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We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well., Comment: A goofy mistake in formulations of Theorem 2 and 4 is fixed

The objective of the paper Aczél and Ostrowski (1973) is to show in an elementary way that any real-valued function f, defined on (0,1) and satisfying he inquality , where n ≧ 3 is fixed, is necessary of the form .

The aim of this short note is to discuss Szisz's paper [1] on rings with almost nil adjoint semigroups and to make some remarks about it. His results should be read as a characterization of rings with almost nit, rather than almost nil adjoint semigroups as stated. All his propositions could be simplified considerably once the word adjoint is dropped. Towards this we state the following result appearing in JACOBSO~ [2].

SynopsisWe give a direct and simple proof of a central result of the above paper.

The result indicated in the title of this paper (Ref. Zh. Mat. 1976, 7A225) is false. The error is corrected.

B. R. Srinivasan has defined a weakly inverse semigroup in his paper "Weakly Inverse Semigroups" El], using the following terminology. An idempotent e in a semigroup S is called a principal idempotent of S if f e f = f e for all idempotents f in S. An element a in S is called a principal element of S if there exists an inverse a' of a such that aa' is a principal idempotent. Let Ea denote the set of principal inverses of the element a (the inverses of a which are principal elements of S). Then a semigroup S is called weakly inverse if for each element a in S, the set Ea is not null, and for an), elements a, b in S

In a recent paper on subreflexive normed linear spaces (see Arch. Math. 8, 444--450 (1957)) and its correction (see Arch. Math. 9,439--440, (1958)) R. R. PHELPS proved the following theorem.

In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

In this short paper, we show that there does not exist a noncompact Type-I [Formula: see text]-solution of the Ricci flow with positive curvature in dimension 3.

In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.

This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions., Comment: 4 pages

Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.

In the above-named paper (this JOURNAL, vol. 77 (1955), pp. 895-913), Theorem 6. 4 should have been stated only for rings of zero characteristic: the argument for the case of prime characteristic breaks down in the last formula line on p. 911. This involves jettisoning Theorem 6. 5 (the implied "proof " of which depended on applying Theorem 6. 4 to homomorphs which cannot be guaranteed to have zero characteristic, even when the given ring has). However, the writer has no evidence that Theorem 6. 4 as stated or Theorem 6. 5 is actually false. And in any case, since the valid part of the proof of Theorem 6. 4 establishes the weaker conclusion [Xm, ynp?] = 0 without characteristic hypothesis, the remarks following Theorem 6. 4 still hold good, provided that we modify the final parenthetical clause so as to read "which would at any rate imply that every weak K-ring R with kc, m, n satisfying (a) or (b) has R/J commutative."

Professor Oystein Ore of Yale University has kindly called my attention to the close relation between my paper and some earlier researches of Dedekind. I should like to correlate, as far as possible, my results with his.