Symmetric Polynomials Research Articles

Regular Polygons and Symmetric Polynomials

Regular Polygons and Symmetric Polynomials

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

PSP-Equivalent Model for Double-Gate and Surrounding-Gate Field Effect Transistors for Circuit Simulation

We introduce a compact core model for double-gate (DGFET) and surrounding-gate (SGFET) MOSFETs designed for circuit simulations. Despite its high precision, the model is crafted to retain the same analytic formulation of the industry standard Pennsylvania State and Philips (PSP). Instead of linearizing the drain current as in the PSP model, we employ a quadratic symmetric polynomial interpolation of the charge in the channel. This eliminates the need for cumbersome derivatives, simplifications, and intricate coding when integrating into a circuit simulator, thereby preventing singularities during numerical iterations. Moreover, thanks to its mathematical formulation equivalent to PSP, this model simplifies the coding of terminal charges, capacitances, potentials, and electric fields in the channel within circuit simulators. We validate the accuracy of the model through comparisons with numerical solutions and experiments from the literature.

A serendipity fully discrete div-div complex on polygonal meshes

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré–Korn-type inequalities for hybrid fields.

The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [ − 1 , 1 ] with parameters α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as β → ∞ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product.

Representation theoretic interpretation and interpolation properties of inhomogeneous spin q-Whittaker polynomials

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=0$$\\end{document} Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of Uq′(sl^2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U'_q({\\widehat{\\mathfrak {sl}}}_2)$$\\end{document}-modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials.

Calculation of symmetric polynomials in the maple computer mathematics system

The article studies the theory of symmetric polynomials. As a result of the study, it was shown that one of the methods for solving systems of higher degree equations is the method using symmetric polynomials. The practical implementation of the solution method has a computer implementation in the Maple system.

Electrical Resistivity Inversion Based on Symmetric Polynomial Constraint

As we seek to make geophysical models more consistent with geological models, more and more a priori information is incorporated into geophysical inverse problems. How we make the most use of a priori information is still a key issue in geophysical inversion, particularly for effectively representing such information in inversion problems. To address this problem, we propose a symmetric polynomial constrained inversion algorithm that can represent multiple known physical properties for the lithology. The electrical resistivity synthetic model tests show that the inversion results constrained with symmetric polynomials can make the model resistivity be in line with a priori information and obtain more realistic sharp boundaries. Finally, we give a practical verification of our inversion algorithm with field data.

Partial isospectrality of a matrix pencil and circularity of the c-numerical range

We study when functions of the eigenvalues of the pencil(1)Re(e−itA)=cos⁡(t)ReA+sin⁡(t)ImA are constant functions of t. The results are then applied to questions regarding the numerical range, the higher rank numerical range and the c-numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs.

A Molev-Sagan type formula for double Schubert polynomials

We give a Molev-Sagan type formula for computing the product Su(x;y)Sv(x;z) of two double Schubert polynomials in different sets of coefficient variables where the descents of u and v satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial Su(x;y) by a factorial elementary symmetric polynomial Ep,k(x;z). Both formulas remain positive in terms of the negative roots when we set y=z, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial sλ(x1,…,xm;y) by a double Schubert polynomial Sv(x1,…,xp;y) such that m≥p. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials.

Explicit calculations for Sono’s multidimensional sieve of 𝐸₂-numbers

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono’s multidimensional sieve of E 2 E_2 -numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple E 2 E_2 -numbers. For example, we show there are infinitely many occurrences of four E 2 E_2 -numbers within a gap size of 94 94 unconditionally and within a gap size of 32 32 assuming the Elliott-Halberstam conjecture for primes and E 2 E_2 -numbers.

Integrable systems and crystals for edge labeled tableaux

We introduce the edge Schur functions Eλ that are defined as a generating series over edge labeled tableaux. We formulate Eλ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of Eλ and show it intertwines with an uncrowding algorithm.

Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations

We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-Sum of Squares (SOS) aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the symmetric integrator and the time-optimal inversion of qubits.

Novel identities for elementary and complete symmetric polynomials with diverse applications

<p>This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n-1}^{(\alpha, \beta)}(x) $. Several illustrative examples are included.</p>

A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

In this paper, we consider the following \(\mathcal{L}\)-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.

A unified core model of double-gate and surrounding-gate MOSFETs for circuit simulation

This paper presents a new core compact model of double-gate (DGFET) and surrounding-gate (SGFET) MOSFETs for circuit simulations. The current and the terminal charges are continuous with high computation efficiency and accuracy. Despite its accuracy, it retains the same simplicity of the industry standard transistors models. The drain current is worked out without invoking the charge-sheet approximation exploiting a quadratic symmetric polynomial interpolation of the charge in the channel. Apart this clear approximation, no other simplification is used to work out the drain current, the terminal charges, the potential, and electric field in the channel. The accuracy of the model is shown by comparison with the exact numerical solution and experimental data of the literature.

Symmetric polynomials in free center-by-metabelian Lie algebras of rank 2

Let [Formula: see text] be the free center-by-metabelian Lie algebra of rank 2 freely generated by the set [Formula: see text]. Let [Formula: see text] be the Lie subalgebra of symmetric polynomials in [Formula: see text], that is, [Formula: see text] consists of all the elements [Formula: see text] in [Formula: see text] such that [Formula: see text]. We give a linear basis and a minimal infinite generating set for [Formula: see text], thus extending results of Ş. Findik and N. Ögüşlü [Palindromes in the free metabelian Lie algebras, Internat. J. Algebra Comput. 29(5) (2019) 885–891].

Dynamic coefficient symmetric polynomial-based secure key management scheme for Internet of Things (IoT) networks.

With the extensive application and continuous expansion of the Internet of Things (IoT), the access of a large number of resource-limited nodes makes the IoT application face a variety of security vulnerabilities and efficiency limitations, and the operating efficiency and security of IoT are greatly challenged. Key management is the core element of network security and one of the most challenging security problems faced by wireless sensor networks. A suitable key management scheme can effectively defend against network security threats. However, among the key management schemes that have been proposed so far, most of them do not take into account the efficiency in terms of connectivity rate and resource overhead, and some of them even have security risks. In this article, based on the symmetric polynomial algorithm, a dynamic coefficient symmetric polynomial key management scheme is proposed to better solve the IoT security problem. In this scheme, the nodes' IDs are mapped into the elements of the shared matrix M by the identity mapping algorithm, and these elements are used to construct polynomials P(x,y) to generate pairwise keys. The communicating nodes have their own coefficients of P(x,y) and thus have higher connectivity. The overall performance evaluation shows that the scheme significantly improves the resilience against node capture and effectively reduces the communication and storage overheads compared to the previous schemes. Moreover, the scheme overcomes the λ-security of symmetric polynomial key management scheme, and is able to provide a large pool of polynomials for wireless sensor networks, facilitating large-scale application of nodes.

Some characterizations of the symmetric q-Dunkl-semiclassical orthogonal q-polynomials

In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion of a class and a criterion for determining the class will be given. As an application, several examples of q-Dunkl-semiclassical forms of the class s ≦ 2 are highlighted.

Real factorization of positive semidefinite matrix polynomials

Suppose Q(x) is a real n×n regular symmetric positive semidefinite matrix polynomial. Then it can be factored asQ(x)=G(x)TG(x), where G(x) is a real n×n matrix polynomial with degree half that of Q(x) if and only if det⁡(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equationXSX−XR+RTX+P=0, where P,R,S are real n×n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization.