Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes

Based on the cavity-mode model, we have developed a fast algorithm for calculating power bus impedance in multilayer printed circuit boards. The fast algorithm is based on a closed-form expression for the impedance Z matrix of a rectangular power bus structure; this expression was obtained by reducing the original double infinite series into a single infinite series under an approximation. The convergence of the single series is further accelerated analytically. The accelerated single summation enables much faster computation, since use of only a few terms is enough to obtain good accuracy. In addition, we propose two ways to compensate for the error due to the approximation involved in the process of reducing the double series to the single series, and have demonstrated that these two techniques are almost equivalent

Dual Actions for Born-Infeld and Dp-Brane Theories

Dual actions with respect to U(1) gauge fields for Born-Infeld and $Dp$-brane theories are reexamined. Taking into account an additional condition, i.e. a corollary to the field equation of the auxiliary metric, one obtains an alternative dual action that does not involve the infinite power series in the auxiliary metric given by ref. \cite{s14}, but just picks out the first term from the series formally. New effective interactions of the theories are revealed. That is, the new dual action gives rise to an effective interaction in terms of one interaction term rather than infinite terms of different (higher) orders of interactions physically. However, the price paid for eliminating the infinite power series is that the new action is not quadratic but highly nonlinear in the Hodge dual of a $(p-1)$-form field strength. This non-linearity is inevitable to the requirement the two dual actions are equivalent.Comment: v1: 11 pages, no figures; v2: explanation of effective interactions added; v3: concision made; v4: minor modification mad

A Mathematical Model of Divine Infinity

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology

Some identities involving the k-th power complements

The main purpose of this paper is using the elementary method to study the calculating problem of one kind infinite series involving the k-th power complements, and obtain several interesting identities

Congruences Among Power Series Coefficients of Modular Forms

Many authors have investigated the congruence relations amongst the coefficients of power series expansions of modular forms $f$ in modular functions $t$. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Ap\'ery numbers associated with the irrationality of $\zeta(3)$. We show that many of the examples of Osburn and Sahu are members of infinite families.Comment: 24 page

Stochastic HJB Equations and Regular Singular Points

IIn this paper we show that some HJB equations arising from both finite and infinite horizon stochastic optimal control problems have a regular singular point at the origin. This makes them amenable to solution by power series techniques. This extends the work of Al'brecht who showed that the HJB equations of an infinite horizon deterministic optimal control problem can have a regular singular point at the origin, Al'brekht solved the HJB equations by power series, degree by degree. In particular, we show that the infinite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a new type of algebraic Riccati equation which we call the Stochastic Algebraic Riccati Equation (SARE). If SARE can be solved then one has a complete solution to this infinite horizon stochastic optimal control problem. We also show that a finite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a Stochastic Differential Riccati Equation (SDRE) that is well known. If these problems are the linear-quadratic-bilinear part of a nonlinear finite horizon stochastic optimal control problem then we show how the higher degree terms of the solutions can be computed degree by degree. To our knowledge this computation is new