arXiv subjects

Mohammad H. M. Rashid

Publications and source records attributed to Mohammad H. M. Rashid.

3 recordsLinked to original sources

arXiv2025

New Properties and Refined Bounds for the $q$-Numerical Range

This paper investigates new properties of $q$-numerical ranges for compact normal operators and establishes refined upper bounds for the $q$-numerical radius of Hilbert space operators. We first prove that for a compact normal operator $T$ with $0 \in W_q(T)$, the $q$-numerical range $W_q(T)$ is a closed convex set containing the origin in its interior. We then explore the behavior of $q$-numerical ranges under complex symmetry, deriving inclusion relations between $W_q(T)$ and $W_q(T^*)$ for complex symmetric operators. For hyponormal operators similar to their adjoints, we provide conditions under which $T$ is self-adjoint and $W_q(T)$ is a real interval. We also study the continuity of $q$-numerical ranges under norm convergence and examine the effect of the Aluthge transform on $W_q(T)$. In the second part, we derive several new and sharp upper bounds for the $q$-numerical radius, incorporating the operator norm, numerical radius, transcendental radius, and the infimum of $\|Tx\|$ over the unit sphere. These bounds unify and improve upon existing results in the literature, offering a comprehensive framework for estimating $q$-numerical radii across the entire parameter range $q \in [0,1]$. Each result is illustrated with detailed examples and comparisons with prior work.

math.FA↗

arXiv2025

Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations

This paper introduces and studies a class of Weyl-type algebras $A_{p,t,\cA} = \Weyl{e^{\pm x^{p} e^{t x}},\; e^{\cA x},\; x^{\cA}}$ constructed over exponential-polynomial rings, where $\FF$ is a field of characteristic zero, $\cA$ is a finitely generated additive subgroup of $\FF$, and $p \in \mathbb{N}^n$, $t \in \FF$. We investigate their structural properties, proving simplicity, establishing faithful infinite-dimensional irreducible representations, and demonstrating the failure of the Noetherian property. A natural filtration by exponential order is introduced, with the associated graded algebra shown to be commutative. We also examine the corresponding Witt-type Lie algebra $\mathfrak{g}_{p,t,\cA} = \Der_{\mathrm{gr}}(R_{p,t,\cA})$ and prove the vanishing of its second cohomology group with adjoint coefficients, implying rigidity under formal deformations. Furthermore, we construct explicit deformation quantizations of the underlying exponential-polynomial rings, compute Hochschild and cyclic homology groups, and relate them to the topology of the parameter space. The deformation rigidity of $A_{p,t,\cA}$ is classified in terms of the rank of $\cA$, and a Gerstenhaber algebra structure on the Hochschild cohomology is described. Several open problems concerning representation classification and geometric realization are proposed.

math.RA↗

arXiv2025

The $Λ$-Set and Its Role in Local Controllability and Necessary Conditions for Free-Time Optimal Control

This paper establishes a unified framework connecting local controllability, necessary conditions for optimality, and attainability in free-time optimal control problems. The central object of our investigation is the $Λ$-set, which governs the relationship between original control systems and their convexifications. Our main results demonstrate that emptiness of the $Λ$-set implies local controllability and guarantees the existence of minimizing sequences achieving the target in reduced time. We derive strengthened necessary conditions for time-optimal control and provide explicit constructive procedures for approximating generalized controls by ordinary trajectories. These results resolve longstanding questions about relaxation phenomena while extending classical theory to address modern challenges in non-convex optimization, establishing foundations for higher-order analysis in free-time problems.

math.OC↗