scholarly journals Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula

10.37236/2651 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Per Alexandersson
Keyword(s):  
Characteristic Equation ◽  
Toeplitz Matrices ◽  
Linear Recurrence ◽  
Schur Polynomials ◽  
Determinant Formula ◽  
Special Case

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot\mathbf{1}^c)/(\mathbf{\lambda} + k\cdot\mathbf{1}^r)}(x_1,\dots,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices.In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

scholarly journals Generalized Alcuin's Sequence

10.37236/2796 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Panario ◽  
Murat Sahin ◽  
Qiang Wang
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Recurrence Equation ◽  
Linear Recurrence ◽  
Fixed Integer ◽  
New Family ◽  
Special Case

We introduce a new family of sequences $\{t_k(n)\}_{n=-\infty}^{\infty}$ for given positive integer $k$. We call these new sequences asgeneralized Alcuin's sequences because we get Alcuin's sequence which has several interesting properties when $k=3$. Also, $\{t_k(n)\}_{n=0}^{\infty}$ counts the number of partitions of $n-k$ with parts being $k, \left(k-1\right), 2\left(k-1\right),$ $3\left(k-1\right)$, $\ldots, \left(k-1\right)\left(k-1\right)$. We find an explicit linear recurrence equation and the generating function for $\{t_k(n)\}_{n=-\infty}^{\infty}$. For the special case $k=4$ and $k=5$, we get a simpler formula for $\{t_k(n)\}_{n=-\infty}^{\infty}$ and investigate the period of $\{t_k(n)\}_{n=-\infty}^{\infty}$ modulo a fixed integer. Also, we get a formula for $p_{5}\left(n\right)$ which is the number of partitions of $n$ into exactly $5$ parts.

On Vandermonde Varieties

2016 ◽  
Vol 119 (1) ◽  
pp. 73 ◽  
Author(s):  
Ralf Fröberg ◽  
Boris Shapiro
Keyword(s):  
Trivial Solution ◽  
Recurrence Relations ◽  
Hilbert Series ◽  
Free Resolution ◽  
Linear Recurrence ◽  
Regular Case ◽  
Schur Polynomials ◽  
Open Problems ◽  
Natural Class ◽  
Determinantal Varieties

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties, which we call Vandermonde varieties. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a Vandermonde variety is the expected one, we present its free resolution, obtain its degree and the Hilbert series. Some interesting relations among Schur polynomials are derived. Many open problems and conjectures are posed.

Möbius transformations in stability theory

1970 ◽  
Vol 68 (1) ◽  
pp. 143-151 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Characteristic Equation ◽  
Stability Theory ◽  
Trivial Solution ◽  
Real Variable ◽  
Continuous Functions ◽  
Asymptotically Stable ◽  
Image Position ◽  
Special Case

1. Introduction: Consider the system of ordinary differential equationswhere the unknown x(t) is a complex m-vector, t is a real variable, D is the operator d/dt and a0, …, an are complex m × m matrices whose elements are continuous functions of t, x, Dx, …, Dn−1x. Furthermore, det a0 ╪ 0. In the special case when a0, …, an are constant matrices the trivial solution x = 0 is asymptotically stable if and only if all the roots of the characteristic equation det f(ζ) = 0 have negative real parts, where

scholarly journals Polynomials from Combinatorial -theory

2019 ◽  
pp. 1-34 ◽  
Author(s):  
Cara Monical ◽  
Oliver Pechenik ◽  
Dominic Searles
Keyword(s):  
Euler Characteristic ◽  
Combinatorial Theory ◽  
Symmetric Polynomials ◽  
Schur Polynomials ◽  
Fundamental Particle ◽  
Ring Of Polynomials ◽  
Special Case

Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$ -theoretic deformation of the quasi-key basis and also a lift of the $K$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.

Synthesis of Spherical Four-Bar Function Generators to Match Two Prescribed Velocity Ratios

1983 ◽  
Vol 105 (4) ◽  
pp. 631-636 ◽  
Author(s):  
C. H. Chiang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Characteristic Equation ◽  
Quadratic Equation ◽  
Present Problem ◽  
Crank Angle ◽  
Function Generators ◽  
Special Case

The present problem is a variation of synthesizing spherical four-bar function generators to coordinate three pairs of finitely separated crank-angle displacements. It is to be stressed here that, since the technique is on the basis of equations of three relative poles, the characteristic equation is simply a quadratic equation in the unknown tan ψo (initial output crank angle). It is not necessary, as in the case of a characteristic equation derived from displacement equations, to resort to techniques such as iterative methods for solving nonlinear equations. Equations are so presented as to facilitate computer programing. The synthesis of spherical crank-rockers is a special case of the present problem.

scholarly journals Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables $z_j = x_j + x_j^{-1}$

10.37236/9354 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Per Alexandersson ◽  
Luis Angel González-Serrano ◽  
Egor Maximenko ◽  
Mario Alberto Moctezuma-Salazar
Keyword(s):  
Toeplitz Matrices ◽  
Symmetric Polynomial ◽  
Laurent Polynomial ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Schur Polynomials ◽  
Power Sums ◽  
Change Of Variables ◽  
Schur Polynomial ◽  
Elementary Symmetric Polynomials

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that\[P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1})=Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}).\] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form $\Phi_n(\operatorname{s}_{\lambda/\mu}^{(2n)})$, where $\operatorname{s}_{\lambda/\mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,\ldots,x_n,x^{-1}_1,\ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

Determination of the phase structure of polymerblends by electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 492-493
Author(s):  
Dr. G. Kaemof
Keyword(s):  
Screw Extruder ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Single Screw Extruder ◽  
Transmission Electron ◽  
The Matrix ◽  
Twin Screw ◽  
Special Case ◽  
Microscopic Investigations

A mixture of polycarbonate (PC) and styrene-acrylonitrile-copolymer (SAN) represents a very good example for the efficiency of electron microscopic investigations concerning the determination of optimum production procedures for high grade product properties.The following parameters have been varied:components of charge (PC : SAN 50 : 50, 60 : 40, 70 : 30), kind of compounding machine (single screw extruder, twin screw extruder, discontinuous kneader), mass-temperature (lowest and highest possible temperature).The transmission electron microscopic investigations (TEM) were carried out on ultra thin sections, the PC-phase of which was selectively etched by triethylamine.The phase transition (matrix to disperse phase) does not occur - as might be expected - at a PC to SAN ratio of 50 : 50, but at a ratio of 65 : 35. Our results show that the matrix is preferably formed by the components with the lower melting viscosity (in this special case SAN), even at concentrations of less than 50 %.

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