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+{"text":"\\section{Introduction}\n\n\\subsection{Isoperimetric problems on the Hamming cube}\n\nThis paper deals with {\\it discrete isoperimetric inequalities} on graphs. Let a graph $G = (V,E)$ be given, and let $A \\subseteq V$ be a set of vertices in $G$. An isoperimetric inequality addresses the question of how small the boundary $\\partial A$ of $A$ can be, given the cardinality of $A$, by lower bounding the size of $\\partial A$ by an appropriate function of $|A|$.  There are various ways do define and measure the boundary of the set, two salient examples being the {\\it vertex boundary} of $A$, consisting of\nthe vertices of $A$ which have neighbors outside $A$, and the {\\it edge\nboundary} of $A$, which is the set of edges crossing from $A$ to its\ncomplement. In both these cases, the size of the boundary is the cardinality of the corresponding set of vertices (or edges). However, one can also think of examples of a somewhat different nature, some of which will be considered below.\n\nIn this paper we deal with\na specific graph - the Hamming cube $\\{0,1\\}^n$. This is a graph with $2^n$ vertices\nindexed by boolean strings of length $n$. Two vertices are\nconnected by an edge if they differ only in one coordinate. The\nmetric defined by this graph is called the Hamming distance. In other words, two vertices $x$\nand $y$ are at distance $d$ if they differ in $d$ coordinates. Let us define two important families of subsets of $\\{0,1\\}^n$. A {\\it Hamming ball} is a ball in the Hamming metric. A {\\it subcube} is a\nsubset of the vertices obtained by fixing the value in some of the\ncoordinates. The number of fixed coordinates is called the co-dimension of\nthe subcube. It turns out \\cite{Harper,Harp, Hart} that a Hamming ball has the smallest vertex boundary for its\nsize, and a subcube has the smallest edge boundary.\n\nWe will consider several versions of the edge-isoperimetric inequality\nfor the cube. Let $|\\partial A|$ be the cardinality of the edge\nboundary of $A$, normalized, for convenience, by $2^{n-1}$.\nThe standard version of the inequality \\cite{Harp, Hart} states that for\nany subset $A \\subseteq \\{0,1\\}^n$ we have\n\\footnote{We use natural logarithms throughout the paper.}\n\\begin{equation}\n\\label{set_isop}\n|\\partial A|~ \\ge~ \\frac{2}{\\log 2} \\cdot \\frac{|A|}{2^n} \\log\n\\frac{2^n}{|A|}.\n\\end{equation}\nThis is tight if $A$ is a subcube of an arbitrary co-dimension $0\\le t\n\\le n$.\n\n\n\\noindent {\\it A logarithmic Sobolev inequality} \\cite {Gr} establishes a relation\nbetween two (appropriately defined) notions: the variation of a function\nand its entropy. For a function $f$ on the Hamming cube $\\{0,1\\}^n$ endowed\nwith the uniform measure, this\ntranslates to\n\\begin{equation}\n\\label{ineq_Sob}\n\\mathbb E_x \\sum_{y \\sim x} (f(x) - f(y))^2 \\ge 2 \\cdot Ent\\(f^2\\) = 2\\cdot \\(\\mathbb E f^2\n\\log f^2 - \\mathbb E f^2 \\log \\mathbb E f^2\\)\n\\end{equation}\nIt is useful to view (\\ref{ineq_Sob}) as an isoperimetric\ninequality. In particular (\\cite{FS}), it implies a functional form of the\nedge-isoperimetric inequality (\\ref{set_isop}). For a non-zero\nfunction $f:~\\{0,1\\}^n \\rightarrow \\mathbb R$ holds\n\\begin{equation}\n\\label{func_isop}\n\\mathbb E_x \\sum_{y \\sim x} (f(x) - f(y))^2 \\ge 2 \\cdot \\mathbb E f^2 \\log \\frac{\\mathbb E\n  f^2}{\\mathbb E^2 |f|}\n\\end{equation}\nChoosing $f$ in (\\ref{func_isop}) to be the\ncharacteristic function of a subset $A$ of the cube, we recover the\nedge-isoperimetric inequality (\\ref{set_isop}) with a somewhat worse\nconstant on the right hand side, replacing $2\/\\log 2$ with $2$. On the\nother hand, there are examples (\\cite{FS}) of real-valued functions $f$ for which the constant $2$ in (\\ref{func_isop}) is tight.\n\nLet us now briefly sketch the results in this paper, before discussing them in fuller detail in subsection~\\ref{subsec:res} below. We will show that the constant $C = 2$ at the right hand side of the logarithmic Sobolev inequality (\\ref{ineq_Sob}) can be replaced by a function $C(\\rho)$ depending on $\\rho = \\frac1n \\frac{Ent\\(f^2\\)}{\\mathbb E f^2}$. The function $C(\\rho)$ will be given explicitly. It is a convex function which increases from $2$ to $2\/\\log 2$ as $\\rho$ goes from $0$ to $\\log 2$. We will also observe that $C(\\rho)$ gives the correct dependence on $\\rho$, describing functions on $\\{0,1\\}^n$ for which the modified inequality is tight.\n\nThis will imply a corresponding modification of the functional isoperimetric inequality (\\ref{func_isop}). Here, as well as in (\\ref{ineq_Sob}), it will be possible to replace the constant $C = 2$ by the function $C(\\rho)$.\n\nThe modified version of (\\ref{func_isop}) is used to derive a discrete analogue of the classical Faber-Krahn inequality\nin $\\mathbb R^n$ for small subsets of the Hamming cube $\\{0,1\\}^n$, answering\\footnote{Up to an error which becomes negligible as the dimension of the cube grows.} a question from \\cite{FT}. We will introduce, following \\cite{FT}, the notion of a fractional edge-boundary size of a subset of $\\{0,1\\}^n$, and show Hamming balls of radius at most $n\/2 - \\tilde{\\Omega}\\(n^{3\/4}\\)$ to be sets with (asymptotically) the smallest fractional edge-boundary for their size.\n\nThe question in \\cite{FT} is a part of an approach to obtain upper bounds on the cardinality of binary error-correcting codes. We will now take a brief detour to the theory of error-correcting codes in order to provide a natural framework in which this question can be discussed.\n\n\\subsection{Bounds on binary error correcting codes}\n\nA binary error-correcting code of length $n$ and minimal distance $d$\nis a subset $C$ of the boolean cube $\\{0,1\\}^n$ such that the distance between\nany two distinct points in $C$ is at least $d$. In other words, the\npoints in $C$ can be taken as centers in a disjoint packing of Hamming\nballs of raduis $\\lceil \\frac{d-1}{2} \\rceil$ into $\\{0,1\\}^n$.\n\nThe question of the maximal possible cardinality $A(n,d)$ of such a packing is\none of the central questions of coding theory. The best known upper\nbounds on $A(n,d)$ were obtained in \\cite{MRRW} following Delsarte's\nlinear programming approach \\cite{dels}. The analysis in \\cite{MRRW}\nuses theory of orthogonal polynomials and is somewhat complicated.\n\nA different approach to obtain some of the bounds in \\cite{MRRW}\nwas presented in \\cite{FT}. The appeal of\nthis new approach is in showing the possibility to work with Delsarte's linear\ninequalities without resorting to language and tools of orthogonal\npolynomial theory. In particular, \\cite{FT} establishes a connection\nbetween packing bounds and isoperimetric questions in the Hamming\ncube.\nTo describe this connection, we need a notion of the {\\it fractional\nedge boundary size} of a subset of\nthe cube \\footnote{This is a reformulation of a closely related notion of the maximal eigenvalue of\na subset introduced in \\cite{FT}.}.\n\nFor $A \\subseteq \\{0,1\\}^n$, the fractional edge boundary size of $A$ is defined as\n\\begin{equation}\n\\label{dfn:frac-bound}\n|\\partial^* A| = \\min\\left\\{\\mathbb E_x \\sum_{y\\sim x} (f(x)-f(y))^2:~\\mbox{supp}(f) \\subseteq\nA,~\\mathbb E f^2 = \\frac{|A|}{2^n}\\right\\}\n\\end{equation}\nThe right hand side of this definition computes a minimum of the variation of $f$\nover a certain family of functions. Note that this family contains the characteristic\nfunction of $A$ whose variation equals the (normalized) cardinality of the edge boundary $\\partial A$.\nConsequently, the fractional boundary of $A$ is at most as large as its boundary.\n\nThe following result is proved in \\cite{NS, NS1}, following the approach of \\cite{FT}. Let us mention that this claim was proved in \\cite{FT} for an important special case of linear codes.\n\n\\begin{theorem}\n\\label{thm:ns}\nLet $A$ be a subset of the boolean cube $\\{0,1\\}^n$ such that\n$$\n|\\partial^* A| \\le (2d+1) \\cdot \\frac{|A|}{2^{n-1}}\n$$\nLet $C$ be a binary error-correcting code with minimal distance $d$. Then\n$$\n|C| \\le n |A|\n$$\n\\end{theorem}\n\nThis suggests a way to obtain upper bounds for codes by finding\nsubsets of the cube with a small fractional boundary. Natural\ncandidates to try are the isoperimetric sets, that is Hamming balls\nand subcubes. Their fractional boundaries are analyzed in\n\\cite{FT}. It turns out that among these two options, a Hamming ball has the smaller fractional\nedge boundary. Note that this pinpoints an intriguing difference between\nthe notions of the fractional edge-boundary and that of the 'ordinary' edge-boundary, for which the subcubes are the optimal sets.\n\nLet $B$ denote a Hamming ball of radius $r$. It is shown in \\cite{FT} that\n\\begin{equation}\n\\label{ineq:ball_bound}\n|\\partial^* B | \\le 4\\(\\frac n2 - \\sqrt{r(n-r)} +o(n)\\) \\cdot \\frac{|B|}{2^n}\n\\end{equation}\nCombined with Theorem~\\ref{thm:ns}, this shows that a\nbinary error-correcting code with minimal distance $d$ is at most as large, up to\nnegligible multiplicative factors, as a Hamming ball of radius $r =\nn\/2 - \\sqrt{d(n-d)}$. This provides an alternative proof of the {\\it first\nlinear programming bound} for binary codes \\cite{MRRW}.\n\nThis concludes our detour into coding theory. We are now ready to state the isoperimetric problem of \\cite{FT}.\n\n\\subsection{An isoperimetric problem for the Hamming cube}\nIn order to obtain the best possible bounds on codes via Theorem~\\ref{thm:ns}, we need to find subsets of the Hamming cube with the smallest possible fractional edge-boundary. In particular, an existence of subsets whose fractional boundary is noticeably smaller than that of Hamming balls of the same cardinality, would imply an improvement on the best currently known bounds. This naturally leads to the following questions \\cite{FT}\n\n{\\bf A fractional edge-isoperimetric problem for the Hamming cube}:\n\\begin{itemize}\n\\item\nWhat is the smallest possible fractional\nboundary of a subset of $\\{0,1\\}^n$ of a given cardinality?\n\\item\nWhich sets have the smallest fractional boundaries?\n\\end{itemize}\n\nThese questions were the starting point of our investigation. Before describing our results, let us mention a connection to the classical Faber-Krahn inequality in $\\mathbb R^n$, as pointed out in \\cite{FT}.\n\nFirst, here is a brief description of the Euclidean space inequality, following \\cite{Chavel}.\nFor an open set $\\Omega$ in $\\mathbb R^n$, consider the functional\n\\begin{equation}\nF[\\phi] = \\frac{\\|grad ~\\phi\\|^2_2}{\\|\\phi\\|^2_2}\n\\label{cont_fk}\n\\end{equation}\nwhere $\\phi$ ranges over smooth functions supported in $\\Omega$, and\nthe associated infimum\n$\n\\lambda^*(\\Omega) = inf_{\\phi} F[\\phi]\n$.\n\n$\\lambda^*(\\Omega)$ is referred to as the fundamental tone of\n$\\Omega$. The Faber-Krahn inequality states that among all sets\n$\\Omega$ of the same measure, Euclidean ball has the minimal\nfundamental tone.\n\nIn the discrete setting of the Hamming cube, a reasonable\ninterpretation of (\\ref{cont_fk}) is to consider the functional\n$$\nF[f] = \\frac{\\mathbb E_x \\sum_{y\\sim x} (f(x) - f(y))^2}{\\mathbb E f^2}\n$$\nwhere $f$ ranges over functions supported in a subset $A$ of $\\{0,1\\}^n$. In\nour terminology, the ``fundamental tone'' of $A$ is\n$$\n\\lambda^*(A) = \\min_f F[f] = \\frac{2^n}{|A|} \\cdot |\\partial^* A|\n$$\nHence, the set with the smallest fractional boundary for its size has the minimal fundamental tone, and vice versa.\n\nFollowing \\cite{FT} we will refer to the fractional edge-isoperimetric\nproblem as the discrete Faber-Krahn problem for the Hamming cube.\n\n\\subsection{Main results}\n\\label{subsec:res}\nOur main technical result is a modified version of the logarithmic Sobolev inequality (\\ref{ineq_Sob}). Let $H(x) = -x\\log x - (1-x) \\log(1-x)$ be the \"natural\" (i.e., using natural logarithms) entropy function.\n\\begin{theorem}\n\\label{thm:log-Sobolev}\n\\begin{itemize}\n\\item\nLet $f:~\\{0,1\\}^n \\rightarrow \\mathbb R$ be a non-zero function, and let $\\rho = \\frac 1n \\frac{Ent\\(f^2\\)}{\\mathbb E f^2}$. Then\n\\begin{equation}\n\\label{ineq:l_sob}\n\\mathbb E_x \\sum_{y \\sim x} (f(x) - f(y))^2 \\ge C(\\rho) \\cdot Ent\\(f^2\\),\n\\end{equation}\nwhere\n$$\nC(x) = \\frac{4}{x} \\cdot \\(\\frac12 - \\sqrt{H^{-1}(\\log 2 - x)\\Big (1 -\n  H^{-1}(\\log 2 - x)\\Big )}\\)\n$$\n\\item\nThe function $C(\\cdot)$ is an increasing convex function, taking $[0, \\log 2]$ to\n$[2,2\/\\log 2]$.\n\\end{itemize}\n\\end{theorem}\n\nInequality (\\ref{ineq:l_sob}) is tight in\nthe following sense: for each $\\rho \\in [0, \\log 2]$ there exists a non-constant\nfunction $f = f_{\\rho}$ such that $Ent\\(f^2\\) \\ge \\rho n \\mathbb E f^2$ and\n\\begin{equation}\n\\label{ineq:sob-tight}\n\\mathbb E_x \\sum_{y \\sim x} (f(x) - f(y))^2 \\le (1 + o_n(1)) \\cdot C(\\rho)\n\\cdot Ent\\(f^2\\)\n\\end{equation}\nThis follows from the tightness of inequality (\\ref{ineq:fk}) below (see the second part of Theorem~\\ref{thm:fk}), since that inequality is a corollary of (\\ref{ineq:l_sob}). The fact that (\\ref{ineq:fk}) is tight for Hamming balls follows from (\\ref{ineq:ball_bound}). The functions $f_{\\rho}$ are the minimal variation functions supported on a Hamming ball of an appropriate radius. They are constructed explicitly in \\cite{FT}.\n\nTheorem~\\ref{thm:log-Sobolev} together with the observation $Ent\\(f^2\\) \\ge \\mathbb E f^2 \\log \\frac{\\mathbb E f^2}{\\mathbb E^2 |f|}$ (\\cite{FS}), implies a corresponding modification of the functional isoperimetric inequality (\\ref{func_isop}).\n\\begin{corollary}\n\\label{cor:ineq:isop}\n\\begin{equation}\n\\label{ineq:isop}\n\\mathbb E_x \\sum_{y \\sim x} (f(x) - f(y))^2 \\ge C(\\rho) \\cdot \\mathbb E f^2 \\log,\n\\frac{\\mathbb E f^2}{\\mathbb E^2 |f|}\n\\end{equation}\n\\end{corollary}\nwhere $\\rho = \\frac 1n \\log \\frac{\\mathbb E f^2}{\\mathbb E^2 |f|}$. Inequality (\\ref{ineq:isop}) is tight in the same sense and for the same reasons (\\ref{ineq:sob-tight}) is tight.\n\nLet us briefly discuss this inequality. It implies, in particular, that as the ratio $\\frac{\\mathbb E f^2}{\\mathbb E^2 f}$ grows (the function $f$ becomes less \"flat\") its edge-isoperimetric constant approaches the isoperimetric constant $C = \\frac{2}{\\log 2}$ in the edge-isoperimetric inequality (\\ref{func_isop}) for $0$-$1$ functions. One possible partial explanation for this phenomenon is that, for functions supported on a small set $A \\subseteq \\{0,1\\}^n$, the main contribution to the variation $\\mathbb E_x \\sum_{y\\sim x} (f(x) - f(y))^2$ is likely to come from edges $(x,y)$ which belong to the edge-boundary of $A$.\n\nIt is now straightforward to derive the fractional edge-isoperimetric inequality (\\ref{ineq:fk}) from Corollary~\\ref{cor:ineq:isop}.\nLet $f$ be a function supported on a subset $A$ of $\\{0,1\\}^n$. Then, by the Cauchy-Schwarz inequality, $\\frac{\\mathbb E f^2}{\\mathbb E^2 f} \\ge \\frac{2^n}{|A|}$. Since the function $C(\\cdot)$ is monotone, we have\n$$\n\\mathbb E_x \\sum_{y\\sim x} (f(x)-f(y))^2 \\ge C\\(\\frac 1n \\log \\frac{2^n}{|A|}\\) \\cdot \\log \\frac{2^n}{|A|} \\cdot \\mathbb E f^2\n$$\nRecalling the definition of the fractional edge boundary (\\ref{dfn:frac-bound}), and substituting the explicit expression for $C(\\cdot)$, gives (\\ref{ineq:fk}).\n\nThe inequality (\\ref{ineq:fk}), together with the second part of Theorem~\\ref{thm:fk},\nprovide an asymptotic solution to the Faber-Krahn problem for the Hamming cube, at least in the range of interest to the coding theory. It turns out that, up to an error which becomes negligible as the dimension $n$ grows, Hamming balls of radius $0 \\le r \\le \\frac n2 -  o(n)$ are the sets with the smallest fractional boundary (fundamental tone) for their size.\n\nFrom the viewpoint of coding theory, this implies that Theorem~\\ref{thm:ns} cannot lead to an improvement on the best currently known bounds for binary codes \\cite{MRRW}. Let us briefly discuss one implication of this fact. The bounds in \\cite{MRRW} are obtained following Delsarte's linear programming approach, and there are claims in coding theory (\\cite{BJ,NS}) which seem to indicate that these are the best bounds attainable with this approach. Since Theorem~\\ref{thm:ns} is derived within the same linear programming framework, inequality (\\ref{ineq:fk}) may be interpreted as an additional evidence in this direction\\footnote{and it is not, in this sense, very surprising.}. It seems worthwhile to point out that, in this manner, the coding theory provides both the question prompting this investigation and an indication of what the answer might be, by suggesting the putative optimality of Hamming balls for the Faber-Krahn problem.\n\n\\begin{theorem}\n\\label{thm:fk}\n\\begin{itemize}\n\\item\nLet $H(x) = -x \\log x - (1-x) \\log(1-x)$ be the entropy function. Then for any subset $A$ of $\\{0,1\\}^n$ holds\n\\begin{equation}\n\\label{ineq:fk}\n|\\partial^* A| \\ge 4n \\(\\frac12 - \\sqrt{H^{-1}\\(\\frac{\\log\n    |A|}{n}\\)\\(1 - H^{-1}\\(\\frac{\\log |A|}{n}\\)\\)}\\) \\cdot \\frac{|A|}{2^n}\n\\end{equation}\n\\item\nOn the other hand, let $B$ be a Hamming ball. Then\n$$\n|\\partial^* B| \\le 4n \\(\\frac12 - \\sqrt{H^{-1}\\(\\frac{\\log\n    |B|}{n}\\)\\(1 - H^{-1}\\(\\frac{\\log |B|}{n}\\)\\)} +o_n(1)\\) \\cdot \\frac{|B|}{2^n}\n$$\n\\end{itemize}\n\\end{theorem}\n\nThe second part of this theorem is due to \\cite{FT}. It follows from (\\ref{ineq:ball_bound}) and the fact that the cardinality of a Hamming ball of radius $r$ is at least $\\exp\\{n H\\(\\frac{r}{n}\\) - o(n)\\}$ \\cite{vLint}.\n\nThe bound in (\\ref{ineq:fk}) is not very good for balanced subsets $A$, for which $\\frac{\\log |A|}{n}$ is close to $\\log 2$. For instance, for $|A| = 2^{n-1}$, the bound gives $|\\partial^* A| \\ge \\log 2$. On the other hand, it is not hard to see that the correct bound in this case is $|\\partial^* A| \\ge 1$. Equality is attained on a subcube of co-dimension one (but not on a Hamming ball of radius $n\/2$).\n\nOn the other hand, (\\ref{ineq:fk}) is interesting, as long as the error term $o(1)$ in the second part of Theorem~\\ref{thm:fk} is of order lower than that of the main term. This error term is of order $n^{-1\/2}$, up to poly-logarithmic terms\\footnote{This could be derived from the computation in \\cite{FT}, or, alternatively, from the estimates on the minimal roots of Krawchouk polynomials \\cite{FS, NS}.}. Therefore, the theorem provides a satisfactory lower bound for the fractional edge boundary size of $A$ as long as\n$$\n\\Big |\\frac{\\log |A|}{n} - \\log 2 \\Big | \\ge \\tilde{\\Omega}\\(n^{-1\/2}\\)\n$$\nIn particular, Hamming balls of radius at most $n\/2 - \\tilde{\\Omega}\\(n^{3\/4}\\)$ have (asymptotically) the smallest fractional edge-boundary for their size.