diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznyli" "b/data_all_eng_slimpj/shuffled/split2/finalzznyli" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznyli" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{introduction}\n\n\n\nLet $f = \\sum_{j = 0}^d a_j X^j\\in \\mathbb{R}[X]$ be a univariate polynomial\nof degree $d \\in \\mathbb{Z}^+$. It is a classical result due to\nNewton (see \\cite{HLP}, \\S2.22 and \\S4.3 for two proofs) that\nwhenever all the roots of $f$ are real, then the coefficients of $f$\nsatisfy the following log-concavity condition:\n\n\\begin{equation} a_i^2 \\geq \\frac{d-i+1}{d-i} \\frac{i+1}{i}\\, a_{i-1} a_{i+1}\n{\\rm \\ for \\ all \\ } i \\in\n\\{1,\\ldots,d-1\\}.\\label{newton}\\end{equation} Moreover, if the roots\nof $f$ are not all equal, these inequalities are strict. When $d =\n2$, condition (\\ref{newton}) becomes $a_1 \\geq 4 a_0 a_2$, which is\nwell known to be a necessary and sufficient condition for all the\nroots of $f$ to be real. Nevertheless, for $d \\geq 3$, the converse\nof Newton's result does not hold any more~\\cite{Kurtz}.\n\n\\medskip\n\nWhen $f \\in \\mathbb{R}^+[X]$, i.e., when $f = \\sum_{j = 0}^d a_j X^j$ with\n$a_j \\geq 0$ for all $j \\in \\{0,\\ldots,d\\}$, a weak converse of\nNewton's result holds true. Namely, a sufficient condition for $f$\nto only have real (and distinct) roots is that\n$$a_i^2\n> 4 a_{i-1} a_{i+1} {\\rm \\ for\\ all}\\ i \\in \\{1,\\ldots,d-1\\}.$$\nWhenever a polynomial fulfills this condition, we\nsay that it satisfies the {\\it Kurtz condition} since this converse\nresult is often attributed to Kurtz~\\cite{Kurtz}.\nNote however that it was obtained some 70 years earlier by Hutchinson~\\cite{Hutchinson}.\n\\medskip\n\nIf $f$ satisfies the Kurtz condition, all of its $d+1$ coefficients\nare nonzero except possibly the constant term. Such a polynomial is\ntherefore very far from being sparse (recall that a polynomial is\ninformally called {\\em sparse} if the number of its nonzero coefficients\nis small compared to its degree).\nOne question that we investigate in this paper is: how can we\nconstruct polynomials satisfying the Kurtz condition using sparse\npolynomials as building blocks?\nMore precisely, consider $f$ a polynomial of the form\n\\begin{equation}\\label{sumprod} f = \\sum_{i = 1}^k \\prod_{j = 1}^m\nf_{i,j}\\end{equation} where $f_{i,j}$ are polynomials with at most\n$t$ monomials each. By expanding the products in~(\\ref{sumprod}) we\nsee that $f$ has at most $k t^m$ monomials.\nAs a result, $d \\leq k t^m$ if $f$ satisfies the Kurtz condition.\nOur goal is to improve this very coarse bound. For the case of\npolynomials $f_{i,j}$ with nonnegative coefficients, we obtain the\nfollowing result.\n\n\\begin{theorem}\\label{bound}\nConsider a polynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f\n= \\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j},$$ where $m \\geq 2$ and the\n$f_{i,j} \\in \\mathbb{R}^+[X]$ have at most $t$ monomials. If $f$ satisfies\nthe Kurtz condition, then $d = \\mathcal O(k m^{2\/3} t^{2m\/3} {\\rm\nlog^{2\/3}}(kt))$.\n\\end{theorem}\nWe prove this result in Section \\ref{kurtzsection}. After that, in\nSection \\ref{strongsection}, we study the following stronger\nlog-concavity condition\n\\begin{equation} \\label{stronglogconcave}a_i^2\n> d^{2d} a_{i-1} a_{i+1} {\\rm \\ for\\ all}\\ i \\in \\{1,\\ldots,d-1\\}.\\end{equation} In\nthis setting we prove the following improved analogue of Theorem\n\\ref{bound}.\n\n\\begin{theorem}\\label{bound2} Consider a\npolynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f = \\sum_{i =\n1}^k \\prod_{j = 1}^m f_{i,j},$$ where $m \\geq 2$ and the $f_{i,j}\n\\in \\mathbb{R}^+[X]$ have at most $t$ monomials. If $f$\nsatisfies~$(\\ref{stronglogconcave})$, then $d \\leq k m t$.\n\\end{theorem}\n\n This\ninvestigation has a complexity-theoretic motivation: we show in\nSection~\\ref{complexity} that a suitable\nextension of\nTheorem~\\ref{bound2} (allowing negative coefficients for the\npolynomials $f_{ij}$) would imply\na separation of the algebraic complexity classes $\\vp$ and $\\vnp$.\nThe classes $\\vp$ of ``easily computable polynomial families'' and\n$\\vnp$ of ``easily definable polynomial families'' were proposed by\nValiant~\\cite{Val79} as algebraic analogues of $\\p$ and $\\np$. As\nshown in Theorem~\\ref{monotone}, Theorem~\\ref{bound2} as it now\nstands is strong enough to provide a new example of a family of\npolynomials in $\\vnp$ which cannot be computed by monotone\narithmetic circuits of polynomial size.\n\n\\section{The Kurtz log-concavity condition}\\label{kurtzsection}\n\nOur main tool in this section is a result of convex geometry\n\\cite{EPRS}. To state this result, we need to introduce some\ndefinitions and notations. For a pair of planar finite sets $R, S\n\\subset \\mathbb{R}^2$, the {\\it Minkowski sum} of $R$ and $S$ is the set $R\n+ S := \\{y + z \\, \\vert \\, y \\in R, z \\in S\\} \\subset \\mathbb{R}^2$. A\nfinite set $C \\subset \\mathbb{R}^2$ is {\\it convexly independent} if and\nonly if its elements are vertices of a convex polygon. The following\nresult provides an upper bound for the number of elements of a\nconvexly independent set contained in the Minkowski sum of two other\nsets.\n\n\\begin{theorem}\\cite[Theorem 1]{EPRS}\\label{convex} Let $R$ and $S$ be two planar point sets with $\\vert\nR \\vert = r$ and $\\vert S \\vert = s$. Let $C$ be a subset of the\nMinkowski sum $R + S$. If $C$ is convexly independent we have that\n$\\vert C \\vert = \\mathcal O(r^{2\/3} s^{2\/3} + r + s)$.\n\\end{theorem}\n\n\n\\medskip From this result the following corollary follows easily.\n\n\\begin{corollary}\\label{maxconvex}Let $R_1,\\ldots,R_k,S_1,\\ldots,S_k,Q_1,Q_2$ be planar point sets with $\\vert\nR_i \\vert = r, \\ \\vert S_i \\vert = s$ for all $i \\in \\{1,\\ldots,k\\}$,\n$\\lvert Q_1\\rvert = q_1$\nand $\\lvert Q_2 \\rvert = q_2$. Let $C$ be a subset of $\\cup_{i = 1}^k (R_i\n+ S_i) + Q_1+Q_2$. If $C$ is convexly independent, then $\\vert C \\vert =\n\\mathcal O(k r^{2\/3} s^{2\/3} q_1^{2\/3}q_2^{2\/3} + k r q_1 + k s q_2)$.\n\\end{corollary}\n\\begin{proof}We observe that $\\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2 =\n\\cup_{i = 1}^k ((R_i+Q_1) + (S_i + Q_2))$. Therefore, we partition $C$ into\n$k$ convexly independent disjoint sets $C_1,\\ldots,C_k$ such that\n$C_i \\subset (R_i+Q_1) + (S_i + Q_2)$ for all $i \\in \\{1,\\ldots,k\\}$. Since\n$\\vert R_i +Q_1\\vert = rq_1$ and $\\vert S_i + Q_2 \\vert \\leq sq_2$, by Theorem\n\\ref{convex}, we get that $\\vert C_i \\vert = \\mathcal O(r^{2\/3}\ns^{2\/3} q_1^{2\/3}q_2^{2\/3}+ rq_1 + sq_2)$ and the result follows.\n\\end{proof}\n\n\\medskip\n\n\n\n\\begin{theorem}\\label{bound2summands}Consider a polynomial $f \\in \\mathbb{R}^+[X]$ of degree $d$ of the form $$f =\n\\sum_{i = 1}^k g_i h_i,$$ where $g_i,h_i \\in \\mathbb{R}^+[X]$, the $g_i$\nhave at most $r$ monomials and the $h_i$ have at most $s$ monomials.\nIf $f$ satisfies the Kurtz condition, then $d = \\mathcal O(k\nr^{2\/3}s^{2\/3}\\,{\\rm log}^{2\/3}(k r)+ k(r+s)\\log^{1\/2}(kr))$.\n\\end{theorem}\n\\begin{proof}We write $f = \\sum_{i = 0}^d c_i X^i$, where $c_i > 0$ for all $i \\in \\{1,\\ldots,d\\}$ and $c_0 \\geq 0$.\nSince $f$ satisfies the Kurtz condition, setting $\\epsilon := {\\rm\nlog}(4)\/2$ we get that\n\\begin{equation}\\label{ineq} 2 {\\rm log}(c_i) > {\\rm log}(c_{i-1}) + {\\rm log}(c_{i+1})\n+ 2 \\epsilon. \\end{equation} for every $i \\geq 2$. For every\n$\\delta_1,\\ldots,\\delta_{d} \\in \\mathbb{R}$, we set\n$C_{(\\delta_1,\\ldots,\\delta_d)} := \\{(i,{\\rm log}(c_i) + \\delta_i)\n\\, \\vert \\, 1 \\leq i \\leq d\\}$. We observe that (\\ref{ineq}) implies\nthat $C_{(\\delta_1,\\ldots,\\delta_d)}$ is convexly independent\nwhenever $0 \\leq \\delta_i < \\epsilon$ for all $i \\in\n\\{1,\\ldots,d\\}$.\n\n\\medskip\nWe write $g_i = \\sum_{j = 1}^{r_i} a_{i,j} X^{\\alpha_{i,j}}$ and\n$h_i = \\sum_{j = 1}^{s_i} b_{i,j} X^{\\beta_{i,j}}$, with $r_i \\leq\nr$, $s_i \\leq s$ and $a_{i,j}, b_{i,j}\n> 0$ for all $i,j$. Then, $c_l = \\sum_{i = 1}^k (\\sum_{\\alpha_{i,j_1} + \\beta_{i,j_2} =\nl} a_{i,j_1} b_{i,j_2})$. So, setting $M_l := {\\rm max} \\{a_{i,j_1}\nb_{i,j_2} \\, \\vert \\, i \\in \\{1,\\ldots,k\\}, \\alpha_{i,j_1} +\n\\beta_{i,j_2} = l\\}$ for all $l \\in \\{1,\\ldots,d\\}$, we have that\n$M_l \\leq c_l \\leq k r M_l$, so ${\\rm log}(M_l) \\leq {\\rm log}(c_l)\n\\leq {\\rm log}(M_l) + {\\rm log}(k r)$.\n\n\\medskip For every $l \\in \\{1,\\ldots,d\\}$, we set\n\\begin{equation}\\label{lambdaepsilon} \\lambda_l := \\left\\lceil \\frac{{\\rm log}(c_l) - {\\rm\nlog}(M_l)}{\\epsilon} \\right\\rceil {\\rm \\ and \\ } \\delta_l := {\\rm\nlog}(M_l) + \\lambda_l \\epsilon - {\\rm log}(c_l),\\end{equation} and\nhave that $0 \\leq \\lambda_l \\leq \\lceil ({\\rm log}(k r))\/\\epsilon\n\\rceil$ and that $0 \\leq \\delta_l < \\epsilon$.\n\n\\medskip\nNow, we consider the sets\n\\begin{itemize}\n\\item $R_i :=\n \\{(\\alpha_{i,j}, {\\rm log}(a_{i,j}))\\, \\vert \\, 1 \\leq j \\leq r_i\\}$\n for $i = 1,\\ldots,k$,\n\\item $S_i := \\{(\\beta_{i,j}, {\\rm\n log}(b_{i,j}))\\, \\vert \\, 1 \\leq j \\leq s_i\\}$ for $i = 1,\\ldots,k$,\n\\item $Q := \\{(0, \\lambda \\epsilon) \\, \\vert \\, 0 \\leq \\lambda\n \\leq \\lceil {\\rm log}(k r) \/ \\epsilon \\rceil \\}$,\n\\item $Q_1 := \\{(0, \\mu \\epsilon) \\, \\vert \\, 0 \\leq \\mu\n \\leq \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\}$, and\n\\item $Q_2 := \\{(0, \\nu \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\epsilon) \\, \\vert \\, 0 \\leq \\nu\n \\leq \\lceil \\sqrt{\\log(k r) \/ \\epsilon} \\rceil \\}$.\n\\end{itemize}\nIf $(0,\\lambda\\epsilon)\\in Q$, then there exist $\\mu$ and $\\nu$ such that\n$\\lambda=\\nu\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil + \\mu$ where\n$\\mu,\\nu\\leq\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil$. We have,\n\\begin{align*}\n (0,\\lambda\\epsilon)=\n (0,\\nu\\lceil\\sqrt{\\log(kr) \/ \\epsilon}\\rceil\\epsilon)+(0,\\mu\\epsilon)\\in Q_1+Q_2,\n\\end{align*}\nso $Q\\subset Q_1+Q_2$.\nThen, we claim that $C_{(\\delta_1,\\ldots,\\delta_d)} \\subset \\cup_{i = 1}^k\n(R_i + S_i) + Q$. Indeed, for all $l \\in \\{1,\\ldots,d\\}$, by\n(\\ref{lambdaepsilon}), $${\\rm log}(c_l) + \\delta_l = {\\rm log}(M_l)\n+ \\lambda_l \\epsilon = {\\rm log}(a_{i,j_1}) + {\\rm log}(b_{i,j_2}) +\n\\lambda_l \\epsilon$$ for some $i \\in \\{1,\\ldots,k\\}$ and some\n$j_1,j_2$ such that $\\alpha_{i,j_1} + \\beta_{i,j_2} = l$; thus\n$$(l,{\\rm log}(c_l) + \\delta_l) = (\\alpha_{i,j_1}, {\\rm log}(a_{i,j_1})) +\n(\\beta_{i,j_2}, {\\rm log}(b_{i,j_1})) + (0, \\lambda_l \\epsilon) \\in\n\\cup_{i = 1}^k (R_i + S_i) + Q.$$ Since\n$C_{(\\delta_1,\\ldots,\\delta_d)}$ is a convexly independent set of\n$d$ elements contained in $\\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2$, a direct\napplication of Corollary \\ref{maxconvex} yields the result.\n\\end{proof}\n\n\\medskip\n\n From this result it is easy to derive an upper bound for the\ngeneral case, where we have the products of $m \\geq 2$ polynomials.\nIf suffices to divide the $m$ factors into two groups of\napproximately $m\/2$ factors, and in each group we expand the product\nby brute force.\n\n\\medskip\n\n\\begin{proof}[Proof of Theorem~\\ref{bound}]\nWe write each of the $k$ products as a product of two\npolynomials $G_i := \\prod_{j = 1}^{\\lfloor m\/2 \\rfloor} f_{i,j}$ and\n$H_i := \\prod_{j = \\lfloor m\/2 \\rfloor + 1}^{m} f_{i,j}$. We can now\napply Theorem \\ref{bound2summands} to $f = \\sum_{i = 1}^k G_i H_i$\nwith $r = t^{\\lfloor m\/2 \\rfloor}$ and $s = t^{m - \\lfloor m\/2\n\\rfloor}$ and we get the result.\n\\end{proof}\n\n\\bigskip\n\n\\begin{remark}We observe that the role of the constant $4$ in the\nKurtz condition can be played by any other constant $\\tau > 1$ in\norder to obtain the conclusion of Theorem \\ref{bound}, i.e., we\nobtain the same result for $f = \\sum_{i = 0}^d a_i X^i$\nsatisfying that $a_i^2 > \\tau a_{i-1} a_{i+1}$ for all $i \\in\n\\{1,\\ldots,d-1\\}$. For proving this it suffices to replace the value\n$\\epsilon = {\\rm log}(4) \/ 2$ by $\\epsilon = {\\rm log}(\\tau) \/ 2$ in\nthe proof of Theorem \\ref{bound2summands} to conclude this more\ngeneral result.\n\\end{remark}\n\n\\bigskip\n\nFor $f = g h$ with $g,h \\in \\mathbb{R}^+[X]$ with at most $t$ monomials,\nwhenever $f$ satisfies the Kurtz condition, then $f$ has only real\n(and distinct) roots and so do $g$ and $h$. As a consequence, both\n$g$ and $h$ satisfy (\\ref{newton}) with strict inequalities and we\nderive that $d \\leq 2t$. Nevertheless, in the similar setting where\n$f = g h + x^i$ for some $i > 0$, the same argument does not apply\nand a direct application of Theorem \\ref{bound} yields $d = \\mathcal\nO(t^{4\/3}\\, {\\rm log^{2\/3}}(t))$, a bound\nwhich seems to be very far from optimal.\n\n\\subsection*{Comparison with the setting of Newton polygons}\n\n\nA result similar to Theorem~\\ref{bound} was obtained in~\\cite{KPTT}\nfor the Newton polygons\nof bivariate polynomials. Recall that the Newton polygon of a\npolynomial $f(X,Y)$ is the convex hull of the points $(i,j)$ such\nthat the monomial $X^iY^j$ appears in $f$ with a nonzero coefficient.\n\\begin{theorem}[Koiran-Portier-Tavenas-Thomass\\'e] \\label{mpolys}\nConsider a bivariate polynomial\n of the form\n\\begin{equation} \\label{bivariateSPS}\nf(X,Y)=\\sum_{i=1}^k \\prod_{j=1}^m f_{i,j}(X,Y)\n\\end{equation}\nwhere $m \\geq 2$ and the $f_{i,j}$ have at most $t$ monomials. The\nNewton polygon of $f$ has $O(k t^{2m\/3})$ edges.\n\\end{theorem}\nIn the setting of Newton polygons, the main issue is how to deal with\nthe cancellations arising from the addition of the $k$ products\nin~(\\ref{bivariateSPS}).\nTwo monomials of the form $cX^iY^j$ with the\nsame pair $(i,j)$ of exponents but oppositive values of the\ncoefficient $c$ will cancel, thereby deleting the point $(i,j)$\nfrom the Newton polygon.\n\nIn the present paper we associate to the monomial $cX^i$ with $c>0$\nthe point $(i,\\log c)$. There are no cancellations since we only\nconsider polynomials $f_{i,j}$ with nonnegative coefficients in\nTheorems~\\ref{bound} and~\\ref{bound2summands}. However, the addition\nof two monomials $cX^i, c'X^i$ with the same exponent will ``move''\nthe corresponding point along the coefficient axis. By contrast, in\nthe setting of Newton polygons points can be deleted but cannot\nmove. In the proof of Theorem~\\ref{bound2summands} we deal with the\nissue of ``movable points'' by an approximation argument, using the\nfact that the constant $\\epsilon=\\log(4)\/2>0$ gives us a little bit\nof slack.\n\n\\section{A stronger log-concavity condition}\\label{strongsection}\n\nThe objective of this section is to improve the bound provided in\nTheorem \\ref{bound} when $f = \\sum_{i = 0}^d a_i X^i \\in \\mathbb{R}^+[x]$\nsatisfies a stronger log-concavity condition, namely, when\n$a_i^2 > d^{2d} a_{i-1} a_{i+1}$ for all $i \\in \\{1,\\ldots,d-1\\}$.\n\nTo prove this bound, we make use of the following\nwell-known lemma (a reference and similar results for polytopes in\nhigher dimension can be found in~\\cite{karavelas2012}).\nFor completeness, we provide a short proof.\n\\begin{lemma}\\label{convexhull} If $R_1,\\ldots,R_s$ are planar sets and $\\vert R_i\n\\vert = r_i$ for all $i \\in \\{1,\\ldots,s\\}$, then the convex hull of\n$R_1 + \\cdots + R_s$ has at most $r_1 + \\cdots + r_s$ vertices.\n\\end{lemma}\n\\begin{proof}We denote by $k_i$ the number of vertices of the\nconvex hull of $R_i$. Clearly $k_i \\leq r_i$. Let us prove that the\nconvex hull of $R_1 + \\cdots + R_s$ has at most $k_1 + \\cdots + k_s$\nvertices. Assume that $s = 2$. We write $R_1 =\n\\{a_1,\\ldots,a_{r_1}\\}$, then $a_i \\in R_1$ is a vertex of the\nconvex hull of $R_1$ if and only if there exists $w \\in S^1$ (the\nunit Euclidean sphere) such that $w \\cdot a_i\n> w \\cdot a_j$ for all $j \\in \\{1,\\ldots,r_1\\} \\setminus \\{i\\}$.\nThus, $R_1$ induces a partition of $S^1$ into $k_1$ half-closed\nintervals. Similarly, $R_2$ induces a partition of $S^1$ into $k_2$\nhalf-closed intervals. Moreover, these two partitions induce a new\none on $S^1$ with at most $k_1 + k_2$ half-closed intervals; these\nintervals correspond to the vertices of $R_1 + R_2$ and; thus, there\nare at most $k_1 + k_2$. By induction we get the result for any\nvalue of $s$.\n\\end{proof}\n\n\\begin{proposition}\\label{SPS}\n Consider a polynomial $f=\\sum_{i=0}^d a_i X^i \\in \\mathbb{R}^+[X]$ of the form\n \\begin{align*}\n f=\\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j}\n \\end{align*}\n where the $f_{i,j} \\in \\mathbb{R}^+[x]$. If $f$ satisfies the\n condition\n \\begin{align*}\n a_i^2 > k^2 d^{2m} a_{i-1} a_{i+1} ,\n \\end{align*}\n then there exists a polynomial $f_{i,j}$ with at least $d \/ km$ monomials.\n\\end{proposition}\n\n\\begin{proof}\nEvery polynomial $f_{i,j} := \\sum_{l = 0}^{d_{i,j}} c_{i,j,l}\\,\nX^l$, where $d_{i,j}$ is the degree of $f_{i,j}$, corresponds to a\nplanar set\n\\begin{align*} R_{i,j} := \\{(l, {\\rm log}(c_{i,j,l}))\\, \\vert \\,\nc_{i,j,l} > 0\\} \\subset \\mathbb{R}^2.\n\\end{align*} We set,\n$C_{i,l} := {\\rm max} \\{0, \\prod_{r = 1}^m c_{i,r,l_r} \\, \\vert \\,\nl_1 + \\cdots + l_m = l \\},$ for all $i \\in \\{1,\\ldots,k\\}$, $l \\in\n\\{0,\\ldots,d\\}$, and $ C_l := {\\rm max}\\{C_{i,l} \\, \\vert \\, 1\n\\leq i \\leq k\\}$ for all $l \\in \\{0,\\ldots,d\\}$. Since the\npolynomials $f_{i,j} \\in \\mathbb{R}^+[X]$ and\n$$a_l = \\sum_{i = 1}^k \\left(\\sum_{l_1 + \\cdots + l_m = l}\\\n\\prod_{r= 1}^m c_{i,r,l_r}\\right)$$ for all $l \\in \\{0,\\ldots,d\\}$,\nwe derive the following two properties:\n\\begin{itemize}\n\\item $C_l \\leq a_l \\leq k d^m C_l$ for all $l \\in \\{0,\\ldots,d\\}$,\n\\item either $C_{i,l} = 0$ or $(l,\n{\\rm log}(C_{i,l})) \\in R_{i,1} + \\cdots + R_{i,m}$ for all $i \\in\n\\{1,\\ldots,k\\}, \\, l \\in \\{0,\\ldots,d\\}$. Since $a_l > 0$ for all $l\n\\in \\{1,\\ldots,d\\}$, we have that $C_l > 0$ and $(l, {\\rm log}(C_l))\n\\in \\bigcup_{i = 1}^k \\left(R_{i,1} + \\cdots + R_{i,m}\\right)$\n\\end{itemize}\n\n\nWe claim that the points in the set $\\{(l, {\\rm log}(C_l)) \\, \\vert\n\\, 1 \\leq l \\leq d\\}$ belong to the upper convex envelope of\n$\\bigcup_{i = 1}^k (R_{i,1} + \\cdots + R_{i,m})$. Indeed, if\n$(a,\\log(b)) \\in \\bigcup_{i = 1}^k (R_{i,1} + \\cdots + R_{i,m})$,\nthen $a \\in \\{0,\\ldots,d\\}$ and $b \\leq C_{a}$; moreover, for all $l\n\\in \\{1,\\ldots,d-1\\}$, we have that $$C_l^2 \\geq a_l^2 \/ (k^2\nd^{2m}) > a_{l-1} \\, a_{l+1} \\geq C_{l-1} C_{l+1}.$$\n\nHence, there exist $i_0 \\in \\{1,\\ldots,k\\}$ and $L \\subset\n\\{1,\\ldots,d\\}$ such that $\\vert L\\vert \\geq d\/k$ and $C_l =\nC_{i_0,l}$ for all $l \\in L$. Since the points in $\\{(l,{\\rm\nlog}(C_{l}))\\, \\vert \\, 1 \\leq l \\leq d\\}$ belong to the upper\nconvex envelope of $\\bigcup_{i = 1}^k (R_{i,1} + \\cdots +\n R_{i,m})$ we easily get that the set $\\{(l, {\\rm\n log}(C_{i_0,l})) \\, \\vert \\, l \\in L\\}$ is a subset of the vertices in the convex hull\nof $R_{i_0,1} + \\cdots + R_{i_0,m}$. By Lemma \\ref{convexhull}, we\nget that there exists $j_0$ such that $\\vert R_{i_0,j_0} \\vert \\geq\n\\lvert L \\rvert \/ m \\geq d \/ km$ points. Finally, we conclude that\n$f_{i_0,j_0}$ involves at least $d \/ km$ monomials.\n\\end{proof}\n\n\\bigskip\n\n\\begin{proof}[Proof of Theorem \\ref{bound2}]If $d \\leq k$ or $d \\leq m$, then $d \\leq kmt$.\nOtherwise, $d^{2d} > k^2 d^{2(d-1)} \\geq k^2 d^{2m}$ and, thus, $f$\nsatisfies (\\ref{stronglogconcave}). A direct application of\nProposition \\ref{SPS} yields the result.\n\\end{proof}\n\n\n\n\\section{Applications to Complexity Theory}\\label{complexity}\n\n\nWe first recall some standard definitions from algebraic complexity\ntheory (see e.g.~\\cite{Burgi} or~\\cite{Val79} for more\ndetails). Fix a field $K$.\nThe elements of the complexity class $\\vp$ are sequences $(f_n)$ of\nmultivariate polynomials with coefficients from $K$. By definition,\nsuch a sequence belongs to $\\vp$ if the degree of $f_n$ is bounded by\na polynomial function of $n$ and if $f_n$ can be evaluated in a\npolynomial number of arithmetic operations (additions and\nmultiplications) starting from variables and from constants in $K$.\nThis can be formalized with the familiar model of {\\em arithmetic\n circuits}.\nIn such a circuit, input gates are labeled by a constant or a\nvariable and the other gates are labeled by an arithmetic operation\n(addition or multiplication). In this paper we take $K = \\mathbb{R}$ since\nthere is a focus on polynomials with nonnegative coefficients. An\narithmetic circuit is {\\em monotone} if input gates are labeled by\nnonnegative constants only.\n\nA family of polynomials\nbelongs to the complexity class $\\vnp$ if it can be\nobtained by summation from a family in $\\vp$.\nMore precisely, $f_n(\\overline{x})$ belongs to $\\vnp$ if there exists\na family $(g_n(\\overline{x},\\overline{y}))$ in $\\vp$ and a polynomial $p$\nsuch that the tuple of variables\n$\\overline{y}$ is of length $l(n) \\leq p(n)$ and\n$$f_n(\\overline{x})=\\sum_{\\overline{y} \\in \\{0,1\\}^{l(n)}} g_n(\\overline{x},\\overline{y}).$$\nNote that this summation over all boolean values of $\\overline{y}$\nmay be of exponential size.\nWhether the inclusion $\\vp \\subseteq \\vnp$ is strict is a major open\nproblem in algebraic complexity.\n\nValiant's criterion~\\cite{Burgi,Val79} shows that ``explicit''\npolynomial families belong to $\\vnp$. One version of it is as follows.\n\\begin{lemma}\nSuppose that the function $\\phi:\\{0,1\\}^* \\rightarrow \\{0,1\\}$ is\ncomputable in polynomial time. Then the family $(f_n)$ of multilinear\npolynomials defined by\n$$f_n=\\sum_{e \\in \\{0,1\\}^n} \\phi(e)x_1^{e_1} \\cdots x_n^{e_n}$$\nbelongs to $\\vnp$.\n\\end{lemma}\nNote that more general versions of Valiant's criterion are know. One\nmay allow polynomials with integer rather than\n0\/1 coefficients~\\cite{Burgi}, but in Theorem~\\ref{monotone}\nbelow we will only have to deal with 0\/1 coefficients.\nAlso, one may allow $f_n$ to depend on any (polynomially bounded)\nnumber of variables rather than exactly $n$ variables and in this case, one may\nallow the algorithm for computing the coefficients of $f_n$ to take as\ninput the index $n$ in addition to the tuple $e$ of exponents\n(see~\\cite{Koi04}, Theorem~2.3).\n\n\n\nReduction of arithmetic circuits to depth~4 is an important\ningredient in the proof of the forthcoming results. This phenomenon\nwas discovered by Agrawal and Vinay \\cite{AV}. Here we will use it\nunder the form of \\cite{Tavenas}, which is an improvement of\n\\cite{Koiran2012}. We will also need the fact that if the original\ncircuit is monotone, then the resulting depth~4 circuit is also\nmonotone (this is clear by inspection of the proof\nin~\\cite{Tavenas}). Recall that a depth~4 circuit is a sum of\nproducts of sums of products of inputs; sum gates appear on layers\n 2 and 4 and product gates on layers 1\nand 3. All gates may have arbitrary fan-in.\n\n\\begin{lemma}\\label{redprof4}Let $C$ be an arithmetic circuit of size $s > 1$\ncomputing a $v$-variate polynomial of degree $d$. Then, there is an\nequivalent depth $4$ circuit $\\Gamma$ of size $2^{\\, \\mathcal\nO\\left(\\sqrt{d \\log (ds) \\log (v)} \\right)}$ with multiplication\ngates at layer $3$ of fan-in $\\mathcal O(\\sqrt{d})$. Moreover, if\n$C$ is monotone, then $\\Gamma$ can also be chosen to be monotone.\n\\end{lemma}\n\nWe will use this result under the additional hypothesis that $d$ is\npolynomially bounded by the number of variables $v$. In this\nsetting, since $v \\leq s$, we get that the resulting depth $4$\ncircuit $\\Gamma$ provided by Lemma \\ref{redprof4} has size\n$s^{\\mathcal O(\\sqrt{d})}$.\n\n\n\\medskip\nBefore stating the main results of this section, we construct an explicit family of log-concave polynomials.\n\\begin{lemma}\\label{lem_concavVn}Let $n, s \\in \\mathbb{Z}^+$ and\nconsider $g_{n,s}(X) := \\sum_{i=0}^{2^n-1} a_i X^i$, with\n\\begin{align*} a_i := 2^{si(2^n-i-1)} {\\text \\ for \\ all \\ } i \\in \\{0,\\ldots,2^n\n- 1\\}. \\end{align*}\n Then, $a_i^2\n> 2^s \\, a_{i-1} \\, a_{i+1}$.\n\\end{lemma}\n\\begin{proof}Take $i \\in \\{1,\\ldots,2^n - 2\\}$, we have that\n \\begin{align*}\n \\log\\left(2^s a_{i-1}a_{i+1}\\right) & = s + s2^n(i-1) - s(i-1)i\n +s2^n(i+1) - s(i+1)(i+2) \\\\\n & = 2s2^n i - 2s i(i+1)- s \\\\\n & < 2s 2^n i -2s i(i+1) \\\\\n & = \\log(a_i^2).\n \\end{align*}\n\\end{proof}\n\nIn the next theorem we start from the family $g_{n,s}$\nof Lemma~\\ref{lem_concavVn} and we set $s=n2^{n+1}$.\n\n\n\\begin{theorem}\\label{ifvpvnp}\nLet $(f_n) \\in \\mathbb{N}[X]$ be the family of polynomials $f_n(x)=g_{n,n2^{n+1}}(x)$.\n\\begin{itemize}\n\\item[(i)] $f_n$ has degree $2^n-1$ and satisfies the log-concavity condition {\\rm (\\ref{stronglogconcave})}.\n\\item[(ii)] If $\\vp=\\vnp$, $f_n$ can be written under form~{\\rm\n (\\ref{sumprod})} with $k=n^{O(\\sqrt{n})}$,\n $m=O(\\sqrt{n})$ and $t=n^{O(\\sqrt{n})}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nIt is clear that $f_n \\in\n\\mathbb{N}[X]$ has degree $2^n - 1$ and, by Lemma \\ref{lem_concavVn}, $f_n$\nsatisfies (\\ref{stronglogconcave}).\n\nConsider now the related family of bivariate polynomials\n$g_n(X,Y)=\\sum_{i=0}^{2^n-1}X^i Y^{e(n,i)},$ where $e(n,i) = s i (2^n - i - 1)$. One can check in time polynomial in $n$ whether\na given monomial $X^iY^j$ occurs in $g_n$: we just need to check\nthat $i<2^n$ and that $j=e(n,i)$. By mimicking the proof of Theorem\n1 in \\cite{KPTT} and taking into account Lemma \\ref{redprof4} we get\nthat, if $\\vp = \\vnp$, one can write\n\\begin{equation} \\label{sumprod2}\ng_n(X,Y)=\\sum_{i=1}^k\\prod_{j=1}^m g_{i,j,n}(X,Y)\n\\end{equation}\nwhere the bivariate polynomials $g_{i,j,n}$ have $n^{O(\\sqrt{n})}$\nmonomials, $k=n^{O(\\sqrt{n})}$ and $m=O(\\sqrt{n})$. Performing the\nsubstitution $Y=2$ in~(\\ref{sumprod2}) yields the required\nexpression for $f_n$.\n\\end{proof}\n\nWe believe that there is in fact no way to write $f_n$ under\nform~(\\ref{sumprod}) so that the parameters $k,m,t$ satisfy the\nconstraints $k=n^{O(\\sqrt{n})}$,\n $m=O(\\sqrt{n})$ and $t=n^{O(\\sqrt{n})}$.\nBy part (ii) of Theorem~\\ref{ifvpvnp}, a proof of this would\nseparate $\\vp$ from $\\vnp$. The proof of Theorem~\\ref{monotone} below\nshows that our belief is actually correct in the special case where\nthe polynomials $f_{i,j}$ in~(\\ref{sumprod}) have nonnegative\ncoefficients.\n\n\\medskip\nThe main point of Theorem \\ref{monotone} is to present an\nunconditional lower bound for a polynomial family $(h_n)$ in $\\vnp$\nderived from $(f_n)$. Note that $(f_n)$ itself is not in $\\vnp$\nsince its degree is too high. Recall that\n\\begin{equation} \\label{fneq}\n f_n(X) := \\sum_{i=0}^{2^n-1} 2^{2n2^ni(2^n-i-1)} X^i.\n\\end{equation}\nTo construct $h_n$ we write down in base 2 the exponents of ``2'' and\n``$X$'' in~(\\ref{fneq}).\nMore precisely, we take $h_n$ of the form:\n\\begin{equation} \\label{hneq}\n h_n :=\\sum_{\\alpha \\in \\{0,1\\}^{n} \\atop \\beta \\in \\{0,1\\}^{4n}}\n\\lambda(n, \\alpha, \\beta)\\, X_0^{\\alpha_0} \\cdots\nX_{n-1}^{\\alpha_{n-1}} Y_0^{\\beta_0} \\cdots\nY_{4n-1}^{\\beta_{4n-1}},\n\\end{equation}\n where $\\alpha =\n(\\alpha_0,\\ldots,\\alpha_{n-1}),\\, \\beta =\n(\\beta_0,\\ldots,\\beta_{4n-1})$ and $\\lambda(n,\\alpha,\\beta) \\in\n\\{0,1\\}$; we set $\\lambda(n,\\alpha,\\beta) = 1$ if and only\nif $\\sum_{j = 0}^{4n-1} \\beta_j 2^j = 2n2^n i (2^n - i - 1) <\n2^{4n},$ where $i := \\sum_{k = 0}^{n-1} \\alpha_{i,k} 2^k$.\nBy construction, we have:\n\\begin{equation} \\label{transfor} f_n(X) = h_n(X^{2^0}, X^{2^1},\\ldots,X^{2^{n-1}},2^{2^0},\n2^{2^1},\\ldots,2^{2^{4n-1}}). \\end{equation}\nThis relation will be useful in the proof of the following lower bound theorem.\n\\begin{theorem}\\label{monotone}\nThe family $(h_n)$ in~{\\rm(\\ref{hneq})} is in $\\vnp$. If $(h_n)$ is\ncomputed by depth $4$ monotone arithmetic circuits of size $s(n)$,\nthen $s(n) = 2^{\\,\\Omega(n)}$. If $(h_n)$ is computed by monotone\narithmetic circuits of size $s(n)$, then $s(n) =\n2^{\\,\\Omega(\\sqrt{n})}$. In particular, $(h_n)$ cannot be computed\nby monotone arithmetic circuits of polynomial size.\n\\end{theorem}\n\\begin{proof}\nNote that $h_n$ is a polynomial in $5n$ variables, of degree at most\n$5n$, and its coefficients $\\lambda(n, \\alpha, \\beta)$ can be\ncomputed in polynomial time. Thus, by Valiant's criterion\nwe conclude that $(h_n)\\in \\vnp$.\n\nAssume that $(h_n)$ can be computed by depth $4$ monotone arithmetic\ncircuits of size $s(n)$. Using (\\ref{transfor}), we get that $f_n =\n\\sum_{i = 1}^k \\prod_{j = 1}^m f_{i,j}$ where $f_{i,j} \\in \\mathbb{R}^+[X]$\n have at most $t$ monomials and $k,m,t$ are $\\mathcal\nO(s(n))$. Since the degree of $f_n$ is $2^n - 1$, by Theorem\n\\ref{bound2}, we get that $2^n - 1 \\leq kmt$. We conclude that $s(n)\n= 2^{\\,\\Omega(n)}$.\n\nTo complete the proof of the theorem, assume that $(h_n)$ can be\ncomputed by monotone arithmetic circuits of size $s(n)$. By\nLemma~\\ref{redprof4}, it follows that the polynomials $h_n$ are\ncomputable by depth~4 monotone circuits of size $s'(n) :=\ns(n)^{\\mathcal O(\\sqrt{n})}$. Therefore $s'(n) = 2^{\\,\\Omega(n)}$\nand we finally get that $s(n) = 2^{\\,\\Omega(\\sqrt{n})}$.\n\\end{proof}\n\n\nLower bounds for monotone arithmetic circuits have been known for a\nlong time (see for instance~\\cite{jerrum82,valiant79negation}).\nTheorem~\\ref{monotone} provides yet another example of a polynomial\nfamily which is hard for monotone arithmetic circuits, with an\napparently new proof method.\n\n\n\\section{Discussion}\n\nAs explained in the introduction, log-concavity plays a role\nin the study of real roots of polynomials.