Exact on-event expressions for discrete potential systems

The properties of systems composed of atoms interacting though discrete potentials are dictated by a series of events which occur between pairs of atoms. There are only four basic event types for pairwise discrete potentials and the square-well/shoulder systems studied here exhibit them all. Closed analytical expressions are derived for the on-event kinetic energy distribution functions for an atom, which are distinct from the Maxwell-Boltzmann distribution function. Exact expressions are derived that directly relate the pressure and temperature of equilibrium discrete potential systems to the rates of each type of event. The pressure can be determined from knowledge of only the rate of core and bounce events. The temperature is given by the ratio of the number of bounce events to the number of disassociation/association events. All these expressions are validated with event-driven molecular dynamics simulations and agree with the data within the statistical precision of the simulations

Solvable rational extensions of the Morse and Kepler-Coulomb potentials

We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using Darboux-B\"acklund transformations based on unphysical regular Riccati-Schr\"odinger functions which are obtained from specific symmetries associated to the considered family of potentials

Random matrix models with log-singular level confinement: method of fictitious fermions

Joint distribution function of N eigenvalues of U(N) invariant random-matrix ensemble can be interpreted as a probability density to find N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic, log-singular level confinement. An effective one-particle Schroedinger equation for wave-functions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson's density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk, and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.Comment: 13 pages (latex), Presented at the MINERVA Workshop on Mesoscopics, Fractals and Neural Networks, Eilat, Israel, March 199

The Role of Capital in Financial Institutions

This paper examines the role of capital in financial institutions. As the introductory article to a conference on the role of capital management in banking and insurance, it describes the authors' views of why capital is important, how market-generated capital requirements' differ from regulatory requirements and the form that regulatory requirements should take. It also examines the historical trends in bank capital, problems in measuring capital and some possible unintended consequences of capital requirements. According to the authors, the point of departure for all modern research on capital structure is the Modigliani-Miller (M&M, 1958) proposition that in a frictionless world of full information and complete markets, a firm s capital structure cannot affect its value. The authors suggest however, that financial institutions lack any plausible rationale in the frictionless world of M&M. Most of the past research on financial institutions has begun with a set of assumed imperfections, such as taxes, costs of financial distress, transactions costs, asymmetric information and regulation. Miller argues (1995) that these imperfections may not be important enough to overturn the M&M Proposition. Most of the other papers presented at this conference on capital take the view that the deviations from M&M s frictionless world are important, so that financial institutions may be able to enhance their market values by taking on an optimal amount of leverage. The authors highlight these positions in this article. The authors next examine why markets require' financial institutions to hold capital. They define this capital requirement' as the capital ratio that maximizes the value of the bank in the absence of regulatory capital requirements and all the regulatory mechanisms that are used to enforce them, but in the presence of the rest of the regulatory structure that protects the safety and soundness of banks. While the requirement differs for each bank, it is the ratio toward which each bank would tend to move in the long run in the absence of regulatory capital requirements. The authors then introduce imperfections into the frictionless world of M&M taxes and the costs of financial distress, transactions costs and asymmetric information problems and the regulatory safety net. The authors analysis suggests that departures from the frictionless M&M world may help explain market capital requirements for banks. Tax considerations tend to reduce market capital requirements , the expected costs of financial distress tend to raise these requirements , and transactions costs and asymmetric information problems may either increase or reduce the capital held in equilibrium. The federal safety net shields bank creditors from the full consequences of bank risk taking and thus tends to reduce market capital requirements . The paper then summarizes the historical evolution of bank capital ratios in the United States and the reasons regulators require financial institutions to hold capital. They suggest that regulatory capital requirements are blunt standards that respond only minimally to perceived differences in risk rather than the continuous prices and quantity limits set by uninsured creditors in response to changing perceptions of the risk of individual banks. The authors suggest an ideal system for setting capital standards but agree that it would be prohibitively expensive, if not impossible. Regulators lack precise estimates of social costs and benefits to tailor a capital requirement for each bank, and they cannot easily revise the requirements continuously as conditions change. The authors continue with suggestions for measuring regulatory capital more effectively. They suggest that a simple risk-based capital ratio is a relatively blunt tool for controlling bank risk-taking. The capital in the numerator may not always control bank moral hazard incentive; it is difficult to measure, and its measured value may be subject to manipulation by gains trading . The risk exposure in the denominator is also difficult to measure, corresponds only weakly to actual risk and may be subject to significant manipulation. These imprecisions worsen the social tradeoff between the externalities from bank failures and the quantity of bank intermediation. To keep bank risk to a tolerable level, capital standards must be higher on average than they otherwise would be if the capital ratios could be set more precisely, raising bank costs and reducing the amount of intermediation in the economy in the long run. Since actual capital standards are, at best, an approximation to the ideal, the authors argue that it should not be surprising that they may have had some unintended effects. They examine two unintended effects on bank portfolio risk or credit allocative inefficiencies. These two are the explosive growth of securitization and the so-called credit crunch by U.S. banks in the early 1990s. The authors show that capital requirements may give incentives for some banks to increase their risks of failure. Inaccuracies in setting capital requirements distort relative prices and may create allocative inefficiencies that divert financial resources from their most productive uses. During the 1980s, capital requirements may have created artificial incentives for banks to take off-balance sheet risk, and changes in capital requirements in the 1990s may have contributed to a credit crunch.

