1,652,223 research outputs found
Analytic structure of solutions to multiconfiguration equations
We study the regularity at the positions of the (fixed) nuclei of solutions
to (non-relativistic) multiconfiguration equations (including Hartree--Fock) of
Coulomb systems. We prove the following: Let {phi_1,...,phi_M} be any solution
to the rank--M multiconfiguration equations for a molecule with L fixed nuclei
at R_1,...,R_L in R^3. Then, for any j in {1,...,M} and k in {1,...,L}, there
exists a neighbourhood U_{j,k} in R^3 of R_k, and functions phi^{(1)}_{j,k},
phi^{(2)}_{j,k}, real analytic in U_{j,k}, such that phi_j(x) =
phi^{(1)}_{j,k}(x) + |x - R_k| phi^{(2)}_{j,k}(x), x in U_{j,k} A similar
result holds for the corresponding electron density. The proof uses the
Kustaanheimo--Stiefel transformation, as applied earlier by the authors to the
study of the eigenfunctions of the Schr"odinger operator of atoms and molecules
near two-particle coalescence points.Comment: 15 page
Distance-two labelings of digraphs
For positive integers , an -labeling of a digraph is a
function from into the set of nonnegative integers such that
if is adjacent to in and if
is of distant two to in . Elements of the image of are called
labels. The -labeling problem is to determine the
-number of a digraph , which
is the minimum of the maximum label used in an -labeling of . This
paper studies - numbers of digraphs. In particular, we
determine - numbers of digraphs whose longest dipath is of
length at most 2, and -numbers of ditrees having dipaths
of length 4. We also give bounds for -numbers of bipartite
digraphs whose longest dipath is of length 3. Finally, we present a linear-time
algorithm for determining -numbers of ditrees whose
longest dipath is of length 3.Comment: 12 pages; presented in SIAM Coference on Discrete Mathematics, June
13-16, 2004, Loews Vanderbilt Plaza Hotel, Nashville, TN, US
Integrality of Homfly (1,1)-tangle invariants
Given an invariant J(K) of a knot K, the corresponding (1,1)-tangle invariant
J'(K)=J(K)/J(U) is defined as the quotient of J(K) by its value J(U) on the
unknot U.
We prove here that J' is always an integer 2-variable Laurent polynomial when
J is the Homfly satellite invariant determined by decorating K with any
eigenvector of the meridian map in the Homfly skein of the annulus.
Specialisation of the 2-variable polynomials for suitable choices of
eigenvector shows that the (1,1)-tangle irreducible quantum sl(N) invariants of
K are integer 1-variable Laurent polynomials.Comment: 10 pages, including several interspersed figure
The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry
This report is concerned with the union of all possible spectra that may emerge when perturbing a large Toeplitz band matrix in the site by a number randomly chosen from some set . The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of as . Also discussed are the cases of small and large sets as well as the "discontinuity of the infinite volume case", which means that in general does not converge to something close to as , where is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. \ud
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The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M1241
Mathematical Theory of Exchange-driven Growth
Exchange-driven growth is a process in which pairs of clusters interact and
exchange a single unit of mass. The rate of exchange is given by an interaction
kernel which depends on the masses of the two interacting clusters. In
this paper we establish the fundamental mathematical properties of the mean
field kinetic equations of this process for the first time. We find two
different classes of behaviour depending on whether is symmetric or
not. For the non-symmetric case, we prove global existence and uniqueness of
solutions for kernels satisfying . This result is optimal in
the sense that we show for a large class of initial conditions with kernels
satisfying ( the solutions cannot exist. On
the other hand, for symmetric kernels, we prove global existence of solutions
for (
while existence is lost for
( In the intermediate regime we can only show
local existence. We conjecture that the intermediate regime exhibits
finite-time gelation in accordance with the heuristic results obtained for
particular kernels.Comment: Mistakes in the uniqueness proofs are fixed. Some typos are
corrected. Some references are adde
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