153 research outputs found

### A Characterization of Convex Functions

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

### Invariance of Ideal Limit Points

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on
the positive integers and let $(x_n)$ be a sequence taking values in a metric
space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is
an $F_\sigma$-set [resp., a closet set]. Let us assume that $X$ is also
separable and the ideal $\mathcal{I}$ 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
$(x_n)$ 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

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

### Convergence Rates of Subseries

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

### A note on primes in certain residue classes

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$
such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic
density relative to the set of all primes which is at least $\prod_{i=1}^k
\left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ 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 $n$ such that $n \not\equiv 0 \bmod a_i$
for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k
\left(1-\frac{1}{a_i}\right)$

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