Weyl Type Research Articles

Unified fractional integral and derivative formulas, integral transforms of incomplete $$\tau $$-hypergeometric function

The objective of this article is to evaluate unified fractional integrals and derivative formulas involving the incomplete $$\tau $$ -hypergeometric function. These integrals and derivatives are further applied in proving theorems on Marichev–Saigo–Maeda operators of fractional integration and differentiation. The results are expressed in terms of the generalized Gauss hypergeometric functions (Fox–Wright function). Corresponding assertions in terms of Saigo, Erdelyi–Kober, Riemann–Liouville, and Weyl type of fractional integrals and derivatives are presented. Also, we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

Anomalous Hall effect in the half-metallic Heusler compound Co2TiX ( X=Si , Ge)

Though Weyl fermions have recently been observed in several materials with broken inversion symmetry, there are very few examples of such systems with broken time reversal symmetry. Various Co$_{2}$-based half-metallic ferromagnetic Heusler compounds are lately predicted to host Weyl type excitations in their band structure. These magnetic Heusler compounds with broken time reversal symmetry are expected to show a large momentum space Berry curvature, which introduces several exotic magneto-transport properties. In this report, we present systematic analysis of experimental results on anomalous Hall effect (AHE) in Co$_2$Ti$X$ ($X$=Si and Ge). This study is an attempt to understand the role of Berry curvature on AHE in Co$_2$Ti$X$ family of materials. The anomalous Hall resistivity is observed to scale quadratically with the longitudinal resistivity for both the compounds. The detailed analysis indicates that in anomalous Hall conductivity, the intrinsic Karplus-Luttinger Berry phase mechanism dominates over the extrinsic skew scattering and side-jump mechanism.

Strichartz estimates for the Schrödinger flow on compact Lie groups

We establish scale-invariant Strichartz estimates for the Schr\"odinger flow on any compact Lie group equipped with canonical rational metrics. In particular, full Strichartz estimates without loss for some non-rectangular tori are given. The highlights of this paper include estimates for some Weyl type sums defined on rational lattices, different decompositions of the Schr\"odinger kernel that accommodate different positions of the variable inside the maximal torus relative to the cell walls, and an application of the BGG-Demazure operators or Harish-Chandra's integral formula to the estimate of the difference between characters.

On the spectral mapping theorem for Weyl spectra of Toeplitz operators

In this paper we study the approximate point spectrum and the surjectivity spectrum of a Toeplitz operators $$T_\phi$$ , defined on Hardy spaces $$H^2(\mathbf{T })$$ of the unit circle $$\mathbf{T }$$ of $${\mathbb {C}}$$ , where the symbol $$\phi \in L^\infty (\mathbf{T })$$ , or $$\phi$$ is a continuous symbol. We also extend some results obtained by Farenick and Lee (Trans Am Math Soc 348(10):4153–4174, 1996). In particular, we study Weyl type theorems and the spectral mapping theorem for the Weyl spectra.

Weyl type f(Q,\xa0T) gravity, and its cosmological implications

We consider an f(Q, T) type gravity model in which the scalar non-metricity Q_{alpha mu nu } of the space-time is expressed in its standard Weyl form, and it is fully determined by a vector field w_{mu }. The field equations of the theory are obtained under the assumption of the vanishing of the total scalar curvature, a condition which is added into the gravitational action via a Lagrange multiplier. The gravitational field equations are obtained from a variational principle, and they explicitly depend on the scalar nonmetricity and on the Lagrange multiplier. The covariant divergence of the matter energy-momentum tensor is also determined, and it follows that the nonmetricity-matter coupling leads to the nonconservation of the energy and momentum. The energy and momentum balance equations are explicitly calculated, and the expressions of the energy source term and of the extra force are found. We investigate the cosmological implications of the theory, and we obtain the cosmological evolution equations for a flat, homogeneous and isotropic geometry, which generalize the Friedmann equations of standard general relativity. We consider several cosmological models by imposing some simple functional forms of the function f(Q, T), and we compare the predictions of the theory with the standard Lambda CDM model.

Cheng–Shen conjecture in Finsler geometry

The well-known Cheng–Shen conjecture says that every [Formula: see text]-quadratic Randers metric on a closed manifold is a Berwald metric. The class of [Formula: see text]-quadratic Randers metrics contains the class of generalized Douglas–Weyl Randers metrics. In this paper, we give a classification of left-invariant Randers metrics of generalized Douglas–Weyl type on three-dimensional Lie groups. Based on our classification theorem, we find a counter-example for the Cheng–Shen conjecture.

On nth roots of normal operators

For n-normal operators A [2,4,5], equivalently n-th roots A of normal Hilbert space operators, both A and A* satisfy the Bishop-Eschmeier-Putinar property (?)?, A is decomposable and the quasinilpotent part H0(A-?) of A satisfies H0(A-?=)-1(0) = (A-?)-1(0) for every non-zero complex ?. A satisfies every Weyl and Browder type theorem, and a sufficient condition for A to be normal is that either A is dominant or A is a class A(1,1) operator.

