130 research outputs found
The Gelfand map and symmetric products
If A is an algebra of functions on X, there are many cases when X can be
regarded as included in Hom(A,C) as the set of ring homomorphisms. In this
paper the corresponding results for the symmetric products of X are introduced.
It is shown that the symmetric product Sym^n(X) is included in Hom(A,C) as the
set of those functions that satisfy equations generalising f(xy)=f(x)f(y).
These equations are related to formulae introduced by Frobenius and, for the
relevant A, they characterise linear maps on A that are the sum of ring
homomorphisms. The main theorem is proved using an identity satisfied by
partitions of finite sets.Comment: 14 pages, Late
Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type
We discuss a family of multi-term addition formulae for Weierstrass functions
on specialized curves of genus one and two with many automorphisms. In the
genus one case we find new addition formulae for the equianharmonic and
lemniscate cases, and in genus two we find some new addition formulae for a
number of curves, including the Burnside curve.Comment: 19 pages. We have extended the Introduction, corrected some typos and
tidied up some proofs, and inserted extra material on genus 3 curve
Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions
The -particle free fermion state for quantum particles in the plane
subject to a perpendicular magnetic field, and with doubly periodic boundary
conditions, is written in a product form. The absolute value of this is used to
formulate an exactly solvable one-component plasma model, and further motivates
the formulation of an exactly solvable two-species Coulomb gas. The large
expansion of the free energy of both these models exhibits the same O(1) term.
On the basis of a relationship to the Gaussian free field, this term is
predicted to be universal for conductive Coulomb systems in doubly periodic
boundary conditions.Comment: 12 page
A note on the Gauss decomposition of the elliptic Cauchy matrix
Explicit formulas for the Gauss decomposition of elliptic Cauchy type
matrices are derived in a very simple way. The elliptic Cauchy identity is an
immediate corollary.Comment: 5 page
Entanglement-Saving Channels
The set of Entanglement Saving (ES) quantum channels is introduced and
characterized. These are completely positive, trace preserving transformations
which when acting locally on a bipartite quantum system initially prepared into
a maximally entangled configuration, preserve its entanglement even when
applied an arbitrary number of times. In other words, a quantum channel
is said to be ES if its powers are not entanglement-breaking for all
integers . We also characterize the properties of the Asymptotic
Entanglement Saving (AES) maps. These form a proper subset of the ES channels
that is constituted by those maps which, not only preserve entanglement for all
finite , but which also sustain an explicitly not null level of entanglement
in the asymptotic limit~. Structure theorems are provided
for ES and for AES maps which yield an almost complete characterization of the
former and a full characterization of the latter.Comment: 26 page
Random transition-rate matrices for the master equation
Random-matrix theory is applied to transition-rate matrices in the Pauli
master equation. We study the distribution and correlations of eigenvalues,
which govern the dynamics of complex stochastic systems. Both the cases of
identical and of independent rates of forward and backward transitions are
considered. The first case leads to symmetric transition-rate matrices, whereas
the second corresponds to general, asymmetric matrices. The resulting matrix
ensembles are different from the standard ensembles and show different
eigenvalue distributions. For example, the fraction of real eigenvalues scales
anomalously with matrix dimension in the asymmetric case.Comment: 15 pages, 12 figure
Random Surfing Without Teleportation
In the standard Random Surfer Model, the teleportation matrix is necessary to
ensure that the final PageRank vector is well-defined. The introduction of this
matrix, however, results in serious problems and imposes fundamental
limitations to the quality of the ranking vectors. In this work, building on
the recently proposed NCDawareRank framework, we exploit the decomposition of
the underlying space into blocks, and we derive easy to check necessary and
sufficient conditions for random surfing without teleportation.Comment: 13 pages. Published in the Volume: "Algorithms, Probability, Networks
and Games, Springer-Verlag, 2015". (The updated version corrects small
typos/errors
Two-Center Integrals for r_{ij}^{n} Polynomial Correlated Wave Functions
All integrals needed to evaluate the correlated wave functions with
polynomial terms of inter-electronic distance are included. For this form of
the wave function, the integrals needed can be expressed as a product of
integrals involving at most four electrons
Quantum chaos, random matrix theory, and statistical mechanics in two dimensions - a unified approach
We present a theory where the statistical mechanics for dilute ideal gases
can be derived from random matrix approach. We show the connection of this
approach with Srednicki approach which connects Berry conjecture with
statistical mechanics. We further establish a link between Berry conjecture and
random matrix theory, thus providing a unified edifice for quantum chaos,
random matrix theory, and statistical mechanics. In the course of arguing for
these connections, we observe sum rules associated with the outstanding
counting problem in the theory of braid groups. We are able to show that the
presented approach leads to the second law of thermodynamics.Comment: 23 pages, TeX typ
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
- …