Powers of the Vandermonde determinant, Schur functions and recursive formulas

International audience Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left. Comme toute puissance paire du déterminant de Vandermonde est un polynôme symétrique, nous voulons comprendre sa décomposition dans la base des fonctions de Schur. Nous allons étudier plusieurs propriétés combinatoires des coefficients de la décomposition. En particulier, nous allons donner une approche récursive pour le calcul du coefficient de la fonction de Schur $s_μ$ dans la décomposition d'une puissance paire du déterminant de Vandermonde en $n+1$ variables, en fonction du coefficient de la fonction de Schur $s_λ$ dans la décomposition de la même puissance paire du déterminant de Vandermonde en $n$ variables, lorsque le diagramme de Young de $μ$ est obtenu à partir du diagramme de Young de $λ$ par l'addition d'une forme de type tetris vers le haut ou vers la gauche.

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The digital function filters – algorithms and applications

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.2478/bpasts-2013-0036 ◽

2013 ◽

Vol 61 (2) ◽

pp. 371-377

Author(s):

M. Siwczyński ◽

A. Drwal ◽

S. Żaba

Keyword(s):

Digital Filters ◽

Phase Shifters ◽

Square Root ◽

Real Power ◽

The Real ◽

Distributed Parameters ◽

Digital Modeling ◽

Analysis And Synthesis ◽

Basic Functions ◽

Recursive Formulas

Abstract The simple digital filters are not sufficient for digital modeling of systems with distributed parameters. It is necessary to apply more complex digital filters. In this work, a set of filters, called the digital function filters, is proposed. It consists of digital filters, which are obtained from causal and stable filters through some function transformation. In this paper, for several basic functions: exponential, logarithm, square root and the real power of input filter, the recursive algorithms of the digital function filters have been determined The digital function filters of exponential type can be obtained from direct recursive formulas. Whereas, the other function filters, such as the logarithm, the square root and the real power, require using the implicit recursive formulas. Some applications of the digital function filters for the analysis and synthesis of systems with lumped and distributed parameters (a long line, phase shifters, infinite ladder circuits) are given as well.

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Optimal Assembly of Systems Using Schur-Functions and Majorization.

10.21236/ada172821 ◽

1986 ◽

Author(s):

Emad El-Neweihi ◽

Frank Proschan ◽

Jayaram Sethuraman

Keyword(s):

Schur Functions ◽

Optimal Assembly

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The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

Fundamenta Informaticae ◽

10.3233/fi-2021-1996 ◽

2021 ◽

Vol 178 (1-2) ◽

pp. 1-30

Author(s):

Florian Bruse ◽

Martin Lange ◽

Etienne Lozes

Keyword(s):

Model Checking ◽

Lower Bounds ◽

Expressive Power ◽

Higher Order ◽

Order Logic ◽

High Complexity ◽

Transition Systems ◽

Recursive Formulas ◽

Second Order Logic ◽

Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

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