Characteristic Polynomial Research Articles

Stoneley waves in isotropic crustal rocks under the theory of generalized thermoelastic diffusion

This paper investigates the propagation of Stoneley waves through bonded and unbonded interfaces between two dissimilar homogeneous transversely isotropic generalized thermo-elastic diffusion rocks. The governing equations for the current problem are obtained and are solved to determine the characteristic polynomial equations for both mediums in terms of penetration depth. These equations are solved using Ferrari’s approach, and the variation in penetration depths with angular frequency and wavenumber is shown. The secular equations of the Stoneley waves are developed using proper boundary conditions and these equations have been solved numerically to determine the phase velocity and attenuation coefficients. We have discovered three modes of the Stoneley waves in thermo-diffusive crustal rocks. We have confirmed that the paths of particle motion for Stoneley waves are elliptic in nature. Some known results [Kumar et al. [2017] “Mathematical modeling of Stoneley wave in a transversely isotropic thermoelastic media”, Applications and Applied Mathematics 12(1), 319–336; Kumar and Kansal [2008b] “Propagation of Rayleigh waves on free surface of transversely isotropic generalized thermoelastic diffusion”, Applied Mathematics and Mechanics29(11), 1451–1462] are recovered from this analysis to validate our investigation.

A METHOD FOR SYNTHESIS OF HIGH PRECISION RADIO TRACKING SYSTEMS WITH SPLIT CONTROL AND FILTERING PROCEDURES

Context. In combined automatic control systems (ACS) with the principle of control by disturbance, there are difficulties in controlling disturbances in some objects of radio engineering systems and somewhat lower accuracy. This is especially noticeable when the object is affected by several equal disturbances. Taking them into account requires increasing the complexity and reducing the reliability of the ACS. And neglect sharply reduces the accuracy of the system. Therefore, there is a need to develop a method of synthesis of radio technical tracking systems that eliminates the indicated shortcomings.
 Objective. The article presents a synthesis method for high precision radio tracking systems, which are equivalent to combined systems with split control and filtering procedures when the entry useful (preset) action, which is not measured and external disturbances and interferences are present simultaneously.
 Method. Methods of automatic control theory were used to achieve the goal of the research.
 Results. It has been demonstrated that there is a conflict between the conditions for split synthesis of the evaluation (smoothing) filter and the control filter (regulator) in automatic tracking control systems operating with deviation.
 The article offers a solution to the problem of control and evaluation in the framework of two-circuit systems, which are equivalent to combined systems. The second circuit can be presented as a product of a reverse transfer function with an error in the first circuit to the transfer function of Controller, which has a positive feedback from the operator, as well as the reverse transfer function of the control object without integrating links. It is proposed to use for evaluation an evaluation filter with closed filter transfer function instead of an open regulator.
 The characteristic polynomial of a two-circuit automatic control system (ACS) excludes the influence of the stable second-circuit evaluation filter on the stability of the entire ACS. The polynomial of the numerator of the transfer function must have the difference of polynomials by error, which ensures the achievement of invariance.
 A double-circuit ACS is equivalent to a combined one, since it provides the following: invariance of the error with respect to the preset action without directly measuring it; stability of the first circuit with a stable second circuit.
 The synthesized double-circuit ACS is equivalent to a combined one. The author has calculated and constructed the evaluation filter, the influence of this filter on ACS astatism (i. e., on its accuracy) has been analyzed.
 Conclusions. The scientific novelty of the developed method of synthesis of high-precision automatic tracking systems with separate control and filtering procedures in conditions where the controlled value is not measured in the presence of disturbances is as follows. Equivalence to combined systems, in contrast to the methods of differential connections, is achieved not by three, but by two control loops. The practical significance lies in the fact that the proposed method is advisable to use for the construction of surveillance radio engineering systems, where the input useful effect is not measured in the presence of external influences and disturbances. In aircraft control systems.

The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.

