6,036 research outputs found
Traveling waves for a bistable equation with nonlocal-diffusion
We consider a single component reaction-diffusion equation in one dimension
with bistable nonlinearity and a nonlocal space-fractional diffusion operator
of Riesz-Feller type. Our main result shows the existence, uniqueness (up to
translations) and stability of a traveling wave solution connecting two stable
homogeneous steady states. In particular, we provide an extension to classical
results on traveling wave solutions involving local diffusion. This extension
to evolution equations with Riesz-Feller operators requires several technical
steps. These steps are based upon an integral representation for Riesz-Feller
operators, a comparison principle, regularity theory for space-fractional
diffusion equations, and control of the far-field behavior
The relation between degrees of belief and binary beliefs: A general impossibility theorem
Agents are often assumed to have degrees of belief (âcredencesâ) and also binary beliefs (âbeliefs simpliciterâ). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the âlottery paradoxâ: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. âLocalityâ means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the âfunctionalityâ restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a âholisticâ one. We explore several concrete forms this âholisticâ relation could take
On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation
We study the existence of stationary solutions for a nonlocal version of the
Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is
a recent study by Berestycki et {al.} [Nonlinearity 22 (2009),
{pp.}~2813--2844] where the nonlocal FKPP equation has been studied and it was
shown for the spatial domain andsufficiently small nonlocality
that there are only two bounded non-negative stationary solutions. Here we
provide a similar result for using a completely different
approach. In particular, an abstract perturbation argument is used in suitable
weighted Sobolev spaces. One aim of the alternative strategy is that it can
eventually be generalized to obtain persistence results for hyperbolic
invariant sets for other nonlocal evolution equations on unbounded domains with
small nonlocality, {i.e.}, to improve our understanding in applications when a
small nonlocal influence alters the dynamics and when it does not.Comment: 24 pages, 1 figure; revised versio
Where do preferences come from?
Rational choice theory analyzes how an agent can rationally act, given his or her preferences, but says little about where those preferences come from. Preferences are usually assumed to be fixed and exogenously given. Building on related work on reasons and rational choice (Dietrich and List, Nous, forthcoming), we describe a framework for conceptualizing preference formation and preference change. In our model, an agentâs preferences are based on certain âmotivationally salientâ properties of the alternatives over which the preferences are held. Preferences may change as new properties of the alternatives become salient or previously salient properties cease to be salient. Our approach captures endogenous preferences in various contexts and helps to illuminate the distinction between formal and substantive concepts of rationality, as well as the role of perception in rational choice
The impossibility of unbiased judgment aggregation
Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. Applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial.mathematical economics;
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