Abstract Let $\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\varepsilon>0$ we call a set $\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$ $\varepsilon$ -approximate progression if for some and d , $|x_i-(a+id)|<\varepsilon d$ holds all $i\in\{0,1\ldots,k-1\}$ . Complementing earlier results of Dumitrescu (2011, J. Comput. Geom. 2 (1) 16–29), in this paper ...

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has $1$. On the other hand, for any $\varepsilon>0$, show there are $>1-\varepsilon$ which $\mathcal{A}\cdot\mathcal{A}$ $<\varepsilon$. This answers questions Hegyv\'{a}ri, Hennecart Pach.

The existence of fixed point in orthogonal metric spaces has been initiated by Eshaghi and et. al [7]. In this paper, we prove existence and uniqueness theorem of fixed point for mappings on $varepsilon$-connected orthogonal metric space. As a consequence of this, we obtain the existence and uniqueness of fixed point for analytic function of one complex variable. The paper concludes with some i...

We consider a blow-up phenomenon for \begin{document}$ { \partial_t^2 u_ \varepsilon} $\end{document} id="M2">\begin{document}$ {- \varepsilon^2 \partial_x^2u_ \varepsilon } id="M3">\begin{document}$ = F(\partial_t \varepsilon)}. The derivative of the solution id="M4">\begin{document}$ \partial_t blows-up...

In this paper, we recover the boundary null controllability for degenerate heat equation by analyzing asymptotic behavior of an eligible family state-control pairs $((u_{\varepsilon}, h_{\varepsilon}))_{\varepsilon >0}$ solving corresponding singularly perturbed internal problems. As in other situations studied literature, our approach relies on Carleman estimates and meticulous weak convergenc...

We prove that the number of fractions $$h_1/h_2$$ integers $$h_1,h_2$$ a subset $$\mathcal {A}\subset \mathcal {H}\cap [1,X]$$ is at least $$ \alpha X/(\log X)^{3/2}$$ , where {H}$$ set $$p-1$$ p being prime such $$p+1$$ sum two coprime squares. So, this $$\gg _\varepsilon ^{1+\varepsilon }|\mathcal {A}|^2$$ $$\varepsilon any positive real number. take opportunity to describe geometrical view s...

Abstract Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of weakly nonlinear Dirac equation (NLDE), where nonlinearity strength is characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We adopt second-order Strang splitting method to discretize NLDE in time, and combine with Fourier pseudospectral space full-discretization....

We study boundary properties of one class periodic functions as $\varepsilon \rightarrow 0$, where $\varepsilon$ is a period periodically perforated domain. show that their weak limit the homothetic mean value such functions.

We show that for any finite set A and an arbitrary $$\varepsilon >0$$ there exists $$k=k(\varepsilon )$$ such the higher energy $$\textsf{E} _k(A)$$ is at most $$|A|^{k+\varepsilon }$$ unless has a very specific structure. As application we obtain subset of real numbers or prime field either contains additive Sidon-type size $$|A|^{1/2+c}$$ multiplicative .

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms local geometry given constraint set $K$ . Our main result shows that risk notation="LaTeX">$\vphantom {_{\int _{\int }}...