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Yuan Zhou

Publications and source records attributed to Yuan Zhou.

At least 325 records · Page 18Linked to original sources
arXiv2014

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

For any open set $Ω\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ =0 \quad \rm in\ \ Ω, $$ where $H$ is given by $$H(x,\,p)= =\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\inΩ, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\barΩ,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.

math.AP↗
arXiv2014

Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and non-approximability results. While no constant factor approximation algorithms are known, even APX-hardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from $1$, even for $Ω(n)$ levels of the hierarchy. This complements recent algorithmic results in Guruswami and Sinop (2011) which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting "polynomial constraints" (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy.

cs.CC↗
arXiv2014

Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs

Building on work of Cai, Fürer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares (SOS) proof of nonisomorphism requires degree $Ω(n)$. In other words, we show an $Ω(n)$-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs $G$ and $H$ which are not even $(1-10^{-14})$-isomorphic. (Here we say that two $n$-vertex, $m$-edge graphs $G$ and $H$ are $α$-isomorphic if there is a bijection between their vertices which preserves at least $αm$ edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem is hard. I.e.\ for every $ε> 0$, there is no efficient algorithm which can distinguish graph pairs which are $(1-ε)$-isomorphic from pairs which are not even $(1-ε_0)$-isomorphic for some universal constant $ε_0$. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest.

cs.CC↗
arXiv2013

Resonance, Fermi surface topology, and Superconductivity in Cuprates

The resonance, a collective boson mode, was usually thought to be a possible glue of superconductivity. We argue that it is rather a natural product of the \emph{d}-wave pairing and the Fermi surface topology. A universal scaling $E_{res}/2Δ^{H}_{S}\sim 1.0$ ($Δ_{S}^{H}$ the magnitude of superconducting gap at hot spot) is proposed for cuprates, irrespective of the hole-/electron-doping, low-/high-energy resonance, monotonic/nonmonotonic \emph{d}-wave paring, and the parameters selected. We reveal that there may exist two resonance peaks in the electron-doped cuprates. The low- and high- energy resonance, originated from the contributions of the different intra-band component, is intimately associated with the Fermi surface topology. By analyzing the data of inelastic neutron scattering, we conclude the nonmonotonic \emph{d}-wave superconducting pairing symmetry in the electron-doped cuprates, which is still an open question

cond-mat.supr-con↗
arXiv2013

Locally Testable Codes and Cayley Graphs

We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over $\F_2^h$: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a shortest-path metric that embeds into $\ell_1$ with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into $\ell_1$. \item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ that has significantly more than $h$ eigenvalues near 1, which have no short linear dependencies among them and which "explain" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues. \end{enumerate}

cs.CC↗
arXiv2013

Intrinsic Geometry and Analysis of Diffusion Processes and $L^\infty$-Variational Problems

The aim of this paper is two-fold: First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (i) for all $n\ge1$, the diffusion matrix $A$ is weak upper semicontinuous on $Ω$ if and only if the intrinsic differential and the local intrinsic distance structures coincide; (ii) if $n=1$, or if $n\ge2$ and $A$ is weak upper semicontinuous on $Ω$, the intrinsic distance and differential structures always coincide; (iii) if $n\ge2$ and $A$ fails to be weak upper semicontinuous on $Ω$, the (non-) coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of $A$. Second, for an arbitrary diffusion matrix $A$, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding $L^\infty$-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix $A$ for which there is an absolute minimizer that is not of class $C^1$. When $A$ is continuous, we also obtain the linear approximation property of the absolute minimizer.

math.CA↗
arXiv2012

Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{not} have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-$d$ SOS proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al.\ can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 ($> .878$) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?

cs.CC↗
arXiv2012

Geometry and Analysis of Dirichlet forms

Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincaré inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Newtonian property, (iii) the intrinsic length structure coincides with the gradient structure. Moreover, for the standard (resistance) Dirichlet form on the Sierpinski gasket equipped with the Kusuoka measure, we identify the intrinsic length structure with the measurable Riemannian and the gradient structures. We also apply the above results to the (coarse) Ricci curvatures and asymptotics of the gradient of the heat kernel.

math.CA↗
arXiv2012

Left Derivations and Strong Commutativity Preserving Maps on Semiprime $Γ$-Rings

In this paper, firstly as a short note, we prove that a left derivation of a semiprime $Γ$-ring $M$ must map $M$ into its center, which improves a result by Paul and Halder and some results by Asci and Ceran. Also we prove that a semiprime $Γ$-ring with a strong commutativity preserving derivation on itself must be commutative and that a strong commutativity preserving endomorphism on a semiprime $Γ$-ring $M$ must have the form $σ(x)=x+ζ(x)$ where $ζ$ is a map from $M$ into its center, which extends some results by Bell and Daif to semiprime $Γ$-rings.

