Consider a convex set S = {x ∈ D : G(x) o 0} where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R is a domain on which G(x) is defined, and G(x) o 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper s...

Given an infinite sequence of positive integers A, we prove that for every nonnegative integer k the number of solutions of the equation n = a1+ · · ·+ak, a1, . . . , ak ∈ A, is not constant for n large enough. This result is a corollary of our main theorem, which partially answers a question of Sárközy and Sós on representation functions for multilinear forms. The main tool used in the argumen...

A three-dimensional object has many possible representations inside a computer. Specifically, constructive solid geometry, boundary representation, and volumetric approximation techniques are commonly used to represent objects in solid geometric modeling (see [4]). Among the volumetric schemes are the octree [3] and the more recently developed switching-function representation [6] in which the ...

A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every ^-dimensional polyhedron admits a representation as the set of solutions of at most d(d + l) /2 polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any boun...

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [9, 6] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation given which the ...

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of 9, 6] to run in polynomial time. We do this by presenting a root-nding algorithm for univariate polynomials over function elds when their coeecients lie in nite-dimensional linear spaces, and proving that there is a polynomial size representation given which the root ndin...

We show that all algebraic-geometric codes possess a succinct representation that allows for the list decoding algorithms of [15, 7] to run in polynomial time. We do this by presenting a root-finding algorithm for univariate polynomials over function fields when their coefficients lie in finite-dimensional linear spaces, and proving that there is a polynomial size representation given which the...

in this paper, we consider a particular class of integral equations of the fourth kind and show that tractability and differentiability index of the given system are 3. tractability and dierentiability index are introduced based on the-smoothing property of a volterra integral operator and index reduction procedure, respectively. using the notion of index, we give sucient conditions for the...

the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

In this paper, we study the lower semilattice of NP-subspaces of both the standard polynomial time representation and the tally polynomial time representation of a countably infinite dimensional vector space V, over a finite field F. We show that for both the standard and tally representation of V, , there exists polynomial time subspaces U and FV such that U + V is not recursive. We also study...