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.Tue, 12 Jun 2018 23:00:00 +0200Opuscula Mathematicahttp://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
http://www.opuscula.agh.edu.pl
Eigenvalue asymptotics for potential type operators on Lipschitz surfaces of codimension greater than 1
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3833.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3833.pdfTue, 12 Jun 2018 22:00:07 +0200 Author(s): Grigori Rozenblum, Grigory Tashchiyan.

Abstract: For potential type integral operators on a Lipschitz submanifold the asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz one. Keywords: integral operators, potential theory, eigenvalue asymptotics. Mathematics Subject Classification: 47G40, 35P20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 733-758, https://doi.org/10.7494/OpMath.2018.38.5.733.

]]>Inverse scattering problems for half-line Schrödinger operators and Banach algebras
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3832.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3832.pdfTue, 12 Jun 2018 22:00:06 +0200 Author(s): Yaroslav Mykytyuk, Nataliia Sushchyk.

Abstract: The inverse scattering problem for half-line Schrödinger operators with potentials from the Marchenko class is shown to be closely related to some Banach algebra of functions on the line. In particular, it is proved that the topological conditions in the Marchenko theorem can be replaced by the condition that the scattering function should belong to this Banach algebra. Keywords: inverse scattering, Schrödinger operator, Banach algebra. Mathematics Subject Classification: 34L25, 34L40, 47L10, 81U40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 719-731, https://doi.org/10.7494/OpMath.2018.38.5.719.

]]>On one condition of absolutely continuous spectrum for self-adjoint operators and its applications
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3831.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3831.pdfTue, 12 Jun 2018 22:00:05 +0200 Author(s): Eduard Ianovich.

Abstract: In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator \(A\) by a sequence of operators \(A_n\) with absolutely continuous spectrum on a given interval \([a,b]\) which converges to \(A\) in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator \(A\) spectrum on the finite interval \([a,b]\) and the condition for that the corresponding spectral density belongs to the class \(L_p[a,b]\) (\(p\ge 1\)). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant \(C\gt 0\) and a positive function \(g(x)\in L_p[a,b]\) (\(p\ge 1\)) such that for all \(n\) sufficiently large and almost all \(x\in[a,b]\) the estimate \(\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C\) holds, where \(P_n(x)\) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and \(b_n\) is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on \([a,b]\) and for the corresponding spectral density \(f(x)\) we have \(f(x)\in L_p[a,b]\). Keywords: self-adjoint operators, absolutely continuous spectrum, equi-absolute continuity, spectral density, Jacobi matrices. Mathematics Subject Classification: 47A10, 47A58. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 699-718, https://doi.org/10.7494/OpMath.2018.38.5.699.

]]>Krein-von Neumann extension of an even order differential operator on a finite interval
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3830.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3830.pdfTue, 12 Jun 2018 22:00:04 +0200 Author(s): Yaroslav I. Granovskyi, Leonid L. Oridoroga.

Abstract: We describe the Krein-von Neumann extension of minimal operator associated with the expression \(\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}\) on a finite interval \((a,b)\) in terms of boundary conditions. All non-negative extensions of the operator \(A\) as well as extensions with a finite number of negative squares are described. Keywords: non-negative extension, Friedrichs' extension, Krein-von Neumann extension, boundary triple, Weyl function. Mathematics Subject Classification: 47A05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 681-698, https://doi.org/10.7494/OpMath.2018.38.5.681.

]]>Small-gain theorem for a class of abstract parabolic systems
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3829.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3829.pdfTue, 12 Jun 2018 22:00:03 +0200 Author(s): Piotr Grabowski.

Abstract: We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense. Keywords: control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems. Mathematics Subject Classification: 49N10, 93B05, 93C25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 651-680, https://doi.org/10.7494/OpMath.2018.38.5.651.

]]>Spectrum of J-frame operators
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3828.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3828.pdfTue, 12 Jun 2018 22:00:02 +0200 Author(s): Juan Giribet, Matthias Langer, Leslie Leben, Alejandra Maestripieri, Francisco Martínez Pería, Carsten Trunk.

Abstract: A \(J\)-frame is a frame \(\mathcal{F}\) for a Krein space \((\mathcal{H},[\cdot,\cdot ])\) which is compatible with the indefinite inner product \([\cdot,\cdot ]\) in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in \(\mathcal{H}\). With every \(J\)-frame the so-called \(J\)-frame operator is associated, which is a self-adjoint operator in the Krein space \(\mathcal{H}\). The \(J\)-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of \(J\)-frame operators in a Krein space by a \(2\times 2\) block operator representation. The \(J\)-frame bounds of \(\mathcal{F}\) are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the \(2\times 2\) block representation. Moreover, this \(2\times 2\) block representation is utilized to obtain enclosures for the spectrum of \(J\)-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all \(J\)-frames associated with a given \(J\)-frame operator. Keywords: frame, Krein space, block operator matrix, spectrum. Mathematics Subject Classification: 47B50, 47A10, 46C20, 42C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 623-649, https://doi.org/10.7494/OpMath.2018.38.5.623.