Orthogonal Polynomials from Hermitian Matrices

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.Comment: 53 pages, no figures. Several sentences and a reference are added. To be published in J. Math. Phy

Crash 2008

Causes and consequences of the current "subprime" financial crisis are analysed. The large number of fraudulent and criminal mortgages obtained by means of false statements unverified by the mortgagees, explains why this crisis was not predicted. The crisis was enhanced by unwise regulations like Marking-to-market of untraded assets and "Basel" capital requirements.

On Superstring Disk Amplitudes in a Rolling Tachyon Background

We study the tree level scattering or emission of n closed superstrings from a decaying non-BPS brane in Type II superstring theory. We attempt to calculate generic n-point superstring disk amplitudes in the rolling tachyon background. We show that these can be written as infinite power series of Toeplitz determinants, related to expectation values of a periodic function in Circular Unitary Ensembles. Further analytical progress is possible in the special case of bulk-boundary disk amplitudes. These are interpreted as probability amplitudes for emission of a closed string with initial conditions perturbed by the addition of an open string vertex operator. This calculation has been performed previously in bosonic string theory, here we extend the analysis for superstrings. We obtain a result for the average energy of closed superstrings produced in the perturbed background.Comment: 15 pages, LaTeX2e; uses latexsym, amssymb, amsmath, slashed macros; (v2): references added, some typo fixes; (v3): reference adde

Upper bounds for packings of spheres of several radii

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.Comment: 31 page

Multiplier Sequences for Simple Sets of Polynomials

In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequence for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward the partitioning of the set of classical multiplier sequences

Average characteristic polynomials in the two-matrix model

The two-matrix model is defined on pairs of Hermitian matrices $(M_1,M_2)$ of size $n\times n$ by the probability measure $$\frac{1}{Z_n} \exp\left(\textrm{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2)\right)\ dM_1\ dM_2,$$ where $V$ and $W$ are given potential functions and \tau\in\er. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices $M_1$ and $M_2$ may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials $p_n(x)$ and $q_n(y)$ associated to the two-matrix model; certain transformed functions $\P_n(w)$ and \Q_n(v); and finally Cauchy-type transforms of the four Eynard-Mehta kernels $K_{1,1}$, $K_{1,2}$, $K_{2,1}$ and $K_{2,2}$. In this way we generalize known results for the $1$-matrix model. Our results also imply a new proof of the Eynard-Mehta theorem for correlation functions in the two-matrix model, and they lead to a generating function for averages of products of traces.Comment: 28 pages, references adde