32,672 research outputs found
A vector partition function for the multiplicities of sl_k(C)
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity
function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition
function. This allows us to apply known results about partition functions to
derive interesting properties of the weight diagrams. We relate this
description to that of the Duistermaat-Heckman measure from symplectic
geometry, which gives a large-scale limit way to look at multiplicity diagrams.
We also provide an explanation for why the weight polynomials in the boundary
regions of the weight diagrams exhibit a number of linear factors. Using
symplectic geometry, we prove that the partition of the permutahedron into
domains of polynomiality of the Duistermaat-Heckman function is the same as
that for the weight multiplicity function, and give an elementary proof of this
for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr
Transcendental extensions of a valuation domain of rank one
Let be a valuation domain of rank one and quotient field . Let
be a fixed algebraic closure of the -adic completion
of and let be the integral closure of in . We describe a relevant class of valuation domains
of the field of rational functions which lie over , which are
indexed by the elements , namely,
.
If is discrete and is a uniformizer, then a valuation domain
of is of this form if and only if the residue field degree
is finite and , for some , where is the maximal ideal
of . In general, for we have
if and only if and are conjugated over
. Finally, we show that the set of
irreducible polynomials over endowed with an ultrametric distance
introduced by Krasner is homeomorphic to the space endowed with the Zariski topology.Comment: accepted for publication in the Proceedings of the AMS (2016);
comments are welcome
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
On -estimates of derivatives of univalent rational functions
We study the growth of the quantity for
rational functions of degree , which are bounded and univalent in the
unit disk, and prove that this quantity may grow as , ,
when . Some applications of this result to problems of regularity
of boundaries of Nevanlinna domains are considered. We also discuss a related
result by Dolzhenko which applies to general (non-univalent) rational
functions.Comment: 16 pages, to appear in Journal d'Analyse Mathematiqu
Effective results for unit equations over finitely generated domains
Let A be a commutative domain containing Z which is finitely generated as a
Z-algebra, and let a,b,c be non-zero elements of A. It follows from work of
Siegel, Mahler, Parry and Lang that the equation (*) ax+by=c has only finitely
many solutions in elements x,y of the unit group A* of A, but the proof
following from their arguments is ineffective. Using linear forms in logarithms
estimates of Baker and Coates, in 1979 Gy\H{o}ry gave an effective proof of
this finiteness result, in the special case that A is the ring of S-integers of
an algebraic number field. Some years later, Gy\H{o}ry extended this to a
restricted class of finitely generated domains A, containing transcendental
elements. In the present paper, we give an effective finiteness proof for the
number of solutions of (*) for arbitrary domains A finitely generated over Z.
In fact, we give an explicit upper bound for the `sizes' of the solutions x,y,
in terms of defining parameters for A,a,b,c. In our proof, we use already
existing effective finiteness results for two variable S-unit equations over
number fields due to Gy\H{o}ry and Yu and over function fields due to Mason, as
well as an explicit specialization argument.Comment: 41 page
Cauchy Type Integrals of Algebraic Functions
We consider Cauchy type integrals with an algebraic function. The main goal is to give
constructive (at least, in principle) conditions for to be an algebraic
function, a rational function, and ultimately an identical zero near infinity.
This is done by relating the Monodromy group of the algebraic function , the
geometry of the integration curve , and the analytic properties of the
Cauchy type integrals. The motivation for the study of these conditions is
provided by the fact that certain Cauchy type integrals of algebraic functions
appear in the infinitesimal versions of two classical open questions in
Analytic Theory of Differential Equations: the Poincar\'e Center-Focus problem
and the second part of the Hilbert 16-th problem.Comment: 58 pages, 19 figure
Non-unique factorization of polynomials over residue class rings of the integers
We investigate non-unique factorization of polynomials in Z_{p^n}[x] into
irreducibles. As a Noetherian ring whose zero-divisors are contained in the
Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring
arbitrary non-zero polynomials into irreducibles to the problem of factoring
monic polynomials into monic irreducibles. The multiplicative monoid of monic
polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to
irreducible polynomials in Z_p[x], and we show that each of these monoids has
infinite elasticity. Moreover, for every positive integer m, there exists in
each of these monoids a product of 2 irreducibles that can also be represented
as a product of m irreducibles.Comment: 11 page
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