This paper is concerned with the following Euler-Lagrange system \[ \begin{cases} \frac{d}{dt} \mathcal{L}_v(t,u(t),\dot{u}(t)) = \mathcal{L}_x(t,u(t),\dot{u}(t)) \quad \textrm{for a.e. $t \in [-T,T]$}, \\ u(-T) u(T), \mathcal{L}_v(-T,u(-T),\dot{u}(-T)) \mathcal{L}_v(T,u(T),\dot{u}(T)), \end{cases} \] where Lagrangian given by $\mathcal{L} F(t,x,v) + V(t,x) \langle f(t), x \rangle$, growth cond...

The aim of this paper is to define an $L^p$ space intuitionistic fuzzy observables. We work in $({\mathcal F}, {\bf m})$ with product, where $\mathcal F$ a family events and ${\bf m}$ state. prove that the corresponding pseudometric $\rho_{IF}$ space.

Let $G$ be a split connected reductive group over $\mathbb{Z}$. $F$ non-archimedean local field. With $K_m: = Ker(G(\mathfrak{O}_F) \rightarrow G(\mathfrak{O}_F/\mathfrak{p}_F^m))$, Kazhdan proved that for field $F'$sufficiently close to $F$, the Hecke algebras $\mathcal{H}(G(F),K_m)$ and $\mathcal{H}(G(F'),K_m')$ are isomorphic, where $K_m'$ denotes corresponding object $F'$. In this article, ...

Let $\mathbb{F}_q[x]$ be the ring of polynomials over a finite field $\mathbb{F}_q$ and $\mathbb{F}_q(x)$ its quotient field. $\mathbb{P}$ set primes in $\mathbb{F}_q[x]$, let $\mathcal{I}$ all $f$ for which $f(\mathbb{P})\subseteq\mathbb{F}_q[x]$. The existence basis is established using notion characteristic ideal; this shows that free $\mathbb{F}_q[x]$-module. Through localization, explicit ...

We give a unified approach to handle the problem of extending functions with values in locally convex Hausdorff space $E$ over field $\mathbb{K}$, which have weak extensions $\mathcal{F}(\Omega,\mathbb{K})$ scalar-valued on set $\Omega$, vector-valued counterpart $\mathcal{F}(\Omega,E)$ $\mathcal{F}(\Omega,\mathbb{K})$. The results obtained are based upon representation as linear continuous ope...

Let K be a discretely valued field with ring of integers $$\mathcal {O}_K$$ perfect residue field. K(x) the rational function in one variable. $${\mathbb {P}}^1_{\mathcal {O}_K}$$ standard smooth model {P}}^1_K$$ coordinate x. $$f(x) \in \mathcal {O}_K[x]$$ squarefree polynomial corresponding divisor zeroes $${{\,\mathrm{div}\,}}_0(f)$$ on . We give an explicit description minimal embedded reso...

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms both width and depth simultaneously. To that end, we first prove multivariate polynomials can be approximated by ReLU $\mathcal{O}(N)$ $\mathcal{O}(L)$ with an $\mathcal{O}(N^{-L})$. Through local Taylor expansions their network approximati...

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on unit circle, when $\chi_1\in \mathcal{A}_1,\dots, \chi_m\in \mathcal{A}_m$ run through arbitrary sets nontrivial multiplicative characters $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1...

Let $mathcal {A}$ be an abelian category with enough projective objects and $mathcal {X}$ be a full subcategory of $mathcal {A}$. We define Gorenstein projective objects with respect to $mathcal {X}$ and $mathcal{Y}_{mathcal{X}}$, respectively, where $mathcal{Y}_{mathcal{X}}$=${ Yin Ch(mathcal {A})| Y$ is acyclic and $Z_{n}Yinmathcal{X}}$. We point out that under certain hypotheses, these two G...