Formulas For Special Cases Research Articles

Analytical potential formulae and fast algorithm for a horn torus resistor network.

In this paper, a (u+1)×v horn torus resistor network with a special boundary is researched. According to Kirchhoff's law and the recursion-transform method, a model of the resistor network is established by the voltage V and a perturbed tridiagonal Toeplitz matrix. We obtain the exact potential formula of a horn torus resistor network. First, the orthogonal matrix transformation is constructed to obtain the eigenvalues and eigenvectors of this perturbed tridiagonal Toeplitz matrix; second, the solution of the node voltage is given by using the famous fifth kind of discrete sine transform (DST-V). We introduce Chebyshev polynomials to represent the exact potential formula. In addition, the equivalent resistance formulae in special cases are given and displayed by a three-dimensional dynamic view. Finally, a fast algorithm of computing potential is proposed by using the mathematical model, famous DST-V, and fast matrix-vector multiplication. The exact potential formula and the proposed fast algorithm realize large-scale fast and efficient operation for a (u+1)×v horn torus resistor network, respectively.

Fast algorithm and new potential formula represented by Chebyshev polynomials for an [Formula: see text] globe network.

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an [Formula: see text] globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.

Graham Higman's PORC theorem

Graham Higman published two important papers in 1960‎. ‎In the first of these‎ ‎papers he proved that for any positive integer $n$ the number of groups of‎ ‎order $p^{n}$ is bounded by a polynomial in $p$‎, ‎and he formulated his famous‎ ‎PORC conjecture about the form of the function $f(p^{n})$ giving the number of‎ ‎groups of order $p^{n}$‎. ‎In the second of these two papers he proved that the‎ ‎function giving the number of $p$-class two groups of order $p^{n}$ is PORC‎. ‎He established this result as a corollary to a very general result about‎ ‎vector spaces acted on by the general linear group‎. ‎This theorem takes over a‎ ‎page to state‎, ‎and is so general that it is hard to see what is going on‎. ‎Higman's proof of this general theorem contains several new ideas and is quite‎ ‎hard to follow‎. ‎However in the last few years several authors have developed‎ ‎and implemented algorithms for computing Higman's PORC formulae in‎ ‎special cases of his general theorem‎. ‎These algorithms give perspective on‎ ‎what are the key points in Higman's proof‎, ‎and also simplify parts of the proof‎. ‎In this note I give a proof of Higman's general theorem written in the light‎ ‎of these recent developments‎.

A class of square-free monomial ideals associated to two integer sequences

ABSTRACTGiven two finite sequences of positive integers α and β, we associate a square-free monomial ideal Iα,β in a ring of polynomials S, and we recursively compute the algebraic invariants of S∕Iα,β. Also, we give precise formulas in special cases.

Reversing Palindromic Enumeration in Rank-Two Free Groups

ABSTRACTThe Gilman–Maskit algorithm for determining the discreteness or non-discreteness of a two-generator subgroup of terminates with a pair of generators that are Farey words [Gilman and Maskit 91]. The Farey words are primitive words that are indexed by rational numbers and infinity. The so-called E-words [Gilman and Keen 11], primitive words with palindromic or palindromic product forms, are also indexed by rational numbers and infinity. We produce a modification of the Gilman–Maskit algorithm so that the stopping generators are E-words and what can be considered a new palindromic enumeration scheme. The original definition of the enumeration scheme can be implemented and run in a machine without any modification. However, every time a recursion calls itself, the state of the previous caller is stored until the recursion stops calling itself. It is often efficient for a recursion to minimize calling itself in order to avoid wasted resources such as time and storage space. The non-recursive formulas for special cases reduce the self-calling of the recursion.

On the number of N-free elements with prescribed trace

In this paper we derive a formula for the number of N-free elements over a finite field Fq with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive several explicit formulae in special cases. In addition we show that if all the prime factors of q−1 divide m, then the number of primitive elements in Fqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,m,N(c), of elements in Fqm with multiplicative order N and having trace c∈Fq.

A generalized linear-quadratic model incorporating reciprocal time pattern of radiation damage repair

It has been conventionally assumed that the repair rate for sublethal damage (SLD) remains constant during the entire radiation course. However, increasing evidence from animal studies suggest that this may not the case. Rather, it appears that the repair rate for radiation-induced SLD slows down with increasing time. Such a slowdown in repair would suggest that the exponential repair pattern would not necessarily accurately predict repair process. As a result, the purpose of this study was to investigate a new generalized linear-quadratic (LQ) model incorporating a repair pattern with reciprocal time. The new formulas were tested with published experimental data. The LQ model has been widely used in radiation therapy, and the parameter G in the surviving fraction represents the repair process of sublethal damage with T(r) as the repair half-time. When a reciprocal pattern of repair process was adopted, a closed form of G was derived analytically for arbitrary radiation schemes. The published animal data adopted to test the reciprocal formulas. A generalized LQ model to describe the repair process in a reciprocal pattern was obtained. Subsequently, formulas for special cases were derived from this general form. The reciprocal model showed a better fit to the animal data than the exponential model, particularly for the ED50 data (reduced χ(2) (min) of 2.0 vs 4.3, p = 0.11 vs 0.006), with the following gLQ parameters: α/β = 2.6-4.8 Gy, T(r) = 3.2-3.9 h for rat feet skin, and α/β = 0.9 Gy, T(r) = 1.1 h for rat spinal cord. These results of repair process following a reciprocal time suggest that the generalized LQ model incorporating the reciprocal time of sublethal damage repair shows a better fit than the exponential repair model. These formulas can be used to analyze the experimental and clinical data, where a slowing-down repair process appears during the course of radiation therapy.