\nIn~\\cite{Koi10a} bounding the number of real roots of sums of products\nof sparse polynomials was suggested as an approach for separating\n$\\vp$ from $\\vnp$. Hrube\\v{s}~\\cite{Hrubes13} suggested to bound the\nmultiplicities of roots, and~\\cite{KPTT} to bound the number of edges\nof Newton polygons of bivariate polynomials.\n\nTheorem~\\ref{ifvpvnp} provides another\n plausible approach to $\\vp \\neq \\vnp$: it\nsuffices to show that if a polynomial $f \\in \\mathbb{R}^+[X]$ under\nform~(\\ref{sumprod}) satisfies the Kurtz condition or the stronger\nlog-concavity condition (\\ref{stronglogconcave}) then its degree is\nbounded by a ``small'' function of the parameters $k,m,t$. A degree\nbound which is polynomial bound in $k,t$ and $2^m$ would be good\nenough to separate $\\vp$ from $\\vnp$. Theorem~\\ref{bound} improves\non the trivial $kt^m$ upper bound when $f$ satisfies the Kurtz\ncondition, but certainly falls short of this goal: not only is the\nbound on $\\deg(f)$ too coarse, but we would also need to allow\nnegative coefficients in the polynomials $f_{i,j}$.\nTheorem~\\ref{bound2} provides a polynomial bound on $k,m$ and $t$\nunder a stronger log-concavity condition, but still needs the extra\nassumption that the coefficients in the polynomials $f_{i,j}$ are\nnonnegative. The unconditional lower bound in Theorem~\\ref{monotone}\nprovides a ``proof of concept'' of this approach for the easier\nsetting of monotone arithmetic circuits.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe intersection cohomology \nof a complex projective variety\nenjoys many of the good properties of the\nordinary cohomology of a smooth variety,\ncollectively known as the \\textit{K\\\"{a}hler package} \n(Poincar\\'{e} duality, weak and hard Lefschetz,\nHodge decomposition and Hodge signature theorem).\nThese properties deal primarily with the\nintersection cohomology group \nthat has attracted most of the attention from algebraic topologists and geometers:\nthe middle-perversity intersection cohomology group.\nHowever, there is additional geometric information carried by other\nintersection cohomology groups, as well as\nby cohomological operations that are defined when allowing other perversities than the middle one\n(such as cup products or Steenrod squares).\nIt is in this context, that Goresky raised the following question in the introduction of \\cite{Goresky}:\n\\begin{quotation}\n{\\small ``It remains as open question whether there is an intersection \nhomology-analogue\nto the rational homotopy theory of Sullivan. For example,\none would like to know when Massey triple products are defined\nin intersection homology\nand whether they always vanish on a (singular) projective algebraic variety''.}\n\\end{quotation}\nThe first part of Goresky's question has been answered by Chataur-Saralegi-Tanr\\'{e}\nin the foundational work \\cite{CST} on rational intersection homotopy theory,\nwhere the \\textit{perverse algebraic model} \nof a topological pseudomanifold is introduced. This is \na perverse commutative differential graded algebra (perverse cdga for short) defined over the rationals,\nwhose cohomology is isomorphic to the rational\nintersection cohomology with all perversities\nand is, when forgetting its multiplicative structure, quasi-isomorphic to the intersection cochains\noriginally introduced by\nGoresky and MacPherson \\cite{GMP1,GMP2}. In general, the perverse algebraic model \ncontains more information than the intersection cohomology ring (for instance, it contains the Massey products) and\ngives rise to a well-defined notion of intersection-formality for \ntopological pseudomanifolds, analogously to the notion of formality \nappearing in the classical rational homotopy theory of Sullivan \\cite{Su}.\n\nOther significant contributions in this direction are the homotopy theory of perverse cdga's\ndeveloped by Hovey \\cite{Hov2} within the context of Quillen model categories,\nthe works of Friedman \\cite{Friedman} and Friedman-McClure \\cite{FMC} \non intersection pairings and cup products in intersection cohomology respectively\nand Banagl's theory of intersection spaces \\cite{Banagl}.\n\n\nThe present work draws its main motivation from the second\npart of Goresky's question, which is almost equivalent to asking whether\nsingular complex projective varieties are intersection-formal.\nThis question is legitimated by a well-known application of the Hodge decomposition to topology: the Formality Theorem \nof Deligne-Griffiths-Morgan-Sullivan \\cite{DGMS}, which states that\nthe rational homotopy type of a compact K\\\"{a}hler manifold\nis entirely determined by its cohomology ring.\n\nIn general, the Hodge decomposition on the intersection cohomology of a singular projective variety\nfails for perversities other than the middle one. Instead, \neach intersection cohomology group carries a mixed Hodge structure. Since the perverse algebraic model\ndepends on all perversities, we do not\nexpect an intersection-analogous statement of the Formality Theorem for singular projective varieties,\nbut of a generalization of this statement involving the weight spectral sequence.\n\nIn this paper, we study the rational intersection homotopy type\nof complex projective varieties with only isolated singularities,\nvia mixed Hodge theory.\n\\\\\n\nWe next explain the contents and main results of this paper. For the rest of this introduction, let $X$ be a complex\nprojective variety with only isolated singularities.\n\nIn Section $\\ref{Section_RIHT}$, we\ncollect preliminary definitions and results\non intersection cohomology and on the homotopy theory of perverse cdga's.\nFollowing \\cite{CST}, we\ndescribe the perverse algebraic \nmodel $I\\mathcal{A}_{\\ov\\bullet}(X)$ of $X$.\nThis can be computed from the morphism\nof rational algebras of piece-wise linear forms $\\mathcal{A}_{pl}(X_{reg})\\to \\mathcal{A}_{pl}(L)$\ninduced by the inclusion $L\\hookrightarrow X_{reg}$\nof the link $L$ of the singularities into the regular part of $X$.\n\nSection $\\ref{Section_MHS}$ is the core of this paper. \nIn this section, we endow the perverse algebraic model $I\\mathcal{A}_{\\ov\\bullet}(X)$\nof $X$ with natural mixed Hodge structures \n(this result is stated in a more technical form in Theorem $\\ref{MHSmodel}$). Our proof relies, first, on the existence of \nmixed Hodge structures on the rational homotopy types of $X_{reg}$ and $L$ due to Morgan \\cite{Mo} and \nDurfee-Hain \\cite{DH} respectively, and second, on the existence of relative models of mixed Hodge diagrams\nproven by Cirici-Guill\\'{e}n in \\cite{CG1}. \nAs an important application of the existence of mixed Hodge structures on the perverse algebraic model,\nwe study the \\textit{perverse weight spectral sequence} \n$IE_{1,\\ov \\bullet}^{*,*}(X)$,\na perverse differential bigraded algebra\nwhose cohomology computes the weight filtration on the intersection cohomology:\n$IE_{2,\\ov \\bullet}^{*,*}(X):=H^{*,*}(IE_{1,\\ov \\bullet}(X))\\cong Gr_{\\bullet}^WIH^*_{\\ov\\bullet}(X;\\mathbb{Q})$.\nIn Theorem $\\ref{IE1formality}$, we prove that \nthe complex intersection homotopy type of $X$ is a direct consequence of \nits perverse weight spectral sequence. In other words: there is a string of quasi-isomorphisms\nof perverse cdga's from\n$I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes\\mathbb{C}$ to \n$IE_{1,\\ov \\bullet}(X)\\otimes\\mathbb{C}$. This result descends to the rationals for perverse cdga's of finite type \nand is the intersection-analogue of the main result of \\cite{CG1},\nwhich in turn is the generalization to singular varieties,\nof the Formality Theorem of \\cite{DGMS}. \nAs in the classical setting,\nthe perverse weight spectral sequence can be described \nin terms of the cohomologies of varieties associated with a resolution of singularities of $X$.\nHence Theorem $\\ref{IE1formality}$ implies that the complex intersection homotopy type of\n$X$ has a finite-dimensional model, determined by\ncohomologies of smooth projective varieties.\n\nThe last two sections contain applications of Theorem $\\ref{IE1formality}$.\nIn Section $\\ref{Section_OIS}$, we prove\nthat if $X$\nadmits a resolution of singularities \nin such a way that the exceptional divisor is smooth, and if the link\nof each singular point is $(n-2)$-connected, where $n$ is the complex dimension of $X$, then $X$ is GM-intersection-formal\nover $\\mathbb{C}$ (the prefix GM accounts for Goresky-MacPherson,\nsince we consider finite perversities only).\nThe main class of examples to which this result applies are varieties with \nordinary multiple points, but it also applies to a large family of\nhypersurfaces and more generally, to complete\nintersections\nadmitting a resolution of singularities with smooth exceptional divisor. \nThis extends a result of \\cite{CST}, where it is shown that any\nnodal hypersurface of $\\mathbb{C}\\mathbb{P}^4$ is intersection-formal.\nLikewise, in Section $\\ref{Section_OIS}$ we prove GM-intersection-formality over $\\mathbb{C}$\nfor every isolated surface singularity.\nIf a variety is (GM)-intersection-formal,\nthen its normalization is formal in the classical sense.\nWe remark that these results generalize our previous work\n\\cite{ChCi1}, where we study the (classical) rational homotopy type of complex \nprojective varieties with normal isolated singularities,\nusing the multiplicative weight spectral sequence.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Rational intersection homotopy types}\\label{Section_RIHT}\nIn this preliminary section, we recall the description \nof the intersection cohomology of a complex projective variety with only isolated singularities\nappearing in \\cite{GMP1}, as well as its main properties. Then, we\nintroduce the notion of rational intersection homotopy equivalence and study its relation with\nthe classical notion of rational homotopy equivalence.\nLastly, we collect the necessary definitions and results\non the homotopy theory of perverse differential graded algebras,\nsuch as the intersection-analogous notions of quasi-isomorphism and formality,\nand describe the perverse algebraic model of a complex\nprojective variety with only isolated singularities, following \\cite{CST}.\n\n\n\\subsection{Intersection cohomology}\\label{sectionintersectioncohomology}\nIntersection cohomology is defined for any topological pseudomanifold and\ndepends on the choice of a multi-index called \\textit{perversity},\nwhich measures how far cycles are allowed to deviate from transversality.\nFor a complex projective variety of dimension $n$ having only isolated singularities,\na perversity $\\ov p$ is determined by a single integer $p$ such that\n$0\\leq p\\leq 2n-2$. We will denote by $\\mathcal{P}$ the totally ordered set of such perversities.\nThere are three distinguished elements in $\\mathcal{P}$ that we shall refer to:\nthe $\\ov{0}$-perversity $\\ov{0}=0$, the middle perversity $\\ov{m}=n-1$\nand the top perversity $\\ov{t}=2n-2$. The complementary perversity of $\\ov p\\in\\mathcal{P}$ is given by $\\ov{t}-\\ov{p}:=2n-2-p$.\nNote that the middle perversity is complementary to itself.\nWe enlarge the set of perversities $\\widehat{\\mathcal{P}}=\\mathcal{P}\\cup \\{\\ov{\\infty}\\}$ \nby adjoining the $\\ov{\\infty}$-perversity. We define\nthe sum of two perversities $\\ov p$ and $\\ov q$ in $\\widehat{\\mathcal{P}}$ by letting\n$\\ov p+\\ov q:=\\ov{p+q}$ if $p+q\\leq 2n-2$ and $\\ov p+\\ov q:=\\ov\\infty$ otherwise.\n\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities. Denote by\n$\\Sigma$ the singular locus of $X$ and by $X_{reg}:=X-\\Sigma$ its regular part.\nThe intersection cohomology of $X$ with perversity $\\ov p$ (and coefficients in a commutative ring $R$) is given by\n(see $\\S$6.1 of \\cite{GMP1})\n$$\nIH^k_{\\ov{p}}(X;R)=\\left\\{\n\\begin{array}{ll}\nH^k(X_{reg};R)&\\text{ if }k\\leq p\\\\\n\\mathrm{Im }\\left(H^k(X;R)\\longrightarrow H^k(X_{reg};R)\\right)&\\text{ if }k=p+1\\\\\nH^k(X;R)&\\text{ if }k>p+1\n\\end{array}\n\\right..\n$$\n\nFor the $\\ov{0}$-perversity we have an isomorphism of graded algebras $IH^*_{\\ov0}(X;R)\\cong H^*(\\overline{X};R)$, \nwhere $\\overline{X}\\to X$ is a normalization of $X$\n(see $\\S$4 of \\cite{GMP1}). \nFor the $\\ov\\infty$-perversity we recover the cohomology ring $IH^*_{\\ov{\\infty}}(X;R)\\cong H^*({X}_{reg};R)$ \nof the regular part of $X$ (see \\cite{CST}).\nA main feature of intersection cohomology is that, when $R=\\mathbb{Q}$,\nfor every finite perversity $\\ov p\\in \\mathcal{P}$ we have a Poincar\\'{e} duality isomorphism (see $\\S$3.3 of \\cite{GMP1})\n$$IH^k_{\\ov{p}}(X;\\mathbb{Q})\\cong (IH^{2n-k}_{\\ov{t}-\\ov{p}}(X;\\mathbb{Q}))^\\vee:=\\mathrm{Hom}(IH^{2n-k}_{\\ov{t}-\\ov{p}}(X;\\mathbb{Q}),\\mathbb{Q}).$$\n\nThe graded objects $IH_{\\ov p}^*(X;R)$ together with the morphisms\n$IH_{\\ov p}^*(X;R)\\longrightarrow IH_{\\ov q}^*(X;R)$ for every pair $\\ov p\\leq \\ov q$,\nand the products\n$IH_{\\ov p}(X;R)\\otimes IH_{\\ov q}(X;R)\\longrightarrow IH_{\\ov p+\\ov q}(X;R)$ \ninduced by the cup products of $H^*(X;R)$ and $H^*(X_{reg};R)$\nfor any pair $\\ov p,\\ov q\\in \\widehat \\mathcal{P}$,\nconstitute the prototypical example of a \\textit{perverse commutative graded $R$-algebra}:\nthis is a commutative monoid in the category\nof functors from $\\widehat\\mathcal{P}$ to the category of graded $R$-modules.\n\nDenote by $\\mathcal{V}_\\mathbb{C}$ the category whose objects are complex projective varieties with only isolated singularities \nand whose morphisms $f:X\\longrightarrow Y$ satisfy $f(X_{reg})\\subset Y_{reg}$.\nThe above formula defines a contravariant functor $IH_{\\ov\\bullet}^*(-;R):\\mathcal{V}_\\mathbb{C}\\longrightarrow \\pga{R}$\nwith values in the category of perverse commutative graded $R$-algebras.\n\n\n\n\n\\subsection{Intersection homotopy equivalence}\nThe consideration of the intersection cohomology ring with all perversities\nleads to a natural notion of rational intersection homotopy equivalence.\n\n\n\\begin{defi}Let\n$f:X\\longrightarrow Y$ be a morphism between simply connected topological pseudomanifolds, such that $f(X_{reg})\\subset Y_{reg}$.\nThen $f$ is said to be a \\textit{rational intersection homotopy equivalence}\nif it induces an isomorphism of perverse graded algebras $f^*:IH^*_{\\ov \\bullet}(Y;\\mathbb{Q})\\longrightarrow IH^*_{\\ov \\bullet}(X;\\mathbb{Q})$.\n\\end{defi}\n\nIf $f:X\\to Y$ is a rational intersection homotopy equivalence then the morphism induced on\nthe normalizations $\\overline f:\\overline X\\longrightarrow \\overline Y$ is a rational\nhomotopy equivalence. The following result exhibits how the notion of rational\nintersection homotopy equivalence is stronger than the classical notion of rational homotopy equivalence.\n\n\\begin{prop}\\label{proj_cones_htp_equiv}\nLet $S$ and $S'$ be two simply connected smooth projective surfaces of $\\mathbb{C}\\mathbb{P}^n$. \nDenote by $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ the projective cones of $S$ and $S'$ respectively.\nThen:\n\\begin{enumerate}[(1)]\n \\item $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally homotopy equivalent if and only if $\\mathcal{X}(S)=\\mathcal{X}(S')$.\n \\item $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally intersection homotopy equivalent if and only if $S$ and $S'$ \n are rationally homotopy equivalent.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nLet $w\\in H^2(S;\\mathbb{Q})$ denote the Poincar\\'{e} dual of the\nhyperplane section of $S\\subset\\mathbb{C}\\mathbb{P}^n$. Since $w^2\\neq 0$, using\nPoincar\\'{e} duality we obtain an orthogonal decomposition \n$H^2(S;\\mathbb{Q})\\cong\\mathbb{Q}\\langle w\\rangle \\oplus^\\bot V.$\nThe projective cone $\\mathbb{P}_cS$ of $S$ is isomorphic to\nthe Thom space of the restriction $S(1)$ of the hyperplane bundle on $\\mathbb{C}\\mathbb{P}^n$ to $S$.\nThe rational cohomology algebra of $\\mathbb{P}_cS$ \ncan be written as $H^*(\\mathbb{P}_cS;\\mathbb{Q})\\cong \\mathbb{Q}\\langle Th\\rangle \\oplus V'$,\nwhere $Th$ has degree $2$ and satisfies $Th^4=0$ and $V'$ is a vector space of degree 4.\nThom's isomorphism $\\cup Th: H^*(S;\\mathbb{Q})\\to \\widetilde H^*(\\mathbb{P}_cS;\\mathbb{Q})$ identifies $w$ with $Th^2$\nand $V$ with $V'$. Furthermore, $Th\\cup V'=0$. This proves (1).\nThe intersection cohomology of $\\mathbb{P}_cS$ can be written as:\n\\begin{equation*}\nIH^s_{\\ov p}(\\mathbb{P}_cS;\\mathbb{Q})\\cong\n\\def1.4{1.4}\n\\begin{array}{| c | c | c | }\n\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov m$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov t$}}}\\\\\n\\hline\n\\mathbb{Q}\\langle Th^3\\rangle &\\mathbb{Q}\\langle Th^3\\rangle&\\mathbb{Q}\\langle Th^3\\rangle \\\\ \\hline\n0&0&0\\\\ \\hline\n\\mathbb{Q}\\langle Th^2\\rangle\\oplus V' &\\mathbb{Q}\\langle Th^2\\rangle\\oplus V' & H^4(S;\\mathbb{Q}) \\\\ \\hline\n0&0&0 \\\\ \\hline\n\\mathbb{Q}\\langle Th\\rangle &\\mathbb{Q}\\langle w\\rangle \\oplus V& \\mathbb{Q}\\langle w\\rangle \\oplus V \\\\\\hline\n0 &0&0 \\\\\\hline\n\\mathbb{Q} &\\mathbb{Q}&\\mathbb{Q} \\\\\\hline\n\\end{array}\n\\def1.4{1.4}\n\\begin{array}{ l }\n\\\\\n\\text{\\tiny{$s=6$}}\\\\\n\\text{\\tiny{$s=5$}}\\\\\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\n\\end{array}\n\\end{equation*}\n\nwhere the product $IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Q})\\otimes IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Q})\\longrightarrow IH_{\\ov t}^4(\\mathbb{P}_cS;\\mathbb{Q})\\cong H^4(S;\\mathbb{Q})=\\mathbb{Q}$\ncorresponds to the product on $H^2(S;\\mathbb{Q})$ and determines the signature of $S$.\nThis proves (2).\n\\end{proof}\n\n\\begin{example}\nLet $S$ be a K3-surface and let $S'$ be the projective plane blown-up at 19 \npoints. Then $\\mathcal{X}(S)=\\mathcal{X}(S')=24$, $Sign(S)=(3,19)$ and $Sign(S')=(1,21)$.\nTherefore $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are rationally homotopy equivalent, but not rationally \nintersection homotopy equivalent.\n\\end{example}\n\n\n\\subsection{Integral intersection cohomology}\nWe prove an analogous statement of Proposition $\\ref{proj_cones_htp_equiv}$\nfor intersection cohomology with integer coefficients.\n\n\\begin{prop}\\label{proj_cone_homeo}\nLet $S$ and $S'$ be two simply connected smooth projective surfaces of $\\mathbb{C}\\mathbb{P}^n$. \nThen their projective cones $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are homeomorphic if and only if\n$IH^*_{\\ov \\bullet}(\\mathbb{P}_cS;\\mathbb{Z})$ and $IH^*_{\\ov \\bullet}(\\mathbb{P}_cS';\\mathbb{Z})$ are isomorphic as perverse graded algebras.\n\\end{prop}\n\\begin{proof}\nWe follow the notation of the proof of Proposition $\\ref{proj_cones_htp_equiv}$.\nThe intersection cohomology algebra of $\\mathbb{P}_cS$ is given by:\n\\begin{equation*}\nIH^s_{\\ov p}(\\mathbb{P}_cS;\\mathbb{Z})\\cong\n\\def1.4{1.4}\n\\begin{array}{| c | c | c | }\n\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov 0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov m$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$\\ov p=\\ov t$}}}\\\\\n\\hline\n\\mathbb{Z}\\langle T\\rangle , Th^3=\\deg(S)\\cdot T&\\mathbb{Z}\\langle T\\rangle&\\mathbb{Z}\\langle T\\rangle \\\\ \\hline\n0&0&0\\\\ \\hline\nH^4(\\mathbb{P}_cS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z}) &H^4(\\mathbb{P}_cS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z}) & H^4(S;\\mathbb{Z})\\cong \\mathbb{Z} \\\\ \\hline\n0&0&0 \\\\ \\hline\n\\mathbb{Z}\\langle Th\\rangle &H^2(S;\\mathbb{Z})& H^2(S;\\mathbb{Z}) \\\\\\hline\n0 &0&0 \\\\\\hline\n\\mathbb{Z} &\\mathbb{Z}&\\mathbb{Z} \\\\\\hline\n\\end{array}\n\\def1.4{1.4}\n\\begin{array}{ l }\n\\\\\n\\text{\\tiny{$s=6$}}\\\\\n\\text{\\tiny{$s=5$}}\\\\\n\\text{\\tiny{$s=4$}}\\\\\n\\text{\\tiny{$s=3$}}\\\\\n\\text{\\tiny{$s=2$}}\\\\\n\\text{\\tiny{$s=1$}}\\\\\n\\text{\\tiny{$s=0$}}\n\\end{array}\n\\end{equation*}\nThe morphism \n$$H^2(\\mathbb{P}_sS;\\mathbb{Z})\\cong IH_{\\ov 0}^2(\\mathbb{P}_sS;\\mathbb{Z})\\longrightarrow IH_{\\ov m}^2(\\mathbb{P}_sS;\\mathbb{Z})\\cong H^2(S;\\mathbb{Z})$$\ndetermines up to sign a class $\\pm w\\in H^2(S;\\mathbb{Z})$ given by the image of a generator of\n$H^2(\\mathbb{P}_sS;\\mathbb{Z})$. We get line bundles $L_s^+$ and $L_s^-$ over $S$\nsatisfying $c_1(L_S^\\pm)=\\pm w$. Since these two bundles are isomorphic as rank 2 vector bundles,\ntheir Thom spaces $Th(L_S^\\pm)\\cong \\mathbb{P}_cS$ are homeomorphic.\n\nAssume that we have an isomorphism $\\Psi:IH_{\\ov\\bullet}(\\mathbb{P}_cS;\\mathbb{Z})\\longrightarrow IH_{\\ov\\bullet}(\\mathbb{P}_cS';\\mathbb{Z})$.\nThen the intersection forms of $S$ and $S'$ are equivalent, and it follows form Freedman's \nTheorem that $S$ and $S'$ are homeomorphic.\nFrom the commutative diagram\n$$\n\\xymatrix{\n\\ar[d]_\\Psi IH_{\\ov 0}^2(\\mathbb{P}_cS;\\mathbb{Z})\\ar[r]&IH_{\\ov m}^2(\\mathbb{P}_cS;\\mathbb{Z})\\ar[d]^\\Psi\\\\\nIH_{\\ov 0}^2(\\mathbb{P}_cS';\\mathbb{Z})\\ar[r]&IH_{\\ov m}^2(\\mathbb{P}_cS';\\mathbb{Z})\n}\n$$\nwe deduce that $\\mathbb{P}_cS$ and $\\mathbb{P}_cS'$ are homeomorphic.\n\\end{proof}\n\n\n\\begin{example}\nLet $S$ be a surface of degree $4$ in $\\mathbb{C}\\mathbb{P}^3$, let $S'$ be the intersection of a quadric\n and a cubic in $\\mathbb{C}\\mathbb{P}^4$, and let $S''$ be the intersection of three quadrics in $\\mathbb{C}\\mathbb{P}^5$.\nAll three surfaces are examples of K3-surfaces with different intersection cohomology algebras.\nHence their projective cones are non-homeomorphic.\n\\end{example}\n\n\nLet $S$ be a simply-connected 4-dimensional\nsmooth manifold and let $w\\in H^2(S;\\mathbb{Z})$.\nTo such a pair $(S,\\pm w)$ one can associate two homeomorphic Thom spaces\n$Th(L_w^\\pm)$.\nThe proof of Proposition $\\ref{proj_cone_homeo}$ is easily generalized \nto this setting. We have:\n\n\\begin{prop}\\label{equiv_pairs}\n Let $(S,\\pm w)$ and $(S',\\pm w')$ be two pairs. The following are equivalent:\n \\begin{enumerate}[(1)]\n \\item The pairs are topologically equivalent: there is a homeomorphism $\\phi:S\\to S'$ \nsuch that $\\phi^*(w)=w'$. \n \\item The line bundles $L_w^\\pm$ and $L_{w'}^\\pm$ are isomorphic as real vector bundles.\n \\item The Thom spaces $Th(L_w^\\pm)$ and $Th(L_{w'}^\\pm)$ are homeomorphic.\n \\item The integral intersection cohomologies $IH^*_{\\ov\\bullet}(Th(L_w^\\pm);\\mathbb{Z})$ and\n $IH^*_{\\ov\\bullet}(Th(L_{w'}^\\pm);\\mathbb{Z})$ are isomorphic as perverse graded algebras.\n \\end{enumerate}\n\\end{prop}\n\n\n\n\\subsection{Perverse differential graded algebras}\\label{perversecdgas}\nAs in the classical rational homotopy theory of Sullivan \\cite{Su}, the study of rational intersection homotopy \ntypes is closely related to the homotopy theory of\nperverse differential graded algebras. We next recall\nthe main definitions. Given our interest in varieties with only isolated singularities, we \nrestrict to the particular case where perversities \nare given by a single integer, and refer \\cite{Hov2} and \\cite{CST} for the general definitions, \nin which perversities are given by multi-indexes.\nFor the rest of this section we let $\\mathbf{k}$ be a field of characteristic 0.\n\n\\begin{defi}\\label{pcdga_def}\nA \\textit{perverse commutative differential graded algebra} \\textit{over $\\mathbf{k}$}\n is a commutative monoid in the category of functors from $\\widehat\\mathcal{P}$\nto the category $C^+(\\mathbf{Vect_\\mathbf{k}})$ of cochain complexes of $\\mathbf{k}$-vector spaces:\nthis is a bigraded $\\mathbf{k}$-vector \nspace $A_{\\ov\\bullet}^*=\\{A^i_{\\ov{p}}\\}$, with $i\\geq 0$ and $\\ov{p}\\in \\widehat\\mathcal{P}$,\ntogether with a linear differential $d:A^i_{\\ov{p}}\\to A^{i+1}_{\\ov{p}}$,\nan associative product $\\mu:A^i_{\\ov{p}}\\otimes A^j_{\\ov{q}}\\to A^{i+j}_{\\ov{p}+\\ov{q}}$ with unit $\\eta:\\mathbf{k}\\to A^0_{\\ov{0}}$ \nand a poset map $A^i_{\\ov{q}}\\to A^i_{\\ov{p}}$ for every $\\ov{q}\\leq \\ov{p}$. \nProducts and differentials satisfy the usual graded commutativity and graded Leibnitz rules, \nand are compatible with poset maps:\nfor all $\\ov{p}\\leq \\ov{p}'$ and $\\ov{q}\\leq \\ov{q}'$ the following diagrams commute:\n$$\\xymatrix{\nA_{\\ov p}\\otimes A_{\\ov q}\\ar[d]\\ar[r]^\\mu& A_{\\ov {p}+\\ov{q}}\\ar[d]\\\\\nA_{\\ov p'}\\otimes A_{\\ov q'}\\ar[r]^\\mu& A_{\\ov{p}'+ \\ov{q}'}\n}\\,\\,\\,\\,\\,\\,;\\,\\,\\,\\,\\,\\,\n\\xymatrix{\nA_{\\ov p}\\ar[d]\\ar[r]^d& A_{\\ov {p}}\\ar[d]\\\\\nA_{\\ov p'}\\ar[r]^d& A_{\\ov {p'}}\n}.\n$$\n\\end{defi}\nThe cohomology of a perverse cdga naturally inherits the structure of a perverse commutative graded algebra.\nDenote by $\\pdga{\\mathbf{k}}$ the category of perverse cdga's over $\\mathbf{k}$.\n\n\\begin{defi}\nA morphism of perverse cdga's $f:A_{\\ov\\bullet}\\to B_{\\ov\\bullet}$ is called \\textit{quasi-isomorphism}\nif for every perversity $\\ov p\\in\\widehat\\mathcal{P}$\nthe induced map $H^*(A_{\\ov p})\\to H^*(B_{\\ov q})$ is an isomorphism.\n\\end{defi}\n\nThe category $\\pdga{\\mathbf{k}}$ admits a Quillen model structure with quasi-isomorphisms as weak equivalences and\nsurjections as fibrations (see \\cite{Hov2}). The existence and uniqueness of minimal models of perverse cdga's\n\\`{a} la Sullivan\nis proven in \\cite{CST}.\nDenote by $\\mathrm{Ho}(\\pdga{\\mathbf{k}})$ the homotopy category of perverse cdga's, defined by inverting quasi-isomorphisms.\n\n\\begin{defi}\n A perverse cdga $A_{\\ov\\bullet}$ is said to be \\textit{intersection-formal} if there is an isomorphism in $\\mathrm{Ho}(\\pdga{\\mathbf{k}})$\nfrom $A_{\\ov\\bullet}$ to $H^*(A_{\\ov \\bullet})$.\n\\end{defi}\nNote that if a perverse cdga $A_{\\ov{\\bullet}}$ is intersection-formal, then both $A_{\\ov{0}}$ and $A_{\\ov{\\infty}}$\nare formal cdga's.\n\nWe shall consider the following weaker notion of intersection-formality, which excludes the infinite perversity.\nDenote by $\\GMpdga{\\mathbf{k}}$ the category of \\textit{GM-perverse cdga's} defined by replacing $\\widehat\\mathcal{P}$ by $\\mathcal{P}$ \nin Definition $\\ref{pcdga_def}$.\nNote that for a GM-perverse cdga $A_{\\ov\\bullet}$ the products\n$A_{\\ov p}\\otimes A_{\\ov q}\\longrightarrow A_{\\ov p + \\ov q}$ need only be defined whenever $\\ov p+\\ov q<\\ov \\infty$.\nThe prefix ``GM'' accounts for Goresky-MacPherson, since only finite perversities are involved.\nDenote by $\\mathrm{U}:\\pdga{\\mathbf{k}}\\longrightarrow \\GMpdga{\\mathbf{k}}$ the forgetful functor.\n\n\\begin{defi}\nA perverse cdga $A_{\\ov\\bullet}$ is said to be \\textit{GM-intersection-formal} if\nthere is an isomorphism in $\\mathrm{Ho}(\\GMpdga{\\mathbf{k}})$ from $A_{\\ov\\bullet}$ to $H^*(A_{\\ov \\bullet})$.\n\\end{defi}\nNote that if a $A_{\\ov{\\bullet}}$ is GM-intersection-formal, then $A_{\\ov{0}}$ is formal, but $A_{\\ov{\\infty}}$ \nneed not be formal.\nWe remark that intersection-formality implies the vanishing of Massey products in intersection cohomology,\nwhile GM-intersection-formality implies the vanishing of Massey products in $\\mathrm{U}(IH^*_{\\ov\\bullet}(A))$.\nWe refer to $\\S$3 of \\cite{CST} for a proof of these statements and further discussion on (GM)-intersection-formality.\n\n\n\n\\subsection{Perverse algebraic model}\nWe next describe the perverse algebraic model of a complex projective variety with only isolated singularities,\nas introduced in $\\S$3.2 of \\cite{CST}.\n\n\nLet us first fix some notation.\nDenote by $\\Lambda(t,dt)=\\mathbf{k}(t,dt)$ the free cdga over $\\mathbf{k}$ generated by $t$ in degree 0 and $dt$ in degree 1.\nFor $\\lambda\\in\\mathbf{k}$ denote by $\\delta_\\lambda:\\Lambda(t,dt)\\to \\mathbf{k}$ the evaluation map defined \nby $t\\mapsto \\lambda$ and $dt\\mapsto 0$.\nGiven a perversity $\\ov p\\in \\widehat{\\mathcal{P}}$, we will denote by\n$\\xi_{\\leq \\ov p}A(t,dt)$\nthe truncation \nof $A(t,dt)=A\\otimes \\Lambda(t,dt)$\nby perverse degree $\\ov p$, given in degree $k$ by:\n$$\\xi_{\\leq \\ov p}A(t,dt)^k=\\left\\{\n\\begin{array}{ll}\nA^k\\otimes\\Lambda(t)\\oplus A^{k-1}\\otimes\\Lambda(t)\\otimes dt&,\\text{ if } kp\n\\end{array}\n\\right..\n$$\nThis truncation is compatible with differentials, products and poset maps: \n$$d(\\xi_{\\leq \\ov p})\\subseteq \\xi_{\\leq \\ov {p}}\\text{ and }\\xi_{\\leq \\ov p}\\times \\xi_{\\leq \\ov q}\\subseteq \\xi_{\\leq \\ov {p}+\\ov{q}}\n\\text{ for all }\\ov p,\\ov q\\in\\widehat\\mathcal{P}, \\text{ and }\n\\xi_{\\leq \\ov {q}}\\subseteq \\xi_{\\leq \\ov p}\\text{ for all }\\ov q\\leq \\ov p.$$\n\n\\begin{defi}\\label{Pullbackpervers}\nLet $f:A\\to B$ be a morphism of cdga's over $\\mathbf{k}$. Given a perversity $\\ov p\\in\\widehat\\mathcal{P}$, consider the pull-back\nin the category of complexes of $\\mathbf{k}$-vector spaces:\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\n\\mathcal{I}_{\\ov{p}}(f)\\ar[r]&\\xi_{\\leq \\ov p}B(t,dt)\\ar[d]^{\\delta_1}\\\\\nA\\ar[r]^{f}&B\n}.\n$$\nSince $\\xi_{\\leq\\ov p}$ is compatible with differentials, products and poset maps, $\\mathcal{I}_{\\ov{\\bullet}}(f)$ with the products\nand differentials\ndefined component-wise, is a perverse cdga, called the \\textit{perverse cdga associated with $f$}.\n\\end{defi}\n\n\nLet $X$ be a complex projective variety with only isolated singularities.\nLet $T$ be a closed algebraic neighborhood of the singular locus $\\Sigma$ in $X$\n(in such a way that the inclusion $\\Sigma\\subset T$ is a homotopy equivalence, see \\cite{Durfee2}).\nThen the link of $\\Sigma$ in $X$ is $L:=\\partial T\\simeq T^*:=T-\\Sigma$.\nThe inclusion $\\iota:L\\hookrightarrow X_{reg}$ of the link into\nthe regular part of $X$ induces a morphism $\\iota^*:\\mathcal{A}_{pl}(X_{reg})\\to \\mathcal{A}_{pl}(L)$\nof cdga's over $\\mathbb{Q}$, between the rational algebras of piecewise linear forms of $X_{reg}$ and $L$. \n\n\\begin{defi}\nThe \\textit{perverse algebraic model for $X$} is the rational perverse cdga\n$I\\mathcal{A}_{\\ov{\\bullet}}(X):=\\mathcal{I}_{\\ov{\\bullet}}(\\iota^*)$ associated with the morphism $\\iota^*$. It is\ngiven by the pull-back diagrams\n$$\n\\xymatrix{\n\\ar@{}[dr]|{\\mbox{\\LARGE{$\\lrcorner$}}}\\ar[d]\nI\\mathcal{A}_{\\ov{p}}(X)\\ar[r]&\\xi_{\\leq \\ov p}\\mathcal{A}_{pl}(L)(t,dt)\\ar[d]^{\\delta_1}\\\\\n\\mathcal{A}_{pl}(X_{reg})\\ar[r]^{\\iota^*}&\\mathcal{A}_{pl}(L)\n}.