Opuscula Mathematica
http://www.opuscula.agh.edu.pl
enA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.Fri, 29 Dec 2017 14:00:00 +0100Opuscula Mathematicahttp://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
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Stochastic differential equations for random matrices processes in the nonlinear framework
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3812.pdf
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3812.pdfFri, 29 Dec 2017 14:00:05 +0100 Author(s): Sara Stihi, Hacène Boutabia, Selma Meradji.

Abstract: In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process \(X_{t}\), where \(X_{t}\) is the solution of a general SDE driven by a \(G\)-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503]. Keywords: \(G\)-Brownian motion matrix, \(G\)-stochastic differential equations, random matrices, eigenvalues, eigenvectors. Mathematics Subject Classification: 60B20, 60H10, 60H05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 2 (2018), 261-283, https://doi.org/10.7494/OpMath.2018.38.2.261.

]]>Trace formulas for perturbations of operators with Hilbert-Schmidt resolvents
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3811.pdf
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3811.pdfFri, 29 Dec 2017 14:00:04 +0100 Author(s): Bishnu Prasad Sedai.

Abstract: Trace formulas for self-adjoint perturbations \(V\) of self-adjoint operators \(H\) such that \(V\) is in Schatten class were obtained in the works of L.S. Koplienko, M.G. Krein, and the joint paper of D. Potapov, A. Skripka and F. Sukochev. In this article, we obtain an analogous trace formula under the assumptions that the perturbation \(V\) is bounded and the resolvent of \(H\) belongs to Hilbert-Schmidt class. Keywords: trace formulas. Mathematics Subject Classification: 47A55, 47A56. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 2 (2018), 253-260, https://doi.org/10.7494/OpMath.2018.38.2.253.

]]>Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3810.pdf
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3810.pdfFri, 29 Dec 2017 14:00:03 +0100 Author(s): Mitsuo Kato, Toshiyuki Mano, Jiro Sekiguchi.

Abstract: A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions. Keywords: flat structure, Painlevé VI equation, algebraic solution, potential vector field. Mathematics Subject Classification: 34M56, 33E17, 35N10, 32S25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 2 (2018), 201-252, https://doi.org/10.7494/OpMath.2018.38.2.201.

]]>Existence results for Kirchhoff type systems with singular nonlinearity
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3809.pdf
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3809.pdfFri, 29 Dec 2017 14:00:02 +0100 Author(s): A. Firouzjai, G. A. Afrouzi, S. Talebi.

Abstract: Using the method of sub-super solutions, we study the existence of positive solutions for a class of singular nonlinear semipositone systems involving nonlocal operator. Keywords: sub-supersolution, infinite semipositone systems, singular weights, Kirchhoff-type. Mathematics Subject Classification: 35J55, 35J65. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 2 (2018), 187-199, https://doi.org/10.7494/OpMath.2018.38.2.187.

]]>Adelic analysis and functional analysis on the finite Adele ring
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3808.pdf
http://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3808.pdfFri, 29 Dec 2017 14:00:01 +0100 Author(s): Ilwoo Cho.

Abstract: In this paper, we study operator theory on the \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\), consisting of all measurable functions on the finite Adele ring \(A_{\mathbb{Q}}\), in extended free-probabilistic sense. Even though our \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\) is commutative, our Adelic-analytic data and properties on \(\mathcal{M}_{\mathcal{P}}\) are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on \(A_{\mathbb{Q}}\), we construct the suitable Hilbert-space representation, and study a \(C^{*}\)-algebra \(M_{\mathcal{P}}\) generated by \(\mathcal{M}_{\mathcal{P}}\) under representation. In particular, we focus on operator-theoretic properties of certain generating operators on \(M_{\mathcal{P}}\). Keywords: representations, \(C^{*}\)-algebras, \(p\)-adic number fields, the Adele ring, the finite Adele ring. Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L54, 47L30, 47L55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 2 (2018), 139-185, https://doi.org/10.7494/OpMath.2018.38.2.139.

]]>Study of ODE limit problems for reaction-diffusion equations
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3807.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3807.pdfMon, 13 Nov 2017 00:00:07 +0100 Author(s): Jacson Simsen, Mariza Stefanello Simsen, Aleksandra Zimmermann.

