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+{"text":"\\section{Introduction}\n Rational approximation has been a research topic for a long time. In \\cite{Wa, Wa1} J. L. Walsh discussed the problem of approximating holomorphic functions by rational functions in one complex variable. A common problem is to find a rational approximation with a given error. Runge's\napproximation theorem (cf. \\cite{Co}) is one of the well-known\nresults in such direction. Best $n$-rational approximation, which is another\nclassical problem in rational approximation, can be formulated as\nfollows. Let $f$ be in the Hardy space $H^2(\\mathbb C_+)$, where\n$\\mathbb C_+=\\{z\\in \\mathbb C; z=x+iy, x\\in \\mathbb R, y>0\\}.$ Find\na pair of co-prime polynomials $p_1$ and $q_1$, where the degrees of $p_1$ and $q_1$\nboth are less or equal to $m,$ and $q_1$ dose not have zeros in $\\mathbb\nC_+$, such that\n\\begin{equation}\\label{best_n}\n\\Vert f-\\frac{p_1}{q_1} \\Vert_{H^2(\\mathbb C_+)}\\leq \\Vert\nf-\\frac{p}{q} \\Vert_{H^2(\\mathbb C_+)}\n\\end{equation}\namong all pairs of co-prime polynomials $p$ and $q$ satisfying\nthe same conditions. Although existence of such $p_1\/q_1$ in\n(\\ref{best_n}) was proved many decades ago (\\cite{Wa}), a\npractical algorithm to find a solution remains as an open problem till now. Some\nconditional solutions have been found (e.g. \\cite{BL,BSW,BW,QW,Q22}). The\nsignificance of studying rational approximation is not only theoretical,\nbut also practical. Rational approximation has direct applications in system\nidentification (e.g. \\cite{AN1,AN2,AN3}). In particular, system\nidentification is to identify the transform function, which itself is often a\nrational function. Identification of transform functions of several complex variables is a recent research topic (e.g. \\cite{Jiao}). For signal processing, people aim to\ndecompose a real-valued function $f\\in L^2(\\mathbb R) $ into a series $\\lq\\lq$ simple \" functions. One can have $f=\nf^+ + f^-$ by the Hardy decomposition, where $f^+$ and $f^-$ are\nnon-tangential boundary limits of certain functions in the respective Hardy\nspaces $H^2(\\mathbb C_+)$ and $H^2(\\mathbb C_{-}).$\nDue to the fact that the real part of $f^+$ is equal\nto $\\frac{1}{2}f,$ the problem is then reduced to decomposing $f^+$ and solved by finding a\nsequence of rational functions in $H^2(\\mathbb C_+)$ approximating\n$f^+.$\n\n Recently, T. Qian et al proposed a function decomposition method called adaptive Fourier decomposition (AFD) that has impacts to both theory and applications. AFD is based on a generalized backward shift process leading to an adaptive Takenaka-Malmquist (TM) system \\cite{Q1,QW}. AFD was originally established in the Hardy spaces of the unit disc and the upper half plane. Here we briefly give a revision of AFD in $H^2(\\mathbb C_+)$. A TM system in the upper half plane is defined as\n\\begin{align}\\label{TM}\nB_k(z)=B_{\\{b_1,...,b_k\\}}(z)=\\sqrt{\\frac{\\beta_k}{\\pi}}\\frac{1}{z-\\overline b_k}\\prod_{j=1}^{k-1}\\frac{z-b_j}{z-\\overline b_j},\\quad b_k=\\alpha_k+i\\beta_k\\in \\mathbb C_+,k=1,2,..., z\\in \\mathbb C_+.\n\\end{align}\nFor $f\\in H^2(\\mathbb C_+)$, and $\\{b_k\\}_{k=1}^m$ being $m$ given points in $\\mathbb C_+$, the main step of AFD is to select $b_{m+1}$ in the upper half complex plane to satisfy\n\\begin{align}\\label{MSP-AFD}\nb_{m+1} := \\arg \\sup_{b\\in \\mathbb C_+}|\\langle f, B_{\\{b_1,...,b_m,b\\}}\\rangle|,\n\\end{align}\nwhere $B_{\\{b_1,...,b_m,b\\}}(z)$ is defined in (\\ref{TM}) corresponding to $\\{b_1,...,b_m,b\\}.$ With a rigorous proof this turns to be realizable and is called the maximal selection principle. AFD offers the fast convergent rational approximation\n$$\n\\lim_{m\\to \\infty}||f-\\sum_{k=1}^m \\langle f, B_k\\rangle B_k||_{H^2(\\mathbb C_+)}=0,\n$$\nwhere each element of $\\{b_k\\}_{k=1}^\\infty$ is selected by the maximal selection principle (\\ref{MSP-AFD}). What AFD gives is rational approximation of functions in $H^2(\\mathbb C_+)$. AFD can be considered as a variation of greedy algorithm, but the main idea of AFD is not identical with any existing greedy algorithm (Pure Greedy Algorithm (PGA) and Orthogonal Greedy Algorithm (OGA), etc., \\cite{DT}).\nBy introducing the complete dictionary concept, Qian \\cite{Q-2D} proposed a new type of greedy algorithm called Pre-Orthogonal Greedy Algorithm (P-OGA) that develops the theory of AFD to abstract Hilbert spaces. As application it is shown that P-OGA is applicable to the Hardy space of the polydisc $H^2(\\mathbb D^2)$ (see \\cite{Q-2D}). Based on the methodology of P-OGA, in \\cite{MQ1} the counterpart of AFD is provided in complex reproducing kernel Hilbert spaces. The higher dimensional analogues of AFD have also been formulated respectively in the quaternionic analysis (see \\cite{QSW}) and Clifford analysis (see \\cite{WQ1}) settings. Among other things, the first setting is a generalization in the spirit of AFD. The second, however, is in the spirit of greedy algorithm, generalizing\n linear combinations of Szeg\\\" o kernels to approximating rational functions. What is in particular achieved in the quaternionic and the Clifford settings is that a global maximal selection of the parameters is attainable\n at each step of the recursive process. The difficulty with the general Clifford setting is caused by the fact that the inner product $\\langle f,f \\rangle$ is not necessarily scalar-valued. Note that in the quaternionic case we obtain rational approximation of functions, but in the Clifford case we can only achieve approximation by linear combination of the shifted Szeg\\\"o kernels: for odd Euclidean dimensions a Szeg\\\"o kernel is not a rational function. The right way of posting the approximation question would be linear combination of Szeg\\\"o kernels, Szeg\\\"o kernel-approximation in short, other than rational approximation.  For more information on AFD and its variations regarding Szeg\\\"o kernel-approximation see \\cite{Q1,QW,QLS,Q22}.\n\n  In this paper, we will study Szeg\\\"o kernel-approximation of functions in the Hardy spaces on the tube over the first octant $H^2(T_{\\Gamma_1})$. Unlike the Clifford case, in $H^2(T_{\\Gamma_1})$ the Cauchy-Szeg\\\"o kernels, as well as their higher order partial derivatives,  are rational functions. By extending the Szeg\\\"o kernel dictionary to its complete dictionary we involve the first and all higher order partial derivatives of the Szeg\\\"o kernels. The introduced complete dictionary is given by using the methodology of P-OGA.\n   Although in $H^2(\\mathbb D^2)$ the P-OGA was well established, the analogous theory in $H^2(T_{\\Gamma_1})$ needs to be independently discussed. Essentially, the reason is that $T_{\\Gamma_1}$ is an unbounded domain while $\\mathbb D^2$ is a bounded domain.  In this paper such rational approximation is called the AFD-type approximation since it can be regarded as AFD in the several complex variables setting. In addition, we will show the convergence of the AFD-type approximation, as well as the rate of convergence of it. As pointed out previously, the study of rational approximation in $L^2(\\mathbb R)$ can be reduced to the study of $H^2(\\mathbb C_+)$ by the decomposition $f=f^++f^-$. Similarly, rational approximation in $L^2(\\mathbb R^n)$ can be reduced to the study of the related $2^n$ Hardy spaces on tubes (also see \\cite{Q-2D}). In the last part of the paper, we will explore the AFD-type approximation in the Hardy spaces on tubes over regular cones $H^2(T_\\Gamma)$.\n\nThe writing plan is as follows. In Section 2, some basic results and\nnotations are given. In Sections 3, we devote to establishing the theory of AFD-type approximation in $H^2(T_{\\Gamma_1})$. In Section 4, several related problems are explored, where the problems include the convergent rate of AFD-type approximation, rational approximation of functions in $L^2(\\mathbb R^n)$, and the exploration of AFD-type approximation in $H^2(T_\\Gamma)$.\n\n\\section{Preliminaries}\nIn this section, we will fast review some basic concepts and properties of $H^2(T_{\\Gamma_1})$. For more information, see e.g. \\cite{SW} and \\cite{DLQ}.\n\nLet $B$ be an open subset in $\\mathbb R^n$. We say that $T_B$ is a\ntube over $B$, if each $z\\in T_B\\subset \\mathbb C^n$ is of the form\n$$\nz=(z_1,z_2,...,z_n)=(x_1+iy_1,x_2+iy_2,...,x_n+iy_n)=x+iy, x\\in\n\\mathbb R^n, y\\in B.\n$$\nIn this paper, we are mainly concerned with the following special tube\n$T_{\\Gamma_1}$, where\n$$\n\\Gamma_1=\\{y\\in \\mathbb R^n: y_1>0, y_2>0,...,y_n>0\\}.\n$$\nDenote by $H^2(T_{\\Gamma_1})$ the Hardy space on $T_{\\Gamma_1}$. We\nsay $F\\in H^2(T_{\\Gamma_1})$, if $F$ is holomorphic on\n$T_{\\Gamma_1}$ and satisfies\n$$\n\\Vert F \\Vert^2=\\sup_{y\\in \\Gamma_1}\\int_{\\mathbb{R}^n}\\vert\nF(x+iy)\\vert^2dx<\\infty.\n$$\n$H^2(T_{\\Gamma_1})$ is a Hilbert space equipped with the inner product\n$$\n\\langle F, G \\rangle =\\int_{\\mathbb R^n}F(\\xi)\\overline {G(\\xi)} d\\xi,\n$$\nwhere $F(\\xi)=\\lim_{\\eta\\in \\Gamma_1,\\eta\\to 0}F(\\xi+i\\eta)$ is the limit function in the $L^2$-norm, so is $G(\\xi)$.\nThroughout this paper, for $F\\in H^2(T_{\\Gamma_1})$ and $\\xi\\in \\mathbb R^n$, $F(\\xi)$ is the limit function in this sense.\n\nWe now present several fundamental properties of functions in $H^2(T_{\\Gamma_1})$ (see e.g. \\cite{SW}).\n\n\\begin{thm}[Paley-Wiener Theorem]\\label{PW}\nSuppose $\\Gamma_1$ is the first octant. Then $F\\in\nH^2(T_{\\Gamma_1})$ if and only if\n$$\nF(z)=\\int_{\\overline \\Gamma_1}e^{2\\pi i z\\cdot t}f(t)dt\n$$\nwhere $f$ is a measurable function on $\\mathbb R^n$ satisfying\n$$\n\\int_{\\overline \\Gamma_1}|f(t)|^2dt <\\infty.\n$$\nFurthermore,\n$$\n||F||=\\left (\\int_{\\overline \\Gamma_1}|f(t)|^2dt \\right\n)^{\\frac{1}{2}}.\n$$\n\\end{thm}\n$H^2(T_{\\Gamma_1})$ is a reproducing kernel Hilbert space whose reproducing kernel is the\nCauchy-Szeg\\\" o kernel\n$$\nK(w,\\overline z)= \\int_{\\overline \\Gamma_1}e^{2\\pi i \\omega \\cdot\nt}\\overline{e^{2\\pi iz \\cdot t}}dt = \\prod_{k=1}^n \\frac{-1}{2\\pi\ni(\\omega_k-\\overline z_k)}, \\quad w,z\\in T_{\\Gamma_1}.\n$$\nThe corresponding Poisson-Szeg\\\"o kernel is given by\n$$\nP_{y}(x)=\\frac{K(z,0)K(0, \\overline z)}{K(z,\\overline z)}=\n\\prod_{k=1}^n \\frac{y_k}{\\pi(x_k^2+y_k^2)}.\n$$\nMoreover, $P_y(x)\\in L^p,$ for $1\\leq p\\leq \\infty$.\\\\\nThe following results are reproducing formulas corresponding to the Cauchy-Szeg\\\"o and\nthe Poisson-Szeg\\\"o kernels, respectively.\n\\begin{thm}\\label{rf_s}\nIf $F\\in H^2(T_{\\Gamma_1})$ then\n$$\nF(z)=\\int_{\\mathbb R^n}F(\\xi) \\overline{K(\\xi, \\overline z)} d\\xi =\n\\int_{\\mathbb R^n}F(\\xi) {K(z,  \\xi)} d\\xi\n$$\nfor all $z=x+iy\\in T_{\\Gamma_1}$, where $F(\\xi)= \\lim_{\\eta\\to 0, \\eta\n\\in \\Gamma_1}F(\\xi+i \\eta)$ is the limit function in the $L^2$-norm.\n\\end{thm}\n\\begin{thm}\\label{rf_p}\nIf $F\\in H^2(T_{\\Gamma_1})$, then\n$$\nF(z)=  \\int_{\\mathbb R^n}F(\\xi) {P_y(x-\\xi)} d\\xi\n$$\nfor all $z=x+iy\\in T_{\\Gamma_1}$, where $F(\\xi)= \\lim_{\\eta\\to 0, \\eta\n\\in \\Gamma_1}F(\\xi+i \\eta)$ is the limit function in the $L^2$-norm.\n\\end{thm}\n\n\\section{Rational Approximation in $H^2(T_{\\Gamma_1})$: AFD-type Approximation}\n\nSuppose that $\\{z^{(k)}\\}_{k=1}^\\infty$ is a sequence of distinct points in $T_{\\Gamma_1}$. Under such assumption, $\\{K(\\cdot,\\overline {z^{(k)}})\\}_{k=1}^\\infty$ are linearly\nindependent.\n One can define\n\\begin{equation}\\label{appro_fun}\nF_{A_m}^{*}(z)=\\sum_{j=1}^m\\sum_{k=1}^m\nF(z^{(j)})\\widetilde{a_{j,k}^{(m)}}K(z,\\overline{z^{(k)}}), \\quad\nz\\in T_{\\Gamma_1},\n\\end{equation}\nwhere $A_m=(a_{j,k})_{m\\times m}$ with elements\n$a_{j,k}=K(z^{(j)},\\overline{z^{(k)}})$, and\n$\\widetilde{a_{j,k}^{(m)}}$s' are elements of\n$\\overline{A^{-1}_m}$.\\\\ By direct calculation, we have\n\\begin{equation}\\label{reprd}\n\\langle F_{A_m}^*, K(\\cdot, \\overline{z^{(k)}}) \\rangle\n=F^*_{A_m}(z^{(k)})= F(z^{(k)}), \\quad k=1,...,m.\n\\end{equation}\nMoreover, for $l>m$,\n\\begin{align*}\n\\Vert F_{A_l}^*-F_{A_m}^* \\Vert^2\n = \\Vert F_{A_l}^* \\Vert^2 - \\Vert F_{A_m}^* \\Vert^2.\n\\end{align*}\n\n Let $\\{ \\mathcal B_k\\}_{k=1}^m$ be the Gram-Schmidt (G-S)\northogonalization of $\\{K(\\cdot, \\overline{z^{(k)}})\\}_{k=1}^m$. The following formula is obvious\n\\begin{align}\\label{F_orth}\nF_{A_m}^*=\\sum_{k=1}^m\\langle F_{A_m}^*, \\mathcal B_k \\rangle\n\\mathcal B_k=\\sum_{k=1}^m\\langle F,\\mathcal  B_k \\rangle \\mathcal  B_k.\n\\end{align}\n(\\ref{F_orth}) indicates that $F_{A_m}^{*}$ is the orthogonal projection of\n$F$ to $\\overline{span\\{K(\\cdot, z^{(k)}), k=1,2,...,m\\}}$.\n\\smallskip\n\n\\begin{remark}\nLet $\\mathcal H(E)$ be a reproducing kernel Hilbert space defined on the set $E$ in $\\mathbb C$, $f\\in \\mathcal H(E)$, and $\\{a_k\\}_{k=1}^\\infty$ be a sequence of distinct points in $E$. In \\cite{Sa10} the so-called Aveiro method was proposed to construct an approximating function of $f$ in terms of $\\{a_k\\}_{k=1}^\\infty$, where the approximating function is with the interpolation property at $\\{a_k\\}_{k=1}^\\infty$. The convergence of such approximating function depends on the assumption that $\\{a_k\\}_{k=1}^\\infty$ is a uniqueness set of $\\mathcal H(E)$ (i.e. if $f\\in\\mathcal H(E)$ satisfies that $f(a_k)=0,k=1,2,...$, then $f\\equiv 0$). The formula of $F_{A_m}^*$ in (\\ref{appro_fun}) can be given by applying the Aveiro method to $H^2(T_{\\Gamma})$. However, the uniqueness set approach in higher dimensions is not easily available. In this study, we will show the convergence of $F_{A_m}^*$ corresponding to $\\{z^{(j)}\\}_{j=1}^\\infty$ that is not necessary being a uniqueness set.\n\\end{remark}\n\n  In the following discussion, we remove the restriction that all elements of $\\{z^{(k)}\\}_{k=1}^\\infty$ are distinct from each other, i.e. there may exist $k\\neq l$ such that $z^{(k)}=z^{(l)}.$ However, the original definition of $F_{A_m}^{*}$ is meaningless in such situation since the inverse of $A_m$ is meaningless. Therefore, we need to define the generalized $F_{A_m}^*$ that is denoted by $\\widetilde F_{A_m}^*$, so that $\\widetilde F_{A_m}^*$ does make sense in such situation. Our generalization is based on the methodology of P-OGA.\n\nSet $\\phi_{k}=K(\\cdot, \\overline{z^{(k)}})$. By the G-S orthogonalization process, we have\n\\begin{align}\\label{G-S}\n\\begin{split}\n\\beta_1&=\\beta_{\\{z^{(1)}\\}}= \\phi_{1},\\\\\n\\beta_l&=\\beta_{\\{z^{(1)},...,z^{(l)}\\}}=\\phi_{l}-\\sum_{k=1}^{l-1}{\\langle \\phi_l, \\frac{\\beta_k}{||\\beta_k||} \\rangle}\\frac{\\beta_k}{||\\beta_k||}, \\quad l\\geq 2\\\\\n\\mathcal B_l &=\\mathcal B_{\\{z^{(1)},...,z^{(l)}\\}}=\\frac{\\beta_l}{||\\beta_l||}.\n\\end{split}\n\\end{align}\nSpecifically, we derive the formula of $\\widetilde F_{A_m}^*$ by studying the property of $\\{\\mathcal B_k\\}_{k=1}^m$ for the case that $z^{(k)}=z^{(l)}, k\\neq l$. In fact, such kind of discussion on $\\{\\mathcal B_k\\}_{k=1}^m$ has been respectively made in the one complex variable \\cite{QWe}, quaternionic analysis \\cite{QSW} and several complex variables \\cite{Q-2D} settings. The general and complete discussion on such property of $\\{\\mathcal B_k\\}_{k=1}^m$ is included in \\cite{Q-2D}.\n  For simplicity, we interpret this for $z=(z_1, z_2)\\in \\mathbb C^2$ and $\\mathcal B_2=\\mathcal B_{\\{z^{(1)},w\\}}$. We identify $\\mathbb C^2$ with $\\mathbb R^4$ and set $w=z^{(1)}+(r\\cos\\theta, r\\sin\\theta \\cos \\zeta, r\\sin\\theta\\sin\\zeta\\cos\\eta,r\\sin\\theta\\sin\\zeta\\sin\\eta)=z^{(1)}+r\\vec l$, where $r>0$, $\\theta, \\zeta\\in [0,\\pi],\\eta\\in [0,2\\pi)$.\n  We consider\n\\begin{align*}\n\\lim_{r\\to 0} \\mathcal B_{\\{z^{(1)},w\\}}&=\\lim_{r\\to 0}\\frac{\\beta_{\\{z^{(1)},w\\}}}{||\\beta_{\\{z^{(1)},w\\}}||}\\\\\n&= \\lim_{r\\to 0}\\frac{\\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}}}{\\sqrt{\\langle \\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}},\\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}}\\rangle}}\\\\\n&= \\lim_{r\\to 0}\\frac{\\frac{\\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}}}{r}}{\\sqrt{\\langle \\frac{\\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}}}{r},\\frac{\\beta_{\\{z^{(1)},w\\}}-\\beta_{\\{z^{(1)},z^{(1)}\\}}}{r}\\rangle}}\\\\\n&= \\lim_{r\\to 0}\\frac{\\bigtriangledown_{\\vec l}\\beta_{\\{z^{(1)},z\\}}|_{z=z^{(1)}}}{||\\bigtriangledown_{\\vec l}\\beta_{\\{z^{(1)},z\\}}|_{z=z^{(1)}}||}\\\\\n&= \\frac{\\bigtriangledown_{\\vec l} \\phi_{z}|_{z=z^{(1)}}-\\langle \\bigtriangledown_{\\vec l} \\phi_{z}|_{z=z^{(1)}},\\frac{\\beta_{1}}{||\\beta_{1}||} \\rangle \\frac{\\beta_{1}}{||\\beta_{1}||} }{||\\bigtriangledown_{\\vec l}\\phi_{z}|_{z=z^{(1)}}-\\langle \\bigtriangledown_{\\vec l} \\phi_{z}|_{z=z^{(1)}},\\frac{\\beta_{1}}{||\\beta_{1}||} \\rangle \\frac{\\beta_{1}}{||\\beta_{1}||}||},\n\\end{align*}\nwhere \\begin{align*}\\bigtriangledown_{\\vec l} \\phi_{z}|_{z=z^{(1)}}&=\\frac{\\partial \\phi_{z}}{\\partial x_1}|_{z=z^{(1)}}\\cos\\theta+\\frac{\\partial \\phi_{z}}{\\partial y_1}|_{z=z^{(1)}}\\sin\\theta\\cos\\zeta\\\\\n& +\\frac{\\partial \\phi_{z}}{\\partial x_2}|_{z=z^{(1)}}\\sin\\theta\\sin\\zeta\\cos\\eta\n+\\frac{\\partial \\phi_{z}}{\\partial y_2}|_{z=z^{(1)}}\\sin\\theta\\sin\\zeta\\sin\\eta,\\end{align*} and it is the directional derivative of $\\phi_{z}$ in the direction $\\vec l$ as a function of $z$. This obversation indicates that the directional derivative of $\\phi_{z}$ should be involved in the G-S orthogonalization process, when $z^{(1)}=z^{(2)}$. Note that\n$\\frac{\\partial \\phi_{z}}{\\partial z_j}=\\frac{1}{2}\\left(\\frac{\\partial }{\\partial x_j}-i\\frac{\\partial}{\\partial y_j}\\right)\\phi_{z}=0,\\quad j=1,2$, which indicate that $\\frac{\\partial \\phi_{z}}{\\partial x_1}$ and $\\frac{\\partial \\phi_{z}}{\\partial y_1}$, as well as $\\frac{\\partial \\phi_{z}}{\\partial x_2}$ and $\\frac{\\partial \\phi_{z}}{\\partial y_2}$, are linearly dependent. We then define $\\mathcal B_{\\{z^{(1)},z^{(1)}\\}}^{\\vec l}$ as the limit in the above sense.\n Let $l_k$ be the cardinality of the set $\\{j:z^{(j)}=z^{(k)}, j\\leq k\\}$. Generally, for $\\frac{(h_k+1)h_k}{2}<l_k\\leq \\frac{(h_k+2)(h_k+1)}{2}$, the $h_k$-th order partial derivative of $\\phi_{z^{(k)}}$ should be involved in the G-S orthogonalization process.\nAs defined in \\cite{Q-2D}, we introduce $\\mathcal A_k,k=1,2,...,$ the function set consisting of all possible directional derivatives of the functions in $\\mathcal A_{k-1}$, where $\\mathcal A_0=\\{K(\\cdot,z): z\\in T_{\\Gamma_1}\\}.$ Denote by $\\mathcal A$ the set\n$$\n\\mathcal A=\\cup_{k=0}^\\infty \\mathcal A_k.\n$$\nThe definition of $\\mathcal A_k$ is slightly different from the one in \\cite{Q-2D} since we do not assume that the elements in $\\mathcal A_k$ are normalized.\nGiven a sequence of points $\\{z^{(k)}\\}_{k=1}^m$ in $T_{\\Gamma_1},$ we can find a sequence of elements $\\{\\Phi_{z^{(k)}}\\}_{k=1}^m$ in $\\mathcal A$ such that $\\{\\mathcal B_k\\}_{k=1}^m$ is the orthonormalization of $\\{\\Phi_{z^{(k)}}\\}_{k=1}^m$ in the above sense. The formula of $\\Phi_{z^{(k)}}$ depends on $l_k$.\nDefine\n\\begin{align}\\label{general_F}\n\\widetilde F_{A_m}^{*}(z)=\\sum_{k=1}^m \\langle F,\\mathcal B_k \\rangle \\mathcal B_k=\\sum_{j=1}^m\\sum_{k=1}^m\n\\langle F, \\Phi_{z^{(j)}}\\rangle\\widetilde{a_{j,k}^{(m)}} \\Phi_{z^{(k)}}(z), \\quad\nz\\in T_{\\Gamma_1},\n\\end{align}\nwhere $A_m$ is the matrix with elements $a_{j,k}=\\langle \\Phi_{z^{(k)}}, \\Phi_{z^{(j)}}  \\rangle$, and $\\widetilde {a_{j,k}^{(m)}}$s' are elements of $\\overline{A_m^{-1}}$.\n(\\ref{general_F}) is well defined since $$\\langle \\widetilde F_{A_m}^*, \\Phi_{z^{(k)}}\\rangle = \\langle F, \\Phi_{z^{(k)}}\\rangle, \\quad k=1,2,...,m.$$\nNote that $F_{A_m}^*=\\widetilde F_{A_m}^*$ if all $z^{(k)}$s' are distinct from each other.\n\n For $\\mathbb C^n,n>2$, we can conclude that if $\\binom{h_k-1+n}{n}<l_k\\leq \\binom{h_k+n}{n}$, the $h_k$-th order partial derivative of $\\phi_{z^{(k)}}$ should be involved in the G-S orthogonalization process. We can similarly define $\\mathcal A$ and $\\widetilde F_{A_m}^*$. Hereafter, we do not distinguish between $F_{A_m}^*$ and $\\widetilde F_{A_m}^*$, and then we adopt the notation $F_{A_m}^*$ for this matter.\n\nThe main procedure for constructing $F_{A_m}^*$ that tends to $F$ ($m\\to \\infty$) in the $H^2$-norm is as follows.\nLet $\\{z^{(j)}\\}_{j=1}^m$ be $m$ given points in $T_{\\Gamma_1}$\nand $A_m$ be the matrix with elements\n\\begin{align}\\label{matrix_A}\na_{j,k}=\\langle \\Phi_{z^{(k)}}, \\Phi_{z^{(j)}} \\rangle.\n\\end{align}\nWe are to select $z^{(m+1)}_*\\in T_{\\Gamma_1}$ satisfying\n\\begin{align}\\label{min_pro_revise}\n\\begin{split}\nz^{(m+1)}_* & :=\\arg \\min_{z^{(m+1)}\\in T_{\\Gamma_1}} ||F- F_{A_{m+1}}^*||^2\\\\\n& := \\arg\\min_{z^{(m+1)}\\in T_{\\Gamma_1}}||F||^2-|| F_{A_{m+1}}^*||^2.\n\\end{split}\n\\end{align}\n\nThe justification of the existence of $z_{*}^{(m+1)}$ is provided later. In fact, it follows from the boundary behaviors of functions in $H^2(T_{\\Gamma_1})$.\nAt this moment, we assume that $z^{(m+1)}_{*}$ can be achieved. The convergence of $F_{A_m}^{*}$ in the\n$H^2$-norm is given by the following result.\n\n\\bigskip\n \\begin{thm}\\label{thm1}\nFor $F\\in H^2(T_{\\Gamma_1})$, we have\n \\begin{align}\\label{convergence}\n ||F -\\sum_{j=1}^{m} \\sum_{k =1}^{m} \\langle F, \\Phi_{z^{(j)}}\\rangle \\widetilde{ a_{j,k}^{(m)}} \\Phi_{z^{(k)}}||\\to 0, \\text{as }  m\\to \\infty,\n \\end{align}\n where each element of $\\{z^{(k)}\\}_{k=1}^{\\infty}$ is selected by the principle $(\\ref{min_pro_revise})$.\n \\end{thm}\n\\begin{proof}\nBy (\\ref{general_F}) and the Riesz-Fischer theorem, we know that there\nexists $F_{A_\\infty}^{*}\\in H^2(T_{\\Gamma_1})$ satisfying\n\\begin{align}\\label{lim1}\nF_{A_\\infty}^{*}=\\lim_{m\\to \\infty}F_{A_m}^{*} \\quad \\text{in the\n$H^2$-norm}.\n\\end{align}\nIf $(\\ref{convergence})$ does not hold, then\n \\begin{align}\n g=F-F_{A_\\infty}^{*}\\not\\equiv 0.\n \\end{align}\n\nBy the Identity theorem, we must have $b\\not\\in \\{z^{(1)},...,z^{(m)}\\}$ such that\n\\begin{align}\\label{g_b}\n|g(b)|= \\delta_0>0.\n\\end{align}\n\nOn one hand,\n\\begin{align}\\label{two term}\n\\delta_0=|g(b)|=| F(b)-F_{A_\\infty}^{*}(b)|\\leq\n|F(b)-F_{A_m}^{*}(b)|+|F_{A_\\infty}^{*}(b)-F_{A_m}^{*}(b)|.\n\\end{align}\nBy $(\\ref{lim1})$, there exists $ N_1>0$ such that when $m>N_1$, the second term of\n$(\\ref{two term})$\n\\begin{align*}\n|F_{A_\\infty}^{*}(b)-F_{A_m}^{*}(b)| &=|\\langle F_{A_\\infty}^{*}(\\cdot)-F_{A_m}^{*}(\\cdot), K(\\cdot, \\overline b) \\rangle|\\\\\n&\\leq ||F_{A_\\infty}^{*}-F_{A_m}^{*}|| ||K(\\cdot, \\overline b)||\\\\\n&<\\frac{\\delta_0}{2},\n\\end{align*}\nwhere the second inequality follows from the Cauchy-Schwarz inequality.\nHence, we have\n$$\n|F(b)-F_{A_m}^{*}(b)|>\\frac{\\delta_0}{2}.\n$$\nOn the other hand, by (\\ref{general_F}) we have\n\\begin{align}\nF(b)=F_{A_{m,b}}^{*}(b),\n\\end{align}\nwhere $A_{m,b}$ is the matrix with elements given by\n$(\\ref{matrix_A})$ corresponding to $(z^{(1)},...,z^{(m)}, b)$.\\\\\nBy $(\\ref{min_pro_revise})$, we have\n\\begin{align}\n||F||^2-||F_{A_{m,b}}^{*}||^2=||F-F_{A_{m,b}}^{*}||^2\\geq\n||F-F_{A_{m+1}}^{*}||^2=||F||^2-||F_{A_{m+1}}^{*}||^2.\n\\end{align}\nTherefore, there exists $N_2>0$ such that when $m>N_2$,\n\\begin{align}\n\\begin{split}\n|F(b)-F_{A_m}^{*}(b)| &= |F^{*}_{A_{m,b}}(b)-F^{*}_{A_{m}}(b)|\\\\\n& \\leq ||F_{A_{m,b}}^{*}-F_{A_{m}}^{*}|| ||K(\\cdot, \\overline b)||\\\\\n& \\leq ||K(\\cdot, \\overline b)||(\\sqrt{||F_{A_{m,b}}^{*}||^2-||F_{A_m}^{*}||^2})\\\\\n& \\leq ||K(\\cdot, \\overline b)||(\\sqrt{||F_{A_{m+1}}^{*}||^2-||F_{A_{m}}^{*}||^2})\\\\\n& \\leq ||K(\\cdot, \\overline b)||{||F_{A_{m+1}}^{*}-F_{A_{m}}^{*}||}\\\\\n& <\\frac{\\delta_0}{2}.\n\\end{split}\n\\end{align}\nIf $m>\\max\\{N_1,N_2\\}$, then we arrive a contradiction. This proves\nthe theorem.\n\\end{proof}\n\nImmediately, we have the following corollary.\n\\begin{cor}\nIf all the conditions in Theorem \\ref{thm1} are fulfilled, then, for\nany compact subset $A$ in $T_{\\Gamma_1}$,\n$$\nF_{A_m}^*(z) =  \\sum_{j=1}^{m} \\sum_{k =1}^{m} \\langle F, \\Phi_{z^{(j)}}\\rangle\n\\widetilde{ a_{j,k}^{(m)}}\\Phi_{z^{(k)}}(z), \\quad z\\in A,\n$$\nuniformly converges to $F(z)$.\n\\end{cor}\n\n\\medskip\n\nDenote by $\\partial T_{\\Gamma_1}$ the boundary of $T_{\\Gamma_1}$. The next\nlemma offers a set of sufficient conditions of the existence of\n$z^{(m+1)}_*$.\n\\begin{lem}\\label{re-ex-lem}\nSuppose that $F\\in H^2(T_{\\Gamma_1})$ and and $z^{(j)}\\in T_{\\Gamma_1},\nj=1,...,m,$ are fixed. If\n\\begin{align}\\label{g-ex-cond}\n\\begin{split}\n\\lim_{z^{(m+1)} \\to \\beta} \\frac{|\\langle F, \\Phi_{z^{(m+1)}}\\rangle|}{||\\Phi_{z^{(m+1)}}||}\n&= 0,\\\\\n\\lim_{z^{(m+1)}\\to \\beta}\\frac{|\\langle \\Phi_{z^{(j)}},\\Phi_{z^{(m+1)}} \\rangle|}{||\\Phi_{z^{(m+1)}}||}&=0,\\quad j=1,2,...,m,\n\\end{split}\n\\end{align}\nwhere $\\beta\\in \\partial T_{\\Gamma_1}$, then\n\\begin{align}\\label{g-ex-boundary}\n\\lim_{z^{(m+1)} \\to \\beta}||F- F^{*}_{A_{m+1}}||=\n||F-F^{*}_{A_{m}}||,\n\\end{align}\nand if\n\\begin{align}\\label{g-ex-cond1}\n\\begin{split}\n\\lim_{|z^{(m+1)}| \\to \\infty}\n\\frac{|\\langle F, \\Phi_{z^{(m+1)}}\\rangle|}{||\\Phi_{z^{(m+1)}}||} &= 0,\\\\\n\\lim_{|z^{(m+1)}|\\to \\infty}\\frac{|\\langle \\Phi_{z^{(j)}},\\Phi_{z^{(m+1)}} \\rangle|}{||\\Phi_{z^{(m+1)}}||}&=0,\\quad j=1,2,...,m,\n\\end{split}\n\\end{align}\nthen\n\\begin{align}\\label{g-ex-infinity}\n\\lim_{|z^{(m+1)}| \\to \\infty}||F- F^{*}_{A_{m+1}}||=\n||F-F^{*}_{A_{m}}||.\n\\end{align}\n\\end{lem}\n\n\n\\begin{proof}\nWe adopt the notation given in (\\ref{G-S}). We know that $\\{\\mathcal B_k =\\frac{\\beta_k}{||\\beta_k||}\\}_{k=1}^{m+1}$ is an orthonormal system. Based on (\\ref{general_F}), we have\n$$\n\\Vert F-F_{A_{m+1}}^* \\Vert^2=\\Vert F-F_{A_m}^*\\Vert^2-|\\langle F, \\mathcal B_{m+1} \\rangle|^2.\n$$\nTo get (\\ref{g-ex-boundary}), we need to show $|\\langle F,\\mathcal B_{m+1} \\rangle|\\to 0$ as $z\\to \\beta$.\nIn fact,\n\\begin{align}\n\\begin{split}\n|\\langle F,\\mathcal B_{m+1}\\rangle| &= \\frac{|\\langle F, {\\beta_{m+1}}\\rangle|}{\\|\\beta_{m+1}\\|}\\\\\n& = \\frac{|\\langle F, \\Phi_{z^{(m+1)}} - \\sum_{k=1}^{m}\\langle \\Phi_{z^{(m+1)}}, \\mathcal B_k\\rangle\\mathcal B_k\\rangle|}{\\|\\Phi_{z^{(m+1)}} - \\sum_{k=1}^{m}\\langle \\Phi_{z^{(m+1)}}, \\mathcal B_k\\rangle\\mathcal B_k\\|}\\\\\n& = \\frac{|\\langle F, \\frac{\\Phi_{z^{(m+1)}}}{\\|\\Phi_{z^{(m+1)}}\\|} - \\sum_{k=1}^{m}\\langle \\frac{\\Phi_{z^{(m+1)}}}{\\|\\Phi_{z^{(m+1)}}\\|}, \\mathcal B_k\\rangle\\mathcal B_k\\rangle|}{1 - \\sum_{k=1}^{m}|\\langle \\frac{\\Phi_{z^{(m+1)}}}{\\|\\Phi_{z^{(m+1)}}\\|}, \\mathcal B_k\\rangle|^2}.\n\\end{split}\n\\end{align}\nBy (\\ref{g-ex-cond}), we can have (\\ref{g-ex-boundary}). We can also conclude (\\ref{g-ex-infinity}) in a similar way.\n\\end{proof}\n\nWe call\n$$\n\\lim_{z^{(m+1)} \\to \\beta} \\frac{|\\langle F, \\Phi_{z^{(m+1)}}\\rangle|}{||\\Phi_{z^{(m+1)}}||}=0\n$$\nand\n$$\n\\lim_{|z^{(m+1)}| \\to \\infty}\n\\frac{|\\langle F, \\Phi_{z^{(m+1)}}\\rangle|}{||\\Phi_{z^{(m+1)}}||} = 0\n$$\nthe \\lq\\lq boundary vanishing condition (BVC)\" in $H^2(T_{\\Gamma_1}),$ where $\\{z^{(k)}\\}_{k=1}^{m+1}$ is a sequence of points in $T_{\\Gamma_1}$. Since $\\{z^{(1)},...,z^{(m)}\\}$ are previously fixed in Lemma \\ref{re-ex-lem}, $z^{(m+1)}$ must be different $z^{(k)}, 1\\leq k\\leq m$ when $z^{(m+1)}$ tends to the points at boundary (including the point at infinity). Thus the conditions (\\ref{g-ex-cond}) and (\\ref{g-ex-cond1}) then follows from\n\\begin{align}\\label{weak-bvc-tube}\n\\begin{split}\n\\lim_{z^{(m+1)} \\to \\beta} \\frac{|\\langle F, \\phi_{z^{(m+1)}}\\rangle|}{||\\phi_{z^{(m+1)}}||} &=0\\\\\n\\lim_{|z^{(m+1)}| \\to \\infty}\n\\frac{|\\langle F, \\phi_{z^{(m+1)}}\\rangle|}{||\\phi_{z^{(m+1)}}||} &= 0,\n\\end{split}\n\\end{align}\nwhere (\\ref{weak-bvc-tube}) is called the weak BVC.\n\nOne can give another proof of Lemma \\ref{re-ex-lem} by using the argument given in  \\cite[Lemma 3.1]{MQ1}.\n Lemma \\ref{re-ex-lem} tells us that (\\ref{min_pro_revise}) can not achieve its minimum value at boundary point $\\beta$ or at infinity.\nTherefore, to justify the existence of $z^{(m+1)}_*$, we only\nneed to prove that the weak BVC holds. In fact, for $H^2(T_{\\Gamma_1})$, we can show the stronger result that the BVC holds.\n\nIn the following discussion we first reduce the BVC to a special case.\nDefine\n\\begin{align}\\label{D-element}\n\\phi_{\\alpha^{(k)},z^{(k)}}=\\frac{\\partial^{|\\alpha^{(k)}|}K(\\cdot, \\overline\n{z})}{\\partial{x_1}^{\\alpha_1^{(k)}}\\partial{x_2}^{\\alpha_2^{(k)}}\\cdots\n\\partial{x_n}^{\\alpha_n^{(k)}}}\\Big |_{z=z^{(k)}} =(\\frac{-1}{2\\pi i})^n \\prod_{j=1}^n\n\\frac{\\alpha_j^{(k)} !}{(w_j-\\overline z_j)^{\\alpha_j^{(k)}+1}}\\Big|_{z=z^{(k)}},\n\\end{align}\nwhere all elements of $n$-tuple $\\alpha^{(k)}=(\\alpha_1^{(k)},...,\\alpha_n^{(k)})$ are non-negative integers and $|\\alpha^{(k)}|=\\sum_{j=1}^n \\alpha_j^{(k)} \\geq 0$.\nNote that $\\frac{\\Phi_{z^{(m+1)}}}{||\\Phi_{z^{(m+1)}}||}$ is a finite linear combination of $\\phi_{\\alpha^{(k)},z^{(k)}}$ with $|\\alpha^{(k)}|=h_{m+1}$, where $h_{m+1}$ is determined by the cardinality of the set $\\{j:z^{(j)}=z^{(m+1)}, j\\leq m+1\\}.$ Specifically,\n$$\n\\frac{\\Phi_{z^{(m+1)}}}{||\\Phi_{z^{(m+1)}}||}=\\sum_{j=1}^{N_{m+1}} \\frac{||\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}||}{||\\Phi_{z^{(m+1)}}||}v_j \\frac{\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}}{||\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}||},\n$$\nwhere $N_{m+1}=\\binom{h_{m+1}+n}{n}-\\binom{h_{m+1}-1+n}{n},$ $|\\alpha^{j,(m+1)}|=h_{m+1}$ and $(v_1,..,v_{N_{m+1}})\\neq 0.$ One can show that $\\frac{||\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}||}{||\\Phi_{z^{(m+1)}}||}$ is not an obstacle while the conditions (\\ref{g-ex-cond}) and (\\ref{g-ex-cond1}) are considered. For instance, for $n=2$, we have\n\\begin{align}\\label{convert-to-K}\n||\\Phi_{z^{(m+1)}}||^2=||\\sum_{j=1}^{h_{m+1}+1}v_j \\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}||^2=||\\phi_{\\alpha^{k,(m+1)},z^{(m+1)}}||^2||\\sum_{j=1}^{h_{m+1}+1}v_j \\frac{\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}}{||\\phi_{\\alpha^{k,(m+1)},z^{(m+1)}}||}||^2.\n\\end{align}\nBy directly calculating, one can get that $||\\sum_{j=1}^{h_{m+1}+1}v_j \\frac{\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}}{||\\phi_{\\alpha^{k,(m+1)},z^{(m+1)}}||}||^2$ can be considered as a polynomial of $\\frac{y_1^{(m+1)}}{y_2^{(m+1)}}$ or $\\frac{y_2^{(m+1)}}{y_1^{(m+1)}}$ of degree $2 h_{m+1}.$ When $y\\to \\partial \\Gamma_1$ or $|y|\\to\\infty$, $\\frac{y_1^{(m+1)}}{y_2^{(m+1)}}$ may be $\\infty$, zero or a positive constant $H$. For the former two cases, $||\\sum_{j=1}^{h_{m+1}+1}v_j \\frac{\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}}{||\\phi_{\\alpha^{k,(m+1)},z^{(m+1)}}||}||^2$ is either $\\infty$ or a nonzero constant. For the last case, $||\\sum_{j=1}^{h_{m+1}+1}v_j \\frac{\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}}{||\\phi_{\\alpha^{k,(m+1)},z^{(m+1)}}||}||^2$ becomes a polynomial of $H$. However, such polynomial can not be zero on $(0,\\infty)$, otherwise, by the left hand-side of (\\ref{convert-to-K}) we get $(v_1,...,v_{h_{m+1}+1})= 0,$ which contradicts to $(v_1,...,v_{h_{m+1}+1})\\neq 0.$ For $\\frac{y_2^{(m+1)}}{y_1^{(m+1)}}$, we can use the same argument. We then conclude that $\\frac{||\\phi_{\\alpha^{j,(m+1)},z^{(m+1)}}||}{||\\Phi_{z^{(m+1)}}||}$ is either zero or a positive constant when $y\\to \\partial \\Gamma_1$ or $|y|\\to\\infty$. One can easily get similar conclusions for general $n>2$ by our argument.\n By the above discussion, it suffices to show that the BVC holds for $\\Phi_{z^{(m+1)}}=\\phi_{\\alpha^{(m+1)},z^{(m+1)}}.$\n\\begin{lem}\nFor $1<p<\\infty$, $z=x+iy\\in T_{\\Gamma_1}$ and $\\alpha=(\\alpha_1,...,\\alpha_n)$,\n$$\\int_{\\mathbb R^n} \\left| \\prod_{j=1}^n\n\\frac{1}{(\\xi_j-\\overline z_j)^{\\alpha_j+1}} \\right|^p d\\xi_1d\\xi_2 \\cdots d\\xi_n= \\pi^{\\frac{n}{2}}\\prod_{j=1}^n \\frac{\\Gamma(-\\frac{1}{2}+\\frac{p(\\alpha_j+1)}{2})}{\\Gamma(\\frac{p(\\alpha_j+1)}{2})}\\left(\\frac{1}{y_j}\\right)^{p(\\alpha_j+1)-1}.$$\n\\end{lem}\n\\begin{proof}\n\\begin{align*}\n\\int_{\\mathbb R^n}\\left|  \\prod_{j=1}^n\n\\frac{1}{(\\xi_j-\\overline z_j)^{\\alpha_j+1}} \\right|^p d\\xi_1d\\xi_2 \\cdots d\\xi_n &= \\int_{\\mathbb R^n}  \\prod_{j=1}^n\\left|\n\\frac{1}{(|\\xi_j-x_j|^2+|y_j|^2)} \\right|^{\\frac{p({\\alpha_j+1})}{2}} d\\xi_1d\\xi_2 \\cdots d\\xi_n\\\\\n &= \\prod_{j=1}^n\\int_{-\\infty}^\\infty  \\left|\n\\frac{1}{(|\\xi_j-x_j|^2+|y_j|^2)} \\right|^{\\frac{p({\\alpha_j+1})}{2}} d\\xi_j\\\\\n&= \\prod_{j=1}^n \\left(\\frac{1}{y_j}\\right)^{p(\\alpha_j+1)-1}\\int_{-\\infty}^\\infty  \\left|\n\\frac{1}{t_j^2+1} \\right|^{\\frac{p({\\alpha_j+1})}{2}} dt_j\\\\\n&= \\pi^{\\frac{n}{2}}\\prod_{j=1}^n \\frac{\\Gamma(-\\frac{1}{2}+\\frac{p(\\alpha_j+1)}{2})}{\\Gamma(\\frac{p(\\alpha_j+1)}{2})}\\left(\\frac{1}{y_j}\\right)^{p(\\alpha_j+1)-1},\n\\end{align*}\nwhere the third equality is due to changing variables by setting $t_j=\\frac{\\xi_j-x_j}{y_j}$, and $\\Gamma(\\cdot)$ is the Gamma function.\n\\end{proof}\\\\\nConsequently, we have\n\\begin{align}\\label{phi-norm}\n\\begin{split}\n||\\phi_{\\alpha,z}||^2 &=\\left(\\frac{1}{2\\pi}\\right)^n\\left(\\prod_{j=1}^n (\\alpha_j!)^2\\right) (\\pi)^{\\frac{n}{2}}\\prod_{j=1}^n \\frac{\\Gamma(\\alpha_j+\\frac{1}{2})}{\\Gamma(\\alpha_j+1)}\\left(\\frac{1}{y_j}\\right)^{2\\alpha_j+1}\\\\\n& = \\prod_{j=1}^n\\frac{(2\\alpha_j)!}{(2y_j)^{2\\alpha_j+1}}.\n\\end{split}\n\\end{align}\nThe following results imply that the conditions (\\ref{g-ex-cond}) and (\\ref{g-ex-cond1}) hold.\n\\begin{thm}\\label{first-app1}\nFor $F\\in H^2(T_{\\Gamma_1})$, and $\\alpha=(\\alpha_1,...,\\alpha_n)$,\n\\begin{align}\\label{app1}\n\\lim_{y\\in \\Gamma_1, y\\to \\beta}\\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0, \\quad z=x+iy\\in T_{\\Gamma_1},\n\\end{align}\nholds uniformly for $x\\in\\mathbb R^n$, where $\\beta\\in \\partial \\Gamma_1$.\n\\end{thm}\n\\begin{proof}\nSince\n$$\\langle F, \\phi_{\\alpha, z} \\rangle=\\int_{\\mathbb R^n} F(\\xi)\\overline {\\phi_{\\alpha, z}(\\xi)}d\\xi,$$ and $F(\\xi)$ is the limit of $F(\\xi+i\\eta)$ in the $L^2$-norm, we can find $G(\\xi)\\in L^2(\\mathbb R^n)\\cap L^p(\\mathbb R^n), 2< p<\\infty,$ such that for any $\\epsilon>0$, $||F-G||_{L^2(\\mathbb R^n)}<\\frac{\\epsilon}{2}$.\n We also have\n \\begin{align}\n \\begin{split}\n \\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha, z}||}&\\leq \\frac{|\\langle F-G,\\phi_{\\alpha,z} \\rangle|}{||\\phi_{\\alpha,z}||} + \\frac{|\\langle G, \\phi_{\\alpha,z}\\rangle| }{||\\phi_{\\alpha,z}||}\\\\\n & \\leq \\frac{||F-G||~||\\phi_{\\alpha,z}||}{||\\phi_{\\alpha,z}||} +\\frac{|\\langle G, \\phi_{\\alpha,z}\\rangle| }{||\\phi_{\\alpha,z}||}\\\\\n & \\leq \\frac{\\epsilon}{2} + \\frac{|\\langle G, \\phi_{\\alpha,z}\\rangle| }{||\\phi_{\\alpha,z}||}.\n \\end{split}\n \\end{align}\nIt suffices to prove that, for $y\\in \\Gamma_1, y\\to \\beta$,\n\\begin{align}\\label{app1-fact1}\n\\frac{\\int_{\\mathbb R^n}|G(\\xi)\\overline{\\phi_{\\alpha, z}(\\xi)}|d\\xi}{||\\phi_{\\alpha,z}||}<\\frac{\\epsilon}{2}.\n\\end{align}\nIndeed, by applying H\\\"older's inequality to $\\int_{\\mathbb R^n}|G(\\xi)\\overline{\\phi_{\\alpha, z}(\\xi)}|d\\xi$, (\\ref{app1-fact1}) follows from\n\\begin{align*}\n\\lim_{y\\in \\Gamma_1,y\\to \\beta}\\frac{(\\int_{\\mathbb R^n}|\\phi_{\\alpha, z}|^q d\\xi)^{\\frac{1}{q}}}{(\\int_{\\mathbb R^n}|\\phi_{\\alpha, z}|^2 d\\xi)^{\\frac{1}{2}}}&=\\lim_{y\\in \\Gamma_1,y\\to \\beta}\\frac{\\pi^{\\frac{n}{2q}}\\prod_{j=1}^n \\left(\\frac{\\Gamma(-\\frac{1}{2}+\\frac{q(\\alpha_j+1)}{2})}{\\Gamma(\\frac{q(\\alpha_j+1)}{2})}\\right)^{\\frac{1}{q}}\\left(\\frac{1}{y_j}\\right)^{(\\alpha_j+1)-\\frac{1}{q}}}{\\pi^{\\frac{n}{4}}\\prod_{j=1}^n \\left(\\frac{\\Gamma(\\alpha_j+\\frac{1}{2})}{\\Gamma(\\alpha_j+1)}\\right)^{\\frac{1}{2}}\\left(\\frac{1}{y_j}\\right)^{\\alpha_j+\\frac{1}{2}}}\\\\\n&=C\\lim_{y\\in \\Gamma_1,y\\to \\beta}\\prod_{j=1}^n y_j^{\\frac{1}{2}-\\frac{1}{p}}\\\\\n&=0,\n\\end{align*}\nwhere $C$ is a constant, and $q$ satisfies $\\frac{1}{p}+\\frac{1}{q}=1.$ The last equality is based on the fact that there exists $y_j\\to 0$ when $y\\to \\beta$.\n\\end{proof}\n\n\\begin{thm}\\label{second-app1}\nFor $F\\in H^2(T_{\\Gamma_1})$, and $\\alpha=(\\alpha_1,...,\\alpha_n)$,\n\\begin{align}\\label{app1-fact2}\n\\lim_{y\\in \\Gamma_1, |y|\\to \\infty} \\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0, \\quad z=x+iy\\in T_{\\Gamma_1},\n\\end{align}\nholds uniformly for $x\\in \\mathbb R^n$.\n\\end{thm}\n\\begin{proof}\nBy Theorem \\ref{rf_p}, for any $\\epsilon>0$, we can find $y^\\prime\\in \\Gamma_1$ such that\n$$\n\\int_{\\mathbb R^n}|F(\\xi)-F(\\xi+iy^\\prime)|^2 d\\xi<\\epsilon.\n$$\n\\begin{align}\\label{app1-fact2-ineq}\n \\begin{split}\n \\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha, z}||}&\\leq \\frac{|\\langle F(\\cdot)-F(\\cdot+iy^\\prime),\\phi_{\\alpha,z} \\rangle|}{||\\phi_{\\alpha,z}||} + \\frac{|\\langle F(\\cdot+iy^\\prime), \\phi_{\\alpha,z}\\rangle| }{||\\phi_{\\alpha,z}||}\\\\\n & \\leq \\frac{||F(\\cdot)-F(\\cdot+iy^\\prime)||~||\\phi_{\\alpha,z}||}{||\\phi_{\\alpha,z}||} +\\frac{|\\langle F, \\phi_{\\alpha,z+iy^\\prime}\\rangle| }{||\\phi_{\\alpha,z}||}\\\\\n & \\leq \\epsilon + \\frac{|\\langle F, \\phi_{\\alpha,z+iy^\\prime}\\rangle| }{||\\phi_{\\alpha,z}||}.