§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the form

where c is a real or complex scalar constant and is thus commutative with both p and q.

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the type

where Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

By using Lebesgue’s dominated convergence theorem and constructing a suitable Lyapunov functional, we study the following almost-periodic Lotka–Volterra model with $M$ predators and $N$ prey of the integro-differential equations

\begin{alignat*}{2} \dot{x}_i(t)\amp=x_i(t)\biggl[b_i(t)-a_{ii}(t)x_i(t)-\sum_{k=1,k\neq i}^{N}a_{ik}(t)\int_{-\infty}^tH_{ik}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma\\ \amp\hskip45mm-\sum_{l=1}^{M}c_{il}(t)\int_{-\infty}^tK_{il}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad i\amp=1,2,\dots,N,\\ \dot{y}_j(t)\amp=y_j(t)\biggl[-r_j(t)-e_{jj}(t)y_j(t) +\sum_{k=1}^{N}d_{jk}(t)\int_{-\infty}^tP_{jk}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma \\ \amp\hskip45mm-\sum_{l=1,l\neq j}^{M} e_{jl}(t)\int_{-\infty}^tQ_{jl}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad j\amp=1,2,\dots,M. \end{alignat*}

Some sufficient conditions are obtained for the existence of a unique almost-periodic solution of this model. Several examples show that the obtained criteria are new, general and easily verifiable.

The purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.

AMS 2000 Mathematics subject classification:Primary 46H05; 46L05

In this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

It is a most implausible fact that a one-to-one operator from c0 into a Banach space which maps the unit ball of c0 onto a closed set is necessarily an isomorphism.

In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semi-embeddings defined on C(X), X compact.

In previous work of the author and M. Culler, contractible simplicial complexes were constructed on which the group of outer automorphisms of a free group of finite rank acts with finite stabilizers and finite quotient. In this paper, it is shown that these complexes are Cohen-Macauley, a property they share with buildings. In particular, the link of a vertex in these complexes is homotopy equivalent to a wedge of spheres of codimension 1.

In this paper certain Turán-type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre–Pólya class of entire functions. Moreover, it is shown that in some cases Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l’Hospital's rule can be used in the proof of the corresponding Turán-type inequalities.

In the following paper, the determination of Green's function, for spaces bounded by surfaces of the cylindrical and spherical polar systems, is effected by what is believed to be a novel process, in which are utilised the properties of cylindrical and spherical harmonics, regarded as functions of their parameters.

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).

It is proved that every space L2 (I1, ∪ I2), where I1 and I2 are finite intervals, has a Riesz basis of complex exponentials , {λk} a sequence of real numbers. A partial result for the corresponding problem for n≧3 finite intervals is also obtained.

The Dirichlet problem for the Laplace equation in a connected-plane region with cuts is studied. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable.

In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequality

how small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?

The operator Cauchy dual to a $2$-hyperexpansive operator $T$, given by $T'\equiv T(T^*T)^{-1}$, turns out to be a hyponormal contraction. This simple observation leads to a structure theorem for the $C^*$-algebra generated by a $2$-hyperexpansion, and a version of the Berger–Shaw theorem for $2$-hyperexpansions.

As an application of the hyperexpansivity version of the Berger–Shaw theorem, we show that every analytic $2$-hyperexpansive operator with finite-dimensional cokernel is unitarily equivalent to a compact perturbation of a unilateral shift.

We will present an investigation of (ε, δ)-Freudenthal–Kantor supertriple systems that are intimately related to Lie supertriple systems and Lie superalgebras. We can also introduce a super analogue of Nijenhuis tensor and almost-complex structure in differential geometry.

We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space (). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

Incomplete Bessel functions of general order are defined. Their recurrence relations, integral representations and asymptotics are derived.

We have constructed Lie superalgebras $D(2,1;\alpha)$, $G(3)$ and $F(4)$ from $U(-1,-1)$-balanced Freudenthal–Kantor triple systems in a natural way.

AMS 2000 Mathematics subject classification: Primary 17A40; 17B60

In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties.