16,840 research outputs found
Quantum Free Yang-Mills on the Plane
We construct a free-probability quantum Yang-Mills theory on the two
dimensional plane, determine the Wilson loop expectation values, and show that
this theory is the limit of U(N) quantum Yang-Mills theory on the
plane.Comment: 24 pages, tikz figure
Upper bounds for packings of spheres of several radii
We give theorems that can be used to upper bound the densities of packings of
different spherical caps in the unit sphere and of translates of different
convex bodies in Euclidean space. These theorems extend the linear programming
bounds for packings of spherical caps and of convex bodies through the use of
semidefinite programming. We perform explicit computations, obtaining new
bounds for packings of spherical caps of two different sizes and for binary
sphere packings. We also slightly improve bounds for the classical problem of
packing identical spheres.Comment: 31 page
Mathematical optimization for packing problems
During the last few years several new results on packing problems were
obtained using a blend of tools from semidefinite optimization, polynomial
optimization, and harmonic analysis. We survey some of these results and the
techniques involved, concentrating on geometric packing problems such as the
sphere-packing problem or the problem of packing regular tetrahedra in R^3.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates
and correction
Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices
We begin with the study of some properties of the radial Dunkl process
associated to a reduced root system . It is shown that this diffusion is the
unique strong solution for all of a SDE with singular drift. Then,
we study , the first hitting time of the positive Weyl chamber : we prove,
via stochastic calculus, a result already obtained by Chybiryakov on the
finiteness of . The second and new part deals with the law of for
which we compute the tail distribution, as well as some insight via stochastic
calculus on how root systems are connected with eigenvalues of standard
matrix-valued processes. This gives rise to the so-called -processes.
The ultraspherical -Jacobi case still involves a reduced root system
while the general case is closely connected to a non reduced one. This process
lives in a convex bounded domain known as principal Weyl alcove and the strong
uniqueness result remains valid. The last part deals with the first hitting
time of the alcove's boundary and the semi group density which enables us to
answer some open questions.Comment: 33 page
Efficient Algorithm for Asymptotics-Based Configuration-Interaction Methods and Electronic Structure of Transition Metal Atoms
Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B.
D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods
for atoms which reproduce, at fixed finite subspace dimension, the exact
Schr\"odinger eigenstates in the limit of fixed electron number and large
nuclear charge. Here we develop, implement, and apply to 3d transition metal
atoms an efficient and accurate algorithm for asymptotics-based CI.
Efficiency gains come from exact (symbolic) decomposition of the CI space
into irreducible symmetry subspaces at essentially linear computational cost in
the number of radial subshells with fixed angular momentum, use of reduced
density matrices in order to avoid having to store wavefunctions, and use of
Slater-type orbitals (STO's). The required Coulomb integrals for STO's are
evaluated in closed form, with the help of Hankel matrices, Fourier analysis,
and residue calculus.
Applications to 3d transition metal atoms are in good agreement with
experimental data. In particular we reproduce the anomalous magnetic moment and
orbital filling of Chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.Comment: 14 pages, 1 figur
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