229,144 research outputs found
The Quantum Dynamics of the Compactified Trigonometric Ruijsenaars-Schneider Model
We quantize a compactified version of the trigonometric
Ruijse\-naars-Schneider particle model with a phase space that is
symplectomorphic to the complex projective space CP^N. The quantum Hamiltonian
is realized as a discrete difference operator acting in a finite-dimensional
Hilbert space of complex functions with support in a finite uniform lattice
over a convex polytope (viz., a restricted Weyl alcove with walls having a
thickness proportional to the coupling parameter). We solve the corresponding
finite-dimensional (bispectral) eigenvalue problem in terms of discretized
Macdonald polynomials with q (and t) on the unit circle. The normalization of
the wave functions is determined using a terminating version of a recent
summation formula due to Aomoto, Ito and Macdonald. The resulting eigenfunction
transform determines a discrete Fourier-type involution in the Hilbert space of
lattice functions. This is in correspondence with Ruijsenaars' observation
that---at the classical level---the action-angle transformation defines an
(anti)symplectic involution of CP^N. From the perspective of algebraic
combinatorics, our results give rise to a novel system of bilinear summation
identities for the Macdonald symmetric functions
Model analysis of thermal UV-cutoff effects on the chiral critical surface at finite temperature and chemical potential
We study the effects of temporal UV-cutoff on the chiral critical surface in
hot and dense QCD using a chiral effective model. Recent lattice QCD
simulations indicate that the curvature of the critical surface might change
toward the direction in which the first order phase transition becomes stronger
on increasing the number of lattice sites. To investigate this effect on the
critical surface in an effective model approach, we use the Nambu-Jona-Lasinio
model with finite Matsubara frequency summation. We find that qualitative
feature of the critical surface does not alter appreciably as we decrease the
summation number, which is unlike the case what is observed in the recent
lattice QCD studies. This may either suggest the dependence of chemical
potential on the coupling strength or due to some additional interacting terms
such as vector interactions which could play an important role at finite
density.Comment: 7 pages, 8 figure
Some Quadratic Transformations and Reduction Formulas associated with Hypergeometric Functions
In this paper, we construct four summation formulas for terminating Gauss’ hypergeometric series having argument “two and with the help of our summation formulas. We establish two quadratic transformations for Gauss’ hypergeometric function in terms of finite summation of combination of two Clausen hypergeometric functions. Further, we have generalized our quadratic transformations in terms of general double series identities as well as in terms of reduction formulas for Kampé de Fériet’s double hypergeometric function. Some results of Rathie-Nagar, Kim et al. and Choi-Rathie are also obtained as special cases of our findings
STABILITY, FINITE-TIME STABILITY AND PASSIVITY CRITERIA FOR DISCRETE-TIME DELAYED NEURAL NETWORKS
In this paper, we present the problem of stability, finite-time stability and passivity for discrete-time neural networks (DNNs) with variable delays. For the purposes of stability analysis, an augmented Lyapunov-Krasovskii functional (LKF) with single and double summation terms and several augmented vectors is proposed by decomposing the time-delay interval into two non-equidistant subintervals. Then, by using the Wirtinger-based inequality, reciprocally and extended reciprocally convex combination lemmas, tight estimations for sum terms in the forward difference of LKF are given. In order to relax the existing results, several zero equalities are introduced and stability criteria are proposed in terms of linear matrix inequalities (LMIs). The main objective for the finite-time stability and passivity analysis is how to effectively evaluate the finite-time passivity conditions for DNNs. To achieve this, some weighted summation inequalities are proposed for application to a finite-sum term appearing in the forward difference of LKF, which helps to ensure that the considered delayed DNN is passive. The derived passivity criteria are presented in terms of linear matrix inequalities. Some numerical examples are presented to illustrate the proposed methodology
Master equation for spin-spin correlation functions of the XXZ chain
We derive a new representation for spin-spin correlation functions of the
finite XXZ spin-1/2 Heisenberg chain in terms of a single multiple integral,
that we call the master equation. Evaluation of this master equation gives rise
on the one hand to the previously obtained multiple integral formulas for the
spin-spin correlation functions and on the other hand to their expansion in
terms of the form factors of the local spin operators. Hence, it provides a
direct analytic link between these two representations of the correlation
functions and a complete re-summation of the corresponding series. The master
equation method also allows one to obtain multiple integral representations for
dynamical correlation functions.Comment: 24 page
Theory of resistor networks: The two-point resistance
The resistance between arbitrary two nodes in a resistor network is obtained
in terms of the eigenvalues and eigenfunctions of the Laplacian matrix
associated with the network. Explicit formulas for two-point resistances are
deduced for regular lattices in one, two, and three dimensions under various
boundary conditions including that of a Moebius strip and a Klein bottle. The
emphasis is on lattices of finite sizes. We also deduce summation and product
identities which can be used to analyze large-size expansions of two-and-higher
dimensional lattices.Comment: 30 pages, 5 figures now included; typos in Example 1 correcte
Mellin Moments of the Next-to-next-to Leading Order Coefficient Functions for the Drell-Yan Process and Hadronic Higgs-Boson Production
We calculate the Mellin moments of the next-to-next-to leading order
coefficient functions for the Drell--Yan and Higgs production cross sections.
The results can be expressed in terms of multiple finite harmonic sums of
maximal weight w = 4. Using algebraic and structural relations between harmonic
sums one finds that besides the single harmonic sums only five basic sums and
their derivatives w.r.t. the summation index contribute. This representation
reduces the large complexity being present in x-space calculations and is well
suited for fast numerical implementations.Comment: 47 pages LATEX, version appearing in journa
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