Walks and eigenvalues of signed graphs

Abstract In this article, we consider the relationships between walks in a signed graph G ˙ \dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues ρ ( G ˙ ) \rho \left(\dot{G}) , known as the spectral radius. Among other results, we derive a sequence of lower bounds for ρ ( G ˙ ) \rho \left(\dot{G}) expressed in terms of walks or closed walks. We also prove that ρ ( G ˙ ) \rho \left(\dot{G}) attains the spectral radius of the underlying graph G G if and only if G ˙ \dot{G} is switching equivalent to G G or its negation. It is proved that the length k k of the shortest negative cycle in G ˙ \dot{G} and the number of such cycles are determined by the spectrum of G ˙ \dot{G} and the spectrum of G G . Finally, a relation between k k and characteristic polynomials of G ˙ \dot{G} and G G is established.

Ambient data-driven SSO online monitoring of type-3 wind turbine generator integrated power systems based on MMPF-KF method

Series compensation grids connected with type-3 wind turbine generator (WTG)-based wind farms have suffered numerous subsynchronous oscillation (SSO) events worldwide. For early alerting of SSO and effective development of protection and control strategies, it is critical to monitor and identify SSO accurately and quickly. Ambient data is continuously available, which is useful for online monitoring. This paper proposes an ambient data-driven SSO online monitoring method based on the Kalman filter (KF) combined with the multi-model partitioning filter (MMPF). The KF is utilized to fit the measured ambient data with an auto regressive (AR) model. Then, the damping factor (or damping ratio) and frequency in the SSO mode can be acquired by solving the roots of the characteristic polynomial corresponding to the AR model. Moreover, the MMPF is an effective model order selection method applied to the KF for better identification. The performance of the MMPF-KF method is demonstrated by simulations and real-time experiments. The results of case studies validate the effectiveness of the proposed method under various conditions.

A non-associative incidence near-ring with a generalized Möbius function

There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall’s Theorem on the Möbius function as an alternating sum of chain counts, Weisner’s Theorem, and Rota’s Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbius function we define analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized Möbius polynomial has − 1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.

Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.

Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs

We show that the maximum cardinality of an equiangular line system in R18 is at most 59. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial (x−22)(x−2)42(x+6)15(x+8)2.

MODELS OF FIRE EXTINGUISHING WHEN FLAMMABLE LIQUID COMBUSTION

The process of extinguishing a class B fire with sprayed water is described by the differential equation of heat conduction with boundary conditions of the second kind. The solution of this differential equation in a dimensionless form allows to make a transition to the operative form of representation of the mathematical model of the quenching process - to the transfer function. The peculiarity of such a fire transfer function when extinguishing it with sprayed water is its irrationality, which causes difficulties in its use. One of the ways out of this situation is to switch to an equivalent representation of an irrational function of a complex variable in the form of a small-rational function of such a variable. Such a transition is carried out with the help of the minimax approximation using the Remez algorithm. Determination of the maximum order of the characteristic polynomial of the fire transfer function is carried out using the Hurwitz stability criterion. It is shown that for the transfer function of class B fire, the order of the Hurwitz polynomials does not exceed four, the approximation error does not exceed 3.6% using the method of undetermined coefficients, the transfer functions of the fire when they are extinguished with sprayed water are constructed in the form of a superposition of the transfer functions of aperiodic links. Numerical values of the parameters of such transfer functions were obtained. For the case when the intensity of the sprayed water supply is described by the Heaviside function, expressions describing the temperature of the surface of the burning liquid were constructed. It is noted that these expressions correspond to the expressions that describe the dynamic properties of class B fire in the time domain. It is shown that the maximum value of the relative error when using such transfer functions does not exceed 4.0%, if the general transfer function represents the superposition of the transfer functions of two aperiodic links. It is noted that when using the Padé approximation, the transfer function in the form of the transfer functions of eight aperiodic links corresponds to this value of the divergence error. Keywords: flammable liquid, fire extinguishing, sprayed water, transfer function.

McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type

Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

AbstractWe study the standing periodic waves in the semidiscrete integrable system modeled by the Ablowitz–Ladik (AL) equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a nonstandard linear Lax system, which is different from the standard Lax representation of the AL equation.

The characteristic polynomials of uniform double hyperstars and uniform hypertriangles

A k-power hypergraph is obtained from a graph by adding k−2 new vertices to each edge of the graph. A k-uniform double hyperstar is the k-power hypergraph of a double star. The k-uniform hypertriangle is the k-power hypergraph of a triangle. We present the characteristic polynomials of uniform double hyperstars by using the Poisson formula. Moreover, we find expressions for the traces of uniform hypertriangles and use these to obtain the characteristic polynomials of uniform hypertriangles.

Machine Learning Seams of Conical Intersection: A Characteristic Polynomial Approach.

The machine learning of potential energy surfaces (PESs) has undergone rapid progress in recent years. The vast majority of this work, however, has been focused on the learning of ground state PESs. To reliably extend machine learning protocols to excited state PESs, the occurrence of seams of conical intersections between adiabatic electronic states must be correctly accounted for. This introduces a serious problem, for at such points, the adiabatic potentials are not differentiable to any order, complicating the application of standard machine learning methods. We show that this issue may be overcome by instead learning the coordinate-dependent coefficients of the characteristic polynomial of a simple decomposition of the potential matrix. We demonstrate that, through this approach, quantitatively accurate machine learning models of seams of conical intersection may be constructed.

Weighted Subspace Designs from q-Polymatroids

The Assmus-Mattson Theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs by Britz et al. in 2009. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also from sets to vector spaces. To achieve this, we study the characteristic polynomial of a q-polymatroid and outline several of its properties. We also derive a MacWilliams duality result and apply this to establish criteria on the weight enumerator of a q-polymatroid for which dependent spaces of the q-polymatroid form the blocks of a weighted subspace design.

The Characteristic Polynomial of Sums of Random Permutations and Regular Digraphs

Abstract Let $A_{n}$ be the sum of $d$ permutation matrices of size $n\times n$, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial $\frac {1}{\sqrt {d}}\det (I_{n} - z A_{n}/\sqrt {d})$ converges when $n\to \infty $ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$.

On (distance) Laplacian characteristic polynomials of power graphs

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are primes, are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

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In this note we show how conjectures and current problems on determinants and eigenvalues of highly structured tridiagonal matrices can be solved using very classical results.

High frequency multi-field continualization scheme for layered magneto-electro-elastic materials

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z-transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory.

On some Toeplitz-type matrices over finite fields

In this paper, we study some arithmetic properties of certain Toeplitz-type matrices over finite fields. For example, suppose {α1,…,αn} are all the nonzero k-th powers in a finite field and for 1⩽i,j⩽n, we setxij={1αi−αjif i≠j,0if i=j. We give an explicit formula for the characteristic polynomial of [xij]1⩽i,j⩽n and confirm several conjectures posed by Zhi-Wei Sun.

Ship collision avoidance control based on improved velocity obstacle method

The velocity obstacle method has extensive research in collision avoidance, which uses circles to represent robots or ships to find a collision-free velocity. To better describe the motion characteristics of ships, an improved velocity obstacle method based on elliptical ship domain is proposed in this paper. In the previous research on collision avoidance, the ship domains of the own ship and the target ship are seldom considered at the same time. This paper considers the ship domains of the own ship and the target ship at the same time and judges whether the ship domains of the two ships overlap based on the generalized characteristic polynomial. The heading angle is discretized to construct the velocity obstacle (VO) by calculating the collision velocity at different heading angles. Velocities on the boundaries of the velocity obstacle are used to find an optimal heading angle. The algorithm is tested in different encounter situations and successfully realizes collision avoidance.