\n \\end{split}\n \\end{align}\nBy applying the argument in Theorem \\ref{first-app1}, we can easily show that, for $y\\in \\Gamma_1$ and $|y|$ large enough, $$\\frac{|\\langle F, \\phi_{\\alpha,z+iy^\\prime}\\rangle| }{||\\phi_{\\alpha,z}||}<C\\epsilon,$$ where $C$ is a constant. The difference is that we need to find $G\\in L^2(\\mathbb R^n)\\cap L^p(\\mathbb R^n), 1<p<2$, such that, for any $\\epsilon>0$, $||F-G||_{L^2(\\mathbb R^n)}<\\epsilon$.\n \\end{proof}\n\n\\begin{thm}\\label{third-app1}\nFor $F\\in H^2(T_{\\Gamma_1})$, and $\\alpha=(\\alpha_1,...,\\alpha_n)$,\n\\begin{align}\\label{app1-fact3}\n\\lim_{|x|\\to \\infty} \\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0, \\quad z=x+iy\\in T_{\\Gamma_1},\n\\end{align}\nholds uniformly for $y\\in \\Gamma_1$.\n\\end{thm}\n\\begin{proof}\nBy Theorems \\ref{first-app1} and \\ref{second-app1}, it suffices to prove that\n \\begin{align}\\label{g-cone-fact2}\n\\lim_{|x|\\to \\infty} \\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0\n \\end{align}\n holds uniformly for $y\\in C_0$, where $C_0$ is a compact subset in $\\Gamma_1$.\n\n\n Since $\\overline{ {\\text span} \\{K(\\cdot,\\overline z), z\\in T_{\\Gamma_1}\\}}=H^2(T_{\\Gamma_1})$, we have $\\{w^{(j)}\\}_{j=1}^N$ in $T_{\\Gamma_1}$ such that\n $$\\|F-G_N\\|<\\frac{\\epsilon}{2},$$\n  where $G_N=\\sum_{j=1}^Nc_j K(\\cdot, \\overline {w^{(j)}})\\in H^2(T_{\\Gamma_1}).$\nHence, we have\n$$\n\\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}\\leq \\frac{|\\langle F-G_N, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}+ \\frac{|\\langle G_N, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}<\\frac{\\epsilon}{2}+ \\frac{|\\langle G_N, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}.\n$$\nIt suffices to show that, for a fixed $w=\\xi+i\\eta\\in T_{\\Gamma_1}$, when $|x|$ is large enough,\n\\begin{align*}\n\\frac{|\\langle K(\\cdot,\\overline w), \\phi_{\\alpha,z}\\rangle|}{||\\phi_{\\alpha,z}||}&=\\prod_{j=1}^n \\frac{(2y_j)^{\\alpha_j+\\frac{1}{2}}\\alpha_j!}{|z_j-\\overline w_j|^{\\alpha_j+1}\\sqrt{(2\\alpha_j)!}}\\\\\n&= \\prod_{j=1}^n \\frac{(2y_j)^{\\alpha_j+\\frac{1}{2}}\\alpha_j!}{|(x_j-\\xi_j)^2+(y_j+\\eta_j)^2|^{\\frac{\\alpha_j+1}{2}}\\sqrt{(2\\alpha_j)!}}\\\\\n& <\\frac{\\epsilon}{2}.\n\\end{align*}\nThe last inequality is based on the fact that there exists $x_j$ satisfying that $|x_j|\\to\\infty$ as $|x|\\to \\infty$.\n\\end{proof}\n\nWe now conclude that the existence of $z^{(m+1)}_*$ is evident by Theorems \\ref{first-app1} - \\ref{third-app1}. Although we can also give another proof of Theorem \\ref{first-app1} as well as Theorem \\ref{second-app1} by using the argument in Theorem \\ref{third-app1}, we will show that the technique used in Theorem \\ref{first-app1} is helpful in proving analogous results of Theorems \\ref{first-app1} - \\ref{third-app1} for $H^p(T_{\\Gamma}), 1<p<\\infty$ (see Section 4).\n\\medskip\n\n\\begin{remark}\nBy using Theorems \\ref{first-app1} - \\ref{third-app1} and the following lemma (Lemma \\ref{MP-first-case}), we can give another kind of rational approximation of functions in $H^2(T_{\\Gamma_1})$. Such rational approximation indeed is analogous to the one given in \\cite{WQ1}. Specifically, we apply the idea of greedy algorithm to $H^2(T_{\\Gamma_1})$ with the dictionary\n\\begin{align}\\label{dictionary}\n\\mathcal D=\\left\n\\{\\psi_{\\alpha,z}(w)=\\frac{\\phi_{\\alpha,z}(w)}{||\\phi_{\\alpha,z}||};\n|\\alpha|=\\sum_{j=1}^n \\alpha_j \\geq 0, z, w\\in T_{\\Gamma_1} \\right\n\\},\n\\end{align}\nwhere $\\phi_{\\alpha,z}(w)$ is defined in (\\ref{D-element}).\nWe briefly give an introduction to greedy algorithm with the dictionary $\\mathcal D$.\n\nLet $F\\in H^2(T_{\\Gamma_1})$. Then, by greedy\nalgorithm, one can have\n\\begin{align}\\label{mp}\nF=\\sum_{l=1}^m \\langle R^l F, \\psi_{\\alpha^{(l)},z^{(l)}}\\rangle\n\\psi_{\\alpha^{(l)},z^{(l)}}  + R^{m+1} F,\n\\end{align}\nwhere $R^l F$ is defined by\n$$\nR^0F=F, R^lF=R^{l-1}F- \\langle R^{l-1} F,\n\\psi_{\\alpha^{(l)},z^{(l)}}\\rangle \\psi_{|\\alpha^{(l)}|,z^{(l)}}, \\quad l\\geq 1,\n$$\n and $\\psi_{\\alpha^{(l)},z^{(l)}}$ satisfies\n\\begin{align}\\label{max-cond}\n|\\langle R^{l}F, \\psi_{\\alpha^{(l)},z^{(l)}} \\rangle|=\n\\sup_{\\psi_{\\alpha,z}\\in \\mathcal D}{ |\\langle R^{l}F,\n\\psi_{\\alpha,z} \\rangle|}.\n\\end{align}\n\nFor our case, (\\ref{max-cond}) does not automatically come into existence.\nIn the following discussion,  we focus on the existence of $\\psi_{\\alpha^{(l)},z^{(l)}}$ in (\\ref{max-cond}) for each $l\\geq 1$.\nTo this end, we show the following result by using the technique in \\cite[Lemma 4.8]{WQ1}.\n\\begin{lem}\\label{MP-first-case}\nFor $F\\in H^2(T_{\\Gamma_1})$,\n\\begin{align}\\label{MP-case1}\n\\lim_{|\\alpha|\\to \\infty}\\frac{|\\langle F, \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0\n\\end{align}\nholds uniformly for $z\\in T_{\\Gamma_1}$.\n\\end{lem}\n\\begin{proof}\nAs shown in Theorem \\ref{third-app1}, for any $\\epsilon>0$, there exists $G_N=\\sum_{j=1}^Nc_j K(\\cdot, \\overline {w^{(j)}})$ such that\n$$\n||F-G_N||<\\frac{\\epsilon}{2}.\n$$\nTherefore,\nwe only need to prove that for any fixed $w=\\xi+i\\eta \\in T_{\\Gamma_1}$\n$$\n\\lim_{|\\alpha|\\to \\infty}\\frac{|\\langle K(\\cdot, \\overline w), \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}=0.\n$$\nIn fact, we have\n\\begin{align}\\label{MP-case1-1}\n\\begin{split}\n\\frac{|\\langle K(\\cdot, \\overline w), \\phi_{\\alpha, z} \\rangle|}{||\\phi_{\\alpha,z}||}&=\\prod_{j=1}^n \\frac{(2y_j)^{\\alpha_j+\\frac{1}{2}}\\alpha_j!}{|z_j-\\overline w_j|^{\\alpha_j+1}\\sqrt{(2\\alpha_j)!}}\\\\\n&\\leq \\prod_{j=1}^n \\frac{(2y_j)^{\\alpha_j+\\frac{1}{2}}\\alpha_j!}{(y_j+\\eta_j)^{\\alpha_j+1}\\sqrt{(2\\alpha_j)!}}\\\\\n&\\leq \\prod_{j=1}^n \\frac{((\\alpha_j+\\frac{1}{2})\\eta_j)^{\\alpha_j+\\frac{1}{2}}2^{2\\alpha_j+1}\\alpha_j!}{((\\alpha_j+1)\\eta_j)^{\\alpha_j+1}2^{\\alpha_j+1}\\sqrt{(2\\alpha_j)!}}\\\\\n&= \\prod_{j=1}^n \\frac{(\\alpha_j+\\frac{1}{2})^{\\alpha_j+\\frac{1}{2}}2^{\\alpha_j}\\alpha_j!}{(\\alpha_j+1)^{\\alpha_j+1}\\sqrt{\\eta_j}\\sqrt{(2\\alpha_j)!}}\\\\\n&\\leq \\prod_{j=1}^n C_j \\eta_j^{-\\frac{1}{2}} \\alpha_j^{-\\frac{1}{4}},\n\\end{split}\n\\end{align}\nwhere $C_j$ is a constant that is independent of $\\alpha_j$. The second inequality is based on the fact that $\\frac{(2y_j)^{\\alpha_j+\\frac{1}{2}}}{(y_j+\\eta_j)^{\\alpha_j+1}}$ arrives its maximum value at $y_j=2(\\alpha_j+\\frac{1}{2})\\eta_j$ for $ 1\\leq j\\leq n$, and the last inequality follows from the Stirling's formula\n$$\\Gamma(h+1)\\sim h^{h+\\frac{1}{2}} e^{-h}\\sqrt{2\\pi}, \\quad h\\in \\mathbb R, h\\to \\infty.$$\n The proof is completed.\n\\end{proof}\n\nCombining Lemma \\ref{MP-first-case} and Theorems \\ref{first-app1} - \\ref{third-app1}, we conclude the existence of $\\psi_{\\alpha^{(l)},z^{(l)}},l\\geq 1.$\nNote that\n$$\n||F||^2=\\sum_{l=1}^m |\\langle R^l F,\n\\psi_{\\alpha^{(l)},z^{(l)}}\\rangle|^2 + ||R^{m+1}F||^2,\n$$\nalthough $\\{\\psi_{\\alpha^{(l)},z^{(l)}}, l=1,2...,m\\}$ is not an orthogonal system. Based on Theorem 1 in \\cite{MZ} and the fact that $\\overline {{\\text span} \\mathcal D}=H^2(T_{\\Gamma_1})$, we have\n\\begin{align}\\label{convergence-MP}\n\\lim_{m\\to \\infty}||R^mF||=0.\n\\end{align}\nFor further discussion on such approximation and greedy algorithm, please see e.g. \\cite{WQ1,DT, MZ,Te}.\n\\end{remark}\n\n\\section{Further Results}\nIn this section, several relevant problems of AFD-type approximation are explored. The rate of convergence is always a point of interest when a certain approximation is considered. Hence, we will show the rate of convergence of AFD-type approximation. We will also give rational approximation of functions in $L^2(\\mathbb R^n)$ based on the known results that describe the relationship between $L^2(\\mathbb R^n)$ and $H^2(T_{\\Gamma_1})$. Lastly, we will explore the AFD-type approximation in $H^2(T_{\\Gamma})$, where $\\Gamma$ is a regular cone.\n\\subsection{Rate of convergence}\nAs in \\cite{DT}, we first introduce the function class\n$$\nH^2(T_{\\Gamma_1},M)= \\left \\{F\\in H^2(T_{\\Gamma_1}): F=\\sum_{j=1}^\\infty c_j\\frac{\\phi_{w^{(j)}}}{||\\phi_{w^{(j)}}||}, w^{(j)}\\in T_{\\Gamma_1}, \\sum_{j=1}^\\infty |c_j|\\leq M \\right \\}.\n$$\nWe give the convergent rate of AFD-type approximation of functions in $H^2(T_{\\Gamma_1},M)$.\nThe result is stated as follows.\n\\begin{thm}\\label{con-rate}\nFor $F\\in H^2(T_{\\Gamma_1},M)$, and $F_{A_m}^*$ corresponding to the sequence $\\{z^{(k)}\\}_{k=1}^m$, where each element of $\\{z^{(k)}\\}_{k=1}^m$ is selected by the principle $(\\ref{min_pro_revise})$, we have\n$$\n\\| F-F^*_{A_m} \\|\\leq \\frac{M}{\\sqrt{m}}.\n$$\n\\end{thm}\n\nTo prove Theorem \\ref{con-rate}, we need the following result.\n\\begin{lem}[\\cite{DT}]\\label{DT-lem}\nLet $\\{d_k\\}_{k=1}^\\infty$ be a sequence of nonnegative numbers satisfying\n$$\nd_1\\leq A, \\quad d_{k+1}\\leq d_k\\left(1-\\frac{d_k}{A}\\right).\n$$\nThen there holds\n$$\nd_k\\leq \\frac{A}{k}.\n$$\n\\end{lem}\n{\\em Proof of Theorem \\ref{con-rate}:}\\\\\nFor $F\\in H^2(T_{\\Gamma_1}, M)$, we have $F=\\sum_{k=1}^\\infty c_k\\frac{\\phi_{w^{(k)}}}{\\|\\phi_{w^{(k)}}\\|}$ and\n$$\n||F||\\leq \\sum_{j=1}^\\infty |c_k|\\leq M.\n$$\nBy (\\ref{general_F}), we have\n$$\n\\|F_{A_{m}}^*\\|^2 =\\sum_{k=1}^{m} |\\langle F, \\mathcal B_k \\rangle|^2,\n$$\nand\n\\begin{align}\\label{F_m}\n\\|F_{m+1}\\|^2= \\|F_m\\|^2-|\\langle F, \\mathcal B_m \\rangle|^2=\\|F_m\\|^2-|\\langle F_m, \\mathcal B_m \\rangle|^2,\n\\end{align}\nwhere $F_{m+1}=F-F_{A_m}^*$.\nBy (\\ref{G-S}),\n\\begin{align}\\label{F_mB_m}\n\\begin{split}\n|\\langle F_m,\\mathcal B_m\\rangle| &= \\frac{|\\langle F_m, {\\gamma_m}\\rangle|}{\\|\\gamma_m\\|}\\\\\n& = \\frac{|\\langle F_m, \\Phi_{z^{(m)}} - \\sum_{k=1}^{m-1}\\langle \\Phi_{z^{(m)}}, \\mathcal B_k\\rangle\\mathcal B_k\\rangle|}{\\|\\Phi_{z^{(m)}} - \\sum_{k=1}^{m-1}\\langle \\Phi_{z^{(m)}}, \\mathcal B_k\\rangle\\mathcal B_k\\|}\\\\\n& = \\frac{|\\langle F_m, \\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|}\\rangle|}{\\|\\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|} - \\sum_{k=1}^{m-1}\\langle \\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|}, \\mathcal B_k\\rangle\\mathcal B_k\\|}\\\\\n&\\geq |\\langle F_m, \\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|}\\rangle|\\\\\n&\\geq |\\langle F_m, \\frac{\\phi_{z^{(m)}}}{\\|\\phi_{z^{(m)}}\\|}\\rangle|,\n\\end{split}\n\\end{align}\nwhere the first inequality is based on $$\\|\\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|} - \\sum_{k=1}^{m-1}\\langle \\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|}, \\mathcal B_k\\rangle\\mathcal B_k\\|^2 = 1-\\sum_{k=1}^{m-1}|\\langle \\frac{\\Phi_{z^{(m)}}}{\\|\\Phi_{z^{(m)}}\\|},\\mathcal B_k\\rangle|^2\\leq 1.$$\nCombining (\\ref{min_pro_revise}), (\\ref{F_m}) and (\\ref{F_mB_m}), we have\n\\begin{align}\\label{F_mB_m1}\n\\begin{split}\n|\\langle F_m, \\mathcal B_m \\rangle| &= \\sup_{z\\in T_{\\Gamma_1}}|\\langle F_m, \\mathcal B_{\\{z^{(1)},z^{(2)},...,z^{(m-1)},z\\}} \\rangle|\\\\\n&\\geq \\sup_{z\\in T_{\\Gamma_1}}|\\langle F_m, \\frac{\\phi_{z}}{\\|\\phi_{z}\\|} \\rangle|\\\\\n&\\geq \\sup_{z\\in \\{w^{(k)}\\}_{k=1}^\\infty}|\\langle F_m, \\frac{\\phi_{z}}{\\|\\phi_{z}\\|} \\rangle|.\\\\\n\\end{split}\n\\end{align}\nNotice that\n\\begin{align*}\n\\|F_m\\|^2&=|\\langle F_m, F\\rangle|\n=|\\langle F_m, \\sum_{k=1}^\\infty c_k\\frac{\\phi_{w^{(k)}}}{\\|\\phi_{w^{(k)}}\\|}\\rangle|\n\\leq M\\sup_{z\\in \\{w^{(k)}\\}_{k=1}^\\infty} |\\langle F_m, \\frac{\\phi_{z}}{\\|\\phi_{z}\\|}\\rangle|.\n\\end{align*}\nHence,\n\\begin{align*}\n\\|F_{m+1}\\|^2&=\\|F_m\\|^2-|\\langle F_m, \\mathcal B_m\\rangle|^2\\\\\n&\\leq \\|F_m\\|^2-\\sup_{z\\in \\{w^{(k)}\\}_{k=1}^\\infty}|\\langle F_m, \\frac{\\phi_{z}}{\\|\\phi_{z}\\|} \\rangle|^2\\\\\n&\\leq \\|F_m\\|^2-\\frac{\\|F_m\\|^4}{M^2}\\\\\n&=\\|F_m\\|^2\\left(1-\\frac{\\|F_m\\|^2}{M}\\right).\n\\end{align*}\nBy Lemma \\ref{DT-lem}, we conclude the desired result.\n\\quad \\hfill$\\Box$\\vspace{2ex}\n\n\\subsection{Rational approximation of functions in $L^2(\\mathbb R^n)$}\nIt is known that, for $f\\in L^2(\\mathbb R)$, one can have $f=f^+ + f^-$,  where $f^+$ and $f^-$ are non-tangential boundary limits of functions contained in $H^2(\\mathbb C_+)$ and $H^2(\\mathbb C_-)$, respectively. Then, rational approximation of functions in $L^2(\\mathbb R)$ can be easily obtained by rational approximations of functions in $H^2(\\mathbb C_+)$ and $H^2(\\mathbb C_-)$. Here we give rational approximation of functions in $L^2(\\mathbb R^n)$ in a similar manner.\nDefine $\\sigma_j=(\\sigma_j(1),\\sigma_j(2),...,\\sigma_j(n)),1\\leq j\\leq 2^n$, whose elements are $+$ and $-$,\nand $$\\Gamma_{\\sigma_j}=\\{y\\in \\mathbb R^n;y_k>0 \\text{ if } \\sigma_j(k)=+ \\text{ and } y_k<0 \\text{ if } \\sigma_j(k)=-, j=1,2,...,n\\}.$$\nObserve that\n $\\mathbb R^n=\\cup_{j=1}^{2^n}\\overline {\\Gamma_{\\sigma_j}}. $\nFor $F\\in L^2(\\mathbb R^n)$, the following result is known.\n\\begin{thm}[\\cite{P}]\\label{Hardy-dec-pro}\nFor $F\\in L^2(\\mathbb R^n),$ if\n\\begin{align}\\label{Hardy-projection}\n\\begin{split}\nF_{\\sigma_j}(z)=\\int_{\\mathbb R^n}F(\\xi)\\overline{K_{\\Gamma_{\\sigma_j}}(\\xi, \\overline z)}\nd\\xi=\\frac{(-1)^{m_j}}{(2\\pi i)^n}\\int_{\\mathbb R^n}F(\\xi)\\prod_{k=1}^n\n\\frac{1}{\\xi_k-z_k}d\\xi_1\\cdots d\\xi_n, \\quad z\\in T_{\\Gamma_{\\sigma_j}},\n\\end{split}\n\\end{align}\nwhere $m_j$ denotes the number of minus signs in $\\sigma_j$,\nthen $F_{\\sigma_j}(z)$ is holomorphic on $T_{\\Gamma_{\\sigma_j}}$, and for $F_{\\sigma_j}(x+iy)$ as a function of $x$,\n\\begin{align}\\label{Riesz-inequality}\n\\Vert F_{\\sigma_j}(\\cdot+iy) \\Vert_{L^2(\\mathbb R^n)}\\leq C \\Vert F\\Vert_{L^2(\\mathbb R^n)},\n\\end{align}\nwhere $C$ is a constant that is independent of $F$ and $y$.\\\\\nFurthermore,\n\\begin{align}\\label{Hardy-decomposition}\nF(x)=\\sum_{j=1}^{2^n}F_{\\sigma_j}(x),\\quad x\\in \\mathbb R^n, \\text{ in the } L^2 \\text{-sense,}\n\\end{align}\nwhere $F_{\\sigma_j}(x)=\\lim_{y\\in \\Gamma_{\\sigma_j},y\\to 0}F_{\\sigma_j}(x+iy)$ is the limit function in the $L^2$-norm.\n\\end{thm}\n\\begin{remark}\nIt is noted that Theorem \\ref{Hardy-dec-pro} is the summary of the results given in \\cite{P}. In fact, the result given in Theorem \\ref{Hardy-dec-pro} holds for $F\\in L^p(\\mathbb R^n), 1<p<\\infty$ (see \\cite{P}). For more information, we also refer to e.g. \\cite{Ti,Till,Vl}. Based on the definition of the Hardy spaces, (\\ref{Riesz-inequality}) implies that $F_{\\sigma_j}\\in H^2(T_{\\Gamma_{\\sigma_j}})$. (\\ref{Hardy-projection}) is called the Hardy projection of $F$. Hence, (\\ref{Hardy-decomposition}) means that $F$ can be decomposed into a sum of boundary limit functions of functions in the Hardy spaces on tubes over octants. Moreover, for $F\\in L^p(\\mathbb R^n), 1<p<\\infty$, $F_{\\sigma_j}(x)$ can be characterized as the Fourier transform of a function supported on $\\Gamma_{\\sigma_j}$ in distribution sense (see \\cite{P}). This can be regard as a generalization of the Paley-Wiener theorem (see Theorem \\ref{PW}). Recently, in \\cite{DLQ} the analogues of the Paley-Wiener theorem have been given for $H^p(T_\\Gamma), 1\\leq p\\leq\\infty$ (for $2<p\\leq\\infty$, the results are shown in distribution sense), where $\\Gamma$ is a regular cone.\n\\end{remark}\n\nDue to Theorem \\ref{Hardy-dec-pro} and the above discussion, we can reduce the relevant study to $F_{\\sigma_j}(1\\leq j\\leq 2^n$) when considering the rational approximation of $F\\in L^2(\\mathbb R^n)$. Therefore, for each $F_{\\sigma_j}$ we can obtain the AFD-type approximation of $F_{\\sigma_j}$. Although the AFD-type approximation is established in the first octant, one can apply it to the other octants without conflicts. Moreover, for the real-valued function $F$ we only need to deal with the related $2^{n-1}$ Hardy spaces.\nFor instance, we interpret this in $\\mathbb C^2$. If $\\{z^{(k)}=(z^{(k)}_1,z^{(k)}_2)\\}_{k=1}^\\infty$ is the sequence such that $F_{A_m,+,+}^{*}\\to F_{+,+}$ as $m\\to \\infty $ in $H^2(T_{\\Gamma_{+,+}})$, then we have that $\\{(\\overline {z^{(k)}_1},\\overline {z^{(k)}_2})\\}_{k=1}^\\infty$ is the sequence such that $F_{A_m,-,-}^{*}\\to F_{-,-}$ as $m\\to \\infty$ in $H^2(T_{\\Gamma_{-,-}})$. Similar conclusions can be drawn for the cases $F_{+,-}$ and $F_{-,+}$.\n\n\\subsection{AFD-type Approximation in $H^2(T_\\Gamma)$}\nIn this part, we will investigate the AFD-type approximation in $H^2(T_\\Gamma)$, where $\\Gamma$ is a regular cone. Based on the discussions in Section 3, we know that Theorems \\ref{first-app1} - \\ref{third-app1} play an important role in studying the AFD-type approximation. The techniques used in Theorem \\ref{thm1} and Lemma \\ref{re-ex-lem} still work for the proposed approximation in $H^2(T_{\\Gamma})$. Therefore, for a regular cone, we only need to consider the analogous results of Theorems \\ref{first-app1} - \\ref{third-app1}. It is difficult to show that the BVC holds in $H^2(T_\\Gamma)$ since we do not have the explicit formula of Cauchy-Szeg\\\"o kernel. However, as mentioned in Section $3$, the weak BVC is sufficient for us to obtain the AFD-type approximation. In this part, under the assumption that (\\ref{CS-infty}) holds, we will give certain properties of the boundary behaviors of functions in $H^p(T_{\\Gamma}),1<p<\\infty$, which can be regarded as special cases of the analogous results of Theorems \\ref{first-app1} - \\ref{third-app1}. When $p=2$, such property is indeed the weak BVC.\n\nWe first recall the definition of regular cones. Open cones are nonempty open subsets $\\Gamma\\in \\mathbb R^n$ satisfying\n(1) $0\\not \\in \\Gamma$, and (2) whenever $x,y \\in \\Gamma$ and $a, b>0$ then $a x +b y \\in \\Gamma.$\nA closed cone is the closure of an open cone. It is clear that if $\\Gamma$ is an open cone then $\\Gamma^{*}=\\{x\\in \\mathbb R^n; x\\cdot t \\geq 0, t\\in \\Gamma\\}$ is closed. Moreover, if $\\Gamma^{*}$ has a non-void interior, then it is a closed cone. In this case we say that $\\Gamma$ is a regular cone, and $\\Gamma^{*}$ is called the cone dual to $\\Gamma.$ Recall that the dual cone of  $\\Gamma_1$ is $\\Gamma_1^*=\\overline \\Gamma_1$. For more information on regular cones, see e.g. \\cite{JFAK}.\n\nWe still denote by $K(w,\\overline z)$ the Cauchy-Szeg\\\"o kernel of $H^2(T_\\Gamma)$. Using the notation $K_\\Gamma(w,\\overline z)$ if we want to emphasize the Cauchy-Szeg\\\"o kernel that corresponds to $\\Gamma$. We need the following results for preparation.\n\\begin{thm}[{\\cite[Theorem 5.6]{SW}}]\\label{g-rf-p}\nSuppose $\\Gamma$ is a regular cone in $\\mathbb R^n$, and $F\\in H^p(T_{\\Gamma}), 1\\leq p<\\infty$, then\n$$\n\\lim_{y\\in\\Gamma,y\\to 0}\\int_{\\mathbb R^n}|F(x+iy)-F(x)|^pdx=0,\n$$\nand\n$$\nF(x+iy)=\\int_{\\mathbb R^n}F(t)P_y(x-t)dt,\n$$\nwhere $F(x)$ is the limit function, whose existence is in the sense of that in \\cite[Theorem 5.5]{SW}.\n\\end{thm}\n\n\\begin{lem}\nSuppose that $\\Gamma$ is a regular cone in $\\mathbb R^n$. For $z=x+iy\\in T_{\\Gamma}$ and $1<p\\leq \\infty$, we have\n\\begin{align}\\label{P-S-estimate}\n\\|P_y(x-\\cdot)\\|_{L^p(\\mathbb R^n)}\\leq \\frac{1}{2^{\\frac{n}{p}}}K(z,\\overline z)^{1-\\frac{1}{p}}.\n\\end{align}\n\\end{lem}\n\\begin{proof}\nFor $2<p<\\infty$ and $z=x+iy\\in T_{\\Gamma}$, we have\n\\begin{align}\n\\int_{\\mathbb R^n}|K(\\xi,\\overline z)|^p d\\xi =\\sup_{\\xi\\in \\mathbb R^n}|K(\\xi,\\overline z)|^{p-2}\\int_{\\mathbb R^n}|K(\\xi,\\overline z)|^2d\\xi \\leq \\frac{1}{2^n}K(z,\\overline z)^{p-2}K(z,\\overline z)=\\frac{1}{2^n}K(z,\\overline z)^{p-1}\n\\end{align}\nHence, for $1<p<\\infty,$\n\\begin{align}\n\\int_{\\mathbb R^n}|P_y(x-\\xi)|^pd\\xi\\leq \\int_{\\mathbb R^n}\\frac{|K(\\xi,\\overline z)|^{2p}}{K(z,\\overline z)^p}d\\xi\\leq \\frac{1}{2^n}K(z,\\overline z)^{2p-1-p}=\\frac{1}{2^n}K(z,\\overline z)^{p-1}.\n\\end{align}\nObviously, when $p=\\infty$, we have $\\|P_y(x-\\cdot)\\|_{L^\\infty(\\mathbb R^n)}\\leq K(z,\\overline z)$.\n\\end{proof}\n\nUnlike the first octant, the estimates of $K(z,\\overline z)$ at points of $\\partial \\Gamma$ and at infinity are not obvious. The next lemma gives the estimate of $K(z,\\overline z)$ on $\\partial \\Gamma$.\n\\begin{lem}[{\\cite[Lemma 2]{AK},\\cite[Proposition I.3.2]{JFAK}}]\\label{AK_lemma}\nLet $\\Gamma$ be a regular cone in $\\mathbb R^n, n\\geq 3$, and $\\beta \\in \\partial \\Gamma$. Then\n\\begin{align}\\label{AK_formula}\n\\lim_{y\\in \\Gamma, y\\to \\beta}\\int_{\\Gamma^{*}}e^{-4y\\cdot t} dt=\n\\infty.\n\\end{align}\n\\end{lem}\n\\begin{remark} It is obvious that $(\\ref{AK_formula})$ is true when $n=1$. For $n=2,$ we only need to verify that (\\ref{AK_formula}) holds for a special class of regular cones in $\\mathbb R^2$\n\\begin{align}\\label{2-cone}\n\\Gamma^\\kappa=\\{y\\in \\mathbb R^2; |y_1|<\\kappa y_2\\}, \\quad 0<\\kappa<\\infty.\n\\end{align}\nChecking that the dual cone of $\\Gamma^\\kappa$ is\n$$\n\\Gamma^{\\kappa,*}=\\{t\\in \\mathbb R^2; |t_1|<\\frac{1}{\\kappa}t_2\\}\n$$\nis easy.\nBy direct calculation, we have $K_{\\Gamma^\\kappa}(z,\\overline z)=\\int_{\\Gamma^{\\kappa,*}}e^{-4\\pi y\\cdot t}dt=\\frac{\\kappa}{8\\pi^2(\\kappa^2y_2^2-y_1^2)}$. Obviously, for $\\beta\\in \\partial \\Gamma^\\kappa$, $\\lim_{y\\in \\Gamma^\\kappa, y\\to \\beta}K_{\\Gamma^\\kappa}(z,\\overline z)=\\infty$. In the following discussion, we write elements in $\\Gamma$ as column vectors.\nIf $\\Gamma$ is a regular cone in $\\mathbb R^2$, then, for $y\\in\n\\Gamma$, there exists a matrix $P\\in SO(2,\\mathbb R)=\\{Q\\in\nGL(2,\\mathbb R); QQ^{T}=Q^{T}Q=I, |Q|=1\\}$ and $\\Gamma^{\\kappa}$\nsuch that\n$$\n(y_1, y_2)^{T}= P(\\xi_1, \\xi_2)^{T}, \\quad\n\\xi=(\\xi_1, \\xi_2)^T\\in \\Gamma^{\\kappa}.\n$$\nThen we\nhave\n$\nK(z,\\overline z)= \\int_{\\Gamma^{*}}e^{-4\\pi y\\cdot t}dt\n= |P|\\int_{\\Gamma^{\\kappa,*}}e^{-4\\pi (P\\xi)\\cdot (Pt^{\\prime})}dt^{\\prime}\n = \\int_{\\Gamma^{\\kappa,*}}e^{-4\\pi \\xi\\cdot\nt^{\\prime}}dt^{\\prime}.\n$\n Since $P$ is nonsingular, $\\xi\\to \\partial \\Gamma^\\kappa$ as $y\\to \\partial \\Gamma$. Hence,\n $\n\\lim_{y\\in\\Gamma, y\\to \\beta} K(z,\\overline z)=\\infty.\n $\n\\end{remark}\n\\medskip\n\nWhat we are concerned with the estimate of Cauchy-Szeg\\\"o kernel at infinity is whether\n\\begin{align}\\label{CS-infty}\\lim_{\\widetilde y\\in \\widetilde{\\Gamma},|\\widetilde y|\\to\\infty}K(x+i\\widetilde y,\\overline{x+i\\widetilde y})=0\\end{align}\nholds, where $\\widetilde \\Gamma=\\overline{y^\\prime+\\Gamma}, y^\\prime\\in \\Gamma.$\nFor such estimate, we first consider the following two classes of cones. One of them is the polygonal cones that are the interior of the convex hull of a finite number of rays meeting at origin. If a polygonal cone is the interior of the convex hull of a finite number of $n$ linearly independent rays meeting at origin, we call it $n$-sided polygonal cone.  A polygonal cone indeed is a finite union of $n$-sided polygonal cones. The other one is the circular cones. We only need to consider the circular cones are of the form $\\Gamma=\\{y\\in\\mathbb R^n;\\sqrt{\\sum_{j=1}^{n-1}|y_j|^2}<\\kappa y_n\\},\\kappa>0,$ since we can move a general circular cone to this form after a rotation. In fact, these two classes of cones can be regarded as two different generalizations of $\\Gamma^\\kappa$ in $\\mathbb R^n,n\\geq 3.$ For simplicity, we show that (\\ref{CS-infty}) holds for $\\Gamma^\\kappa$ by two different ways. One can easily conclude that $(\\ref{CS-infty})$ holds for the polygonal and the circular cones.\n\nWe first show the way that can be utilized in proving that (\\ref{CS-infty}) holds for the circular cones. As shown previously, for $y\\in \\overline {\\Gamma^\\kappa}$,\n\\begin{align*}\nK_{\\Gamma^\\kappa}(x+i(y+y^\\prime),\\overline {x+i(y+y^\\prime)}) &=\\frac{\\kappa}{8\\pi^2(\\kappa^2(y_2+y_2^\\prime)^2-(y_1+y_1^\\prime)^2)}\\\\\n&=\\frac{\\kappa}{8\\pi^2[\\kappa(y_2+y_2^\\prime)-|y_1+y_1^\\prime|][\\kappa(y_2+y_2^\\prime)+|y_1+y_1^\\prime|]}\\\\\n&\\leq \\frac{\\kappa}{8\\pi^2\\delta_0[\\kappa(y_2+y_2^\\prime)+|y_1+y_1^\\prime|]},\n\\end{align*}\nwhere $\\delta_0=dist(\\widetilde{\\Gamma^\\kappa},\\Gamma^{\\kappa,c})>0$, and $\\Gamma^{\\kappa,c}$ is the complement of $\\Gamma^\\kappa$. Then, we get the desired conclusion.\nNote that there exist a linear transformation $Q$ that maps the first octant onto ${\\Gamma^{\\kappa}}$ (see e.g. \\cite{SW, Rudin}).\nHence we have\n$$\nK_{\\Gamma^\\kappa}(x+i\\widetilde y, \\overline{x+i\\widetilde y})=\\int_{\\Gamma^{\\kappa,*}} e^{-4\\pi\\widetilde y\\cdot t} dt = \\frac{1}{|Q|}\\int_{\\overline\\Gamma_1}e^{-4\\pi\\widetilde\\xi\\cdot t^\\prime} dt^\\prime= \\frac{1}{|Q|}K_{\\Gamma_1}(i\\widetilde\\xi,\\overline{\ni\\widetilde \\xi}).\n$$\nSince $|\\widetilde\\xi|\\to\\infty$ as $|\\widetilde y|\\to \\infty$, we conclude the desired result again. Since the interior of the dual cone of a polygonal cone $\\Gamma$ is polygonal, there exist $n$-sided polygonal cones $\\Gamma_{(k)},k=1,...,N,$ such that\n$$K_{\\Gamma}(w,\\overline z)=\\sum_{k=1}^N \\int_{\\Gamma_{(k)}^*}e^{-4\\pi(w-\\overline z)\\cdot t}dt=\\sum_{k=1}^N K_{\\Gamma_{(k)}}(w,\\overline z),$$\nwhere $\\Gamma^*=\\cup_{k=1}^N \\Gamma_{(k)}^*$, and for each $\\Gamma_{(k)}$ there exists a linear transformation mapping the first octant onto $\\Gamma_{(k)}$. Therefore, we can easily get that (\\ref{CS-infty}) holds for the polygonal cones.\n\nIn the following part, we would like to discuss whether (\\ref{CS-infty}) holds in general. For general regular cones, (\\ref{CS-infty}), however, does not seem to be so obvious. The following lemma gives a partial answer to this question.\n\n\\begin{lem}\\label{infty-lem}\nSuppose that $\\Gamma_0$ is a regular cone whose closure is contained in $\\Gamma\\cup \\{0\\}$. Then\n$$\n\\lim_{y\\in \\overline \\Gamma_0,|y|\\to \\infty}K(iy,\\overline {iy})=0.\n$$\n\n\\end{lem}\n\\begin{proof}\nWe first show that\n\\begin{align*}\n\\lim_{y\\in \\overline\\Gamma_0,|y|\\to\\infty}e^{-4\\pi y\\cdot t}=0.\n\\end{align*}\n\nWe claim that if $\\eta\\in \\overline \\Gamma_0$ and $t\\in \\Gamma^{*}$, then there\nexists a $\\delta>0$ such that $\\delta|\\eta||t|\\leq \\eta\\cdot t$. Denote by $\\Sigma$ the set $\\{\\xi\\in\\mathbb R^n;|\\xi|=1\\}$. Define a\nfunction $H(\\eta, t)=\\eta\\cdot t$, $\\eta\\in \\overline \\Gamma_0\\cap \\Sigma, t\\in\n\\Gamma^{*}\\cap \\Sigma$. From the definition of $\\Gamma^{*}$ and $\\overline\\Gamma_0$\nwe have $0< \\eta\\cdot t$. Since $\\overline\\Gamma_0\\cap \\Sigma$ and\n$\\Gamma^{*} \\cap \\Sigma$ are both compact, the existence of $\\delta\n>0$ follows from the fact that $0<\\eta\\cdot t=H(\\eta,t)$ and $H(\\eta,t)$ is a\ncontinuous function. Consequently, we have\n$$\n\\lim_{y\\in \\overline\\Gamma_0, |y|\\to \\infty}e^{-4\\pi y\\cdot t}\\leq \\lim_{y\\in \\overline\\Gamma_0,\n|y|\\to \\infty}e^{-4\\pi \\delta |y||t|}=0, \\quad t\\in \\Gamma^{*}\n$$\nand $$e^{-4\\pi\\delta |y||t|}\\leq e^{-4\\pi\\delta |t|},\\quad |y|\\geq 1,$$ where $\\int_{\\Gamma^*}e^{-4\\pi\\delta|t|}dt<\\infty.$\nTherefore, by the Lebesgue's dominated convergence theorem we have\n$$\n\\lim_{y\\in \\overline\\Gamma_0,|y|\\to\\infty}K(iy,\\overline{iy})=0.\n$$\n\\end{proof}\n\n\\begin{remark}\nOn one hand, the argument used in Lemma \\ref{infty-lem} can not be applied to $\\widetilde\\Gamma$ since the key point of such argument is that the union of all dilations of $\\overline \\Gamma_0\\cap\\Sigma$ is $\\overline\\Gamma_0$ while this is not the fact of $\\widetilde\\Gamma\\cap \\Sigma.$ On the other hand, Lemma \\ref{infty-lem} shows that (\\ref{CS-infty}) holds in most situations. The unsolved situation can be almost concluded as that $\\widetilde y\\in \\partial \\widetilde \\Gamma,|\\widetilde y|\\to \\infty.$ The case that $ \\widetilde y $ meets $\\partial \\widetilde\\Gamma$ at infinity can be considered as a special case of the above unsolved situation. The formula (\\ref{CS-infty}) should hold for more cones other than those discussed in this paper. For instance, one can easily check that (\\ref{CS-infty}) holds for the symmetric cone in $\\mathbb R^3$ given by $\\{y=(y_1,y_2,y_3)\\in\\mathbb R^3; y_1>0,y_1y_2-y_3^2>0 \\}$ (see \\cite{BBGNPR}).\n\\end{remark}\n\nUnder the assumption that (\\ref{CS-infty}) holds, the main result of this part is stated as follows.\n\\begin{thm}\\label{g-cone}\nSuppose that $\\Gamma$ is a regular cone such that (\\ref{CS-infty}) holds. For $F\\in H^p(T_\\Gamma), 1< p<\\infty,$ and $z=x+iy\\in T_{\\Gamma}$, we have the following results.\n\\begin{align}\\label{case1-general-cone}\n\\lim_{y\\in \\Gamma,y\\to\\beta}\\frac{|F(z)|}{K(z,\\overline z)^{\\frac{1}{p}}}=0\n\\end{align}\nholds uniformly for $x\\in \\mathbb R^n$, where $\\beta\\in \\partial \\Gamma$.\n\\begin{align}\\label{case2-general-cone}\n\\lim_{y\\in \\Gamma, |y|\\to \\infty}\\frac{|F(z)|}{K(z,\\overline z)^{\\frac{1}{p}}}=0\n\\end{align}\nholds uniformly for $x\\in \\mathbb R^n$.\n\\begin{align}\\label{case3-general-cone}\n\\lim_{|x|\\to \\infty}\\frac{|F(z)|}{K(z,\\overline z)^{\\frac{1}{p}}}=0\n\\end{align}\nholds uniformly for $y\\in \\Gamma$.\\\\\nIn particular, we do not need the assumption (\\ref{CS-infty}) for all regular cones in $\\mathbb R^2$, the polygonal and the circular cones in $\\mathbb R^n.$\n\\end{thm}\n\\begin{proof}\nBy Theorem \\ref{g-rf-p}, we have\n$$\nF(z)=\\int_{\\mathbb R^n}F(\\xi)P_y(x-\\xi)d\\xi\n$$\nwhere $F(\\xi)\\in L^p(\\mathbb R^n)$.\\\\\nTherefore, for any $\\epsilon>0$, we can find $G\\in L^r(\\mathbb R^n)\\cap L^p(\\mathbb R^n), p<r<\\infty,$ such that $$\\|F-G\\|_{L^p(\\mathbb R^n)}<\\epsilon.$$\nHence, for $q$ satisfying $\\frac{1}{p}+\\frac{1}{q}=1$ and $h$ satisfying $\\frac{1}{r}+\\frac{1}{h}=1$,\n\\begin{align}\\label{g-cone-fact}\n\\begin{split}\n\\frac{|F(z)|}{K(z,\\overline z)^{\\frac{1}{p}}}&\\leq \\frac{\\int_{\\mathbb R^n}|F(\\xi)-G(\\xi)|P_y(x-\\xi)d\\xi}{K(z,\\overline z)^{\\frac{1}{p}}}+\\frac{\\int_{\\mathbb R^n}|G(\\xi)|P_y(x-\\xi)d\\xi}{K(z,\\overline z)^{\\frac{1}{p}}}\\\\\n&\\leq \\|F-G\\|_{L^p(\\mathbb R^n)}\\frac{\\|P_y(x-\\cdot)\\|_{L^q(\\mathbb R^n)}}{K(z,\\overline z)^{\\frac{1}{p}}}+\\|G\\|_{L^r(\\mathbb R^n)}\\frac{\\|P_y(x-\\cdot)\\|_{L^h(\\mathbb R^n)}}{K(z,\\overline z)^{\\frac{1}{p}}}\\\\\n&< \\epsilon\\frac{\\frac{1}{2^{\\frac{n}{q}}}K(z,\\overline z)^{1-\\frac{1}{q}}}{K(z,\\overline z)^{\\frac{1}{p}}}+\\|G\\|_{L^r(\\mathbb R^n)}\\frac{\\frac{1}{2^{\\frac{n}{h}}}K(z,\\overline z)^{1-\\frac{1}{h}}}{K(z,\\overline z)^{\\frac{1}{p}}}\\\\\n&= \\frac{\\epsilon}{{2^{\\frac{n}{q}}}}+\\frac{\\|G\\|_{L^r(\\mathbb R^n)}}{2^{\\frac{n}{h}}}K(z,\\overline z)^{\\frac{1}{r}-\\frac{1}{p}},\n\\end{split}\n\\end{align}\nwhere the second inequality follows from H\\\" older's inequality.\\\\\nBy Lemma \\ref{AK_lemma} and the fact that $\\frac{1}{r}-\\frac{1}{p}<0$, we have $K(z,\\overline z)^{\\frac{1}{r}-\\frac{1}{p}}\\to 0$ as $y\\to \\beta$. Therefore, we complete the proof of (\\ref{case1-general-cone}).\n\nWe know that\n$$\nF(z)=\\int_{\\mathbb R^n} F(\\xi) P_y(x-\\xi)d\\xi.\n$$\nThen, for any given $\\epsilon>0$, we can find $y^\\prime\\in \\Gamma$ such that\n$$\n\\int_{\\mathbb R^n} |F(\\xi+iy^\\prime)-F(\\xi)|^p d\\xi <\\epsilon.\n$$\nSo\n\\begin{align*}\n\\frac{|F(z)|}{{K(z,\\overline z)}^{\\frac{1}{p}}} &= \\frac{|\\int_{\\mathbb R^n} \\left(F(\\xi)-F(\\xi+iy^\\prime)\\right)P_y(x-\\xi)d\\xi|+|\\int_{\\mathbb R^n} F(\\xi+iy^\\prime)P_y(x-\\xi)d\\xi|}{{K(z,\\overline z)}^{\\frac{1}{p}}}\\\\\n& < \\frac{\\epsilon}{2^{\\frac{n}{q}}}+ \\frac{|\\int_{\\mathbb R^n} F(\\xi)P_{y+y^\\prime}(x-\\xi)d\\xi|}{K(z,\\overline z)^{\\frac{1}{p}}}.\n\\end{align*}\nSimilar to (\\ref{g-cone-fact}), we have\n\\begin{align}\\label{g-cone-fact2-ineq}\n\\begin{split}\n\\frac{|\\int_{\\mathbb R^n} F(\\xi)P_{y+y^\\prime}(x-\\xi)d\\xi|}{K(z,\\overline z)^{\\frac{1}{p}}}\\leq \\frac{\\epsilon}{2^\\frac{n}{q}} \\frac{K(z+iy^\\prime,\\overline{z+iy^\\prime})^{1-\\frac{1}{q}}}{K(z,\\overline z)^{\\frac{1}{p}}}+\\frac{||G||_{L^r(\\mathbb R^n)}}{2^\\frac{n}{h}}\\frac{K(z+iy^\\prime,\\overline{z+iy^\\prime})^{1-\\frac{1}{h}}}{K(z,\\overline z)^{\\frac{1}{p}}},\n\\end{split}\n\\end{align}\nwhere $G\\in L^r(\\mathbb R^n)\\cap L^p(\\mathbb R^n), 1<r<p$, $\\frac{1}{p}+\\frac{1}{q}=1$ and $\\frac{1}{r}+\\frac{1}{h}=1$.\\\\\nNotice that $K(z+iy^\\prime,\\overline{z+iy^\\prime})<K(z,\\overline z)$. Hence, the first term in (\\ref{g-cone-fact2-ineq}) is strictly less than $\\frac{\\epsilon}{2^{\\frac{n}{q}}}$. For the second term in (\\ref{g-cone-fact2-ineq}), by $K(z+iy^\\prime,\\overline{z+iy^\\prime})<K(z,\\overline z),$ $\\lim_{y\\in\\Gamma,|y|\\to \\infty}K(z+iy^\\prime,\\overline {z+iy^\\prime})=0$ and $\\frac{1}{r}-\\frac{1}{p}>0,$ we can easily conclude that, for $|y|$ large enough,\n$$\n\\frac{K(z+iy^\\prime,\\overline{z+iy^\\prime})^{1-\\frac{1}{h}}}{K(z,\\overline z)^{\\frac{1}{p}}}<\\epsilon.\n$$\n\nBy the above discussions, the proof of (\\ref{case2-general-cone}) is completed.\\\\\n To prove (\\ref{case3-general-cone}), because of (\\ref{case1-general-cone}) and (\\ref{case2-general-cone}), we only need to show that\n \\begin{align}\\label{g-cone-fact2}\n \\lim_{|x|\\to\\infty}\\frac{|F(z)|}{K(z,\\overline z)^{\\frac{1}{p}}}=0\n \\end{align}\n holds uniformly for $y\\in C_0$, where $C_0$ is a compact subset in $\\Gamma$.\n Notice that\n$$\nK(z, \\overline z)= \\int_{\\Gamma^*}e^{-4\\pi y\\cdot\nt}dt=K(iy, -iy).\n$$\nand $y\\in C_0$. It suffices to show\n\\begin{align}\\label{suff-cond-third-case}\n\\lim_{|x|\\to \\infty} {|F(z)|}=0.\n\\end{align}\nSince $C_0$ is compact, there exists a constant $\\rho>0$\nsuch that $d(C_0,{\\Gamma}^{c})=\\inf\\{|y-\\xi|;y\\in C_{0}, \\xi\n\\not\\in \\Gamma\\}\\geq {\\rho}$, where ${\\Gamma}^c$ is the complement\nof $\\Gamma$. Let $C_1=\\overline {\\cup_{y\\in C_0}\\{\\eta;\n|\\eta-y|<\\frac{\\rho}{2}\\}}$. Obviously, $d(C_1,{\\Gamma}^c)\\geq\n\\frac{\\rho}{2}$ and $C_1$ is also compact. Based on the fact that $ \\int_{C_1}\\int_{\\mathbb\nR^n}|F(x+iy)|^p dx dy<\\infty$, and the definition of\nfunctions in $H^p(T_{\\Gamma}),$ we have\n\\begin{align}\\label{case3-fact1}\n\\int_{C_1}\\int_{\\mathbb |x|>N}|F(x+iy)|^p dx dy\\to 0, \\quad N\\to\n\\infty.\n\\end{align}\n Recall that $|F|^p$ is subharmonic.\nFor $z\\in T_{\\Gamma}$, we have\n\\begin{align}\\label{case3-fact2}\n|F(x+iy)|^p\\leq\n\\frac{1}{V(B_z(\\frac{\\rho}{4}))}\\int_{B_z(\\frac{\\rho}{4})}|F(\\xi+i\\eta)|^p d\\xi\nd\\eta,\n\\end{align}\nwhere $V(B_z(\\frac{\\rho}{4}))$ is the volume of the ball\n$B_z(\\frac{\\rho}{4})$ centered at $z$ with radius $\\frac{\\rho}{4}$. From (\\ref{case3-fact2}), for $y\\in C_0$, we\nhave\n\\begin{align}\\label{case3-fact3}\n\\begin{split} |F(x+iy)|^p &\\leq\n\\frac{1}{V(B_z(\\frac{\\rho}{4}))}\\int_{B_z(\\frac{\\rho}{4})}|F(\\xi+i\\eta)|^pd\\xi\nd\\eta\\\\\n&\\leq \\frac{1}{V(B_z(\\frac{\\rho}{4}))}\\int_{\\{\\eta; |\\eta-y|\\leq\n\\frac{\\rho}{4}\\}}\\int_{\\{\\xi;|\\xi-x|\\leq\n\\frac{\\rho}{4}\\}}|F(\\xi+i\\eta)|^pd\\xi\nd\\eta\\\\\n&\\leq\n\\frac{1}{V(B_z(\\frac{\\rho}{4}))}\\int_{C_1}\\int_{\\{\\xi;|\\xi-x|\\leq\n\\frac{\\rho}{4}\\}}|F(\\xi+i\\eta)|^pd\\xi\nd\\eta.\\\\\n\\end{split}\n\\end{align}\nSince $|\\xi-x|\\leq \\frac{\\rho}{4}$, we have $|x|-\\frac{\\rho}{4}\\leq\n|\\xi| \\leq |x|+\\frac{\\rho}{4}$. Therefore,\nwhen $|x|>N+\\frac{\\rho}{4}$, by (\\ref{case3-fact1}) we have\n(\\ref{case3-fact3}) tends to $0$ uniformly for $y\\in C_0$.\n\\end{proof}\n\n\\begin{remark}\n(1) When $p=2$, Theorem \\ref{g-cone} implies the existence of $z^{(m+1)}_*$ in the following minimization problem\n$$\nz^{(m+1)}_* :=\\arg \\min_{z^{(m+1)}\\in \\Gamma}\\|F-F_{A_{m+1}}^*\\|^2,\n$$\nHence, we obtain the AFD-type approximation in $H^2(T_\\Gamma)$ if $\\Gamma$ is one of the following cases: a regular cone in $\\mathbb R^2$; a polygonal cone in $\\mathbb R^n$; a circular cone in $\\mathbb R^n.$\\\\\n(2) Since $|F|^p$ is still subharmonic for $0<p\\leq 1$ and the technique used in Theorem \\ref{g-cone} still works, (\\ref{g-cone-fact2}) holds for $0<p<\\infty$. Note that Theorem \\ref{g-cone} plays an essential role in studying the AFD-type approximation. Moreover, Theorem \\ref{g-cone} indeed is analogous to the following known result in the Hardy spaces on the unit ball $H^p(\\mathbb B_n), 1<p<\\infty$ (cf. \\cite[page 123]{Z1}),\n$$\n\\lim_{|z|\\to 1^{-}}(1-|z|^2)^{\\frac{n}{p}}|F(z)|=0, \\quad F\\in H^p(\\mathbb B_n),\n$$\nwhere $K_{\\mathbb B_n}(w,\\overline z)=\\frac{1}{(1-\\sum_{k=1}^n w_k\\overline {z_k})^n}$ is the Cauchy-Szeg\\\"o kernel for $H^2(\\mathbb B_n).$\n\\end{remark}\n\n\n\\bigskip\n\n\n\\addcontentsline{toc}{section}{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{T}{he} use of deep neural network (DNN) techniques in intelligent vehicles has expedited the development of self-driving vehicles in research and industry. Self-driving cars  can operate automatically because equipped perception, planning, and control modules operate  cooperatively \\cite{Gri20, Tam20, Yur20}. The most common perception components used in autonomous vehicles include cameras and radar\/lidar devices; cameras are combined with DNN  to recognize relevant objects, and radars\/lidars  are mainly used for distance measurement \\cite{Yur20, YLi20}. Because of limitations related to sensor cost and size, current active driving assistance systems primarily rely on camera-based perception modules with supplementary radars  \\cite{Con20}.