\n\n\\noindent {\\bf Questions}:\n\\begin{itemize}\n\\item\nHow small can the fractional edge boundary size of a balanced subset $A$ of $\\{0,1\\}^n$ be?\n\\item\nWhich balanced sets have the smallest fractional edge-boundary?\n\\end{itemize}\n\nFinally, let us briefly mention an additional application of inequality (\\ref{ineq:l_sob}), of a somewhat different nature. The 'standard' logarithmic Sobolev inequality (\\ref{ineq_Sob}) is used in \\cite{Gr} to derive a hypercontractive inequality for functions on the discrete cube (see also \\cite{Bonami, Beckner}).\nIn order to describe this inequality, let $\\{w_S\\}_{S\\in \\{0,1\\}^n}$ be the character basis in the space of real-valued functions on the Hamming cube.\nLet $0 \\le t \\le 1$ and let $T = T_t$ be\na linear operator taking a function $f = \\sum_S \\hat{f}(S) w_S$ to $Tf\n= \\sum_S t^{|S|} \\hat{f}(S) w_S$. Then (\\cite{Gr})\n\\begin{equation}\n\\label{ineq:beckner}\n\\|Tf\\|_2 \\le \\|f\\|_{1+t^2}\n\\end{equation}\n\nSubstituting  (\\ref{ineq:l_sob}) instead of  (\\ref{ineq_Sob}) in the proof in \\cite{Gr} leads to a modified version of (\\ref{ineq:beckner}). It turns out that the exponent $2$ on the right hand side of the inequality can be replaced by a function $e(\\rho)$, depending on $\\rho = \\frac1n \\frac{Ent\\(f^2\\)}{\\mathbb E f^2}$. The function $e(\\rho)$ is decreasing, with $e(0) = 2$.\\footnote{Hence, for functions with high entropy, this gives a strengthening of (\\ref{ineq:beckner}).} This modified inequality might be useful in coding theory, following the applications of (\\ref{ineq:beckner}) in \\cite{KL, ACKL}.\n\n\n\\section{The proof of Theorem~\\ref{thm:log-Sobolev}}\nLet us start with a brief overview.\nThe main goal of this section is to prove the logarithmic Sobolev inequality (\\ref{ineq:l_sob}).\nOur proof follows the outline of the proof of (\\ref{ineq_Sob}) in \\cite{Gr}. We will prove an inequality (\\ref{ineq:tech}), which will imply (\\ref{ineq:l_sob}) as a corollary, first for the base case $n=1$, and then for general $n$, using subadditivity of entropy. Compared to \\cite{Gr}, we need to prove a bit more for the base case (see Remark~\\ref{rem:diff} below) and to carry this additional information along to the general case. This seems to complicate things somewhat, making it necessary to go through the intermediate inequality (\\ref{ineq:tech}).\n\nWe will prove the second claim of the theorem, on the properties of the function $C(\\rho)$ in (\\ref{ineq:l_sob}), along the way, in Lemma~\\ref{lem:functions}.\n\nWe may and will assume from now on that we deal only with nonnegative functions on $\\{0,1\\}^n$, since substituting $|f|$ instead of $f$ in (\\ref{ineq:l_sob}) decreases the left hand side and does not affect the right hand side.\n\nSeveral functions on the real line play an important role in the proof. We start by defining these functions and stating some of their properties. Let $\\psi$ be defined on $[0,1]$ by\n\\begin{equation}\n\\label{def:psi}\n\\psi(t) = \\frac12 (1-\\sqrt t)^2 \\log (1-\\sqrt t)^2 + \\frac12\n(1+\\sqrt t)^2 \\log (1+\\sqrt t)^2 - (1+t)\\log(1+t)\n\\end{equation}\nIn other words, $\\psi(t) = Ent\\(f^2\\)$, where $f$ is a function on\n$\\{0,1\\}$ with $f(0) = 1 - \\sqrt t$, $f(1) = 1 + \\sqrt t$.\n\nThe following main technical lemma lists the relevant properties of $\\psi$ and several derived functions, including the function $C$. The proof of the lemma is rather long and is postponed till the next section.\n\\begin{lemma}\n\\label{lem:functions}\n\\begin{enumerate}\n\\item\nThe function $\\psi$ is strictly increasing and concave on $[0,1]$, taking this interval onto $[0, 2\\log 2]$. This allows us to define the inverse function $\\phi = \\psi^{-1}$. This is a strictly increasing convex function taking $[0,2\\log 2]$ onto $[0,1]$.\n\\item\nThe function $\\psi(t)\/(1 + t)$ is strictly increasing and concave on $[0,1]$, taking this interval onto $[0, \\log 2]$. This allows us to define the inverse function $\\alpha(t) = \\(\\frac{\\psi(t)}{1 + t}\\)^{-1}$. This is a strictly increasing convex function taking $[0, \\log 2]$ to $[0,1]$.\n\\item\nThe function $c(t) = \\frac{4 \\alpha(t)}{t(1 + \\alpha(t))}$ is strictly increasing and convex on $[0, \\log 2]$, taking this interval onto $[2, 2\/\\log 2]$.\\footnote{Here, as usual, we take $c(0) = \\lim_{t\\rightarrow 0} \\frac{4 \\alpha(t)}{t(1 + \\alpha(t))} = 2$.}\n\\item\nThe function $c(t)$ has an explicit representation\n$$\nc(t) = \\frac{4}{t} \\cdot \\(\\frac12 - \\sqrt{H^{-1}(\\log 2 - t)\\Big (1 -\n  H^{-1}(\\log 2 - t)\\Big )}\\).\n$$\nIn other words $c = C$, where $C$ is the function in (\\ref{ineq:l_sob}).\n\\end{enumerate}\n\\end{lemma}\nNote that the second claim of Theorem~\\ref{thm:log-Sobolev} follows from the third and the fourth claims of this lemma.\n\n\\begin{remark}\n\\label{rem:diff}\nWe mentioned a difference between the proof of the base case $n=1$ here and in \\cite{Gr}. Let us give some details. In \\cite{Gr},  (\\ref{ineq_Sob}) is shown for $\\{0,1\\}$, which, in our notation, amounts to proving an inequality $\\psi(t) \\le 2t$ on $[0,1]$. We show, in addition, that $\\psi$ is concave on $[0,1]$. This additional convexity property turns out to be crucial for our proof.\n\\vrule height7pt width4pt depth1pt\\end{remark}\n\nNext, we introduce additional notation, and prove a simple auxiliary inequality.\n\nFor a function $f$ on $\\{0,1\\}^n$ let $D^2(f) = \\mathbb E_x \\sum_{y \\sim x} (f(x) -\nf(y))^2$, and let $K^2(f) = \\frac 14 \\cdot \\mathbb E_x \\sum_{y \\sim x} (f(x) +\nf(y))^2$. Note that $K^2(f) = n \\mathbb E f^2 - \\frac14 \\cdot D^2(f)$. Note\nalso that for a non-zero nonnegative function $f$, $K^2(f)$ is strictly positive.\n\n\\begin{lemma}\n\\label{lem:ent_and_k}\nFor a nonnegative function $f$ on $\\{0,1\\}^n$ holds\n$$\nEnt\\(f^2\\) \\le 2\\log 2 \\cdot K^2(f)\n$$\n\\end{lemma}\n\\noindent{\\bf Proof:} \nWe will prove the claim for the base case $n=1$ and then use subadditivity of entropy to deduce it for the general case.\n\nThe case $n = 1$. We may assume $f$ is non-zero, since the claim is trivially true otherwise. Both sides of the inequality are\n$2$-homogeneous, and consequently we may assume $\\mathbb E f = 1$. Without loss of generality,  $f(0) = 1 - s$,\n$f(1) = 1 + s$ for some $0 \\le s \\le 1$. We need to show\n$$\nEnt\\(f^2\\) = \\psi\\(s^2\\) \\le 2 \\log 2 \\cdot K^2(f) = 2\\log 2,\n$$\nand this is indeed true by the first claim of Lemma~\\ref{lem:functions}, since $\\psi$ is increasing and $\\psi(1) = 2\\log 2$.\n\nThe general case $n \\ge 1$. Let $1 \\le i \\le n$ be an index of a coordinate and let $x \\in \\{0,1\\}^n$. Fixing all the coordinates $j \\not = i$ to be $x_j$ we obtain a copy of $\\{0,1\\}$. Let $f^{(x)}_i$ be the restriction of $f$ to this one-dimensional cube.\n\nRecall (\\cite{Ledoux}) that entropy is subadditive, namely\n$$\n\\sum_{i=1}^n \\mathbb E_{x} Ent\\(f^{(x)}_i\\) \\ge Ent(f),\n$$\nwhile $D^2(f)$ is additive, that is\n$$\n\\sum_{i=1}^n \\mathbb E_{x} D^2\\(f^{(x)}_i\\) = D^2(f).\n$$\n$K^2(f)$ is also\nadditive, since $\\sum_{i=1}^n \\mathbb E_x K^2\\(f^{(x)}_i\\) = \\sum_{i=1}^n\n\\mathbb E_x \\(\\mathbb E \\(f^{(x)}_i\\)^2 - 1\/4 D^2\\(f^{(x)}_i\\)\\) = n \\mathbb E f^2 - 1\/4 D^2(f) = K^2(f)$.\n\nFrom this and the base case, we have\n$$\nEnt\\(f^2\\) \\le \\sum_{i=1}^n \\mathbb E_x Ent\\(\\(f^{(x)}_i\\)^2\\) \\le 2\\log 2 \\cdot \\sum_{i=1}^n \\mathbb E_x K^2\\(f^{(x)}_i\\) = 2\\log 2\n\\cdot K^2(f)\n$$\n\\vrule height7pt width4pt depth1pt\n\nLet us now pass to the main technical claim in this section. Note that the right hand side in (\\ref{ineq:tech}) is well-defined, due to the preceding lemma.\n\n\\begin{proposition}\n\\label{thm:technical}\nLet $f$ be a non-zero nonnegative function on $\\{0,1\\}^n$. Then\n\\begin{equation}\n\\label{ineq:tech}\nD^2(f) \\ge 4 \\cdot K^2(f) ~\\phi\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\)\n\\end{equation}\n\\end{proposition}\nThe proof follows the same outline. First we prove the base case $n = 1$.\nIn this case, we will show equality\n$$\nD^2(f) = 4 \\cdot K^2(f) ~\\phi\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\).\n$$\nIndeed, since $f$ is non-zero, we may assume $\\mathbb E f = 1$. This implies $K^2(f) = 1$. Thus we have to prove $D^2(f) = 4\n\\phi\\(Ent\\(f^2\\)\\)$. Let $0 \\le s \\le 1$, and $f(0) = 1 - s$, $f(1) =\n1 + s$. Then $D^2(f) = 4s^2$ and\n$4\\phi\\(Ent\\(f^2\\)\\) = 4\\phi \\(\\psi\\(s^2\\) \\) = 4s^2$,\nverifying the base case.\n\nWe also need to deal with a slight technicality, the case $f$ is the zero function, because in the general case below, some of the one-dimensional restrictions of $f$ might be zero. In this case, we formally define $Ent\\(f^2\\)\/K^2(f)$ to be zero. Then (\\ref{ineq:tech}) remains valid (as equality) in the one-dimensional case. It is easy to see that this formal definition does not affect the computation below.\n\nThe general case. Let $n \\ge 1$. Then, by the base case, by\nsubadditivity of the entropy, and by convexity and monotonicity of $\\phi$:\n$$\nD^2(f) = \\sum_{i=1}^n \\mathbb E_x D^2\\(f^{(x)}_i\\)  =\n4 \\cdot \\sum_{i=1}^n \\mathbb E_x K^2\\(f^{(x)}_i\\)\n\\phi\\(\\frac{Ent\\(\\(f^{(x)}_i\\)^2\\)}{K^2\\(f^{(x)}_i\\)}\\) =\n$$\n$$\n4 K^2(f) \\cdot \\sum_{i=1}^n \\mathbb E_x \\frac{K^2\\(f^{(x)}_i\\)}{K^2(f)}\n\\phi\\(\\frac{Ent\\(\\(f^{(x)}_i\\)^2\\)}{K^2\\(f^{(x)}_i\\)}\\) \\ge\n$$\n$$\n4 K^2(f) \\cdot\n\\phi\\( \\frac{1}{K^2(f)} \\cdot \\sum_{i=1}^n \\mathbb E_x Ent\\(\\(f^{(x)}_i\\)^2\\)\\) \\ge\n4 K^2(f) \\cdot \\phi\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\)\n$$\n\\vrule height7pt width4pt depth1pt\n\nWe proceed to derive (\\ref{ineq:l_sob}) from (\\ref{ineq:tech}). By homogeneity, we may assume $\\mathbb E f^2 = 1$. Let $0 \\le \\rho\n\\le \\log 2$. Consider the functional\n$$\nR[f] = \\frac{D^2(f)}{Ent\\(f^2\\)}\n$$\nwhere $f$ ranges over the non-empty\\footnote{Recall $Ent\\(f^2\\) \\ge \\mathbb E f^2 \\log \\frac{\\mathbb E f^2}{\\mathbb E^2 f}$ (\\cite{FS}).} compact set of nonnegative non-zero functions satisfying $\\mathbb E f^2 = 1$ and $Ent\\(f^2\\) \\ge \\rho n$, and the associated minimum\n$$\nm(\\rho) = \\min_f R[f]\n$$\nTo complete the proof of (\\ref{ineq:l_sob}) and of Theorem~\\ref{thm:log-Sobolev}, we will show\n\\begin{equation}\n\\label{ineq:mc}\nm(\\rho) \\ge c(\\rho),\n\\end{equation}\nwhere $c$ is the function defined in the third part of Lemma~\\ref{lem:functions}.\n\nIndeed, fix $\\rho$ and let $m = m(\\rho)$. Let $f$ be a function at which $R[f]$ attains its minimum, that is: $\\mathbb E f^2 = 1$, $Ent\\(f^2\\) \\ge \\rho n$ and $D^2(f) = m Ent\\(f^2\\)$. Then\n$$\nD^2(f) = m Ent\\(f^2\\) \\ge m \\rho n.\n$$\nThis means $K^2(f) = n \\mathbb E f^2 - 1\/4 D^2(f) \\le \\( 1 - (m\\rho)\/4\\) n$, and therefore $\\frac{Ent\\(f^2\\)}{K^2(f)} \\ge \\frac{4\\rho}{4 - m\\rho}$.\n\nRecall $\\phi$ is an increasing convex function on $[0,2 \\log 2]$ with\n$\\phi(0) = 0$. Therefore the function $\\tau(y) = \\phi(y)\/y$ is\nincreasing. This implies\n$$\n\\phi\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\) = \\frac{Ent\\(f^2\\)}{K^2(f)} \\tau\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\) \n\\ge \\frac{Ent\\(f^2\\)}{K^2(f)}\n\\tau\\(\\frac{4\\rho}{4 - m\\rho}\\)\n$$\nand, by Proposition~\\ref{thm:technical},\n$$\nD^2(f) \\ge 4 K^2(f) \\phi\\(\\frac{Ent\\(f^2\\)}{K^2(f)}\\) \\ge 4\n\\tau\\(\\frac{4\\rho}{4 - m\\rho}\\) Ent\\(f^2\\)\n$$\nwhich means\n$$\nm \\ge 4 \\tau\\(\\frac{4\\rho}{4 - m\\rho}\\)\n$$\nThe rest is simple algebra. Recall $\\tau = \\phi(y)\/y$ and $\\psi\n= \\phi^{-1}$. Since $\\psi$ is increasing, the last inequality is equivalent to\n$$\n\\psi\\(\\frac{m\\rho}{4 - m\\rho}\\) \\ge \\frac{4\\rho}{4 - m\\rho}.\n$$\nSubstituting  $y = \\frac{m\\rho}{4 - m\\rho}$, this translates to\n$\\psi(y)\/(y+1) \\ge \\rho$. Recall $\\alpha = \\(\\frac{\\psi(y)}{1 +\n  y}\\)^{-1}$. Since $\\alpha$ is increasing, we obtain\n$$\n\\frac{m\\rho}{4 - m\\rho} \\ge \\alpha(\\rho)\n$$\nThis is the same as\n$$\nm \\ge \\frac{4 \\alpha(\\rho)}{\\rho(\\alpha(\\rho) + 1)}.\n$$\nThis is equivalent to (\\ref{ineq:mc}), completing the proof of (\\ref{ineq:l_sob}) and of Theorem~\\ref{thm:log-Sobolev}.\n\\vrule height7pt width4pt depth1pt\n\n\\section{Proof of Lemma~\\ref{lem:functions}}\nLet\n\\begin{equation}\nh(t) = \\frac12 (1-t)^2 \\log (1-t)^2 + \\frac12 (1+t)^2 \\log (1+t)^2 - \\(1+t^2\\)\\log\\(1+t^2\\).\n\\label{def:h}\n\\end{equation}\nIn other words, $h(t) = \\psi\\(t^2\\)$. We will start with some useful properties of the function $h$.\n\n\\begin{lemma}\n\\label{lem:h-prop}\n\\begin{enumerate}\n\\item\n$$\nh' \\ge h \\ge 0\n$$\n\\item\n$$\n\\(1-t^2\\) h' \\ge t h''\n$$\n\\end{enumerate}\n\\end{lemma}\n\\noindent{\\bf Proof:} \nWe have $h'(t) = 2\\cdot\\((1+t)\\log(1+t) - (1-t)\\log(1-t) - t \\log\\(1+t^2\\)\\)$ and $h''(t) = \\frac{4}{1+t^2} - 2\\log\\(\\frac{1+t^2}{1-t^2}\\)$.\n\nThe first claim of the lemma is easy. Nonnegativity of $h$ follows from nonnegativity of $\\psi$, and\n$$\nh'(t) - h(t) = \\(1-t^2\\) \\log(1+t) - (1-t)(3-t) \\log(1-t) + \\(1-t^2\\) \\log\\(1+t^2\\) \\ge 0\n$$\nThe second claim is somewhat harder. We have, rearranging and simplifying:\n$$\n\\frac12 \\cdot \\(\\(1-t^2\\) h' - t h''\\) = t^3 \\log\\(\\frac{1+t^2}{1-t^2}\\) + \\(1-t^2\\)\\log\\(\\frac{1+t}{1-t}\\) - \\frac{2t}{1+t^2}\n$$\nThat is, we need to show\n$$\n\\(1+t^2\\) t^3 \\log\\(\\frac{1+t^2}{1-t^2}\\) + \\(1-t^4\\) \\log\\(\\frac{1+t}{1-t}\\) \\ge 2t\n$$\nIn fact, an even stronger inequality\n$$\nt^3 \\log\\(\\frac{1+t^2}{1-t^2}\\) + \\(1-t^4\\) \\log\\(\\frac{1+t}{1-t}\\) \\ge 2t\n$$\nis valid for $t \\in [0,1]$. This is easy to check for $t=1$. To see this for $0 \\le t < 1$, recall that for $-1 < x < 1$ holds $\\log\\(\\frac{1+x}{1-x}\\) = 2\\sum_{k=0}^{\\infty} \\frac{x^{2k+1}}{2k+1}$. Substituting these series in the inequality above, we need to show\n$$\nt^3 \\cdot \\sum_{k=0}^{\\infty} \\frac{t^{4k+2}}{2k+1} + \\(1-t^4\\) \\cdot \\sum_{k=0}^{\\infty} \\frac{t^{2k+1}}{2k+1} \\ge t,\n$$\nand this is easily verified by observing that all the higher coefficients of the power series on the left hand side are nonnegative.\n\\vrule height7pt width4pt depth1pt\n\nWe pass to the proof of Lemma~\\ref{lem:functions}\n\n\n\\noindent {\\bf Claim 1}\n\n\\noindent\n\nWe will prove $\\psi$ is concave by showing $\\psi''$ is negative on $(0,1)$. We have for $0 < t < 1$:\n$$\n\\psi''(t) = -\\frac{1}{4t^{3\/2}} \\cdot h'\\(\\sqrt t\\) + \\frac{1}{4t}\n\\cdot h''\\(\\sqrt t\\) < 0\n$$\nThe last inequality follows from the second claim of Lemma~\\ref{lem:h-prop}.\n\nTo see that $\\psi$ is increasing, it suffices to verify $\\psi' \\ge 0$ at\n$1$. Indeed,\n$\n\\psi'(1) = \\frac{1}{2} h'(1) = \\log 2 > 0\n$,\ncompleting the proof of Claim 1.\n\n\\noindent {\\bf Claim 2}\n\n\\noindent Let $\\xi(t) = \\frac{\\psi}{1+t}$. We will verify $\\xi' > 0$ on $(0,1)$. Indeed,\n$$\n\\xi' = \\frac{(1+t)\\psi' - \\psi}{(1+t)^2}\n$$\nso we need to check that $\\psi' > \\frac{\\psi}{1+t}$. Substituting\n$\\psi(t) = h(\\sqrt t)$, this amounts to checking\n$$\nh'(\\sqrt t) \\ge \\frac{2\\sqrt t}{1 + t} \\cdot h(\\sqrt t)\n$$\nSince $\\frac{2\\sqrt t}{1 + t} < 1$ on $(0,1)$, it suffices to\nshow $h' \\ge h$, which is true by the first claim of Lemma~\\ref{lem:h-prop}.\n\nTo show concavity of $\\xi$, we will prove that\n\\begin{equation}\n-\\xi'' > 2\\xi,\n\\label{about-xi}\n\\end{equation}\nimplying $\\xi'' < 0$ on $(0,1)$. Direct calculation gives\n$$\n\\xi''(t) = \\frac{(1+t)^2 \\psi'' - 2\\((1+t)\\psi' - \\psi\\)}{(1+t)^3}\n$$\nSimplifying, $-\\xi'' > 2 \\xi'$ reduces to\n$$\n2t \\psi > 2t(1+t) \\psi' + (1+t)^2 \\psi''\n$$\nSince $\\psi'' < 0$ we have $(1+t)^2 \\psi'' < 4t \\psi''$. Therefore, it suffices to prove\n$$\n\\psi \\ge (1+t) \\psi' + 2 \\psi''\n$$\nAgain, since $\\psi$ is concave with $\\psi(0) = 0$, we have $\\psi \\ge t \\psi'$. Therefore, we only need to prove\n$$\n-2 \\psi'' \\ge \\psi'\n$$\nWriting this in terms of $h$, this is equivalent to $\\(1-s^2\\) h' \\ge s h''$ which is given by the second claim of Lemma~\\ref{lem:h-prop}.\n\n\n\\noindent {\\bf Claim 4}\n\nWe will show this claim before the third claim of the lemma. That claim is somewhat more involved, and its proof is relegated to the end of this section.\n\nHere we need to verify $$\\frac{\\alpha(t)}{1+\\alpha(t)} = \\frac12 - \\sqrt{H^{-1}\\(\\log 2 - t\\)\\(1-H^{-1}\\(\\log 2 -t\\)\\)}$$ for all $t \\in [0,\\log 2]$. This is equivalent to\n$$\nH^{-1}\\(\\log 2 - t\\) = \\frac12 - \\sqrt{\\frac{\\alpha}{1+\\alpha}\\(1 - \\frac{\\alpha}{1+\\alpha}\\)} = \\frac{\\(1-\\sqrt{\\alpha}\\)^2}{2(1+\\alpha)}\n$$\nRecall that $\\alpha = \\alpha(t)$ is defined to satisfy\n$t = \\frac{\\psi(\\alpha)}{1 + \\alpha}$, where $\\psi(\\alpha) =\nEnt\\(f^2\\)$, and $f$ is a function on $\\{0,1\\}$ with $g(0) = 1\n- \\sqrt \\alpha$, $g(1) = 1 + \\sqrt \\alpha$. It is not hard to verify the identity\n$$\n\\frac{\\psi(\\alpha)}{1 + \\alpha} = \\ln 2 - H\\(\\frac{\\(1-\\sqrt \\alpha\\)^2}{2(1 + \\alpha)}\\)\n$$\nfor all $\\alpha \\in [0,1]$, and we are done.\n\n\\noindent {\\bf Claim 3}\n\nFirst, we show that $c$ is increasing. Direct computation gives that $c'$ is positive on $(0,\\log 2)$ iff $t \\alpha' > \\alpha + \\alpha^2$ on this interval. Both sides of this inequality are $0$ at\n$0$, and we compare derivatives, that is, show  $t \\alpha'' > 2 \\alpha \\alpha'$.\n\nSince $\\alpha$ is convex with $\\alpha(0) = 0$, we have $\\alpha \\le t \\alpha'$ in the interval.\nHence, it suffices to show $\\alpha'' > 2\\(\\alpha'\\)^2$.\n\nRecall $\\alpha = \\xi^{-1}$. Consequently, $\\alpha'(\\xi(t)) =\n\\frac{1}{\\xi'(t)}$, and $\\alpha''(\\xi(t)) =\n-\\frac{\\xi''(t)}{\\(\\xi'(t)\\)^3}$. Therefore, $\\alpha'' >\n2\\(\\alpha'\\)^2$ is equivalent to $-\\xi'' > 2\\xi'$, which is given by (\\ref{about-xi}).\n\nIt remains to show that $c$ is convex. This turns out to be significantly harder than the other proofs in this Section. We provide a somewhat sketchy argument below.\n\nDirect computation shows that $c'' >0$ on $(0,\\log 2)$ iff\n\\begin{equation}\nt^2 (1 + \\alpha) \\alpha'' + 2\\alpha (1+\\alpha)^2 >\n2t^2\\(\\alpha'\\)^2 + 2t (1+\\alpha)\\alpha'\n\\end{equation}\nFirst, we rewrite this inequality in terms of $\\xi = \\alpha^{-1}$. Let $t = \\xi(x)$, that is $\\alpha(t) = x$, $\\alpha'(t) = \\frac{1}{\\xi'(x)}$, $\\alpha''(t) = -\\frac{\\xi''(x)}{\\(\\xi'(x)\\)^3}$. Substituting and simplifying, one gets\n$$\n-(1+x) \\xi^2 \\xi'' + 2x(1+x)^2 \\(\\xi'\\)^3 > 2\\xi^2 \\xi' + 2(1+x) \\xi \\(\\xi'\\)^2,\n$$\nwhich has to hold for all $x$ in $(0,1)$.