]]>The spectral theorem for locally normal operators
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3827.pdf
http://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3827.pdfTue, 12 Jun 2018 22:00:01 +0200 Author(s): Aurelian Gheondea.

Abstract: We prove the spectral theorem for locally normal operators in terms of a locally spectral measure. In order to do this, we first obtain some characterisations of local projections and we single out and investigate the concept of a locally spectral measure. Keywords: locally Hilbert space, locally \(C^*\)-algebra, locally normal operator, local projection, locally spectral measure. Mathematics Subject Classification: 47B15, 46A13, 46C05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 5 (2018), 597-621, https://doi.org/10.7494/OpMath.2018.38.5.597.

]]>Toeplitz versus Hankel: semibounded operators
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3826.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3826.pdfWed, 11 Apr 2018 18:00:06 +0200 Author(s): Dmitri R. Yafaev.

Abstract: Our goal is to compare various results for Toeplitz \(T\) and Hankel \(H\) operators. We consider semibounded operators and find necessary and sufficient conditions for their quadratic forms to be closable. This property allows one to define \(T\) and \(H\) as self-adjoint operators under minimal assumptions on their matrix elements. We also describe domains of the closed Toeplitz and Hankel quadratic forms. Keywords: semibounded Toeplitz, Hankel and Wiener-Hopf operators, closable and closed quadratic forms. Mathematics Subject Classification: 47B25, 47B35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 573-590, https://doi.org/10.7494/OpMath.2018.38.4.573.

]]>Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3825.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3825.pdfWed, 11 Apr 2018 18:00:05 +0200 Author(s): Qingkai Kong, Thomas E. St. George.

Abstract: We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions. Keywords: nodal solutions, integral boundary value problems, Sturm-Liouville problems, eigenvalues, matching method. Mathematics Subject Classification: 34B10, 34B15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 557-571, https://doi.org/10.7494/OpMath.2018.38.4.557.

]]>On the non-existence of zero modes
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3824.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3824.pdfWed, 11 Apr 2018 18:00:04 +0200 Author(s): Daniel M. Elton.

Abstract: We consider magnetic fields on \(\mathbb{R}^3\) which are parallel to a conformal Killing field. When the latter generates a simple rotation we show that a Weyl-Dirac operator with such a magnetic field cannot have a zero mode. In particular this allows us to expand the class of non zero mode producing magnetic fields to include examples of non-trivial smooth compactly supported fields. Keywords: Weyl-Dirac operator, zero modes. Mathematics Subject Classification: 35J46, 35P20, 35Q40, 81Q10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 537-556, https://doi.org/10.7494/OpMath.2018.38.4.537.

]]>Banach *-algebras generated by semicircular elements induced by certain orthogonal projections
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3823.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3823.pdfWed, 11 Apr 2018 18:00:03 +0200 Author(s): Ilwoo Cho, Palle E. T. Jorgensen.

Abstract: The main purpose of this paper is to study structure theorems of Banach \(*\)-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a \(C^{*}\)-probability space. Keywords: free probability, orthogonal projections, weighted-semicircular elements, semicircular elements. Mathematics Subject Classification: 46L10, 46L54, 47L15, 47L30, 47L55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 501-535, https://doi.org/10.7494/OpMath.2018.38.4.501.

]]>On spectra of quadratic operator pencils with rank one gyroscopic linear part
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3822.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3822.pdfWed, 11 Apr 2018 18:00:02 +0200 Author(s): Olga Boyko, Olga Martynyuk, Vyacheslav Pivovarchik.

Abstract: The spectrum of a selfadjoint quadratic operator pencil of the form \(\lambda^2M-\lambda G-A\) is investigated where \(M\geq 0\), \(G\geq 0\) are bounded operators and \(A\) is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator \(G\) the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator \(G\). Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators \(A\) are given. Keywords: quadratic operator pencil, gyroscopic force, eigenvalues, algebraic multiplicity. Mathematics Subject Classification: 47A56, 47E05, 81Q10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 483-500, https://doi.org/10.7494/OpMath.2018.38.4.483.

]]>Positive definite functions and dual pairs of locally convex spaces
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3821.pdf
http://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3821.pdfWed, 11 Apr 2018 18:00:01 +0200 Author(s): Daniel Alpay, Saak Gabriyelyan.

Abstract: Using pairs of locally convex topological vector spaces in duality and topologies defined by directed families of sets bounded with respect to the duality, we prove general factorization theorems and general dilation theorems for operator-valued positive definite functions. Keywords: positive definite function, locally convex space, dual pair, the (strong) factorization property, dilation theory. Mathematics Subject Classification: 42A82, 47A20, 47A68. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 4 (2018), 463-482, https://doi.org/10.7494/OpMath.2018.38.4.463.

]]>Graphons and renormalization of large Feynman diagrams
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3820.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3820.pdfMon, 19 Mar 2018 17:00:08 +0100 Author(s): Ali Shojaei-Fard.

Abstract: The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions. Keywords: graph functions, Dyson-Schwinger equations, Connes-Kreimer renormalization Hopf algebra. Mathematics Subject Classification: 05C05, 05C63, 81T16, 81T18. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 427-455, https://doi.org/10.7494/OpMath.2018.38.3.427.

