Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics??a generalization of a game called Tug of War that has been used in Peres et?al. (J Am Math Soc 22(1):167?C210, 2009) to study real-valued AMLEs.     

Edge singularities and structure of the 3-D Williams expansion

The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202]. To cite this article: T. Apel et al., C. R. Mecanique 336 (2008).     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Efficient paging and multicast schemes for Mobile Virtual Private Networks

The issue of providing a mobile virtual private network (MVPN) service by a cellular network is addressed in this study. The main focus of this study is on two aspects of MVPN handling: a multicast transmission, and location management of the MVPN members. Virtual private networks (VPNs) already exist in wired networks. The growing number of mobile users increases the demand for mobile services. It is expected that future wireless networks will have to provide wireless internet services, and in particular, to support MVPNs. In order to maintain an MVPN service offering, the system must support an efficient information delivery to the MVPN members. In order to achieve this goal, the physical locations of the MVPN members must be known to the system. In this study, we present methods to efficiently route and transmit information to the MVPN members, without affecting the quality of the personal service provided to each and every MVPN member. It is shown that the wireless MVPN bandwidth requirement is significantly less than the equivalent bandwidth required by a wired VPN. The offered bandwidth utilization is significantly better than the efficiency of IP‐based multicast transmission in the Internet, and the proposed IP‐based multicast transmission in cellular networks. Copyright © 2005 John Wiley & Sons, Ltd.     

Functional polymers. XVI. Synthesis and polymerization of 2(2-hydroxy-5-isopropenylphenyl)-2H-benzotriazole and a new synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole

2(2-Hydroxy-5-isopropenylphenyl)2H-benzotriazole was synthesized in 40% overall yield starting from o-nitroaniline. Diazotization in aqueous hydrochloric acid gave o-nitrophenyl diazonium chloride which was condensed with p-hydroxyacetophenone; the azo compound was reduced to 2(2-hydroxy-5-acetylphenyl) 2H-benzotriazole with zinc powder in sodium hydroxide solution and the 2-hydroxy group of the compound was acetylated. Treatment of the acetyl compound with methyl Grignard reagent resulted in the methylation of the 5-acetyl group to 2[2-acetoxy-5(2-hydroxy-2-propyl)phenyl]2H-benzotriazole which was then dehydrated with potassium hydrogen sulfate to the desired 2(2-hydroxy-5-isopropenylphenyl)2H-benzotriazole. This monomer did not homopolymerize, but was copolymerized readily with styrene, methyl methacrylate, and n-butyl acrylate with azobisisobutyronitrile as the initiator. 2(2-Acetoxy-5-acetylphenyl)2H-benzotriazole was also reduced with sodium borohydride to form 2[2-acetoxy-5-(1-hydroxyethyl)phenyl]2H-benzotriazole which was dehydrated and hydrolyzed to the known 2(2-hydroxy-5-vinylphenyl)-2H-benzotriazole. This route provides a novel and simpler synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole.     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

Trees and Markov Convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors. The research of the first author was conducted while he was at U. C. Berkeley and the Institute for Advanced Study.     

Some applications of Ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice , equipped with the metric, in any -uniformly convex Banach space is of order .

     

Sequent Systems for Negative Modalities

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.     

Cut problems in graphs with a budget constraint

We study budgeted variants of classical cut problems: the Multiway Cut problem, the Multicut problem, and the k-Cut problem, and provide approximation algorithms for these problems. Specifically, for the budgeted multiway cut and the k-cut problems we provide constant factor approximation algorithms. We show that the budgeted multicut problem is at least as hard to approximate as the sparsest cut problem, and we provide a bi-criteria approximation algorithm for it.