\n$$\n\\end{defi}\nWe have an isomorphism of perverse commutative graded algebras $H^*(I\\mathcal{A}_{\\ov \\bullet}(X))\\cong IH^*_{\\ov\\bullet}(X;\\mathbb{Q})$.\nFor the $\\ov{0}$-perversity\nwe have a quasi-isomorphism of cdga's $I\\mathcal{A}_{\\ov0}(X)\\simeq \\mathcal{A}_{pl}(\\overline{X})$, where $\\overline{X}\\to X$ is a normalization of $X$.\nFor the $\\ov\\infty$-perversity we recover the rational homotopy type\n$I\\mathcal{A}_{\\ov{\\infty}}(X)\\simeq \\mathcal{A}_{pl}({X}_{reg})$ of the regular part of $X$.\n\n\nThe above construction defines a contravariant functor $I\\mathcal{A}_{\\ov\\bullet}:\\mathcal{V}_\\mathbb{C}\\longrightarrow \\mathrm{Ho}(\\pdga{\\mathbb{Q}})$\nfrom the category $\\mathcal{V}_\\mathbb{C}$ of complex projective varieties with only isolated singularities \nand stratified morphisms, to the the homotopy category of perverse cdga's over $\\mathbb{Q}$.\n\n\\begin{defi}Let $\\mathbb{Q}\\subset \\mathbf{K}$ be a field.\nA complex projective variety $X$ with isolated singularities is called \\textit{(GM)-intersection-formal\nover $\\mathbf{K}$} if and only if $I\\mathcal{A}_{\\ov\\bullet}(X)\\otimes \\mathbf{K}$ is (GM)-intersection-formal.\n\\end{defi}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Mixed Hodge Structures and Perverse Weight Spectral Sequence}\\label{Section_MHS}\nIn this section, we endow the perverse algebraic model of a complex projective variety $X$ with only isolated singularities,\nwith natural mixed Hodge structures.\nWe then study the perverse weight spectral sequence of $X$\nand prove that\nthe complex intersection homotopy type of $X$ is a direct consequence of \nits perverse weight spectral sequence.\nLastly, we describe the perverse weight spectral sequence in terms of the cohomologies\nof the varieties associated with a resolution of $X$.\n\n\\subsection{Mixed Hodge structures on intersection cohomology}\nDeligne showed that the rational cohomology ring\nof every complex algebraic variety $X$ is endowed with \\textit{mixed Hodge structures}:\nfor every $k\\geq 0$, there is an increasing filtration\n$W$ of the rational cohomology $H^k(X;\\mathbb{Q})$,\ncalled the \\textit{weight filtration}, together with a decreasing filtration \n$F$ of the complex cohomology $H^k(X;\\mathbb{C})$, called the \\textit{Hodge filtration}, \nin such a way that the filtration induced by $F$ and its complex conjugate $\\overline{F}$ on\nthe graded objects\n$Gr_m^WH^k(X;\\mathbb{C})\\cong Gr_m^WH^k(X;\\mathbb{Q})\\otimes\\mathbb{C}$\ndefine a Hodge decomposition of pure weight $m$.\nFurthermore, these filtrations are functorial and\ncompatible with products of varieties (we refer to \\cite{DeHII}, \\cite{DeHIII} or the book \\cite{PS} for details).\n\nIf $X$ is a complex projective variety with only isolated singularities,\nthe compatible mixed Hodge structures on the cohomologies\nof $X$ and $X_{reg}$ define canonical mixed Hodge structures on\n $IH^k_{\\ov{p}}(X;\\mathbb{Q})$, which are compatible with products and\nposet maps. In particular, for every $k\\geq 0$ the morphism\n $IH^k_{\\ov{0}}(X;\\mathbb{Q})\\to IH^k_{\\ov{\\infty}}(X;\\mathbb{Q})$ induced by the inclusion\n$X_{reg}\\hookrightarrow \\ov{X}$ preserves mixed Hodge structures.\n\nA well-known result on the mixed Hodge theory of projective varieties with isolated singularities\nis that for the middle perversity, \nthe weight filtration $W$ on $IH^k_{\\ov{m}}(X;\\mathbb{Q})$ is pure of weight $k$, for all $k\\geq 0$, that is:\n$0=W_{k-1}\\subset W_k=IH^k_{\\ov{m}}(X;\\mathbb{Q}).$\nThis is a consequence of Gabber's purity theorem and the decomposition theorem of\nintersection homology (see \\cite{Ste}. A direct proof using Hodge theory appears in \\cite{Na2}).\nWe next give the bounds on the weight filtration $W$ for an arbitrary perversity.\n\\begin{lem}\nLet $X$ be a complex projective variety of dimension $n$ with only isolated singularities.\n\\begin{enumerate}[(1)]\n\\item If $\\ov pn$, then the weight filtration $W$ on $IH^k_{{\\ov p}}(X;\\mathbb{Q})$ is pure of weight $k$.\n \\item If $\\ov p=n-1$ then the weight filtration $W$ on $IH^k_{{\\ov p}}(X;\\mathbb{Q})$ is pure of weight $k$ for, all $k\\geq 0$.\n\\item If $\\ov p>n-1$ then $0=W_{-1}\\subset W_0\\subset \\cdots \\subset W_{k}=IH^k_{{\\ov p}}(X;\\mathbb{Q})$.\nIf in addition, $kn$, the filtration $W$ on $H^k(X;\\mathbb{Q})$ is pure of weight $k$,\nwhile for $k0$, \ndenote by $D^{(r)}=\\bigsqcup_{|I|=r}D_I$ the disjoint union of all \n$r$-fold intersections \n$D_I:=D_{i_1}\\cap\\cdots \\cap D_{i_r}$ where $I=\\{i_1,\\cdots,i_r\\}$ denotes an ordered subset of $\\{1,\\cdots,N\\}$.\nSince $D$ has simple normal crossings, it follows that $D^{(r)}$ is a smooth projective variety of dimension $n-r$. \nFor $1\\leq k\\leq r$, denote by $j_{I,k}:D_I\\hookrightarrow D_{I\\setminus \\{i_k\\}}$ \nthe inclusion and let $j_{r,k}:=\\bigoplus_{|I|=r} j_{I,k}:D^{(r)}\\hookrightarrow D^{(r-1)}$.\nThese maps define a simplicial resolution\n$D_\\bullet=\\{D^{(r)}, j_{r,k}\\}$.\n\nLet $r\\geq 1$. For every $1\\leq k\\leq r$ we\nwill denote by $j_{r,k}^*:=(j_{r,k})^*:H^*(D^{(r-1)})\\to H^*(D^{(r)})$ the restriction morphism induced by the inclusion $j_{r,k}$ and\nby $\\gamma_{r,k}:=(j_{r,k})_!:H^{*-2}(D^{(r)})\\to H^*(D^{(r-1)})$ the corresponding Gysin map.\nWe have combinatorial restriction morphisms\n$$j_{(r)}^s:=\\sum_{k=1}^r (-1)^{k-1} (j_{r,k})^{*}:H^{s}(D^{(r-1)};\\mathbb{Q})\\longrightarrow H^s(D^{(r)};\\mathbb{Q})$$\nand combinatorial Gysin maps\n$$\\gamma_{(r)}^s:=\\sum_{k=1}^r (-1)^{k-1} (j_{r,k})_{!}:H^{s-2r}(D^{(r)};\\mathbb{Q})\\longrightarrow H^{s-2(r-1)}(D^{(r-1)};\\mathbb{Q}).$$\n\nWith this notation, the weight spectral sequence for $X_{reg}$ can be written as:\n$$\nE_1^{r,s}(X_{reg})=\n\\def1.4{1.6}\n\\begin{array}{c c c c c c c c c}\n\\multicolumn{1}{c}{}\\\\\n\\cdots&\\longrightarrow &H^{s-4}(D^{(2)};\\mathbb{Q})&\\xra{\\gamma^s_{(2)}}&H^{s-2}(D^{(1)};\\mathbb{Q})&\\xra{\\gamma^s_{(1)}}&H^s(\\widetilde X;\\mathbb{Q})&\\longrightarrow&0\\\\ \n\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}\n\\end{array}.\n$$\nIts algebra structure \nis given by the maps\n$H^{m}(D^{(p)};\\mathbb{Q})\\otimes H^{l}(D^{(q)};\\mathbb{Q})\\longrightarrow H^{m+l}(D^{(p+q)};\\mathbb{Q})$\ninduced by combinatorial restriction morphisms, for $p+q\\leq n$ (see \\cite{Mo}).\n\nWe next describe the multiplicative weight spectral sequence of the link $L\\simeq L(D,\\widetilde X)$.\nIn \\cite{Durfee}, Durfee endows the cohomology of the link of an\nisolated singularity with mixed Hodge structures, and describes\nits weight spectral sequence in terms of a resolution of singularities.\nHowever, such spectral sequence is not multiplicative,\nsince it is the spectral sequence associated with a mixed Hodge complex for $L$.\nTo describe the multiplicative weight spectral sequence of the link we\nanalyze the construction due to Durfee-Hain \\cite{DH} \nof a mixed Hodge diagram of cdga's for $L$.\n\nFor all $i\\in \\{1,\\cdots,N\\}$ define \n$$L_i:=L(D_i-\\bigsqcup_{i\\neq j} D_i\\cap D_j,\\widetilde X).$$\nFor all $r>0$ denote by $L^{(r)}=\\bigsqcup_{|I|=r}L_I$ the disjoint union of all \n$r$-fold intersections \n$L_I:=L_{i_1}\\cap\\cdots \\cap L_{i_r}$ where $I=\\{i_1,\\cdots,i_r\\}$ denotes an ordered subset of $\\{1,\\cdots,N\\}$.\nWe have\n$$L^{(1)}:=\\bigsqcup_i L_i,\\,\\, L^{(2)}:=\\bigsqcup_{i\\neq j}L_i\\cap L_j,\\cdots$$\nWe obtain a\nsimplicial manifold\n$L_\\bullet=\\{L^{(r)}, i_{r,k}\\}$,\nwhere $i_{r,k}:L^{(r)}\\hookrightarrow L^{(r-1)}$, for\n$1\\leq k\\leq r$, denote the natural inclusions.\n\nThe multiplicative weight spectral sequence for $L^{(r)}$ is given by:\n$$E_1^{*,*}(L^{(r)})=\\bigoplus_{I=\\{i_1,\\cdots,i_r\\}} E_1(D_I-\\mathrm{Sing}(D_I))\\widetilde\\otimes \\Lambda(\\theta_{i_1},\\cdots,\\theta_{i_r}).$$\nwhere $\\theta_k$ are generators of bidegree $(-1,2)$ \nand $\\widetilde \\otimes$ accounts for the fact that the differential of $\\theta_k$ is given by\n$d(\\theta_{k})=c_k$, where $c_k\\in H^2(D_{k};\\mathbb{Q})$ is the Chern class of $D_{k}$.\n\nThe multiplicative weight spectral sequence for $L$ is then given by the end\n$$E_1^{p,q}(L):=\\int_\\alpha \\bigoplus_mE_1^{p-m,q}(L^{(\\alpha)})\\otimes \\Omega_\\alpha^m,$$\nwhere $\\Omega_\\alpha$\nis the simplicial cdga given by\n$\\Omega_\\alpha:={{\\Lambda(t_0,\\cdots,t_\\alpha,dt_0,\\cdots,dt_\\alpha)}\/{\\sum t_i-1,\\sum dt_i}}$,\nwith $t_i$ of degree 0 and $dt_i$ of degree 1.\n\nIn Sections $\\ref{perverseordinary}$ and $\\ref{perversesurface}$ we provide a description\nof the morphism $E_1(X_{reg})\\longrightarrow E_1(L)$ in the particular cases of\nordinary isolated singularities and isolated surface singularities respectively,\nthus giving an explicit description of the perverse weight spectral sequence in these cases.\n\n\n\n\n\n\n\n\n\n\n\\section{Ordinary Isolated Singularities}\\label{Section_OIS}\nFor the rest of this section, let $X$ be a complex projective variety of dimension $n$ with isolated singularities. \nWe will show that if $X$ admits a resolution of singularities \nin such a way that the exceptional divisor is smooth, and if the link\nof each singular point is $(n-2)$-connected, then $X$ is GM-intersection-formal over $\\mathbb{C}$.\nThe main class of examples to which this result applies are varieties with \nordinary multiple points, but it also applies to a large family of\nhypersurfaces with isolated singularities and more generally, to complete\nintersections with isolated singularities\nadmitting a resolution of singularities with smooth exceptional divisor.\n\n\\subsection{Notation}\nDenote by $\\Sigma$ the singular locus of $X$ and by $X_{reg}=X-\\Sigma$ its regular part.\nDenote by $L:=L(\\Sigma,X)$ the link of $\\Sigma$ in $X$, and by $\\iota:L\\hookrightarrow X_{reg}$ the natural inclusion.\nSince $\\Sigma$ is discrete, the link $L$ can be\nwritten as a disjoint union $L=\\sqcup L_\\sigma$, where $L_\\sigma=L(\\sigma,X)$ is the link of $\\sigma\\in\\Sigma$\nin $X$.\n\nAssume that $X$ admits a resolution of singularities \n$f:\\widetilde X\\longrightarrow X$ of $X$ such that the exceptional\ndivisor $D:=f^{-1}(X)$ is smooth.\nDenote by\n$$j^k:H^k(\\widetilde X)\\longrightarrow H^k(D)\\text{ and }\\gamma^k:H^{k-2}(D)\\longrightarrow H^k(\\widetilde X)$$\nthe restriction morphisms and the Gysin maps \ninduced by the inclusion $j:D\\hookrightarrow \\widetilde X$.\nFor all $k\\geq 2$, define $j_{\\#}^k:=j^k\\circ \\gamma^k:H^{k-2}(D)\\longrightarrow H^{k}(D)$.\n\nUnless stated otherwise, all cohomologies are taking with rational coefficients.\n\n\\subsection{Perverse weight spectral sequence}\\label{perverseordinary}\nThe morphism $E_1(\\iota^*):E_1^{*,*}(X_{reg})\\longrightarrow E_1^{*,*}(L)$ of weight spectral sequences\ninduced by the inclusion $\\iota:L\\hookrightarrow X_{reg}$ can be written as:\n\n$$\n\\xymatrix@R=8pt@C=36pt{\nE_1^{r,s}(X_{reg})=\\ar[ddd]&\\ar[ddd]^{Id} H^{s-2}(D)\\ar[r]^{\\gamma^s}&\\ar[ddd]^{j^s} H^{s}(\\widetilde X)\\\\\n\\\\\n\\\\\nE_1^{r,s}(L)=& H^{s-2}(D)\\ar[r]^{j_{\\#}^s}& H^{s}(D)\\\\\n&\\text{\\tiny{$r=-1$}}&\\text{\\tiny{$r=0$}}&\n}\n$$\nThe algebra structure of $E_1^{*,*}(X_{reg})$ is induced by the \ncup product of $H^*(\\widetilde X)$, together with the maps\n$H^s(\\widetilde X)\\times H^{s'}(D)\\longrightarrow H^{s+s'}(D)$\ngiven by $(x,a)\\mapsto j^*(x)\\cdot a$.\nSince $\\gamma(a\\cdot j^*(x))=\\gamma(a)\\cdot x$,\nthis algebra structure is compatible with the differential $\\gamma$.\nThe non-trivial products of $E_1^{*,*}(L)$ are the maps $E_1^{0,s}(L)\\times E_1^{r,s'}(L)\\longrightarrow E_1^{r,s+s'}(L)$,\nwith $r\\in\\{0,1\\}$ and $s,s'\\geq 0$, induced by the cup product of $H^*(D)$.\n\n\nThe perverse weight spectral sequence $IE^{*,*}_{1,\\ov \\bullet}(X):=\\mathcal{I}_{\\ov{\\bullet}}(E_1(i^*))$ for $X$ \n can be written as:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{p}}(X)=\n\\def1.4{1.6}\n\\begin{array}{| c c c c c |}\n\\hline\nH^{s-2}(D)\\otimes\\Lambda(t)\\otimes t&\\longrightarrow&\\mathcal{J}^s_{1}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline\n\\mathrm{Ker }(j_{\\#}^s)\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes t&\\longrightarrow&\\mathcal{J}^s_{1}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline\nH^{s-2}(D)\\otimes\\Lambda(t)&\\longrightarrow&\\mathcal{J}^s_{0}\\oplus H^{s-2}(D)\\otimes\\Lambda(t)\\otimes dt&\\longrightarrow&H^{s}(D)\\otimes\\Lambda(t)\\otimes dt\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}\n\\end{array}\n\\begin{array}{ l }\n\\text{\\tiny{$s>p+1$}}\\\\\n\\text{\\tiny{$s=p+1$}}\\\\\n\\text{\\tiny{$sp+1$}}\\\\\n\\text{\\tiny{$s=p+1$}}\\\\\n\\text{\\tiny{$sp+1\n\\end{array}\n\\right..\n$$\n\n\nThe following is straightforward.\n\\begin{lem}\\label{PD}\nFor all $0\\leq s\\leq n$ have Poincar\\'{e} duality isomorphisms\n$$\\mathrm{Coker }(\\gamma^{n+s})\\cong \\mathrm{Ker }(j^{n-s})^\\vee\\text{ and }\\mathrm{Ker }(\\gamma^{n+s})\\cong \\mathrm{Coker }(j^{n-s})^\\vee.$$\n\\end{lem}\n\n\n\\subsection{Conditions on the cohomology of the link}\nSince $\\dim(\\Sigma)=0$, the weight filtration on the cohomology of the link\nis semi-pure: the weights on $H^k(L)$ are less than or equal to $k$ for $kn+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$sn+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s1$.\nWe will define $M_{\\ov\\bullet}$ step by step, for the perversities $\\ov 0$, $\\ov m$, $\\ov n$, $\\ov t$ and $\\ov \\infty$.\nWe begin with the $\\ov 0$-perversity. Let\n$M_{\\ov{0}}$ be the bigraded complex with trivial differential given by\n$$\nM_{\\ov{0}}^{r,s}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c | }\n\\hline\n\\,\\,\\,\\,\\,\\,0\\,\\,\\,\\,\\,\\,&$H^{2n}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n-1})$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s\\geq n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s\\leq n$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine\n$M_{\\ov{n}}$ as the bigraded sub-complex of $IE_{1,\\ov{n}}(X)$ given by\n$$\nM^{r,s}_{\\ov{n}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes t$&$H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\ \\hline\n$\\mathrm{Ker }(j_{\\#}^{n+1})$&$\\mathrm{Ker }(j^{n+1})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes (t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s>}[ddd]& \\\\\n\\\\\n\\\\\nIE_{2,\\ov{n}}^{*,n+1}(X)=&\\mathrm{Ker }(\\gamma^{n+1})\\ar[r]^{0}&\\mathrm{Coker }(\\gamma^{n+1})&\n}.\n$$\nIn degree $2n$ we have a commutative diagram\n$$\n\\xymatrix@R=5pt{\nM^{*,2n}_{\\ov{n}}=&\\ar@{->>}[ddd]_{}H^{2n-2}(D)\\otimes t\\ar[r]^-{d}&H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt\\ar@{->>}[ddd]^\\pi& \\\\\n\\\\\n\\\\\nIE_{2,\\ov{n}}^{*,2n}(X)=&0\\ar[r]^{0}&H^{2n}(\\widetilde X)&\n}\n$$\nwhere \n$d(a\\cdot t)=(\\gamma^{2n}(a), a\\cdot dt)$ and \n$\\pi(x,a\\cdot dt)=\\gamma^{2n}(a)-x$.\nThis gives quasi-isomorphisms of complexes \n$IE_{1,\\ov{n}}(X)\\stackrel{\\sim}{\\longleftarrow} M_{\\ov{n}}\\stackrel{\\sim}{\\longrightarrow} IE_{2,\\ov{n}}(X)$\ncompatible with the inclusion $M_{\\ov m}\\to M_{\\ov n}$.\n\nThe $\\ov t$-perversity weight spectral sequence for $X$ is given by:\n\\begin{equation*}\\resizebox{1\\hsize}{!}{$\nIE^{r,s}_{1,\\ov{t}}(X)=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes \\Lambda(t)\\otimes t$&$H^{2n}(X)\\oplus H^{2n-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$0$\\\\ \\hline\n$H^{s-2}(D)\\otimes \\Lambda(t)$&$\\left(H^s(\\widetilde X)\\oplus_{H^s(D)}H^s(D)\\Lambda(t)\\right)\\oplus H^{s-2}(D)\\otimes \\Lambda(t)\\otimes dt$&$H^{s}(D)\\otimes \\Lambda(t)\\otimes dt$\\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s<2n$}\\\\\n\\multicolumn{1}{c}{\\tiny{}}\n\\end{tabular}\n$}\\end{equation*}\n\nDefine\n$M_{\\ov{t}}$ as the bigraded sub-complex of $IE_{1,\\ov{t}}(X)$ given by\n$$\nM^{r,s}_{\\ov{t}}=\n\\arraycolsep=18pt\\def1.4{1.6}\n\\begin{tabular}{| c | c | c |}\n\\hline\n$H^{2n-2}(D)\\otimes t$&$H^{2n}(\\widetilde X)\\oplus H^{2n-2}(D)\\otimes dt$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^s)$&0\\\\ \\hline\n$H^{n-1}(D)$&$H^{n+1}(\\widetilde X)$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n})$&0\\\\ \\hline\n0&$\\mathrm{Ker }(j^{n+1})^\\vee\\oplus \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\otimes (t-1)$&$\\mathrm{Ker }(\\gamma^{n+1})^\\vee \\otimes dt$\\\\ \\hline\n0&$\\mathrm{Ker }(j^{s})$&0\\\\ \\hline\n0&$H^0(\\widetilde X)$&0 \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\tiny{$r=-1$}}&\\multicolumn{1}{c}{\\tiny{$r=0$}}&\\multicolumn{1}{c}{\\tiny{$r=1$}}\n\\end{tabular}\n\\,\\,\\,\\,\\,\n\\begin{tabular}{ l }\n\\tiny{$s=2n$}\\\\\n\\tiny{$s>n+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$sn+1$}\\\\\n\\tiny{$s=n+1$}\\\\\n\\tiny{$s=n$}\\\\\n\\tiny{$s=n-1$}\\\\\n\\tiny{$s0$, the only \nnon-trivial product in $M_{\\ov 0}^{1,*}$ is\n$H^{0}(\\widetilde X)\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\longrightarrow \\mathrm{Ker }(\\gamma^{n+1})^\\vee$.\nAlso, from the multiplicative structure of $IE_{2,\\ov 0}(X)$ we have\n$\\mathrm{Ker }(\\gamma^{n+1})^\\vee\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee=0.$\nThis proves that $M_{\\ov 0}\\times M_{\\ov 0}\\subseteq M_{\\ov 0}$ and that the map $\\varphi_{\\ov 0}:M_{\\ov 0}\\longrightarrow IE_{1,\\ov 0}(X)$ is a morphism of cdga's.\nTo see that $M_{\\ov 0}\\times M_{\\ov m}\\subseteq M_{\\ov m}$ it suffices to prove that\n$\\mathrm{Ker }(j^{n+1})^\\vee\\cdot \\mathrm{Ker }(\\gamma^{n+1})^\\vee=0.$\nThis follows from the algebra structure of $IE_{2,\\ov \\bullet}(X)$ together with the\ncorresponding Poincar\\'{e} duality isomorphisms.\nWe now show that $M_{\\ov 0}\\times M_{\\ov n}\\subseteq M_{\\ov n}$.\nNote that for all $s>0$ we have\n$\\mathrm{Ker }(j^s)\\cdot \\mathrm{Ker }(j_{\\#}^{n+1})=0.$\nThe remaining inclusions are trivial.\nTherefore $M_{\\ov\\bullet}$ is a perverse cdga and the inclusion $\\varphi_{\\ov \\bullet}:M_{\\ov \\bullet}\\longrightarrow IE_{1,\\ov \\bullet}(X)$ \nis a quasi-isomorphism of perverse cdga's.\n\nLastly, we show that for\nevery pair of perversitites $\\ov p,\\ov q\\in\\mathcal{P}$ such that $\\ov{p}+\\ov{q}<\\ov{\\infty}$, the diagram\n$$\n\\xymatrix{\n\\ar[d]^{\\psi_{\\ov p}\\otimes \\psi_{\\ov q}}M_{\\ov p}\\otimes M_{\\ov p}\\ar[r]&M_{\\ov p+\\ov q}\\ar[d]^{\\psi_{\\ov p+\\ov q}}\\\\\nIE_{2,\\ov p}(X)\\otimes IE_{2,\\ov q}(X)\\ar[r]&IE_{2,\\ov p+\\ov q}(X)\n}\n$$\ncommutes.\nThe only non-trivial cases are when $\\ov p=\\ov 0$ and $\\ov q=\\ov n$ or $q=\\ov t$.\nWe show that the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times H^0(\\widetilde X)\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov 0}}\\ar[r]^-{\\mu}&\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]_{\\psi_{\\ov n}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times H^0(\\widetilde X)\\ar[r]^-{\\mu}&\\mathrm{Ker }(\\gamma^{n+1})\n}\n$$\ncommutes, where $\\mu(a,x)=a\\cdot j^*(x)$. Recall that the morphism $\\psi_{\\ov n}:\\mathrm{Ker }(j_{\\#}^{n+1})\\longrightarrow \\mathrm{Ker }(\\gamma^{n+1})$ is defined by\ntaking a direct sum decomposition $\\mathrm{Ker }(j_{\\#}^{n+1})=\\mathrm{Ker }(\\gamma^{n+1})\\oplus C$ and choosing the projection to the first component.\nLet $(a,x)\\in \\mathrm{Ker }(j_{\\#}^{n+1})\\times H^0(\\widetilde X)$, and decompose $a=\\overline{a}+c$ with $\\overline{a}\\in \\mathrm{Ker }(\\gamma^{n+1})$ and $c\\in C$.\nThen $\\mu(a,x)=(\\overline{a}+c)\\cdot j^*(x)$. Since $\\gamma(\\overline{a}\\cdot j^*(x))=\\gamma(\\overline{a})\\cdot x=0$,\nit suffices to show that $c\\cdot j^*(x)\\in C$. Since $x=1\\in H^0(\\widetilde X)$ and $\\gamma(c)\\neq 0$, it follows that $\\gamma(c\\cdot j^*(x))=\\gamma(c)\\cdot x\\neq 0$.\nHence $c\\cdot j^*(x)\\in C$, and the above diagram commutes.\nWe next show that the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times \\mathrm{Ker }(\\gamma^{n+1})^\\vee\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov 0}}\\ar[r]^-{\\mu}&H^{2n-2}(D)dt\\ar[d]_{\\psi_{\\ov n}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times \\mathrm{Coker }(j^{n-1})\\ar[r]^-{\\mu}&H^{2n}(\\widetilde X)\n}\n$$\ncommutes.\nLet $(a,b)\\in \\mathrm{Ker }(j_{\\#}^{n+1})\\times \\mathrm{Ker }(\\gamma^{n+1})^\\vee$. Then\n$\\psi_{\\ov t}(\\mu(a,b))=\\gamma^{2n}(a\\cdot b).$\nOn the other hand we have \n$\\mu(\\psi_{\\ov t}(a),\\psi_{\\ov 0} b)=\\gamma^{2n}(\\overline{a}\\cdot b),$\nwhere $a=\\overline{a}+c$ is a decomposition such that $\\ov{a}\\in \\mathrm{Ker }(\\gamma^{n+1})$ and $c\\in C$.\nHence to prove that the above diagram\ncommutes, it suffices to see that $c\\cdot b=0$. This follows from the fact that $C\\cap \\mathrm{Ker }(\\gamma^{n+1})=\\{0\\}$ and $b\\in \\mathrm{Ker }(\\gamma^{n+1})^\\vee$.\nThis proves that \n$\\psi_{\\ov{0}}\\cdot \\psi_{\\ov{n}}=\\psi_{\\ov n}$.\nThe same arguments allow us to prove that $\\psi_{\\ov{0}}\\cdot \\psi_{\\ov{t}}=\\psi_{\\ov t}$.\nTherefore \n$\\psi_{\\ov \\bullet}$ is multiplicative for finite perversities, and \n$X$ is GM-intersection-formal over $\\mathbb{C}$.\n\nAssume now that $X$ has only one isolated singularity. Then $\\mathrm{Ker }(\\gamma^{2n})=0$ and the diagram\n$$\n\\xymatrix{\n\\mathrm{Ker }(j_{\\#}^{n+1})\\ar[d]\\times \\mathrm{Ker }(j^{n+1})^\\vee\\ar[d]_{\\psi_{\\ov n}\\times \\psi_{\\ov m}}\\ar[r]^-{\\mu}&H^{2n-2}(D)dt\\ar[d]_{\\psi_{\\ov \\infty}}\\\\\n\\mathrm{Ker }(\\gamma^{n+1})\\times \\mathrm{Coker }(j^{n-1})\\ar[r]^-{\\mu}&\\mathrm{Ker }(\\gamma^{2n})\n}\n$$\ncommutes. This proves that\n$\\psi_{\\ov{p}}\\cdot \\psi_{\\ov{q}}=\\psi_{\\ov p + \\ov q}$ for all $p,q\\in\\widehat\\mathcal{P}$. Hence in this case, $X$ is intersection-formal over $\\mathbb{C}$.\n\\end{proof}\n\n\\subsection{Applications}\nA singular point $\\sigma\\in X$ is called \\textit{ordinary} if there exists a neighborhood of $\\sigma$\nisomorphic to an affine cone $C_\\sigma$ with vertex $\\sigma$, over a\nsmooth hypersurface $S_\\sigma$ of $\\mathbb{C}\\mathbb{P}^n$.\nIn such case, the link $L_\\sigma$ of $\\sigma$ in $X$\nis a smooth real manifold of dimension $(2n-1)$ which is $(n-2)$-connected\nHence we have:\n\n\\begin{cor}\n Let $X$ be a complex projective variety with only ordinary isolated singularities.\n Then $X$ is GM-intersection-formal over $\\mathbb{C}$.\n Furthermore, if $X$ has only one singular point, then $X$ is intersection-formal over $\\mathbb{C}$.\n\\end{cor}\n\n\\begin{example}[Segre cubic]\nLet $S$ denote the set of points $(x_0:x_1:x_2:x_3:x_4:x_5)$ of $\\mathbb{C}\\mathbb{P}^5$\nsatisfying\n$x_0+x_1+x_2+x_3+x_4+x_5=0$ and $x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=0$.\nThis is a normal projective threefold with 10 isolated ordinary singular points,\nknown as the \\textit{Segre cubic}.\nA resolution of $S$ is given by the moduli space $f:\\ov{\\mathcal{M}}_{0,6}\\longrightarrow S$ of stable rational curves with 6 marked points,\nand $D:=f^{-1}(\\Sigma)=\\bigsqcup_{i=1}^{10} \\mathbb{C}\\mathbb{P}^1\\times\\mathbb{C}\\mathbb{P}^1$, where $\\Sigma=\\{\\sigma_1,\\cdots,\\sigma_{10}\\}$\ndenotes the singular locus of $S$.\nFor each $0\\leq i\\leq 10$ the link of $\\sigma_i$ in $S$ is homeomorphic to a product\nof spheres $L_i\\simeq S^2\\times S^3$. In particular, $L_i$ is simply connected.\nHence $S$ is GM-intersection-formal over $\\mathbb{C}$.\nThe intersection homotopy type of $S$ is determined\nby the perverse graded algebra $IH_{\\ov\\bullet}^*(S;\\mathbb{Q})$, which we next describe.\nThe rational cohomology of $\\ov{\\mathcal{M}}_{0,6}$ is well-known, with non-trivial Betti numbers:\n$b_0(\\ov{\\mathcal{M}}_{0,6})=b_6(\\ov{\\mathcal{M}}_{0,6})=1$ and \n$b_2(\\ov{\\mathcal{M}}_{0,6})=b_4(\\ov{\\mathcal{M}}_{0,6})=16$.\nLet $j^s:H^s(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})\\to H^s(D;\\mathbb{Q})$ denote the restriction map induced by the inclusion\n$j:D\\hookrightarrow \\ov{\\mathcal{M}}_{0,6}$, and $\\gamma^s:H^{s-2}(D;\\mathbb{Q})\\to H^s(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})$\nthe corresponding Gysin map.\nThe rational cohomology of $S$ is:\n$$\nH^*(S;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^4)\\cong \\mathbb{Q}^6$\\\\ \\hline\n$\\mathrm{Coker }(j^2)\\cong \\mathbb{Q}^5$\\\\ \\hline\n$\\mathrm{Ker }(j^2)\\cong \\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline\n\\end{tabular}\n$$\nNote that for $k\\neq 3$, the weight filtration on $H^k(S;\\mathbb{Q})$ is pure of weight $k$,\nwhile for $k=3$ we have a non-trivial weight filtration, with $Gr^W_3H^3(S;\\mathbb{Q})\\cong \\mathrm{Ker }(j^3)=0$ and\n$Gr^W_2H^3(S;\\mathbb{Q})\\cong H^3(S;\\mathbb{Q})\\cong \\mathbb{Q}^5.$\n\nDenote by $Van:=\\mathrm{Coker }(j^2)\\cong \\mathbb{Q}^5$ and let $Exc\\cong \\mathbb{Q}^5$ be defined via the direct sum decomposition\n$H^2(\\ov{\\mathcal{M}}_{0,6};\\mathbb{Q})\\cong \\mathrm{Ker }(j^2)\\oplus \\mathrm{Coker }(\\gamma^2)\\oplus Exc$.\nThe rational intersection cohomology of $S$ is given by:\n$$\nIH^*_{\\ov p}(S;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{0}\\leq \\ov{p}\\leq \\ov{1}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee\\oplus Exc^\\vee$\\\\ \\hline\n$Van$\\\\ \\hline\n$H^2(S;\\mathbb{Q})$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c |}\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov 2$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee \\oplus Exc^\\vee$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})\\oplus Exc$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{3}\\leq \\ov{p}\\leq \\ov{4}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$H^2(S;\\mathbb{Q})^\\vee$\\\\ \\hline\n$Van^\\vee$\\\\ \\hline\n$H^2(S;\\mathbb{Q})\\oplus Exc$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline \n\\end{tabular}\n$$\nNote that the weight filtration of $IH^*_{\\ov \\bullet}(S;\\mathbb{Q})$ is non-trivial, since\n$Gr^W_2IH^3_{\\ov 0}(S;\\mathbb{Q})\\cong Van\\neq 0$.\n\nSince $S$ is simply connected, and $IH_{\\ov 0}(S;\\mathbb{Q})\\cong H^*(S;\\mathbb{Q})$,\none may compute the rational homotopy groups $\\pi_*(S)\\otimes\\mathbb{Q}$ from a minimal model of\n$IH_{\\ov 0}^*(S;\\mathbb{Q})$, as done in Example 4.7 of \\cite{ChCi1}.\nLikewise, a perverse minimal model (in the sense of \\cite{CST}) of the perverse cdga $IH_{\\ov \\bullet}^*(S;\\mathbb{Q})$ would give\nthe ``rational intersection homotopy groups'' of $S$.\n\\end{example}\n\n\n\nFor a complete intersection $X$ with isolated singularities, the link of each singular point in $X$ is $(n-2)$-connected\n(this result is due to Milnor \\cite{Milnor} in the case of hypersurfaces\nand to Hamm \\cite{Hamm} for general complete intersections).\nAs another direct consequence of Theorem $\\ref{intersformal}$ we have:\n\n\\begin{cor}\nLet $X$ be a complete intersection with singular locus $\\Sigma$ of dimension 0.\nAssume that there exists a resolution of singularities $f:\\widetilde X\\to X$ such that\n$D=f^{-1}(\\Sigma)$ is smooth. Then $X$ is GM-intersection-formal over $\\mathbb{C}$.\n\\end{cor}\n\n\n\n\n\n\n\n\n\\section{Isolated Surface Singularities}\\label{Section_ISS}\nIn this last section we prove that isolated surface singularities are GM-intersection-formal over $\\mathbb{C}$.\n\n\\subsection{Notation}\nLet $X$ be a complex projective surface with only isolated singularities and denote by $\\Sigma$ the singular locus of $X$.\nLet $f:\\widetilde X\\longrightarrow X$ be a resolution of singularities of $X$ such that\n$D:=f^{-1}(\\Sigma)=D_1\\cup\\cdots\\cup D_N$ is a\nsimple normal crossings divisor.\nLet $\\widetilde D:=D^{(1)}=\\sqcup_i D_i$ and $Z:=D^{(2)}=\\sqcup_{i\\neq j} D_i\\cap D_j$. Then\n$\\widetilde D$ is a disjoint union of smooth projective curves and $Z$ is a finite collection of points.\nDenote by $j:\\widetilde D\\longrightarrow \\widetilde X$ the natural inclusion.\nLet $i_1:Z\\to \\widetilde D$ be the inclusion defined by $D_i\\cap D_j\\mapsto D_i$, for every $i>}[d]&\\ar@{->>}[d]H^3(\\widetilde X)&\\\\\nIE_{2,\\ov 2}^{*,3}(X)=&\\mathrm{Ker }(\\gamma^3)\\ar[r]^-0&\\mathrm{Coker }(\\gamma^3)\n}\n$$\nFor $s=4$ we have a commutative diagram\n$$\n\\xymatrix{\nM_{\\ov 2}^{*,4}=&\\mathrm{Ker }(\\eta)\\ar[r]^-{0}\\ar[d]^{Id}&\\ar[d]H^2(\\widetilde D)\\otimes t\\ar[r]^-{d}&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt\\ar@{->>}[d]^\\pi\\\\\nIE_{2,\\ov 2}^{*,4}(X)=&\\mathrm{Ker }(\\eta)\\ar[r]&0\\ar[r]&T\n}\n$$\nwhere $d(a\\cdot t)=(\\gamma^4(a),a\\cdot dt)$ and $\\pi(x,a\\cdot dt)=\\gamma^4(a)-x$.\nHence we have quasi-isomorphisms of complexes\n$IE_{1,\\ov 2}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov2}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov2}(X)$\ncompatible with the inclusion $M_{\\ov 1}\\longrightarrow M_{\\ov 2}$.\n\n\nThe $\\ov \\infty$-perversity weight spectral sequence $IE_{1,\\ov \\infty}(X)$ for $X$ is:\n\\begin{equation*}\n\\resizebox{1\\hsize}{!