Let $$f_1(n), \ldots , f_k(n)$$ be polynomial functions of n. For fixed $$n\in \mathbb {N}$$ let $$S_n\subseteq the numerical semigroup generated by $$f_1(n),\ldots ,f_k(n)$$ . As n varies, we show that many invariants $$S_n$$ are eventually quasi-polynomial in n, most notably Betti numbers, but also type, genus, and size $$\Delta $$ -set. The tool use is expressibility logical system parametri...

In this paper, based on Serre’s p-adic family of Eisenstein series, we prove a general congruences for series $$G_k$$ in the form $$\begin{aligned} \sum _{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)(mod\,p^N), \end{aligned}$$ where $$f_1(t),\ldots ,f_n(t)\in {\mathbb Z}[t]$$ are non-constant integer polynomials with positive leading coefficients and $$g_0(t),\ldots ,g_n(t)\in Q}(t)$$ rational function...

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite of \alpha-lattice points for fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring algebraic integers \mathbb{K}. These arise from equations form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), F is rational function, g r are polynomials \mathbb{K}...

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of stable manifold zero solution. In particular, for $n$ dimensional case, solution globally asymptotically if and only this $n.$ Fixed $n,$ let $X$ be random variable that assigns to each system index, $p_k$ with $k=0,1,\ldots,n,$ denote probabilities $P(X=k)$. paper we obtain eith...

Abstract The paper contains the study of weighted $$L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\rightarrow L^p$$ L p 1 × 2 … m → ...

We propose the Lie-algebraic interpretation of poly-analytic functions in $$L_2({{\mathbb {C}}},d\mu )$$ , with Gaussian measure $$d\mu $$ based on a flag structure formed by representation spaces $$\mathfrak {sl}(2)$$ -algebra realized differential operators z and $${\bar{z}}$$ . Following pattern one-dimensional situation, we define poly-Fock d complex variables way, as invariant for action g...

Let $r>0, n\in\mathbb N, {\bf k}\in\mathbb N$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\ldots,x(t-d_{{\bf k}}(Lx_t))) for $g:( \mathbb R^n)^{{\bf k}}\supset V\to\mathbb R^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\mathbb R^n)$ into finite-dimensional vector-space $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, and $x_t(s)=x(t...

We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set $$s_1, \ldots , s_n \in {\mathbb {R}}^p$$ with corresponding responses $$t_1,\ldots ,t_n {R}}^q$$ fitting k-layer network $$\nu _\theta : {R}}^p \rightarrow involves estimation of weights $$\theta {R}}^m$$ via an ERM: $$\begin{aligned} \inf _{\theta {R}}^m} \ \sum _{i=...

Edmonds, Lov\'asz, and Pulleyblank showed that if a matching covered graph has nontrivial tight cut, then it also ELP-cut. Carvalho et al. gave stronger conjecture: cut $C$, ELP-cut does not cross $C$. Chen, al proof of the conjecture. This note is inspired by paper We give simplified conjecture, prove following result which slightly than $C$ $G$ an ELP-cut, there sequence $G_1=G, G_2,\ldots,G_...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...