Abstract Let $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ A ⊂ B ( H ) be a row contraction and $\Phi _{\mathcal {A}}$ Φ determined by {A}$ completely positive map on ${\mathcal . ...

‎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...

Abstract Let $\mathcal {X}$ X be an infinite-dimensional complex Banach space, {B}(\mathcal {X})$ B ( ) the algebra of all bounded linear operators on . Denote spectral domain by $\sigma _{\gamma}(T)=\{\lambda \in \sigma _{a}(T): T { \text{ that is semi-Fredh...

Previous computer-assisted problem-solving systems have incorporated all the problem-solving steps within a single stage, making it difficult to diagnose stages at which errors occurred when a student encounters difficulties, and imposing a too-high cognitive load on students in their problem solving. This study proposes a computer-assisted system named MathCAL, whose design is based on four pr...

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota the type \[ \|A f\|_{\mathcal{X}}^2 \leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in dom\big(A^2\big), \] recall that under exceedingly stronger hypotheses on operator $A$ and/or Banach space $\mathcal{X}$, optimal constant $C$ in these diminishes from $4$ (e.g., w...

‎Let ${X_{alpha}:alphainLambda}$ be a collection of topological spaces‎, ‎and $mathcal {G}_{alpha}$ be a grill on $X_{alpha}$ for each $alphainLambda$‎. ‎We consider Tychonoffrq{}s type Theorem for $X=prod_{alphainLambda}X_{alpha}$ via the above grills and a natural grill on $X$ ‎related to these grills, and present a simple proof to this theorem‎. ‎This immediately yields the classical theorem...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,topological space), in order to obtain Egoroff and Lusin type theorems forset multifunctions in the Vietoris hypertopology. Some mathematicalapplications are established and several physical implications of thema...

We discover a new minimality property of the absolute minimizers supremal functionals, whose variational problems are also known as $L^\infty$ problems. In particular for every minimizer $v$ quasi-convex functional $$ {\rm ess.sup}\,{\Omega}H(x,Dv(x)), we consider set \mathcal{A}(v)=\{x: H (x,Dv(x))={\rm ess.sup}\,{\Omega}H(x,Dv(x))\}, suitably defined. If $u$ is an minimizer, give structure re...

Letting $A \subset \mathbb{R}^n$ be Borel measurable and $W_0 : A \to \mathbb{G}(n,m)$ Lipschitzian, we establish that \begin{equation*} \limsup_{r 0^+} \frac{\mathcal{H}^m \left[ \cap B(x,r) (x+ W_0(x))\right]}{\alpha(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for $\mathcal{L}^n$-almost every $x \in A$. In particular, it follows $A$ is $\mathcal{L}^n$-negligible if only $\mathcal{H}^m(A (x+W_0...

In this paper we show that the only sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ satisfying $$\begin{aligned} \phi (x){\mathcal {D}}_q P_{n}(x)=a_n{\mathcal {S}}_q P_{n+1}(x) +b_n{\mathcal P_n(x) +c_n{\mathcal P_{n-1}(x), \end{aligned}$$ ( $$c_n\ne 0$$ ) where $$\phi $$ is a well chosen polynomial degree at most two, $${\mathcal {D}}_q$$ Askey-Wilson operator and {S}}_q$$ averaging op...