}{$\n\\def1.4{2}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)\\Lambda(t)&\nH^2(\\widetilde D)\\Lambda(t)\\oplus H^0(Z)\\Lambda(t)dt&\nH^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^1(\\widetilde D)\\Lambda(t)&\nH^3(\\widetilde X)\\oplus H^1(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\nH^0(\\widetilde D)\\Lambda(t)&\nH^2(\\widetilde X)\\oplus_{H^2(\\widetilde D)}\\left(H^2(\\widetilde D)\\Lambda(t)\\oplus H^0(\\widetilde D)\\Lambda(t)dt\\right)&\nH^2(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^1(\\widetilde X)\\oplus_{H^1(\\widetilde D)}H^1(\\widetilde D)\\Lambda(t)&\nH^1(\\widetilde D)\\Lambda(t)dt&\n\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&\nH^0(\\widetilde X)\\oplus_{H^0(\\widetilde D)}H^0(\\widetilde D)\\Lambda(t)t&\nH^0(Z)\\Lambda(t)(t-1)\\oplus H^0(\\widetilde D)\\Lambda(t)dt&\nH^0(Z)\\Lambda(t)dt\\\\\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{-2}}}&\\multicolumn{1}{c}{\\text{\\tiny{-1}}}&\\multicolumn{1}{c}{\\text{\\tiny{0}}}&\\multicolumn{1}{c}{\\text{\\tiny{1}}}&\\multicolumn{1}{c}{\\text{\\tiny{2}}}\n\\end{array}$}\n\\end{equation*}\n\nLet $M_{\\ov \\infty}$ be the bigraded sub-complex of $IE_{1,\\ov \\infty}(X)$ given by:\n\\begin{equation*}\nM_{\\ov \\infty}=\n\\def1.4{1.6}\n\\begin{array}{| c | c | c | c | c |}\n\\hline\nH^0(Z)&H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&H^1(\\widetilde D)&H^3(\\widetilde X)&\\cellcolor{black!8}&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&0&\\mathrm{Ker }(j^2)&0&\\cellcolor{black!8}\\\\\\hline\n\\cellcolor{black!8}&\\cellcolor{black!8}&H^1(\\widetilde X)\\oplus \\mathrm{Ker }(\\gamma^3)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\gamma^3)^\\vee\\otimes dt&\\cellcolor{black!8}\\\\\\hline\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\cellcolor{black!8}&H^0(\\widetilde X)&\\mathrm{Ker }(\\eta)^\\vee\\otimes (t-1)&\\mathrm{Ker }(\\eta)^\\vee\\otimes dt\\\\\\hline\\hline\n\\multicolumn{1}{c}{\\text{\\tiny{$r=-2$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=-1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=0$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=1$}}}&\\multicolumn{1}{c}{\\text{\\tiny{$r=2$}}}\n\\end{array}\n\\end{equation*}\nNote that for $s<4$ we have $M_{\\ov\\infty}^{*,s}=M_{\\ov t}^{*,s}$.\nIn degree $s=4$, the differential of $M_{\\ov\\infty}$ is given by the map\n$H^0(Z)\\to H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t$ defined by\n$z\\mapsto (\\eta(z),0)$ and the map $H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t\\to H^4(\\widetilde X)\\to H^2(\\widetilde D)\\otimes dt$\ndefined by $(a,b\\cdot t)\\mapsto (\\gamma^4(a)+\\gamma^4(b),b\\cdot dt)$.\nTo define a surjective morphism of complexes $\\psi_{\\ov \\infty}:M_{\\ov \\infty}\\to IE_{2,\\ov \\infty}(X)$ it suffices to define\n$\\psi_{\\ov \\infty}:M^{*,4}_{\\ov \\infty}\\to IE^{*,4}_{2,\\ov \\infty}(X)$.\nChoose a decomposition $H^0(Z)\\cong \\mathrm{Ker }(\\eta)\\oplus C_0$ and\nconsider the projection $H^0(Z)\\to \\mathrm{Ker }(\\eta)$ to the first component. Also,\nchoose a decomposition\n$H^2(\\widetilde D)\\cong \\mathrm{Ker }(\\gamma^4)\\oplus C_2$ and consider the composition\n$\\rho:H^2(\\widetilde D)\\twoheadrightarrow \\mathrm{Ker }(\\gamma^4)\\twoheadrightarrow \\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)$.\nThis gives a commutative diagram\n$$\n\\xymatrix{\nM_{\\ov \\infty}^{*,4}=&H^0(Z)\\ar[r]^-{d}\n\\ar@{->>}[d]&\\ar[d]^{(\\rho,0)}H^2(\\widetilde D)\\oplus H^2(\\widetilde D)\\otimes t\\ar[r]^-{d}&H^4(\\widetilde X)\\oplus H^2(\\widetilde D)\\otimes dt\\ar@{->>}[d]\\\\\nIE_{2,\\ov \\infty}^{*,4}(X)=&\\mathrm{Ker }(\\eta)\\ar[r]^-0&\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)\\ar[r]&0\n}.\n$$\nHence we have\nquasi-isomorphisms of complexes\n$IE_{1,\\ov \\infty}(X)\\stackrel{\\sim}{\\longleftarrow}M_{\\ov\\infty}\\stackrel{\\sim}{\\longrightarrow}IE_{2,\\ov\\infty}(X)$\ncompatible with the inclusion $M_{\\ov 2}\\longrightarrow M_{\\ov \\infty}$.\n\nConsider on $M_{\\ov\\bullet}$ the multiplicative structure induced by the inclusion \ninclusion $\\varphi_{\\ov\\bullet}:M_{\\ov\\bullet}\\to IE_{1,\\ov\\bullet}(X)$. It is a matter of verification to\nsee that this structure is closed in $M_{\\ov\\bullet}$, so that $\\varphi_{\\ov\\bullet}$\n is a morphism of perverse cdga's, which is a quasi-isomorphism.\n\nWe next show that for every pair of perversities $\\ov p$ and $\\ov q$ such that $\\ov p+\\ov q<\\ov \\infty$, the diagram\n$$\n\\xymatrix{\nM_{\\ov p}\\times M_{\\ov q}\\ar[d]_{(\\psi_{\\ov p}, \\psi_{\\ov q})}\\ar[r]^-{\\mu}&M_{\\ov p+\\ov q}\\ar[d]^-{\\psi_{\\ov p+\\ov q}}\\\\\nIE_{2,\\ov p}(X)\\times IE_{2,\\ov q}(X)\\ar[r]^-{\\ov \\mu}&IE_{2,\\ov p+\\ov q}(X)\n}\n$$\ncommutes, so that $\\psi_{\\ov\\bullet}$ is multiplicative for finite perversities. The only non-trivial case is\n$$\n\\xymatrix{\nH^1(\\widetilde D)\\times \\mathrm{Ker }(\\gamma^3)^\\vee\\ar[d]_-{(\\psi_{\\ov 2},\\psi_{\\ov 0})}\\ar[r]^-{\\mu}&H^2(\\widetilde D)\\ar[d]^-{\\psi_{\\ov 2}}\\\\\n\\mathrm{Ker }(\\gamma^3)\\times \\mathrm{Ker }(\\gamma^3)^\\vee\\ar[r]^-{\\ov \\mu}&H^4(\\widetilde X)\n}.\n$$\nLet $(a,b)\\in H^1(\\widetilde D)\\times \\mathrm{Ker }(\\gamma^3)^\\vee$. Then $\\psi_{\\ov 2}\\mu(a,b)=\\gamma^4(a\\cdot b)$.\nLet $a=\\ov a+c$ be a decomposition of $a$ such that $a\\in \\mathrm{Ker }(\\gamma^3)$ and $c\\in C_1$.\nThen $\\ov \\mu(\\psi_{\\ov 2}(a),\\psi_{\\ov 0}(b))=\\gamma^4(\\ov a\\cdot b)$. Hence to prove that the above diagram commutes it suffices to show that $\\gamma^4(c\\cdot b)=0$.\nThis follows from the fact that $\\mathrm{Ker }(\\gamma^3)\\cap C_1=\\{0\\}$.\nThis proves that $X$ is GM-intersection-formal over $\\mathbb{C}$.\n\nAssume now that $X$ has only one isolated singularity. Then $\\mathrm{Ker }(\\gamma^4)\/\\mathrm{Im }(\\eta^4)=0$ and $X$ is\nintersection-formal over $\\mathbb{C}$.\n\\end{proof}\n\n\n\\subsection{An example}\nWe end with an example of a projective surface with an \nisolated singularity and non-trivial weight filtration on its intersection cohomology.\n\n\\begin{example}[Cusp singularity]\nLet $C$ be a nodal cubic curve in $\\mathbb{C}\\mathbb{P}^2$. Choose a smooth plane quartic $C'$\nintersecting $C$ transversally, so that $|C\\cap C'|=12$.\nConsider the blow-up $\\widetilde X=Bl_{C\\cap C'}\\mathbb{C}\\mathbb{P}^2$ of $\\mathbb{C}\\mathbb{P}^2$ at the $12$ points of $C\\cap C'$.\nThen the proper transform $\\widetilde C$ of $C$ has negative self-intersection, and we may consider the blow-down $X$ of $\\widetilde C$ to a point. \nThen $X$ is a projective surface with a normal isolated singularity (see $\\S7$ of \\cite{ToChow},\nsee also Example 4.2 of \\cite{ChCi1} for a more general construction).\nTo make $\\widetilde C$ into a simple normal crossings divisor\nwe blow-up $2$ further times at the node of $\\widetilde C$.\nThis gives a resolution $f:Y\\to X$ where $Y\\simeq \\#_{15} \\mathbb{C}\\mathbb{P}^2$\nand the exceptional divisor $D$ is a cycle of three rational curves, so that\n$D^{(1)}=\\sqcup_{i=1}^3\\mathbb{C}\\mathbb{P}^1$ and $D^{(2)}=\\sqcup_{i=1}^3p_i$.\nLet $j^s:H^s(Y;\\mathbb{Q})\\longrightarrow H^s(D^{(1)};\\mathbb{Q})$ and $i^*:H^0(D^{(1)};\\mathbb{Q})\\to H^2(D^{(2)};\\mathbb{Q})$ denote the restriction morphisms.\nThe rational intersection cohomology of $X$ is given by:\n$$\nIH^*_{\\ov p}(X;\\mathbb{Q})\\cong \n\\def1.4{1.4}\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov{0}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)\\oplus Van$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline \\hline\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c |}\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov 1$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline \\hline\n\\end{tabular}\n\\,\\,\\,;\\,\\,\\,\n\\begin{tabular}{| c | }\n\\multicolumn{1}{c}{\\tiny{$\\ov{p}=\\ov{2}$}}\\\\\n\\hline\n$\\mathbb{Q}$\\\\ \\hline\n0\\\\ \\hline\n$\\mathrm{Ker }(j^2)^\\vee\\oplus Van^\\vee$\\\\ \\hline\n0\\\\ \\hline\n$\\mathbb{Q}$ \\\\ \\hline\n\\end{tabular}\n$$\nwhere $Van:=\\mathrm{Coker }(i^*)\\cong \\mathbb{Q}$ and $\\mathrm{Ker }(j^2)\\cong \\mathrm{Ker }(j^2)^\\vee\\cong \\mathbb{Q}^{12}$.\nThe weight filtration on $IH^*_{\\ov \\bullet}(X;\\mathbb{Q})$ is non-trivial, with\n$Gr^W_2IH^2_{\\ov 0}(X;\\mathbb{Q})\\cong \\mathbb{Q}^{12}$, $Gr^W_1IH^2_{\\ov 0}(X;\\mathbb{Q})=0$ and \n$Gr^W_0IH^2_{\\ov 0}(X;\\mathbb{Q})\\cong \\mathbb{Q}$.\n\\end{example}\n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{results -- to dos}\n\n\\begin{comment}\n\\section*{Carlos notes (to address)}\n\n\\begin{itemize}\n\t\\item talk about reservoir papers in the introduction\n\t\\item in the trajectory difference part want to include 50\/50 autoencoders, have to also show Power diss and Re Im plot in some section, also explain why I'm using these, what to they show us, should consider changing terms from hibernating to quiescent. \n\t\\item in the hibernating and bursting time section I think we might\/should also be able to take the initial peak out, by setting a threshold of the data we want to include in this PDF\n\t\\item maybe talk about neural ODEs in the conclusion\n\\end{itemize}\n\\end{comment}\n\\section{Introduction} \\label{sec:Intro}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\nDevelopment of reduced order dynamical models for complex flows is an issue of long-standing interest, with applications in improved understanding, as well as control, of flow phenomena. The classical approach for dimension reduction of these systems consists of extracting dominant modes from data via principal component analysis (PCA), also known as proper orthogonal decomposition (POD) and Karhunen-Lo\\'{e}ve decomposition \\cite{holmes2012turbulence}. PCA determines a set of basis vectors ordered by their contribution to the total variance (fluctuating kinetic energy) of the flow. Given $N_s$ data vectors (``snapshots\") $x_i\\in \\reals^N$, one can obtain these basis vectors by performing singular value decomposition (SVD) on the data matrix $X=[ x_1,x_2,\\cdots ] \\in \\mathbb{R}^{N \\times N_s}$ such that $X=U \\Sigma V^T$. Projecting the data onto the first $d_h$ basis vectors (columns of $U$) then gives a low-dimensional representation -- a projection onto a linear subspace of the full state space. To find a reduced order model (ROM), a Galerkin approximation of the Navier-Stokes Equations (NSE) using this basis can be implemented; these have shown some success in capturing the dynamics of coherent structures \\cite{noack1994low, aubry1988dynamics}. Previous research has also used POD as well as a filtered version thereof \\cite{sieber2016spectral}, which are linear reduction techniques, to reduce dimensions and learn a time evolution map from data with the use of neural networks (NNs) \\cite{lui2019construction}.\n\nAlthough PCA provides the best linear representation of a data set in $d_h$ dimensions, in general the long-time dynamics of a general nonlinear dynamical systems are not expected to lie on a linear subspace of the state space. For a primer and more details on data-driven dimension reduction methods for dynamical systems refer to Linot \\& Graham \\cite{linot2022data}. For dissipative systems, such as the NSE, it is expected that the long-time dynamics will lie on an invariant manifold $\\mathcal{M}$, which can be represented \\emph{locally} with Cartesian coordinates, but may have a complex global topology. In fluid mechanics, this manifold is often called an \\emph{inertial manifold}\n\\cite{foias1988modelling, temam1989inertial, zelik2022attractors}. Hence to find a high-fidelity low-dimensional model one desires to find $\\mathcal{M}$ and the dynamical system on it. The present work will consider discrete-time models, though differential equation models could be found as well \\cite{linot2022data}. \n\nIn general one can think of breaking up $\\mathcal{M}$ into overlapping regions that cover the domain, to find the local representation. These are called charts and are equipped with a coordinate domain and a coordinate map \\cite{lee2013smooth}. The strong Whitney's embedding theorem states that any smooth manifold of dimension $d_\\mathcal{M}$ can be embedded into a Euclidean space of so-called \\emph{embedding} dimension $2d_\\mathcal{M}$ \\cite{lee2013smooth, whitney1944self}. This means that in the worst case we can expect in principle to be able to find a $2d_\\mathcal{M}$-dimensional Euclidean space in which the dynamics lie. To find a $d_\\mathcal{M}$-dimensional Euclidean space one would in general need to develop overlapping local representations and evolution equations -- this avenue is not pursued in the present work but has been done elsewhere \\cite{floryan2021charts}. In this work we aim to find a high-fidelity low-dimensional dynamical model using data from simulations of two-dimensional Kolmogorov flow. In this work, the governing Navier-Stokes Equations will only be used to generate the data -- the models will only use this data, not the equations that generated it. Neural networks (NNs) will be used to map between the full state space and the manifold, as well as for the dynamical system model on the manifold. \n\n\n\n\nA number of previous studies have focused on finding {data-driven} models for fluid flow problems with the use of NNs. Srinivasan \\textit{et al.} \\cite{srinivasan2019predictions} developed NN models to attempt to predict the time evolution of the Moehlis-Faisst-Eckhardt (MFE) model \\cite{moehlis2004low}, which is a nine-dimensional model for turbulent shear flows. They used two approaches to finding discrete-time dynamical systems. The first is to simply use a neural netowrk as a discrete-time map, yielding a Markovian representation of the time evolution. The second is to use a long short-term memory (LSTM) network, which yields a non-Markovian evolution equation. Despite the fact that the dynamics are in fact Markovian, the LSTM approach worked better, yielding reasonable agreement with the Reynolds stress profiles. Page \\textit{et al.} used deep convolutional autoencoders (CAEs) to learn low-dimensional representations for two-dimensional (in physical space) Kolmogorov flow, showing that these networks retain a wide spectrum of lengthscales and capture meaningful patterns related to the embedded invariant solutions \\cite{page2021revealing}. They considered the case where bursting dynamics is obtained at a Reynolds number of $\\text{Re}=40$ and $n=4$ wavelengths in the periodic domain. Nakamura \\textit{et al.} used CAEs for dimension reduction combined with LSTMs and applied it to minimal turbulent channel flow for $\\text{Re}_\\tau=110$ where they showed to capture velocity and Reynolds stress statistics \\cite{nakamura2021convolutional}. They studied various degrees of dimension reduction, showing good performance in terms of capturing the statistics; however for drastic dimension reduction they showed how only large vortical structures were captured. Hence, the selection of the minimal dimension to accurately represent the state becomes a challenging task. Reservoir networks have also shown great potential in learning nonlinear models for time evolution. For example, Doan \\textit{et al.} trained what they call an Auto-Encoded Reservoir-Computing (AE-RC) framework where the latent space is fed into an Echo State Network (ESN) to model evolution in discrete time \\cite{doan2021auto}. By considering the two-dimensional Kolmogorov flow for $\\text{Re}=30$ and $n=4$ good performance was obtained when comparing the kinetic energy and dissipation evolution in time. They also showed how the model captures the velocity statistics. However, the nature of the reservoir in the ESN stores past history, making the model non-Markovian. \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\\begin{comment}\nThe Navier-Stokes Equations (NSE) are known to be infinite dimensional partial differential equations and when solved numerically many dimensions are usually needed to capture the correct dynamics. This mainly arises due to the need of capturing the smallest scales that are present in the system, which are responsible for dissipating turbulent kinetic energy \\cite{moin1998direct}. Attempts to reduce the domain in turbulent channel flow have been accomplished by finding the smallest domain in which turbulence is sustained in a channel flow \\cite{jimenez1991minimal}. Other attempts to find low-dimensional representations of the state consist in extracting dominant patterns by the use of principal component analysis (PCA), also known as POD and Karhunen\u2013Lo\\'{e}ve. This state can subsequently be evolved in this basis using, for example, Galerkin methods \\cite{noack1994low} which have shown success in capturing the dynamics of coherent structures \\cite{aubry1988dynamics}. However dynamics on a linear transformation require many modes, or dimensions, to resolve all of the relevant scales. Recent studies have focused on finding nonlinear projections with the use of Neural Networks (NNs) \\cite{murata2020nonlinear}. By learning a nonlinear reduced order model (ROM) a more natural representation of the state can be found due to the nonlinearities that appear in the governing equations, which combined with spatio-temporal evolution has shown success in prediction of turbulent flows \\cite{san2018extreme, srinivasan2019predictions}. \n\\end{comment}\n\n\n\n\n Although previous research has found data-driven ROMs for fluid flow problems, the focus on these has not been to find the minimal dimension required to capture the data manifold and dynamics. Linot \\& Graham have addressed this issue for the Kuramoto-Sivashinsky equation (KSE) \\cite{linot2020deep, linot2022data}. They showed that the mean squared error (MSE) of the reconstruction of the snapshots using an AE for the domain size of $L=22$ exhibited an orders-of-magnitude drop when the dimension of the inertial manifold is reached. Furthermore, modeling the dynamics with a dense NN at this dimension either with a discrete time map \\cite{linot2020deep} or a system of ordinary differential equations (ODE) \\cite{linot2022data} yields excellent trajectory predictions and long-time statistics. Increasing domain size to $L=44$ and $L=66$, which makes the system more chaotic, affects the drops of MSE significantly. However a drop is still seen, and when obtaining the dynamics and calculating long time statistics, good agreement with the true data is obtained. This work, denoted ``Data-driven manifold dynamics\" (DManD) has been extended to incorporate reinforcement learning control for reduction of dissipation in the KSE, yielding a very effective control policy \\cite{Zeng.2022}.\n \n\n\nWe aim to extend this approach to the NSE, specifically to the two-dimensional Kolmogorov flow, where an external forcing drives the dynamics. As $\\text{Re}$ increases, the trivial state becomes unstable, giving rise to periodic orbits (POs), relative periodic orbits (RPOs) and eventually chaos. Relative periodic orbits correspond to periodic orbits in in a moving reference frame, such that in a fixed frame, the pattern at time $t+T$ is a phase-shifted replica of the pattern at time $t$. The nature of the weakly turbulent dynamics at a Reynolds number of $\\operatorname{Re} = 14.4$, and connections with RPO solutions are the focus of this study. Due to the symmetries of the system the chaotic dynamics travels between unstable RPOs \\cite{crowley2022turbulence} through bursting events \\cite{armbruster1996symmetries} that shadow heteroclinic orbits connecting the RPOs. A past study \\cite{armbruster1992phase} shows that low-dimensional representations can be found with PCA for two-dimensional Kolmogorov flow where in the case of weakly turbulent data, the first two PCA basis in the streamfunction formulation capture most of the energetic content when filtering out the bursting events before the analysis, and including a third basis function captures the bursting information. This point hints at the low-dimensional nature of this system, where a low number of PCA basis functions can energetically represent the data. However, even though the energy can be contained in a low number of basis functions, this does not imply that these will properly capture the dynamics \\cite{rowley2017model}. In \\cite{armbruster1992phase}, development of a model of time-evolution was not considered. \n\n\n\nReturning to the aims of the present work, our focus is twofold. We aim to learn a minimal-dimensional high fidelity data-driven model for the long-time dynamics of two-dimensional Kolmogorov flow with the use of an autoencoder (AE), and a discrete-time map, in the form a dense NN, of the dynamics on the manifold. In this map, the future time prediction only depends on the present state (on the manifold), in keeping with the Markovian nature of the dynamics on the manifold. We will evaluate model predictions as a function of dimension, considering short-time trajectories, long-time statistics, quiescent and bursting time distributions, and predictions of bursting events. This paper is structured as follows: in Section \\ref{sec:Framework} we present the governing equations together with the symmetries of the system. We also present the dynamics at the two values of $\\operatorname{Re}$ considered and the connections of the RPOs with the chaotic regime. In Section \\ref{sec:AEs} we show the methodology for data-driven dimension reduction and dynamic modeling, which includes the AE architecture and the time map NN. Section \\ref{sec:Results} shows the results, and concluding remarks are given in Section \\ref{sec:Conclusion}.\n\n\\newpage\n\n\\section{Kolmogorov flow formulation and dynamics} \\label{sec:Framework}\n\nThe two-dimensional Navier-Stokes equations (NSE) with Kolmogorov forcing are\n\\begin{gather}\n\\frac{\\partial \\boldsymbol{u}}{\\partial t}+\\boldsymbol{u} \\cdot \\nabla \\boldsymbol{u}+\\nabla p=\\frac{1}{\\operatorname{Re}} \\nabla^{2} \\boldsymbol{u}+\\sin (n y) \\hat{\\boldsymbol{x}} \\\\\n\\nabla \\cdot \\boldsymbol{u}=0\n\\end{gather}\nwhere $\\boldsymbol{u}=[u,v]$ is the velocity vector, $p$ is the pressure, $n$ is the wavenumber of the forcing, and $\\hat{\\boldsymbol{x}}$ is the unit vector in the $x$ direction. Here $\\operatorname{Re}=\\frac{\\sqrt{\\chi}}{v}\\left(\\frac{L_{y}}{2 \\pi}\\right)^{3 \/ 2}$ where $\\chi$ is the dimensional forcing amplitude, $\\nu$ is the kinematic viscosity, and $L_y$ is the size of the domain in the $y$ direction. We consider the periodic domain $[0,2 \\pi \/ \\alpha] \\times[0,2 \\pi]$ with $\\alpha=1$. Vorticity is defined as $\\omega = \\nabla \\times \\boldsymbol{u}$. The equations are invariant under several symmetry operations \\cite{chandler2013invariant}, namely a shift (in $y$)-reflect (in $x$), a rotation through $\\pi$, and a continuous translation in $x$:\n\\begin{gather}\n\\mathscr{S}:[u, v, \\omega](x, y) \\rightarrow[-u, v,-\\omega]\\left(-x, y+\\frac{\\pi}{n}\\right), \\\\\n\\mathscr{R}:[u, v, \\omega](x, y) \\rightarrow[-u,-v, \\omega](-x,-y), \\\\\n\\mathscr{T}_{l}:[u, v, \\omega](x, y) \\rightarrow[u, v, \\omega](x+l, y) \\quad \\text { for } 0 \\leqslant l<\\frac{2 \\pi}{\\alpha}.\n\\end{gather}\n\\begin{figure}\n\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KE_po_v2.pdf}\n\t\n\t\t\\caption{}\n\t\t\\label{fig:sub1}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KE_bur_hib_norm_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub2}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=1\\linewidth]{figures\/I_D_Re14d4_v6.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subID}\n\t\\end{subfigure}\n\t\\caption{(a) Time evolution of $KE$ at $\\operatorname{Re}=13.5$. (b) Time evolution of $KE$ at $\\operatorname{Re}=14.4$. (c) Time evolution of $D$ and $I$ at $\\operatorname{Re}=14.4$.}\n\t\\label{fig:test}\n\\end{figure}\nThe total kinetic energy for this system ($KE$), dissipation rate ($D$) and power input ($I$) are \n\\begin{equation}\nKE =\\frac{1}{2}\\left\\langle\\boldsymbol{u}^{2}\\right\\rangle_{V}, D=\\frac{1}{\\operatorname{Re}}\\left\\langle|\\nabla \\boldsymbol{u}|^{2}\\right\\rangle_{V}, \\quad I=\\langle u \\sin (n y)\\rangle_{V}\n\\end{equation}\nwhere subscript $V$ corresponds to the average taken over the domain. For the case of $n=1$ the trivial solution is linearly stable at all $\\text{Re}$ \\cite{iudovich1965example}. It is not until $n=2$ that the laminar state becomes unstable, with a critical value of $\\operatorname{Re}_c=n^{3\/2}2^{1\/4}$\\cite{meshalkin1961investigation, green1974two, thess1992instabilities}.\n\n\nThe NSE are evolved numerically in time in the vorticity representation on a $[d_x \\times d_y]=[32 \\times 32]$ grid following the pseudo-spectral scheme given by Chandler \\& Kerswell \\cite{chandler2013invariant}, which is based on the code by Bartello \\& Warn \\cite{bartello1996self}. We show here time series results for the two dynamical regimes considered in this work, an RPO regime at $\\mathrm{Re}=13.5$ and a chaotic regime at $\\mathrm{Re}=14.4$. Figure \\ref{fig:sub1} shows the $KE$ evolution for an RPO obtained at $\\operatorname{Re} = 13.5$. Due to the discrete symmetries of the system, there are several RPOs \\cite{armbruster1996symmetries}, as we further discuss below. Figure \\ref{fig:sub2} shows the $KE$ evolution for a trajectory at $\\operatorname{Re} = 14.4$. The dynamics are characterized by quiescent intervals where the trajectories are close to RPOs (which are now unstable), punctuated by heteroclinic-like excursions between the RPOs, which are indicated by the intermittent increases of the $KE$. The RPOs are all related by the symmetries $\\mathscr{S}$ and $\\mathscr{R}$ \\cite{armbruster1996symmetries, platt1991investigation, nicolaenko1990symmetry}. This behavior can also be seen in Figure \\ref{fig:subID}, where the black curve corresponds to the time evolution of $D$ and the blue curve to the time evolution of $I$. Figure \\ref{re14d4re13d5}, shows a state-space projection of a trajectory onto the plane $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ where $a(k_x,k_y,t) = a_{k_x,k_y}(t)=\\mathcal{F}\\{\\omega(x,y,t)\\}$ is the discrete Fourier transform in $x$ and $y$. The grey curve corresponds to $\\operatorname{Re} = 14.4$ and the different blue curves show four different RPOs related by the shift-reflect symmetry $\\mathscr{S}$ at $\\operatorname{Re} = 13.5$. \n\n\n\n\n \\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/Re13d5_Re14d4_v3.pdf}\n\t\\caption{Evolution of the real and imaginary components corresponding to the $a_{0,1}(t)$ Fourier mode for $\\operatorname{Re}=13.5$ and $\\operatorname{Re}=14.4$.}\n\t\\label{re14d4re13d5}\n\\end{figure}\n \n\n\n\\section{Data-driven dimension reduction and dynamic modeling} \\label{sec:AEs}\n\n\\subsection{Dimension reduction with autoencoders}\\label{sec:aesub}\n\n\n\nTo learn a minimal dimension high-fidelity model for the two-dimensional Kolmogorov flow we first have to find a low-dimensional nonlinear mapping from the full state to the reduced representation. For this purpose we consider a common machine learning architecture known as an undercomplete autoencoder (AE), whose purpose is to learn a reduced representation of the state such that the reconstruction error with respect to the true data is minimized. The AE consists of an encoder, $\\mathcal{E}(\\cdot)$, that maps from the full space $\\mathbb{R}^{N}$ to the lower dimensional latent space $h(t) \\in \\mathbb{R}^{d_h}$ (i.e., coordinates on the manifold $\\mathcal{M}$), and a decoder, $\\mathcal{D}(\\cdot)$, that maps back to the full space. Flattened versions of $\\omega (x,y,t)$ are used, which we refer from this point on as $\\omega(t)$, so $N = 32 \\times 32 = 1024$. We shall see that the latent space dimension $d_h$ will be much smaller than the dimension $N$ of the full spatially-resolved state. The encoder $\\mathcal{E}(\\omega(t))$ is a coordinate mapping from $\\mathbb{R}^{N}$ to $\\mathcal{M}$, and the decoder $\\mathcal{D}(h(t))$ is the mapping back from $\\mathcal{M}$ to $\\mathbb{R}^{N}$.\n\n\n\n\nWe train the AEs with $\\omega(t)$ obtained from the evolution of NSE for the original data as well as accounting for the discrete and continuous symmetries. By accounting for the symmetries it is expected that the networks will perform better, by not having to learn the symmetries in the latent space mapping. We account for the continuous symmetry in $x$, $\\mathscr{T}_{l}$, with the method of slices \\cite{budanur2015periodic, budanur2015reduction}. The $k_x=1, k_y=0$ Fourier mode is used to find the spatial phase: $\\phi_x(t)=\\operatorname{atan} 2\\left\\{\\operatorname{Im}\\left[a_{1,0}(t)\\right], \\operatorname{Re}\\left[a_{1,0}(t)\\right]\\right\\}$. This can then be used to phase-align the vorticity snapshots such that this mode is a pure cosine: $\\hat{\\omega}(x,y,t)=\\mathcal{F}^{-1}\\left\\{\\mathcal{F}\\{\\omega(x,y,t)\\} e^{-i k \\phi_x(t)}\\right\\}$. Doing this ensures that the snapshots lie in a reference frame were no translation happens in the $x$ direction. We will learn evolution equations for both $\\hat{\\omega}(t)$ and $\\phi_x(t)$, which we will denote as the pattern dynamics and phase dynamics, respectively. We also consider the shift-reflect (SR) symmetry, $\\mathscr{S}$, as well as the rotation through $\\pi$, $\\mathscr{R}$. To account for the SR symmetry the goal is to collapse the phase-aligned snapshots to the same common state. We can define two indicator functions such that the SR subspace is specified. The first one, $I_{Even}=\\operatorname{sgn}(\\phi_y)$, where $\\phi_y(t)=\\operatorname{atan} 2\\left\\{\\operatorname{Im}\\left[a_{0,1}(t)\\right], \\operatorname{Re}\\left[a_{0,1}(t)\\right]\\right\\}$ is the spatial phase in $y$. The second indicator function is $I_{odd}=\\operatorname{sgn}(\\operatorname{Re}[a_{2,0} (t)])$, the sign of the real part of the second Fourier mode in $x$. We can then map the vorticity snapshots in such a way that $I_{Even},I_{Odd}>0$ by applying SR operations to the state. The rotation symmetry is accounted for, on top of the SR symmetry, by minimizing the $L_2$ norm of the data with respect to a template snapshot. This is done by applying the discrete operation that rotates and shift-reflects the vorticity snapshots and selecting the snapshot that minimizes the norm. We note that we take a different approach for reducing the symmetries compared to previous research on symmetry-aware AEs \\cite{kneer2021symmetry}.