Δελτίο της Ελληνικής Μαθηματικής Εταιρίας: Τεύχος 45

This article describes the q-numerical range of matrix polynomials. This is defined for a matrix polynomial P(lambda) = A_m lambda^m+cdots +A_1lambda+A_0 where A is a square matrix and lambda is complex as W_q(P) = {lambda in C:x^*P(lambda)y=0, with x^*x=t^*y=1 and x^*y=0} for any q in [0,1]. The elementary properties of W_q(P) are presented as well as some applications of the q-numeric range to the factorization of matrix polynomials.

This article studies two-sided Banach algebras. Some properties of general commutative algebras are presented, followed by an overview of two-sided algebras. In the second part of the article, the Banach case is studied and some results are presented, namely that they are commutative modulo the Jacobson radical.

This article shows the random equivalent of the Edelstein theorem for Banach fixed points. The main theorem proven is:If f:Omega times X right arrow X is a contractive random mapping and if for each w in Omega, an orbit O_w(xi) has a cluster point then f has a unique random fixed point.'' The last part of the article deals with a generalization of this theorem to random multivalued maps.

This paper describes an exterior mixed boundary problem for the Helmholtz equation in three dimensions. The first two sections describe the statement and the formulation of the problem. The authors investigate an exterior boundary value problem for a domain consisting of N-bodies. the unique solvability of the problem is discussed in section 4, followed by the proof of the existence of the solution in section 5. The article concludes with a summary of future applications of this problem (section 6).

This article considers the far-field equations for a rigid body, the cavity and the transmission case in two-dimensional linear elasticity. The scattering problem is presented in its differential and integral forms. The authors derive a pair of integral equations of the first kind for the far-field region, which hold regardless of the boundary conditions. The assumptions for these equations are that the incident field is produced by a superposition of plane waves in all directions and polarizations. Finally the authors present some conditions for solvability using properties of the Hergoltz functions.

This article discusses properties of the dual Euler totient function defined for n =ge 1 and n integer by a(n) = n =i(n) where i is the Euler (totient) function. Some interesting properties are proven about this function, such as that every positive rational number can be written as a ratio a(m)/a(n) for suitable m,n. Also the length of increasing or decreasing sub sequences of a(n) is studied. In the last parts of the paper a necessary and sufficient condition for a(n) to divide n is given and some results regarding the divisibility of a(n) by n-1.

This short article extends the work known in the field of transcendence of numbers whose continued fraction expansion contains enough repetitions.

This article considers the Hecke Algebra H of endomorphisms of the Gel'fand-Graev module of GL_n(q) (the finite general linear group consisting of all invertible nxn matrices with entries in the finite field of q elements). The basis elements of the above can be expressed using certain monomial matrices, N_. The authors provide a reduced form of the matrices in N_ W using simple reflections of the Weyl group W. The article concludes with an application of the results to the action of G on the cuspidal modules.

This article deals with semi-completeness of the permutational wreath product W of two groups A and B. Some necessary and sufficient condition s are known; the author however considers the case where A and B are finite but A is not abelian, for which some interesting results are obtained.

The authors present some sufficient conditions for the controllability of neutral functional differential inclusions in Banach spaces. The controllability is studied on semi-infinite intervals. The second part of the article considers, in the same lines, the controllability of neutral functional integrodifferential inclusions. The paper concludes with a short example.

This short article presents an overview of the Hans Lewy operator L. This operator is useful in proving the existence of first order linear partial differential equations with C^infty coefficients which possess no smooth solution in any neighbourhood. A geometric interpretation is presented in the sense that L(p), bar L(p) and [L,L](p) form a vector basis for the tangent vector space T_p(S^3-{q}), where S^3-{q} is the unit sphere with one point q removed.

This article considers some properties of locally multiplicative pseudo-convex algebras E, which are endowed with a vector involuion x -> x* such that (xy)*=x*y* for every element x, y in E. The authors show that if such an algebra is Hermitian and a Q-algebra, then it is also commutative modulo the Jacobson radical.

This article considers the motion of a charged particle moving under the Lorentz force in the plane of motion of two parallel rotating magnetic dipoles, in the case where the particle lies in the region behind the two magnetic sources. The families of FI and FIII of symmetric and periodic orbits of the particles are studied, and the stability of the orbits is also investigated (section 3). Finally the paper concludes with some numerical results for various initial values of the parameters.