22,812 research outputs found
Alternating sums of binomial quotients
By combining telescoping and the linearization method, a class of alternating sums of binomial quotients are investigated. Several summation and transformation formulae are established. Asymptotic behavior for these sums is also examined
Classifying Flow-kick Equilibria: Reactivity and Transient Behavior in the Variational Equation
In light of concerns about climate change, there is interest in how sustainable management can maintain the resilience of ecosystems. We use flow-kick dynamical systems to model ecosystems subject to a constant kick occurring every Ļ time units. We classify the stability of flow-kick equilibria to determine which management strategies result in desirable long-term characteristics. To classify the stability of a flow-kick equilibrium, we classify the linearization of the time-Ļ map given by the time-Ļ map of the variational equation about the equilibrium trajectory. Since the variational equation is a non-autonomous linear differential equation, we conjecture that the asymptotic stability classification of each instantaneous local linearization along the equilibrium trajectory indicates the stability of the variational time-Ļ map. In Chapter 3, we prove this conjecture holds when all of the asymptotic and transient behavior of the instantaneous local linearizations is the same. To explore whether the conjecture holds in general, we ask: To what degree can transient behavior differ from asymptotic behavior? Under what conditions can this transient behavior accumulate asymptotically? In Chapter 4, we develop the radial and tangential velocity framework to characterize transient behavior in autonomous linear systems. In Chapter 5, we use this framework to construct an example of a non-autonomous linear system whose time-Ļ map has asymptotic behavior that differs from the asymptotic behavior of each instantaneous linear system that composes it. Future work seeks to determine whether this constructed example can arise as a variational equation, and thus provide a counterexample for our conjecture
Selection of the ground state for nonlinear Schroedinger equations
We prove for a class of nonlinear Schr\"odinger systems (NLS) having two
nonlinear bound states that the (generic) large time behavior is characterized
by decay of the excited state, asymptotic approach to the nonlinear ground
state and dispersive radiation. Our analysis elucidates the mechanism through
which initial conditions which are very near the excited state branch evolve
into a (nonlinear) ground state, a phenomenon known as {\it ground state
selection}.
Key steps in the analysis are the introduction of a particular linearization
and the derivation of a normal form which reflects the dynamics on all time
scales and yields, in particular, nonlinear Master equations.
Then, a novel multiple time scale dynamic stability theory is developed.
Consequently, we give a detailed description of the asymptotic behavior of the
two bound state NLS for all small initial data. The methods are general and can
be extended to treat NLS with more than two bound states and more general
nonlinearities including those of Hartree-Fock type.Comment: Revision of 2001 preprint; 108 pages Te
On the asymptotic uniqueness of bargaining equilibria
The paper studies the model of multilateral bargaining over the alternatives representedby points in the mĆ¢dimensional Euclidean space. Proposers are chosen randomly and the acceptance of a proposal requires the unanimous approval of it by all the players. The focus of the paper is on the asymptotic behavior of subgame perfect equilibria in pure stationary strategies (called bargaining equilibria) as the breakdown probability tends to zero. Bargaining equilibria are said to be asymptotically unique if the limit of a sequence of bargaining equilibria as the breakdown probability tends to zero is independent of the choice of the sequence and is uniquely determined by the primitives of the model. We show that the limit of any sequence of bargaining equilibria is a zero point of the soĆ¢called linearization correspondence. The asymptotic uniqueness of bargaining equilibria is then deduced in each of the following cases: (1) m = nā1, where n is the number of players, (2) m = 1, and (3) in the case where the utility functions are quadratic, for each 1 ā¤ m ā¤ nā1. In each case the linearization correspondence is shown to have a unique zero. Result 1 hasbeen established earlier in Miyakawa and Laruelle and Valenciano. Result 2 is subsumed by the result in Predtetchinski. Result 3 is new.microeconomics ;
Linearization models for parabolic dynamical systems via Abel's functional equation
We study linearization models for continuous one-parameter semigroups of
parabolic type. In particular, we introduce new limit schemes to obtain
solutions of Abel's functional equation and to study asymptotic behavior of
such semigroups. The crucial point is that these solutions are univalent
functions convex in one direction. In a parallel direction, we find analytic
conditions which determine certain geometric properties of those functions,
such as the location of their images in either a half-plane or a strip, and
their containing either a half-plane or a strip. In the context of semigroup
theory these geometric questions may be interpreted as follows: is a given
one-parameter continuous semigroup either an outer or an inner conjugate of a
group of automorphisms? In other words, the problem is finding a fractional
linear model of the semigroup which is defined by a group of automorphisms of
the open unit disk. Our results enable us to establish some new important
analytic and geometric characteristics of the asymptotic behavior of
one-parameter continuous semigroups of holomorphic mappings, as well as to
study the problem of existence of a backward flow invariant domain and its
geometry
Parameter estimation in softmax decision-making models with linear objective functions
With an eye towards human-centered automation, we contribute to the
development of a systematic means to infer features of human decision-making
from behavioral data. Motivated by the common use of softmax selection in
models of human decision-making, we study the maximum likelihood parameter
estimation problem for softmax decision-making models with linear objective
functions. We present conditions under which the likelihood function is convex.
These allow us to provide sufficient conditions for convergence of the
resulting maximum likelihood estimator and to construct its asymptotic
distribution. In the case of models with nonlinear objective functions, we show
how the estimator can be applied by linearizing about a nominal parameter
value. We apply the estimator to fit the stochastic UCL (Upper Credible Limit)
model of human decision-making to human subject data. We show statistically
significant differences in behavior across related, but distinct, tasks.Comment: In pres
On the phase structure of vector-matrix scalar model in four dimensions
The leading-order equations of the -- expansion for a vector-matrix
model with interaction in four dimensions are
investigated. This investigation shows a change of the asymptotic behavior in
the deep Euclidean region in a vicinity of a certain critical value of the
coupling constant. For small values of the coupling the phion propagator
behaves as free. In the strong-coupling region the asymptotic behavior
drastically changes -- the propagator in the deep Euclidean region tend to some
constant limit. The phion propagator in the coordinate space has a
characteristic shell structure. At the critical value of coupling that
separates the weak and strong coupling regions, the asymptotic behavior of the
phion propagator is a medium among the free behavior and the constant--type
behavior in strong--coupling region. The equation for a vertex with zero
transfer is also investigated. The asymptotic behavior of the solutions shows
the finiteness of the charge renormalization constant. In the strong-coupling
region, the solution for the vertex has the same shell structure in coordinate
space as the phion propagator. An analogy between the phase transition in this
model and the re-arrangement of the physical vacuum in the supercritical
external field due to the "fall-on-the-center" phenomenon is discussed.Comment: 7 page
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