63,046 research outputs found

### 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 $\alpha$-cigar domain with
$\alpha\in(0,\,1)$ if and only if \dot F^s_{p,\,\infty}({\mathbb
R}^2)|_\boz=\dot M^{s,\,p}_\ball(\boz) for some/all $s\in[\alpha,\,1)$ and
p=(2-\az)/(s-\alpha), 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.Comment: submitte

### 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?Comment: 34 page

### 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 $(1, 2)$-Poincar\'e
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.Comment: Advance in Mathematics, to appear,51p

### New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for
the Gomory--Johnson infinite group problem. They lead to the discovery of new
extreme functions, whose existence settles several open questions.Comment: 54 pages, many figure

### On the Outage Probability of Localization in Randomly Deployed Wireless Networks

This paper analyzes the localization outage probability (LOP), the
probability that the position error exceeds a given threshold, in randomly
deployed wireless networks. Two typical cases are considered: a mobile agent
uses all the neighboring anchors or select the best pair of anchors for
self-localization. We derive the exact LOP for the former case and tight bounds
for the LOP for the latter case. The comparison between the two cases reveals
the advantage of anchor selection in terms of LOP versus complexity tradeoff,
providing insights into the design of efficient localization systems

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