\n\n\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[scale=0.35]{overview.pdf}}\n\\caption{Proposed vision-based automated driving framework. The system comprises the following modules: a multi-task DNN for surrounding perception, a lateral\/longitudinal CILQR controller for automobile motion planning and adherence to driving commands (steering, acceleration, and brake), and a  proportional and integral (PI)  controller combined with the longitudinal CILQR algorithm for velocity  tracking.  These modules  take data from a front-facing camera  and a few cost-effective radars as inputs, and they operate collaboratively to achieve automated vehicle operation. The DNN and lateral\/longitudinal CILQR algorithms are run efficiently every 24.52 and 0.58\/0.65 (ms), respectively.\n}\n\\end{figure}\n\n\n\n\nTo understand complex driving scenes, multi-task DNN (MTDNN) models that output multiple predictions simultaneously are often applied in autonomous vehicles to reduce inference time and device power consumption. In \\cite{Tei18}, street classification, vehicle detection, and road segmentation problems were solved using a single MultiNet model. In \\cite{Piz19}, the researchers trained a MTDNN to detect drivable areas and road classes for vehicle navigation. DLT-Net, presented in \\cite{Qia20}, is a unified neural network for the simultaneous detection of drivable areas, lane lines, and traffic objects; it aims to provide vehicle' localization when an HD map is unavailable. The context tensors between subtask decoders in DLT-Net share mutual features learned from different tasks. A lightweight multi-task semantic attention network was proposed in \\cite{Lai21} to achieve object detection and semantic segmentation simultaneously; this network  boosts detection performance and reduces computational costs through the use of a semantic attention module. YOLOP \\cite{Don21} is a panoptic driving perception network that simultaneously performs traffic object detection, drivable area segmentation, and lane detection on an NVIDIA  TITAN XP  GPU at a speed of 41 FPS.\n\n\n\n\\begin{figure*}[!t]\n\\centerline{\\includegraphics[scale=0.5]{network.pdf}}\n\\caption{Overview of proposed MTUNet architecture. The input RGB image of size  228 $\\times$ 228 is fed into the model, which then performs lane line segmentation, ego vehicle's pose estimation,  and traffic object detection at the same time. The backbone-seg-subnet is a UNet-based network; three variants of UNet (UNet$\\_$2$\\times$ \\cite{Lee21a}, UNet$\\_$1$\\times$ \\cite{Nab20}, and MResUNet \\cite{Nab20}) are compared in this work. The ReLU  activation functions in  pose and det subnets are not shown for simplicity.\n}\n\\end{figure*}\n\n\n\n \n \n\nIn a modular self-driving system, the perceived results can be sent to a model predictive control (MPC) planner to generate spatio-temporal trajectories over a time horizon.  The  system then reactively selects a short interval of optimal solutions as control inputs to minimize the gap between target and current states \\cite{Pad16}. The MPC model, which can be applied using various methods [e.g., active set, conjugated gradient, gradient project, augmented Lagrangian, interior point, and sequential quadratic programming (SQP)] \\cite{Lim21,Gut17,Ma22}, seems to be a promising tool for dealing with  vehicle optimal control problems. In \\cite{Tur13}, a linear MPC control model was proposed that addresses vehicle lane-keeping and obstacle avoidance problems by using lateral automation. In \\cite{Kat13}, an MPC control scheme that combines longitudinal and lateral  dynamics was designed for velocity-trajectory-following problems. \\cite{SLi15}  proposed a scale reduction method for reducing the online computational efforts of MPC controllers, and they applied it to longitudinal vehicle automation, achieving an average  computational time of approximately 5 ms. In \\cite{Gut17}, a linear time-varying MPC scheme was proposed for solving automobile lateral  trajectory optimization problems. The cycle time required for the optimized trajectory to be communicated to the feedback controller was 10 ms. \\cite{Lim22} investigated automatic weight determination for car-following control, and the corresponding linear MPC algorithm was implemented using CVXGEN \\cite{Mat12}, which solves  the relevant problem  within 1 ms. \n\n\n\nThe constrained iterative linear quadratic regulator (CILQR) method  was proposed to solve online trajectory optimization problems with nonlinear system dynamics and general constraints \\cite{Che17,Che19}. The CILQR algorithm is also an MPC method, while it is constructed on the basis of differential dynamic programming (DDP) \\cite{Jac70}. The computational burden of the well-established SQP solver is higher than that of  DDP \\cite{Ma22}. Thus, the CILQR solver, which uses analytical derivations, outperforms  the SQP approach in terms of computational efficiency; SQP requires  40.4 times more computation time per iteration to solve automobile motion planning problems \\cite{Che19}.\n\n\n\n\nWe proposed an MTDNN in \\cite{Lee21} to directly perceive the ego vehicle's heading angle ($\\theta $) and its distance  to the lane centerline ($\\Delta $) for autonomous driving. However, this end-to-end driving approach performs poorly in environments that are not shown during the training phase \\cite{Yur20, Li19}. In \\cite{Lee21a}, we  proposed  an improved control algorithm based on a multi-task  UNet architecture (MTUNet) that comprises lane line segmentation and pose estimation subnets. A Stanley controller \\cite{Thr06} and an MPC controller that uses the SQP solver were then designed to respectively control the lateral and longitudinal automation of an automobile. The Stanley controller takes  $\\theta $ and  $\\Delta $ yielding from the network as it's input for lane-centering, and the MPC controller uses radar data for car-following. The improved algorithm outperforms the model in \\cite{Lee21} and is comparable to the reinforcement learning model in \\cite{Li19}. However, the algorithm still has some problems. For example, the Stanley controller does not consider vehicle dynamic models, the computational expense of the SQP algorithm is high, and the DNN module lacks object detection capabilities.\n\n \nTo address the aforementioned deficiencies in existing approaches, we propose a new integrated algorithm for real-time automated driving based on developments described in a previous work \\cite{Lee21a}. First, a YOLOv4  detector  \\cite{Ale20} was added to the MTUNet for object detection. Second, we increased the MTUNet inference speed by reducing the  input size without sacrificing network performance. Third, we used CILQR algorithms to perform fast motion planning and automobile control in both lateral and longitudinal directions. The main goal of this work was to design an  automated driving  system that can operate efficiently in real-time lane-keeping and car-following scenarios. The contributions of this paper are as follows:\n\\begin{enumerate}\n  \\item The running speed of the proposed MTDNN scheme achieves  40 FPS for executing multiple driving perception tasks, including lane line segmentation, ego vehicle's heading angle regression, road type classification, and traffic object detection.\n  \\item The proposed lateral and longitudinal CILQR controllers can operate with low online computational load,  efficiently responding to driving commands  within 1 ms.\n  \\item The proposed framework integrates MTDNN and CILQR controllers to operate a vehicle autonomously in challenging simulation environments, demonstrating the effectiveness of the proposed method.\n\\end{enumerate}\n\n\n\n\\section{Methodology}\nAs depicted in Fig. 1, the proposed self-driving system  is composed of several modules. The DNN is a  multi-task UNet network that can solve multiple perception problems simultaneously. The CILQR controllers  then take the data from the DNN and radar to compute driving commands in lateral and longitudinal directions. \n\n\n\n\\subsection{MTUNet network}\nAs indicated in Fig. 2, the proposed MTDNN is a neural network with an MTUNet architecture featuring a common backbone encoder and three  subnets for completing multiple tasks at the same time. The following sections describes each part.\n\n\n\n\\subsubsection{Backbone and segmentation subnet}\nThe shared backbone and segmentation (seg) subnet employ encoder-decoder UNet-based networks for  pixel-level lane line classification tasks. Two classical  UNets (UNet$\\_$2$\\times$ \\cite{Lee21a} and UNet$\\_$1$\\times$ \\cite{Nab20})  and one enhanced version (MultiResUNet \\cite{Nab20}, denoted as MResUNet throughout the paper)   were used to investigate the effects of model size and complexity on task performance. For UNet$\\_$2$\\times$ and UNet$\\_$1$\\times$, each repeated block includes two convolutional (Conv) layers, and the first UNet has twice as many filters as the second. For MResUNet, each modified block consists of three 3 $\\times$ 3 Conv layers and one 1 $\\times$ 1 Conv layer. Table I summarizes the  filter number and related kernel size of the Conv layers used in these models. The resulting total number of parameters of UNet$\\_$2$\\times$\/UNet$\\_$1$\\times$\/MResUNet is 31.04\/7.77\/7.26 M, and the corresponding total number of multiply-and-accumulates (MACs) is 38.91\/9.76\/12.67 G. All 3 $\\times$ 3 Conv layers we used are padded with one pixel to preserve the  spatial resolution after convolution operations \\cite{Sim15}. This setting  reduces the network input size from 400 $\\times$ 400 to 228 $\\times$ 228 but preserves  model performance and achieves faster inference compared with in our previous work \\cite{Lee21a} (the experimental results are presented in Section IV), where the network utilizes unpadded 3 $\\times$ 3 Conv layers \\cite{Ron15}, and  zero padding must be applied to the input to equalize the input-output resolutions \\cite{Bru18}. In the training phase, the weighted cross-entropy loss is adopted to deal with the lane detection sample imbalance problem  \\cite{Xie15, Zou20} and is represented as\n\\begin{eqnarray}\nL_S = &-& \\frac{N^{-}}{N^{+}+N^{-}}\\sum\\limits_{\\tilde y=1} \\log \\left(\\sigma(y) \\right)\\notag \\\\\n&-& \\frac{N^{+}}{N^{+}+N^{-}} \\sum\\limits_{\\tilde y=0} \\log \\left(1-\\sigma(y) \\right) ,\n\\end{eqnarray}\n where  $N^{+}$ and $N^{-}$ are numbers of foreground and background samples in a batch of images, respectively; $y$ is a predicted score; $\\tilde y$ is the corresponding label; and $\\sigma$ is the sigmoid function. \n\n\n\n\n\\begin{table}[!t]\n\\caption{Conv layers used in the UNet-based networks}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\n\\multirow{2}{*}{Conv-block}  &\\multirow{2}{*}{UNet$\\_$2$\\times$ \\cite{Lee21a}}   &\\multirow{2}{*}{UNet$\\_$1$\\times$ \\cite{Nab20}}  &\\multirow{2}{*}{MResUNet \\cite{Nab20}}   \\\\  \n & & &   \\\\  \\hline\n\\multirow{2}{*}{Block$1^{a}$\/9}  & 64 (3 $\\times$ 3)$^{b}$  & 32 (3 $\\times$ 3)  & 8 (3 $\\times$ 3), 17 (3 $\\times$ 3),   \\\\ \n   &  64 (3 $\\times$ 3)  & 32 (3 $\\times$ 3)   &  26 (3 $\\times$ 3), 51 (1 $\\times$ 1)  \\\\ \n\\multirow{2}{*}{Block2\/8}  & 128 (3 $\\times$ 3)  & 64 (3 $\\times$ 3)  & 17 (3 $\\times$ 3), 35 (3 $\\times$ 3),   \\\\ \n   &  128 (3 $\\times$ 3)  & 64 (3 $\\times$ 3)   &  53 (3 $\\times$ 3), 105 (1 $\\times$ 1)  \\\\ \n\\multirow{2}{*}{Block3\/7}  & 256 (3 $\\times$ 3)  & 128 (3 $\\times$ 3)  & 35 (3 $\\times$ 3), 71 (3 $\\times$ 3),   \\\\ \n   &  256 (3 $\\times$ 3)  & 128 (3 $\\times$ 3)   &  106 (3 $\\times$ 3), 212 (1 $\\times$ 1)  \\\\ \n\\multirow{2}{*}{Block4\/6}  & 512 (3 $\\times$ 3)  & 256 (3 $\\times$ 3)  & 71 (3 $\\times$ 3), 142 (3 $\\times$ 3),   \\\\ \n   &  512 (3 $\\times$ 3)  & 256 (3 $\\times$ 3)   &  213 (3 $\\times$ 3), 426 (1 $\\times$ 1)  \\\\ \n\\multirow{2}{*}{Block5}  & 1024 (3 $\\times$ 3)  & 512 (3 $\\times$ 3)  & 142 (3 $\\times$ 3), 284 (3 $\\times$ 3),   \\\\ \n   &  1024 (3 $\\times$ 3)  & 512 (3 $\\times$ 3)   &  427 (3 $\\times$ 3), 853 (1 $\\times$ 1)  \\\\ \\hline\n\n\\multicolumn{4}{l}{$^{a}$\\scriptsize{Block1 of UNet$\\_$2$\\times$\/UNet$\\_$1$\\times$ only contains one 3 $\\times$ 3 Conv layer}}  \\\\ \n\\multicolumn{4}{l}{$^{b}$\\scriptsize{The notation $n$ ($k$ $\\times$ $k$) represents a Conv layer with $n$ filters of kernal size $k$ $\\times$ $k$  }}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\begin{table}[!t]\n\\caption{Conv and FC layers used in the pose subnet of various MTUNets}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n\\multirow{2}{*}{Layer}  & \\multirow{2}{*}{MTUNet$\\_$2$\\times$}   & MTUNet$\\_$1$\\times$ \\\\  \n &  & MTMResUNet  \\\\  \\hline\n\n Conv1  &  1024 (3 $\\times$ 3)  & 512 (3 $\\times$ 3)     \\\\ \n Conv2  &  1024 (3 $\\times$ 3)  & 512 (3 $\\times$ 3)     \\\\ \n FC1-2  &  256, 256  & 256, 256   \\\\ \n FC3-4  &  1, 3  & 1, 3   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[!t]\n\\caption{Conv layers used in the detection subnet  of various MTUNets}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n\\multirow{2}{*}{Layer}   & \\multirow{2}{*}{MTUNet$\\_$2$\\times$}   & MTUNet$\\_$1$\\times$ \\\\  \n &  & MTMResUNet  \\\\  \\hline\n Conv1-6  &  512 (3 $\\times$ 3)  & 384 (3 $\\times$ 3)   \\\\ \n Conv7-9  &  512 (1 $\\times$ 1)  & 384 (1 $\\times$ 1)     \\\\ \n Conv10-12  &  256 (1 $\\times$ 1)  & 256 (1 $\\times$ 1)     \\\\ \n Conv13-15  &  256 (1 $\\times$ 1)  & 256 (1 $\\times$ 1)     \\\\ \n Conv16-18  &  6 (1 $\\times$ 1)  & 6 (1 $\\times$ 1)     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Pose subnet} This subnet is mainly responsible for whole-image angle regression and road type classification problems, where the road involves three categories (left turn, straight, and right turn) designed to prevent the angle estimation from mode collapsing \\cite{Lee21a,Cui19}. The network architecture of the pose subnet is presented in Fig. 2; the pose subnet takes the fourth Conv-block output feature maps of the backbone as its input. Subsequently, the input maps are fed into shared parts including two consecutive Conv layers and one global average pooling (GAP) layer to extract general features. Lastly, the resulting vectors are passed  separately through two fully connected (FC) layers before being mapped into a sigmoid\/softmax activation layer for regression\/classification task. Table II summarizes the number of filters  and output units of the corresponding Conv and FC layers, respectively. The expression   MTUNet$\\_$2$\\times$\/MTUNet$\\_$1$\\times$\/MTMResUNet in Table II represents a multi-task UNet scheme in which subnets are built on the UNet$\\_$2$\\times$\/UNet$\\_$1$\\times$\/MResUNet model throughout the article. The pose task loss function, including L2 regression loss ($L_R$) and cross-entropy loss ($L_C$), is employed for network training; this function is represented as follows:\n\\begin{subequations}\n\\begin{equation}\nL_R  = \\frac{1}{{2B}}\\sum\\limits_{i = 1}^B {\\left| {\\sigma(\\tilde{\\theta}_i)- \\sigma(\\theta_i) } \\right|^2 },\n\\end{equation}\n\\begin{equation}\nL_C  =  \\frac{-1}{B}\\sum\\limits_{i = 1}^B \\sum\\limits_{j = 1}^3 \\tilde p_{ij} \\log(p_{ij}),\n\\end{equation}\n\\end{subequations}\nwhere $\\tilde \\theta $ and $\\theta$ are the ground truth and estimated value, respectively; $B$ is the input batch size; and $\\tilde p$ and $p$ are true and softmax estimation values, respectively.\n\n\n\n\n\n\n\\subsubsection{Detection subnet}\nThe detection (det) subnet takes advantage of a simplified YOLOv4 detector \\cite{Ale20} for real-time traffic object (leading car) detection. This fully-convolutional subnet that has three branches for multiscale detection takes the output feature maps of the backbone as its input, as illustrated in Fig. 2. The initial part of each branch is composed of single or consecutive 3 $\\times$ 3 filters for extracting contextual information at different scales, and a  shortcut connection with one 1 $\\times$ 1 filter from the input layer for residual mapping \\cite{Nab20}. The top of the addition layer contains sequential 1 $\\times$ 1 filters for reducing the number of channels. The resulting feature maps of each branch have six channels (five for bounding box offset and confidence score predictions, and one for class probability estimation) with a size of $K$ = 15 $\\times$ 15 to divide the input image into $K$ grids. In this article, we select $M$ = 3 anchor boxes, which are then shared between three branches according to the context size. Ultimately, spatial features from three detecting scales are concatenated together and sent to the output layer. Table  III presents the design of detection subnet of MTUNets. The overall  loss function for training comprises objectness ($L_{obj}$), classification ($L_{cls}$), and complete intersection over union (CIoU) losses ($L_{CIoU}$) \\cite{Zhe20, Ce21, Jun22}; these losses are constructed as follows:\n\\begin{subequations}\n\\begin{align}\n\\begin{split}\n L_{obj}  =  &- \\sum\\limits_{i = 0}^K {\\sum\\limits_{j = 0}^M {I_{ij}^{obj} \\left[ {\\tilde Q_i \\log \\left( {Q_i } \\right)} \\right.} }  \\\\ &\n \\left. { + \\left( {1 - \\tilde Q_i } \\right)\\log \\left( {1 - Q_i } \\right)} \\right] \\\\ &\n  - \\sum\\limits_{i = 0}^K {\\sum\\limits_{j = 0}^M {\\lambda_{noobj} I_{ij}^{noobj} \\left[ {\\tilde Q_i \\log \\left( {Q_i } \\right)} \\right.} }  \\\\ &\n \\left. { + \\left( {1 - \\tilde Q_i } \\right)\\log \\left( {1 - Q_i } \\right)} \\right], \\\\ \n\\end{split}\n\\end{align} \n\\begin{align}\n\\begin{split}\n L_{cls}   = &- \\sum\\limits_{i = 0}^{K} {    I_{ij}^{obj} \\sum\\limits_{n \\in classes} {\\left[ {\\tilde p_i \\left( n \\right)\\log \\left( {p_i \\left( n \\right)} \\right)} \\right.} }  \\\\ &\n \\left. { + \\left( {1 - \\tilde p_i \\left( n \\right)} \\right)\\log \\left( {1 - p_i \\left( n \\right)} \\right)} \\right],  \n\\end{split}\n\\end{align} \n\\begin{equation}\nL_{CIoU}  = 1 - IoU + \\frac{{E ^2 \\left( {{\\bf \\tilde o},{\\bf o}} \\right)}}{{\\beta ^2 }} + \\alpha \\gamma,\n\\end{equation}\n\\end{subequations}\nwhere $I_{ij}^{obj\/noobj}$ = 1\/0 or 0\/1 indicates that the $i$-th grid does or does not contain an object, respectively; $\\tilde Q_i$\/$Q_i$ and $\\tilde p_i$\/$p_i$ are the true\/estimated objectness and class scores corresponding to each grid, respectively; and $\\lambda_{noobj}$ is a hyperparameter intended for balancing positive and negative samples. With regard to CIoU loss, ${\\bf \\tilde o}$ and ${\\bf o} $ are the central points of the prediction ($B_p$) and ground truth ($B_{gt}$) boxes, respectively; $E$ is the related Euclidean distance; $\\beta $ is the diagonal distance of the smallest enclosing box covering $B_p$ and $B_{gt}$; $\\alpha$ is a  tradeoff hyperparameter; and $\\gamma$ is used to  measure  aspect ratio  consistency \\cite{Zhe20}.\n\n\n\n\n\\subsection{CILQR algorithm}\n\nThis section first briefly describes the concept behind  CILQR and related approaches based on Ref. \\cite{Che17,Che19,Tas12,Tas14}; it then presents the lateral\/longitudinal CILQR control algorithm  that takes the MTUNet inference and radar data as its inputs to yield driving decisions. \n\n\n\\subsubsection{Problem formulation}\nProvided  a sequence of states ${\\bf X} \\equiv \\left\\{ {{\\bf x}_0 ,{\\bf x}_1 ,...,{\\bf x}_N } \\right\\}$ and the corresponding control sequence ${\\bf U} \\equiv \\left\\{ {{\\bf u}_0 ,{\\bf u}_1 ,...,{\\bf u}_{N - 1} } \\right\\}$  are within the preview horizon $N$,  the  system's discrete-time dynamics ${\\bf f}$ are satisfied, with\n\\begin{equation}\n{\\bf x}_{i + 1}  = {\\bf f}\\left( {{\\bf x}_i ,{\\bf u}_i } \\right)\n\\end{equation}\nfrom time $i$ to $i+1$. The total cost denoted by $\\mathcal{J}$, including running costs $\\mathcal{P}$ and the final cost $\\mathcal{P}_f$, is presented as follows:\n\\begin{equation}\n\\mathcal{J}\\left( {{\\bf x}_0 ,{\\bf U}} \\right) = \\sum\\limits_{i = 0}^{N - 1} {\\mathcal{P}\\left( {{\\bf x}_i ,{\\bf u}_i } \\right) + \\mathcal{P}_f \\left( {{\\bf x}_N } \\right)}. \n\\end{equation}\nThe  optimal  control sequence is then written as \n\\begin{equation}\n{\\bf U}^* \\left( {{\\bf x}^* } \\right) \\equiv \\mathop {\\arg \\min }\\limits_{\\bf U} \\mathcal{J}\\left( {{\\bf x}_0 ,{\\bf U}} \\right)\n\\end{equation}\nwith an optimal trajectory ${{\\bf x}^* }$. The partial sum of $\\mathcal{J}$ from any time step $t$\nto $N$ is represented as\n\\begin{equation}\n\\mathcal{J}_t \\left( {{\\bf x},{\\bf U}_t } \\right) = \\sum\\limits_{i = t}^N {\\mathcal{P}\\left( {{\\bf x}_i ,{\\bf u}_i } \\right) + \\mathcal{P}_f \\left( {{\\bf x}_N } \\right)}, \n\\end{equation}\nand the optimal value function $\\mathcal{V}$ at time $t$ starting at ${\\bf x}$ takes the form\n\\begin{equation}\n\\mathcal{V}_t \\left( {\\bf x} \\right) \\equiv \\mathop {\\arg \\min }\\limits_{{\\bf U}_t } \\mathcal{J}_t \\left( {{\\bf x},{\\bf U}_t } \\right)\n\\end{equation}\nwith  the final time step value function  $\\mathcal{V}_N \\left( {\\bf x} \\right) \\equiv \\mathcal{P}_f \\left( {{\\bf x}_N } \\right)$.  \n\nIn practice, the  final step value function $\\mathcal{V}_N\\left( {\\bf x} \\right)$ is obtained by executing a forward pass using the current control sequence. Subsequently,  local control signal minimizations are performed in the proceeding backward pass using the following Bellman equation:\n\\begin{equation}\n\\mathcal{V}_i \\left( {\\bf x} \\right) = \\mathop {\\min }\\limits_{\\bf u} \\left[ {\\mathcal{P}\\left( {{\\bf x},{\\bf u}} \\right) + \\mathcal{V}_{i + 1} \\left( {{\\bf f}\\left( {{\\bf x},{\\bf u}} \\right)} \\right)} \\right].\n\\end{equation}\nTo compute the optimal  trajectory,  the perturbed function around the $i$-th state-control pair in Eq. (9) is used; this function is written as follows: \n\\begin{align}\n\\begin{split}\n\\mathcal{O}\\left( {\\delta {\\bf x},\\delta {\\bf u}} \\right) =& \\mathcal{P}_i \\left( {{\\bf x} + \\delta {\\bf x},{\\bf u} + \\delta {\\bf u}} \\right) - \\mathcal{P}_i \\left( {{\\bf x},{\\bf u}} \\right) \\\\ &+ \\mathcal{V}_{i + 1} \\left( {{\\bf f}\\left( {{\\bf x} + \\delta {\\bf x},{\\bf u} + \\delta {\\bf u}} \\right)} \\right) - \\mathcal{V}_{i + 1} \\left( {{\\bf f}\\left( {{\\bf x},{\\bf u}} \\right)} \\right)\n\\end{split}\n\\end{align} \nThis equation can  be  approximated to a quadratic function by employing a second-order Taylor expansion with the  following n coefficients:\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{O}_{\\bf x}  = \\mathcal{P}_{\\bf x}  + {\\bf f}_{\\bf x}^{\\rm T} \\mathcal{V}_{\\bf x}, \n\\end{equation}\n\\begin{equation}\n\\mathcal{O}_{\\bf u}  = \\mathcal{P}_{\\bf u}  + {\\bf f}_{\\bf u}^{\\rm T} \\mathcal{V}_{\\bf x},\n\\end{equation}\n\\begin{equation}\n\\mathcal{O}_{{\\bf xx}}  = \\mathcal{P}_{{\\bf xx}}  + {\\bf f}_{\\bf x}^{\\rm T} \\mathcal{V}_{{\\bf xx}} {\\bf f}_{\\bf x}  + \\mathcal{V}_{\\bf x}  \\cdot {\\bf f}_{{\\bf xx}},\n\\end{equation}\n\\begin{equation}\n\\mathcal{O}_{{\\bf ux}}  = \\mathcal{P}_{{\\bf ux}}  + {\\bf f}_{\\bf u}^{\\rm T} \\mathcal{V}_{{\\bf xx}} {\\bf f}_{\\bf x}  + \\mathcal{V}_{\\bf x}  \\cdot {\\bf f}_{{\\bf ux}}, \n\\end{equation}\n\\begin{equation}\n\\mathcal{O}_{{\\bf uu}}  = \\mathcal{P}_{{\\bf uu}}  + {\\bf f}_{\\bf u}^{\\rm T} \\mathcal{V}_{{\\bf xx}} {\\bf f}_{\\bf u}  + \\mathcal{V}_{\\bf x}  \\cdot {\\bf f}_{{\\bf uu}}. \n\\end{equation}\n\\end{subequations}\nThe second-order  coefficients of the system dynamics (${\\bf f}_{\\bf xx}$, ${\\bf f}_{\\bf ux}$, and ${\\bf f}_{\\bf uu}$) are omitted  to reduce computational effort \\cite{Ma22,Tas12}. The values of these  coefficients are zero for linear systems [e.g., Eq. (19) and Eq. (25)], leading to fast convergence in trajectory optimization. \n\nThe optimal control signal modification can  be obtained by minimizing the quadratic\n $\\mathcal{O}\\left( {\\delta {\\bf x},\\delta {\\bf u}} \\right)$:\n\\begin{equation}\n\\delta {\\bf u}^*  = \\mathop {\\arg \\min }\\limits_{\\delta {\\bf u}} \\mathcal{O}\\left( {\\delta {\\bf x},\\delta {\\bf u}} \\right) = {\\bf k} + {\\bf K}\\delta {\\bf x},\n\\end{equation}\nwhere \n\\begin{subequations}\n\\begin{equation}\n{\\bf k} =  - \\mathcal{O}_{{\\bf uu}}^{ - 1} \\mathcal{O}_{\\bf u},\n\\end{equation}\n\\begin{equation}\n{\\bf K} =  - \\mathcal{O}_{{\\bf uu}}^{ - 1} \\mathcal{O}_{{\\bf ux}}  \n\\end{equation}\n\\end{subequations}\nare optimal control gains. If the optimal control indicated in Eq. (12) is plugged into  the approximated $\\mathcal{O}\\left( {\\delta {\\bf x},\\delta {\\bf u}} \\right)$  to recover the quadratic value function, the corresponding coefficients can  be obtained \\cite{Pla17}:\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{V}_{\\bf x}  = \\mathcal{O}_{\\bf x}  - {\\bf K}^{\\rm T} \\mathcal{O}_{{\\bf uu}} {\\bf k},\n\\end{equation}\n\\begin{equation}\n\\mathcal{V}_{{\\bf xx}}  = \\mathcal{O}_{{\\bf xx}}  - {\\bf K}^{\\rm T} \\mathcal{O}_{{\\bf uu}} {\\bf K}.\n\\end{equation}\n\\end{subequations}\nControl gains at each state (${\\bf k}_i$, ${\\bf K}_i$) can then be estimated by recursively computing Eq. (11), (13), and (14) in a backward process. Subsequently, the modified  control and state sequences can be evaluated through a renewed forward pass: \n\\begin{subequations}\n\\begin{equation}\n{\\bf \\hat u}_i  = {\\bf u}_i  +\\lambda {\\bf k}_i  + {\\bf K}_i \\left( {{\\bf \\hat x}_i  - {\\bf x}_i } \\right),\n\\end{equation}\n\\begin{equation}\n{\\bf \\hat x}_{i + 1}  = {\\bf f}\\left( {{\\bf \\hat x}_i ,{\\bf \\hat u}_i } \\right),\n\\end{equation}\n\\end{subequations}\nwhere  ${\\bf \\hat x}_0  = {\\bf x}_0 $. Here $\\lambda$ is the backtracking parameter for line search; it is set to 1 in the beginning and  designed to be reduced gradually in  the   forward-backward propagation loops  until convergence is reached.\n\n\nIf the system has the constraint \n\\begin{equation}\n \\mathcal{C}\\left( {x,u} \\right) < 0,\n\\end{equation}\nwhich can be shaped using an exponential barrier function \\cite{Che17,Pan20}\n\\begin{equation}\n\\mathcal{B}\\left( {\\mathcal{C}\\left( {x,u} \\right)} \\right) = q_1 \\exp \\left( {q_2 \\mathcal{C}\\left( {x,u} \\right)} \\right)\n\\end{equation}\nor a logarithmic barrier function \\cite{Che19}, then \n\\begin{equation}\n\\mathcal{B}\\left( {\\mathcal{C}\\left( {x,u} \\right)} \\right) =  - \\frac{1}{t}\\log \\left( { - \\mathcal{C}\\left( {x,u} \\right)} \\right),\n\\end{equation}\nwhere $q_1$, $q_2$, and $t > 0$ are parameters. The barrier function can be added to the cost function as a penalty. Eq. (18) converges toward the ideal indicator function as $t$ increases iteratively. \n\n\n\n\n\n\\subsubsection{Lateral CILQR controller}\nThe lateral vehicle dynamic model \\cite{Lee19} is employed for steering control. The state variable and control input are defined as $\n{\\bf x} = \\left[ {\\begin{array}{*{20}c}\n   \\Delta  & {\\dot \\Delta } & \\theta  & {\\dot \\theta }  \\\\\n\\end{array}} \\right]^{\\rm T} \n$ and ${\\bf u} = \\left[  \\delta  \\right]$, respectively, where $\\Delta$ is the lateral offset, $\\theta $ is the angle between the ego vehicle's heading and the tangent of the road, and $\\delta$ is the steering angle. As described in  our previous work \\cite{Lee21a, Lee22},  $\\theta $ and $\\Delta$ can be obtained from MTUNets and related post-processing methods, and it is assumed that $\\dot \\Delta  = \\dot \\theta  = 0$. The corresponding discrete system model is written as follows: \n\\begin{equation}\n{\\bf x}_{t + 1}  \\equiv {\\bf f}\\left( {{\\bf x}_t ,{\\bf u}_t } \\right) = {\\bf Ax}_t  + {\\bf Bu}_t,\n\\end{equation}\nwhere\n\\[\n{\\bf A} = \\left[ {\\begin{array}{*{20}c}\n{\\alpha _{11} } & {\\alpha _{12} } & 0 & 0 \\\\\n0 & {\\alpha _{22} } & {\\alpha _{23} } & {\\alpha _{24} } \\\\\n0 & 0 & {\\alpha _{33} } & {\\alpha _{34} } \\\\\n0 & {\\alpha _{42} } & {\\alpha _{43} } & {\\alpha _{44} } \\\\\n\\end{array}} \\right],\\quad{\\bf B} = \\left[ {\\begin{array}{*{20}c}\n0 \\\\\n{\\beta _1 } \\\\\n0 \\\\\n{\\beta _2 } \\\\\n\\end{array}} \\right],\n\\]\nwith coefficients\n\\[\n\\begin{array}{l}\n\\alpha _{11} = \\alpha _{33} = 1, \\quad\\alpha _{12} = \\alpha _{34} = dt, \\\\\n\\alpha _{22} = {1 - \\frac{{2\\left( {C_{\\alpha f} + C_{\\alpha r} } \\right)dt}}{{mv }}},\\quad\\alpha _{23} = {\\frac{{2\\left( {C_{\\alpha f} + C_{\\alpha r} } \\right)dt}}{m}}, \\\\\n\\alpha _{24} = {\\frac{{2\\left( { - C_{\\alpha f} l_f + C_{\\alpha r} l_r } \\right)dt}}{{mv }}},\\quad\\alpha _{42} = {\\frac{{2\\left( {C_{\\alpha f} l_f - C_{\\alpha r} l_r } \\right)dt}}{{I_z v }}} , \\\\\n\\alpha _{43} = {\\frac{{2\\left( {C_{\\alpha f} l_f - C_{\\alpha r} l_r } \\right)dt}}{{I_z }}},\\quad\\alpha _{44} = {1 - \\frac{{2\\left( {C_{\\alpha f} l_f^2 - C_{\\alpha r} l_r^2 } \\right)dt}}{{I_z v }}}, \\\\\n\\beta _1 = {\\frac{{2C_{\\alpha f} dt}}{m}} ,\\quad\\beta _2 = {\\frac{{2C_{\\alpha f} l_f dt}}{{I_z }}}. \\\\\n\\end{array}\n\\]\nHere, $v$ is the ego vehicle's current speed along the heading direction and $dt$ is the sampling time. The model parameters for the experiments are as follows: vehicle mass $m$ = 1150 (kg), cornering stiffness ${C_{\\alpha f} }$ = 80\\thinspace000 (N\/rad), ${C_{\\alpha r} }$ = 80\\thinspace000 (N\/rad), center of gravity point $l_f$ = 1.27 (m), $l_r$ = 1.37 (m), and moment of inertia $I_z$ = 2000 (kgm$^{2}$). \n\nThe objective function $\\mathcal{J}$ containing the iterative linear quadratic regulator (ILQR) and constraint terms can be represented as\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{J} = \\mathcal{J}_{ILQR}  + \\mathcal{J}_{c}, \n\\end{equation}\n\\begin{equation}\n\\mathcal{J}_{ILQR}  = \\sum\\limits_{i = 0}^{N - 1} {\\left( {{\\bf x}_i  - {\\bf x}_{r} } \\right)^{\\rm T} {\\bf Q}\\left( {{\\bf x}_i  - {\\bf x}_{r} } \\right) + {\\bf u}_i^{\\rm T} {\\bf Ru}_i }, \n\\end{equation}\n\\begin{equation}\n\\mathcal{J}_{c}  = \\sum\\limits_{i = 0}^{N - 1} {\\mathcal{B} \\left( u_i \\right)  + \\mathcal{B} \\left( \\Delta_i  \\right) }.\n\\end{equation}\n\\end{subequations}\nHere,  the reference  state ${\\bf x}_{r}$ = $\\mathbf{0}$,  ${\\bf Q}$\/${\\bf R}$ is the weighting matrix, and $\\mathcal{B} \\left( u_i \\right)$ and $\\mathcal{B} \\left( \\Delta_i  \\right)$ are the corresponding barrier functions:\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{B} \\left( u_i \\right) =  - \\frac{1}{t}\\left[ {\\log \\left( {u_i  - \\delta_{\\min } } \\right) + \\log \\left( {\\delta_{\\max }  - u_i } \\right)} \\right],\n\\end{equation}\n\\begin{equation}\n\\mathcal{B}\\left( {\\Delta _i } \\right) = \\left\\{ \\begin{array}{l}\n\\exp \\left( {\\Delta _i - \\Delta _{i - 1} } \\right)\\quad\\text{for}\\quad \\Delta _0 \\ge 0, \\\\\n\\exp \\left( {\\Delta _{i - 1} - \\Delta _i } \\right)\\quad\\text{for}\\quad \\Delta _0 < 0, \\\\\n\\end{array} \\right.\n\\end{equation}\n\\end{subequations}\nwhere $\\mathcal{B}$($u_i$) is used to limit control inputs and the high (low) steer bound is $\\delta_{\\max } $ $\\left( {\\delta_{\\min } } \\right)$ = ${\\pi  \\mathord{\\left\/\n {\\vphantom {\\pi  6}} \\right.\n \\kern-\\nulldelimiterspace} 6}$ $\\left( { - {\\pi  \\mathord{\\left\/\n {\\vphantom {\\pi  6}} \\right.\n \\kern-\\nulldelimiterspace} 6}} \\right)$ (rad). The objective of  $\\mathcal{B} \\left( \\Delta _i \\right)$ is to control the ego vehicle moving toward the lane center. \n \n \n \n \n\n\\begin{table*}[!t]\n\\caption{Summary of data sets used in our experiments}\n\\begin{center}\n\\begin{tabular}{l|c|c|c|c}\n\\hline\nDataset &  No. of images&  Labels  & No. of traffic objects & Source   \\\\ \\hline\n \\multirow{2}{*}{LLAMAS}  & \\multirow{2}{*}{22714} & ego-lane lines, &  \\multirow{2}{*}{29442} & \\cite{Beh19},  \\\\\n  &  &  bounding boxes &  & this work \\\\\\hline\n \\multirow{4}{*}{TORCS}  & \\multirow{4}{*}{42747} & ego-lane lines, & \\multirow{4}{*}{30189}  & \\multirow{4}{*}{\\cite{Lee21a}} \\\\\n  &  & bounding boxes,  &   & \\\\\n &  & ego's heading,  &   & \\\\\n  &  & road type  &   & \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n \n \n \nThe first element of the optimal steering  sequence is then selected to define the normalized steering command at a given time  as\nfollows:\\begin{equation}\n{\\rm SteerCmd} = \\frac{{\\delta _0^*  }}{{{\\pi  \\mathord{\\left\/\n {\\vphantom {\\pi  6}} \\right.\n \\kern-\\nulldelimiterspace} 6}}}.\n\\end{equation}\n\n\n\n\n\n\\subsubsection{Longitudinal CILQR controller}\nIn the longitudinal direction, a proportional-integral (PI) controller \\cite{Sam21} \n\\begin{equation}\nPI(v) = k_P e  + k_I \\sum\\limits_i {e_i } \n\\end{equation}\nis first applied to the ego car for tracking  reference speed $v_r$ under cruise conditions,  \nwhere  $e=v-v_r$  and $k_P$\/$k_I$ are  the  tracking error and the proportional\/integral gain, respectively. The normalized acceleration command is then given as follows:  \n\\begin{equation}\n{\\rm AcclCmd} = \\tanh (PI(v)).\n\\end{equation}\nWhen a slower preceding vehicle is encountered, the AccelCmd must be updated to maintain a safe distance from that vehicle to avoid a collision; for this purpose, we use the following longitudinal CILQR algorithm.\n\nThe state variable and control  input for longitudinal inter-vehicle dynamics are defined as $\n{\\bf x'} = \\left[ {\\begin{array}{*{20}c}\n   D & v & a  \\\\\n\\end{array}} \\right]^{\\rm T}\n$ and $\n{\\bf u'} = \\left[ j \\right]$, respectively, where $a$, $j = \\dot a$, and $D$ are the ego vehicle's acceleration, jerk, and distance to the preceding car, respectively. The corresponding discrete-time system model is written as\n\\begin{equation}\n{\\bf x'}_{t + 1}  \\equiv {\\bf f'}\\left( {{\\bf x'}_t ,{\\bf u'}_t } \\right) = {\\bf A'x'}_t  + {\\bf B'u'}_t  + {\\bf C'w'}, \n\\end{equation}\nwhere\n\\[\n\\begin{array}{l}\n{\\bf A'} = \\left[ {\\begin{array}{*{20}c}\n1 & { - dt} & { - \\frac{1}{2}dt^2 } \\\\\n0 & 1 & {dt} \\\\\n0 & 0 & 1 \\\\\n\\end{array}} \\right],\\quad{\\bf B'} = \\left[ {\\begin{array}{*{20}c}\n0 \\\\\n0 \\\\\n{dt} \\\\\n\\end{array}} \\right], \\\\\n{\\bf C'} = \\left[ {\\begin{array}{*{20}c}\n0 & {dt} & {\\frac{1}{2}dt^2 } \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n\\end{array}} \\right],\\quad{\\bf w'} = \\left[ {\\begin{array}{*{20}c}\n0 \\\\\n{v_l } \\\\\n{a_l } \\\\\n\\end{array}} \\right]. \\\\\n\\end{array}\n\\]\nHere, $v_l$\/$a_l$ is the preceding car's speed\/acceleration, and ${\\bf w'}$ is the measurable disturbance input \\cite{Qiu15}. The values of $D$ and $v_l$ are measured by the radar; $v$ is known; and $a = a_l  = 0$ is assumed. Here, MTUNets are used to recognize traffic objects,  and the radar is responsible for providing precise  distance measurements.\n\n\n\n\n\nThe objective function $\\mathcal{J}'$ for the longitudinal  CILQR controller can be written as,\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{J}' = \\mathcal{J}'_{ILQR}  + \\mathcal{J}'_{c},\n\\end{equation}\n\\begin{equation}\n\\mathcal{J}'_{ILQR}  = \\sum\\limits_{i = 0}^{N - 1} {\\left( {{\\bf x'}_i  - {\\bf x'}_r } \\right)^{\\rm T} {\\bf Q'}\\left( {{\\bf x'}_i  - {\\bf x'}_r } \\right) + {\\bf u'}_i^{\\rm T} {\\bf R'u'}_i }, \n\\end{equation}\n\\begin{equation}\n\\mathcal{J}'_{c}  = \\sum\\limits_{i = 0}^{N - 1} {\\mathcal{B}' \\left( {u'_i} \\right)  + \\mathcal{B}' \\left( D_i \\right) + \\mathcal{B}'\\left( {a_i } \\right)}.\n\\end{equation}\n\\end{subequations}\nHere,  the reference state ${\\bf x'}_r  = \\left[ {\\begin{array}{*{20}c}\n   {D_r } & {v_l } & {a_l }  \\\\\n\\end{array}} \\right]$, and $D_r$ is the reference distance for safety.  ${{\\bf Q'}}$\/${\\bf R'}$ is the weighting matrix, and $\\mathcal{B}' \\left( {u'_i} \\right)$, $\\mathcal{B}' \\left( D_i \\right),$ and $\\mathcal{B}'\\left( {a_i } \\right)$ are related  barrier functions:\n\\begin{subequations}\n\\begin{equation}\n\\mathcal{B}' \\left( {u'_i} \\right) =  - \\frac{1}{t'}\\left[ {\\log \\left( {u'_i  - j_{\\min } } \\right) + \\log \\left( {j_{\\max }  - u'_i } \\right)} \\right],\n\\end{equation}\n\\begin{equation}\n\\mathcal{B}' \\left( D_i \\right) = \\exp \\left( {D_r  - D_i } \\right),\n\\end{equation}\n\\begin{equation}\n\\mathcal{B}'\\left( {a_i } \\right) =  \\exp \\left( {a_{\\min }  - a_i } \\right) + \\exp \\left( {a_i  - a_{\\max } } \\right),\n\\end{equation}\n\\end{subequations}\nwhere $\\mathcal{B}' \\left( D_i  \\right)$  is used for  maintaining a safe distance, and $\\mathcal{B}'$($u'_i$) and $\\mathcal{B}'$($a_i$) are used to limit the ego vehicle's jerk and acceleration to [$-$1, 1] (m\/s$^3$) and  [$-$5, 5] (m\/s$^2$), respectively.\n\nThe first element of the optimal jerk sequence is then chosen to update AccelCmd in the car-following scenario as \n\\begin{equation}\n{\\rm AcclCmd} = \\tanh \\left( {PI\\left( v \\right)} \\right) + j_0^*.\n\\end{equation}\nThe brake command (BrakeCmd) gradually increases in value from 0 to 1 when $D$ is smaller than a certain critical value $D = D_c < D_r $ in times of emergency.\n\n\n\n\n\n\\begin{figure*}[t]\n\\centerline{\\includegraphics[scale=0.225]{example.jpg}}\n\\caption{Example traffic object and lane line detection results of the MTUNet$\\_$1$\\times$ network on  LLAMAS (first row) and TORCS (second row) images.