The adaptive computation of far-field patterns by a posteriori error estimations of linear functionals

This paper is concerned with the derivation of a priori and a posteriori error bounds for a class of linear functionals arising in electromagnetics which represent the far-field pattern of the scattered electromagnetic field. The a posteriori error bound is implemented into an adaptive finite element algorithm, and a series of numerical experiments is presented

On the graph limit question of Vera T. S\'os

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. S\'os proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.Comment: 4 page

Loading Surface in the Course of Mechanical- Thermal Treatment and Steady-State Creep of Metals

Abstract: Kinetics of the loading surface of a material gives precious information on the level of the hardening of the material. This paper is concerned with the evolution of the loading surface during successive actions, such as: (i) plastic deformation, (ii) annealing of the pre-strained specimen, and (iii) secondary creep of the treated material. The analysis of the loading surface is carried out in terms of the synthetic theory of irrecoverable deformation. Keywords: loading surface; mechanical-thermal treatment; creep and plastic strain; synthetic theory of irrecoverable deformatio

Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page

Finite element approximation of high-dimensional transport-dominated diffusion problems

High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud \ud (Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.

Random local algorithms

Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which can use any local statistics about the graph, and produces an almost optimal structure, then the same can be achieved by a random local algorithm using no statistics.Comment: 9 page

Independent sets and cuts in large-girth regular graphs

We present a local algorithm producing an independent set of expected size $0.44533n$ on large-girth 3-regular graphs and $0.40407n$ on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with $1.34105n$ edges on large-girth 3-regular graphs. These decrease the gaps between the best known upper and lower bounds from $0.0178$ to $0.01$, from $0.0242$ to $0.0123$ and from $0.0724$ to $0.0616$, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of $1 / 0.44533 \approx 2.24554$ and $1 / 0.40407 \approx 2.4748$ and fractional edge coloring number $1.5 / 1.34105 \approx 1.1185$. Our algorithms are applications of the technique introduced by Hoppen and Wormald

A Gagliardo-Nirenberg inequality, with application to duality-based a posteriori error estimation in the L1 norm

We establish the Gagliardo-Nirenberg-type multiplicative interpolation inequality \[ \|v\|_{{\rm L}1(\mathbb{R}^n)} \leq C \|v\|^{1/2}_{{\rm Lip}'(\mathbb{R}^n)} \|v\|^{1/2}_{{\rm BV}(\mathbb{R}^n)}\qquad \forall v \in {\rm BV}(\mathbb{R}^n), \] where $C$ is a positive constant, independent of $v$. We then use a local version of this inequality to derive an a posteriori error bound in the ${\rm L}1(\Omega')$ norm, with $\bar\Omega' \subset\Omega=(0,1)^n$, for a finite-element approximation to a boundary value problem for a first-order linear hyperbolic equation, under the limited regularity requirement that the solution to the problem belongs to ${\rm BV}(\Omega)$.\ud \ud Dedicated to Professor Boško S Jovanovic on the occasion of his sixtieth birthda

An undecidability result on limits of sparse graphs

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (G_i) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of G_i is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the case where the sequence (G_i) is replaced by a unimodular random graph.Comment: 6 page