\n\nNext, we rewrite this in terms of $\\psi = (1+x) \\xi$, obtaining\n$$\n(1+x)\\psi^2\\(-\\psi''\\) > 2\\((1+x)\\psi' - \\psi\\)^2 \\(\\psi- x\\psi'\\)\n$$\nNote that all the expressions in the brackets are positive, since $\\psi$ is concave and $\\xi$ is increasing, as we saw in the proofs of Claims 1 and 2 above. We simplify this inequality, replacing $\\psi$ with $x \\psi'$ on the left hand side and in the first term on the right hand side, and arriving to the stronger inequality\n$$\nx(1+x)\\psi\\(-\\psi''\\) > 2 \\psi' \\(\\psi - x \\psi'\\)\n$$\nWe rewrite this in terms of the function $h$, defined in (\\ref{def:h}) above. As in the proof of Claim 1, expressing $\\psi$ and its derivatives in terms of $h$, leads to the following equivalent inequality:\n\\begin{equation}\n2x(h')^2 > \\(3-x^2\\)hh' + x\\(1+x^2\\) h h''\n\\label{interms:h}\n\\end{equation}\nFrom now on we concentrate on the proof of (\\ref{interms:h}). It will be convenient to write $h$ and its derivatives in terms of two new functions $L_1(x) = \\log{\\frac{1+x}{1-x}}$ and $L_2(x) = \\log{\\frac{1+x^2}{1-x^2}}$. Recalling the expressions for $h$ and its derivatives (as in the proof of Lemma~\\ref{lem:h-prop} above, we have\n\\begin{itemize}\n\\item\n$\nh(x) = 2xL_1 - \\(1+x^2\\)L_2\n$\n\\item\n$\nh'(x) = 2L_1 - 2xL_2\n$\n\\item\n$\nh''(x) = \\frac{4}{1+x^2} - 2L_2\n$\n\\end{itemize}\n\nRewriting (\\ref{interms:h}) in terms of $L_1$ and $L_2$, and simplifying, one arrives to\n$$\n\\(3-x^2\\)\\(1+x^2\\)L_1 L_2 + 2x\\(1+x^2\\)L_2 > 2x\\(1-x^2\\)L^2_1 + 4xL^2_2 + 4x^2L_1\n$$\nWe expand both sides of this inequality as power series for $x \\in (0,1)$. Recall that $L_1(x) = 2\\sum_{k=0}^{\\infty} \\frac{1+x^{2k+1}}{2k+1}$, and, consequently, $L_2(x) = 2\\sum_{k=0}^{\\infty} \\frac{1+x^{4k+2}}{2k+1}$. Therefore, both sides of this inequality have only odd terms.\n\nLet the left hand side be equal to\n$$\nF(x) = 4 \\cdot \\sum_{k=0}^{\\infty} \\ell_{2k+1} x^{2k+1}\n$$\nand the right hand side be equal to $$\nG(x) = 4 \\cdot \\sum_{k=0}^{\\infty} r_{2k+1} x^{2k+1}\n$$\nWe will argue that\n\\begin{enumerate}\n\\item\nAll the coefficients $\\ell_{2k+1}$ and $r_{2k+1}$ are nonnegative.\n\\item\n$\\ell_1 = r_1 = 0$, $\\ell_3 = r_3 = \\ell_5 = r_5 = 4$.\n\\item\nFor all odd $k$ starting from $k = 3$:\n$$\n\\ell_{2k+1} > r_{2k+1}~~~and~~~\\ell_{2k+1} + \\ell_{2k+3} > r_{2k+1} + r_{2k+3}\n$$\n\\end{enumerate}\n\n\\noindent This will imply\n$$\nF(x) - G(x) = 4\\cdot \\sum_{k=3}^{\\infty} \\(\\ell_{2k+1} - r_{2k+1}\\) x^{2k+1} = 4\\cdot \\sum_{odd~k \\ge 3} \\(\\(\\ell_{2k+1} - r_{2k+2}\\) - \\(\\ell_{2k+3} - r_{2k+3}\\)x^2 \\) \\cdot x^{2k+1} >\n$$\n$$\n4\\cdot \\sum_{odd~k \\ge 3} \\(\\ell_{2k+1} - r_{2k+2}\\)\\(1-x^2\\) \\cdot^{2k+1} > 0,\n$$\ncompleting the proof of (\\ref{interms:h}) and of Claim 3. Hence it remains to prove the properties of the coefficients.\n\nIn fact, the coefficients can be computed explicitly, which makes it possible to verify the required properties. We omit the (easy but cumbersome) details. For completeness sake, we do list explicit expressions for the coefficients below\\footnote{Our apologies to the reader.}.\n\\begin{itemize}\n\\item\nFor an odd $k \\ge 3$:\n$$\n\\ell_{2k+1} = \\(\\frac{8k - 20}{(2k-3)(2k+1)}\\) \\cdot \\sum_{m=1}^{(k-1)\/2} \\frac{1}{4m-3} ~~~+~~~ \\frac{4}{2k-1} \\cdot \\sum_{m = 1}^{k-2} \\frac{1}{2m+1} ~~~+~~~\n$$\n$$\n\\(\\frac{3}{2k+1} +\n\\frac{2}{2k-1} - \\frac{1}{2k-3}\\) \\cdot \\sum_{m = 1}^{(k-1)\/2} \\frac{1}{2m-1} ~~~+~~~ \\(\\frac{1}{k} + \\frac{3}{k(2k+1)} + \\frac{6}{(2k-1)(2k+1)}\\)\n$$\nand\n$$\nr_{2k+1} = \\frac{2k+2}{k(2k-1)} ~~~-~~~ \\frac{2}{k(k-1)} \\cdot \\sum_{m = 1}^{k-1} \\frac{1}{2m-1}\n$$\n\\item\nFor an even $k \\ge 4$:\n$$\n\\ell_{2k+1} = \\(\\frac{8k - 20}{(2k-3)(2k+1)}\\) \\cdot  \\sum_{m = 1}^{k-2} \\frac{1}{2m+1} ~~~+~~~ \\frac{4}{2k-1} \\cdot  \\sum_{m=1}^{k\/2} \\frac{1}{4m-3}~~+~~\n$$\n$$\n\\(\\frac{3}{2k+1} + \\frac{2}{2k-1} - \\frac{1}{2k-3}\\) \\cdot \\sum_{m = 1}^{(k-2)\/2} \\frac{1}{2m-1} ~+~ \\(\\frac{1}{k-1} + \\frac{6}{(2k-1)(2k+1)} +  \\frac{10k-1}{(k-1)(2k-1)(2k+1)} \\)\n$$\nand\n$$\nr_{2k+1} = \\frac{8}{k} \\cdot \\sum_{m=1}^{k\/2} \\frac{1}{2m - 1} ~~~+~~~ \\frac{2k+2}{k(2k-1)} ~~~-~~~ \\frac{2}{k(k-1)} \\cdot \\sum_{m = 1}^{k-1} \\frac{1}{2m-1}\n$$\n\\end{itemize}\n\nIt remains to compute the coefficients for $k = 1, 2$. This is easily done directly, verifying the property 2 above.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $R$ be a root system in $\\R^d$, $d\\geq1$, and we fix a positive subsystem $R_+$ of $R$ and a nonnegative multiplicity function $k:R\\to\\R_+$.\nFor every $\\alpha\\in R$, let $H_\\alpha$ be the hyperplane  orthogonal to $\\alpha$ and $\\sigma_\\alpha$ be the reflection with respect to~$H_\\alpha$, that is, for every~$x\\in\\R^d$,\n$$\n\\sigma_\\alpha x=x-2 \\frac{\\langle x,\\alpha\\rangle}{|\\alpha|^2}\\alpha\n$$\nwhere $\\langle\\cdot,\\cdot\\rangle$ denotes the Euclidean inner product of $\\R^d$.\nThe Dunkl Laplacian $\\Delta_k$ is defined \\cite{dunkl1},  for  $f\\in C^2(\\R^d)$, by\n$$\n\\Delta_kf(x)=\\Delta f(x)+2\\sum_{\\alpha\\in R_+}k(\\alpha)\\left(\\frac{\\langle\\nabla f(x),\\alpha\\rangle}{\\langle\\alpha,x\\rangle}-\\frac{|\\alpha|^2}2\\frac{f(x)-f(\\sigma_\\alpha x)}{\\langle\\alpha,x\\rangle^2}\\right),\n$$\nwhere $\\nabla$ denotes the  gradient on~$\\R^d$. Obviously, $\\Delta_k=\\Delta$ when $k\\equiv0$.\n\nGiven a bounded open subset $D$ of $\\R^d$, we consider the following Dirichlet problem~:\n\\begin{equation}\\label{ddp}\n\\displaystyle\\left\\{\n\\begin{array}{rcll}\n \\Delta_kh & = & 0 & \\mbox{on }\\;D, \\\\\n  h & = & f& \\mbox{on }\\; \\R^d\\setminus D,\n\\end{array}\n\\right.\n\\end{equation}\nwhere\n  $f$ is a continuous function on $\\R^d\\setminus D$. When $D$ is invariant under all reflections $\\sigma_\\alpha$, it was shown in \\cite{mbke}, using probabilistic  tools from potential theory, that  there exists a unique continuous function $h$ on $\\R^d$, twice differentiable on~$D$ and such that both equations in~(\\ref{ddp}) are pointwise fulfilled.  In this paper, we shall investigate problem (\\ref{ddp}) for a bounded domain $D$  which is not invariant. Let $D$ be a bounded open set such that its closure $\\overline{D}$ is in some Domain of $\\R^d\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha$.\nWe mean by a solution of problem (\\ref{ddp}), every function $h : \\R^d\\to\\R$ which is continuous on $\\R^d$ such that $h=f$ on $\\R^d\\setminus D$ and\n$$\n\\int_{\\R^d}h(x)\\Delta_k\\varphi(x) w_k(x) dx =0\\quad \\textrm{ for every }\\; \\varphi \\in C^\\infty_c(D),\n$$\nwhere $C^\\infty_c(D)$ denotes the space of infinitely differentiable functions on $D$  with compact support  and $w_k$ is the invariant weight function defined on $\\R^d$ by\n$$\nw_k(x)=\\prod_{\\alpha\\in\nR_+}\\langle x,\\alpha\\rangle^{2k(\\alpha)}.\n$$\nThe set $D$ is called $\\Delta_k$-regular if, for every continuous function $f$ on $\\R^d\\setminus D$, problem (\\ref{ddp}) admits one and only one solution; this solution will be denoted by $H_D^{\\Delta_k}f$.\nBy transforming  problem (\\ref{ddp})  to a  boundary value problem associated with Schr\\\"{o}dinger's operator $\\Delta-q$, we show that $D$ is $\\Delta_k$-regular provided it is $\\Delta$-regular. We also give an analytic formula characterizing the solution $H_D^{\\Delta_k}f$ (see Theorem \\ref{t1} below). We derive from this formula that, for every $x\\in D$, $H_D^{\\Delta_k}f(x)$ depends only on the values of $f$ on $\\cup_{\\alpha\\in R_+}\\sigma_\\alpha(D)$ and on $\\partial D$ the Euclidean boundary of $D$.\nIf, in addition, we assume that $f$ is locally H\\\"{o}lder continuous on $\\cup_{\\alpha\\in R_+}\\sigma(D)$ then  $H_D^{\\Delta_k}f$ is continuously twice differentiable on $D$ and therefore the first equation in~(\\ref{ddp}) is  fulfilled by $H_D^{\\Delta_k}f$ not only in the sense of distributions but also pointwise.\n\nIt was shown in \\cite{kh,hmkt} that the operator $\\Delta_k$ is hypoelliptic on all invariant open subset $D$ of $\\R^d$. However, if $D$ is not invariant, the question whether $\\Delta_k$ is hypoelliptic on $D$ or not remaind open. For $\\Delta_k$-regular open set $D$, we show that if $D$ is not invariant then $\\Delta_k$ is not hypoelliptic in $D$. Hence the condition \" $D$ is invariant\" is necessary and sufficient for the hypoellipticity of  $\\Delta_k$ on $D$.\n\\section{Main results}\nWe first present various facts on the Dirichlet boundary value problem associated with Schr\\\"{o}dinger's operator which are needed for our approach.\nWe refer to \\cite{abwhhh,kczz} for  details.\nLet $G$ be the Green function on $\\R^d$, but without the constant factors :\n$$\nG(x,y)=\n\\left\\{\n  \\begin{array}{ll}\n    |x-y|^{2-d} & \\hbox{if}\\; d\\geq 3; \\\\\n    \\ln\\frac{1}{|x-y|} & \\hbox{if}\\; d=2; \\\\\n    |x-y| & \\hbox{if}\\; d=1.\n  \\end{array}\n\\right.\n$$\nLet $D$ be a bounded domain of $\\R^d$ and let $q\\in J(D)$ the Kato class on $D$, i.e., $q$ is a Borel measurable function on $\\R^d$ such that $G(1_D|q|)$ the Green potential of $1_D|q|$ is continuous on $\\R^d$.\nNote that the Kato class $J(D)$ contains all bounded Borel measurable functions on $D$.\nAssume that  $D$ is $\\Delta$-regular. Then, for every continuous function $f$ on  $\\partial D$, there exists a unique continuous function $h$ on  $\\overline{D}$  such that $h=f$ on $\\partial D$ and\n\\begin{equation}\\label{shr}\n\\int h(x)(\\Delta-q)\\varphi(x) dx=0 \\quad \\textrm{ for every}\\; \\varphi\\in C^\\infty_c(D).\n\\end{equation}\nIn the sequel, we denote $H_D^{\\Delta-q}f$ the unique continuous extension on $\\overline{D}$ of $f$ which satisfies the Schr\\\"{o}dinger's equation (\\ref{shr}).\nLet $G_D^\\Delta$ and $G_D^{\\Delta-q}$ denotes, respectively, the Green potential operator in $D$ of  $\\Delta$ and  $\\Delta-q$. The operator $G_D^{\\Delta-q}$ acts as a right inverse of the Schr\\\"{o}dinger's operator $-(\\Delta-q)$, i.e., for every Borel bounded function $g$ on $D$, we have\n$$\n\\int G_D^{\\Delta-q}g(x)(\\Delta-q)\\varphi(x) dx= -\\int g(x)\\varphi(x) dx \\quad \\textrm{ for every}\\; \\varphi\\in C^\\infty_c(D).\n$$\nThen the unique continuous function $h$ on $\\overline{D}$ such that $h=f$ on $\\partial D$ and\n\\begin{equation}\\label{shrg}\n\\int h(x)(\\Delta-q)\\varphi(x) dx= -\\int g(x)\\varphi(x) dx \\quad \\textrm{ for every}\\; \\varphi\\in C^\\infty_c(D)\n\\end{equation}\nis given, for $x\\in D$, by\n\\begin{equation}\\label{ss}\nh(x)= H_D^{\\Delta-q}f(x)+G_D^{\\Delta-q}g(x).\n\\end{equation}\nThe function $G_D^{\\Delta-q}g$ is continuous on $\\overline{D}$, vanishing on $\\R^d\\setminus D$ and, for every $x\\in D$,\n\\begin{equation}\\label{vgd}\nG_D^{\\Delta-q}g(x)=G_D^\\Delta g(x)- G_D^\\Delta(qG_D^{\\Delta-q}g)(x).\n\\end{equation}\nMoreover, if, in addition, we assume  that $q\\in C^\\infty(D)$ then, proceeding by induction, it follows from (\\ref{vgd}) that $G_D^{\\Delta-q}g\\in C^n(D)$ if and only if $G_D^\\Delta g\\in C^n(D),\\; n\\in \\N$.\n\nNow we are ready to establish our first main result giving a characterization of solutions of the Dirichlet boundary value problem associated with the Dunkl Laplacian $\\Delta_k$.\n\n\\begin{theorem}\\label{t1}\nLet $D$ be a bounded open set such that $\\overline{D}$ is in some Domain of $\\R^d\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha$. If $D$ is $\\Delta$-regular then $D$ is $\\Delta_k$-regular. Moreover, for every continuous function $f$ on $\\R^d\\setminus D$ and for every $x\\in D$,\n\\begin{equation}\\label{solhar}\nH_D^{\\Delta_k}f(x)=\\frac{1}{\\sqrt{w_k(x)}}\\left(H_D^{\\Delta-q}(f\\sqrt{w_k})(x)+G_D^{\\Delta-q}\\left(\\sqrt{w_k}Nf\\right)(x)\\right),\n\\end{equation}\nwhere $q$ and $Nf$ are the functions defined, for $x\\in D$, by\n$$\nq(x):=\\sum_{\\alpha\\in R_+}\\left(\\frac{|\\alpha|k(\\alpha)}{\\langle x,\\alpha\\rangle}\\right)^2\n$$\nand\n$$\nNf(x):=\\sum_{\\alpha\\in R_+}\\frac{|\\alpha|^2k(\\alpha)}{\\langle x,\\alpha\\rangle^2}f(\\sigma_\\alpha x).\n$$\n\\end{theorem}\n\n\\begin{proof}\nLet $f$ be a continuous function on $\\R^d\\setminus D$. We intend to prove existence and uniqueness of a continuous function $h$ on $D$ such that $h=f$ on $\\R^d\\setminus D$ and\n\\begin{equation}\\label{har}\n\\int h(x)\\Delta_k\\varphi(x) w_k(x) dx=0 \\quad \\textrm{ for every}\\; \\varphi\\in C^\\infty_c(D).\n\\end{equation}\nIt is clear that\n$$\n\\nabla\\left(\\sqrt{w_k}\\right)(x)= \\sqrt{w_k(x)}\\sum_{\\alpha\\in R_+}\\frac{k(\\alpha)}{\\langle x,\\alpha\\rangle}\\alpha.\n$$\nThen, using the fact that \\cite{dunkl1}\n$$\n\\sum_{\\alpha, \\beta\\in R_+}k(\\alpha)k(\\beta)\\frac{\\langle\\alpha , \\beta\\rangle}{\\langle x , \\alpha\\rangle\\;\\langle x , \\beta\\rangle}=\\sum_{\\alpha\\in R_+}\\frac{|\\alpha|^2k^2(\\alpha)}{\\langle x,\\alpha\\rangle^2},\n$$\ndirect computation  shows that\n$$\n\\Delta\\left(\\sqrt{w_k}\\right)(x)= \\sqrt{w_k(x)}\\sum_{\\alpha\\in R_+}|\\alpha|^2\\frac{k^2(\\alpha)-k(\\alpha)}{\\langle x,\\alpha\\rangle^2}.\n$$\nThus, for every $\\varphi\\in C^\\infty_c(D)$,\n\\begin{eqnarray*}\n\\Delta\\left(\\varphi\\sqrt{w_k}\\right)(x)&=& \\sqrt{w_k(x)}\\left(\\Delta\\varphi(x)+ 2\\sum_{\\alpha\\in R_+}k(\\alpha)\\left(\\frac{\\langle\\nabla \\varphi(x),\\alpha\\rangle}{\\langle\\alpha,x\\rangle}-\\frac{|\\alpha|^2}{2}\\frac{\\varphi(x)}{\\langle\\alpha,x\\rangle^2}\\right)\\right)\\\\\n   & & +\\; q(x)\\varphi(x)\\sqrt{w_k(x)},\n\\end{eqnarray*}\nand thereby\n\\begin{equation}\\label{ddk}\n\\sqrt{w_k(x)}\\Delta_k\\varphi(x)=\\left(\\Delta\\left(\\varphi\\sqrt{w_k}\\right)(x)- q(x)\\varphi(x)\\sqrt{w_k(x)}\\right)+ \\sqrt{w_k(x)}N\\varphi(x).\n\\end{equation}\nSince the map $ \\varphi\\to\\varphi\\sqrt{w_k}$ is invertible on the space $C^\\infty_c(D)$ and the function $x\\to\\frac{w_k(x)}{\\langle x,\\alpha\\rangle^2}$ is invariant under the reflection $\\sigma_\\alpha$, equation (\\ref{har}) is equivalent to the following Schr\\\"{o}dinger's equation : For every $\\psi\\in C^\\infty_c(D)$,\n$$\n\\int h(x)\\sqrt{w_k(x)}\\left(\\Delta-q\\right)\\psi(x) dx= -\\int \\sqrt{w_k(x)}Nf(x)\\psi(x) dx.\n$$\nFinally, since $q$ is bounded on $D$ and therefore is in $J(D)$, the statements follow from (\\ref{shrg}) and (\\ref{ss}).\n\\end{proof}\n\nTo construct a $\\Delta$-regular set $D$, it suffices to choose $D$ such that its Euclidean boundary $\\partial D$  satisfies the the geometric assumption known as \" cone condition\", i.e.,   for every $z\\in \\partial D$ there exists a cone $C$ of vertex $z$ such that $C\\cap B(z,r)\\subset \\R^d\\setminus D$ for some $r>0$, where $B(z,r)$ is the ball of center $z$ and radius $r$  (see, for example, \\cite{kczz}).\n\\begin{remark}\\rm\nNote that, in order to obtain $q\\in J(D)$, the hypothesis of the above theorem \"$\\overline{D}\\subset\\R^d\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha$\" is nearly optimal. Indeed, assume that there exists a cone $C_z$ of vertex $z\\in \\overline{D}\\cap H_\\alpha$ for some $\\alpha\\in R_+$ with $k(\\alpha)\\neq 0$ such that $C_z^r := C_z\\cap B(z,r)\\subset D$ for some $r>0$. Then,\n\\begin{eqnarray*}\nG(1_Dq)(z) & \\geq & |\\alpha|^2k^2(\\alpha)\\int_{C_z^r}G(z,y)\\frac{1}{\\langle y,\\alpha\\rangle^2} dy\\\\\n& = & |\\alpha|^2k^2(\\alpha)\\int_{C_z^r}G(z,y)\\frac{1}{\\langle z-y,\\alpha\\rangle^2} dy\\\\\n&\\geq& k^2(\\alpha) \\int_{C_z^r-z}G(0,y)\\frac{1}{|y|^2} dy \\\\\n&=& \\infty.\n\\end{eqnarray*}\n\\end{remark}\nIt is easy to see  that for every $x\\in D$ the map $f\\to H_D^{\\Delta_k}f(x)$ defines a positive Radon measure on $\\R^d\\setminus D$. We denote this measure by $H_D^{\\Delta_k}(x,dy)$. The following results are obtained in a convenient way by using formula (\\ref{solhar}) of the above theorem.\n\\begin{corollary}\n For every $x\\in D$, $H_D^{\\Delta_k}(x,dy)$ is a probability measure supported by\n$$\n\\partial D\\cup \\left(\\cup_{\\alpha\\in R_+}\\sigma_\\alpha(D)\\right)\n$$\nand satisfies\n$$\n\\frac{\\sqrt{w_k(x)}}{\\sqrt{w_k(y)}}H_D^{\\Delta_k}(x,dy)=H_D^{\\Delta-q}(x,dy)+\\sum_{\\alpha\\in R_+}\\frac{|\\alpha|^2k(\\alpha)}{\\langle y,\\alpha\\rangle^2}G_D^{\\Delta-q}(x,\\sigma_\\alpha y) dy.\n$$\n\\end{corollary}\n\n\\begin{corollary}\nLet $D$ be a $\\Delta$-regular bounded open set such that $\\overline{D}$ is in some Domain of $\\R^d\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha$.\nLet $f$ be a continuous function on $\\partial D\\cup \\left(\\cup_{\\alpha\\in R_+}\\sigma_\\alpha(D)\\right)$. If $f$ is locally H\\\"{o}lder continuous on $\\cup_{\\alpha\\in R_+}\\sigma(D)$ then $H_D^{\\Delta_k}f\\in C^2(D)$ and, for every $x\\in D$,\n$$\n\\Delta_k\\left(H_D^{\\Delta_k}f\\right)(x)=0.\n$$\n\\end{corollary}\n\n\\begin{proof}\nSince $H_D^{\\Delta-q}(f\\sqrt{w_k})$ is a solution of the Schr\\\"{o}dinger's equation (\\ref{shr}), the hypoellipticity of the operator $\\Delta-q$ on $D$ implies that $H_D^{\\Delta-q}(f\\sqrt{w_k})\\in C^\\infty(D)$. Moreover, since $Nf$ is locally H\\\"{o}lder continuous on $D$, $G_D^\\Delta\\left(\\sqrt{w_k}Nf\\right) \\in C^2(D)$  and consequently $G_D^{\\Delta-q}\\left(\\sqrt{w_k}Nf\\right)\\in C^2(D)$. Then it follows from (\\ref{solhar}) that $H_D^{\\Delta_k}f\\in C^2(D)$. For every $\\varphi\\in C^\\infty_c(D)$, direct computation using  (\\ref{ddk}) yields\n$$\n \\int \\Delta_k\\left(H_D^{\\Delta_k}f\\right)(x)\\varphi(x) w_k(x) dx=\\int H_D^{\\Delta_k}f(x)\\Delta_k\\varphi(x) w_k(x) dx.\n$$\nThis completes the proof.\n\\end{proof}\n\nLet $D$ be an open subset of $\\R^d$. The operator $\\Delta_k$ is said to be hypoelliptic on $D$ if, for every $f\\in C^\\infty(D)$, every continuous function $h$ on $\\R^d$ which satisfies\n$$\n\\int_{\\R^d}h(x)\\Delta_k\\varphi(x) w_k(x) dx =\\int  f(x)\\varphi(x) w_k(x) dx \\quad \\textrm{ for every }\\; \\varphi \\in C^\\infty_c(D)\n$$\nis infinitely differentiable on $D$. We note that the problem of the hypoellipticity of $\\Delta_k$  is discussed in \\cite{kh,hmkt}, where the authors show that  $\\Delta_k$ is hypoelliptic on $D$ provided  $D$ is  invariant under all reflections $\\sigma_\\alpha$. However, if $D$ is not invariant, the question whether $\\Delta_k$ is hypoelliptic on $D$ or not remaind open.\n\n\\begin{theorem}\nLet $D$ be a $\\Delta_k$-regular open set. Then $\\Delta_k$ is  hypoelliptic on $D$ if and only if $D$ is invariant.\n\\end{theorem}\n\\begin{proof}\nIt is obviously sufficient to prove that $\\Delta_k$ is not hypoelliptic on $D$ provided $D$ is not invariant. Assume that $D$ is not invariant. Since\n the open set $D\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha$ is also not invariant, there exists a nonempty  open ball $B$\nsuch that\n$$\n\\overline{B}\\subset D\\setminus\\cup_{\\alpha\\in R_+}H_\\alpha\\quad\\textrm{and}\\quad\\sigma_\\alpha(B)\\subset \\R^d\\setminus D\\;\\textrm{ for some }\\;\\alpha\\in R_+.\n$$\n We also choose the ball $B$ small enough such that, for every $\\alpha\\in R_+$,\n$$\n\\sigma_\\alpha(B)\\subset D \\quad \\textrm{ or } \\quad \\sigma_\\alpha(B)\\subset\\R^d\\setminus D.\n$$\nLet $I :=\\{\\alpha\\in R_+;\\; \\sigma_\\alpha(B)\\subset\\R^d\\setminus D\\}$ and $J :=R_+\\setminus I$.\nLet $f$ be a continuous function on $\\R^d\\setminus D$ and denote $H_D^{\\Delta_k}f$ by $h$.\nSince $B$ is $\\Delta$-regular and $h$ satisfies\n$$\n\\int h(x)\\Delta_k\\varphi(x) w_k(x) dx=0 \\quad \\textrm{ for every}\\; \\varphi\\in C^\\infty_c(B),\n$$\n it follows from Theorem \\ref{t1} that $B$ is $\\Delta_k$-regular and, for every $x\\in B$,\n\\begin{equation}\\label{hyp}\nh(x)=\\frac{1}{\\sqrt{w_k(x)}}\\left(H_B^{\\Delta-q}(h\\sqrt{w_k})(x)+G_B^{\\Delta-q}\\left(\\sqrt{w_k}Nh\\right)(x)\\right).