Optimal partial hedging of options with small transaction costs

AbstractThis study uses asymptotic analysis to derive optimal hedging strategies for option portfolios hedged using an imperfectly correlated hedging asset with small fixed and/or proportional transaction costs, obtaining explicit formulae in special cases. This is of use when it is impractical to hedge using the underlying asset itself. The hedging strategy holds a position in the hedging asset whose value lies between two bounds, which are independent of the hedging asset's current value. For low absolute correlation between hedging and hedged assets, highly risk‐averse investors and large portfolios, hedging strategies and option values differ significantly from their perfect market equivalents. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 31:855–897, 2011

Optimal Partial Hedging of Options with Small Transaction Costs

This paper uses asymptotic analysis to derive optimal hedging strategies for option portfolios hedged using an imperfectly correlated hedging asset with small fixed and/or proportional transaction costs, obtaining explicit formulae in special cases. This is of use when it is impractical to hedge using the underlying asset itself. The hedging strategy holds a position in the hedging asset whose value lies between two bounds, which are independent of the hedging asset's current value. For low absolute correlation between hedging and hedged assets, highly risk-averse investors and large portfolios, hedging strategies and option values differ significantly from their perfect market equivalents.

WE‐D‐M100J‐02: A Closed Form of Linear‐Quadratic Model with Reciprocal‐Time for Radiation Damage Repair

Purpose: Repair of sublethal damage plays an important role in radiation therapy. Conventional radiobiology concepts are based on the assumption that the repair rate remains constant during the entire radiation course. However, increasing evidence from animal experiments report that the repair process may slow down with time and the data does not fit an exponential pattern. To address this dilemma, we developed a new closed form of Linear‐Quadratic (LQ) model based on the repair pattern with a reciprocal time. The new formulas were tested with published experimental data. Methods and Materials: The LQ model has been widely used in radiation therapy, . The parameter G represents the repair process of sublethal damage with Tr as the repair half‐time: in the reciprocal pattern. A closed form of G was derived analytically for arbitrary radiation schemes: dose rate . A set of published animal data was adopted to test the reciprocal formulas, in which rat foot skin was irradiated in split‐doses with increasing time intervals (0–22h). The complication data of moist desquamation were analyzed using the generalized LQ model. Results: A closed form of the LQ model to describe the repair process in a reciprocal pattern was obtained: , where function . Formulas for special cases were derived from this general form. The reciprocal model showed a better fit to the animal data than the exponential model, especially for the ED50 data (reduced of 2.0 vs. 4.3, p=0.11 vs. 0.006), with the following LQ parameters: α/β = 2.6–4.8Gy, Tr= 3.2–3.9h. Conclusions: The repair process following a reciprocal‐time has been investigated in this study and a closed form of the generalized LQ model was presented and validated. These formulas can be used to analyze the experimental and clinical data, where a slowing‐down repair process appears during the radiation course.

Strength and deformation of arbitrary beam sections using adaptive FEM

Still today, structural design is often based on simplified hand-calculation formulas for special cases, which is tedious to apply and do not cover more general situations. To improve the situation, the paper presents a FE-program for analysis of strength and deformation of arbitrary beam cross sections of steel and concrete loaded by all six force and moment components. The program is based on (1) an enhanced beam theory with three-dimensional state of stress, (2) multi-surface elasto-plasticity for modeling of the materials and (3) efficient numerical algorithms (mesh adaptive FEM, arc-length method, etc.). The paper presents the theory behind the program and some initial numerical testing to show how the program works. The initial numerical testing on steel and reinforced concrete beam sections loaded in bending, shear and torsion show promising results.

Comment on “Eigenfunction expansion of the dyadic Green’s function in a gyroelectric chiral medium by cylindrical vector wave functions”

Several conceptual errors found in the paper by Cheng [Phys. Rev. E 55, 1950 (1997)] are corrected, and we provide general expansion formulas for the dyadic Green's function using cylindrical vector wave functions, which could be reduced to the exact formulas in special cases (i.e., anisotropic medium, isotropic chiral medium, and isotropic medium).

Accurate computation of individual and tables of 3-j and 6-j symbols

By rewriting the formulas for 3-j and 6-j symbols in terms of several possible alternating binomial sums, it is possible to calculate these quantities quickly and accurately, often exactly, using floating point operations. The binomial sums can be calculated by direct summation or by recursion. A simple method for uniquely parameterizing the well-known Regge symmetries of the 3-j and 6-j symbols makes it possible to systematize the choice of the smallest magnitude binomial sum (which enhances the accuracy of floating point calculations and speeds up exact calculations using large integer routines). Formulas for special cases of the 3-j symbols enable the construction of recursion sequences which are often substantially faster than direct summation, especially for very large angular momentum arguments. For both 3-j and 6-j symbols, recursion offers several advantages over direct summation in exact calculations and for calculating tables.

Strategic analysis of contingent claims

Traditional contingent claims analysis provides an elegant and complete model of the financing of the dynamic firm but has difficulty fitting the observations in the market. Recent work in corporate finance suggests this is because of its failure to model financial distress realistically. We survey recent efforts to address this issue by deriving valuation formulae from the analysis of non-cooperative equilibria in extensive form games. We illustrate how a simple static bankruptcy model can be incorporated in a dynamic game. We then show how to find the continuous time equivalent of this game. This we use to find closed-form bond formulae in special cases and to exploit efficient numerical techniques generally.

General theory of Maker fringes in crystals of low symmetry

A set of general equations for Maker fringes, applicable to all crystal structures and orientations, has been derived. Alternative forms of the equations are given, suitable for numerical work or for analytical reduction in cases where symmetry or other factors reduce their complexity. Formulae for special cases have been obtained from the general equations and the results compared with previously published work. The case of monoclinic symmetry has been discussed in more detail and a numerical example of the application of the formulae to the study of the d-coefficients is presented.