scholarly journals On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$

2020 ◽  
Author(s):  
Rebecca E. Garcia ◽  
Pamela E. Harris ◽  
Marissa Loving ◽  
Lucy Martinez ◽  
David Melendez ◽  
...  
Keyword(s):  
Multiplicity Formula

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

scholarly journals Computing subschemes of the border basis scheme

2020 ◽  
Vol 30 (08) ◽  
pp. 1671-1716
Author(s):  
Martin Kreuzer ◽  
Le Ngoc Long ◽  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
General Problem ◽  
Hilbert Scheme ◽  
Affine Space ◽  
Rational Points ◽  
Open Covering ◽  
Natural Question ◽  
Quadratic Equations ◽  
Multiplicity Formula

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

Smoothness of refinable function vectors on ℝn

2017 ◽  
Vol 15 (05) ◽  
pp. 1750051
Author(s):  
Ramazan Tinaztepe ◽  
Denise Jacobs ◽  
Christopher Heil
Keyword(s):  
Scaling Function ◽  
Joint Spectral Radius ◽  
Refinement Equation ◽  
Dilation Matrix ◽  
Finite Set ◽  
Expansive Matrix ◽  
Refinable Function Vectors ◽  
Multiplicity Formula ◽  
Compactly Supported Solutions ◽  
Order Smoothness

Let [Formula: see text] be a dilation matrix, an [Formula: see text] expansive matrix that maps [Formula: see text] into itself. Let [Formula: see text] be a finite subset of [Formula: see text] and for [Formula: see text] let [Formula: see text] be [Formula: see text] complex matrices. The refinement equation corresponding to [Formula: see text] and [Formula: see text] is [Formula: see text] A solution [Formula: see text] if one exists, is called a refinable vector function or a vector scaling function of multiplicity [Formula: see text] This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the [Formula: see text]-norm joint spectral radius of a finite set of finite matrices determined by the coefficients [Formula: see text]

On the Kostant multiplicity formula

1988 ◽  
Vol 5 (4) ◽  
pp. 721-750 ◽  
Author(s):  
V. Guillemin ◽  
E. Lerman ◽  
S. Sternberg
Keyword(s):  
Multiplicity Formula
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