scholarly journals Finite sum Cauchy identity for dual Grothendieck polynomials

2014 ◽  
Vol 90 (7) ◽  
pp. 87-91 ◽  
Author(s):  
Alain Lascoux ◽  
Hiroshi Naruse
Keyword(s):  
Finite Sum ◽  
Cauchy Identity ◽  
Grothendieck Polynomials

scholarly journals Set-valued Rothe tableaux and Grothendieck polynomials

2021 ◽  
Vol 128 ◽  
pp. 102203
Author(s):  
Neil J.Y. Fan ◽  
Peter L. Guo
Keyword(s):  
Grothendieck Polynomials

scholarly journals Identities on factorial Grothendieck polynomials

2019 ◽  
Vol 111 ◽  
pp. 101933 ◽  
Author(s):  
Peter L. Guo ◽  
Sophie C.C. Sun
Keyword(s):  
Grothendieck Polynomials

On the effects of thermal properties structure and water bottom temperature variation on temperature gradients in lake sediments

1986 ◽  
Vol 23 (9) ◽  
pp. 1257-1264 ◽  
Author(s):  
K. Wang ◽  
P. Y. Shen ◽  
A. E. Beck
Keyword(s):  
Heat Flow ◽  
Temperature Variation ◽  
Stratified Medium ◽  
Depth Interval ◽  
Flow Density ◽  
Detailed Knowledge ◽  
Bottom Temperature ◽  
Temperature Variations ◽  
Density Values ◽  
Finite Sum

In heat flow determinations, it is customary to treat the surface temperature variation as a finite sum of Fourier components. The medium is assumed to be homogeneous or horizontally stratified with each layer having a constant conductivity and diffusivity. This allows the effect of each periodic component to be calculated analytically. We extend this formulation to include cases where thermal conductivities in some layers of a stratified medium may vary linearly with depth as have been found in the sediments of some continental lakes. The application of this formalism to temperature measurements in Lake Greifensee and Lac Leman shows that even with excellent records of bottom temperature variations over several years, failure to take into account the conductivity variation leads to errors as high as 20% in heat flow density values, depending on the depth interval used. The combined effects of lack of detailed knowledge of conductivity structure and the use of too short and (or) inaccurate records of bottom temperature variations, leading to very significant errors, are also discussed, with particular reference to the problems arising from a lack of recognition of the existence of nonannual terms in the bottom temperature variation and the use of probes that do not penetrate the sediments deeply enough.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

2012 ◽  
Vol 08 (05) ◽  
pp. 1257-1270
Author(s):  
M. A. GÓMEZ-MOLLEDA ◽  
JOAN-C. LARIO
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Elliptic Functions ◽  
Number Fields ◽  
Jacobi Elliptic Functions ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Finite Sums ◽  
Finite Sum

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Philip D. Cha ◽  
Siyi Hu
Keyword(s):  
Linear Structure ◽  
Frequency Equation ◽  
Governing Equations ◽  
Linear Structures ◽  
Combined System ◽  
Digamma Function ◽  
Simply Supported ◽  
Lumped Elements ◽  
Finite Sum ◽  
Infinite Sum

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Rationale for a Localization Formula

2020 ◽  
pp. 232-238
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Closed Form ◽  
Blow Up ◽  
Torus Action ◽  
Exact Form ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Localization Formula ◽  
Point Set ◽  
Finite Sum

This chapter offers a rationale for a localization formula. It looks at the equivariant localization formula of Atiyah–Bott and Berline–Vergne. The equivariant localization formula of Atiyah–Bott and Berline–Vergne expresses, for a torus action, the integral of an equivariantly closed form over a compact oriented manifold as a finite sum over the fixed point set. The central idea is to express a closed form as an exact form away from finitely many points. Throughout his career, Raoul Bott exploited this idea to prove many different localization formulas. The chapter then considers circle actions with finitely many fixed points. It also studies the spherical blow-up.

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