\n\\end{equation}\nLet $g_1$ and $g_2$ be the functions defined on $B$ by\n\n$$\ng_1(x)=\\sum_{\\alpha\\in J}\\frac{|\\alpha|^2k(\\alpha)}{\\langle x,\\alpha\\rangle^2}h(\\sigma_\\alpha x)\\quad\n\\textrm{and}\n\\quad\ng_2(x)=\\sum_{\\alpha\\in I}\\frac{|\\alpha|^2k(\\alpha)}{\\langle x,\\alpha\\rangle^2}f(\\sigma_\\alpha x).\n$$\nIt is clear that the function $g_2$ is not trivial and $Nh=g_1+g_2$.\nNow, assume that $h\\in C^\\infty(D)$. Then $g_1\\in C^\\infty(B)$ and therefore $G_B^{\\Delta-q}\\left(\\sqrt{w_k}g_1\\right)\\in C^\\infty(B)$. Furthermore, since $H_B^{\\Delta-q}(h\\sqrt{w_k})\\in C^\\infty(B)$, it follows from  (\\ref{hyp}) that $G_B^{\\Delta-q}\\left(\\sqrt{w_k}g_2\\right)\\in C^\\infty(B)$. Thus $-(\\Delta-q)G_B^{\\Delta-q}\\left(\\sqrt{w_k}g_2\\right)=\\sqrt{w_k}g_2\\in C^\\infty(B)$ and therefore $g_2\\in C^\\infty(B)$, a contradiction.\nHence $h$ is not infinitely differentiable on $D$ and consequently the Dunkl Laplacian $\\Delta_k$ is not hypoelliptic on $D$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nProstate cancer detection is a longstanding challenge in medical imaging for which, many algorithms have been proposed using conventional machine learning algorithms~\\cite{Khalvati2015a,Khalvati2016,Khalvati2018,Hussain2018}. With the advent of deep Convolutional Neural Networks (CNNs) and the promising results achieved by CNNs in computer vision tasks, there has been a shift in designing computer-aided detection algorithms for prostate cancer toward CNN architectures~\\cite{Kwak2017,Song2018,Wang2017,Yang2017,Yoo2019}. From Machine learning perspective, prostate cancer detection is a binary classification task. To evaluate performance of such a binary classification model, Area Under Receiver Operating Characteristic (ROC) curve (AUC) is usually used. In medical imaging in particular, AUC is widely used as a performance measure~\\cite{Park2004}.\n\nConventional approach for training a CNN is backpropagation~\\cite{nref1}. For a loss function to work in backpropagation, it must be differentiable~\\cite{nref2}. However, AUC is not differentiable and therefore, CNNs are usually trained using a loss functions based on other performance metrics such as cross entropy. During the training process, while loss is being minimized, AUC is monitored and the best performing model is selected based on the highest AUC~\\cite{Wang2017,Yang2017,Yoo2019}. The challenge here is that a model optimized for minimum loss may not necessarily produce the best possible AUC. To address this issue, we propose an evolution-based method to fine-tune a CNN that has been trained for prostate cancer detection. \n\nGenetic algorithms (GAs) are a class of evolutionary methods which have been used for optimization in machine learning for a number of years~\\cite{ref9, DBLP:journals\/corr\/XieY17}. GAs do not rely on the derivative of the loss function (called fitness function in evolutionary algorithms domain). High computational cost of GAs has limited their application in CNN optimization~\\cite{Castillo_g-prop-iii:global}. Nevertheless, there are efforts for using GAs for optimizing CNNs for image classification~\\cite{ref10,Sun2018}. \n\nIn this paper, we use a GA to fine-tune a CNN, which has been trained for prostate cancer detection using Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI). The GA is applied to the fully connected (FC) layers of the CNN, thus the computational cost is significantly reduced. Although more sophisticated CNN architectures have been used for prostate cancer detection~\\cite{Wang2017,Yang2017,Yoo2019}, we developed a simple CNN with 3 convolutional layers and 3 FC layers to demonstrate capability of the proposed method in improving the performance of CNN architectures. The proposed evolutionary fine-tuning algorithm improves AUC of the CNN by 9.3\\% in the test set, which includes 1,334 slices of DW-MRI images of prostate.\n\n\\section{Methods}\n\nDW-MRI images from 414 prostate cancer patients (5,706 2D slices) were used as the dataset. Institutional review board approval was obtained for this study. Six DWI sequences were included for each slice: apparent diffusion coefficient (ADC) map, and five different b-value images (0, 100, 400, 1000, and 1600 $sm^{-2}$). Images were preprocessed and prostate regions were cropped using manual contours of the prostate. Each prostate region was resized to $64 \\times 64$ pixels. The dataset was divided into training (217 patients, 2,955 slices), validation (102 patients, 1,417 slices), and test sets (95 patients, 1,334 slices). Label for each slice was generated based on the targeted biopsy results where a clinically significant prostate cancer (Gleason score>6) was considered a positive label.\n\nFigure~\\ref{figure:1} shows the CNN architecture that we used for the experiments. The configuration of the CNN is shown in Table~\\ref{table:1}. Padding was not used in the architecture and stride was equal to one. Weights of CNN layers were initialized by Xavier method~\\cite{Glorot10understandingthe}. All biases and weights of FC layers were randomly initialized from a uniform distribution over [0, 1]. The model was trained based on Cross Entropy loss function and optimized by Stochastic Gradient Descent (SGD)~\\cite{Ruder2016}. We used Python 3.7.3, PyTorch 1.1.0, and Ubuntu 19.04 for the experiments.\n\n\\begin{figure}[htp]\n  \\centering\n  \\includegraphics[scale=0.45]{Fig1.JPG}\n  \\caption{The Proposed GA-CNN Architecture}\n  \\label{figure:1}\n\\end{figure}\n\n\\begin{table}\n  \\caption{Configuration of the CNN}\n  \\label{table:1}\n  \\centering\n  \\begin{tabular}{ll}\n    \\toprule\n    Layer     & Configuration \\\\\n    \\midrule\n    CNN-1\t&input channels=6, output channels=16, kernel size=7 \\\\\n    Max Pooling-1\t&kernel size=2 \\\\\n    Dropout-1\t&probability=0.1\\\\\n    CNN-2\t&input channels=16, output channels=32, kernel size=5\\\\\n    Max Pooling-2\t&kernel size=2\\\\\n    Dropout-2\t&probability=0.1\\\\\n    CNN-3\t&input channels=32, output channels=64, kernel size=4\\\\\n    Max Pooling-3\t&kernel size=2\\\\\n    Dropout-3\t&probability=0.1\\\\\n    Fully Connected-1\t&input size=1024, output size=256\\\\\n    Fully Connected-2\t&input size=256, output size=64\\\\\n    Fully Connected-3\t&input size=64, output size=2\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\nConvolution layers can be considered as feature extractors, which are optimized by SGD. However, FC layers are in fact classifiers, which may not reach an optimum point in terms of AUC by SGD. Thus, we hypothesize that by introducing a GA to FC layers, the classifier portion of the CNN is further optimized for AUC. Therefore, our proposed approach is similar to freezing shallow layers (feature extractors) of CNNs in Transfer Learning~\\cite{Shie2015}.\n\nThe initial population of our GA includes 512 instances of the classifier (3 FC layers of the CNN). One instance is extracted from the trained CNN model while the remaining 511 instances are randomly initialized. Classifiers (instances) are ranked based on their AUC performance for the training set. Top half of the instances are transferred to the next generation. They are then crossed over and mutated to produce two remaining quarters of the generation. Mutation occurs with probability of 1\\%. As long as the maximum AUCs on the validation and training sets are improved, this process continues.\n\nEven with targeting FC layers, computational cost and memory requirements of the GAs are still high. To mitigate this, we applied the crossover and mutation operations at layer level, instead of individual nodes (Figure~\\ref{figure:2}). In other words, we do not optimize each individual weight and bias of the classifier and instead, parents and offsprings are in the form of an entire layer.\n\\begin{figure}[htp]\n  \\centering\n  \\includegraphics[scale=0.25]{Fig2.JPG}\n  \\caption{High-level GA application: (a) Crossover (b) Mutation}\n  \\label{figure:2}\n\\end{figure}\n\n\\section{Evaluation and Results}\nBased on a grid search performed for selecting optimum hyper-parameters, we set learning rate to 0.001, and momentum equal to 0.8. L2 penalty of 0.001 and batch size of 1 were used. Although the maximum epoch number was 50, the best AUC was achieved in 10$^{th}$ epoch. Once he CNN was optimized by SGD, the model attained an AUC of 0.794 on the validation set and 0.707 on the test set. Our best model after applying the GA was achieved in the third generation. The AUC performance was 0.815 and 0.773 on the validation and test sets, respectively, which is a 9.3\\% of AUC improvement on the test set. Table~\\ref{table:2} lists the results in detail. We ran the GA algorithm on GeForce GTX 1060 and it took 10 min to reach the optimal AUC result.\n\n\n\\begin{table}[h!]\n  \\caption{AUC performance for SGD and proposed GA-based Method}\n  \\label{table:2}\n  \\centering\n  \\begin{tabular}{lll}\n    \\toprule\n\t& SGD\t&GA\\\\\n    \\midrule\n    AUC on train set\t&0.867\t&0.877\\\\\n    AUC on validation set\t&0.794\t&0.815\\\\\n    AUC on test set\t&0.707\t&0.773\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\n\n\\section{Conclusion}\nIn this work, we proposed a GA-based method to fine-tune CNNs for prostate cancer detection. Monitoring validation set AUC during conventional training of CNNs results in a sub-optimal model. By applying a GA to FC layers, and performing crossover and mutation on the entire layer instead of individual coefficients, the proposed evolution-based fine-tuning procedure becomes feasible even for low-end GPUs such as GeForce GTX 1060. We demonstrated that for a simple CNN architecture with 3 convolutional layers and 3 FC layers, our proposed evolutionary algorithm can improve the AUC of test dataset by 9.3\\%.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe question of the stability of the shock structure has regained recent interest with the advent of multipoint measurements at the Earth's bow shock. Numerical simulations have indicated for a long time that even under steady upstream conditions and for a broad range of parameters collisionless shocks can exhibit dramatic structural changes and eventually self-reform quasi-periodically \\citep{BW72JGR,Bur89GRL,LGS04SSR}. However experimental evidence of this kind of behaviour in space plasmas have been limited. For instance, \\citet{TGB90JGRa} and \\citet{TGO93JGR} have observed quasi-periodic variations on a time-scale of approximately two upstream proton gyroperiods in the ion velocity distributions downstream of the quasi-parallel Earth's bow shock which they have attributed to shock reformation. Using multipoint Cluster measurements, \\citet{LHD08JGR} have studied the scales and growth of large amplitude pulsations (Short Large Amplitude Magnetic Structures, SLAMS) thought to be part of the quasi-parallel shock transition region. At the quasi-perpendicular shock, \\citet{HCL01AG} have noticed on some Cluster crossings a significant variability in magnetic field profiles, particularly in the foot, despite a relatively small spacecraft separation. A large variability in magnetic and electric field time series was also observed by \\citet{LKB07GRL} at a high Mach number quasi-perpendicular shock crossing, complemented with bursty variations in reflected ions occuring on a time scale comparable to a proton gyroperiod. Another form of non-stationarity was found by \\citet{MBH06JGR} who have identified coherent oscillations with a wavelength of a few tens of upstream ion inertial length confined to the shock layer and propagating along it.\n\nThe general problem of the shock stationarity is complicated by the importance of the shock geometry. Quasi-parallel and quasi-perpendicular shocks are indeed distinguished by significantly different structure, scales and dissipative processes\n(e.g. \\citep{Sch06SSR}). At least for supercritical shocks resistivity is insufficient to provide the required dissipation, which is then provided in a first step by reflecting upstream part of incoming ions. Assuming a specular reflection process and a step-like shock transition one finds out that the limit between quasi-perpendicular and quasi-parallel shocks $\\theta_{Bn}=45^{\\circ}$ corresponds to the angle below which the guiding-centre of the reflected ions is directed upstream instead of being directed back towards the shock \\citep{GTB82GRL}. However specularly reflected ions at shocks only slightly below $\\theta_{Bn}=45^{\\circ}$ may still re-encounter the shock front during the course of their gyromotion unless the shock angle is actually lower than $\\sim 40^{\\circ}$ \\citep{STG83JGR}. Only then are these reflected ions able to escape upstream, and by backstreaming against the solar wind flow they excite upstream waves (e.g. \\citep{Que88JGR}). As these low-frequency waves grow to large amplitudes they interact with the incoming solar wind, giving quasi-parallel shocks a more extended and visibly more complex and dynamic structure than quasi-perpendicular ones. Clearly there is more to collisionless shock dissipation than simple specular reflection. In the case of curved shocks such as the Earth's bow shock this picture is further complicated by ions streaming from other parts of the shock which also contribute to instabilities near the quasi-parallel shock. \\citet{BLS05SSR} provide a recent review of observations of quasi-parallel shock structure and processes (see also \\citep{Bur95ASR}).\n\nObservations have therefore shown that the bow shock can significantly deviate from the textbook picture of a locally planar and stationary structure and that it may eventually reform. However the scales and complexity of shock reformation combined with the limited number of measurement points have not provided so far direct evidence by following sequentially the reformation of a shock front.\n\nThis paper presents a case study of a crossing by the Cluster fleet \\citep{EFG01AG} of a shock with $\\theta_{Bn}\\approx 45^{\\circ}$. The region upstream of the shock exhibits a low-frequency quasi-monochromatic wave as more commonly observed upstream of quasi-parallel shocks. The wave cycle nearest to the shock ramp is found to steepen and grow into a pulse-like structure which we argue corresponds to the formation of a new shock-ramp.  The four Cluster spacecraft were able to sequentially observe this process which we interpret to be shock reformation.\n\n\n\\section{Data}\n\nThis study concentrates on a shock crossing on 16 March 2005 around 1530 UT by Cluster occurring during a data burst mode interval. The quartet was in a tetrahedron configuration with inter-spacecraft separations of the order of 1300 km.\nThe magnetic field data from the Flux-Gate Magnetometer (FGM) \\citep{BCA01AG}  used in this study was averaged at a resolution of one vector per spin-period (4 s) or 10 vectors per second, averaged from a 67 vectors\/second sampling frequency in burst mode. The electric field data from the Electric Field and Wave (EFW) \\citep{GBH97SSR} instrument used here has a temporal resolution of approximately 2 ms. Ion data was provided by the Hot Ion Analyzer (HIA) of the Cluster Ion Spectrometer (CIS) \\citep{RAB01AG} which measures fluxes of positive ions irrespective of species in the energy range 0.005-26 keV\/e and takes a spin period to build a full 3d distribution (transmitted to the ground every spin in burst mode) and has an angular resolution of $22.5^{\\circ}\\times 22.5^{\\circ}$. However during this interval the instrument was in a magnetospheric mode not optimized for the solar wind, and CIS\/HIA data were available on spacecraft 1 and 3 only. Electron data came from the Low Energy Electrostatic Analyzer (LEEA) of the Plasma Electrons And Current Experiment (PEACE) \\citep{JAB97SSR}. In this burst mode, a 3d distribution with reduced polar resolution (3DX1) was transmitted to the ground every spin, and consists of 26 energy levels in the range 0.007-1.7 keV, 6 polar and 32 azimuthal angular bins. This data was then reduced on the ground to pitch-angle distributions using high-resolution FGM data, and corrected for the spacecraft potential effect using spin-resolution EFW data. Magnetic field and particle from the MAG and SWEPAM instruments onboard the Advanced Composition Explorer (ACE) spacecraft \\citep{SFM98SSR} were used to compute the upstream solar wind parameters. The time delay taken by the solar wind to travel from the Lagrange point to the Earth's bow shock was taken into account assuming it moves with a constant velocity parallel to the GSE $x$-axis. \n\n\n\\section{Shock observations}\n\n\\subsection{Global shock properties and Cluster configuration}\n\n\\begin{table*}[ht]\n\\caption{\n  Main shock parameters at the asymptotic upstream and downstream locations. Upstream parameters were taken from ACE data.\n  Frame-dependent quantities are given in the Normal Incidence frame (the shock rest frame where the upstream \n  solar wind velocity is directed along the shock normal). \n}\n\\begin{tabular}{llll}\n    \\tableline\n    Parameters                                               & units          & upstream                   & downstream               \\\\  \\hline\n    $B$                                                      & nT & 9     & 27   \\\\ \n    $n_p$                                                    & cm$^{-3}$      & 5.6                      & 19.6     \\\\\n    Proton ram energy                   & eV             & 946                    & 78                    \\\\ \n    $T_p$                                                    & eV             & 4                      & 198                   \\\\ \n    $\\beta_p$                                                & --             & 0.1                      & 2.2                     \\\\ \n\n    $T_e$                                                    & eV             & 17                     & 54                    \\\\ \n    $\\beta_e$                                                & --             & 0.5                      & 0.6                     \\\\  \n    Proton gyrofrequency, $f_{gp}$                           & Hz             & 0.14                      & 0.41                     \\\\ \n    Proton inertial length, $\\lambda_{p}$                   & km             & 96                     & 51                    \\\\ \n    Thermal proton gyroradius, $\\rho_{p}$                    & km             & 22                     & 54                    \\\\ \n    Convected proton gyroradius, $v_{p1}\/\\Omega_{gp}$        & km             & 493                    & 167                   \\\\ \n    Specularly-reflected proton gyroradius                   & km             & 4536                   & --                       \\\\ \n    Phase-standing whistler wavelength, $\\lambda_w$          & km             & 75                     & --                       \\\\ \n    $\\theta_{Bn}$                                            & deg            & 47                     & 77                    \\\\ \n    Alfv\\'en velocity, $c_A$                                 & km\/s           & 77                     & 122                   \\\\ \n    Alfv\\'en Mach number, $M_A$                              & --             & 5.5                      & 1.00                     \\\\ \n    Magnetosonic Mach number, $M_{ms}$                       & --             & 4.6                      & 0.8                     \\\\ \n    \\\\ \\tableline\n\\end{tabular}\\label{tab:shock_params}\n\\end{table*}\n\n\n\\begin{figure}\n  \\includegraphics[width=18pc]{2009ja014268-p01_orig.eps}\n  \\caption{Shock crossing configuration. Top panels show the upstream magnetic field lines and the location of the crossing (black square) in the GSE $xy$ and $xz$ planes. The main panels show the asymptotic magnetic fields, solar wind velocity in the normal incidence shock frame, and projection onto the coplanarity (NL) plane of spacecraft locations and direction of the upstream wavector. The grey area corresponds to the main shock ramp, and distances are normalized to the upstream proton inertial length $\\lambda_p$. Spacecraft in this plane are approximately aligned along the shock normal for C1 (black), 3 (green) and 4 (blue), while C2 (red) crosses the shock nearly simultaneously to C1 but approximately $10 \\lambda_{p}$ away along the shock front. The spacecraft travel from downstream to upstream.