The integrable coupling system of a 3 × 3 discrete matrix spectral problem

Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system.     

Full-rank representations of {2, 4}, {2, 3}-inverses and successive matrix squaring algorithm

We present the full-rank representations of {2, 4} and {2, 3}-inverses (with given rank as well as with prescribed range and null space) as particular cases of the full-rank representation of outer inverses. As a consequence, two applications of the successive matrix squaring (SMS) algorithm from [P.S. Stanimirovi?, D.S. Cvetkovi?-Ili?, Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput. 203 (2008) 19-29] are defined using the full-rank representations of {2, 4} and {2, 3}-inverses. The first application is used to approximate {2, 4}-inverses. The second application, after appropriate modifications of the SMS iterative procedure, computes {2, 3}-inverses of a given matrix. Presented numerical examples clarify the purpose of the introduced methods.     

On solutions to the twisted Yang-Baxter equation

Solutions to the twisted Yang-Baxter equation given by intertwiners for cyclic representations of are described via two coupled lattice current algebras. Bibliography: 4 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 301–308. Translated by V. O. Tarasov.     

An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N = 581).In addition to using geometric figures and manipulatives, teachers used natural language such as the words nanny and house to characterize mathematical procedures or algorithms. Some teachers used the words or phrases bigger, smaller, doubling, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers and students, specifically equations with missing equals signs and not multiply/dividing by one to find equivalent fractions. No evidence though of a relationship between representational forms and degree of correctness of solutions was found on student work. However, when students exhibited misconceptions, those misconceptions were linked to teachers’ use of idiosyncratic representations.     

Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.     

Counting SL2（F2^s） Representations of Torus Knot Groups

In this paper,we count the number of SL2(F2^s)-representations of torus knot groups up to a conjugacy.For the finite field F2^s with character 2,the counting method is similar to that in out previous work[1].Explicit formulae of the effective counting are given in this paper.Twisted Alexander polynomials related to those reprsentations are discussed.     

Non-integrable variants of Boussinesq equation with two solitons

Three variants of the Boussinesq equation, namely, the (2 + 1)-dimensional Boussinesq equation, the (3 + 1)-dimensional Boussinesq equation, and the sixth-order Boussinesq equation are studied. The Hirota bilinear method is used to construct two soliton solutions for each equation. The study highlights the fact that these equations are non-integrable and do not admit N-soliton solutions although these equations can be put in bilinear forms.     

The common Re-nnd and Re-pd solutions to the matrix equations AX = C and XB = D

In this article, we consider common Re-nnd and Re-pd solutions of the matrix equations AX = C and XB = D with respect to X, where A, B, C and D are given matrices. We give necessary and sufficient conditions for the existence of common Re-nnd and Re-pd solutions to the pair of the matrix equations and derive a representation of the common Re-nnd and Re-pd solutions to these two equations when they exist. The presented examples show the advantage of the proposed approach.     

Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems

The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm-Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler-Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k = 1, 2, 3, … , can be obtained by using minimally 2(k + 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies.     

Comments on some recent results concerning {2, 3} and {2, 4}-generalized inverses

We give a comment on some recent results concerning the representations of generalized {2, 3} and {2, 4}-inverses. Shorter proofs of some previous results are presented.     

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.     

Existence of a positive solution to a system of discrete fractional boundary value problems

We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by -Δ^ν_iy_i(t)=λ_ia_i(t+ν_i-1)f_i(y₁(t+ν₁-1),y₂(t+ν₂-1)), for t∈[0,b]_N0, subject to y_i(ν_i − 2) = ψ_i(y_i) and y_i(ν_i + b) = ?_i(y_i), for i = 1, 2, where ψ_i,?_i:R^b+3→R are given functionals. We also assume that ν_i ∈ (1, 2], for each i. Although we assume that both a_i and f_i(y₁, y₂) are nonnegative for each i, we do not necessarily presume that each ψ_i(y_i) and ?_i(y_i) is nonnegative for each i and each y_i ? 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case.     

A pseudo-differential calculus on the Heisenberg group

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4], [2] and [3] of pseudo-differential calculi on graded groups. The relation between the Weyl quantisation and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.     

Least-squares solutions to the matrix equations AX = B and XC = D

The least-squares solution and the least-squares symmetric solution with the minimum-norm of the matrix equations AX = B and XC = D are considered in this paper. By the matrix differentiation and the spectral decomposition of matrices, an explicit representation of such solution is given.     

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.     

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.     

Elliptic complex numbers with dual multiplication

Investigated is a number system in which the square of a basis number: (w)², and the square of its additive inverse: (−w)², are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C, then changing a defining conjugacy relation from conj(z) + z = 0 in the complexes to conj(z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.     

Canonical representations on two-sheeted hyperboloids

The two-sheeted hyperboloid in ℝⁿ can be identified with the unit sphere Ω in ℝⁿ with the equator removed. Canonical representations of the group G = SO ₀(n − 1, 1) on are defined as the restrictions to G of the representations of the overgroup = SO ₀(n, 1) associated with a cone. They act on functions and distributions on the sphere Ω. We decompose these canonical representations into irreducible constituents and decompose the Berezin form. Bibliography: 12 titles. To the memory of my teacher F. A. Berezin __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 91–124.     

A preemptive discrete-time priority buffer system with partial buffer sharing

This paper considers a Geo/Geo/1 discrete-time queue with preemptive priority. Both the arrival and service processes are Bernoulli processes. There are two kinds of customers: low-priority and high-priority customers. The high-priority customers have a preemptive priority over low-priority customers. If the total number of customers is equal or more than the threshold (k), the arrival of low-priority customers will be ignored. Hence the system buffer size is finite only for the low-priority customers. A recursive numerical procedure is developed to find the steady-state probabilities. With the aid of recursive equations, we transform the infinite steady-state departure-epoch equations set to a set of (k + 1) × (k + 2)/2 linear equations set based on the embedded Markov Chain technique. Then, this reduced linear equations set is used to compute the steady-state departure-epoch probabilities. The important performance measures of the system are calculated. Finally, the applicability of the solution procedure is shown by a numerical example and the sensitivity of the performance measures to the changes in system parameters is analyzed.     

The improved linear multistep methods for differential equations with piecewise continuous arguments

This paper deals with the convergence of the linear multistep methods for the equation x′(t) = ax(t) + a₀x([t]). Numerical experiments demonstrate that the 2-step Adams-Bashforth method is only of order p = 0 when applied to the given equation. An improved linear multistep methods is constructed. It is proved that these methods preserve their original convergence order for ordinary differential equations (ODEs) and some numerical experiments are given.