \\mathbb Z^d$,\n\\item[(iii)] $\\phi^\\epsilon(x,y)=-\\phi^\\epsilon(y,x)$ for all $x,y\\in \\mathbb Z^d$.\n\\end{enumerate}\nSuch operators are usually called \\emph{discrete magnetic Schr\\\"odinger operators}.\nNote that condition (i) ensures that $H^\\epsilon$ extends continuously to a bounded operator in $\\mathcal H$, while conditions (ii) and (iii)\nimply that the corresponding operator is self-adjoint. In the sequel a map $\\phi:\\mathbb Z^d\\times\\mathbb Z^d\\to \\mathbb R$ satisfying $\\phi(x,y)=-\\phi(y,x)$ for any $x,y\\in \\mathbb Z^d$\nwill simply be called a \\emph{magnetic potential}.\n\nLet us consider a compact Hausdorff space $\\Omega$ and assume that $\\epsilon \\in \\Omega$.\nA natural question in this setting is the following: Under which regularity conditions on the maps $\\epsilon\\mapsto h^\\epsilon$\nand $\\epsilon \\mapsto \\phi^\\epsilon$ can one get some continuity for the spectra of the family of operators $\\{H^\\epsilon\\}_{\\epsilon\\in \\Omega}$,\nand what kind of continuity can one expect on these sets ?\nAs already mentioned above, we shall consider the notion of inner and outer continuity, borrowed from \\cite{AMP} but originally inspired by \\cite{Bel}.\n\n\\begin{Definition}\nLet $\\Omega$ be a compact Hausdorff space, and let $\\{\\sigma_\\epsilon\\}_{\\epsilon\\in\\Omega}$ be a family of closed subsets of $\\mathbb R$.\n\\begin{enumerate}\n\\item The family $\\{\\sigma_\\epsilon\\}_{\\epsilon\\in \\Omega}$ is \\emph{outer continuous at $\\epsilon_0\\in \\Omega$} if for any compact subset $\\mathcal K$ of $\\mathbb R$\nsuch that $\\mathcal K\\cap \\sigma_{\\epsilon_0}=\\emptyset$ there exists a neighbourhood $\\mathcal N=\\mathcal N(\\mathcal K,\\epsilon_0)$ of $\\epsilon_0$ in $\\Omega$ such that\n$\\mathcal K\\cap \\sigma_{\\epsilon}=\\emptyset$ for any $\\epsilon\\in \\mathcal N$,\n\\item The family $\\{\\sigma_\\epsilon\\}_{\\epsilon\\in \\Omega}$ is \\emph{inner continuous at $\\epsilon_0\\in \\Omega$} if for any open subset $\\mathcal O$ of $\\mathbb R$\nsuch that $\\mathcal O\\cap \\sigma_{\\epsilon_0}\\neq \\emptyset$ there exists a neighbourhood $\\mathcal N=\\mathcal N(\\mathcal O,\\epsilon_0)$ of $\\epsilon_0$ in $\\Omega$ such that\n$\\mathcal O\\cap \\sigma_{\\epsilon}\\neq\\emptyset$ for any $\\epsilon\\in \\mathcal N$.\n\\end{enumerate}\n\\end{Definition}\n\nLet us now present a special case of our main result which will be stated in Theorem \\ref{thm_main}.\nThe following statement is inspired from \\cite{Nen} and a comparison with the existing literature will be established just afterwards.\n\n\\begin{Theorem}\\label{thm_Nenciu}\nFor each $\\epsilon \\in \\Omega:=[0,1]$ let $h^\\epsilon: \\mathbb Z^d\\times \\mathbb Z^d\\to \\mathbb C$ satisfy the above conditions (i) and (ii).\nAssume that the family $\\{h^\\epsilon\\}_{\\epsilon\\in \\Omega}$ satisfies for any $y\\in \\mathbb Z^d$ the condition\n$$\n\\lim_{\\epsilon'\\to \\epsilon}\\sup_{q\\in \\mathbb Z^d}|h^{\\epsilon'}(q;y)-h^\\epsilon(q;y)|=0\n$$\nand $|h^\\epsilon(q;y)|\\leq f(y)$ for some $f\\in l^1(\\mathbb Z^d)$, all $q\\in \\mathbb Z^d$ and all $\\epsilon \\in \\Omega$.\nLet also $\\phi$ be a magnetic potential which satisfies\n$$\n\\big|\\phi(x,y)+\\phi(y,z)+\\phi(z,x)\\big| \\leq \\hbox{ area } \\triangle (x,y,z),\n$$\nwhere $\\triangle (x,y,z)$ means the triangle in $\\mathbb R^d$ determined by the three points $x,y,z\\in \\mathbb Z^d$.\nThen for $H^\\epsilon$ defined on $u\\in \\mathcal H$ by\n$$\n[H^\\epsilon u](x):=\\sum_{y\\in \\mathbb Z^d} h^\\epsilon(x;y-x)\\mathop{\\mathrm{e}}\\nolimits^{i\\epsilon\\phi(x,y)}u(y)\n$$\nthe family of spectra $\\sigma(H^\\epsilon)$ forms an outer and an inner continuous family at every points $\\epsilon\\in \\Omega$.\n\\end{Theorem}\n\nObserve that if one considers a function $h\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ independent of $\\epsilon$ and which satisfies $\\overline{h(q+x;-x)}=h(q;x)$ for any $q,x\\in \\mathbb Z^d$,\nthen the various assumptions on the family $\\{h^\\epsilon\\}_{\\epsilon \\in \\Omega}$ are easily checked.\nIn \\cite{Nen} the case $d=2$ is considered for a fixed symbol $h$ satisfying a decay of the form $\\sup_{q\\in \\mathbb Z^d}|h(q;x)|\\leq C\\mathop{\\mathrm{e}}\\nolimits^{-\\beta|x|}$, where $0<\\beta\\leq 1$ and $|x|$ denotes the Euclidean norm in $\\mathbb Z^2$.\nIn this framework, stronger continuity properties of the family of spectra are obtained, but these results deeply depend on the parameter $\\beta$.\nOn the other hand our results are somewhat weaker but hold for a much more general class of symbols. In addition, more general $\\epsilon$-dependent magnetic potentials\nare considered in our main result.\n\nLet us now emphasize that the framework presented in Section \\ref{sec_field} does not allow us to get any quantitative estimate,\nas emphasized in the recent paper \\cite{BB}. Indeed, the very weak continuity requirement we impose on the $\\epsilon$-dependence on our objets\ncan not lead to any Lipschitz or H\\\"older continuity. More stringent assumptions are necessary for that purpose, and such estimates\ncertainly deserve further investigations.\n\nOur approach relies on the concepts of twisted crossed product $C^*$-algebras and on a field of such algebras, mainly borrowed from \\cite{Rie,Zel}.\nIn the discrete setting, such algebras have already been used, for example in \\cite{Bel,Kot,Sun}. However, instead of considering\na $2$-cocycle with scalar values, which is sufficient for the case of a constant magnetic field, our $2$-cocycles take values\nin the group of unitary elements of $l^\\infty(\\mathbb Z^d)$. This allows us to consider arbitrary magnetic potential on $\\mathbb Z^d$ and to encompass\nall the corresponding operators in a single algebra.\n\nLet us finally describe the content of this paper. In Section \\ref{sec_magn_alg} we introduce the framework for a single magnetic system,\n\\ie~for a fixed $\\epsilon$. For that reason, no $\\epsilon$-dependence is indicated in this section. In Section \\ref{sec_field} the $\\epsilon$-dependence\nis introduced and the continuous dependence on this parameter is studied. Our main result is presented in Theorem \\ref{thm_main}.\nIn the last section, we provide the proof of Theorem \\ref{thm_Nenciu}.\n\n\\section{Discrete magnetic systems}\\label{sec_magn_alg}\n\\setcounter{equation}{0}\n\nThis section is divided into three parts. First of all, we motivate the introduction of the algebraic formalism by showing\nthat any magnetic potential leads naturally to the notion of a normalized $2$-cocycle with one additional property.\nBased on this observation, we introduce in the second part of the section a special instance of a twisted crossed product $C^*$-algebra.\nA faithful representation of this algebra in $l^2(\\mathbb Z^d)$ is also provided.\nIn the third part, we draw the connections of this abstract construction with the initial magnetic system.\n\n\\subsection{From magnetic potentials to $2$-cocycles}\n\nWe start by recalling that a magnetic potential consists in a map $\\phi:\\mathbb Z^d\\times \\mathbb Z^d\\to \\mathbb R$ satisfying for any $x,y\\in \\mathbb Z^d$ the relation\n\\begin{equation}\\label{eq_phi}\n\\phi(x,y)=-\\phi(y,x).\n\\end{equation}\nThen, given such a magnetic potential $\\phi$ let us introduce and study a new map\n$$\n\\omega: \\mathbb Z^d\\times \\mathbb Z^d \\times \\mathbb Z^d\\to \\mathbb T\n$$\ndefined for $q,x,y\\in \\mathbb Z^d$ by\n\\begin{equation}\\label{def_omega}\n\\omega(q;x,y):=\\exp\\big\\{i\\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\}.\n\\end{equation}\nNote that the distinction between the variable $q$ and the variables $x$ and $y$ is done on purpose.\nIndeed, for fixed $x,y\\in \\mathbb Z^d$ we shall also use the notation $\\omega(x,y)$ for the map\n\\begin{equation*}\n\\omega(x,y):\\mathbb Z^d\\ni q \\mapsto [\\omega(x,y)](q):=\\omega(q;x,y) \\in  \\mathbb T.\n\\end{equation*}\n\nSince $\\mathbb Z^d$ acts on itself by translations, let us introduce the action $\\theta$ of $\\mathbb Z^d$ on any $f\\in l^\\infty(\\mathbb Z^d)$ by\n\\begin{equation}\\label{def_theta}\n\\theta_xf(y)=f(x+y).\n\\end{equation}\nIn particular, since $\\omega(x,y)\\in l^\\infty(\\mathbb Z^d)$ we have\n\\begin{equation*}\n\\big[\\theta_z\\omega(x,y)\\big](q):=[\\omega(x,y)](q+z)= \\omega(q+z;x,y).\n\\end{equation*}\nBased on these definitions, the following properties for $\\omega$ can now be proved:\n\n\\begin{Lemma}\\label{lem_2_cocycle}\nLet $\\phi$ be a magnetic potential and let $\\omega$ defined by \\eqref{def_omega}.\nThen for any $x,y,z\\in \\mathbb Z^d$ the following properties hold:\n\\begin{enumerate}\n\\item[(i)] $\\omega(x+y,z)\\;\\omega(x,y)  = \\theta_x\\omega(y,z)\\;\\omega(x,y+z)$,\n\\item[(ii)] $\\omega(x,0)=\\omega(0,x)=1$,\n\\item[(iii)] $\\omega(x,-x)=1$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nThe proof consists only in simple computations. Indeed by taking \\eqref{eq_phi} into account one gets that for any $q,x,y,z\\in \\mathbb Z^d$\n\\begin{align*}\n&[\\omega(x+y,z)](q)\\;[\\omega(x,y)](q) \\\\\n& =\\omega(q;x+y,z)\\;\\omega(q;x,y) \\\\\n& = \\exp\\big\\{i\\big[\\phi(q,q+x+y)+\\phi(q+x+y,q+x+y+z)+\\phi(q+x+y+z,q)\\big]\\big\\}  \\\\\n& \\quad \\ \\exp\\big\\{i\\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\} \\\\\n& =  \\exp\\big\\{i\\big[\\phi(q+x,q+x+y)+\\phi(q+x+y,q+x+y+z)+\\phi(q+x+y+z,q+x)\\big]\\big\\}  \\\\\n& \\quad \\ \\exp\\big\\{i\\big[\\phi(q,q+x)+\\phi(q+x,q+x+y+z)+\\phi(q+x+y+z,q)\\big]\\big\\} \\\\\n& =\\omega(q+x;y,z)\\;\\omega(q;x,y+z) \\\\\n& = [\\theta_x\\omega(y,z)](q)\\;[\\omega(x,y+z)](q)\n\\end{align*}\nwhich proves (i). Similar computations lead to (ii) and (iii) once\nthe equality $\\phi(x,x)=0$ for any $x\\in \\mathbb Z^d$ is taken into account.\n\\end{proof}\n\nLet us now make some comments about the previous definitions and results.\nFor fixed $x,y$ the map $\\omega(x,y):\\mathbb Z^d\\to \\mathbb T$ can been seen as an element of the unitary group of the algebra $l^\\infty(\\mathbb Z^d)$.\nFor simplicity, we set $\\mathscr U(\\mathbb Z^d)$ for this unitary group, \\ie\n$$\n\\mathscr U(\\mathbb Z^d)=\\{f:\\mathbb Z^d\\to \\mathbb T\\}.\n$$\nIn addition, property (i) of the previous lemma is usually considered as a \\emph{$2$-cocycle property}\nwhile property (ii) corresponds to a normalization of this $2$-cocycle.\nIn the second part of this section, we shall come back to these definitions.\nFor the time being, let us just mention that this $2$-cocycle will be at the root of the definition of a\ntwisted crossed product $C^*$-algebra.\nHowever, before recalling the details of this construction, let us still show that $\\omega$ depends only\non equivalent classes of magnetic potentials.\n\n\\begin{Lemma}\\label{lem_equi_class}\nLet $\\phi$ be a magnetic potential and let $\\varphi:\\mathbb Z^d\\to \\mathbb R$. Then the map $\\phi':\\mathbb Z^d\\times \\mathbb Z^d\\to \\mathbb R$\ndefined by\n$$\n\\phi'(x,y)=\\phi(x,y)+\\varphi(y)-\\varphi(x).\n$$\nis a magnetic potential. In addition, by formula \\eqref{def_omega} the two magnetic potentials $\\phi$ and $\\phi'$\ndefine the same $2$-cocycle.\n\\end{Lemma}\n\n\\begin{proof}\nClearly, $\\phi'(x,y)=-\\phi'(y,x)$ which means that $\\phi'$ is a magnetic potential.\nIf we denote by $\\omega$ (resp. $\\omega'$) the $2$-cocycle defined by \\eqref{def_omega} for the magnetic potential $\\phi$ (resp. $\\phi'$) we get\n\\begin{align*}\n\\omega'(q;x,y)&:=\\exp\\big\\{i\\big[\\phi'(q,q+x)+\\phi'(q+x,q+x+y)+\\phi'(q+x+y,q)\\big]\\big\\} \\\\\n& = \\exp\\big\\{i\\big[\\phi(q,q+x)+\\varphi(q+x)-\\varphi(q)+\\phi(q+x,q+x+y)+ \\varphi(q+x+y)-\\varphi(q+x) \\\\\n& \\quad \\ +\\phi(q+x+y,q)+\\varphi(q)-\\varphi(q+x+y)\\big]\\big\\} \\\\\n&  = \\exp\\big\\{i\\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\} \\\\\n& = \\omega(q;x,y).\\qedhere\n\\end{align*}\n\\end{proof}\n\nOne could argue that the $2$-cocycle $\\omega$ depends only\non the \\emph{magnetic field} as introduced in \\cite{CTT}, and not on the choice of a magnetic potential. However, this would lead us too far from our purpose\nsince we would have to consider $\\mathbb Z^d$ as a graph endowed with edges between every pair of vertices.\n\n\\subsection{Twisted crossed product algebras and their representations}\\label{subsec_abs}\n\nLet us adopt a very pragmatic point of view and recall only the strictly necessary information on twisted\ncrossed product $C^*$-algebras. More can be found in the fundamental papers \\cite{PR1,PR2} or in the review paper \\cite{MPR}.\nSince the group we are dealing with is simply $\\mathbb Z^d$, most of the necessary information can also be found in \\cite{Zel}.\n\nConsider the group $\\mathbb Z^d$ and the algebra $l^\\infty(\\mathbb Z^d)$ endowed with the action $\\theta$ of $\\mathbb Z^d$ by translations, as defined in \\eqref{def_theta}.\nAs suggested by the notation, the vector space $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ is endowed with the following norm\n\\begin{equation}\\label{eq_norm}\n\\|f\\|_{1,\\infty}:=\\sum_{x\\in \\mathbb Z^d}\\sup_{q\\in \\mathbb Z^d}|f(q;x)| \\qquad f\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big),\n\\end{equation}\nwhere $x$ is the variable in the $l^1$-part and $q$ is the variable in the $l^\\infty$-part.\nThis set also admits an action of $\\mathbb Z^d$ defined for any $f\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ by\n$$\n[\\theta_y f(x)](q):=[f(\\cdot +y;x)](q) = f(q+y;x).\n$$\n\nIn order to endow $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ with a twisted product,\nlet $\\omega$ be any normalized $2$-cocycle on $\\mathbb Z^d$ with values in the unitary group of $l^\\infty(\\mathbb Z^d)$,\nor in other words let $\\omega:\\mathbb Z^d\\times \\mathbb Z^d\\to \\mathscr U(\\mathbb Z^d)$ satisfy for any $x,y,z\\in \\mathbb Z^d$:\n\\begin{equation}\\label{eq_2}\n\\omega(x+y,z)\\;\\omega(x,y) = \\theta_x\\omega(y,z)\\;\\omega(x,y+z)\n\\end{equation}\nand\n\\begin{equation}\\label{eq_n}\n\\omega(x,0)=\\omega(0,x)=1.\n\\end{equation}\nBecause of the point (iii) of Lemma \\ref{lem_2_cocycle}, we shall also assume that the $2$-cocycle $\\omega$\nsatisfies an additional property, namely for any $x\\in \\mathbb Z^d$:\n\\begin{equation}\\label{eq_add}\n\\omega(x,-x)=1.\n\\end{equation}\n\nWe can now define the twisted product and an involution: for any $f,g\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$\none sets\n\\begin{equation}\\label{eq_produit}\n[f\\diamond g](x):=\\sum_{y\\in \\mathbb Z^d} f(y)\\; \\theta_y g(x-y) \\;\\omega(y,x-y)\n\\end{equation}\nand\n\\begin{equation}\\label{eq_involution}\nf^{\\diamond}(x)= [\\theta_x f(-x)]^*=\\overline{f(\\cdot+x;-x)}.\n\\end{equation}\nBoth operations are continuous with respect to the norm introduced in \\eqref{eq_norm}.\n\nThe enveloping $C^*$-algebra of $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$, endowed with the above product and involution,\nwill be denoted by $\\mathfrak C(\\omega)\\equiv\\mathfrak C$. Recall that this algebra corresponds to the completion of $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$\nwith respect to the $C^*$-norm defined as the supremum over all the faithful representations of $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$.\nAs a consequence, $l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$\nis dense in $\\mathfrak C$ and the new $C^*$-norm $\\|\\cdot\\|$ satisfies $\\|f\\|\\leq \\|f\\|_{1,\\infty}$.\n\n\\begin{Remark}\nIn \\cite{MPR} an additional ingredient is introduced in the previous construction, namely an endomorphism $\\tau$ of $\\mathbb Z^d$.\nIn the continuous case, when $\\mathbb Z^d$ is replaced by $\\mathbb R^d$, this additional degree of freedom allows one\nto encompass in a single framework the formulas for the Weyl quantization and for the Kohn-Nirenberg quantization.\nIn the discrete setting, we stick to the case $\\tau=0$ since the other choices\ndo not seem to be relevant.\n\\end{Remark}\n\nLet us now look at a faithful representation of the algebra $\\mathfrak C$ in the Hilbert space $\\mathcal H=l^2(\\mathbb Z^d)$.\nFirst of all, by \\cite[Lem.~2.9]{MPR} there always exists a \\emph{$1$-cochain} $\\lambda$, \\ie~a map $\\lambda: \\mathbb Z^d\\to \\mathscr U(\\mathbb Z^d)$, such that\n\\begin{equation}\\label{eq_1_cochain}\n\\lambda(x)\\; \\theta_x\\lambda(y)\\; \\lambda(x+y)^{-1} = \\omega(x,y).\n\\end{equation}\nIn fact, an example of such a $1$-cochain can be defined by the following formula:\n\\begin{equation}\\label{eq_lambda_t}\n\\lambda_t(q;x)\\equiv[\\lambda_t(x)](q):=\\omega(0;q,x).\n\\end{equation}\nIndeed, it easily follows from the $2$-cocycle property \\eqref{eq_2} that\n\\begin{align*}\n\\lambda_t(q;x)\\;\\!\\lambda_t(q+x;y)\\;\\!\\lambda_t(q;x+y)^{-1}\n& = \\omega(0;q,x)\\;\\!\\omega(0;q+x,y)\\;\\! \\omega(0;q,x+y)^{-1} \\\\\n&= \\theta_q\\omega(0;x,y)\\\\\n& = \\omega(q;x,y).\n\\end{align*}\nNote that in the continuous case this choice corresponds to the transversal gauge for the magnetic potential, and this is why the index $t$ has been added.\n\nSince the $2$-cocycle $\\omega$ has been chosen normalized and with the additional property \\eqref{eq_add}, the\n$1$-cochains satisfying \\eqref{eq_1_cochain} also share some additional properties, namely:\n\n\\begin{Lemma}\\label{lem_sur_lambda}\nLet $\\lambda$ be a $1$-cochain satisfying \\eqref{eq_1_cochain} for $\\omega$ satisfying \\eqref{eq_2}-\\eqref{eq_add}. Then,\n\\begin{enumerate}\n\\item[(i)] $\\lambda(q;0)=1$ for any $q\\in \\mathbb Z^d$,\n\\item[(ii)] $\\lambda (y;x-y)=\\lambda(x;y-x)^{-1}$ for any $x,y\\in \\mathbb Z^d$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{proof}\nOne infers from \\eqref{eq_1_cochain} for $y=0$ and from \\eqref{eq_n} that\n$$\n\\lambda(q;x)\\; \\lambda(q+x;0)\\; \\lambda(q;x)^{-1} = \\lambda(q+x;0) = \\omega(q;x,0)=1.\n$$\nSimilarly, from \\eqref{eq_1_cochain} for $y=-x$ and from \\eqref{eq_add} one gets that\n\\begin{equation*}\n\\lambda(x)\\;\\!\\theta_x\\lambda(-x)\\;\\!\\lambda(0)=\\omega(x,-x)=1,\n\\end{equation*}\nfrom which one deduces that $\\lambda(q;x) = \\lambda(q+x;-x)^{-1}$. Finally, by replacing $q$ by $y$ and $x$ by $x-y$ in the previous equality\none deduces the statement.\n\\end{proof}\n\nOnce a $1$-cochain satisfying \\eqref{eq_1_cochain} has been chosen, a representation of $\\mathfrak C$ in $\\mathcal H$ can be defined, as shown in  \\cite[Sec.~2.4]{MPR}.\nMore precisely, for any $h\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$, any $u\\in \\mathcal H$ and any $x\\in \\mathbb Z^d$ one sets\n\\begin{equation*}\n[{\\mathfrak{Rep}}^\\lambda(h)u](x) := \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\lambda(x;y-x) \\;\\!u(y).\n\\end{equation*}\nThe main properties of this representation are gathered in the following statement, which corresponds to \\cite[Prop.~2.16 \\& 2.17]{MPR}\nadapted to our setting. In (i) the operator $\\varphi(X)$ denotes the operator of multiplication by the function $\\varphi$.\n\n\\begin{Proposition}\\label{prop_MPR}\nLet $\\lambda$ and $\\lambda'$ be two $1$-cochains satisfying \\eqref{eq_1_cochain} for the same $\\omega$ that satisfies \\eqref{eq_2}-\\eqref{eq_add}. Then,\n\\begin{enumerate}\n\\item[(i)] There exists $\\varphi:\\mathbb Z^d\\to \\mathbb R$ such that\n$$\\lambda'(q;x)=\\mathop{\\mathrm{e}}\\nolimits^{i\\theta_x \\varphi(q)}\\mathop{\\mathrm{e}}\\nolimits^{-i\\varphi(q)}\\lambda(q;x) .$$\nIn addition one has for any  $h\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$\n$$\n{\\mathfrak{Rep}}^{\\lambda'}(h) = \\mathop{\\mathrm{e}}\\nolimits^{-i\\varphi(X)}\\;\\! {\\mathfrak{Rep}}^\\lambda(h)\\;\\!\\mathop{\\mathrm{e}}\\nolimits^{i\\varphi(X)},\n$$\n\\item[(ii)] The representation ${\\mathfrak{Rep}}^\\lambda$ is irreducible,\n\\item[(iii)] The representation ${\\mathfrak{Rep}}^\\lambda$ is faithful.\n\\end{enumerate}\n\\end{Proposition}\n\nLet us end this abstract part with a result about self-adjointness. The following statement shows that\nif $h\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ satisfies $h^\\diamond=h$, with the involution defined in \\eqref{eq_involution},\nthen the corresponding operator ${\\mathfrak{Rep}}^\\lambda(h)$ is self-adjoint.\n\n\\begin{Lemma}\nLet $\\lambda$ be any $1$-cochain satisfying \\eqref{eq_1_cochain} with $\\omega$ satisfying \\eqref{eq_2}-\\eqref{eq_add},\nand let $h\\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$. Then ${\\mathfrak{Rep}}^\\lambda(h)$ is self-adjoint if $h^\\diamond=h$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $u,v$ be elements of the Hilbert space $l^2(\\mathbb Z^d)$ with compact support, let $\\langle \\cdot,\\cdot\\rangle$ denote its scalar product\nand let $\\langle\\cdot,\\cdot\\rangle_\\mathbb C$ denote the scalar product in $\\mathbb C$.\nLet us also observe that with a simple change of variables the equality $h^\\diamond = h$ is equivalent to $h(y;x-y)=\\overline{h(x;y-x)}$.\nThen by taking Lemma \\ref{lem_sur_lambda}.(ii) into account one gets\n\\begin{align*}\n\\big\\langle v, {\\mathfrak{Rep}}^\\lambda(h)u\\big\\rangle\n= & \\sum_{x\\in \\mathbb Z^d} \\Big\\langle v(x), \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\lambda(x;y-x) \\;\\!u(y)\\Big\\rangle_\\mathbb C \\\\\n= &  \\sum_{y\\in \\mathbb Z^d}  \\Big\\langle  \\sum_{x\\in \\mathbb Z^d} \\overline{h(x;y-x)} \\;\\!\\overline{\\lambda(x;y-x)}\\;\\!v(x), u(y)\\Big\\rangle_\\mathbb C \\\\\n= &  \\sum_{x\\in \\mathbb Z^d}  \\Big\\langle  \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\lambda(x;y-x)\\;\\!v(y), u(x)\\Big\\rangle_\\mathbb C \\\\\n= & \\big\\langle {\\mathfrak{Rep}}^\\lambda(h)v,u\\big\\rangle.\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\subsection{Back to magnetic systems}\n\nLet us now come back to a magnetic potential $\\phi$ and to the \\emph{magnetic $2$-cocycle} $\\omega$ defined by \\eqref{def_omega}.\nBy Lemma \\ref{lem_2_cocycle}, the three conditions \\eqref{eq_2}-\\eqref{eq_add} are satisfied for such a $2$-cocycle, and thus\nthe construction of Section \\ref{subsec_abs} is at hand. Let us thus list some relations between this abstract section and\nsome magnetic objects considered before.\n\nFirst of all, the relation between $\\lambda_t$ introduced in \\eqref{eq_lambda_t} and $\\phi$ can be explicitly computed, namely\n\\begin{align}\\label{eq_lambda_phi}\n\\nonumber \\lambda_t(q;x) & = \\omega(0;q,x) \\\\\n\\nonumber & = \\exp\\big\\{i\\big[\\phi(0,q)+\\phi(q,q+x)+\\phi(q+x,0)\\big]\\big\\} \\\\\n\\nonumber & = \\exp\\big\\{i\\big[\\phi(q,q+x)+ \\phi(q+x,0)-\\phi(q,0)\\big]\\big\\} \\\\\n& = \\exp\\big\\{i\\big[\\phi(q,q+x)+ \\varphi(q+x)-\\varphi(q)\\big]\\big\\}\n\\end{align}\nwith $\\varphi:\\mathbb Z^d\\to \\mathbb R$ defined by $\\varphi(x):=\\phi(x,0)$.\nOn the other hand, the obvious choice\n\\begin{equation}\\label{eq_obvious}\n\\lambda_\\phi(q;x):=\\mathop{\\mathrm{e}}\\nolimits^{i\\phi(q,q+x)}\n\\end{equation}\nis also a $1$-cochain satisfying \\eqref{eq_1_cochain}, as a consequence of \\eqref{def_omega} and $\\phi(x,y)=-\\phi(y,x)$.\n\nAt the level of the representations, for the $1$-cochain $\\lambda_\\phi$ one gets\n\\begin{equation}\\label{eq_Rep_phi}\n[{\\mathfrak{Rep}}^{\\lambda_\\phi}(h)u](x) = \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\mathop{\\mathrm{e}}\\nolimits^{i\\phi(x,y)} \\;\\!u(y).\n\\end{equation}\nClearly, this expression corresponds to the one provided in \\eqref{eq_def_H} which\nwas the starting point of our investigations.\nIt is precisely the equality of these two expressions which makes the algebraic formalism useful\nfor the study of magnetic operators.\n\nOn the other hand for the $1$-cochain $\\lambda_t$ and if \\eqref{eq_lambda_phi} is taken into account one obtains\n\\begin{align*}\n[{\\mathfrak{Rep}}^{\\lambda_t}(h)u](x) & = \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\exp\\{i\\phi(x,y) + \\varphi(y)-\\varphi(x)\\} \\;\\!u(y) \\\\\n& = \\mathop{\\mathrm{e}}\\nolimits^{-i\\varphi(x)} \\sum_{y\\in \\mathbb Z^d} h(x;y-x) \\;\\!\\mathop{\\mathrm{e}}\\nolimits^{i\\phi(x,y)}\\;\\!\\mathop{\\mathrm{e}}\\nolimits^{i\\varphi(y)} \\;\\!u(y) \\\\\n& = \\big[\\mathop{\\mathrm{e}}\\nolimits^{-i\\varphi(X)} {\\mathfrak{Rep}}^{\\lambda_\\phi}(h) \\mathop{\\mathrm{e}}\\nolimits^{i\\varphi(X)}u\\big](x).\n\\end{align*}\nThese equalities mean that the representations provided by ${\\mathfrak{Rep}}^{\\lambda_\\phi}$ and ${\\mathfrak{Rep}}^{\\lambda_t}$ are unitarily equivalent,\nas it could already be inferred from Proposition \\ref{prop_MPR}.(i).\n\nIn summary, any magnetic potential defines a magnetic $2$-cocycle, and subsequently a twisted crossed product $C^*$-algebra\nwhich can be represented faithfully in $\\mathcal H$. This algebra depends on an equivalence class of magnetic potentials, as emphasized\nin Lemma \\ref{lem_equi_class}. Reciprocally, any normalized $2$-cocycle on $\\mathbb Z^d$ with values in $\\mathscr U(\\mathbb Z^d)$ and which satisfies\nthe additional relation \\eqref{eq_add} comes from a magnetic potential, as shown in the following lemma.\n\n\\begin{Lemma}\nLet $\\omega:\\mathbb Z^d\\times\\mathbb Z^d\\to \\mathscr U(\\mathbb Z^d)$ satisfy conditions \\eqref{eq_2}-\\eqref{eq_add}.\nThen there exists a magnetic potential which satisfies the relation \\eqref{def_omega}.\n\\end{Lemma}\n\n\\begin{proof}\nFirst of all, observe that the equality\n\\begin{equation}\\label{eq_rel_1}\n\\omega(x,y)=\\omega(x+y,-y)^{-1}\n\\end{equation}\nis a direct consequence of \\eqref{eq_2}-\\eqref{eq_add}\ntaking $z=-y$ in \\eqref{eq_2}.\n\nFor any $x,y\\in \\mathbb Z^d$ with $\\omega(0;x,y-x)\\neq -1$ let us set $\\phi(x,y)\\in (-\\pi,\\pi)$ by\n$$\n\\mathop{\\mathrm{e}}\\nolimits^{i \\phi(x,y)}:=\\omega(0;x,y-x).\n$$\nBy \\eqref{eq_rel_1} one infers that\n$$\n\\mathop{\\mathrm{e}}\\nolimits^{i\\phi(y,x)}=\\omega(0;y,x-y) = \\omega(0;x,y-x)^{-1} = \\big(\\mathop{\\mathrm{e}}\\nolimits^{i\\phi(x,y)}\\big)^{-1} = \\mathop{\\mathrm{e}}\\nolimits^{-i\\phi(x,y)}\n$$\nwhich means that $\\phi(x,y)=-\\phi(y,x)$. If $\\omega(0;x,y-x)= -1$, then one sets $\\phi(x,y):=-\\pi$ if $x<y$ (lexicographic order on $\\mathbb Z^d$)\nwhile $\\phi(x,y):=\\pi$ if $y<x$. With this convention and by the same argument one obtains that  $\\phi(x,y)=-\\phi(y,x)$ which is thus\nproved for any $x,y\\in \\mathbb Z^d$. As a consequence, $\\phi$ is indeed a magnetic potential.\n\nIn order to show \\eqref{def_omega}, it is enough to observe that\n\\begin{align*}\n& \\exp\\big\\{i\\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\} \\\\\n& = \\omega(0;q,x)\\;\\!\\omega(0;q+x,y)\\;\\!\\omega(0;q+x+y,-x-y) \\\\\n& = \\omega(0;q,x)\\;\\!\\omega(0;q+x,y)\\;\\!\\omega(0;q,x+y)^{-1} \\\\\n& = \\omega(q;x,y)\n\\end{align*}\nwhere \\eqref{eq_rel_1} has again been used for the second equality, and where the $2$-cocycle property \\eqref{eq_2}\nhas been taken into account for the last equality.\n\\end{proof}\n\n\\section{A continuous field of $C^*$-algebras}\\label{sec_field}\n\\setcounter{equation}{0}\n\nIn this section we consider a family of discrete magnetic systems which are parameterized by the elements $\\epsilon$\nof a compact Hausdorff space $\\Omega$. The necessary continuity relations between\nthe various objects is encoded in the structure of a field of twisted crossed product $C^*$-algebras,\nas introduced in \\cite{Rie} and already used in a similar context in \\cite{AMP}.\n\nOur first aim is to recall the notion of a continuous field of $2$-cocycles \\cite[Def.~2.1]{Rie}.\nIn our framework, with the locally compact group $\\mathbb Z^d$ and the algebra $l^\\infty(\\mathbb Z^d)$, we get the following definition.\n\n\\begin{Definition}\\label{def_cont_omega}\nA \\emph{continuous field over $\\Omega$ of $2$-cocycles on $\\mathbb Z^d$} is a map\n$$\n\\boldsymbol \\omega: \\Omega\\times \\mathbb Z^d\\times \\mathbb Z^d \\to \\mathscr U(\\mathbb Z^d)\n$$\nsuch that\n\\begin{enumerate}\n\\item[(i)] For any $x,y,z\\in \\mathbb Z^d$ and $\\epsilon \\in \\Omega$ the following relations hold:\n\\begin{equation}\\label{eq_cont_field}\n\\boldsymbol \\omega(\\epsilon;x+y,z)\\;\\boldsymbol \\omega(\\epsilon;x,y)  = \\theta_x\\boldsymbol \\omega(\\epsilon;y,z)\\;\\boldsymbol \\omega(\\epsilon;x,y+z)\n\\end{equation}\nand\n\\begin{equation}\\label{eq_norm_field}\n\\boldsymbol \\omega(\\epsilon;x,0) =\\boldsymbol \\omega(\\epsilon;0,x)=1,\n\\end{equation}\n\\item[(ii)] For any fixed $(x,y)\\in \\mathbb Z^d\\times \\mathbb Z^d$ the map\n\\begin{equation}\\label{cont_cocycle}\n\\Omega\\ni \\epsilon \\mapsto \\boldsymbol \\omega(\\epsilon;x,y)\\in \\mathscr U(\\mathbb Z^d)\n\\end{equation}\nis continuous.\n\\end{enumerate}\n\\end{Definition}\n\nNote that in equation \\eqref{eq_cont_field} the shift $\\theta_x$ acts on\n$\\boldsymbol \\omega(\\epsilon;x,y)\\in\\mathscr U(\\mathbb Z^d)$, as in the previous section.\nClearly, for each fixed $\\epsilon$ the relation \\eqref{eq_cont_field} corresponds to \\eqref{eq_2} in the abstract framework or to the statement (i) of Lemma \\ref{lem_2_cocycle}\nin the magnetic case.\nSimilarly, \\eqref{eq_norm_field} is a reminiscence of \\eqref{eq_n} or of the normalization property (ii) of Lemma \\ref{lem_2_cocycle}.\nOn the other hand, the new assumption \\eqref{cont_cocycle} is the one which provides the necessary continuity condition.\n\nFor the subsequent algebraic construction we shall consider the algebra $C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)$\nwhich is going to replace the algebra $l^\\infty(\\mathbb Z^d)$ of the previous section.\nNote that for any $f\\in C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)$ the action of $\\mathbb Z^d$ by translations is defined by $\\theta_x f(q,\\epsilon)=f(q+x,\\epsilon)$\nfor any $q,x\\in \\mathbb Z^d$ and $\\epsilon \\in \\Omega$.\nSo, let us consider a continuous field $\\boldsymbol \\omega$ over $\\Omega$ of $2$-cocycles on $\\mathbb Z^d$ and observe that its definition is made such that it can be\ninterpreted as a normalized $2$-cocycle on $\\mathbb Z^d$ taking values in the unitary group of the Abelian algebra $C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)$. Indeed,\nfor fixed $x,y\\in \\mathbb Z^d$ one can set\n\\begin{equation*}\n[\\omega(x,y)](q,\\epsilon):=[\\boldsymbol \\omega(\\epsilon;x,y)](q)\\equiv\\boldsymbol \\omega(q,\\epsilon;x,y).\n\\end{equation*}\nThen, by condition (ii) of Definition \\ref{def_cont_omega} one infers that\n$$\n\\omega(x,y)\\in C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\qquad \\hbox{ and } \\qquad [\\omega(x,y)](q,\\epsilon)\\in \\mathbb T.\n$$\nIn addition, \\eqref{eq_norm_field} implies that $\\omega$ is a normalized $2$-cocycle.\nNote that the additional property\n\\begin{equation}\\label{eq_cond_add}\n\\omega(x,-x)=1\n\\end{equation}\nholds if and only if $\\boldsymbol \\omega(\\epsilon;x,-x)=1$ for every $\\epsilon \\in \\Omega$.\nSince this property is satisfied by magnetic $2$-cocycles we shall assume it in the sequel.\n\nIn summary, starting from a continuous field $\\boldsymbol \\omega$ over $\\Omega$ of $2$-cocycles on $\\mathbb Z^d$ which satisfies the additional property\n$\\boldsymbol \\omega(\\epsilon;x,-x)=1$ for any $\\epsilon\\in \\Omega$ and $x\\in \\mathbb Z^d$, we end up with the $2$-cocycle $\\omega$\ntaking values in the unitary group of the algebra $C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)$\nand having the additional property \\eqref{eq_cond_add}.