}\n  \\label{fig:config}\n\\end{figure}\n\nThe solar wind as monitored by ACE, time-shifted by the solar wind travel time from the Lagrange point to the shock, remains steady and quiet during the whole time interval and for nearly half an hour before, in particular with a velocity around 400 km\/s and no important change in magnetic field direction. Because of the presence of large amplitude fluctuations extending far upstream of the shock (and of the CIS instrument mode), the asymptotic upstream field and particle parameters are estimated from the ACE measurements (taking into account the solar wind travel time to the shock).\n\nThe crossing occurs at $(12.7, 0, 4.6)Re$ (GSE), from downstream to upstream (outbound crossing). The shock timing analysis (which assume a planar surface moving at constant speed) \\citep{Sch00Book} yields a normal $\\hat{\\mathbf{n}} = (0.93, -0.12, 0.35)$ (GSE), a shock angle between the upstream magnetic field and the normal $\\theta_{Bn} = 45^{\\circ}$ and a velocity along normal of -13 km\/s in the spacecraft frame. The Abraham-Shauner method \\citep{Sch00Book} which makes use of the MHD jump (Rankine-Hugoniot) conditions for the magnetic and velocity fields (and assumes as well a planar surface and shock stationarity) yields a similar result, $\\hat{\\mathbf{n}} = (0.93, -0.10,  0.35)$ and $\\theta_{Bn} = 46.5^{\\circ}$. The shock is therefore within experimental errors at the formal limit of quasi-perpendicular and quasi-parallel shocks, and we shall generically call it an oblique shock.\n\nThe projected spacecraft locations onto the coplanarity plane show that Cluster-1 (C1), 3 and 4 are approximately aligned along the shock normal while C1 and 2 are perpendicular to it and cross the ramp nearly simultaneously (fig.\\ \\ref{fig:config}). The shock is crossed first by C4, then by C1 and 2 together and finally by C3.\n\nThe shock is super-critical with an Alfv{\\'e}n Mach number $M_A=5.5$. The proton thermal to magnetic pressure ratio for this shock is low, $\\beta_p=0.1$. Other shock and plasma parameters are summarized in Table \\ref{tab:shock_params}.\n\n\\begin{figure}\n  \\includegraphics[width=18pc]{2009ja014268-p02_orig.eps}\n  \\caption{4s-resolution magnetic field intensity (top panel) and components in shock normal coordinates on all four spacecraft. Data are time-shifted to allow comparisons of the shock structure and upstream wavetrain measured by the spacecraft. Time information is translated into distance assuming the whole structure travels at the constant shock velocity estimated from the timings.}\n  \\label{fig:lowresshock}\n\\end{figure}\n\n\n\\subsection{The upstream low-frequency wave}\n\nAn upstream ultra-low frequency wave (ULF) with a period $\\sim$42 s ($f=24$ mHz) in the spacecraft frame is observed for over 10 minutes from the ramp (fig.\\ \\ref{fig:lowresshock}). The time-series appear quasi-monochromatic at 4 s-resolution, but considerable broadband higher-frequency fluctuations are seen in the higher-resolution data. The ULF oscillations can be seen in the magnetic field intensity as well as all three components and strongly increase in amplitude in the vicinity of the ramp. For instance, at the shock ramp the magnetic field fluctuations along the shock normal reach $\\delta B_N \/ B_N\\approx 1.7$, and the fluctuations in the non-coplanar component of the magnetic field $\\delta B_M$ are nearly as large as the overall difference between upstream and downstream magnetic field component in the maximum variance direction $B_L$. \nThe general shock profile, ULF wave characteristics and its relative location with respect to the shock seem remarkably similar on all four spacecraft despite the large spacecraft separation and temporal spread of the crossings (the first and last shock crossings are approximately 80 s apart). This gives to the wave an unexpected appearance of phase-stationarity with respect to the shock (to be discussed in the final section). Oscillations extend over one or two periods downstream of the ramp with a shorter period $\\sim$36 s, although it is not entirely clear whether they correspond to a transmitted wave or to the overshoot-undershoot cycle.\n\nMinimum Variance Analysis (MVA, see e.g. \\citep{SR09SSR}) applied suggests that the wavevector is roughly aligned with the magnetic field (the deviation, $\\theta_{kB}\\approx 24^{\\circ}$, is mainly in the out of coplanarity plane direction), \n$\\mathbf{k}\/k = \\pm (0.63, 0.38, 0.67)$ (NML). \nThe wave has a high degree of polarization (0.92), is left-hand polarized (with respect to the magnetic field) in the spacecraft frame with ellipticity $\\varepsilon = -0.81$ and has a low compression ratio $C_e=\\left(\\delta n\/n_0\\right)^2(B_0^2\/\\delta B^2)=0.12$ (tab. \\ref{tab:wave_params}).\n\nMore properties can be derived from the time delays between spacecraft assuming a planar wavefront. Although in general for a monochromatic wave the time delays can only be determined modulo the wave period, we find that the smallest delays determined from the time-series provide a direction close to that of the MVA, namely \n$\\mathbf{k}\/k = (0.66,  0.37,  0.65)$ in NML coordinates\n(2$^{\\circ}$ away from the MVA result). The corresponding phase velocity in the spacecraft frame is 142 km\/s which yields a wavelength of 5930 km. From the wavevector and the mean upstream plasma velocity, the frequency in the plasma frame is found to be 0.01 Hz and the plasma frame phase velocity is 56 km\/s, lower than but comparable to the upstream Alfv\\'en speed (76 km\/s). In this frame, the wave propagates against the solar wind flow and therefore is right-hand polarized. The properties summarized in table \\ref{tab:wave_params} such as wavelength $\\sim R_E$, upstream propagation direction and right-hand polarization are fairly typical of ULF waves studied using ISEE \\citep{HR83JGR} or Cluster \\citep{EBD02GRL,AHL05JGR} datasets, and are thought to result from an electromagnetic ion\/ion right-hand resonant instability due to ions backstreaming from the shock. Finally, the phase velocity along the shock normal in any shock rest frame is estimated to be $\\mathbf{v}_{\\varphi}^{\\mathrm{(shock)}}\\cdot\\hat{\\mathbf{n}}\\approx -80$ km\/s showing that as expected the wave is not phase-standing but convected towards the shock by the solar wind.\n\nApplying MVA to shorter time intervals (2 wave periods) however reveals that these properties change closer to the shock front. As shown in fig.\\ \\ref{fig:running_mva} the wavevector approximately aligns itself with the shock normal, the polarization becomes less circular and the compression ratio increases. These changes appear during intervals which do not yet include the shock ramp, corresponding to specific properties of the cycle nearest to the ramp which may be affected by the shock foot and reflected ions. Since the wave is assumed to be planar and has a high degree of polarization, an effect of the alignment of the wavevector with the shock normal should be to diminish the perturbations due to the wave to the planarity of the shock surface. In addition the oscillations seem to remain in-phase with the shock as if the ramp was part of one cycle and other cycles were phase-standing next to it. Indeed, the timings of the pulse nearest to the shock ramp confirm the wavevector alignment with the normal (consequently $\\theta_{kB}\\approx 40^{\\circ}$) and a velocity nearly identical to that of the shock, about -13 km\/s in the spacecraft frame. This yields a wavelength an order of magnitude lower than further upstream, $\\lambda\\approx 500 km \\approx 5 \\lambda_p$. \n\nBesides changes in wavevector and velocity, the wave experiences a strong amplification near the shock. The cycle standing nearest to the ramp indeed nearly reaches shock-like amplitudes and displays an interesting behaviour detailed in the next section.\n\n\n\\begin{table*}[ht]\n\\caption{Properties of the upstream ultra-low frequency wave in spacecraft and plasma frames.}\n\\begin{tabular}{lll}\n    \\tableline\n              & Spacecraft & Plasma \\\\\n    \\tableline\n    Frequency $\/f_{gp}$ & 0.14  &  0.07  \\\\\n    Wavelength $\/\\lambda_p$ & 62 & 62 \\\\\n    Phase velocity $\/c_A$ &  1.8 & 0.74 \\\\\n    Polarization degree & 0.92 & 0.92 \\\\\n    Polarization & Left-hand & Right-hand \\\\\n    Ellipticity & -0.81 & + 0.81 \\\\\n    Compression ratio & 0.12 & 0.12 \\\\\n    \\tableline\n\\end{tabular}\\label{tab:wave_params}\n\\end{table*}  \n\n\n\\begin{figure}\n  \\includegraphics[width=18pc]{2009ja014268-p03_orig.eps}\n  \\caption{Wave properties in spacecraft frame derived from MVA on 84s intervals of 4s resolution data. The top panel shows the magnetic field intensity. Next panel shows the polar angle $\\theta$ between $\\mathbf{k}$ and $\\hat{\\mathbf{n}}$, and $\\phi$ which is the azimuthal angle in the LM plane with respect to the L axis. The dotted lign indicates $\\theta_{Bn}$. Lower panels show the ellipticity and electron compression ratio.}\n  \\label{fig:running_mva}\n\\end{figure}\n\n\n\n\\subsection{The growing and steepening pulse upstream of the shock ramp}\n\n\\begin{figure}\n  \\includegraphics[width=19pc]{2009ja014268-p04_orig.eps}\n  \\caption{Main shock ramp and upstream pulse, shown with 4 s resolution FGM data. The pulse growth rate is approximately $0.03f_{cpu}$.}\n  \\label{fig:pulse_growth}\n\\end{figure}\n\nAs noted in the previous paragraph, upstream magnetic field fluctuations reach their largest amplitude at the wave cycle nearest to the shock front. This large amplitude pulse-like structure is crossed by the spacecraft in the same order as the shock ramp (C4 first, followed nearly simultaneously by C1 and C2 and then C3), showing that it is not a partial recrossing of the shock front but a distinct upstream structure. The time delay between the crossings of the shock ramp and the feature is nearly the same for all four spacecraft (about 20s), suggesting that this structure extends parallel to the shock ramp and remains at a constant distance from it.\n\nAs shown in fig.\\ \\ref{fig:pulse_growth} the magnetic field intensity of the structure is the lowest at the crossing by C4, larger at C1 and C2 (and slightly more so at C2 than C1) and largest at C3 where its amplitude is comparable to that of shock itself. The same observation applies to $B_L$. The structure is therefore growing in time. An exponential fit to the peak amplitudes yields a growth rate of $\\gamma\\approx 4\\cdot 10^{-3}$ Hz $\\approx 0.03 f_{gpu}$. The amplitudes are equally well fit by a linear curve with slope $0.011B_u$ s$^{-1}$ $\\approx 0.08B_uf_{gpu}$. Both models estimate that it takes up to $\\sim 35$ upstream proton gyroperiods for the pulse to grow to shock-like amplitudes.\n\n\\begin{figure}\n  \\includegraphics[width=19pc]{2009ja014268-p05_orig.eps}\n  \\caption{High-resolution (10 vectors per-second) magnetic field measurements of the growing structure in shock NML coordinates. The structure is not only found to grow in amplitude but also to steepen while emitting whistlers, and measured electric field intensities correspondingly increase to reach ramp-like values.}\n  \\label{fig:bump_highres}\n\\end{figure}\n\nHigh-resolution magnetic field data show that besides growing in amplitude the structure is found to steepen (fig.\\ \\ref{fig:bump_highres}). The steepening is most visible in $B_L$ and occurs on the upstream edge of the pulse (apart perhaps from C2 on which the \"downstream\" side of the pulse seems at least as steep as its upstream side). As the steepening proceeds quasi-periodic whistler-like fluctuations (at $\\approx 0.15$ Hz, best seen on $B_M$) are emitted upstream and the measured electric field magnitude $(E_x^2+E_y^2)^{1\/2}$ increases too and reaches on C3 values comparable to those in the ramp. Furthermore, high-resolution data reveal that the pulse is seen slightly earlier on C2, consistent with its position slightly upstream of C1 as sketched in Fig 1. However, that data also shows the pulse's growth and steepening to be more advanced at C2. This shows that at this separation scale the pulse's structure and growth is not perfectly homogeneous along the shock plane.\n\nFinally one notes on C1 and C2 in between the pulse and the shock ramp localized dips in $B_L$ reaching negative values, similar to that observed  during the reformation cycle in 1d \\citep{Bur89GRL,WTO90JGR} and 2d \\citep{SFK93JGR} quasi-parallel shock hybrid simulations.\n\n\n\\subsection{Particles and cross-shock potential}\n\n\n\\begin{figure}\n  \\includegraphics[width=19pc]{2009ja014268-p06_orig.eps}\n  \\caption{Energy spectrograms of ions from CIS\/HIA, summed over all angles. Upstream, both solar wind and energetic ions are strongly affected by the wave. At the shock, a distinct population of reflected ions around 3 keV is observed at the shock ramp on C1 but not on C3, where it is observed on the upstream steepened structure instead.}\n  \\label{fig:hia_spectro}\n\\end{figure}\n\nIf the growing feature is indeed becoming a new shock front, then it should affect the heating and reflection of incoming solar wind particles and correspondingly develop an electric potential jump. fig.\\ \\ref{fig:hia_spectro} shows magnetic intensity profile and ion energy spectra (from CIS\/HIA, all species and summed over all view angles) on C1 and C3. Two distinct ion populations are clearly observed in the solar wind. The lowest energy one ($\\approx 1$ keV) is the incoming solar wind beam which undulates under the effect of the wave, yielding large oscillations of the velocity moment. It seems however that the wave results in little or no ion heating (which is nevertheless difficult to check quantitatively in the absence of reliable temperature estimates due to the particular CIS instrument mode). \n\nThere is also a higher energy ($\\approx 3$ keV), less dense but hotter population which is strongly modulated by the wave suggesting that some of these energetic ions could be trapped by the large-amplitude wave. The highest count rate of energetic ions on C1 appears during the shock ramp crossing around 15:28:40 UT. This distinct ion population has an energy around 3 keV and should consist of gyrating specularly-reflected protons. A similar but lower count-rate group of ions is observed on the next peak of magnetic intensity. The situation on C3 is slightly different however. There is no distinct energetic population of energetic gyrating ions observed at the \"old\" shock ramp. These are only seen at the upstream steepened structure, as if it became a new shock front where most of the ion reflection occurs. One notes however that there is little appreciable ion heating between this structure and the old ramp, suggesting that some solar wind plasma is not processed by a full ramp structure but caught in between. \n\n\\begin{figure}\n  \\includegraphics[width=19pc]{2009ja014268-p07_orig.eps}\n  \\caption{Magnetic field intensity (top) and estimated electric potential (bottom) time-shifted in order to make the magnetic ramps coincide. The downstream cross-shock potential values are very similar on all four spacecraft. However the location of main potential jump seems to move upstream of the main magnetic ramp.}\n  \\label{fig:cspot}\n\\end{figure}\n\n\nSpecular ion reflection being associated to the cross-shock potential at least part of the potential jump must occur at the new front. One of the most reliable ways to estimate the electric potential across a shock is to use electron data combined with Liouville mapping (\\citep{LSF07JGR} and references therein). Based on assumptions of conservation of electron energy in the de Hoffmann-Teller frame and first adiabatic invariant, this technique relates the changes in electron velocity distributions between two locations to the unknown potential difference. A variant of this technique is used which takes into account the field maxima in between the two locations \\citep{LSF07JGR}.\n\nThe estimated cross-shock potentials on all four-spacecraft in the de Hoffmann-Teller frame are shown in fig.\\ \\ref{fig:cspot}, where time-series have been shifted in order to make the main magnetic ramps coincide. Their downstream values are remarkably similar on all four spacecraft despite the shock non-stationarity. However the location of the main potential jump with respect to the main magnetic ramp seems to differ from spacecraft to spacecraft. During the C4 crossing, when the new ramp was just starting to form, the potential jump occurs at the magnetic ramp. It is then observed further upstream on C1 and C2 crossings, and near the newly formed ramp on C3. \n\n\n\\subsection{Large-amplitude downstream perturbations and indications of a reformation cycle}\n\n\\begin{figure*}\n  \\includegraphics[width=0.7\\textwidth]{2009ja014268-p08_orig.eps}\n  \\caption{Time-shifted high-resolution magnetic field intensity and electric field ($|E_{xy}|=(E_x^2+E_y^2)^{1\/2}$). Spacecraft are ordered in the order by which they cross the shock. Besides the growth of the upstream pulse which becomes a new ramp, data show strong perturbations approximately one wavelength downstream of the main ramp. This can be interpreted as remnants of an old shock front from a previous reformation cycle.