Remarks on the rightmost critical value of the triple product L-function

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.

Complex band structure with non-orthogonal basis set: analytical properties and implementation in the SIESTA code

The complex band structure, although not directly observable, determines many properties of a material where the periodicity is broken, such at surfaces, interfaces and defects. Furthermore, its knowledge helps in the interpretation of electronic transport calculations and in the study of topological materials. Here we extend the transfer matrix method, often used to compute the complex bands, to electronic structures constructed using an atomic non-orthogonal basis set. We demonstrate that when the overlap matrix is not the identity, the non-orthogonal case, spurious features appear in the analytic continuation of the band structure to the complex plane. The properties of these are studied both numerically and analytically and discussed in the context of existing literature. Finally, a numerical implementation to extract the complex band structure from periodic calculations carried out with the density functional theory code SIESTA is presented. This is constructed as a simple post-processing tool, and it is therefore amenable to high-throughput studies of insulators and semiconductors.

Elliptic Ruijsenaars Difference Operators on Bounded Partitions

By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials.

A novel matrix technique for multi-order pantograph differential equations of fractional order

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted by S n ( x ) and called special polynomials , were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class.

The quarter median

We introduce and discuss a multivariate version of the classical median that is based on an equipartition property with respect to quarter spaces. These arise as pairwise intersections of the half-spaces associated with the coordinate hyperplanes of an orthogonal basis. We obtain results on existence, equivariance, and asymptotic normality.