49,627 research outputs found
Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem
A new approach to the analytic theory of difference equations with rational
and elliptic coefficients is proposed. It is based on the construction of
canonical meromorphic solutions which are analytical along "thick paths". The
concept of such solutions leads to a notion of local monodromies of difference
equations. It is shown that in the continuous limit they converge to the
monodromy matrices of differential equations. New type of isomonodromic
deformations of difference equations with elliptic coefficients changing the
periods of elliptic curves is constructed.Comment: 38 pages, no figures; typos remove
Fredholm factorization of Wiener-Hopf scalar and matrix kernels
A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape
Basic Polyhedral Theory
This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of
Operations Research and Management Science) describing parts of the theory of
convex polyhedra that are particularly important for optimization. The topics
include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem,
faces of polyhedra, projections of polyhedra, integral polyhedra, total dual
integrality, and total unimodularity.Comment: 14 page
Rational fixed points for linear group actions
Let be a finitely generated field, let be an algebraic variety and
a linear algebraic group, both defined over . Suppose acts on
and every element of a Zariski-dense semigroup has a
rational fixed point in . We then deduce, under some mild technical
assumptions, the existence of a rational map , defined over ,
sending each element to a fixed point for . The proof makes use of
a recent result of Ferretti and Zannier on diophantine equations involving
linear recurrences. As a by-product of the proof, we obtain a version of the
classical Hilbert Irreducibility Theorem valid for linear algebraic groups.Comment: 35 pages, Plain Tex. A gap in the previous proof of Theorem 1.2
overcome, plus minor changes. Thanks to J. Bernik and the refere
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
This paper gives -dimensional analogues of the Apollonian circle packings
in parts I and II. We work in the space \sM_{\dd}^n of all -dimensional
oriented Descartes configurations parametrized in a coordinate system,
ACC-coordinates, as those real matrices \bW with \bW^T
\bQ_{D,n} \bW = \bQ_{W,n} where is the -dimensional Descartes quadratic
form, , and \bQ_{D,n} and
\bQ_{W,n} are their corresponding symmetric matrices. There are natural
actions on the parameter space \sM_{\dd}^n. We introduce -dimensional
analogues of the Apollonian group, the dual Apollonian group and the
super-Apollonian group. These are finitely generated groups with the following
integrality properties: the dual Apollonian group consists of integral matrices
in all dimensions, while the other two consist of rational matrices, with
denominators having prime divisors drawn from a finite set depending on the
dimension. We show that the the Apollonian group and the dual Apollonian group
are finitely presented, and are Coxeter groups. We define an Apollonian cluster
ensemble to be any orbit under the Apollonian group, with similar notions for
the other two groups. We determine in which dimensions one can find rational
Apollonian cluster ensembles (all curvatures rational) and strongly rational
Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings
beginning with math.MG/0010298. Revised and extended. Added: Apollonian
groups and Apollonian Cluster Ensembles (Section 4),and Presentation for
n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200
Associated primes of local cohomology modules and of Frobenius powers
We construct normal hypersurfaces whose local cohomology modules have
infinitely many associated primes. These include unique factorization domains
of characteristic zero with rational singularities, as well as F-regular unique
factorization domains of positive characteristic. As a consequence, we answer a
question on the associated primes of Frobenius powers of ideals, which arose
from the localization problem in tight closure theory
Uniform Mixing and Association Schemes
We consider continuous-time quantum walks on distance-regular graphs of small
diameter. Using results about the existence of complex Hadamard matrices in
association schemes, we determine which of these graphs have quantum walks that
admit uniform mixing.
First we apply a result due to Chan to show that the only strongly regular
graphs that admit instantaneous uniform mixing are the Paley graph of order
nine and certain graphs corresponding to regular symmetric Hadamard matrices
with constant diagonal. Next we prove that if uniform mixing occurs on a
bipartite graph X with n vertices, then n is divisible by four. We also prove
that if X is bipartite and regular, then n is the sum of two integer squares.
Our work on bipartite graphs implies that uniform mixing does not occur on
C_{2m} for m >= 3. Using a result of Haagerup, we show that uniform mixing does
not occur on C_p for any prime p such that p >= 5. In contrast to this result,
we see that epsilon-uniform mixing occurs on C_p for all primes p.Comment: 23 page
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