594 research outputs found
Existence and Stability of Standing Pulses in Neural Networks : I Existence
We consider the existence of standing pulse solutions of a neural network
integro-differential equation. These pulses are bistable with the zero state
and may be an analogue for short term memory in the brain. The network consists
of a single-layer of neurons synaptically connected by lateral inhibition. Our
work extends the classic Amari result by considering a non-saturating gain
function. We consider a specific connectivity function where the existence
conditions for single-pulses can be reduced to the solution of an algebraic
system. In addition to the two localized pulse solutions found by Amari, we
find that three or more pulses can coexist. We also show the existence of
nonconvex ``dimpled'' pulses and double pulses. We map out the pulse shapes and
maximum firing rates for different connection weights and gain functions.Comment: 31 pages, 29 figures, submitted to SIAM Journal on Applied Dynamical
System
On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system
The Camassa-Holm equation and its two-component Camassa-Holm system
generalization both experience wave breaking in finite time. To analyze this,
and to obtain solutions past wave breaking, it is common to reformulate the
original equation given in Eulerian coordinates, into a system of ordinary
differential equations in Lagrangian coordinates. It is of considerable
interest to study the stability of solutions and how this is manifested in
Eulerian and Lagrangian variables. We identify criteria of convergence, such
that convergence in Eulerian coordinates is equivalent to convergence in
Lagrangian coordinates. In addition, we show how one can approximate global
conservative solutions of the scalar Camassa-Holm equation by smooth solutions
of the two-component Camassa-Holm system that do not experience wave breaking
Electronic stress tensor analysis of hydrogenated palladium clusters
We study the chemical bonds of small palladium clusters Pd_n (n=2-9)
saturated by hydrogen atoms using electronic stress tensor. Our calculation
includes bond orders which are recently proposed based on the stress tensor. It
is shown that our bond orders can classify the different types of chemical
bonds in those clusters. In particular, we discuss Pd-H bonds associated with
the H atoms with high coordination numbers and the difference of H-H bonds in
the different Pd clusters from viewpoint of the electronic stress tensor. The
notion of "pseudo-spindle structure" is proposed as the region between two
atoms where the largest eigenvalue of the electronic stress tensor is negative
and corresponding eigenvectors forming a pattern which connects them.Comment: 22 pages, 13 figures, published online, Theoretical Chemistry
Account
Mean-Field- and Classical Limit of Many-Body Schr\"odinger Dynamics for Bosons
We present a new proof of the convergence of the N-particle Schroedinger
dynamics for bosons towards the dynamics generated by the Hartree equation in
the mean-field limit. For a restricted class of two-body interactions, we
obtain convergence estimates uniform in the Planck constant , up to an
exponentially small remainder. For h=0, the classical dynamics in the
mean-field limit is given by the Vlasov equation.Comment: Latex 2e, 18 page
Renormalization: the observable-state model
The usual mathematical formalism of quantum field theory is non-rigorous
because it contains divergences that can only be renormalized by non-rigorous
mathematical methods. The purpose of this paper is to present a method of
subtraction of this divergences using the formalism of decoherence. This is
achieved by replacing the standard renormalization method by a projector on a
well defined Hilbert subspace. In this way a list of problems of the standard
formalism disappears while the physical results of QFT remains valid. From it
own nature, this formalism can be used in non-renormalizable theories.Comment: 23 page
On Hirschman and log-Sobolev inequalities in mu-deformed Segal-Bargmann analysis
We consider a deformation of Segal-Bargmann space and its transform. We study
L^p properties of this transform and obtain entropy-entropy inequalities
(Hirschman) and entropy-energy inequalities (log-Sobolev) that generalize the
corresponding known results in the undeformed theory.Comment: 42 pages, 3 figure
Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes
We construct a class of solutions to the Cauchy problem of the Klein-Gordon
equation on any standard static spacetime. Specifically, we have constructed
solutions to the Cauchy problem based on any self-adjoint extension (satisfying
a technical condition: "acceptability") of (some variant of) the
Laplace-Beltrami operator defined on test functions in an -space of the
static hypersurface. The proof of the existence of this construction completes
and extends work originally done by Wald. Further results include the
uniqueness of these solutions, their support properties, the construction of
the space of solutions and the energy and symplectic form on this space, an
analysis of certain symmetries on the space of solutions and of various
examples of this method, including the construction of a non-bounded below
acceptable self-adjoint extension generating the dynamics
Wigner Functions and Separability for Finite Systems
A discussion of discrete Wigner functions in phase space related to mutually
unbiased bases is presented. This approach requires mathematical assumptions
which limits it to systems with density matrices defined on complex Hilbert
spaces of dimension p^n where p is a prime number. With this limitation it is
possible to define a phase space and Wigner functions in close analogy to the
continuous case. That is, we use a phase space that is a direct sum of n
two-dimensional vector spaces each containing p^2 points. This is in contrast
to the more usual choice of a two-dimensional phase space containing p^(2n)
points. A useful aspect of this approach is that we can relate complete
separability of density matrices and their Wigner functions in a natural way.
We discuss this in detail for bipartite systems and present the generalization
to arbitrary numbers of subsystems when p is odd. Special attention is required
for two qubits (p=2) and our technique fails to establish the separability
property for more than two qubits.Comment: Some misprints have been corrected and a proof of the separability of
the A matrices has been adde
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