A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ? 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.     

On two perturbation estimates of the extreme solutions to the equations X ± A*XA = Q

Two perturbation estimates for maximal positive definite solutions of equations X + A*X⁻¹A = Q and X − A*X⁻¹A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X ± A*X⁻¹A = Q, Linear Algebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples.     

Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

Ternary analogues of Lie and Malcev algebras

We consider two analogues of associativity for ternary algebras: total and partial associativity. Using the corresponding ternary associators, we define ternary analogues of alternative and assosymmetric algebras. On any ternary algebra the alternating sum [a, b, c] = abc − acb − bac + bca + cab − cba (the ternary analogue of the Lie bracket) defines a structure of an anticommutative ternary algebra. We determine the polynomial identities of degree ?7 satisfied by this operation in totally and partially associative, alternative, and assosymmetric ternary algebras. These identities define varieties of ternary algebras which can be regarded as ternary analogues of Lie and Malcev algebras. Our methods involve computational linear algebra based on the representation theory of the symmetric group.     

Eigenvector-free solutions to AX = B with PX = XP and X = sX constraints

The matrix equation AX = B with PX = XP and X^H = sX constraints is considered, where P is a given Hermitian involutory matrix and s = ±1. By an eigenvalue decomposition of P, we equivalently transform the constrained problem to two well-known constrained problems and represent the solutions in terms of the eigenvectors of P. Using Moore-Penrose generalized inverses of the products generated by matrices A, B and P, the involved eigenvectors can be released and eigenvector-free formulas of the general solutions are presented. Similar strategy is applied to the equations AX = B, XC = D with the same constraints.     

On absolute valued algebras with involution

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA_s = A_s, where e denotes the unique nonzero self-adjoint idempotent of A, and A_s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay’s algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.     

Integrable coupling hierarchy and Hamiltonian structure for a matrix spectral problem with arbitrary-order

We presented an integrable coupling hierarchy of a matrix spectral problem with arbitrary order zero matrix r by using semi-direct sums of matrix Lie algebra. The Hamiltonian structure of the resulting integrable couplings hierarchy is established by means of the component trace identities. As an example, when r is 2 × 2 zero matrix specially, the integrable coupling hierarchy and its Hamiltonian structure of the matrix spectral problem are computed.     

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2β ∈ (0, 1) and 2α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax-Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.     

A max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n × n nonnegative matrices in max algebra. In this paper we propose the notions of the max algebra version of the generalized spectral radius μ(Ψ) of Ψ, and the max algebra version of the joint spectral radius η(Ψ) of Ψ. The max algebra version of the generalized spectral radius theorem μ(Ψ) = η(Ψ) is established. We propose the relationship between the generalized spectral radius ρ(Ψ) of Ψ (in the sense of Daubechies and Lagarias) and its max algebra version μ(Ψ). Moreover, a generalization of Elsner and van den Driessche’s lemma is presented as well.     

Probability of unique integer solution to a system of linear equations

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1}ⁿ. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.     

Factoring and decomposing a class of linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M′, where M (resp., M′) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R′z = 0). These morphisms define applications sending solutions of the system R′z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R = R₂R₁ of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R′z = 0, where R′ is a block-triangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R₁y₁ = 0 and R₂y₂ = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R′z = 0, where R′ is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in the Maple package Morphisms based on the library oremodules.     

The representations of the Drazin inverse of differences of two matrices

In this paper, we give explicit representations of (P ± Q)^D, (P ± PQ)^D and (PQ)^# of two matrices P and Q, as a function of P, Q, P^D and Q^D, under the conditions P³Q = QP and Q³P = PQ.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

New types of 3-D systems of quadratic differential equations with chaotic dynamics based on Ricker discrete population model

The wide class of 3-D autonomous systems of quadratic differential equations, in each of which either there is a couple of coexisting limit cycles or there is a couple of coexisting chaotic attractors, is found. In the second case the couple consists of either Lorentz-type attractor and another attractor of a new type or two Lorentz-type attractors. It is shown that the chaotic behavior of any system of the indicated class can be described by the Ricker discrete population model: z_i+1 = z_i exp(r − z_i), r > 0, z_i > 0, i = 0, 1, … . The values of parameters, at which in the 3-D system appears either the couple of limit cycles or the couple of chaotic attractors, or only one limit cycle, or only one sphere-shaped chaotic attractor, are indicated. Examples are given.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

Orbits in max-min algebra

Properties of orbits in max-min algebra are described, mainly the properties of periodic orbits. An O(n³) algorithm computing the period of a periodic orbit is presented. As a consequence, an O(n³ log n) algorithm computing the period of arbitrary orbit is obtained, as the pre-periodic part of the orbit has length at most (n − 1)² + 1.     

An adaptive Huber method for non-linear systems of weakly singular second kind Volterra integral equations

Numerical methods for systems of weakly singular Volterra integral equations are rarely considered in the literature, especially if the equations involve non-linear dependencies between unknowns and their integrals. In the present work an adaptive Huber method for such systems is proposed, by extending the method previously formulated for single weakly singular second kind Volterra equations. The method is tested on example systems of integral equations involving integrals with kernels K(t, τ) = (t − τ)^−1/2, K(t, τ) = exp[−λ(t − τ)](t − τ)^−1/2 (where λ > 0), and K(t, τ) = 1. The magnitude of the errors, and practical accuracy orders, observed for IE systems, are comparable to those for single IEs. In cases when the solution vector is not differentiable at t = 0, the estimation of errors at t = 0 is found somewhat less reliable for IE systems, than it was for single IEs. The stability of the IE systems solved appears to be sufficient, in practice, for the numerical stability of the method.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).