889 research outputs found
q-Extensions of certain Erdélyi type integrals
AbstractRecently we discovered several new Erdélyi type integrals. In the present paper, it is shown how the q-extensions of all those integrals involving and representing certain q-hypergeometric functions can be developed. The well-known special cases and applications of these q-integrals are also pointed out
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Dynamical aspects of mean field plane rotators and the Kuramoto model
The Kuramoto model has been introduced in order to describe synchronization
phenomena observed in groups of cells, individuals, circuits, etc... We look at
the Kuramoto model with white noise forces: in mathematical terms it is a set
of N oscillators, each driven by an independent Brownian motion with a constant
drift, that is each oscillator has its own frequency, which, in general,
changes from one oscillator to another (these frequencies are usually taken to
be random and they may be viewed as a quenched disorder). The interactions
between oscillators are of long range type (mean field). We review some results
on the Kuramoto model from a statistical mechanics standpoint: we give in
particular necessary and sufficient conditions for reversibility and we point
out a formal analogy, in the N to infinity limit, with local mean field models
with conservative dynamics (an analogy that is exploited to identify in
particular a Lyapunov functional in the reversible set-up). We then focus on
the reversible Kuramoto model with sinusoidal interactions in the N to infinity
limit and analyze the stability of the non-trivial stationary profiles arising
when the interaction parameter K is larger than its critical value K_c. We
provide an analysis of the linear operator describing the time evolution in a
neighborhood of the synchronized profile: we exhibit a Hilbert space in which
this operator has a self-adjoint extension and we establish, as our main
result, a spectral gap inequality for every K>K_c.Comment: 18 pages, 1 figur
Geometric phases for generalized squeezed coherent states
A simple technique is used to obtain a general formula for the Berry phase
(and the corresponding Hannay angle) for an arbitrary Hamiltonian with an
equally-spaced spectrum and appropriate ladder operators connecting the
eigenstates. The formalism is first applied to a general deformation of the
oscillator involving both squeezing and displacement. Earlier results are shown
to emerge as special cases. The analysis is then extended to multiphoton
squeezed coherent states and the corresponding anholonomies deduced.Comment: 15 page
Naked Singularity Formation In f(R) Gravity
We study the gravitational collapse of a star with barotropic equation of
state in the context of theories of gravity.
Utilizing the metric formalism, we rewrite the field equations as those of
Brans-Dicke theory with vanishing coupling parameter. By choosing the
functionality of Ricci scalar as , we
show that for an appropriate initial value of the energy density, if
and satisfy certain conditions, the resulting singularity would be naked,
violating the cosmic censorship conjecture. These conditions are the ratio of
the mass function to the area radius of the collapsing ball, negativity of the
effective pressure, and the time behavior of the Kretschmann scalar. Also, as
long as parameter obeys certain conditions, the satisfaction of the
weak energy condition is guaranteed by the collapsing configuration.Comment: 15 pages, 4 figures, to appear in GR
Fertility, Living Arrangements, Care and Mobility
There are four main interconnecting themes around which the contributions in this book are based. This introductory chapter aims to establish the broad context for the chapters that follow by discussing each of the themes. It does so by setting these themes within the overarching demographic challenge of the twenty-first century – demographic ageing. Each chapter is introduced in the context of the specific theme to which it primarily relates and there is a summary of the data sets used by the contributors to illustrate the wide range of cross-sectional and longitudinal data analysed
A Model for the Development of the Rhizobial and Arbuscular Mycorrhizal Symbioses in Legumes and Its Use to Understand the Roles of Ethylene in the Establishment of these two Symbioses
We propose a model depicting the development of nodulation and arbuscular mycorrhizae. Both processes are dissected into many steps, using Pisum sativum L. nodulation mutants as a guideline. For nodulation, we distinguish two main developmental programs, one epidermal and one cortical. Whereas Nod factors alone affect the cortical program, bacteria are required to trigger the epidermal events. We propose that the two programs of the rhizobial symbiosis evolved separately and that, over time, they came to function together. The distinction between these two programs does not exist for arbuscular mycorrhizae development despite events occurring in both root tissues. Mutations that affect both symbioses are restricted to the epidermal program. We propose here sites of action and potential roles for ethylene during the formation of the two symbioses with a specific hypothesis for nodule organogenesis. Assuming the epidermis does not make ethylene, the microsymbionts probably first encounter a regulatory level of ethylene at the epidermis–outermost cortical cell layer interface. Depending on the hormone concentrations there, infection will either progress or be blocked. In the former case, ethylene affects the cortex cytoskeleton, allowing reorganization that facilitates infection; in the latter case, ethylene acts on several enzymes that interfere with infection thread growth, causing it to abort. Throughout this review, the difficulty of generalizing the roles of ethylene is emphasized and numerous examples are given to demonstrate the diversity that exists in plants
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