\n}\n\\end{figure*}\n\n\n\n\n\n\\section{Experimental setup}\nThe proposed MTUNets extract  local and global contexts from input images to solve segmentation, detection, and pose tasks simultaneously. Because the learning rates of these tasks are different, the proposed MTUNets are trained in a stepwise manner rather than in an end-to-end manner to help  the backbone network gradually learn common features. Setups of the training strategy, image data, and dynamic validation are given as follows:  \n\n\\subsection{Network training strategy}\nThe MTUNets are trained through three stages. The pose subnet is first trained through stochastic gradient descent (SGD) with a batch size (bs) of 20, momentum (mo) of 0.9, and learning rate (lr) starting from $10^{ - 2}$ and decreasing by a factor of 0.9 every 5 epochs for a total of 100 epochs. Detection and pose subnets are then trained jointly based on the trained  parameters from the first stage using the SGD optimizer with bs = 4, mo = 0.9, and lr = $10^{ - 3}$, $10^{ - 4}$, and $10^{ - 5}$ for the first 60 epochs, the 61st to 80th epochs, and the last 20 epochs, respectively. All subnets (detection, pose, and segmentation) were trained together in the last stage using the  pretrained  model from the previous stage using the Adam optimizer. Bs and mo were set to 1 and 0.9, respectively. Lr was set to $10^{ - 4}$ for the first 75 epochs and $10^{ - 5}$ for the last 25 epochs. The total loss in each stage is a weighted sum of the corresponding losses \\cite{Lee17}.\n\n\n\\begin{table*}[!t]\n\\caption{Performance of trained MTUNets on test data}\n\\begin{center}\n\\begin{tabular}{l|c|c|c|c|c|c|c|c}\n\\hline\n\\multirow{3}{*}{Network} & \\multirow{3}{*}{Dataset} & \\multirow{3}{*}{Tasks}  &\\multicolumn{2}{c|}{Det}  &\\multicolumn{2}{c|}{Seg} & \\multicolumn{2}{c}{Pose} \\\\ \\cline{4-9}\n& &  & \\multirow{2}{*}{Recall} & \\multirow{2}{*}{AP (\\%)} & \\multirow{2}{*}{Recall} & \\multirow{2}{*}{F1 Score} & Heading & Road Type \\\\ \n& &  &         &  &                 &                           & MAE (rad)     & Accuracy (\\%)  \\\\ \\hline\nMTUNet$\\_$2$\\times$ & \\multirow{3}{*}{LLAMAS}  & \\multirow{3}{*}{Det+Seg} &0.942& 64.42 &0.935& 0.827 & - & - \\\\ \nMTUNet$\\_$1$\\times$ &   & & 0.946  &  59.96& 0.936& 0.831 & - & - \\\\ \nMTMResUNet  &   & & 0.950  & 57.40 & 0.736 & 0.748 & - & - \\\\  \\hline\n\nMTUNet$\\_$2$\\times$ & \\multirow{9}{*}{TORCS}  & \\multirow{3}{*}{Pose} &-& - &-& - & 0.004 & 90.42 \\\\ \nMTUNet$\\_$1$\\times$ &   &  &-& - &-& - & 0.005 & 90.48 \\\\ \nMTMResUNet &   &   &-& - &-& -& 0.006 & 83.77 \\\\ \\cline{3-9}\\cline{1-1}\n\n\nMTUNet$\\_$2$\\times$ &   & \\multirow{3}{*}{Det+Seg} &0.976& 71.51 &0.905& 0.889 & - & - \\\\ \nMTUNet$\\_$1$\\times$ &   &  &0.974& 66.14 &0.904& 0.894 & - & - \\\\ \nMTMResUNet &   &   &0.968& 66.12 &0.833& 0.869 & - & - \\\\ \\cline{3-9}\\cline{1-1}\nMTUNet$\\_$2$\\times$ &   & \\multirow{3}{*}{Det+Seg+Pose} &0.952& 65.83 &0.922& 0.883 & 0.005 & 87.08 \\\\ \nMTUNet$\\_$1$\\times$ &   &  &0.956& 59.25 &0.901& 0.882 & 0.004 & 94.30 \\\\ \nMTMResUNet &   &   &0.959& 51.88 &0.830& 0.855 & 0.007 & 80.46 \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n\n\n\\begin{table}\n\\caption{Results for MTUNets in terms of parameters (Params), multiply-and-accumulates (MACs), and frames per second (FPS)}\n\\begin{center}\n\\begin{tabular}{l|c|c|c|c}\n\\hline\nNetwork  & Tasks & Params & MACs & FPS \\\\ \\hline\nMTUNet$\\_$$2\\times$ & \\multirow{3}{*}{Det+Seg+Pose} &  83.31 M & 50.55 B &  23.28  \\\\ \nMTUNet$\\_$$1\\times$  & &  25.50 M & 13.69 B & 40.77  \\\\\nMTMResUNet & & 26.56 M  & 16.95 B & 27.30  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Image data sets}\nWe conducted experiments on artificial TORCS \\cite{Lee21a} and real-world LLAMAS \\cite{Beh19} data sets; the corresponding statistics are summarized in Table IV. The customized TORCS data set has joint labels for all tasks. The highway scenario of the LLAMAS data set is close to that of TORCS;  however, the original LLAMAS data set only contained lane line labels. Thus, we annotated each LLAMAS image with traffic object  bounding boxes to imitate the TORCS data. The resulting numbers of labeled traffic objects in the TORCS and LLAMAS data sets were approximately 30\\thinspace000  and 29\\thinspace000, respectively. To determine  anchor boxes  for  the detection task, the $k$-means algorithm \\cite{Mac67} was applied to partition  ground truth boxes. Nevertheless, the LLAMAS data set still lacked the ego vehicle's angle labels; therefore, this data set could only be used for comparing  segmentation and detection tasks. The ratio of the number of images used in the training phase to that used in the test  phase was approximately 10 for both data sets, as in our previous works \\cite{Lee21a, Lee22}. Recall\/average precision (AP; IoU was set to 0.5) \\cite{Lia22}, recall\/F1 score \\cite{Zou20}, and accuracy\/mean absolute error (MAE) \\cite{Lee21a} were used to evaluate  model performance in detection, segmentation, and pose tasks, respectively.\n\n\n\n\\subsection{Autonomous driving simulation}\nThe open-source driving environment TORCS provides sophisticated physics and graphics engines, making it ideal for not only visual processing but also vehicle dynamics research \\cite{Wym00}. Thus, the ego vehicle controlled by our integrated algorithms comprising trained MTUNets and CILQR controllers was designed to drive autonomously in unseen TORCS roads (e.g., Track A and B in Fig. 4) to verify the effectiveness of our approach. All experiments, including MTUNets training, testing, and  driving simulation were conducted  on a PC equipped with an Intel i9-9900K CPU, 64 GB of RAM, and  an NVIDIA RTX 2080 Ti GPU. The control frequency for the ego vehicle in TORCS was approximately 150 Hz on our computer.\n\n\n\n\n\\section{Results and discussions}\nTable V presents the performance results of MTUNets on the testing data with different task configurations. MTUNet$\\_$2$\\times$\/MTUNet$\\_$1$\\times$ outperformed  MTMResUNet in both data sets when the models jointly performed detection and segmentation tasks (see the first and third row of Table V); this finding differs from single segmentation task analysis results for biomedical images \\cite{Nab20}. Because interference to task gradients  often lowers  MTDNN performance \\cite{Sta20, Kok17}, the  MTUNet$\\_$2$\\times$\/MTUNet$\\_$1$\\times$ model outperformed the complicated  MTMResUNet network owing to its elegant architecture. When the pose task is included (see the last row of Table V), MTUNet$\\_$2$\\times$\/MTUNet$\\_$1$\\times$ can also outperform  MTMResUNet in all evaluation metrics; the  descending AP scores for the detection task result from an increase in false positive (FP) detections. However, the recall scores of detection task of all models only decreased by approximately 0.02 after the pose task was added (see the last two rows of Table V); approximately 95$\\%$  of ground truth boxes can be detected when all tasks were considered yet. Additionally, as described in Section II,  although we reduced the input size of  MTUNets by using padded 3 $\\times$ 3 Conv layers, which do not influence model performance, MTUNet$\\_$2$\\times$\/MTUNet$\\_$1$\\times$ still had similar measurements to our previous model in segmentation and pose tasks \\cite{Lee21a}. For a comparison of computational efficiency, Table VI presents the number of parameters, the computational complexity, and the computation speed of all schemes; the MTUNet$\\_$1$\\times$ model had the fastest inference by 40.77 FPS (24.52 ms\/frame). This speed is comparable to that of the YOLOP model \\cite{Don21}. Because MTUNet$\\_$2$\\times$ slightly outperformed MTUNet$\\_$1$\\times$ in several metrics, as presented in Table V, we conclude that MTUNet$\\_$1$\\times$ is the most efficient model for collaborating with controllers to achieve automated driving. Fig. 3. presents example outputs for traffic object and lane detection using the MTUNet$\\_$1$\\times$ network on TORCS and LLAMAS data sets.\n\n\n\n\n\n\nTo objectively evaluate the dynamic performance  of algorithms related to autonomous driving, lane-keeping (lateral) and car-following (longitudinal) maneuvers were performed on challenging tracks A and B, as shown in  Fig. 4. MTUNet$\\_$1$\\times$ was then integrated with controllers to drive the ego vehicle and ensure it performed these maneuvers; we implemented SQP algorithms using  the ACADO toolkit \\cite{Hou11} to conduct comparisons with the CILQR controllers. The setting of both algorithms were the same, and the relevant parameters are summarized in Table VII. For lateral control experiments, automatic vehicles were designed to drive at a cruise speed of 70 and 50 (km\/h) on track A and B, respectively. The corresponding validation results of the CILQR\/SQP algorithm for $\\theta$ and $\\Delta$  are shown in Fig. 5\/6.  Both approaches with MTUNet$\\_$1$\\times$ model are able to  guide the ego car such that it drives along the lane center and completes one lap on Track A and B; this function is similar to that of a reinforcement learning model \\cite{Li19}. The discrepancy in the heading $\\theta$ between the MTUNet$\\_$1$\\times$ estimation  and  the ground truth trajectory was caused by curvy or shadowy road segments, which may induce vehicle jittering (see the $\\theta$ data on Track A of Fig. 5\/6) \\cite{Li19}. Nevertheless, $\\Delta$ estimated from lane line segmentation results was more robust to difficult scenarios than obtaining it via the end-to-end method  \\cite{Che15}. Therefore,  $\\Delta$ can help controllers effectively correct $\\theta$ errors, and bring ego car to the road center (see the $\\Delta$ data on Track A of Fig. 5\/6). Moreover, the resulting mean absolute error (MAE) of  $\\theta$ and $\\Delta$ are shown in Table VIII, where  lateral CILQR controller outperformed  SQP method in terms of $\\theta$- and $\\Delta$-MAE on both tracks. \n\n\n\n\\begin{figure}[!t]\n\\centerline{\\includegraphics[scale=0.5]{tracks.jpg}}\n\\caption{Tracks A (left) and B (right) for dynamically evaluating integrated MTUNet and control models. The  total length of Track A\/B was 2843\/3919 (m) with lane width 4 (m), and the maximum curvature was approximately 0.05\/0.03 (1\/m), which was curvier than a typical road \\cite{Fit94}. The self-driving car drove in a counterclockwise direction, and the starting locations are marked by green filled circle symbols. A self-driving vehicle \\cite{Li19} could not finish a lap on Track A using the direct perception approach  \\cite{Che15}.}\n\\end{figure}\n\n\n\n\\begin{table}[!t]\n\\caption{Dynamic system models and parameters for CILQR\/SQP controllers implementation}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n  & Lateral & Longitudinal \\\\ \n  & CILQR\/SQP & CILQR\/SQP \\\\ \\hline\nDynamic model    & Eq. (19) & Eq. (25) \\\\ \nSampling time ($dt$)  & 0.05 (s)  & 0.1 (s)\\\\ \nPred. horizon ($N$) & 30  & 30 \\\\ \nRef. dist. ($D_r$) & -  &  11 (m)  \\\\ \nWeighting matrixes  & ${\\bf Q}$ = ${\\rm diag}\\left( {20 ,1 ,20 ,1 } \\right)$  & ${{\\bf Q'}}$ = ${\\rm diag}\\left( {20 ,20 ,1 } \\right)$ \\\\ \n  & ${\\bf R} = \\left[ 1 \\right]$ &${\\bf R'} = \\left[ 1 \\right]$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[!t]\n\\centerline{\\includegraphics[scale=0.33]{la_cilqr.pdf}}\n\\caption{Dynamic performance of lateral CILQR algorithm with MTUNet$\\_$1$\\times $ model in ego vehicle's heading $\\theta$ and lateral offset $\\Delta$ for lane-keeping maneuver in central lane of Track A\/B at speed 70\/50 (km\/h).}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\centerline{\\includegraphics[scale=0.33]{la_sqp.pdf}}\n\\caption{Dynamic performance of lateral SQP algorithm with MTUNet$\\_$1$\\times $ model for lane-keeping maneuver in  central lane of  Track A\/B at the same speed as that in Fig. 5.}\n\\end{figure}\n\n\n\n\\begin{figure}[!t]\n\\centerline{\\includegraphics[scale=0.33]{lo_new.pdf}}\n\\caption{Experiment results of CILQR and SQP algorithms in car-following scenario after ego car travels 1075 (m) on Track B; $v$ and  $D$ are speed and inter-vehicle distance, respectively.}\n\\end{figure}\n\n\n\nFigure 7 presents the longitudinal experimental results related to the CILQR and SQP controllers. This maneuver was performed  on a section of Track B when the ego vehicle cruised at 76 (km\/h) initially and approached a slower preceding car with speed in the range of [63, 64] (km\/h). The ego vehicle with a CILQR\/SQP controller was able to regulate its speed, track the preceding vehicle, and maintain a safe distance from it. However, the uncertainty in optimal solutions led  to differences between  reference  and response trajectories \\cite{Lim22}. For the longitudinal  CILQR\/SQP controller, the $v$-MAE  was \n0.1885\/0.2605 (m\/s), and the $D$-MAE  was 0.4029\/0.4634 (m) (see Table VIII). CILQR outperformed SQP again in this experiment. Table IX presents the average time to arrive at a solution for lateral and longitudinal CILQR and SQP controllers; the iterative period of the SQP was longer than the ego vehicle control period (6.66 ms), and the SQP required 16.7$\\times$ and 21.5$\\times$ longer computation times per cycle to accomplish lane-keeping and car-following tasks, respectively. Therefore, the inferior performance of the SQP methods may be ascribed to slow reaction times, whereas the CILQR algorithms had higher computational efficiency in these cases. A supplementary video featuring lane-keeping and car-following simulations can be found at https:\/\/youtu.be\/pqQzEp1hKuQ.\n\n\n\n\n  \n\n\n\n\n\n\\begin{table}[!t]\n\\caption{Performance of CILQR\/SQP algorithm with MTUNet$\\_$1$\\times $ in terms of the mean absolute error (MAE) of parameters presented in Fig. 5\/6\/7 on Track A\/B }\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\n & \\multicolumn{2}{c}{CILQR} & \\multicolumn{2}{c}{SQP} \\\\\\hline\n  & $\\theta$ (rad) & $\\Delta $ (m) & $\\theta$ (rad) & $\\Delta $ (m) \\\\ \nTrack A    &  0.0087 &  0.1266   & 0.0104  &  0.1338 \\\\ \nTrack B    &  0.0081 &  0.0946   & 0.0089  &  0.1061 \\\\ \n\\hline\n  & $v$ (m\/s) & $D$ (m) & $v$ (m\/s) & $D$ (m) \\\\ \nTrack B$^{a}$    &  0.1885 &  0.4029   & 0.2605  &  0.4634 \\\\ \n\\hline\n\\multicolumn{4}{l}{$^{a}$\\scriptsize{Computation  from 1150 to 1550 (m)}} \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!t]\n\\caption{Average computation time of CILQR and SQP controllers}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n & CILQR & SQP \\\\ \\hline\nLateral    & 0.58 (ms) & 9.70 (ms)\\\\ \nLongitudinal  & 0.65 (ms) & 14.01 (ms) \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, we propose a vision-based self-driving framework that involves using a front-facing camera and  radars to collect sensing data; the framework comprises a MTUNet network and CILQR modules for  environment perception and  motion planning, respectively. The proposed MTUNet model is an improvement on our previous model \\cite{Lee21a}; we added a YOLOv4 detector and increased the network efficiency by reducing the network input size  for use with LLAMAS \\cite{Beh19} and TORCS data \\cite{Lee21a}. The most efficient MTUNet model, namely MTUNet$\\_$1$\\times $, achieved an inference speed of 40.77 FPS during the simultaneous operation of lane line segmentation, ego vehicle's pose estimation, and traffic object detection tasks. For vehicular automation, a lateral CILQR controller was designed to plan vehicle motion over a horizon based on ego's heading $\\theta$ and lateral offset $\\Delta$  produced by MTUNet$\\_$1$\\times$; then, the optimal steering angle was applied to guide the ego vehicle along the lane centerline; at this time, the longitudinal CILQR controller was activated when a slower preceding car was detected. The optimal jerk was then applied to regulate the ego vehicle's speed to avoid a collision. The MTUNet$\\_$1$\\times $ model and CILQR controllers can collaborate to operate the ego vehicle on challenge tracks in a TORCS environment; this model is comparable to an autonomous driving model that combines deep learning and reinforcement learning methods \\cite{Li19}. The CILQR modules can solve lane-keeping and car-following problems at a cycle time of 0.58 and 0.65 ms, respectively, outperforming SQP algorithms in not only computation speed  but also MAEs. These experiments validate the applicability of the proposed  system, which integrates  perception, planning, and control algorithms and is suitable for real-time autonomous vehicle applications.\n\n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:1}\n\\input{sections\/introduction}\n\\section{Governing equations of fluid dynamics}\n\\label{sec:2}\n\\input{sections\/governingEquationsAndNumericalMethods}\n\\section{The phantom domain mesh deformation method}\n\\label{sec:3}\n\\input{sections\/thePhantomDomainDeformationMeshUpdateMethod}\n\\section{Computational results}\n\\label{sec:4}\n\\input{sections\/computationalResults}\n\\section{Discussion}\n\\label{sec:5}\n\\input{sections\/discussion}\n\n\\begin{acknowledgement}\nThis work was supported by the German Research Foundation under the Cluster of Excellence \"Integrative production technology for high-wage countries\" (EXC128) as well as the  German Research Foundation under the Cluster of Excellence \"Internet of Production\". Computing resources were provided by the AICES graduate school and RWTH Aachen University Center for Computing.\n\\end{acknowledgement}\n\n\\input{referenc}\n\\end{document}\n\n\n\n\n\\subsection{Governing equations of fluid dynamics}\nConsider an incompressible fluid covering the deformable fluid domain $\\Omega_t^f \\subset  \\mathrm{R}^{n_{sd}}$, with $n_{sd}$ indicating the number of spatial dimensions. At every time instant $t\\in[0,T]$, the fluid's unknown velocity  $\\mathbf{u}(\\mathbf{x},t)$ and pressure $p(\\mathbf{x},t)$ are governed by the Navier-Stokes equations for incompressible fluids:\n\\begin{subequations}\n\\begin{align}\n\\rho^f \\left(\\frac{\\partial \\mathbf{u}^f}{\\partial t} \\,+\\, \\mathbf{u}^f \\cdot \\boldsymbol{\\nabla}  \\mathbf{u}^f \\,-\\, \\mathbf{f}^f \\right)\\,-\\, \\boldsymbol{\\nabla}\\cdot \\boldsymbol{\\sigma}^f\\,=\\, \\mathbf{0} &\\qquad\\text{on} ~\\Omega_t^f, \\forall t \\in \\left(0,T\\right),\\\\\n\\boldsymbol{\\nabla} \\cdot  \\mathbf{u}^f\\,=\\, 0 &\\qquad\\text{on}~ \\Omega_t^f, \\forall t \\in \\left(0,T\\right),\n\\end{align}\n\\label{Eq:NS}\n\\end{subequations}\nwith $\\rho^f$ denoting the fluid density and $\\mathbf{f}^f$ representing all external body forces per unit mass.\nFor Newtonian fluids, the stress tensor $\\boldsymbol{\\sigma}^f$ is defined as\n\\begin{equation}\n\\boldsymbol{\\sigma}^f\\,=\\,-p^f \\mathbf{I}\\,+\\,2\\rho^f\\nu^f \\boldsymbol{\\varepsilon}^f(\\mathbf{u}^f),\n\\end{equation}\nwith\n\\begin{equation}\n\\boldsymbol{\\varepsilon}^f(\\mathbf{u}^f)\\, =\\, \\frac{1}{2} \\left( \\boldsymbol{\\nabla} \\mathbf{u}^f + \\left(\\boldsymbol{\\nabla} \\mathbf{u}^f \\right)^T \\right),\n\\end{equation}\nwhere $\\nu^f$ denotes the dynamic viscosity. A well-posed system is obtained when boundary conditions are imposed on the external boundary  $\\Gamma^{f}_{t}$. Here, we distinguish between Dirichlet and Neumann boundary conditions given by:\n\\begin{subequations}\n\t\\begin{align}\n\t\\mathbf{u}^f \\,=\\,\\mathbf{g}^f &\\qquad\\text{on}~\\Gamma^{f}_{t,g},\\\\\n\t\\mathbf{n}^f\\cdot \\boldsymbol{\\sigma}^f \\,=\\,\\mathbf{h}^f &\\qquad\\text{on}~\\Gamma^{f}_{t,h},\n\t\\end{align}\n\\end{subequations}\nwhere $\\mathbf{g}^f$ and $\\mathbf{h}^f$ prescribe the velocity and stress values on complementary subsets of $\\Gamma^{f}_{t}$.\nWith regard to deformation of the fluid domain $\\Omega_t^f$ in time, the DSD\/SST method is applied to solve the Navier-Stokes equations.\n\\subsection{Deforming-spatial-domain\/stabilized space-time method}\\label{sec:DSDSST}\nThe DSD\/SST method is a space-time-based finite-element (FE) method, i.e., a FE discretization is applied to space and time.\nIt was first applied to flow problems with moving boundaries in \\cite{TezduyarBehrLiou1992,TezduyarEtAl1992}.\\\\\n\\\\\nThe advantage of the DSD\/SST method is, that the variational form of the governing equations implicitly incorporates the deformations of the domain.\nIn order to construct the interpolation and weighting function spaces used in the variational formulation of the problem, the time interval $(0,T)$ is split into $N$ subintervals $I_n=\\left[t_n,t_{n+1}\\right]$, where $t_n$  and $t_{n+1}$ belong to an ordered series of time levels.\nThus, the space-time continuum is divided into $N$ space-time slabs $Q_n$ as depicted in Figure \\ref{fig:SpaceTimeSlab}, bounded by the spatial configurations $\\Omega_t$ at time $t_n$ and $t_{n+1}$, and $P_n$ describing the course of the spatial boundary $\\Gamma^{f}_t$ as $t$ traverses $I_n$.\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics{Figures\/spaceTimeSlab}\n\t\\caption{Space-time slab.}\n\t\\label{fig:SpaceTimeSlab}\n\\end{figure}\nThe boundary $P_n$ can be decomposed into two complementary subsets $(P_n)_g$ and $(P_n)_h$, representing the Dirichlet and Neumann boundary conditions of $\\Gamma^f_{t}~\\forall t \\in I_n$.\nThe space-time slabs are weakly coupled along their interfaces using jump terms.\nFor the spatial approximation $\\Omega^{f}_{t,h}$ of the domain $\\Omega^f_t$, the following finite element trial and weighting function spaces are constructed:\n\\begin{subequations}\n\t\\begin{align}\n\t\\mathcal{H}^{1h}(Q_n)&:= \\left\\{ \\mathbf{w}^h \\in \\mathcal{H}^1 \\left(Q_n\\right) \\left| \\mathbf{w}^h_T\\right| \\text{is a first-order polynominal }\\forall T\\in \\mathcal{T}^h \\right\\},\\\\\n\t\\mathcal{S}^h_u &:= \\left\\{ \\mathbf{u}^h | \\mathbf{u}^h \\in \\left[\\mathcal{H}^{1h}\\left(Q_n\\right) \\right]^{nsd}, \\mathbf{u}^h =\\mathbf{g} ~\\text{on} ~\\left(P_n\\right)_g \\right\\},\\\\\n\t\\mathcal{V}^h &:= \\left\\{ \\mathbf{w}^h | \\mathbf{w}^h \\in \\left[\\mathcal{H}^{1h} \\left(Q_n\\right) \\right]^{nsd}, \\mathbf{w}^h= \\mathbf{0} ~\\text{on} ~\\left(P_n\\right)_g \\right\\},\\\\\n\t\\mathcal{S}^h_p &=  \\mathcal{V}^h_p :=\\left\\{ {q}^h | {q}^h \\in \\mathcal{H}^{1h}\\left(Q_n\\right)  \\right\\}.\n\t\\end{align}\n\\end{subequations}\nThe interpolation functions are globally continuous in space, but discontinuous in time.\nUsing the following notational convention,\n\\begin{subequations}\n\\begin{align}\n\\left(\\mathbf{u}^h\\right)^{\\pm}_n\\,=\\,\\underset{\\epsilon \\rightarrow 0}{lim}~  \\mathbf{u}\\left(t_n \\pm \\epsilon \\right)\\\\\n\\int_{Q_n} \\cdots \\text{d} Q = \\int_{I_n} \\int_{\\Omega_t} \\cdots \\text{d}\\Omega \\text{d}t,\\\\\n\\int_{(P_n)} \\cdots \\text{d} P = \\int_{I_n} \\int_{\\Gamma_t} \\cdots \\text{d}\\Gamma \\text{d}t,\n\\end{align}\n\\end{subequations}\nand following references \\cite{TezduyarBehrLiou1992, HughesFrancaHulbert1989, PauliBehr2017}, the stabilized variational formulation of the Navier Stokes equations is obtained:\nGiven $\\left(\\mathbf{u}^h\\right)_n^-$ with $\\left(\\mathbf{u}^h\\right)_0^- = \\mathbf{u}_0$, find $\\mathbf{u}^h \\in \\mathcal{S}^h_\\mathbf{u}$ and $p^h\\in \\mathcal{S}^h_p$ such that $\\forall \\mathbf{w}^h \\in \\mathcal{V}^h_\\mathbf{u}$, $\\forall q\\in \\mathcal{V}^h_P$:\n\\begin{align}\n\\int_{Q_n} \\mathbf{w}^h \\cdot \\rho^f \\left( \\frac{\\partial \\mathbf{u}^h}{\\partial t} + \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\cdot \\mathbf{u}^h - \\mathbf{f}\\right) \\text{d}Q +\n\\int_{Q_n} \\boldsymbol{\\nabla}\\mathbf{w}^h : \\boldsymbol{\\sigma} ( p^h, \\mathbf{u}^h ) \\text{d}Q \\nonumber \\\\\n+ \\int_{Q_n} q^h \\boldsymbol{\\nabla}\\cdot \\mathbf{u}^h \\text{d}Q  +\n\\int_{\\Omega_n} \\left(\\mathbf{w}^h\\right)^+_n \\cdot \\rho^f \\left(\\left(\\mathbf{u}^h\\right)^+_n - \\left(\\mathbf{u}^h\\right)^-_n\\right)\\text{d}\\Omega \\nonumber \\\\\n+ \\sum_{e=1}^{n_{el}}\\int_{Q^e_n} \\frac{1}{\\rho^f} \\tau_{MOM}  \\left[\\rho^f \\mathbf{u}^h \\cdot \\boldsymbol{\\nabla} \\mathbf{w}^h + \\boldsymbol{\\nabla} q^h\\right] \\nonumber \\\\\n \\cdot \\left[\\rho^f \\left(\\frac{\\partial \\mathbf{u}^h}{\\partial t} + \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\cdot \\mathbf{u}^h - \\mathbf{f} \\right) - \\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}(p^h, \\mathbf{u}^h) \\right]\\text{d}\\Omega \\nonumber\\\\\n+ \\sum_{e=1}^{n_{el}}\\int_{Q_n^e} \\boldsymbol{\\nabla} \\cdot \\mathbf{w}^h \\rho^f \\tau_{CONT} \\boldsymbol{\\nabla} \\cdot \\mathbf{u}^h \\text{d}\\Omega \\nonumber\\\\\n= \\int_{\\left(P_n\\right)_h} \\mathbf{w}^h \\cdot \\mathbf{h}^h\\text{d}P. \\label{eq:NS_WEAK}\n\\end{align}\nIn Equation \\eqref{eq:NS_WEAK}, the first three terms and the last term directly result from the variational formulation of Equation \\eqref{Eq:NS}, whereas the fourth term denotes the jump terms between the space-time slabs.\nTerms five and six result from a Galerkin-Least Squares (GLS) stabilization applied to the Navier-Stokes equations.\nThe stabilization approach used within this work and the choice of the stabilization parameters $\\tau_{CONT}$ and $\\tau_{MOM}$ are described in detail in \\cite{PauliBehr2017}.\\\\\n\\\\\nThough the DSD\/SST method implicitly accounts for the domain deformations in one time slab, a deformation rule is needed to deform the FE mesh according to the boundary movements.\n\\subsection{Elastic mesh update method}\\label{sec:EMUM}\nOne approach for the automatic mesh update in boundary conforming meshes is the elastic mesh update method (EMUM) introduced by \\cite{JohnsonTezduyar1994}, where the mesh is understood as an elastic body occupying the bounded region \\mbox{$\\Omega^\\# \\subset \\mathcal{R}^{n_{sd}}$} with boundary $\\Gamma^\\#$.\nThus, the deformation of the mesh is expressed in terms of the nodal displacements $\\mathbf{d}^\\#$ governed by the equilibrium equation of elasticity:\n\\begin{equation}\n\\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\sigma}^\\#\\,=\\,\\mathbf{0},\n\\end{equation}\nwhere $\\boldsymbol{\\sigma}^\\#$ corresponds to the Cauchy stress tensor,\n\\begin{equation}\n\\boldsymbol{\\sigma}^\\# \\,=\\, \\lambda \\left( tr \\boldsymbol{\\epsilon}^\\# \\right) \\mathbf{I}\\,+\\,2\\mu\\boldsymbol{\\epsilon}^\\# ~,\\qquad\n\\boldsymbol{\\epsilon}^\\#\\,=\\, \\frac{1}{2} \\left( \\boldsymbol{\\nabla} \\mathbf{d}^\\# + \\left(\\boldsymbol{\\nabla} \\mathbf{d}^\\# \\right)^T \\right).\n\\end{equation}\nThe imposition of Dirichlet and Neumann boundary conditions yields a well-posed problem for the mesh deformation:\n\\begin{align}\n\\mathbf{d}^\\# \\,=\\,\\mathbf{g}^\\# &\\qquad\\text{on}~ \\left(\\Gamma\\right)^\\#_g,\\\\\n\\mathbf{n}\\cdot \\boldsymbol{\\sigma}^\\# \\,=\\,\\mathbf{h}^\\# &\\qquad\\text{on}~ \\left(\\Gamma\\right)^\\#_h,\n\\end{align}\nwhere $\\mathbf{g}^\\#$ and $\\mathbf{h}^\\#$ prescribe the displacements and normal stresses on the mesh boundaries.\\\\\n\\\\\nThe elasticity problem is solved with the Galerkin FE method and the resulting displacements are applied to the mesh nodes representing the upper mesh configuration of the current space-time slab.\n\n\n\n\\subsection{2D Poiseuille flow on moving background mesh}\nIn the first test case we examine the influence of the PD-DMUM on a flow problem with a well-known solution.\nFor this purpose, we consider a two-dimensional Poiseuille flow in a tube.\nThe topology of the fluid domain remains unchanged, yet a predefined motion is applied to the underlying mesh.\nThe PD-DMUM is used to perform the mesh update, but should not affect the flow field within the tube.\\\\\n\\\\\nThe geometric dimensions of the tube are chosen  according to Figure \\ref{fig:PoiseuilleGeometry}.\nIn the middle of the domain, we position a mesh section $\\Gamma_T$ by means of which the predefined mesh motion is imposed as a Dirichlet boundary condition.\nThe boundary $\\Gamma_T$ has no physical impact with respect to the flow problem.\nThe additional phantom domains required within the PD-DMUM are positioned along the upper and lower boundary of the tube.\n\\begin{figure}[h]\n\t\t\\resizebox {\\textwidth} {!}{\n\t\\includegraphics{Figures\/sketch2dChannel}}\n\t\\centering\n\t\\caption{Tube geometry for Poiseuille flow.}\n\t\\label{fig:PoiseuilleGeometry}\n\\end{figure}\nThe material properties of the fluid are chosen according to Table \\ref{tab:PoiseuilleFlow}.\n\\begin{table}[h]\n\t\\begin{tabular}{ccc}\n\t\t\\hline\n\t\tParameter & Identifier  & Value  \\\\\n\t\t\\hline\n\t\tdensity & $\\rho$  &  $1.0$ [kg\/m$^3$]\\\\\n\t\tviscosity & $\\nu$  & $0.001$ [kg\/m$\\cdot $s] \\\\\n\t\tmean velocity& $U$  & $2.5$ [m\/s]  \\\\\n\t\t\\hline\\\\\n\t\\end{tabular}\n\t\\caption{Properties of fluid in 2D Poiseuille flow.}\n\t\\label{tab:PoiseuilleFlow}\n\\end{table}\nRegarding boundary conditions of the flow, we impose no-slip condition along the walls of the tube.\nThis also applies to the boundary section $\\Gamma_{PF}$ at the interface between the phantom domain and the fluid domain.\nA parabolic inflow profile for the velocity is given at the inlet of the tube:\n\\begin{equation}\n\\mathbf{u}(y) \\,=\\,\\left( \\frac{4Uy(H-y)}{H^2}, 0 \\right).\n\\end{equation}\nWith respect to the mesh update, the position of the nodes at the inlet, the outlet, and the tube walls are fixed.\nHowever, this does not apply to $\\Gamma_{PF}$ and the remaining boundaries of the phantom domain, as these nodes should be able to move freely.\nFor the boundary $\\Gamma_T$ we prescribe the following sinusoidal movement:\n\\begin{equation}\n\\mathbf{d}(t) \\,=\\,\\left(0~,~ 0.1\\cdot\\text{sin}\\left(\\frac{2\\,\\pi\\, t}{T} \\right) \\right).\n\\end{equation}\nThe mesh deformation is examined for a period of $T=8$[s]. The time step size is $\\Delta t = 0.02$ [s].\nInitially, a fully developed flow profile is already present in the pipe.\\\\\n\\\\\nThe Poiseuille flow is computed on four mesh configurations with the PD-DMUM and for the purpose of comparison for one configuration by the EMUM.\nFor the comparison of the solutions we use the flow velocity.\nThe velocity is measured at a probe positioned at point $(1.1\\,,\\,0.2)$ inside the tube.\nTogether with the given analytical solution of the Poiseille flow, the relative error can be computed for the different mesh configurations.\\\\\n\\\\\nIn a first step, the relative error of the computed velocity is evaluated for the probe position.\nIn Figure \\ref{fig:RelativErrorPoiseuille} it can be observed that the relative error decreases as the mesh is refined.\nThe comparison between the solution of the EMUM and the PD-DMUM on similar grids shows that the relative error for the calculated velocity is of the same order of magnitude.\nThe fluctuations that can be observed for all computations can be explained by the linear interpolation of the parabolic velocity profile at the probe position.\nIn Figure \\ref{fig:ConvergencePoiseuille}, we can observe that the numerical solution converges for the PD-DMUM towards the analytic solution of the Poiseuille problem.\nBoth, the convergence of the PD-DMUM and the comparable results to the EMUM for moderate mesh deformations indicate that the PD-DMUM provides a valid mesh update.\n\\begin{figure}[h]\n\t\\resizebox{\\textwidth}{!}{\n\t\\includegraphics{Figures\/derivationVelocity}}\n\t\\centering\n\t\\caption{Relative error of velocity at probe position.}\n\t\\label{fig:RelativErrorPoiseuille}\n\\end{figure}\n\\begin{figure}[h]\n\t\\resizebox{\\textwidth}{!}{\n\t\\includegraphics{Figures\/convergence}}\n\t\\centering\n\t\\caption{Average relative error for different mesh resolutions.}\n\t\\label{fig:ConvergencePoiseuille}\n\\end{figure}\n\\subsection{Falling ring in a fluid-filled container}\nThe second test case is used to illustrate possible applications of the PD-DMUM.\nFor this purpose, we consider a fluid-structure interaction with large translational boundary movement.\nMore precisely, we simulate an elastic ring that falls inside a fluid-filled container until it hits the ground and rebounds.\nConcerning the mesh deformation, this is a demanding process, since the number of mesh cells, which are initially positioned between the ring and the bottom, must be reduced to zero by the time of contact.\nUsing previous mesh update methods it is not possible to simulate this process on boundary conforming meshes without frequent remeshing of the fluid domain.\\\\\n\\\\\nThe geometric dimensions of the container and the ring are chosen according to Figure \\ref{fig:SketchCylinder}.\nThe ring is represented by a non-uniform rational B-spline (NURBS) \\cite{PieglTiller1997} with 721 elements and second-order basis functions.\nIn total 13448 elements are used to discretize the fluid domain and the additional phantom domains.\nIn the flow problem no-slip conditions are prescribed along the walls and the bottom of the container, whereas the top of the container is assumed to be open.\nThe fluid velocity at the ring surface corresponds to the  structural velocity.\nIn terms of the mesh deformation problem the mesh nodes on the container and walls of the phantom domains are restricted to a vertical movement.\nThe structural deformation is prescribed as a Dirichlet value for the ring boundary.\\\\\n\\begin{figure}[!h]\n\t\t\\resizebox {0.48\\textwidth} {!}{\n\t\\includegraphics{Figures\/sketchCylinder}}\n\t\\centering\n\t\\caption{Geometry of container with ring.}\n\t\\label{fig:SketchCylinder}\n\\end{figure}\n\\\\\nThe FSI problem is solved in a partitioned solution approach \\cite{FelippaParkFarhat1998}.\nOn the structural side, the deformation of the ring are represented by a linear elastic problem solved with isogeometric analysis (IGA) \\cite{HughesCottrellBazilevs2004}.\nThe contact interaction between the ring and the bottom of the container is considered via the penalty method \\cite{TemizerWriggersHughes2011}.\nThe flow field induced by the motion of the ring is described by the Navier-Stokes equations which are solved by the DSD\/SST approach in combination with the presented PD-DMUM.\nThe two field problems are strongly coupled in time \\cite{Wall1999}, and for the spatial coupling we apply a NURBS-based coupling following \\cite{HostersEtAl2017}.\\\\\n\\\\\nIn Figures \\ref{fig:Ball1} to \\ref{fig:Ball5}, we present snapshots of the simulation at different points in time, starting from the initial position of the ring, via the moment when the ring is in contact with the bottom of the container, up to the point of maximal altitude after the first contact interaction.\nAs it can be guessed from the snapshot in Figure \\ref{fig:Ball3}, one element remains between the bottom of the container and the falling ring.\nThis element will not be removed because we cannot exactly comply with the contact conditions using the penalty method.\nNevertheless, it can be observed in every snapshot, that mesh cells experience large displacements but only little deformations.\nDue to the application of the PD-DMUM, the entire FSI problem was solved without remeshing.\n\\begin{figure}[h!]\n\t\\centering\n\t\\begin{minipage}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Figures\/ringT1.eps}\n\t\t\\caption{Velocity at $t=0 ~\\text{s}$.}\n\t\t\\label{fig:Ball1}\n\t\\end{minipage}%\n\t\\begin{minipage}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Figures\/ringT2.eps}\n\t\t\\caption{Velocity at $t=0.55~\\text{s}$.}\n\t\t\\label{fig:Ball2}\n\t\\end{minipage}\n\t\\\\\n\t\\begin{minipage}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Figures\/ringT3.eps}\n\t\t\\caption{Velocity at $t=0.75~ \\text{s}$.}\n\t\t\\label{fig:Ball3}\n\t\\end{minipage}%\n\t\\begin{minipage}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Figures\/ringT4.eps}\n\t\t\\caption{Velocity at $t=1.0 ~\\text{s}$.}\n\t\t\\label{fig:Ball4}\n\t\\end{minipage}\\\\\n\t\\centering\n\\end{figure}\n\\begin{figure}[h!]\n\t\\centering\n\t\\ContinuedFloat\n\t\\begin{minipage}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Figures\/ringT5.eps}\n\t\t\\caption{Velocity at $t=1.45~\\text{s}$.}\n\t\t\\label{fig:Ball5}\n\t\\end{minipage}\n\\end{figure}\n\\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\protect\\bigskip Introduction}\nThe literature contains numerous studies on generalisations of the familiar Einstein-Hilbert action of general relativity (GR) to more complicated functions of the curvature and higher-order invariants \\cite{ottewill, schmidt1, schmidt2}. Motivation for these studies comes from several sources, including astronomical phenomena which are currently inadequately explained by the standard model of general relativity, such as providing a natural source of inflation in the early universe \\cite{starob1}, or dark energy and the late-time acceleration of the universe's expansion \\cite{Carroll:2004de, Nojiri:2006be, Nojiri:2006su, Nojiri:2006jy, Nojiri:2006gh, Amendola:2006kh}, and also attempts to include quantum behaviour in the gravitational theory \\cite{stelle}. A review of one of the most common extensions to general relativity, the so-called $f(R)$ models, in which the Lagrangian is allowed to be a general function of the scalar curvature, may be found in \\cite{sot}. \n\nIt is of particular interest to discover the behaviour of these higher-order theories at high curvatures and it is in this limit when we might expect the influence of quantum corrections to become important. Therefore, where initial singularities are expected to involve infinities in one or more of the curvature invariants of the space-time, we expect that the addition\nof higher-order terms to the Lagrangian might produce a new dominant behaviour to such singularities. The (past) stability properties of special initial isotropic cosmological singularities were investigated in \\cite{midd2} for higher-order theories where the dominant term in the Lagrangian took the form $(R_{ab}R^{ab})^n$.