Binary Darboux transformations andN-wave systems in rings

The covariance theorems for elementary and binary Darboux transformations in rings are formulated and proved for generalized Zakharov-Shabat problems. The definition of the elementary Darboux transformation is extended to an arbitrary number of orthogonal idempotents. The binary transformation is defined as a sequence of elementary transformations for direct and conjugate problems. The heredity property for the reduction constraints is established for some UV pairs in rings; hence, the transformation generates solutions and infinitesimal symmetries of the corresponding zero-curvature equations. The explicit expressions for the transformations, solitons, and infinitesimals are given in the general case and in physically significant cases of extended non-AbelianN-wave equations (with linear terms added). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 239–250, February, 2000.     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.     

Moving space curve equations and a family of coupled KdV type systems

It is known that the Hasimoto transformation of Heisenberg ferromagnetic system yields nonlinear Schrödinger equation. In this paper we investigate a series of moving space curve equations whose Hasimoto transformations are KdV type equations. In this paper we also propose their Lax pairs.     

On the stability of the J transformation

The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the _p {J} transformations, the Weniger S transformation, the Levin transformation and a special case of the generalized Richardson extrapolation process of Sidi are S-stable. An efficient algorithm for the calculation of stability indices is presented. A numerical example demonstrates the validity of the approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Lax pairs and Bäcklund transformations for a coupled Ramani equation and its related system

A coupled Ramani equation and its related system are proposed. By dependent variable transformation, they are transformed into bilinear equations. Lax pairs and Bäcklund transformations are presented for these two systems. Soliton solutions and rational solutions to the systems could be obtained.     

A block encryption algorithm based on dynamic sequences of multiple chaotic systems

A block encryption algorithm using dynamic sequences generated by multiple chaotic systems is proposed in this paper. In this algorithm, several one-dimension chaotic maps generate pseudo-random sequences, which are independent and approximately uniform. After a series of transformations, the sequences constitute a new pseudo-random sequence uniformly distributing in the value space, which covers the plaintext by executing Exclusive-OR and shifting operations some rounds to form the cipher. This algorithm makes the pseudo-random sequence possess more concealment and noise like characteristic, and overcomes the periodic malpractice caused by the computer precision and single chaotic system. Simulation results show that the algorithm is efficient and useable for the security of communication system.     

Limit theorems for the multivariate binomial distribution

Nonsingular limit distributions are determined for sequences of affine transformations of random vectors whose distributions are multivariate binomial. Each of these limit distributions is that of an affine transformation of a random vector having a multivariate normal distribution or a multivariate Possion distribution or a joint distribution of two independent random vectors, one normal and the other Poisson.     

Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

A new algorithm for pole assignment of single-input linear systems using state feedback

In this paper we present a new algorithm for the single-input pole assignment problem using state feedback. This algorithm is based on the Schur decomposition of the closed-loop system matrix, and the numerically stable unitary transformations are used whenever possible, and hence it is numerically reliable.The good numerical behavior of this algorithm is also illustrated by numerical examples.     

Accuracy of Gram–Schmidt orthogonalization and Householder transformation for the solution of linear least squares problems

Accuracy of a Gram–Schmidt algorithm for the solution of linear least squares equations is compared with accuracy of least squares subroutines in three highly respected mathematical packages that use Householder transformations. Results from the four programs for 13 test problems were evaluated at 16 digit precision on four different desktop computers using four different compilers. Singular values obtained from the different programs are compared and the effect of pivoting to improve the accuracy is discussed. Solution vectors from the program using the Gram–Schmidt algorithm were generally more accurate or comparable to solution vectors from the programs using the Householder transformations. © 1997 John Wiley & Sons, Ltd.     

Interval versions of Milne’s multistep methods

In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski–Durbin–Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented.     