\n\n\n\n\n\nPrevious work \\cite{linot2020deep} has shown that training a NN to learn the difference between the data and the projection onto the leading PCA basis vectors improved reconstruction performance compared to learning a latent space directly from the full data. To present the framework, we will use the phase-aligned and flattened vorticity $\\hat{\\omega} (t)$, since that is what we use for the time-evolution. Below, however, we will present some results where other versions of the data are used -- e.g.~the data with phase-shifting. The autoencoder aspect of the analysis is identical. \n\nWe begin he process by computing the projection of the data onto the first $d_h$ basis vectors, $P_{d_{h}} U^{T}\\hat{\\omega}(t)$. We then seek to learn a $d_h$-dimensional correction to that projection, $E\\left(U^{T} \\hat{\\omega}(t)\\right)$ -- the sum of these is the latent-space representation $h(t)$. In other words, the encoding step learns the deviation from PCA\n\\begin{equation}\nE\\left(U^{T} \\hat{\\omega}(t)\\right)=h(t)-P_{d_{h}} U^{T} \\hat{\\omega}(t).\n\\label{encoder}\n\\end{equation}\nWe emphasize that this step \\emph{is not} simply a projection onto a linear subspace defined by $d_h$ PCA modes-- rather it is an approach that learns the deviation of the data from that projection.\nSimilarly the decoding section learns the difference \n\\begin{equation}\nD(h(t))=U^{T} \\tilde{\\hat{\\omega}}(t)-\\left[\\begin{array}{c}h(t) \\\\ 0\\end{array}\\right],\n\\label{decoder}\n\\end{equation}\nwhere $\\tilde{\\hat{\\omega}}(t)$ corresponds to the reconstruction of $\\hat{\\omega} (t)$. Inserting Equation \\ref{encoder} into Equation \\ref{decoder} and noting that by definition $\\tilde{\\hat{\\omega}}(t) = U[P_{d_{h}} U^{T} \\hat{\\omega}(t), P_{d-d_{h}} U^{T} \\hat{\\omega}(t)]^{T}$ we get that the exact solution satisfies $E\\left(U^{T} \\hat{\\omega}(t)\\right)+D_{d_{h}}((h(t))=0$. To satisfy this constraint we add it to the loss function as a penalty to obtain\n\\begin{equation}\nL=\\|\\hat{\\omega}(t)-\\tilde{\\hat{\\omega}}(t)\\|^{2}+\\alpha_L \\left\\|E(U^{T}\\hat{\\omega}(t))+D_{d_{h}}(h(t))\\right\\|^{2}\n\\end{equation}\nwhere $\\| \\cdot \\|$ is the $l^2$-norm and we select $\\alpha_L=1$. We can now train the AEs by minimizing $L$ via stochastic gradient descent. We train 4 AEs at each of several values of $d_h$ to study the MSE of the reconstruction of $\\hat{\\omega} (t)$. All models were trained for 300 epochs with an Adam optimizer using Keras. The training data consists of long time series from the direct simulations, with initial transients removed. We use a total of $10^5$ snapshots separated by $\\tau=5$ time units for $\\text{Re}=14.4$, and $10^4$ snapshots separated by $\\tau=5$ for $\\text{Re}=13.5$. We do an $80\\%\/20\\%$ split for training and testing respectively. Figure \\ref{framework1}\\textcolor{blue}{a} shows a summary of the AE and Table \\ref{tablenn} gives information on the layer dimensions, and activations used in each layer of the encoder and decoder. At each value of $d_h$, the model with the smallest MSE over a test data set from the phase-aligned data is then selected for the discrete time map. We will show in Section \\ref{sec:autoencoders} that factoring out the phase dramatically increases AE performance.\n\n\n\n\n\\subsection{Time evolution via a dense NN}\\label{sec:aetime}\n\n\nAfter finding $h(t)$ from the AEs, we seek a discrete-time map\n\\begin{equation}\n\th(t+\\tau)=F(h(t))\n\\end{equation}\nthat evolves $h(t)$ from time $t$ to $t+\\tau$. We fix $\\tau=5$. The function $F$ is also expressed as a dense NN. Here we train 5 NNs for the different $d_h$ cases with the following loss\n\\begin{equation}\nL_t=\\|\\tilde{h}(t+\\tau)-h(t+\\tau)\\|^{2},\n\\end{equation}\nwhere $h(t+\\tau)$ comes from true data and $\\tilde{h}(t+\\tau) = F(h(t))$ from the prediction, and select the one with the best performance. For the discrete time map we trained for 600 epochs with the use of a learning rate scheduler. In this case we noticed an increase in performance when dropping the learning rate hyperparameter by an order of magnitude after 300 epochs. Figure \\ref{framework1}\\textcolor{blue}{b} shows a summary the framework just described, and Table \\ref{tablenn} gives information on the layer dimensions and activations used in each layer.\n\n\n\nAs discussed previously, the time evolution is done in the phase-aligned space. To complete the dynamical picture we seek a discrete-time map for the phase evolution \n\\begin{equation}\n\t\\Delta \\tilde{\\phi}_x(t + \\tau) = G(h(t)),\\label{eq:phaseevolution}\n\\end{equation}\nwhere $\\Delta \\phi_x(t + \\tau)=\\phi_x(t + \\tau)-\\phi_x(t)$. Because of translation equivariance, the actual phase is only unique to within a constant. We train 5 NNs for the the different $d_h$ cases with the following loss\n\\begin{equation}\nL_p=\\|\\Delta \\tilde{\\phi}_x(t + \\tau)-\\Delta \\phi_x(t + \\tau)\\|^{2},\n\\end{equation}\nsuch that $\\Delta \\tilde{\\phi}_x(t + \\tau) = G(h(t))$. Figure \\ref{framework1}\\textcolor{blue}{c} shows a summary of the framework we have described, and Table \\ref{tablenn} gives information on the layer dimensions and activations used in each layer.\n\n\\begin{table}[h]\n\\caption{Neural network layer dimensions and activations used in each layer. Sigmoid function are denoted 'S'.}\n$$\n\\begin{array}{lccc}\\hline \\hline & \\text { Function } & \\text { Shape } & \\text { Activation } \\\\ \\hline \\text { Encoder } & E & 1024: 5000:1000:d_{h} & \\text { S:S:S } \\\\ \\text { Decoder } & D & d_{h}: 1000: 5000: 1024 & \\text { S:S:linear } \\\\ \\text { Evolution } & F & d_{h}: 500: 500: d_{h} & \\text { S:S:linear } \\\\ \\text { Phase Prediction } & G & d_{h}: 500: 500:500: 1 & \\text { S:S:S:linear } \\\\ \\hline \\hline\\end{array}\n$$\n\\label{tablenn}\n\\end{table}\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{figures\/NNarch10.pdf}\n\t\\caption{Neural network frameworks for (a) autoencoder (b) discrete-time map for pattern prediction and (c) discrete-time for phase prediction.}\n\t\\label{framework1}\n\\end{figure}\n\n\n\n\n\\section{Results} \\label{sec:Results}\nWe present results as follows. First we will show the AE performance for the various $d_h$ and symmetries considered. We then report results for time evolution models, again studying performance as a function of the number of dimensions. Both evolution of the pattern and phase dynamics are considered. {We wrap up the results by predicting bursting events based on the low-dimensional representation.} \n\n\n\\subsection{Dimension reduction with autoencoders}\\label{sec:autoencoders}\n\n\n\n\nWe begin by showing results for $\\operatorname{Re} = 13.5$. In Figure \\ref{fig:subMSEre13d5} we see the MSE versus $d_h$ trend where the grey curve corresponds to the PCA reconstruction for the original data ($\\tilde{\\omega}(t)=U_{d_h}U^T_{d_h} \\omega (t)$), the black curve to the AE with the original data, and the blue curve to the AE with the phase factored out before training. The MSE is calculated over the test data set. Notice that, as expected, the AEs perform better than PCA. This is because of the nonlinearities that are added to the linear optimal latent space found in PCA in combination with the nonlinear decoder. The blue curve exhibits a sharp drop in the MSE at a dimension of $d_h=2$, which is the correct embedding dimension for a limit cycle. This happens because the phase is accounted for; the dynamics of the system in the phase-aligned reference frame corresponds to a PO and the autoencoder does not have to learn all the possible phases due to the continuous translation in $x$. The overall embedding dimension is $d_h+1 = 3$, where 1 corresponds to the phase. Hence we are able to estimate the dimension for this system by looking at the drop in the MSE curve. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.7\\linewidth]{figures\/MSE_Re13d5.pdf}\n\t\\caption{MSE for test data corresponding to $\\operatorname{Re}=13.5$}\n\t\\label{re13d5intro}\n\\end{figure}\n\\end{comment}\n\nWe now consider the $\\operatorname{Re} = 14.4$ case, where the dynamics are chaotic, moving between the regions near the now unstable RPOs. In Figure \\ref{fig:subMSEre14d4} we show the same curves as in Figure \\ref{fig:subMSEre13d5} but we also include the green and magenta curves, which in addition factor out the SR and the SR-Rotation symmetries respectively before training the AEs. These are included due to the added complexity of $\\operatorname{Re}=14.4$, where the chaotic trajectory travels in the vicinity of the RPOs related by the symmetry groups previously discussed. A monotonic decrease in MSE can be seen for the different symmetries considered in the blue, green, and magenta curves, but no sharp drop is apparent. Instead we notice that the MSE drops at different rates in different regions. For example, in the blue curve corresponding to the phase aligned data, we see a sharp drop from $d_h=1-6$ followed by a more gradual drop from $d_h=6-13$. In the following sections we couple the dimension-reduction analysis with models for prediction of time evolution for the phase aligned data. We expect that this combination will help us determine how many dimensions are needed to correctly represent the state. \n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.7\\linewidth]{figures\/MSE_Re14d4.pdf}\n\t\\caption{MSE for test data corresponding to $\\operatorname{Re}=14.4$}\n\t\\label{re14d4auto}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/MSE_Re13d5_leg3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subMSEre13d5}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/MSE_Re14d4_leg4.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subMSEre14d4}\n\t\\end{subfigure}\n\n\n\n\n\n\t\\caption{MSE versus dimension $d_h$ over the test data corresponding to (a) $\\operatorname{Re}=13.5$ and (b) $\\operatorname{Re}=14.4$. The PCA curve corresponds to the MSE of the reconstruction for the test data set with respect to the true data $\\omega (t)$, with no symmetries factored out, using the truncated $U$ into $d_h$ dimensions such that $\\tilde{\\omega}(t)=U_{d_h}U^T_{d_h} \\omega (t)$ ; the `Original', `Phase', `Phase-SR', and `Phase-SR-Rotation' curves correspond to the MSEs of the reconstruction for the test data set with respect to the true data using AEs. In the curve labeled `Original', no symmetries are factored out and in the other curves the corresponding symmetries in the labels are factored out. }\n\t\\label{re13d5_14d4_auto}\t\n\n\\end{figure}\n\n\n\\subsection{Time evolution as a function of dimension - Short time predictions}\\label{sec:dimredevshort}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/traj_true_re13d5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipdtruere13d5}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/traj_dh2_re13d5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipd2_re13d5}\n\t\\end{subfigure}\n\t\\caption{Trajectory of $I(t)$ vs $D(t)$ corresponding to $\\operatorname{Re}=13.5$ for (a) true and (b) predicted data corresponding to dimensions $d_h=2$.}\n\t\\label{re13d5PD}\t\n\\end{figure}\n\nThe focus of this work is the chaotic dynamics at $\\mathrm{Re}=14.4$. Before considering that case, for completeness we briefly present results for $\\operatorname{Re}=13.5$. In Figure \\ref{re13d5PD} we see $D(t)$ versus $I(t)$ for the true and predicted dynamics at $d_h=2$; they are indistinguishable. At $d_h=1$, which is not shown, the model fails and the dynamics can not be captured. The reason for this is simple -- the embedding dimension for a limit cycle is two.\n\nNow we return to the case of $\\mathrm{Re}=14.4$, focusing first on short-time trajectory predictions. The Lyapunov time $t_L$ for this system is approximately $t_L \\approx 20$ \\cite{inubushi2012covariant}, hence $t_L \\approx 4\\tau$. We take initial conditions $h(t) \\in \\mathbb{R}^{d_{h}}$ to evolve recurrently with the discrete time map $F(\\cdot)$, such that $\\tilde{h}(t+\\tau) = F(h(t))$, $\\tilde{h}(t+2\\tau) = F(\\tilde{h}(t+\\tau))$, $\\tilde{h}(t+3\\tau) = F(\\tilde{h}(t+2\\tau))$ and so on. After evolving in time the data is then decoded to get $\\tilde{\\hat{\\omega}}_h (t)$ and compared with $\\hat{\\omega} (t)$. \nWe consider trajectories with ICs starting in the quiescent as well as in the bursting regions. The nature of the intermittency of the data makes it challenging to assign either bursting or quiescent labels. We consider a window of past and future snapshots and a criterion on $ \\| \\hat{\\omega} (t) \\| $ to make this decision, using the algorithm described in Algorithm \\ref{alg:bursting}. \n\n\\begin{algorithm}[H]\n\t\\caption{Quiescent\/Bursting labeling of vorticity snapshots}\\label{alg:cap}\n\t\\begin{algorithmic}\n\t\t\\State $W \\gets [\\hat{\\omega}(t_1),\\hat{\\omega}(t_2) \\cdots]$ \\Comment{Matrix with $N_s$ vorticity snapshots, $W \\in \\mathbb{R}^{N \\times N_s}$}\n\t\t\\State $S$ \\Comment{Initialize label array $S$}\n\t\t\\State $W_{l2} \\gets \\| W \\|$ \\Comment{Calculate $l^2$-norm of snapshots, $W_{l2} \\in \\mathbb{R}^{ N_s}$ }\n\t\t\\State $b \\gets 10$ \\Comment{Number of past snapshots in time to consider }\n\t\t\\State $f \\gets 10$ \\Comment{Number of future snapshots in time to consider }\n\t\t\\For {$i=b$, $b+1,\\ldots N_s-f$} \\Comment{$i$ is snapshot I.D.}\n\t\t\\If{$W_{l2}[i]<60$ }\n\t\t\\State $d_p \\gets \\operatorname{abs}(W_{l2}[i-b:i]-W_{l2}[i])$ \\Comment{Difference between current and past snapshots}\n\t\t\\State $b_p \\gets \\operatorname{sum}(d_p>5)$ \\Comment{Sums values that exceed a threshold of 5 (user defined)}\n\t\t\\State $d_f \\gets \\operatorname{abs}(W_{l2}[i:i+f]-W_{l2}[i])$ \\Comment{Difference between current and future snapshots}\n\t\t\\State $b_f \\gets \\operatorname{sum}(d_f>5)$ \\Comment{Sums values that exceed a threshold of 5 (user defined)}\n\t\t\\If{$b_p = 0$ or $b_f = 0$ }\n\t\t\\State $S[i-b] \\gets 0$\n\t\t\\Else \n\t\t\\State $S[i-b] \\gets 1$\n\t\t\\EndIf\n\t\t\\Else\n\t\t\\State $S[i-b] \\gets 1$\n\t\t\\EndIf\n\t\t\\EndFor\n\t\\end{algorithmic}\\label{alg:bursting}\n\\end{algorithm}\n\\noindent Doing this we ensure that snapshots that are contained in the bursting events and have a value of $ \\| \\hat{\\omega} (t) \\| $ similar to quiescent snapshots are correctly classified. We use a threshold on $ \\| \\hat{\\omega} (t) \\| $ to determine if a check is needed. For the classification strategy any snapshot above a threshold of 60 is classified as bursting with a label of 1, below 60 we enter a loop as shown in Algorithm \\ref{alg:cap} to determine if it should be classified as bursting or quiescent, where quiescent corresponds to a label of 0. This check is needed to correctly label snapshots that have comparable $ \\| \\hat{\\omega} (t) \\| $ but are still in the bursting regime. Figure \\ref{re14d4labels} shows a short time trajectory where the black line corresponds to $ \\| \\hat{\\omega} (t) \\| $ and the red to the 0\/1 labels. Notice that, as shown in Algorithm \\ref{alg:cap}, some of the data at the beginning and at the end of the time series will not be labeled, there are no past or future snapshots to compare to, and can be removed.\n\nAfter labeling the data as quiescent or bursting, we then consider the time evolution from ICs of $h(t)$ using the models of various dimensions. We will first show sample trajectories from ICs starting in the two regions, then show the ensemble-averaged prediction error as a function of time. Figure \\ref{KEhib} shows the KE evolution for an IC starting in the quiescent region. The black curve corresponds to the true data and the colored curves to the different $d_h$ models. At a dimension of $d_h=3$ we see that the predicted $KE$ diverges quickly with respect to the true $KE$. In the case of $d_h=4$ the $KE$ seems to fall on top of the unstable RPO with good agreement in the oscillatory behavior; however the bursting that occurs at $t \\approx 180$ is not captured. It is not until we reach $d_h=5$ that we see that the bursting event is correctly captured. In the case of $d_h=6$ we see that the bursting event is captured but with a time delay, and $d_h=7$ does not capture the bursting in this time frame considered. Figure \\ref{KEbur} shows the KE evolution for an IC starting in the bursting region. The black curve corresponds to the true data and the colored curves to the different $d_h$ models. At a dimension of $d_h=3$ the $KE$ stays bursting and does not show agreement with the true $KE$. However the cases $d_h=4$ and $d_h=5$ show better agreement and are also capable of closely predicting the end of the bursting event. In this case $d_h=6$ and $d_h=7$ agree closely with the $KE$ evolution, and specifically $d_h=7$ seems to track better before traveling to the quiescent region. \n\n\n\nTurning from examples of individual trajectories to ensemble averages, Figure \\ref{re14d4ICs} shows ensemble averages of the difference between the true and predicted trajectories, separately considering ICs in the bursting and quiescent regions. Blue curves correspond to quiescent ICs and red curves to bursting ICs. Starting from $d_h=3$ (lightest curve) we increase up to $d_h=7$ (darkest). We selected $10^4$ ICs in total where approximately 1\/3 of the ICs correspond to bursting. As expected, predictions at $d_h=3$ diverge quickly from the true dynamics in both quiescent and bursting IC scenarios. With increasing $d_h$, trajectories track better for both types of ICs. We can also notice that the three darkest curves, corresponding to $d_h=5,6,7$, perform best, and in the case of the quiescent ICs there is not much increase in performance between the three. We also notice that the trajectories for the quiescent ICs track almost perfectly for approximately two Lyapunov times. In Figure \\ref{re14d4ICstotal} we show ensemble averages of the difference between the true and predicted dynamics based on all ICs. The same trend is obtained as discussed for Figure \\ref{re14d4ICs}, and as expected the errors increase for all of the curves due to the divergence of the bursting ICs. We can conclude that models $d_h=5,6,7$ are very good at capturing trajectories the quiescent regions, which happens through the accurate prediction of the oscillatory behavior of the unstable RPO right before a bursting occurs. Prediction from bursting ICs is harder, due to the complex dynamics involved in this region. \n\n\n\n\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.55\\linewidth]{figures\/hibbur_labels_v3.pdf}\n\t\\caption{Labeling of $\\hat{\\omega} (t)$ snapshots in a short time series where 1 corresponds to bursting and 0 to quiescent.}\n\t\\label{re14d4labels}\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KEshorthib_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{KEhib}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KEshortbur_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{KEbur}\n\t\\end{subfigure}\n\t\n\t\\caption{Trajectory of $KE$ at different $d_h$ for (a) quiescent and (b) bursting ICs.}\n\t\\label{KEtots}\n\t\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/hibbur_ICs_v5diffcolors.pdf}\n\t\t\\caption{}\n\t\t\\label{re14d4ICs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/total_ICs_v5diffcolors.pdf}\n\t\t\\caption{}\n\t\t\\label{re14d4ICstotal}\n\t\\end{subfigure}\n\t\n\t\\caption{Difference between true vorticity evolution and vorticity evolution obtained from the time map $F$ from $h(t)$ where (a) correspond to averages taken over bursting and quiescent ICs and (b) averages over all the data. Lightest color curve corresponds to $d_h=3$ and darkness increases until $d_h=7$.}\n\t\\label{totalsplitICs}\n\t\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure\n\\centering\n\\includegraphics[width=.6\\linewidth]{figures\/hibbur_ICs.pdf}\n\\caption{Difference between predicted and real vorticity evolution. Green curves correspond to averages taken over bursting ICs and black over hibernating ICs. Lightest curve corresponds to $d_h=3$, $d_h$ is increased until reaching the darkest curve corresponding to $d_h=7$}\n\\label{re14d4ICs}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\n\\subsection{Time evolution as a function of dimension - Long time predictions}\\label{sec:dimredev}\n\nIn this section we present long time statistics for the models and true data at $\\mathrm{Re}=14.4$. From ICs on the attractor, we evolve for $2 \\times 10^5$ time units, yielding to get $4 \\times 10^4$ snapshots of data. This duration is sufficient to densely sample the quiescent and bursting regions. We note that long time statistics did not change if the IC was in a bursting or quiescent region.\n\n\n\nFigure \\ref{re14d4PD} shows the joint probability density function (PDF) of $I$ and $D$ for true and predicted data from models with $d_h$ from 3 to 7 -- note the logarithmic scale, here and below. We notice that at $d_h=3$ the different areas corresponding to quiescent and bursting regions are populated similarly in terms of the probability intensity compared with the true PDF shown, but the shape of the predicted PDF takes a curved form that is not seen in the true PDF. When we get to $d_h=4$ the shape of $D$ and $I$ event region approximates better the true data. However scattered points of high $D$ do not compare with the true data. It is not until $d_h=5$ is reached that the $D$ and $I$ events are better captured, and similarly for $d_h=6,7$. We also compute the joint PDF of $\\operatorname{Re}\\left[a_{0,1}\\right]$ and $\\operatorname{Im}\\left[a_{0,1}\\right]$, shown in Figure \\ref{re14d4PD01}. From this quantity we can observe the heteroclinic-like connections between the unstable RPOs, which correspond to the four ribbon-like regions of high probability. Here we see similar trends as in the joint PDF for $I$-$D$: $d_h=3,4$ show poor qualitative reconstruction compared with higher dimensions, and once $d_h\\geq 5$, the joint PDFs from the model prediction are virtually indistinguishable from the true PDFs. To further quantify the relationship of the PDFs from the models to the true data, we calculate the Kullback-Leibler (KL) divergence, \n\\begin{equation}\nD_{KL}(\\tilde{P}||P)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\tilde{P}\\{a,b\\} \\text{ln}\\dfrac{\\tilde{P}\\{a,b\\} }{P\\{a,b\\}}da \\; db,\n\\end{equation}\nwhere $\\tilde{P}$ corresponds to the predicted PDF and $P$ to the true PDF. Due to the approximation of the integral to discrete data we ignore areas where either the true or predicted PDFs are zero. Let us first consider the case $a=I$ and $b=D$. Figure \\ref{fig:subKLIPD} shows $D_{KL}$ calculated with varying $d_h$. The dashed grey line corresponds to $D_{KL}$ calculated over different true data sets. This serves as a baseline for comparison to the predicted PDFs. A significant decrease happens at $d_h=4$ followed by small decreases at higher dimensions. We see that after $d_h=5$ no significant information is gained. We can also look at the case where $a= \\;$Re $\\left[a_{0,1}\\right]$ and $b= \\;$Im $\\left[a_{0,1}\\right]$ in Figure \\ref{fig:subKLF01}. We notice that errors of the joint PDF in Figure \\ref{fig:subKLF01} show the same trend as Figure \\ref{fig:subKLIPD}. We can infer from these results that the embedding dimension of this system lies in the range $d_h=5-7$, and furthermore that the data-driven model can reproduce the long-time statistics with very high fidelity. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=0.85\\linewidth]{figures\/ip_diss_true_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipdtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh3_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh4_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh5_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh6_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd6}\n\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/ip_diss_dh7_pdf_bin0025_notime.pdf}\n\t\\caption{}\n\t\\label{fig:subipd7}\n\\end{subfigure}\n\t\\caption{$\\operatorname{Re}=14.4$: Joint PDFs of $I$-$D$ corresponding to $\\operatorname{Re}=14.4$ for (a) true and (b)-(f) predicted data corresponding to dimensions $d_h=3-7$.}\n\t\\label{re14d4PD}\t\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/ipdpdferror.pdf}\n\t\\caption{Joint PDFs difference between true and predicted data corresponding to $Re=14.4$ of power input and dissipation.}\n\t\\label{re14d4pdferror}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.85\\linewidth]{figures\/01_true_pdf_bin01_notime.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01dtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh3_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh4_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh5_pdf_bin01_notime.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01d5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh6_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d6}\n\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\n\t\\includegraphics[width=.85\\linewidth]{figures\/01_dh7_pdf_bin01_notime.pdf}\n\t\\caption{}\n\t\\label{fig:sub01d7}\n\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: Joint PDFs of $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ corresponding to $\\operatorname{Re}=14.4$ for (a) true and (b)-(f) predicted data corresponding to dimensions $d_h=3-7$.}\n\t\\label{re14d4PD01}\n\t\n\\end{figure}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KLDiv_IP_D_Re14d4_noerror_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLIPD}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/KLDiv_F01_Re14d4_noerror_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLF01}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: $D_{KL}$ vs dimension $d_h$ for (a) $I$-$D$ and (b) $\\operatorname{Re}\\left[a_{0,1}\\right]-\\operatorname{Im}\\left[a_{0,1}\\right]$ predicted vs true joint PDFs. Dashed grey line corresponds to $D_{KL}$ calculated over true data sets.}\n\t\\label{re14d4KLDiv}\n\t\n\\end{figure}\n\n\n\\begin{comment}\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/01pdferror.pdf}\n\t\\caption{Joint PDFs difference between true and predicted data corresponding to $Re=14.4$ of real and imaginary part for Fourier coefficient (1,0)}\n\t\\label{re14d4PD01error}\n\\end{figure}\n\\end{comment}\n\n\n\\begin{comment}\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/ip_diss_true_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipdtrueRe20}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/ip_diss_dh10_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subipd10Re20}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/01_true_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01dtruRe20e}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/01_dh10_pdf_Re20.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub01d10Re20}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/KLDiv_IP_D_Re20_werror.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLIPDRe20}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/KLDiv_F01_Re20_werror.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subKLF01Re20}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=20$: Joint PDFs of $I(t)$-$D(t)$ for (a) True and (b) Predicted data corresponding to dimension $d_h=10$. Joint PDFs of $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ for (c) True and (d) Predicted data corresponding to dimension $d_h=10$. $D_{KL}$ vs dimension $d_h$ corresponding to the difference between (e) $I(t)$-$D(t)$ and (f) $\\operatorname{Re}\\left[a_{0,1}(t)\\right]-\\operatorname{Im}\\left[a_{0,1}(t)\\right]$ predicted vs true joint PDFs. }\n\t\\label{re20PD}\t\n\\end{figure}\n\\end{comment}\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/MSE_Re20.pdf}\n\t\\caption{MSE of HNN-AE vs dimension $d_h$ over the test data corresponding to $\\operatorname{Re}=20$}\n\t\\label{re20}\n\\end{figure}\n\\end{comment}\n\n\n\nThe above PDFs yield no information about the temporal behavior of the system. One temporal feature of significant interest in problems with intermittency is the probability density of the durations of time intervals with different behavior. To address this, we consider the PDFs of time spent in bursting ($t_b$) and in quiescent ($t_q$) regions. The labeling method discussed in the previous section is used. For this calculation we take a trajectory of $10^5$ snapshots from an arbitrary IC. The PDF for the true data is shown in Figure \\ref{fig:subhibburtrue} followed by the PDFs that come from the $d_h=3-7$ models in Figures \\ref{fig:subhibburd3} - \\ref{fig:subhibburd7}. The true data shows that $t_q$ is mostly concentrated between $t \\approx 200-300$ with a high intensity peak shown at $t=5$. We attribute this peak to a small fraction of snapshots in the bursting region that get mislabeled as quiescent due to the weakly chaotic nature of the data. We do not expect for this to drastically change our conclusions because the same labeling system is used for the true data and the models. In the case of $t_b$ we notice that these are mostly concentrated between $t \\approx 0-200$. We also report the means of these time durations as well as the standard error of the mean (SE) in Table \\ref{tablehibburtime}. Looking at both the PDFs and averages of the times we see that $d_h=3$ fails to correctly capture the shape of the PDF and also underpredicts $ \\langle t_q \\rangle$ and $ \\langle t_b \\rangle$. At $d_h=4$, $ \\langle t_q \\rangle$ and $ \\langle t_b \\rangle$ get closer to the true values, but the shape of the PDF still looks different from the true data. At $d_h=5$ we start getting better agreement where we see that the PDFs clearly show the two regions where $t_b$ and $t_q$ are concentrated. Performance is similar at $d_h=6$, however at $d_h=7$ we can see that the quiescent PDF spreads into regions with higher $t_q$. Figure \\ref{re14d4tqtb} shows $D_{KL}$ with varying $d_h$ for these PDFs. As expected from observing the PDFs we see that $D_{KL}$ decreases up until $d_h=5$ for both cases. In the case of $t_q$ we see an increase in the error after $d_h=5$ which agrees with the above observation of the PDFs. For $t_q$, $D_{KL}$ seems to keep slightly decreasing after $d_h=5$. In short, while the low-dimensional models do not achieve the same agreement with the true results for these duration statistics as we do for the static quantities considered above, they nevertheless capture the key features of the distributions and capture their means with reasonable accuracy, within about $20\\%$. \n\n\n \n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=0.85\\linewidth]{figures\/hibburtime_true_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburtrue}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh3_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd3}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh4_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd4}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh5_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd5}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh6_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd6}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=.85\\linewidth]{figures\/hibburtime_dh7_pdf_v3.