\n\nIt is well known that contributions to the Lagrangian from terms dependent on the scalar curvature only are conformally equivalent to the presence of a minimally coupled scalar field in general relativity \\cite{conformal}, however, in theories in which the gravitational Lagrangian contains higher-order curvature invariants, a much richer diversity of anisotropic behaviour is possible. For example, specific counterexamples were found in \\cite{hervik} which demonstrate that the ``cosmic no-hair theorem'' of general relativity may be violated in higher-order theories. Furthermore, anisotropies diverge faster than isotropies at high curvatures and will tend to dominate the cosmological behaviour at early times. Thus, whilst the majority of previous studies of these modified theories of gravity have focussed on the behaviour of isotropic cosmologies, it is the role of anisotropy on approach to the initial singularity which we wish to investigate here.\n\nSome of the most important anisotropic cosmological solutions in general relativity are the vacuum Kasner solutions of Bianchi type I \\cite{kasner}. Since they are characterised by just a single free constant, they are geometrically special, but nevertheless they provide us with a very useful insight into the dynamics of anisotropies, since they give a good description of the evolution of more general anisotropic cosmological models over finite time intervals. The chaotic oscillatory behaviour of the spacetime on approach to the initial singularity exhibited by the Bianchi type VIII and type IX (``Mixmaster'') cosmologies can be approximated by a sequence of different Kasner epochs \\cite{bkl,Misner,PhysRevLett.46.963,Barrow:1981sx,Chernoff:1983zz,Rendall:1997dc}. Provided that at least one of the Kasner exponents is negative, inhomogeneities and perturbations from the Bianchi I anisotropies will grow as the singularity is approached and force the solution to switch from one set of Kasner exponents to another. If the solution must always have at least one negative Kasner exponent, as is the situation in general relativity, then these oscillations will continue infinitely as the singularity is approached. However, in some higher-order theories of gravity \\cite{deruelle, clifton1, clifton2} it may be possible for all of the Kasner indices to take positive values, whence after a sufficient (finite) number or permutations, the indices will reach such a configuration and the oscillatory behaviour will cease. Since all spatial directions will be contracting, the initial singularity will be reached monotonically.\n \n \nPreviously, Kasner-type cosmological models in quadratic gravity and in Lovelock theories of gravity in higher dimensions were investigated by Deruelle \\cite{deruelle}. Clifton and Barrow \\cite{clifton1, clifton2} discovered the conditions for the existence of Kasner-like solutions and the exact forms of these solutions for the particular cases where the Lagrangian is an arbitrary power of one of the curvature invariants, $R, R_{ab}R^{ab}$ or $R_{abcd}R^{abcd}$. One might expect that the dynamics and the asymptotic behaviour of any solution in a more general higher-order theory would be controlled by the highest powers of the curvature in the past, and the lowest powers of the curvature in the future. However, this assumption is not necessarily accurate \\cite{clifton3} and we wish to extend the investigation of \\cite{clifton1, clifton2} to include more general Lagrangians. \n\nThus, in this work, we wish to investigate, within this class of higher-order theories of gravity, the constraints on the Lagrangian for the existence of some simple anisotropic but homogeneous solutions of Bianchi Type I and to find all such solutions. In particular, our main focus will be to consider the possibility of anisotropic Kasner-like solutions in vacuum, and in the presence of a cosmological constant. We will also consider the properties of these solutions with respect to their relation to the behaviour of the more general Bianchi type VIII and IX cosmologies. In addition, we will also discover all solutions in these higher-order theories which are expanding anisotropically but exponentially in each of the three spatial directions.\n\n\\section{Field Equations}\n\nIn this paper we will consider theories of gravitation in which the field equations are derived from an arbitrary analytic function of the three possible linear and quadratic contractions of the Riemann curvature tensor; $R, R_{ab}R^{ab}$ and $R_{abcd}R^{abcd}$. The relevant action\nis given by\n\\begin{equation*}\nS=\\int d^{4}x\\sqrt{-g}\\left[ \\frac{1}{\\chi }f(X,Y,Z)+L_{m}%\n\\right] ,\n\\end{equation*}\nwhere $f(X,Y,Z)$ is an arbitrary function of $X, Y$ and $Z$ which are defined\n$X \\equiv R, Y \\equiv R^{ab}R_{ab}$ and $Z \\equiv R^{abcd}R_{abcd}$. Spacetime indices run from $0$ to $3$ and are denoted by roman letters, whilst Greek letters are used to denote purely spatial indices.\n\nUnlike the situation in general relativity, which may be recovered by choosing $f=R$, the Palatini and metric formalisms are not equivalent for a general choice of $f$. In what follows, we shall restrict attention to the metric formalism. The field equations obtained by\nvarying the action with respect to the metric are \\cite{clifton2}:\n\\begin{equation}\n        P_{b}^{a}= \\frac{\\chi }{2}T_{b}^{a}\\, ,\n\\end{equation}\nwhere\n\\begin{eqnarray*}\nP^{ab} & = &-\\frac{1}{2}f g^{ab}+f_X R^{ab}+2f_{Y} R^{c(a}R^{b)} \\, _{c}\n+2f_{Z} R^{edc(a}R^{b)}\\,_{cde}+(g^{ab}g^{cd}-g^{ac}g^{bd})f_{X;cd} \\\\\n&& +\\Box (f_{Y}R^{ab}) +g^{ab}(f_{Y} R^{cd})_{;cd} -2(f_{Y}R^{c(a})_;\n\\,^{b)}_{c} - 4(f_{Z}R^{d(ab)c})_{;cd} \\: ,\n\\end{eqnarray*}%\nwhere $f_X \\equiv \\frac{\\partial f}{\\partial X}$, $f_Y \\equiv \\frac{\\partial\nf}{\\partial Y}$ and $f_Z \\equiv \\frac{\\partial f}{\\partial Z}$. Whilst the focus of this work is on solutions in vacuum, with $T^a_b=0$, the study can be extended in a simple way to allow the possibility of a non-zero cosmological constant, $\\Lambda$, by including any such term in the gravitational part of the Lagrangian, $f(X,Y,Z)$. Furthermore, we will also consider the effects of including a perfect fluid for $f(R)$ theories of gravity.\n\n\\section{The Kasner Model}\nIn this paper, we consider homogeneous but anisotropic Bianchi I models, and in particular our main focus will be on those spacetimes described by the line element\n\\begin{equation}\nds^{2}=-dt^{2}+t^{2p_{1}}dx^{2}+t^{2p_{2}}dy^{2}+t^{2p_{3}}dz^{2} ,  \\label{kas}\n\\end{equation} where the Kasner exponents $p_{\\alpha}$ are constants and assumed to be real in order for the metric to be of physical significance. We define the useful quantities \\begin{eqnarray*}\nH & \\equiv & p_{1}+p_{2}+p_{3}  \\: ,\\\\\nJ & \\equiv & p_{1}\\!^2 + p_{2}\\!^2 + p_{3}\\!^2  \\quad \\text{and} \\\\\nK & \\equiv & p_{1}\\!^3 + p_{2}\\!^3 + p_{3}\\!^3\n\\end{eqnarray*}\n\nThe line element (\\ref{kas}) also describes the limiting case in which all three Kasner exponents are equal, $p_{1}=p_{2}=p_{3}$, and corresponds to isotropic, spatially flat Friedmann-Robertson-Walker solutions for which the scalefactor is a power-law in time. These solutions will be included for completeness. It is useful to note that for real-valued choices of $p_{\\alpha}$, $J\\geq \\frac{H^2}{3}$, with equality if and only if the solution is isotropic.  The three relevant curvature scalars with which we will be working take the values\n\\begin{eqnarray}\nX \\equiv R & = & \\frac{J-2H+H^2}{t^2} \\: , \\label{eq:x} \\\\\nY \\equiv R_{ab}R^{ab} &=& \\frac{J(H-1)^2+(J-H)^2}{t^4} \\: , \\label{eq:y}\\\\\nZ \\equiv R_{abcd}R^{abcd} &=& \\frac{(3J-H^2)^2 +12J+8K(H-3)}{3t^4}  \\label{eq:z} \\: .\n\\end{eqnarray}\n\nIn general relativity, the Kasner exponents for the vacuum solution must satisfy $H=J=1$ \\cite{kasner}. One solution of this is the Milne model \\cite{Milne}, which without loss of generality can be described using the choice of axes such that $p_{1}=1$, $p_{2}=p_{3}=0$. However, this solution is related to Minkowski space by a coordinate transformation; if we introduce new coordinates $\\tau =  t \\cosh{x}, \\chi=t \\sinh{x}$, then the usual form of the flat Minkowski metric is explicitly recovered \\cite{clifton2}. The Riemann tensor $R_{abcd}$ vanishes for Minkowski space, and therefore this is a vacuum solution in any higher-order theory of gravity of the form $f(X,Y,Z)$ for which $f(0,0,0)=0$.\n\nFor all other general relativistic Kasner solutions, one Kasner exponent must be negative, whilst the other two are positive. Thus, although the spacetime volume is expanding to the future, one of the spatial directions is contracting. Moreover, as the initial singularity is approached, inhomogeneities and deviations from the Bianchi type I anisotropies in the more general Bianchi type IX solution will grow and cause the Kasner exponents to be permuted to different values, leading to an infinite series of chaotic BKL oscillations between different Kasner epochs \\cite{bkl,Misner,PhysRevLett.46.963,Barrow:1981sx,Chernoff:1983zz,Rendall:1997dc}. \n\nHowever, this is not true in general for the higher-order theories considered in this paper. In \\cite{clifton1, clifton2}, Clifton and Barrow found some exact solutions, a subset of which permits all the Kasner exponents to take positive values. We will see that this such solutions exist in a much wider class of higher-order theories. In this scenario, it is possible that after a finite number of transitions between different Kasner epochs, the solution will reach a state in which all of the Kasner exponents are positive. Once this occurs, the perturbations from the Bianchi I model would not grow and the chaotic oscillations will cease. Thus, the solution will remain in this epoch and the initial singularity is then approached monotonically.\n\nBy considering those classes of theories in which only one curvature scalar - $\\Phi$, say - contributes a time scale to the Lagrangian, one can solve for the time coordinate $t$ in terms of the scalar $\\Phi$. In this way, time derivatives may be eliminated and the field equations may be re-written in terms of $\\Phi$, the Lagrangian $f(\\Phi)$, and its derivatives with respect to $\\Phi$. The resultant differential equation(s) can then be solved to find all possible forms of $f$. This technique, which was also used by Dunsby et al. in $f(R)$ theories \\cite{dunsby}, allows us to find all possible exact Kasner-like vacuum solutions within this general class of Lagrangians, with one exception, which may be dealt with separately. This exception is when quantity which usually determines the time scale becomes independent of time for some special choice of the parameters $p_{\\alpha}$. For example, in general for the metric (\\ref{kas}), the scalar curvature, $R$, is proportional to $t^{-2}$, and so in the context of $f(R)$ models one can substitute for the time coordinate $t$ using $R$, but it is necessary to consider separately those possible solutions with $R=0$.\n\nAs a consequence, we will find that the subset of metrics for which the curvature scalars are independent of time plays an important role and it is useful to discuss those metrics briefly now. Note from the expressions (\\ref{eq:x}), (\\ref{eq:y} and (\\ref{eq:z}), the scalars $R, R_{ab}R^{ab}$ and $R_{abcd}R^{abcd}$ are only constants if and when they are zero. A special case within this class of metrics is flat Minkowski space, which may be respresented by the metric (\\ref{kas}) with $p_{\\alpha}=0$, that is to say $g_{ab}=\\eta_{ab}$, for which the Riemann tensor, $R_{abcd}$, is identically zero, and thus $X=Y=Z=0$. This is a solution in all theories for which $f(0,0,0)=0$ and the discussion in subsequent sections will be concentrated on metrics describing curved spacetimes.\n\nBy expressing the Kretschmann scalar, $Z \\equiv R_{abcd}R^{abcd}$, as \\begin{equation*}\nZ=\\frac{4}{t^4}\\left(p_{1}\\!^{2}(p_{1}-1)^{2}+p_{2}\\!^{2}(p_{2}-1)^{2}+p_{3}\\!^{2}(p_{3}-1)^{2}+p_{1}\\!^2 p_{2}\\!^2+p_{1}\\!^2 p_{3}\\!^2+p_{2}\\!^2 p_{3}\\!^2 \\right) \\:,\n\\end{equation*}it can be seen that there are two possible real solutions to $Z=0$, given by Minkowski space, $p_{\\alpha}=0$, and $p_{1}=1, p_{2}=p_{3}=0$ (plus permutations). However, as we have seen, the latter solution is the Milne model, and is related to the former by the coordinate transformation $\\tau =  t \\cosh{x}, \\chi=t \\sinh{x}$.\nIn order to satisfy $Y \\equiv R_{abcd}R^{abcd}=0$, real solutions must have either $H=J=0$ or $H=J=1$. The only possibility in the former case is Minkowski space. The latter are the well-known Kasner solutions of general relativity \\cite{kasner}. Thus, $Z=0$ implies that $Y=0$.\n\n\nSolutions for which the scalar curvature, $X \\equiv R$, is zero require $J=2H-H^{2}$ and thus $Y=0$ implies that $X=0$. From this equation for $J$, one can see that real solutions must satisfy the constraint $J \\leq 1$. Furthermore, from our definitions, $J\\geq \\frac{H^2}{3}$ for all real-valued choices of the constants $p_{\\alpha}$, and so it is also necessary that $0\\leq H \\leq 3\/2$. According to these constraints, we see that the spacetime volume of any solution of this form must expand no faster than $t^{3\/2}$, and that the expansion rate in any one direction can be no faster than $t$. Finally, we note that solutions with $X=0$ and $H \\leq 1$ cannot have all three Kasner exponents positive and, except for Minkowski space and the Milne model, at least one exponent must be negative. This is the situation in general relativity and, except for the two cases above, implies that although the spacetime volume is expanding, space must be contracting in one direction. In contrast, if $X=0$ but $1 < H <3\/2$, then it is possible for all the Kasner exponents to be distinct and positive. Thus, as we have discussed, these solutions can avoid the infinite series of chaotic oscillations between different Kasner epochs on approach to the initial cosmological singularity seen in the BKL picture. The only solution with $X=0$ and $H=3\/2$ is the isotropic solution with $p_{1}=p_{2}=p_{3}=1\/2$.\n\nIn what follows, we shall consider in turn several commonly-studied classes of higher-order theories of gravity, obtaining the forms of the Lagrangian within these classes for which solutions of the form (\\ref{kas}) exist, and the conditions which the Kasner exponents are subject to in each case.\n\n\\subsection{$f=f(R)$}\nIf the Lagrangian depends on the scalar curvature only, $f=f(R)$, then it is well-known to be conformally equivalent to general relativity with a minimally coupled scalar field \\cite{conformal}. Thus, one does not expect a large range of anisotropic behaviour to be possible in such models; indeed for the case of quadratic corrections to the Einstein-Hilbert Lagrangian it is precisely the presence of the Ricci term, $R_{ab}R^{ab}$, which permits the existence of the anisotropic Bianchi I solutions found by Barrow and Hervik \\cite{hervik}.\n\nThe vacuum field equations are $P^a_b=0$, where, for the Kasner-like metric given by (\\ref{kas}), we have\n\\begin{eqnarray}\nP^0_0 &=& -\\frac{t^2 f+2(H - J)f_{R} + 2Ht \\dot{f_{R}}}{2t^2} \\: ,\\\\\nP^{\\alpha}_{\\alpha} &=& \\frac{-3t^2 f+2 H(H-1)f_R -4Ht\\dot{f_R}-6t^2 \\ddot{f_R}}{2t^2} \\:  ,\\\\\nP^{\\mu}_{\\mu}-P^{\\nu}_{\\nu} &=& \\frac{1}{t^2}(p_{\\mu} -\np_{\\nu}) \\left( (H-1)f_{R} + t\\dot{f_R} \\right )  \\qquad \\mbox{and} \\label{eq:anisor}\\\\\nR &=& \\frac{J-2H+H^2}{t^2} \\: ,    \\label{eq:r}\n\\end{eqnarray}where $\\mu, \\nu$ in equation (\\ref{eq:anisor}) for the anistropic stress are not indices to be summed over, but instead are used to label the Kasner exponents, taking values from $1$ to $3$. Otherwise summation convention is used as normal. Recall that we may allow $f$ to contain a cosmological constant term, $f_{R}$ is used to denote $\\frac{df}{dR}$ and overdots represent derivatives with respect to the coordinate time, $t$.\n\nGiven the form of equation (\\ref{eq:anisor}), in what follows we shall consider separately the situations where the metric is isotropic and where it is anisotropic, both for vacuum and with the inclusion of a perfect fluid.\n\n\\subsubsection{Isotropic power-law vacuum solutions}\nIf all the Kasner exponents are equal, the solution is isotropic, with $p_{1}=p_{2}=p_{3}=\\frac{H}{3}$,\n$R=\\frac{2H(2H-3)}{3t^2}$ and there is only one independent field equation:\n\\begin{equation} 3t^2 f-2H(H-3) f_{R}+ 6Ht\\dot{f_{R}} = 0 \\: .\n\\end{equation}\nFor a particular theory, that is to say a particular choice of $f(R)$, this equation\nmight appear to provide a constraint on $H$, the one free constant remaining.\nHowever, in general, it is algebraic in $H$ \\textit{and} $t$ and therefore one\nwill not always be able to find \\textit{constant} solutions of this equation\nfor $H$; in fact such solutions are rare.\n\nFor models with Lagrangians of the form $f=f(R)$, it is possible to summarise the full set of possible isotropic power-law vacuum solutions, and the conditions on the function $f$ for these to be valid, using the classification in table \\ref{tab:isoRv}.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n    \\begin{tabular}{ l || l | p{9.5cm} }   \nClass & Solution & Validity \\\\[2pt] \\hline \\hline \n\\textbf{0} & $H=0$ & Minkowski space is a vacuum solution in any model with $f(0)=0$. \\\\[4pt]\n\\textbf{I} & $H=3\/2$ & This solution, analagous to the radiation-dominated Friedmann universe of General Relativity, is a solution of the \\textit{vacuum} field equations for any model satisfying $f(0)=0$ and $f_{R}(0)=0$. \\\\[4pt]\n\\textbf{II} & $H=\\frac{3(2n-1)(n-1)}{2-n}$ & This class of solutions is possible if and only if the Lagrangian takes the form\n$f=\\alpha_{n} R^{n}+\\alpha_{m} R^{m}$, where $n \\notin \\{0,5\/4,2 \\}$, $m=\\frac{4n-5}{2(n-2)}$ and $\\alpha_{n}, \\alpha_{m}$ are constants, with $\\alpha_n \\neq 0$. These solutions are expanding to the future if $n< 1\/2$ or $1<n<2$. \\\\\n\\end{tabular}\\caption{Isotropic power-law vacuum solutions in $f(R)$ gravity.}\\label{tab:isoRv}\n\\end{center}\n\\end{table}\n\nThe class II solutions were found previously in \\cite{clifton1} in the context of models where the Lagrangian is a power of the scalar curvature, $f=R^n$. However, it is important to note that for all such models with $n>1$, the solution of class I, which has zero scalar curvature, also solves the vacuum field equations, a fact which was not stressed in \\cite{clifton1}. Furthermore, for any model in which the Lagrangian may be written as a power series in $R$, a non-zero linear term (i.e. an Einstein-Hilbert term) in the series precludes the possibility of isotropic power-law vacuum solutions other than Minkowski space.\n\n\\subsubsection{Isotropic power-law solutions with a perfect fluid}\nLet us now consider a universe filled with a comoving perfect fluid, with equation of state $p= w \\rho$, and energy density evolving as $\\rho (t) = \\rho_0 t^{-H(1+w)}$, where $\\rho_0$ is a constant. The possible isotropic power-law solutions of the field equations for a Lagrangian of the form $f=f(R)$ are similar to those found in the vacuum case and are summarised in table \\ref{tab:isoRf}.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n    \\begin{tabular}{ l || l | l | p{6cm} } \nClass & Solution & Equation of state & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $H=3\/2$ & $w=1\/3$ & This is a solution in any model satisfying $f(0)=0$ and $f_{R}(0)= constant \\neq 0$.  \\\\[4pt]\n\\textbf{II} & $H=\\frac{2n}{1+w}$ & $w \\neq -1$  & This class of solution exists iff the Lagrangian is a power of the scalar curvature, $f=\\alpha_{n} R^{n}$, where $n \\neq 0$ and $\\alpha_{n}$ is a constant. \\\\[4pt]\n\\textbf{III} & $H=\\frac{3(2n-1)(n-1)}{2-n}$ & $w=-1+\\frac{2p(2-n)}{3(2n-1)(n-1)}$ & This is a solution iff the Lagrangian takes the form\n$f=\\alpha_{n} R^{n}+\\alpha_{m} R^{m}+\\alpha_{p}R^{p}$, where $n \\notin \\{0,5\/4,2 \\}$, $m=\\frac{4n-5}{2(n-2)}$ and $\\alpha_{p}\\neq0 , \\alpha_{n}$ and $\\alpha_{m}$ are constants. For power-law Lagrangians ($\\alpha_m=\\alpha_n=0$), this solution reduces to a subset of the class II solutions. \\\\\n\\end{tabular}\\caption{Isotropic power-law solutions in $f(R)$ gravity with a comoving perfect fluid.} \\label{tab:isoRf}\n\\end{center}\n\\end{table}\n\nThe solution of class I is analagous to the radiation-dominated Friedmann universe of General Relativity, and also to the class I \\textit{vacuum} solution when an extra Einstein-Hilbert term is added to the Lagrangian.\n\nIt is apparent that the vacuum solutions found in the previous section correspond to fluid-filled solutions with an appropriate extra curvature term in the Lagrangian.\n\n\\subsubsection{Anisotropic vacuum solutions}\nWe have seen that if the Lagrangian is a function of the scalar curvature only, $f=f(R)$, there is a strong constraint on its form for the\nexistence of exact isotropic power-law solutions. We shall now turn our attention to the situation\nfor anisotropic Kasner-like solutions within these models, the main focus of this paper. In this case, the vacuum field equations can be reduced to \\begin{eqnarray}\n(H-1)f_{R}+t\\dot{f_{R}} &=& 0 \\label{c1} \\: , \\\\\nf &=& 0 \\label{c2} \\: ,\\\\\nRf_{R} &=& 0  \\label{c3} \\: .\n\\end{eqnarray}\n\nIf the arbitrary function $f$ is non-trivial and algebraic in the scalar curvature, $R$, then the only possible solutions for $R$ of $f(R)=0$ must be constants, and therefore any such solution satisfies $R=0$ and thus $J=2H-H^2$. Recall that from our definitions, solutions with zero scalar curvature for which the constants $p_{\\alpha}$ take real values must satisfy the constraints $0\\leq H \\leq 3\/2$ and $J \\leq 1$. According to these constraints, we see that the spacetime volume of any solution of this form must expand no faster than $t^{3\/2}$, and that the expansion rate in any one direction can be no faster than $t$.\n\n\nWhilst $f(0)=0$ is a necessary condition for a $f(R)$ model to contain anisotropic Kasner-like solutions in vacuum, and this also implies that equation (\\ref{c3}) is satisfied at $R=0$, it is not sufficient, since $f_R(0)$ is model-dependent. Thus it is not guaranteed that there will be solutions of equation (\\ref{c1}) with \\textit{constant} $H$ in models where $f_R(0)$ diverges; in this case, solutions will only exist if the divergent part of the Lagrangian is a power of the scalar curvature. If, however, $f_{R}(0)$ is zero, then equation (\\ref{c1}) is satisfied trivially. If $f_R(0)$ is a non-zero constant, as is the case in general relativity, defined by $f(R)=R$, then this gives the extra constraint $H=1$, whence $J=1$ and the only possible solutions are those of general relativity.\n\nTo exemplify the situation, we could consider a Lagrangian which can be expanded as a power series about $R=0$. A non-zero linear term - an effective Einstein-Hilbert term - in the series precludes the possibility of anisotropic Kasner-like vacuum solutions other than the general relativistic one, with $H=J=1$, whilst a non-zero constant term - an effective cosmological constant - would preclude the possibility of these solutions altogether.\n\nA summary of the conditions that must be satisfied by the model in order for solutions of this kind to exist and classification of the possible solutions is given in table \\ref{tab:kasR}.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n    \\begin{tabular}{ l || p{3cm} | p{9cm} } \nClass & Solution constraints & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $J=2H-H^2$ & These solutions are possible if $f(0)=0$ and $f_{R}(0)=0$. The exponents $p_{\\alpha}$ are real provided $0\\leq H \\leq 3\/2$. \\\\[4pt]\n\\textbf{II} & $H=J=1$  & If the model satisfies $f(0)=0$, but $f_{R}(0)$ is a non-zero constant, then the extra constraint from equation (\\ref{c1}) means that only this subset of the first class of solutions is possible. These are the solutions for the case of general relativity, $f(R)=R$. \\\\[4pt]\n\\textbf{III} & $H=2n-1$, \\newline $J=(2n-1)(3-2n)$ & This subset of the first class of solutions is relevant if $f(0)=0$, but $f_{R}(0)$ diverges due to a term which is a power of the scalar curvature. Thus, the Lagrangian is required to be of the form $f=\\alpha R^n+ \\hat{f}(R)$, with $0<n<1$ and $\\hat{f}(0)=\\hat{f}_R(0)=0$, although $\\hat{f}$ need not be identically zero. These solutions are complex and of limited physical interest if $n < 1\/2$. \\\\\n\\end{tabular} \\caption{Kasner-like vacuum solutions in $f(R)$ gravity.} \\label{tab:kasR}\n\\end{center}\n\\end{table}\n\nThe first class of solutions contains a subset for which all the Kasner indices may be positive; this is possible if $1<H<3\/2$.  We have seen that the existence of such solutions means that the infinite series of chaotic oscillations on approach to the initial singularity in the more general Bianchi type IX Mixmaster universe can be avoided. \n\nThe third class of these solutions are relevant for models in which the Lagrangian is an arbitrary power of the scalar curvature, $f=\\alpha R^n$, and were found in that context by Clifton and Barrow \\cite{clifton1}. However, for $n>1$, we have found here that there are additional exact Kasner-like solutions corresponding to $R=0$ which do not necessitate that $H=2n-1$ and are subject only to $J=2H-H^2$.\n\n\n\\subsubsection{Anisotropic solutions with perfect fluid}\nIt is interesting to also include the possibility of a comoving perfect fluid with a barotropic equation of state, $p=w \\rho$. The energy density of the fluid is given by\n\\begin{equation}\n \\rho = \\rho_0 t^{-H(1+w)} \\: .\n\\end{equation}\nIf the fluid is comoving, the spatial part of the energy-momentum tensor is isotropic and so equation (\\ref{c1}) still holds. This can be solved to give $f_R \\propto t^{1-H}$ and the remaining field equations simplify to\n\\begin{eqnarray}\n \\rho(1+w) &=& -R f_R  \\: ,\\\\\nw \\rho &=& -f\/2 \\:  .\n\\end{eqnarray}\nThe only solutions of these equations with constant non-zero energy density correspond to a cosmological constant, which was considered together with the vacuum case in the previous section. For all other fluid-filled solutions, $\\rho \\propto t^{-H(1+w)}$ and $R \\propto t^{-2}$, and thus the field equations can only have a solution of this sort if the Lagrangian is a power of the scalar curvature, $f(R)=R^n$, for some $n \\in \\mathbb{R} \\backslash \\{0\\}$, and $R \\neq 0$. For $f=R^{1\/2}$, the right hand sides of the above equations are identically equal, hence to allow a non-zero energy density, we further require $n \\neq 1\/2$.\n\nFor all $n \\neq 1\/2$, these equations have the solution $H=2n-1, w=(2n-1)^{-1}$. Since these solutions need $R \\neq 0$, it is also required that $J \\neq (2n-1)(3-2n)$, but otherwise it is unrestricted. \n\n\\subsubsection{Examples of solutions for specific choices of $f(R)$}\nTo summarise the results of the previous sections, in table \\ref{tab:existR} we consider some of the more commonly-studied choices for the Lagrangian, $f(R)$, and state whether these models contain Minkowski space, isotropic power-law solutions, and exact anisotropic Kasner-like solutions, both in vacuum and in the presence of a comoving perfect fluid, according to the classifications given in the preceeding sections. We include the choices of power-law Lagrangians \\cite{clifton1}, and two exact $f(R)$ models which have recently been proposed, by Starobinsky \\cite{starob2}, and by Hu and Sawicky \\cite{hu}. These models are of particular interest since they provide viable cosmologies and evade the known constraints on the form of $f(R)$ from solar system tests. They are given by\n\\begin{eqnarray}\nf(R) = f_{\\text{Star}}(R) & \\equiv & R-\\lambda R_{0}\\left(1-\\left(\\frac{1}{1+(R\/R_{0})^2}\\right)^{n}\\right) \\quad \\text{and}\\\\\nf(R) = f_{\\text{Hu}}(R) & \\equiv & R+\\lambda R_{0}\\frac{(R\/R_{0})^n}{1+\\alpha (R\/R_{0})^n}\n\\end{eqnarray}\nrespectively. The parameter $n$ is taken to be positive for both the Starobinsky and the Hu-Sawicky models, in order to ensure their viability with observations.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n\\begin{tabular}[ht]{l||c|c|c|c|c}\n $f(R)$ & Minkowski & \\multicolumn{2}{c|}{Power-law} & \\multicolumn{2}{c}{Kasner}\n\\\\\n & & in vacuum & with fluid & in vacuum & with fluid \\\\ \\hline\\hline\n $R$ & $\\checkmark$ & $\\times$ & Class I, II & Class II & $\\checkmark$  \\\\\n $R+\\Lambda$ & $\\times$ & $\\times$  & $\\times$    & $\\times$ & $\\times$ \\\\\n $R+\\alpha R^{2}$ & $\\checkmark$ & $\\times$ & Class I,III  & Class II & $\\times$ \\\\\n $R+\\alpha \/R$ & $\\times$ & $\\times$ & Class III  & $\\times$ & $\\times$ \\\\\n $R^{n}, n<0$ & $\\times$ & $\\times$ & Class II & $\\times$ & $\\checkmark$  \\\\\n $R^{n}, 0<n<\\frac{1}{2}$ & $\\checkmark$ & Class II & Class II & $\\times$ & $\\checkmark$ \\\\\n $R^{1\/2}$ & $\\checkmark$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ \\\\\n $R^{n}, \\frac{1}{2}<n<1$ & $\\checkmark$ & Class II & Class II & Class III & $\\checkmark$  \\\\\n $R^{n}, n>1$ & $\\checkmark$ & Class I, II & Class II & Class I & $\\checkmark$  \\\\\n   $\\exp{(R\/R_{0})}$ & $\\times$ & $\\times$ & $\\times$     & $\\times$ & $\\times$ \\\\\n $f_{\\text{Star}}$ & $\\checkmark$ & $\\times$ & Class I     & Class II & $\\times$ \\\\\n $f_{\\text{Hu}}, n \\geq 1$ & $\\checkmark$ & $\\times$ & Class I     & Class II & $\\times$ \\\\\n   $f_{\\text{Hu}}, 1>n>0$ & $\\checkmark$ & $\\times$ & $\\times$     & $\\times$ & $\\times$ \\\\\n   $f_{\\text{Hu}}, n<0$ & $\\times$ & $\\times$ & $\\times$     & $\\times$ & $\\times$    \n\\end{tabular} \\caption{The existence of power-law and Kasner-like solutions in various models of $f(R)$ gravity.}   \\label{tab:existR}\n \\end{center}\n\\end{table}\n\n\n\\subsubsection{Remarks}\nFriedmann-Robertson-Walker power-law solutions in $f(R)$ gravity with a perfect fluid were investigated in \\cite{dunsby}. There, it was claimed that the only possible form of Lagrangian which admits these solutions and has the correct general relativistic limit is a power-law, $f=R^n$. In fact, there is another class of isotropic power-law solution (type I in our classification) corresponding to a radiation-filled universe, which is possible in any theory for which $f(0)=0$ and $f_{R}(0)$ is constant, however comoving perfect fluids with other equations of state are not possible. Furthermore, we have seen here that it is only the $R^{n}$ Lagrangians which allow anisotropic Kasner-like solutions with a perfect fluid. We can see, therefore, that these $R^{n}$ theories of gravity are special in admitting this sort of solution. Some Bianchi type I, III and Kantowski-Sachs solutions in $f(R)$ gravity have also been investigated recently by Farasat Shamir \\cite{Shamir:2010ee}.\n\nAnisotropic singularities of Bianchi Type I were recently studied in the context of more general Lagrangians of the type $f(R, \\phi, \\chi)$, with $\\chi = -\\frac{1}{2} g^{ab} \\partial _{a} \\phi \\partial _{b} \\phi$ \\cite{saa}, with particular focus on the anistropic instabilities related to the existence of the hypersurface $\\frac{\\partial f}{\\partial R} =0$ and solutions being able to cross this surface, leading to questions about the viability of these models. Here, it has been shown that exact anisotropic Kasner-like solutions in $f(R)$ theories must have $R=0$ and although they can live on the hypersurface defined by $\\frac{\\partial f}{\\partial R} =0$, they cannot cross it.\n\n\\subsection{$f=f(R^{ab}R_{ab})$}\nLet us now consider the case where the Lagrangian is a function of the Ricci invariant only. For isotropic cosmologies, the contributions to the field equations from the simplest such term, $R_{ab}R^{ab}$, are proportional to those from a term quadratic in the scalar curvature, $R^2$. However, this is not true more generally, and the Ricci term allows much more diverse anisotropic behaviour \\cite{hervik}.\n\n  For this class of theories, the relevant field equations for the metric (\\ref{kas}) in vacuum are $P^{a}_{b} = 0 $, where\n\\begin{eqnarray}\nP^0_0 & = & \\frac{1}{2t^4}\\left(-t^4 f +\n 2 (-H^2 + H^3 - HJ +2J^2-J) f_{Y} + 2(H^2 + J - 2 HJ) t \\dot{f_{Y}} \\right)\\: ,\\label{eqY1}\n\\\\\nP^{\\mu}_{\\mu}-P^{\\nu}_{\\nu} &=& \\frac{1}{t^4}(p_{\\mu} -\np_{\\nu}) \\left(2 (H-3)(J - 1) f_{Y} + (-4 + 2 J + 3 H - H^2)t\n\\dot{f_Y}+(1-H) t^2 \\ddot{f_{Y}} \\right)  \\label{eqY3} \\: , \\end{eqnarray}\nwhere as before $\\mu, \\nu$ in equation (\\ref{eqY3}) for the anistropic stress are not indices to be summed over, but are used here as labels, taking values from $1$ to $3$. Otherwise summation convention is used as normal. Recall that $f_{Y}$ is used to denote $\\frac{df}{dY}$ and overdots represent derivatives with respect to the time coordinate $t$. The Ricci term is given by\\begin{equation}\nY \\equiv R_{ab}R^{ab} = \\frac{J(H-1)^2+(J-H)^2}{t^4}  \\, .\\label{Y}\n\\end{equation}\n\nIt is useful to recall our earlier observation that for real-valued choices of the constants $p_{\\alpha}$, the Ricci term, $Y$, takes non-negative values, and is zero if and only if $H=J=1$ or $H=J=0$.\n\n\\subsubsection{Isotropic power-law solutions}\nFor $f=f(Y)$, and the special case of an isotropic metric, $p_{1}=p_{2}=p_{3}=H\/3$, so that $J=H^2 \/3$, the\nequations reduce to:\n\\begin{equation}\n-9 t^4 f + 4H^2((-6 + 3 H + H^2) f_Y - 3 (H-2) t \\dot{f_Y}) = 0 \\:  .\n\\end{equation}\nBy an argument analagous to that used before for the $f(R)$ theories, $f(Y)$ can be zero for all times only if $Y$ is a constant, and therefore zero. Using (\\ref{Y}), it is clear that Minkowski space is the only real isotropic power-law solution for which $Y=0$. If $Y$ is non-zero, then one can eliminate the time variable $t$ and instead consider this equation as a differential equation in $Y$. It may then be integrated to find that, in order to admit solutions of this sort, the Lagrangian is required to be of the form of a power of the Ricci invariant, or possibly a sum of two such terms.\n\nWe can now summarise the existence conditions for vacuum solutions of this type for theories with Lagrangians of the form $f=f(Y)$, using the definitions $H_{\\pm}(m)\\equiv \\frac{3- 9 m + 12 m^2 \\pm \\sqrt{3(-1 + 10 m - 5 m^2 - 40 m^3 + 48m^4)}}{2(1-m)}$ and $n_{\\pm}(m) \\equiv\n\\frac{2m-1}{2(m-1)}+\\frac{1}{2 + 10 m - 24 m^2 \\mp\n 2\\sqrt{3(-1 + 10 m - 5 m^2 - 40 m^3 + 48 m^4)}}$. This is found in table \\ref{tab:isoY}.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n    \\begin{tabular}{ l || l | p{9cm} } \nClass & Solution & Validity \\\\[2pt] \\hline \\hline \n\\textbf{0} & $H=0$ & Minkowski space is a solution in any model with $f(0)=0$. \\\\[4pt]\n\\textbf{I} & $H=3\/2$ & This is a solution if the Lagrangian is linear in $Y$, $f(Y) = \\alpha Y$, where $\\alpha$ is a non-zero constant.  \\\\[4pt]\n\\textbf{IIa} & $H=H_{+}(m)$ & This solution is possible if and only if the Lagrangian takes the form\n$f=\\alpha_{m} Y^{m}+\\alpha_{n} Y^{n}$, where $m \\neq 1$, $n=n_{+}(m$ and $\\alpha_{n}, \\alpha_{m}$ are constants, with $\\alpha_m \\neq 0$. These solutions are expanding to the future if $m<1$ \\\\[4pt]\n\\textbf{IIb} & $H=H_{-}(m)$ & This solution is possible if and only if the Lagrangian takes the form\n$f=\\alpha_{m} Y^{m}+\\alpha_{n} Y^{n}$, where $m \\neq 1$, $n=n_{-}(m)$ and $\\alpha_{n}, \\alpha_{m}$ are constants, with $\\alpha_m \\neq 0$. These solutions are expanding to the future if $m<1\/4$ or $m>1\/2$.\\\\\n\\end{tabular} \\caption{Isotropic power-law vacuum solutions in $f(R_{ab}R^{ab})$ gravity.} \\label{tab:isoY}\n\\end{center}\n\\end{table}\n\nAs with the situation for the Lagrangian $f=R^2$, we see that the radiation-dominated Friedmann universe of General Relativity is a vacuum solution of the higher-order quadratic theory. \n\nWhilst $H_{+}$ is unbounded as $n\\rightarrow \\pm \\infty$, $H_{-}$ is bounded and tends to $2$ in both these limits. For power-law Lagrangians in the Ricci term, both class IIa and IIb solutions are valid; these were found in \\cite{clifton2} and their stability on approach to the initial singularity under small perturbations of the metric was previously studied in \\cite{midd2}.\n\n\n\\subsubsection{Anisotropic vacuum solutions}\nFor anisotropic Kasner-like solutions, we again consider separately the cases $Y=0$ and $Y \\neq 0$, so that in the latter scenario, we can replace the time variable $t$ and treat the field equations (\\ref{eqY1}-\\ref{eqY3}) as differential equations in the Ricci term, $Y \\equiv R_{ab}R^{ab}$. Similarly to the situation for isotropic power-law metrics in these theories, anisotropic solutions of the field equations with $Y \\neq 0$ can only exist if the Lagrangian for the theory is a power of the Ricci term, with $f=\\alpha Y^n$.\n\nThe conditions for the existence of anisotropic Kasner-like solutions within this class of theories and the constraints that must be satisfied by the Kasner indices may be summarised as in table \\ref{tab:kasY}.