]]>On domination multisubdivision number of unicyclic graphs
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3819.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3819.pdfMon, 19 Mar 2018 17:00:07 +0100 Author(s): Joanna Raczek.

Abstract: The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems. Keywords: domination number, domination subdivision number, domination multisubdivision number, trees, unicyclic graphs. Mathematics Subject Classification: 05C69, 05C05, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 409-425, https://doi.org/10.7494/OpMath.2018.38.3.409.

]]>On the boundedness of equivariant homeomorphism groups
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3818.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3818.pdfMon, 19 Mar 2018 17:00:06 +0100 Author(s): Jacek Lech, Ilona Michalik, Tomasz Rybicki.

Abstract: Given a principal \(G\)-bundle \(\pi:M\to B\), let \(\mathcal{H}_G(M)\) be the identity component of the group of \(G\)-equivariant homeomorphisms on \(M\). The problem of the uniform perfectness and boundedness of \(\mathcal{H}_G(M)\) is studied. It occurs that these properties depend on the structure of \(\mathcal{H}(B)\), the identity component of the group of homeomorphisms of \(B\), and of \(B\) itself. Most of the obtained results still hold in the \(C^r\) category. Keywords: principal \(G\)-manifold, equivariant homeomorphism, uniformly perfect, bounded, \(C^r\) equivariant diffeomorphism. Mathematics Subject Classification: 57S05, 58D05, 55R91. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 395-408, https://doi.org/10.7494/OpMath.2018.38.3.395.

]]>On expansive and anti-expansive tree maps
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3817.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3817.pdfMon, 19 Mar 2018 17:00:05 +0100 Author(s): Sergiy Kozerenko.

Abstract: With every self-map on the vertex set of a finite tree one can associate the directed graph of a special type which is called the Markov graph. Expansive and anti-expansive tree maps are two extremal classes of maps with respect to the number of loops in their Markov graphs. In this paper we prove that a tree with at least two vertices has a perfect matching if and only if it admits an expansive cyclic permutation of its vertices. Also, we show that for every tree with at least three vertices there exists an expansive map with a weakly connected (strongly connected provided the tree has a perfect matching) Markov graph as well as anti-expansive map with a strongly connected Markov graph. Keywords: maps on trees, Markov graphs, Sharkovsky's theorem. Mathematics Subject Classification: 37E25, 37E15, 05C20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 379-393, https://doi.org/10.7494/OpMath.2018.38.3.379.

]]>Forbidden configurations for hypohamiltonian graphs
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3816.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3816.pdfMon, 19 Mar 2018 17:00:04 +0100 Author(s): Igor Fabrici, Tomáš Madaras, Mária Timková.

Abstract: A graph \(G\) is called hypohamiltonian if \(G\) is not hamiltonian, but \(G-x\) is hamiltonian for each vertex \(x\) of \(G\). We present a list of 331 forbidden configurations which do not appear in hypohamiltonian graphs. Keywords: hypohamiltonian graph, forbidden configuration, long cycle. Mathematics Subject Classification: 05C38, 05C45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 357-377, https://doi.org/10.7494/OpMath.2018.38.3.357.

]]>Improved iterative oscillation tests for first-order deviating differential equations
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3815.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3815.pdfMon, 19 Mar 2018 17:00:03 +0100 Author(s): George E. Chatzarakis, Irena Jadlovská.

Abstract: In this paper, improved oscillation conditions are established for the oscillation of all solutions of differential equations with non-monotone deviating arguments and nonnegative coefficients. They lead to a procedure that checks for oscillations by iteratively computing \(\lim \sup\) and \(\lim \inf\) on terms recursively defined on the equation's coefficients and deviating argument. This procedure significantly improves all known oscillation criteria. The results and the improvement achieved over the other known conditions are illustrated by two examples, numerically solved in MATLAB. Keywords: differential equation, non-monotone argument, oscillatory solution, nonoscillatory solution. Mathematics Subject Classification: 34K06, 34K11. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 327-356, https://doi.org/10.7494/OpMath.2018.38.3.327.

]]>Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3814.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3814.pdfMon, 19 Mar 2018 17:00:02 +0100 Author(s): Dariusz Borkowski, Katarzyna Jańczak-Borkowska.

Abstract: We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral. Keywords: backward stochastic differential equation, fractional Brownian motion, backward stochastic variational inequalities, subdifferential operator. Mathematics Subject Classification: 60H05, 60H07, 60H22. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 307-326, https://doi.org/10.7494/OpMath.2018.38.3.307.

]]>Solutions to p(x)-Laplace type equations via nonvariational techniques
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3813.pdf
http://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3813.pdfMon, 19 Mar 2018 17:00:01 +0100 Author(s): Mustafa Avci.

Abstract: In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent. Keywords: Leray-Lions type operator, nonlinear monotone operator, approximation, variable Lebesgue spaces. Mathematics Subject Classification: 35J60, 35J70, 35J92, 58E05, 76A02. Journal: Opuscula Mathematica. Citation: Opuscula Math. 38, no. 3 (2018), 291-305, https://doi.org/10.7494/OpMath.2018.38.3.291.

]]>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.