math.RA↗
arXiv2012

Transversal Propagation Behaviors of Helical Edge States In a QSH System

The transversal propagation of the edge states in a two-dimensional quantum spin Hall system are classified by decay characteristic quantity $λ$. Two different modes of the helical edge states exhibit distinct behaviors. The penetration depth is momentum dependent in normal edge states. The finite size gap decays monotonously with sample width, leading to the normal size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states with much shorter length. The finite size gap decays non-monotonously with width, leading to the anomalous finite size effect. Real materials are compared in the phase diagram, which explicitly demonstrates their differences. We also propose an intuitive way to search the special edge states in two-dimensional quantum spin Hall system.

cond-mat.mes-hall↗
arXiv2011

Approximation Algorithms and Hardness of the k-Route Cut Problem

We study the k-route cut problem: given an undirected edge-weighted graph G=(V,E), a collection {(s_1,t_1),(s_2,t_2),...,(s_r,t_r)} of source-sink pairs, and an integer connectivity requirement k, the goal is to find a minimum-weight subset E' of edges to remove, such that the connectivity of every pair (s_i, t_i) falls below k. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (k-1) edge-disjoint paths connecting s_i to t_i in G \ E', while in the vertex-connectivity version, NC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k >= 3, but no non-trivial approximation algorithms were known for any value k>3, except in the single-source setting. We show an O(k log^{3/2}r)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and NC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that NC-kRC is hard to approximate to within a factor of k^{eps} for some fixed eps>0. We then turn to study a simpler version of NC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of NC-kRC has no constant-factor approximation, assuming Feige's Random k-AND assumption.

cs.DS↗
arXiv2011

Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph

The densest k-subgraph (DkS) problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for DkS: the current best algorithm gives an ~ O(n^{1/4}) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P != NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of densest k-subgraph and its variants. Thus, understanding the approximability of DkS is an important challenge. In this work, we give evidence for the hardness of approximating DkS within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving densest k-subgraph. Our results include: * A lower bound of Omega(n^{1/4}/log^3 n) on the integrality gap for Omega(log n/log log n) rounds of the Sherali-Adams relaxation for DkS. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are in fact Erdos-Renyi random graphs. * For every epsilon > 0, a lower bound of n^{2/53-eps} on the integrality gap of n^{Omega(eps)} rounds of the Lasserre SDP relaxation for DkS, and an n^{Omega_eps(1)} gap for n^{1-eps} rounds. Our construction proceeds via a reduction from random instances of a certain Max-CSP over large domains. In the absence of inapproximability results for DkS, our results show that even the most powerful SDPs are unable to beat a factor of n^{Omega(1)}, and in fact even improving the best known n^{1/4} factor is a barrier for current techniques.

cs.DS↗
arXiv2010

New Properties of Besov and Triebel-Lizorkin Spaces on RD-Spaces

An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $ε\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.

math.CA↗
arXiv2010

Anisotropic Singular Integrals in Product Spaces

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by ${\mathcal A}_\infty(\rnm; \vec A)$ the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.

math.CA↗
arXiv2010

Tunable striped-patterns by lattice anisotropy and magnetic impurities in d-wave superconductors

The pattern transition induced by lattice anisotropy (LA) and magnetic impurities is computationally observed in near-optimally doped d-wave superconductors (DSCs). For the single impurity case, a transition from the checkerboard to stripe pattern can be induced even with a very weak LA. Moreover, the modulation period of eight lattice constants (8$a$) in the spin order coincides with neutron scattering data. For the two-impurity case, an orientation transition from the longitudinal impurity-pinned stripe into the transverse pattern is observed when the LA ratio reaches some critical value. At the critical point, it is found that the structures around magnetic impurities could restore checkerboard patterns. These results indicate that the formation of stripes in DSCs might induced by various effects, and could be tunable experimentally.

cond-mat.supr-con↗
arXiv2010

Haj\lasz-Sobolev Imbedding and Extension

The author establishes some geometric criteria for a Haj\lasz-Sobolev $\dot M^{s,\,p}_\ball$-extension (resp. $\dot M^{s,\,p}_\ball$-imbedding) domain of ${\mathbb R}^n$ with $n\ge2$, $s\in(0,\,1]$ and $p\in[n/s,\,\infty]$ (resp. $p\in(n/s,\,\infty]$). In particular, the author proves that a bounded finitely connected planar domain $\boz$ is a weak $α$-cigar domain with $α\in(0,\,1)$ if and only if $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz=\dot M^{s,\,p}_\ball(\boz)$ for some/all $s\in[α,\,1)$ and $p=(2-\az)/(s-α)$, where $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz$ denotes the restriction of the Triebel-Lizorkin space $\dot F^s_{p,\,\infty}({\mathbb R}^2) $ on $\boz$.

math.CA↗
arXiv2010

Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces and Quasiconformal Mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.

math.CA↗