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subhibburd7}\n\t\\end{subfigure}\n\t\n\t\\caption{PDFs of $t_q$ and $t_b$ at $\\operatorname{Re}=14.4$ for (a) true and (b)-(f) predicted data for dimensions $d_h=3-7$.}\n\t\\label{re14d4PDhibbur}\n\t\n\\end{figure}\n\n\\begin{table}[h]\n\t\\caption{Average and standard error of the mean of $t_q$ and $t_b$ for true and dimensions $d_h=3-7$.}\n\t$$\n\t\\begin{array}{lcccc}\\hline \\hline & \\text { $ \\langle t_q \\rangle$ } \\; \\; \\; & \\text { $\\langle t_b \\rangle$ } \\; \\; \\; & \\text { SE($ t_q$) } \\; \\; \\; & \\text { SE($ t_b$) } \\; \\; \\; \\\\ \\hline \\text { True } & 176& 97 & 3 & 2 \\\\ \\text { $d_h=3$ } & 37 & 73 & 1 & 2 \\\\ \\text { $d_h=4$ } & 174 & 85 & 5 & 3 \\\\ \\text { $d_h=5$ } & 160 & 105 & 3 & 3\\\\ \\text { $d_h=6$ } & 185 & 106 & 3 & 3\\\\ \\text { $d_h=7$ } & 202 & 101 & 4 & 3 \\\\ \\hline \\hline\\end{array}\n\t$$\n\t\\label{tablehibburtime}\n\\end{table}\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/tqKL_nostdev_v2.pdf\n\t\t\\caption{}\n\t\t\\label{fig:subtq}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/tbKL_nostdev_v2.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subtb}\n\t\\end{subfigure}\n\t\n\t\\caption{$\\operatorname{Re}=14.4$: $D_{KL}$ vs dimension $d_h$ corresponding to PDFS for (a) $t_q$ (b) $t_b$. Dashed grey line corresponds to $D_{KL}$ calculated over different true data sets.}\n\t\\label{re14d4tqtb}\n\t\n\\end{figure}\n\n\n\\subsection{Phase prediction}\\label{phasepred}\n\n\n\n\n\nRecall that we gain substantial accuracy in dimension reduction by factoring out the spatial phase $\\phi_x(t)$ of the data. Here we complete the dynamical picture of the model predictions at $\\operatorname{Re}=14.4$ by illustrating the predictions of phase evolution, as given by the learned phase evolution equation \\eqref{eq:phaseevolution}. Figure \\ref{re14d4phasecompare} shows a short time evolution of $\\phi_x(t)$ corresponding to the true and predicted data for the $d_h=5$ model. We select $d_h=5$ because previous short time tracking and long time statistics show agreement with respect to the true data. The smooth increases and decreases in Figure \\ref{re14d4phasecompare} correspond to trajectories during time intervals where they are near an RPO and thus are traveling in the $x$-direction. The intervals where the phase flucuates rapidly are the bursts during which the trajectories are moving between the RPO regions. This behavior is well-captured at $d_h=5$. Notice that although the trajectories diverge, for short times we get around two $t_L$ of prediction horizon where the model still captures the correct dynamics, and Figure \\ref{re14d4phasecompare} provides a clear visual indications that the loss of predictability occurs during the bursts. \n\nWe now take an approach to quantify how well the model performs with respect to the true data. Taking a look at the drops and increases for $\\phi_x(t)$ we can observe that after every burst the trajectory will either travel, essentially randomly, in the positive (increasing $\\phi_x$) or negative (decreasing $\\phi_x$) $x$ direction. This behavior is essentially a run and tumble or random walk behavior in the sense that the long periods of positive or negative phase drift correspond to ``runs\" that are separated by ``tumbles\" that correspond to the bursts, in which the direction of phase motion is reset. Hence, a natural analysis of quantification for this type of dynamics consists of calculating the mean squared displacement (MSD) of the phase: \n\\begin{equation}\n\\mbox{MSD} (t)= \\langle (\\phi_x (t) - \\phi_x (0))^2 \\rangle.\n\\end{equation}\nFigure \\ref{re14d4phaseMSD} shows the time evolution of MSD of true and predicted data. The black line corresponds to the true data and the black and blue dashed lines serve as references with slopes of 1 and 1.5, respectively. The colored lines correspond to models with various dimensions. Looking at the true curve we notice a change from superdiffusive (slope = 1.5) to diffusive (slope = 1) scaling that happens around $t \\approx 200$, which corresponds to the mean duration of the quiescent intervals, as discussed above: i.e., to the average time the trajectories travel along the RPOs before bursting. The trajectory then bursts and reorients which is captured by the long time diffusive trend. Looking at the performance of the models we observe that $d_h=3,4$ do a good job at capturing the short time scaling, however these are not to able capture the change in slope that is observed in the true data. It is not until $d_h=5,6,7$ that the correct behavior at long times is observed -- indeed the predictions agree very well with the data, the slight upward shift upward at long times corresponding to the slight overprediction of the mean duration of the quiescent periods.\n\n\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/phasecorr_withdh5_moretime_v2.pdf}\n\t\\caption{Time evolution of $\\phi_x$ corresponding to the true data and $d_h=5$.}\n\t\\label{re14d4phasecompare}\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/phasecorr_withdh5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subphase}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.5\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/deltaphase_withdh5.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:subdeltaphase}\n\t\\end{subfigure}\n\t\n\t\\caption{(a) Time evolution of $\\phi_x$ corresponding to the True data and $d_h=5$. (b) Time evolution of $\\Delta \\phi_x(t + \\tau)$ corresponding to the True data and $d_h=5$.\\MDG{Aren't these just short- and long-time trajectories of the same thing? You don't seem to say so anywhere. Why are you labeling one $\\phi_x$ and the other $\\Delta\\phi_x$? Rethink this plot -- do you really need two separate ones?} }\n\t\\label{re14d4phasevis}\n\t\n\\end{figure}\n\\end{comment}\n\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/MSD_phase_v3.pdf}\n\t\\caption{MSD of $\\phi_x (t)$ corresponding to true data and models with dimensions $d_h=3-7$.}\n\t\\label{re14d4phaseMSD}\n\\end{figure}\n\n\\begin{comment}\n\\subsection{Short and long -time vorticity prediction}\\label{sec:dataspacing}\n\n\\MDG{Show some time evolution predictions. I think the presentation order here is backwards. You have long-time statistics before short-time trajectory predictions. And for the trajectory predictions, you only show averages (figure 10), not any examples of actual predictions as a function of time. I know you have these figures -- you have shown them to me. Ask yourself what the reviewer\/reader is going to want to see. } \n\n\\MDG{Observation that linear encoder (PCA) gives good representation of dynamics as long as nonlinear decoder is used}\n\n\\MDG{Anything at higher Reynolds number? Look back at older results -- let's revisit.}\n\\end{comment}\n\n\n\n\n\n\n\n\n\\subsection{Bursting prediction}\\label{burpred}\n\n\n\n\nPrevious research has focused on finding indicators that guide predictions of when a burst will occur. It has been shown for the Kolmogorov flow that before a burst there is a depletion of the content in the $(1,0)$ Fourier mode, which then feeds into the forcing mode $(0,n)$ \\cite{nicolaenko1990symmetry}. Figure \\ref{indandKE} shows how this looks for $\\operatorname{Re} = 14.4$, $n=2$. By considering a variational framework and finding solutions to a constrained optimization problem it was also found that examination of these modes can lead to predictions of when a burst will occur \\cite{farazmand2017variational}.\n\n\n\n\\begin{figure\n\n\n\n\n\t\\includegraphics[width=0.6\\linewidth]{figures\/KE_02_10_v4.pdf\n\n\n\n\t\\begin{comment}\n\t\\begin{subfigure}{0.5\\textwidth}\n\n\t\\includegraphics[width=.9\\linewidth]{figures\/Indicator_v2.pdf\n\t\\caption{}\n\t\\label{Indicators}\n\t\\end{subfigure}\n\t\\end{comment}\n\t\n\t\\caption{Time evolution of $KE$ and amplitudes corresponding to $(1,0)$ and $(0,2)$ Fourier mode for $\\operatorname{Re}=14.4$. }\n\t\\label{indandKE}\n\t\n\\end{figure}\n\nWith our framework, natural indicators are the latent variables $h$, which we will consider here along with some variations, including the indicators used in previous work. To predict bursting events based on a given indicator, we will use a simple binary classifier in the form of a support vector machine (SVM) with a radial basis function kernel \\cite{boser1992training}. These have shown success in predicting extreme events for problems such as extreme rainfall \\cite{nayak2013prediction}. With this approach, data at time $t$ is used to learn a function that outputs a binary label of bursting\/not bursting at time $t+\\tau_b$. For all of the cases considered we use the $d_h=5$ model, taking a dataset of $5 \\times 10^4$ snapshots to train the SVM and another $5 \\times 10^4$ as a test set. \n\n\nFigure \\ref{Pburs} shows the percent correct classification of bursting events with varying time $\\tau_b$ in the future. The black curve corresponds to predicting the events based on the PCA projection of the data, $P_{d_{h}} U^{T} \\omega$, into the first $d_h=5$ coefficients and the cyan curve corresponds to $h$ of dimension $d_h=5$. We notice that the PCA and $h$ curve fall on top of another and have a high probability of correct classification when considering prediction horizons less than one $t_L$. For this purpose we see that PCA is enough to predict bursting events. Figure \\ref{Pbursind} shows the percent correct classification of bursting at time $\\tau_b$ in the future for the previous discussed indicators. None of these work nearly as well as $P_{d_{h}} U^{T} \\omega$ or $h$. The blue curve corresponds to $(1,0)$ amplitude of the original true data, the green curve to the forcing $(0,2)$ amplitude, and we also consider $\\Delta \\phi$ in the purple curve. In the case of $\\Delta \\phi$ we see some predictability at times longer than one $t_L$ and less than two. This also happens for the case of $(1,0)$, however there seems to be no decrease or increase in the probability of correct classification. We can see from Figure \\ref{indandKE} that even though a depletion from $(1,0)$ mode is seen, the part corresponding to the bursting in the amplitude oscillates closely in terms of magnitude to the values corresponding to the quiescent region, which might be the reason of the poor prediction. The amplitude $(0,2)$ shows to be the better predictor for bursting events. At small $\\tau_b$ its predictions outperform $(1,0)$ and $\\Delta \\phi$, however at times larger than one $t_L$, $\\Delta \\phi$ performs better. \n\n\n\\begin{figure\n\n\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/Pburs_model_v7.pdf}\n\t\t\\caption{}\n\t\t\\label{Pburs}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\n\t\t\\includegraphics[width=1\\linewidth]{figures\/Pburs_ind_v5.pdf}\n\t\t\\caption{}\n\t\t\\label{Pbursind}\n\t\\end{subfigure}\n\n\t\n\n\n\n\n\n\t\n\n\n\n\n\t\\caption{Percent of correctly classified bursting events at $\\tau_b$ forward in time for: (a) $P_{d_{h}} U^{T} \\omega$ and $h$ at $d_h=5$, (b) and indicators $\\Delta \\phi$, $(1,0)$, and $(0,2)$. Note that the vertical scales on (a) and (b) are very different.}\n\t\\label{Pbursall}\n\t\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{figures\/KE_02_10_v2.pdf}\n\t\\caption{Time evolution of $KE$ and amplitudes corresponding to $(1,0)$ and $(0,2)$ Fourier mode for $\\operatorname{Re}=14.4$}\n\t\\label{KE0210}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure\n\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{figures\/Pburs_v2.pdf}\n\t\\caption{Probability of correctly predicting a burst at $\\tau_b$ forward in time}\n\t\\label{Pburs}\n\\end{figure}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion} \\label{sec:Conclusion}\n\n\n\nThe nonlinearity of the NSE poses challenges when using ROMs, where the dynamics are expected to evolve on an invariant manifold that will not lie in a linear subspace. Neural networks have proven to be powerful tools for learning efficient ROMs solely from data, however finding and exploiting a \\emph{minimal}-dimensional model has not been emphasized. We present a data-driven methodology to learn an estimate of the embedding dimension of the manifold for chaotic Kolmogorov flow and the time evolution on it. An autoencoder is used to find a nonlinear low-dimensional subspace and a dense neural network to evolve it in time.\n\n\nOur autoencoders are trained on vorticity data from two cases: a case where the dynamics show a relative periodic orbit solution ($\\operatorname{Re}=13.5$), and a case with chaotic dynamics ($\\operatorname{Re}=14.4$). The chaotic regime we consider comes with challenges due to the intermittent behavior observed where the trajectory travels in between quiescent intervals and bursting events. We factor out the rich symmetries of Kolmogorov flow before training of the autoencoders, which dramatically improves reconstruction error of the snapshots. This improves training efficiency by not having to learn a compression of the full state. Specifically, factoring out the translation symmetry decreases the mean-squared reconstruction error by an order of magnitude compared to the case where phase is not factored out, and several orders of magnitude compared to PCA. The phase-aligned low-dimensional subspace is then used for time evolution where the RPO dynamics is learned essentially perfectly at $d_h=2$ for $\\operatorname{Re}=13.5$ and very good agreement for short and long time statistics is obtained at $d_h=5$ for $\\operatorname{Re}=14.4$. For comparison, the full state space of the numerical simulation data is $N=1024$.\n\n\nWe also show phase prediction evolution results based on the low-dimensional subspace learned. The time evolution of the true phase exhibits a superdiffusive scaling at short times and a diffusive scaling at long times which we attribute to the traveling near an RPO and the reorientation due to bursting. \nFinally, using the low-dimensional representation enables accurate prediction of bursting events based on conditions about a Lyapunov time ahead of the event.\n This work opens new avenues for data-driven reduced order models with applications such as control for drag reduction. One important challenge that remains is more effective treatment of systems with intermittent dynamics like those described here. A recent study \\cite{floryan2021charts} has introduced a method that uses the differential topology formalism of charts and atlases to develop \\emph{local} manifold representations and dynamical model that can be stitched together to form a global dynamical model. One attractive feature of that formalism is that it enables use of separate representations for regions of state space with very different dynamics, and has already shown in specific cases to provide dramatically improved results for dynamics with intermittency. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\n\n\n\nchallenges are higher Re, symmetries, \n\n\\MDG{Rewrite to be half to two-thirds as long. Pick out the important points rather than trying to summarize everything (this isn't a summary, it's a conclusion). You have missed half the point of the modeling, which is not just dimension reduction but also time evolution. You didn't say anything about data-driven, or how K-flow is challenging because of its intermittency. Once the summary is finished, draw some conclusions. What works about the approach so far and what doesn't? What future work might be done to improve the predictions?} \n\nWe present an AE architecture and apply it to two-dimensional Kolmogorov flow. Specifically for the onset on which chaotic dynamics is sustained which happens for $n=2$, $\\operatorname{Re}=14.4$. We combine results from MSE, short time predictions, as well as PDFs of $I(t)$-$D(t)$ and Re $\\left[a_{0,1}(t)\\right]$ - Im $\\left[a_{0,1}(t)\\right]$ obtained from time maps at $d_h \\ll N$ to guide us in the estimation of the embedding dimension. The difference between the PDFs is calculated using KL divergence which guides us to selecting $d_h=5$. After this dimension no significant information is gained. This is further reassured with the short time tracking with respect to the true data as well as with the calculation of the quiescent and bursting fractions. We then complete the picture by predicting $\\Delta \\phi_x (t)$ from $h(t)$. Two behaviors are observed when calculating the MSD of $\\phi_x (t)$. At short times MSD scales in a super diffusive fashion while at long times diffusive scaling is obtained. The change to super diffusive happens at approximately $t \\approx 200$ which agrees with the quiescent time scale. Here we see that starting at $d_h=5$ the change to diffusive scaling is obtained. We further ask the question: can we predict bursting events from our model at $d_h=5$? To this end we use SVM to predict if a bursting will occur. We find that predicting with the first 5 PCA modes gives similar results as predicting with $h$, which we attribute to the similarity of these two projections. Using the mode $(1,0)$ shows no predictive capabilities which is not the case at higher Re and $n$. However $\\Delta \\phi_x$ and $(0,2)$ amplitude show to better predict bursting events. With varying dimensions we see that after $d_h=5$ is reached, no further improvement is obtained in the bursting prediction and it outperforms the different indicators considered. The selection of $d_h=5$ arises naturally in this case. Furthermore we consider the encoder section of the AE to predict bursting events. Using the full state PCA$-h$ (not shown) did not improve the event prediction, however predicting based on individual units gives some interesting outcomes. As shown previously, we see that PCA$_2-h_2$ does a better job at predicting bursting events than PCA$_2$ or $h_2$. The correction of the encoder shows to come into play in the bursting events where the difference increases in magnitude as seen in Figure \\ref{fig:diffh2dh6}. When taking the discrete Fourier transform of PCA$_2$ we see that most of its content lies in the $(0,1)$ Fourier mode. Recall that this mode corresponds to the projection in which the traveling between the different symmetric subspaces is observed. We find that using the amplitude of $(0,1)$ alone as an indicator does not provide good bursting prediction compared to the other indicators considered in this manuscript. The nonlinearity introduced by the encoder in PCA$_2$ seems to split quiescent and bursting events in such a way that prediction is improved. This work motivates new avenues such as flow control based on minimal model. Finding these models in experiments could also be powerful tools for controls. Extensions of this work also include charting the manifold such that the inertial manifold dimension can be estimated.\\MDG{what would be the advantage of this? Not just determining manifold dimension!!!! Cite Daniel's paper here!}\n\n\n\\end{comment}\n\n\n\n\\begin{comment}\n\nWe can argue that features extracted by the NNs are the necessary ones to correctly evolve the state in time and capture long time statistics. All models evolved in such a way that the turbulence was sustained, with no apparent blow up of the trajectory evolution for the models showed. \n\nFor bursting at lower n and Re delta phi is a better predictor\n\n\\end{comment}\n\n\\begin{comment}\n\n\\subsection{Trajectory Difference}\\label{sec:dataspacing}\n\nShort and long-time prediction of the trajectory is compared with the true data. We first take a look at the calculation of the KE for $d_h = 5$. Figure \\ref{re14d4ke} shows the trajectory comparison with respect to the true data. We notice that predicted trajectory travels close to the true trajectory for about a Lyapunov time which in this case is $\\approx 93$ time units \\hl{[cite]}. \n\n\\begin{figure}[H]\n\n\t\\centering\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\linewidth]{figures\/kecomparisonshort.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub7}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{figures\/kecomparisonlong.pdf}\n\t\t\\caption{}\n\t\t\\label{fig:sub8}\n\t\\end{subfigure}\n\n\t\\caption{(a) Short-time prediction comparison of KE with true data (b) Long-time prediction comparison of KE with true data. Notice that model is able to capture quiescent and bursting events.}\n\t\\label{re14d4ke}\n\t\n\\end{figure}\n\nWe also compare the vorticity snapshots obtained from the model with the true data. To this end we calculate the MSE between the two trajectories. Similar to the KE results we see that the trajectory travels well for about a Lyapunov time. \n\n\\begin{figure}[H]\n\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{figures\/wdifference.pdf}\n\t\\caption{Vorticity difference between predicted data from the model and true data.}\n\t\\label{re14d4wdiff}\n\\end{figure}\n\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\\begin{acknowledgments}\nThis work was supported by AFOSR FA9550-18-1-0174 and ONR N00014-18-1-2865 (Vannevar Bush Faculty Fellowship). We also want to thank the Graduate Engineering Research Scholars (GERS) program and funding through the Advanced Opportunity Fellowship (AOF) as well as the PPG Fellowship.\n\\end{acknowledgments}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nAnti-de Sitter (AdS) gravity coupled to scalar fields only, has received recently considerable \nattention.\nThis simple model admits both soliton \n\\cite{Hertog:2004rz}, \n\\cite{Hertog:2004ns}, \n\\cite{Radu:2005bp},\n\\cite{Faulkner:2010fh},\nand black hole solutions (see $e.g.$\n\\cite{Torii:2001pg},\n\\cite{Sudarsky:2002mk},\n\\cite{Winstanley:2002jt},\n\\cite{Hertog:2004dr},\n\\cite{Martinez:2004nb},\n\\cite{Park:2008zzb})\nwith interesting properties, the resulting picture being in strong contrast\nwith the one found \nin the absence of a cosmological constant.\nMoreover, since scalar fields generically enter the gauge supergravity\nmodels, the study of such solutions is also relevant\nto the AdS\/CFT conjecture.\n \nHowever, most of the studies in the literature assume that the scalar fields \nare real and have the same symmetries as the underlying spacetime. \nIt is interesting to ask if the known solutions in \\cite{Hertog:2004rz}-\\cite{Martinez:2004nb} \ncan be generalized \nfor a complex scalar field. \nIn formulating a scalar field ansatz in this case, it is natural to take \na separation of variables \n\\begin{equation}\n\\label{Phi}\n\\Phi(\\vec x,t)=e^{-i \\omega t}\\phi(\\vec x),\n\\end{equation}\n(with $t,\\vec x$ the time and space coordinates, respectively) \nsuch that the energy-momentum tensor is time independent\n(note that in general, $\\phi(\\vec x)$ is a complex function as well).\nThis model possesses a conserved current \n\\begin{eqnarray}\nj^{\\mu}= - i \\left( \\Phi^* \\partial^{\\mu} \\Phi \n - \\Phi \\partial^{\\mu}\\Phi ^* \\right) ,~~~\nj^{\\mu} _{\\ ; \\, \\mu} = 0 \\ ,\n\\end{eqnarray}\nand a conserved Noether charge $Q$, which is the integral over a $t=const.$ hypersurface\nof the $j^t$ component of the current.\n \nScalar field configurations carrying a global U(1) Noether charge have been extensively studied in the literature, \nfor a Minkowski spacetime background and four spacetime dimensions.\nWhile black hole solutions are rather difficult to find in this case \n\\cite{Pena:1997cy},\n\\cite{Kleihaus:2010ep},\nthe spectrum of smooth horizonless solutions with harmonic \n time dependence is very rich.\nFor example, one finds (non-topological) soliton solutions even\nin the absence of gravity--the so-called $Q-$balls \n\\cite{Friedberg:1976me},\n\\cite{Coleman:1985ki}, \n\\cite{Lee:1991ax}. \nThere, the scalar field possesses a potential which\nis necessarily non-renormalizable\\footnote{It is interesting to note that $Q-$ball solutions appear in supersymmetric \ngeneralizations of the standard model \n\\cite{Kusenko:1997zq}.\nThey may be responsible for the generation of\nbaryon number or may be regarded as candidates for dark matter \n\\cite{Kusenko:1997si}. \n}.\nWhen gravity is coupled to $Q-$balls, boson stars arise \n(see the review work \\cite{Jetzer:1991jr}).\nMoreover, in this case, solutions with rather similar properties are found even for a\npotential consisting in a mass term only\n\\cite{Kaup:1968zz}, \n\\cite{Ruffini:1969qy}, \n\\cite{Mielke:1980sa}.\nThe self-gravity of such objects is balanced by the dispersive effect\ndue to the wave character of the complex scalar field.\n\n The study of boson stars and Q-balls in \n AdS spacetime has received relatively little attention, only spherically symmetric solutions\n being discussed so far \n \\cite{Astefanesei:2003qy}, \n \\cite{Prikas:2004yw}, \n \\cite{Hartmann:2012wa} \n (see, however, the planar solitons with a complex scalar field in \\cite{Horowitz:2010jq}).\n While the results in \\cite{Astefanesei:2003qy} for a gravitating \n massive scalar field without self-interaction are rather similar to those valid in \n the asymptotically flat limit, \n the recent study \n \\cite{Hartmann:2012wa}\n has shown the existence of some new features\n in the case of Q-ball solutions. \n\n\nFor a vanishing cosmological constant, $\\Lambda=0$, the scalar solitons admit also \nrotating generalizations \n\\cite{Schunck:1996wa}, \n\\cite{Yoshida:1997qf}, \n\\cite{Volkov:2002aj},\n\\cite{Kleihaus:2005me}, \n\\cite{Kleihaus:2007vk}.\nThese are stationary localized solutions possessing a finite mass and angular momentum.\nInterestingly, their angular momentum is quantized\n$J=n Q$ (with $n$ an integer), \nand the energy density exhibits a toroidal distribution.\n \nThe main purpose of this work is to investigate the existence of spinning scalar solitons \nfor the case of a four dimensional AdS background\\footnote{Rotating AdS boson stars were found \nhowever in $d=3$ \\cite{Astefanesei:2003rw} and also in $d=5$ \\cite{Dias:2011at}\ndimensions, where a special ansatz proposed in \\cite{Hartmann:2010pm} allows to deal with ODEs.}.\nThese solutions are found by solving numerically\na set of partial differential equations with suitable boundary conditions. \nIrrespective of the scalar field potential, they exhibit the same quantization of the angular momentum\nas for $\\Lambda=0$.\nWe find that the spinning \n solutions emerge as perturbations of the AdS spacetime\nfor a critical value of the frequency which is fixed by the scalar field mass and the cosmological constant.\nIn the absence of a scalar field self-interaction, the basic\nproperties of the solutions\nare rather similar to those of the asymptotically flat counterparts.\nNew features are found once we allow for self-interaction terms in the potential\nleading to a violation of the positive energy condition.\nFor example, we find a class of solutions with a smooth $\\omega=0$\nlimit, describing static axially symmetric solitons. \n \n\n\\section{The model}\n\\subsection{The action and field equations}\n We consider the action of a self-interacting complex scalar field \n$\\Phi$ coupled to Einstein gravity with a negative cosmological constant\n$\\Lambda=-3\/\\ell^2$,\n\\begin{equation}\n\\label{action}\nS=\\int d^4x \\sqrt{-g}\\left[ \\frac{1}{16\\pi G}(R-2 \\Lambda)\n -\\frac{1}{2} g^{\\mu\\nu}\\left( \\Phi_{, \\, \\mu}^* \\Phi_{, \\, \\nu} + \\Phi _\n{, \\, \\nu}^* \\Phi _{, \\, \\mu} \\right) - U( \\left| \\Phi \\right|) \n \\right] , \n\\end{equation}\nwhere $R$ is the curvature scalar,\n$G$ is Newton's constant,\nthe asterisk denotes complex conjugation,\nand $U$ denotes the scalar field potential.\n\nVariation of the action with respect to the metric\nleads to the Einstein equations\n\\begin{equation}\n\\label{Einstein-eqs}\nE_{\\mu\\nu}= R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R+\\Lambda g_{\\mu\\nu} - 8 \\pi G T_{\\mu\\nu}=0\\ , \n\\end{equation} \nwhere $T_{\\mu\\nu}$ is the\n stress-energy tensor of the scalar field\n\\begin{eqnarray}\n\\label{tmunu} \nT_{\\mu \\nu} \n&=&\n\\left(\n \\Phi_{, \\, \\mu}^*\\Phi_{, \\, \\nu}\n+\\Phi_{, \\, \\nu}^*\\Phi_{, \\, \\mu} \n\\right )\n-g_{\\mu\\nu} \\left[ \\frac{1}{2} g^{\\alpha\\beta} \n\\left( \\Phi_{, \\, \\alpha}^*\\Phi_{, \\, \\beta}+\n\\Phi_{, \\, \\beta}^*\\Phi_{, \\, \\alpha} \\right)+U(\\left|\\Phi\\right|)\\right]\n \\ .\n\\end{eqnarray}\nVariation with respect to the scalar field\nleads to the matter field equation,\n\\begin{eqnarray}\n\\label{scalar-eq}\n\\frac{1}{\\sqrt{-g}} \\partial_\\mu \\big(\\sqrt{-g} \\partial^\\mu\\Phi \\big)=\\frac{\\partial U}{\\partial\\left|\\Phi\\right|^2} \\Phi.