Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Culture and Development: An Analytical Framework

This paper develops a framework which analyzes how a population's culture affects the decisions of rational profit maximizing firms, while simultaneously exploring how the actions of these firms in turn affect the population's culture.By endogenizing culture as well as the more usual economic variables, it shows how an economically valuablebehavioural trait can be sustained as part of a competitive equilibrium.It is shown that, for given primitives, an economy can be in either a 'good' steady state, in which the valuable cultural trait is present, or a welfare dominated 'bad' one in which the valuable cultural trait disappears.Starting from the 'good' steady state and implementing productivity improvements raises welfare, but if changes are too rapid this steady state will not be reached from the old one.Instead, the unique trajectory is to the bad steady state where welfare is reduced.culture;development;inequality;technological change

Trust, Social Capital and Economic Development

Many argue that elements of a society s norms, culture or social capital are central to understanding its development.However, these notions have been difficult to capture in economic models.Here we argue that trustworthiness is the economically relevant component of a society s culture and hence comprises its social capital.Individuals are trustworthy when they perform actions they have promised, even if these do not maximize their payoffs.The usual focus on incentive structures in motivating behaviour plays no role here.Instead, we emphasize more deep-seated modes of behaviour and consider that trustworthy agents are socialized to act as they do.To model this socialization, we borrow from a relatively new process of preference evolution pioneered by Bisin and Verdier (2001).The model developed endogenously accounts for social capital and explores its role in the process of economic development.It captures in a simple, formal way the interaction between social capital and the economy s productive process.The results obtained caution against rapid reform, provide an explanation for why late developing countries cannot easily transplant the modes of production that have proved useful in the West, and suggest an explanation for the pattern of reform experiences in ex-communist countries.technological change;public finance;economic development

Characterization of the size and position of electron-hole puddles at a graphene p-n junction

The effect of an electron-hole puddle on the electrical transport when governed by snake states in a bipolar graphene structure is investigated. Using numerical simulations we show that information on the size and position of the electron-hole puddle can be obtained using the dependence of the conductance on magnetic field and electron density of the gated region. The presence of the scatterer disrupts snake state transport which alters the conduction pattern. We obtain a simple analytical formula that connects the position of the electron-hole puddle with features observed in the conductance. Size of the electron-hole puddle is estimated from the magnetic field and gate potential that maximizes the effect of the puddle on the electrical transport.Comment: This is an author-created, un-copyedited version of an article published in Nanotechnology. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi:10.1088/0957-4484/27/10/10520

Gender Wage Differentials in a Competitive Labor Market: The Household Interaction Effect

We present a theoretical explanation of the gender wage gap which turns on the interaction between men and women in households.In equilibria where men are over-represented in full-time work, we show that firms rationally choose to hire women only at strictly lower wages than men.The model developed predicts a gap even controlling for education, occupation and industry of workers and does so in a competitive labor market where there exist no inherent gender differences. We test our theory using CPS data over the period 1979-98 and find it is strongly supported by the data.gender discrimination;household models;wage gap

The relationship between noise and annoyance around Orly

The extent to which annoyance estimated by an isopsophic index is a good forecaster for annoyance perceived near airport approaches was investigated. An index of sensed annoyance is constructed, and the relationship between the annoyance index and the isopsophic index is studied

Co-movement, Capital and Contracts: 'Normal' Cycles Through Creative Destruction

We develop a unified theory of endogenous business cycles in which expansions are neoclassical growth periods driven by productivity improvements and capital accumulation, while downturns are the result of Keynesian contractions in aggregate demand below potential output. Recessions allow skilled labor to be reallocated to growth promoting activities which fuel subsequent expansions. However, rigidities in production and contractual limitations, inherent to the process of creative destruction, leave capital severely underutilized. A key feature of our equilibrium is the endogenous emergence of long term supply contracts between capitalist owners and producers.Long-term contracting;investment irreversibility;putty-clay technology;asset- specificity;Endogenous cycles and growth

A Definition of Metastability for Markov Processes with Detailed Balance

A definition of metastable states applicable to arbitrary finite state Markov processes satisfying detailed balance is discussed. In particular, we identify a crucial condition that distinguishes genuine metastable states from other types of slowly decaying modes and which leads to properties similar to those postulated in the restricted ensemble approach \cite{pen71}. The intuitive physical meaning of this condition is simply that the total equilibrium probability of finding the system in the metastable state is negligible. As a concrete application of our formalism we present preliminary results on a 2D kinetic Ising model.Comment: 5 pp. 1 Figure, presented in News, Expectations and Trends in Statistical Physics-3rd International Conference, 13-18 August 2005, Kolymbari Cret

870 micron Imaging of a Transitional Disk in Upper Scorpius: Holdover from the Era of Giant Planet Formation?

We present 880 micron images of the transition disk around the star [PZ99] J160421.7-213028, a solar-mass star in the nearby Upper Scorpius association. With a resolution down to 0.34 arcsec, we resolve the inner hole in this disk, and via model fitting to the visibilities and spectral energy distribution we determine both the structure of the outer region and the presence of sparse dust within the cavity. The disk contains about 0.1 Jupiter masses of mm-emitting grains, with an inner disk edge of about 70 AU. The inner cavity contains a small amount of dust with a depleted surface density in a region extending from about 20-70 AU. Taking into account prior observations indicating little to no stellar accretion, the lack of a binary companion, and the presence of dust near 0.1 AU, we determine that the most likely mechanism for the formation of this inner hole is the presence of one or more giant planets.Comment: 12 pages, 7 figures. To appear in the Astrophysical Journa