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Vol.1 No.1
Abstract
In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)
Abstract
Classically, valuation theory is closely related to the theory of divisors and conversely. If D is a Dedekined ring and K is its quotient field, then we can clearly construct the theory of divisors on D (or K), and then we can induce all the valuations on K ([3]). In particular, if K is a number field and A is the ring of algebraic integers, then since Z is Dedekind, A is a Dedekind rign and K is the field of fractions of A.(omitted)
Abstract
The study of the integral of the scalar curvature, <TEX>$A(g)\;=\;{\int}_M\;RdV_9$</TEX> as a functional on the set M of all Riemannian metrics of the same total volume on a compact orient able manifold M is now classical, dating back to Hilbert [6] (see also Nagano [8]). Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric.(omitted)
Abstract
Let P be a probability measure on the real line with Lebesque-density f. The usual estimator of the distribution function (≡df) of P for the sample <TEX>$\chi$</TEX><TEX>$_1$</TEX>,…, <TEX>$\chi$</TEX><TEX>$\_$</TEX>n/ is the empirical df: F<TEX>$\_$</TEX>n/(t)=(equation omitted). But this estimator does not take into account the smoothness of F, that is, the existence of a density f. Therefore, one should expect that an estimator which is better adapted to this situation beats the empirical df with respect to a reasonable measure of performance.(omitted)
Abstract
Suppose that X is a Banach space with continuous dual <TEX>$X^{**}$</TEX>, (<TEX>$\Omega$</TEX>, <TEX>$\Sigma$</TEX>, <TEX>${\mu}$</TEX>) is a finite measure space. f : <TEX>$\Omega\;{\longrightarrow}$</TEX> <TEX>$X^{*}$</TEX> is a weakly measurable function such that <TEX>$\chi$</TEX><TEX>$^{**}$</TEX> f <TEX>$\in$</TEX> <TEX>$L_1$</TEX>(<TEX>${\mu}$</TEX>) for each <TEX>$\chi$</TEX><TEX>$^{**}$</TEX> <TEX>$\in$</TEX> <TEX>$X^{**}$</TEX> and <TEX>$T_{f}$</TEX> : <TEX>$X^{**}$</TEX> \longrightarrow <TEX>$L_1$</TEX>(<TEX>${\mu}$</TEX>) is the operator defined by <TEX>$T_{f}$</TEX>(<TEX>$\chi$</TEX><TEX>$^{**}$</TEX>) = <TEX>$\chi$</TEX><TEX>$^{**}$</TEX>f. In this paper we study the properties of bounded <TEX>$X^{*}$</TEX> - valued weakly measurable functions and bounded <TEX>$X^{*}$</TEX> - valued weak＊ measurable functions.(omitted)
Abstract
There are various aspects of the solution of boundary-value problems for second-order linear elliptic equations in two independent variables. One useful method of solving such boundary-value problems for Laplace's equation is by means of suitable integral representations of solutions and these representations are obtained most directly in terms of particular singular solutions, termed Green's functions.(omitted)