\n\\end{eqnarray} \nThe potential $U$ can be decomposed according to\n\\begin{eqnarray}\n\\label{pot1}\nU(|\\Phi|)= \\mu^2 |\\Phi|^2 +U_{int}(|\\Phi|),\n\\end{eqnarray}\n where $\\mu$ is the mass of the field, and $U_{int}$ is a self-interaction potential.\nAs discussed in \\cite{Astefanesei:2003qy} for $\\mu^2>0$, this model possesses finite mass solutions \neven in the absence of self-interaction,\n$U_{int}=0$, the so-called 'mini-boson stars'.\n\nHowever, the inclusion\nof an interaction potential may lead\nto a more complex picture (see $e.g.$ \\cite{Colpi:1986ye} for the $\\Lambda=0$ case). \nOur choice of $U_{int}$ was guided by the requirement that nontopological solitons\nexist also in a fixed AdS background.\nMoreover, we are interested in the case when the solutions would possess a nontrivial static limit. \nAs discussed in \\cite{Hertog:2004rz}, this \nrequires the occurrence of negative energy densities,\n $i.e.$ $U(|\\Phi|)<0$\nin some region.\nSince, in order \nto make contact with the previous work on mini-boson stars, we restrict the numerical part of our\nstudy to the case $\\mu^2>0$, this implies that $U_{int}$ is not strictly positive. \n\nWe have found that the simplest choice of the interaction potential \nsatisfying these conditions is $U_{int}=-\\lambda|\\Phi|^{2k}$, with $k>1$ and $\\lambda>0$.\nMost of the results in this work are found\\footnote{Note that the exact solution with spherical symmetry\n in Section 3 covers a more general range of $k$.} for $k=2$,\nsuch that \n\\begin{eqnarray}\n\\label{U}\n U(|\\Phi|)=\\mu^2 |\\Phi|^2 -\\lambda |\\Phi|^{4}.\n\\end{eqnarray} \nAlthough the action (\\ref{action}) together with (\\ref{U}) does not seem to correspond to any supergravity \nmodel, it is likely that some features of its solutions are generic.\n In particular, we have found the same general picture \n for a more general potential (which was used in the previous studies \n \\cite{Volkov:2002aj},\n\\cite{Kleihaus:2005me}, \n\\cite{Kleihaus:2007vk}\n on $\\Lambda=0$ spinning Q-balls and boson stars)\n \\begin{eqnarray}\n\\label{Un}\n U(|\\Phi|)=\\mu^2 |\\Phi|^2 -\\lambda |\\Phi|^{4}+\\nu |\\Phi|^{6},\n\\end{eqnarray} \n\n provided that\n the new coupling constant $\\nu>0$ is small enough.\n \n \n\n\\subsection{The Ansatz }\n\nWe are interested in stationary axially symmetric configurations, \nwith a spacetime geometry admiting two Killing vectors \n$\\partial_t$\nand \n$\\partial_{\\varphi}$,\nin a system of adapted coordinates $\\{t, r, \\theta, \\varphi\\}$.\nThus the line element can be written as\n\\begin{eqnarray}\n\\label{ansatzg}\nds^2 =- F_0 N dt^2 \n+ F_1 \\left( \\frac{dr^2}{N} + r^2 \\, d\\t^2 \\right) \n + F_2 r^2 \\sin^2 \\t \\left( d \\varphi\n- \\frac{W}{r} dt \\right)^2 . \n\\end{eqnarray}\nThe metric functions $F_0$, $F_1$, $F_2$ and $W$\ndepend on the variables $r$ and $\\theta$ only, while\n\\begin{eqnarray}\nN=1+\\frac{r^2}{\\ell^2}\n\\end{eqnarray}\nis a suitable 'background' function.\n \n\nFor the scalar field $\\Phi$ we adopt the stationary ansatz \n\\begin{eqnarray}\n\\label{ansatzp}\n\\Phi (t,r,\\t, \\varphi)= \\phi (r, \\t)\n e^{ i( n \\varphi-\\omega t )} , \n\\end{eqnarray}\nwhere $\\phi (r, \\theta)$ is a real function,\nand $\\omega $ and $n$ are real constants.\nSingle-valuedness of the scalar field requires\n$\\Phi(\\varphi)=\\Phi(2\\pi + \\varphi)$;\nthus the constant $n$ must be an integer,\n$i.e.$, $n \\, = \\, 0, \\, \\pm 1, \\, \\pm 2, \\, \\dots$~.\nIn what follows, we shall take $n\\geq 0$ and $\\omega \\geq 0$,\nwithout any loss of generality. \n\n\n The spherically symmetric limit is found for $n=0$, in which case the\n functions $F_0,F_1,F_2$ and $\\phi$\n depend only on $r$, with $F_1=F_2$ and $W=0$. \n \n\\subsection{The boundary conditions}\nThe solutions in this work describe horizonless, particle-like configurations. \nA study of an approximate form of the solutions as a power series around $r=0$ leads to the following\n boundary conditions at the origin\\footnote{For spherically\nsymmetric solutions, the scalar field is nonvanishing ar $r=0$.}:\n\\begin{eqnarray}\n\\label{bc0} \n\\partial_r F_i|_{r=0}=0, ~~\nW|_{r=0}=0,~~\n\\phi| _{r =0}=0~,\n\\end{eqnarray}\n (with $i=0,1,2$).\nAt infinity, the AdS background is approached, while the scalar field vanishes.\nWithout any loss of generality,\nwe are choosing a frame in which the solutions do not rotate at infinity,\nthe conformal boundary being a static Einstein universe $R\\times S^2$.\nThe boundary conditions compatible with these requirement are\n\\begin{eqnarray}\n\\label{bcinf} \nF_i|_{r \\rightarrow \\infty} =1,~~\nW|_{r \\rightarrow \\infty} =0, ~~\n\\phi| _{r \\rightarrow \\infty}=0 \\ .\n\\end{eqnarray}\nFor $\\t=0,\\pi$ \nwe require the boundary conditions\n\\begin{eqnarray}\n\\label{bct0} \n\\partial_{\\t} F_i|_{\\t=0,\\pi}=0, ~~\n\\partial_{\\t} W |_{\\t=0,\\pi}=0,~~\n\\f |_{\\t=0,\\pi}=0.\n\\end{eqnarray}\n The absence of conical singularities\n imposes on the symmetry axis the supplementary condition \n$F_1|_{\\theta=0,\\pi}=F_2|_{\\theta=0,\\pi},$\nwhich is used to verify the accuracy of the solutions.\n\nAlso, all solutions in this work \nare invariant under the parity transformation $\\theta \\to\\pi-\\theta $.\nWe make use of this symmetry to integrate the equations for $0\\leq \\theta\\leq \\pi\/2$ only, the\nfollowing boundary conditions being imposed in the equatorial plane\n\\begin{eqnarray}\n\\label{bctpi2} \n\\partial_{\\t} F_i|_{\\t=\\pi\/2}=0 ,~~\n\\partial_{\\t} W |_{\\t=\\pi\/2}=0 \\ ,~~\n\\partial_{\\t} \\phi |_{\\t=\\pi\/2}=0 \\ .\n \\end{eqnarray} \n\n\n\\subsection{The far field asymptotics and global charges}\nFor solutions\nwith $\\mu^2>0$ (the only case considered in the numerics),\nthe scalar field decays asymptotically as \n\\begin{eqnarray}\n\\label{asym-scalar0}\n\\phi\\sim \n\\frac{c_1(\\theta)}{r^{\\frac{3}{2}\\left( 1+\\sqrt{1+\\frac{4}{9}\\mu^2 \\ell^2} \\right)}}+\\dots~.\n\\end{eqnarray}\nThe physical interpretation of $c_1(\\theta)$\nis that it corresponds, up to a normalization,\nto the expectation value of some scalar operator in the dual theory.\n\nWithout entering into details, \nwe mention that the picture is more complicated \\cite{Henneaux:2006hk} if one allows for a tachyonic\nmass of the scalar field, $\\mu^2<0$.\nFor $-9\/4< \\mu^2\\ell^2<-5\/4$,\nthe general asymptotic behaviour of the scalar field \nis more complicated\\footnote{For\na field which saturates the Breitenlohner-Freedman bound $\\mu^2\\ell^2=-9\/4$,\none finds $\\Delta_+=\\Delta_-=\\Delta$, and the second solution asymptotically behaves \nlike $\\log r\/r^\\Delta$.\n}, with the existence of a second mode apart from (\\ref{asym-scalar0}): \n\\begin{eqnarray}\n\\label{asym-scalar}\n\\phi\\sim \n\\frac{c_1(\\theta)}{r^{\\Delta_+}}+\n\\frac{c_2(\\theta)}{r^{\\Delta_-}},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Deltapm}\n\\Delta_{\\pm}=\\frac{3}{2}\\left( 1\\pm \\sqrt{1+\\frac{4}{9}\\mu^2 \\ell^2} \\right). \n\\end{eqnarray}\nFor this range of $\\mu^2<0$, both modes above are normalizable in the sense that the spatial integral of $j^t$ is finite, $i.e.$\nthe scalar field possesses a finite Noether charge.\nTo have a well defined theory,\none must specify a boundary condition at infinity, $i.e.$\nto choose a relation between $c_1$ and $c_2$, the standard choice being $c_2=0$. \nHowever, as discussed $e.g.$ in \\cite{Hertog:2004dr}, \\cite{Henneaux:2004zi},\nthe solutions with a slower decay at infinity, $c_2 \\neq 0$,\nare also physically acceptable, the AdS charges involving in this\ncase a scalar field contribution (thus the \nexpression (\\ref{EJ}) below would not be valid).\n\n \nRestricting to the case $\\mu^2>0$, the scalar field decays asymptotically faster than $1\/r^3$ and\nthus the Einstein equations (\\ref{Einstein-eqs}) imply the following form of the metric functions\nas $r\\to \\infty$\n \\begin{eqnarray}\n\\nonumber\n&&F_0=1+ \\frac{f_{03}(\\theta)}{r^3}+O(1\/r^5),\n~~~F_1=1+ \\frac{f_{13}(\\theta)}{r^3}+O(1\/r^5), \n\\\\\n\\label{asym1}\n&&F_2=1+ \\frac{f_{23}(\\theta)}{r^3}+O(1\/r^5),\n~~W=\\frac{w_2(\\theta)}{r^2}+O(1\/r^4),\n\\end{eqnarray}\nin terms of two functions $f_{13}(\\theta)$ and $w_2(\\theta)$ which result from the numerics, \nwith \n\\begin{eqnarray}\nf_{03}(\\theta)=-3 f_{13}(\\theta)-\\frac{4}{3}\\tan\\theta f_{13}'(\\theta),~~{\\rm and}~~\nf_{23}(\\theta)= f_{13}(\\theta)+\\frac{4}{3}\\tan\\theta f_{13}'(\\theta).\n\\end{eqnarray} \n\nA straightforward computation based on the formalism in \\cite{Balasubramanian:1999re} \nleads to the following expression for \nthe mass-energy $E$ and angular momentum $J$ of the configurations\n\\begin{eqnarray}\n\\label{EJ}\nE= \\frac{1}{8 G \\ell^2}\\int_{0}^\\pi d\\theta \\sin\\theta \\bigg(5 f_{13}(\\theta)+3 f_{23}(\\theta)\\bigg),~~\nJ=-\\frac{3}{8 G }\\int_{0}^\\pi d\\theta \\sin^3\\theta ~w_2(\\theta) .\n\\end{eqnarray} \n(Note that the same result can be derived by using\n the Ashtekar-Magnon-Das conformal mass definition \\cite{Ash}).\n Moreover, \nthe same expression for the angular momentum is found from the Komar integral:\n\\begin{eqnarray}\n\\label{j1}\n &J=\\frac{1}{8 \\pi G }\\int R_{\\varphi}^t \\sqrt{-g} dr d\\theta d\\varphi\n =\n\n\n\n -\\frac{1}{8 G }\\int_{0}^{\\infty}dr \\int_0^\\pi d \\theta\n \\bigg [\n \\bigg(r^4 \\sqrt{\\frac{F_2^4}{F_0}}\\sin^3 \\theta \\big(\\frac{W}{r} \\big)_{,r} \\bigg)_{,r}\n + \\bigg(\\frac{r}{N} \\sqrt{\\frac{F_2^3}{F_0}}\\sin^3 \\theta W_{,\\theta} \\bigg)_{,\\theta}\n \\bigg ].~{~~}\n\\end{eqnarray} \nThe solutions possess also a conserved Noether charge ($i.e.$ the total\nparticle number)\n\\begin{eqnarray}\n\\label{Q1}\nQ= \\int j^t \\sqrt{-g} dr d\\theta d\\varphi=\n2 \\pi \\int_0^\\infty dr \\int_0^\\pi d\\theta~\n2r \\sin \\theta F_1\\sqrt{\\frac{F_2}{F_0}}\n\\frac{ \\phi^2}{N}(\\omega r-n W).\n\\end{eqnarray} \nSince $T_\\varphi^t=n j^t$, from (\\ref{j1}) \nand the Einstein equation $R_{\\varphi}^t=8\\pi G T_{\\varphi}^t$\nwe find that the generic relation\n\\begin{eqnarray}\n\\label{JQ}\nJ= n Q~,\n\\end{eqnarray} \n(which was proven in \\cite{Schunck:1996wa}\n for asymptotically flat spinning solutions)\nholds also in the AdS case.\nAs a result, the spinning solitons\ndo not emerge as perturbations\nof the spherically symmetric\nconfigurations, $i.e.$\nthere are no slowly rotating\nsolutions in this model (note also that the expression of the scalar field\npotential was not used in the derivation of (\\ref{JQ})). \n\nThese scalar solitons have no horizon and therefore they are zero entropy objects,\nwithout an intrinsic temperature.\nThe first law of thermodynamics\nreads in this case \\cite{Lee:1991ax}\n$dE= \\omega dQ=\\frac{\\omega }{n} dJ$.\n\n\\subsection{The numerical scheme}\nThere are not many studies on numerical \nsolutions of Einstein gravity with negative cosmological constant\ncoupled with matter fields,\n describing stationary, axially symmetric configurations.\nThe model in this work provides perhaps \nthe simplest test ground for \ninvestigating various approaches and developing numerical techniques for elliptic \nproblems with AdS asymptotics\\footnote{Because the ansatz (\\ref{ansatzp}) has an explicit dependence \n on both $\\varphi$ and $t$,\n the scalar field is neither static nor axisymmetric.\n However, all physical quantities, such as the current $j^\\mu$\n and the energy-momentum tensor $T_{\\mu\\nu}$,\n will exhibit no dependence on $\\varphi$ and $t$.}.\n\n\n\nThe solutions of the field equations (\\ref{Einstein-eqs}), (\\ref{scalar-eq}) \nare found by using an approach originally proposed in \\cite{Kleihaus:1996vi}\nfor $d=4$ asymptotically flat solutions of Einstein \ngravity coupled with Yang-Mills gauge fields. \nThe equations for the metric functions $ \\mathcal F=(F_0,F_1,F_2,W)$ \n employed in the numerics,\nare found by using a suitable combination of the Einstein equations (\\ref{Einstein-eqs}),\n$E_t^t =0,~E_r^r+E_\\theta^\\theta =0$, $E_\\varphi^\\varphi=0$\nand $E_{\\varphi}^{t} =0$,\n which diagonalizes them $w.r.t.$ $\\nabla^2 \\mathcal F$ \n (where $\\nabla^2=\\partial_{rr}+\\frac{1}{r}\\partial_{r}+\\frac{1}{r^2 N}\\partial_{\\theta\\theta}$).\n An important issue here concerns the status of the \nremaining equations $E_\\theta^r =0,~E_r^r-E_\\theta^\\theta =0$,\n which\nyield two constraints. \nFollowing \\cite{Wiseman:2002zc}, one can show that\nthe identities $\\nabla_\\mu E^{\\mu r} =0$ and $\\nabla_\\mu E^{\\mu \\theta}=0$, \nimply the Cauchy-Riemann relations\n$\n\\partial_{\\bar r} {\\cal P}_2 +\n\\partial_\\theta {\\cal P}_1 \n= 0 ,~~\n \\partial_{\\bar r} {\\cal P}_1 \n-\\partial_{\\theta} {\\cal P}_2\n~= 0 ,\n$\nwith ${\\cal P}_1=\\sqrt{-g} E^r_\\theta$, ${\\cal P}_2=\\sqrt{-g}r \\sqrt{N}(E^r_r-E^\\theta_\\theta)\/2$\nand $d\\bar r=\\frac{dr}{r \\sqrt{N}}$.\nTherefore the weighted constraints still satisfy Laplace equations, and the constraints \nare fulfilled, when one of them is satisfied on the boundary and the other \nat a single point\n\\cite{Wiseman:2002zc}. \nFrom the boundary conditions (\\ref{bc0})-(\\ref{bct0}) we are imposing,\nit turns out that this is the case for our solutions,\n $i.e.$ the numerical scheme is consistent.\n \nTo obtain spinning boson star solutions,\nwe solve numerically the set of five coupled non-linear\nelliptic partial differential equations for $(\\mathcal F,\\phi)$,\nsubject to the boundary conditions (\\ref{bc0})-(\\ref{bct0}).\nWe employ a compactified radial coordinate $\\bar r= r\/(1+ r)$\n which maps spatial infinity to the finite value $\\bar r=1$.\n Then the equations are discretized on a non-equidistant grid in\n$\\bar r$ and $\\theta$.\nTypical grids used have sizes $250 \\times 30$,\ncovering the integration region\n$0\\leq \\bar r \\leq 1$ and $0\\leq \\bar \\theta \\leq \\pi\/2$.\n(See \\cite{Kleihaus:1996vi} and \\cite{schoen} \nfor further details and examples for the numerical procedure.) \nThe numerical calculations are based on the Newton-Raphson method\nand are performed with help of the software package FIDISOL \\cite{schoen},\nwhich provides also an error estimate for each unknown function.\nThe typical relative error for the solutions \nin this work is smaller that\n$10^{-3}$.\n \n\\section{The solutions in the probe limit}\n\nWe shall start with a discussion of the solutions in a fixed AdS\nbackground, $i.e.$ \nwithout backreaction, $F_0=F_1=F_2=1,~W=0$ in the metric ansatz (\\ref{ansatzg}).\nThe problem is much easier to study in this limit and\nthe solutions exhibit already some basic features of\nthe gravitating configurations.\n\nThe usual Derick-type scaling argument (see $e.g.$\nthe discussion in \\cite{Radu:2008pp} \nfor the $\\Lambda=0$ limit) implies that\nthe Q-balls in a fixed AdS background satisfy\nthe following virial identity\n\\begin{eqnarray}\n\\label{virial}\n\\int_0^\\infty dr \\int_0^\\pi d\\theta \\sin \\theta\n\\bigg [\nN\\phi_{,r}^2\n+\\frac{\\phi_{,\\theta}^2}{r^2}\n+\\frac{n^2\\phi^2}{r^2\\sin^2\\theta}\n+3\\left(U(\\phi)-\\frac{\\omega^2\\phi^2}{N}\\right)\n+\\frac{2r}{\\ell^2}\\left(\\phi_{,r}^2+\\frac{ \\omega^2 \\phi^2}{N^2}\\right)\n\\bigg]=0,\n\\end{eqnarray}\nwhich was used as a further test of the numerical accuracy.\nIt is clear that the \nsolutions with a strictly positive potential, $U(\\phi)>0$,\nowe their existence to the harmonic time dependence of the scalar field.\nAlso, the solutions may exist in the $ \\omega\\to 0$ limit \nas long as $U(\\phi)$ is allowed to take negative values.\n\nAs usual in the absence of back reaction, \nthe total mass-energy $E$ and angular momentum of the configurations\nare found by integrating over the entire space\nthe components $-T_t^t$ and $T_\\varphi^t$ of the energy \nmomentum tensor.\n\n\n\n\\subsection{Spherically symmetric configurations.\nAn exact solution}\n\nIn the spherically symmetric limit, the equation (\\ref{scalar-eq}) with a self-interaction\npotential $U_{int}=-\\lambda \\phi^{2k}$\nadmits the following simple exact solution, which to our knowledge,\nwas not yet discussed in the literature: \n\\begin{eqnarray}\n\\label{ex-sol}\n\\Phi(r,t)=\\left(\n\\frac{\\mu^2}{\\lambda}\\frac{\\Delta^2-\\ell^2 \\omega^2}{(\\Delta-3)(\\Delta+1))}\n\\right)^{\\frac{\\Delta}{2}} {\\left(1+\\frac{r^2}{\\ell^2} \\right)^{-\\frac{\\Delta}{2}}} e^{-i \\omega t}, \n\\end{eqnarray}\nwith $\\Delta=\\Delta_{\\pm}$, as given by (\\ref{Deltapm}).\n \n \nFor this exact solution, \nthe coefficient $k$ of the self-interaction term \nin the scalar field potential (\\ref{U}) is fixed by $\\Delta$,\n\\begin{eqnarray}\n\\label{k}\nk=1+\\frac{1}{\\Delta}.\n\\end{eqnarray}\nThis is a one parameter family of solutions which, for given input parameters $\\mu^2,\\lambda$ and $\\ell$,\ncan be parametrized by the frequency of the field.\nChoosing $\\Delta=\\Delta_{+}$ leads to a range $10$\nin this work (including the spinning ones with $n\\neq 0$).\n The \noccurrence of an extra-branch of solutions with $ \\omega> \\omega_c$ (see Figure 1b)\nis a feature of the $\\phi^3$ potential.\n(Note that this branch does not appear for the exact solution with a $\\phi^4$ potential.) \n\n\nFor the same value of the scalar field mass, when choosing instead\n$\\Delta=\\Delta_-=1$ in (\\ref{ex-sol}), (\\ref{k}), one recovers the model with\na $\\phi^4$ potential. \nUnfortunately, the total mass-energy of these solutions, as defined\nin the usual way as the integral of $T_t^t$,\ndiverges linearly\\footnote{However, the mass can be regularized\nby supplementing the action with a suitable scalar field boundary counterterm.},\n$E=E_{div}+E_0$, with $E_{div}=- V_2\\frac{ (1-\\ell^2 \\omega^2) }{2\\lambda \\ell}r_c$ \n(where $r_c\\to \\infty$), \nwhile the charge is finite:\n\\begin{eqnarray}\n\\label{phi4}\nE_0=V_2\\frac{\\pi (1-\\ell^2 \\omega^2)(1+3 \\ell^2 \\omega^2)}{16\\lambda \\ell},~~\nQ=V_2\\frac{\\pi \\ell \\omega(1- \\omega^2 \\ell^2)}{4 \\lambda}.\n\\end{eqnarray} \n \n\nSolutions beyond the framework (\\ref{ex-sol}), (\\ref{k})\nare found by using a numerical approach,\nfor a generic ansatz $\\Phi = \\phi (r) e^{-i \\omega t} $. \nHere, for simplicity we restrict ourselves to the case of nodeless solutions.\nThen, for $\\mu^2>0$ and a self-interaction\npotential $U_{int}=-\\lambda \\phi^{4}$,\nit turns out that the picture in Figure 1a is generic.\nFor any $\\Lambda$, the solutions exist for a limited range of frequencies,\n$0\\leq \\omega< \\omega_c= \\Delta_+\/\\ell$.\nThe mass-energy and Noether charge are bounded and approach a maximum\nfor some $ \\omega$ around $ \\omega_c\/2$.\nThe same pattern is recovered \n in the presence of an extra $\\nu \\phi^6$ self-interaction term (with $\\nu>0$), provided that\n the new coupling constant is small enough.\n \n \n \n\n\\subsection{Spinning scalar solitons in an AdS background}\n\nThe spinning generalizations of these solutions are found by taking $n\\neq 0$\nin the general ansatz (\\ref{ansatzp}).\nFor a $\\phi^4$ self-interaction potential,\napart from the winding number $n$,\nthe input parameters are $\\ell$, $ \\omega$, $\\mu$ and $\\lambda$.\n\\\\\n \n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=var-cosm.eps,width=8cm}}\n\\put(8,0.0){\\epsfig{file=fi6.eps,width=8cm}}\n\\end{picture}\n\\\\\n\\\\\n{\\small {\\bf Figure 2.} {\\it Left:} The mass-energy $E$ and angular momentum $J$\n of $n=1$ spinning nongravitating Q-ball solutions \n are shown as a function of the cosmological constant $\\Lambda$ for two values of\n the frequency $ \\omega$.\n {\\it Right:}\n The mass-energy $E$ \n of $n=1$ spinning nongravitating Q-ball solutions with a potential \n $U(\\phi)=\\mu^2 \\phi^2-\\lambda \\phi^4+\\nu \\phi^6$\n are shown as a function of the frequency for several values of the coefficient of the\n $\\phi^6$ term.\n }\n\\vspace{0.5cm}\n\n \\\\\nHowever, the model admits the scaling symmetry\n$r \\to r\/\\mu$, $ \\omega\\to \\omega \\mu$, $\\ell \\to \\ell\/\\mu$,\ntogether with the scalar field redefinition $\\phi\\to \\phi\/c$, $\\lambda\\to \\lambda c^2$,\nwhich allows us to set $\\mu=\\lambda=1$ without any loss of generality.\n\nThe following picture appears to be generic for the numerical solutions in this work:\nfirst, no solutions exist \nfor frequencies above a critical value $ \\omega= \\omega_c$.\nAs $\\omega\\to \\omega_c$, the solution emerges \n as a perturbation around the ground state $\\phi=0$, with\n\\begin{eqnarray}\n\\label{pert2}\n\\delta \\phi(r,\\theta) \\simeq \\frac{(r \\sin\\theta)^n}{\\left(1+\\frac{r^2}{\\ell^2}\\right)^{\\frac{1}{2} \\omega_c \\ell}} \n\\end{eqnarray}\nbeing the regular solution of the linearized Klein-Gordon equation in a fixed AdS background\n(here we restrict our discussion to nodeless configurations).\nThe critical frequency is given by\n\\begin{eqnarray}\n\\label{pert1}\n \\omega_{c}=\\frac{n+\\Delta_{+}}{\\ell},\n\\end{eqnarray}\nbeing found by requiring the perturbation $\\delta \\phi(r,\\theta)$\nto be regular at both $r\\to 0$ and $r\\to \\infty$. \nThis upper bound on the frequency\nis universal, \nand holds also in the presence of gravity. \n\n\n\nWhen decreasing the frequency, the nonlinear term in $\\phi$\nstarts to contribute\n and the mass-energy and the angular momentum of the \nsolitons start to increase.\nBoth $E$ and $J$ approach a maximum at some intermediate value of $ \\omega$,\ndecreasing afterwards.\nAs $ \\omega\\to 0$, \nthe solutions describe finite mass, static, axially symmetric (for $n\\neq 0$) solitons, \nthough\nwith a vanishing Noether charge.\n\n\nThe dependence of the solutions on the cosmological constant is shown in Figure 2 (left).\nOne can see that the pattern depends on the value of the frequency.\nFor $\\omega>\\mu$, the solutions stop to exist for a maximal value of the cosmological constant,\n $\\Lambda=-\\frac{4(\\mu^2-\\omega^2)^2}{ 3( \\omega(1+2n\/3)+\\sqrt{ \\omega^2+4\\mu^2n(n+3)\/9})^2}$ \n (which results from (\\ref{pert1})),\n where both $E$ and $Q$ vanish in that limit.\nFor $\\omega<\\mu$,\n one finds solutions for all values of cosmological constant, including $\\Lambda=0$.\n \n \n \n We have also constructed AdS generalizations of the flat spacetime Q-balls\n with the usual potential choice (\\ref{Un}).\nAs one can see in Figure 2 (right), the picture found\nfor the $\\phi^4$-model is recovered for small enough values of $\\nu$.\nHowever, for $\\nu$ above a critical value (around $0.67$ for those parameters), the solutions exist\nonly for $ \\omega_{min}< \\omega< \\omega_c$ (with $\\omega_{min}>0$).\nSimilar to the flat space solutions \\cite{Kleihaus:2005me}, both the \nmass-energy and Noether charge diverge\nas $ \\omega\\to \\omega_{min}$.\n\n \n\\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=w-BS.eps,width=8cm}}\n\\put(8,0.0){\\epsfig{file=w-Q.eps,width=8cm}}\n\\end{picture}\n\\\\\n\\\\\n{\\small {\\bf Figure 3.} \nThe mass-energy $E$ and the angular momentum $J$ are shown as a function of the frequency\n$\\omega$\nfor (mini-)boson star solutions without a self-interaction potential (left)\nand for gravitating Q-balls with a $\\phi^4$ interaction potential (right).\n }\n\\vspace{0.1cm}\n\n \n \n\\section{Gravitating spinning scalar soliton}\n\nWe have found that all Q-balls in a fixed AdS background \nallow for gravitating generalizations.\nMoreover, nontrivial solutions are found in this case\neven in the absence of a self-interaction term in the scalar potential (\\ref{U}),\ngeneralizing for $J\\neq 0$ the static AdS\nboson stars in \\cite{Astefanesei:2003qy}.\n\nLet us start with a brief discussion of those configurations with \n $U=\\mu^2 \\phi^2$ \n(usually called\nmini-boson stars in the literature), \nwhich play an important role in the limiting behaviour of \nthe configurations with a $\\phi^4$ term in the potential.\nRestricting again to a real mass of the scalar field, $\\mu^2>0$, the usual \nrescaling $r\\to r\\mu$ implies that the model depends on two\ndimensionless parameters, $\\ell \\mu$ and $ \\omega\/\\mu$, only, the factor $1\/\\sqrt{8\\pi G}$\nbeing absorbed in $\\phi$. \nOur results show that similar to the spherically symmetric case,\nthe spinning solutions exist for a limited range of frequencies\n$0< \\omega_{min}< \\omega< \\omega_c$, emerging\nas a perturbation of AdS spacetime \nfor a critical frequency $ \\omega_c$ as given by (\\ref{pert1}). \nThus, as expected, the value $ \\omega=0$ is not approached in the absence of a \nscalar field self-interaction\\footnote{Finding\nAdS\nstatic solitons requires a violation of the energy conditions \\cite{Hertog:2004rz}, which is not the case for\nthe boson stars with $U(|\\Phi|)=\\mu^2 |\\Phi|^2>0$.}.\nAs $ \\omega\\to \\omega_{min}$,\na backbending towards larger values of $ \\omega$ is observed, see Figure 3 (left).\nWe conjecture that, similar to the spherically symmetric case, \nthis backbending would lead to an inspiraling of the solutions\ntowards a limiting configuration with $ \\omega_0> \\omega_{min}$.\nNote also that the mass-energy and the angular momentum of these mini-boson star solutions stay finite\nin the allowed range of frequencies.\n\n\nThe picture is more complicated for solitons\nwith a self-interaction term in the potential.\nFor simplicity, we shall restrict\\footnote{However, we have also constructed \n gravitating solutions with a $\\phi^6$-term in the potential,\nin which case we did not find new qualitative features.} our discussion to the case of \na $\\phi^4$ potential (\\ref{U}).\nWithout any loss of generality,\n one can set $\\mu=\\lambda=1$ for the two parameters in (\\ref{U}).\nThis choice is achieved by using the rescaling \n$r \\to r\/\\mu$, $\\ell \\to \\ell\/\\mu $, $\\omega \\to \\omega \\mu$, \ntogether with a redefinition of the scalar field\n$\\phi \\to \\phi \\mu\/\\sqrt{\\lambda}$.\nThis reveals the existence of a dimensionless parameter $\\alpha^2=4 \\pi G \\mu^2\/\\lambda$,\nsuch that the\nEinstein equations read $R_{ij}-\\frac{1}{2}g_{ij}R+\\Lambda g_{ij}=2\\alpha^2 T_{ij}$,\nwith $\\alpha=0$ corresponding \nto the probe limit discussed in Section 3.\n \n Starting with the dependence of the solutions on the frequency, we have found \nthat for a given $\\Lambda$, this is fixed by the parameter $\\alpha$.\n The spinning solutions with large enough values of $\\alpha$ exhibit the same pattern as in the \nabsence of a self-interaction term \n(although with a smaller value of $ \\omega_{min}$),\nand the picture in Figure 3 (left) is recovered \n(the same result was found also for $\\Lambda=0$ solutions \\cite{Kleihaus:2005me}).\n However, for values of $\\alpha$ below a critical value ($i.e.$ for large enough $\\lambda$)\nthe picture for $Q-$balls in an AdS background is recovered. \n \n \n \\setlength{\\unitlength}{1cm}\n\\begin{picture}(8,6) \n\\put(-0.5,0.0){\\epsfig{file=Ttt.eps,width=8cm}}\n\\put(8.1,0.0){\\epsfig{file=var-alfa.eps,width=8cm}}\n\\end{picture}\n\\\\\n{\\small {\\bf Figure 4.} \n {\\it Left:}\n The energy density of a static axially symmetric gravitating soliton\n with $ \\omega=0$, $n=2$\n is shown for several different angles as a function of the radial coordinate.\n{\\it Right:} \n The mass-energy $E$ and the angular momentum $J$ are shown as a function of the coupling constant\n$\\alpha=\\sqrt{4\\pi G \\mu^2\/\\lambda}$\nfor boson star solutions \nwith a $\\phi^4$ self interaction potential for two different frequencies.\n}\n\\vspace{0.5cm}\n\\\\\n Thus, for any $n$, the solutions exist for a range of frequencies $0\\leq \\omega< \\omega_c$, see Figure 3 (right).\nIn both cases, the configurations with small $E$, $J$ are just \nperturbations of AdS.\nThese solitons branch off from the\nAdS spacetime for the specific value of the frequency given by (\\ref{pert1}).\n\n\n\nHowever, different from the case discussed above with $U(|\\Phi|)=\\mu^2 |\\Phi|^2$,\na self-interaction term in the potential leading to negative energy densities,\nallows for a nontrivial limiting solution with $\\omega=0$.\nThis is a new type of soliton, which is different from other solutions with\ngravitating scalar fields in the literature \n\\cite{Hertog:2004rz}, \n\\cite{Hertog:2004ns}, \n\\cite{Radu:2005bp}.\nAlthough it is static ($\\partial\/\\partial t$ being a Killing vector of the configuration), \nthe geometry has axial symmetry only, being\n regular everywhere, in particular at $r=0$ and on the symmetry axis.\nAlso, this solution has a \nvanishing Noether charge; however, its mass-energy is finite and nonzero\n(see Figure 4 (left) for a plot of the energy density for a typical configuration\nwith $n=2$; one notices the existence of a region \nwith negative energy density, $\\rho=-T_t^t<0$).\n\n\nConcerning the dependence on $\\alpha^2=4 \\pi G \\mu^2\/\\lambda$, \na central role is played here by the solutions with a $|\\Phi|^2$ potential only.\nWe have found that for a range of the frequency $ \\omega_{min}< \\omega< \\omega_c$ \n(with $ \\omega_{min}$ the minimal allowed value\nof the frequency for the boson stars without a self-interaction term),\nthe solutions exist for arbitrarily large values of $\\alpha$.\nSimilar to the asymptotically flat case \\cite{Kleihaus:2005me},\nthe limit $\\alpha \\to \\infty$\ncorresponds (after a rescaling) to the solution of the $|\\Phi|^2$-model.\nThe picture is different for smaller frequencies, $ \\omega< \\omega_{min}$,\nin which case there are no solutions in the $|\\Phi|^2$ model.\nThe numerical calculations indicate that, in this case, the range of $\\alpha$ is bounded from above,\nand a critical configuration is approached for $\\alpha \\to \\alpha_c$,\nwith $\\alpha_c$ depending on $ \\omega$ and $\\ell$.\nThis limiting soliton has a finite mass-energy and Noether charge.\n(These two cases are illustrated in Figure 4 (right).)\n Unfortunately, \n the numerical accuracy does not allow to clarify the limiting behaviour \nat the critical value of $\\alpha$.