\nWith this $2$-cocycle one defines in analogy with \\eqref{eq_produit} the product for any\n$\\boldsymbol f,\\boldsymbol g\\in l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$ by\n\\begin{align*}\n\\big[[\\boldsymbol f\\diamond \\boldsymbol g](x)\\big](q,\\epsilon) & \\equiv [\\boldsymbol f\\diamond \\boldsymbol g](q,\\epsilon;x)\\\\\n\\nonumber & := \\sum_{y\\in \\mathbb Z^d} [\\boldsymbol f(y)](q,\\epsilon)\\; [\\boldsymbol g(x-y)](q+y,\\epsilon) \\;[\\omega(y,x-y)](q,\\epsilon) \\\\\n\\nonumber & \\equiv \\sum_{y\\in \\mathbb Z^d} \\boldsymbol f(q,\\epsilon;y)\\; \\boldsymbol g(q+y,\\epsilon;x-y) \\;\\boldsymbol \\omega(q,\\epsilon;y,x-y), \\quad \\forall \\;\\! q,x\\in \\mathbb Z^d, \\epsilon\\in \\Omega.\n\\end{align*}\nWe also endow $l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$ with the involution\n\\begin{equation*}\n[\\boldsymbol f^\\diamond(x)](q,\\epsilon)\\equiv \\boldsymbol f^{\\diamond}(q,\\epsilon;x):= \\overline{\\boldsymbol f(q+x,\\epsilon;-x)}\n\\end{equation*}\nand with the norm\n$$\n\\|\\boldsymbol f\\|_{1,\\infty}:=\\sum_{x\\in \\mathbb Z^d}\\sup_{q\\in \\mathbb Z^d}\\sup_{\\epsilon\\in \\Omega}|\\boldsymbol f(q,\\epsilon;x)| \\qquad \\boldsymbol f\\in l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big),\n$$\nmaking it a unital Banach $^*$-algebra. The enveloping $C^*$-algebra of $l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$ will be denoted by $\\mathfrak C_\\Omega$.\n\nLet us now emphasize the main point of all this construction: there exists an \\emph{evaluation map}\n$$\ne_\\epsilon: l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big) \\to l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)\n$$\ndefined for any $\\boldsymbol f\\in l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$ by $[e_\\epsilon (\\boldsymbol f)](q;x);=\\boldsymbol f(q,\\epsilon;x)$\nfor any $q,x\\in \\mathbb Z^d$ and $\\epsilon\\in\\Omega$.\nThis map is clearly norm-decreasing and surjective, and extends continuously to a norm-decreasing $*$-homomorphism\n$e_\\epsilon:\\mathfrak C_\\Omega\\to \\mathfrak C_\\epsilon$, with $\\mathfrak C_\\epsilon:=\\mathfrak C(\\omega_\\epsilon)$ the $C^*$-algebra constructed in the previous section\nwith the $2$-cocycle $\\omega_\\epsilon:=\\boldsymbol \\omega(\\epsilon;\\cdot,\\cdot):\\mathbb Z^d\\times\\mathbb Z^d\\to \\mathscr U(\\mathbb Z^d)$.\n\nIn this framework, the main result borrowed from \\cite{Rie} reads:\n\n\\begin{Proposition}\\label{prop_Rieffel}\nLet $\\boldsymbol \\omega$ be a continuous field over $\\Omega$ of $2$-cocycles on $\\mathbb Z^d$ satisfying $\\boldsymbol \\omega(\\epsilon;x,-x)=1$ for\nevery $\\epsilon \\in \\Omega$ and $x\\in \\mathbb Z^d$.\nThen the following properties hold:\n\\begin{enumerate}\n\\item[(i)] The map $e_\\epsilon: \\mathfrak C_\\Omega\\to \\mathfrak C_\\epsilon$ is surjective,\n\\item[(ii)] For any $\\boldsymbol f\\in \\mathfrak C_\\Omega$ one has $\\|\\boldsymbol f\\|_{\\mathfrak C_\\Omega} = \\sup_{\\epsilon\\in\\Omega}\\|e_\\epsilon(\\boldsymbol f)\\|_{\\mathfrak C_\\epsilon}$,\n\\item[(iii)] For any $\\boldsymbol f\\in \\mathfrak C_\\Omega$ the map $\\Omega\\ni \\epsilon\\mapsto \\|e_\\epsilon(\\boldsymbol f)\\|_{\\mathfrak C_\\epsilon} \\in \\mathbb R_+$ is continuous.\n\\end{enumerate}\n\\end{Proposition}\n\n\\begin{proof}\nBefore mentioning the precise arguments borrowed from \\cite{Rie}, let us stress that part of the proofs in that reference\nrelies on the existence of an bounded approximate identity. However, this technical point is automatically satisfied in our framework,\nas shown in the seminal paper \\cite[Sec.~2.28 \\& 2.29]{Zel}.\n\nOnce this preliminary observation is taken into account, statement (i) is a direct consequence of \\cite[Prop.~2.3]{Rie}.\nFor (ii) it is enough to observe that the map $\\boldsymbol f\\mapsto \\oplus_\\epsilon e_\\epsilon(\\boldsymbol f)$  is injective.\nThe upper semi-continuity of the map mentioned in (iii) follows from \\cite[Thm.~2.4]{Rie} while the lower semi-continuity\nof the same map is a consequence of \\cite[Thm.~2.5]{Rie} together with the equality between the twisted crossed product algebra $\\mathfrak C_\\Omega$\nand its reduced version, see \\cite[Thm.~5.1]{Zel}.\n\\end{proof}\n\nLet us now state and prove our main result:\n\n\\begin{Theorem}\\label{thm_main}\nLet $\\boldsymbol \\omega$ be a continuous field over $\\Omega$ of $2$-cocycles on $\\mathbb Z^d$ satisfying $\\boldsymbol \\omega(\\epsilon;x,-x)=1$ for any $\\epsilon\\in \\Omega$ and $x\\in \\mathbb Z^d$,\nand let $\\omega_\\epsilon$ be defined by $\\boldsymbol \\omega(\\epsilon;\\cdot,\\cdot)$.\nConsider a family $\\{h^\\epsilon\\}_{\\epsilon\\in \\Omega}\\subset l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ such that the following conditions are satisfied:\n\\begin{enumerate}\n\\item[(i)] For any $y\\in \\mathbb Z^d$, $\\sup_{q\\in \\mathbb Z^d} \\big|h^\\epsilon(q;y)-h^{\\epsilon'}(q;y)\\big|\\to 0$ as $\\epsilon'\\to \\epsilon$ in $\\Omega$,\n\\item[(ii)] $\\sum_{y\\in \\mathbb Z^d} \\sup_{\\epsilon\\in \\Omega}\\sup_{q\\in \\mathbb Z^d}|h^\\epsilon(q;y)|<\\infty$.\n\\item[(iii)]  $(h^\\epsilon)^\\diamond=h^\\epsilon$.\n\\end{enumerate}\nThen, for any family of $1$-cochains $\\lambda^\\epsilon: \\mathbb Z^d\\to \\mathscr U(\\mathbb Z^d)$ satisfying\n$\\lambda^\\epsilon(x)\\; \\theta_x\\lambda^\\epsilon(y)\\; \\lambda^\\epsilon(x+y)^{-1} = \\omega_\\epsilon(x,y)$,\nthe family of spectra $\\Big\\{\\sigma\\big({\\mathfrak{Rep}}^{\\lambda^\\epsilon}(h^\\epsilon)\\big)\\Big\\}_{\\epsilon \\in \\Omega}$ forms an outer and an inner continuous family\nat every point of $\\Omega$.\n\\end{Theorem}\n\nLet us stress that condition (ii) is satisfied for example if there exists $f\\in l^1(\\mathbb Z^d)$ such that $|h^\\epsilon(q;y)|\\leq f(y)$\nfor any $\\epsilon \\in \\Omega$ and $q\\in \\mathbb Z^d$.\nIn the following proof, we use the notation $C_0(\\mathbb R)$ for the set of continuous functions on $\\mathbb R$ which vanish at infinity.\nRecall also that $\\mathcal H=l^2(\\mathbb Z^d)$.\n\n\\begin{proof}\na) Let us first observe that the conditions on $\\{h^\\epsilon\\}_{\\epsilon\\in \\Omega}$ have been chosen such that the function $\\boldsymbol h:\\Omega\\times\\mathbb Z^d\\times \\mathbb Z^d\\to \\mathbb C$\ndefined by $\\boldsymbol h(\\epsilon,q;x):=h^\\epsilon(q;x)$ satisfies $\\boldsymbol h\\in l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$.\nAs a consequence, $\\boldsymbol h\\in \\mathfrak C_\\Omega$ and the statements of Proposition \\ref{prop_Rieffel} hold for $\\boldsymbol h$ instead of $\\boldsymbol f$.\nIn particular, since $e_\\epsilon(\\boldsymbol h)=h^\\epsilon$ one infers that the map\n$$\n\\Omega\\ni \\epsilon\\mapsto \\|h^\\epsilon\\|_{\\mathfrak C_\\epsilon} \\in \\mathbb R_+\n$$\nis continuous. Furthermore, since $\\boldsymbol h^{\\diamond}=\\boldsymbol h$ and since the algebra $l^1\\big(\\mathbb Z^d;C\\big(\\Omega;l^\\infty(\\mathbb Z^d)\\big)\\big)$ is unital,\none infers that for $z\\in \\mathbb C\\setminus\\mathbb R$ the element $\\boldsymbol h-z$ is invertible in the $C^*$-algebra $\\mathfrak C_\\Omega$, with respect to the product $\\diamond$.\nIts inverse is simply denoted by $(\\boldsymbol h-z)^{-1}$.\nSince $e_\\epsilon$ is a $*$-homomorphism, one gets that $e_\\epsilon\\big((\\boldsymbol h-z)^{-1}\\big)=(h^\\epsilon-z)^{-1}$, and as a consequence the map\n$$\n\\Omega \\ni \\epsilon \\mapsto \\big\\|(h^\\epsilon-z)^{-1}\\big\\|_{\\mathfrak C_\\epsilon}\\in \\mathbb R_+\n$$\nis also continuous.\n\nNow, since ${\\mathfrak{Rep}}^{\\lambda^\\epsilon}$ defines a faithful representation of $\\mathfrak C_\\epsilon$ in $\\mathscr B(\\mathcal H)$ it follows that\n$$\n\\|h^\\epsilon\\|_{\\mathfrak C_\\epsilon} = \\big\\|{\\mathfrak{Rep}}^{\\lambda^\\epsilon}(h^\\epsilon)\\big\\|\n$$\nwhere the norm on the r.h.s.~corresponds to the usual norm in $\\mathscr B(\\mathcal H)$. If we set $H^\\epsilon:= {\\mathfrak{Rep}}^{\\lambda^\\epsilon}(h^\\epsilon)$,\nwhich is a self-adjoint element of the $C^*$-algebra\n${\\mathfrak{Rep}}^{\\lambda^\\epsilon}(\\mathfrak C_\\epsilon)\\subset \\mathscr B(\\mathcal H)$, one then deduces that the map\n$$\n\\Omega\\ni \\epsilon\\mapsto \\|(H^\\epsilon-z)^{-1}\\|\\in \\mathbb R_+\n$$\nis continuous for any $z\\in \\mathbb C\\setminus \\mathbb R$.\nFinally, by a density argument of the linear span of $\\{(\\cdot-z)^{-1}\\}_{z\\in \\mathbb C\\setminus \\mathbb R}$, one\ninfers that for any $\\eta\\in C_0(\\mathbb R)$, the map\n\\begin{equation}\\label{eq_cont_eta}\n\\Omega\\ni \\epsilon \\mapsto \\|\\eta(H^\\epsilon)\\|\\in \\mathbb R_+\n\\end{equation}\nis continuous, see \\cite[p.~364]{ABG}.\n\nb) It remains to show that the continuity \\eqref{eq_cont_eta} implies the inner and the outer continuity for the family of spectra\n$\\sigma(H^\\epsilon)$. The following argument is directly borrowed from the proof of \\cite[Prop.~2.5]{AMP} which we present here for the sake of completeness.\nFor the outer continuity, let $\\epsilon_0\\in \\Omega$ and let $\\mathcal K$ be a compact set in $\\mathbb R$ such that $\\mathcal K\\cap \\sigma(H^{\\epsilon_0})=\\emptyset$.\nBy Urysohn's lemma there exists $\\eta\\in C_0(\\mathbb R)$ with $\\eta\\geq 0$ such that $\\eta|_\\mathcal K = 1$ and $\\eta|_{\\sigma(H^{\\epsilon_0})}=0$, and therefore\n$\\eta(H^{\\epsilon_0})=0$. By the continuity of \\eqref{eq_cont_eta} one can then chose a neighbourhood $\\mathcal N$ of $\\epsilon_0$ in $\\Omega$ such that\nfor any $\\epsilon\\in \\mathcal N$ one has $\\|\\eta(H^\\epsilon)\\|\\leq 1\/2$.\nBy contradiction, if for some $\\epsilon \\in \\mathcal N$ one would have $\\nu\\in \\mathcal K\\cap \\sigma(H^\\epsilon)$ then  it would follow that\n$$\n1=\\eta(\\nu)\\leq \\sup_{\\mu\\in \\sigma(H^\\epsilon)}\\eta(\\mu) \\leq \\|\\eta(H^\\epsilon)\\|\\leq 1\/2\n$$\nwhich is absurd. Thus, the family $\\{\\sigma(H^\\epsilon)\\}_{\\epsilon\\in \\Omega}$ is outer continuous at every points of $\\Omega$.\n\nFor the inner continuity, let $\\mathcal O$ be an open subset of $\\mathbb R$ such that there exists $\\nu\\in \\mathcal O\\cap \\sigma(H^{\\epsilon_0})$.\nBy Urysohn's lemma there exists $\\eta\\in C_0(\\mathbb R)$ with $\\eta(\\nu)=1$ and $\\mathrm{supp}\\;\\! \\eta\\subset \\mathcal O$. As a consequence, one has\n$\\|\\eta(H^{\\epsilon_0})\\|\\geq 1$. By contradiction, assume now that for any neighbourhood $\\mathcal N$ of $\\epsilon_0$ in $\\Omega$ there exists $\\epsilon \\in \\mathcal N$ such that\n$\\mathcal O\\cap \\sigma(H^\\epsilon)=\\emptyset$. It follows that $\\eta(H^\\epsilon)=0$. However, this clearly contradicts the continuity provided in \\eqref{eq_cont_eta}.\nAs a consequence, the family $\\{\\sigma(H^\\epsilon)\\}_{\\epsilon\\in \\Omega}$ is inner continuous at every points of~$\\Omega$.\n\\end{proof}\n\n\\section{The scaling example}\\label{sec_Nenciu}\n\\setcounter{equation}{0}\n\nIn this section we provide the proof of Theorem \\ref{thm_Nenciu}. On the way we also show that the setting considered in \\cite{Nen}\nis covered by our formalism. Note that only $d=2$ is considered in that reference, but that the extension for\narbitrary $d\\in \\mathbb N$ is harmless in our framework.\n\nLet us now fix $\\Omega=[0,1]$. In the following statement, $\\triangle (x,y,z)$ denotes the triangle in $\\mathbb R^d$ defined\nby the three points $x,y,z\\in \\mathbb Z^d$.\n\n\\begin{Proposition}\\label{prop_triangle}\nLet $\\phi$ be a magnetic potential which satisfies\n$$\n\\big|\\phi(x,y)+\\phi(y,z)+\\phi(z,x)\\big| \\leq \\hbox{ area } \\triangle (x,y,z).\n$$\nThen the map $\\boldsymbol \\omega$ defined for $\\epsilon \\in [0,1]$ and $q,x,y\\in \\mathbb Z^d$ by\n\\begin{equation}\\label{eq_def_tomega}\n\\boldsymbol \\omega(q,\\epsilon;x,y) :=\\exp\\big\\{i\\epsilon \\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\}\\in \\mathbb T\n\\end{equation}\nis a continuous field over $[0,1]$ of $2$-cocycles on $\\mathbb Z^d$.\n\\end{Proposition}\n\n\\begin{proof}\nClearly, for each fixed $\\epsilon \\in [0,1]$ the map $\\omega(\\epsilon;\\cdot,\\cdot)$ satisfied the normalized $2$-cocycle requirements as\nmentioned in the point (i) of Definition \\ref{def_cont_omega}.\nFor the condition (ii) of this definition, let $x,y\\in \\mathbb Z^d$ be fixed, and let $\\epsilon,\\epsilon'\\in [0,1]$ and $q\\in \\mathbb Z^d$.\nThen one has\n\\begin{align*}\n& \\big|\\boldsymbol \\omega(q,\\epsilon';x,y)-\\boldsymbol \\omega(q,\\epsilon;x,y)\\big| \\\\\n& \\leq \\Big|1-\\exp\\big\\{i(\\epsilon-\\epsilon') \\big[\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\big]\\big\\}\\Big| \\\\\n& \\leq \\sum_{n=1}^\\infty \\frac{|\\epsilon-\\epsilon'|^n}{n!}\\;\\! \\Big|\\phi(q,q+x)+\\phi(q+x,q+x+y)+\\phi(q+x+y,q)\\Big|^n \\\\\n& = \\sum_{n=1}^\\infty \\frac{|\\epsilon-\\epsilon'|^n}{n!}\\;\\!\\big|\\hbox{ area } \\triangle (q,q+x,q+x+y)\\big|^n \\\\\n& = \\sum_{n=1}^\\infty \\frac{|\\epsilon-\\epsilon'|^n}{n!}\\;\\!\\big|\\hbox{ area } \\triangle (0,x,x+y)\\big|^n.\n\\end{align*}\nSince the last expression is independent of $q$ one directly infers that\n$$\n\\lim_{\\epsilon'\\to\\epsilon} \\big\\|\\boldsymbol \\omega(\\epsilon';x,y)-\\boldsymbol \\omega(\\epsilon;x,y)\\big\\|_{\\mathscr U(\\mathbb Z^d)}=\n\\lim_{\\epsilon'\\to\\epsilon} \\sup_{q\\in \\mathbb Z^d}\\big|\\boldsymbol \\omega(q,\\epsilon';x,y)-\\boldsymbol \\omega(q,\\epsilon;x,y)\\big|=0\n$$\nwhich corresponds to the required continuity property.\n\\end{proof}\n\nWe can now show that Theorem \\ref{thm_Nenciu} is a special case of Theorem \\ref{thm_main}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm_Nenciu}]\nLet us first observe that the initial conditions (i) and (ii) imposed on $h^\\epsilon$ are equivalent to $h^\\epsilon \\in l^1\\big(\\mathbb Z^d;l^\\infty(\\mathbb Z^d)\\big)$ and to\n$(h^\\epsilon)^\\diamond=h^\\epsilon$.\nThese two conditions together with the other two conditions imposed on $h^\\epsilon$\nimply that all assumptions on $h^\\epsilon$ required by Theorem \\ref{thm_main} are satisfied.\n\nOn the other hand, the condition imposed on the magnetic potential $\\phi$ are precisely the one already used\nin Proposition \\ref{prop_triangle} for defining the continuous field $\\boldsymbol \\omega$ of $2$-cocycles in \\eqref{eq_def_tomega}.\nObserve also that the additional condition $\\boldsymbol \\omega(\\epsilon;x,-x)=1$ holds for any $x\\in \\mathbb Z^d$.\nThus, we have checked so far that all assumptions of Theorem \\ref{thm_main} are satisfied. It only remains to exploit its consequences.\n\nFor that purpose let us choose the $1$-cochain $\\lambda^\\epsilon$ defined as in \\eqref{eq_obvious} by\n$$\n\\lambda^\\epsilon(q;x):=\\mathop{\\mathrm{e}}\\nolimits^{i\\epsilon\\phi(q,q+x)}.\n$$\nWith this choice and as in \\eqref{eq_Rep_phi} one obtains on any $u\\in \\mathcal H$\n\\begin{equation*}\n[{\\mathfrak{Rep}}^{\\lambda^\\epsilon}(h^\\epsilon)u](x) = \\sum_{y\\in \\mathbb Z^d} h^\\epsilon(x;y-x) \\;\\!\\mathop{\\mathrm{e}}\\nolimits^{i\\epsilon\\phi(x,y)} \\;\\!u(y) \\equiv [H^\\epsilon u](x)\n\\end{equation*}\nor equivalently ${\\mathfrak{Rep}}^{\\lambda^\\epsilon}(h^\\epsilon)=H^\\epsilon$.\nThe statement of Theorem \\ref{thm_Nenciu} can now be directly deduced from Theorem \\ref{thm_main}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{aba:sec1}\n\n\\emph{Entropy}, originally introduced in thermodynamics, is nowadays\nrecognized as a rather universal concept with a variety of uses,\nranging from physics and chemistry to information theory and the\ntheory of computation. Besides the role it plays for foundational\nquestions, it is also relevant for applications. For example, entropy\nis used to study the efficiency of steam engines, but it also occurs\nin formulae for the data transmission capacity of optical fibres.\n\nWhile entropy can be defined in various ways, a very common form\nemployed for the study of classical systems is the \\emph{Gibbs\n  entropy} or, in the context of information theory, the \\emph{Shannon\n  entropy}~\\cite{shannon_mathematical_1948}. It is defined for any\nprobability distribution $P$ as\n\\begin{align*}\n  H(P) = - \\sum_x P(x) \\log P(x)\n\\end{align*}\n(up to an unimportant proportionality factor). This definition has\nbeen generalized to the von Neumann entropy~\\cite{neumann_mathematical_1996}, which is\ndefined for density\noperators,\n\\begin{align*}\n  H(\\rho) = -\\operatorname{tr}(\\rho \\log \\rho).\n\\end{align*}\nWhile these entropy measures have a wide range of\napplications, it has recently become apparent that they are not suitable for correctly\ncharacterizing operationally relevant quantities in general scenarios\n(as explained below). This has led to the development of\nextensions~\\cite{ogawa_strong_2000}, among them the \\emph{information spectrum\n  approach}~\\cite{Nagaoka2007,han_information-spectrum_2002,Bowen2006}  and\n\\emph{smooth entropies}~\\cite{Renner2004,Renner2005} (where the former can be obtained as an asymptotic limit of the latter~\\cite{DatRen09}).\n\nThe aim of this work is to study an alternative measure of entropy\nthat generalizes von Neumann entropy. The generalized entropy is\nclosely related to smooth entropies, which, in turn, are connected to\na variety of operational quantities.\n\n\n\\subsection{Axiomatic and operational approach to\n  entropy} \\label{sec_opapproach}\n\nThe variety of areas and applications where entropies are used is\nimpressive, and one may wonder what it is that makes entropy such a\nversatile concept.\n\nOne could attempt to answer the question from an \\emph{axiomatic}\nviewpoint. Here, the idea is to consider (small) sets of axioms that\ncharacterize the nature of entropy. There is a vast amount of\nliterature devoted to the specification of such axioms and their\nstudy~\\cite{shannon_mathematical_1948,jaynes_information_1957,aczel_why_1974,ochs_new_1975,lieb_guide_1998,lieb_fresh_2000,csiszar_axiomatic_2008,baumgartner_characterizing_2012}. While\nthe choice of a set of axioms is ultimately a matter of taste, we\nsketch in the following some of the most popular axioms. We do this\nfor the case of entropies defined on quantum systems, i.e., we\nconsider functions $H$ from the set of density operators (denoted by\n$\\rho$) to the real numbers.\n\\begin{itemize}\n\\item \\emph{Positivity:} $H(\\rho) \\geq 0$.\n\\item \\emph{Invariance under isometries:} $H(U \\rho U^{\\dagger}) =\n  H(\\rho)$.\n\\item \\emph{Continuity:} $H$ is a continuous function of $\\rho$.\n\\item \\emph{Additivity:} $H(\\rho_A \\otimes \\rho_B) = H(\\rho_A) +\n    H(\\rho_B)$.\n\\item \\emph{Subadditivity:} $H(\\rho_{A B}) \\leq H(\\rho_A) +\n  H(\\rho_B)$.\\footnote{Here $\\rho_{A B}$ denotes a density operator on\n    a bipartite system and $\\rho_A$ and $\\rho_B$ are obtained by\n    partial traces over the second and first subsystem, respectively.}\n\\end{itemize}\nThe (special) case of classical entropies is obtained by replacing the\ndensity operators by probability distributions. Note that the second\naxiom then reduces to the requirement that the entropy is invariant\nunder permutations.\n\nIt is easy to verify that the von Neumann entropy satisfies the above\naxioms. Furthermore, it can be shown that (up to a constant factor,\nwhich may be fixed by an additional normalization axiom) the von\nNeumann entropy is essentially the only function satisfying the above\npostulates~\\cite{ochs_new_1975}.  This result -- as well as similar\nresults based on slightly different sets of axioms -- nicely expose the\nuniversal nature of entropy. Note, in particular, that the above\naxioms do not refer specifically to thermodynamic or\ninformation-theoretic properties of a system.\n\nAn alternative to this axiomatic approach is to relate entropy to\n\\emph{operational} quantities. In thermodynamics, examples for such\noperational quantities include measures for heat flow or the amount of\nwork that is transformed into heat during a given process. In\ninformation theory, operational quantities are, for instance, the\nminimum size to which the information generated by a source can be\ncompressed, or the amount of uniform randomness that can be extracted\nfrom a non-uniform source.\n\nGiven the very different nature of these operational quantities, it is\nnot obvious that this approach can lead to a reasonable notion of\nentropy. One would rather expect an entire family of entropy\nmeasures -- possibly as large as the number of different operational\nquantities one considers. However, there exist remarkable connections,\neven relating thermodynamic and information-theoretic quantities. For\nexample, it follows from Landauer's\nprinciple~\\cite{landauer_irreversibility_1961,bennett_logical_1973}\nthat the amount of work that can be extracted from a system is\ndirectly related to the size to which the information contained in it\ncan be compressed~\\cite{RARDV11,Dahlsten2011,Faist2012}.\n\nRecent work has shown that a large number of operational quantities\ncan be characterized with one single class of entropy\nmeasures. \\emph{Smooth entropies} (denoted by $H_{\\min}^\\epsilon$ and\n$H_{\\max}^\\epsilon$), which were developed mostly within quantum\ninformation theory, are an example of such a class. For instance,\n$H_{\\min}^\\epsilon$ quantifies the number of uniformly random\n(classical) bits that can be deterministically extracted from a weak\nsource of randomness\\cite{Renner2005,RenKoe2005} and\n$H_{\\max}^\\epsilon$ quantifies the number of bits needed to encode a\ngiven (classical) value\\cite{Renes12}. More generally,\n$H_{\\min}^\\epsilon$ can be used to characterize\n\\emph{decoupling}\\cite{Dupuis2009}, a quantum version of randomness\nextraction\\cite{Dupuis2010}, and \\emph{state\n  merging}\\cite{Horodecki2005,Horodecki2006}, which can be seen as the\nfully quantum analogue of coding\\cite{Berta2009}. Also, a combination\nof $H_{\\min}^\\epsilon$ and $H_{\\max}^\\epsilon$ gives an expression for\nthe classical capacity of a classical\\cite{Renner2006} or a\nquantum\\cite{Renes2011} channel, as well as its ``reverse''\ncapacity\\cite{Berta2011}. Additional applications can be found particularly in quantum cryptography (see, e.g.,\n~\\cite{Renner2005,Damgard2007,Scarani2008a}). Smooth entropies also\nhave operational interpretations within thermodynamics. For example,\nthey can be used in a single-shot version of Landauer's principle to quantify the amount of work required by an\noperation that moves a given system into a pure\nstate\\cite{RARDV11,Dahlsten2011,Faist2012}.\n\nHowever, smooth entropies are generally different from the von Neumann\nentropy except in special cases. This implies that many operational\nquantities, characterized by smooth entropies, are not in general\naccurately described by the von Neumann entropy (e.g. the amount of\nextractable randomness or the encoding length). In particular, it follows that some of\nthe axioms considered above must be incompatible with the operational\napproach.\n\nThis can also be seen directly, for example, for the\n(classical) task of randomness extraction. Let $C(X)$ be the number of uniform bits that can be obtained by applying a function to\na random variable $X$ distributed according to $P_X$. Then the\nquantity $C$ automatically has the properties one would expect from an\nuncertainty measure: it equals $0$ if $X$ is perfectly known, and it\nincreases as $X$ becomes more uncertain. One may therefore interpret\n$C$ as an (operationally defined) entropy measure for classical random variables.\n\nHowever, while $C$ is indeed positive, invariant under permutations,\nand additive, it is not subadditive. To see this, consider a random\nvariable $R$ uniformly distributed over the set $\\{1, \\ldots,\n2^\\ell\\}$, for some large $\\ell \\in I}{\\mathbb{I}{N}$. Furthermore, define\nthe random variables $X$ and $Y$ by\n\\begin{align*}\n  X & =\n  \\begin{cases}\n    R & \\text{if $R \\leq 2^{\\ell-1}$} \\\\ 0 & \\text{otherwise},\n  \\end{cases}\n  \\\\\n  Y & =\n  \\begin{cases}\n    R & \\text{if $R > 2^{\\ell-1}$} \\\\ 0 & \\text{otherwise.}\n  \\end{cases}\n\\end{align*}\nSince $\\Pr[X=0] = \\Pr[Y=0] = \\frac{1}{2}$, it is not possible to\nextract more than $1$ bit from either of $X$ or $Y$ separately, i.e.,\n$C(X) = C(Y) \\leq 1$. However, since the pair $(X,Y)$ is in one-to-one\nrelation to $R$, we have $C(X Y) = C(R) = \\ell$. Hence, subadditivity,\n$C(X Y) \\leq C(X) + C(Y)$ can be violated by an arbitrarily large\namount.\\footnote{However, an inequality of similar form can be recovered --- this is known as the \\emph{entropy splitting lemma}~\\cite{Wullschleger2007,Damgaard07}.}\n\n\\subsection{Generalized entropy measure}\n\nThe above considerations show that an operational approach to\nentropies necessitates the use of entropy measures that are more\ngeneral than those obtained by the usual axiomatic approaches. The aim\nof this paper is to investigate such a generalization, which is\nmotivated by previous\nwork~\\cite{buscemi_quantum_2010,brandao_one-shot_2011,wang_one-shot_2012,tomamichel_hierarchy_2012}.\nWe derive a number of properties of this measure and relate it back to\nthe better-studied family of smooth entropies.\n\nOur generalized entropy measure is, technically, a family of\nentropies, denoted $H_H^\\epsilon$, and parametrized by a real number\n$\\epsilon$ from the interval $[0,1]$. $H_H^\\epsilon$ is defined via a\nrelative-entropy type quantity, i.e., a function that depends on two\ndensity operators, $\\rho$ and $\\sigma$, similarly to the\nKullback-Leibler divergence~\\cite{Kullback1951,Wehrl1978}. This\nquantity, denoted $D_H^\\epsilon$, has a simple interpretation in the\ncontext of quantum hypothesis testing~\\cite{Helstrom1969}. Consider a\nmeasurement for distinguishing whether a system is in state $\\rho$ or\n$\\sigma$. $D_H^\\epsilon(\\rho\\|\\sigma)$ then corresponds to the\nnegative logarithm of the failure probability when the system is in\nstate $\\sigma$, under the constraint that the success probability when\nthe system is in state $\\rho$ is at least $\\epsilon$ (see\nSection~\\ref{sec_relentrdef} below).\n\nStarting from $D_H^\\epsilon(\\rho\\|\\sigma)$, it is possible to directly\ndefine a \\emph{conditional entropy}, $H_H^\\epsilon(A|B)$, i.e., a\nmeasure for the uncertainty of a system $A$ conditioned on a system\n$B$ (see Section~\\ref{sec_entrdef} below). We note that, while the\nconditional von Neumann entropy may be defined analogously using the\nKullback-Leibler divergence, the standard expression for conditional\nvon Neumann entropy~\\cite{Nielsen2000},\n\\begin{align} \n  \\label{eq_condvN} H(A|B) = H(\\rho_{A B}) -H(\\rho_B)\\ ,\n\\end{align}\ncannot be generalized directly. However, as shown in\nSection~\\ref{sec_chainrule}, $H_H^\\epsilon$ satisfies a \\emph{chain\n  rule}, i.e., an inequality which resembles~\\eqref{eq_condvN}. In\naddition, we show that $H_H^\\epsilon$ has many desirable properties\nthat one would expect an entropy measure to have (see\nSection~\\ref{sec_basicproperties}), for instance that it reduces to the von Neumann entropy in the asymptotic limit (Asymptotic Equipartition Property).\n\nApart from deriving the chain rule for the considered entropy measure, the\n  main contribution of this paper is to\n \nestablish direct relations\nto the \\emph{smooth entropy measures} $H_{\\min}^\\epsilon$ and\n$H_{\\max}^{\\epsilon}$ (Section~\\ref{sec_smooth}).  As explained above,\nit has been shown that these accurately characterize a number of\noperational quantities, such as information compression, randomness\nextraction, entanglement manipulation, and channel\ncoding. Furthermore, they are also relevant in the context of\nthermodynamics, e.g., for quantifying the amount of work that can be\nextracted from a given system.  The bounds derived in\nSection~\\ref{sec_smooth} imply that $H_H^\\epsilon$ has a similar\noperational significance.\n\n\\section{Preliminaries}\n\n\\subsection{Notation and Definitions}\n\nFor a finite-dimensional Hilbert space $\\mathcal{H}$, let\n$\\mathcal{L}(\\mathcal{H})$ and $\\mathcal{P}(\\mathcal{H})$ be the\nlinear and positive semi-definite operators on $\\mathcal{H}$,\nrespectively. On $\\mathcal{L}(\\mathcal{H})$ we employ the\nHilbert-Schmidt inner product $\\left<X,Y\\right>:=\\operatorname{Tr}(X^\\dagger\nY)$. Quantum states form the set\n$\\mathcal{S}(\\mathcal{H})=\\{\\rho\\in\\mathcal{P}(\\mathcal{H}):\\operatorname{Tr}(\\rho)=1\\}$,\nand we define the set of subnormalized states as\n$\\mathcal{S}_\\leq(\\mathcal{H})=\\{\\rho\\in\\mathcal{P}(\\mathcal{H}):0<\\operatorname{Tr}(\\rho)\\leq1\\}$.\nTo describe multi-partite quantum systems on tensor product spaces we\nuse capital letters and subscripts to refer to individual subsystems\nor marginals. We call a state $\\rho_{XB}$ {\\it classical-quantum (CQ)}\nif it is of the form $\\rho_{XB}=\\sum_x\np(x)\\left|{x}\\right>\\left<{x}\\right|\\otimes \\rho^x_B$ with\n$\\rho_B^x\\in\\mathcal{S}(\\mathcal{H}_B)$, $p(x)$ a probability\ndistribution and $\\{\\left|{x}\\right>\\}$ an orthonormal basis of\n$\\mathcal{H}_X$.\n\nA map $\\mathcal{E}:\\mathcal{L}(\\mathcal{H})\\rightarrow\n\\mathcal{L}(\\mathcal{H'})$ for which $\\mathcal{E}\\otimes\\mathcal{I}$, for any $\\mathcal{H''}$, maps\n$\\mathcal{P}(\\mathcal{H}\\otimes\\mathcal{H''})$ to\n$\\mathcal{P}(\\mathcal{H'}\\otimes\\mathcal{H''})$ is called a completely\npositive map (CPM). It is called trace-preserving if\n$\\operatorname{Tr}(\\mathcal{E}[X])=\\operatorname{Tr}(X)$ for any\n$X\\in\\mathcal{P}(\\mathcal{H})$. A unital map satisfies\n$\\mathcal{E}(I}{\\mathbb{I}{Id})=I}{\\mathbb{I}{I}$, and a map is sub-unital if\n$\\mathcal{E}(I}{\\mathbb{I}{I})\\leq I}{\\mathbb{I}{I}$.  The adjoint $\\mathcal{E}^*$\nof $\\mathcal{E}$ is defined by\n$\\operatorname{Tr}\\left(\\mathcal{E}^*(Y)\\,X\\right) =\n\\operatorname{Tr}\\left(Y\\,\\mathcal{E}(X)\\right)$.\n\nWe employ two distance measures on subnormalized states: the purified\ndistance\n$P(\\rho,\\sigma)$ \\cite{gilchrist_distance_2005,rastegin_sine_2006,tomamichel_duality_2010}\nand the generalized trace distance $D(\\rho,\\sigma)=\\frac{1}{2}\\Vert\n\\rho-\\sigma\\Vert_1+\\tfrac12|\\operatorname{Tr}\\rho-\\operatorname{Tr}\\sigma|$ (where\n$\\vert\\vert\\rho\\vert\\vert_1=\\operatorname{Tr}(\\sqrt{\\rho^\\dagger\\rho})$).\nThe purified distance is defined in terms of the generalized fidelity $F(\\rho,\\sigma)=\\Vert\\sqrt{\\rho}\\sqrt{\\sigma}\\Vert_1+\\sqrt{(1-\\operatorname{Tr}\\rho)(1-\\operatorname{Tr}\\sigma)}$ by $P(\\rho,\\sigma)=\\sqrt{1-F(\\rho,\\sigma)^2}$. (The fidelity itself is just the first term in the expression.) The purified and trace distances obey\nthe following relation~\\cite{fuchs_cryptographic_1999}:\n$D(\\rho,\\sigma)\\leq P(\\rho,\\sigma)\\leq \\sqrt{2D(\\rho,\\sigma)}$.\n\nFinally, the operator inequalities $A\\leq B$ and $A<B$ are taken to mean that $B-A$\nis positive semi-definite and positive definite respectively, and when comparing a matrix to a scalar we\nassume that the scalar is multiplied by the identity matrix. Note also\nthat all logarithms taken in the calculations are base 2.