The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total...

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addit...

In the present investigation, inspired by work on Yamaguchi type class of analytic functions satisfyingthe criteria $\mathfrak{Re}\{\frac{f (z)}{z}\} > 0, $ in openunit disk $\Delta=\{z \in \mathbb{C}\colon |z|<1\}$ and making use S\v{a}l\v{a}gean-difference operator, which is a special Dunkl operator with constant $\vartheta$ $\Delta$ , wedesignate definite new classes $\mathcal{R}_{\lam...

Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Rapid Communications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Report may be no longer than four printed pages and must be accompanied by an abstract. The same publication schedule as for regular articles is fol...

Abstract We study the Li–Yau inequality for heat equation corresponding to Dunkl harmonic oscillator, which is a nonlocal Schrödinger operator parameterized by reflections and multiplicity functions. In particular case when reflection group isomorphic ℤ 2 d {\mathbb{Z}...

algebraic frames are generalizations of fourier transforms on locally compact abelian groups in the sense that the family of vectors forming the frame are replaced by a family of unbounded linear functionals. the paper studies the indexing measure space of the algebraic frames; as the investigation narrows down to the class of lower semiframes, more relations are revealed between the discret...

suppose $t$ and $s$ are moore-penrose invertible operators betweenhilbert c*-module. some necessary and sufficient conditions are given for thereverse order law $(ts)^{ dag} =s^{ dag} t^{ dag}$ to hold.in particular, we show that the equality holds if and only if $ran(t^{*}ts) subseteq ran(s)$ and $ran(ss^{*}t^{*}) subseteq ran(t^{*}),$ which was studied first by greville [{it siam rev. 8 (1966...