Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

In this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.     

Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.     

An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations

We propose an efficient time-splitting Chebyshev-Tau spectral method for the Ginzburg-Landau-Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg-Landau-Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev-Tau spectral discretization in space and Crank-Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O(Nlog(N)) compared with that of the non-optimized one, which is O(N²). By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg-Landau-Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg-Landau-Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg-Landau-Schrödinger equation are also presented.     

Lump solutions to a (2+1)-dimensional extended KP equation

With the aid of a computer algebra system, we present lump solutions to a ($2+1$)-dimensional extended Kadomtsev–Petviashvili equation (eKP) and give the sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions. We plot a few solutions for some specific values of the free parameters involved and finally derive one of the lump solutions of the Kadomtsev–Petviashvili (KP) equations from the lump solutions of the eKP equation.     

On generalized Melvin’s solutions for Lie algebras of rank 2

We consider a class of solutions in multidimensional gravity which generalize Melvin’s well-known cylindrically symmetric solution, originally describing the gravitational field of a magnetic flux tube. The solutions considered contain the metric, two Abelian 2-forms and two scalar fields, and are governed by two moduli functions H₁(z) and H₂(z) (z = ρ², ρ is a radial coordinate) which have a polynomial structure and obey two differential (Toda-like) master equations with certain boundary conditions. These equations are governed by a certain matrix A which is a Cartan matrix for some Lie algebra. The models for rank-2 Lie algebras A₂, C₂ and G₂ are considered. We study a number of physical and geometric properties of these models. In particular, duality identities are proved, which reveal a certain behavior of the solutions under the transformation ρ → 1/ρ; asymptotic relations for the solutions at large distances are obtained; 2-form flux integrals over 2-dimensional regions and the corresponding Wilson loop factors are calculated, and their convergence is demonstrated. These properties make the solutions potentially applicable in the context of some dual holographic models. The duality identities can also be understood in terms of the Z₂ symmetry on vertices of the Dynkin diagram for the corresponding Lie algebra.     

Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls

This paper investigates nonlocal problems for a class of fractional integrodifferential equations via fractional operators and optimal controls in Banach spaces. By using the fractional calculus, Hölder inequality, p-mean continuity and fixed point theorems, some existence results of mild solutions are obtained under the two cases of the semigroup T(t), the nonlinear terms f and h, and the nonlocal item g. Then, the existence conditions of optimal pairs of systems governed by a fractional integrodifferential equation with nonlocal conditions are presented. Finally, an example is given to illustrate the effectiveness of the results obtained.     

Positive solutions to a nonlinear abel-type integral equation on the whole line

We consider the existence of positive solutions to the nonlinear integral equation

where g is a continuous, nondecreasing function such that g(0) = 0. We show that the equation always has nontrivial solutions and we give a necessary and sufficient condition for the existence of solutions u such that u(x) > − ∞. We also provide a condition which ensures that all the nontrivial solutions experience the blow-up behaviour.     

To a nonstationary group pursuit problem with phase constraints

We consider two linear nonstationary pursuit-evasion problems with one evader and a group of pursuers under the conditions that the players have equal dynamic abilities and that the evader cannot leave a certain set. We prove that if the number of pursuers is less than the space dimension, then the evader can avoid capture in the interval [t ₀,∞).     

Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

Let X = Cⁿ. In this paper we present an algorithm that computes the de Rham cohomology groups H_dRⁱ(U,C ) where U is the complement of an arbitrary Zariski-closed set Y in X. Our algorithm is a merger of the algorithm given inOaku and Takayama (1999), who considered the case where Y is a hypersurface, and our methods from Walther (1999) for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with supports H_{dR, Z}ⁱ(U,C ) where again U is an arbitrary Zariski-open subset of X and Z is an arbitrary Zariski-closed subset of U. Our main tool is a generalization of the restriction process from Oaku and Takayama (in press) to complexes of modules over the Weyl algebra. The restriction rests on an existence theorem onV_d -strict resolutions of complexes that we prove by means of an explicit construction via Cartan–Eilenberg resolutions. All presented algorithms are based on Gröbner basis computations in the Weyl algebra and the examples are carried out using the computer system Kan by Takayama (1999).     

