153 research outputs found

    A Characterization of Convex Functions

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    Let DD be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function f:DR{+}f: D\to \mathbf{R}\cup \{+\infty\} is convex if and only if for all x,yDx,y \in D there exists α=α(x,y)(0,1)\alpha=\alpha(x,y) \in (0,1) such that f(αx+(1α)y)αf(x)+(1α)f(y)f(\alpha x+(1-\alpha)y) \le \alpha f(x)+(1-\alpha)f(y)

    Invariance of Ideal Limit Points

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    Let I\mathcal{I} be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let (xn)(x_n) be a sequence taking values in a metric space XX. First, it is shown that the set of ideal limit points of (xn)(x_n) is an FσF_\sigma-set [resp., a closet set]. Let us assume that XX is also separable and the ideal I\mathcal{I} satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of (xn)(x_n) is equal to the set of ideal limit points of almost all its subsequences.Comment: 11 pages, no figures, to appear in Topology App

    Characterizations of the Ideal Core

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    Given an ideal I\mathcal{I} on ω\omega and a sequence xx in a topological vector space, we let the I\mathcal{I}-core of xx be the least closed convex set containing {xn:nI}\{x_n: n \notin I\} for all III \in \mathcal{I}. We show two characterizations of the I\mathcal{I}-core. This implies that the I\mathcal{I}-core of a bounded sequence in Rk\mathbf{R}^k is simply the convex hull of its I\mathcal{I}-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and ee-convergence of double sequences.Comment: 10 pages, to appear in Journal of Mathematical Analysis and Application

    Convergence Rates of Subseries

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    Let (xn)(x_n) be a positive real sequence decreasing to 00 such that the series nxn\sum_n x_n is divergent and lim infnxn+1/xn>1/2\liminf_{n} x_{n+1}/x_n>1/2. We show that there exists a constant θ(0,1)\theta \in (0,1) such that, for each >0\ell>0, there is a subsequence (xnk)(x_{n_k}) for which kxnk=\sum_k x_{n_k}=\ell and xnk=O(θk)x_{n_k}=O(\theta^k).Comment: 5 pp. To appear in The American Mathematical Monthl

    A note on primes in certain residue classes

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    Given positive integers a1,,aka_1,\ldots,a_k, we prove that the set of primes pp such that p≢1modaip \not\equiv 1 \bmod{a_i} for i=1,,ki=1,\ldots,k admits asymptotic density relative to the set of all primes which is at least i=1k(11φ(ai))\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right), where φ\varphi is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer nn such that n≢0modain \not\equiv 0 \bmod a_i for i=1,,ki=1,\ldots,k admits asymptotic density which is at least i=1k(11ai)\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)
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