Abstract: In this work we study ODE limit problems for reaction-diffusion equations for large diffusion and we study the sensitivity of nonlinear ODEs with respect to initial conditions and exponent parameters. Moreover, we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents when the exponents go to 2 in \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity. Keywords: ODE limit problems, shadow systems, reaction-diffusion equations, parabolic problems, variable exponents, attractors, upper semicontinuity. Mathematics Subject Classification: 35B40, 35B41, 35K57, 35K59. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 117-131, https://doi.org/10.7494/OpMath.2018.38.1.117.

]]>On the stability of some systems of exponential difference equations
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3806.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3806.pdfMon, 13 Nov 2017 00:00:06 +0100 Author(s): N. Psarros, G. Papaschinopoulos, C. J. Schinas.

Abstract: In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations. Keywords: difference equations, asymptotic behaviour, global stability, centre manifold, biological dynamics. Mathematics Subject Classification: 39A10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 95-115, https://doi.org/10.7494/OpMath.2018.38.1.95.

]]>Wiener index of strong product of graphs
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3805.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3805.pdfMon, 13 Nov 2017 00:00:05 +0100 Author(s): Iztok Peterin, Petra Žigert Pleteršek.

Abstract: The Wiener index of a connected graph \(G\) is the sum of distances between all pairs of vertices of \(G\). The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph \(G\) of constant eccentricity with a cycle are derived. Keywords: Wiener index, graph product, strong product. Mathematics Subject Classification: 05C12, 05C76. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 81-94, https://doi.org/10.7494/OpMath.2018.38.1.81.

]]>Asymptotic profiles for a class of perturbed Burgers equations in one space dimension
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3804.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3804.pdfMon, 13 Nov 2017 00:00:04 +0100 Author(s): F. Dkhil, M. A. Hamza, B. Mannoubi.

Abstract: In this article we consider the Burgers equation with some class of perturbations in one space dimension. Using various energy functionals in appropriate weighted Sobolev spaces rewritten in the variables \(\frac{\xi}{\sqrt\tau}\) and \(\log\tau\), we prove that the large time behavior of solutions is given by the self-similar solutions of the associated Burgers equation. Keywords: Burgers equation, self-similar variables, asymptotic behavior, self-similar solutions. Mathematics Subject Classification: 35B20, 35B40, 35C20, 35K55, 35L05, 35L60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 41-80, https://doi.org/10.7494/OpMath.2018.38.1.41.

]]>Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3803.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3803.pdfMon, 13 Nov 2017 00:00:03 +0100 Author(s): Amar Chidouh, Delfim F. M. Torres.

Abstract: We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained. Keywords: fractional difference equations, Lyapunov-type inequalities, fractional boundary value problems, positive solutions. Mathematics Subject Classification: 26A33, 26D15, 39A12. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 31-40, https://doi.org/10.7494/OpMath.2018.38.1.31.

]]>The spectrum problem for digraphs of order 4 and size 5
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3802.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3802.pdfMon, 13 Nov 2017 00:00:02 +0100 Author(s): Ryan C. Bunge, Steven DeShong, Saad I. El-Zanati, Alexander Fischer, Dan P. Roberts, Lawrence Teng.

Abstract: The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let \(D\) be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for \(D\), we establish necessary and sufficient conditions on \(n\) for the existence of a \((K^{*}_{n},D)\)-design. Partial results are given for the remaining 6 possibilities for \(D\). Keywords: spectrum problem, digraph decompositions. Mathematics Subject Classification: 05C20, 05C51. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 15-30, https://doi.org/10.7494/OpMath.2018.38.1.15.

]]>Upper bounds for the extended energy of graphs and some extended equienergetic graphs
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3801.pdf
http://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3801.pdfMon, 13 Nov 2017 00:00:01 +0100 Author(s): Chandrashekar Adiga, B. R. Rakshith.

Abstract: In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on \(n\) vertices for \(n\equiv 0(\text{mod } 8)\) starting with a pair of extended equienergetic non regular graphs on \(8\) vertices and also we construct a pair of extended equienergetic graphs on \(n\) vertices for all \(n\geq 9\) starting with a pair of equienergetic regular graphs on \(9\) vertices. Keywords: energy of a graph, extended energy of a graph, extended equienergetic graphs. Mathematics Subject Classification: 05C50. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 1 (2018), 5-13, https://doi.org/10.7494/OpMath.2018.38.1.5.