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n    \\begin{tabular}{ l || p{4cm} | p{8cm} } \nClass & Solution constraints & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $H=J$ & There is a family of anisotropic solutions of this sort if $f(Y)=\\alpha Y^{1\/2}$.  The Kasner exponents $p_{\\alpha}$ are real provided that $0\\leq H \\leq 3$. \\\\[4pt]\n\\textbf{II} & $H=J=1$  & These solutions require that $f(0)=0$, and also that $f_{Y}(0)$ either converges to a constant or diverges slower than $Y^{-1\/2}$. \\\\[4pt]\n\\textbf{III} & $H=(1-2n)^2$, \\newline $J=(1-2n)(1-6n+4n^2)$ & These are solutions for Lagrangians of the form $f(Y) = \\beta Y^n$. In order for the Kasner exponents to be real, it is needed that $(1-2n) (1- 6n + 4n^3) \\geq 0$. \\\\\n\\end{tabular} \\caption{Kasner-like vacuum solutions in $f(R_{ab}R^{ab})$ gravity.} \\label{tab:kasY}\n\\end{center}\n\\end{table}\n\nThe first class of solutions have real Kasner exponents provided $0\\leq H \\leq 3$. The volume of these solutions can thus expand no faster than $t^{3}$, but the individual exponents must lie in the range $\\frac{1}{2}\\left( 1-\\sqrt{3}\\right) \\leq p_{\\alpha} \\leq \\frac{1}{2}\\left( 1+\\sqrt{3}\\right)$, and so the expansion may be faster than the speed of light in a particular direction.\n\nThe second class of solutions are the general relativistic solutions, with $H=J=1$; we have seen that these are the only anisotropic metrics of Kasner type for which the Ricci term is zero. It is interesting to compare these with the class II anisotropic vacuum solutions found in the previous section for $f(R)$ models, which also have $H=J=1$. In that context the solutions require that $f_{R} (0)$ is constant, and thus $f \\sim R$ as $R \\rightarrow 0$, whilst here $f \\sim Y^{1\/2}$ (or higher powers) as $Y \\rightarrow 0$.\n\nThe third class are the solutions found in \\cite{clifton2}, in which theories where the Lagrangian is a power of the Ricci term were previously investigated. The condition that the third class of solutions have real exponents is satisfied if either $n_{1} \\leq n \\leq n_{2}$ or $1\/2 \\leq n \\leq n_{3}$, where $n_{1}, n_{2}$ and $n_{3}$ are the roots of $(1- 6n + 4n^3)=0$, chosen such that $n_{1}<n_{2}<n_{3}$. These roots have approximate numerical values $n_{1}\\approx -1.30,  n_{2} \\approx 0.17$ and $n_{3} \\approx 1.13$. These boundaries correspond to isotropic solutions and one can see their relevance for the existence of isotropic solutions in \\cite{midd2}. For $n>0$, both $H$ and $J$ must be less than unity, and so for positive $n$ the expansion of the solution cannot accelerate. However, $H>1$ for these solutions if $n<0$, and in particular for models with $ n_{1} \\leq n \\leq \\frac{1}{2}\\left( 1-\\sqrt{3}\\right)$, $H$ will be greater than $3$ and so the volume of the spacetime must increase faster than $t^{3}$ and undergo an accelerated expansion. Whilst this third class of solutions includes the special case of quadratic gravity defined by $n=1$, it is easy to see that the conditions give the second class of solutions, $H=J=1$, and there is no additional set of solutions in this case. \n\n\n\\subsubsection{Examples for specific $f(R^{ab}R_{ab})$}\nWe are now able to summarise the possible vacuum Kasner-like solutions for various choices of Lagrangian of the form $f(Y)$. In table  \\ref{tab:existRic}, we consider several more common examples of this type of model and detail whether they allow isotropic power-law or Kasner-like solutions in vacuum, according to the classification systems we have used in the preceeding sections. In this table, use is also made of the definitions given in the previous section; recall that $n_{1}, n_{2}$ and $n_{3}$ are the roots of $(1- 6n + 4n^3)=0$, chosen such that $n_{1}<n_{2}<n_{3}$, and have approximate numerical values $n_{1}\\approx -1.30,  n_{2} \\approx 0.17$ and $n_{3} \\approx 1.13$.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n\\begin{tabular}[ht]{l||c|c|c}\n\n {$f(Y)$} & Minkowski & Power-law & Anisotropic Kasner \\\\ \\hline\\hline\n $Y$ & $\\checkmark$ & Class I & Class II \\\\\n $Y+\\Lambda$ & $\\times$ & $\\times$      & $\\times$ \\\\\n $Y^{n}, n<n_{1} $ & $\\times$ & Class II & $\\times$ \\\\\n $Y^{n}, n_{1}<n<0 $ & $\\times$ & Class II & Class III \\\\\n $Y^{n}, 0<n<n_{2} $ & $\\checkmark$ & Class II & Class II,III \\\\\n $Y^{n}, n_{2}<n<\\frac{1}{2} $ & $\\checkmark$ & Class II & Class II \\\\\n $Y^{1\/2}$ & $\\checkmark$ & Class II   & Class I \\\\\n $Y^{n}, \\frac{1}{2}<n<n_{3}, n\\neq 1 $ & $\\checkmark$ & Class II & Class II, III \\\\\n $Y^{n}, n_{3}<n$ & $\\checkmark$ & Class II  & Class II \\\\\n $Y+\\alpha Y^2$ & $\\checkmark$ & $\\times$      & Class II \\\\\n $exp(Y\/Y_{0})$ & $\\times$ & $\\times$      & $\\times$ \\\\\n $sin(Y\/Y_{0})$ & $\\checkmark$ & $\\times$      & Class II \n\\end{tabular}\\caption{The existence of power-law and Kasner-like vacuum solutions in various models of $f(R_{ab}R^{ab})$ theories of gravity.}     \\label{tab:existRic}\n\\end{center}\n\\end{table}\n\n\\subsection{$f=f(Z)$}\nFor functions of the Kretschmann scalar, $Z \\equiv R_{abcd}R^{abcd}$, only, the field equations for the metric (\\ref{kas}) in vacuum reduce to $P^{a}_{b}=0$, where\n\\begin{eqnarray}\n P^0_0\n &=& -\\frac{1}{2}f +2Z f_Z+\\frac{4}{t^4}(J-K)\\left((H-3)f_Z+t \\dot{f_Z}\\right)\\\\\nP^{\\mu}_{\\mu}-P^{\\nu}_{\\nu} &=& 2 \\frac{(p_{\\mu}-p_{\\nu})}{t^4} \\biggl(2 \\left(H-p_{\\mu}-p_{\\nu}\\right) t \\left((H-2) \\dot{f_Z}+t \\ddot{f_Z}\\right) + (3-H) (4 -2H + H^2- 3 J)f_Z \\nonumber \\\\\n&& + (-8 + 8H - 3 H^2+ 3 J)t\\dot{f_Z} + 2 (1 - H) t^2 \\ddot{f_Z}\\biggr) \\: , \\qquad \\text{with} \\label {eq:zaniso}\\\\\nZ\n&=& \\frac{(3J-H^2)^2 +12J+8K(H-3)}{3t^4} \\:  .\n\\end{eqnarray} Once again $\\mu, \\nu$ in equation (\\ref{eq:zaniso}) for the anisotropic stress are not indices to be summed, but are used to label the Kasner exponents, taking values from $1$ to $3$. Otherwise summation convention is used as normal. Recall that $f_{Z}$ denotes $\\frac{df}{dZ}$ and overdots are used to represent derivatives with respect to the time coordinate $t$.\n\nRecall also that, by expressing $Z$ explicitly in terms of the Kasner exponents, it can be seen that the only possible real solutions of $Z=0$ are Minkowski space and the Milne model, which may be defined by $p_{1}=1, p_{2}=p_{3}=0$ and is related to the Minkowski solution by the coordinate transformation $\\tau =  t \\cosh{x}, \\chi=t \\sinh{x}$. Thus, for all other Kasner-like metrics with $Z \\neq 0$, it is possible as before to substitute for the time variable $t$ using $Z$ and integrate to find the requirements on the Lagrangian $f=f(Z)$ in order to permit such metrics as solutions of the field equations. Using this method, it is found that in order for Kasner-like solutions to exist in vacuum, the Lagrangian must be a power of the scalar $Z$, or possibly a sum of two terms if they permit the same values of the Kasner exponents $p_{\\alpha}$. Such power-law Lagrangians were first investigated in \\cite{clifton2}.\n\n\\subsubsection{Anisotropic solutions}\nFor the previous two types of models that were considered, with $f=f(R)$ or $f=f(R^{ab}R_{ab})$, there was only one independent equation for the anisotropic stress. In contrast, for models with Lagrangians that are functions of the Kretschmann scalar, $f=f(R^{abcd}R_{abcd})$, there are in general \\emph{two} independent equations for the anisotropic stress. This appears to be due to the fact that the Kretschmann scalar cannot be expressed solely in terms of the parameters $H$ and $J$, unlike the scalar curvature and Ricci scalar. For the cases of interest, where the Lagrangian is specified by $f= \\alpha Z^n$, with $n \\neq 0$, these simplify to\n\\begin{eqnarray}\n0 &=& (p_{1}-p_{2}) (1+H-4n)(-4-3J + H(6+H-8n)+ 8n+8p_{3} (n-1)) \\: ,\\\\\n0 &=& (p_{2}-p_{3}) (1+H-4n)(-4-3J + H(6+H-8n)+ 8n+8p_{1} (n-1)) \\: .\n\\end{eqnarray}\nTogether, these imply that \n\\begin{equation}\n0 = (p_{1}-p_{2})(p_{1}-p_{3})(p_{2}-p_{3})(1+H-4n)(n-1) \\: , \\end{equation} and so fully anisotropic solutions with no two Kasner indices equal can be possible only if either $n=1$ or $H=4n-1$. However, substituting $H=4n-1$ into the remaining independent field equation $P^1_1=0$ leads to the further requirement that $Z=0$, for which we have seen that the only real solutions are Minkowski spacetime and the Milne model. The only other possibility for a fully anisotropic solution with $p_{\\alpha}$ all real and distinct is in the theory with $n=1$, i.e. $f=\\alpha Z$, a particular case of the commonly studied class of quadratic gravity theories, which will be discussed in more generality in the next section. Using the remaining independent field equation, such anisotropic solutions must have $H=J=1$. Recall that these were also solutions for the other special cases of quadratic gravity considered in the previous sections; $f=\\alpha R^{2}$ and $f=\\alpha Y$.\n\n\\subsubsection{Locally rotationally symmetric solutions}\nUnlike the situation for the previous two classes of model, with $f=f(R)$ or $f=f(R^{ab}R_{ab})$, there are, in general, two equations for the anisotropic stress in those models with $f=f(R^{abcd}R_{abcd})$. Consequently, in addition to the isotropic and fully anisotropic metrics one must also consider the scenario of locally rotationally symmetric (LRS) solutions which without loss of generality are described by $p_{2}=p_{3}=(H-p_{1})\/2$, $H\\neq 3p_{1}$.\n\nFor these LRS Kasner-like metrics, in the context of power-law Lagrangians, $f=\\alpha Z^n$, the field equations may be reduced to the two independent equations;\n\\begin{eqnarray}\n0&=& 8 - 8 p_{1} + 9 p_{1}\\!^2 - 4 H - 6 p_{1} H + H^2 + 8n(H+p_{1}-2) \\label{eq:lrs1}\\\\\n0 &=&  3 H^3 (n-1) - H^2 (4-23n-8n^2) +\n     2 H (1-2n)(20 -25n + 32 n^2) \\label{eq:lrs2}\\\\ && + (2n-1) (47-148n+128n^2) \n-     p_{1} (n-1) ((5 - 8 n) (13 - 16 n) - H (51 - 60 n)) \\: . \\nonumber \\end{eqnarray}\n\nFor all $n$, these equations are solved by $H=p_{1}=1$, giving $p_{2}=p_{3}=0$; this is the subset of the anisotropic solutions found in the previous section with $H=J=1$ which corresponds to the Milne model. For $n=1$ there are no other LRS solutions of this form. For $n \\neq 1$, the second of these equations is linear in $p_{1}$; substituting the solution one obtains for $p_{1}$ into the first equation gives a sixth order polynomial equation for $H$ in terms of $n$. However, factoring out the root $H=1$  leaves the fifth order polynomial:\n\\begin{eqnarray*}\n 0 &=& H^5 (n-1)^2 + H^4 (1-n) (15-41n + 8 n^2) +\n   H^3 (38 - 216 n + 283 n^2 + 56 n^3 - 80 n^4) \\\\ \n& & +\n   H^2 (-346 + 2156 n - 5421 n^2 + 7032 n^3 - 4560 n^4 + 896 n^5) \\\\\n&& + H (1-2n)^2 (653 - 2594 n + 3848 n^2 - 2688 n^3 + 1024 n^4)  \\\\\n&& - (1-2n)^2 (361 - 2104 n + 4768 n^2 - 4992 n^3 + 2048 n^4) \\:  . \n\\end{eqnarray*}\n\nConsequently, in addition to the Milne solution, there must always be at least one locally rotationally symmetric Kasner-like solution with real values for $p_{1}$ and $H$ for any real choice of $n$, other than $n=1$. For example, when $n=1\/2$, there is a solution given by $p_{1}=1, H=5$, giving $p_{2}=p_{3}=2$, whilst for $n=7\/8$ there are three real solutions, one of which is given by $p_{1}=-1\/2, H=1\/2$, so that $p_{2}=p_{3}=1\/2$. Thus it is dependent upon the choice of $n$ as to whether the signs of the Kasner exponents must all be positive or not. These solutions are new and were not found in \\cite{clifton2}.\n\n\n\\subsubsection{Isotropic power-law solutions}\nIn order to complete the discussion of this section, we consider the isotropic metrics, with $p_{1}=p_{2}=p_{3}=H\/3$. Except for Minkowski spacetime, which is a solution in any theory for which $f(0)=0$, the relevant equation for these metrics in vacuum in the context of a power-law Lagrangian, $f=\\alpha Z^n$, is\n\\begin{equation}\n0=-1+6n-8 n^2 + \\frac{2}{3}(1 - 2 n + 4 n^2) H + \\frac{2}{9} (n - 1) H^2 \\: . \\label{eq:isoz}\n\\end{equation} \nFor $n=1$, there is only one solution, with $H=3\/2$. For $n \\neq 1$, there are two solutions, given by\n\\begin{equation*}\nH=H_{\\pm} \\equiv \\frac{ \\left(-1 + 2 n -4n^2 \\pm \\sqrt{-1 +10n - 16 n^2 + 16n^4}\\right)}{6(n-1)} \\: ,\n\\end{equation*} which are real if $n\\leq n_{1} \\equiv -\\left(\\sqrt{5} + \\sqrt{3 + 2 \\sqrt{5}}\\right)\/4 \\approx -1.24$ or $n\\geq n_{2}\\equiv \\left(-\\sqrt{5} + \\sqrt{3 + 2 \\sqrt{5}}\\right)\/4 \\approx 0.12$.\n\nFinally, we note that if $H \\neq 3$, equation (\\ref{eq:isoz}) is quadratic in $n$ and so two values of $n$ correspond to the same value of $H$ and so Lagrangians which are a sum of two powers related by this equation also contain an isotropic power-law solution. Now $H=3$ for $n=\\frac{1}{4}$ and thus, for $f=\\alpha_{n} Z^n+\\alpha_{m} Z^m$, with $\\alpha_{n},\\alpha_{m} \\neq 0, (4n-1)(n-1)\\neq 0$ and $m=m_{\\mp} \\equiv \\frac{5-16n+20n^2 \\pm 3\\sqrt{-1+10n-16n^2+16n^4}}{8(4n-1)(n-1)}$, then there is a solution with $H=H_{\\mp}$.\n\n\\subsubsection{Summary}\n\nWe have seen that anisotropic Kasner-like solutions in higher-order theories of gravity where the Lagrangian depends only on the Kretschmann scalar, $Z \\equiv R_{abcd}R^{abcd}$, are possible only if the Lagrangian is of the form $f(Z)=Z^{n}$. Whilst it was claimed in an earlier work by Clifton and Barrow \\cite{clifton2} that there are no exact anisotropic solutions of the form (\\ref{kas}) in these theories, this is not quite correct. Firstly, for $n=1$, the Kasner solutions of general relativity, for which $H=J=1$, remain valid. This is to be expected, since the theory with $f=\\alpha Z$ is a special case of quadratic gravity and all vacuum solutions of general relativity must also be solutions of the pure quadratic theory \\cite{hervik2}. However, for all other values of $n$, fully anisotropic solutions of this form with the three Kasner exponents all real and distinct are not possibl\n. Nevertheless, for choices of $n$ other than $1$, we have seen that solutions which have local rotational symmetry but exhibit an anisotropic third spatial coordinate \\emph{are} possible and the Kasner indices are subject to equations (\\ref{eq:lrs1}) and (\\ref{eq:lrs2}). \n\nIn table \\ref{tab:existRiem}, these results are summarised to show whether isotropic power-law, locally rotationally symmetric and fully anisotropic Kasner-like solutions are possible for various choices of the Lagrangian $f(Z)$. Use is made of the definitions given in the previous section; $n_{1} \\equiv -\\left(\\sqrt{5} + \\sqrt{3 + 2 \\sqrt{5}}\\right)\/4 \\approx -1.24$ and $n_{2}\\equiv \\left(-\\sqrt{5} + \\sqrt{3 + 2 \\sqrt{5}}\\right)\/4 \\approx 0.12$.\n\n\\begin{table}[ht]\\small \n\\begin{center}\n\\begin{tabular}[ht]{l||c|c|c|c}\n$f(Z)$ & Minkowski & Power-law & LRS Kasner & Anisotropic Kasner \\\\ \\hline\\hline\n $Z$ & $\\checkmark$ & $\\checkmark$ & $\\times$ & $\\checkmark$ \\\\\n $Z+\\Lambda$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ \\\\\n $Z^{n}, n <n_{1} $ & $\\times$ & $\\checkmark$ & $\\checkmark$ & $\\times$ \\\\\n  $Z^{n}, n_{1}<n <0 $ & $\\times$ & $\\times$ & $\\checkmark$ & $\\times$ \\\\\n   $Z^{n}, 0<n<n_{2} $ &$\\checkmark$ & $\\times$ & $\\checkmark$& $\\times$ \\\\\n  $Z^{n}, n>n_{2}, n\\neq 1 $ & $\\checkmark$ & $\\checkmark$ & $\\checkmark$ & $\\times$ \\\\\n $\\alpha Z^{n}+\\beta Z^{m_{\\pm}}, n>0, n\\neq \\frac{1}{4},1$ & $\\checkmark$ & $\\checkmark$ & $\\times$ & $\\times$ \n\\end{tabular} \\caption{The existence of power-law and Kasner-like vacuum solutions in various models of $f(R_{abcd}R^{abcd})$ theories of gravity.} \\label{tab:existRiem}\n\\end{center}\n\\end{table}\n\n\\subsection{Quadratic gravity; $f=\\kappa R+\\alpha R^2 +\\beta R_{ab}R^{ab} +\\gamma R_{abcd}R^{abcd} +\\Lambda$ \\label{quad}}\nThe theory of quadratic gravity, in which the Einstein-Hilbert Lagrangian of general relativity is supplemented by quadratic Riemann, Ricci and scalar curvature corrections, is a particularly interesting special case to consider. The Lagrangian is given by \\begin{equation}\nf=\\kappa R+\\alpha R^2 +\\beta R_{ab}R^{ab} +\\gamma R_{abcd}R^{abcd} +\\Lambda \\, . \\label{quadlag} \\end{equation} It was shown by Starobinsky \\cite{starob1} that addition of quadratic curvature corrections to the Einstein-Hilbert Lagrangian leads to the emergence of inflation. Furthermore, in contrast to general relativity, fourth-order gravity is renormalisable \\cite{stelle} and thus it is often motivated as a first-order quantum correction to Einstein's theory. A review of the history of the study of these models of gravity may be found in \\cite{schmidt2}. \n\nWithout loss of generality, one may immediately set $\\gamma=0$, since the Gauss-Bonnet term, defined by $G \\equiv  R^2-4R_{ab}R^{ab}+R_{abcd}R^{abcd}$, is a total divergence in four dimensions, so its variational derivative with respect to the metric does not contribute to the field equations. If the Ricci term is not present in the Lagrangian, i.e. $\\beta = 0$, then the theory is a special of the $f(R)$ theories studied earlier and, moreover, it is conformally equivalent to that of a minimally coupled scalar field in general relativity. Thus, one expects that the presence of the Ricci term in the Lagrangian might permit much more diverse anisotropic behaviour, as was found in \\cite{hervik}. The quadratic theory is scale-invariant iff $\\kappa \\Lambda = 0$ and conformally invariant iff $\\kappa = \\Lambda = 0$ and $3\\alpha+\\beta =0$.\n\nIn order to obtain the correct Newtonian limit in the slow-motion weak-field limit, it is necessary that the fourth-order terms contributed by the quadratic parts of the Lagrangian are exponentially vanishing, rather than oscillatory. To ensure that this is the case, the parameters must satisfy $\\beta \/ \\kappa \\leq 0$ and $(3\\alpha+\\beta)\/\\kappa \\geq 0$, with $\\kappa \\neq 0$.\n\n\\subsubsection{Field equations}\nFor the line element given by (\\ref{kas}), the vacuum field equations for the theory of quadratic gravity defined by the quadratic Lagrangian in equation (\\ref{quadlag}) are $P^{a}_{b}=0$, and the relevant independent quantities are given by \\begin{eqnarray}\nP^{a}_{a} &=& \\frac{1}{t^{4}}\\left((H^{2}-2H + J) (4 (H-3) (3 \\alpha + \\beta) - \n    \\kappa  t^2 ) + 4\\Lambda t^{4}\\right) \\\\\nP^{0}_{0}    &=& \\frac{1}{6t^{4}}\\bigl((3 J-H^2)\\left(3(J+H^{2})\\alpha +(3J-3+2H)\\beta \\right)+4H^{2}(2H-3)\\left(3\\alpha+\\beta \\right)  \\nonumber \\\\ && + 3(J-H^2) \\kappa t^2   + 6\\Lambda t^4  \\bigr) \\\\\n P^{\\mu}_{\\mu}- P^{\\nu}_{\\nu} &=& \\frac{1}{t^{4}}(p_{\\mu}-p_{\\nu})\\left((2 (H-3) ((H^{2}-2H + J) \\alpha + (J-1) \\beta) + (H-1)\\kappa t^2 \\right) \\, \\label{eq:qaniso} .\n\\end{eqnarray} \nAs before, $\\mu, \\nu$ in equation (\\ref{eq:qaniso}) for the anisotropic stress are not indices to be summed, but are used there as labels, taking values from $1$ to $3$. Otherwise summation convention is used as normal and overdots represent derivatives with respect to the coordinate time, $t$.\n\nOne can immediately see that the cosmological constant, $\\Lambda$, must be zero in order for any solutions of this sort to exist.\n\n\\subsubsection{$\\kappa \\neq 0$}\nIf the Einstein-Hilbert term is present in the Lagrangian, then the only possible isotropic power-law solution is Minkowski space, which requires $\\Lambda=0$. There is one family of anistropic solutions, that of general relativity, given by $H=J=1$, provided $\\Lambda =0$. In light of the results of the previous sections, this is not unexpected.\n\n\\subsubsection{$\\kappa = 0$}\nIn pure quadratic theories, with $\\kappa =0$, a non-zero energy-momentum tensor gives rise to a strong gravitational field and consequently spacetime is not asymptotically flat \\cite{pech}, but we include them for completeness of this discussion.\n\nAny isotropic solutions of the field equations require $\\Lambda =0$ and in general there are two such solutions; Minkowski space and the radiation-like solution with $H=3\/2$. For the special case of Weyl gravity, $3\\alpha +\\beta =0$, \\emph{all} power-law isotropic solutions are possible, since in the isotropic case, the contributions to the field equations from the $R^2$ and the Ricci terms in the Lagrangian are the same up to a constant multiple.\n\nThere are several possible families of anisotropic solutions; as we have pointed out they all require the cosmological constant, $\\Lambda$, to be zero. These solutions and the constraints that must be satisfied are summarised in table \\ref{tab:quad}\n\n\\begin{table}[ht]\\small\n\\begin{center}\n\\begin{tabular}{ l || p{5.5cm} | p{6.5cm} } \nClass & Solution constraints & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $H=J=1$ & This is a vacuum solution for all values of $\\alpha, \\beta$ and $\\kappa$. \\\\[4pt]\n\\textbf{II} & $J=\\frac{1}{2}(3-2H+H^2)$ & These are solutions for the Weyl theory of gravity only, that is if $3\\alpha+\\beta=0=\\kappa$. \\\\[4pt]\n\\textbf{III} & $H=3$, \\newline $J=\\frac{1}{\\alpha+\\beta}(-3\\alpha+\\beta-2\\sqrt{-2\\beta(3\\alpha+\\beta)})$ & These are valid with real exponents if $\\kappa = 0$ and $-\\beta \/3 < \\alpha < -\\beta$. \\\\[4pt]\n\\textbf{IV} & $H=3$, \\newline $J=\\frac{1}{\\alpha+\\beta}(-3\\alpha+\\beta+2\\sqrt{-2\\beta(3\\alpha+\\beta)})$ & These are valid with real exponents if $\\kappa = 0$ and $-\\beta  < \\alpha < -\\beta \/3$. \\\\[4pt]\n\\end{tabular}\n\\caption{Kasner-like vacuum solutions in pure quadratic gravity.} \\label{tab:quad}\n\\end{center}\n\\end{table}\n\n\n\nRecall that the first class of solutions are vacuum solutions in general relativity and so have vanishing Ricci tensor. As a consequence, they must also be vacuum solutions of the quadratic theory for all values of the parameters $\\alpha, \\beta$ and $ \\kappa$. Indeed, we have already seen that they are solutions for the special cases of quadratic Lagrangians within the more general models we have previously investigated. The constraint on the solutions of class II implies that $p_{3}= p_{1}+p_{2}-1 \\pm 2 \\sqrt{(p_{1}-1)(p_{2}-1)}$, and so all three Kasner exponents may be positive and greater than unity. Classes I-III were previously found by Deruelle \\cite{deruelle}.\n\n\\subsection{Gauss-Bonnet theories; $f=f(G), f=R+\\hat{f}(G)$}\nIn four dimensions, the Gauss-Bonnet term is a topological invariant and its variation does not contribute to the field equations, though it may give rise to interesting cosmological effects in higher dimensions \\cite{nojiri1}. These terms arise in the low energy effective actions of string theory, however modified Gauss-Bonnet theories have also been proposed as a form of gravitational dark energy capable of successfully describing cosmology at late times \\cite{nojiri2}. Here, we shall consider the situation where the Lagrangian is a general function of the Gauss-Bonnet invariant, i.e. $f=f(G)$ and also the case $f=R+\\hat{f}(G)$, where the Gauss-Bonnet term is defined by $G\\equiv R^2-4R_{ab}R^{ab}+R_{abcd}R^{abcd}$, and takes the form \\begin{equation}\nG=\\frac{4p_{1}}{t^4} (H-3) (2 p_{1}\\!^2 - 2 H p_{1} + H^2 - J)\n\\end{equation} for the metric given by (\\ref{kas}). \n\nThe contributions $\\hat{P}^a_b$ to the field equations due to a function $f(G)$ in the Lagrangian are given by \\begin{eqnarray}\n\\hat{P}^{0}_{0} &=& -\\frac{1}{2}f + 4\\frac{p_{1}p_{2}p_{3} }{t^4}\\left((H-3) f_G - 3 t \\dot{f}_{G} \\right) \\: ,\\\\\n\\hat{P}^{i}_{i} &=& -2f +\\frac{2}{t^4}(8 p_{1}p_{2}p_{3} (H-3) f_G + (J-H^2) ((H-2)t \\dot{f}_{G} + t^2 \\ddot{f}_{G}) \\: , \\\\\n\\hat{P}^{\\mu}_{\\mu}-\\hat{P}^{\\nu}_{\\nu} &=& \\frac{4}{t^3}\\left(p_{\\mu}-p_{\\nu}\\right)\\left(H-p_{\\mu}-p_{\\nu}\\right)\\left((H-2) \\dot{f}_G + t \\ddot{f}_{G} \\right) \\: , \\label{eq:gbaniso}\n\\end{eqnarray} where once again, $\\mu, \\nu$ in equation (\\ref{eq:gbaniso}) are not indices to be summed, but labels taking values from $1$ to $3$. Otherwise summation convention is used as normal. Here, $f_{G}$ is used to denote $ \\frac{df}{dG}$ and overdots represent derivatives with respect to the time coordinate $t$.\n\nThe anisotropic stress will vanish independently of $f$ both for isotropic solutions, ie $p_{1}=p_{2}=p_{3}$, and solutions in which two of the Kasner exponents are zero, and also in theories in which the Lagrangian is a power of the Gauss-Bonnet term, provided that $H$ is suitably chosen in this case.\n\n\\subsubsection{$f=f(G)$}\nExcept for the trivial case of $f(G)=G$, the only possible real anisotropic solutions have $G=0$. Furthermore, for any Kasner solutions to be possible, it is required that $f(0)=0$. Provided $f_{G}(0)$ does not diverge, then any solution of $G=0$ defines a two-parameter family of exact Kasner-like solutions. If, on the other hand, $f_{G}(0)$ is divergent, then for general such $f(G)$, there is a one-parameter family of solutions where the three Kasner indices are given by $p_{1}=p_{2}=0$, with the third index free, plus permuations. In the special case $f(G)=\\alpha G \\log{G}$, there are additional families of solutions given by $p_{1}=0$, with $p_{2}$ and $p_{3}$ free (and permutations), or $J=2 p_{1}\\!^2 - 2 H p_{1} + H^2$.\n\n\\subsubsection{$f=R+\\hat{f}(G)$}\nThe only real solutions with $G=0$ and $R_{ab}=0$ are Minkowski space and the Milne model, corresponding to $p_{1}=1, p_{2}=p_{3}=0$ (plus permutations), which are vacuum solutions whenever $\\hat{f}(0)=0$. Consequently, only these solutions can separately satisfy both the vacuum Einstein equations, $G_{ab}=0$, and the vacuum equations due to the purely Gauss-Bonnet terms, $\\hat{P}_{ab}=0$. Other solutions must have $G \\neq 0$ and the Einstein-Hilbert terms must balance the Gauss-Bonnet terms, that is to say they must be of the same order in time. Therefore, the possible real Kasner-like vacuum solutions in these theories may be summarised as in table \\ref{tab:gb}.\n\n\\begin{table}[ht]\\small \n\\begin{center}   \n \\begin{tabular}{ l || p{6cm} | p{6cm} } \nClass & Solution constraints & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $p_{1}=p_{2}=p_{3}=-1$ & This is a vacuum solution if $f=\\alpha \\sqrt {G}+\\frac{\\sqrt{3G}}{2} \\log {G}$. \\\\[4pt]\n\\textbf{II} & $p_{1}=p_{2}=p_{3}=-3$ & This is a vacuum solution if $f=\\alpha G \\log{G}- 3\\sqrt{2G}$. \\\\[4pt]\n\\textbf{III} & $p_{1}=p_{2}=p_{3}=\\frac{1}{6-\\alpha^2}(3+\\alpha^2 \\pm \\sqrt{12\\alpha^2+9})$ & This is a vacuum solution if $f=\\alpha \\sqrt{G}$. \\\\[4pt]\n\\textbf{IV} & $p_{1}=1, p_{2}=p_{3}=1+\\alpha^2$ \\newline (and permutations) & These are vacuum solutions if $f=\\alpha \\sqrt{G}$. \\\\[4pt]\n\\textbf{V} & $H=1, J=1 + 4 p_{1}\\alpha^2 + 4p_{1}\\alpha \\sqrt{1-p_{1}+\\alpha^2}$ & These are vacuum solutions if $f=\\alpha \\sqrt{G}$. \\\\[4pt]\n\\textbf{VI} &$H=1, J=1 + 4 p_{1}\\alpha^2 - 4p_{1}\\alpha \\sqrt{1-p_{1}+\\alpha^2}$ & These are vacuum solutions if $f=\\alpha \\sqrt{G}$. \\\\[4pt]\n\\end{tabular} \\caption{Power-law and Kasner-like vacuum solutions in a modified Gauss-Bonnet gravity.} \\label{tab:gb}\n\\end{center}\n\\end{table}\n\nWe can see that although there are several different and interesting types of Kasner-like solutions in these theories, they are only valid for a small set of Lagrangians. This is because only the Minkowski and Milne universes solve both the vacuum Einstein equations, $G_{ab}=0$, and the vacuum field equations due to the purely Gauss-Bonnet terms in the action. Thus, terms in the field equations due to the Einstein-Hilbert term must be balanced by those due to $\\hat{f}(G)$ and so these terms must be of the same order in time. \n\n\n\\subsection{Weyl theories of gravity}\nWe now consider the situation where the Lagrangian is a general function of the Weyl invariant, i.e. $f=f(W)$, where the Weyl term is defined by $W\\equiv \\frac{1}{3}R^2-2R_{ab}R^{ab}+R_{abcd}R^{abcd}$. We may write \\begin{equation*}\nW=\\frac{4}{3t^{4}}\\left(  (p_{1} - p_{3}) (p_{1} - p_{2})(p_{1}-1)^2  -(p_{2}-p_{3})(p_{1}-p_{2})(p_{2}-1)^2+(p_{1} - p_{3}) (p_{2}-p_{3})(p_{3} - 1)^2 \\right) \\end{equation*} If we assume, without loss of generality, $p_{1}\\geq p_{2} \\geq p_{3}$, then we see that the first and third terms are positive, but the second is negative. However, the middle term must be no greater in absolute magnitude than the third term if $p_{2}<1$ and similarly it must be no greater in magnitude than the first term if $p_{2}\\geq1$. Thus the Weyl tensor is non-negative and is zero only if each term vanishes separately. This requires that either the metric is locally rotationally symmetric with wlog $p_{2}=p_{3}$ and $p_{1}=1$, or it is isotropic, $p_{1}=p_{2}=p_{3}$. The vacuum field equations are $P^{a}_{b}=0$, with \\begin{eqnarray}\nP^{i}_{i} &=& -2f + 2W f_{W} \\: , \\\\\nP^{\\alpha}_{\\alpha} &=& -\\frac{3}{2} f + \n \\frac{2}{3t^4} (18 p_{1}p_{2}p_{3} + H^2 - 2 H^3 - 3 J + 4 H J) ((H-3) f_{W} + t \\dot{f}_{W})) \\: ,\\\\\n\\hat{P}^{\\mu}_{\\mu}-\\hat{P}^{\\nu}_{\\nu} &=& \\frac{1}{3t^4}\\left(p_{\\mu}-p_{\\nu}\\right)\\biggl(12t (H-p_{\\mu}-p_{\\nu}) ((H-2) \n\\dot{f}_{W} + t \n\\ddot{f}_{W})  + \n 4 (H - 3) (2 (J - 1) - (H - 1)^2) f_{W} \\nonumber \\\\ && + \n 2 (3 (H - 1)- 5 (H - 1)^2 + 4 (J - 1) ) t  \n\\dot{f}_{W} - 6 (H - 1) t^2 \n\\ddot{f}_{W} \\biggr) \\: , \\label{eq:weylaniso}\n\\end{eqnarray} where once again, $\\mu, \\nu$ in equation (\\ref{eq:gbaniso}) are not indices to be summed, but labels taking values from $1$ to $3$. Otherwise summation convention is used as normal, $f_{W}$ is used to denote $f'(W) \\equiv \\frac{df}{dW}$ and overdots represent derivatives with respect to the time coordinate $t$. The equations (\\ref{eq:weylaniso}) for the anisotropic stress may be combined to obtain \\begin{equation}\n0=\n \\frac{4}{t^3}\\left(p_{1}-p_{2}\\right)\\left(p_{1}-p_{3}\\right)\\left(p_{2}-p_{3}\\right) \\left((H-2) \\dot{f}_{W} + t \\ddot{f}_{W}\\right) \\: .\n\\end{equation}\n\nFor fully anisotropic solutions, the Weyl tensor is non-zero, and as before one can treat the field equations as differential equations for $f$, solving them to find the required form of Lagrangian for the higher-order theory to possess such solutions. It is found that the only possible choice is that of Weyl gravity, $f(W) = \\alpha W$, and that solutions require $p_{3}=p_{1}+p_{2}-1 \\pm 2\\sqrt{(p_{1}-1)(p_{2}-1)}$; this is a special case of the quadratic Lagrangians considered before in section \\ref{quad}, corresponding to $3\\alpha+\\beta=0$.\n\n\nFor solutions of the vacuum field equations where the Weyl tensor vanishes, it is necessary that $f(0) = 0$. In fact, if $f(0)=0$ and $f'(W)$ does not diverge at $W=0$, then all solutions of $W=0$ are solutions of the vacuum field equations. However, if $f(0)=0$ and $f'(0)$ diverges then we must consider both the isotropic and locally rotationally symmetric types of solution in turn as the existence of such solutions will depend upon the manner of this divergence.\n\nThese results are summarised in table \\ref{tab:weyl}.\n\n\\begin{table}[ht]\\small \n\\begin{center}   \n \\begin{tabular}{ l || p{5.5cm} | p{7cm} } \nClass & Solution constraints & Validity \\\\[2pt] \\hline \\hline \n\\textbf{I} & $p_{1}=p_{2}=p_{3}$ & These are vacuum solutions for all values of $p_{1}$ if $f(0)=0$ and either $f'(0)$ converges or diverges due to terms in $f(W)$ of the form $W^{n}$ with $1\/2 < n < 1$. \\\\[4pt]\n\\textbf{IIa} & $p_{1}=p_{2}=p_{3}=2n-1$ & This subset of the class I solutions remains a vacuum solution if $f(0)=0$ and the divergence in $f'(0)$ is due to terms in $f(W)$ of the form $W^{n}$ with $0 < n < 1\/2$.\n \\\\[4pt]\n \\textbf{IIb} & $p_{1}=p_{2}=p_{3}=(4n-1)\/3$ & This subset of the class I solutions remains a vacuum solution if $f(0)=0$ and the divergence in $f'(0)$ is due to terms in $f(W)$ of the form $W^{n}$ with $0 < n < 1\/2$.\n \\\\[4pt]\n \\textbf{III} & $p_{1}=1, p_{2}=p_{3}$ \\newline (and permutations) & These are vacuum solutions if $f(0)=0$ and $f'(0)$ either converges or diverges due to terms in $f(W)$ of the form $W^{n}$ with $1\/2 < n < 1$. \\\\[4pt]\n\\textbf{IV} & $p_{1}=1, p_{2}=p_{3}=2n-1$ \\newline (and permutations) & This subset of the class III solutions remains a vacuum solution if $f(0)=0$ and the divergence in $f'(0)$ is due to terms in $f(W)$ of the form $W^{n}$ with $0 < n < 1\/2$.\n \\\\[4pt]\n \\textbf{V} & $p_{3}=p_{1}+p_{2}-1 \\pm 2\\sqrt{(p_{1}-1)(p_{2}-1)}$ & This is a vacuum solution in Weyl gravity, $f(W)=\\alpha W$.\n \\\\[4pt]\n\\end{tabular} \\caption{Power-law and Kasner-like vacuum solutions in Weyl gravity.} \\label{tab:weyl}\n\\end{center}\n\\end{table}\nNote that if $f(W)$ contains a term $\\sqrt{W}$, i.e. $n=1\/2$, then the only solution is that of the Milne model, with metric given by\n\\begin{equation*}\n ds^2 = -dt^2+t^2 dx^2 + dy^2 + dz^2 \\: ,\n\\end{equation*} which we have seen is Minkowski space in a different coordinate system. Finally we point out that if the $f'(W)$ diverges at $W=0$ due to a type of term other than a power law, there are no vacuum Kasner-like solutions in that theory.\n\n\\subsection{Homogeneous Lagrangians \\label{sec:homogeneous}}\nThus far, this study has considered models in which the Lagrangian is a general function of one of the curvature scalars, $X\\equiv R$, $Y \\equiv R^{ab}R_{ab}$, or $Z \\equiv R^{abcd}R_{abcd}$, or of a particular combination of these. Consequently, in each case there has been only one timescale in the problem, and, by substituting for the time variable using the appropriate curvature invariant, it has proved possible to derive all possible solutions and their existence conditions within these wide classes of models.\n\nHowever, it is useful to also consider more general Lagrangians depending on more than just one variable. We may also conisder Lagrangians which are homogeneous in $X^{2}, Y$ and $Z$, that is to say for all values of $\\lambda$, $f(\\lambda X^2, \\lambda Y, \\lambda Z) = \\lambda^{n}f(X^2, Y, Z)$ for some $n$. $X^{2}$ is chosen as the argument here, rather than $X$, in order that each term in the function will be of the same order in time. Thus, functions of this form will be relevant where the dominant terms at early or late times are more complicated than in those Lagrangians studied previously, such as a monomial in $X,Y,Z$. We may write \n\\begin{equation}\nf\\left(X,Y,Z\\right)= X^{2n} \\alpha \\left(\\theta, \\phi \\right)  \\label{homogeneous},\n\\end{equation} where $\\alpha$ is a general differentiable function of $\\theta \\equiv \\frac{X^{2}}{Z}$ and $ \\phi \\equiv \\frac{Y}{Z}$, and further define \\begin{eqnarray}\n\\beta & \\equiv & \\frac{\\partial \\alpha}{\\partial \\theta} \\: ,\\\\\n\\gamma & \\equiv & \\frac{\\partial \\alpha}{\\partial \\phi}  \\: . \\end{eqnarray} \nIt is important to note that $\\alpha, \\beta$ and $\\gamma$ take constant values which are dependent upon the exact form of the Lagrangian $f$ and also the values of $\\theta$ and $\\phi$ for a particular solution. For example, in the case of the monomial given by \\begin{equation}\nf=\\xi X^{\\sigma} Y^{\\mu} Z^{\\nu} ,\n\\end{equation} we have\n\\begin{eqnarray}\n2n &=& \\sigma+2\\mu+2\\nu \\: , \\\\\n\\alpha &=& \\xi \\left(\\frac{X^{2}}{Z}\\right)^{-(\\mu+\\nu)}\\left(\\frac{Y}{Z}\\right)^{\\mu} \\: , \\\\\n\\beta &=& -(\\mu+\\nu)\\frac{Z}{X^{2}} \\alpha \\: , \\\\\n\\gamma &=& \\mu \\frac{Z}{Y} \\alpha \\: .\n\\end{eqnarray}\n\nInserting the general form (\\ref{homogeneous}) for the homogeneous Lagrangian into the vacuum field equations for the metric (\\ref{kas}), one obtains the equation\n\\begin{eqnarray}\n0 &=& \\left(p_{2}-p_{3}\\right)\\left(P^{1}_{1}-P^{3}_{3}\\right)-\\left(p_{1}-p_{3}\\right)\\left(P^{2}_{2}-P^{3}_{3}\\right) \\nonumber \\\\\n&=&\n\\frac{16 t^{4} X^{2n}}{Z^{2}}\\left(p_{1}-p_{2} \\right)\\left(p_{1}-p_{3}\\right)\\left(p_{2}-p_{3}\\right)(n-1)(1+H-4n)\\left(\\beta X^{2}+\\gamma Y \\right) \\: . \\label{eq:aniso}\n\\end{eqnarray}\nWe will consider in turn the solutions of this equation in the remaining field equations.\n\n\\subsubsection{Isotropic solutions}\nIn the isotropic case, $p_{1}=p_{2}=p_{3}$, there is only one independent field equation, given by\n\\begin{eqnarray*}0 & = & p_{1}\\left(p_{1}(2p_{1}-1)\\right)^{2n-1}\\bigl(\\alpha(2p_{1}^2-2p_{1}+1)^2(1+2p_{1}(n-1)-6n+8n^2)\\\\\n& & -(6\\beta+\\gamma) p_{1}(1-2p_{1})^2(1+3p_{1}-4n)\\bigr) \\: .\n\\end{eqnarray*}\n\nRecall that $\\alpha, \\beta$ and $\\gamma$ are constants which will depend upon $p_{1}$ according to the exact form of the Lagrangian $f$. Thus, given a choice of $f$, this equation provides a necessary and sufficient condition for there to be a isotropic power-law solution expanding as $t^{p_{1}}$.\n\n\\subsubsection{Locally rotationally symmetric solutions}\nIn the case of locally rotationally symmetric (LRS) solutions, there are two independent field equations. Without loss of generality, we set $p_{2}=p_{3} \\neq p_{1}$. \n\nLRS solutions with $X=0$ are a special case of the fully anisotropic ones of that kind and will satisfy the same existence conditions. Similarly, if $p_{1}+2p_{2}=4n-1$, then there are LRS solutions provided $\\alpha X^{2n}=0$ can be solved simultaneously. This requires either the additional constraint $\\alpha=0$ to be satisfied, or if $1\/4<n<5\/8$ then there are real solutions of this form with $X=0$.  Both these types of LRS solution satisfy the same constraints as in the fully anisotropic case, and so these will be discussed at greater length in the next section.