A simultaneous projections method for linear inequalities

An iterative method for solving general systems of linear inequalities is considered. The method, a relaxed generalization of Cimmino's scheme for solving linear systems, was first suggested by Censor and Elfving. Each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities. The algorithm is particularly suitable for implementation on computers with parallel processors. We prove convergence from any starting point for both consistent and nonconsistent systems (to a feasible point in the first case, and to a weighted least squares type solutions in the second).     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

Doubly nilpotent numbers in the 2D plane

Dual numbers, split-quaternions, split-octonions, and other number systems with nilpotent spaces have received sporadic yet persistent interest, beginning from their roots in the 19th century, to more recent attention in connection with supersymmetry in physics. In this paper, a number system in the 2D plane is investigated, where the squares of its basis elements p and q each map into the coordinate origin. Modeled similarly to an original concept by C. Musès, this new system will be termed “PQ space” and presented as a generalization of nilpotence and zero. Compared to the complex numbers, its multiplicative group and underlying vector space are equipped with as little as needed modifications to achieve the desired properties. The locus of real powers of basis elements p^α and q^α resembles a four-leaved clover, where the coordinate origin at (0, 0) will not only represent the additive identity element, but also a map of “directed zeroes” from the multiplicative group. Algebraic and geometric properties of PQ space are discussed, and its naturalness advertised by comparison with other systems. The relation to Musès’ “p and q numbers” is shown and its differences defended. Next to possible applications and extensions, a new butterfly-shaped fractal is generated from a recursion algorithm of Mandelbrot type.     

Generalized Zernike or disc polynomials: An application in quantum optics

An application of the “generalized Zernike or disc polynomials”, recently introduced in the literature, is shown, resorting to the Lie algebra based investigation of the dynamics of quantum systems driven by two-mode interaction Hamiltonians. Further properties of the associated “disc functions” are deduced. Also, a generalization of the disc polynomials towards the Hahn polynomials is suggested.     

Weyl groupoids with at most three objects

We adapt the generalization of root systems by the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields the result that there exist infinitely many “standard”, but only 9 “exceptional” cases.     

Global properties of the triangular systems in the singular case

We introduce a new class of the triangular (multi-input and multi-output) control systems, of O.D.E., which are not feedback linearizable, and investigate its global behavior. The triangular form introduced is a generalization of the classes of triangular systems, considered before. For our class, we solve the problem of global robust controllability. Combining our main result with that of [F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag, A.I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control 42 (1997) 1394-1407], we obtain a corollary on the global discontinuous sampled stabilization (an example showing that global smooth stabilization can be irrelevant to the singular case is considered). To prove our main result, we apply a certain “back-stepping” algorithm and combine the technique proposed in [V.I. Korobov, S.S. Pavlichkov, W.H. Schmidt, Global robust controllability of the triangular integro-differential Volterra systems, J. Math. Anal. Appl. 309 (2005) 743-760] with solving a specific problem of global “practical stabilization” by means of a discontinuous, time-varying feedback law.     

Characterizations of the optimal stable allocation mechanism

The stable allocation problem is the generalization of (0,1)-matching problems to the allocation of real numbers (hours or quantities) between two separate sets of agents. The same unique-optimal matching (for one set of agents) is characterized by each of three properties: “efficiency”, “monotonicity”, and “strategy-proofness”.     

Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Vertex-weightings for distance moments and thorny graphs

Valence-weightings are considered for shortest-path distance moments, as well as related weightings for the so-called “Wiener” polynomial. In the case of trees the valence-weighted quantities are found to be expressible as a combination of unweighted quantities. Further the weighted quantities for a so-called “thorny” graph are considered and shown to be related to the weighted and unweighted quantities for the underlying parent graph.     

General Expa-function method for nonlinear evolution equations

This work generalizes the exponential function method in considering an arbitrary base “a” as opposed to the conventional base “e” for the exponential function. The combined KdV-mKdV equation is considered to reveal the effectiveness and convenience of the proposed generalization. The study highlights the power of the proposed method on constructing solutions expressed in terms of exponential, hyperbolic, periodic, symmetrical Fibonacci, symmetrical Lucas, and k-Fibonacci functions. Some of the obtained solitary wave solutions are sketched graphically.     

