Characteristic Polynomial Research Articles

Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

Polynomial reconstruction problem for hypergraphs

We show that, in general, the characteristic polynomial of a hypergraph is not determined by its “polynomial deck”, the multiset of characteristic polynomials of its vertex-deleted subgraphs, thus settling the “polynomial reconstruction problem” for hypergraphs in the negative. The proof proceeds by showing that a construction due to Kocay of an infinite family of pairs of 3-uniform hypergraphs which are non-isomorphic but share the same hypergraph deck, in fact, have different characteristic polynomials. The question remains unresolved for ordinary graphs.

New Insights into Aromaticity through Novel Delta Polynomials and Delta Aromatic Indices

We have developed novel polynomials called delta polynomials, which are, in turn, derived from the characteristic and matching polynomials of graphs associated with polycyclic aromatic compounds. Natural logarithmic aromatic indices are derived from these delta polynomials, which are shown to provide new insights into the aromaticity of polycyclic aromatic compounds, including the highly symmetric C60 buckminsterfullerene, several other fullerenes, graphene, kekulene series and other cycloarenes, such as polycyclic circumcoronaphenes and coronoids. The newly developed aromatic index yields a value of 6.77 for graphene, 6.516865 for buckminsterfullerene C60(Ih), 5.914023 for kekulene (D6h symmetry), 6.064420 for coronene (D6h), 6.137828 for circumcoronene (D6h), 6.069668 for dicronylene and so forth. Hence, the novel scaled logarithmic aromatic delta indices developed here appear to provide good quantitative measures of aromaticity, especially when they are used in conjunction with other aromatic indicators.

Singular graphs and the reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3], the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.

Convolution formulas for multivariate arithmetic Tutte polynomials

The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid.

Rational orthogonal matrices and isomorphism of graphs

Let On(Q) and Sn(Z) be the set of all n by n orthogonal matrices with rational entries and the set of all n by n symmetric matrices with integer entries, respectively. We consider the following question: Given A∈Sn(Z). When does QTAQ∈Sn(Z) with Q∈On(Q) imply that Q is a signed permutation matrix? Wang and Yu [15] show that the answer to the above question is yes whenever the discriminant Δ(A) of the characteristic polynomial of A is odd and square-free. However, the above result cannot be applied for the adjacency matrix of a simple graph, since Δ(A) can never be odd in this situation. We give a new condition under which the answer is still yes for the adjacency matrices of graphs. The result was achieved by employing some tools from our recent work on the generalized spectral characterizations of graphs, which is somewhat unexpected. Some applications of the above result are also provided.

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues.

Generalized Inversion of Nonlinear Operators

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

Deletion-contraction for a unified Laplacian and applications

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees.

Moments of zeta and correlations of divisor‐sums: Stratification and Vandermonde integrals

AbstractWe refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta‐function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r

An integral quadratic form q : ℤn → ℤ is usually identified with a bilinear form 𝕓∨q:ℤn×ℤn→ℤ satisfying q(x)=𝕓∨q(x,x) for any vector x in ℤn, and such that its Gram matrix with respect to the canonical basis of ℤn is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type 𝔸r (r ≥ 1) and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49–82, 2021] and [Fundamenta Informaticae 185(3):221–246, 2022].