\n\nIf we define $\\hat{X}\\equiv t^2 X, \\hat{Y}\\equiv t^4 Y$ and $\\hat{Z} \\equiv t^4 Z$ for convenience, then, for $X\\neq 0$, these may be reduced to\n\\begin{eqnarray*}\n0 &=& \\biggl(n \\alpha  \\hat{Z}^2 - 2 \\hat{X} (2-3p_{1}-2p_{2} +(p_{1}-p_{2})^2 + 4 n(p_{1}+p_{2}-1)) (\\beta \\hat{X}^2 + \\gamma \\hat{Y}) + \\beta \\hat{X}^2 \\hat{Z} \\\\ \n&& + \\gamma \\hat{X}\\hat{Z} (p_{1}\\!^2 + 2 p_{2}\\!^2 - 1 + 2(p_{1}+2p_{2}-1)(n-1)) \\biggr)(1+p_{1}+2p_{2}-4n) \\: , \\\\\n0&=& \\alpha \\hat{Z}^2\\left(4n \\left(p_{1}\\!^2 + 2p_{2}\\!^2+ (4n-3)(p_{1}+2p_{2})\\right) - \\hat{X} \\right) \\\\ \n&& +16\\hat{X} p_{2}\\!^2(1+p_{1}+2p_{2}-4n)  \\biggl(\\beta \\hat{X}\\left(p_{1}\\!^3 -2p_{1}\\!^2p_{2}+ p_{1}p_{2}\\!^2 -5p_{1}\\!^2-2p_{1}p_{2} - 2p_{2}\\!^2\\right) + \\gamma \\bigl( (2-3p_{2}) p_{2}\\!^3 \\\\\n&& +p_{1}\\!^5 + 2 p_{1} (p_{2}-1) p_{2}\\!^3 - 2 p_{1}\\!^4 (1 + p_{2}) + \n   p_{1}\\!^2 p_{2} (6 - 7 p_{2} - 4 p_{2}\\!^2) + p_{1}\\!^3 (1 - 4 p_{2} + 3 p_{2}\\!^2)\\bigr)\\biggr) \\: .\n\\end{eqnarray*}\n\nIn this way, the field equations give relationships between the theory-dependent constants $\\alpha, \\beta$ and $\\gamma$ which must be satisfied if the theory is to contain any other LRS solutions.\n\n\\subsubsection{Anisotropic solutions}\n\nWe now consider those solutions of equation (\\ref{eq:aniso}) which are fully anisotropic with $p_{\\alpha}$ all distinct. We have seen that for metrics described by the line element (\\ref{kas}), $Z=0$ only for Minkowski space and the Milne model. Thus for fully anisotropic solutions, $Z$ will be non-zero. The only real Kasner-like solutions with $X=Y=0$ are those of general relativity, which satisfy $H=J=1$. Since the leading order terms in the field equations as $X,Y \\rightarrow 0$ behave like $\\alpha(2n-1)X^{2n-1}$, then if $\\alpha$ diverges at $X=0$, faster than $X^{1-2n}$, there will be no solutions. However, if $\\alpha$ is finite at $Y=0$, one either requires that $n \\geq 1\/2$ or that $\\alpha=0$ and $n>0$ so that the next leading order terms do not diverge. For solutions with $X=0$ but $Y\\neq 0$, the leading order terms in the field equations as $X \\rightarrow 0$ behave like $\\alpha(H-4n+1) X^{2n-1}$, so if $\\alpha$ remains finite at $X=0$, one either requires that $n \\geq 1\/2$  or that either $H=4n-1$ or $\\alpha=0$ and $n>0$ so that the next leading order terms do not diverge.\n\nIf there are configurations of the Kasner exponents such that $\\alpha=\\beta=\\gamma=0$, then these will be solutions of the theory. Recall that $\\alpha, \\beta$ and $\\gamma$ depend on the explicit form of the Lagrangian.\n\nSubstituting $H=4n-1$ into the field equations leads to the necessary and sufficient condition $\\alpha X^{2n}=0$. For $X=0$ with $H=4n-1$, this implies that $J=16n(1-n)-3$ and requires that $1\/4 \\leq n \\leq 5\/8$ for these solutions to be real. Again, since $\\alpha$ depends on the explicit form of the Lagrangian and the values of the Kasner exponents and so $\\alpha=0$ defines an extra constraint on the solution.\n\nIn addition to those solutions which are a special case of the ones with $H=4n-1$ and $\\alpha=0$, there is another class of solutions if $n=1$. This class of solutions exists if\n\\begin{eqnarray} \\alpha &=&-\\frac{\\gamma}{4 Z}\\left(X^2-4Y+Z\\right)  \\quad \\text {and}\\\\\n\\beta &=&-\\frac{\\gamma}{4 X^2}\\left(4Y-Z\\right) \\: . \\end{eqnarray}\nAgain, $\\alpha, \\beta$ and $\\gamma$ will depend on the values of the Kasner exponents through the curvature scalars $X, Y$ and $Z$ and so these equations will typically give two further constraints on the possible set of solutions. However, if for example, $\\alpha=\\alpha(X^{2}\/(4Y-Z))$, then the second of these equations will be trivial and if $\\alpha=\\lambda(X^2-4Y+Z)\/X^{2}$ for some constant $\\lambda$, then both of these will be trivial.\n\nFinally, there are several additional solutions if $\\beta X^2 + \\gamma Y=0$. This equation is satisfied trivially if the Lagrangian $f$ is independent of $Z$. Otherwise, it provides a constraint on the solutions. The remaining field equations then show that if $n=1$ there are solutions subject to $H=1$ and \\begin{equation}\n\\alpha=\\frac{4\\beta}{Z t^{4}}(1-J) \\: .\n\\end{equation} If $n=1\/2$, there are solutions subject to \\begin{equation}\n\\alpha = 2\\beta \\frac{X^{2}}{t^{4}Y Z} (1-H) (H^2 - J) \\: . \\label{nhalf} \\end{equation} Finally, for general $n$ there are solutions of this sort if \\begin{eqnarray}\n\\alpha &=& 2\\beta \\frac{X^2}{n t^{4}Y Z}  (H-1) \\left(H-H^2 +n\\left(H^{2}-2H+J \\right)\\right) \\quad \\text{and}\\\\ \nJ &=& \\frac{1}{2n}\\left(1 - 2 H + H^2 - 6 n + 8 H n + 8 n^2 - 8 H n^2\\right)  \\: .\n\\end{eqnarray}\n\n\\subsubsection{An explicit example}\nAs an explicit example, we shall consider fully anisotropic Kasner-like solutions in the model of Br\\\"{u}ning, Coule and Xu \\cite{coule}, which is described by the Lagrangian \\begin{equation}\nf = X +\\lambda \\frac{Y}{X}+\\tau \\frac{Z}{X}\n\\: , \\label{f:coule} \\end{equation} where $\\lambda$ and $\\tau$ are constants, and was studied there in the context of FRW solutions. Note that the model reduces to general relativity in the limit  $\\lambda=\\tau=0$, so in the discussion that follows we will assume that $\\lambda$ and $\\tau$ are not both zero. In terms of our notation, \n\\begin{eqnarray*}\nn &=& \\frac{1}{2} \\: ,\\\\\n\\alpha &=& 1+\\lambda \\frac{Y}{X^{2}}+\\tau \\frac{Z}{X^{2}} \\: ,\\\\ \n\\beta &=& -\\frac{Z}{X^{4}}\\left(\\lambda Y+\\tau Z \\right) \\quad \\text{and} \\\\\n\\gamma &=& \\lambda \\frac{Z}{X^{2}} \\: .\n\\end{eqnarray*} Thus we can explicitly observe the dependence of $\\alpha, \\beta$ and $\\gamma$ upon the values of the Kasner parameters $p_{\\alpha}$ and in particular for this model we note that $\\alpha$ diverges for the general relativistic solution, $H=J=1$. \n\nFor anisotropic solutions in this model, with the Kasner exponents $p_{\\alpha}$ all distinct, equation (\\ref{eq:aniso}) reduces to \\begin{equation}\n0=\\tau \\left(\\frac{H-1}{H^2-2H+J}\\right) \\: ,\n\\end{equation}and so the possible solutions depend on the value of $\\tau$. For the special case when $\\tau$ is zero, the field equations reduce to the single equation \\begin{equation}\n\\frac{2 H^3 -3H^2+ J + 2 H J -H^{2}J- J^2 }{(H^2-2H+J)^{2}}\\lambda = 1 \\: .\n\\end{equation} This equation allows a two-parameter family of solutions to be found with real-valued Kasner exponents for all values of $\\lambda$ except those in the range $-1<\\lambda < 0$.\n\nFor the general case with $\\tau$ non-zero, solutions require $H=1$ and \n\\begin{equation}\n\\xi(J-1)^2 +8 p_{1}(J-1) - 16  p_{1}\\!^{2} (p_{1} - 1)=0 \\: ,\\label{ineq1} \\end{equation}\nwhere we have defined $\\xi \\equiv (1+3\\lambda+\\tau)\/\\tau$. These solutions therefore are described by the single free paramter $p_1$. Seeking solutions for which the Kasner exponents $p_{\\alpha}$ take real values, it is necessary to choose $p_{1}$ such that \\begin{equation}\n1+\\xi(p_{1}-1) \\geq  0 \\end{equation} and either \\begin{eqnarray}\n3p_{1}\\!^{2}+2\\left(\\frac{4}{\\xi}-1\\right)p_{1}-1 &<& 0 \\qquad \\text{or} \\label{ineq2}\\\\\n\\xi\\left(\\xi(1+3p_{1})^{2}-16p_{1}\\right) & < & 0 \\: . \\label{ineq3}\n\\end{eqnarray}It is possible to find $p_{1}$ such that the inequalities (\\ref{ineq1}) and (\\ref{ineq2}) may be satisfied simultaneously if $\\xi <5\/4$, whilst the inequalities (\\ref{ineq1}) and (\\ref{ineq3}) may be satisfied simultaneously provided $\\xi <4\/3$. Thus, this bound provides the restriction on the allowable choices of $\\lambda$ and $\\tau$ such that the general model (\\ref{f:coule}) will contain anisotropic Kasner-like solutions of the form (\\ref{kas}).\n\nIf we compare these findings with those in the previous section, when the general case of a homogeneous Lagrangian was discussed, we see that these correspond to the solutions of equation (\\ref{nhalf}). When $\\tau=0$, the equation $\\beta X^2 + \\gamma Y=0$ is trivial, but for non-zero $\\tau$, it provides an additional constraint and reduces the number of free parameters in the solution from two to one. There are no anisotropic solutions corresponding to $X=0$, since (except in the limiting case of general relativity) $\\alpha X$ diverges there.\n\n \n\\subsubsection{Remarks}\nIn this section, we have studied those Lagrangians which are homogeneous in time for the metrics (\\ref{kas}), and which may be written in the form (\\ref{homogeneous}). We have found the existence conditions for all possible Kasner-like solutions in this class of theories. Although it is impossible to find the exact forms of these solutions without explicitly defining the Lagrangian, and the dependence of the functionals $\\alpha, \\beta$ and $\\gamma$ on the Kasner parameters may be complicated, we have developed a general framework which can be used to obtain all possible solutions once the gravitational theory has been defined. This framework was then explicitly demonstrated for the model of Br\\\"{u}ning, Coule and Xu \\cite{coule}.\n \nSince the terms in the field equations must vanish at each order in time, then unless there exists a particular configuration of Kasner exponents such that the Lagrangian $f(X,Y,Z)$ is a constant, it is reasonable to expect that the solutions found in this section will be the \\emph{only} Kasner-like solutions with real Kasner indices that are possible in higher-order metric theories of gravity derived from Lagrangians with the general form $f(R, R_{ab}R^{ab}, R_{abcd}R^{abcd})$. \n\n\n\\section{Anisotropically Inflating Solutions}\nIn the preceeding sections, we have studied higher-order theories of gravity in which the Lagrangian is dependent upon the scalar curvature, $X \\equiv R$, the Ricci invariant, $Y \\equiv R_{ab}R^{ab}$, and the Kretschmann scalar, $Z \\equiv R_{abcd}R^{abcd}$, to find the conditions required on the form of the Lagrangian for a particular theory to contain Kasner-like vacuum solutions, where the line element is described by (\\ref{kas}). Within the context of this class of higher-order gravitational theories, it is interesting to also study the possibility of other simple anisotropic exact solutions. A natural yet simple extension is to consider a Bianchi type I solution which describes exponential but anisotropic expansion, where the metric is of an anisotropic deSitter-like form;\n\\begin{equation}\n ds^2=-dt^2+e^{2p_{1}t} dx^2 +e^{2p_{2}t} dy^2 +e^{2p_{3}t} dz^2 , \\label{ds}\n\\end{equation}\nand the exponents $p_{\\alpha}$ are real. \n\nIn this section, we shall investigate whether such metrics solve the vacuum field equations of these theories and we include also the possibility of a cosmological constant. Such solutions were found to be possible in theories with quadratic Lagrangians, provided that the Ricci term is present \\cite{hervik} and their existence demonstrates that the cosmic no-hair theorem of general relativity, which states that the presence of a positive cosmological constant drives the solution towards the deSitter one at late times, cannot be extended to higher-order theories in general.\n\nThe three curvature scalars $X,Y,Z$ are constants, and take the values\n\\begin{eqnarray*}\nX &=& H^2+J \\: , \\\\\nY &=& (H^2+J)J \\: ,\\\\\nZ &=& 2(J^2+M) \\: \n\\end{eqnarray*}\nwhere the useful definitions $H \\equiv p_{1}+p_{2}+p_{3}, J \\equiv p_{1}\\!^2+p_{2}\\!^2+p_{3}\\!^2$ and $M \\equiv p_{1}\\!^4+p_{2}\\!^4+p_{3}\\!^4$ have been made. Note that all three curvature scalars $X,Y$ and $Z$ are constant and also positive unless $p_{1}=p_{2}=p_{3}=0$, corresponding to Minkowski space. Thus, unlike the situation for the Kasner solutions discussed previously, it is not necessary for example that $R=0$ for $f(R)$ to be zero. Furthermore, since the curvature scalars are constant, the field equations simplify substantially \\cite{clifton4}, and for the isotropic case they reduce to the single equation\n\\begin{equation}\n\\frac{1}{2} f = \\frac{H^2}{3} f_{X}+\\frac{2}{9}H^4 f_{Y} +\\frac{4}{27}H^4 f_{Z} \\: ,\n\\end{equation} which must be satisfied by $f$ if the de Sitter universe is to be a solution in a particular higher-order theory of gravity of the form $f(X,Y,Z)$.\n\nFor all vacuum \\emph{anisotropic} solutions of the form (\\ref{ds}), they become\n\\begin{eqnarray}\nf &=& 8H p_{1}p_{2}p_{3} f_{Z} \\: ,  \\label{eq:ds1} \\\\\nf_X+ 2J f_Y &=& 2(H^2-3J) f_Z \\: , \\label{eq:ds2}\n\\end{eqnarray}\nwhere, as before, subscripts are used to denote differentiation of $f$ with respect to that curvature scalar, so that, for example, $f_{X} \\equiv \\frac{\\partial f}{\\partial X}$. Any cosmological constant is to be included in the function $f$.\n\nIf the Lagrangian is a function of only one of the three curvature invariants, $f=f(\\xi)$ say, where $\\xi \\in \\{ X, Y, Z \\}$, then the field equations for anisotropic solutions simplify further to\n\\begin{eqnarray}\nf &=& 0 \\: , \\\\\nf_{\\xi} &=& 0 \\: .\n\\end{eqnarray}\nThus the theory will contain a two-parameter family of exact anisotropic deSitter-like solutions if the Lagrangian, $f$, has a \\textit{positive} double root, $\\xi=\\xi_{0}>0$, the simplest example of which being a quadratic Lagrangian, $f=\\alpha(\\xi-\\xi_{0})^2$. It should be noted that for the case where $\\xi \\equiv R$, the scalar curvature, choosing the sign of the Lagrangian to give the correct Newtonian limit would give a negative cosmological constant due to the required positivity of $R_0$. \n\nIn this way, some degree of fine-tuning is required in the Lagrangian for such solutions to exist. However, more general types of model do not require such fine-tuning for solutions of this sort to exist. In particular, if the Lagrangian is of the form $f=f(Y\/X^{2})$, then equation (\\ref{eq:ds2}) is identically satisfied and so it is only required that there exist positive roots of $f=0$.\n\n\n\n\\section{Conclusions}\nWe have studied some aniosotropic cosmological Bianchi type I solutions to a wide class of higher-order theories of gravity derived from the three curvature invariants $R, R_{ab}R^{ab}$ and $R_{abcd}R^{abcd}$. Although general relativity is well-supported by solar system tests, in the high curvature limit, such as on approach to an initial cosmological singularity, we expect quantum effects to become important and to cause deviations from the standard behaviour in general relativity. At high curvatures, anisotropies diverge faster than isotropies and will tend to dominate the cosmological behaviour at early times. Furthermore, it has previously been shown \\cite{hervik} that anisotropies in these higher-order theories may display significantly different behaviour to that found in general relativity. Thus, in this work we have investigated the role of anisotropy on approach to the initial singularity.\n\nIn particular, we have found all Kasner-like solutions given by the Bianchi type I line element (\\ref{kas}) for several wide classes of higher-order theories of gravity, and also all of the similar Bianchi I solutions which are described by the line element (\\ref{ds}). Previously, Kasner-like solutions were known only for quadratic gravity \\cite{deruelle} and higher-order Lagrangians which were powers of one of the curvature invariants $R, R_{ab}R^{ab}$ and $R_{abcd}R^{abcd}$. We have extended this to much more general Lagrangians including those of the form $f(R), f(R_{ab}R^{ab})$ and $f(R_{abcd}R^{abcd})$, and \nin the course of this investigation, we have also found additional solutions to the previously-studied power-law Lagrangians which were not found in \\cite{clifton1, clifton2}. \n\nWe have further widened this study to include also those Lagrangians which are homogeneous in time for the metrics (\\ref{kas}), which may be written in the form (\\ref{homogeneous}), using the model of Br\\\"{u}ning, Coule and Xu \\cite{coule} as a particular example. Since the terms in the field equations must vanish at each order in time, then unless there exists a particular configuration of Kasner exponents such that the Lagrangian $f(X,Y,Z)$ is a constant, we expect that the solutions found in section \\ref{sec:homogeneous} will be the \\emph{only} Kasner-like solutions with real Kasner indices that are possible in higher-order metric theories of gravity derived from Lagrangians with the general form $f(R, R_{ab}R^{ab}, R_{abcd}R^{abcd})$. \n\nAlthough they are geometrically special, the Kasner-like solutions given by the Bianchi type I line element (\\ref{kas}) provide us with a very useful insight into the dynamics of anisotropies, and also they give a good description of the evolution of more general anisotropic cosmological models over finite time intervals. This is of particular interest when considering the behaviour on approach to the initial singularity exhibited by the Bianchi type VIII and type IX (``Mixmaster'') cosmologies, which can be approximated by a sequence of different Kasner epochs. We have considered the properties of the solutions found in relation to the behaviour of these more general anisotropic cosmologies and found that in general it is model-dependent as to whether the universe will experience an infinite sequence of oscillations between Kasner regimes as the singularity is approached. A more detailed analysis is required to understand the extent of the validity of these vacuum  solutions in the presence of a non-comoving perfect fluid.\n\nThe conditions for the existence of de Sitter solutions in higher-order theories of gravity were already known \\cite{clifton4}, and some examples of anistropically inflating solutions had previously been discovered for particular theories \\cite{hervik}. However, we have extended this work and explicitly discovered the existence conditions for anistropically inflating solutions of the form (\\ref{ds}) in all higher-order metric theories of gravity derived from Lagrangians with the general form $f(R, R_{ab}R^{ab}, R_{abcd}R^{abcd})$. For the simpler models in which the Lagrangian depends on just one of the curvature invariants $R, R_{ab}R^{ab}$, and $R_{abcd}R^{abcd}$, we found that some degree of fine-tuning is required in the Lagrangian for such solutions to exist, however this is not required in more general theories. These solutions explicitly demonstrate that the cosmic ``no-hair'' theorem of general relativity does not hold in general in higher-order theories.\n\n\n\\section*{Acknowledgements}The author wishes to thank John D. Barrow for useful discussions and acknowledges a STFC studentship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nActive matter describes systems comprising individual units that exert propelling forces on their environment~\\cite{marchetti_hydrodynamics_2013,Bechinger2016RMP,o2022time}.  Examples  extend across scales, from molecular motors~\\cite{schaller2010polar,sanchez2012spontaneous} to animals~\\cite{ballerini2008interaction,calovi2014swarming}, including both biological~\\cite{bi2016motility,matoz2017nonlinear} and artificial systems~\\cite{geyer2019freezing,van2019interrupted}. Active systems have attracted a lot of interest recently due to their rich collective behaviors~\\cite{vicsek_novel_1995,cates_motility-induced_2015,marchetti_hydrodynamics_2013} and to their non-trivial interactions with passive boundaries and objects~\\cite{galajda_wall_2007,Tailleur2009EPL,Sokolov2010PNAS,di_leonardo_bacterial_2010,Solon2015NatPhys,granek2021anomalous,paul2022force}.\nUnlike in equilibrium settings, asymmetric objects generically induce long-ranged currents in active fluids which, in turn, mediate long-range interactions between inclusions~\\cite{baek_generic_2018,granek2020bodies}. These currents have been shown to play an important role in the context of motility-induced phase separation~\\cite{cates_motility-induced_2015} where random obstacles placed in the bulk of a system suppress phase separation in $d<4$ dimensions~\\cite{ro2021disorder}. Surprisingly, disordered obstacles localized on the boundaries also destroy phase separation in $d<3$ dimensions~\\cite{bendor2021far}, something impossible in equilibrium. \n\n\n\\begin{figure}\n\\includegraphics[width=8.6 cm]{trajectories_complete_uniform_same_enlarged4.eps}\n\t\\caption{Probability density of a semicircular mobile object surrounded by $N=10^3$ non-interacting run-and-tumble particles confined to a circular cavity of unit diameter. Particle speed and tumbling rate are set to $v=10^{-2}$ and $\\alpha=1$, respectively. The diameter of the semicircular object is equal to the particle persistence length $\\ell_p=10^{-2}$. The dynamics of the object is an overdamped Brownian motion at zero temperature with translational mobility set to unity and rotational mobility set to $\\gamma=10^2$ in (a) and $\\gamma=10^5$ in (b). The gray lines show typical trajectories of the object. The latter is displayed in black and enlarged by a factor of six. See Appendix~\\ref{app:numerics} for numerical details.}\n \t\\label{fig:body prob dist}\n\\end{figure}\n\nIn this article, we study another surprising role of boundaries. In Figure~\\ref{fig:body prob dist}, we show numerical simulations of an active fluid confined in a circular cavity in which a mobile asymmetric object has been inserted. Depending on the parameters, the object is either localized close to the cavity walls or in the middle of the cavity. As we show below, this is a direct consequence of the ratchet current induced by the object in the active bath and its interactions with the cavity walls. Note that, on general symmetry grounds, an isotropic object cannot be localized in a diffusive fluid. Indeed, the sole symmetry breaking field in the vicinity of an isotropic object is the gradient of the fluid density, $\\nabla \\rho$. The force ${\\bf F}$ exerted on the object thus satisfies ${\\bf F} \\propto \\nabla \\rho$. If the fluid is diffusive, it satisfies $\\nabla^2 \\rho = 0$ in the steady state so that $\\nabla \\cdot {\\bf F}=0$~\\cite{rohwer2020activated}. In analogy to Earnshaw's theorem in electrostatics, this rules out the possibility of a stable equilibrium for the passive tracer. In contrast, an asymmetric polar object introduces a symmetry-breaking vector along which it generically generates a ratchet current~\\cite{Galajda2007,Tailleur2009EPL,di_leonardo_bacterial_2010,Sokolov2010PNAS,baek_generic_2018}. This current is directly related to the non-vanishing mean force ${\\bf p}$ exerted by the object on the surrounding fluid~\\cite{nikola_active_2016}. Due to Newton's third law, one thus generically expects a contribution to $\\mathbf{F}$ along $-\\mathbf{p}$, which opens up the possibility of a localization transition. Figure~\\ref{fig:body prob dist} shows that this is indeed the case.\n\nTo uncover the mechanism behind this localization transition, we study the influence of boundaries on the coupling between asymmetric objects and active fluids. We start in Section~\\ref{subsec:density and current} by considering the case of an asymmetric polar object in the presence of a flat confining boundary. We show that the latter alters the far-field current and density modulation induced by the polar object, and that this effect can be rationalized using a generalized image theorem. As we show in Section~\\ref{subsec:force}, this leads to a repulsive force, which decays as a power-law, between the object and the wall. In Section~\\ref{sec:circular cavity} we then generalize our approach to the case of a polar object confined by a circular cavity. Finally, in Section~\\ref{sec:dynamics}, we consider a mobile object and show the existence of a localization transition. We note that our results could be tested experimentally by adapting a recent setup in which a \\textit{symmetric} object was immersed in a circular cavity confining active colloids~\\cite{paul2022force}. In this case, as expected on symmetry ground, no localization transition was observed and the interaction between the object and the wall is short ranged. We predict that employing a polar object should lead to rich physics. All derivations below are presented in two space dimensions but can easily be generalized to higher dimensions.\n\n\n\\section{An asymmetric object next to a flat wall}\\label{sec:flat wall derivation}\n\n\\begin{center}\n\\begin{figure}[t]\n    \\begin{tikzpicture}\n        \\fill[teal!20!white] (0,-1.5) rectangle (4,1.5);\n        \\fill[orange!65!white] (-0.2,-1.5) rectangle (0,1.5);\n        \\fill[pattern=north west lines, pattern color=black] (-0.2,-1.5) rectangle (0,1.5);\n        \n        \\fill[purple!70!white] (2.59611+0.15,0.00139995+0.4) arc(158:338:0.375) -- (3.17561+0.15,-0.217276+0.4) arc(338:158:0.25) -- cycle;\n        \n        \\draw[-stealth,thick] (0,0) -- (3,0) node[anchor=east, yshift=-3mm, xshift=3mm] {${\\bf r}_0=(d,0)$};\n        \\draw[-stealth,thick] (3,0) -- (3+0.26222,0.649029) node[anchor=east] {${\\bf p}$};\n        \\draw[thick] (3.35 ,0) arc(0:68:0.35) node[anchor=west, xshift=1.5mm] {$\\phi\\ $};\n        \n        \\draw[thick,->] (-.5,0) -- (4,0) node[anchor=north west] {$\\!x$};\n        \\draw[thick,->] (0,-1.5) -- (0,1.5) node[anchor=south east] {$y$};\n    \\end{tikzpicture}\n    \\caption{An asymmetric passive object in an active fluid next to a flat wall at $x=0$. The object is located at $\\vec{r}_0=(d,0)$. Due to its asymmetric shape, it experiences a force $-\\vec{p}$ from the active bath and thus exerts the opposite force $\\vec{p}$ on the active medium, whose orientation we denote by $\\phi$.}\\label{fig:configuration}\n\\end{figure}\n\\end{center}\n\nWe start by studying the influence of an infinite flat wall at $x = 0$ on an asymmetric object embedded inside the system in the neighborhood of ${\\bf r}_0{= (d,0)}$ (see Fig.~\\ref{fig:configuration}). We first determine in Sec. \\ref{subsec:density and current} how the presence of the wall influences the ratchet current and the density modulation induced by the asymmetric object in the active bath. Then in Section~\\ref{subsec:force}, we show how the density modulation translates into a net nonconservative force exerted on the object. We characterize the force in the far field limit and show its magnitude to depend on the distance from the wall and on the orientation of the object. \n\nIn most of what follows, we focus on a dilute active bath. We thus solve for a single active particle that interacts with the obstacle and the boundaries. The average density for a bath comprising $N$ active particles is then simply $\\rho_N(\\bfr)=N\\rho(\\bfr)$ where $\\rho(\\bfr)$ is the probability density of finding the active particle at position $\\bfr$. For the flat-wall case discussed in this section, our results are generalized to particles interacting via pairwise forces in Appendix~\\ref{app:PFAPs}.\n\nTo proceed, consider the master equation for the probability density $P_a(\\vec{r},\\theta)$ to find an Active Brownian Particle (ABP) or a Run-and-Tumble particle (RTP) at $\\vec{r}=(x,y)$ with orientation $\\vec{u}(\\theta)=(\\cos\\theta,\\sin\\theta)$:\n\\begin{align}\\label{eq:FP}\n    \\partial_t P_a({\\bf r},\\theta)= & -\\!\\nabla\\!\\cdot\\!\\left[-\\mu P_a \\nabla U + v{\\bf u}P_a \\!-\\!{D}_t\\nabla P_a\\right] \\\\\n\t&+{D}_r\\partial_{\\theta}^{2}P_a -\\alpha P_a +\\frac{\\alpha}{2\\pi} \\intop d\\theta' \\, P_a(\\vec{r},\\theta') \\;.\\nonumber\n\\end{align}\nHere $v$ is the self-propulsion speed of the active particle, $\\mu$ its mobility, and ${D}_t$  a translational diffusivity. The particle undergoes random reorientations with a (tumbling) rate $\\alpha$ and rotational diffusion with an angular diffusivity $D_r$. The object is described by the external potential $U(\\vec r)$. In what follows we denote by $\\tau=1\/(D_r+\\alpha)$ and $\\ell_p\\equiv v\\tau$ the particle's persistence time and length, respectively. The hard wall at $x=0$ imposes a zero-flux condition: \\begin{equation}\\label{eq:BCJ}\n    -\\mu\\Pa \\partial_x U  + v \\cos\\theta \\Pa - D_t\\partial_x \\Pa=0\\;.\n\\end{equation}\n\nIntegrating Eq. \\eqref{eq:FP} over $\\theta$ leads to a conservation equation for the density field $\\rho(\\bfr)=\\int d\\theta \\,P_a(\\vec{r},\\theta)$:\n\\begin{subequations}\\label{eq:rho}\n\\begin{align}\n    \\partial_t \\rho &= -\\nabla\\cdot\\vec{J}\\;, \\\\\n    \\vec{J} &= -\\mu \\rho \\nabla U + v\\vec{m} -{D}_t \\nabla \\rho\\ ,\\label{eq:current}\n\\end{align}\n\\end{subequations}\nwhere $\\vec{m} = \\int d\\theta\\,\\vec{u}(\\theta)P_a(\\vec{r},\\theta)$ is the polarization of the active particle and $\\vec{J}$ is the particle current in position space. The boundary condition~\\eqref{eq:BCJ} then translates into $J_x(x=0,y)=0$.\n\nThe dynamics of $\\vec{m}$ is then obtained by multiplying Eq.~\\eqref{eq:FP} by $\\vec{u}(\\theta)$ and integrating over $\\theta$, which gives:\n\\begin{equation}\\label{eq:m}\n        \\tau\\partial_t \\vec{m} = \\frac{\\mu}{v} \\nabla \\cdot \\boldsymbol{\\sigma^a} -  \\vec{m}\\;, \n\\end{equation}\nwhere we have introduced the active stress tensor $\\boldsymbol{\\sigma^a}$~\\cite{takatori_swim_2014,yang_aggregation_2014,Solon2015NatPhys,solon_pressure_2015-3,fily_mechanical_2017}:\n\\begin{subequations}\\label{eq:sigmas}\n\\begin{align}\n    \\sigma^a_{ij} &= - \\frac{v^2\\tau}{2\\mu}\\rho \\delta_{ij} + \\Sigma_{ij}\\;, \\label{eq:sigma}\\\\\n    \\Sigma_{ij} &= -\\frac{v\\tau }{\\mu}\\left[v Q_{ij}-\\left(\\mu \\partial_j U + D_t\\partial_j\\right) m_i\\right]\\;.\\label{eq:Sigma}\n\\end{align}\n\\end{subequations}\nHere $Q_{ij} = \\int d\\theta\\,\\left(u_iu_j - \\delta_{ij}\/2\\right)P_a(\\vec{r},\\theta)$ is the nematic tensor and we have singled out the contribution of the ideal gas pressure $v^2\\tau \\rho\/(2\\mu)$ in the active stress tensor. \n\nTo determine the steady-state density profile, we first note that, on large length scales and long times, far from both the confining wall and the asymmetric object, the motion of the active particle is diffusive with a diffusion coefficient ${D}_{\\rm eff}={D}_{t}+v^2\\tau\/2$. The corresponding probability current is then given by $\\vec{J} \\simeq - {D}_{\\rm eff} \\nabla\\rho$. As one moves closer to the object or the wall, this behavior is modified, which motivates us to define a residual field: the deviation $\\boldsymbol{\\mathcal{J}}$ from a diffusive current~\\cite{bendor2021far}\n\\begin{equation}\\label{eq:def diff J}\n    \\boldsymbol{\\mathcal{J}} \\equiv \\vec{J} + D_{\\rm eff} \\nabla\\rho\\;.\n\\end{equation}\nUsing Eqs.~\\eqref{eq:rho}-\\eqref{eq:m} in the steady state where $\\nabla\\cdot\\vec{J}=0$, one finds that the density $\\rho$ satisfies\n\\begin{subequations}\\label{eq:Poisson}\n\\begin{align}\n    D_{\\rm eff}\\nabla^2\\rho &= \\nabla\\cdot\\boldsymbol{\\mathcal{J}}\\;, \\\\\n    \\mathcal{J}_i &= -\\mu \\rho \\partial_i U + \\mu\\partial_j\\Sigma_{ij}\\;. \\label{eq:Poisson equation diff J}\n\\end{align}\n\\end{subequations}\nThe zero-flux boundary condition on the current at $x=0$ then reads\n\\begin{equation}\\label{eq:BC}\n    J_{x}(x=0,y) = \\left(\\mathcal{J}_{x}-D_{\\rm eff}\\partial_x \\rho\\right)\\big|_{x=0} = 0\\;.\n\\end{equation}\n\n\n\\subsection{Density profile and current}\\label{subsec:density and current}\n\nIn the absence of the obstacle, the solution $\\rho\\fw(\\bfr)$ to Eq.~\\eqref{eq:Poisson} with the boundary condition~\\eqref{eq:BC} is a homogeneous bulk complemented by a finite-size boundary layer near the wall where active particles accumulate on a scale comparable to the persistence length $\\ell_p$~\\cite{elgeti2009self}. In what follows, we denote by $\\boldsymbol{\\mathcal{J}}\\fw$ the corresponding source term in Eq.~\\eqref{eq:Poisson equation diff J} and by $\\boldsymbol\\Sigma\\fw$ the contribution~\\eqref{eq:Sigma} to the active stress.\nBy itself, determining $\\rho\\fw$ is already a difficult problem, whose exact solution is not known~\\cite{elgeti_wall_2013,lee2013active,Fily2015SM,ezhilan2015distribution,wagner_steady-state_2017,wagner2022steady}. To proceed, we thus work in the far field limit away from both the wall and the object, which is itself assumed to be far from the wall.\n\n\nWe first decompose the density field as $\\rho(\\bfr)=\\rho\\fw(\\bfr)+\\delta\\rho(\\bfr)$~\\footnote{In the semi-infinite system we consider here, $\\delta\\rho(\\bfr)$ vanishes at infinity. In a finite system, one would need to consider $\\rho=z \\rho\\fw +\\delta\\rho$ to ensure the proper normalization of the density field.}. Thanks to the linearity of Poisson's equation~\\eqref{eq:Poisson}, $\\delta\\rho$ satisfies\n\\begin{subequations}\\label{eq:Poisson difference}\n\\begin{align}\n    &D_{\\rm eff}\\nabla^2\\delta\\rho = \\nabla\\cdot\\boldsymbol{\\delta}\\boldsymbol{\\mathcal{J}}\\;, \\\\\n    &\\delta\\mathcal{J}_i = -\\mu \\rho \\partial_i U + \\partial_j \\delta\\Sigma_{ij}\\;,\n\\end{align}\n\\end{subequations}\nwhere we have defined $\\boldsymbol{\\delta}\\boldsymbol{\\mathcal{J}} \\equiv \\boldsymbol{\\mathcal{J}}-\\boldsymbol{\\mathcal{J}}\\fw$, and $\\delta\\boldsymbol{\\Sigma} \\equiv \\boldsymbol{\\Sigma}- \\boldsymbol{\\Sigma}\\fw$. The corresponding boundary conditions read:\n\\begin{equation}\\label{eq:BC difference}\n    D_{\\rm eff}\\partial_x \\delta\\rho\\big|_{x=0} = \\delta\\mathcal{J}_{x}(0,y)\\;.\n\\end{equation} \nEquations~\\eqref{eq:Poisson difference}-\\eqref{eq:BC difference} describe the density modulation created by the asymmetric object on the density profile induced by a flat wall. \nTo solve for $\\delta\\rho(\\bfr)$, we introduce the Neumann-Green's function in the right half-plane:\n\\begin{equation}\\label{eq:Neumann-Green's function flat wall}\n    G_N(\\vec{r};\\vec{r}') = -\\frac{1}{2\\pi} \\left[\\ln\\frac{|\\vec{r}-\\vec{r}'|}{\\ell_p} + \\ln\\frac{|\\vec{r}^\\perp-\\vec{r}'|}{\\ell_p}\\right]\\;.\n\\end{equation}\nHere the term involving $\\vec{r}^\\perp = (-x,y)$ can be interpreted as a mirror image created on the other side of the wall.\nNote that the Neumann-Green's function~\\eqref{eq:Neumann-Green's function flat wall} does not satisfy the boundary condition specified by Eq.~\\eqref{eq:BC difference}, since its $x$--derivative vanishes on the boundary. Using Green's second identity~\\cite{kevorkian1990partial,jackson_classical_1999}, this means that the solution $\\delta\\rho$ also includes a surface integral to enforce the correct boundary condition. All in all, it reads\n\\begin{subequations}\\label{eq:formal solution}\n\\begin{align}\n    \\delta \\rho(x,y) =& -\\frac{1}{D_{\\rm eff}}\\intop_0^\\infty dx'\\intop_{-\\infty}^\\infty dy'\\,G_N(x,y;x',y')\\nabla'\\cdot\\boldsymbol{\\delta}\\boldsymbol{\\mathcal{J}}'\\nonumber\\\\\n    & - \\intop_{-\\infty}^\\infty dy'\\,G_N(x,y;0,y')\\partial_x'\\delta\\rho'\\bigg|_{x'=0} \\nonumber \\\\\n    = &  -\\frac{\\mu}{D_{\\rm eff}}\\intop_0^\\infty dx'\\intop_{-\\infty}^\\infty dy'\\,\\rho'\\nabla'U \\cdot \\nabla' G_N(x,y;x',y')\\label{eq:rho2}\\\\\n    & -\\frac{\\mu}{D_{\\rm eff}}\\intop_0^\\infty dx'\\intop_{-\\infty}^\\infty dy'\\, G_N(x,y;x',y') \\partial_i'\\partial_j'\\delta\\Sigma_{ij}'\\label{eq:rho3} \\\\\n    & - \\frac{\\mu}{D_{\\rm eff}}\\intop_{-\\infty}^\\infty dy'\\,G_N(x,y;0,y')\\partial_j'\\delta\\Sigma_{xj}'\\bigg|_{x'=0}\\;, \\label{eq:rho4}\n\\end{align}\n\\end{subequations}\nwhere primed derivatives are taken with respect to $x'$ and $y'$. To obtain Eq.~\\eqref{eq:formal solution} we use Eqs. \\eqref{eq:Poisson difference} and \\eqref{eq:BC difference} and an integration by parts. As we now show, the leading-order contribution to $\\delta\\rho$ in the far field is given by~\\eqref{eq:rho2}. Noting that $\\nabla' U $ is localized at $\\bfr_0=(d,0)$, we  approximate the Green's function as $\\nabla' G_N(x,y;x',y')\\simeq \\nabla' G_N(x,y;x',y')|_{x'=d,y'=0}$ in the first integral. In the far field, where ${|\\vec{r}-\\vec{r}_0|,d\\gg a,\\ell_p}$ with $a$ the size of the object, this leads to\n\\begin{align}\\label{eq:density force monopole flat wall}\n    \\rho(\\vec{r}) \\simeq& \\rho_b + \\frac{\\mu }{2\\pi D_{\\rm eff}}\\left[\\frac{\\vec{p}\\cdot (\\vec{r}-\\vec{r}_0)}{\\left|\\vec{r}-\\vec{r}_0\\right|^2} + \\frac{\\vec{p}^\\perp \\cdot (\\vec{r}-\\vec{r}_0^\\perp)}{\\left|\\vec{r}-\\vec{r}_0^\\perp\n    \\right|^2}\\right] \\nonumber \\\\\n    & + \\mathcal{O}\\left(\\left|\\vec{r}-\\vec{r}_0\\right|^{-2},d^{-2}\\right)\\;,\n\\end{align}\nwhere $\\vec{p}^\\perp=(-p_x,p_y)$ and we have used both that \n$\\rho\\fw\\simeq \\rho_b$ far from the wall and that ${\\bf u} \\cdot {\\bf v} = {\\bf u}^\\perp \\cdot {\\bf v}^\\perp$. In Eq.~\\eqref{eq:density force monopole flat wall},  $\\vec{p}$ is a force monopole defined by\n\\begin{equation}\\label{eq:force monopole p flat wall}\n    \\vec{p} = -\\intop d\\vec{r}\\,\\rho\\nabla U\\;.\n\\end{equation}\nIt measures the force exerted on the active fluid by the object in a system without a wall, whose exact value depends on microscopic details of $U$.\nGoing back to Eq.~\\eqref{eq:formal solution}, we show in appendix~\\ref{app:multipole} that~\\eqref{eq:rho3} and~\\eqref{eq:rho4} are indeed negligible compared to~\\eqref{eq:density force monopole flat wall}. Intuitively, this relies both on the extra derivatives in Eq.~\\eqref{eq:rho3} and on the fact that we can use self-consistently the far-field approximation to $\\delta \\boldsymbol\\Sigma$ far away from the object.