相关论文: On $\mathcal I(<q)$- and $\mathcal I(\leq q)$-conv…

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries…

The so called $q$-triplets were conjectured in 2004 and then found in nature in 2005. A relevant further step was achieved in 2005 when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of…

We study the best approximation problem: \[ \displaystyle \min_{\alpha\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf x}_i) \right|. \] Here: $\Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\}$ is a list of…

Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb P}^{s-1}$, over the field $\mathbb{F}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of…

Full-sampling (e.g., Q-learning) and pure-expectation (e.g., Expected Sarsa) algorithms are efficient and frequently used techniques in reinforcement learning. Q$(\sigma,\lambda)$ is the first approach unifies them with eligibility trace…

In this paper we have studied on $\mathcal{I}$-density function using the notion of $\mathcal{I}$-density, introduced by Banerjee and Debnath \cite{banerjee 4} where $\mathcal{I}$ is an ideal of subsets of the set of natural numbers. We…

This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean…

Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < \lambda(A) < 1 < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, or the other way…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

We call a positive real number $\lambda$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $\alpha$ such that $\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of $\alpha$ -…

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…

For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c…

Let $\mathcal{I}$ be an ideal on $\omega$. For $f,g\in\omega^\omega$ we write $f \leq_{\mathcal{I}} g$ if $f(n) \leq g(n)$ for all $n\in\omega\setminus A$ with some $A\in\mathcal{I}$. Moreover, we denote…

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties…

For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ for which there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q…

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given…

Let $1<q<2$ and \[ \Lambda(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $\Lambda(q)$ is dense in $\mathbb{R}$. We give several…

This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly…

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the…

Let $I$ and $J$ be two intervals, and let $f, g: I \rightarrow \mathbb{R}$. If for any points $a$ and $b$ in $I$ and any positive numbers $p$ and $q$ such that $p + q = 1$, we have \begin{align} \nonumber p f(a) + q f(b) + g(pa + qb) \in J,…