Regularization is a method used in mathematics and statistics to deal with insufficient information. The reader must supply additional information in the form of an operator. From where is this operator to come, and what does it mean? It amounts to us, as practitioners, specifying a “style” of model. Where the model is a signal or an image, it amounts to specifying one weighting function in phy...

We study the inversion of weighted Radon transforms in two dimensions,Rρf (L) = ∫ L f (·) ρ(L, ·) ds, where the weight function ρ(L, x), L a line and x ∈ L, has a special form. It was an important breakthrough when R.G. Novikov recently gave an explicit formula for the inverse of Rρ when ρ has the form (1.2); in this case Rρ is called the attenuated Radon transform. Here we prove similar result...

This paper studies the problem of allocating utility losses among n agents with cardinal non-comparable utility functions. This problem is referred to as the Nash rationing problem, as it can be regarded as the translation of the Nash bargaining problem to a rationing scenario. We show that there is no single-valued solution satisfying the obvious reformulation of Nash’s axioms, nor a multivalu...

A Euclidean t-design, as introduced by Neumaier and Seidel (1988), is a finite set X ⊂ Rn with a weight function w : X → R+ for which

We consider the partition of a finite Coxeter group W into left cells with respect to a weight function L. In the equal parameter case, Lusztig has shown that the representations carried by the left cells are precisely the so-called constructible ones. We show that this holds for general L, if the conjectural properties (P1)–(P15) in Lusztig’s book on Hecke algebras with unequal parameters hold...

We are interested in the tail behavior of the randomly weighted sum ∑n i=1 θiXi, in which the primary random variables X1, . . . , Xn are real valued, independent and subexponentially distributed, while the random weights θ1, . . . , θn are nonnegative and arbitrarily dependent, but independent of X1, . . . , Xn. For various important cases, we prove that the tail probability of ∑n i=1 θiXi is ...

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), a...

Abstract. The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler’s formula was o...

*Correspondence: [email protected] Department of Mathematics, Inner Mongolia University, Hohhot, 010021, China Abstract The present paper deals with the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. The eigenvalues depend not only continuously but also smoothly on the problem. Some new differential expressions of eigenvalues with respect to an endpoi...