\n\n\n\\subsection{Semi-Definite Programs}\n\\label{sec:sdp}\n\nWatrous has given an elegant formulation of semidefinite programs\nespecially adapted to the present\ncontext~\\cite{watrous_semidefinite_2009}. Here we follow his notation;\nsee also~\\cite{boyd_convex_2004} for a more extensive treatment. A\nsemidefinite program over $\\mathcal{X}=I}{\\mathbb{I}{C}^n$ and\n$\\mathcal{Y}\\inI}{\\mathbb{I}{C}^m$ is specified by a triple $(\\Psi,A,B)$,\nfor $A$ and $B$ Hermitian operators in $\\mathcal{L}(\\mathcal{X})$ and\n$\\mathcal{L}(\\mathcal{Y})$ respectively, and\n$\\Psi:\\mathcal{L}(\\mathcal{X})\\rightarrow \\mathcal{L}(\\mathcal{Y})$ a\nlinear, Hermiticity-preserving operation.\n\nThis semidefinite program corresponds to two optimization problems, the so-called ``primal'' and ``dual'' problems:\\\\\n\n\\begin{minipage}[t]{0.23\\textwidth}\n  PRIMAL\\\\\n\n  minimize\\\\\n  subj. to\\\\\n\\end{minipage}\n\\begin{minipage}[t]{0.23\\textwidth}\n  \\text{}\\\\\n  \\\\\n  $\\left<A,X\\right>$\\\\\n  $\\Psi(X)\\geq B$\\\\\n  $X\\in\\mathcal{P}(\\mathcal{X})$\n\n\\end{minipage}\n\\begin{minipage}[t]{0.23\\textwidth}\n  DUAL\\\\\n\n  maximize\\\\\n  subj. to\\\\\n\\end{minipage}\n\\begin{minipage}[t]{0.23\\textwidth}\n  \\text{}\\\\\n  \\\\\n  $\\left<B,Y\\right>$\\\\\n  $\\Psi^*(Y)\\leq A$\\\\\n  $Y\\in\\mathcal{P}(\\mathcal{Y})$\n\n\\end{minipage}\\\\\n\\text{}\\\\\n\\\\\nWith respect to these problems, one can define the primal and dual\nfeasible sets $\\mathcal{A}$ and $\\mathcal{B}$ respectively:\n\\begin{align}\n  \\mathcal{A}&=\\{X\\in\\mathcal{P}(\\mathcal{X}) :  \\Psi(X)\\leq B\\},\\\\\n  \\mathcal{B}&=\\{Y\\in\\mathcal{P}(\\mathcal{Y}) : \\Psi^*(Y)\\geq A\\}.\n\\end{align}\nThe operators $X\\in\\mathcal{A}$ and $Y\\in\\mathcal{B}$ are then called\nprimal and dual feasible (solutions) respectively.\n\nTo each of the primal and dual problems, the associated optimal values\nare defined as:\\footnote{If $\\mathcal{A}=\\emptyset$ or\n  $\\mathcal{B}=\\emptyset$, we define $\\alpha=\\infty$ or\n  $\\beta=-\\infty$ respectively}\n\\begin{equation*}\n  \\alpha=\\inf_{X\\in\\mathcal{A}}\\left<A,X\\right>\n  \\quad\\text{and}\\quad\\beta=\\sup_{Y\\in\\mathcal{B}}\\left<B,Y\\right>.\n\\end{equation*}\nSolutions to the primal and dual problems are related by the following\ntwo duality theorems:\n\\begin{theorem}\n  (Weak duality). $\\alpha\\leq\\beta$ for every semidefinite program\n  $(\\Psi, A, B)$.\n\\end{theorem}\n\n\\begin{theorem}\n  (Slater-type condition for strong duality). For every semi-definite\n  program $(\\Psi, A, B)$ as defined above, the following two\n  statements hold:\n  \\begin{enumerate}\n  \\item Strict primal feasibility: If $\\beta$ is finite and there\n    exists an operator $X> 0$ s.t. $\\Psi(X)> B$, then $\\alpha=\\beta$\n    and there exists $Y\\in\\mathcal{B}$ s.t. $\\left< B,Y\\right>=\\beta$.\n  \\item Strict dual feasibility: If $\\alpha$ is finite and there\n    exists an operator $Y> 0$ s.t. $\\Psi^*(Y)<A$, then $\\alpha=\\beta$\n    and there exists $X\\in\\mathcal{A}$ s.t. $\\left<\n      A,X\\right>=\\alpha$.\n  \\end{enumerate}\n\\end{theorem}\nGiven strict feasibility, we obtain \\emph{complementary slackness}\nconditions linking the optimal $X$ and $Y$ for the primal and the dual\nproblem:\n\\begin{equation}\n  \\Psi(X)Y=BY\\quad\\text{and}\\quad \\Psi^*(Y)X=AX.\n\\end{equation}\n\nSemidefinite programs can be solved efficiently using the ellipsoid\nmethod~\\cite{Grotschel1993}. There exists an algorithm that, under\ncertain stability conditions and bounds on the primal feasible and\ndual feasible sets, finds an approximation for the optimal value of\nthe primal problem. The running time of the algorithm is bounded by a\npolynomial in $n$, $m$, and the logarithm of the desired accuracy\n(see~\\cite{watrous_semidefinite_2009} for more details).\n\n\\section{Relative and Conditional Entropies}\nWe will now introduce the new family of entropy measures, as well as\nthe smooth entropies, and the set of relative entropies that they are\nbased on.\n\n\\subsection{Definition of relative entropies} \\label{sec_relentrdef}\n\nWe define the $\\epsilon$-relative entropy $D^{\\epsilon} (\\rho\\vert\n\\vert \\sigma)$ of a subnormalized state $\\rho\\in\\mathcal{S}_\\leq(\\mathcal{H})$\nrelative to $\\sigma\\in\\mathcal{P}(\\mathcal{H})$ as\\footnote{Note that\n  this differs slightly from both the definitions used by Wang and\n  Renner~\\cite{wang_one-shot_2012}, Tomamichel and\n  Hayashi~\\cite{tomamichel_hierarchy_2012}, and Matthews and Wehner~\\cite{matthews_finite_2012}. Similar formulations\n  specific to mutual information and entanglement were previously\n  given respectively by Buscemi and Datta~\\cite{buscemi_quantum_2010}\n  and Brand\\~ao and Datta~\\cite{brandao_one-shot_2011}.\\mbox{}}\n\\begin{equation} \\label{eq_Depsdef}\n  2^{-D^{\\epsilon}(\\rho\\vert\\vert\\sigma)}:=\\tfrac{1}{\\epsilon}\\min\\{\\left<Q,\\sigma\\right>\\vert\n  0\\leq Q\\leq 1 \\land \\left< Q,\\rho\\right> \\geq \\epsilon\\} .\n\\end{equation}\nThis corresponds to minimizing the probability that a strategy $Q$ to\ndistinguish $\\rho$ from $\\sigma$ produces a wrong guess on input\n$\\sigma$ while maintaining a minimum success probability $\\epsilon$ to\ncorrectly identify $\\rho$. In particular, for $\\epsilon=1$,\n$D_H^\\epsilon(\\rho\\vert\\vert\\sigma)$ is equal to R\\'enyi's\nentropy\\cite{Renyi1961} of order $0$, and $D_0(\\rho\\vert\\vert\n\\sigma)=-\\log\\operatorname{Tr}(\\rho^0\\sigma)$, with $\\rho^0$ the projector on\nthe support of $\\rho$~\\cite{tomamichel_hierarchy_2012}.\n\nThe relative min- and max-entropies $D_{\\min}$ and $D_{\\max}$\nfor $\\rho\\in\\mathcal{S}_\\leq (\\mathcal H)$ and $\\sigma\\in\\mathcal{P}(\\mathcal{H})$ are defined as follows:\\footnote{The relative max-entropy was introduced in~\\cite{datta_min_2009}, but our definition of the relative min-entropy differs from the one used therein.}\n\\begin{align}\n  2^{-D_{\\min}(\\rho\\vert\\vert\\sigma)}&=\\left\\|\\sqrt{\\rho}\\sqrt{\\sigma}\\right\\|_1^2\\\\\n  D_{\\max}(\\rho\\vert\\vert\\sigma)&=\\min\\{\\lambda\\in I}{\\mathbb{I}{R}:\n  2^\\lambda\\sigma\\geq\\rho\\}.\n\\end{align}\nWe also define the corresponding smoothed quantities:\n\\begin{align}\n  D_{\\min}^\\epsilon(\\rho\\vert\\vert\\sigma)&=\\max_{\\tilde\\rho\\in\\mathcal{B}_\\epsilon(\\rho)}D_{\\min}(\\tilde\\rho\\vert\\vert\\sigma),\\\\\n  D_{\\max}^\\epsilon(\\rho\\vert\\vert\\sigma)&=\\min_{\\tilde\\rho\\in\\mathcal{B}_\\epsilon(\\rho)}D_{\\max}(\\tilde\\rho\\vert\\vert\\sigma),\n\\end{align}\nwith\n$\\mathcal{B}_\\epsilon(\\rho)=\\{\\tilde\\rho\\in\\mathcal{S}_\\leq(\\mathcal{H})\\vert\nP(\\tilde\\rho,\\rho)\\leq\\epsilon\\}$ the purified-distance-ball around\n$\\rho$ so that the optimization is over all subnormalized states\n$\\tilde\\rho$ $\\epsilon$-close to $\\rho$ with respect to the purified\ndistance. The latter is given by\n$P(\\rho,\\sigma)=\\sqrt{1-F^2(\\rho,\\sigma)}$.\n\n\\subsection{Definition of the conditional\n  entropies} \\label{sec_entrdef}\n\nWe define the new entropy $H_H^\\epsilon(A\\vert B)_\\rho$, in terms of\nthe relative entropy we have already introduced, as follows:\n\\begin{align}\n  H_H^\\epsilon(A\\vert\n  B)_\\rho&:=-D_H^{\\epsilon}(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes\\rho_B)\n\\end{align}\nIn the smooth entropy framework, two variants of the min- and max- entropies are given\nby:~\\cite{tomamichel_duality_2010,Koenig2009IEEE_OpMeaning,TSSR11}\n\\begin{align}\n  {H^\\epsilon_{\\min}(A\\vert B)_{\\rho|\\sigma}} &:=\n  -D_{\\max}^\\epsilon(\\rho_{AB}\\VertI}{\\mathbb{I}{I}_A\\otimes\\sigma_B) ,\\\\\n  H^\\epsilon_{\\max}\\left(A\\vert B\\right)_{\\rho|\\sigma} &:=\n  {-D_{\\min}^\\epsilon(\\rho_{AB}\\VertI}{\\mathbb{I}{I}_A\\otimes\\sigma_B)}\\ ,\n\\end{align}\n\\begin{align}\n  {H^\\epsilon_{\\min}(A\\vert B)_\\rho} &:=\n  \\max_{\\tilde\\rho\\in\\mathcal{B}_\\epsilon(\\rho)} \\max_{\\sigma_B\\in\\mathcal S_\\leq (\\mathcal H_B)}\\;\n  -D_{\\max}(\\tilde\\rho_{AB}\\VertI}{\\mathbb{I}{I}_A\\otimes\\sigma_B) ,\\\\\n  H^\\epsilon_{\\max}\\left(A\\vert B\\right)_\\rho &:=\n  \\min_{\\tilde\\rho\\in\\mathcal{B}_\\epsilon(\\rho)} \\max_{\\sigma_B\\in\\mathcal S_\\leq (\\mathcal H_B)}\\;\n  {-D_{\\min}(\\tilde\\rho_{AB}\\VertI}{\\mathbb{I}{I}_A\\otimes\\sigma_B)}\\ .\n\\end{align}\nThe non-smoothed versions $H_{\\min}(A|B)$ and $H_{\\max}(A|B)$\nare given by setting $\\epsilon=0$. In both cases, the optimal $\\sigma$ is a normalized state, i.e.\\ it is sufficient to restrict the maximization to $\\sigma_B\\in\\mathcal S(\\mathcal H_B)$.\n\n\nFor the special case when $\\epsilon\\rightarrow 0$,\n$H_H^\\epsilon(A\\vert B)$ converges to $H_{\\min}(A\\vert\nB)_{\\rho|\\rho}$ since for the optimal solutions to the semi-definite program as defined below $X\\rightarrow 0$. In\nthe case where one is also not conditioning on any B-system (i.e. take\n$B$ to be a trivial system, or take\n$\\rho_{AB}=\\rho_{A}\\otimes\\rho_{B}$), then $H_H^\\epsilon$ reduces to\nthe min-entropy:\n\\begin{equation}\n  \\lim_{\\epsilon\\rightarrow0}H_H^\\epsilon(A)_\\rho=H_{\\min}(A)_\\rho=-\\log \\vert\\vert\\rho_A\\vert\\vert_\\infty.\n\\end{equation}\nNote also that $H_H^\\epsilon$ is monotonically increasing in\n$\\epsilon$: to see this, observe that the dual optimal $\\{\\mu,X\\}$ for\n$2^{H_H^\\epsilon}$ (see below) is also feasible for $2^{H_H^{\\epsilon'}}$ with\n$\\epsilon'\\geq\\epsilon$.\n\n\\subsection{Elementary Properties} \\label{sec_basicproperties} As we\nare going to show in this section, the quantities $D_H^\\epsilon$ and\n$H_H^\\epsilon$ we introduced satisfy many desirable properties one\nwould expect from an entropy measure.\n\n\\subsubsection{Properties of $D_H^\\epsilon$}\n\n$D_H^\\epsilon$ can be expressed in terms of a semi-definite program, meaning it can be efficiently approximated. Due to strong duality we obtain two equivalent expressions with optimal solutions linked by complementary slackness conditions~\\cite{boyd_convex_2004}.\nThe semi-definite program for $2^{-D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)}$ reads:\\\\\n\\\\\n\\begin{minipage}[t]{0.24\\textwidth}\n  PRIMAL\\\\\n\n  minimize\\\\\n  subj. to\\\\\n\\end{minipage}\n\\begin{minipage}[t]{0.24\\textwidth}\n  \\text{}\\\\\n  \\\\\n  $\\frac{1}{\\epsilon}$Tr[Q$\\sigma$]\\\\\n  Q$\\leqI}{\\mathbb{I}{I}$\\\\\n  Tr[Q$\\rho$]$\\geq\\epsilon$\\\\\n  $Q\\geq 0$\n\n\\end{minipage}\n\\begin{minipage}[t]{0.24\\textwidth}\n  DUAL\\\\\n\n  maximize\\\\\n  subj. to\\\\\n\\end{minipage}\n\\begin{minipage}[t]{0.24\\textwidth}\n  \\text{}\\\\\n  \\\\\n  $\\mu-\\frac{\\text{Tr} [X]}{\\epsilon}$\\\\\n  $\\mu\\rho\\leq \\sigma + $X\\\\\n  $X\\geq 0$\\\\\n$\\mu\\geq 0$\n\n\\end{minipage}\n\n\\text{}\\\\\n\\\\\nThis yields the following complementary slackness conditions for\nprimal and dual optimal solutions $\\{Q\\}$ and $\\{ \\mu,X \\}$:\n\\begin{align}\n  (\\mu\\rho-X)Q&=\\sigma Q\\\\\n  \\operatorname{Tr}[Q\\rho]&=\\epsilon\\\\\n  QX&=X\n\\end{align}\nfrom which we can infer that $[Q,X]=0$, as well as the fact that the positive part of $(\\mu\\rho-\\sigma)$ is in the eigenspace of $Q$ with eigenvalue 1.\\\\\n\\\\\nFurther properties include\n\\begin{prop}[Positivity]\n  For any $\\rho, \\sigma\\in\\mathcal{S}(\\mathcal{H})$,\n  \\begin{equation}\n    D_H^\\epsilon(\\rho\\vert\\vert\\sigma)\\geq 0,\n  \\end{equation}\n  with equality if $\\rho=\\sigma$.\n\\end{prop}\n\\begin{proof}\n  Positivity follows immediately from the definition of $D_H^\\epsilon$\n  by choosing $Q=\\nobreak \\epsilonI}{\\mathbb{I}{I}$.  Equality is achieved if\n  $\\rho=\\sigma$ because\n  $\\frac1\\epsilon\\min_{\\operatorname{Tr}(Q\\rho)\\geq\\epsilon}\\operatorname{Tr}(Q\\rho)=1$.\n\\end{proof}\nNote that $D_H^\\epsilon\\left(\\rho\\Vert\\sigma\\right)=0$ does not\ngenerally imply $\\rho=\\sigma$: for example, consider the case where\n$\\epsilon=1$ and where $\\rho$ and $\\sigma$ have same support.\\\\\n\nThe following property relates the hypothesis testing relative entropy to the Trace Distance. Both the proposition and its proof are due to Marco Tomamichel \\cite{EmailMarcoTom}.\n\\begin{prop}[Relation to trace distance]\nFor any $\\rho, \\sigma\\in\\mathcal{S}(\\mathcal{H})$, $0<\\epsilon<1$ and $\\delta=D(\\rho,\\sigma)$ the trace distance between $\\rho$ and $\\sigma$,\n\\begin{equation}\n\\log\\frac{\\epsilon}{\\epsilon-(1-\\epsilon)\\delta}\\leq D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)\\leq \\log\\frac{\\epsilon}{\\epsilon-\\delta}.\n\\end{equation}\nIn particular, we have the Pinsker-like inequality $\\frac{1-\\epsilon}{\\epsilon}\\cdot D(\\rho,\\sigma)\\leq D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)$. Furthermore, the proposition implies that for $0<\\epsilon<1$, $D_H^\\epsilon(\\rho\\vert\\vert\\sigma)= 0$ if and only if $\\rho=\\sigma$, inheriting this property from the trace distance.\n\\end{prop}\n\\begin{proof}\nThe trace distance can be written as\n\\begin{equation}\nD(\\rho,\\sigma)=\\max_{0\\leq Q\\leq 1} \\operatorname{Tr}(Q(\\rho-\\sigma))=\\operatorname{Tr}(\\{\\rho>\\sigma\\}(\\rho-\\sigma)),\n\\end{equation}\nwhere $\\{\\rho>\\sigma\\}$ denotes the projector onto the positive part of $(\\rho-\\sigma)$. We thus immediately have that $\\operatorname{Tr}(Q(\\rho-\\sigma))\\leq\\delta=D(\\rho,\\sigma)$ for all $0\\leq Q\\leq I}{\\mathbb{I}{I}$, and so $\\operatorname{Tr}(Q\\sigma)\\geq\\operatorname{Tr}(Q\\rho)-\\delta\\geq \\epsilon-\\delta$ for $Q$ the optimal choice in $D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)$. This directly implies that $2^{-D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)}\\geq \\frac{\\epsilon-\\delta}{\\epsilon}$. This proves the upper bound.\\\\\n\\\\\nFor the lower bound, we may choose $0\\leq \\tilde Q\\leq I}{\\mathbb{I}{I}$ as\n\\begin{equation}\n\\tilde Q=(\\epsilon-\\mu)I}{\\mathbb{I}{I}+(1-\\epsilon+\\mu)\\{\\rho>\\sigma\\},  \\quad\\text{where } \\mu=\\frac{(1-\\epsilon)\\operatorname{Tr}(\\{\\rho>\\sigma\\}\\rho)}{1-\\operatorname{Tr}(\\{\\rho>\\sigma\\}\\rho)}.\n\\end{equation}\nHence, $\\mu=(1-\\epsilon+\\mu)\\operatorname{Tr}(\\rho\\{\\rho>\\sigma\\})$ and thus\n\\begin{equation}\n\\operatorname{Tr}(\\tilde Q\\rho)=(\\epsilon-\\mu)+(1-\\epsilon+\\mu)\\operatorname{Tr}(\\rho\\{\\rho>\\sigma\\})=\\epsilon.\n\\end{equation}\nMoreover,\n\\begin{equation}\n\\operatorname{Tr}(\\tilde Q\\sigma)=\\epsilon-\\mu+(1-\\epsilon+\\mu)\\operatorname{Tr}(\\{\\rho>\\sigma\\}\\sigma)=\\epsilon-\\frac{(1-\\epsilon)\\delta}{1-\\operatorname{Tr}(\\{\\rho>\\sigma\\}\\rho)}\\leq \\epsilon-(1-\\epsilon)\\delta.\n\\end{equation}\nHence, $D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)\\geq\\log\\frac{\\epsilon}{\\epsilon-(1-\\epsilon)\\delta}$. For the Pinsker-like inequality, observe that $\\log\\frac{\\epsilon}{\\epsilon-(1-\\epsilon)\\delta}=-\\log(1-\\frac{(1-\\epsilon)\\delta}{\\epsilon})\\geq\\delta\\frac{1-\\epsilon}{\\epsilon}$.\n\\end{proof}\n\n\\begin{prop}[Data Processing Inequality (DPI)]\n  For any completely positive, trace non-increasing map $\\mathcal{E}$,\n  \\begin{equation}\n    D_H^\\epsilon(\\rho\\vert\\vert\\sigma)\\geq D_H^\\epsilon(\\mathcal{E}(\\rho)\\vert\\vert\\mathcal{E}(\\sigma)).\n  \\end{equation}\n\\end{prop}\n\\begin{proof}\n  For a proof of this DPI, see~\\cite{wang_one-shot_2012}.\n\\end{proof}\n\\begin{prop}[Asymptotic Equipartition Property]\n  Let\n  \\begin{align*}\n    D(\\rho\\vert\\vert\\sigma)=\\operatorname{Tr}[\\rho(\\log\\rho-\\log\\sigma)]\n  \\end{align*}\n  be the relative entropy between $\\rho$ and\n  $\\sigma$\\cite{Wehrl1978}. Then, for any $0<\\epsilon < 1$,\n  \\begin{align}\n    \\lim_{n\\rightarrow \\infty}\\tfrac{1}{n}\\,\n    D_H^{\\epsilon}(\\rho^{\\otimes n}\\vert\\vert\\sigma^{\\otimes\n      n})&=D(\\rho\\vert\\vert\\sigma).\n  \\end{align}\n\\end{prop}\n\\begin{proof}\n  From Stein's lemma\\cite{ogawa_strong_2000,hiai_proper_1991} it\n  immediately follows that\n  \\begin{align}\n    \\lim_{n\\rightarrow \\infty}\\tfrac{1}{n}\\,\n    D_H^{\\epsilon}(\\rho^{\\otimes n}\\vert\\vert\\sigma^{\\otimes\n      n})&=\\lim_{n\\rightarrow \\infty}-\\tfrac{1}{n} \\log\n    \\min\\tfrac{1}{\\epsilon} \\operatorname{Tr}\\{\\sigma^{\\otimes n}Q\\},\n    \\\\\n    &=D(\\rho\\vert\\vert\\sigma)-\\lim_{n\\rightarrow \\infty}\\tfrac{1}{n}\n    \\left(\\log\\tfrac{1}{\\epsilon}\\right)\n    \\\\\n    &=D(\\rho\\vert\\vert\\sigma),\n  \\end{align}\n  where the minimum is taken over $0\\leq Q \\leq 1$ such that\n  $\\operatorname{Tr}Q\\rho\\geq \\epsilon$.\n\\end{proof}\n\n\n\\subsubsection{Properties of $H_H^\\epsilon$}\n\\begin{prop}[Bounds]\n  \\label{prop:bounds}\n  For $\\rho_{AB}$ an arbitrary normalized quantum state and\n  $\\rho_{XB}$ a classical-quantum state,\n  \\begin{align}\n    -\\log|A|\\leq &H_H^\\epsilon(A|B)_\\rho\\leq \\log|A|,\\\\\n    0\\leq &H_H^\\epsilon(X|B)_\\rho\\leq \\log|X|.\n  \\end{align}\n  For classical-quantum states, $H_H^\\epsilon(X\\vert B)=0$ if $X$ is\n  completely determined by $B$ (so that\n  $\\operatorname{Tr}(\\rho_B^x\\rho_B^{x'})=0$ for any $x'\\neq x$), and the\n  entropy is maximal if X is completely mixed and independent of B\n  (i.e. $\\rho_{XB}=\\frac{1}{\\vert X \\vert}I}{\\mathbb{I}{I}_X\\otimes\\rho_B$).\n\\end{prop}\n\\begin{proof}\n  Start with the upper bound on $H_H^\\epsilon$, and choose\n  $\\epsilonI}{\\mathbb{I}{I}$ as a feasible $Q$:\n  \\begin{align}\n    2^{H_H^\\epsilon(A|B)_\\rho}\n    &=\\min_{\\operatorname{Tr}[Q_{AB}\\rho_{AB}]\\geq\n      \\epsilon}\\tfrac1\\epsilon\\operatorname{Tr}[Q_{AB}I}{\\mathbb{I}{I}_A\\otimes\n    \\rho_B]\n    \\\\\n    &\\leq\n    \\tfrac1\\epsilon\\operatorname{Tr}[\\epsilonI}{\\mathbb{I}{I}_{AB}I}{\\mathbb{I}{I}_A\\otimes\n    \\rho_B]\n    \\\\\n    &=|A|.\n  \\end{align}\nFor the lower bound we use the inequality $|A|I}{\\mathbb{I}{I}_A\\otimes \\rho_B\\geq\n\\rho_{AB}$, which holds for arbitrary quantum states $\\rho_{AB}$.\n To establish this inequality, define the superoperator $\\mathcal{E}$ as $\\mathcal{E}(\\rho)=\\frac1{d^2}\\sum_{j,k}(U^jV^k)\\rho(U^jV^k)^{\\dagger}$. Here, $d={\\rm dim}(\\mathcal{H})$ while $U$ and $V$ are unitary operators defined by $\\left|{j}\\right\\rangle=\\left|{j+1}\\right\\rangle$ and $V\\left|{k}\\right\\rangle=\\omega^{k}\\left|{k}\\right\\rangle$, for an orthonormal basis set $\\{\\left|{j}\\right\\rangle\\}_{j=0}^{d-1}$, $\\omega=e^{2\\pi i\/d}$, and where arithmetic inside the ket is taken modulo $d$. (The operators $U$ and $V$ are often called the discrete Weyl-Heisenberg operators, as they generate a discrete projective representation of the Heisenberg algebra.) Then it is easy to work out that $\\mathcal{E}\\otimesI}{\\mathbb{I}{I}[\\rho^{AB}]=\\frac{1}{|A|}I}{\\mathbb{I}{I}_A\\otimes \\rho_B$, which by the form of $\\mathcal{E}$ implies the sought-after inequality. Then, for the optimal $Q_{AB}$ in $H_H^\\epsilon(A|B)_\\rho$,  \n\\begin{align}\n2^{H_H^\\epsilon(A|B)_\\rho} &=\\frac1\\epsilon\\operatorname{Tr}[Q_{AB}\\,I}{\\mathbb{I}{I}_A\\otimes\\rho_B]\\\\\n&\\geq \\frac1{\\epsilon|A|}\\operatorname{Tr}[Q_{AB}\\rho_{AB}]\\\\\n&\\geq \\frac{1}{|A|}.\n\\end{align}\nClassical-quantum states $\\rho_{XB}$ obey $I}{\\mathbb{I}{I}_X\\otimes \\rho_B\\geq \\rho_{XB}$, as $\\sum_{x'} p_{x'} \\rho^{x'}_B\\geq p_x\\rho^x_B$ for all $x$. This  implies $H_H^\\epsilon(X|B)_\\rho\\geq  0$ by the same argument.\n\n  That the extremal cases are reached for the described cases follows\n  immediately from the respective definitions of $\\rho_{XB}$ and\n  $H_H^\\epsilon$.\n\\end{proof}\n\nSimilarly to $D_H^\\epsilon$, $H_H^\\epsilon$ also satisfies a data\nprocessing inequality\\footnote{This proof is adapted from the DPI\n  proof for a differently defined $H^\\epsilon$ in Tomamichel and\n  Hayashi~\\cite{tomamichel_hierarchy_2012}. }.\n\\begin{prop}[Data Processing Inequality]\n\n  \\label{prop:DataProcessingInequalityLea} \n  For any $\\rho_{AB}\\in\\mathcal{S}(\\mathcal{H_{AB}})$, let\n  $\\mathcal{E}:A\\rightarrow A'$ be a sub-unital TP-CPM, and\n  $\\mathcal{F} :B\\rightarrow B'$ be a TP-CPM. Then, for\n  $\\tau_{A'B'}=\\mathcal{E}\\circ\\mathcal{F}(\\rho_{AB})$,\n  \\begin{equation}\n    H_H^\\epsilon(A\\vert B)_\\rho \\leq H_H^\\epsilon(A'\\vert B')_\\tau\n  \\end{equation}\n\\end{prop}\n\\begin{proof}\n  Let $\\{\\mu,X_{AB}\\}$ be dual-optimal for $H_H^\\epsilon(A\\vert\n  B)_\\rho$. Starting from $\\mu\\rho_{AB}\\leq\n  I}{\\mathbb{I}{I}_A\\otimes\\rho_B+X_{AB}$ and applying\n  $\\mathcal{E}\\circ\\mathcal{F}$ to both sides of the inequality\n  yields:\n  \\begin{equation}\n    \\mu\\tau_{AB}\\leq \\mathcal{E}(I}{\\mathbb{I}{I}_A)\\otimes\\tau_{B'}+\\mathcal{E}\\circ\\mathcal{F}(X_{AB})\\leq I}{\\mathbb{I}{I}_{A'}\\otimes\\tau_{B'}+\\mathcal{E}\\circ\\mathcal{F}(X_{AB}).\n  \\end{equation}\n  Hence, $\\{\\mu,\\mathcal{E}\\circ\\mathcal{F}(X_{AB})\\}$ is dual\n  feasible for $H_H^\\epsilon(A'\\vert B')_\\tau$ and\n  $2^{H_H^\\epsilon(A'\\vert B')_\\tau}\\geq\n  \\mu-\\operatorname{Tr}(\\mathcal{E}\\circ\\mathcal{F}(X_{AB})\/\\epsilon)=2^{H_H^\\epsilon(A\\vert\n    B)_\\rho}$.\n\\end{proof}\n\\begin{prop}[Asymptotic Equipartition Property]\n  For any $0<\\epsilon < 1$, it holds that\n  \\begin{align}\n    \\lim_{n\\rightarrow \\infty}\\tfrac{1}{n}\\, H_H^{\\epsilon}(A^{n}\\vert\n    B^{n})_{\\rho^{\\otimes n}}&=H(A\\vert B)_\\rho ,\n  \\end{align}\n  where $H(A\\vert B)$ refers to the conditional von Neumann entropy.\n\\end{prop}\n\\begin{proof}\n  Using the asymptotic property of $D_H^\\epsilon$ derived from Stein's\n  lemma above, we can show for $H_H^\\epsilon(A\\vert B)$:\n  \\begin{align}\n    \\lim_{n\\rightarrow \\infty}\\tfrac{1}{n} (H_H^{\\epsilon}(A^{\\otimes\n      n}\\vert B^{\\otimes n})_\\rho)&=\\lim_{n\\rightarrow\n      \\infty}\\tfrac{1}{n} (-D_H^{\\epsilon}(\\rho^{\\otimes\n      n}\\vert\\vert(I}{\\mathbb{I}{I}_A\\otimes \\rho_B)^{\\otimes n}))\n    \\\\\n    &=-D(\\rho_{AB}\\vert \\vert I}{\\mathbb{I}{I}_A\\otimes \\rho_B)\n    \\\\\n    &=-\\operatorname{Tr}\\rho_{AB}(\\log\\rho_{AB}-\\logI}{\\mathbb{I}{I}_A\\otimes\\rho_B)\n    \\\\\n    &=H(AB)-\\operatorname{Tr}(\\rho_B\\log\\rho_B)\n    \\\\\n    &=H(AB)-H(B)\n    \\\\\n    &=H(A\\vert B) .\n  \\end{align}\n\\end{proof}\n\n\n\n\n\\section{Relation to (relative) min- and max-entropies} \\label{sec_smooth}\n\nThe following propositions relate the new quantities to smooth\nentropies. This guarantees an operational significance for\n$D_H^\\epsilon$ and $H_H^\\epsilon$ (see Section~\\ref{sec_opapproach}).\\footnote{Note that the lower bound on $D_H$ in \\eqref{eq:maxvsH} is similar to Lemma 17 of~\\cite{datta_strong_2011}.}\n\n\\begin{prop} {\\it Let $\\rho\\in\\mathcal{S}(\\mathcal{H_{AB}})$,\n    $\\sigma\\in\\mathcal{P}(\\mathcal{H_{AB}})$ and\n    $0<\\epsilon\\leq1$. Then,}\n  \\begin{align}\n    D_{\\max}^{\\sqrt {2\\epsilon}}(\\rho\\vert\\vert\\sigma)&\\leq D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)\\leq D_{\\max}(\\rho\\vert\\vert\\sigma)\\label{eq:maxvsH}\\\\\n    H_{\\min}^{\\sqrt{2\\epsilon}}(A\\vert B)_\\rho&\\geq\n    H_H^\\epsilon(A\\vert B)_\\rho\\geq H_{\\min}(A\\vert\n    B)_{\\rho\\vert\\rho}\n  \\end{align}\n\\end{prop}\n\\begin{proof}\n  The upper bound for $D_H^\\epsilon$ follows immediately from the fact\n  that $\\mu=2^{-D_{\\max}(\\rho\\vert\\vert\\sigma)}$ and $X=0$ are\n  feasible for $2^{-D_H^\\epsilon(\\rho\\vert\\vert\\sigma)}$ in the dual\n  formulation. For the lower bound, let $\\mu$ and X be dual-optimal\n  for $2^{-D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)}$. Now define\n  $G:=\\sigma^{1\/2}(\\sigma+X)^{-1\/2}$ and let $\\tilde\\rho:=G\\rho\n  G^\\dagger$. It thus follows that $\\mu \\tilde\\rho\\leq \\sigma$, and\n  hence $2^{-D_{\\max}(\\tilde\\rho\\vert\\vert\\sigma)}\\geq\\mu$. Since\n  $\\operatorname{Tr}[X]\\geq 0$, it holds that $\\mu\\geq\n  2^{-D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)}$, which implies that\n  $2^{-D_H^{\\epsilon}(\\rho\\vert\\vert\\sigma)}\\leq\n  2^{-D_{\\max}(\\tilde\\rho\\vert\\vert\\sigma)}$.\n\n  It is now left to prove that the purified distance between\n  $\\tilde\\rho$ and $\\rho$ does not exceed $\\sqrt{2\\epsilon}$: For this\n  we employ Lemma~\\ref{lem:aeplem}, from which we obtain the upper\n  bound $\\sqrt{\\smash[b]{\\frac{2}{\\mu}}\\operatorname{Tr}[X]}$. Together with $0\\leq\n  \\epsilon\\mu-\\operatorname{Tr}[X]$, this implies that\n  $P(\\rho,\\tilde\\rho)\\leq\\sqrt{2\\epsilon}$, which concludes the proof.\n\n  These bounds can now be rewritten to relate $H_H^\\epsilon$ to\n  $H_{\\min}^\\epsilon$. We have\n  \\begin{equation}\n    H_{\\min}^{\\sqrt{2\\epsilon}}(A\\vert B)_\\rho\\geq -D_{\\max}^{\\sqrt{2\\epsilon}}(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes\\rho_B)\\geq -D_H^\\epsilon(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes \\rho_B)= H_H^\\epsilon(A\\vert B)_\\rho.\n  \\end{equation}\n  In the other direction we find:\n  \\begin{equation}\n    H_H^\\epsilon(A\\vert B)_\\rho= -D_H^\\epsilon(\\rho_{AB}\\vert\n    \\vertI}{\\mathbb{I}{I}_A\\otimes \\rho_B)\\geq \n    -D_{\\max}(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes\\rho_B)\n    :=H_{\\min}(A\\vert B)_{\\rho\\vert\\rho}.\n  \\end{equation}\n\\end{proof}\n\n\\begin{prop} {\\it Let $\\rho\\in\\mathcal{S}(\\mathcal{H})$ and\n    $\\sigma\\in\\mathcal{P}(\\mathcal{H})$ have intersecting support, and\n    $0<\\epsilon\\leq1$. Then, }\n  \\begin{gather}\n    D_{\\min}(\\rho\\vert\\vert\\sigma)-\\log\\frac{1}{\\epsilon^2}\\leq D_H^{1-\\epsilon}(\\rho\\vert\\vert\\sigma)\\leq D_{\\min}^{\\sqrt{2\\epsilon}}(\\rho\\vert\\vert\\sigma)-\\log\\frac{1}{(1-\\epsilon)}\n    \\\\\n    H_{\\max}(A\\vert B)_{\\rho}+\\log\\frac{1}{\\epsilon^2}\\geq H_H^{(1-\\epsilon)}(A\\vert B)_\\rho\n  \\end{gather}\n\\end{prop}\n\\begin{proof}\n  We begin with the lower bound for $D_H^{1-\\epsilon}$. Let $\\mu$, Q,\n  and X be optimal for the primal and dual programs for\n  $2^{-D_H^{1-\\epsilon}(\\rho\\vert\\vert\\sigma)}$ and define\n  $Q^\\perp:=1-Q$.  Complementary slackness implies\n  $\\operatorname{Tr}[Q^\\perp\\rho]=\\epsilon$, $QX=X$ and\n  $Q(\\mu\\rho-\\sigma-X)=0$. Thus,\n  \\begin{equation}\n    Q(\\mu\\rho-\\sigma-X)=Q(\\mu\\rho-\\sigma)-X,\n  \\end{equation}\n  meaning $Q(\\mu\\rho-\\sigma)$ is hermitian and positive\n  semidefinite. This implies that $Q^\\perp(\\mu\\rho-\\sigma)$ is also\n  hermitian and $Q^\\perp(\\mu\\rho-\\sigma)\\leq 0$. Since\n  $Q+Q^\\perp=I}{\\mathbb{I}{I}$, this gives a decomposition of\n  $(\\mu\\rho-\\sigma)$ into positive and negative parts, and thus\n  $\\vert\\mu\\rho-\\sigma\\vert=Q(\\mu\\rho-\\sigma)-Q^\\perp(\\mu\\rho-\\sigma)$. We\n  can now proceed:\n  \\begin{align}\n    2^{-\\frac{1}{2}D_{\\min}(\\rho\\vert\\vert\\sigma)}&=\\left\\|\\sqrt{\\rho}\\sqrt{\\sigma}\\right\\|_1\\\\\n    &=\\frac{1}{\\sqrt\\mu}\\left\\|\\sqrt{\\mu\\rho}\\sqrt{\\sigma}\\right\\|_1\\\\\n   \n    &\\geq\\frac{1}{2\\sqrt\\mu}\\operatorname{Tr}[\\mu\\rho+\\sigma-\\vert\\mu\\rho-\\sigma\\vert]\\\\\n    &=\\frac{1}{2\\sqrt\\mu}\\operatorname{Tr}[\\mu\\rho+\\sigma-Q(\\mu\\rho-\\sigma)+Q^\\perp(\\mu\\rho-\\sigma)]\\\\\n    &=\\frac{1}{\\sqrt{\\mu}}\\operatorname{Tr}[Q\\sigma+\\mu Q^\\perp\\rho]\\\\\n    &\\geq \\sqrt\\mu\\operatorname{Tr}[Q^\\perp\\rho]\\\\\n    &=\\sqrt\\mu\\epsilon\\\\\n    &\\geq\\epsilon\\sqrt{\\mu-\\operatorname{Tr}[X]\/(1-\\epsilon)}\\\\\n    &=\\epsilon 2^{-\\frac{1}{2}D_H^{1-\\epsilon}(\\rho\\vert\\vert\\sigma)}.\n  \\end{align}\n  We have used that $\\vert\\vert\\sqrt A\\sqrt B\\vert\\vert_1\\geq\n  \\operatorname{Tr}[A+B-\\vert A-B\\vert]\/2$ for positive semidefinite\n  A, B (a variation of the trace distance bound on the fidelity; see\n  Lemma~A.2.6 of~\\cite{Renner2005}).\n\n  Now we prove the upper bound. Let Q be primal-optimal for\n  $2^{-D_H^{1-\\epsilon}(\\rho\\vert\\vert\\sigma)}$, define\n  $\\tilde\\rho:=Q^\\frac{1}{2}\\rho Q^\\frac{1}{2}$, and let $\\rho_{AB}$\n  be an arbitrary purification of $\\rho_A$. Conjugating both sides of\n  $\\rho_{AB}\\leq I}{\\mathbb{I}{I}$ by $Q^\\frac{1}{2}$, we obtain\n  $\\tilde\\rho_{AB}\\leq Q_A\\otimes I}{\\mathbb{I}{I}_B$.\n\n  The square of the fidelity between two subnormalized states $\\zeta$ and $\\eta$ can be written also in terms of an SDP, with $\\zeta_{AB}$ an arbitrary purification of $\\zeta_A$ \\cite[Corollary 7]{watrous_semidefinite_2009}:\\footnote{Note that this formulation can be brought into the standard form defined in Section~\\ref{sec:sdp} by negating the objective functions and interchanging minimization with maximization.}\\\\\n\n \n  \\begin{minipage}[t] {0.23\\textwidth}\n    PRIMAL\\\\\n\n    maximize\\\\\n    subj. to\\\\\n  \\end{minipage}\n  \\begin{minipage}[t] {0.23\\textwidth}\n    \\text{}\\\\\n    \\\\\n    $\\text{Tr}[\\zeta_{AB}X_{AB}]$\\\\\n    $\\text{Tr}_B[X_{AB}]=\\eta_A$\\\\\n    $X_{AB}\\geq 0$\n\n  \\end{minipage}\n  \\begin{minipage}[t] {0.23\\textwidth}\n    DUAL\\\\\n\n    minimize\\\\\n    subj. to\\\\\n  \\end{minipage}\n  \\begin{minipage}[t] {0.23\\textwidth}\n    \\text{}\\\\\n    \\\\\n    $\\text{Tr}[Z\\eta]$\\\\\n    $\\zeta_{AB}\\leq Z_A\\otimes I}{\\mathbb{I}{I}_B$\\\\\n    $Z\\geq 0$\n\n  \\end{minipage}\n\n  \\text{}\\\\\n  \\\\\n  Hence, we see that $Q$ is a feasible $Z_A$ in the SDP for $\\left\\|\\sqrt{\\tilde\\rho}\\sqrt{\\sigma}\\right\\|_1^2$. Hence,\n  \\begin{align}\n    2^{-D_{\\min}(\\tilde\\rho\\vert\\vert\\sigma)}&=\\left\\|\\sqrt{\\tilde\\rho}\\sqrt{\\sigma}\\right\\|_1^2\t\\\\\n    &\\leq \\operatorname{Tr}[Q\\sigma]\\\\\n    &=(1-\\epsilon)2^{-D_H^{(1-\\epsilon)}(\\rho\\vert\\vert\\sigma)},\n  \\end{align}\n  and so $D_{\\min}(\\tilde\\rho\\vert\\vert\\sigma)\\geq\n  D_H^{(1-\\epsilon)}(\\rho\\vert\\vert\\sigma)+\\log\\frac{1}{1-\\epsilon}$.\n\n  From complementary slackness we get that\n  $\\operatorname{Tr}[Q\\rho]=1-\\epsilon$. Using Lemma~\\ref{lem:unclem}\n  we obtain $P(\\tilde\\rho,\\rho)\\leq\\sqrt{1-\\operatorname{Tr}[Q\\rho]^2}\n  \\leq\\sqrt{2\\epsilon}$, and the first part of the proposition\n  follows.\\\\\n\\\\\n \n\n  Rewriting this for $H_{\\max}$ and $H_H^{(1-\\epsilon)}$\n  yields:\n  \\begin{align}\n    H_{\\max}(A\\vert B)_{\\rho}&\\geq H_{\\max}(A\\vert B)_{\\rho\\vert\\rho}\\\\\n    &= -D_{\\min}(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes\\rho_{B})\\\\\n    &\\geq -D_H^{1-\\epsilon}(\\rho_{AB}\\vert\\vertI}{\\mathbb{I}{I}_A\\otimes\\rho_{B})-\\log\\frac{1}{\\epsilon^2}\\\\\n    &= H_H^{(1-\\epsilon)}(A\\vert B)_\\rho-\\log\\frac{1}{\\epsilon^2}\n   \n   \n  \\end{align}\n\\end{proof}\n\n\\section{Decomposition of Hypothesis Tests \\& Entropic Chain Rules}\n\\label{sec:decomp}\n\\label{sec_chainrule}\nIn this section we prove a bound on hypothesis testing between arbitrary states $\\rho$ and states $\\sigma$ invariant under a group action, in terms of hypothesis tests between $\\rho$ and its group symmetrized version $\\xi$ and $\\xi$ and $\\sigma$. This bound yields a chain rule for the hypothesis testing entropy. \n For a group $G$ and unitary representation $U_g$, let $\\mathcal{E}_G(\\rho)=\\frac1{|G|}\\sum_{g\\in G}U_g\\rho U^\\dagger_g$, which is a quantum operation.  (For simplicity of presentation we assume the group is finite, but the argument applies to continuous groups as well.)\n\n\\begin{prop}\nFor any $\\rho,\\sigma\\in\\mathcal{S}(\\mathcal{H})$ and group $G$ such that $\\sigma=\\mathcal{E}_G(\\sigma)$, let $\\xi=\\mathcal{E}_G(\\rho)$. Then, for $\\epsilon,\\epsilon'> 0$,\n\\begin{align}\nD_H^{\\epsilon+\\sqrt{2\\epsilon'}}(\\rho||\\sigma)\\leq D_H^{\\epsilon}(\\rho||\\xi)+D_H^{\\epsilon'}(\\xi||\\sigma)+\\log\\frac{\\epsilon+\\sqrt{2\\epsilon'}}{\\epsilon}.