Modified Laguerre operational matrices for fractional calculus and applications

The modified Laguerre polynomial is defined with an additional parameter from the conventional one and is applied to approach the problems of fractional calculus. First, the operational matrices for the integration and the differentiation of the modified Laguerre polynomials are derived. The generalized operational matrices corresponding to s, 1/s, s/(s₂ + 1) ^1/2 exp [ ? s/(s + 1)] are derived as examples. Comparison of the modified Laguerre series approximate inversions of irrational Laplace transforms with exact solution shows that the present modification method is much better than the conventional one. In addition, the present proposed modified Laguerre polynomials can also be used to approximate the solution of fractional calculus which cannot be obtained from conventional Laguerre polynomials.     

Generalization of Cayley-Hamilton theorem for n-D polynomial matrices

The Cayley-Hamilton theorem is extended to real polynomial matrices in n variables. The known extensions of the classical Cayley-Hamilton theorem are particular cases of the proposed extension.     

Bayesian generalized probability calculus for density matrices

One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions can be seen as a special case when the density matrix is restricted to be diagonal. We develop a probability calculus based on these more general distributions that includes definitions of joints, conditionals and formulas that relate these, including analogs of the Theorem of Total Probability and various Bayes rules for the calculation of posterior density matrices. The resulting calculus parallels the familiar “conventional” probability calculus and always retains the latter as a special case when all matrices are diagonal. We motivate both the conventional and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the conventional Bayes rule and Umegaki’s quantum relative entropy the new Bayes rule for density matrices. Whereas the conventional Bayesian methods maintain uncertainty about which model has the highest data likelihood, the generalization maintains uncertainty about which unit direction has the largest variance. Surprisingly the bounds also generalize: as in the conventional setting we upper bound the negative log likelihood of the data by the negative log likelihood of the MAP estimator.     

Derivation of operational matrices of differentiation for orthogonal polynomials

A direct method for deriving an operational matrix of differentiation for Legendre polynomials is presented. The general case is also considered. The presented formulas are formally proved. Results are compared with the other known method. The new method is accurate for all matrix dimensions, while the other method may fail for higher dimensions.     

The generalized Sylvester equation in polynomial matrices

Jordan chains of polynomial matrices are used to characterize the solvability of the generalized Sylvester equation     

Generalization of Leverrier's algorithm to polynomial matrices of arbitrary degree

Leverrier's algorithm to find the determinant of (sI_{r}-A) is generalized to the polynomial matrices of arbitrary degree which are in the formI_{r}s^{n} + H_{n-1}s^{n-1} + ... + H_{1}s + H_{0}.     

System identification via generalized block pulse operational matrices

Owing to the elegant properties of generalized block pulse operational matrices, a SID (system identification) matrix can be derived to identify transfer function coefficients of a linear time-invariant system from the unit step response data. The identification algorithm performs in a least-squares sense and can be recursive according to previous work. This approach not only yields satisfactory results but also provides a more precise and efficient method for the system identification problem.     

Application of generalized block-pulse operational matrices for the approximation of continuous-time systems

The central theme of this paper is to apply generalized block-pulse operational matrices to approximate continuous-time systems. Generalized block-pulse operational matrices and the Routh approximation method are used together to find a low-order transfer function to approximate the original high-order transfer function. The Routh approximation method is used to preserve the stability of the original system by first determining the denominator coefficients of the reduced-order system. Generalized block-pulse operational matrices are then applied to determine numerator coefficients of the reduced-order system by optimally matching the unit step responses of the original and reduced-order systems. This new constrained time-domain matching approach not only yields more satisfactory results than previous methods, but also provides a more straightforward and efficient method for the approximation of continuous-time systems.     

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

Many of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Lower bounds for the polynomial calculus

We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle must have degree at least over any field. This is the first non-trivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function in n variables, every polynomial calculus proof of the statement “ cannot be computed by any circuit of size t,” must have degree . Loosely speaking, this means that low degree polynomial calculus proofs do not prove . Received: January 15, 1997.     

Homogenization and the polynomial calculus

In standard implementations of the Gröbner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the proof complexity of refutations of polynomials derived from Boolean formulas in both the Polynomial Calculus (PC) and Nullstellensatz systems. We show that the PC refutations of homogenized formulas give crucial information about the complexity of the original formulas. The minimum PC refutation degree of homogenized formulas is equal to the Nullstellensatz refutation degree of the original formulas, whereas the size of the homogenized PC refutation is equal to the size of the PC refutation for the originals. Using this relationship, we prove nearly linear ((n/log n) vs. O(1)) separations between Nullstellensatz and PC degree, for a family of explicitly constructed contradictory 3CNF formulas. Previously, an (n ^1/2) separation had been proved for equations that did not correspond to any CNF formulas, and a log n separation for equations derived from kCNF formulas.     

豪斯道夫微积分和分数阶微积分模型的分形分析

清晰解读豪斯道夫微积分和分数阶微积分阶数的分形维意义,并比较这2种微积分建模方法的区别与联系.这是首次清晰定量地导出分数阶微积分的分形几何基础.提供豪斯道夫导数模型描述历史依赖过程的几何解释,即初始时刻依赖性问题,并与分数阶导数模型对比.基于本文作者的早期工作,详细描述非欧几里得距离的豪斯道夫分形距离定义——豪斯道夫导数扩散方程的基本解就是基于该豪斯道夫分形距离.该基本解实质上就是目前广泛使用的伸展高斯分布和伸展指数衰减统计模型.     

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

A division algorithm for polynomial matrices

A new algorithm is presented for directly determining both the quotient and the remainder associated with the division of one polynomial matrix by another. The procedure requires that the denominator matrix be column proper with nonzero column degrees. However, unlike earlier algorithms, only real matrix multiplications are employed; i.e., there is no need for matrix inversions, either explicit or implicit.     

Stability of polynomial matrices

A necessary and sufficient condition for the stability of a polynomial matrix is presented and is illustrated by an example.     

D-stability of polynomial matrices

Necessary and sufficient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing quadratic and semidefinite programming, LFRs and rank-one LMIs. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the D-stability conditions can be combined with other LMI conditions arising in robust stability analysis.     

An operational semantics for a calculus for wireless systems

In wireless systems, the communication mechanism combines features of broadcast, synchrony, and asynchrony. We develop an operational semantics for a calculus of wireless systems. We present different Reduction Semantics and a Labelled Transition Semantics and prove correspondence results between them. Finally, we apply CWS to the modelling of the Alternating Bit Protocol, and prove a simple correctness result as an example of the kind of properties that can be formalized in this framework.A major goal of the semantics is to describe the forms of interference among the activities of processes that are peculiar of wireless systems. Such interference occurs when a location is simultaneously reached by two transmissions. The Reduction Semantics differ on how information about the active transmissions is managed.We use the calculus to describe and analyse a few properties of a version of the Alternating Bit Protocol.     

Algorithm for obtaining the proper relatively prime matrices of polynomial matrices

An algorithm for obtaining the proper relatively prime matrices of polynomial matrices by means of the concept of coefficient-matrix row (column) position is introduced.