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Transformations of measures via their generalized densities
No full textIn this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range
The principal series of p-adic groups with disconnected centre
Get PDFLet G be a split connected reductive group over a local non-archimedean field.
We classify all irreducible complex G-representations in the principal series,
irrespective of the (dis)connectedness of the centre of G. This leads to a local
Langlands correspondence for principal series representations, which satisfies all expected properties. We also prove that the ABPS conjecture about the geometric structure of Bernstein components is valid throughout the principal series of G
Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices
Get PDFTo approximate f(A)b---the action of a matrix function on a vector---by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating f on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contains important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices A and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guarantees convergence for the same classes of functions and any restart length if A is not necessarily Hermitian but positive real
An Algorithm for the Matrix Lambert W Function
Get PDFAn algorithm is proposed for computing primary matrix Lambert functions of a square matrix , which are solutions of the matrix equation . The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert function in a numerically reliable way
The principal angles and the gap
No full textIn this note we provide proofs for some known results on the principal angles and the gap between two subspaces of . Both the principal angles and the gap are introduced with respect to an arbitrary positive definite inner product. We show that the principal angles between two subspaces and are unique and prove that the largest one, , satisfies and
when
Twists of rational Cherednik algebras
No full textThe main result of the paper is that braided Cherednik algebras introduced by the first two authors are cocycle twists of rational Cherednik algebras.
This gives a new construction of mystic reflection groups and a new proof that such groups have Artin-Schelter regular rings of quantum polynomial invariants.
Furthermore, the main result leads to a construction of
finite-dimensional representations of braided Cherednik
algebras.
In this first version of the paper, we give a full proof of the main result and sketch the application to representations of braided Cherednik algebras
Free centre-by-nilpotent-by-abelian Lie rings of rank 2
Get PDFWe study the free Lie ring of rank in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length . This is the quotient with where is the free Lie ring of rank , is the -th term of the lower central series of the derived ideal of , and is the third term of the derived series of . We show that the quotient is a direct sum of a free abelian group and a torsion group of exponent . We exhibit an explicit generating set for the torsion subgroup
Three-Dimensional Transient Electromagnetic Modeling Using Rational Krylov Methods
Get PDFA computational method is given for solving the forward modeling problem for transient electromagnetic exploration. Its key features are discretization of the quasi-static Maxwell's equations in space using the first-kind family of curl-conforming Nédélec elements combined with time integration using rational Krylov subspace methods. We show how rational Krylov subspace methods may be used to solve the same problem in the frequency domain followed by a synthesis of the transient solution using the fast Hankel transform, arguing that the pure time-domain is more efficient. We also propose a simple method for selecting the pole parameters of the rational Krylov subspace method which leads to convergence within an a priori determined number of iterations independent of mesh size and conductivity structure. These poles are repeated in a cyclic fashion, which, in combination with direct solvers for the discrete problem, results in significantly faster solution times than previously proposed schemes
Point vortices on the hyperbolic plane
Get PDFWe investigate the dynamical system of point vortices on the hyperboloid. This system has noncompact
symmetry SL(2,R) and a coadjoint equivariant momentum map. The relative equilibrium
conditions are found and the trajectories of relative equilibria with non-zero momentum
value are described. We also provide the classification of relative equilibria and the stability criteria
for a number of cases, focusing on 2 and 3 vortices. Unlike the systemon the sphere, this system
has relative equilibria with non-compact momentum isotropy subgroup, and these are used to illustrate
the different stability types of relative equilibria
Convergence rate of extremes of generalized Maxwell distribution
No full textLet be a sequence of independent and identically distributed random variables with common
distribution following the generalized Maxwell distribution. In this paper, we obtain the exact uniform convergence rate of the distribution of the maximum to its extreme value distribution