Semiclassical Sampling and Discretization of Certain Linear Inverse Problems

We study sampling of functions $f$ and their images $Af$ under Fourier integral operators $A$ at rates $sh$ with $s$ fixed and $h$ a small parameter. We show that the Nyquist sampling limit of $Af$ and $f$ are related by the canonical relation of $A$ using semiclassical analysis. We apply this analysis to the Radon transform in the parallel and the fan-beam coordinates. We explain and illustrate the optimal sampling rates for $Af$, the aliasing artifacts, and the effect of averaging (blurring) the data $Af$. We prove a Weyl type of estimate on the minimal number of sampling points to recover $f$ stably in terms of the volume of its semiclassical wave front set.

Operator matrices and their Weyl type theorems

We denote the collection of the 2 x 2 operator matrices with (1,2)-entries having closed range by S. In this paper, we study the relations between the operator matrices in the class S and their component operators in terms of the Drazin spectrum and left Drazin spectrum, respectively. As some application of them, we investigate how the generalized Weyl?s theorem and the generalized a-Weyl?s theorem hold for operator matrices in S, respectively. In addition, we provide a simple example about an operator matrix in S satisfying such Weyl type theorems.

Spectral properties for some extensions of isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, \(S^{*}\) are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\). Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point \(\lambda \) of a 2-isometric operator T is self-adjoint and satisfies \({\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}\). Further, we show that quasi-2-isometric operator satisfies Bishop’s property (\(\beta \)). Finally, we prove Weyl type theorems for \(f(d_{TS})\), where \( d_{TS}\) denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and \(S^{*}\) and \(f \in H(\sigma (d_{TS}))\), the set of analytic functions which are defined on an open neighborhood of \(\sigma (d_{TS})\).

Scattering of acoustic waves from a point source over an impedance wedge

In this work we study diffraction of a spherical acoustic wave due to a point source, by an impedance wedge In the exterior of the wedge the acoustic pressure satisfies the stationary wave (Helmholtz) equation and classical impedance boundary conditions on two faces of the wedge, as well as Meixner’s condition at the edge and the radiation conditions at infinity. Solution of the boundary value problem is represented by a Weyl type integral and its asymptotic behavior is discussed. On this way, we derive various components in the far field interpreting them accordingly and discussing their physical meaning.

English

English

Property $$(UW_\Pi )$$ and Perturbations

Property \((UW_\Pi )\) holds for a bounded linear operator \(T \in B(X),\) defined on a complex infinite dimensional Banach space X, if the poles of the resolvent of T are exactly the spectral points \(\lambda \) for which \(\lambda I-T\) is upper semi-Weyl. In this paper, we discuss the relationship between property \((UW_\Pi )\) and other Weyl type theorems. The stability of property \((UW_\Pi )\) is also studied under nilpotent, quasi-nilpotent, finite-dimensional or compact perturbations commuting with T.

A Study on Generalized Multivariable Mittag-Leffler Function via Generalized Fractional Calculus Operators

We study some properties of generalized multivariable Mittag-Leffler function. Also we establish two theorems, which give the images of this function under the generalized fractional integral operators involving Fox’s H-function as kernel. Relating affirmations in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some known special cases have also been mentioned in the concluding section.

Polaroid operators and Weyl type theorems

Let [Formula: see text] be a [Formula: see text]-quasiposinormal operator on a complex Hilbert space [Formula: see text]. In this paper, we give basic properties for [Formula: see text] and we show that a [Formula: see text]-quasiposinormal operator [Formula: see text] is polaroid. We also prove that all Weyl type theorems (generalized or not) hold and are equivalent for [Formula: see text], where [Formula: see text] is an analytic function defined on a neighborhood of [Formula: see text].

Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

Properties (BR) and (BgR) for bounded linear operators

In this paper we introduce the notion of property (BR) and property (BgR) for bounded linear operators defined on an infinite-dimensional Banach space. These properties in connection with Weyl type theorems and in the frame of polaroid operators are investigated. Moreover, we study the stability of these properties under perturbations by commuting finite-dimensional, quasi-nilpotent, Riesz and algebraic operators.

Einstein-Maxwell fields with vanishing higher-order corrections

We obtain a full characterization of Einstein-Maxwell $p$-form solutions $(\boldsymbol{g},\boldsymbol{F})$ in $D$-dimensions for which all higher-order corrections vanish identically. These thus simultaneously solve a large class of Lagrangian theories including both modified gravities and (possibly non-minimally coupled) modified electrodynamics. Specifically, both $\boldsymbol{g}$ and $\boldsymbol{F}$ are fields with vanishing scalar invariants and further satisfy two simple tensorial conditions. They describe a family of gravitational and electromagnetic plane-fronted waves of the Kundt class and of Weyl type III (or more special). The local form of $(\boldsymbol{g},\boldsymbol{F})$ and a few examples are also provided.

Almost universal spacetimes in higher-order gravity theories

We study almost universal spacetimes - spacetimes for which the field equations of any generalized gravity with the Lagrangian constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order reduce to one single differential equation and one algebraic condition for the Ricci scalar. We prove that all d-dimensional Kundt spacetimes of Weyl type III and traceless Ricci type N are almost universal. Explicit examples of Weyl type II almost universal Kundt metrics are also given. The considerable simplification of the field equations of higher-order gravity theories for almost universal spacetimes is then employed to study new Weyl type II, III, and N vacuum solutions to quadratic gravity in arbitrary dimension and six-dimensional conformal gravity. Necessary conditions for almost universal metrics are also studied.

Constructing Bach flat manifolds of signature (2, 2) using the modified Riemannian extension

We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate this phenomena in the context of homogeneous affine surfaces.