140,587 research outputs found
Pseudo-centrosymmetric matrices, with applications to counting perfect matchings
We consider square matrices A that commute with a fixed square matrix K, both
with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda
defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we
define A to be pseudo-centrosymmetric with respect to K; we show that the
determinant of every even-order pseudo-centrosymmetric matrix is the sum of two
squares over F, as long as -1 is not a square in F. When a
pseudo-centrosymmetric matrix A contains only integral entries and is
pseudo-centrosymmetric with respect to a matrix with rational entries, the
determinant of A is the sum of two integral squares. This result, when
specialized to when K is the even-order alternating exchange matrix, applies to
enumerative combinatorics. Using solely matrix-based methods, we reprove a weak
form of Jockusch's theorem for enumerating perfect matchings of 2-even
symmetric graphs. As a corollary, we reprove that the number of domino tilings
of regions known as Aztec diamonds and Aztec pillows is a sum of two integral
squares.Comment: v1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures.
Reworked so that linear algebraic results are over a field not of
characteristic 2, not over the real numbers. Accepted, Linear Algebra and its
Application
Character Expansion Methods for Matrix Models of Dually Weighted Graphs
We consider generalized one-matrix models in which external fields allow
control over the coordination numbers on both the original and dual lattices.
We rederive in a simple fashion a character expansion formula for these models
originally due to Itzykson and Di Francesco, and then demonstrate how to take
the large N limit of this expansion. The relationship to the usual matrix model
resolvent is elucidated. Our methods give as a by-product an extremely simple
derivation of the Migdal integral equation describing the large limit of
the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a
number of models solvable by traditional means. We then proceed to solve a new
model: a sum over planar graphs possessing even coordination numbers on both
the original and the dual lattice. We conclude by formulating equations for the
case of arbitrary sets of even, self-dual coupling constants. This opens the
way for studying the deep problem of phase transitions from random to flat
lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into
the text in Pictex commands. (Two minor math typos corrected.
Acknowledgements added.
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
The loop expansion of the Kontsevich integral, the null move and S-equivalence
This is a substantially revised version. The Kontsevich integral of a knot is
a graph-valued invariant which (when graded by the Vassiliev degree of graphs)
is characterized by a universal property; namely it is a universal Vassiliev
invariant of knots. We introduce a second grading of the Kontsevich integral,
the Euler degree, and a geometric null-move on the set of knots. We explain the
relation of the null-move to S-equivalence, and the relation to the Euler
grading of the Kontsevich integral. The null move leads in a natural way to the
introduction of trivalent graphs with beads, and to a conjecture on a rational
version of the Kontsevich integral, formulated by the second author and proven
in joint work of the first author and A. Kricker.Comment: AMS-LaTeX, 20 pages with 31 figure
Leading RG logs in theory
We find the leading RG logs in theory for any Feynman diagram with 4
external edges. We obtain the result in two ways. The first way is to calculate
the relevant terms in Feynman integrals. The second way is to use the RG
invariance based on the Lie algebra of graphs introduced by Connes and Kreimer.
The non-RG logs, such as , are discussed.Comment: 24 pages LaTeX, 12 figure
Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula
We study the large-N limit of a class of matrix models for dually weighted
triangulated random surfaces using character expansion techniques. We show that
for various choices of the weights of vertices of the dynamical triangulation
the model can be solved by resumming the Itzykson-Di Francesco formula over
congruence classes of Young tableau weights modulo three. From this we show
that the large-N limit implies a non-trivial correspondence with models of
random surfaces weighted with only even coordination number vertices. We
examine the critical behaviour and evaluation of observables and discuss their
interrelationships in all models. We obtain explicit solutions of the model for
simple choices of vertex weightings and use them to show how the matrix model
reproduces features of the random surface sum. We also discuss some general
properties of the large-N character expansion approach as well as potential
physical applications of our results.Comment: 37 pages LaTeX; Some clarifying comments added, last Section
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