}\n  \\label{fig:cycle}\n\\end{figure*}\n\nLarge perturbations are also found downstream of the main shock ramp as shown in fig. \\ref{fig:cycle}, localised at approximately the same distance from the main ramp as the upstream pulse. On all four spacecraft a depression in the mean magnetic field intensity is observed. Large magnetic field and electric field fluctuations are present within this depression. The amplitudes of the electric field fluctuations are comparable to or even larger than at the main ramp. The depth of the depression and amplitude of the fluctuations are also decreasing in time, being the lowest on C3 which is the last to cross the shock. The decrease occurs on time scales comparable to the growth of the upstream pulse. These perturbations can be interpreted as remnants of an older shock ramp which decays in time, showing that the previously described formation of a new ramp is not an isolated event but part of a quasi-periodic reformation cycle. Based on the growth rate of the upstream pulse and the decay of downstream perturbations, the reformation period can be estimated as several tens of upstream proton gyroperiods, and could be a few periods of the upstream low-frequency wave. This is significantly longer than the period of variations of the downstream ion populations ($\\sim 2 f_{pu}^{-1}$) observed by \\citet{TGB90JGRa}, although the shocks studied by these authors have slightly different parameters and in particular higher Mach number.\n\n\n\n\\section{Summary and discussion}\n\nThe shock analysed in this paper is oblique ($\\theta_{Bn}\\approx 45^{\\circ}$), moderate Mach number ($M_A\\approx 5.5$) and low-$\\beta$ ($\\beta_i\\approx 0.1$).\n\nUpstream of the shock a long-wavelength ($\\lambda\\approx 62\\lambda_p$), low-frequency ($f\\approx 0.07f_p$ in the plasma frame) and right-hand polarized in the plasma frame quasi-monochromatic wave propagates against the solar wind flow approximately parallel to the upstream magnetic field ($\\theta_{kB}\\approx 24^{\\circ}$). Its properties are consistent with a magnetosonic-like wave excited by a weak ($n_b\\ll n$) and cool ($v_{Tb}<v_b$) ion beam backstreaming from the shock through an electromagnetic ion\/ion right-hand resonant instability \\citep{Gar91SSR}. Such an instability has its maximum growth rate for field-aligned wavevectors and obeys the linear dispersion relation in the cold plasma approximation $\\omega = k_{\\parallel}v_b - \\Omega_p$ (real part). In the low frequency limit $\\omega\\ll \\Omega_p$ one therefore has $k_{\\parallel}\\approx\\Omega_p\/v_b$. Assuming that the energetic ion population seen in fig.\\ \\ref{fig:hia_spectro} ($\\sim$3 keV in the spacecraft frame) corresponds to the backstreaming ions yields an estimate for the phase velocity $\\omega\/k_{\\parallel}\\approx (\\omega\/\\Omega_p)v_b\\approx 80$ km\/s in good agreement with the timing results, but the wavelength is found to be $\\lambda\\approx 8300$ km which is larger than the observed wavelength of 5900 km. It is therefore likely that the 3 keV population (which actually spreads from 2 to 10 keV) is modulated by the wave but not its source. Instead the wave may have been generated further upstream by a beam of ions coming from another part of the shock. Indeed the wave properties described in this paragraph are also similar to those of ULF waves in the ion foreshock \\citep{HR83JGR,EBD02GRL}, waves which are generally not observed simultaneously to the field-aligned beams but in association to diffuse-ions or gyrating beams in cyclotron resonance with the wave \\citep{MML03PSS}. The observed high energy ion population modulated by the wave is therefore more likely to be either a gyrating beam of reflected ions which managed to escape from the shock or gyrating ions trapped by the wave. \n\n\\begin{figure}\n  \\includegraphics[width=12pc]{2009ja014268-p09_orig.eps}\n  \\caption{Illustration of the wave refraction near the shock. The wavefronts (dashed lines) become nearly parallel to the shock surface (thick grey line), which requires a lower phase velocity in the shock frame at location 1 (near the shock) than at 2 (further upstream).}\n  \\label{fig:alignment}\n\\end{figure}\n\nDue to its small phase velocity the wave is swept back by the solar wind, and closer to the shock the wave properties change significantly. The wavevector aligns itself with the shock normal, which could be due to an increase of refraction index in the foot region. A similar alignment was found in the 2D hybrid simulations of \\citet{SFK93JGR}, which they attributed to an increase in density of the diffuse ions near the shock. This alignment makes the perturbations of the shock due to the wave more homogeneous along the shock surface. It is this greater deviation from parallel propagation which allows the wave to become more compressional and steepen, an effect which was conjectured by \\citet{HKT87JGR} to happen as ULF waves are convected into areas of the foreshock with higher densities of diffuse ions (see also \\citep{BCO06JGR,EBL05JGR}). Besides, the wave has all the appearance of phase-standing with respect to the shock. From a velocity identical to that of the shock in this region, the wavelength is found to be 10 times shorter than further upstream. This slowing down of the wave in the shock frame is entirely consistent with its wavevector alignment with the shock normal, as illustrated by fig.\\ \\ref{fig:alignment}.\n\nIn addition to refractive effects an instability could be operating near the shock front, partly accounting for the changes in wave properties and its strong amplification. This could involve, as dicussed by \\citet{WTO90JGR}, a relatively dense and cold beam of specularly reflected ions observed near the shock front during the reformation process in the simulations, or an instability localized at the shock front resulting from the interaction between upstream and some of downstream heated ions. This \"interface instability\" generates waves on the magnetosonic branch with wavelength intermediate between whistlers and ULF waves. For $M_A=5$ the maximum growth rate corresponds to $k\\lambda_p\\approx 0.5$ or $\\lambda\\approx4\\pi\\lambda_p$. This wavelength ($\\lambda\\approx 5\\lambda_p$) estimated near the shock is smaller, but still in the range where the growth rate is close to its maximum value. The instability is weakly dependant on $\\theta_{Bn}$ and tends to have its wavector aligned with the shock normal. However the exact mechanism(s) operating on the wave near the shock front is (are) certainly more complicated since we are dealing with very large-amplitude (nonlinear) waves and a highly inhomogeneous situation. One can however note that because the wavelength near the shock is about 6 times the theoretical wavelength of the phase-standing whistler the mechanism leading to reformation is distinct from the whistler-mediated reformation discussed for oblique and quasi-perpendicular shocks by \\citet{BW72JGR,KLS02POP,SB07POP}.\n\nDue to the strong amplification of the wave near the shock, its last cycle before the shock acquires an amplitude comparable to that of the shock itself. This pulsation grows in time, steepens and in the process it emits whistlers, reflects more and more ions and contributes more significantly to the cross-shock potential jump. Reasons for this growth and steepening can be the change in refractive index, a localized instability near the shock front such as the interface instability or nonlinear dynamics of the pulse and its interaction with the shock. Furthermore the amplification mechanism may not operate steadily in time. The negative dips in $B_L$ seen only on C1 and C2 during the intermediate growth stage of the pulse may indeed be the signature of a transient dense ion beam reflected from the shock at this stage of the shock reformation (e.g. as in the simulations of \\citet{TWO90JGR}). At least this indicates an unusual and transient processing of ions in between the main ramp and the pulse during the pulse growth.\n\nWhatever mechanism operates here, these observations give strong evidence that the wave cycle standing nearest to the shock is forming a new shock front on a time scale $\\sim 35f_{gpu}^{-1}$ or less, comparable to that observed in hybrid simulations of quasi-parallel shocks (e.g. \\citep{Bur89GRL,TWO90JGR,SFK93JGR}). The reformation scale (distance between the shock front and pulsation) is $\\sim 5\\lambda_p$, also in good agreement with simulation results. Assuming that the real shock front thus jumps forward such a distance in a time of $35f_{gpu}^{-1}$, its velocity is found to be only a few km\/s larger than that of the \"former\" shock front in the solar wind frame, which does not induces a significant change in effective Mach number. At approximately the same distance from the main shock ramp but on the downstream side there are magnetic and electric field fluctuations with amplitudes comparable to those at the main ramp. These can be naturally interpreted as remnants of an older ramp which decays on a time scale of at least a few wave periods, which is comparable to the one of the new front formation. These remnants show that the observed reformation is not an isolated event but part of a quasi-periodic reformation cycle.\n\nThe reformation process appears quite coherent along the shock surface at the spacecraft separation scale, $\\sim 10\\lambda_p$, thanks to the wavevector alignment with the shock normal. Some differences in the pulse shape between C1 and C2 measurements are nevertheless observed. The cause of this inhomogeneity could be the finite transverse correlation length of the upstream ULF wave, which is typically of the order of 1$R_E$ according to \\citet{LR90JGR} or between 8 and 18 $R_E$ according to \\citet{AHL05JGR}. This is however a bit large to account for the inhomogeneity, even when projected on the shock front. It is nevertheless possible that the ULF waves are unstable to transverse perturbations above some threshold, which would reduce their perpendicular correlation length as their amplitude grows. Another reason could be fluctuations or ripples of the shock surface and the inhomogeneity of the ion reflection\/wave amplification process. The scale is indeed more consistent with the ripple wavelength (a few tens of $\\lambda_p$) estimated by \\citet{MBH06JGR} at the quasi-perpendicular bow shock. In any case this inhomogeneity along the shock front is small since the differences between C1 and C2 observations are much smaller than differences with the other two spacecraft. To a first approximation the reformation thus occurs coherently along the shock front on scales of at least $10\\lambda_p$.\n\n\n\\section{Conclusion}\n\nThe global picture is thus in general agreement with the scenario of quasi-parallel shock reformation proposed by \\citet{Bur89GRL}: as some of the specularly reflected ions escape far upstream they generate waves on the magnetosonic branch which are then convected back into the shock and destabilize it. The major difference between the 1d simulations which inspired this scenario and the observations lies in the changes in the wave properties as it nears the shock, where wave refraction, an additional localized instability mechanism such as the interface instability and probably nonlinear phenomena participate in deflecting, slowing down and amplifying the wave cycle which ultimately forms a new shock front. The wave refraction results in a more coherent reformation along the shock surface. On a longer time-scale, this reformation process has no reason to stop unless there is a major change in the upstream conditions and may be repeated quasi-periodically. In this picture the elementary reformation cycle consists of a shock front and wave locally receding at the same speed while the pulse nearest to the \"old\" shock grows to actually become the \"new\" shock front upstream of the previous one, the ramp effectively making a jump forward. The old shock slowly vanishes and a new pulse grows upstream of the new one, repeating the reformation cycle.\n\nFinally one may list two key elements which conspire to create this particular shock structure and dynamics lending themselves unusually well to analysis.\n\nFirst, there is the shock angle $\\theta_{Bn}\\approx 45^{\\circ}$. It means that relatively small changes in this angle have important consequence on the kinematics of the reflected ions, which either cross the shock after half a gyration and thermalise downstream as in quasi-perpendicular shocks or escape upstream and feed ULF waves as in quasi-parallel shocks. Since on average very few of the reflected ions are expected to escape upstream, this angle may be the reason why there is such a large gradient in the reflected ion density near the shock front and therefore why the wave undergoes such large changes in this region. In some sense this leads to a simpler and compacted version of lower-angle shocks. For instance the growing pulse shares some characteristics with SLAMS \\citep{SB91GRL}. Indeed evidence suggests that SLAMS grow very rapidly, picking up energy from shock-produced energetic ions, and very possibly only in close proximity to the shock \\citep{LHD08JGR}. In addition, they are well-defined entities that propagate from upstream into the nominal shock and give rise to the reformation of the local shock transition. As such, they are\nmanifestations of the dynamic, structured nature of collisionless shocks under oblique geometries. However their growth mechanism differs from that of the pulse described in this paper, in that they are found in more turbulent ULF wave fields and grow off of gradients in diffuse ion populations rather than the quite mono-energetic ions observed here.\n\nSecond, there is the low $\\beta_i$. A consequence of this is that specularly reflected ions form a relatively cold beam well distinct from the background ions, helping to shape unstable ion distributions. But the low $\\beta$ also makes the specular reflection itself more delicate by making it more difficult for the shock to adjust the electric potential in order to reflect the right amount of ions required for the dissipation of the incoming solar wind kinetic energy. In the zero temperature limit specular reflection even results in a \"all or nothing\" situation which clearly cannot be steadily sustained, similarly to the 100\\% ion reflection or transmission occuring in the high Mach number perpendicular shock simulations of \\cite{Que85PRL}. The excess of reflected ions near the shock front is again a likely possibility to explain the wave changes in this region, leading to the formation of a new shock front. \n\nThese two factors show that the shock is in a situation where small changes in $\\theta_{Bn}$ or electric potential have major consequences, making the shock unstable and poorly adaptable to changes in the incoming plasma. As it is perturbed major variations in reflected ions strongly modify the energy flow and structure of the shock. The dynamical modifications of the shock scales and structure may then be viewed as a first order attempt by the shock to correct an inappropriate reflection of ions while trying to maintain on average the asymptotic downstream state.\n\n\n\\begin{acknowledgments}\nThe authors thank M. Fujimoto, S. P. Gary, T. Horbury and I. Shinohara for fruitful discussions.\nParts of this work were supported by a grant to Imperial College from the UK Science and Technology Facilities Council. BL also acknowledges partial support from DOE grant DE-FG02-07ER54941.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tMonolayer graphene is a thoroughly investigated material and boasts exotic applications due to its fundamentally unique physical electronic and optical properties \\cite{Novoselov2004}. For example graphene possesses a gapless Dirac-type band structure \\cite{Tombros2007}, high carrier mobilities and universal broadband optical conductivities \\cite{Koppens2011}. Recently, staggered 2D semiconductors materials \\cite{Castellanos-Gomez2017,Zhao2016} have also attracted intensive attention due to their unusual topological properties \\cite{Zhao2016,Grazianetti2016}. These materials include silicene \\cite{Molle2017}, germanene \\cite{Vogt2012}, stanene \\cite{Davila2014}, and plumbene \\cite{Saxena2016}. Graphene in general shares some common properties with these materials. For example, these 2D materials have stable honeycomb lattice structures \\cite{Cahangirov2009,Cai2015} and are therefore called the graphene family or graphene's siblings. Second, due to larger ionic size of silicon, germanium and tin atoms, they have buckled structures and therefore possess large band gaps. Third, the strong spin-orbit interaction (SOI) is responsible for endowing mass to the Dirac fermions. The strengths of this interaction in silicene \\cite{Ezawa2012,Liu2011}, germanene \\cite{Vogt2012} and tinene \\cite{Xu2013}, are of the order of 1.55\\textendash7.9, 24\\textendash93, and 100 meV respectively. Further, a static external electric field can also tune the band gap for each spin and valley rendering the Dirac mass controllable in both the $K$ and $K'$ valleys. Under the influence of an electric field, the staggered materials are also known to exhibit different topological phase transitions \\cite{Ezawa2013}. Last, analogous to electronics and spintronics, valleytronics provides another degree of freedom (DOF) in these systems, the valley pseudospin, which can be used to store and carry information with implications for quantum information processing \\cite{Schaibley2016,Xiao2007}. These materials therefore provide an accessible playground for valleytronics and the possibility to realize novel tuneable magneto-optic (MO) devices \\cite{Xiao2012,Zeng2012}.\n\t\n\tGoos-H\\\"{a}nchen (GH) and Imbert-Fedorov (IF) shifts are special beam shift phenomena which occur when a finite extent polarized beam of light is incident on the interface of two dielectric media and the reflected beam undergoes longitudinal (parallel to the plane) and transverse (perpendicular to the plane) shifts, respectively \\cite{Goos1947,Imbert1972,Fedorov2013}. These effects are shown in Fig.~\\ref{fig1}. During the last decade, the GH shift has been extensively studied in many different systems, such as photonics \\cite{He2006,Soboleva2012}, plasmonics \\cite{Yin2004,Yin2006}, chiral materials \\cite{Huang2017}, metamaterials \\cite{Longhi2011} and quantum systems \\cite{Lee2014}. The potential applications of GH shift are in biosensors \\cite{Yin2006}, optical measurement and optical heterodyne sensors \\cite{Hashimoto1989}. For example, controlling the Fermi energy in THz region, L. Jiang \\emph{et. al} theoretically studied electrically tunable GH shifts of monolayer graphene \\cite{Fan2016}. The magnetic field and Fermi energy modulated giant quantized GH effect on the surface of graphene in the quantum Hall regime was recently predicted \\cite{Wu2017}. T. Tang \\emph{et. al} proposed an experimental scheme based on a prism-graphene coupling structure for MO tunable GH effect sensing \\cite{Tang2014,Tang12018}. Recently the GH shift on the surface of silicene \\cite{Wu2018}, TMDCs \\cite{Das2018} and Weyl semimetals \\cite{Ye2019} has also been investigated. From an application perspective, Farmani \\emph{et. al} designed a bidirectional and tunable graphene plasmonic switch in a modified Kretchmann configuration in THz range \\cite{Farmani2018}. In the THz to mid-infrared range, the observation of GH effect in graphene may find significant new and interesting applications in extremely sensitive optical sensors \\cite{Li2013,Li2014}. Many works are devoted to study the electronic analog of the valley and spin polarized GH shift in silicene and gapped graphene structures \\cite{Azarova2017, Wu2011}.