\n\\end{align}\n\\end{prop}\n\\begin{proof}\nLet $\\mu_1$ and $X_1$ be optimal in the dual program of $D_H^{\\epsilon}(\\rho||\\xi)$ and, similarly, $\\mu_2$ and $X_2$ be optimal in $D_H^{\\epsilon'}(\\xi||\\sigma)$. Thus,\n$\\mu_1\\rho\\leq \\xi+X_1$ and $\\mu_2\\xi\\leq \\sigma+X_2$. Observe that $X_2$ can be chosen $G$-invariant without loss of generality, since $\\mu_2\\xi\\leq \\sigma+\\mathcal{E}_G(X_2)$ and $\\operatorname{Tr}[X_2]=\\operatorname{Tr}[\\mathcal{E}_G(X_2)]$. \n\n\nChaining the inequalities gives\n\\begin{align}\n\\mu_1\\mu_2\\rho\\leq \\sigma+X_2+\\mu_2 X_1.\n\\end{align}\nNext, define $T=\\sigma^{\\frac12}(\\sigma+X_2)^{-\\frac12}$ and conjugate both sides of the above by $T$. This gives \n\\begin{align}\n\\mu_1\\mu_2T\\rho T^\\dagger \\leq \\sigma+\\mu_2 TX_1T^\\dagger.\n\\end{align}\nThus, the pair $\\mu_1\\mu_2$, $\\mu_2TX_1T^\\dagger$ is feasible for $D_H^\\epsilon(T\\rho T^\\dagger||\\sigma)$. \nSince $T$ is a contraction ($TT^\\dagger\\leq I}{\\mathbb{I}{I}$), we can proceed as follows:\n\\begin{align}\n2^{-D_H^\\epsilon(T\\rho T^\\dagger \\vert\\vert\\sigma)}&\\geq\\mu_1\\mu_2-\\frac{\\mu_2\\operatorname{Tr}[TX_1T^\\dagger]}{\\epsilon}\\\\\n&\\geq \\mu_1\\mu_2-\\frac{\\mu_2\\operatorname{Tr} X_1}{\\epsilon}\\\\\n&=\\mu_2 2^{-D_H^\\epsilon(\\rho||\\xi)}\\\\\n&\\geq 2^{-D_H^{\\epsilon'}(\\xi||\\sigma)}2^{-D_H^\\varepsilon(\\rho||\\xi)}.\n\\label{eq:halfchainrule}\n\\end{align}\nNow we show that $P(\\rho,T\\rho T^\\dagger)\\leq \\sqrt{2\\epsilon'}$, in order to invoke Lemma~\\ref{lem:smoothing}. Let the isometry $V:{\\mathcal{H}_A\\rightarrow \\mathcal{H}_A\\otimes\\mathcal{H}_R}$ be a Stinespring dilation of $\\mathcal{E}_G$, so that\n$\\overline{\\xi}_{AR}=V_{A\\rightarrow AR}\\rho_A V_{A\\rightarrow AR}^\\dagger=\\frac1{|G|}\\sum_{g,g'\\in G}U_g \\rho U^\\dagger_{g'}\\otimes\\left|{g}\\right\\rangle\\left\\langle{g'}\\right|$. \nThe state $\\overline{\\xi}_{AR}$ is an extension of $\\xi_A$ since $\\xi_A=\\operatorname{Tr}_R[\\overline{\\xi}_{AR}]$. Clearly $T_A\\overline{\\xi}_{AR}T_A^\\dagger$ is an extension of $T\\xi T^\\dagger$. We now apply Lemma~\\ref{lem:aeplem} to the inequality $\\xi\\leq \\sigma\/\\mu_2+X_2\/\\mu_2$, noting that the contraction in the lemma is just the operator $T$, to find \n\\begin{align}\n\tP(\\bar{\\xi}_{AR}, T_A \\bar{\\xi}_{AR} T_A^{\\dagger}) &\\leqslant \\sqrt{\\frac{\\operatorname{Tr}[X_2]}{\\mu_2}\\left( 2 - \\frac{\\operatorname{Tr}[X_2]}{\\mu_2} \\right)}\\\\\n\t&\\leq \\sqrt{2 \\epsilon'}.\n\\end{align}\n\nThis entails that\n\\begin{align}\n\tP(\\rho, T\\rho T^{\\dagger}) &= P(V \\rho_A V^{\\dagger}, V T \\rho T^{\\dagger} V^{\\dagger})\\\\\n\t&= P(V \\rho_A V^{\\dagger}, T V \\rho V^{\\dagger}T^{\\dagger} )\\\\\n\t&= P(\\bar{\\xi}_{AR}, T_A \\bar{\\xi}_{AR} T_A^{\\dagger})\\\\\n\t&\\leqslant \\sqrt{2 \\epsilon'},\n\\end{align}\nwhere we have used the fact that $T_A$ commutes with $V_{AR}$. This then implies that $\\tfrac12||\\rho - T\\rho T^\\dagger ||_1\\leq \\sqrt{2\\epsilon'}$. Lemma~\\ref{lem:smoothing} and (\\ref{eq:halfchainrule}) then yields the proposition: \n\\begin{align}\nD_H^{\\epsilon+\\sqrt{2\\epsilon'}}(\\rho||\\sigma)+\\log\\frac{\\epsilon}{\\epsilon+\\sqrt{2\\epsilon'}}&\\leq\nD_H^\\epsilon(T\\rho T^\\dagger ||\\sigma)\\\\\n&\\leq D_H^\\epsilon(\\rho||\\xi)+D_H^{\\epsilon'}(\\xi||\\sigma).\n\\end{align}\n\\end{proof}\n\n\n\\begin{corollary}[Chain rule for $H_H^\\epsilon$]\n {\\it Let $\\rho_{ABC}\\in\\mathcal{S}(\\mathcal{H})$ be an arbitrary normalized state, and $\\epsilon,\\epsilon'>0$. Then, }\n\\begin{equation}\nH_H^{\\epsilon+\\sqrt{8\\epsilon'}}(AB\\vert C)_\\rho\\geq H^\\epsilon(A\\vert BC)_\\rho+H^{\\epsilon'}(B\\vert C)_\\rho-\\log\\frac{\\epsilon+\\sqrt{2\\epsilon'}}{\\epsilon}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nLet $G$ be the Weyl-Heisenberg group representation (as in the proof of Prop~\\ref{prop:bounds}) acting on $A$, for which $\\mathcal{E}_G(\\rho_{ABC})=\\pi_A\\otimes \\rho_{BC}$, where $\\pi_A=I}{\\mathbb{I}{I}\/{\\rm dim}(\\mathcal{H}_A)$. Applied to the hypothesis test between $\\rho_{ABC}$ and $\\pi_{AB}\\otimes \\rho_C$, we find\n\\begin{align}\n&\\!\\!D_H^{\\epsilon+\\sqrt{8\\epsilon'}}(\\rho_{ABC}||\\pi_{AB}\\otimes\\rho_C)\\nonumber\\\\\n&\\leq D_H^{\\epsilon}(\\rho_{ABC}||\\pi_A\\otimes\\rho_{BC})+D_H^{\\epsilon'}(\\pi_A\\otimes \\rho_{BC}||\\pi_{AB}\\otimes\\rho_C)+\\log\\frac{\\epsilon+\\sqrt{2\\epsilon'}}{\\epsilon}\\\\\n&\\leq D_H^{\\epsilon}(\\rho_{ABC}||\\pi_A\\otimes\\rho_{BC})+D_H^{\\epsilon'}( \\rho_{BC}||\\pi_{B}\\otimes\\rho_C)+\\log\\frac{\\epsilon+\\sqrt{2\\epsilon'}}{\\epsilon}.\n\\end{align}\nAs $H_H^\\epsilon(A|B)_\\sigma=\\log d_A-D_H^\\epsilon(\\sigma_{AB}||\\pi_A\\otimes \\sigma_B)$, this is equivalent to the desired result.\n\\end{proof}\n\n\\section*{Acknowledgements}\nWe acknowledge discussions with Marco Tomamichel. Research leading to\nthese results was supported by the Swiss National Science Foundation\n(through the National Centre of Competence in Research `Quantum\nScience and Technology' and grant No. 200020-135048) and the European\nResearch Council (grant No. 258932).\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Introduction}\nThe dynamics of granular systems in general and granular gases in particular is of much current\ninterest \\cite{IGreview}. Except for the physical  dimensions of typical macroscopic grains, the\nmain difference between granular and molecular many-body systems is the dissipative nature of the\ninteractions in the former. This property has far reaching consequences, many of which are a-priori\ncounterintuitive.\n\nWhile much of the theory of granular solids and quasi-static granular flows is of phenomenological\nnature, granular gases seem to be well described by kinetic theory  \\cite{IGreview},  i.e. the\nBoltzmann \\cite{Sela_Gold,Dufty}, or Enskog-Boltzmann equations \\cite{Gol'd_Sok,densegrangases},\nwith the possible exception of strongly inelastic systems. Attempts to go beyond the Boltzmann\nlevel of description  are noted \\cite{GKIG,GK2,vannoije}, but so far these directions have not been\nfully exploited.\n\nThe Bogolyubov method for deriving kinetic equations for many-body systems \\cite{Bog,Bogol} is\nbased on an assumption known as the functional hypothesis. According to it, for long times\n$t\\gg\\tau_0$, many-particle distribution functions become functionals of the corresponding\none-particle distribution function. Characteristic time $\\tau_{0}$ is of the order of the typical\nduration of a collision (for hard sphere collisions, $\\tau_{0}$ is the time in which a particle\ntraverses a distance that equals its diameter). The Bogolyubov functional hypothesis can be\nconsidered as a generalization of the Chapman-Enskog method for deriving hydrodynamic equations on\nthe basis of kinetic equation. It should also be emphasized that the term ``functional hypothesis\"\nis not, in some measure, adequate because this statement was proved for some important cases (see,\ne.g., \\cite{Akh_Pel}). In fact, the first proof of the functional hypothesis was given by Gilbert\nin discussion of solutions of the Boltzmann equation.\n\nAn important component of the Bogolyubov method is the principle of spatial correlations weakening,\nwhich reflects statistical independence of physical values at distant spatial points. One can\nconsider it as a reasonable mixing property of many-particle distribution functions. The impact of\nthe Bogolyubov approach on kinetic theory is described in \\cite{Cohen}, and a detailed exposition\nof his ideas in this field and some of their applications can be found in the monograph\n\\cite{Akh_Pel}, whose techniques we generalize here to render them applicable to dissipative\nsystems.\n\nThe present article has two main goals. The first is to present a (rather straightforward)\ngeneralization of the standard models of granular gas collisions by introducing a dissipation\nfunction in conjunction with a Hamiltonian formulation of classical mechanics for dissipative\nsystems. The use of dissipation functions is of course not new, but their application to granular\nsystems seems to be novel. On the basis of this formulation we develop a BBGKY hierarchy for\ndissipative systems. A BBGKY hierarchy for a system of hard inelastically colliding spheres, which\nis based on a pseudo-Liouville equation, is presented e.g., in \\cite{vannoije,Dufty-1}. However, in\nthis approach there is a problem of adequacy of description of many-body dynamics by using a\npseudo-Liouville equation.\n\nThe second goal of this article is to implement the Bogolyubov method to the derivation of kinetic\nequations to the case of dissipative many-body classical systems. As mentioned, this approach is\nbased on the functional hypothesis. This conjecture seems to be borne out by all studied\nnonequilibrium systems we are aware of. In some non-trivial cases, such as the properties of\nnon-equilibrium steady states, it was shown to successfully reproduce results obtained by other\nmethods \\cite{SokOpp}. Moreover, the Bogolyubov method enables to study, for example, the problem\nof convergence and non-analyticity arising in a perturbation theory \\cite{Sok}. Whether in the\nrealm of granular systems it will yield novel results, which the commonly used methods are\nincapable of producing, is at present unclear. However, given the difficulties one faces when\nstudying granular systems we believe it is important to explore the possibilities afforded by an\nalternate formulation. Therefore, this article is devoted to the exposition of the Bogolyubov\nformulation of the kinetics of dissipative gases. As a simple test, it is shown that in the limit\nof a dilute collections of monodisperse spheres interacting by collisions characterized by a fixed\ncoefficient of normal restitution,  the present formulation reproduces the corresponding\n(inelastic) Boltzmann equation. Other cases, which can be treated perturbatively are presented\nbelow.\n\n\nThe structure of this paper is as follows. In section \\ref{formulation} we formulate a dissipative\ndynamics on the basis of Hamilton equations of motion and dissipation function. Then we derive the\ncorresponding Liouville equation and BBGKY hierarchy and formulate the Bogolyubov principle of\nspatial correlation weakening for many-particle distribution functions. Section \\ref{weak}\nintroduces the Bogolyubov functional hypothesis and boundary condition as necessary concept for\nsolving BBGKY hierarchy. It also formulates the basic equations for deriving kinetic descriptions\nin different limiting cases. In particular, it is shown how a kinetic equation in the limit of weak\ninteractions can be derived. Section \\ref{dilute} deals with kinetic theory in the low density\nlimit. Here a generalized Boltzmann equation for gases with dissipative interactions is obtained.\nSection \\ref{weakdisskin} is devoted to a derivation of a generalized Boltzmann equation for the\ncase of weak dissipation. At the end of this section a sketch of the theory of homogeneous cooling\nstates is presented. Section \\ref{spheres} discusses the connection of the proposed kinetic theory\nof gases in the presence of dissipative interaction with the Bolztmann equation for inelastic rigid\nspheres. Finally, Section \\ref{conclusion} comprises a brief summary and outlook.\n\n\n\n\\section{Formulation, the  Liouville equation and the BBGKY\nhierarchy}\\label{formulation}\n\nConsider a system composed of $N$ identical classical particles of\nmass $m$ each.  Their reversible interactions are assumed to be\nderivable from a Hamiltonian, $H$, and their dissipative\ninteractions are assumed to be determined by a dissipation\nfunction, $R$.  Both $H$ and $R$ are assumed to  depend on the\nspatial  coordinates of the centers of mass of the particles, $\\{\n{\\bf x}_{i};\\  1\\leqslant i \\leqslant N \\}$ and the respective\nmomenta  $\\{ \\vec p_i;\\ 1\\leqslant i \\leqslant N \\} $.   The\ngeneralized Hamilton equations are given by:\n\\begin{equation} \\lab{eq:2.1}\n\\dot p_{in}=-{\\partial H\\over\\partial\nx_{in}}-{\\partial R\\over\\partial p_{in}},\\qquad \\dot\nx_{in}={\\partial H\\over\\partial p_{in}},\n\\end{equation}\nwhere we assume for simplicity that the particles experience only\nbinary interactions:\n\\begin{equation}\\lab{eq:2.2}\nH=H_{0}+V=\\sum_{1\\leqslant i\\leqslant N}{{p}_{i}^{2}\\over\n2m}+\\sum_{1\\leqslant i<j\\leqslant N} V_{ij}, \\quad V_{ij}\\equiv\nV(\\vec x_{ij}),\n\\end{equation}\nand where $H_0$ denotes the kinetic part of the Hamiltonian. The\ndissipation function $R$ is taken to depend on the coordinate\nand momentum differences (to preserve Galilean invariance). For\nsake of simplicity it assumed to equal a sum of two-particle\ninteractions:\n\\begin{equation}\\lab{eq:2.3}\nR=\\sum_{1\\leqslant i<j\\leqslant N}R_{ij}, \\qquad R_{ij}\\equiv\nR(\\vec x_{ij},\\vec p_{ij}),\n\\end{equation}\nwhere ${\\bf x}_{ij}\\equiv {\\bf x}_{i}-{\\bf x}_{j}$ and ${\\bf p}_{ij} \\equiv {\\bf p}_{i}-{\\bf\np}_{j}$. As the entity  $R_{ij}$ is a scalar, it depends on  $\\vec p^2_{ij}$, $\\vec x^2_{ij}$ and\n${\\bf p}_{ij}{\\bf x}_{ij}$. The resulting equations of motion are invariant under Galilean\ntransformations, as they should, and conserve momentum but not energy. The force exerted by\nparticle $j$ on particle $i$ is defined as\n\\begin{equation}\\lab{eq:2.4}\n{F}_{ij,n}\\equiv{F_n}({\\vec x}_{ij},{\\vec p}_{ij})=-{\\partial\nV_{ij}\\over\\partial{x}_{in}}-{\\partial\nR_{ij}\\over\\partial{p}_{in}}\n\\end{equation}\nThe time derivative of total energy of\nthe system is given by\n\\begin{equation}\\lab{eq:2.4'}\n{dH\\over dt}=-\\sum_{1\\leqslant i\\leqslant N}{\\partial\nR\\over\\partial  p_{in}}{\\partial H\\over\\partial p_{in}}=\n-\\sum_{1\\leqslant i\\leqslant N}{p_{in}\\over m}\n{\\partial R\\over\\partial p_{in}}.\n\\end{equation}\n\nWe shall assume that the nature of dissipation is associated with friction of macroscopic\nparticles. Therefore, as a model dissipation function, we take $R_{ij}$ as follows \\cite{Landau}:\n\\begin{equation}\\lab{eq:2.4''}\nR_{ij}={1\\over 2}\\gamma({\\bf x}_{ij}){\\bf p}_{ij}^2,\n\\end{equation}\nmoreover, $\\gamma(\\vec x_{ij})$=0 when $|\\vec x_{ij}|\n\\gtrsim r_0$, where $r_{0}$ is the radius of dissipative\ninteraction, i.e. the dissipative force acts at the moment\nof contact of particles. According to (\\ref{eq:2.4'}),\n(\\ref{eq:2.4''}), one finds\n$$\n{dH\\over dt}=-{2\\over m}\\sum_{1\\leqslant i<j\\leqslant N}R_{ij}<0.\n$$\nSince $\\gamma({\\bf x}_{ij})>0$ (see \\cite{Landau}), the energy\ndissipation occurs.\n\n\nAs a precursor to the derivation of the corresponding Liouville\nequation for the system described by Eqs. \\p{eq:2.1}, we first\npresent some rather well known results concerning general systems\nof ordinary differential equations of the form:\n\\begin{equation} \\lab{eq:2.5}\n\\dot{x}_{i}(t)=h_{i}(x_{1}(t),...,x_{N}(t)), \\qquad 1\\leqslant i\\leqslant N.\n\\end{equation}\nIn particular, if $h_{i}$ is a random field, then Eqs. (\\ref{eq:2.5}) can be used to derive the kinetic description of stochastic systems \\cite{LaskPelPr}.\nDenote by $X_i(t,x)$ the solution of the Cauchy problem of this equation with initial condition\n$x\\equiv (x_1,x_2,...,x_N)$ ($X_i(x,0)\\equiv x_i$). It is well known that Eqs.~\\p{eq:2.5} admit the\nfollowing formal solution:\n\\begin{equation}\\lab{eq:2.6}\nX_{i}(t,x)=e^{t \\Lambda(x)}x_{i},\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:2.6'}\n\\Lambda(x)=\\sum_{1\\leqslant i\\leqslant\nN}h_{i}(x){\\partial\\over\\partial x_{i}}.\n\\end{equation}\nWhen the evolution operator $e^{t\\Lambda(x)}$ acts on an arbitrary function $\\varphi(x)$ the result\nis as follows:\n\\begin{equation}\\lab{eq:2.7}\ne^{\\tau{\\Lambda}(x)}\\varphi(x)= \\varphi(e^{\\tau{\\Lambda}(x)}x).\n\\end{equation}\nNotice that since Eqs. (\\ref{eq:2.5}) are autonomous, the solution (\\ref{eq:2.6}) can be inverted:\n\\begin{equation}\\lab{eq:2.8}\nx'_i\\equiv X_i(t,x) \\Rightarrow  x=X(-t,x').\n\\end{equation}\n\nDefine ${\\cal D}(x,0)$ to denote the probability distribution of the initial conditions $x$ (see\nEqs. (\\ref{eq:2.5})). Normalization requires that\n\\begin{equation}\\label{8.5}\n\\int dx {\\cal D}(x,0)=1\\quad (dx\\equiv dx_1...dx_N).\n\\end{equation}\nThe distribution function at time $t$ is, therefore, given by\n\\begin{equation}\\lab{eq:2.9}\n{\\cal D}(x,t)=\\int dx'{\\cal D}(x',0)\\prod_{1\\leqslant i\\leqslant\nN}\\delta(x_{i}-X_{i}(t,x')).\n\\end{equation}\nChanging the integration variables from $x'$ to $y=X(t,x')$ and using the relation \\p{eq:2.8}, one\nobtains\n\\begin{equation}\\lab{eq:2.10}\n{\\cal D}(x,t)=I(x,t){\\cal D}(X(-t,x),0),\n\\end{equation}\nwhere\n\\begin{equation} \\lab{eq:2.11}\nI(x,t)=\\bigg\\vert{\\partial X(-t,x)\\over\\partial x}\\bigg\\vert\n\\end{equation}\nis the Jacobian of the transformation $x\\to X(-t,x)$.\n\nFollowing \\p{eq:2.6}-\\p{eq:2.7} and expression \\p{eq:2.10}, it can be shown that the distribution\nfunction ${\\cal D}(x,t)$ satisfies the equation:\n\\begin{equation}\\lab{eq:2.13}\n{\\partial{\\cal D}\\over\\partial t}=\\left({\\partial I\\over\\partial t}-I\\sum_{1\\leqslant i\\leqslant\nN}h_{i}{\\partial\\over\\partial x_{i}}\\right)I^{-1}{\\cal D}.\n\\end{equation}\nThe equation of motion for the Jacobian $I(x,t)$ has the form\n\\begin{equation}\\lab{eq:2.14}\n{\\partial I(x,t)\\over\\partial t}+\\tilde \\Lambda (x)I(x,t)=0,\n\\qquad I(x,0)=1,\n\\end{equation}\nwith operator $\\tilde \\Lambda (x)$ defined by\n\\begin{equation}\\lab{eq:2.16'} \\tilde \\Lambda\n(x)\\varphi (x)\\equiv \\sum_{1\\leqslant i\\leqslant N}{\\partial\\over\\partial\nx_{i}}\\left(h_{i}(x)\\varphi(x)\\right).\n\\end{equation}\nIn deriving \\p{eq:2.14} we have employed the fact that $\\int dx{\\partial{\\cal D}\/\\partial t}=0$\n(see \\p{eq:2.9}) and that this relation should hold for any allowed initial distribution function.\nUpon elimination of $\\partial I\/\\partial t$ on the right-hand side of \\p{eq:2.13}, one obtains\n\\begin{equation}\\lab{eq:2.15}\n{\\partial{\\cal D}(x,t)\\over\\partial t}+\\tilde\\Lambda(x) {\\cal\nD}(x,t)=0.\n\\end{equation}\nA comparison of \\p{eq:2.10} with the solution of Eq. \\p{eq:2.15} gives the following operator\nrelation:\n\\begin{equation}\\lab{eq:2.17}\ne^{-t\\tilde\\Lambda(x)}=I(x,t)e^{-t\\Lambda(x)}.\n\\end{equation}\n\n\nAt this stage we return to the dynamical model \\p{eq:2.1}--\\p{eq:2.3}. In this case, the\ncorresponding Liouville equation \\p{eq:2.15} and the evolution equation \\p{eq:2.14} for the\nJacobian (where $x_{i}=({\\bf x}_{i},{\\bf p}_{i})$) assume the form:\n\\begin{equation}\\lab{eq:2.18}\n{\\partial{\\cal D}\\over\\partial t}-\\{H,{\\cal D}\\}=\\sum_{1\\leqslant\ni\\leqslant N} {\\partial\\over\\partial p_{in}} \\biggl({\\cal\nD}{\\partial R\\over\\partial p_{in}}\\biggr),\n\\end{equation}\n\\begin{equation}\\lab{eq:2.19}\n{\\partial I\\over\\partial t}-\\{H,I\\}=\\sum_{1\\leqslant i\\leqslant N}\n{\\partial\\over\\partial p_{in}} \\biggl(I{\\partial R\\over\\partial\np_{in}}\\biggr),\n\\end{equation}\nwhere $\\{A,B\\}$ is a Poisson bracket,\n$$\n\\{A,B\\}=\\sum_{1\\leqslant i\\leqslant N}\\biggl({\\partial\nA\\over\\partial x_{in}}{\\partial B\\over\\partial p_{in}}-{\\partial\nA\\over\\partial p_{in}}{\\partial B\\over\\partial x_{in}}\\biggr).\n$$\nThe operators $\\Lambda(x)$, $\\tilde\\Lambda(x)$, which are defined\nby \\p{eq:2.6'}, \\p{eq:2.16'}, become for the  case of Eqs.\n\\p{eq:2.1}:\n\\begin{equation}\\lab{eq:2.20}\n\\Lambda(x)=\\sum_{1\\leqslant i\\leqslant N}{p_{in}\\over m}\n{\\partial\\over\\partial x_{in}}+\\sum_{1\\leqslant i,j\\leqslant N}\nF_{ij,n} {\\partial\\over\\partial p_{in}},\n\\end{equation}\n\\begin{equation}\\lab{eq:2.21}\n\\tilde\\Lambda(x)=\\sum_{1\\leqslant i\\leqslant N}{p_{in}\\over m}\n{\\partial\\over\\partial x_{in}}+\\sum_{1\\leqslant i,j\\leqslant N}\n{\\partial\\over\\partial p_{in}} F_{ij,n}.\n\\end{equation}\n\nThe generalized Liouville equation \\p{eq:2.18} represents a basis for studying kinetics of systems\nwith dissipative interaction. The next step is to derive the corresponding generalized BBGKY\nhierarchy, which is a starting point for derivation of kinetic equations. The $s$-particle\ndistribution function is defined by\n\\begin{equation} \\lab{eq:2.22}\nf_{s}(x_{1},...,x_{s};t)=V^{s}\\int dx_{s+1}...dx_{N}{\\cal\nD}(x_{1},...,x_{N},t),\n\\end{equation}\nwhere $V$ is the volume of the system and $x_{i}=({\\bf x}_{i},{\\bf p}_{i})$. Using the Liouville\nequation \\p{eq:2.18}, one now straightforwardly obtains the desired hierarchy:\n\\begin{equation} \\lab{eq:2.24}\n{\\partial f_{s}\\over\\partial t}=-\\tilde\\Lambda_{s}f_{s}-\\sum_{1\\leqslant i\\leqslant\ns}{\\partial\\over\\partial p_{in}}{1\\over v}\\int dx_{s+1}f_{s+1}F_{i\\,s+1,n},\n\\end{equation}\nwhere $\\tilde\\Lambda_{s}$ is given by Eq.(\\ref{eq:2.21}) with $N=s$ and $1\/v \\equiv N\/V$ is the\nparticle number density. The generalized BBGKY hierarchy \\p{eq:2.24} reduces to the standard BBGKY\nhierarchy when the dissipation function vanishes.\n\nOur next goal is to solve the hierarchy \\p{eq:2.24} employing a perturbative approach. To this end\nwe shall take the Bogolyubov principle of spatial correlation weakening \\cite{Bogol} as a basis of\nour consideration. In terms of the many-particle distribution functions $f_{s}(x_{1},...,x_{s};t)$,\nthis principle states that\n\\begin{equation}\\lab{eq:2.26}\nf_{s}(x_{1},...,x_{s},t)\\xrightarrow[r\\to\\infty]{}\nf_{s'}(x_{1}',...,x_{s'}',t)f_{s''}(x_{1}'',...,x_{s''}'',t),\n\\end{equation}\nwhere two groups of $s'$ and $s''$ ($s=s'+s''$) particles are formed from $x_{1},...,x_{s}$ and $r$\nis the minimal distance between the particles from different groups. The relation \\p{eq:2.26} has a\nsimple physical meaning: the phase variables of particles are statistically independent at large\ndistance between particles. The property (\\ref{eq:2.26}) of $f_{s}(x_{1},...,x_{s},t)$ holds in the\nthermodynamic limit and it specifies a set of functions, in terms of which one should seek a\nsolution of the BBGKY hierarchy.\n\n\\section{ Kinetic stage of evolution} \\label{weak}\n\nThe present section is devoted to a description of a kinetic stage of evolution for the dissipative\nsystem under consideration. Following Bogolyubov, we assume his functional hypothesis as a basis of\nour investigation \\cite{Akh_Pel}. According to this hypothesis, for sufficiently large times,\nmany-particle distribution functions depend on time and initial distribution functions only through\none-particle distribution function \\cite{Bog,Bogol}:\n\\begin{equation}\\lab{eq:3.1}\nf_{s}(x_{1},...,x_{s},t)\\xrightarrow[t\\gg\\tau_{0}]{}\nf_{s}(x_{1},...,x_{s};f(t)),\n\\end{equation}\nwhere\n$$\nf_1(x_1,t)\\xrightarrow[t\\gg\\tau_{0}]{}f(x_1,t).\n$$\nHere $f_{s}(x_{1},...,x_{s};f)$ are functionals of the one-particle distribution function, $\\tau_0$\nis a microscopic time, usually estimated as a collision time. One can consider the above functional\nhypothesis \\p{eq:3.1} as a generalization of the Chapman-Enskog approach to the derivation of\nhydrodynamic equations proceeding from the Boltzmann equation.\n\nAccording to the functional hypothesis (\\ref{eq:3.1}), the functionals $f_{s}(x_{1},...,x_{s};f)$\nare universal because they do not depend on initial conditions for the many-particle distribution\nfunctions $f_{s}(x_{1},...,x_{s};t=0)$. These functionals can be calculated in a perturbative\napproach. To illustrate the subsequent steps in this direction, we first study a more simple\nperturbative approach for the case of weak interactions. The next section deals with the low\ndensity limit, which is adequate to the situation of granular systems, in particular granular\nfluids.\n\nFollowing Eqs. \\p{eq:2.24}, the single-particle distribution function satisfies the kinetic\nequation of the form\n\\begin{equation}\\lab{eq:3.5}\n{\\partial f(x_{1},t)\\over\\partial t}+{p_{1n}\\over m}{\\partial\nf(x_{1},t)\\over\\partial x_{1n}}=L(x_{1}; f(t)),\n\\end{equation}\nwhere the functional $L(x_{1},f)$ represents the generalized\ncollision integral,\n\\begin{equation}\\lab{eq:3.6}\nL(x_{1};f)=-{\\partial\\over\\partial p_{1n}}{1\\over v}\\int\ndx_{2}f_{2}(x_{1},x_{2};f)F_{12,n}.\n\\end{equation}\nThe Bogolyubov functional hypothesis \\p{eq:3.1}, in self-evident shorthand notation, gives:\n\\begin{equation}\\lab{eq:3.4}\n{\\partial f_{s}(f(t))\\over\\partial t}=\\int dx{\\left.{\\delta\nf_{s}(f)\\over\\delta f(x)}\\right |}_{f\\to f(t)}{\\partial\nf(x,t)\\over\\partial t},\n\\end{equation}\nwhere $\\delta f_{s}\/\\delta f$ denotes  a functional derivative. This relation and Eqs.\\p{eq:2.24}\nyield the following equation for the functional $f_s(f)$:\n\\begin{equation}\\lab{eq:3.7}\n-\\int dx{\\delta f_{s}(f)\\over\\delta f(x)}{p_n \\over m} {\\partial\nf(x)\\over\\partial x_n}+\\sum_{1\\leqslant i\\leqslant s}{ p_{in}\\over\nm}{\\partial f_{s}(f)\\over\\partial x_{in}}=K_{s}(f)\n\\end{equation}\nwhere we have introduced an auxiliary functional $K_s(f)$,\n$$\nK_{s}(f)=-\\sum_{1\\leqslant i,j\\leqslant s}{\\partial\\over\\partial\np_{in}}(f_{s}(f)F_{ij,n})-\n$$\n$$\n-\\sum_{1\\leqslant i\\leqslant s}{\\partial\\over\\partial p_{in}}{1\\over v}\\int\ndx_{s+1}f_{s+1}(f)F_{i\\,s+1,n}-\n$$\n$$\n-\\int dx{\\delta f_{s}(f)\\over\\delta f(x)}L(x;f).\n$$\n\nNext, following Bogolyubov again \\cite{Bog,Bogol}, in order to obtain an unambiguous solution to\nEq. \\p{eq:3.7} we need to formulate a asymptotical condition (\"boundary condition\") for the\nfunctionals $f_{s}(f)$. This condition should reflect the principle of spatial correlation\nweakening \\p{eq:2.26} and it should be written taking into account the evolution of the system in\nphysical direction of time \\cite{Bog,Bogol}. To this end, we introduce an auxiliary parameter\n$\\tau$, which has dimensions of time but does not represent physical time, and we use it in the\nfollowing manner:\n$$\ne^{-\\tau \\Lambda^0_s}f_s(x_1,...,x_s;f)=\n$$\n$$\n=f_{s}\\left({\\bf x}_{1}-{{\\bf p}_{1}\\over m}\\tau, {\\bf\np}_{1},...,{\\bf x}_{s}-{{\\bf p}_{s}\\over m}\\tau, {\\bf\np}_{s};f\\right)\\xrightarrow[\\tau \\sim +\\infty]{}\n$$\n\\begin{equation} \\lab{eq:3.8}\n\\prod_{1\\leqslant i\\leqslant s}f\\left({{\\bf x}_{i}-{{\\bf p}_{i}\\over m}\\tau}, {\\bf\np}_{i}\\right)=e^{-\\tau \\Lambda^0_s}\\prod_{1\\leqslant i\\leqslant s}f(x_i).\n\\end{equation}\nHere $e^{-t \\Lambda^0_s}$ is the evolution operator of $s$ free particles and $\\Lambda^0_s$ is\ngiven by the first term in \\p{eq:2.20} (see also \\p{eq:2.21}):\n\\begin{equation} \\lab{eq:3.9}\n\\Lambda_{s}^{0}=\\sum_{1\\leqslant i\\leqslant s}{p_{in}\\over m}{\\partial\\over\\partial x_{in}}.\n\\end{equation}\nThe asymptotical condition \\p{eq:3.8} can be written in a more compact form,\n\\begin{equation} \\lab{eq:3.10}\n\\lim_{\\tau \\to +\\infty}e^{-\\tau\n\\Lambda^0_s}f_s(x_1,...,x_s;e^{\\tau\n\\Lambda^0_1}f)=f^0_s(x_1,...,x_s;f),\n\\end{equation}\nwhere\n\\begin{equation} \\lab{eq:3.11}\nf^0_s(x_1,...,x_s;f)\\equiv \\prod_{1\\leqslant i\\leqslant s}f(x_i).\n\\end{equation}\n\nIn order to solve Eqs.\\p{eq:3.7} we recast them in the form\n\\begin{equation}\\label{evol}\n{\\partial\\over\\partial\\tau}e^{-\\tau\\Lambda_{s}^{0}}f_{s}(e^{\\tau\n\\Lambda^0_1}f)=-e^{-\\tau\\Lambda_{s}^{0}}K_{s}(e^{\\tau \\Lambda^0_1}f).\n\\end{equation}\n(The straightforward differentiation of (\\ref{evol}) gives Eqs. (\\ref{eq:3.7})). Upon integrating\nEq.~ \\p{evol} over $\\tau$ from $0$ to $+\\infty$, and using the above boundary condition\n\\p{eq:3.10}, one obtains the following chain of integral equations for the distribution functions:\n\\begin{equation} \\lab{eq:3.12}\nf_{s}(f)=f^0_{s}(f)+\\int_{0}^{+\\infty}d\\tau\ne^{-\\tau\\Lambda_{s}^{0}} K_{s}(e^{\\tau \\Lambda^0_1}f).\n\\end{equation}\nEquations~\\p{eq:3.12} are solvable in a perturbative theory in weak interaction. In the leading\napproximation in small parameter $\\lambda$ ($F_{ij,n}\\sim\\lambda$), one obtains:\n$$\nf_{s}^{(0)}(f)=f_{s}^0(f),\n$$\n$$\nL^{(1)}(x_{1};f)=-{1\\over v}{\\partial\\over\\partial\np_{1n}}f(x_{1})\\int dx_{2}f(x_{2})F_{12,n}.\n$$\nThis yields the following kinetic equation (see \\p{eq:3.5}), correct to linear order in the\ninteraction strengh:\n$$\n{\\partial f(x_{1})\\over\\partial t}+{p_{1n}\\over m}{\\partial f(x_{1})\\over\\partial x_{1n}}=\n$$\n\\begin{equation}\\lab{eq:3.13}\n={1\\over v}{\\partial\\over\\partial p_{1n}}f(x_{1})\\left(\\int\ndx_{2}f(x_{2}){\\partial V_{12}\\over\\partial x_{1n}}+\\int\ndx_{2}f(x_{2}){\\partial R_{12}\\over\\partial  p_{1n}}\\right)\n\\end{equation}\nIn the absence of dissipative forces (i.e., in the case $R=0$) this kinetic equation reduces to a\nkinetic equation of Vlasov type with a self-consistent field $U({\\bf x}_{1})$ given by:\n$$\nU({\\bf x}_{1})=\\int d{\\bf x}_{2}V_{12}\\int d{\\bf p}_{2}f({\\bf\nx}_{2},{\\bf p}_{2}).\n$$\nAnother simple case is that of spatial homogeneity (but in the presence of dissipation). Then, Eq.\n\\p{eq:3.13} transforms to:\n\\begin{equation}\\lab{eq:3.14}\n{\\partial f({\\bf p}_{1})\\over\\partial t}={1\\over\nv}{\\partial\\over\\partial p_{1n}}f({\\bf\np}_{1}){\\partial\\over\\partial p_{1n}}\\int d{\\bf p}_{2}f({\\bf\np}_{2})R_0({\\bf p}_{12}),\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:3.15}\nR_0({\\bf p})\\equiv \\int d{\\bf x}R({\\bf x};{\\bf p}).\n\\end{equation}\nWhen the dissipation function is given by \\p{eq:2.4''}, one obtains from \\p{eq:3.14}, \\p{eq:3.15}\nthe following equations for the densities of energy, momentum, and particle number:\n$$\n{\\partial\\over\\partial t}\\int d{\\bf p}{p^{2}\\over 2m} f({\\bf\np})=-{\\gamma_0\\over 2vm}\\int d{\\bf p}_{1} d{\\bf p}_{2}f({\\bf\np}_{1})f({\\bf p}_{2}){\\bf p}^2_{12}<0,\n$$\n$$\n{\\partial\\over\\partial t}\\int d{\\bf p}{\\bf p}f({\\bf p})=0, \\quad\n{\\partial\\over\\partial t}\\int d{\\bf p}f({\\bf p})=0,\n$$\nwhere\n$$\n\\gamma_0\\equiv\\int d{\\bf x}\\gamma({\\bf x}).\n$$\nWe see that the system becomes cool (the kinetic energy decreases) during its time evolution as it\nshould be. The chain of integral equations \\p{evol} allows of studying the higher order\napproximations in interaction without any principal difficulties.\n\n\n\\section{ Kinetic equation for dilute gases with\ndissipative interaction}\\label{dilute}\n\n\nThe present section is devoted to the case of small density with arbitrary in strength short-range\ninteraction. In addition, we do not allow for the possibility of formation of complexes of\nparticles, so as to avoid the necessity to introduce additional distribution functions. However,\nthe Bogolyubov method can be applied here as well.\n\nIn principle, we can start from the chain of integral equations \\p{eq:3.12}, as in the previous\nsection. However, this is not convenient as the density expansion would require the use of\nnontrivial resummation techniques applied to the pertinent virial expansion (see, for example,\n\\cite{Akh_Pel}). Therefore, we choose to employ here an alternate approach, similar to those,\ndeveloped by Bogolyubov to derive the Boltzmann equation.\n\nClearly, the expansion of the distribution functions $f_{s}(x_{1},...x_{s};f)$ in Taylor functional\nseries in the one-particle distribution,  $f(x)$, is equivalent to a density expansion (in powers\nof  $1\/v$). Moreover, it is easy to see, on the basis of the structure of Eqs.