Ten commandments for good default expression simplification

This article provides goals for the design and improvement of default computer algebra expression simplification. These goals can also help users recognize and partially circumvent some limitations of their current computer algebra systems. Although motivated by computer algebra, many of the goals are also applicable to manual simplification, indicating what transformations are necessary and sufficient for good simplification when no particular canonical result form is required.After motivating the ten goals, the article then explains how the Altran partially factored form for rational expressions was extended for Derive and for the computer algebra in Texas Instruments products to help fulfill these goals. In contrast to the distributed Altran representation, this recursive partially factored semi-fraction form:

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does not unnecessarily force common denominators,

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discovers and preserves significantly more factors,

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can represent general expressions, and

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can produce an entire spectrum from fully factored over a common denominator through complete multivariate partial fractions, including a dense subset of all intermediate forms.

     

On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces

In this paper, we investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. The results are obtained by using the α-resolvent family (S_α(t))_t≥0 on a complex Banach space X combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.     

On a new class of analytic functions associated with conic domain

The aim of this paper is to generalize the conic domain defined by Kanas and Wisniowska, and define the class of functions which map the open unit disk E onto this generalized conic domain. A brief comparison between these conic domains is the main motivation of this paper. A correction is made in selecting the range interval of order of conic domain.     

Algorithms to Separate $\{0,\frac{1}{2}\}$ -Chvátal-Gomory Cuts

Chvátal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or , such cuts are known as -cuts. It has been proven by Caprara and Fischetti (Math. Program. 74:221–235, 1996) that separation of -cuts is -hard. In this paper, we study ways to separate -cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated -cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.     

Maximally equidistributed pseudorandom number generators via linear output transformations

This article discusses construction of maximally equidistributed (ME) linear feedback shift register generators by using linear output transformations. We introduce a new strategy to find linear output transformations attaining the ME property, and apply it to the WELL generators and the Mersenne Twister. As a result, we found a modified version of the WELL generators with the ME property, having fewer operations than the original ME WELL generators. We also found a ME Mersenne Twister which is faster than the WELL generators with a period of 2¹⁹⁹³⁷−1$2^{19937}-1$ on some platforms. The C codes of the obtained generators are available at http://www3.ocn.ne.jp/∼harase/megenerators.html.     

On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors

In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations.     

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.     

数字示波器参数测量算法的设计与优化

在对原有参数测量算法进行误差分析的基础上,该文提出了”所见即所测”的测量原则。并根据此原则重新设计了参数测量的软件方案以及相关映射算法,从而扩大了测量范围,提高了测量精度,并减小了测量误差。采用本参数测量方案的产品已投放市场。     

An alternative Kalman-Yakubovich-Popov lemma and some extensions

This paper introduces an alternative formulation of the Kalman-Yakubovich-Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this new formulation encompasses previous generalizations of the KYP Lemma which hold in the case the coefficient matrix of the FDI does not depend on frequency. In addition, it allows the coefficient matrix of the frequency domain inequality to vary affinely with the frequency parameter. One application of this results is illustrated in an example of computing upper bounds to the structured singular value with frequency-dependent scalings.