Definition and properties of characteristic frequencies of a linear equation

The definition of the characteristic frequencies of zeroes and changes of sign for solutions is given. It is equal to the upper medium (with respect to the time half-axis) of their number on the half-interval of length π. We also define the main frequencies for a linear homogeneous equation of order n. These main frequencies for an equation with constant coefficients coincide with the absolute values of the imaginary parts of the roots of the corresponding characteristic polynomial. It is proved that for the second-order equation the main frequencies are the same for all solutions and that they are stable with respect to uniformly small and infinitely small perturbations of the coefficients. For the third-order equation they can be different, and for any of the main frequencies an example of nonstability is given. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 249–294, 2005.     

Properties of Characteristic Frequencies of Linear Equations of Arbitrary Order

New characteristics of oscillation properties of solutions are suggested, namely, the frequencies equal to the upper or the lower time-average of the null-points, sign alteration points, or roots (their multiplicity being taken into account). We examine several definitions of the principal values of frequencies of a linear homogeneous equation; for an equation with constant coefficients, all these values coincide with the moduli of the imaginary parts of the roots of the corresponding characteristic polynomial. It is shown that for equations of an arbitrary (> 2) order, these definitions yield different values, which are, in general, unstable with respect to uniformly small, or even infinitesimal, perturbations of coefficients, and the extreme values belong to precisely the second Baire class, both in the uniform and the compact-open topologies.     

Linear relations among algebraic solutions of differential equations

For any univariate polynomial with coefficients in a differential field of characteristic zero and any integer, q, there exists an associated nonzero linear ordinary differential equation (LODE) with the following two properties. Each term of the LODE lies in the differential field generated by the rational numbers and the coefficients of the polynomial, and the qth power of each root of the polynomial is a solution of this LODE. This LODE is called a qth power resolvent of the polynomial. We will show how one can get a resolvent for the logarithmic derivative of the roots of a polynomial from the αth power resolvent of the polynomial, where α is an indeterminate that takes the place of q. We will demonstrate some simple relations among the algebraic and differential equations for the roots and their logarithmic derivatives. We will also prove several theorems regarding linear relations of roots of a polynomial over constants or the coefficient field of the polynomial depending upon the (nondifferential) Galois group. Finally, we will use a differential resolvent to solve the Riccati equation.     

An application of the Hurwitz theorem to the root analysis of the characteristic equation

In this work we show that a classical result of A. Hurwitz is still very effective in studying the root analysis of the characteristic equation for a linear functional differential equation. A conjecture was made by Funakubo et al. (2006) [3] regarding the asymptotic stability condition of the zero solution of a linear integro-differential equation of Volterra type. We applied the Hurwitz theorem to the characteristic equation in question and showed the existence of a root with positive real part and solved the conjecture. The Hurwitz theorem is expected to work well for the root analysis in critical cases.     

Symmetric complete intersections

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals and describe formulas for the graded characters of the corresponding quotient rings.     

Detection of positive roots of a polynomial with five parameters

We give a complete set of discriminatory criteria for a polynomial with five parameters to have a positive root. This polynomial arises from the characteristic equation of a difference equation Δⁿ(x_k+px_k−τ)+qx_k−σ=0, which is used to model population dynamics. Our investigations are self-contained and based on the theory of envelopes.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

Direct solution methods for singular integral equations with nonnegative indices

In this paper we consider a complete singular integral equation with the Cauchy kernel on the real axis and a bisingular integral equation on a plane with a degenerate characteristic part. We theoretically substantiate the polynomial methods of moments and collocation in the case of nonnegative indices. We also prove the convergence of the method of mechanical quadratures for the corresponding one-dimensional equation.     

一类新的极小谱任意符号模式

若给定任意一个$n$次首一实系数多项式$f(\lambda)$,都存在一个实矩阵$B\in Q(A)$, 使得$B$的特征多项式为$f(\lambda)$,则称$A$为谱任意符号模式. 如果一个谱任意符号模式的任意非零元被零取代后所得到的符号模式不是谱任意,那么这个谱任意符号模式称为极小谱任意符号模式.本文证明一类极小谱任意符号模式.     

Instability in the critical case of pair of pure imaginary roots for a class of systems with aftereffect

The stability of a system described by Volterra integrodifferential equations is investigated in the critical case when the characteristic equation has a pair of pure imaginery roots. Conditions for instability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was established previously in [2] for integrodifferential equations of simpler structure with integral kernels of exponential-polynomial type). For the proof, several manipulations are used to simplify the original equation and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogous approach was used to analyse instability for Volterra integrodifferential equations in the critical case of zero root in [3, 4]). As an example, the sign of the Lyapunov constant in the problem of the rotational motion of a rigid body with viscoelastic supports is calculated.     

Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.     

Stably computing the multiplicity of known roots given leading coefficients

We show that a monic univariate polynomial over a field of characteristic zero, with k distinct nonzero known roots, is determined by precisely k of its proper leading coefficients. Furthermore, we give an explicit, numerically stable algorithm for computing the exact multiplicities of each root over $\mathbb{C}$. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Then, we demonstrate how these results can be used to obtain the full homogeneous spectra of symmetric tensors—in particular, complete characteristic polynomials of hypergraphs.     

二次自伴矩阵多项式阵的特征值

系统地论证了二次自伴矩阵多项式特征值,特征向量的性质.给出了二次自伴矩阵多项式特征值与任一非零向量所对应的二次多项式根之间的大小关系;精确地给出了二次自伴矩阵多项式是负定时参数的界;简化了二次自伴矩阵多项式的符号特征是正(负)的特征值对应特征向量间可以是线性无关等定理的证明.     

On stability of annth-order equation in a critical case

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomousnth-order equation in the case where the root of the boundary characteristic equation is equal to zero and has multiplicity greater than one.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 47, No. 8, pp. 1138–1143, August, 1995.     

On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions \(\Gamma \) with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.     

On the instability of one nonautonomous essentially nonlinear equation of thenth order

We establish sufficient conditions for the Lyapunov instability of the trivial solution of a nonautonomous equation of thenth order in the case where its limit characteristic equation has a multiple zero root. The instability is determined by nonlinear terms. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 835–841, June, 1999.     

On the central charge of a factorizable Hopf algebra

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel’d element differs from the value that it takes on the Drinfel’d element itself by at most a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ by at most a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.     

Necessary and sufficient conditions of polynomial growth of varieties of Leibniz-Poisson algebras

Leibniz-Poisson algebras are generalizations of Poisson algebras. We give equivalent conditions of polynomial growth of a variety of Leibniz-Poisson algebras over a field of characteristic zero. We find all varieties of Leibniz-Poisson algebras with almost polynomial growth belonging to a certain class of varieties.     

Critical cases of stability of one nonautonomous essentially nonlinear equation of the nth order

We establish sufficient conditions of the Lyapunov stability of the trivial solution of a nonautonomous ordinary differential equation of the nth order in the case where its characteristic equation has a multiple zero root. The stability is determined by nonlinear terms. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii, Zhurnal, Vol. 49, No. 5, pp. 720–724, May, 1997.     

Stability and bifurcation in a discrete system of two neurons with delays

In this paper, we consider a simple discrete two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark–Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field.