Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic

(WP 2010-10) Assessing the Predictive Power of Labor-Market Indicators of Inflation

This paper examines two different measures of wages as predicators of prices in a vector error-correction framework using quarterly data for the U.S. for the period from 1947.Q1 through 2008.Q1. Based on cointegration and a series of exogeneity tests, it is found that: 1) there is a stable, long-run relationship between the Consumer Price Index (CPI) and the Personal Consumption Expenditure Deflator (PCED) on the one hand and unit labor costs (ULC) and average earnings per unit of output (AHE) on the other; 2) ULC is weakly exogenous for both price indices while the two price indices are weakly exogenous for AHE; 3) ULC is strongly exogenous for CPI but not for AHE; 4) ULC is super exogenous for CPI. Taken together, these findings lead to the conclusion that ULC is a reliable indicator of price inflation but productivity-adjusted hourly earnings is not. Thus monetary policymakers are justified in using information about the behavior of ULC in formulating policy actions for achieving the goal of price stability

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

Finite difference lattice Boltzmann model with flux limiters for liquid-vapor systems

In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme involving flux limiter techniques is proposed to minimize them and guarantee a better numerical stability at very low viscosity. The phase separation kinetics is investigated and we find evidence of two different growth regimes depending on the value of the fluid viscosity as well as on the liquid-vapor ratio.Comment: 10 pages, 10 figures, to be published in Phys. Rev.

Sources of pro-cyclicality in east Asian financial systems

Procyclicality is a normal feature of economic systems, but financial sector weaknesses can exacerbate it sufficiently to pose a threat to macroeconomic and financial stability. These include shortcomings in bank risk management and governance, in supervision and in terms of dependence on volatile sources of funds. The paper tests econometrically for the importance of such features leading to pro-cyclicality in the financial systems of 11 East Asian countries. This analysis makes it possible to identify specific policy measures for East Asian countries that could limit the extent to which financial systems exacerbate pro-cyclicality

Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes

We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.Comment: Revised WIAS Preprin