In this paper we consider two recovery problems based on information given with an error. First is the problem of optimal class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\in \ell_q\,:\,\|h\|_q\leqslant 1\}$, where $1\le q < \infty$ and $t_1\geqslant t_2\geqslant \ldots \geqslant 0$, in space $\ell_q$ when capacity inexact know either first $n\in\mathbb{N}$ elements a sequence error measured finite seque...

Given integers k ≥ 2 $k\ge 2$ and a 1 , … ${a}_{1},\ldots ,{a}_{k}\ge 1$ let ≔ ( ) ${\boldsymbol{a}}:= ({a}_{1},\ldots ,{a}_{k})$ n + ⋯ $n:= {a}_{1}+\cdots \,\,+{a}_{k}$ . An a-multiset permutation is string of length $n$ that contains exactly i ${a}_{i}$ symbols $i$ for each = $i=1,\ldots ,k$ In this work we consider the problem exhaustively generating all ${\boldsymbol{a}}$ -multiset permutat...

Abstract Fix a poset Q on $\{x_1,\ldots ,x_n\}$ . A -Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is whose monomials are closed under the Borel-like moves induced by I principal ideal, denoted $I=Q(m)$ , if there m such that all minimal generators of can be obtained via from In this paper we study powers ideals. Among our results, show $Q(m)$ agree with their symbolic powers,...

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. $H$ a $k$-graph with vertex set $[n]$, $e(H)$ denote the number of edges $H$. $\nu(H)$ $\tau(H)$ size largest matching minimum cover in $H$, respectively. Define $A^k_i(n,s):=\{e\in\binom{[n]}{k}:|e\cap[(s+1)i-1]|\geq i\}$ for $2\leq i\leq k$ $HM^k_{n,s}:=\big\{e\in\binom{[n]}{k}:e\cap[s-1]\neq...

Given a fixed k ∈ Z and λ ∈ Z, the objective of a λ-L(k, k − 1, . . . , 2, 1)-labeling of a graph G is to assign nonnegative integers (known as labels) from the set {0, . . . , λ − 1} to the vertices of G such that the adjacent vertices receive values which differ by at least k, vertices connected by a path of length two receive values which differ by at least k− 1, and so on. The vertices whic...

A permutation $\pi$ over alphabet $\Sigma = {1,2,3,\ldots,n}$, is a sequence where every element $x$ in $\Sigma$ occurs exactly once. $S_n$ the symmetric group consisting of all permutations length $n$ defined $\Sigma$. $I_n$ $(1, 2, 3,\ldots, n)$ and $R_n =(n, n-1, n-2,\ldots, 1)$ are identity (i.e. sorted) reverse respectively. An operation, that we call as an $LRE$ has been OEIS with A186752...

‎let $f$ be a proper $k$-coloring of a connected graph $g$ and‎ ‎$pi=(v_1,v_2,ldots,v_k)$ be an ordered partition of $v(g)$ into‎ ‎the resulting color classes‎. ‎for a vertex $v$ of $g$‎, ‎the color‎ ‎code of $v$ with respect to $pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_pi}(v)=(d(v,v_1),d(v,v_2),ldots,d(v,v_k))$‎, ‎where $d(v,v_i)=min{d(v,x):~xin v_i}‎, ‎1leq ileq k$‎. ‎if‎ ‎distinct...

Let $f$ be a positive multiplicative function and let $k\geq 2$ an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in sense $\sum_{p}|f(p)-1|=\infty$, there exists real number $c>0$ such $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense hypercube $[0,c]^k$ or $[c,+\infty)^k$. In particular, $f(p+1),\ldots,f(p+k)$ can put any increasin...

Abstract For positive integers $K$ and $L$, we introduce study the notion of $K$-multiplicative dependence over algebraic closure ${\overline{{\mathbb{F}}}}_p$ a finite prime field ${\mathbb{F}}_p$, as well $L$-linear points on elliptic curves in reduction modulo primes. One our main results shows that, given non-zero rational functions $\varphi _1,\ldots ,\varphi _m, \varrho ,\varrho _n\in{\ma...