An axiomatic semantics for nested concurrency

We give transformation rules for the concurrent parts of Hoare's language CSP, transforming concurrent CSP programs into nondeterministic, sequential programs.On the basis of these transformations we define an axiomatic semantics for CSP with nested concurrency.This axiomatic system includes a rule for binary, associative process composition, enabling modular verification dealing with parts of concurrent systems as well as full programs.The proof system is fully abstract, in the sense that the internal structure of processes is irrelevant in the specification inasmuch it is not externally observable.An outline of a verification of a recursive, concurrent sorter is given as an example.     

On certain properties of linear iterative equations

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator that generates an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. An expression for the parameters of the source equation of the transformed equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form is obtained in terms of the simplest determining equation, and several examples of application are discussed.     

VPAStab: Stabilised Vector-Padé Approximation with Application to Linear Systems

An algorithm called VPAStab is given for the acceleration of convergence of a sequence of vectors. It combines a method of vector-Padé approximation with a successful technique for stabilisation. More generally, this algorithm is designed to find the fixed point of the generating function of the given sequence of vectors, analogously to the way in which ordinary Padé approximants can accelerate the convergence of a given scalar sequence. VPAStab is justified in the context of its application to the solution of a large sparse system of linear equations. The possible breakdowns of the algorithm are listed. Numerical experiments indicate that these breakdowns can be classified either as pivot-type (type L) or as ghost-type (type D).     

Positivity of linear transformations of mean-starshaped sequences

In this paper, we give the necessary and sufficient conditions for a linear transformation of a mean-starshaped sequence to be positive. Using this result, we obtain the necessary and sufficient conditions for a lower triangular matrix to preserve the mean-starshape of a sequence and we discuss some special cases of linear transformations. Our next result deals with the convergence of a sequence of mean-starshaped sequences to any given mean-starshaped sequence and the positivity of a linear operator on the set of mean-starshaped sequences.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

An indirect adaptive pole place for MIMO systems based on multirate sampling of the plant output

The use of multirate-output controllers in order to achieveadaptive pole placement in linear multiple-input multiple-outputsystems with unknown parameters is investigated for the firsttime. Multirate-output controllers contain a multirate samplingmechanism with different sampling period at each system output.Such {acute} control allows us to assign the poles of the sampledclosed-loop system arbitrarily in desired locations, and doesnot make assumptions on the plant other than controllability,observability, and the knowledge of two sets of structural indices,namely the controllability and the observability indices. Anindirect adaptive control scheme based on these sampled-datacontrollers is proposed, which estimates the controller parameterson line. Using the proposed adaptive algorithm, the problemof adaptive pole placement is reduced to the determination ofa fictitious static state-feedback controller, due to the meritsof dynamic multirate-output controllers. Known techniques resortto the direct computation of dynamic controllers. The controllerdeterminations here is based on the transformation of the discreteanalogue of the systems under control to a phase-variable canonicalform prior to the applications of the control design procedure.The solution of the problem can be obtained by a quite simpleutilization of the concept of state similarity transformation,whereas known techniques usually require the solution of matrixpolynomial Diophantine equations. Moreover, persistent excitationof the continuous-time plant is provided without assuming anyspecial richness of the reference signals.     

A Newton-type curvilinear search method for constrained optimization

An algorithm is presented that minimizes a nonlinear function in many variables under equality constraints by generating a monotonically improving sequence of feasible points along curvilinear search paths obeying an initialvalue system of differential equations. The derivation of the differential equations is based on the idea of a steepest descent curve for the objective function on the feasible region. Our method for small stepsize behaves as the generalized reduced gradient algorithm, whereas for large enough stepsize the constrained equivalent of Newton's method for unconstrained minimization is obtained.     

Acceleration property for the E-algorithm and an application to the summation of series

The E-algorithm is the most general sequence transformation actually known, since it contains as particular cases almost all the sequence transformations discovered so far: Richardson polynomial extrapolation, Shanks’ transformation, summation processes, Germain-Bonne transformation, Levin’s generalized transformations, the processp and rational extrapolation. In [10] some results concerning the columns of the E-algorithm were proved. In this paper, by adding conditions about determinants, we prove that the diagonal of this algorithm also accelerates the convergence of the initial sequence.     

Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.