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 }}...

We consider the homogenization of compressible Navier-Stokes-Fourier equations in a randomly perforated domain $\mathbb{R}^3$. Assuming that particle size scales like $\varepsilon^\alpha$, where $\varepsilon>0$ is their mutual distance and $\alpha>3$, we show limit $\varepsilon\to 0$, velocity, density, temperature converge to solution same system. follow methods Lu Pokorn\'{y} [https://doi.org...

in this paper,  we provide two different kinds of fixed pointtheorems in  fuzzy metric spaces. the first kind is for the fuzzy$varepsilon$-contractive type mappings and  the second kind is forthe fuzzy order $psi$-contractive type mappings. they improve thecorresponding  conclusions in the literature.

In this paper, we determine the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of bilinear Hilbert transform $H_{\gamma}(f,g)$ along a convex curve $\gamma$ $$H_{\gamma}(f,g)(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x-t)g(x-\gamma(t)) \,\frac{\textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, $r>\frac{1}{2}$, $p>1$, $q>1$...

in this paper, a certain new connectedness of l-fuzzy subsets inl-topological spaces is introduced and studied by means of preclosed sets. itpreserves some fundamental properties of connected set in general topology.especially the famous k. fan’s theorem holds.

In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with doubling measure, and the Newton--Sobolev uniform domain $X_\varepsilon$. This is obtained as uniformization (Gromov) hyperbolic filling $Z$. To do so, construct family fillings in style work Bonk Kleiner Bourdon Pajot. Then for each parameter $\beta>0$ lift $\mu_\beta$ measure $\nu...