5,937 research outputs found
New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential
Integrable N-particle systems have an important property that the associated
Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not
apply to the most interesting class of elliptic and double-elliptic systems.
Studying the commutativity conjecture for theta-functions on the families of
associated spectral curves, we derive some other non-linear equations for the
perturbative Seiberg-Witten prepotential, which turn out to have exactly the
double-elliptic system as their generic solution. In contrast with the WDVV
equations, the new equations acquire non-perturbative corrections which are
straightforwardly deducible from the commutativity conditions. We obtain such
corrections in the first non-trivial case of N=3 and describe the structure of
non-perturbative solutions as expansions in powers of the flat moduli with
coefficients that are (quasi)modular forms of the elliptic parameter.Comment: 25 page
Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model
We advocate a new approach to the study of unitary matrix models in external
fields which emphasizes their relationship to Generalized Kontsevich Models
(GKM) with non-polynomial potentials. For example, we show that the partition
function of the Brezin-Gross-Witten Model (BGWM), which is defined as an
integral over unitary matrices, , can also be considered as a GKM with potential . Moreover, it can be interpreted as the generating functional for
correlators in the Penner model. The strong and weak coupling phases of the
BGWM are identified with the "character" (weak coupling) and "Kontsevich"
(strong coupling) phases of the GKM, respectively. This sort of GKM deserves
classification as one (i.e. or ) when in the Kontsevich
phase. This approach allows us to further identify the
Harish-Chandra-Itzykson-Zuber (IZ) integral with a peculiar GKM, which arises
in the study of theory and, further, with a conventional 2-matrix model
which is rewritten in Miwa coordinates. Inspired by the considered unitary
matrix models, some further extensions of the GKM treatment which are inspired
by the unitary matrix models which we have considered are also developed. In
particular, as a by-product, a new simple method of fixing the Ward identities
for matrix models in an external field is presented.Comment: FIAN/TD-16/93, ITEP-M6/93, UBC/S-93/93 (39 pages
Phase behaviour of block copolymer melts with arbitrary architecture
The Leibler theory [L. Leibler, Macromolecules, v.13, 1602 (1980)] for
microphase separation in AB block copolymer melts is generalized for systems
with arbitrary topology of molecules. A diagrammatic technique for calculation
of the monomeric correlation functions is developed. The free energies of
various mesophases are calculated within the second-harmonic approximation.
Model highly-branched tree-like structures are considered as an example and
their phase diagrams are obtained. The topology of molecules is found to
influence the spinodal temperature and asymmetry of the phase diagrams, but not
the types of phases and their order. We suggest that all model AB
block-copolymer systems will exhibit the typical phase behaviour.Comment: Submitted to J. Chem. Phys., see also
http://rugmd4.chem.rug.nl/~morozov/research.htm
Continuum Limits of ``Induced QCD": Lessons of the Gaussian Model at d=1 and Beyond
We analyze the scalar field sector of the Kazakov--Migdal model of induced
QCD. We present a detailed description of the simplest one dimensional
{()} model which supports the hypothesis of wide applicability of the
mean--field approximation for the scalar fields and the existence of critical
behaviour in the model when the scalar action is Gaussian. Despite the
ocurrence of various non--trivial types of critical behaviour in the
model as , only the conventional large- limit is
relevant for its {\it continuum} limit. We also give a mean--field analysis of
the model in {\it any} and show that a saddle point always exists in
the region . In it exhibits critical behaviour as
. However when there is no critical
behaviour unless non--Gaussian terms are added to the scalar field action. We
argue that similar behaviour should occur for any finite thus providing a
simple explanation of a recent result of D. Gross. We show that critical
behaviour at and can be obtained by adding a
term to the scalar potential. This is equivalent to a local
modification of the integration measure in the original Kazakov--Migdal model.
Experience from previous studies of the Generalized Kontsevich Model implies
that, unlike the inclusion of higher powers in the potential, this minor
modification should not substantially alter the behaviour of the Gaussian
model.Comment: 31 page
On non existence of tokamak equilibria with purely poloidal flow
It is proved that irrespective of compressibility tokamak steady states with
purely poloidal mass flow can not exist in the framework of either
magnetohydrodynamics (MHD) or Hall MHD models. Non-existence persists within
single fluid plasma models with pressure anisotropy and incompressible flows.Comment: The conclusion reported in the last sentence of the first paragraph
of Sec. V in the version of the paper published in Physics of Plasmas is
incorrect. The correct conclusion is given here (15 pages
Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication
We employ partial integro-differential equations to model trophic interaction
in a spatially extended heterogeneous environment. Compared to classical
reaction-diffusion models, this framework allows us to more realistically
describe the situation where movement of individuals occurs on a faster time
scale than the demographic (population) time scale, and we cannot determine
population growth based on local density. However, most of the results reported
so far for such systems have only been verified numerically and for a
particular choice of model functions, which obviously casts doubts about these
findings. In this paper, we analyse a class of integro-differential
predator-prey models with a highly mobile predator in a heterogeneous
environment, and we reveal the main factors stabilizing such systems. In
particular, we explore an ecologically relevant case of interactions in a
highly eutrophic environment, where the prey carrying capacity can be formally
set to 'infinity'. We investigate two main scenarios: (i) the spatial gradient
of the growth rate is due to abiotic factors only, and (ii) the local growth
rate depends on the global density distribution across the environment (e.g.
due to non-local self-shading). For an arbitrary spatial gradient of the prey
growth rate, we analytically investigate the possibility of the predator-prey
equilibrium in such systems and we explore the conditions of stability of this
equilibrium. In particular, we demonstrate that for a Holling type I (linear)
functional response, the predator can stabilize the system at low prey density
even for an 'unlimited' carrying capacity. We conclude that the interplay
between spatial heterogeneity in the prey growth and fast displacement of the
predator across the habitat works as an efficient stabilizing mechanism.Comment: 2 figures; appendices available on request. To appear in the Bulletin
of Mathematical Biolog
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