\n\n     The novelty of this current work lies in obtaining large spin and valley resolved GH shifts simultaneously in staggered 2D materials in the presence of applied electric and magnetic fields, a scenario not comprehensively covered in previous works. The present study details the impact of magnetic field, chemical potential, incident frequency, electric field and incident angle modulation of the valley- and spin-polarized GH shifts in staggered 2D semiconductors materials.\n\n\n    Furthermore, the Brewster effect is a fundamental electromagnetic and optical phenomenon in which $p$ polarized light experiences zero reflection \\cite{Brewster1815,Lakhtakia1989}. Controlling the Brewster angle is extremely important in broadband devices such as the solid-state modulator \\cite{Chen2018}. We also show that the Brewester effect in the silicenic atomic layer is strongly influenced by magnetic fields. Last, we find that angular and spatial GH shifts can be significantly enhanced by tuning the chemical potential, incident beam frequency and the electric field which can render the system in various regimes such as the topological insulator (TI), valley-spin polarized metal (VSPM) and band insulator (BI).\n\t\n\t\\subsection{Model and Theory}\n\t\\begin{figure*}[!htb]\n\t\t\\centering\n\t\t\\begin{tabular}{cc}\n\t\t\t\\includegraphics[width=0.5\\linewidth]{GH1-v3.eps}\n\t\t\t\t\\includegraphics[width=0.5\\linewidth]{GH2-v3.eps}\n\t\t\\end{tabular}\n\t\t\\caption{Schematic representation of the beam reflection at a\n\t\t\t2D staggered material-substrate interface in the presence of an external electric and magnetic fields are shown for partial reflection (PR) and total internal reflection (TIR) conditions. The spatial and angular GH shifts for (a) PR and for (b) TIR. The incident, classically pridected and reflected beams are denoted by a, b and c respectively.}\n\t\t\\label{fig1}\n\t\\end{figure*}\n\tWe consider a well-collimated, monochromatic Gaussian beam of frequency $\\Omega$ with a finite beam width impinging from a medium of refractive index $n_{1}$ onto a staggered 2D semiconductor-substrate system as shown in Fig.~\\ref{fig1}. The incident angle is $\\theta_{1}$ at the interface with a medium of refractive index $n_{2}$. Consider that the $z$ axis of the laboratory Cartesian frame $(x,y,z)$ is normal to the air-staggered 2D monolayer-substrate interface (at $z=0$). A static magnetic field may be applied perpendicularly to the monolayer-substrate system in the $z$ direction. The wave vectors are $k_1$ and $k_2$ where $k_{n}=\\Omega\\sqrt{\\mu_{n}\\varepsilon_{n}}$, $Z_n=Z_0\\sqrt{\\mu_{n}\/\\varepsilon_{n}}$ and $Z_0=\\sqrt{\\mu_{0}\/\\varepsilon_{0}}$, where $\\mu_{0}$ and $\\varepsilon_{0}$ are the vacuum permeability and permittivity respectively and $n=$1, 2 denote the incident and refracted media.\n\tThe low-energy effective Dirac Hamiltonian for the staggered graphene system can be written as \\cite{Tabert2013}\n\t\\begin{equation}\\label{a11}\n\t\\hat{H}_{\\xi \\sigma}=\\hbar v_{F}(\\xi k_x \\hat{\\tau}_{x}+k_y \\hat{\\tau}_{y})-\\frac{1}{2}\\xi \\sigma \\Delta_{so}\\hat{\\tau}_{z}+\\frac{1}{2}\\Delta_{z}\\hat{\\tau}_{z}\\cdot\n\t\\end{equation}\n\tHere, $\\xi=\\pm 1$ denotes the $K$ or $K'$ valley, $v_{F}$ is the Fermi velocity, $\\Delta_{so}$ is the intrinsic spin-orbit coupling and $\\Delta_z=aE_z$ is the staggered electric potential. The vector operators $\\vec{\\sigma}=(\\hat{\\sigma}_{x},\\hat{\\sigma}_{y},\\hat{\\sigma}_{z})$ and $\\vec{\\tau}=(\\hat{\\tau}_{x},\\hat{\\tau}_{y},\\hat{\\tau}_{z})$ represent Pauli matrices of the true and pseudo spin lattice DOF respectively. For Landau level (LL) quantization imposed by an external magnetic field $B$, we introduce the Landau gauge so that the magnetic vector potential $A = (-yB, 0,0)$, and the energy spectrum take the form,\n\t\\begin{equation}\\label{a12}\n\tE(\\xi,\\sigma, n,t)=\\begin{cases}\n\tt\\sqrt{2 v_{F}^{2} \\hbar e B |n|+\\Delta_{\\xi\\sigma}^{2}}, & \\text{if $n\\neq 0$}.\\\\\n\t-\\xi \\Delta_{\\xi\\sigma}, & \\text{if $n=0$}.\n\t\\end{cases}\n\t\\end{equation}\n\tHere, $t=\\textrm{sgn(n)}$ denotes the electron\/hole band, $\\Delta_{\\xi\\sigma}=-\\frac{1}{2}\\xi \\sigma \\Delta_{so}+\\frac{1}{2}\\Delta_{z}$ and $n$ is an integer, the quantum number denoting Landau quantization.\tThe Fresnel reflection coefficients of the staggered 2D monolayer-substrate system in the presence of magnetic field have been spelled out in several previous works and can be expressed as \\cite{Kort-Kamp2015,Wu2017}\n\t\\begin{eqnarray}\\label{b1}\n\n\tr_{pp} &=& \\frac{\\alpha_{+}^{T}\\alpha_{-}^{L}+\\beta}{\\alpha_{+}^{T}\\alpha_{+}^{L}+\\beta}, \\\\\n\t\\label{b2}\n\tr_{ss} &=& -\\bigg(\\frac{\\alpha_{-}^{T}\\alpha_{+}^{L}+\\beta}{\\alpha_{+}^{T}\\alpha_{+}^{L}+\\beta}\\bigg), \\\\\n\t\\label{b3}\n\tr_{sp}=r_{ps}&=&\\frac{-2Z_{0}^{2}\\mu_{0}\\mu_{1}\\mu_{2}k_{1z}k_{2z}(\\sigma_{xy}^{antisym}+\\sigma_{xy}^{sym})}{Z_{1}(\\alpha_{+}^{T}\\alpha_{+}^{L}+\\beta)},\n\t\\end{eqnarray}\n\twhere,\n\t\\begin{eqnarray}\\label{b4}\n\n\t\\alpha_{\\pm}^{L} &=& (k_{1z}\\varepsilon_{2}\\pm k_{2z}\\varepsilon_{1}+k_{1z}k_{2z}\\sigma_{L}\/(\\varepsilon_{0}\\Omega)), \\\\\n\t\\label{b5}\n\t\\alpha_{\\pm}^{T} &=& (k_{2z}\\mu_{1}\\pm k_{1z}\\mu_{2}+\\mu_{0}\\mu_{1}\\mu_{2}\\sigma_{T}\\Omega), \\\\\n\t\\label{b6}\n\t\\beta &=& Z_{0}^{2}\\mu_{1}\\mu_{2}k_{1z}k_{2z}[(\\sigma_{xy}^{antisym})^2-(\\sigma_{xy}^{sym})^{2}] \\cdot\n\t\\end{eqnarray}\n\tHere, $k_{1z}=k_{1}\\cos(\\theta_{1})$ and $k_{2z}=k_{2}\\cos(\\theta_{2})$. The conductivities $\\sigma_{L}(\\sigma_{T})$ are the longitudinal (transverse) components. In fact, the Hall conductivity $\\sigma_{xy}$ has symmetric $\\sigma_{xy}^{sym}$ and asymmetric $\\sigma_{xy}^{antisym}$ parts. Here, we are interested only in the antisymmetric part of the conductivity since the symmetric $\\sigma_{xy}^{sym}$=0. The corresponding real and imaginary parts of the longitudinal and transverse magneto-optical conductivities of the graphene family computed at $T=0$ K are given by\n\t\\begin{widetext}\n\t\t\\begin{multline}\\label{a6}\n\t\t\\frac{\\textrm{${\\textrm{Re} \\above 0pt \\textrm{Im}}$}\\big\\} \\big(\\sigma_{xx}(\\Omega)\\big)}{\\sigma_{0}}=\\frac{2 v_{F}^{2}\\hbar e B}{\\pi}\\sum_{\\xi, \\sigma}\\sum_{m,n}\\frac{\\Theta(E_{n}-\\mu_{F})-\\Theta(E_{m}-\\mu_{F})}{E_{n}-E_{m}}\\\\\\times\\bigg[(A_{m}B_{n})^{2}\n\t\t\\delta_{|m|-\\xi,|n|}+(B_{m}A_{n})^{2}\\delta_{|m|+\\xi,|n|}\\bigg]\\bigg\\{\\textrm{${\\textrm{F} \\above 0pt \\textrm{G}}$},\n\t\t\\end{multline}\n\t\twhere, $\\sigma_{0}=e^2\/4\\hbar$, $F=\\Gamma\/\\bigg((\\hbar\\Omega-(E_n-E_m))^{2}+\\Gamma^{2}\\bigg)$ and $G=\\big(\\hbar\\Omega-(E_n-E_m)\\big)\/\\bigg((\\hbar\\Omega-(E_n-E_m))^{2}+\\Gamma^{2}\\bigg)$. In these expressions, the Kronecker deltas ensure the rules for electric dipole transitions between the LL's are satisfied. The absolute values of $m$ and $n$ inside the Kronecker deltas generalizes the case in Ref. \\cite{Tabert2013} to both positive and negative chemical potentials $\\mu_{F}$, \\emph{i.e.} n and p-type 2D materials.\n\t\tThe Heaviside functions $\\Theta(E_{n}-\\mu_{F})$ ensure that transitions across the Fermi level are possible, hence they effectively account for the so called Pauli blocking \\cite{Grigorenko2012}. Similarly, the real and imaginary parts of the transverse conductivity are\n\t\t\\begin{multline}\\label{a7}\n\t\t\\frac{\\textrm{${\\textrm{Re} \\above 0pt \\textrm{Im}}$}\\big\\} \\big(\\sigma_{xy}(\\Omega)\\big)}{\\sigma_{0}}=\\frac{2 v_{F}^{2}\\hbar e B}{\\pi}\\sum_{\\xi, \\sigma}\\sum_{m,n}\\xi\\frac{\\Theta(E_{n}-\\mu_{F})-\\Theta(E_{m}-\\mu_{F})}{E_{n}-E_{m}}\\\\\\times\\bigg[(A_{m}B_{n})^{2}\n\t\t\\delta_{|m|-\\xi,|n|}-(B_{m}A_{n})^{2}\\delta_{|m|+\\xi,|n|}\\bigg]\\bigg\\{\\textrm{${\\textrm{-G} \\above 0pt \\textrm{F}}$}\\cdot\n\t\t\\end{multline}\n\t\\end{widetext}\n\tIn the limit $\\Delta_{so}=\\Delta_{z}=0$, we recover graphene's Hall conductivity \\cite{Gusynin2005}, as expected.\n\t\n\tThe $p$ polarized angular and spatial GH of the reflected light from the staggered 2D monolayer-substrate system can be written as \\cite{Wu2017}\n\t\\begin{equation}\\label{d2}\n\t\\Theta_{GH}=-\\frac{2(R_{pp}^{2}\\rho_{pp}+R_{ps}^{2}\\rho_{ps})}{2k_{1}(R_{ps}^{2}+R_{pp}^{2})\\Lambda_{R}+\\chi_{pp}+\\chi_{ps}},\n\t\\end{equation}\n\t\\begin{equation}\\label{d3}\n\t\\Delta_{GH}=\\frac{2(R_{pp}^{2}\\varphi_{pp}+R_{ps}^{2}\\varphi_{ps})\\Lambda_{R}}{2k_{1}(R_{ps}^{2}+R_{pp}^{2})\\Lambda_{R}+\\chi_{pp}+\\chi_{ps}}.\n\t\\end{equation}\n\twhere, $\\Lambda_{R}=\\pi w_{0}^2\/\\lambda$ is the Rayleigh range,  $r_\\lambda=R_\\lambda\\exp(i\\phi_{\\lambda})$, $\\lambda \\in (pp,ps,ss,sp)$, $\\rho_\\lambda=\\textrm{Re}(\\partial \\ln r_{\\lambda}\/\\partial\\theta_{1})$, $\\varphi_\\lambda=\\textrm{Im}(\\partial \\ln r_{\\lambda}\/\\partial\\theta_{1})$ $\\chi_\\lambda=R_{\\lambda}^{2}(\\varphi_{\\lambda}^{2}+\\rho_{\\lambda}^{2})$, $R_\\lambda$ is the amplitude and $\\phi_{\\lambda}$ is the phase of reflection coefficients.\n\t\n\\section{Results and Discussion}\n\\subsection{Magnetic field modulated GH under PR and TIR conditions}\n\\begin{figure*}[!t]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.9\\linewidth]{LLS.eps}\n\t\\end{tabular}\n\t\\caption{Schematic representation of the allowed transitions between LL's for three\n\t\tdifferent magnetic fields in the $K$ valley at (a) 1 T (b) 2 T (c) 3 T. Blue lines represent Landau levels for spin up ($\\sigma=\\uparrow$) and red lines represent Landau level for spin down ($\\sigma=\\downarrow$). The same color scheme applies for the Landau levels transitions. (d) Longitudinal conductivity as a function of photon frequency. The parameters used are $\\Delta_{z}=0.5\\Delta_{so}$ and chemical potential $\\mu_{F}=0$.}\n\t\\label{fig2}\n\\end{figure*}\nWe quickly recount the effect of the magnetic field on the energy level structure and subsequently the MO response. Schematic diagrams showing the allowed transitions between Landau levels (LL's) for three different magnetic field ($B=$1, 3 and 5 T) all in the topological insulator regime $(\\Delta_{z} < \\Delta_{so})$ and in the $K$ valley are shown in Fig.~\\ref{fig2}(a)\\textendash(c). The transition energy is determined from the energy difference between LL's obeying certain selection rules namely $|n|-|m|=\\pm1$ and the conservation of real spin implying that transitions between $\\sigma=+1$ and $-1$ levels are spin forbidden. The excitation energies corresponding to the different transitions, $E_{m,K (K'),\\uparrow(\\downarrow)}\\rightarrow E_{n,K (K'),\\uparrow(\\downarrow)}$ are labelled as $\\Delta_{mn,K (K'),\\uparrow(\\downarrow)}$. Blue lines represent Landau levels for spin up ($\\sigma=\\uparrow$) and red lines represent Landau levels for spin down ($\\sigma=\\downarrow$). In each of the depicted transitions, one of the participating levels is an $n$=0 level. For example for $B=1$ T, the first and second magneto-excitation energies correspond to the $\\Delta_{-10,K, \\uparrow}$ and $\\Delta_{01,K, \\downarrow}$ for spin up and spin down respectively and are calculated at 20.3 meV (4.9 THz) and 25.1 meV (6.1 THz) respectively. For higher $n$, the LL's are closely spaced, we ignore these transitions for the time being. Similarly for $B=3$ and 5 T, the allowed transitions between LL's are shown in Fig.~\\ref{fig2}(b) and (c). In fact, Table I summarizes magneto-excitation frequencies within the $n=-1,0,1$ manifold for the magnetic fields considered. In Fig.~\\ref{fig2}(d), we have shown the variation of the longitudinal conductivity as a function of photon frequency for three different magnetic fields also shown exclusively for the TI regime. We can see resonant peaks when the incident $\\hbar\\Omega$ hits the magneto-excitation energy. As we increase the strength of the applied magnetic field, the MO excitations shift towards higher frequencies. For practicality and simplicity, we will restrict ourselves to the lowest magneto-excitation transition frequency, originating only from the $\\Delta_{-10,K, \\uparrow}$ transition while investigating the GH shift in this article, unless otherwise specified.\n\n\\begin{figure*}[!ht]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.9\\linewidth]{Rss.eps}&\n\t\\end{tabular}\n\t\\caption{Modulus and phase of the $s$ and $p$ polarized reflection coefficients for 2D staggered graphene-substrate system as a function of incident angle for different magnetic fields in $K$ valley for PR: (a) $R_{ss}$, (b)$R_{pp}$, (c) $\\phi_{pp}$ and (d) $\\phi_{ss}$. The parameters used are $\\Gamma=0.2\\Delta_{so}$, refractive index $n_{2}=1.84$ and chemical potential $\\mu_{F}=0$.}\n\t\\label{fig3}\n\\end{figure*}\n\nWith knowledge of the LL transitions and magneto optical conductivities of the 2D materials for different magnetic fields at hand, we are now capable to studying the spatial and angular GH shifts. We first discuss the reflectivity and phases of the reflected $s$ and $p$ polarized waves for incident light. Clearly, if the impinging Gaussian beam frequencies are smaller than all Dirac gaps, then no electrons can be excited from the valence to the conduction band. On the other hand, a resonant Gaussian wave will excite a electron-hole pair. As the resonance condition is spin-dependent, so spins of only one kind (say up spin) electrons will be excited. The resonant excitation energies (frequencies) for the $\\Delta_{-10,K, \\uparrow}$ transition are 20.3 meV (4.9 THz) at $B = 1$ T; 33.5 meV (8.1 THz) at $B = 3$ T and 42.6 meV (10.3 THz) at $B = 5$ T. Fig.~\\ref{fig3} illustrates the magnetic field modulated reflectivity ($R_{ss}$ and $R_{pp}$) and their phases ($\\phi_{ss}$ and $\\phi_{pp}$) as a function of incident angle in the TI regime. In Fig.~\\ref{fig3}(a) and (b), the moduli of $R_{pp}$ and $R_{ss}$ are shown for various magnetic fields indicating that for $p$ polarized incident light, $R_{pp}$ achieves a minimum value at a certain $\\theta_{1}$ and rises again. This is called the pseudo-Brewster angle $\\theta_{B}=\\tan^{-1}(n_{2}\/n_{1})$, whereas $R_{ss}$ increases smoothly as the angle of incidence is increased.  At $\\theta_{B}$, the magnitude of reflection coefficient $R_{pp}$ intensity for the uncoated surface (without 2D material) reaches zero, as shown in Fig.~\\ref{fig3}(a) by dashed line. For the 2D staggered atomic layer, $R_{pp}$ at the Brewster angle $\\theta_{B}$ is non-zero, because the MO conductivity of the graphene-family is complex. The phase $\\phi_{pp}$ shows transition from $0$ to $-\\pi$ in the vicinity of $\\theta_{B}$ for different magnetic fields. A similar variation has been reported in 2D-TMDC \\cite{Das2018}. The phase $\\phi_{ss}$ shows an increasing trend with $\\theta_{1}$ in the TI regime, as shown in Fig.~\\ref{fig3}(d). Note that we haven't shown results in the VSPM state and BI regime, for which $R_{ss}$ and $R_{pp}$ don't appreciably change for magnetic fields. In the reminder of this section, we will only discuss the magnetically induced spatial and angular shifts for the interesting case of $p$ polarized light. The shifts for the charge neutral graphene-family ($\\mu_{F}=0$ meV) are plotted as a function of incident angle for different magnetic fields in the TI regime in Fig.~\\ref{fig4}(a), whereas the angular GH shifts are represented in Fig.~\\ref{fig4}(b). Since $\\partial\\phi_{p}\/\\partial\\theta_{1}\\ne 0$ and is accentuated in the vicinity of $\\theta_{B}$, the spatial shifts are dominant in the vicinity of the pseudo-Brewster angle.\n\n\t\\begin{table*}[!htbp]\n\t\\textbf{\\caption{Table of allowed transitions in $K$ valley in the $n=-1,0,1$ subspace, for different magnetic fields in the TI regime with $\\Delta_{so}=0.5\\Delta_{z}$.}}\n\t\\centering\n\t\\begin{tabular}{lllll}\n\t\t\\hline\n\t\t$\\Delta_{mn, K(K'),\\uparrow(\\downarrow)}$  &~~~~~~ $B$ (T)  &~~~~~~~~~~~~Frequency (THz) \\\\ \\hline\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~1 &~~~~~~~~~~~~~~~~~~~~~$4.9$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~1 &~~~~~~~~~~~~~~~~~~~~~$6.1$  \\\\\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~3 &~~~~~~~~~~~~~~~~~~~~~$8.1$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~3 &~~~~~~~~~~~~~~~~~~~~~$9.2$  \\\\\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~5 &~~~~~~~~~~~~~~~~~~~~~$10.3$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~5 &~~~~~~~~~~~~~~~~~~~~~$11.4$  \\\\\n\t\\end{tabular}\n\\end{table*}\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.90\\linewidth]{GHPR.eps}\n\t\\end{tabular}\n\t\\caption{The $p$ polarized spatial and angular GH shifts for charge neutral staggered graphene-substrate system as a function of incident angle for different magnetic fields in the $K$ valley in the TI regime for PR and TIR. (a) The $p$ polarized spatial GH shifts for PR, (b) the $p$ polarized angular GH shifts for PR, (c) the $p$ polarized spatial GH shifts for TIR and (d) the $p$ polarized angular GH shifts for TIR. The dashed lines represents the values of $\\theta_{B}$ and $\\theta_{C}$ for the native dielectric substrate. The parameters used are identical across all figures, unless stated otherwise. (A color version of this figure can be viewed online).}\n\t\\label{fig4}\n\\end{figure*}\n\nTo better visualize the dependence of the GH shifts with respect to the magnetic field consider the $p$ polarized spatial GH shifts in Fig.~\\ref{fig4}(a), for the three different magnetic fields. We observe giant negative spatial beam shifts in the TI regime. If we look at the $\\phi_{pp}$ spectra in Fig.~\\ref{fig3}(c) together with these shifts, we observe that $\\phi_{pp}$ shows a change of $-\\pi$ resulting in negative spatial GH shifts. The magnitude of spatial GH shifts are larger for smaller magnetic fields and smaller for large magnetic fields values; however, the extent of the spatial GH shifts for $p$ polarization on a 2D staggered monolayer graphene-substrate system is larger compared to the case of the graphene-coated surface \\cite{Grosche2015}. It is also worth noting that the Brewster angle is also sensitive to the applied magnetic field. From the results shown in Fig.~\\ref{fig4}(a), we observe that the Brewster angle of the spatial GH shifts can be tuned from $62^{\\circ}$ to $63^{\\circ}$ by modulating magnetic field for the three proposed magnetic fields.\n\nSimilarly in Fig.