First order linear fuzzy differential equations under generalized differentiability

First order linear fuzzy differential equations are investigated. We interpret a fuzzy differential equation by using the strongly generalized differentiability concept, because under this interpretation, we may obtain solutions which have a decreasing length of their support (which means a decreasing uncertainty). In several applications the behaviour of these solutions better reflects the behaviour of some real-world systems. Derivatives of the H-difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations, using different versions of the variation of constants formula. Some examples show the rich behaviour of the solutions obtained.     

Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing

This paper proposes a method for finding solutions of arbitrarily nonlinear systems of functional equations through stochastic global optimization. The original problem (equation solving) is transformed into a global optimization one by synthesizing objective functions whose global minima, if they exist, are also solutions to the original system. The global minimization task is carried out by the stochastic method known as fuzzy adaptive simulated annealing, triggered from different starting points, aiming at finding as many solutions as possible. To demonstrate the efficiency of the proposed method, solutions for several examples of nonlinear systems are presented and compared with results obtained by other approaches. We consider systems composed of n   equations on Euclidean spaces ?n${?}^n$ (n variables: x₁, x₂, x₃, ? , x_n).     

New Riccati equations for well-posed linear systems

We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative self-adjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the infinite-dimensional generalization of this Riccati equation is not always well-defined. We show that X always satisfies alternative Riccati equations, which are more suitable for algebraic and numerical computations.     

Some Bounds on the Computational Power of Piecewise Constant Derivative Systems

We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the (d-2) th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is Σ _d-2 -complete. Received May 1997, and in final form May 1998.     

Lion and Man Game and Fixed Point-Free Maps

This paper is dedicated to the pursuit-evasion game in which both players (Lion and Man) move in a metric space, have equal maximum speeds and complete information about the location of each other. We assume that evasion is successful if, for some initial positions of players, there exists a positive number p and an evader’s non-anticipative strategy guaranteeing that the distance between the players is always greater than p. We consider connection between successful evasion and such properties of the phase space as geodesics behavior and the existence of non-expanding fixed point-free self-maps.     

微粒群算法在求解方程组中的应用

研究了利用微粒群算法求解线性、非线性方程组解的问题。对于线性、非线性方程组可以在指定的搜索区间内获得方程组的实数解。最后在计算机上进行了实验,证明了方法的正确性。     

Quantum Erasure of Decoherence

We consider the classical algebra of observables that are diagonal in a given orthonormal basis, and define a complete decoherence process as a completely positive map that asymptotically converts any quantum observable into a diagonal one, while preserving the elements of the classical algebra. For quantum systems in dimension two and three any decoherence process can be undone by collecting classical information from the environment and using such an information to restore the initial system state. As a relevant example, we illustrate the quantum eraser of Scully et al. [Nature 351, 111 (1991)] as an example of environment-assisted correction, and present the generalization of the eraser setup for d-dimensional systems. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

A Note on the Numerical Solution of the Heavy Top Equations

In Multibody System Dynamics 2, 71–88, wedescribed the Munthe-Kaas and Crouch–Grossman methods for integratingordinary differential equations numerically on Lie groups. We used theheavy top as a special test problem, and showed that the numericalsolution respects the configuration space TSO(3). We were, however, notable to generate numerical solutions that preserved the first integralsof the top. In this paper, we formulate the heavy top equations on amore natural configuration space, and show that both the Munthe-Kaas andthe Crouch–Grossman methods with suitable coefficient sets can generatenumerical solutions that render first integrals to machine accuracy. Asa partial answer to the comment in Concluding Remarks inMultibody System Dynamics 2, 71–88, we also argue that forHamiltonian systems on the dual space of a Lie algebra, theinfinitesimal generator map describing the differential equation for thecoadjoint action is the functional derivative of the Hamiltonian.     

A B-spline Galerkin method for the Dirac equation

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y^″=−λ²y, that can also be written as a pair of first-order equations y^′=λz, z^′=−λy. Expanding both y(r) and z(r) in the Bk basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the Bk basis and z(r) in the dBk/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method (Bk,Bk±1) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.     

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

We investigate the periodic nature of the positive solutions of the fuzzy difference equation , where k, m are positive integers, A₀, A₁, are positive fuzzy numbers and the initial values x_i, i = −d, −d + 1, … , −1, d = max{k, m}, are positive fuzzy numbers. In addition, we give conditions so that the solutions of this equation are unbounded.     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.