A generalization of parameterized inexact Uzawa method for generalized saddle point problems

Recently, a class of parameterized inexact Uzawa methods has been proposed for generalized saddle point problems by Bai and Wang [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932], and a generalization of the inexact parameterized Uzawa method has been studied for augmented linear systems by Chen and Jiang [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. (2008)]. This paper is concerned about a generalization of the parameterized inexact Uzawa method for solving the generalized saddle point problems with nonzero (2, 2) blocks. Some new iterative methods are presented and their convergence are studied in depth. By choosing different parameter matrices, we derive a series of existing and new iterative methods, including the preconditioned Uzawa method, the inexact Uzawa method, the SOR-like method, the GSOR method, the GIAOR method, the PIU method, the APIU method and so on. Numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact Uzawa methods.     

The Robin problem for the Laplace equation in a three-dimensional starlike domain

The internal and external Robin problems for the Laplace equation in bounded starlike domains are addressed. We show how to derive the relevant solutions by using a suitable Fourier series-like method. Numerical results are specifically obtained considering three-dimensional domains whose boundary is defined by a generalization of the so-called “superformula” introduced by Gielis. By using the computer algebra code Mathematica©, truncated series approximations of the solutions are determined. Our findings are in good agreement with the theoretical results on the Fourier series due to Carleson.     

The inequality-satisfiability problem

We define a generalization of the satisfiability problem (SAT) where each “clause” is an or-list of inequalities in n variables. This problem is NP-complete even when containing only two inequalities per “clause”, but solvable in polynomial time when either the number of variables or the number of “clauses” is fixed.     

How ordinary elimination became Gaussian elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — which Euler did not recommend, which Legendre called “ordinary,” and which Gauss called “common” — is now named after Gauss: “Gaussian” elimination. Gauss’s name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least-squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.     

Local radial point interpolation method for the fully developed magnetohydrodynamic flow

In this paper, a local radial point interpolation method (LRPIM) is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight duct of rectangular section with arbitrary wall conductivity and orientation of applied magnetic field. Local weak forms are developed using weighted residual method locally for the governing equations of fully developed MHD flow. The shape functions from LRPIM possess the delta function property. Therefore, essential boundary conditions can be applied as easily as that in the finite-element method. The implementation procedure of LRPIM method is node based, and it doesn’t need any “mesh” or “element”. Computations have been carried out for different Hartmann numbers, wall conductivities and orientations of applied magnetic field.     

线性方程组迭代方法残差光滑技术的一个推广

1　引　　言我们考虑求解线性方程组Ax=b,A∈Rn×n,b,x∈Rn.(1)的迭代方法.迭代序列{xk}的性态常常由与之对应的残差范数序列{‖rk‖}的特性来决定.人们自然希望{‖rk‖}光滑地(单调地)收敛到0.在所有Krylov子空间方法中,GMRES[7]方法因为可使{‖rk‖}最优地趋于0,故是一个较为成功的方法.但是,GMRES方法的工作量和存贮量却随着迭代步数的增加而迅速增加.而BCG[4]和CGS[10]等方法具有运算量小,收敛快等突出优点.但它们的残差范数性态却很不规则,{‖rk‖}振荡不定.这给判断收敛性及何时停机带来很大的不便.残差光滑技术是一个行之有…     

Generalization of Stanley's monster reciprocity theorem

By studying the reciprocity property of linear Diophantine systems in light of Malcev-Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.     

Exchange lemmas 1: Deng's lemma

Deng's lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium. We prove a generalization in which “partially hyperbolic equilibrium” is replaced by “normally hyperbolic invariant manifold.”     

A method for efficiently computing the number of codewords of fixed weights in linear codes

The problem of computing the number of codewords of weights not exceeding a given integer in linear codes over a finite field is considered. An efficient method for solving this problem is proposed and discussed in detail. It builds and uses a sequence of different generator matrices, as many as possible, so that the identity matrix takes disjoint places in them. The efficiency of the method is achieved by optimizations in three main directions: (1) the number of the generated codewords, (2) the check whether a given codeword is generated more than once, and (3) the operations for generating and computing these codewords. Since the considered problem generalizes the well-known problems “Weight Distribution” and “Minimum Distance”, their efficient solutions are considered as applications of the algorithms from the method.     

An efficient algorithm for solving rank one perturbed linear Diophantine systems using Rosser��s approach

Recently, we described a generalization of Rosser’s algorithm for a single linear Diophantine equation to an algorithm for solving systems of linear Diophantine equations. Here, we make use of the new formulation to present a new algorithm for solving rank one perturbed linear Diophantine systems, based on using Rosser’s approach. Finally, we compare the efficiency and effectiveness of our proposed algorithm with the algorithm proposed by Amini and Mahdavi-Amiri (Optim Methods Softw 21:819–831, 2006).