\nWe notice only that, as $\\alpha \\to \\alpha_c$, \nthe metric function $F_0$ almost reaches zero at $r=0$,\nwhile the other functions remain finite and nonzero \n(although $F_1$ and $F_2$ take large values at the origin). \n\n\n\n We remark also that for all spinning solutions,\nthe distributions of the mass-energy density $-T_t^t$\nare very different from those of the spherically symmetric configurations,\n$i.e.$ the typical energy density isosurfaces have a toroidal shape.\nHowever, although\n the violation of the positive energy condition is a generic feature of \n the solutions with $U_{int}=-\\lambda|\\Phi|^4<0$ (at least for small enough values of $\\omega$),\n the mass-energy of all our solutions as given by (\\ref{EJ}) is strictly positive. \n \n We close this Section by noticing that, similar to the $\\Lambda=0$ \n case \\cite{Kleihaus:2007vk}, the AdS rotating boson stars possess ergoregions \nin a large part of their domain of existence.\n The ergoregion resides inside the ergosurface \n defined by the condition $g_{tt}=-F_0N+F_2\\sin^2\\theta W^2=0$,\n in the metric parametrization (\\ref{ansatzg}). \nThis type of configurations are typically found for large enough values of $\\omega$, $n$ and $\\alpha$.\n \n\\section{Further remarks}\nIn this work we have initiated a preliminary\ninvestigation of spinning Q-balls and boson stars in four dimensional AdS spacetime. \nThis study was partially motivated by the recent interest in \nsolutions of AdS gravity coupled to scalar fields only. \nThe picture we have found has some interesting new features as compared to the \nwell-known case of solutions with a vanishing cosmological constant.\nPerhaps the most interesting new result is the existence of \naxially symmetric solitons with a smooth static limit possessing a vanishing Noether charge.\n Also, all solutions have an upper bound on frequencies,\n which is fixed by the scalar field mass and the cosmological constant.\n\nMoreover,\nwe expect the existence of a much richer set of spinning scalar solitons apart from the \nsolutions reported in this work.\nFor example, the $\\Lambda=0$\nQ-balls and boson stars with odd parity with respect to \na reflection in the equatorial plane reported in \\cite{Kleihaus:2007vk},\nshould allow for AdS generalizations.\nIn particular, it would be interesting \nto construct AdS `{\\it twisted}' Q-balls and boson stars,\nwhich combine features of both even and odd parity solutions \\cite{Radu:2008pp}.\nFor such configurations, the scalar field is endowed with a $(r,\\theta)$-dependent phase,\n$\\Phi=\\phi (r,\\theta)e^{i(n\\varphi- \\omega t+\\Psi(r,\\theta))}=(X(r,\\theta)+i Y(r,\\theta))e^{i(n\\varphi- \\omega t)}$,\nsuch that the amplitude of the scalar field\nvanishes in the equatorial plane.\nThis would lead to a 'topological charge' of the solutions \n(see \\cite{Radu:2008pp} for the details of this\nconstruction in the flat spacetime case).\nAlso, the issue of AdS vortons, $i.e.$\nspinning\nvortex loops stabilized by the centrifugal force,\nstill remains to be investigated.\nThe results in this work suggest that the \nAdS picture may be very different as compared to \nthe one found in the flat spacetime limit \\cite{Radu:2008pp}, \\cite{Battye:2008mm}. \n\n\nA natural question which arises concerns the issue \nof higher dimensional counterparts of the solutions discussed in this paper.\nWorking in the probe limit,\nwe have found that the general picture we have presented for $d=4$\nremains valid \nfor spinning solitons \nwith a single angular momentum in $d=5,6$ dimensions.\nTherefore we expect it to be generic for any $d\\geq 4$.\nMoreover, for the same self-interaction potential $U_{int}=-\\lambda \\phi^{2k}$ \n(with $k$ still given by (\\ref{k})), the exact Q-ball solution (\\ref{ex-sol})\nadmits a straightforward generalization\\footnote{Here we consider a fixed AdS background,\nwith $ds^2=\\frac{dr^2}{1+r^2\/\\ell^2}+r^2d\\Omega_{d-2}^2-(1+r^2\/\\ell^2)dt^2$.}\n for any $d\\geq 3$, with \n$\n\\Phi(r,t)= (\n\\frac{\\mu^2}{\\lambda}\\frac{\\Delta^2-\\ell^2 \\omega^2}{(\\Delta-(d-1))(\\Delta+1))}\n )^{\\frac{\\Delta}{2}} {\\left(1+\\frac{r^2}{\\ell^2} \\right)^{-\\frac{\\Delta}{2}}} e^{-i \\omega t}, \n$\n and $\\Delta=\\frac{1}{2}\\left( (d-1)\\pm \\sqrt{(d-1)^2+4\\mu^2 \\ell^2} \\right)$.\n\nAlso, based on some preliminary results, \nwe conjecture that it is possible to add\na small black hole in the center of the $d=4$ solitons with a harmonic time dependence studied in this work.\nTherefore the inclusion of rotation would allow to circumvent the\nno-hair results in \\cite{Pena:1997cy}, \\cite{Astefanesei:2003qy}.\nIndeed, such solutions \nwere constructed recently in \\cite{Dias:2011at} for $d=5$\nand a complex doublet scalar field,\nin which case a special ansatz \\cite{Hartmann:2010pm} allows to deal with ODEs.\n\n \n It would be desirable to study all these solutions\n also\nfrom an AdS\/CFT perspective and\nto see what they correspond to\nin the dual theory.\n\nWe close by remarking that\nthe study of Q-balls and boson stars\nis interesting from yet another point of view.\nThis type of relatively simple \nconfigurations provide an ideal ground for \ninvestigating various numerical techniques\non axially symmetric problems with AdS asymptotics,\nwhich thereafter can be applied to more complex models.\n\\\\\n\\\\\n\\noindent{\\textbf{~~~Acknowledgements.--~}} \nWe are grateful to Jutta Kunz\nfor her careful reading of the manuscript and many helpful comments.\n We also thank Burkhard Kleihaus for collaboration in the initial stages of this work. \nWe gratefully acknowledge support by the DFG,\nin particular, also within the DFG Research\nTraining Group 1620 ''Models of Gravity''. \n\n \\begin{small}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Article}\n\nThe Kondo effect~\\cite{DeHaas36,Kondo64} has captured the attention of experimentalists and theorists alike for decades because of its complex many-body physics. In metals with dilute magnetic impurities, the experimental signature of Kondo effect is the low temperature increase of resistivity, which is attributed to the many-body antiferromagnetic s-d exchange interaction between the impurity spin and the conduction electron spins of the host metal. The Kondo effect has also been observed in semiconductor quantum dot (QD) systems where an unpaired spin in a QD is coupled to the surrounding electron reservoirs.~\\cite{Goldhaber98} A popular but controversial physical picture of the Kondo effect is the Kondo screening cloud, which is an electron cloud surrounding the impurity site with an overall spin polarization opposite to the impurity spin. At temperatures well below the Kondo temperature $ T_K $, the net spin of the Kondo cloud completely screens the impurity spin forming a Kondo singlet state. The spatial extent $ \\xi_K $ of the Kondo cloud is given by $ \\hbar{v_F}\/k_BT_K $ in ballistic transport regime and $ \\sqrt{{\\hbar}D\/k_{B}T_{K}} $ in diffusive regime,~\\cite{Chandr00book,Affleck09} where $ v_F $ is the Fermi velocity, $ k_B $ is the Boltzmann constant, and $ D $ is the diffusion constant. Experimental evidence for the screening cloud is scarce and therefore its physical existence has been questioned.~\\cite{Boyce74} Recently Borzenets $et~al.$~\\cite{Borzenets20} found convincing evidence for micrometer-sized Kondo clouds in a QD system. In diffusive metals, $ \\xi_K $ is expected to be $\\sim 100$ nm, but has not yet been experimentally confirmed. \n\n\nIn recent years, the Kondo effect crosses paths with spintronics. In the Cu channels of nonlocal spin valves (NSLVs)~\\cite{Johnson85,Jedema01} with dilute Fe impurities, the spin relaxation rate $ {\\tau_{s}}^{-1} $ is found to increase at low temperatures complementing Kondo effect's low temperature increase of the momentum relaxation rate $ {\\tau_{e}}^{-1} $.~\\cite{Obrien14,Batley15,Hamaya16,Watts19} Here $ \\tau_s $ and $ \\tau_e $ are the spin relaxation time and momentum relaxation time, respectively. For spin relaxation in general, Elliott-Yafet (EY)~\\cite{Elliott54,Yafet83} and Dyakonov-Perel (DP)~\\cite{Dyakonov72} models give explicit relations between $ {\\tau_s}^{-1} $ and $ {\\tau_e}^{-1} $. The EY spin relaxation is caused by weak spin-orbit coupling between energy bands and $ {\\tau_s}^{-1} $ is proportional to $ {\\tau_e}^{-1} $. The ratio $ \\tau_e\/\\tau_s $ is the spin flip probability $ \\alpha $. The DP spin relaxation originates from spin-orbit coupling, caused by inversion symmetry breaking, between two spin subbands within the same energy band and the $ {\\tau_s}^{-1} $ is inversely proportional to $ {\\tau_e}^{-1} $. The Kondo spin relaxation, however, is caused by s-d exchange interaction instead of spin orbit effects. The relation between the Kondo spin relaxation rate $ {\\tau_{sK}}^{-1} $ and Kondo momentum relaxation rate $ {\\tau_{eK}}^{-1} $, to the best of our knowledge, has not yet been explored.\n\n\nIn this work, we extract values of $ {\\tau_{sK}}^{-1} $ and $ {\\tau_{eK}}^{-1} $ from Cu-based NLSVs fabricated by 2-step electron beam lithography. Each NLSV includes a spin injector $\\mathrm{F_1}$, a spin detector $\\mathrm{F_2}$, and a Cu channel, as shown in Figure 1 (a). Magnetic electrodes $\\mathrm{F_1}$ and $\\mathrm{F_2}$, made of $\\mathrm{Ni_{81}Fe_{19}}$ alloy (permalloy or Py), are patterned in the first step and Cu channels are patterned in the second step. The materials are deposited by electron beam evaporation. Before the deposition of Cu, low energy ion milling is performed to clean the surface of Py and a 3 nm $\\mathrm{AlO_x}$ layer is deposited. The Py\/$\\mathrm{AlO_x}$\/Cu interface has been shown to provide a higher effective spin polarization than the ohmic Py\/Cu interfaces.~\\cite{Wang09,Cai16} The distance $L$ between $\\mathrm{F_1}$ and $\\mathrm{F_2}$ varies from 1 to 5 $\\mu$m with 1 $ \\mu $m increment. All Cu channels are 500 nm wide and 300 nm thick to prevent the suppression of Kondo clouds.~\\cite{Chen91,Blachly95} This work involves data from two sample substrates (chip 11 and chip 12) with 10 devices on each. Devices on the same substrate undergo identical fabrication conditions. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure1.eps}\n\t\\caption{\\label{fig1}(a) SEM image of a NLSV. Plots of (b) $R_s$ versus $B$, (c) $ \\Delta{R}_s $ versus $T$, and (d) $ \\rho_{cu} $ versus $T$ for device 11-43 ($ L=3.0\\,\\mu $m).}\n\\end{figure}\n\n\nThe measurement configuration is shown in Figure 1 (a). A low frequency excitation alternating current (AC) $ I_e $ is driven from $ \\mathrm{F_1} $ to the upper end of the Cu channel, and the spin accumulation is detected by measuring the nonlocal voltage $V_{nl}$ between $ \\mathrm{F_2} $ and the lower end of the channel. Figure 1 (b) shows the nonlocal resistance $ R_s={V_{nl}}\/{I_e} $ as a function of magnetic field $B$ applied parallel to $\\mathrm{F_1}$ and $\\mathrm{F_2}$ stripes. The high and low states of $R_s$ correspond to the parallel and antiparallel states of $\\mathrm{F_1}$ and $\\mathrm{F_2}$ magnetizations, respectively. The difference is the spin signal~\\cite{Johnson93} \n\\begin{equation} \n\\Delta{R_s}=\\frac{{P_e}^2\\rho_{cu}\\lambda_{cu}}{A_{cu}}e^{-\\frac{L}{\\lambda_{cu}}}, \\label{Rs} \n\\end{equation}\nwhere $ P_e $ is the effective spin polarization of $\\mathrm{F_1}$ and $\\mathrm{F_2}$, $ \\rho_{cu} $ the Cu resistivity, $ \\lambda_{cu} $ the Cu spin relaxation length, and $ A_{cu} $ the Cu channel cross sectional area. $ \\Delta{R_s}(T) $ of each NLSV is measured from 5 K to 100 K and Figure 1 (c) shows the data of device 11-43 (device 43 on chip 11). As $ T $ decreases, $ \\Delta{R_s} $ initially increases, reaching its maximum at 30 K, and then decreases. This feature is well documented~\\cite{Kimura08,Mihajlovic10,Zou12APL,Villamor13} for NLSVs and convincingly attributed to the Kondo effect.~\\cite{Obrien14,Batley15,Hamaya16,Watts19,Obrien16,Kim17}\n\n\n\nThe resistivity $ \\rho_{cu} $ of a given NLSV is deduced from its Cu channel resistance $ R_{cu} $, which is obtained by sending in a current through the channel and measuring the voltage difference between $ \\mathrm{F_1} $ and $\\mathrm{F_2}$. The $ \\rho_{cu}(T) $ for device 11-43 is shown in Figure 1 (d) with $ \\rho_{cu}=0.43 \\,\\mu\\Omega\\cdot $cm at 5 K and $ \\rho_{cu}=2.60 \\,\\mu\\Omega\\cdot $cm at 295K. The ratio of the two values (6.1) is the residual resistivity ratio (RRR). The inset of Figure 1 (d) shows the low temperature portion of $ \\rho_{cu}(T) $. The low $T$ increase of $ \\rho_{cu} $ indicates Kondo effect from dilute magnetic impurities in Cu. \n\nNext, we extract the average $ P_e $ and $ \\lambda_{cu} $ values of devices on the same substrate. $ \\Delta{R_s} $ versus $ L $ is plotted for 10 devices on chip 11 at 30 K in Figure 2 (a). Fitting Eq.~(\\ref{Rs}) to the plot yields $ \\lambda_{cu}=2.6\\pm0.1 \\,\\mu$m and $ P_e=0.066\\pm0.003 $. The average $ \\rho_{cu} $ used in this process is deduced from the linear fitting of the $R_{cu}$ versus $L$ data in Figure 2 (b). In this manner, the average $P_e$ and $ \\lambda_{cu} $ are obtained between 5 K and 100 K and shown in Figure 2 (c) and its inset, respectively. $ \\lambda_{cu}(T) $ resembles $ \\Delta{R_s}(T) $ in Figure 1(c) and reaches its maximum of 2.6 $ \\mu $m at 30 K. $ \\lambda_{cu} $ decreases to 2.2 $ \\mu $m at 5 K because of the enhanced Kondo spin relaxation. The plot of $ P_e(T) $ shows a rather flat trend around 0.07 within the temperature range of our measurements. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure2.eps}\n\t\\caption{\\label{fig2}(a) Spin signal $\\Delta{R}_s$ and (b) Cu resistance $R_{cu}$ versus channel length $L$ for NLSVs on chip 11 at 30 K. (c) Fitted average $P_e$ and $ \\lambda_{cu} $ (inset) as a function of $T$. (d) $ \\lambda_{cu} $ versus $T$ for device 11-43. }\n\\end{figure}\n\nAs suggested by previous works on Py\/Cu NLSVs, the Kondo effect originates from Fe impurities.~\\cite{Obrien14,Batley15,Hamaya16,Obrien16,Kim17} The maximum $ \\lambda_{cu} $ occurs at 30 K, which is the Kondo temperature $T_K$ for Fe impurities in Cu host. Data analysis of $ {\\tau_s}^{-1}(T) $ and $ {\\tau_e}^{-1}(T) $ later in the text is also consistent with $T_K = 30$ K. The Fe impurities are likely introduced in the fabrication processes. When the Py surface is ion milled, Fe atoms are removed and deposited on the side walls of the resist. When Cu is evaporated, the vapor flux of Cu transfers momentum to the Fe atoms on the side walls and redeposits them into the Cu channel. In some of the previous works,~\\cite{Obrien14,Watts19,Obrien16,Kim17} Fe impurities are concentrated near the ohmic Py\/Cu interfaces, and as a result the spin polarization $ P_e(T) $ is suppressed at low $T$. In our devices, the Fe impurities are located throughout the Cu channel. This is evident from the low $T$ upturn of $ \\rho_{cu}(T) $, the low $T$ downturn of $ \\lambda_{cu}(T) $, and the flat trend of $ P_e(T) $.\n\n\n\nIt is noticeable that data points disperse around the fitted lines in Figure 2 (a) and (b). For the two devices with $ L=3\\,\\mu $m, for example, data points of $ \\Delta{R_s} $ are above the fitted line and those of $ R_{cu} $ are below. The two devices with $ L=4\\,\\mu $m have $ \\Delta{R_s} $ below the fitted line and $ R_{cu} $ above. These indicate variations of $ \\lambda_{cu} $ and $ \\rho_{cu} $ between devices. Assuming a common $P_e$ (the fitted $ P_e $) for all devices on the same substrate at a specific $T$, we deduce $ \\lambda_{cu} $ for each individual NLSV from its $ \\Delta{R_s} $ and $ \\rho_{cu} $ by using Eq.~(\\ref{Rs}). $ \\lambda_{cu}(T) $ for device 11-43 is shown in Figure 2 (d) with a maximum $ \\lambda_{cu}=3.0\\pm0.1 \\,\\mu $m at 30 K. In this manner $ \\lambda_{cu}(T) $ are obtained for all 20 NLSVs. The spin relaxation rate $ {\\tau_s}^{-1}(T) $ is then calculated from $ \\lambda_{cu}(T) $ by using the relation $ \\lambda_{cu}=\\sqrt{D\\tau_s} $ and shown in Figure 3 (a) and (b) for devices 11-33 and 12-32, respectively. $ D=\\frac{1}{3}{v_F}^2\\tau_e $ is the diffusion constant and $ v_F=1.57\\times10^6 $ m\/s is the Fermi velocity of Cu. $ \\tau_e $ can be derived from $ \\rho_{cu} $ by using the Drude model $ \\rho_{cu}=m\/(\\tau_ene^2) $, where $ n=8.47\\times10^{28}\\:\\mathrm{m^{-3}} $ is the Cu electron density and $m$ and $e$ are electron mass and charge, respectively. With a decreasing $T$, $ {\\tau_s}^{-1} $ initially decreases, reaches its minimum around 30 K, and then increases upon further cooling. This resembles Kondo effect's low temperature increase of $ \\rho_{cu} $ as shown in the insets of Figure 3 (a) and (b). The low $T$ increase of $ \\rho_{cu} $ of 11-33 is much smaller than that of 12-32, indicating a lower impurity concentration in 11-33. However, the low $T$ increase of $ {\\tau_s}^{-1}$ of the two devices are surprisingly comparable. This provides the first hint for an unusual relation between Kondo momentum relaxation and Kondo spin relaxation. \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure3.eps}\n\t\\caption{\\label{fig3}Spin relaxation rate $ {\\tau_s}^{-1} $ versus $T$ for (a) device 11-33 and (b) device 12-32. $ \\rho_{cu}(T) $ plots are shown in the insets. (c) $ {\\tau_s}^{-1} $ versus $ {\\tau_{eK}}^{-1} $ for $ T \\le 30 $ K for the two devices. The slopes of the linear fittings are compared with $ \\alpha_K $ values obtained from fittings with Eq.~(\\ref{taus}). (d) $ {\\tau_{s,ph}}^{-1} $ versus $ {\\tau_{e,ph}}^{-1} $ plots. }\n\\end{figure}\n\n\nApplying Matthiessen's rule to spin relaxation, the total $ {\\tau_s}^{-1} $ is given by $ {\\tau_s}^{-1}={\\tau_{s,def}}^{-1}+{\\tau_{s,ph}}^{-1}+{\\tau_{sK}}^{-1} $, where $ {\\tau_{s,def}}^{-1} $, $ {\\tau_{s,ph}}^{-1} $ , and $ {\\tau_{sK}}^{-1} $ are the spin relaxation rates attributed to defects, phonon, and Kondo effects, respectively. Defining $ {\\tau_{e,def}}^{-1} $, $ {\\tau_{e,ph}}^{-1} $, and $ {\\tau_{eK}}^{-1} $ as the corresponding momentum relaxation rates and $ \\alpha_{def} $, $ \\alpha_{ph} $, and $ \\alpha_K $ as the associated spin flip probabilities, we have\n\\begin{equation}\n\t\\frac{1}{\\tau_s(T)}=\\alpha_{def}\\frac{1}{\\tau_{e,def}}+\\alpha_{ph}\\frac{1}{\\tau_{e,ph}(T)}+\\alpha_{K}\\frac{1}{\\tau_{eK}(T)}. \\label{taus} \n\\end{equation} \nIt is well justified to assume a linear relation between $ {\\tau_s}^{-1} $ and $ {\\tau_e}^{-1} $ for defects and phonons, because EY mechanism is dominant in these processes. We will show later that $ {\\tau_{sK}}^{-1} $ is also proportional to $ {\\tau_{eK}}^{-1} $ under varying $T$.\n\n\n\nThe $ {\\tau_e}^{-1} $ of each type (total, defect, phonon, or Kondo) is linked to the corresponding $ \\rho $ by the Drude model $ \\rho=m\/(\\tau_ene^2) $. The defect resistivity $ \\rho_{def} $ is $T$ independent and the phonon resistivity can be described as $ \\rho_{ph}(T)=AT^5 $ at low $T$, where $A$ is constant related to the Debye temperature.~\\cite{Zimanbook} The Kondo resistivity can be described by a phenomenological formula~\\cite{Goldhaber98} \n\\begin{equation}\n\t\\rho_K(T)=\\rho_{K0}{\\left({\\frac{{T_K'}^2}{T^2+{T_K'}^2}}\\right)}^s, \\label{Kondo}\n\\end{equation}\nwhere $ T_K'=T_K\/\\sqrt{2^{1\/s}-1} $, $s=0.225$ and $T_K = 30$ K. From $ {\\tau_e}^{-1}={\\tau_{e,def}}^{-1}+{\\tau_{e,ph}}^{-1}+{\\tau_{sK}}^{-1} $, the total resistivity is \n\\begin{equation}\n\t\\rho_{cu}(T)=\\rho_{def}+AT^5+\\rho_K(T). \\label{rho}\n\\end{equation}\nFitting Eq.~(\\ref{rho}) along with Eq.~(\\ref{Kondo}) to the measured $ \\rho_{cu}(T) $ data below 20 K yields $ \\rho_{def} $, $A$, and $ \\rho_{K0} $. Note that the fitting does not work well for $ T> $ 20 K, because $ \\rho_{ph}(T)=AT^5 $ is an approximation valid at low $T$. For the data of 11-33 and 12-32 in the insets of Figure 3 (a) and (b), the fitted values of $ \\rho_{K0} $ are 0.0013 $ \\mu\\Omega\\cdot $cm and 0.0067 $ \\mu\\Omega\\cdot $cm, respectively. $ \\rho_{K0} $ or $ {\\tau_{eK0}}^{-1} $ represents the $ \\rho_K $ or $ {\\tau_{eK}}^{-1} $ value at $T << T_K$. \n\nTo extract $ \\alpha_{def} $, $ \\alpha_{ph} $, and $ \\alpha_K $, we fit Eq.~(\\ref{taus}) to the $ {\\tau_s}^{-1}(T) $ data by using the empirical data of $ {\\tau_{e,def}}^{-1} $, $ {\\tau_{e,ph}}^{-1}(T) $, and $ {\\tau_{eK}}^{-1}(T) $ obtained from the measured $ \\rho_{cu}(T) $ and fitting. More specifically, $ {\\tau_{e,def}}^{-1} $ can be obtained from the fitted $ \\rho_{def} $ and $ {\\tau_{eK}}^{-1}(T) $ from the fitted $ \\rho_{K0} $ and Eq.~(\\ref{Kondo}). For $ {\\tau_{e,ph}}^{-1}(T) $ we use the relation $ \\rho_{ph}(T)=\\rho_{cu}(T)-\\rho_{def}-\\rho_K(T) $. We do not use $ \\rho_{ph}(T)=AT^5 $ because it significantly deviates from experimental data when $ T>20 $ K. The best fits for $ \\alpha_K $ are 0.30 $\\pm$ 0.03 and 0.066 $\\pm$ 0.006 and the best fits for $ \\alpha_{ph} $ are $(8.4 \\pm 0.3) \\times 10^{-4}$ and $ (9.3\\pm0.4) \\times 10^{-4}$ for devices 11-33 and 12-32, respectively. While $ \\alpha_{ph} $ values are comparable, $ \\alpha_K $ values are quite different. Again, the results point to the unusual scaling for Kondo spin relaxation. \n\n\n\nWe should justify the assumed linear relation $ {\\tau_{sK}}^{-1}(T)=\\alpha_K\\cdot{{\\tau_{eK}}^{-1}(T)} $ under varying $T$ in Eq.~(\\ref{taus}). In Figure 3 (c), $ {\\tau_s}^{-1} $ is plotted versus $ {\\tau_{eK}}^{-1} $ between 5 K and 30 K for the two NLSVs and we observe clear linear dependences. At $T\\leq$ 30 K, the variation of $ {\\tau_s}^{-1} $ should be dominated by $ {\\tau_{sK}}^{-1} $ , because $ {\\tau_{s,def}}^{-1} $ is $T$ independent and $ {\\tau_{s,ph}}^{-1} $ is negligible compared to $ {\\tau_{sK}}^{-1} $. Therefore, Figure 3 (c) confirms the linear relation between $ {\\tau_{sK}}^{-1}(T) $ and $ {\\tau_{eK}}^{-1}(T) $ under varying $T$. In addition, the slopes of the linear fittings to the $ {\\tau_s}^{-1} $ versus $ {\\tau_{eK}}^{-1} $ data are very close to the fitted $ \\alpha_K $ values using Eq.~(\\ref{taus}). Similarly, linear relation for phonons between $ {\\tau_{s,ph}}^{-1}(T) $ and $ {\\tau_{e,ph}}^{-1}(T) $ is also verified in Figure 3 (d). The data of $ {\\tau_{s,ph}}^{-1} $ is obtained by subtracting $ \\alpha_{def}\\cdot{\\tau_{e,def}}^{-1} $ and $ \\alpha_{K}\\cdot{\\tau_{eK}}^{-1} $ from the total $ {\\tau_s}^{-1} $. The slopes of the fitted lines are the same as the fitted $ \\alpha_{ph} $ values by using Eq.~(\\ref{taus}). \n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure4.eps}\n\t\\caption{\\label{fig4}(a) Kondo spin flip probability $ \\alpha_K $ versus Kondo resistivity $ \\rho_{K0} $. (b) Phonon spin flip probability $ \\alpha_{ph} $ versus 100 K phonon resistivity $ \\rho_{ph,100K} $.}\n\\end{figure}\n\n\nNext, we demonstrate the unusual relation between $ {\\tau_{sK}}^{-1} $ and $ {\\tau_{eK}}^{-1} $ under a varying impurity concentration $ C_{Fe} $ which is approximately proportional to $ \\rho_{K0} $ or $ {\\tau_{eK0}}^{-1} $. Figure 4 (a) shows $ \\alpha_K $ versus $ \\rho_{K0} $ extracted from all NLSVs. Strikingly, $ \\alpha_K $ decreases drastically from $ 0.44 \\pm 0.05 $ to $ 0.045 \\pm 0.004 $ as $ \\rho_{K0} $ ($ \\propto{\\tau_{eK0}}^{-1} $) increases from $ < 0.001 \\,\\mu\\Omega\\cdot $cm to $> 0.009 \\,\\mu\\Omega\\cdot $cm. As a comparison, Figure 4 (b) shows $ \\alpha_{ph} $ versus $ \\rho_{ph,100K} $, which is the $ \\rho_{ph} $ at 100K, for all NLSVs. $ \\alpha_{ph} $ remains nearly a constant and independent of $ \\rho_{ph,100K} $ as expected for processes governed by EY mechanism. The average value of $ \\alpha_{ph} $ ($\\sim 8.5\\times10^{-4} $) is in good agreement with previous works.~\\cite{Watts19,Villamor13,Monod79} The average value of $ \\alpha_{def} $ is $ 3.2\\times10^{-4} $ and the data are shown in the Supplementary Materials (Note S1). The decreasing trend in Figure 4 (a) suggests that the relation between $ {\\tau_{sK0}}^{-1} $ and $ {\\tau_{eK0}}^{-1} $ is not linear, where $ {\\tau_{sK0}}^{-1} $ is the value of $ {\\tau_{sK}}^{-1} $ at $T \\ll T_K$. Figure 5 (a) shows $ {\\tau_{sK0}}^{-1} $ , obtained by using the definition $ {\\tau_{sK0}}^{-1}=\\alpha_K\\cdot{\\tau_{eK0}}^{-1} $, versus $ {\\tau_{eK0}}^{-1} $. While $ {\\tau_{eK0}}^{-1} $ varies by a factor of 10, $ {\\tau_{sK0}}^{-1} $ stays nearly constant clearly defying a linear dependence. In contrast, the few previous theoretical treatments of Kondo spin relaxation assume a linear relation and yield a constant $ \\alpha_K $ of $2\/3$.~\\cite{Kondo64,Kim17} The dependences shown in Figure 4 (a) and 5 (a) have been neither anticipated nor addressed previously. These plots with horizontal error bars are available in the Supplementary Materails (Note S2). The $ C_{Fe} $ for each NLSV can be extracted from the temperature $ T_{min} $ that corresponds to the minimum of the fitted $ \\rho_{cu}(T) $ curve.~\\cite{Obrien16,Franck61} Figure 5 (b) shows the extracted $ C_{Fe} $ versus $ \\rho_{K0} $ for all NLSVs. \n\n\n\\begin{figure}\n\t\\includegraphics[width=8.6cm]{Figure5.eps}\n\t\\caption{\\label{fig5}(a) Kondo spin relaxation rate $ {\\tau_{sK0}}^{-1} $ versus Kondo momentum relaxation rate $ {\\tau_{eK0}}^{-1} $ from 20 NLSVs. (b) Fe impurity concentration $ C_{Fe} $ versus $ \\rho_{K0} $. (c) Illustration of the Kondo medium. The gray scale indicates the spin density, and the white arrows indicate the polarization directions of the domains. }\n\\end{figure}\n\nTo address this unusual scaling between the Kondo momentum and spin relaxation, the physical picture of the Kondo cloud becomes appealing. If Kondo clouds exist, it is valid to consider them as momentum scattering barriers as well as spin scattering barriers for conduction electrons passing through them.~\\cite{Simon03} The $ {\\tau_{eK}}^{-1} $ should be proportional to the average charge density of the cloud. The $ {\\tau_{sK}}^{-1} $ should be proportional to the average spin density of the cloud. It may also be related to the relative orientation between the conduction electron spin and the polarization direction of the cloud. The observed unusual scaling arises when the Kondo clouds of adjacent impurities overlap. \n\nTwo relevant length scales are the size of a single Kondo cloud $ \\xi_K $ and the average distance $ d_{Fe} $ between Fe impurities. The former is estimated to be $ \\xi_K=\\sqrt{{\\hbar}D\/k_BT_K}\\approx 100 $ nm for diffusive Cu channels. The latter is 10 nm $ < d_{Fe} <$ 20 nm, estimated from the $ C_{Fe} $ of our NLSVs, and obviously $ \\xi_K > d_{Fe} $. Therefore, the Kondo clouds from adjacent impurities overlap and the conduction electrons associated with the clouds form a continuous medium in the Cu channel. The medium can be characterized by its local charge density, spin density, and polarization direction with some spatial variations. The charge density of overlapping clouds should simply add up. However, the spin density of overlapping clouds may cancel out each other. Because impurity spin directions are random and so are the polarization directions of the clouds. Such cancellation effect of spin density has important implications on the $ {\\tau_{sK}}^{-1} $. Figure 5 (c) is a qualitative illustration of the spin density distribution and polarization directions of the Kondo medium. Domains with random polarization directions are formed in the medium around impurity sites. \n\n\nWhen a conduction electron traverses through the medium, the spin and momentum relaxation occur through the interaction between the electron and the Kondo medium. The $ {\\tau_{eK0}}^{-1} $ or $ {\\tau_{sK0}}^{-1} $ should be proportional to the average charge density or the average spin density of the medium, respectively, along the electron's path. The influence of the polarization directions on $ {\\tau_{sK0}}^{-1} $ can be neglected, because the traversing electron passes through many ($ \\approx 10^4 $) randomly oriented Kondo domains within the time of $ \\tau_{sK0} $. As $ C_{Fe} $ increases, more electrons are added to the Kondo medium, leading to a higher charge density and a higher $ {\\tau_{eK0}}^{-1} $. However, the spin density may not increase, because a higher $ C_{Fe} $ enhances cloud overlapping and the cancellation effect. The exact trend is challenging to predict, because it requires precise knowledge of the spatial distributions of spin and charge densities of Kondo clouds and how overlapping clouds interact. From experimental results in Figure 5 (a), we infer that the average spin density of the medium maintains a nearly constant value within the range of 1 ppm $ < C_{Fe} < $ 12 ppm, corresponding to 10 nm $ < d_{Fe} < $ 20 nm. The red curve in Figure 5 (a) is a guide to the eye with a reasonable assumption that $ {\\tau_{sK0}}^{-1}\\rightarrow 0 $ as $ {\\tau_{eK0}}^{-1}\\rightarrow 0 $. We speculate that the initial slope of the curve, representing $ \\alpha_K $ in the limit of $ {\\tau_{eK0}}^{-1}\\rightarrow 0 $, should be the theoretically predicted $ 2\/3 $.~\\cite{Kondo64,Kim17} \n\nIn conclusion, we extract the Kondo momentum relaxation rate $ {\\tau_{eK0}}^{-1} $ and the Kondo spin relaxation rate $ {\\tau_{sK0}}^{-1} $ from Cu-based nonlocal spin valves with Fe impurities. While $ {\\tau_{eK0}}^{-1} $ is tuned by a factor of 10 by varying Fe concentrations, $ {\\tau_{sK0}}^{-1} $ remains nearly constant and defies a more intuitive linear dependence on $ {\\tau_{eK0}}^{-1} $. Such a relation can be understood by considering a continuous Kondo medium formed by overlapping Kondo clouds. Spin relaxation occurs through interaction between a conduction electron spin and the medium. As the impurity concentration increases, the polarized spins of overlapping Kondo clouds partially cancel each other, and the average spin density of the Kondo medium reaches a stable value giving rise to a nearly constant $ {\\tau_{sK0}}^{-1} $. Our experimental results provide evidence for the physical existence of the elusive Kondo screening clouds.\n\n\\subsection{}\n\\subsubsection{}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}