~\\p{eq:3.7}, that the\nleading contribution to $f_{s}(f)$ is proportional to $f^s$. In accordance with this, it is\nconvenient to rewrite Eqs. \\p{eq:3.7} in the form:\n\\begin{equation}\\lab{eq:4.1}\n-\\int dx{\\delta f_{s}(f)\\over\\delta f(x)}{p_n\\over m}{\\partial\nf(x)\\over\\partial x_n}+\\tilde\\Lambda_{s}f_{s}(f)=Q_{s}(f),\n\\end{equation}\nwhere we have introduced a new auxiliary functional $Q_{s}(f)$,\n$$\nQ_{s}(f)\\equiv -\\sum_{1\\leqslant i\\leqslant s}{\\partial\\over\\partial p_{in}}{1\\over v}\\int\ndx_{s+1}f_{s+1}(f)F_{i\\,s+1,n}-\n$$\n$$\n-\\int dx{\\delta f_{s}(f)\\over\\delta f(x)}L(x;f).\n$$\nThe differential operator $\\tilde{\\Lambda}_{s}$ is defined by \\p{eq:2.21} with $N=s$. The expansion\nof the left-hand side of Eqs.~\\p{eq:4.1} includes terms that are proportional to $f^{s}$ and higher\norder contributions, whereas the right-hand side is, at least, of order $f^{s+1}$.\n\nThe next step is to formulate a boundary condition for (\\ref{eq:4.1}) taking into account the\nevolution of the system in physical direction of time. This boundary condition reflects the\nprinciple of spatial correlation weakening. From (\\ref{eq:2.17}), (\\ref{eq:2.7}), (\\ref{eq:2.6}) we\nhave\n$$\ne^{-\\tau\\tilde\\Lambda_{s}}f_{s}(f)=I_s(x,\\tau)e^{-\\tau \\Lambda_{s}} f_{s}(f)=\n$$\n$$\n=I_s(x,\\tau)f_{s}\\left(X_1(-\\tau,x),...,X_s(-\\tau,x);f\\right),\n$$\nwhere $I_{s}(x,\\tau)$ denotes the Jacobian (\\ref{eq:2.11}) for $s$-particle dynamics. The\napplication of the principle of spatial correlation weakening (\\ref{eq:2.26}) now yields\n$$\ne^{-\\tau\\tilde\\Lambda_{s}}f_{s}(f)\\xrightarrow[\\tau \\sim +\\infty]{}\n$$\n\\begin{equation} \\lab{eq:4.2}\nI_s(x,\\tau)\\prod_{1\\leqslant i\\leqslant s}f\\left(\\vec X_i^*(x)-{\\tau \\over m}\\vec P_i^*(x),\\vec\nP_i^*(x)\\right).\n\\end{equation}\nHere, following \\cite{Bog,Bogol,Akh_Pel}, we have introduced the asymptotic coordinates and momenta\n$X_{i}^{*}(x)=({\\bf X}_{i}^{*}(x),$ ${\\bf P}_{i}^{*}(x))$ ($i=1,...,s$),\n$$\n{\\bf X}_i(t,x)\\xrightarrow[t \\sim -\\infty]{}\\vec X_i^*(x)+{t \\over m}\\vec P_i^*(x),\n$$\n\\begin{equation}\\lab{eq:4.3}\n{\\bf P}_i(t,x)\\xrightarrow[t \\sim -\\infty]{}\\vec P_i^*(x)\n\\end{equation}\n($X_{i}(t,x)\\equiv({\\bf X}_{i}(t,x),{\\bf P}_{i}(t,x))$). The asymptotic coordinates and momenta\n$X_{i}^{*}(x)$ do exist, because, for long times in the past, the particles of the system with\ninteraction under consideration are in a state of free motion.\nNow, according to (\\ref{eq:4.2}), we need to find the limiting value of Jacobian, $I_{s}(x,t)$ as\n$t \\to +\\infty$. Making use the definition (\\ref{eq:2.11}), one obtains\n$$\nI_s(x,t)=\\bigg\\vert{\\partial X(-t,x)\\over\\partial x}\\bigg\\vert=\\bigg\\vert{\\partial\nX(-t,x)\\over\\partial X^{*}(x)}\\bigg\\vert\\bigg\\vert{\\partial X^{*}(x)\\over\\partial x}\\bigg\\vert,\n$$\nwhence, exploiting \\p{eq:4.3}, we have\n\\begin{equation}\\lab{eq:4.9}\nI_s(x,t)\\xrightarrow[t \\to+\\infty]{}I_s^{*}(x)\\equiv\\bigg\\vert{\\partial X^{*}(x)\\over\\partial\nx}\\bigg\\vert.\n\\end{equation}\nAs a result, \\p{eq:4.2} can be written in the final form\n\\begin{equation}\\lab{eq:4.10}\n \\lim_{\\tau \\to\n+\\infty}e^{-\\tau\\tilde\\Lambda_{s}}f_{s}(x;e^{\\tau\\Lambda_1^0}f)=f_s^{(s)}(x;f),\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:4.11}\nf_s^{(s)}(x;f)\\equiv I_s^*(x)\\prod_{1\\leqslant i\\leqslant s}f( X_i^*(x))\n\\end{equation}\n(above, in \\p{eq:3.10}, \\p{eq:3.11}, we have used a more detailed notation $(x_{1},...,x_{s})\\equiv\nx$). To solve Eqs. \\p{eq:4.1} taking into account the obtained boundary condition \\p{eq:4.10}, we\nrewrite it as follows:\n\\begin{equation}\\label{eq:4.12'}\n{\\partial\\over\\partial\\tau}e^{-\\tau\\tilde\\Lambda_{s}}f_{s}(e^{\\tau\n\\Lambda^0_1}f)=-e^{-\\tau\\tilde\\Lambda_{s}}Q_{s}(e^{\\tau \\Lambda^0_1}f).\n\\end{equation}\n(The straightforward differentiation of \\p{eq:4.12'} gives Eqs. \\p{eq:4.1}). Integration of this\nchain of equations over $\\tau$ from $0$ to $+\\infty$ yields the following integral equations for\nthe many-particle distribution functions:\n\\begin{equation} \\lab{eq:4.12''}\nf_{s}(f)=f_{s}^{(s)}(f)+\\int_{0}^{+\\infty}d\\tau\ne^{-\\tau\\tilde\\Lambda_{s}}Q_{s}(e^{\\tau \\Lambda^0_1}f),\n\\end{equation}\nEquations \\p{eq:4.12''} are solvable in a perturbative approach in density. Similar integral\nequations were obtained in \\cite{Akh_Pel} for Hamiltonian systems. The difference between both\nequations consists in the presence of the asymptotical value of Jacobian $I_s^*(x)$ in Eqs.\n\\p{eq:4.11}.\n\nIn the leading order in density, Eqs. \\p{eq:4.12''} give the two-particle distribution function,\nwhich is proportional to the squared density,\n\\begin{equation} \\lab{eq:4.12}\nf_{2}^{(2)}(x_{1},x_{2};f)=I_2^{*}(x_{1},x_{2})\\prod_{1\\leqslant\ni\\leqslant 2}f({\\bf X}_{i}^{*}(x_{1},x_{2}),{\\bf\nP}_{i}^{*}(x_{1},x_{2})).\n\\end{equation}\nNext, using  \\p{eq:3.5} and \\p{eq:3.6}, one obtains the  following kinetic equation:\n\\begin{equation}\\lab{eq:4.13}\n{\\partial f(x_{1},t)\\over\\partial t}+{p_{1n}\\over m}{\\partial\nf(x_{1},t)\\over\\partial x_{1n}}=L^{(2)}(x_{1};f(t)),\n\\end{equation}\nwhere the collision integral $L^{(2)}(x_{1};f)$ is determined by\n\\begin{equation} \\lab{eq:4.14}\nL^{(2)}(x_{1};f)=-{1\\over v}{\\partial\\over\\partial p_{1n}}\\int dx_{2}f_{2}^{(2)}(x_{1},x_{2};f)\nF_{12,n}.\n\\end{equation}\nThe kinetic equation \\p{eq:4.13} is a generalization of the\nBoltzmann kinetic equation to non-Hamiltonian systems. The\ncollision integral \\p{eq:4.14} is written in the Bogolyubov form\nand expressed through the asymptotic coordinates and momenta,  and\nthe  Jacobian corresponding to two-particle dynamics.\n\nIn the case of weakly nonuniform states, when the gradients of the one one-particle distribution\nfunction $f(\\vec x,\\vec p)$ are small, the collision integral \\p{eq:4.14} can be further\nsimplified. For these states, the range $r_{0}$ of the interpartical forces is small compared to\nthe characteristic scale of inhomogeneity $a$, $r_{0}\\ll a$, i.e. in comparison to those distances\nover which the one-particle distribution function $f(\\vec x,\\vec p)$ changes substantially. Also,\nwe take into account that\n\\begin{equation}\\lab{eq:4.15}\n|{\\bf X}_{i}^{*}(x_1,x_2)-{\\bf x}_{i}|\\sim r_{0}, \\quad {\\bf P}_{i}^{*}(x_{1},x_{2})\\equiv {\\bf\nP}_{i}^{*}({\\bf x}_{21},{\\bf p}_{1},{\\bf p}_{2}),\n\\end{equation}\n($i=1,2$). Following Eq.~(\\ref{eq:4.12}) these asymptotic\ncoordinates and momenta of the two particles problem determine the\ntwo-particle distribution function $f_{2}^{(2,0)}(x_{1},x_{2};f)$\nto  second order in the particle density and  zeroth  order in the\ngradients. Using \\p{eq:4.1} one obtains:\n$$\n{{p_{12,n}}\\over m}{\\partial\nf_{2}^{(2,0)}(x_{1},x_{2};f)\\over\\partial{\nx_{1n}}}+{\\partial\\over\\partial\np_{1n}}(f_{2}^{(2,0)}(x_{1},x_{2};f)F_{12,n})+\n$$\n$$\n+{\\partial\\over\\partial p_{2n}}(f_{2}^{(2,0)}(x_{1},x_{2};f)\nF_{21,n})=0.\n$$\nIntegration of this equation over $x_{2}$ leads, using \\p{eq:4.14}, to an expression for the\ncollision integral $L^{(2,0)}(x;f)$,\n\\begin{equation} \\lab{eq:4.15'}\nL^{(2,0)}(x_{1};f)={1\\over v}\\int dx_{2}{{p_{21,n}}\\over m}{\\partial\nf_{2}^{(2,0)}(x_{1},x_{2};f)\\over\\partial x_{2n}},\n\\end{equation}\nwhere\n$$\nf_{2}^{(2,0)}(x_{1},x_{2};f)=\n$$\n\\begin{equation} \\lab{eq:4.16}\n=I_2^{*}(x_{1},x_{2})f({\\bf x}_{1},{\\bf\nP}_{1}^{*}(x_{1},x_{2}))f({\\bf x}_{1},{\\bf\nP}_{2}^{*}(x_{1},x_{2}))\n\\end{equation}\n(see \\p{eq:4.12}, \\p{eq:4.15}).\n\nNow, we evaluate integral over ${\\bf x}_{2}$ ($dx_{2}=d{\\bf x}_{2}d{\\bf p}_{2}$) in \\p{eq:4.15'}.\nThe integration can be replaced by an integration over the difference ${\\bf x}_{21}$ (see\n\\p{eq:4.12}, \\p{eq:4.15}). In performing the integral over ${\\bf x}_{21}$, we employ cylindrical\ncoordinates $z$, $b$ and $\\varphi$ with the origin at the point ${\\bf x}_{1}$ and the $z$-axis\ndirected along the vector ${\\bf p}_{21}$:\n$$\nL^{(2,0)}(x_{1};f)={1\\over v}\\int d{\\bf\np}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\,\\,b\\,\\,{|{\\bf p}_{21}|\\over m}\\times\n$$\n\\begin{equation} \\lab{eq:4.17}\n\\times f_{2}^{(2,0)}(x_{1},x_{2}; f)\\vert_{z=-\\infty}^{z=+\\infty}\n\\end{equation}\nwhere $f_{2}^{(2,0)}(x_{1},x_{2};f)$ is given by \\p{eq:4.16}. The asymptotic momenta\n$$\n{\\bf P}_{i}^{*}({\\bf x}_{\\perp},z,{\\bf p}_{1},{\\bf p}_{2})\\equiv {\\bf P}_{i}^{*}({\\bf x}_{21},{\\bf\np}_{1},{\\bf p}_{2}),\n$$\nwhich determine $f_{2}^{(2,0)}(x_{1},x_{2};f)$, have the following properties:\n$$\n{\\bf P}_{i}^{*}({\\bf x}_{\\perp},z,{\\bf p}_{1},{\\bf\np}_{2},)\\vert_{z\\to +\\infty}={\\bf p}^{\\prime}_{i}(b,\\varphi,{\\bf\np}_{1},{\\bf p}_{2}),\n$$\n\\begin{equation} \\lab{eq:4.19}\n{\\bf P}_{i}^{*}({\\bf x}_{\\perp},z,{\\bf p}_{1},{\\bf\np}_{2})\\vert_{z\\to-\\infty}={\\bf p}_{i},\n\\end{equation}\nwhere ${\\bf x}_{\\perp}=({\\bf x}_{21})_{\\perp}=(b,\\varphi)$, $z={\\bf x}_{21} {\\bf p}_{21}\/|{\\bf\np}_{21}|$. Indeed, according to \\p{eq:4.3}, ${\\bf P}_i^*(x_1,x_2)$ are the momenta of two particles\nat the moment of time $t=-\\infty$, if at $t=0$ they have phase variables $\\vec x_1,\\vec p_1, \\vec\nx_2, \\vec p_2$. Then, the relationship\n$$\n\\left|(\\vec x_1+{\\vec p_1 \\over m}t)-(\\vec x_2+{\\vec p_2 \\over m}t)\\right|=|\\vec x_{12}|+{z \\over\nt_0}t+O(t^2)\n$$\n($t_0 \\equiv {|\\vec x_{21}|m \/ |\\vec p_{21}|}$), which is valid for small $t$, shows the following:\nwhen $z>0$, the collision of particles precedes the moment $t=0$, whereas when $z<0$, the collision\noccurs after $t=0$. This reasoning explains the relations \\p{eq:4.19}, where ${\\bf\np}^{\\prime}_{1}(b,\\varphi,{\\bf p}_{1},{\\bf p}_{2})$, ${\\bf p}^{\\prime}_{2}(b,\\varphi,{\\bf\np}_{1},{\\bf p}_{2})$ are the momenta of particles before the collision (precollisional  momenta)\nafter which the particles have momenta ${\\bf p}_1$, ${\\bf p}_2$.\n\nWith these observation we can now find the following expression\nfor the generalized Boltzmann collision integral determined by\n\\p{eq:4.16} and \\p{eq:4.17}:\n$$\nL^{(2,0)}(x_{1};f)={1\\over v}\\int d{\\bf p}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\,b\\,{|{\\bf\np}_{21}|\\over m}\\times\n$$\n\\begin{equation}\\lab{eq:4.20}\n\\times\\{I_2'({\\bf x}_{\\perp},{\\bf p}_{1},{\\bf p}_{2})f({\\bf\nx}_{1},{\\bf p}_{1}')f({\\bf x}_{1},{\\bf p}_{2}')- f({\\bf\nx}_{1},{\\bf p}_{1})f({\\bf x}_{1},{\\bf p}_{2})\\},\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:4.21}\nI_2'({\\bf x}_{\\perp},{\\bf p}_{1},{\\bf\np}_{2})=I^{*}(x_{1},x_{2})\\vert_{z=+\\infty}.\n\\end{equation}\nThe calculation of the collision integral $L^{(2,0)}(x_{1};f)$\n(much like the calculation of the two-particle distribution\nfunction)  involves only the solution of the two-particle\ndynamics. This is elaborated in the subsection that appears\nimmediately below.\n\n\\section{Dissipative dynamics}\\label{disdin}\n\n\\subsection{Relative motion in two-particle dynamics}\n\nFor the Hamiltonian systems the two-particle problem is reduced to study of relative motion of the\nparticles. The same situation takes place in the presence of dissipative forces.\n\nTo obtain this result and some its consequences, let us consider the equations of motion for two\nparticles in the presence of dissipative forces. According to Eqs.~\\p{eq:2.1}-\\p{eq:2.4}, these\nequations have the form\n\\begin{equation}\\lab{eq:5.1}\nm\\dot{\\bf x}_{1}={\\bf p}_{1}, \\qquad m\\dot{\\bf x}_{2}={\\bf p}_{2},\n\\end{equation}\n\\begin{equation}\\lab{eq:5.1'}\n\\dot{\\bf p}_{1}={\\bf F}({\\bf x}_{12};\\,{\\bf p}_{12}), \\qquad\n\\dot{\\bf p}_{2}=-{\\bf F}({\\bf x}_{12};\\,{\\bf p}_{12}),\n\\end{equation}\nwhere\n$$\nF_n({\\bf x};{\\bf p})=-{{\\partial V({\\bf x})}\\over\\partial\nx_{n}}-{{\\partial R({\\bf x};{\\bf p})}\\over\\partial p_{n}}.\n$$\nLet us introduce the following new phase variables ${\\bf x}$, ${\\bf p}$, ${\\bf x_c}$, ${\\bf p_c}$\nwith clear meaning:\n$$\n{\\bf x}={\\bf x}_{1}-{\\bf x}_{2}, \\qquad {\\bf p}={{{\\bf p}_{1}-{\\bf\np}_{2}}\\over 2},\n$$\n\\begin{equation}\\lab{eq:5.2}\n{\\bf x_c}={{{\\bf x}_{1}+{\\bf x}_{2}}\\over 2}, \\qquad {\\bf\np_c}={\\bf p}_{1}+{\\bf p}_{2}.\n\\end{equation}\nThen, the equations of motion \\p{eq:5.1}, \\p{eq:5.1'} are separated into equations for the center\nof mass and relative motion:\n\\begin{equation}\\lab{eq:5.5}\n2m\\dot{\\bf x}_c={\\bf p_c}, \\qquad \\dot{\\bf p}_c=0,\n\\end{equation}\n\\begin{equation}\\lab{eq:5.5'}\nm\\dot{\\bf x}=2{\\bf p}, \\qquad \\dot{\\bf p}={\\bf F}({\\bf x},2{\\bf\np}).\n\\end{equation}\n\nLet ${\\bf x}(t,{\\bf x},{\\bf p})$, ${\\bf p}(t,{\\bf x},{\\bf p})$ be a solution of Eqs.~\\p{eq:5.5'}\nwith initial conditions ${\\bf x}$, ${\\bf p}$. According to \\p{eq:4.3}, \\p{eq:5.2}, the\ncorresponding asymptotic coordinates and momenta are given by\n$$\n{\\bf p}(t,{\\bf x},{\\bf p})\\xrightarrow[t \\sim -\\infty]{}{\\bf\np}^{*}({\\bf x},{\\bf p}),\n$$\n\\begin{equation}\\lab{eq:5.6}\n{\\bf x}(t,{\\bf x},{\\bf p})\\xrightarrow[t\\sim -\\infty]{}{\\bf\nx}^{*}({\\bf x},{\\bf p})+{2t\\over m}{\\bf p}^{*}({\\bf x},{\\bf p}),\n\\end{equation}\nwhere\n$$\n\\vec p^*(\\vec x,\\vec p)\\equiv {1\\over 2}\\left(\\vec P_1^*(x_1,x_2)-\\vec P_2^*(x_1,x_2)\\right)\n$$\n\\begin{equation} \\label{eq:5.6'}\n\\vec x^*(\\vec x,\\vec p)\\equiv \\vec X_1^*(x_1,x_2)-\\vec X_2^*(x_1,x_2).\n\\end{equation}\nIntegrating Eqs. \\p{eq:5.5}, we can also come to the following identities:\n$$\n\\vec p_1+\\vec p_2=\\vec P^*_1(x_1,x_2)+\\vec P^*_2(x_1,x_2),\n$$\n\\begin{equation}\\label{eq:5.6''}\n\\vec x_1+\\vec x_2=\\vec X^*_1(x_1,x_2)+\\vec X^*_2(x_1,x_2).\n\\end{equation}\n\nThe comparison of Eqs. \\p{eq:5.6}-\\p{eq:5.6''} enables to express the asymptotic coordinates and\nmomenta ${\\bf {X}}_{i}^{*}(x_{1}, x_{2})$ and ${\\bf P}_{i}^{*}(x_{1},x_{2})$ through the functions\n$\\vec x^*(\\vec x,\\vec p)$, $\\vec p^*(\\vec x,\\vec p)$\n$$\n{\\bf P}_{1}^{*}(x_{1},x_{2})={{\\bf p_c}\\over 2}+{\\bf p}^{*}({\\bf\nx},{\\bf p}), \\,\\,{\\bf P}_{2}^{*}(x_{1},x_{2})={{\\bf p_c}\\over\n2}-{\\bf p}^{*}({\\bf x},{\\bf p}),\n$$\n\\begin{equation}\\lab{eq:5.10}\n{\\bf X}_{1}^{*}(x_{1},x_{2})={\\bf x_c}+{1\\over 2}{\\bf x}^{*}({\\bf\nx},{\\bf p}), \\,\\, {\\bf X}_{2}^{*}(x_{1},x_{2})={\\bf x_c}-{1\\over\n2}{\\bf x}^{*}({\\bf x},{\\bf p}).\n\\end{equation}\nWe can see that the calculation of asymptotic phase variables for the two-particle dynamics is\nreduced to the calculation of asymptotic phase variables for the relative motion.\n\nNext, consider the  Jacobian $I_2(x_{1},x_{2},t)$ that corresponds to the dynamics of two particles\nand determines the collision integral \\p{eq:4.20}. According to Eq.\\p{eq:2.19}, this Jacobian\nsatisfies the following equation:\n$$\n{\\partial I_2\\over\\partial t}-\\{H_{2},I_2\\}={\\partial\\over\\partial p_{1n}}\\biggl(I_2{{\\partial\nR}\\over\\partial p_{1n}}\\biggr)+{\\partial\\over\\partial p_{2n}}\\biggl(I_2{\\partial R\\over\\partial\np_{2n}}\\biggr),\n$$\nwhere $H_{2}$ is the  two-particle Hamiltonian (see \\p{eq:2.2} for $N=2$). Changing the independent\nvariables in this equation to ${\\bf x}$, ${\\bf p}$, ${\\bf x}_c$, ${\\bf p}_c$ (see \\p{eq:5.2}), one\nfinds:\n$$\n{\\partial I_2\\over\\partial t}+{p_{cn}\\over 2m}{\\partial\nI_2\\over\\partial x_{cn}}+{2p_n\\over m}{\\partial I_2\\over\\partial\nx_n}-{\\partial V\\over\\partial x_n}{\\partial I_2\\over\\partial\np_n}={1\\over 2}{\\partial\\over\\partial p_n}\\left(I_2{\\partial\nR\\over\\partial p_n}\\right).\n$$\nSince $I_2(x_1,x_2,0)=1$, it follows from the latter equation that the Jacobian does not depend on\n${\\bf x}_c$ and ${\\bf p}_c$, i.e. $I_2(x_1,x_2,t)\\equiv I_2(\\vec x, \\vec p,t)$. The Jacobian\n$I_2(\\vec x, \\vec p,t)$ satisfies equation\n\\begin{equation}\\lab{eq:5.11}\n{\\partial I_2\\over\\partial t}-\\{h,I_2\\}={1\\over\n2}{\\partial\\over\\partial p_n}\\biggl(I_2{\\partial R\\over\\partial\np_n}\\biggr), \\qquad I_2\\vert_{t=0}=1,\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:5.12}\n\\{h,I_2\\}={\\partial h\\over\\partial x_n}{\\partial I_2\\over\\partial p_n}-{\\partial h\\over\\partial\np_{n}}{\\partial I_2\\over\\partial x_n}, \\qquad h\\equiv{p^{2}\\over m}+V({\\bf x}).\n\\end{equation}\nNext,  introduce the Jacobian $\\tilde I_2({\\bf x},{\\bf p},t)$ corresponding to the dynamics defined\nby  \\p{eq:5.5'}. It can be easily seen, using Eqs. \\p{eq:2.19}, \\p{eq:5.5'}, that this Jacobian\nsatisfies the same equation and initial condition as $I_2({\\bf x},{\\bf p},t)$ (see Eqs.\n\\p{eq:5.11}, \\p{eq:5.12}). Therefore, these two Jacobians are equal to each other,\n\\begin{equation}\\lab{eq:5.12'}\nI_2({\\bf x},{\\bf p},t)=\\tilde{I}_2({\\bf x},{\\bf p},t).\n\\end{equation}\nTaking into account this result and the definition \\p{eq:4.9} of the limiting Jacobian, we obtain\n\\begin{equation}\\lab{eq:5.12''}\nI^*_2(x_1,x_2)=I^*_2(\\vec x,\\vec p)={\\partial (\\vec x^*,\\vec\np^*)\\over\\partial (\\vec x,\\vec p)}.\n\\end{equation}\n\n\n\\subsection{Two-particle dynamics with weak\ndissipation}\n\nThe present  section is devoted to the study of the case of weak dissipation, namely the kinetics\nfor which it is sufficient to consider only the linear order in an expansion of collision integral\nin powers of the dissipation function. This case is similar to the corresponding expansion in\npowers of the degree of inelasticity \\cite{Sela_Gold}.\n\nIn accordance with Eqs. \\p{eq:4.9}, \\p{eq:4.19}-\\p{eq:4.21}, in order to derive a kinetic equation\nin the case of weak dissipation, we have to calculate the asymptotic coordinates ${\\vec\nX}^*_i(x_1,x_2)$ and momenta ${\\vec P}^*_i(x_1,x_2)$ $(i=1,2)$ in a perturbative approach in the\ndissipation function $R(\\vec x,\\vec p)$. However, in the previous sub-section, we have showed that\nit is sufficient to find the asymptotic coordinates ${\\vec x}^*(\\vec x, \\vec p)$ and momenta ${\\vec\np}^*(\\vec x, \\vec p)$ for relative motion. This motion is described by the solution ${\\vec\nx}(t,x)\\equiv{\\vec x}(t,\\vec x, \\vec p)$, ${\\vec p}(t,x)\\equiv{\\vec p}(t,\\vec x, \\vec p)$  of Eqs.\n\\p{eq:5.5'}, which can be written in the form\n\\begin{equation}\\lab{eq:6.1}\n{\\vec x}(t,\\vec x, \\vec p)=e^{t(\\lambda_0+\\lambda_1)}\\vec x,\\quad {\\vec p}(t,\\vec x, \\vec\np)=e^{t(\\lambda_0+\\lambda_1)}\\vec p,\n\\end{equation}\nwhere the operators $\\lambda_0$, $\\lambda_1$ have the following structure:\n\\begin{equation}\\lab{eq:6.2}\n\\lambda_0\\equiv {2p_n \\over m}{\\partial \\over\n\\partial x_n}-{\\partial V(\\vec x)\\over \\partial x_n}{\\partial\n\\over \\partial p_n}, \\quad \\lambda_1\\equiv -{1\\over 2}{\\partial\nR(\\vec x, 2\\vec p)\\over \\partial p_n}{\\partial \\over \\partial p_n}\n\\end{equation}\n(see Eqs. \\p{eq:2.5}-\\p{eq:2.6'}).\n\nIn the sequel, while calculating ${\\vec x}(t,\\vec x, \\vec p)$, ${\\vec p}(t,\\vec x, \\vec p)$ we\nshall consider $\\lambda_1$ as a small perturbation. The unperturbed relative motion is expressed as\n\\begin{equation}\\lab{eq:6.3}\n{\\bf p}^{(0)}(t,\\vec x, \\vec p)=e^{t\\lambda_0}{\\bf p}, \\quad {\\bf\nx}^{(0)}(t,\\vec x, \\vec p)=e^{t\\lambda_0}{\\bf x}.\n\\end{equation}\nSince the operators $\\lambda_0$ and $\\lambda_1$ do not commute, we\nuse the following well known expansion:\n\\begin{equation}\\lab{eq:6.4}\ne^{t(\\lambda_0+\\lambda_1)}=e^{t\\lambda_0}+\\int_{0}^{t}dt'\ne^{t'\\lambda_0}\\lambda_1e^{(t-t') \\lambda_0}+...\n\\end{equation}\nThis formula, in conjunction with  \\p{eq:6.1}-\\p{eq:6.3}, gives\n$$\n{\\bf p}(t,x)={\\bf p}^{(0)}(t,x)+\\int_{0}^{t}dt'\\left.\\lambda_1\\,{\\bf p}^{(0)}(t-t',x)\\right|_{x\\to\nx^{(0)}(t',x)},\n$$\n$$\n{\\bf x}(t,x)={\\bf x}^{(0)}(t,x)+\\int_{0}^{t}dt'\\left.\\lambda_1\\,{\\bf x}^{(0)}(t-t',x)\\right|_{x\\to\nx^{(0)}(t',x)}\n$$\n$(x^{(0)}(t,x)\\equiv ({\\vec x}^{(0)}(t,x), {\\vec p}^{(0)}(t,x))$. Using \\p{eq:5.6}, it is easy to\nfind the asymptotic momentum ${\\bf p}^{*}(x)$ and coordinate ${\\bf x}^{*}(x)$ by letting $t$ to\n$-\\infty$:\n$$\n{\\bf p}^{*}(x)={\\bf p}^{*(0)}(x)-\\int_{-\\infty}^{0}dt\\left.\\lambda_1\\,{\\bf p}^{*(0)}(x)\\right\n|_{x\\to x^{(0)}(t,x)},\n$$\n$$\n{\\bf x}^{*}(x)={\\bf x}^{*(0)}(x)-\\int_{-\\infty}^{0}dt\\,\\lambda_1\\,\\bigg\\{ {\\bf x}^{*(0)}(x)-\n$$\n\\begin{equation} \\lab{eq:6.9}\n-{2t\\over m}{\\bf p}^{*(0)}(x)\\bigg\\}\\bigg\\vert_{x\\to x^{(0)}(t,x)}.\n\\end{equation}\n\nConsider now the Jacobian $I_2(x,t)$ in the linear order in the dissipation function $R$. To this\norder in $R$, it follows from Eq. \\p{eq:5.11} (with $I_{2}(x,t)=1$ at $R=0$) that\n$$\n{\\partial I_2\\over\\partial t}+\\lambda_0 I_2={1\\over 2}{\\partial^{2}R({\\bf x},2{\\bf p})\\over\\partial\np_n\\partial p_n},\n$$\nwhence\n$$\nI_{2}(x,t)=1+{1\\over 2}\\int_{-t}^{0}dt'\\left.{\\partial^{2}R({\\bf x},2{\\bf p})\\over\\partial\np_n\\partial p_n}\\right |_{x\\to x^{(0)}(t',x)}.\n$$\nAccording to Eq. \\p{eq:4.9}, the asymptotic value of the Jacobian $I_2(x,t)$ is given by the\nformula:\n\\begin{equation}\\lab{eq:6.10}\nI^{*}_2(x)=1+{1\\over 2}\\int_{-\\infty}^{0}dt\\left.{\\partial^{2}R({\\bf x},2{\\bf p})\\over\\partial\np_n\\partial p_n}\\right |_{x\\to x^{(0)}(t,x)}.\n\\end{equation}\n\n\\section{Kinetic equation in the weak\ndissipation approximation}\\label{weakdisskin}\n\nIn this section we study the kinetic equation \\p{eq:3.5} with the collision integral \\p{eq:4.20} in\nthe weak dissipation approximation. In the spatially homogeneous case and in the linear\napproximation in $R$, the collision integral Eq. \\p{eq:4.20} assumes the form:\n\\begin{equation}\\lab{eq:6.11}\nL^{(2,0)}({\\bf p}_{1};f)=L_{0}^{(2,0)}({\\bf p}_{1};f)+L_{1}^{(2,0)}({\\bf p}_{1};f),\n\\end{equation}\nwhere $L_{0}^{(2,0)}({\\bf p}_{1},f)$ is the Boltzmann collision\nintegral, which accounts only for the reversible (potential)\ninteractions:\n$$\nL_{0}^{(2,0)}({\\bf p}_{1};f)={1\\over v}\\int d{\\bf\np}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\,b{{|{\\bf p}_{21}|}\\over m}\\times\n$$\n\\begin{equation}\\lab{eq:6.11'}\n\\times\\{f({\\bf p}_{10}')f({\\bf p}_{20}')-f({\\bf p}_{1})f({\\bf\np}_{2})\\}\n\\end{equation}\n(${\\vec p}_{i0}'\\equiv {\\vec p_i'}^{(0)}$ are the Boltzmann precollisional momenta). The second\nterm in \\p{eq:6.11} is a correction to the Boltzmann collision integral, which accounts for\ndissipation to linear order in $R$:\n$$\nL_{1}^{(2,0)}({\\bf p}_{1};f)={1\\over v}\\int d{\\bf\np}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\,b{{|{\\bf p}_{21}|}\\over m}\\times\n$$\n$$\n\\times\\delta[I'({\\bf x}_{\\perp},{\\bf p}_{1},{\\bf p}_{2})f({\\bf\np}_{1}')f({\\bf p}_{2}')],\n$$\nwhere ${\\bf x}_{\\perp}=({\\bf x}_{21})_{\\perp}=(b,\\varphi)$ (see section 4). It is clear that $I'=1$\nfor the reversible dynamics (when $R=0$). Thus, taking into account that $\\delta{\\bf\np}_{1}'=-\\delta{\\bf p}_{2}'$ (see \\p{eq:4.19}, \\p{eq:5.10}) and using \\p{eq:6.9}, \\p{eq:6.10} for\nthe asymptotic  values of the momentum and Jacobian, we find\n$$\nL_{1}^{(2,0)}({\\bf p}_{1};f)={1\\over v}\\int d{\\bf\np}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\,b{{|{\\bf p}_{21}|}\\over m}\\times\n$$\n$$\n\\times\\bigg\\{a({\\bf x}_{\\perp},{\\bf p}) +b_n({\\bf x}_{\\perp},{\\bf p})\\biggl({\\partial\\over\\partial\np_{1n}}-{\\partial\\over\\partial p_{2n}}\\biggr)\\bigg\\}\\times\n$$\n\\begin{equation}\\lab{eq:6.12}\n\\times\\left. f({\\bf p}_{1})f({\\bf p}_{2})\\right|_{{\\bf p}_{1},{\\bf\np}_{2}\\to{{\\bf p}_{10}^{\\prime},{{\\bf p}_{20}^{\\prime}}}},\n\\end{equation}\nwhere\n\\begin{equation}\\lab{eq:6.13}\na({\\bf x}_{\\perp},{\\bf p})={1\\over\n2}\\int_{-\\infty}^{0}dt\\left.{\\partial^{2} R\\over{\\partial p_n\n\\partial p_n}} \\right |_{x\\to x^{(0)}(t,x),\\,z\\to+\\infty},\n\\end{equation}\n\\begin{equation}\\lab{eq:6.14}\nb_n({\\bf x}_{\\perp},{\\bf p})={1\\over 2}\\int_{-\\infty}^{0}dt\\left.{\\partial R\\over\\partial\np_{m}}{\\partial p_{n}^{*(0)}\\over\\partial p_{m}} \\right |_{x\\to x^{(0)}(t,x),\\,z\\to+\\infty}\n\\end{equation}\n($z=({\\bf x}{\\bf p})\/|{\\bf p}|$, see section 4).\n\nNext we wish to evaluate  \\p{eq:6.12} for the  case in which the\nparticle interactions vanish ($V=0$).  In this case the  solution\nof the equations of motion assumes  of the form\n$$\n{\\bf x}^{(0)}(t,x)={\\bf x}+{2t\\over m}{\\bf p}\n$$\nand, as expected, the asymptotic  momentum and space coordinate\ncoincide with their respective initial values, ${\\bf\np}^{*(0)}={\\bf p}$, ${\\bf x}^{*(0)}={\\bf x}$. Noting that\n$$\n\\int_{-\\infty}^{0}dtg\\biggl({\\bf x}+{2t\\over m}{\\bf\np}\\biggr)=\\int_{-\\infty}^{0}dtg\\biggl({\\bf x}_{\\perp},z+{2t\\over\nm}p\\biggr)=\n$$\n$$\n={m\\over 2p}\\int_{-\\infty}^{z}dz'g({\\bf x}_{\\perp},z')\n$$\nis valid for an arbitrary function $g({\\bf x})=g({\\bf x}_{\\perp},z)$ (the $z$-axis of the\ncylindrical coordinates we employ  is chosen to point in the direction of  ${\\bf p}$), one finds,\nusing \\p{eq:6.13}, \\p{eq:6.14}, that:\n$$\na({\\bf x}_{\\perp},{\\bf p})={m\\over\n4p}\\int_{-\\infty}^{\\infty}dz{\\partial^{2} R(\\vec x\n_{\\perp},z,2\\vec p)\\over\\partial p_n\\partial p_n},\n$$\n$$\nb_n({\\bf x}_{\\perp},{\\bf p})={m\\over\n4p}\\int_{-\\infty}^{\\infty}dz{\\partial R(\\vec x _{\\perp},z,2\\vec\np)\\over\\partial p_n}.\n$$\nFinally, upon substituting these expressions into \\p{eq:6.12} and noting  that ${\\bf p}_{10}'={\\bf\np}_{1}$, ${\\bf p}_{20}'={\\bf p}_{2}$ for $V=0$, one obtains\n\\begin{equation}\\lab{eq:6.15}\nL_{1}^{(2,0)}({\\bf p}_{1};f)={1\\over v}{\\partial\\over\\partial\np_{1n}}f({\\bf p}_{1}){\\partial\\over\\partial p_{1n}}\\int d{\\bf\np}_{2}f({\\bf p}_{2})R_0({\\bf p}_{12}),\n\\end{equation}\nwhere $R_0({\\bf p})$ is defined by Eq. \\p{eq:3.15}. Formula\n\\p{eq:6.15} coincides with \\p{eq:3.14} obtained within the weak\ninteraction approximation.\n\nIn conclusion of this section we briefly concern the question of the evolution of the system\ndescribed by the kinetic equation\n\\begin{equation}\\lab{eq:6.15'}\n{\\partial f({\\bf p},t)\\over\\partial t}=L_{0}^{(2,0)}( {\\bf\np};f(t))+L_{1}^{(2,0)}({\\bf p};f(t)).\n\\end{equation}\nLet $\\tau_{r}$ be the relaxation time defined by the usual Boltzmann term in Eq. \\p{eq:6.15'}. In\nthe absence of dissipative interaction this relaxation leads to the Maxwellian distribution for\n$f({\\bf p},t)$. However, in the presence of small dissipative interaction described by the second\nterm in Eq. \\p{eq:6.15'}, we shall observe a weak relaxation of temperature of the system, i.e. the\nhomogeneous cooling state (see, for example, \\cite{IGreview}). The description of this state can be\nbased on the functional hypothesis of the form\n\\begin{equation}\\lab{eq:6.16}\nf(\\vec p,t)\\xrightarrow[t\\gg\\tau_{r}]{}f(\\vec p,\\varepsilon(t)),\n\\end{equation}\nwhere the asymptotic value of the energy density $\\varepsilon(t)$ is defined by\n$$\n\\int d\\vec p f(\\vec p,t){\\vec p^2 \\over 2m}\\xrightarrow\n[t\\gg\\tau_{r}]{}\\varepsilon(t).\n$$\nThis functional hypothesis results to the equation for $\\varepsilon(t)$,\n$$\n{\\partial \\varepsilon(t)\\over \\partial t}=L(\\varepsilon(t))\n$$\nwith the following right-hand side:\n$$\nL(\\varepsilon)\\equiv \\int d\\vec p {\\vec p^2 \\over 2m}L_{1}^{(2,0)}({\\bf p}_{1};f(\\varepsilon))\n$$\n(the Boltzmann collision integral $L_{0}^{(2,0)}({\\bf p}_{1};f)$ does not contribute to\n$L(\\varepsilon)$).\n\nThe distribution function $f(\\varepsilon)$, according to Eq. \\p{eq:6.15'} and the functional\nhypothesis \\p{eq:6.16}, satisfies the equation\n\\begin{equation}\\lab{eq:6.17}\n{\\partial f(\\varepsilon)\\over \\partial\\varepsilon }L(\\varepsilon)=\nL_{0}^{(2,0)}( {\\bf p};f(\\varepsilon))+L_{1}^{(2,0)}({\\bf\np};f(\\varepsilon)).\n\\end{equation}\nThis equation is solvable in a perturbative approach in powers of the dissipation function. We\nshall not discuss here the study of the homogeneous cooling state based on the obtained equations.\nThis can be done similar to those theory developed for spatially nonuniform states in\n\\cite{Sela_Gold,Gol'd_Sok}. Finally, we note that in another terminology, the sketched theory is\nthe application of the Chapman-Enskog method to the solution of the kinetic equation \\p{eq:6.15'}.\n\n\n\\section{Connection to the Boltzmann equation for inelastic\nrigid spheres }\\label{spheres}\n\nIn this section we compare our kinetic equation \\p{eq:4.20} with that obtained by considering a\nsystem of rigid particles experiencing instantaneous inelastic collisions characterized by a fixed\ncoefficient of normal restitution (see, e.g. \\cite{Sela_Gold,Gol'd_Sok}):\n$$\nL'(\\vec p_1;f)={d^{2}\\over mv}\\int_{ \\vec k \\vec p_{12}>0}d{\\bf p}_{2}\\int d^{2}{\\bf k}(\\vec k {\\bf\np}_{12})\\times\n$$\n\\begin{equation}\\lab{eq:7.1}\n\\times \\{{1\\over\\varepsilon^{2}}f({\\bf p}_{1}^{\\prime})f({\\bf p}_{2}^{\\prime})-f({\\bf p}_{1})f({\\bf\np}_{2})\\},\n\\end{equation}\nwhere ${\\bf k}$ is a unit vector pointing  from the center of sphere $1$ to that of sphere $2$ at\nthe moment of contact ($d^{2}{\\bf k}=\\sin{\\theta}d\\theta d\\varphi$; the polar axis $z$ is directed\nalong ${\\bf p}_{21}={\\bf p}_{2}-{\\bf p}_{1}$), $d$ is the diameter of a sphere, and $\\varepsilon$\nis the coefficient of normal restitution. Using the identity\n$$\n\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}b\\,db|{\\bf p}_{12}|...=d^{2}\\int_{{\\bf k}{\\bf\np}_{21}>0}d^{2}{\\bf k}({\\bf k}{\\bf p}_{21})...\n$$\n($b=d\\sin\\theta$, $0\\leqslant\\theta\\leqslant\\pi\/2$), one obtains\nfrom \\p{eq:7.1}:\n$$\nL'(\\vec p_1;f)={1\\over v}\\int d{\\bf p}_{2}\\int_{0}^{2\\pi}d\\varphi\\int_{0}^{\\infty}db\\ b\\ {{|{\\bf\np}_{21}|}\\over m}\\times\n$$\n\\begin{equation}\\lab{eq:7.2}\n\\times\\bigg\\{{1\\over\\varepsilon^{2}}f({\\bf p}_{1}')f({\\bf p}_{2}')-f({\\bf p}_{1})f({\\bf\np}_{2})\\bigg\\}.\n\\end{equation}\nHere ${\\bf p}_{1}'$, ${\\bf p}_{2}'$ are the precollisional momenta are determined by\n\\begin{equation}\\lab{eq:7.3}\n{\\bf p}_{1}'={\\bf p}_{1}+{{1+\\varepsilon}\\over\\varepsilon}{\\bf\nk}({\\bf p}{\\bf k}), \\quad {\\bf p}_{2}'={\\bf\np}_{2}-{{1+\\varepsilon}\\over\\varepsilon}{\\bf k}({\\bf p}{\\bf k}),\n\\end{equation}\nwhere ${\\bf p}=({\\bf p}_1-{\\bf p}_2)\/2$. The collision integral \\p{eq:4.20} is determined by the\nasymptotic ($t\\to -\\infty$) values of the momenta, coordinates, and Jacobian, which specify the\ntwo-particle dynamics. In terms of the relative momentum ${\\bf p}$, the collision law \\p{eq:7.3}\ncan be written as\n\\begin{equation}\\lab{eq:7.4}\n{\\bf p}'={\\bf p}+{{1+\\varepsilon}\\over\\varepsilon}{\\bf k}({\\bf\npk}).\n\\end{equation}\nWithin the framework of the formalism developed in this article, one needs to know the relation\nbetween the asymptotic values  of coordinate ${\\bf x}'$ and the initial coordinate ${\\bf x}$. We\nestablish this relation in the terms orf the relative coordinate $\\vec x=\\vec x_1- \\vec x_2$ as\nfollows:\n\\begin{equation}\\lab{eq:7.4a}\n{\\bf x}^{\\prime}={\\bf x}+{{1+\\varepsilon}\\over\\varepsilon}{\\bf k}({\\bf x k}).\n\\end{equation}\nUsing the above  ${\\bf x}^{\\prime}$ (we assume ${\\bf k}={\\rm\nconst}$) we obtain the Jacobian:\n$$\n{\\partial({\\bf x}',{\\bf p}')\\over\\partial({\\bf x},{\\bf\np})}={1\\over\\varepsilon^{2}}.