A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $\nu$, of the eigenfunctions are considered. The billiards for which the time-independent Schr\"odinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $\nu$ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.Comment: 13 pages, 5 figure

Nodal domains of the equilateral triangle billiard

We characterise the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.Comment: 26 pages, 13 figure

The relationship between Hispanic residential location and homeownership

This article asks two basic questions. First, is homeownership more or less likely for Hispanics who choose to reside in an ethnic location; and second, is the location decision jointly or endogenously made with the homeownership decision? The findings suggest that, indeed, the location and homeownership decisions are jointly made. Furthermore, the decision to reside in a Hispanic enclave has a positive, significant influence on the likelihood of owning a home.Home ownership ; Hispanic Americans - Population

Tenure choice with location selection: the case of Hispanic neighborhoods in Chicago

A notable feature of immigration into the U.S. is the high degree of spatial concentration of different immigrant groups. We ask the question whether residing in areas with a large proportion of a co-ethnic group influence the decision to own a home for Hispanics in the Chicago Metropolitan area. The results show that Hispanics choose to live in Hispanic enclaves based on relatively homogeneous characteristics such as recent migration, less English language fluency, and lower income. More years in the U.S., higher education attainment and English language fluency remain strong predictors of homeownership. Individuals are less likely to be homeowners in communities with a larger co-ethnic concentration, foreign-born residents, or lower-incomeImmigrants ; Income ; Mortgage loans

Prepaid cards: an important innovation in financial services

This paper describes the characteristics of closed-system and open-system prepaid cards. Of particular interest is a class of open-system programs that offer a set of features similar to conventional deposit accounts using card-based payment applications. The benefits that open-system prepaid cards offer for consumers, providers, and issuing banks contribute to the increased adoption of these payment applications. Using these cards, consumers can pay bills, make purchases, and get cash from ATM networks. At the same time, consumers who hold prepaid cards need not secure a traditional banking relationship nor gain approval for a deposit account or revolving credit. By offering prepaid cards, issuing banks may meet the financial needs of consumers who may not otherwise qualify for more traditional banking products and these banks may do so with a card-based electronic payment application that essentially eliminates credit risk for the bank.Payment systems

On the size and growth of government

The size of the U.S. federal government, as well as state and local governments, increased dramatically during the 20th century. This paper reviews several theories of government size and growth that are dominant in the public choice and political science literature. The theories are divided into two categories: citizen-over-state theories and state-over-citizen theories. The relationship between the 16th Amendment to the U.S. Constitution and the timing of government growth is also presented. It is likely that portions of each theory can explain government size and growth, but the challenge facing economists is to develop a single unifying theory of government growth.Economic development ; Expenditures, Public ; Local government ; Federal government

How effective were the financial safety nets in the aftermath of Katrina?

This paper describes the U.S. financial system’s response to the destruction caused by Hurricane Katrina and examines how financial safety nets helped meet consumers’ needs in the aftermath of the storm. Overall, we find that consumers who hold deposit accounts at financial institutions are less vulnerable to financial disruptions than individuals who do not have either a checking or a savings account (the unbanked). The federal banking regulators’ and financial institutions’ responses to Hurricane Katrina, the financial vulnerability of unbanked families to this unexpected catastrophic event, and how the American Red Cross, FEMA, and the Gulf States’ relief efforts supplied financial assistance to Katrina’s victims are also addressed. Finally, we present several strategies that can be pursued to further safeguard the U.S. population and the financial community against extraordinary events.Hurricane Katrina, 2005 ; Unbanked