5,949 research outputs found
A note on a problem in communication complexity
In this note, we prove a version of Tarui's Theorem in communication
complexity, namely . Consequently, every
measure for leads to a measure for , subsuming a result of
Linial and Shraibman that problems with high mc-rigidity lie outside the
polynomial hierarchy. By slightly changing the definition of mc-rigidity
(arbitrary instead of uniform distribution), it is then evident that the class
of problems with low mc-rigidity equals . As , this rules out the possibility, that had been
left open, that even polynomial space is contained in
Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
We study convex integration solutions in the context of the modelling of
shape-memory alloys. The purpose of the article is two-fold, treating both
rigidity and flexibility properties: Firstly, we relate the maximal regularity
of convex integration solutions to the presence of lower bounds in variational
models with surface energy. Hence, variational models with surface energy could
be viewed as a selection mechanism allowing for or excluding convex integration
solutions. Secondly, we present the first numerical implementations of convex
integration schemes for the model problem of the geometrically linearised
two-dimensional hexagonal-to-rhombic phase transformation. We discuss and
compare the two algorithms from [RZZ16] and [RZZ17].Comment: 35 pages, 14 figure
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
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