\n$$\nSubstitution of this Jacobian into the collision integral \\p{eq:4.20} gives the collision integral\n\\p{eq:7.2}. Therefore, when  \\p{eq:7.3}, \\p{eq:7.4} are satisfied, the collision integrals\n\\p{eq:4.20}, \\p{eq:7.1} coincide, as they should.\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\nWe have shown that the Bogolyubov method of derivation of kinetic\nequations can be applied to dissipative many-body systems with the\ncorresponding modifications. In the case of inelastically\ncolliding hard spheres we reproduce the inelastic Boltzmann\nequation. The reader may be justified  in asking whether yet\nanother formulation is needed to study dissipative systems. We\nbelieve that the answer is that given the difficulties encountered\nby other approaches, in particular the problems  emanating from\nthe lack of scale separation in granular systems, it is\nadvantageous to consider a powerful approach such as that of\nBogolyubov. The application of this approach to dense systems, for\ninstance, would not only serve to complement the results obtained\nby using the Enskog corrected Boltzmann equation, but may also\nenable the study of systems (such as binary granular mixtures)\nwhere a naive application of the Enskog-Boltzmann equation has\nbeen shown to be invalid even in the framework of elastically\ninteracting particles \\cite{mixtures}. Much like any other\napproach to many-body systems, the present one is not directly\nuseful: perturbative expansions need to be implemented to obtain\nphysically significant results. However, as the formulation is\nrather different from e.g.,  those directly based on the Boltzmann\nequation or its ring corrections, one may be able to study\nhitherto inaccessible cases (or limits), e.g. when gradients are\nlarge (typical of granular systems) or many-body contacts are of\nimportance. Whether the present approach will indeed provide\nuseful results for these and other cases of dissipative systems\nremains to be seen. \\vspace{0,3cm}\n\n\\noindent\n\n{\\bf Acknowledgments}\n\n\\vspace{0,2cm}\n\n\nThe authors gratefully acknowledge the useful discussions with Yu.V. Slyusarenko \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $(M,d)$ be a metric space with a distinguished point $0\\in M$. Denote $\\operatorname{Lip}_0(M)$ the space of all Lipschitz functions $f:M\\to\\R$ with the property $f(0)=0$. Such a space can be equipped with the Lipschitz norm $\\| f\\|=\\operatorname{sup}_{x\\neq y}\\frac{|f(x)-f(y)|}{d(x,y)}$, which turns it into a Banach space. We see that each point in $M$ can be naturally embedded into $\\operatorname{Lip}_0(M)^*$ via the Dirac mapping $\\delta$: $\\delta_x(f)=f(x)$, $f\\in\\operatorname{Lip}_0(M)$, $x\\in M$. The norm-closure of the subspace generated by functionals $\\delta_x$, $x\\in M$, i.e.\n$$\\overline{\\operatorname{span}}^{\\operatorname{Lip}_0(M)^*}\\{\\delta_x|x\\in M\\}$$\nis the Lipschitz Free space over $M$, denoted $\\mathcal F(M)$. Lipschitz Free spaces were introduced already by Arens and Eells in \\cite{AE}, although the authors did not use the name Lipschitz Free spaces. Free spaces are called Arens-Eells spaces in \\cite{W}, where a lot of results regarding the topic is presented.\n\nLipschitz Free spaces gained a lot of interest in last decades, connecting nonlinear theory with the linear one. Given two pointed metric spaces $M,N$, every Lipschitz mapping $\\varphi:M\\to N$ which fixes the point $0$ extends to a bounded linear map $F:\\mathcal F(M)\\to\\mathcal F(N)$, making the following diagram commute:\n$$\\begin{CD}\n\\mathcal F(M) @>{F}>> \\mathcal F(N)\\\\\n@A{\\delta_M}AA @AA{\\delta_N}A\\\\\nM @>{\\varphi}>> N\n\\end{CD}$$\nWe focus on structural properties of Lipschitz Free spaces. It is well-known that $\\operatorname{Lip}_0(\\R)= L_\\infty$, which yields $\\mathcal F(\\R)= L_1$ isometrically and similarly $\\mathcal F(\\N)=\\ell_1$. In \\cite{CDW}, the authors prove that $\\mathcal F(M)$ contains a complemented copy of $\\ell_1(\\N)$ if $M$ is infinite (has at least cardinality $\\aleph_0$), which was further extended from $\\N$ to all cardinalities in \\cite{HN}. However, $\\mathcal F(\\R^2)$ cannot be embedded in $\\mathcal F(\\R)=L_1$ (see \\cite{NS}).\n\nCertain results were obtained concerning approximation properties in Free spaces, including \\cite{PS},\\cite{LP},\\cite{HLP},\\cite{K},\\cite{Godefroy},\\cite{GO} and of course \\cite{GK}. However, not much is known yet about Schauder bases in Free spaces. H\\'{a}jek and Perneck\\'{a} \\cite{HP} constructed a Schauder basis for the Free spaces $\\mathcal F(\\ell_1)$ and $\\mathcal F(\\R^n)$. From \\cite{Kaufmann} we have $\\mathcal F(M)$ is isomorphic to $\\mathcal F(\\R^n)$ for every $M$ with non-empty interior, which gives existence of Schauder basis on such $\\mathcal F(M)$.\n\nThis article follows up the article \\cite{HN}, where the authors proved existence (and in the case of $c_0$ constructively) of a Schauder basis on $\\mathcal F(N)$, for any net $N$ in spaces $C(K)$ for $K$ metrizable compact (hence for $c_0$ and $\\R^n$). In section \\ref{nonexistence} we show that the same construction as in \\cite{HN} cannot be used for constructing bases in $\\mathcal F(N)$ for arbitrary uniformly discrete subset $N$. In section \\ref{unconditionality} we prove that bases constructed in \\cite{HN} are not unconditional and that for nets in $\\R^n$, no Schauder basis on $\\mathcal F(N)$ arising from the technique using retractions can be unconditional.\n\\section{Preliminaries}\nAs we mentioned, we are interested in constructing a Schauder basis on Lipschitz Free space. However, constructing such basis directly on the Free space is rather complicated, wherefore we prefer to work with its adjoint space and transfer the results to the Free space. The next theorem shows a way to construct a Schauder basis through operators on $\\operatorname{Lip}_0(M)$.\n \n\\begin{theorem} \\label{operator} Let $M$ be a pointed metric space. Suppose there exists a sequence of linear operators $E_n:\\operatorname{Lip}_0(M)\\to \\operatorname{Lip}_0(M)$, which satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\dim E_n \\left(\\operatorname{Lip}_0(M)\\right)=n$ for every $n\\in\\N$,\n\\item There exists $K>0$ such that $E_n$ is $K$-bounded for every $n\\in\\N$,\n\\item $E_m E_n=E_n E_m=E_n$ for every $m,n\\in\\N$, $n\\leq m$,\n\\item \\label{weak}For every $n$, the operator $E_n$ is continuous with respect to topology of pointwise convergence on $\\operatorname{Lip}_0(M)$,\n\\item \\label{continuity} For every $f\\in\\operatorname{Lip}_0(M)$ the function sequence $E_n f$ converges pointwise to $f$.\n\\end{enumerate}\nThen the space $\\mathcal F(M)$ has a Schauder basis with the basis constant at most $K$.\n\\end{theorem}\n\\begin{proof}\nNote first that the topology of pointwise convergence coincides with the $w^*$-topology on bounded subsets of $\\operatorname{Lip}_0(M)$. Therefore, from the condition (\\ref{weak}), the operators $E_n$ are $w^*$ to $w^*$ continuous on bounded subsets of $\\operatorname{Lip}_0(M)$ and hence there exist linear operators $P_n:\\mathcal F(M)\\to\\mathcal F(M)$ such that $P^*_n=E_n$ for every $n\\in\\N$. It is now clear that $\\| P_n\\|\\leq K$, $\\dim P_n \\left(\\mathcal F(M)\\right)=n$ and that $P_m P_n=P_n P_m=P_n$ for every $m,n\\in\\N$, $n\\leq m$. Furthermore (\\ref{continuity}) together with the fact that the topology of pointwise convergence coincides with the $w^*$-topology on bounded subsets of $\\operatorname{Lip}_0(M)$ means, that for every $f\\in \\operatorname{Lip}_0(M)$ the sequence $E_n f$ converges $w^*$ to $f$, and that for every $\\mu\\in\\mathcal F(M)$ the sequence $P_n\\mu$ converges weakly to $\\mu$. But that means $\\| P_n\\mu-\\mu\\|\\to 0$ for every $\\mu\\in\\mathcal F(M)$. Indeed, if there were $\\mu\\in\\mathcal F(M)$, $c>0$ and a subsequence $P_{n_k}$, such that $\\| P_{n_k}\\mu-\\mu\\|>c$ for all $k\\in\\N$, then for every $n\\geq n_1$, there exists a $k\\in\\N$ such that $n\\leq n_k$, which yields\n\n$$c<\\| P_{n_k}\\mu-\\mu\\|\\leq\\| P_{n_k}\\mu-P_n\\mu\\| + \\| P_{n}\\mu-\\mu\\|\\leq (K+1)\\| P_n\\mu-\\mu\\|.$$\n\nFrom $P_1(\\mathcal F(M))\\subseteq P_2(\\mathcal F(M))\\subseteq P_3(\\mathcal F(M))\\subseteq...$ we get $E=\\bigcup_{n=1}^{\\infty}P_n(\\mathcal F(M))$ is a convex set and as all $P_n$ are commuting projections, we have that $\\mu\\notin\\overline E$. Indeed, if $\\mu \\in \\overline E$, then there is a sequence $\\{x_k\\}_{k=1}^\\infty\\subseteq E$, such that $x_k\\to \\mu$. If we choose an increasing sequence of numbers $l_k\\in\\N$, $l_k>n_1$, which satisfy $P_{l_k}x_k=x_k$, we get that\n$$\\| P_{l_k}x_k-\\mu\\|\\geq \\| P_{l_k}\\mu-\\mu\\|-\\| P_{l_k}\\mu-P_{l_k}x_k\\|\\geq \\frac{c}{K+1}-K\\| \\mu-x_k\\|.$$\nLimiting $k\\to\\infty$ yields $0\\geq \\frac{c}{K+1}$, which is a contradiction. Therefore $\\mu\\notin\\overline E$. Hence Hahn-Banach theorem gives us the existence of a linear functional $f\\in\\operatorname{Lip}_0(M)$, $\\| f\\|=1$ with $f|_E=0$ and $f(\\mu)>0$. But that is a contradiction as $P_n\\mu\\overset{w}\\to\\mu$. Therefore $P_n \\mu\\to \\mu$.\n\\end{proof}\n\nThe following corollary appears already in \\cite{HN}, p.12. It gives us a way to construct the Schauder basis on $\\mathcal F(M)$ only by using the metric space $M$.\n\n\\begin{corollary}\\label{one} Let $M$ be a metric space with a distinguished point $0$. Suppose there exists a sequence of distinct points $\\{\\mu_n\\}_{n=0}^{\\infty}\\subseteq M$, $\\mu_0=0$, together with a sequence of retractions\n$\\{\\varphi_n\\}_{n=0}^\\infty$, $\\varphi_n:M\\to M$, $n\\in\\N_0$ which satisfy the following conditions:\n\\begin{enumerate}[(i)]\n\\item $\\varphi_n(M)=\\{\\mu_j\\}_{j=0}^{n}$ for every $n\\in\\N_0$,\\label{bed1}\n\\item $\\overline{\\bigcup_{j=0}^{\\infty}\\{\\mu_j\\}}=M$, \\label{bed2}\n\\item There exists $K>0$ such that $\\varphi_n$ is $K$-Lipschitz for every $n\\in\\N_0$,\\label{bed3}\n\\item $\\varphi_m\\varphi_n=\\varphi_n\\varphi_m=\\varphi_n$ for every $m,n\\in\\N_0$, $n\\leq m$. \\label{bed4}\n\\end{enumerate}\nThen the space $\\mathcal F(M)$ has a Schauder basis with the basis constant at most $K$.\n\\end{corollary}\n\\begin{proof}\nIt is not difficult to see that for each $n\\in\\N$ the formula $E_nf=f\\circ\\varphi_n$, $f\\in\\operatorname{Lip}_0(M)$ defines a linear \noperator $E_n:\\operatorname{Lip}_0(M)\\to\\operatorname{Lip}_0(M)$, such that the sequence $E_n$ satisfies the assumptions of Theorem \\ref{operator}.\n\\end{proof}\n\nThe last two theorems lead us to the following definition.\n\n\\begin{definition} Let $M$ be an infinite metric space such that $\\mathcal F(M)$ has a Schauder basis $E$ with projections $P_n$, $n\\in\\N$. We say $E$ is an extensional Schauder basis if there exist finite sets $\\{0\\}=M_0\\subseteq M_1\\subseteq M_2\\subseteq...$ such that $\\bigcup_{n=1}^\\infty M_n$ is dense in $M$ and we have that for every $n\\in\\N$ the adjoint $P_n^*$ is a linear extension operator $P_n^*:\\operatorname{Lip}_0(M_n)\\to\\operatorname{Lip}_0(M)$ with $P_n^*f|_{M_n}=f$ (or equivalently $P_n$ is a projection onto $\\mathcal F(M_n)$). We say $E$ is a retractional Schauder basis, if there exist retractions $\\{\\varphi_n\\}_{n=0}^\\infty$, $\\varphi_n:M\\to M$ which satisfy the conditions of Corollary \\ref{one} and such that they give rise to the basis $E$, i.e. the adjoints $P_n^*$ satisfy $P_n^*f=f\\circ\\varphi_n$, $f\\in\\operatorname{Lip}_0(M)$.\n\\end{definition}\nIt is clear that in the definition we actually have $|M_n\\setminus M_{n-1}|=1$ for every $n\\in\\N$. Note also that every retractional Schauder basis is a special case of an extensional Schauder basis. The next lemma shows in more detail what form the basis vectors take.\n\\begin{lemma}\\label{char}\nLet $M$ be a metric space such that there is a sequence of distinct points $0=\\mu_0,\\mu_1,\\mu_2,...\\in M$ such that $\\bigcup_{n=1}^\\infty \\{\\mu_0,\\mu_1,...,\\mu_n\\}$ is dense in $M$. For every $n\\in\\N_0$ denote $M_n=\\{\\mu_0,...,\\mu_n\\}$. Suppose $\\mathcal F(M)$ has a Schauder basis $B=\\{e_n\\}_{n=1}^\\infty$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{extb} $B$ is an extensional Schauder basis with extension operators $E_n:\\operatorname{Lip}_0(M_n)\\to\\operatorname{Lip}_0(M)$.\n\\item\\label{basf} For every $n\\in\\N$, there are constants $0\\neq c_n,a^n_i\\in\\R$, $i\\in\\{1,...,n-1\\}$ such that we have $c_n e_n=\\delta_{\\mu_n}-\\sum_{i=1}^{n-1}a_i^{n}\\delta_{\\mu_i}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n$(\\ref{basf})\\Rightarrow(\\ref{extb})$. Note first that for every $n\\in\\N$, we have $e_n\\in \\operatorname{Im} P_n\\cap\\ker P_{n-1}$. From that it follows inductively for every $n\\in\\N$ that $\\operatorname{Im} P_n=\\operatorname{span}\\{\\delta_{\\mu_1},...,\\delta_{\\mu_n}\\}$.\n\n$(\\ref{extb})\\Rightarrow(\\ref{basf})$ The fact that $E_n=P^*_n$ is a bounded linear extension from $M_n$ to $M$ implies that each $P_n$ maps $\\mathcal F(M)$ onto $\\mathcal F(M_n)$, which means each basis vector $e_n$ is a linear combination of Dirac functionals at the points of $M_n$, such that the coefficients at $\\delta_{\\mu_n}$ do not vanish.\n\\end{proof}\nKeeping the notation from previous lemma, we see that for each $n\\in\\N$ we may define a finite dimensional operator $R_{n}:\\operatorname{Lip}_0(M_{n-1})\\to\\operatorname{Lip}_0(M_{n})$ via\n$$R_nf(\\mu_{j})=\\begin{cases}\nf(\\mu_j) & j\\in\\{0,...,n-1\\}\\,,\\\\\n\\sum_{i=1}^{n-1}a_i^{n}f(\\mu_{i}) & j=n\\,.\\\\\n\\end{cases}$$\nThe operator $E_n=P^*_n$ can be then reconstructed through a $w^*$-limit of operator composition $\\lim_k R_kR_{k-1}...R_{n+1}$. The constants $c_n$ were in the lemma only for scaling of the basis vectors $e_n$.\n\nIn case of a retractional basis, the basis vectors take form of two-point molecules: For every $n\\in\\N$ and $i\\in\\{1,...,n-1\\}$ exactly one of the coefficients $a_i^n$ is non-zero, namely has the value $1$. If for example $a_j^n=1$, then $\\varphi_j(\\mu_n)=\\mu_j$, which means $e_n=\\delta_{\\mu_n}-\\delta_{\\mu_{j}}$.\n\nThroughout this article, given a metric space $M$, $d$ will denote its metric. If $M$ is a countable (even finite) uniformly discrete metric space with $\\mathcal F(M)$ having a retractional Schauder basis, by symbols $\\mu_0,\\mu_1,\\mu_2...$, resp. $\\varphi_0,\\varphi_1,\\varphi_2,...$ we will always mean points $\\mu_i\\in M$, resp. retractions $\\varphi_i:M\\to M$ which satisfy Corollary \\ref{one}. Obviously the finite analogues of Corollary \\ref{one} and Theorem \\ref{operator} also hold. We are going to look in more detail on some properties of retractional Schauder basis.\n\nFollowing the notation of Corollary \\ref{one} (or the proof of Lemma 14 in \\cite{HN}) we find useful to denote the set-valued functions $F_i=\\varphi_i^{-1}:M\\to 2^M$, $F_i(x)=\\{y|\\ \\varphi_i(y)=x\\}$, $i\\in\\N_0$. Clearly $F_0(0)=M$. From the commutativity of the $\\varphi_i$'s further follows that for any $i<j$ we have\n$$F_i(\\mu_i)\\cap F_j(\\mu_j)\\in\\{\\emptyset,F_j(\\mu_j)\\}.$$\n\\begin{definition} Let $M$ and $\\mu_i$, $\\varphi_i$, $i\\in\\N_0$ satisfy the assumptions from Corollary \\ref{one} and $M=\\{\\mu_i\\}_{i=0}^\\infty$. A finite or infinite sequence of points $(\\mu_{k_1},\\mu_{k_2},\\mu_{k_3},...)$ is called a chain whenever $k_1<k_2<k_3<...$ and $\\varphi_{k_i-1}(\\mu_{k_i})=\\mu_{k_{i-1}}$ for every $i\\in\\{2,3,...\\}$.\n\\end{definition}\nNote that for every chain $(\\mu_{k_1},\\mu_{k_2},\\mu_{k_3}...)$ we have $F_{k_1}(\\mu_{k_1})\\supseteq F_{k_2}(\\mu_{k_2})\\supseteq F_{k_3}(\\mu_{k_3})\\supseteq ...$. We can also introduce partial order on $M$ by $\\mu_i\\prec\\mu_j$ if and only if there exist $n\\in\\N_0$ points $\\mu_{k_1},...,\\mu_{k_n}\\in M$ such that $(\\mu_i,\\mu_{k_1},...,\\mu_{k_n},\\mu_j)$ is a chain. Note also that for two chains $S,T$ the difference $S\\setminus T$ and intersection $S\\cap T$ are also chains, if nonempty. For a finite chain $(x_1,x_2,...,x_n)$ we call the point $x_1$ its initial point and $x_n$ its final point.\n\nEvery chain can be viewed as a path or its segment from $0\\in M$ to a given point $x\\in M$. Indeed, for every $x\\in M$ there exists $n\\in\\N$ such that for every $i\\geq n$ one has $\\varphi_i(x)=x$. Assuming $n$ is the least number with that property we can define the set $T^{x}_0=\\bigcup_{i=0}^{n}\\{\\varphi_i(x)\\}$ which contains exactly the points of the chain with initial point $0$ and final point $x$. Regarding $T_0^x$ as an ordered set (the order $\\prec$ is linear on $T_0^x$), it is clear that given $x\\in M$, there exists exactly one chain $T_0^x$ from $0$ to $x$.\n\nNote also that for every chain $(\\mu_{n_1},...,\\mu_{n_k})$, $k\\geq 2$ there exist constants $c_{n_i}$ (the constants from Lemma \\ref{char}) such that for basis vectors $e_{n_1},...,e_{n_k}$ we have\n$$\\sum_{i=2}^kc_{n_i}e_{n_i}=\\delta_{\\mu_{n_k}}-\\delta_{\\mu_{n_1}}.$$\nThe following lemma says that, if the space is not too \"porous\", basis vectors can be made only of two-point molecules in points which are not too far from each other.\n\n\n\\begin{lemma}[Step lemma]\\label{step} Let $M$ be a countable metric space, $\\alpha>0, K\\geq 1$ and $\\varphi_n:M\\to M$ a system of retractions from Corollary \\ref{one}. If $(\\mu_{i_1},...,\\mu_{i_j})$, $j>1$ is a chain and there exist distinct points $x_1,...,x_k\\in M$ with $d(x_l,x_{l+1})\\leq\\alpha$, $l\\in\\{1,...,k-1\\}$, $x_1=\\mu_{i_j}$, $x_k=\\mu_{i_1}$ and $\\operatorname{sup}_{i_1\\leq n\\leq i_j}\\operatorname{Lip}\\varphi_n\\leq K$, then $d(\\mu_{i_{m-1}},\\mu_{i_{m}})\\leq 2K\\alpha$ for all $m\\in\\{2,...,j\\}$.\n\\end{lemma}\n\\begin{proof}\nSuppose $d(\\mu_{i_{m-1}},\\mu_{i_{m}})> 2K\\alpha$ for some $m\\in\\{2,...,j\\}$. We know $\\varphi_{i_m}(\\mu_{i_j})=\\mu_{i_m}$ and $\\varphi_{i_m-1}(\\mu_{i_j})=\\mu_{i_{m-1}}$. We prove by induction for all $l\\in\\{1,...,k\\}$ that $\\varphi_{i_m}(x_l)=\\mu_{i_m}$ and $\\varphi_{i_m-1}(x_l)=\\mu_{i_{m-1}}$, which is a contradiction as $x_k=\\mu_{i_1}$ and $\\varphi_{i_m}(\\mu_{i_1})=\\mu_{i_1}\\neq \\mu_{i_m}$. For $l=1$ we have $x_l=\\mu_{i_j}$ and the statement clearly holds. Suppose it holds for all $l=1,...,s-1<k$. From $d(x_{s-1},x_{s})\\leq\\alpha$ it follows that $d(\\mu_{i_m},\\varphi_{i_m}(x_{s}))\\leq K\\alpha$ and $d(\\mu_{i_{m-1}},\\varphi_{i_{m}-1}(x_{s}))\\leq K\\alpha$. From commutativity of all $\\varphi_n$'s follows that either $\\varphi_{i_{m}-1}(x_s)=\\varphi_{i_m}(x_s)\\notin\\{\\mu_{i_m}\\}$ holds or $\\varphi_{i_m}(x_s)=\\mu_{i_m}$ and $\\varphi_{i_m-1}(x_s)=\\mu_{i_{m-1}}$ is true. Since $B_{K\\alpha}(\\mu_{i_m})\\cap B_{K\\alpha}(\\mu_{i_{m-1}})=\\emptyset$ we conclude the latter is true, which completes the induction step and the contradiction is obtained.\n\\end{proof}\n\nIn the following section, we are going to prove that there are spaces $\\mathcal F(M)$ with no retractional Schauder basis yet having Schauder basis, moreover extensional.\n\n\n\n\\section{Nonexistence of retractional Schauder bases}\\label{nonexistence}\n\n\\begin{definition} Let $x_0,x_1,...,x_n$, $n\\in\\N$ be distinct points. The set $C^0_n=\\{x_0,x_1,x_2...,x_n\\}$ with the (standard graph) metric $d(x_k,x_0)=n$, $k\\neq 0$, $d(x_k,x_l)=\\min\\{|k-l|,n-|k-l|\\}$, $k,l>0$ we call a circle or a circle of radius $n$ with centre $x_0$.\n\\end{definition}\nIn the following, we regard the centre $x_0$ as the base point in the pointed metric space $(C^0_n,d,x_0)$ and denote it $0$.\n\nWe are also going to use an uncentered circle, i.e. a subgraph $C_n=\\{x_1,x_2,...,x_n\\}$ with the induced metric. On $C_n$, we define orientation: We say point $x_l$ lies to the left of the point $x_k$, $k,l\\in\\{1,...,n\\}$, if one of these situations happens:\n\\begin{enumerate}\n\\item $k>\\frac{n-1}{2}$ and $l\\in\\{k,k-1,...,k-\\lfloor\\frac{n+1}{2}\\rfloor+1\\}$,\n\\item $k\\leq \\frac{n-1}{2}$ and $l\\in\\{k,k-1,...,1\\}\\cup\\{n,n-1,...,n-\\lfloor\\frac{n+1}{2}\\rfloor+k+1\\}$.\n\\end{enumerate}\nAnalogously, we say $x_l$ lies to the right of $x_k$ if one of the following conditions is satisfied:\n\\begin{enumerate}\n\\item $k\\leq\\frac{n-1}{2}$ and $l\\in\\{k,k+1,...,k+\\lfloor\\frac{n+1}{2}\\rfloor\\}$,\n\\item $k>\\frac{n-1}{2}$ and $l\\in\\{k,k+1,...,n\\}\\cup\\{1,2,...,\\lfloor\\frac{n+1}{2}\\rfloor-(n-k+1)\\}$.\n\\end{enumerate}\nWe show that every retractional Schauder basis on $\\mathcal F(C_n^0)$ has a basis constant which is increasing with $n$.\n\n\\begin{theorem}\\label{circle}\nLet $n\\in\\N$, $n\\geq 10$ and let $\\{\\varphi_i\\}_{i=0}^n$ be a system of retractions on a circle $C^0_n$ satisfying the conditions of Corollary \\ref{one} . Then there is an $s\\in\\{1,...,n\\}$ such that $$\\operatorname{Lip}\\varphi_s\\geq\\frac{\\sqrt{8n+1}-1}{8}.$$\n\\end{theorem}\n\\begin{proof}\nLet us fix $n\\geq 10$ and denote $K=\\frac{\\sqrt{8n+1}-1}{8}$. We have $\\mu_0=0$ and $\\mu_1\\in C_n$ with $\\varphi_1(x)=\\mu_1$ for all $x\\in C_n$ and $\\varphi_1(0)=0$. Indeed, if $\\varphi_1(y)=0$ for some $y\\in C_n$, then the sets $F_1(0)$ and $F_1(\\mu_1)$ have distance $1$. Since they are finite, there exist $w\\in F_1(0)$, $z\\in F_1(\\mu_1)$ such that $d(w,z)=1$ and clearly $d(\\varphi_1(w),\\varphi_1(z))=n$, which trivially yields the result, as $n>K$. We prove the theorem by contradiction and assume therefore, $\\operatorname{Lip}\\varphi_i< K$ for all $i\\in\\{1,...,n\\}$.\n\nFor every point $x\\in C_n$ there exists a $k\\in\\{1,...,n\\}$ such that $\\{x\\}=\\{\\mu_k\\}=\\varphi_k(C_n)\\setminus\\varphi_{k-1}(C_n)$ and therefore there exists exactly one chain $S_x=(\\mu_{k_1},\\mu_{k_2},...,\\mu_{k_l})$, such that $\\mu_{k_1}=\\mu_1$ and $\\mu_{k_l}=x$ (equivalently $k_1=1$ and $k_l=k$).\n\nLet us introduce sets \n$$A=\\{y|\\ y\\in C_n\\setminus\\{\\mu_1\\},d(y,\\mu_1)\\leq 3K,\\text{$y$ lies to the left of $\\mu_1$}\\}$$\n$$B=\\{y|\\ y\\in C_n\\setminus\\{\\mu_1\\},d(y,\\mu_1)\\leq 3K,\\text{$y$ lies to the right of $\\mu_1$}\\}$$\nand a mapping $f:C_n\\setminus\\{\\mu_1\\}\\to\\{A,B\\}$,\n$$f(w)=\\begin{cases}\nA & \\text{there is a $z\\in S_w\\cap A$ such that for every $y\\in S_w$, $z\\prec y$, we have $y\\notin A\\cup B$}\\,, \\\\ \nB & \\text{there is a $z\\in S_w\\cap B$ such that for every $y\\in S_w$, $z\\prec y$, we have $y\\notin A\\cup B$}\\,. \\\\\n\\end{cases}$$\nNote that the definitions of $A,B$ make perfect sense, as $3K<\\frac{n}{2}$. Also, the mapping $f$ is well-defined, as for every $w\\in C_n\\setminus\\{\\mu_1\\}$ the intersection $S_w\\cap(A\\cup B)$ is nonempty. Indeed, according to Step lemma \\ref{step} applied on the $C_n$, the distance between any two adjacent points in a chain is smaller than $2K$ and therefore for the second element $z\\in S_w$ (meaning $S_w=(\\mu_1,z,...,w)$) we have $d(\\mu_1,z)\\leq 2K$ and thus $z\\in A$ or $z\\in B$.\n\nObserve that $f(w)=A$ for every $w\\in A$ and $f(w)=B$ for every $w\\in B$. We prove there exist two points $a,b\\in C_n\\setminus\\left(\\{\\mu_1\\}\\cup A\\cup B\\right)$ such that $d(a,b)=1$, $f(a)=A$ and $f(b)=B$.\n\nLet us assume for contradiction that $f(w)=A$ for all points $w\\in C_n\\setminus\\left(\\{\\mu_1\\}\\cup A\\cup B\\right)$. Denote $z$ the closest point to the right of the set $B$, i.e. the only point with $3K <d(z,\\mu_1)\\leq 3K+1$ and $d(z,B)=1$. We have $f(z)=A$, which means the chain $S_z=(\\mu_1,\\mu_{k_2},...,z)$ leaves the set $A$ and goes to the left around (meaning omitting the set $A\\cup B$) the circle to the point $z$, with steps smaller than $2K$. Therefore there exists a point $\\mu_l\\in S_z$ such that $d(\\mu_1,\\mu_l)\\geq\\frac{n-2K}{2}$. But then we have $\\varphi_{l}(\\mu_1)=\\mu_1$, $\\varphi_{l}(z)=\\mu_l$, which yields\n$$\\operatorname{Lip}\\varphi_l\\geq \\frac{d(\\mu_1,\\mu_l)}{d(\\mu_1,z)}\\geq\\frac{n-2K}{2(3K+1)}\\geq K,$$\nas $n\\geq 10$, which contradicts our assumption.\n\nTherefore, let there exist two points $a,b\\in C_n\\setminus\\left(\\{\\mu_1\\}\\cup A\\cup B\\right)$ such that $d(a,b)=1$, $f(a)=A$ and $f(b)=B$. Consider the two chains $S_a=(\\mu_1,\\mu_{k_1},\\mu_{k_2}...,a)$ and $S_b=(\\mu_1,\\mu_{l_1},\\mu_{l_2}...,b)$ and let $i$ and $j$ be such that $\\mu_{k_i}\\in A$, $\\mu_{l_j}\\in B$ and we have $\\mu,\\nu\\notin A\\cup B\\cup\\{\\mu_1\\}$ for every $\\mu\\in S_a$, $\\mu_{k_i}\\prec \\mu$ and every $\\nu\\in S_b$, $\\mu_{l_j}\\prec \\nu$. Note that $d(\\mu_1,\\mu_{l_j})\\geq K+1$ and $d(\\mu_1,\\mu_{k_i})\\geq K+1$.\n\nWithout loss of generality suppose $k_i<l_j$. Then $\\varphi_{l_j}(b)=\\mu_{l_j}$ and $d(\\varphi_{l_j}(a),\\mu_{l_j})\\leq K$. This implies $u_a:=\\varphi_{l_j}(a)$ has distance at most $K$ from the set $B$, which yields $d(\\mu_1,u_a)\\leq 4K$ and the chain $S=(\\mu_{k_i},...,u_a)$ must go from the set $A$ to the left around the circle closer to the set $B$. Thus there must exist a point $v=\\mu_s\\in S$ such that $d(v,\\mu_1)\\geq\\frac{n-2K}{2}$. It follows that $$\\operatorname{Lip} \\varphi_s\\geq \\frac{d(\\varphi_s(u_a),\\varphi_s(\\mu_1))}{d(u_a,\\mu_1)}=\\frac{d(v,\\mu_1)}{d(u_a,\\mu_1)}\\geq\\frac{n-2K}{8K}=K,$$ which is again a contradiction. We conclude there exists an $s\\in\\{1,...,n\\}$ such that $\\operatorname{Lip}\\varphi_s\\geq K=\\frac{\\sqrt{8n+1}-1}{8}$.\n\\end{proof}\n\\begin{corollary}\\label{no basis} There exists a uniformly discrete set $N\\subseteq\\R^2$ such that the Free space $\\mathcal F(N)$ has no retractional Schauder basis.\n\\end{corollary}\n\\begin{proof}\nLet $N=\\bigcup_{n=1}^\\infty C^0_{4^n}$ be a union of circles with the same centre $0$ and with radii $4^n$, $n\\in\\N$. Suppose $d_n$ is the metric on $C^0_{4^n}$. Let us define a metric on $N$ in the following way: \n$$d(x,y)=\\begin{cases}\nd_n(x,y) & \\text{ if } x,y\\in C^0_{4^n}\\,,\\\\\n\\max\\{4^i,4^j\\} & \\text{ if } x\\in C_{4^i},y\\in C_{4^j},i\\neq j\\,.\\\\\n\\end{cases}$$\nIt is clear that $d$ is indeed a metric on $N$ and one has no difficulties to embed $N$ into $\\R^2$ in a bilipschitz way, actually with distortion not worse than $2\\pi$.\nWe show that every sequence of retractions $\\varphi_i:N\\to N$ satisfying conditions $(i)$ and $(iv)$ from Corollary \\ref{one} cannot satisfy the condition $(iii)$ of that corollary.\n\nLet therefore $\\varphi_i:N\\to N$ be a commuting sequence of retractions such that $\\varphi_0(0)=0$ and $|\\varphi_i(N)|=i+1$. We show that for every $k\\in\\N$, $k\\geq 4$, there exists an $n=n_k\\in\\N$ such that $\\operatorname{Lip}\\varphi_{n_k}\\geq k.$ Pick therefore $k\\in\\N$, $k\\geq 4$, and find the smallest $n$ such that $\\mu_n\\in C_{4^k}$. Then $\\varphi_i(\\mu_n)=\\mu_n$ for every $i\\geq n$. If there exist $j\\geq n$ and $x\\in C_{4^k}$ such that $\\varphi_j(x)\\notin C_{4^k}$, we have $\\operatorname{Lip}\\varphi_j\\geq 4^k\\geq k$ and the proof is finished. Indeed, if we take $x\\in C_{4^k}$ such that $\\varphi_j(x)\\notin C_{4^k}$ and without loss of generality we assume $x$ is such that $d(x,\\mu_n)$ is minimal among all $x\\in C_{4^k}$ with $\\varphi_j(x)\\notin C_{4^k}$, we have $d\\left(\\varphi_j(y),\\varphi_j(x)\\right)\\geq 4^k\\geq k$ for one of $x$'s neighbours $y$ (i.e. $d(x,y)=1$). This means $\\operatorname{Lip}\\varphi_j\\geq k$.\nIf, on the contrary, we have $\\varphi_i(x)\\in C_{4^k}$ for all $i\\geq n$ and all $x\\in C_{4^k}$, we find ourselves in the case of Theorem \\ref{circle}. Indeed, if we view the circle $C^0_{4^k}$ as a set $C^0_{4^k}=\\{0,\\mu_{s_1},\\mu_{s_2},...,\\mu_{s_{4^k}}\\}$ (for some eligible $s_1,s_2,...,s_{4^k}\\in\\N$) and look only at retractions $\\varphi_0,\\varphi_{s_1},\\varphi_{s_2},...,\\varphi_{s_{4^k}}$ restricted to the circle $C^0_{4^k}$, we apply \\ref{circle} and conclude $\\max\\{\\operatorname{Lip}\\varphi_{s_1},\\operatorname{Lip}\\varphi_{s_2},...,\\operatorname{Lip}\\varphi_{s_{4^k}}\\}\\geq\\frac{\\sqrt{8\\cdot 4^k+1}-1}{8}\\geq k$.\n\\end{proof}\nWe see it is impossible to build a retractional Schauder basis on $\\mathcal F(N)$. However, the space $\\mathcal F(N)$ has an extensional Schauder basis as we are going to show in the next proposition:\n\\begin{proposition} Let $N=\\bigcup_{n=1}^\\infty C^0_{4^n}$ be the metric space from Corollary \\ref{no basis}. Then $\\mathcal F(N)$ has an extensional monotone Schauder basis.\n\\end{proposition}\n\\begin{proof} First, note that we have orientation of every $C_{4^n}$, $n\\in\\N$. For every $i\\in\\N$, define $k=k(i)$ as the unique integer such that $\\frac{4^{k}-1}{3}\\leq i< \\frac{4^{k+1}-1}{3}$. Let $N=\\{0,x_1,x_2,x_3,...\\}$ be enumerated in such way that for every $i\\in\\N$ we have $x_i\\in C_{4^{k}}$ and that the enumeration respects the orientation on every circle $C_{4^k}$. Namely, if $x_i,x_{i+1}\\in C_{4^k}$, we have that $d(x_i,x_{i+1})=1$ and $x_{i+1}$ lies to the right of $x_i$. Denote $D_i=\\{0,x_1,x_2,...x_{i}\\}$.\n\nWe are going to define a sequence of extension operators $P_i:\\operatorname{Lip}_0(D_i)\\to\\operatorname{Lip}_0(N)$ and prove they satisfy the assumptions of Theorem \\ref{operator}. In order to do that, let us define some preparatory notions. Define the left and the right \"$D_i$-neighbour\" functions $\\nu^l_i,\\nu^r_i:\\bigcup_{n=1}^{k(i)} C_{4^n}^0\\to D_i$ as follows: For each $n\\in\\{1,2,...,k(i)\\}$ and $x\\in C_{4^{n}}$, let $\\nu^l_i(x)\\in D_i$ be the closest point to the left of $x$ and let $\\nu^r_i(x)\\in D_i$ be the closest point to the right of $x$. We set $\\nu^l_i(0)=\\nu^r_i(0)=0$. Note that $\\nu^l_i(x)=\\nu^{r}_i(x)=x$ if and only if $x\\in D_i$. Further we need to define \"right-\" and \"left-\" metric function (not proper metrics) on every circle $C_{4^n}$. For points $x,y\\in C_{4^n}$ we set the value $d^l(x,y)$ as the length of the path (in the graph $C_{4^n}$) going from $x$ to the left up to $y$. Analogously, we set $d^r(x,y)$ as the length of the path going from $x$ to the right up to $y$. It is clear that for $x,y\\in C_{4^n}$ we have $d^l(x,y)=d^r(y,x)$ and $d(x,y)=\\min\\{d^r(x,y),d^l(x,y)\\}$.\n\nFurther we define for every $i\\in\\N$ the $i$-th interpolation function $I_i:\\operatorname{Lip}_0(D_i)\\times \\bigcup_{n=1}^{k(i)} C_{4^n}^0\\to\\R$ via\n$$I_i(f,x)=\\frac{d^r(x,\\nu^r_i(x))f(\\nu^l_i(x))+d^l(x,\\nu^l_i(x))f(\\nu^r_i(x))}{d^l(x,\\nu^l_i(x))+d^r(x,\\nu^r_i(x))}\\ \\ \\  \\text{if } x\\neq \\nu^l_i(x) \\text{ or } x\\neq \\nu^r_i(x)$$\nand $I_i(f,x)=f(x)$ for $x=\\nu^l_i(x)=\\nu^r_i(x)$. Clearly, $I_i(f,x)$ is the value of linear interpolation of the function $f$ between closest points of $x$ to the left and to the right from the set $D_i$, given we take $x$ itself to be the closest point to $x$ in any direction if $x\\in D_i$. Let now $f\\in\\operatorname{Lip}_0(D_i)$. Then we define our (extension) operators $P_i$, $i\\in\\N$ simply as\n$$P_if(x)=\\begin{cases}\nI_i(f,x) & x\\in\\bigcup_{n=1}^{k(i)} C_{4^n}^0\\,,\\\\\n0 & x\\in\\bigcup_{n=k(i)+1}^{\\infty} C_{4^n}\n\\end{cases}$$\nand of course, $P_0=0$. Clearly $P_i$ is a linear operator for every $i\\in\\N$ and the function $P_if$ is Lipschitz with the same constant as $f$. Indeed, if we take $x\\in C_{4^{n}}$ and $y\\in C_{4^{m}}$ with $m<\tn$, we see from the definition of $I_i$ that $\\min_{z\\in C_{4^{n}}}f(z)\\leq P_if(x)\\leq\\max_{z\\in C_{4^{n}}}f(z)$ and $\\min_{w\\in C_{4^{m}}}f(w)\\leq P_if(y)\\leq\\max_{w\\in C_{4^{m}}}f(w)$. From that and from the fact that $d(z,w)=d(x,y)=4^n$ holds for all $z\\in C_{4^n}$ and $w\\in C_{4^m}$, we get\n$$|P_if(x)-P_if(y)|\\leq \\max_{\\substack{\n            z\\in C_{4^{n}}\\\\\n            w\\in C_{4^{m}}}}\n|f(z)-f(w)|\\leq 4^n \\| f\\|=d(x,y)\\| f\\|.