Antisymmetric Orbit Functions

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.

Method of good matrices for multi-dimensional numerical integrations—An extension of the method of good lattice points

Stable solutions of [formula omitted] on [formula omitted

Under certain conditions, we obtain global pointwise estimates for Neumann series associated with an integral operator with general quasimetric kernel. The estimates involve the first and second iterations of the original kernel. As a consequence we deduce sharp bilateral bounds of Green's function for the fractional Schrödinger operator (− Δ)α/2 −q with general attractive potential q ≥ 0 on the entire Euclidean space \(\mathbb{R}^n\) for 0 < α < n, or a bounded nontangentially accessible domain \(\Omega \subseteq \mathbb{R}^n \) for 0 < α ≤ 2, under a certain “smallness condition” on q. Most of our results are new even in the classical case α = 2.KeywordsBounded DomainMeasure SpaceLipschitz DomainEuclidean BallDoubling MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsShif BerhanuSHIF BERHANU graduated with a B.Sc. degree from Addis Ababa University, Ethiopia, in 1980. In 1987 he received his Ph.D. from Rutgers University under the direction of Francois Trevès. He is a professor of mathematics at Temple University. Berhanu's main area of research involves complex vector fields, that is, firstorder partial differential operators with complex-valued coefficients. Currently, he, P. Cordaro, and J. Hounie are collaborating to write a book on the subject of complex vector fields.Ahmed MohammedAHMED MOHAMMED received his B.Sc. in 1983 and M.Sc. in 1986 from Addis Ababa University, Ethiopia. He obtained his Ph.D. in 1995 from Temple University under the guidance of Professor Cristian E. Gutierrez. Between 1995 and 2000, he taught in the Department of Mathematics and Computer Science at Kuwait University, Kuwait. He held a visiting position at Temple University from 2000 to 2002. Currently he is an associate professor in the Department of Mathematical Sciences at Ball State University. If he is not doing mathematics in his office or teaching a class, chances are you'll find him at the Starbucks in the university book store.

Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac–Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type Ã2 on a Euclidean space V≅ℝ2 to study the Bruhat order on W. We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of Boe and Graham (which is a conjectural simplification of the Carrell–Peterson criterion for rational smoothness) to type Ã2. Computational evidence suggests that the only Schubert varieties in type Ã2 where the "nontrivial" case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety X(w) of type Ã2. The proof uses descriptions we obtain of the elements x≤w and of the rationally smooth locus of X(w) in terms of the W-action on V. As a consequence we describe the maximal nonrationally smooth points of X(w). The results of this paper are used in a sequel to describe the smooth locus of X(w), which is different from the rationally smooth locus.

This paper deals with uncertainty propagation of general stochastic hybrid systems (GSHS) where the continuous state space is a compact Lie group. A computational framework is proposed to solve the Fokker-Planck (FP) equation that describes the time evolution of the probability density function for the state of GSHS. The FP equation is split into two parts: the partial differential operator corresponding to the continuous dynamics, and the integral operator arising from the discrete dynamics. These two parts are solved alternatively using the operator splitting technique. Specifically, the partial differential equation is solved by the spectral method where the density function is decomposed into a linear combination of a complete orthonormal function basis brought forth by the Peter-Weyl theorem, thereby resulting an ordinary differential equation. Next, the integral equation is solved by approximating the integral by a finite summation using a quadrature rule. The proposed method is then applied to a three-dimensional rigid body pendulum colliding with a wall, evolving on the product of the three-dimensional special orthogonal group and the Euclidean space. It is illustrated that the proposed method exhibits numerical results consistent with a Monte Carlo simulation, while explicitly generating the density function that carries the complete stochastic information of the hybrid state.

An asymptotic cell category for cyclic groups

We find the fundamental solution to the P P -Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. A solution to the 2-Laplace equation in a wider class of spaces is presented.

Boundary identification for a 3D Laplace equation using a genetic algorithm

Effacement des dérivations et spectres premiers des algèbres quantiques

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

This monograph is devoted to the development of the theory of pseudo-di?erential n operators on spaces with symmetries. Such spaces are the Euclidean space R ,the n torus T , compact Lie groups and co

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.