58 research outputs found
Multilateral inversion of A_r, C_r and D_r basic hypergeometric series
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic
hypergeometric matrix inverse with applications to bilateral basic
hypergeometric series. This matrix inversion result was directly extracted from
an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and
involves two infinite matrices which are not lower-triangular. The present
paper features three different multivariable generalizations of the above
result. These are extracted from Gustafson's A_r and C_r extensions and of the
author's recent A_r extension of Bailey's 6-psi-6 summation formula. By
combining these new multidimensional matrix inverses with A_r and D_r
extensions of Jackson's 8-phi-7 summation theorem three balanced
very-well-poised 8-psi-8 summation theorems associated with the root systems
A_r and C_r are derived.Comment: 24 page
Positive trigonometric polynomials for strong stability of difference equations
We follow a polynomial approach to analyse strong stability of linear
difference equations with rationally independent delays. Upon application of
the Hermite stability criterion on the discrete-time homogeneous characteristic
polynomial, assessing strong stability amounts to deciding positive
definiteness of a multivariate trigonometric polynomial matrix. This latter
problem is addressed with a converging hierarchy of linear matrix inequalities
(LMIs). Numerical experiments indicate that certificates of strong stability
can be obtained at a reasonable computational cost for state dimension and
number of delays not exceeding 4 or 5
A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks
Motivated by problems from Chemical Reaction Network Theory, we investigate
whether steady state ideals of reversible reaction networks are generated by
binomials. We take an algebraic approach considering, besides concentrations of
species, also rate constants as indeterminates. This leads us to the concept of
unconditional binomiality, meaning binomiality for all values of the rate
constants. This concept is different from conditional binomiality that applies
when rate constant values or relations among rate constants are given. We start
by representing the generators of a steady state ideal as sums of binomials,
which yields a corresponding coefficient matrix. On these grounds we propose an
efficient algorithm for detecting unconditional binomiality. That algorithm
uses exclusively elementary column and row operations on the coefficient
matrix. We prove asymptotic worst case upper bounds on the time complexity of
our algorithm. Furthermore, we experimentally compare its performance with
other existing methods
Arithmetic Levi-Civita connection
This paper is part of a series of papers where an arithmetic analogue of
classical differential geometry is being developed. In this arithmetic
differential geometry functions are replaced by integer numbers, derivations
are replaced by Fermat quotient operators, and connections (respectively
curvature) are replaced by certain adelic (respectively global) objects
attached to symmetric matrices with integral coefficients. Previous papers were
devoted to an arithmetic analogue of the Chern connection. The present paper is
devoted to an arithmetic analogue of the Levi-Civita connection
Symmetric algebras of modules arising from a fixed submatrix of a generic matrix
AbstractWe analyze symmetric algebras which arise from rather âbadâ ideals and modules. For example, the ideals are mixed, and every value â 0 occurs as the projective dimension of one of the modules. We are interested in the Cohen-Macaulay property, the canonical module, normality, and the divisor class group. The symmetric algebras under consideration can be defined as residue class rings modulo determinantal ideals covered by the theory of Hochster-Eagon. Part of the results can be regarded as an extension of work of Andrade and Simis
A new multivariable 6-psi-6 summation formula
By multidimensional matrix inversion, combined with an A_r extension of
Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7
summation is derived. By a polynomial argument this 8-phi-7 summation is
transformed to another multivariable 8-phi-7 summation which, by taking a
suitable limit, is reduced to a new multivariable extension of the
nonterminating 6-phi-5 summation. The latter is then extended, by analytic
continuation, to a new multivariable extension of Bailey's very-well-poised
6-psi-6 summation formula.Comment: 16 page
A Graph Theoretical Approach for Testing Binomiality of Reversible Chemical Reaction Networks
We study binomiality of the steady state ideals of chemical reaction
networks. Considering rate constants as indeterminates, the concept of
unconditional binomiality has been introduced and an algorithm based on linear
algebra has been proposed in a recent work for reversible chemical reaction
networks, which has a polynomial time complexity upper bound on the number of
species and reactions. In this article, using a modified version of
species--reaction graphs, we present an algorithm based on graph theory which
performs by adding and deleting edges and changing the labels of the edges in
order to test unconditional binomiality. We have implemented our graph
theoretical algorithm as well as the linear algebra one in Maple and made
experiments on biochemical models. Our experiments show that the performance of
the graph theoretical approach is similar to or better than the linear algebra
approach, while it is drastically faster than Groebner basis and quantifier
elimination methods
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