$$\nFor $x\\in C_{4^{n}}$ and $0$ we have clearly $|P_if(x)-P_if(0)|\\leq\\max_{z\\in C_{4^{n}}}|f(z)|\\leq d(x,0)\\| f\\|$.\n\nThe only nontrivial case to prove is the case $x,y\\in C_{4^{k(i)}}$. Let therefore $x,y\\in C_{4^{k(i)}}$. There are three cases. If $x,y\\in D_i$, then $P_if(x)=f(x)$ and $P_if(y)=f(y)$, which is trivial. Let $x,y\\notin D_i$ and $\\nu^r_i(x)=\\nu^r_i(y)=a$, $\\nu_i^l(x)=\\nu_i^l(y)=b$. We can assume $d^r(b,x)\\leq d^r(b,y)$, for the roles of $x$ and $y$ are symetrical. From that we have $d^r(y,a)\\leq d^r(x,a)$. If $d(x,y)=d^r(x,y)$, we have\n\\begin{align*}\n|P_if(x)-P_if(y)|&=\\left|\\frac{f(b)d^r(x,a)+f(a)d^r(b,x)}{d^r(b,a)}-\\frac{f(b)d^r(y,a)+f(a)d^r(b,y)}{d^r(b,a)}\\right|\\\\\n&=\\left|\\frac{f(b)d^r(x,y)-f(a)d^r(x,y)}{d^r(b,a)}\\right|\\\\\n&\\leq \\frac{\\| f\\| d(a,b)}{d^r(b,a)}d^r(x,y)\\leq \\| f\\| d(x,y).\n\\end{align*}\nIf $d(x,y)=d^l(x,y)$, then $d(x,y)=d^r(y,a)+d^r(a,b)+d^r(b,x)$ and then from $d^r(b,a)-d^r(x,y)=d^r(b,x)+d^r(y,a)$ we have by triangle inequality\n\\begin{align*}\n|P_if(x)-P_if(y)|&=\\left|\\frac{f(b)d^r(x,y)-f(a)d^r(x,y)}{d^r(b,a)}\\right|\\\\\n&=\\left|\\frac{f(b)\\left(d^r(x,y)-d^r(b,a)\\right)+(f(b)-f(a))d^r(b,a)+f(a)\\left(d^r(b,a)-d^r(x,y)\\right)}{d^r(b,a)}\\right|\\\\\n&=\\left|\\frac{\\left(d^r(b,x)+d^r(y,a)\\right)\\left(f(a)-f(b)\\right)+(f(b)-f(a))d^r(b,a)}{d^r(b,a)}\\right|\\\\\n&\\leq \\| f\\|\\left(\\frac{d(a,b)}{d^r(a,b)}\\left(d^r(b,x)+d^r(y,a)\\right)+d(a,b)\\right)\\leq\\| f\\| d(x,y)\\\\\n\\end{align*}\nThe case $x\\in D_i$, $y\\notin D_i$ is proved in a similar way.\n\nWe see that the functions $f_j=P_i\\left(\\chi_{\\{x_j\\}}\\right)$, $1\\leq j\\leq i$ create a basis of each space $P_i(\\operatorname{Lip}_0(N))$, hence $\\dim P_i(\\operatorname{Lip}_0(N))=i$ for every $i\\in\\N$.\n\nTo prove the commutativity it suffices to prove $P_{i+1}P_i=P_iP_{i+1}=P_i$ for every $i\\in\\N$. While $P_{i}P_{i+1}=P_i$ is clear, we prove for every $f\\in\\operatorname{Lip}_0(N)$ we have $P_{i+1}P_if=P_if$. Fix $f\\in\\operatorname{Lip}_0(N)$. If $k(i+1)>k(i)$, then $D_i=\\bigcup_{n=1}^{k(i)} C_{4^n}^0$ and $P_{i+1}P_i f(x)=f(x)=P_i f(x)$ for all $x\\in \\bigcup_{n=1}^{k(i)} C_{4^n}^0$ and $P_{i+1}P_i f(x)=0=P_i f(x)$ for all $x\\notin \\bigcup_{n=1}^{k(i)} C_{4^n}^0$. Let therefore $k(i+1)=k(i)$.\n\nDenote $a=x_i=\\nu^l_{i}(x_{i+1})$ and $b=\\nu^r_{i}(x_{i+1})$. All we need to check is $P_{i+1}P_i f(y)=P_i f(y)$ holds for all $y\\in C_{4^{k(i)}}\\setminus D_i$. Indeed, for all other points $x$ we have $P_if(x)=P_{i+1}f(x)$. Take therefore a point $y\\neq x_{i+1}$ (otherwise it is trivial). Note that $\\nu^l_{i}(y)=a$, $\\nu^r_{i}(y)=b$ and that $d^r(a,x_{i+1})=1$. Then we have\n\\begin{align*}\nP_{i+1}(P_i f)(y)&=\\frac{d^r(x_{i+1},y)P_if(b)+d^r(y,b)P_if(x_{i+1})}{d^r(x_{i+1},b)}\\\\\n&=\\frac{d^r(x_{i+1},y)f(b)+d^r(y,b)\\cdot\\frac{f(b)+d^r(x_{i+1},b)f(a)}{d^r(a,b)}}{d^r(x_{i+1},b)}\\\\\n&=\\frac{d^r(x_{i+1},y)d^r(a,b)+d^r(y,b)}{d^r(x_{i+1},b)d^r(a,b)}\\cdot f(b)+\\frac{d^r(y,b)}{d^r(a,b)}\\cdot f(a)\\\\\n&=\\frac{d^r(x_{i+1},y)d^r(x_{i+1},b)+d^r(x_{i+1},y)+d^r(y,b)}{d^r(x_{i+1},b)d^r(a,b)}\\cdot f(b)+\\frac{d^r(y,b)}{d^r(a,b)}\\cdot f(a)\\\\\n&=\\frac{d^r(x_{i+1},b)\\left(1+d^r(x_{i+1},y)\\right)}{d^r(x_{i+1},b)d^r(a,b)}\\cdot f(b)+\\frac{d^r(a,y)}{d^r(a,b)}\\cdot f(a)\\\\\n&=\\frac{d^r(a,y)}{d^r(a,b)}\\cdot f(b)+\\frac{d^r(a,y)}{d^r(a,b)}\\cdot f(a)\\\\\n&=P_i f(y)\\\\\n\\end{align*}\nand the commutativity is proved.\n\nLet $i\\in\\N$. If $f_{\\alpha}\\to f$ pointwise, then for every $x\\in D_i$ we have $P_if_{\\alpha}(x)=f_{\\alpha}(x)\\to f(x)=P_if(x)$ and for every $x\\in\\bigcup_{l=k(i)+1}^\\infty C_{4^{l}}$ we have $P_if_{\\alpha}(x)=0=P_if(x)$. Finally, for every $x\\in C_{4^{k(i)}}\\setminus D_i$ we have $P_if_{\\alpha}(x)=\\gamma_x f_{\\alpha}(a_x)+(1-\\gamma_x)f_{\\alpha}(b_x)$, for some eligible $\\gamma_x\\in [0,1]$, $a_x,b_x\\in D_i$ and the choice of these points depends only on $x$ (and $i$ of course). Therefore $P_if_{\\alpha}\\to P_if$ pointwise, which means that every operator $P_i$ is continuous with respect to topology of pointwise convergence. \n\nFinally the sequence $P_if$ converges pointwise to $f$. Indeed, for every $y\\in N$ there exists $i\\in\\N$ such that $y\\in D_i\\subseteq D_{i+1}\\subseteq D_{i+2}\\dots$, which yields $P_i f(y)=P_{j}f(y)=f(y)$ for all $j\\geq i$. Hence $P_i f\\to f$ pointwise.\n\nSince the operators $P_i$ meet all assumptions from Theorem \\ref{operator}, we get that there is a sequence of operators $T_i:\\mathcal F(N)\\to\\mathcal F(N)$, $i\\in\\N_0$ with $T_i^*=P_i$ which build a monotone Schauder basis for $\\mathcal F(N)$.\n\\end{proof}\n\\begin{remark*} It was not necessary for the construction of $P_i$'s to enumerate the set $N$ with respect to orientation on every circle $C_{4^k}$. Actually any enumeration which satisfies $x_i\\in C_{4^{k}}$ for every $i\\in\\N$ works. Our choice only slightly simplifies the proof.\n\\end{remark*}\n\\section{Unconditionality of retractional Schauder bases}\\label{unconditionality}\nAs we construct a Schauder basis on $\\mathcal F(M)$ via sequence of retractions, as described in Corollary \\ref{one}, properties of such a basis depend also on properties of the metric space $M$. Naturally it leads us to the question: What can $M$ be like such that there is an unconditional retractional Schauder basis on $\\mathcal F(M)$? The next lemma sets a condition on the chains under which the acquired basis is conditional. It is further used in Theorem \\ref{main}, which shows that retractional bases on Free spaces of nets in finite-dimensional spaces are conditional.\n\\begin{lemma}\\label{alligned chains} Let $\\alpha,\\beta>0$ and let $N$ be an $\\alpha$-separated metric space, such that there exist retractions $\\varphi_i:N\\to N$ satisfying the conditions from Corollary \\ref{one}. Suppose there exists $n_0\\in\\N$ such that for every $n\\in\\N$, $n\\geq n_0$ there exist chains $S=(\\mu_{0},\\mu_{k_1},...,\\mu_{k_s})$ and $T=(\\mu_0,\\mu_{l_1},...,\\mu_{l_m})$, $s,m\\in\\N$ with $d(\\mu_{k_s},\\mu_{l_m})\\leq\\beta$ and $|S\\setminus T|\\geq n$. Then the retractional Schauder basis on $\\mathcal F(N)$ corresponding to the retractions $\\varphi_i$ is conditional.\n\\end{lemma}\n\\begin{proof}\nLet now $P_i$ be the associated Schauder projection to the mapping $\\varphi_i$ for each $i\\in\\N_0$, i.e. the projection to the subspace $\\operatorname{span}\\{\\delta_{\\mu_0},\\delta_{\\mu_1},...,\\delta_{\\mu_i}\\}$. Instead of working directly with $P_0,P_1,P_2,...$ we will use their adjoints $P_0^*,P_1^*,P_2^*,...$ and for every $n\\in\\N$, $n\\geq n_0$ we construct a function $f_n\\in\\operatorname{Lip}_0(N)$ with $\\| f_n\\|\\leq 1$ and find a sequence of signs $ \\varepsilon_0, \\varepsilon_1,..., \\varepsilon_{k_s}$ for some $s\\geq n$ such that the following inequality holds \n$$\\left\\Vert\\sum_{i=0}^{k_s} \\varepsilon_i(P_{i+1}^*-P_{i}^*)f_n\\right\\Vert\\geq \\frac{\\alpha (n-1)}{\\beta}.$$\nFix $n\\in\\N$ and chains $S=(\\mu_0,\\mu_{k_1},...,\\mu_{k_s})$, $T=(\\mu_0,\\mu_{l_1},...,\\mu_{l_m})$ for which we have $d(\\mu_{k_s},\\mu_{l_m})<\\beta$ and $|S\\setminus T|\\geq n$. Suppose now $t\\in\\{0,1,2,...,s-n\\}$ is such that $\\mu_{k_t}\\in T$ and $\\mu_{k_{t+1}}\\notin T$ (we set $\\mu_{k_0}=\\mu_0$). We define the function $f_n$ on $N$ via the formula\n\n$$f_n(x)=\\begin{cases}\n\\frac{\\alpha}{2} & x=\\mu_{k_j}\\text{ for } j \\text{ odd},j>t\\,,\\\\\n\\frac{-\\alpha}{2} & x=\\mu_{k_j}\\text{ for } j \\text{ even},j> t\\,,\\\\\n0 & \\text{else}\\,.\n\\end{cases}$$\nClearly, $f_n(\\mu_0)=0$ and $\\| f_n\\|\\leq 1$. For the following choice of sings $ \\varepsilon_0=1$,\n$$ \\varepsilon_i=\\begin{cases}\n- \\varepsilon_{i-1} & i=k_j \\text{ for some }j\\in\\N\\,,\\\\\n \\varepsilon_{i-1} & \\text{else}\\,,\n\\end{cases}$$\nwe have\n$$\\sum_{i=0}^{k_s} \\varepsilon_i(P_{i+1}^*-P_{i}^*)=-P_0^{*}+2\\sum_{j=1}^s(-1)^{j+1}P^*_{k_j}+(-1)^sP^{*}_{k_s+1}=:P$$\nand then\n\\begin{align*}\n\\left\\Vert P\\right\\Vert&\\geq \\left\\Vert P f_n\\right\\Vert\\geq \\left\\Vert \\frac{Pf_n(\\mu_{k_s})-Pf_n(\\mu_{l_m})}{d(\\mu_{k_s},\\mu_{l_m})}\\right\\Vert\\geq\\frac{1}{\\beta}\\left\\Vert Pf_n(\\mu_{k_s})-Pf_n(\\mu_{l_m})\\right\\Vert=\\\\\n&=\\frac{1}{\\beta}\\left\\vert -f_n(0)+2\\sum_{j=1}^s(-1)^{j+1}f_n(\\mu_{k_j})+(-1)^sf_n(\\mu_{k_s})+0\\right\\vert\\\\\n&=\\frac{1}{\\beta}\\left\\vert 2\\sum_{j=t+1}^s\\frac{\\alpha}{2}-\\frac{\\alpha}{2}\\right\\vert\\geq\\frac{\\alpha(s-t-1)}{\\beta}\\geq\\frac{\\alpha (n-1)}{\\beta}.\\\\\n\\end{align*}\n\\end{proof}\nRecall that a subset $S$ of a metric space $M$ is called an $\\alpha,\\beta$-net whenever $S$ is $\\alpha$-separated and $\\beta$-dense in $M$, i.e. $\\inf_{x\\neq y}d(x,y)\\geq \\alpha$, $x,y\\in S$ and $\\operatorname{sup}_{x\\in M} d(x,S)\\leq \\beta$.\n\nIn \\cite{HN}, the authors constructed a system of retractions on the integer lattice in $c_0$ which satisfies the conditions of Corollary \\ref{one}. Through suitable homomorphisms they further showed the existence of a basis on any Free space of a net in a separable $C(K)$ space or a net in $c_0^+$, the positive cone in $c_0$.\n \n\\begin{corollary} Let  $N$ be a net in any of the following metric spaces: $C(K)$, $K$ metrizable compact, or\n$c_0^+$ (the subset of $c_0$ consisting of elements with non-negative coordinates). The basis on $\\mathcal F(N)$ constructed in \\cite{HN} is conditional.\n\\end{corollary}\n\\begin{proof}\nFirst we consider the case $N=\\mathbb{Z}^{<\\omega}\\subseteq c_0$, the integer lattice in $c_0$. Following the proof of Lemma $14$ in \\cite{HN} we see, there are chains which go parallelly along the first coordinate axis (or any other coordinate axis). Every such two chains hence satisfy the conditions of the previous lemma, which yields that a basis arising from these retractions cannot be unconditional. As the existence of bases in other cases than $N$ being the integer lattice in $c_0$ was proven only by isomorphisms, we conclude that none of them are unconditional.\n\\end{proof}\n\\begin{theorem}\\label{main}\nLet $N$ be an $\\alpha,\\beta$-net in a finite-dimensional normed space $X$ with $\\dim X\\geq 2$. Let $E=\\{e_i\\}_{i=1}^\\infty$ be a retractional Schauder basis on $\\mathcal F(N)$. Then $E$ is conditional.\n\\end{theorem}\nIn the following, $B_{ \\varepsilon}(x)$ denotes closed ball of radius $ \\varepsilon>0$ and centre $x\\in X$, $B_{ \\varepsilon}^{\\circ}(x)$ denotes its interior. In the same way $B_{ \\varepsilon}:=B_{ \\varepsilon}(0)$ and $S_{ \\varepsilon}$ denotes sphere of radius $ \\varepsilon$ and centre $0$.\n\\begin{proof}\nLet $\\varphi_i:N\\to N$ be the corresponding retractions to the basis $E$. We prove the theorem by showing that the assumptions of Lemma \\ref{alligned chains} are met. Denote $\\operatorname{sup}_{i\\in\\N}\\operatorname{Lip}\\varphi_i=K<\\infty$. Pick $n\\in\\N$, such that $n>8K$. Define annulus with radii $r$ and $w$, $w<r$ as $A(r,w)=B_{r+w}(0)\\setminus B^{\\circ}_{r-w}(0)$. Our aim is to prove there exist chains $T,Z$ with final points $t,z\\in A(3K\\beta n+\\beta,\\beta)\\cap N$ with $d(t,z)\\leq 2\\beta$ such that $x\\in B_{K\\beta n}$ holds for $x=x_{t,z}$, the final point of the chain $T\\cap Z$. Then we have $d(t,x)\\geq 3K\\beta n-K\\beta n=2K\\beta n$ and Step lemma \\ref{step} yields $|T\\setminus Z|\\geq n$, which by Lemma \\ref{alligned chains} concludes the proof.\n\nFor the following, for every two points $x,y\\in N$ with $x\\prec y$ denote $T_x^y$ the chain with initial point $x$ and final point $y$. Assume now for contradiction, for every pair of points $t,z\\in A(3K\\beta n+\\beta,\\beta)\\cap N=:A$ with $d(t,z)\\leq 2\\beta$ the final point $x_{t,z}$ of $T_0^t\\cap T_0^z$ lies outside the ball $B_{K\\beta n}$. That means there exists a point $\\mu_m\\in N$ for some $m\\in\\N$ with $d(0,\\mu_m)>K\\beta n$ such that $\\varphi_m(t)=\\mu_m$ for every $t\\in A$. To prove this, note that $0\\in\\bigcap_{t\\in A}T_0^t$ and as $\\bigcap_{t\\in A}T_0^t$ is a chain, it has a final point which we denote $\\mu_m$ and prove that $d(0,\\mu_m)>K\\beta n$. We show that for every two points $t,z\\in A$ the final point $x_{t,z}$ of the chain $T_0^t\\cap T_0^z$ is of greater norm than $K\\beta n$. Clearly, if $d(t,z)\\leq 2\\beta$, the statement holds as assumed. If $d(t,z)> 2\\beta$, we can find a finite sequence of points $y_1,...,y_l\\in A$,  $l\\in\\N$ such that $d(y_i,y_{i+1})\\leq 2\\beta$ for every $i\\in\\{1,...,l-1\\}$ and that $y_1=t$ and $y_l=z$. Then $x_{t,z}\\in\\{x_{y_i,y_{i+1}}|\\ i\\in\\{1,...,l-1\\}\\}$, which means $\\| x_{t,z}\\|> K\\beta n$. Note that for any three points $s,t,z\\in A$ the final point $x_{s,t,z}$ of the chain $T_0^s\\cap T_0^t\\cap T_0^z$ is equal to one of the points $x_{s,t},x_{t,z},x_{s,z}$. Indeed, as $x_{s,t},x_{t,z}\\in T_0^t$, we have that either $x_{s,t}\\prec x_{t,z}$ or $x_{s,t}\\succ x_{t,z}$. If $x_{s,t}\\prec x_{t,z}$, then $x_{s,z}=x_{s,t}=x_{s,t,z}$ and the other case follows symmetrically. But from that we get inductively that for any finite number of points $t_1,...,t_v$, there are indices $i,j\\in\\{1,...,v\\}$, such that the final point $x_{t_1,...,t_v}$ of the chain $\\bigcap_{l=1}T_0^{t_l}$ equals $x_{t_i,t_j}$. Because for each two $t,z\\in A$ we have $\\| x_{t,z}\\|>K\\beta n$ and $A$ is finite, we have $\\| \\mu_m\\|> K\\beta n$.\n\nObserve further, that $T_{\\mu_m}^t\\cap B_{\\beta n}=\\emptyset$ holds for every chain $T_{\\mu_m}^t$ with initial point $\\mu_m$ and final point $t\\in A$. Indeed, if $\\mu_p\\in T_{\\mu_m}^t$, $p\\in\\N$ is such that $\\|\\mu_p\\|\\leq \\beta n$, we have $\\operatorname{Lip}\\varphi_m\\geq \\frac{\\|\\varphi_m(0)-\\varphi_m(\\mu_p)\\|}{\\| \\mu_p\\|}=\\frac{\\|\\mu_m\\|}{\\|\\mu_p\\|}>\\frac{K\\beta n}{\\beta n}=K$, which is not possible.\n\n\n\nLet us denote $S=\\bigcup_{t\\in A}T_{\\mu_m}^{t}$ the set of all chains from $\\mu_m$ to points of $A$. Let $S=\\{\\mu_{k_1},...,\\mu_{k_q}\\}$ for some $k_1<k_2<...<k_q$, $q\\in\\N$. Note that $\\mu_{k_1}=\\mu_m$. For every chain $T=(t_1,...,t_l)$, $l\\in\\N$ define a trajectory of the chain $\\operatorname{Tr}(T)$ as the union of the line segments $\\bigcup_{i=1}^{l-1} [t_i,t_{i+1}]$. Denote $D=\\bigcup_{t\\in A}\\operatorname{Tr}\\left(T_{\\mu_m}^{t}\\right)$. Define now a function $F:[1,q]\\times A\\to D$ via \n$$F(t,x)=(t-i)\\varphi_{k_{i+1}}(x)+(1-(t-i))\\varphi_{k_{i}}(x), x\\in A, t\\in [i,i+1), i\\in\\{1,...,q-1\\}$$\nand $F(q,x)=x$, $x\\in A$. We see that for each $t\\in [1,q]$, the function $F(t,\\cdot)$ is $K$-Lipschitz and that we have $F(1,x)=\\mu_m$ for all $x\\in A$.\n\nLet $ \\varepsilon=3K\\beta n+\\beta$ and consider $B=\\{B^{\\mathrm{o}}_{2\\beta}(x)\\cap S_{ \\varepsilon}\\}_{x\\in A}$ as an open cover of $S_{ \\varepsilon}$ and find a partition of unity $\\{\\psi_a\\}_{a\\in A}$ subordinated to the cover $B$. Define a function $R:[1,q]\\times S_{ \\varepsilon}\\to X$ by\n\n$$R(t,x)=\\sum_{a\\in A}\\psi_a(x)F(t,a),\\ \\ t\\in [1,q],\\ x\\in S_{ \\varepsilon}.$$\n\nWe see, that $R(1,x)=\\mu_m$ for all $x\\in S_{ \\varepsilon}$ and that $$\\operatorname{sup}_{x\\in S_{ \\varepsilon}}|R(q,x)-x|\\leq 2\\beta.$$ Of course, $R$ is continuous on $[1,q]\\times S_{ \\varepsilon}$. Our goal is to prove there is a continuous deformation of  $S_{ \\varepsilon}$ into one point $\\mu_m$ avoiding the origin, which is a contradiction. For that we define a straight-line homotopy between identity and $R(q,\\cdot)$ by $W:[0,1]\\times S_{ \\varepsilon}\\to A( \\varepsilon,2\\beta)$, $W(t,x)=tR(q,x)+(1-t)x$. Joining mappings $W$ and $R$ we get a mapping $Z:[0,q]\\times S_{ \\varepsilon}\\to X$ precisely defined by\n$$Z(t,x)=\\begin{cases}\nW(t,x) & t\\in[0,1), x\\in S_{ \\varepsilon}\\,,\\\\\nR\\left(\\frac{q}{t},x\\right) & t\\in[1,q], x\\in S_{ \\varepsilon}\\,.\\\\\n\\end{cases}$$\nAll there is left to prove is that $R([1,q]\\times S_{ \\varepsilon})\\cap \\{0\\}=\\emptyset$. To see that, note that the value $R(t,x)$ is a convex combination of values $F(t,a)$, where $a\\in A$ are such that $d(x,a)<2\\beta$. Fix therefore $x\\in S_{ \\varepsilon}$ and let $\\mu_{l_1},\\mu_{l_2},...,\\mu_{l_p}\\in A$ be such that $d(x,\\mu_{l_i})<2\\beta$ for all $i$. From the preceding paragraphs it follows that the trajectory $\\operatorname{Tr}(T_{\\mu_m}^{\\mu_{l_i}})$ of each chain from $\\mu_m$ to $\\mu_{l_i}$ has no intersection with $B_{\\beta \\frac{n}{2}}$. Indeed, as the chain $T_{\\mu_{m}}^{\\mu_{l_i}}$ avoids the ball $B_{\\beta n}$ and the distance between two consecutive points in a chain is bounded by $2\\beta K$ and $n>2K$, we get the result. From the fact that $d(\\mu_{l_j},\\mu_{l_i})\\leq 4\\beta$ for all $i,j$, we have that $\\| F(t,\\mu_{l_i})-F(t,\\mu_{l_j})\\|\\leq 4K\\beta$ for all $t\\in [1,q]$. But as $n>8K$ we get $R(t,x)\\neq 0$ for any $t\\in [0,1]$. Altogether we obtain $Z([0,q]\\times S_{ \\varepsilon})\\cap \\{0\\}=\\emptyset$, which was to prove.\n\\end{proof}\nOne could ask in general what are the metric spaces $M$ such that $\\mathcal F(M)$ has an unconditional Schauder basis. It is clear that if $M$ contains a line segment, then $ L_1$ is contained in $\\mathcal F(M)$ and therefore $\\mathcal F(M)$ cannot have an unconditional Schauder basis. The only interesting cases are then topologically discrete spaces $M$. Our guess is that if $\\mathcal F(M)$ has an unconditional Schauder basis, it is isomorphic to $\\ell_1$.\n\\\\\n\\\\\\textbf{Open problem 1} \\textit{Suppose $\\mathcal F(M)$ has an unconditional Schauder basis. Is it isomorphic to $\\ell_1$?}\n\nIn \\cite{Gd}, one sees that $\\mathcal F(M)$ is a complemented subspace of $L_1$ if and only if $M$ can be bi-Lipschitzly embedded into an $\\R$-tree. A complemented subspace of $ L_1$ with unconditional basis is isomorphic to the space $\\ell_1$ due to \\cite{LiPe}. One can therefore restate the conjecture above into: Suppose $\\mathcal F(M)$ has a Schauder basis $B$. If $M$ cannot be embedded into an $\\R$-tree, is it true that $B$ is conditional?\n\\\\\n\\\\\\textbf{Open problem 2} \\textit{Is it true that for every uniformly discrete set $N\\subseteq\\R^2$ the space $\\mathcal F(N)$ has a Schauder basis?}\n\nIt follows from Corollary \\ref{no basis} the answer is no if we restrict ourselves only to retractional Schauder bases. However, we don't know if, supposed the answer is yes, we can find for every uniformly discrete set $N\\subseteq\\R^2$ an extensional Schauder basis on $\\mathcal F(N)$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAll graphs considered in this work are finite graphs with no parallel edges. Later, \nwe will further restrict ourselves to loopless graphs. For standard notions of\nGraph Theory we refer the reader to \\cite{bondy2008}. In particular, for $n\\ge 3$,\n we denote by $P_n$  (resp.\\ $C_n$) the path (resp.\\ cycle) on $n$ vertices.\n\nGiven a pair of graphs $G$ and $H$ a \\textit{full-homomorphism}\n$\\varphi\\colon G\\to H$ is a vertex mapping such that for each pair of\nvertices $x,y\\in V(G)$ there is an edge $xy\\in E(G)$ if and only if\n$\\varphi(x)\\varphi(y)\\in E(G)$. In particular, if $H$ is a simple graph, then\nadjacent vertices in $G$ are mapped to different vertices in $H$. Moreover,\nif $\\varphi(x)  = \\varphi(y)$, then $x$ and $y$ have the same neighbourhood\nin $G$. \n\nFor a fixed graph $H$, a \\textit{full $H$-colouring} of a graph $G$\nis a full-homomorphism of $G$ to $H$. A \\textit{minimal $H$-obstruction} is a graph\nthat does not admit a full $H$-colouring, such that every proper induced subgraph\nof $G$ admits a full $H$-colouring. We denote by $\\obs(H)$ the set of minimal\n$H$-obstructions. In~\\cite{federDM308}, Feder and Hell showed that for a graph $H$\nwith $l$ vertices with loops, and $k$ vertices without loops, every graph in $\\obs(H)$\nhas at most $(k+1)(l+1)$ vertices, and this bound is tight. Later, Hell and\nHern\\'andez-Cruz showed that the same tight bound holds in the case of\ndigraphs~\\cite{hellDM338}. Independently and in a more \ngeneral setting, Ball, Ne\\v{s}et\\v{r}il, and Pultr~\\cite{ballEJC31}, proved that\nfor each relational structure $A$, there are a finite number\nof minimal $A$-obstructions. Each of these results imply\nthat for every simple graph $H$ there are finitely many minimal $H$-obstructions.\n\n\\begin{proposition}\\label{prop:finite-minimal}\\cite{ballEJC31,federDM308,hellDM338}\nFor each graph $H$ there is a finite number of minimal $H$-obstructions.\n\\end{proposition}\n\nFurthermore, Ball, Ne\\v{s}et\\v{r}il, and Pultr \\cite{ballEJC31} describe the\nconnected minimal obstructions of paths and cycles, i.e., the connected\ngraphs in $\\obs(C_n)$ and in $\\obs(P_n)$. They also propose a\nrecursive description of disconnected minimal $P_n$-obstructions,\nbut the ``lists corresponding to the paths do not seem to be more transparent\nthan those in the connected case'' \\cite{ballEJC31}. In this work, we propose\na transparent description of the list of disconnected minimal obstructions\nof paths. We do so by means of positive solutions to integer equations. In\nparticular,  we list all minimal $P_n$-obstructions, and we build on this\ndescription to propose the complete list of minimal obstructions for cycles.\n\nThe rest of this work is structured as follows.  First, in\nSection~\\ref{sec:paths} we propose a description of minimal\n$P_n$-obstructions. In Section~\\ref{sec:cycles}, we make some\ngeneral observations regarding minimal obstructions of regular graphs,\nand use these to propose a description of minimal $C_n$-obstructions\nin terms of minimal $P_{n-1}$-obstructions. We conclude\nthis work in Section~\\ref{sec:conclusions} where we propose some problems\nthat arise from observations in Section~\\ref{sec:cycles}.\nThe rest of this section contains some preliminary results needed for this work.\n\nFrom this point onwards, we only consider loopless finite graphs.\n A pair of vertices $x$ and $y$ of a graph $G$ are called\n\\textit{false twins}  if $N(x) = N(y)$, and \\textit{true twins} of $N[x] = N[y]$. In\nparticular, every pair of true twins are adjacent, while every pair of false twins are\nnon-adjacent.  In \\cite{sumnerDM5}, Sumner defined a \\textit{point determining} graph\nas a graph for which non adjacent vertices have distinct neighbourhoods, i.e., a graph\n$G$ is point determining if it has no pair of false twins. \n\n\n\\begin{proposition}\\cite{sumnerDM5}\\label{prop:rem-v}\nFor every non trivial point determining graph $G$ there is a vertex $v\\in V(G)$\nsuch  that $G-v$ is point determining. Moreover, if $G$ is connected, then \nthere are two distinct vertices with that property.\n\\end{proposition}\n\nA pair of graphs $G$ and $H$ are \\textit{full-homomorphically equivalent} if\n$G$ admits a full $H$-colouring and $H$ admits a full $G$-colouring.\nA \\textit{core} in the category of graphs with full-homomorphisms, is a graph\n$G$ such that every full-homomorphism $\\varphi\\colon G\\to G$ is surjective.\nIt is not hard to notice that for each graph $H$, there is a unique (up to isomorphism)\ncore $G$ full-homomorphically equivalent to $H$. In this case, we\nsay that $G$ is the \\textit{full-core} of $H$. \n\nPoint determining graphs play an important role in the category of graphs with\nfull-homomorphisms. In particular, every core in the full-homomorphism \ncategory of graphs is a point determining graph. Indeed, suppose that $x$ and\n$y$ are a pair of false twins in a graph $G$. By mapping $x$ to $y$, we obtain a\nfull-homomorphism of $G$ onto a proper subgraph, which implies that $G$ is not\na core.  Moreover, the same argument also implies that if $G$ \nis a minimal $H$-obstruction for some graph $H$, then $G$ is a point\ndetermining graph. Finally,  it is also straightforward to notice that if $G$ is\na point determining graph,  then each full-homomorphism whose domain\nis $G$ is an injective mapping. \nThe following statement captures two of the facts argued in this paragraph.\n\n\\begin{lemma}\\label{lem:point-det}\nThe following statements hold for any pair of graphs $G$ and $H$:\n\\begin{enumerate}\n\t\\item If $G$ is point determining, then every full-homomorphism\n\t$\\varphi\\colon G\\to H$ is injective.\n\t\\item If $G\\in \\obs(H)$, then $G$ is a point determining graph.\n\\hfill $\\square$\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\n\n\n\n\\section{Path obstructions}\n\\label{sec:paths}\n\n\nIn this section, we describe the minimal $P$-obstructions when $P$ is a path.\nWe begin by describing some particular minimal $P$-obstructions. To do so,\nwe introduce the graphs $A$, $B$ and $E$ depicted in \\cref{fig:ABE}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tikzpicture}\n[every circle node\/.style ={circle,draw,minimum size= 5pt,inner sep=0pt, outer sep=0pt},\nevery rectangle node\/.style ={}];\n\n\\begin{scope}[xshift = 4cm, scale=0.4]\n\\node [circle] (1) at (-1.5,0)[label=left:$v_0$]{};\n\\node [circle] (2) at (-1.5,3)[label=left:$v_1$]{};\n\\node [circle] (3) at (-1.5,6)[label=left:$v_2$]{};\n\\node [circle] (4) at (1.5,6)[label=right:$v_3$]{};\n\\node [circle] (5) at (1.5,3)[label=right:$v_4$]{};\n\\node [circle] (6) at (1.5,0)[label=right:$v_5$]{};\n\\foreach \\from\/\\to in {1\/2, 2\/3, 1\/6, 2\/5, 3\/4}\n\\draw [-, shorten <=1pt, shorten >=1pt, >=stealth, line width=.7pt] (\\from) to (\\to);\n\\node [rectangle] at (0,-1.5){Graph $E$};\n\n\\end{scope}\n\n\\begin{scope}[xshift=-4cm, scale = 0.4]\n\\node [circle] (1) at (-1.5,0)[label=left:$v_0$]{};\n\\node [circle] (2) at (-1.5,3)[label=left:$v_1$]{};\n\\node [circle] (3) at (-1.5,6)[label=left:$v_2$]{};\n\\node [circle] (4) at (1.5,6)[label=right:$v_3$]{};\n\\node [circle] (5) at (1.5,3)[label=right:$v_4$]{};\n\\node [circle] (6) at (1.5,0)[label=right:$v_5$]{};\n\\foreach \\from\/\\to in {1\/2,2\/3,3\/4,4\/5,5\/6,2\/5}\n\\draw [-, shorten <=1pt, shorten >=1pt, >=stealth, line width=.7pt] (\\from) to (\\to);\n\\node [rectangle] (1) at (0,-1.5){Graph $A$};\n\n\\end{scope}\n\\begin{scope}[xshift=0cm, scale=0.4]\n\\node [circle] (1) at (-1.5,0)[label=left:$v_0$]{};\n\\node [circle] (2) at (-1.5,3)[label=left:$v_1$]{};\n\\node [circle] (3) at (-1.5,6)[label=left:$v_2$]{};\n\\node [circle] (4) at (1.5,6)[label=right:$v_3$]{};\n\\node [circle] (5) at (1.5,3)[label=right:$v_4$]{};\n\\node [circle] (6) at (1.5,0)[label=right:$v_5$]{};\n\\foreach \\from\/\\to in {1\/2,2\/3,3\/4,4\/5,5\/6,2\/5,1\/6}\n\\draw [-, shorten <=1pt, shorten >=1pt, >=stealth, line width=.7pt] (\\from) to (\\to);\n\\node [rectangle] (1) at (0,-1.5){Graph $B$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{For a path $P$, every minimal $P$-obstruction that is not a linear forest\nnor a cycle, is one of these graphs (\\cref{lem:3possibilties}).}\n\\label{fig:ABE}\n\\end{figure}\n\nRecall that for $n \\ge 3$, we denote by $C_n$ (resp.\\ by $P_n$) the cycle\n(resp.\\ path) on $n$ vertices; we denote by $K_1$ and $K_2$ the paths on\none and two vertices, respectively. In general,\nwe denote by $K_n$ the complete graph on $n$ vertices.\n\n\\begin{lemma}\\label{lem:Cn-On}\nFor every positive integer $n$, the following statements hold:\n\\begin{enumerate}\n\t\\item The graph $A$ is a minimal $P_n$-obstruction if and only if $n \\ge 6$.\n\t\\item The graph $B$ is a minimal $P_n$-obstruction if and only if $n \\ge 5$.\n\t\\item The graph $E$ is a minimal $P_n$-obstruction if and only if $n \\ge 7$.\n\t\\item The $m$-cycle is a minimal $P_n$-obstruction if and only if\n\t$m = 3$ or $5\\le m \\le n+1$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAll graphs in statements 1--3 are point determining graphs that do not\nadmit a full $P$-colouring for any path $P$. By removing $v_4$ from $A$, we\nobtain $K_1+P_4$ which is not full $P_5$-colourable, so $A$ is not a minimal\n$P_n$ obstruction for any $n\\le 5$. On the other hand, any induced subgraph\nof $A$ admits a full $P_6$-colouring, and thus, it is a minimal $P_n$-obstruction\nfor every $n\\ge 6$.  Similarly, $A-v_1$ is not full $P_5$-colourable, and\n$E-v_1$ is not full $P_6$-colourable. Also,\nevery proper induced subgraph of $A$ is full $P_6$-colourable, and\nevery proper induced subgraph of $E$ is full $P_7$-colourable. Hence,\n$A$ is a minimal $P_n$-obstruction if and only if $n\\ge 6$, and \n$E$ is a minimal $P_n$-obstruction if and only if $n\\ge 7$.  \nThe last statement is clearly true.\n\\end{proof}\n\nNow, we observe that each path minimal obstructions is either a graph\nmentioned in \\cref{lem:Cn-On} or a linear forest.\n\n\n\\begin{lemma}\\label{lem:3possibilties}\nConsider a path $P$ and a graph $G$. If $G\\in \\obs(P)$ then one of the\nfollowing statements holds,\n\\begin{enumerate}\n\t\\item either $G$ is a cycle,\n\t\\item or $G$ is a linear forests,\n\t\\item or $G$ is one of the graphs $A$, $B$ or $E$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $n$ be a positive integer such that  $G\\in \\obs(P_n)$. We\nshow that if $G$ is neither a cycle nor a linear forest, then is one of the graphs \n$A$, $B$ or $E$. First, suppose that $G$ is not a forest. By minimality of $G$, and\nby the fourth statement of \\cref{lem:Cn-On}, we know that $G$ does not contain\na triangle nor a cycle of length $m$ with $5\\le m \\le n+1$. It is not hard to notice that\nthe path on $n+1$ vertices is not full $P_n$-colourable, thus $G$ does not contain\nan induced path on $n+1$ vertices, and so, it does not contain a cycle of length\n$m\\ge n+2$. Putting both of these observations together we conclude that\n$G$ contains no triangle nor an induced cycle of length $m\\ge 5$. Since\n$G$ is not a forest, there is an induced $4$ cycle $C$,  $C = v_1,v_2,v_3,v_4$,\nin $G$. \nBy the choice of $G$ and by second part of \\cref{lem:point-det}, it is the case that\n$G$ is a point determining graph. In particular, $N(v_1) \\neq N(v_3)$ and\n$N(v_2)\\neq N(v_4)$ so, without loss of generality\nwe assume that $v_1$ has a neighbour $v_0\\not\\in\\{v_2,v_4\\}$ and $v_4$ has a\nneighbour $v_5\\not\\in\\{v_1,v_3\\}$. Since $G$ has no triangles,  the unique neighbour\nof $v_0$ (resp.\\ $v_5$) in $C$ is $v_1$ (resp.\\ $v_4$). \nLet $H$ be the subgraph of $G$ induced by $\\{v_0,\\dots, v_5\\}$. This graph\nis isomorphic to either $A$ or $B$. Clearly, neither of $A$ nor $B$ admit a full\n$P_n$-colouring, and thus, by minimality of $G$ we conclude that\n$G = H$.\n\nIn the paragraph above, we showed that if $G\\in \\obs(G)$ and $G$ is not a forest,\nthen either $G$ is a cycle  or $G \\in\\{A,B\\}$. To conclude the proof, suppose\nthat $G$ is a forest but not a linear forest. In this case, $G$ contains an induced \nclaw $C$. With a similar procedure to the paragraph above, we extend $C$\nto an induced subgraph $H$ of $G$ such that $H\\cong E$. \nSince $E$ does not admit a full $P_n$-colouring, we conclude that $G = H \\cong E$, \nand the claim follows.\n\\end{proof}\n\n\nIn order to complete the characterization of $\\obs(P_n)$,  we study minimal \n$P_n$-obstructions that are linear forests. Since linear forests are\ndisjoint unions of paths, we will denote a linear forests $L$ as\n$\\sum_{k=1}^mP_{n_k}$, where the $k$th component of $L$\nis the path on $n_k$ vertices. Notice that if\n$\\varphi\\colon L\\to P$ is an injective full-homomorphism from $L$ to a path\n$P$, then  the image $\\varphi[P_{n_i}]$ and $\\varphi[P_{n_j}]$ of every pair of\ncomponents must be at distance at least two in $P$.  