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in subring of R and thus by hypothesis there exists a right identity for S, say e, = ~ a~ x~. i=1 Then xes=x, and if we set P~(t)= ~a~t ~-1 we have x2P~(x)=x. i=1 For each associator d=(u, v, w)ER there exists a polynomial Pe(t) such that d=d2Pa(d) and Slater's theorem implies that R is associative. At this point we could apply the result of Fuchs and Szele to finish; however we take a somewhat different route by next applying Herstein's theorem to see that R is commutative. Any ideal of R is of the form Re for some idempotent e, since it has an identity as a subring. Thus if 1 is the identity of R, R=Re| --e) and R is completely reducible and hence a finite direct sum of fields F 1 .... , F,. No F~ is of characteristic

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In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

Let I be the Stanley–Reisner ideal of a simplicial complex Δ. In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I(r) in the case Δ is a union of disjoint matroids, where I(r) is the rth symbolic power of I. As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127–140, 2014) for the k-Buchsbaumness of S/I(r).

My proof of Kellogg's theorem for solutions of Plateau's problem (these Inventiones math. 8, 3 1 3 333 (1969)) can still be shortened: The relation ~p(e ia) ~'(x(eia))= 0 from p. 328 is available in almost all boundary points as soon as it is known that the functions fj(w) belong to class C o, l(p). The proof of this fact is accomplished on the bottom of p. 324. Let now w = 1 be the point described in the middle ofp. 325. From

We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.

It was shown by F. Wilhelm that Gromov’s filling radius of any positively curved closed Riemannian manifolds are less than that of the round sphere unless they are isometric to each other. In this short paper, we adapt his proof to see that the same is true for any positively curved closed Alexandrov spaces as well.

This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23–27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with Chaichian, Tureanu, Sun, Wang, Xie and Zhang.

In this short paper, we study the existence of common universal series for uncountable families of specific linear operators. In particular we deal with some derived forms of Seleznev’s theorem and we obtain common universal elements in the space of formal power series in several complex variables.

In an unpublished paper, Araque, Hall and Magnanti considered polyhedra associated with the Capacitated Vehicle Routing Problem (CVRP) in the special case of unit demands. Among the valid and facet-inducing inequalities presented in that paper were the so-called multistar and partial multistar inequalities, each of which came in several versions. Some related inequalities for the case of general demands have appeared subsequently and the result is a rather bewildering array of apparently different classes of inequalities. The main goal of the present paper is to present two relatively simple procedures that can be used to show the validity of all known (and some new) multistar and partial multistar inequalities, in both the unit and general demand cases. The procedures provide a unifying explanation of the inequalities and, perhaps more importantly, ideas that can be exploited in a cutting plane algorithm for the CVRP. Computational results show that the new inequalities can be useful as cutting planes for certain CVRP instances.

In this short paper we prove that even though the latest transitive operator has only one negative entry, we cannot get rid of it by simple operations. The research is motivated by, still unsolved, invariant subspace problem for positive operators.

Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/Op(G)| < bcl(G) in the case where P is abelian.

In this short paper, we study Peck’s method in positive characteristic. Curiously enough, the simultaneous approximations this method provides are better than in the real field.