Construction of a Fuchs equation with four singular points and with given reducible 2 × 2 monodromy matrices on the complex projective line

On the complex projective line, we construct a Fuchs equation with four singular points and with given reducible 2 × 2 monodromy matrices such that the fundamental solution matrix of this equation realizes a given homomorphism.     

Partition,Construction, and Enumeration of M–P Invertible Matrices over Finite Fields

A necessary and sufficient condition for an m×n matrix A over F_q having a Moor–Penrose generalized inverse (M–P inverse for short) was given in (C. K. Wu and E. Dawson, 1998, Finite Fields Appl. 4, 307–315). In the present paper further necessary and sufficient conditions are obtained, which make clear the set of m×n matrices over F_q having an M–P inverse and reduce the problem of constructing M–P invertible matrices to that of constructing subspaces of certain type with respect to some classical groups. Moreover, an explicit formula for the M–P inverse of a matrix which is M–P invertible is also given. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing results in geometry of classical groups over finite fields (Z. X. Wan, 1993, “Geometry of Classical Groups over Finite Fields”, Studentlitteratur, Chatwell Bratt).     

The Zero Divisor Graph of 2 × 2 Matrices Over a Field

A zero divisor graph, Γ(R), is formed from a ring R by having each element of Z(R) \ {0} to be a vertex in the graph and having two vertices u and v adjacent if the corresponding elements from the ring are nonequal and have product equal to zero. In this paper, the structure of the zero-divisor graph of 2 × 2 matrices over a field, Γ(M2(F)), are completely determined.     

Long-Time Behavior of Finite Difference Solutions of a Nonlinear Schrödinger Equation with Weakly Damped

A weakly damped Schrödinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

Exposed and Strongly Exposed Points of Finite Order in Some Banach Spaces

in this paper we study the exposed and strongly exposed points of finite order inl(D,X) as well as in L(ll (D,X), Y) and L(X, l(D,Y) ), where X and Y are real Banachspaces while D is a non empty index set. In particular, we characterize the exposed andstrongly exposed points in these spaces.     

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S ³ and H ³ to S ³ × R and H ³ × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S ³ × R and H ³ × R. We have complete minimal surfaces in S ³ × R and H ³ × R. Also we obtain minimal 2-tori in S ³ × S ¹, some of which are embedded.     

Bounds for the Extreme Eigenvalues of Block 2 × 2 Hermitian Matrices

Let an n × n Hermitian matrix A be presented in block 2 × 2 form as , where A₁₂ ≠ 0, and assume that the diagonal blocks A₁₁ and A₂₂ are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds

Simplest Singular Points of 1–forms Invariant with Respect to an Action of Group of Order Three

Singular points that cannot be removed by small deformations of 1–forms and invariant with respect to the action of a cyclic group of order 3 are classified. It is proved that for ?₃–invariant 1–forms the equivariant index of a singular point, as an element of the representation ring of the group, coincides with the class of the representation on the space of germs of forms of highest order factored by the subspace of forms divisible by the given 1–form.     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.     

The Singular Upper Triangle Type Solutions with Spin 1/2 of Quantum Yang-Baxter Equation （I）

In these two papers （I） and （II）, the singular upper triangle type solutions with spin 1/2 of quantum Yang-Baxter equation are given. In the Paper （I）, we give the Yang-Baxter equation and give the general solutions of some function equations for the next paper.     

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

Abstract

We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

Initial Value Problem for a Generalized Korteweg－de Vries Equation with Singular Integral－Differential Terms

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A Necessary and Sufficient Condition for the Solvability of the Nonlinear Schr?dinger Equation on a Finite Interval

The admissibility of the initial-boundary data,which characterizes the existence of solution for the initial-boundary value problem,is important.Based on the Fokas method and the inverse scattering transformation,an approach is developed to solve the initial-boundary value problem of the nonlinear Schrodinger equation on a finite interval.A necessary and sufficient condition for the admissibility of the initial-boundary data is given,and the reconstruction of the potential is obtained.