"Let $\mathcal{S}^*_e$ and $\mathcal{S}^*_B$ denote the class of analytic functions $f$ in open unit disc normalized by $f(0)=0=f'(0)-1$ satisfying, respectively, following subordination relations: $$ \frac{zf'(z)}{f(z)}\prec e^z\quad{\rm and}\quad\frac{zf'(z)}{f(z)}\prec e^{e^z-1}.$$ In this article, we investigate majorization problems for classes without acting upon any linear or nonlinear o...

In this article, we study the space $\mathcal B_\mu(B_X,Y)$ of $Y$-valued Bloch-type functions on unit ball $B_X$ an infinite dimensional Hilbert $X$ with $\mu$ is a normal weight and $Y$ Banach space. We also investigate characterizations $\mathcal{WB}_\mu(B_X)$ $Y$-valued, locally bounded, weakly holomorphic associated B_\mu(B_X)$ scalar-valued in sense that $f\in \mathcal{WB}_\mu(B_X)$ if $w...

‎We define hyperbolic harmonic $omega$-$alpha$-Bloch space‎ ‎$mathcal{B}_omega^alpha$ in the unit ball $mathbb{B}$ of ${mathbb R}^n$ and‎ ‎characterize it in terms of‎ ‎$$frac{omegabig((1-|x|^2)^{beta}(1-|y|^2)^{alpha-beta}big)|f(x)-f(y)|}{[x,y]^gamma|x-y|^{1-gamma}‎},$$ where $0leq gammaleq 1$‎. ‎Similar results are extended to‎ ‎little $omega$-$alpha$-Bloch and Besov spaces‎. ‎These obtained‎...

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...