DYNAMIC CHARACTERISTICS OF TIME-VARYING CONTROL SYSTEM OF THE ROCKET’S ROTATIONAL MOVEMENT

The rocket motion control system is time-varying, because its parameters during flight depend on the trajectory point and fuel consumptions. In the available sources, due attention is not paid to the development of a mathematical apparatus of an applied value for the quan- titative assessment of the dynamic characteristics of a time-varying system.
 The purpose of the work is to justify the possibility of an algorithm building for calcu- lating the parameters of a link with constant parameters, which is equivalent to a time- varying system in terms of dynamic characteristics at the selected trajectory section.
 The link’s parameters are found by using the criterion of equivalence of the array of values of the motion model’soutput signal and the analytical solution of the link’s differential equation for a given sequence of input signals. This makes it possible to use the mathematical apparatus of stationary systems to determine the indicators of the disturbances compensation.
 The model of the control system of the rocket’s rotational movement in one plane is taken as a linear differential equation with time-varying parameters without taking into ac- count the executive device inertia and other disturbances. The link with constant parameters is a fractional-rational function of the second order, for the determination of which a se- quence of signals is applied to the input of the system model, the duration of which depends on the desired stability margin on the roots plane of the characteristic polynomial.
 Using the example of a time-varying system for controlling the rocket’srotational movement in the yaw plane, the possibility of determining the parameters of the transition process of compensation for the disturbances characteristic of it is shown for the selected tra- jectory section.
 The obtained results can be used in the design of a motion control system with time- varying parameters.
 The next stage of the research is an assessment of the complexity level of the calcula- tion algorithm when increasing the order of the system’smathematical model.

New constructions of nonregular cospectral graphs

Abstract We consider two types of joins of graphs G 1 {G}_{1} and G 2 {G}_{2} , G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} – the neighbors splitting join and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} – the nonneighbors splitting join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of these joins. When G 1 {G}_{1} and G 2 {G}_{2} are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} , and the normalized Laplacian spectrum of G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} . We use these results to construct nonregular, nonisomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian, signless Laplacian and normalized Laplacian.

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra R ∪ { − ∞ } is a semiring with the two operations: addition a ⊕ b := max ( a , b ) and multiplication a ⊗ b := a + b . Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that algebraic eigenvectors of symmetric matrices with respect to distinct algebraic eigenvalues are orthogonal to each other.

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

Vertex-weighted digraphs and freeness of arrangements between Shi and Ish

We introduce and study a digraph analogue of Stanley’s ψ-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in the literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish, recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight ψ, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

Stabilization policy and lags

Macroeconomic stabilization policy is notoriously subject to inside lags (which delay the reaction of policy to the state of the economy) and outside lags (which delay the effects of policy on the economy). In a broad class of dynamic rational-expectations models, I show that neither inside lags nor outside lags of any length restrict the ability of the policymaker to ensure local-equilibrium determinacy and to control the anticipation and convergence rates, under a weak condition on the model and the policymaker’s observation set. To establish this result, I invert the problem usually addressed in the literature: I start from a targeted characteristic polynomial, and I derive a corresponding policy-instrument rule. For any lags, this method offers some degrees of freedom that can be exploited to design rules with additional properties; I illustrate this possibility by designing non-superinertial rules, which the literature suggests may be more robust under model uncertainty.

More on characteristic polynomials of Lie algebras

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation.

Optimal control strategy for traffic platoon longitudinal coordination around equilibrium state enabled by partially automated vehicles

This paper presents an internal model-Kalman filtering-based optimal hybrid feedforward/feedback control strategy for traffic platoon control coordination enabled by SAE Automation Level 2 or Level 3 vehicles, i.e., partially automated vehicles (PAVs). Based on the Helly linear car-following model, a PAV platoon is established. Taking each vehicle’s characteristic polynomial as the dominant internal model polynomial, an augmented system state model with filtered inputs is formed, and then both the system states and the external disturbance/internal perturbation from previous vehicle can be estimated by a delicately designed Kalman filter. By using a linear quadratic regulation approach, a distributed hybrid optimal feedforward/feedback controller utilizing the local estimated states is constructed to achieve a practically decoupled platoon regulation around an equilibrium state. String stability of the control system thus obtained is further analyzed. Finally, extensive numerical simulations verified the theoretical analysis in decoupling between successive vehicles, in suppressing the influence of various external disturbances, and in maintaining string stability.

Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion in Sobolev space

We prove that the convergence of the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion toward Gaussian free fields on the cylinder, as the matrix dimension goes to infinity, holds in certain suitable Sobolev spaces, whose regularity we prove to be optimal. Our result can be seen as the natural dynamical analogue to the stationary result for a fixed time by Hughes et al (2001 Commun. Math. Phys. 220 429–51). Further our result is related to the work of Spohn (1998 Markov Processes and Related Fields vol 4), from which the identification of the above limit as the Gaussian free field first followed, albeit in a different function space.