A study to deduce some fundamental properties of power series.

This note extends a certain combinatorics problem of I. Tomescu.

The purpose of this paper is to develop a general method for using Finite Elements in the Steepest Descent Method. The main application is to a partial differential equation for a Transonic Flow Problem. It is also applied to Burger's equation, Laplace's equation and the minimal surface equation. The entire method is tested by computer runs which give satisfactory results. The validity of certain of the procedures used are proved theoretically. The way that the writer handles finite elements is quite different from traditional finite element methods. The variational principle is not needed. The theory is based upon the calculation of a matrix representation of operators in the gradient of a certain functional. Systematic use is made of local interpolation functions.

This article discusses mixed Grassmann manifolds. Abstract: Se construieşte varietea grassmanniană modelată intr-un spaţiu Banach mixt, situaţie ce generalizează simultan conceptele grassmanniene real şi complex.

Every continuous convex function defined on a separable Banach space is Gateaux differentiable on a dense G^ subset of the space E [Mazur]. Suppose we are given a sequence (xn) that Is dense in E. Can we always find a Gateaux differentiable point x such that x = z^=^anxn.for some sequence (an) with infinitely many non-zero terms so that Ση∞=1||anxn|| < co ? According to this paper, such points are called of "simple type," and shown to be dense in E. Mazur's theorem follows directly from the result and Rybakov's theorem (A countably additive vector measure F: E -* X on a cr-field is absolutely continuous with respect to |x*F] for some x* e Xs) can be shown without deep measure theoretic Involvement.

A set-theoretical point of view to study algebraic numbers has been introduced. We extend a result of Navarro-Bermudez concerning shift invariant measures in the Cantor space which are topologically equivalent to shift invariant measures which correspond to some algebraic integers. It is known that any transcendental numbers and rational numbers in the unit interval are not binomial. We proved that there are algebraic numbers of degree greater than two so that they are binomial numbers. Algebraic integers of degree 2 are proved not to be binomial numbers. A few compositive relations having to do with algebraic numbers on the unit interval have been studied; for instance, rationally related, integrally related, binomially related, B1-related relations. A formula between binomial numbers and binomial coefficients has been stated. A generalized algebraic equation related to topologically equivalent measures has also been stated.

Article discussing the Levi problem on mixed manifolds. This article gives a new proof that the problem of Levi, to mixed varieties have a solution in the sense that every pseudoconvex closely mixed variety, with enough real global morphisms, is a variety of Cartan.

The method of dual steepest descent is used to solve ordinary differential equations with nonlinear boundary conditions. A general boundary condition is B(u) = 0 where where B is a continuous functional on the nth order Sobolev space Hn[0.1J. If F:HnCO,l] —• L2[0,1] represents a 2 differential equation, define *(u) = 1/2 IIF < u) li and £(u) = 1/2 l!B(u)ll2. Steepest descent is applied to the functional 2 £ a * + £. Two special cases are considered. If f:lR —• R is C^(2), a Type I boundary condition is defined by B(u) = f(u(0),u(1)). Given K: [0,1}xR—•and g: [0,1] —• R of bounded variation, a Type II boundary condition is B(u) = ƒ1/0K(x,u(x))dg(x).

Article proposing a new direct proof of the fact that L^1 verifies the strong maximum principle, i.e., any analytic map from the complex unit disk into L^1, constant in norm, must be constant.

Article on an abstract index theorem on non-compact Riemannian manifolds.

This article characterizes the infinitesimal generators of semigroups of linear transformations.

This article studies special infinite iterated function systems derived from complex continued fraction expansions with restricted entries.

This article considers solutions to the wave equation in 3+1 spacetime dimensions whose data is compactly supported at some initial time.

Article on the constructibility with ruler and compass of a minimum chord in a parabola. The author gives a constructive description of all the points inside a parabola admitting a minimum chord of fixed direction.

This article gives results of numerical calculations of asymptotic behavior of critical points of a Ginzburg-Landau functional.

Article on SL₂-polynomial invariance. The purpose of this article is to describe completely, in a constructive way, the structure of these invariant polynomials in the particular case n = 2, for an arbitrary field k.

This article discusses divergences of a sequence of functions and its set of subsequences.

Article discussing Bär's conformal lower bound for the spectrum of generalized Dirac operators.

This article discusses the first Vafa-Witten bound for two-dimensional tori.

Article generalizing a certain geometric locus problem due to A. Dafina.

This article demonstrates a relationship between the Dunford-Pettis property and the DP1 property with a similar relationship for vector-valued continuous function spaces.

Article on Morley's miracle theorem. The authors use work by A. Connes to explore possible generalizations of Morley's trisector theorem to triangles in arbitrary value fields.

This article investigates an efficient way of evaluating sums and series, based on a result of Abel applied to associated power series.

Article addressing, from a historical perspective, Newton's sums of like powers of natural numbers.