In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like $S_{2}^i$ and $T_{2}^i$. In correspondence with polynomial induction, we consider a rather new notion of cut that we call p-cut. We also consider small cuts, i.e. cuts that are bounded above by a small element. We study the basic properties of p-cuts and small cuts. In particula...

the topological index of a graph g is a numeric quantity related to g which is invariant underautomorphisms of g. the vertex pi polynomial is defined as piv (g)  euv nu (e)  nv (e).then omega polynomial (g,x) for counting qoc strips in g is defined as (g,x) =cm(g,c)xc with m(g,c) being the number of strips of length c. in this paper, a new infiniteclass of fullerenes is constructed. the ...

discoverer, developed under Maple, is a package aiming at discovering and proving inequality-type theorems. It can discover new inequalities automatically, without requiring us to put forward any conjectures beforehand. It is also applied to the classification of the real physical solutions of geometric constraint problems. Many inequalities with various backgrounds have been discovered or re-d...

This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we est...

We investigate the coloured Tutte polynomial in Valiant’s algebraic framework of NP-completeness. Generalising the well known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynomial, thus showing that its evaluation is a VNP−complete problem.

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial in a region of positive volume. Our res...

We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a, q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or th...

Abstract—A novel multichannel spectral factorization algorithm is illustrated in this paper. This new algorithm is based on an iterative method for polynomial eigenvalue decomposition (PEVD) called the second order sequential best rotation (SBR2) algorithm [1]. By using the SBR2 algorithm, multichannel spectral factorization problems are simply broken down to a set of single channel problems wh...

A graph polynomial is an algebraic object associated with a graph that is usually invariant at least under graph isomorphism. As such, it encodes information about the graph, and enables algebraic methods for extracting this information. This chapter surveys a comprehensive, although not exhaustive, sampling of graph polynomials. It concludes Graph Polynomials and their Applications I: The Tutt...

Theorem 1.1 [Zaslavsky 1975]. Let A be a hyperplane arrangement in Rn . The number of regions into which A dissects Rn is equal to (−1)χA(−1). The number of regions which are relatively bounded is equal to (−1)χA(1). Theorem 1.2 [Orlik and Solomon 1980]. Let A be a hyperplane arrangement in Cn , and let MA =Cn− ⋃ H∈A H be its complement. Then the Poincaré polynomial of the cohomology ring of MA...