~\\ref{fig4}(b), we also plot the angular GH shifts as a function of the incident angle. The angular GH shift $\\Theta_{GH}^{p}$ is positive and gradually increases with incident angles, but as the incident angle reaches $\\theta_{B}$, $\\Theta_{GH}^{p}$ decreases rapidly and results in a negative GH shift. By tuning the angle of incidence of the $p$ polarized THz beam and the applied magnetic field, one can therefore control the polarity of the angular GH shift as well as its amplitude. Similar to the spatial counterpart, the amplitude of the angular GH decreases for increasing magnetic fields and this finding is in good agreement with purely graphene-coated surfaces \\cite{Grosche2015}. Once again, the tuning of the Brewster angle $\\theta_{B}$ with magnetic field is also clearly observable.\n\n\nSimilarly we plot the $p$ polarized spatial and angular GH shifts under the total internally reflected (TIR) condition in Figs~\\ref{fig4}(c) and (d) respectively. Notable deviations from external reflection and certain additional features are vividly observable here. From Fig.~\\ref{fig4}(c) we observe that for magnetic fields of 1, 3 and 5 T, we have peaks in the vicinity of $\\theta_{B}$ and sharply precipiced dips at the critical angle $\\theta_{C}=\\sin^{-1}(n_{2}\/n_{1})$, in the TI regime. Once again, the amplitude of the spatial GH shift is attenuated for larger magnetic fields. Similarly the magnetic field modulated $p$ polarized angular GH shifts for TIR geometry are depicted in Fig~\\ref{fig4}(d) wherein we expect unusual behavior of the GH shifts in the vicinity of both the Brewster and the critical angle. The angular GH shift $\\Theta_{GH}^{p}$ is positive when the incidence angle is smaller than the Brewster angle and negative beyond. Similarly, just prior to the critical angle, the angular shift acquires a large negative spike and diminishes immediately after. Furthermore, near $\\theta_{B}$, the angular GH decreases in size with increasing magnetic field values.\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.90\\linewidth]{GHE.eps}\n\t\\end{tabular}\n\t\\caption{(a) Longitudinal conductivity as a function of incident photon frequency and the $p$ polarized spatial and angular GH shifts for the staggered graphene-substrate system as a function of incident angle for magnetic field $B=1$ T in the $K$ valley in three distinct topological regimes and for four different chemical potentials. The incidence is external PR, while (b) and (c) show the $p$ polarized spatial and angular GH shifts with modulation of the external electric field, for the TI, VSPM and BI at a magnetic field of 1 T. (d) Schematic representation of the allowed transitions between LL's for three different values of chemical potential $\\mu_{F}$=0, 10 and 22 meV, and (e) and (f) are the $p$ polarized spatial and angular GH shifts with modulation of the chemical potential in the TI and classical regimes for a magnetic field of 1 T.}\n\t\\label{fig5}\n\\end{figure*}\n\n\n\\begin{table*}[!htb]\n\t\\textbf{\\caption{Table of allowed transitions in $K$ valley in the $n=-1,0,1$ subspace, for $B=1$ T in three different topological regimes for $\\Delta_{so}=8$ meV.}}\n\t\\centering\n\t\\begin{tabular}{lllll}\n\t\t\\hline\n\t\t$\\Delta_{mn, K(K'),\\uparrow(\\downarrow)}$  &~~~~~~ $\\Delta_{z}$ (meV) &~~~~~~ Regime &~~~~~~~~~~~~Frequency (THz) \\\\ \\hline\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~0 &~~~~~~~~~TI &~~~~~~~~~~~~~~~~~~~~~$5.6$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~0 &~~~~~~~~~TI &~~~~~~~~~~~~~~~~~~~~~$5.6$  \\\\\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~4 &~~~~~~~~~TI &~~~~~~~~~~~~~~~~~~~~~$4.9$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~4 &~~~~~~~~~TI &~~~~~~~~~~~~~~~~~~~~~$6.1$  \\\\\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~8 &~~~~~~~VSPM &~~~~~~~~~~~~~~~~~~~~~$4.4$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~8 &~~~~~~~VSPM &~~~~~~~~~~~~~~~~~~~~~$6.7$  \\\\\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~16 &~~~~~~~~~BI &~~~~~~~~~~~~~~~~~~~~~$5.5$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~16 &~~~~~~~~~BI &~~~~~~~~~~~~~~~~~~~~~$8.2$  \\\\\n\t\\end{tabular}\n\t\\label{11}\n\\end{table*}\n\n\\subsection{Electric field modulated GH shift}\nA static electric field $E_{z}$, controls the electronic band structure of the 2D staggered graphene family by generating a staggered sublattice potential $\\Delta_{z}$. An increase in the electric field triggers a well known quantum phase transition occurs from topological insulator to band insulator state \\cite{Ezawa2012}. Fig.~\\ref{fig5}(a) shows the longitudinal conductivity verses photon energy for various values of $\\Delta_{z}$. As we increase the applied electric field $\\Delta_{z}$, each interband peak splits into two spin-polarized peaks in the TI regime $(\\Delta_{z} < \\Delta_{so})$. Concomitantly, due to a redistribution of spectral weight, the intensity of the peaks is reduced for larger fields values. When $\\Delta_{z} = \\Delta_{so}$, the gap of one of the spin-split bands closes and a new type of metallic phase emerges called the valley-spin polarized metal (VSPM) state. At the VSPM point, the lowest frequency peaks, move apart: the $\\Delta_{-10,K, \\uparrow}$ peak is red shifted while the $\\Delta_{01,K, \\downarrow}$ peak is blue shifted. The excitation energies corresponding to the first two peaks at the VSPM point are now 18.2 meV (4.4 THz) and 27.8 meV (6.7 THz) for spin up and down, respectively. Further increasing $\\Delta_{z}$ results in re-opening of the gaps and the system transitions from the VSPM to the band insulator (BI) state. In BI regime all interband peaks move to higher energies. The magneto-excitations frequencies are presented in Table II for the first two transitions in the three different regimes. The main role of the electric field is that it controls the band structure and is responsible for spin and valley polarized responses, and therefore, similar to magnetic fields, also controls the magneto-optic excitation energies.\n\nIn Figs.~\\ref{fig5}(b) and (c), we show the $p$ polarized spatial and angular GH shifts as a function of the incident angle in the TI, VSPM and BI regimes, illustrating selective excitation of spin up carriers in the $K$ valley. From Fig.~\\ref{fig5}(b), it is obvious that $p$ polarized incident light produces giant negative spatial GH shifts. For example, in the TI regime when all Dirac gaps are open and at the magneto-excitation frequency of 4.9 THz, a maximum value of $\\Delta_{GH}^{p}\\approx222$ $\\mu$m is attained for scattering rate of $\\Gamma=0.2\\Delta_{so}$. Further increasing $\\Delta_{z}$ results in reopening the lowest energy gaps and the system undergoes a topological phase transition from VSPM state to the BI regime. After band inversion, we observed smaller shifts, $\\Delta_{GH}^{p}=130$ $\\mu$m. By increasing the electric field, the locations of the Brewster angle does not change but the spatial GH shifts diminish in size.\n\n\nThe $p$ polarized angular GH shift as a function of the incident angle in the TI, VSPM and BI regimes can also be seen in Fig.~\\ref{fig5}(c), showing first, the sign inversion of the angular displacement across Brewster's angle and second, illustrating that the impact of the electric field is to suppress the size of the angular shifts.\n\n\\subsection{Controlling the magnitude, sign and the position of GH shifts through the  chemical potential}\nThe THz magneto-optical conductivity is heavily influenced by the chemical potential $\\mu_{F}$. We examine the $p$ polarized beam shifts by modulating the chemical potential solely in the $K$ valley. Results for the $K'$ valley are analogous. We observe that the magnitude, sign and position of the Brewster's angle can all be controlled by this external stimulus. For this purpose we choose three different values of chemical potential of $\\mu_{F}$=0, 10 and 22 meV, while keeping the magnetic field 1 T positioning the system in the TI regime. In contrast to neutral staggered graphene ($\\mu_{F}$=0 meV), we now have interband as well as intraband transitions. For $\\mu_{F}$=10 meV the chemical potential lies in between the $n$=0 and $n$=1 LL's and for $\\mu_{F}$=22 meV the chemical potential is in between the $n$=1 and $n$=2 LL's. A schematic diagram which helps us to understand the possible LL's transitions for this chemical potential adaptation is shown in Fig.~\\ref{fig5}(d). Moving from left to right, we can see the LL's interband transitions for $\\mu_{F}=0$ meV are energies $\\Delta_{-10,K,\\uparrow}=20.3$ meV and $\\Delta_{01,K,\\downarrow}=25.1$ meV. As $\\mu_{F}$ increases to $10$ meV, then according to selection rules, certain transitions become Pauli blocked. For instance, the interband transition $\\Delta_{-10,K,\\uparrow}$ becomes forbidden which is shown by the dashed upward pointing arrow in the middle. In its lieu, the intra-band transition $\\Delta_{01,K,\\uparrow}=16.0$ meV emerges whereas for the spin down electron, we still have the allowed interband transition $\\Delta_{01,K,\\downarrow}=25.1$ meV, shown by a solid arrow. If the chemical potential $\\mu_{F}$ is further enhanced to $22$ meV, so that it lies between the $n$=1 and $n$=2 manifolds, then both of the transitions $\\Delta_{-10,K,\\uparrow}$ and $\\Delta_{01,K,\\downarrow}$ become Pauli blocked, which are again indicated by the dashed arrows in the rightmost part of Fig.~\\ref{fig5}(d). In this case, the excitation energies (frequencies) corresponding to the first two intra-band transitions $\\Delta_{12,K,\\uparrow}$ and $\\Delta_{12,K,\\downarrow}$ are 7.5 meV (1.8 THz) and 7.2 meV (1.6 THz), respectively. Table III summarizes these results.\n\n\\begin{table*}[!htb]\n\t\\textbf{\\caption{Table of allowed transitions in $K$ valley in the $n=-1,0,1$ subspace, for different chemical potentials in the TI regime with $\\Delta_{so}=8$ meV.}}\n\t\\centering\n\t\\begin{tabular}{lllll}\n\t\t\\hline\n\t\t$\\Delta_{mn, K(K'),\\uparrow(\\downarrow)}$  &~~~~~~ $\\mu_{F}$ (meV)  &~~~~~~  Inter\/intraband &~~~~~~~~~~~~Frequency (THz) \\\\ \\hline\n\t\t$\\Delta_{-10, K, \\uparrow}$ &~~~~~~~~~0 &~~~~~~  Inter &~~~~~~~~~~~~~~~~~~~~~$4.9$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~~0 &~~~~~~  Inter &~~~~~~~~~~~~~~~~~~~~~$6.1$  \\\\\n\t\t$\\Delta_{01, K, \\uparrow}$ &~~~~~~~~10 &~~~~~~  Intra &~~~~~~~~~~~~~~~~~~~~~$4.0$  \\\\\n\t\t$\\Delta_{01, K, \\downarrow}$ &~~~~~~~~10 &~~~~~~  Inter &~~~~~~~~~~~~~~~~~~~~~$6.1$  \\\\\n\t\t$\\Delta_{12, K, \\uparrow}$ &~~~~~~~~22 &~~~~~~  Intra &~~~~~~~~~~~~~~~~~~~~~$1.8$  \\\\\n\t\t$\\Delta_{12, K, \\downarrow}$ &~~~~~~~~22 &~~~~~~ Intra &~~~~~~~~~~~~~~~~~~~~~$1.6$  \\\\\n\t\\end{tabular}\n\\end{table*}\n\n\nFigs.~\\ref{fig5}(e) and (f) depicts the impact of altering the chemical potential (e.g. by chemical doping) on the GH shifts.  Only the $K$ valley is demonstrated for the sake of brevity. For these results, we selected the resonance frequency that excites only the first allowed LL transition for the spin up electrons. The $p$ polarized spatial GH shifts as a function of incident angle for different chemical potentials are shown in Fig.~\\ref{fig5}(e). For $\\mu_{F}$=0 meV, we have negative spatial GH shifts originating from the $\\Delta_{-10,K,\\uparrow}$ transition, which is purely interband. When the chemical potential $\\mu_{F}$=10 meV, we have the intra-band transition and the magnitude of the $p$ polarized spatial GH shift is larger. As we further increase to $\\mu_{F}$=22 meV, we have purely intraband transitions. Here the sign of GH shift switches which is a remarkable demonstration of chemical potential modulated angular GH shifts. Note that an intraband transition is responsible for the giant positive GH shift, precisely $\\Delta_{12, K, \\uparrow}$ and whose resonant frequency is 1.8 THz. With a further increase of chemical potential, the magnitude of the $p$ polarized spatial GH shift decreases while its full width half maximum increases. Furthermore, we conclude that the pseudo-Brewster angles shifts to larger incidence angles due to the chemical modification. The inversion of sign of the GH shift from negative to positive has significant applications in optoelectronic devices and chemical potential measurement.\n\n\n\\begin{figure*}[!htb]\n\t\\centering\n\t\\begin{tabular}{c}\n\t\t\\includegraphics[width=0.90\\linewidth]{SGHV.eps}\n\t\\end{tabular}\n\t\\caption{The valley and spin polarized spatial and angular GH shifts for staggered 2D material-substrate system as a function of incident angle for $K$ and $K'$ valleys in the TI regime. (a) The spatial GH shifts for both spins and for both valleys, (b) the $p$ polarized angular GH shifts for both spins and valleys. The spatial and angular GH shifts for staggered 2D material-substrate system as a function of photon energy in the $K$ valley for different magnetic and electric fields. (c) the $p$ polarized spatial and angular GH shifts for three different magnetic fields in the TI regime, (d) the spatial and angular GH shifts for $B=1$ T in three distinct topological regimes. The $p$ polarized response is shown only.}\n\t\\label{fig6}\n\\end{figure*}\n\nIn the semiclassical approximation the LL spacing becomes unimportant, which occur when $|\\mu_{F}|\\gg|E_{0}|$, or another words when $\\mu_{F}$ is high up in the conduction band or deep down in the valance band \\cite{Shah2019}. In this regime, the conductivities can be modeled by classical Drude-like peaks. For example we suppose that $\\mu_{F}$=55 meV which places the chemical potential between the $n$=9 and $10$ LL's. According to the selection rules, only two transitions $\\Delta_{-9~10}$ and $\\Delta_{9~10}$ are allowed \\cite{Shah2019}. The magneto excitation frequencies for $\\Delta_{-9~10}$ and $\\Delta_{9~10}$ are 27.1 and 0.71 THz respectively. We plot the spatial and angular GH shifts for this semi-classical scenario in Figs.~\\ref{fig5}(e) and (f). We can see in Figs.~\\ref{fig5}(e) a giant negative and minuscule spatial GH shift for the incident beam exciting the $\\Delta_{-9~10}$ and $\\Delta_{9~10}$ transitions respectively. We also observe that the Brewster angle is strongly influenced by the chemical potential and dramatically changes in the classical regime for the two different transitions. Similarly we plot the $p$ polarized angular GH shifts in Fig.~\\ref{fig5}(f) for the two mentioned transitions. Remarkably, we get giant angular shift for $\\Delta_{9~10}$ transition as compared to a small effect for the $\\Delta_{-9~10}$ transition.\n\n\n\\subsection{Spin and valley polarized GH shifts}\nIn addition to charge and spin degree of freedom, the Dirac electrons in staggered 2D atomic layers possess a valley degree of freedom which acts like a fictitious spin $1\/2$ particle. The valley information for our system is actually embodied in the variable $\\xi$, which is described in the system Hamiltonian in Eq.~(\\ref{a11}). Since the MO excitation energies are spin and valley dependent, one can selectively address the valley pseudospin in these materials to achieve fully spin and valley-polarized GH shifts. For instance, by impinging right or left handed circularly polarized light of the correct frequency, we can selectively excite spin up or spin down electrons in either of the $K$ or $K'$ valleys. Hence we have full freedom to choose spin or valley by tuning the frequency and helicity of the impinging radiation.\n\nIn this vein, the valley and spin-dependence of the spatial GH shifts are depicted in Fig~\\ref{fig6}(a), indicating positive and negative GH shifts for both $K$ and $K'$ valleys. It is clear that this valley-dependent response is quite interesting. The sign of the lateral GH shift is inverted across the two valleys for the same spin identity. However, the angular shift does \\emph{not} undergo sign-inversion across the two valleys. Therefore for experimental verification, the direction of the lateral shift can allow probing of the valley polarization tuning it into a prospective readout modality for the valley qubit in quantum information processing schemes.\n\n\n\n\\subsection{Incident photon frequency dependence of GH shifts}\nAs we vary the magnetic field imposed on the 2D atomic layer, the energy and hence the MO excitation energies manifold change. It is therefore instructive, to examine the dependence of the spatial and angular GH shifts on the frequency (energy) of the incident THz beam. In Fig.~\\ref{fig6}, we plot the beam shifts as a function of incident photon energy. In Fig.~\\ref{fig6}(c) we show the response for three different values of $B=$1, 3 and 5 T, while keeping $\\mu_{F}=0$ and $\\theta_{1}=61^{\\circ}$ (near the Brewster angle) and keeping the electric field fixed to assign the system to the TI regime. Only the $K$ valley spin up polarized response is demonstrated. The beam shifts display an oscillating dependence moving gradually to higher frequencies as the magneto excitation energy is increased. This is shown in Fig.~\\ref{fig6}(c). Likewise for a fixed magnetic field ($B=1$ T), as the electric field is changed, rendering the system into various topological regimes, the position as well as the magnitudes of the beam shifts change. From the TI$\\rightarrow$VSPM$\\rightarrow$BI progression, the GH shifts increase and then decreases. This is shown in Fig.~\\ref{fig6}(d).\n\n\\section{Conclusion}\n\nIn summary, we have systematically investigated the novel spatial and angular mechanical Goos-Hanchen shifts by impinging a Gaussian beam on the surface of a staggered 2D monolayer of the graphene family. The lateral shifts are modulated by electric and magnetic fields and are analyzed in the THz frequency range. By taking into account the fascinating tunable electro-optic and MO properties of these materials, we have studied the effect of chemical potential, spin, valley and incident photon frequency dependence on the silicenic layer, with the particular enhancement of the GH shift in the vicinity of Brewster angle. Furthermore, we have found that the Brewster angle is sensitive to changing magnetic field and chemical potential.\n\nThe MO controlled Brewster angle can be used to develop a highly tunable solid-state modulator. More interestingly, the valleys and spins indices can be used for the switching of the GH shift from negative to positive and vice versa.  In conclusion, the GH shift in staggered 2D materials paves the way of realizing spin and valley dependent devices and systems that can be useful optical readout markers for experiments in quantum information processing, biosensing, spintronics and valleytronics in the THz regime. Moreover, the demonstrated electro-optic and MO tunable GH shifts can be used for the detection of electric field and magnetic fields sensing and also for the determination of the doping level of these 2D staggered layers.\n\n\\section*{Acknowledgements}\nThe authors would like to acknowledge financial support from the National Research program for Universities (NRPU), scheme number 10375 funded by the Higher Education Commission of Pakistan.\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}