We count algebraic points of bounded height and degree on the graphs certain functions analytic unit disk, obtaining a bound which is polynomial in logarithm multiplicative height. combine this work with $p$-adic methods to obtain, for each positive $\varepsilon$, an upper form $cD^{3n/4 + \varepsilon n}$ number irreducible factors $P^{\circ n}(X)-P^{\circ n}(\alpha)$ over $K$, where $K$ field,...

An expander code is a binary linear whose parity-check matrix the bi-adjacency of bipartite graph. We provide new formula for minimum distance such codes. also proof result that $2(1-\varepsilon) \gamma n$ lower bound given by an $(m,n,d,\gamma,1-\varepsilon)$

Milne-like spacetimes are a class of hyperbolic FLRW which admit continuous spacetime extensions through the big bang, $\tau = 0$. The existence extension follows from writing metric in conformal Minkowskian coordinates and assuming that scale factor satisfies $a(\tau) \tau + o(\tau^{1+\varepsilon})$ as \to 0$ for some $\varepsilon > This asymptotic assumption implies o(\tau)$. In this paper, w...

Abstract We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula number integers $n=p_1p_2 \leq X$ such $n+h$ is a product exactly two which holds all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$ , under restriction size one factors n and . Additionally, we co...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of mean curvature flow in $${\mathbb {R}}^{n+1}$$ all $$n\ge 3$$ : show that if a $$\{M_t\}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such $$M_t\cap B(x_0,\varepsilon )$$ is $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$...

in this paper, we introduce the concepts of right, left and product stabilizers on hoops and study some properties and the relation between them.  and we try to find that how they can be equal and investigate that under what condition they can be filter, implicative filter, fantastic and positive implicative filter. also, we prove that  right and product stabilizers are filters and if they are ...

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit one-parameter family of continuous maps $\mathbf{f}_{\varepsilon} \colon \mathbb{S}^{d-1} \to K$ which, for $\varepsilon > 0$, are defined on the sphere have pro...

In this work we apply the unfolding operator method to analyze asymptotic behavior of solutions $p$-Laplacian equation with Neumann boundary condition set in a bounded thin domain type $R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon g\left({x}/{\varepsilon^\alpha}\right)\right\rbrace$ where $g$ is positive periodic function. We study three cases $0<\alpha<1$...