Thus, with a simple \ncomputation, and using the fact that $\\varphi$ is injective, we conclude that the\nfollowing statement holds.\n\n\\begin{lemma}\\label{lem:LF-injective}\nLet $L=\\sum_{k=1}^mP_{n_k}$ be a linear forest and $n$ a positive integer.\nThere is an injective full-homomorphism $\\varphi\\colon L\\to P_n $ if and only if the following inequality\nholds\n\\[\n(m-1) + \\sum_{k=1}^mn_k\\le n.\\vspace{-0.7cm} \n\\]\n\\hfill $\\square$\n\\end{lemma}\n\nWe denote by $c(G)$ the number of connected components of a graph $G$.\nNotice that for a linear forest $L$ and a vertex $x\\in V(L)$, the \nequality $c(L) - 1 = c(L-x)$ holds if and only if $x$ is an isolated vertex, and\notherwise  $c(L)\\le c(L-x) \\le c(L)+1$.\n\n\\begin{lemma}\\label{lem:LF-bounds}\nLet $L = \\sum_{k=1}^mP_{n_k}$ be a linear forest and $n$ a positive integer.\nIf $L\\in \\obs(P_n)$, then\n\\[\nn+1 \\le (m-1) + \\sum_{k=1}^mn_k = (c(L)-1)  + |V(L)| \\le n+2.\n\\]\nMoreover, if  $L$ has no isolated vertices, then  $(m-1)+|V(L)|-1 =n+1$ equivalently, \n$c(L) + |V(L)| = n +3$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:LF-injective}, if $(m-1)+\\sum_{k=1}^mn_k\\le n$ then there\nis a full $P_n$-colouring of $L$. This shows that the inequality $n+1 \\le (m-1)+|V(L)|$\nholds. To prove that the second inequality holds, recall that by\n\\cref{prop:rem-v} there is  a vertex $x\\in V(L)$ such that $L-x$ is point determining.\nBy minimality of $L$, there is an injective full-homomorphism\n$\\varphi\\colon L-x\\to P_n$. Again,\nby Lemma~\\ref{lem:LF-injective},\n\\[\nc(L-x)-1+|V(L)|-1=c(L-x)-1+|V(L-x)|\\le n.\n\\]\nBy substituting $c(L) -1 \\le c(L-x)$ in the inequality above, we observe that\n$c(L)-2+|V(L)|- 1\\le n$ and thus, $c(L) + |V(L)| -1 \\le n+2$. The first statement\nis now proved. Suppose that $L$ has no \nisolated vertices. In this case, $c(L)\\le c(L-x)$, which implies that\n$c(L)-1+|V(L)| -1 \\le n+1$. The claim follows.\n\\end{proof}\n\n\nConsider a linear forest  $L=\\sum_{k=1}^mP_{n_k}$. In order to simplify our writing, \nwe define $m_i$ to be the number of  components of length $i$ in $L$.\nIn other words, $m_i$ is the cardinality of the set $\\{k\\in\\{1,\\dots,m\\}:n_k=i\\}$.\nIn particular, $m_i=0$ for all $i > |V(L)|$. \n \n\\begin{lemma}\\label{lem:LF-components}\nLet $L=\\sum_{k=1}^mP_{n_k}$ be a linear forest and $n$ a positive integer. \nIf $L\\in \\obs(P_n)$, then the following statements hold,\n\\begin{enumerate}\n      \\item $n_k\\in\\{1,2,4,6\\}$ for all $k\\in\\{1,\\dots,m\\}$,\n      \\item $m_1\\le 1$, \n      \\item if $n_k\\in\\{4,6\\}$ for some $k$, then $m_1=1$, and\n      \\item if $n_k = 6$ for some $k$, then $|V(L)| +m -1 = n+1$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nBy the second part of \\cref{lem:point-det}, $L$ is a point determining graph,\nand since $P_3$ is not, then no component of $L$ is isomorphic\nto $P_3$.  Now, we show that every component of $L$ has at most\n$6$ vertices but not $5$. Anticipating a contradiction, suppose that \nthere is a path $P_{n_k} = v_1,v_2,\\dots, v_{n_k}$ with $n_k = 5$ or\n$n_k\\ge 7$, for some $k\\in\\{1,\\dots, m\\}$. In such case, $L-v_2$ is a point determining \ngraph and $c(L-v_3)=c(L)+1$, so the following equalities hold,\n\\[\n|V(L-v_3)|+c(L-v_3)-1=|V(L)|-1+c(L)+1-1=|V(L)|+c(L)-1.\n\\]\nBy the choice of $L$, there is a full $P_n$-colouring of $L-v_3$ which by\nthe first part of \\cref{lem:point-det} is injective. So, by \\cref{lem:LF-injective},\n$|V(L-v_3)|+c(L-v_3)-1\\le n$, and by the previous equalities we\nconclude that $|V(L)|+c(L)-1\\le  n$. Thus, using again \\cref{lem:LF-injective},\nwe conclude that $L$ admits a full $P_n$-colouring which contradicts the fact\nthat $L$ is not full $P_n$-colourable. Therefore, if $L\\in \\obs(P_n)$ then\n$n_k\\in\\{1,2,4,6\\}$ for every $k\\in\\{1,\\dots, m\\}$.\n\nThe second statement follows because every graph\nwith two isolated vertices is not a point determining graph, but every minimal\n$P_n$-obstruction is a point determining graph (second part of \\cref{lem:point-det}).\n\nTo prove the third statement, suppose  that $P_{n_k}=v_1,\\dots,v_{n_k}$ with \n$n_k\\in\\{4,6\\}$ for some  $k\\in\\{1, 2, \\dots, m\\}$. In this case, $c(L-v_2) = c(L)+1$.\nSo, if $L-v_2$ is a point determining graph, by using a similar argumentation\nas in the first paragraph of this proof, we conclude that\n$L$ admits a full $P_n$-colouring, contradicting the fact that\n$L\\in\\obs(P_n)$. Hence, $L-v_2$ is not a point determining linear forest. Since\nevery component of $L-v_2$ is either a component\nof $L$, or $v_1$, or the path $v_3,\\dots, v_{n_k}$, it must be the case that\nthere is an isolated vertex in $L-v_2$ other than $v_1$. Hence, $L$ has at least\none isolated vertex so, by the second statement of this lemma, we conclude that\n$m_i = 1$.\n\nThe final statement follows with similar arguments as above.  Clearly, \nif $P_{n_k} = v_1,\\dots, v_6$ for some $k\\in\\{1,\\dots, m\\}$, then $L-v_1$ is a\npoint determining graph with $c(L-v_1) = c(L)$. Hence, by the first part of \n\\cref{lem:point-det},\nany full-homomorphism from $L-v_1$ to $P_n$ is an injective mapping, and\nthus $c(L-v_1) -1 + |V(L-x)| \\le n$.\nTherefore, $c(L) -1 + |V(L)| -1 \\le n$ which is equivalent to the inequality\n$|V(L)| +m -1 \\le n+1$. We conclude that $c(L) -1 + |V(L)| -1 = n$ using \nthe previous inequality and the leftmost inequality of \\cref{lem:LF-bounds}.\n\\end{proof}\n\n\nIt turns out the necessary conditions stated in \\cref{lem:LF-bounds,%\nlem:LF-components} are also sufficient. We put these conditions together\nin the following statement. Recall that $m_1$ denotes the number of \nisolated vertices in a linear forest $L = \\sum_{k=1}^mP_{n_k}$.\n\n\\begin{proposition}\\label{prop:LF-suf-nec}\nLet $L=\\sum_{k=1}^mP_{n_k}$ be a linear forest and $n$ a positive integer. \nIn this case, $L\\in\\obs(P_n)$ if and only if  one of the following statements holds,\n\\begin{enumerate}\n     \\item either $|V(L)| + m=n+2$ and $n_k=2$ for all $k\\in\\{1,\\dots,m\\}$,\n     \\item  or $|V(L)| + m=n+2$ and $n_k\\in\\{1,2,4,6\\}$ with $m_1 = 1$,\n     \\item or $|V(L)| + m=n+3$ and $n_k\\in\\{1,2,4\\}$ with $m_1 = 1$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe fact that $L\\in\\obs(P_n)$ implies that one of the three conditions holds,\nfollows from \\cref{lem:LF-bounds,lem:LF-components}. Here, we prove that either\nof conditions 1--3 is sufficient for $L$ to be a minimal $P_n$-obstruction. In\neach possible\ncase, $L$ is a point determining graph so, every full-homomorphism\n$\\varphi\\colon L\\to P_n$ is an injective mapping (first part of \\cref{lem:point-det}).\nAnd thus, $L$ is not $P_n$-colourable\nbecause $n$ does not satisfy  inequalities of \\cref{lem:LF-injective}. To conclude\nthe proof, we argue that $L-x$ admits a full $P_n$-colouring for each $x\\in V(L)$. \nThe first case is immediate so we only consider the last two cases.\n\nWe proceed to prove that if $|V(L)| +m = n+2$\nwith $n_k\\in\\{1,2,4,6\\}$, and $L$ has exactly one isolated vertex,\nthen $L-x$ admits a full $P_n$-colouring for each $x\\in V(L)$. \nThere are two possible cases for such a vertex $x$, either $c(L-x) \\le c(L)$\nor $c(L-x) = c(L)+1$. In the former case, the claim follows since by the\nchoice of $L$, the equality  $|V(L)| + c(L)=n+2$ holds, and so,\n$|V(L-x)| + c(L) \\le n +1$. By solving the last equation for $n$, and using\n\\cref{lem:LF-injective}, we conclude that $L-x$ admits a full $P_n$-colouring. \nFor the second case, when $c(L-x) = c(L) +1$, notice that $L-x$ \ncontains either two isolated vertices or a component on three vertices. \nIn either case, $L-x$ is not a point determining graph. So, by identifying a pair\nof false twins in $L-x$ we obtain a full homomorphism of $L-x$ into a proper\nsubgraph $L'$ which has one less vertex than $L-x$ and $c(L') \\le c(L-x) \\le\nc(L)+1$. Therefore, $c(L') + |V(L')| -1 \\le (c(L) +1)  + (|V(L)| -2) -1$ and since\n$|V(L)| + m=n+2$, we conclude that $c(L') + |V(L')| -1 \\le n+2 +1 -3 = n$.\nThus, by \\cref{lem:LF-injective}, $L'$ is full $P_n$-colourable, and by composing\nfull-homomorphism, we see that $L-x$ is full $P_n$-colourable.\n\nFinally, when  $|V(L)| +m = n+3$ it suffices to consider two cases. \nFirst, when $x$ is an isolated vertex so $c(L-x) = c(L) -1$, and in the\nother case, $L-x$ contains a pair of false twins. In both cases, we conclude\nthat $L-x$ is full $P_n$-colourable by following similar arguments as in the\nprevious paragraph.\n\\end{proof}\n\nThe following proposition is a restatement of \\cref{prop:LF-suf-nec}\nin terms of solutions to integer equations. Recall that given a linear\nforest $L$, we denote by $m_i$ the number of connected components of\n$L$ with exactly $i$ vertices. In particular, $m_2$, $m_4$ and $m_6$ denote\nthe number of components of $L$ which are paths on $2$, $4$ and $6$ vertices,\nrespectively. \n\n\\begin{proposition}\\label{prop:LF-equations}\nConsider a linear forest $L$ and a positive integer $n$. In this case,\n$L\\in \\obs(P_n)$ if and only if one of the following statements holds,\n\\begin{enumerate}\n     \\item either $L = m_2K_2$ where $m_2$ is a non-negative integer solution\n     \tto the equation $3m_2=n+2$,\n     \\item or $L = K_1 + m_2K_2 + m_4P_4$ where $m_2$ and $m_4$ are non-negative \n              integer solutions to the equation $3m_2+5m_3=n+1$,\n     \\item or $L = K_1+m_2K_2+m_4P_4 + m_6P_6$ where $m_2$, $m_4$,\n     and $m_6$ are non-negative integer solutions to the equation\n     $3m_2 + 5m_4 + 7m_6 = n$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nConsider a linear forest $L = \\sum_{k=1}^mP_{n_k}$ where each connected\ncomponent has at most $l$ vertices. All we have to do is \nuse \\cref{prop:LF-suf-nec},  and notice that $|V(L)|+m = \\sum_{i=1}^l im_i +\n\\sum_{i=1}^l m_i$. Having made these observations it is immediate that,\nfrom 1--3 in \\cref{prop:LF-suf-nec}, 1 is equivalent\nto the first statement of this proposition, while 2 is equivalent to the\nthird one of this proposition, and 3 is equivalent to the second item\nof this proposition.\n\\end{proof}\n\nWe are ready to propose a description of all minimal $P_n$-obstructions.\nTo do so, we introduce three sets $C(n)$, $LF(n)$ and $O(n)$,  which depend\non $n$ --- $C$ stands for cycles, $LF$ for linear forests, and $O$ for other. \nWe begin with the simplest, \n\\[\nC(n):= \\big\\{C_m:~m = 3 \\text{ or } 5\\le m\\le n+1\\big\\}.\n\\]\n\\noindent\nSecondly, we define $O(n)$ as follows\n\\[\nO(n):=\n\\begin{cases}\n\\varnothing \\text{ if } n \\le 4,\\\\ \n\\{B\\} \\text{ if } n = 5,\\\\ \n\\{A,B\\} \\text{ if } n = 6,\\\\ \n\\{A,B, E\\} \\text{ if } n \\ge 7.\\\\ \n\\end{cases}\n\\]\n\\noindent\nFinally, $LF(n)$ is the union $LF_1(n)\\cup LF_2(n)\\cup LF_3(n)$\nwhere\n\\[\nLF_1(n):=\\big\\{m_2K_2:~ 3m_2 = n+2\\big\\}\n\\]\n\\[\nLF_2(n) :=  \\big\\{K_1 + m_2K_2 + m_4P_4:~ 3m_3 + 5m_4  = n+1\\big\\}, \\text{ and }\n\\]\n\\[\nLF_3(n) := \\big\\{K_1 + m_2K_2 + m_4P_4+ m_6P_6:~3m_2 + 5m_4 + 7m_6 = n\\big\\}.\n\\]\n\\noindent\nWe describe the set $\\obs(P_n)$ of minimal $P_n$-obstructions in terms of\nthe previously defined sets. \n\n\\begin{theorem}\\label{thm:paths}\nFor every positive integer $n$ the set $\\obs(P_n)$ of minimal $P_n$-obstructions is the\nunion $ C(n)\\cup LF(n) \\cup O(n)$.\n\\end{theorem}\n\\begin{proof}\nThis statement is a consequence of\n\\cref{lem:3possibilties,lem:Cn-On,prop:LF-equations}.\n\\end{proof}\n\nTo conclude this section, allow us to discuss an implication of \\cref{thm:paths}.\nSince all paths are linear forests, any graph that admits a full-homomorphism\nto some path, admits a full-homomorphism to some linear forest. On the other\nhand, each linear forest admits a full-homomorphism to a large enough path. \nThus, a graph $G$ admits a full-homomorphism to a path if and only if it admits\na full-homomorphism $G$ to a linear forest.\n\nA \\textit{blow-up} of a graph $G$ is obtained by addition of false twins ---\nintuitively, by ``blowing up'' some vertices of $G$ to an independent set. \nClearly, a graph $G$ admits a full $H$-colouring if and only if $G$ is \na blow-up of some induced graph of $H$. Since the class of linear forest\nis closed under induced subgraphs, we use the observation in the paragraph\nabove to prove the following statement.\n\n\\begin{corollary}\nA graph $G$ is a blow-up of a linear forest if and only if $G$ is\nan $\\{A,B,E\\}$-free graph such that all induced cycles have length four.\n\\end{corollary}\n\\begin{proof}\nIf $G$ is a blow-up of some linear forest, then $G$ admits a full\n$P_n$-colouring for some large enough $n$. Thus, by \\cref{thm:paths},\n$G$ is an $\\{A,B,E\\}$-free graph such that all induced cycles have length four.\nOn the other hand, notice that the number of vertices of the smallest graph in \n$LF(n)$ increases with respect to $n$. Thus, for every a graph $G$ \nthere is a positive integer $N$ such that $G$ is an $LF(N)$-free graph.\nHence, if $G$ is\nan $\\{A,B,E\\}$-free graph such that all induced cycles have length four,\nthen $G$ is an $(O(N)\\cup C(N) \\cup LF(N))$-free graph. \nTherefore, $G$ admits a full $P_N$-colouring, and so, $G$ is\na blow-up of a linear forest.\n\\end{proof}\n\n\n\n\n\\section{Cycle obstructions}\n\\label{sec:cycles}\n\nThe aim of this section is listing all minimal obstructions of cycles.\nTo do so, we first make some general observations\nregarding minimal obstructions of regular graphs. \\cref{prop:rem-v}\nasserts that for each point determining graph $G$, there is a vertex\n$x\\in V(G)$ such that $G-x$ is point determining.  We begin by noticing that\nthis can be strengthen in the case of regular graphs.\n\n\\begin{proposition}\\label{prop:nucleus-regular}\nLet $H$ be a point determining graph. If $H$ is a regular graph, then\nfor each $x\\in V(H)$ the graph $H-x$ is point determining.\n\\end{proposition}\n\\begin{proof}\nProceeding by contrapositive, suppose that there is a vertex $x\\in V(H)$ such that\n$H-x$ is not point determining. Let $r,s\\in V(H-x)$ be a pair of false twins, i.e.,\n$rs\\not\\in E(H-x)$ and $N_{H-x}(r) = N_{H-x}(s)$. \nSince $H$ is a point determining graph and $rs\\not\\in E(H)$, it must be the case\nthat $xr\\in E(H)$ and $xs\\notin E_H$ (or viceversa). Hence, $d_H(s) = d_{H-x}(s) =\nd_{H-x}(r) =  d_H(r) -1$. Thus, $H$ is not a regular graph.\n\\end{proof}\n\n\nConsider a graph $H$ and a minimal $H$-obstruction $G$. By the second\npart of \\cref{lem:point-det}, $G$ is a point determining graph so, by\n\\cref{prop:rem-v}, there is a vertex $v\\in V(G)$ such that\n$G-v$ is a point determining graph, and $G-v$ admits a full $H$-colouring\nby minimality of $G$.\nAlso, by  the first part\nof  \\cref{lem:point-det}, each full-homomorphism from $G-v$ to $H$ is injective,\nand thus $|V(G-v)| \\le |V(H)|$. Therefore, every graph $G\\in \\obs(H)$ has at\nmost $|V(H)|+1$ vertices. We denote by $\\obs^\\ast(H)$ the set of minimal\n$H$-obstructions on $|V(H)|+1$ vertices. The following statement was\nproved in \\cite{federDM308}.\n\n\\begin{proposition}\\cite{federDM308}\\label{prop:obs*}\nFor any graph $H$, there are at most two non-isomorphic graphs in $\\obs^*(H)$.\n\\end{proposition}\n\n\nBy similar arguments as in the paragraph above, we observe\nthat if $G\\in \\obs^\\ast(H)$, then there is a vertex $v\\in V(G)$ such that\n$G-v\\cong H$. \n\n\\begin{observation}\\label{obs:iso}\nConsider a pair of graphs $G$ and $H$. If $G\\in \\obs^\\ast(H)$, then\nthere is a vertex $v\\in V(G)$ such that $G-v\\cong H$.\n\\end{observation}\n\nThis observation about the structure of graphs in $\\obs^\\ast(H)$\ncan be strengthen when $H$ is a regular graph. Recall that a pair of\nvertices $u$ and $v$ in a graph $G$ are true twins if $N[u] = N[v]$.\n\n\\begin{lemma}\\label{lem:true-twins}\nLet $H$ be a non-complete regular connected graph.\nFor every graph $G\\in \\obs^\\ast(H)$ there is a pair of true twins\n$u,v\\in V(G)$ ($u\\neq v$)  such that $G-v\\cong H$ and $G-u \\cong H$.\n\\end{lemma}\n\\begin{proof}\nSince $H$ is non-complete, it is not isomorphic to $K_2$, and since it is connected,\nit is not a matching. Thus, $H$ is a $k$-regular graph with\n$k \\ge 2$. Let $G\\in \\obs^\\ast(H)$. By \\cref{obs:iso}, there is a vertex $x\\in V(G)$ such that $G-x\\cong H$.\nFor this proof, it will be convenient to identify $H$ with the subgraph of $G$\ninduced by $V(G)-\\{x\\}$. We fix $x$ and use this identification\nthroughout the proof.  We proceed to show $x$ is not an isolated vertex.\nSince $k \\ge 2$, there\nare no leaves in $H$. Consider a vertex $v\\in V(G)-\\{x\\}$ and let\n$\\varphi\\colon G- v \\rightarrow H$ be a full-homomorphism.\nSince $H$ is a regular graph, by \\cref{prop:nucleus-regular} we know that\n$H-v$ is point determining  so, by the first part of \\cref{lem:point-det}, \nthe restriction of $\\varphi$ fo $H-v$ is an injective mapping. Let $L$ be the image\n$\\varphi[H-v]$ of $H-v$. In particular,  $|V(L) | = |V(H)| -1$. Since $H$ is connected,\neither $\\varphi(x)$ has a neighbour in $L$ or $\\varphi(x)$ belongs to $L$.\nRecall, that  $L\\cong H-v$ and $H$ has no leaves so, $L$ has no isolated vertices.\nTherefore, if $\\varphi(x)$ belongs to $L$, then $\\varphi(x)$ has a neighbour in $L$,\nand since $\\varphi$  is a full-homomorphism, $x$ cannot be an isolated vertex\nin $G$.\n\nIn the paragraph above, we proved that $x$ is an isolated vertex. Since\n$G-x$  is connected (recall that\n$G-x = H$), then $G$ is a  connected graph. By \\cref{prop:rem-v}, there is a vertex\n$y\\in V(G)-\\{x\\}$ such that $G-y$ is point determining, and so, $G-y\\cong H$.\nWe conclude the proof by showing that $x$ and $y$ are true twins in $G$.\nSince $H$ is a $k$-regular graph, for each $v\\in V(G-y)$ the equality $d_{G-y}(v)=k$ \nholds. Also,  since $H = G-x$, the equality $d_{H-y}(v)=k-1$ holds if and only if\n$v\\in N_H(y) = N_{G-x}(y)$. On  the other hand, $k-1 = d_{(G-y)-x}(v)-1$ if and only if $v\\in N_{G-y}(x)$. Clearly, $(G-y)-x = H- y$, and so,\n$v\\in N_{G-y}(x)$ if and only if  $v\\in N_{G-x}(y)$ for any $v\\in V_{G-\\{x,y\\}}$. Thus, \n$N_G(x)-y=N_{G-y}(x) = N_{G-x}(y)=N_G(y)-x$ so in particular,\n$N_G(x)-y = N_G(y)-x$. Since $G$ is point determining,\n$x$ and $y$ are not false twins in $G$ so, $xy\\in E(G)$, and thus \n$x$ and $y$ are true twins. The claim follows. \n\\end{proof}\n\n\n\\cref{prop:obs*} asserts that $|\\obs^\\ast(H)|\\le 2$ for every graph $H$. Using\n\\cref{lem:true-twins}, we show that in the case of regular non-complete graphs\n $\\obs^\\ast(H) = \\varnothing$.\n\n\\begin{proposition}\\label{prop:obs*-regular}\nFor a connected regular graph $H$, the\nfollowing equalities hold\n\\[\n\\obs^\\ast(H) =\n\\begin{cases}\n\\{K_1+K_2, K_3\\} \\text{ if } H \\cong K_2,\\\\\n\\{K_{n+1}\\} \\text{ if } H \\cong K_n \\text{ and }n\\neq 2,\\\\\n\\varnothing \\text{ otherwise.}\\\\\n\\end{cases}\n\\]\n\\end{proposition}\n\\begin{proof}\nSince the class of complete multipartite graphs is the class of $K_1+K_2$-free graphs,\nthe class of full $K_n$-colourable graphs is the class of $\\{K_1+K_2,K_{n+1}\\}$-free\ngraphs. \nNow, suppose  that $H$ is a regular non-complete connected graph and\nlet $G\\in \\obs^\\ast(H)$. By \\cref{lem:true-twins}, there is a pair \nof true twins $x$ and $y$ of $G$, such that $G-x\\cong H \\cong G-y$. Again, we identify\n$H$ with the subgraph of $G$ induced by $V(G)-x$. Notice that if $x$ and\n$y$ are universal vertices in $G$, then $y$ is a universal vertex in $H$ and so,\n$H$ is a complete graph (because $H$ is a regular graph). So, by the choice of $H$, \nthere is a vertex $z\\in V(G)$ such that $zy\\notin E(G)$, and since $x$ and $y$\nare true twins, it is the case that $xz\\notin E(G)$. By the choice of $G$, there is a\nfull-homomorphism\n$\\varphi\\colon G-z\\rightarrow H$. Let $k$ be the degree of every vertex in $H$\nso, $d_G(x) = d_G(y)=k+1$. Since $zx,zy\\not\\in E(G)$, it is the case that\n$d_{G-z}(x) = d_{G-z}(y)=k+1$. But $d_H(\\varphi(y)) = k$ so,\nthere are two vertices $r,s\\in N_{G-z}(y)$ such that \n$\\varphi(r) = \\varphi(s)$. Hence $N_{G-z}(r) = N_{G-z}(s)$ and $rs\\not \\in E(G-z)$.\nRecall that $H = G -x$, so $N_{H-z}(r) = N_{H-z}(s)$ and $rs\\not \\in E(G-z)$, i.e.,\n$r$ and $s$ are false twins in $H-z$. Thus, $H-z$ is not a point determining graph\nwhich contradicts the fact that $H$ is a regular graph and\n\\cref{prop:nucleus-regular}.\n\\end{proof}\n\nThe following statement shows that if a graph $G$ is a minimal $H$-obstruction\nof size $|V(H)|+1$, then every minimal $G$-obstruction $F$ is either a minimal\n$H$-obstruction or $|V(F)| = |V(G)| +1$. Conversely, every minimal $H$-obstruction\nother than $G$ is a minimal $G$-obstruction.\n\n\\begin{theorem}\\label{thm:obsH-obsG}\nConsider a pair of graphs $H$ and $G$. If $G\\in \\obs^\\ast(H)$, then\n\\[\n\\obs(G) = (\\obs(H) \\setminus\\{G\\})\\cup \\obs^\\ast(G).\n\\]\n\\end{theorem}\n\\begin{proof}\nTo simplify notation, let $S = (\\obs(H) \\setminus\\{G\\})\\cup \\obs^\\ast(G)$. \nWe need to prove that a graph $F$ belongs to $S$ if and only if \nit belongs to $\\obs(G)$. Clearly, every graph in $S$ has at most $|V(G)| + 1$\nvertices and so does every graph in $\\obs(G)$. Moreover, by definition\nof $\\obs^\\ast(G)$, a graph on $|V(G)| +1$ vertices belongs to $\\obs(G)$\nif and only if it belongs to $\\obs^\\ast(G)$. Thus, it suffices to prove the claim\nfor graphs on at most $|V(G)|$ vertices, and by the second part of\n\\cref{lem:point-det} it suffices to consider point determining graphs.\nWe begin by showing that the claim holds for graphs on at most\n$|V(G)| -1$ vertices (recall that $|V(G)| = |V(H)| +1$).\nSince $G$ is a minimal $H$-obstruction, every proper induced subgraph of $G$\nadmits a full-homomorphism to $H$. Thus, any graph  that admits a\nnon-surjective full homomorphism to $G$, admits a full $H$-colouring. \nHence, a graph on at most $|V(G)| -1$ vertices admits a  full $G$-colouring\nif and only if it admits a full $H$-colouring. Therefore a graph on at most\n$|V(G)-1|$ vertices belongs to $S$ if and only if it belongs to $\\obs(G)$. \n\nFinally, consider a point determining graph $L$ on $|V(G)|$ vertices.\nBy the first part of \\cref{lem:point-det}, every full-homomorphism from\n$L$ to $G$ is injective, thus $L$ admits a full $G$-colouring if and only\nif $L\\cong G$.  By similar arguments as in the paragraph above,\nwe conclude that $L$ is a minimal $G$-obstruction \nif and only if it is a minimal $H$-obstruction.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:obs*-determine}\nConsider a pair of graphs $H_1$ and $H_2$. If $\\obs^\\ast(H_1) \\cap \\obs^\\ast(H_2)\n\\neq\\varnothing$, then $H_1\\cong H_2$. \n\\end{corollary}\n\\begin{proof}\nLet $G\\in \\obs^\\ast(H_1) \\cap \\obs^\\ast(H_2)$. By applying \\cref{thm:obsH-obsG}\nto $H_1$ and $G$, and to $H_2$ and $G$, we conclude that a graph is\na minimal $H_1$-obstruction if and only if it is a minimal $H_2$-obstruction.\nThe claim follows.\n\\end{proof}\n\nIn other words, \\cref{cor:obs*-determine} asserts that if a graph $G$ is\na minimal obstruction of two smaller graphs, then these graphs are isomorphic.\nAnother immediate implication of \\cref{thm:obsH-obsG} is the following one.\n\n\\begin{corollary}\nConsider a  pair of  graphs $H$ and $G$. If $G\\in \\obs^\\ast(H)$, then there\nis at most one minimal $G$-obstruction on $|V(G)|$ vertices.\n\\end{corollary}\n\n\nRecall that the orbit of a vertex $y\\in V(G)$ is the set of vertices $x\\in V(G)$ such\nthat there  is an automorphism $\\varphi\\colon G\\to G$ such that $\\varphi(y) = x$.\nWe denote the orbit of $y$ by $o(y)$. Clearly, if  $x\\in o(y)$, then $G-x\\cong G-y$. \n\n\\begin{proposition}\\label{prop:v-transitive}\nLet $H$ be a non-complete connected vertex transitive graph. For any vertex $x$ \nof $H$, the following equalities hold\n\\[\nobs(H) = obs(H-x) \\setminus\\{H\\} \\text{ and } obs(H-x) = obs(H) \\cup \\{H\\}\n\\]\n\\end{proposition}\n\\begin{proof}\nSince $H$ is vertex transitive, $H-x\\cong H-y$ for any pair of vertices $x,y\\in V(H)$. So,\nevery proper induced subgraph of $H$ admits a full $(H-x)$-colouring.  Since\n$H$ is a point determining graph, $H$ is not full $(H-x)$-colourable, so,\n$H\\in \\obs^\\ast(H-x)$. By \\cref{thm:obsH-obsG}, we conclude that\n$\\obs(H) = \\obs(H-x) \\setminus\\{H\\} \\cup \\obs^\\ast(H)$ so, using \\cref{prop:obs*-regular}\nwe observe that $\\obs^\\ast(H) = \\varnothing$.\n\\end{proof}\n\nBy applying \\cref{prop:v-transitive} to cycles, we see that minimal\n$C_n$-obstructions are determined by minimal $P_{n-1}$-obstructions,\nand viceversa.\n\n\\begin{corollary}\\label{cor:cycles}\nFor every positive integer $n$, $n\\ge 5$, the following equalities hold\n\\[\n\\obs(C_n) = \\obs(P_{n-1})\\setminus\\{C_n\\} \\text{ and } \n\\obs(P_{n-1}) = \\obs(C_n) \\cup\\{C_n\\}.\n\\]\n\\end{corollary}\n\n\nThe following characterization of minimal $C_n$-obstructions follows\nfrom \\cref{cor:cycles,thm:paths}.\n\n\\begin{theorem}\\label{thm:cycles}\nFor every positive integer $n$ the set $\\obs(C_n)$ of minimal $C_n$-obstructions\nis the union $C(n-2)\\cup LF(n-1) \\cup O(n-1)$.\n\\end{theorem}\n\\begin{proof}\nBy \\cref{cor:cycles}, the equality $\\obs(C_n) = \\obs(P_{n-1})\\setminus \\{C_n\\}$ holds. \nBy \\cref{thm:paths}, the set of  minimal $P_n$-obstructions is \n$C(n-1)\\cup LF(n-1) \\cup O(n-1)$. Finally, by definition of $C(n)$, the\nequality $C(n-2) = C(n-1)\\setminus\\{C_n\\}$ holds, and so, the\nclaim follows.\n\\end{proof}\n\n\\begin{corollary}\nA graph $G$ is full $C_5$-colourable if and only if it is\n$\\{C_3,~K_1+P_4,~2K_2\\}$-free.\n\\end{corollary}\n\nTo conclude this section, we list all $C_n$-minimal obstructions\nfor small integers  $n$ in Table~\\ref{tab:small-n}.\n\n\\begin{table}[ht!]\n\\begin{center}\n    \\begin{tabular}{| c | l | l |}\n    \\hline\n    $n$ & Linear forests in $\\obs(C_n)$ & Other minimal $C_n$-obstructions\\\\ \\hline\n    $5$ & $K_1+P_4$ and $2K_2$ & $C_3$\\\\ \\hline\n    $6$ &  $K_1+P_4$ and $K_1 + 2K_2$ & $C_3,~C_5$ and $B$\\\\ \\hline\n    $7$ &  $K_1+2K_2$ & $C_3,~C_5,~C_6,~A$ and $B$\\\\ \\hline\n    $8$ &   $3K_2$, $K_1+K_2+P_4$, and $K_1+P_6$ & $C_3,~C_5,~C_6,~C_7,~A,~B$\n     and $E$\\\\ \\hline\n    $9$ &   $K_1+3K_2$ and $K_1+ K_2 +P_4$ & $C_3,~C_5,~C_6,~C_8,~A,~B$\n     and $E$\\\\ \\hline\n    $10$ & $K_1 + 2P_4$ and $K_1 + 3K_2$ &\n    $C_3,~C_5,~C_6,~C_7,~C_8,~C_9,~A,~B$ and $E$ \\\\ \\hline\n    \\end{tabular}\n    \\caption{To the left, the number of vertices in a cycle $C$. In the middle, \n    the linear forests which are minimal $C$-obstructions. To the right, all\n    minimal $C$-obstructions that are not linear forests.}\n    \\label{tab:small-n}\n    \\end{center}\n  \\end{table}\n  \n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\\cref{prop:obs*-regular} asserts that for a  connected regular graph $H$ the set\n$\\obs^\\ast(H)$ is empty if and only if $H$ is not a complete graph. Also, if $H$ is\nobtained from a vertex-transitive graph $G$ by removing one vertex, then\n$G\\in \\obs^\\ast(H)$ so, $\\obs^\\ast(H)\\neq \\varnothing$. A possible\ninteresting question to investigate is the following one. \n\n\\begin{question}\nIs there a meaningful  characterization of those graphs $H$ for which\n$\\obs^\\ast(H) \\neq \\varnothing$? \n\\end{question}\n\n\\cref{thm:obsH-obsG} suggests that there is a close relation between \na graph $H$ such that $\\obs^\\ast(H)\\neq \\varnothing$ and a graph\n$G\\in \\obs^\\ast(H)$. \nFor this reason, we believe that another possible interesting problem\nis determining which graphs $G$ are a minimal $H$-obstruction of\nsize $|V(H)|+1$ for some graph $H$. \n\n\\begin{question}\\label{qst:2}\nFor which graphs $G$ there is a graph $H$ such that\n$G$ is a minimal $H$-obstruction in $\\obs^\\ast(H)$?\n\\end{question} \n\nWe briefly observe that this problem is not interesting if we remove the\nrestriction that $|V(G)| = |V(H)|+1$.\n\n\\begin{proposition}\nFor every point determining connected graph $G$, there is a graph $H$\nsuch that $G$ is a minimal $H$-obstruction. \n\\end{proposition}\n\\begin{proof}\nLet $G$ be as in the hypothesis, and for each vertex $x\\in V(G)$ let\n$H_x$ be the full-core of $G-x$. Finally, let $H$ be the disjoint union\n$\\sum_{x\\in V(G)}H_x$. Since every connected component of $H$\nhas less than $|V(G)|$ vertices, there is no injective full-homomorphism\nfrom $G$ to $H$. By the first part of \\cref{lem:point-det}, and since\n$G$ is a point determining graph, we conclude that $G$ does not admit\na full $H$-colouring. It is not hard to notice that, by the choice of \n$H_x$, for every vertex $x\\in V(G)$ there is a full $H$-colouring\nof $G-x$. The claim follows.\n\\end{proof}\n\nThe following observation shows that, in the case of regular graphs,\n\\cref{qst:2} has a meaningful answer. \n\n\\begin{proposition}\nLet $G$ be a point determining regular graph. There is a graph $H$ such that \n$G\\in\\obs^\\ast(H)$ if and only if $G$ is a vertex transitive graph.\n\\end{proposition}\n\\begin{proof}\nBy \\cref{prop:nucleus-regular}, if $G$ is a point determining regular graph,\nthen for each $x\\in V(G)$ the induced subgraph $G-x$ is point determining.\nSo, if $|V(G)| = |V(H)|+1$, then by the first part of \\cref{lem:point-det}, for\neach $x\\in V(G)$, every full-homomorphism from $G-x$ to $H$ is an\nisomorphism. Hence,  all vertex-deleted subgraphs of $G$ are isomorphic,\nand thus $G$ is a vertex transitive graph.\n\\end{proof}\n\n \nAs a final implication of this work, notice that for every positive\ninteger $n$, there are at most three graphs in $O(n)$, at most $n-2$ graphs\nin $C(n)$, and as many graphs in $LF(n)$ as non-negative solutions\nto the diophantine equations, $3x = n+2$, $3x + 5y = n+1$, and\n$3x +5y + 7z = n$. It is not hard to observe that there are $O(n^{k-1})$\nsolutions to each of these equations, where $k$ is the number of variables\nin the corresponding equation. \nHence, there are quadratically many linear forests in $LF(n)$. \nThese arguments, together with \\cref{thm:paths,thm:cycles}, imply\nthat the following statement holds.\n\n\\begin{corollary}\nFor every positive integer $n$, there are quadratically\nmany (with respect to $n$) minimal $P_n$-obstructions and\nminimal $C_n$-obstructions.\n\\end{corollary}\n\nThe well-defined and simple structure of paths and cycles might be the reason\nwhy their number of minimal obstructions is polynomially bounded (with respect\nto $n$). Nonetheless, having made this observation, it is natural to ask about\nthe cardinality  of $\\obs(H)$ in terms of the  cardinality of the vertex set of $H$.\n\n\\begin{question}\nIs there a polynomial $p(n)$ such that the size of $\\obs(H)$ is bounded\nby $p(|V(H)|)$ for each graph $H$?\n\\end{question}\n\n\n\\section*{Acknowledgements}\nThe author is deeply grateful to C\\'esar Hern\\'andez-Cruz for several \ndiscussions that helped developing this work. In particular, \nC\\'esar proposed the problems of describing the minimal path and\ncycle obstructions.\n\n\\vspace{2mm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}