Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

Liouvillian first integrals for the planar Lotka-Volterra system

We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.     

A note on the dimension of an orbital subspace

This note contains a dimension formula for an orbital subspace in a symmetry class of tensors corresponding to an irreducible character λ of a subgroup G of S_m. An algorithm for choosing a basis is also described.     

Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method

Here we present a semi-algorithmic method to deal with rational first-order ordinary differential equations, with Liouvillian solutions. This method is based on the knowledge of the general structure for the integrating factor for such equations.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.     

On a class of second order Fuchsian hyperbolic equations

Summary We study a class of second order Fuchsian hyperbolic operators. The well-posedness of the Cauchy problem in a space of regular distributions is proved, together with results on the propagation of singularities of the solution. Moreover we give a representation formula for the distribution solutions of the homogeneous equation.     

On the fundamental solutions of a class of hyperbolic quartic operators in dimension 3

The fundamental solutions of the linear hyperbolic partial differential operators with constant coefficients of the form are represented by elliptic integrals of the first kind. Mathematics Subject Classification (2000) 35A08, 35A20, 35E05     

On a subspace perturbation problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let and be bounded self-adjoint operators. Assume that the spectrum of consists of two disjoint parts and such that 0$">. We show that the norm of the difference of the spectral projections

for and is less than one whenever either (i) or (ii) and certain assumptions on the mutual disposition of the sets and are satisfied.

     

The maximum dimension of a subspace of nilpotent matrices of index 2

A matrix M is nilpotent of index 2 if M²=0. Let V be a space of nilpotent n×n matrices of index 2 over a field k where and suppose that r is the maximum rank of any matrix in V. The object of this paper is to give an elementary proof of the fact that . We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to find the maximum dimension of spaces of anti-commuting matrices and zero subalgebras of special Jordan Algebras.     

On the Galois group of some Fuchsian systems

In this paper, we give a new result ofn the differential Galois theory of linear ordinary differential equations. In particular, we compute the differential Galois group for a special type of nonresonant Fuchsian system.     

拟Fuchs群的收敛指数

根据H2上的双曲距离在拟共形变换下的拟不变性,给出了K-拟共形抛物循环Fuchs群的收敛指数的估计．     

On Sets of Generators of Fuchsian Groups

We give applications of the discontinuity function of a discrete group for Fuchsian groups which act on the unit disk and which are finitely generated. We obtain two sets of theorems: One set corresponds to the Euclidean metric. The second set corresponds to the hyperbolic metric. These theorems state inequalities that involve combinatorial quantities (such as counting functions of the elements of the group with a given bounded length, or the order of growth of the group) and geometric quantities (such as distances of the images of a point under a fixed set of generators to the unit circle, or hyperbolic areas of certain disks). This paper is a sequel to the paper “The Discontinuity Function of Discrete Groups and Radius of Schlichtness” by the author, that recently appeared in this journal.     

On the functional dimension of the space of solutions of partial differential equations

Given a differential polynomial P(D) in Rⁿ with constant coefficients, consider the functional dimension df$\text{N}$ of the space $\text{N}$ = {u∈C(Rⁿ):P(D)u = 0} endowed with the topology of uniform convergence on compact subsets of Rⁿ. If P(D) is elliptic then df$\text{N}$ = n, by a theorem of Y. Kōmura. We prove the converse: If df$\text{N}$ = n then the differential polynomial P(D) must be elliptic.     

On the Cauchy problem for a class of iterated Fuchsian partial differential equations

In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.