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For a prime p and an absolutely irreducible modulo p polynomial ƒ(U, V) L 1\U, V] we obtain an asymptotic formula for the number of solutions to the congruence ƒ(x,y) = a (modp) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a...
A categorical generalization of the notion of movability from inverse systems and shape theory was given by the first author who defined the notion of movable category and used it to interpret the movability of topological spaces. In this paper the authors define the notion of uniformly movable category and prove that a topological space is uniformly movable in the sense of shape theory if and only...
Suppose that K is a CW-complex, X is an inverse sequence of stratifiable spaces, and X = lim X. Using the concept of semi-sequence, we provide a necessary and sufficient condition for X to be an absolute co-extensor for K in terms of the inverse sequence X and without recourse to any specific properties of its limit. To say that X is an absolute co-extensor for K is the same as saying that K is an...
On each nonreflexive Banach space X there exists a positive continuous convex function ƒ such that 1/ƒ is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of...
We investigate the subadditivity property also known as the tensorization property of (φ-entropy functionals and their iterations. In particular we show that the only iterated φ-entropies with the tensorization property are iterated variances. This is a complement to the result due to Latala and Oleszkiewicz on characterization of the standard φ-entropies with the tensorization property.
We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in R^k. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions...
Suppose ƒ = (ƒn), g = (gn) are martingales with respect to the same filtration, satisfying |ƒn-ƒn-i| ≤ |gn -gn-1|, n = 1,2,..., with probability 1. Under some assumptions on ƒo, go and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of ƒ and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic...
We prove that if some power of a space X is rectiflable, then X[sup]πω(x) is rectifiable. It follows that no power of the Sorgenfrey line is a topological group and this answers a question of Arhangeliskiî. We also show that in Mal'tsev spaces of point-countable type, character and π-character coincide.
Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Holder spaces illustrates the abstract result.
We prove that generating relations between the elements [r] = r² - r of a commutative ring are the following: [r + s] = [r] + [s] + rs[2] and [rs] = r²[s] + s[r].
Working in the framework of reverse mathematics, we consider representations of reals as rapidly converging Cauchy sequences, decimal expansions, and two sorts of Dedekind cuts. Converting single reals from one representation to another can always be carried out in RCA[sub]0. However, the conversion process is not always uniform. Converting infinite sequences of reals in some representations to other...
In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".
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