\n\n\\begin{center}\n\\begin{figure}[t]\n    \\begin{tikzpicture}\n        \\fill[teal!20!white] (-3,-1.5) rectangle (3,1.5);\n        \n        \\fill[purple!70!white] (2.59611-1.15,0.00139995+0.4) arc(158:338:0.375) -- (3.17561-1.15,-0.217276+0.4) arc(338:158:0.25) -- cycle;\n        \n        \\fill[purple!70!white] (-2.59611+1.15,0.00139995+0.4) arc(22:-158:0.375) -- (-3.17561+1.15,-0.217276+0.4) arc(-158:22:0.25) -- cycle;\n        \n        \\draw[-stealth,thick] (1.7,0) -- (1.7+0.26222,0.649029) node[anchor=east] {${\\bf p}$};\n        \\draw[thick] (2.05 ,0) arc(0:68:0.35) node[anchor=west, xshift=1.5mm] {$\\phi\\ $};\n        \n        \\draw[-stealth,thick] (-1.7,0) -- (-1.7-0.26222,0.649029) node[anchor=west, xshift=1mm] {${\\bf p}^\\perp$};\n        \\draw[thick] (-2.05 ,0) arc(180:180-68:0.35) node[anchor=east, xshift=-1mm] {$\\phi\\ $};\n        \n        \\draw[thick,->] (-3,0) -- (3,0) node[anchor=north west] {$\\!x$};\n        \\draw[thick,->] (0,-1.5) -- (0,1.5) node[anchor=south east] {$y$};\n    \\end{tikzpicture}\n    \\caption{The asymmetric object and the flat wall shown in Fig.~\\ref{fig:configuration} generate  density modulations and currents in the active medium, far away from both the object and the wall, equivalent to those generated by two force monopoles $\\bf p$ and $\\bf p^\\perp$ placed symmetrically with respect to the $x=0$ plane.}\\label{fig:configuration image}\n\\end{figure}\n\\end{center}\n\nThe far-field currents can then be obtained from the above result. We first note that, outside the object, $\\boldsymbol{\\mathcal{J}}$ is negligible compared to the diffusive current $-D_{\\rm eff} \\nabla \\rho$ (See Eq.~\\eqref{eq:def diff J}). One thus has that $\\vec{J}=\\boldsymbol{\\mathcal{J}}-D_{\\rm eff} \\nabla \\rho \\simeq -D_{\\rm eff} \\nabla \\rho$ so that, to leading order:\n\\begin{align}\n    \\vec{J} &\\underset{|\\vec{r}-\\vec{r}_0|,d\\gg a,\\ell_p}{\\simeq} \\frac{\\mu}{2\\pi |\\vec{r}-\\vec{r}_0|^2}\\left[\\frac{2[(\\vec{r}-\\vec{r}_0)\\cdot\\vec{p}](\\vec{r}-\\vec{r}_0)}{|\\vec{r}-\\vec{r}_0|^2} - \\vec{p}\\right]\\nonumber \\\\\n    & + \\frac{\\mu }{2\\pi |\\vec{r}^\\perp\n    -\\vec{r}_0|^2}\\left[\\frac{2[(\\vec{r}-\\vec{r}_0^\\perp)\\cdot\\vec{p}^\\perp\n    ](\\vec{r}-\\vec{r}_0^\\perp)}{|\\vec{r}^\\perp-\\vec{r}_0|^2} - \\vec{p}^\\perp\n    \\right]\\;.\n\\end{align}\n\nIn summary, to this order of the multipole expansion, the problem of finding the steady-state density in the far field is reduced to a much simpler problem\n\\begin{subequations}\\label{eq:reduced Poisson's equation}\n\\begin{align}\n    D_{\\rm eff}\\nabla^2  \\rho &= \\mu \\nabla\\cdot\\left[\\vec{p}\\delta(\\vec{r}-\\vec{r}_0)\\right]\\;,\\\\\n    \\partial_x \\rho\\big|_{x=0} &= 0\\;,\n\\end{align}\n\\end{subequations}\nwhich amounts to Eqs.~\\eqref{eq:Poisson difference} and~\\eqref{eq:BC difference} with $\\boldsymbol{\\mathcal{J}}\\simeq\\mu \\vec{p}\\delta(\\vec{r}-\\vec{r}_0)$. In the far field of both the object and the wall, the object thus appears as a force monopole ${\\bf p}$ at position $\\bfr_0$ driving the fluid while the wall is equivalent to an image monopole ${\\bf p}^\\perp$ at position $\\bfr_0^\\perp$, as can be read directly in Eq.~\\eqref{eq:density force monopole flat wall} (see Fig.~\\ref{fig:configuration image}).\n\n\n\\subsection{Nonconservative force induced on the object}\\label{subsec:force}\n\nAccording to Newton's third law, the object experiences a force $-{\\bf p}$ from the active fluid. Equation~\\eqref{eq:force monopole p flat wall} shows ${\\bf p}$ to depend on the local density of active particles $\\rho(\\bfr)$, which in turn depends on the distance $d$ from the wall through Eq.~\\eqref{eq:density force monopole flat wall}. It is thus convenient to decompose ${\\bf p}$ as: \n\\begin{equation}\\label{eq:F definition}\n    \\vec{p} \\equiv \\vec{p}_b - \\vec{F}\\;,\n\\end{equation}\nwith $\\vec{p}_b$ defined as the value of ${\\bf p}$ when $d \\to \\infty$. Then $\\vec{F}$ measures the change in the force due to the presence of the wall. Namely, $\\vec{F}$ is the force induced on the object by the wall, which is mediated by the active bath. \n\nSince the interaction with the wall is equivalent, to leading order, to the interaction with an image object, we can use the results of Refs.~\\cite{baek_generic_2018,granek2020bodies} to derive $\\vec{F}$. In the setting considered there, two objects, referred to as object 1 and object 2, are placed at positions $\\vec{r_1}$ and $\\vec r_2$, with  $\\vec{r}_{12}\\equiv\\vec{r}_1-\\vec{r}_2$. When $|\\vec{r}_{12}| \\to \\infty$ the objects experience forces $-\\vec{p}_1$ and $-\\vec{p}_2$ from the fluid, respectively. When $|\\vec{r}_{12}|$ is finite the force experienced by object 1 is $-\\vec{p}_1+\\vec{F}_{12}$, with $\\vec{F}_{12}$ the force exerted on object 1 due to the presence of object 2. In~\\cite{baek_generic_2018}, it was shown that, to leading order in the far field, the interaction force arises due to a density modulation $\\Delta \\rho (\\vec{r}_1)$ near object 1 due to object 2. This non-reciprocal interaction force takes the form:\n\\begin{equation} \\label{eq:F12}\n    \\vec{F}_{12} = -\\frac{\\Delta\\rho(\\vec{r}_1)}{\\rho_b}\\vec{p}_1\\;, \n\\end{equation}\nwith\n\\begin{equation} \\label{eq:Deltarho1}\n     \\Delta \\rho(\\vec{r}_1) = \\frac{\\mu}{2\\pi D_{\\rm eff}} \\frac{\\vec{r}_{12}\\cdot\\vec{p}_2}{|\\vec{r}_{12}|^2} + \\mathcal{O}(|\\vec{r}_{12}|^{-2})\\;.\n\\end{equation}\nHere, $\\bfr_{12}=(2d,0)$ and ${\\bf p}_2={\\bf p}_b^\\perp$ (see Fig.~\\ref{fig:configuration image}), leading to\n\\begin{equation} \\label{eq:F12-bare}\n    \\vec{F} = \\frac{\\mu} {2 \\pi D_{\\rm eff} \\rho_{b}} \\frac{p_{b,x} }{2 d}  \\vec{p}_b+\\mathcal{O}(d^{-2})\\;.\n\\end{equation}\nDenoting by $\\phi$ the orientation of ${\\bf p}_b$ relative to the $\\vec x$ axis then leads to:\n\\begin{align}\\label{eq:force flat wall}\n    \\vec{F} \n    = &\\frac{\\mu p_b^2}{8\\pi D_{\\rm eff} \\rho_b d} \\binom{1+\\cos(2\\phi)}{\\sin(2\\phi)}+{\\cal O}\n    (d^{-2})\\;,\n\\end{align} \nwith $p_b = |\\vec{p}_b|$. Note that this result implies that the wall \\textit{always repels the object}, irrespective of its orientation $\\phi$. It is easy to check that $\\partial_x F_y-\\partial_y F_x\\neq 0$, except when $\\phi\\in\\{0,\\pi\\}$, so that the interaction force is not conservative~\\footnote{The divergence of the force is negative throughout the domain, $\\nabla\\cdot{\\bf F}<0$, in principle allowing for possible stable fixed points if ${\\bf F}=0$.}.\n\n\\begin{center}\n\\begin{figure}[t]\n    \\begin{tikzpicture}[]\n    \n        \\fill[teal!20!white] (0,-1.5) rectangle (4,1.5);\n        \\fill[orange!65!white] (-0.2,-1.5) rectangle (0,1.5);\n        \\fill[pattern=north west lines, pattern color=black] (-0.2,-1.5) rectangle (0,1.5);\n        \n        \\node[color=purple!70!white, font=\\fontsize{50}{22.4}] at (2.5,0) {$\\uptau$};\n        \n        \\node[] at (2.55,1) {$\\tau_b$};\n        \\draw[thick,->] (1.9,0.3) arc(160:20:.7);\n        \n        \\draw[thick,->] (-.5,0) -- (4,0) node[anchor=north west] {$\\!x$};\n        \\draw[thick,->] (0,-1.5) -- (0,1.5) node[anchor=south east] {$y$};\n    \\end{tikzpicture}\n    \\caption{A $\\uptau$-shaped object generically experiences a non-zero self-torque $\\tau_b$.}\\label{fig:torque}\n\\end{figure}\n\\end{center}\n\nFinally, an asymmetric object may also experience a torque from the surrounding active fluid~\\cite{di_leonardo_bacterial_2010,Sokolov2010PNAS} (See Fig.~\\ref{fig:torque}). In two dimensions this torque is given by\n\\begin{equation}\\label{eq:torque}\n    \\boldsymbol\\tau = \\intop_\\Omega d\\vec{r}\\, \\rho(\\vec{r}) (\\vec{r}-\\vec{r}_{\\scriptstyle \\rm CM}) \\times \\nabla U\\;,\n\\end{equation}\nwhere $\\vec{r}_{\\scriptstyle \\rm CM}$ is the object's center of mass. Denoting the magnitude of $\\boldsymbol\\tau$ when $d\\to\\infty$ as $\\tau_b$ and using the image object along with the results of Refs.~\\cite{baek_generic_2018,granek2020bodies}, we find that the interaction torque $M$ due to the wall, defined through \n$\\tau=\\tau_b+M$, is given by\n\\begin{equation}\n    M = \\frac{\\mu p_b}{4\\pi D_{\\rm eff}\\rho_b} \\frac{\\cos(\\phi)}{d}\\tau_b + \\mathcal{O}(d^{-2})\\;.\n\\end{equation}\n\nNote that when the object is not chiral, $\\tau_b$ vanishes and there is no torque to order $\\mathcal{O}(d^{-1})$. Higher order contributions are, however, expected from symmetry considerations: the density modulation along $\\hat x$ due to the presence of the wall indeed breaks the chiral symmetry when $\\bf p$ is not along $\\hat x$. \n\n\n\\section{A object inside a circular cavity}\\label{sec:circular cavity}\n\n\nIn the previous section, we showed that, far from the object and away from a boundary layer created by a flat wall, the steady-state distribution and current of active particles are equivalent to those induced by two force monopoles placed symmetrically with respect to the plane of the wall. In turn, we showed that the object interacts with its mirror image, with an interaction force given by Eq.~\\eqref{eq:F12-bare}. \nWe now consider a different setup of an asymmetric object placed in a circular cavity (See Fig.~\\ref{fig:configuration circle}). We first determine in Section~\\ref{sec:CCdens} the long-ranged density modulation and current induced by object. Then, in Section~\\ref{sec:CCforce}, we compute the contribution of the force experienced by the object due to the circular confining boundary.\n\n\\begin{center}\n\\begin{figure}[t]\n    \\begin{tikzpicture}\n        \\clip (2-0.3,3.5) rectangle (5.5,-0.35);\n        \\filldraw[color=black,fill=orange!65!white] (2,0) circle (2.65);\n        \\filldraw[pattern=north east lines, pattern color=black] (2,0) circle (2.65);\n        \n        \\filldraw[color=black,fill=teal!20!white] (2,0) circle (2.5);\n    \n        \\draw[thick,->] (-1.1,0) -- (5.1,0) node[anchor=north west] {$\\!x$};\n        \\draw[thick,->] (2,-3) -- (2,3) node[anchor=south east] {$y$};\n        \n        \\fill[purple!70!white] (2.59611+.4,0.00139995+1) arc(158:338:0.375) -- (3.17561+.4,-0.217276+1) arc(338:158:0.25) -- cycle;\n        \n        \\draw[-stealth,thick] (3+.3,0+.6) -- (3+.3+0.26222,0.649029+.6) node[anchor=east, xshift=-1mm] {${\\bf p}$};\n        \\draw[-stealth,thick] (2,0) -- (3+.3,0+.6) node[anchor=south, xshift=-7mm, yshift=-3mm] {${\\bf r}_0$};\n        \\draw[dashed,thick] (3+.3,0+.6) -- (4.2699,1.04765);\n        \\draw[dashed,thick] (3+.3,0+.6) -- (4.35,.6);\n        \\draw[line width=1.2pt,blue] (3.3 - 0.02 + 0.363184 ,.6 + 0.167623) arc(25:68:0.4) node[anchor=west, xshift=1.5mm, yshift=1.5mm] {$\\psi\\; $};\n        \n        \\draw[thick] (3.3+.3 ,.6) arc(0:68:0.3) node[anchor=west, xshift=-2mm, yshift=-5mm] {$\\phi\\; $};\n        \n        \\draw[thick] (2+.7 ,0) arc(0:68:0.27) node[anchor=west, xshift=2mm, yshift=-.6mm] {$\\theta_0$};\n\n    \\end{tikzpicture}\n    \\caption{An asymmetric passive object in an active fluid placed inside a circular cavity of radius $R$. The object is located at $\\vec{r}_0$ at an angle $\\theta_0$ relative to the $\\boldsymbol{\\hat{x}}$ axis. The corresponding force monopole $\\vec{p}$  is directed along $\\psi=\\phi-\\theta_0$ relative to $\\hat {\\bf r}$.}\\label{fig:configuration circle}\n\\end{figure}\n\\end{center}\n\n\\subsection{Density profile and current}\n\\label{sec:CCdens}\nConsider a passive asymmetric object placed inside an active fluid confined by a circular cavity of radius $R$. To make progress we assume that the far-field density modulation is given, to leading order, by the solution of Eq.~\\eqref{eq:reduced Poisson's equation} together with the Neumann boundary condition $\\hat r \\cdot \\nabla \\rho(\\bfr)|_{|\\bfr|=R}=0$.\nThe Neumann-Green's function in this geometry can be obtained in several ways, for example, by using conformal transformations or by using the polar symmetry of the domain. It is given by~\\cite{kevorkian1990partial}\n\\begin{align}\n    G^{\\rm disk}_N(\\vec{r};\\vec{r}_0) = -\\frac{1}{2\\pi}&\\left[\\ln\\left(|\\vec{r}-\\vec{r}_0|\/\\ell_p\\right)\\right. \\nonumber \\\\\n    +&\\left. \\ln\\left(|\\vec{r}-\\tilde{\\vec{r}}_0|\/\\ell_p\\right) + \\ln\\left(r_0\/\\ell_p\\right)\\right]\\;,\n\\end{align}\nwith $\\tilde{\\vec{r}}_0 \\equiv (R\/r_0)^2\\vec{r}_0$. Again, we write  $\\rho=\\rho_b + \\delta\\rho$, with $\\rho_b$ the average density in the cavity. The leading order density modulation $\\delta\\rho(\\vec{r})$ is then given in the far field by\n\\if{\\begin{equation}\\label{eq:density disk}\n    \\delta\\rho \\simeq - \\frac{\\mu}{2\\pi D_{\\rm eff}}\\left[\\left(\\frac{\\vec{r}-\\vec{r}_0}{\\left|\\vec{r}-\\vec{r}_0\\right|^2} - \\frac{\\vec{r}_0}{r_0^2}\\right)\\cdot \\vec{p} - \\frac{(\\vec{r}-\\tilde{\\vec{r}}_0)\\cdot \\tilde{\\vec{p}}}{\\left|\\vec{r}-\\tilde{\\vec{r}}_0\\right|^2}\\right]\\;,\\nonumber\n\\end{equation}}\\fi\n\\begin{equation}\\label{eq:density disk}\n    \\delta\\rho \\simeq - \\frac{\\mu}{2\\pi D_{\\rm eff}}\\left[\\frac{(\\vec{r}-\\vec{r}_0)\\cdot \\vec{p}}{\\left|\\vec{r}-\\vec{r}_0\\right|^2} - \\frac{(\\vec{r}-\\tilde{\\vec{r}}_0)\\cdot \\tilde{\\vec{p}}}{\\left|\\vec{r}-\\tilde{\\vec{r}}_0\\right|^2}\\right]+\\frac{\\mu \\,\\bfr_0\\cdot \\vec{p}}{2 \\pi D_{\\rm eff}r_0^2}\\;,\n\\end{equation}\nwhere $\\tilde{\\vec{p}}\\equiv (R\/r_0)^2 p \\vec{u}(2\\theta_0-\\phi)$. The diffusive current is then obtained using $\\vec{J}\\simeq -D_{\\rm eff}\\nabla \\rho$, leading to\n\\begin{align}\n    \\vec{J} \\simeq \\frac{\\mu}{2\\pi} &\\left[\\frac{1}{|\\vec{r}-\\vec{r}_0|^2}\\left(\\frac{2[(\\vec{r}-\\vec{r}_0)\\cdot\\vec{p}](\\vec{r}-\\vec{r}_0)}{|\\vec{r}-\\vec{r}_0|^2} - \\vec{p}\\right) \\right. \\nonumber \\\\\n    - &\\;\\, \\left. \\frac{1}{|\\vec{r}-\\tilde{\\vec{r}}_0|^2}\\left(\\frac{2[(\\vec{r}-\\tilde{\\vec{r}}_0)\\cdot\\tilde{\\vec{p}}](\\vec{r}-\\tilde{\\vec{r}}_0)}{|\\vec{r}-\\tilde{\\vec{r}}_0|^2} - \\tilde{\\vec{p}}\\right) \\right]\\;.\\label{eq:Jcircular}\n\\end{align}\nAgain, the current in Eq.~\\eqref{eq:Jcircular} is equivalent to that generated by the force monopole and an image monopole $\\tilde {\\bf p}$  placed at $\\tilde \\bfr_0$. The same applies to the density modulation, which also experiences an additional uniform contribution that enforces mass conservation.\n\nWe verified our predictions using numerical simulations of RTPs which are shown in Figure~\\ref{fig:density and current}. Both density modulations and currents  are well described by Eqs.~\\eqref{eq:density disk} and~\\eqref{eq:Jcircular}. \n\n\\begin{figure}[h!]\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{density_and_current.eps}\n\t\\caption {Density and current profiles surrounding an asymmetric object inside a circular cavity. The object, shaped as a semicircular arc of diameter $d_{\\rm arc}=\\ell_p$, is located at $\\vec{r}_0=(0.45R,0)$ with an orientation making an angle $\\phi=0.6\\pi$ with the $\\boldsymbol{\\hat{x}}$ axis. \n\tThe object is displayed in orange and enlarged by a factor of six.\n\t(a) Steady-state density modulation relative to the bulk density, $\\delta\\rho\/\\rho_b$, compared with the analytical expression~\\eqref{eq:density disk} in gray. (b) Streamlines of the steady-state current. The measurement (in light blue) is compared with the theory (in gray), for the same parameters as in (a).\n    In both panels, the parameters were set as follows:  $N=10^5$ RTPs travel with speed $v=10^{-4}$ and tumble at rate $\\alpha = 10^{-2}$. \n    See Appendix~\\ref{app:numerics} for details.}\n \t\\label{fig:density and current}\n\\end{figure}\n\n\\subsection{Interaction force}\\label{sec:CCforce}\nNext we turn to derive the force induced on the object by the circular wall. To do this we first note that the presence of the wall leads to a density modulation\n\\begin{equation}\n    \\Delta\\rho(\\vec{r}_0) \\approx \\frac{\\mu}{2\\pi D_{\\rm eff}}\\left.\\left[ \\frac{\\vec{r}_0}{r_0^2}\\cdot \\vec{p} + \\frac{(\\vec{r}-\\tilde{\\vec{r}}_0)\\cdot \\tilde{\\vec{p}}}{\\left|\\vec{r}-\\tilde{\\vec{r}}_0\\right|^2}\\right]\\right|_{\\vec{r}=\\vec{r_0}}\\;,\n\\end{equation}\nwhen compared to the situation in an infinite space. The force due to the presence of the wall is then given by  Eq.~\\eqref{eq:F12}, which leads to:\n\\begin{equation}\\label{eq:force circular cavity}\n    \\vec{F} \\approx -\\frac{\\mu p_b^2}{2\\pi D_{\\rm eff}\\rho_b}\\frac{r_0\\cos(\\phi-\\theta_0)}{R^2-r_0^2}\\binom{\\cos(\\phi)}{\\sin(\\phi)}\\;,\n\\end{equation}\nwith $p_b$ the magnitude of the force monopole measured either in the center of the cavity or equivalently for $\\tilde{ r}_0 \\to \\infty$. As in the case of a flat wall, the force always repels the object away from the wall, as can be seen by setting $\\theta_0=0$. Figure~\\ref{fig:force collapse} shows a collapse of the force measured on the object for various orientations and distances from the wall, showing good agreement with the theory~\\eqref{eq:force circular cavity}.\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{force_collapse.eps}\n\t\\caption{Collapse of the interaction force exerted on the object. The data displayed here shows magnitude of the interaction force~\\eqref{eq:force circular cavity} divided by its angular dependence $\\cos(\\psi)=\\cos(\\phi-\\theta_0)$ relative to the strength of the force monopole $p$. \n\tThe solid black line shows the theoretical prediction with no parameter fitting.\n\tNote that the deviation from the theory near the walls is expected, due to higher-order interactions.\n\t}\n \t\\label{fig:force collapse}\n\\end{figure}\n\nFinally, as in the case of the flat wall, we can compute the interaction torque $M$ acting on the object, which is given by\n\\begin{equation}\n    M \\approx \\frac{\\mu p_b}{2\\pi D_{\\rm eff}\\rho_b}\\frac{r_0\\cos(\\phi-\\theta_0)}{R^2-r_0^2}\\tau_b\\;,\n\\end{equation}\nwhere $\\tau_b$ is the object's self-torque measured at $r_0=0$.\n\n\\section{Dynamics of an asymmetric object inside a circular cavity}\\label{sec:dynamics}\n\nIn the previous section, we computed  the density modulation and current induced by a polar object held fixed in a circular cavity. The presence of confining walls leads to a renormalization of the force felt by the object which depends on its position and orientation. When the object is mobile, it is thus endowed with a non-uniform propulsion force. In this section we use a toy model to capture the corresponding dynamics and characterize its steady-state distribution. We find that the interaction with the wall leads to a transition between two distinct behaviors: the object is  localized either in the center of the cavity or near the edges, as observed in Fig.~\\ref{fig:body prob dist}. \n\nTo lighten the notations, we drop the subscript \"0\" when refering to the object so that its position reads ${\\bf r}=r {\\bf u}(\\theta)$ and its orientation makes an angle $\\phi$ with $\\hat x$. We model the object's dynamics as an effective Langevin equation:\n\\begin{subequations}\\label{eq:body Langevin}\n\\begin{align}\n    \\dot{\\vec{r}} =& \\mu_0 p \\vec{u}(\\phi) + \\mu_0 \\vec{F}(r,\\theta,\\phi)  + \\sqrt{2D_t^e}\\boldsymbol{\\eta}(t) \\;,\\label{eq:dynpos}  \\\\\n    \\dot{\\phi} =& \\sqrt{2D_r^e} \\xi(t)\\;,\n\\end{align}\n\\end{subequations}\nwhere $\\mu_0$ is the mobility of the object, $p$ is the magnitude of $-{\\bf p}_b$, $D_t^e$ and $D_r^e$ are effective translational and rotational diffusivities, and $\\eta_i(t)$ and $\\xi(t)$ are Gaussian white noises of zero mean and unit variance. For simplicity, we consider a symmetric object whose self-torque is zero. \n\nWe now use the explicit expression of ${\\bf F}$ given in Eq.~\\eqref{eq:force circular cavity} and the angle $\\psi=\\phi-\\theta$ between the object and $\\hat {\\bf r}$ (see Fig.~\\ref{fig:configuration circle}) to rewrite  Eq.~\\eqref{eq:body Langevin} as a dynamics for $r$, $\\theta$ and $\\psi$. Since $r={\\bf r}\\cdot \\hat {r}$, It\\=o calculus implies that $\\dot r=\\dot{\\bf r}\\cdot \\hat r + D_t^e\/r$. Similar to the case of a particle in a harmonic well~\\cite{solon_active_2015}, the equations for $r$ and $\\psi$ decouple from the dynamics of $\\theta$, and read:\n\\begin{subequations}\\label{eq:body Langevin 2}\n\\begin{align}\n    \\dot r =& \\mu_0 p \\cos\\psi \\left[1 -\\frac{q r R \\cos(\\psi)}{ R^2-r^2}\\right]+\\frac{D_t^e}{r}+ \\sqrt{2D_t^e} \\eta_r(t) \\;,\\label{eq:dynposr}  \\\\\n    \\dot{\\psi} =& -\\frac{\\mu_0 p \\sin\\psi}{r} \\left[1 -\\frac{q r R \\cos(\\psi)}{ R^2-r^2}\\right]\\nonumber \\\\ &\\quad+\\sqrt{2\\left(\\frac{D_t^e}{r^2}+D_r^e\\right)} \\xi_\\psi(t)\\;\\label{eq:dynpospsi},\n\\end{align}\n\\end{subequations}\nwhere $\\eta_r$ and $\\xi_\\psi$ are Gaussian white noises of zero mean and unit variance and $q=\\mu p\/(2 \\pi D_{\\rm eff} \\rho_b R)$ is a dimensionless parameter.\n\n\\if{\nThis Langevin dynamics is equivalent to the Fokker-Planck equation:\n\\begin{subequations}\\label{eq:FP object}\n\\begin{align}\n    \\partial_t P =& - \\frac{1}{r_0}\\frac{\\partial}{\\partial r_0}\\left(r_0 J_r\\right) - \\frac{1}{r_0}\\frac{\\partial J_\\psi}{\\partial \\psi}\\;,\\\\\n    J_r =& \\mu_0 p \\cos(\\psi) \\left(1-\\frac{q R^2 r_0 \\cos(\\psi)}{R^2 - r_0^2}\\right) P - D_t^e \\partial_{r_0} P\\;,\\\\\n    J_\\psi =& -\\mu_0 p \\sin(\\psi) \\left(1-\\frac{q R^2 r_0 \\cos(\\psi)}{R^2 - r_0^2}\\right)P \\nonumber \\\\\n    &- \\left(\\frac{D_t^e}{r_0} + r_0 D_r^e\\right) \\partial_\\psi P\\;.\n\\end{align}\n\\end{subequations}\nHere $J_r$ is the radial current flowing in the system, $J_\\psi$ the current associated with the relative angle $\\psi$, and $q\\equiv \\mu p\/(2\\pi D_{\\rm eff}\\rho_b R^2)$ an inverse length scale associated with the interaction with the walls. It should further be noted that the probability distribution $P(r_0,\\psi)$ is normalized such that $\\int dr_0 d\\psi\\,r_0 P(r_0,\\psi)=1$.}\\fi\n\nSolving for the steady-state probability distribution $P(r,\\psi)$ remains a hard task. Instead, we study the dynamics~\\eqref{eq:body Langevin 2} in two limits: first when the object reorients so quickly that it is no longer persistent, $D_r^e\\to\\infty$, resulting in an effective equilibrium dynamics; and second, in the opposite limit, $D_r^e\\to0$, when the object is highly persistent. These two regimes lead to very different behaviors that explain the transition observed in Fig.~\\ref{fig:body prob dist}.\n\n\\textit{Effective equilibrium limit.}\nIn the large $D_r^e$ limit, the dynamics of $\\psi$ is dominated by the rotational diffusion, which leads to $P(r,\\psi)\\simeq P(r)\/(2\\pi)$. Taking the average of Eq.~\\eqref{eq:dynposr} with respect to $\\psi$ then leads to:\n\\begin{equation}\\label{eq:body Langevin 4}\n\\dot r = - \\frac{ \\mu_0 p  q r R}{2( R^2-r^2)}+ \\frac{D_t^e}{r}+\\sqrt{2D_t^e} \\eta_r(t) \\;.\n\\end{equation}\nThe steady-state distribution of $r$ is then given by\n\\begin{equation}\\label{eq:P diff scalar}\n P(r)\\propto r \\left(1-\\frac{r^2}{R^2}\\right)^{\\frac{\\mu_0 p q R}{4 D_t^e}}\\;,\n\\end{equation}\nwhere $P(r)$ is normalized as $\\int_0^R dr\\, P(r)=1$. Going back to the original ${\\bf r}$ variable, one thus gets\n\\begin{equation}\\label{eq:P diff}\n    P({\\bf r})\\propto  \\left(1-\\frac{|{\\bf r}|^2}{R^2}\\right)^{\\frac{\\mu_0 p q R}{4 D_t^e}}\\;,\n\\end{equation}\nwhose normalization in two dimensions reads $\\int d{\\bf r}\\,P({\\bf r})=1$. Importantly, as can be seen in Fig.~\\ref{fig:diffusive P}, the distribution is peaked around ${\\bf r}=0$ and perfectly matches microscopic simulations of Eq.~\\eqref{eq:body Langevin 2}. The result is reminiscent of the steady-state distribution of a run-and-tumble particle in a harmonic well in one space dimension~\\cite{Tailleur2008PRL,dhar2019run}. \n\n\nFinally, we note that this effective equilibrium regime allows for the localization of the object in the bulk of a non-equilibrium diffusive fluid. As mentioned in the introduction, this would be impossible in equilibrium due to Earnshaw's theorem. Here, when going from Eq.~\\eqref{eq:body Langevin 2} to Eq.~\\eqref{eq:body Langevin 4}, the `bare' self-propulsion force of the object has cancelled out and we are only left with the contribution from its image. The reason why the latter does not lead to a vanishing contribution is the strong anti-correlation between ${\\bf p}$ and its image.\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{diffusive_bfr.eps}\n\t\\caption{Steady-state probability distribution $P({\\bf r})$ of the dynamics~\\eqref{eq:body Langevin 2} in the large $D_r^e$ limit. Direct simulations of the Langevin dynamics~\\eqref{eq:body Langevin 2} (blue dots) agree perfectly with the analytic prediction of Eq.~\\eqref{eq:P diff} (orange solid line).\n\t}\n \t\\label{fig:diffusive P}\n\\end{figure}\n\n\\textit{Large-persistence regime}.\nWe now consider the opposite limit of a very small rotational diffusivity and set, for simplicity, $D_t^e=0$. In this limit, the dynamics~\\eqref{eq:body Langevin 2} reduce to:\n\\begin{subequations}\\label{eq:body Langevin 3}\n\\begin{align}\n    \\dot r =& \\mu_0 p \\cos\\psi \\left[1 -\\frac{q r R \\cos(\\psi)}{ R^2-r^2}\\right]\\;,  \\\\\n    \\dot{\\psi} =& -\\frac{\\mu_0 p \\sin\\psi}{r} \\left[1 -\\frac{q r R \\cos(\\psi)}{ R^2-r^2}\\right]\\;.\\label{eq:psi dynamics persistent}\n\\end{align}\n\\end{subequations}\nFollowing~\\cite{solon_active_2015}, we expect that, in this noiseless limit, the object's position and orientation remain close to the stable fixed points of the dynamics~\\eqref{eq:body Langevin 3}, found by requiring $\\partial_t r=\\partial_t \\psi = 0$.\nDirect inspection shows that all fixed points $(r^*,\\psi^*)$ satisfy\n\\begin{equation}\n    1 -\\frac{q r^* R \\cos(\\psi^*)}{ R^2-(r^*)^2} = 0\\;.\n\\end{equation}\nThere is thus a continuous line of fixed points, which can be parameterized as $r^*=r^*(\\psi^*)$:\n\\begin{equation}\\label{eq:r*}\n    r^*(\\psi^*) = \\frac R 2 \\left(\\sqrt{(q \\cos\\psi^*)^2+4} - q \\cos\\psi^*\\right)\\;,\n\\end{equation}\nwith $\\psi^*\\in[-\\pi\/2,\\pi\/2]$. The minimal value of $r^*(\\psi^*)$ corresponds to $\\psi^*=0$ and $r^*(0)=\\frac{R}2(\\sqrt{q^2+4}-q)>0$. This demonstrates that, in the steady state, the object is positioned at a finite distance from the origin, unlike in the effective equilibrium limit. By changing the rotational diffusion of the object, one can thus shift its most probable localization from the center of the cavity to its periphery.\nNote that Eq.~\\eqref{eq:body Langevin 3} relies on the far-field approximation, which is only valid far from the walls of confining boundaries. \\if{Ultimately, the only stable position of the object is at a distance close enough to the boundary where the continuum limit is no longer valid.}\\fi\nUltimately, the only stable position of the object is at a distance close enough to the boundary that the modulation of the density of active particles is of order of $\\rho_b$.\n\nWhile the above discussion already proves the existence of the localization transition, we characterize, for completeness, the stability of the line of fixed points of the large persistence regime. As a first step, we linearize the dynamics~\\eqref{eq:body Langevin 3} about $r^*(\\psi^*)$. This yields a dynamical system that can be written as\n\\begin{equation}\n    \\partial_t \\begin{pmatrix}\n                    \\delta r\\\\\n                    \\delta \\psi\n                \\end{pmatrix} = M(\\psi^*) \\begin{pmatrix}\n                    \\delta r\\\\\n                    \\delta \\psi\n                \\end{pmatrix} + \\mathcal{O}(\\delta r^2,\\delta\\psi^2,\\delta r \\delta\\psi)\\;,\n\\end{equation}  \nfor $\\delta r\\equiv r - r^*(\\psi^*)$ and $\\delta \\psi \\equiv \\psi - \\psi^*$, where \n\\begin{subequations}\n\\begin{align}\n    M_{11} =& -\\frac{\\mu_0p}{2q R}\\left[4 +q \\cos\\psi^*\\times \\right. \\\\ \n    & \\quad\\times\\left. \\left( \\sqrt{(q \\cos\\psi^*)^2+4}+q \\cos\\psi^*\\right)\\right]\\;,\\nonumber \\\\\n    M_{12} =& \\mu_0p\\sin\\psi^*\\;,\\\\\n    M_{21} =& \\frac{mu_0p}{2 q R^2}\\left[2 \\tan\\psi^* \\sqrt{(q \\cos\\psi^*)^2+4}+4 q \\sin\\psi^*\\right. \\nonumber \\\\\n    &\\left.+ \\frac{q^2}2 \\sin(2\\psi^*) \\left(\\sqrt{(q \\cos\\psi^*)^2+4}+q \\cos\\psi^*\\right)\\right]\\\\\n    M_{22} =& -\\frac{2 \\mu_0 p}{R}\\frac{ \\sin\\psi^*\\tan\\psi^*}{\\sqrt{(q \\cos\\psi^*)^2+4}-q \\cos\\psi^*}\\;.\n\\end{align}\n\\end{subequations}\nNote that the matrix $M$ depends on the value of $\\psi^*$. For a given value of $\\psi^*$, $M$ can be diagonalized. Its eigenvectors point in two different directions: ${\\bf v}_1(\\psi^*)$ is tangent to the curve $r^*(\\psi^*)$ and corresponds to a zero eigenvalue $\\lambda_1=0$;  ${\\bf v}_2(\\psi^*)$ points in a different direction and is associated to a negative eigenvalue $\\lambda_2$, given by\n\\begin{equation}\n    \\lambda_2(\\psi^*) = -\\frac{\\mu_0p}{2 q R}\\left(4 + q^2 + q\\frac{\\sqrt{(q\\cos\\psi^*)^2+4}}{\\cos\\psi^*}\\right)\\;.\n\\end{equation}\n\\if{It should be noted that when $\\psi^*=0$ the eigenvectors point along the directions of $\\delta r$ and $\\delta \\psi$, namely, ${\\bf v}_1(\\psi^*=0)=(0,1)$ and ${\\bf v}_2(\\psi^*=0)=(1,0)$.}\\fi The direction along the line of fixed point is thus, as expected, marginally stable, whereas the transverse direction is linearly stable. To find the most probable value of $\\psi^*$ and $r^*$, we thus need to go beyond the linear stability analysis and consider the non-linear dynamics of the perturbation along the line of fixed points. \n\n\\if{The linear stability of the fixed points can be studied using the eigenvalues and the eigenvectors of $M$. Since ${\\bf v}_2$ never points along $r^*(\\psi^*)$ and since $\\lambda_2(\\psi^*)$ is always negative, the direction of ${\\bf v}_2$ is stable, with a `restoring force' making the dynamics\n\\begin{equation}\n    \\partial_t {\\bf v}_2 = -\\lambda_2 {\\bf v}_2\\;.\n\\end{equation}\nA general perturbation will thus have a component parallel to ${\\bf v}_2$, that will take it back to the line $r^*(\\psi^*)$.\n\nThe dynamics along the other eigenvector is different. As the eigenvector ${\\bf v}_1(\\psi^*)$ has a null eigenvalue, it is only marginally stable: it is not clear whether the object moves along the line $r^*(\\psi^*)$, changing $\\psi^*$, or stays put. To study this point, one needs to go beyond the linear stability analysis and consider the non-linear effects of the perturbation. }\\fi\n\nTo do so, we expand the evolution of $\\delta\\psi$, given by Eq.~\\eqref{eq:psi dynamics persistent}, to second order in $\\delta r $ and $\\delta \\psi$.\nImposing a perturbation tangent to the curve $r^*(\\psi^*)$ then couples $\\delta r$ and $\\delta \\psi$. This leads to a closed dynamics for $\\delta\\psi$ given by\n\\begin{equation}\n    \\partial_t \\delta \\psi = \\Gamma(\\psi^*) \\delta \\psi^2\\;,\n\\end{equation}\nwhere $\\Gamma(\\psi^*)$ is given in Appendix~\\ref{app:numerics}. Figure~\\ref{fig:dtpsi} shows that $\\Gamma(\\psi^*)$ and $\\psi^*$ have opposite signs so that $\\psi^*=0$ is the sole stable fixed point.\n\nIn the steady state, we thus expect the object to point towards the wall, with $\\psi=0$, at a distance $r=\\frac{R}{2} (\\sqrt{q^2+4} - q)$ from the center of the cavity. Small deviations from this solution are expected mainly along the line of fixed points $r^*(\\psi^*)$. This behavior is verified by a direct numerical simulation of Eq.~\\eqref{eq:body Langevin 2} presented in Fig.~\\ref{fig:persistent P}.\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{Gamma_psi.eps}\n\t\\caption{ $\\Gamma(\\psi^*)$ as a function of $\\psi^*$, for $q=0.5$. The exact expression is given in Appendix~\\ref{app:numerics}. For $\\psi^*>0$, any perturbation $\\delta\\psi$ causes $\\psi^*$ to decrease. For $\\psi^*<0$, the opposite happens, leading to $\\psi^*=0$ as the sole stable fixed point.\n\t}\n \t\\label{fig:dtpsi}\n\\end{figure}\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{persistent.eps}\n\t\\caption{Steady-state behavior of the object in the small $D_r^e$ limit (with $D_t^e=0$).\n\t(a) The probability distribution $P({\\bf r},\\psi)$ measured by direct simulations of the Langevin dynamics~\\eqref{eq:body Langevin 2}. The dotted orange line corresponds to the line of fixed points $r^*(\\psi^*)$ given in Eq.~\\eqref{eq:r*}.\n\t(b) The corresponding marginal probability density $P({\\bf r})$. The dotted orange line marks the expected position of the object when $D_r^e=0$: $r=r^*(\\psi=0)$.\n\t}\n \t\\label{fig:persistent P}\n\\end{figure}\n\n\\if{The dynamics near the stable fixed point $(r^*,\\psi^*)=\\left(\\frac R 2 \\left[\\sqrt{q^2+4} - q\\right],0\\right)$ can be further described using a time-dependent Landau approach.}\\fi\n\n\nAll in all, the two limits of large and small rotational diffusivity show that there is a localization transition from a distribution where the object is localized close to the walls of the cavity to a distribution where the object is localized in its center. The latter occurs when the rotational diffusivity is large. \nFigure~\\ref{fig:body prob dist2} shows the probability distributions of a polar object in a bath of active particles, measured numerically for the two regimes illustrated in Fig.~\\ref{fig:body prob dist}, which indeed exhibits the corresponding transition.\n\n\\if{Importantly the distribution is either concave or convex near the origin depending on whether the object's persistence time $1\/D_r^e$ is larger or smaller than a characteristic time $1\/(2\\mu_0 p q)$, which is controlled by the active bath, the object's shape and the size of the cavity. \nFigure~\\ref{fig:body prob dist2} shows the probability distributions measured numerically for the two regimes that were illustrated in Fig.~\\ref{fig:body prob dist}.\n\nThe underlying physics can be qualitatively understood by inspection of the Langevin dynamics~\\eqref{eq:body Langevin}. In the limit of fast rotational diffusion, the `bare' self propulsion $p {\\bf u}(\\phi)$ averages out, while the repulsion from the wall does not. This effectively localizes the object in the center of the cavity. On the contrary, when the object is very persistent and ${\\rm Pe}\\ll1$, the weak correction ${\\bf F}$ to the self-propulsion does not prevent the object from reaching the wall.\n}\\fi\n\n\\begin{figure}\n \t\\centering\n\t\\includegraphics[width=8.6 cm]{prob_dist26_new_norm.eps}\n\t\\caption{Comparison between the probability density of the object's position $P(\\bfr)$ and a uniform distribution $P_U(\\bfr)=1\/(\\pi R^2)$ distribution. The two panels differ by the object's rotational mobility  $\\gamma=10^2$ in (a) and $\\gamma=10^5$ in (b). The panels correspond to Figs.~\\ref{fig:body prob dist} (a) and (b), respectively. The shaded region indicates scales smaller than the object's diameter, where the sampling is expected to fail. See Appendix~\\ref{app:numerics} for numerical details.\n\t}\n \t\\label{fig:body prob dist2}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this manuscript we have considered the influence of boundaries on the motion of an asymmetric tracer in an active bath. Specifically, we have shown that the tracer experiences a non-conservative force mediated by the active medium,  whose magnitude depends on the object's orientation. We then demonstrated how this force can be used to control the position of the object far in the bulk of the system.\n\nTo leading order, we have shown that the interaction with the walls can be accounted for using a generalized image theorem, which states that the passive object experiences long-range forces from its image. This holds despite the non-trivial boundary condition for the active fluid near the boundary. Using this result, we showed that, inside a circular cavity, two regimes can be observed depending on the parameters: either the object is confined in the center of the cavity or it is localized close to its boundaries. All the results above are in sharp contrast to the case of a symmetric object where no stable minima can be found. \n\nFrom a broader point of view, numerous mechanisms were suggested in the past to localize objects in the center of closed domains. This was studied extensively~\\cite{ierushalmi2020centering,wuhr2009does,mitchison2012growth,xie2020cytoskeleton}, in particular in the context of cell division~\\cite{grill2005spindle,gundersen2013nuclear}.\nOur work offers a new, simple and generic mechanism to localize a passive object in the center of a circular region without requiring any exotic interactions. While simplistic in nature, this robust mechanism might play a role in such processes. It is tempting to search for additional applications of these forces, for instance to engineer passive objects that could be controlled by modifying the boundaries of the confining system. \nFinally, it would be interesting to generalize our approach to domains of arbitrary shapes. The only non-trivial step seems to be the calculation of the Green's function, whose form can be derived using conformal mappings. \nThis, however, is harder than it might seem since the Neumann boundary condition may introduce fictitious sources through the conformal mapping. A full generalization of our approach to general domains thus remains an open challenging problem.\n\n\\acknowledgements{}\n\nYBD and YK are supported by an Israel Science Foundation grant (2038\/21). YBD, YK and MK are supported by an NSF5-BSF Grant No. DMR-170828008. JT acknowledges the financial support of ANR grant THEMA.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}