ON THE ACCURACY OF THE LEAST SQUARES AND THE TOTAL LEAST SQUARES METHODS

Consider solving an overdetermined system of linear algebraic equations by both the least squares method (LS) and the total least squares method (TLS). Extensive published computational evidence shows that when the original system is consistent. one often obtains more accurate solutions by using the TLS method rather than the LS method. These numerical observations contrast with existing analytic perturbation theories for the LS and TLS methods which show that the upper bounds for the LS solution are always smaller than the corresponding upper bounds for the TLS solutions. In this paper we derive a new upper bound for the TLS solution and indicate when the TLS method can be more accurate than the LS method.Many applied problems in signal processing lead to overdetermined systems of linear equations where the matrix and right hand side are determined by the experimental observations (usually in the form of a lime series). It often happens that as the number of columns of the matrix becomes larger, the

Sharp Lower Bound of the Least Eigenvalue of Graphs with Given Diameter

The eigenvalues of graphs play an important role in the fields of quantum chemistry, physics, computer science, communication network, and information science. Particularly, they can be interpreted in some situations as the energy levels of an electron in a molecule or as the possible frequencies of the tone of a vibrating membrane. The diameter of a graph, the maximum distance between any two vertices of a graph, has great impact on the service quality of communication networks. So we were motivated to investigate the sharp lower bound of the least eigenvalue of graphs with given diameter. Let gn. d be the set of graphs on n vertices with diameter d. For any graph G ∈ gn, d, by considering the least eigenvalue of its connected spanning bipartite subgraph, we obtained the sharp lower bound of the least eigenvalue of graph G. Furthermore, an upper bound of Laplacian spectral radius of graph G was given.

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation {ut/｜x｜^s-△pu=u^q,（x,t）∈Ω×（0,T）,u（x,t）=0, （x,t）∈δΩ×（0,T） ,u（x,t）=u0（x）,u0（x）≥0,u0（x） absolotely unequalto 0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, where Ω is a smooth bounded domain in R^N（N〉p）,0∈Ω,△pu=div（｜△↓｜^P-2 △↓u）,0≤s≤2,p≥2,p-1 〈q〈 ^N-p -N＋p/N-p.

EXACT NULL CONTROLLABILITY OF NON-AUTONOMOUS FUNCTIONAL EVOLUTION SYSTEMS WITH NONLOCAL CONDITIONS

In this article, by using theory of linear evolution system and Schauder fixed point theorem, we establish a sufficient result of exact null controllability for a non-autonomous functional evolution system with nonlocal conditions. In particular, the compactness condition or Lipschitz condition for the function g in the nonlocal conditions appearing in various literatures is not required here. An example is also provided to show an application of the obtained result.

Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation

The effects of constant excitation on the recently proposed smooth-and-discontinuous(SD)oscillator are investigated,which may lead to the variation of equilibrium and the property of phase portrait.By solving a quartic algebraic equation,the transition set and bifurcation for SD oscillator under constant excitation(CSD)are presented,while the number of equilibria depends on the values of the smoothness parameter and the constant excitation.Complicated structures of Kolmogorov–Arnold–Moser(KAM)structures on the Poincarésection are depicted for the driven system without dissipation.Chaotic behaviour is also demonstrated numerically for the system perturbed by both viscous-damping and external excitation.The results show that CSD is an unsymmetrical system that displays different dynamical behaviours from an SD oscillator and will enrich the range of the SD oscillator in research and application.

The Existence and Uniqueness of Periodic Solutions for a Kind of Lienard Equation with a Deviating Argument

By means of Mawhin's continuation theorem,we study a kind of lie'nard functional differential equations:x"(t)+f(x(t))x'(t)+g(t,x(t-τ(t))) = e(t).Some new results on the existence and uniqueness of periodic solutions are obtained.

The Reverse Order Law for Moore-Penrose Inverse of Closed Operators

In this paper, the reverse order law for the Moore-Penrose inverse of closed linear operators with closed range is investigated by virtue of the Norm-preserving extension of the bounded linear operators. The results generalize some results obtained by S Izumino in [12].

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED HYBRID DYNAMICAL SYSTEMS

In this paper, we consider a class of optimal control problem for the singularly perturbed hybrid dynamical systems. By means of variational method, we obtain the necessary conditions of the hybrid dynamical systems. Meanwhile, the existence of solution for the hybrid dynamical system is proved by the sewing method and the uniformly valid asymptotic expansion of the optimal trajectory is constructed by the boundary function method. Finally,an example is presented to illustrate the result.

A PERTURBATION ANALYSIS FOR THE PROJECTION OF A STIFFLY SCALED MATRIX

In this paper we study the perturbation bound of the projection ( W A ) ( W A )+,where both the matrices A and W are given with W positive diagonal and severely stiff.When the perturbed matrix (A)= A + δA satisfy several row rank preserving conditions,we derive a new perturbation bound of the projection.

Some New Results for Perturbation of g-frames

Some new results for perturbation of g-frames are established,and it is shown that the existing partial results are the corollaries of our results.

Effect of Time-Dependent Ionization on Propagation of a Few-Cycle Circularly Polarized Laser Pulse in Two-Level Medium

Influence of multiphoton ionization on the propagation and spectrum of few-cycle circularly （elliptically） polarized laser pulses in an open two-level medium （two-level plus continuum model） is investigated based on the conventional two-level model proposed by Slavcheva and Hess （Phys. Rev. A 72 （2005） 053804）, and the propagation dynamics of an arbitrary elliptically polarized laser pulse is reduced into that of right and left circularly polarized laser pulses. When the laser intensity is high enough to cause ionization, there are significant impacts of ionization on the pulse reshaping and on the higher order spectral components, and the impacts for the open two-level model are different from those for the dosed two-level model.

On Nonlinear Stability Theorems of 3D Quasi-geostrophic Flow

Nonlinear stability criteria for quasi-geostrophic zonally symmetric flow are improved by establishing an optimal Poincard inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605... times its channel length （which is the geophysically relevant case）.

PERIODIC SOLUTION TO A KIND OF SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE IN CHEMICAL KINETICS

This paper deals with the problem on the periodic solution for the singularly perturbed differential equations of parabolic type originating from chemical kinetics stratified media. A uniformly valid asymptotic solution is constructed and the related asymptotic estimate is given.

CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this article, we give sufficient conditions for controllability of some partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result.

Global behavior of dynamical agents in directed network

This paper investigates the global behavior of controlled dynamical agents in directed networks. The agents are Lyapunov stable, are distributed in a line, and communicate through a directed network. The communication topology of the network is characterized by a directed graph and the control protocol is designed in simple linear decentralized feedback law. We study the different conditions under which agents will achieve aggregation, and critical and divergent trajectories, respectively. Our investigation on the dynamical agent system under network is extended to the time-delay network case. Furthermore, we study the case with two pre-specified virtual leaders in the system. Numerical simulations are given and demonstrate that our theoretical results are effective.

Laplacian Spectra Characterization of Some Special θ-Graphs

A graph G is said to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.We consider θ-graphs,that is,graphs obtained by subdividing the edges of the multigraph consist of three parallel edges.In this paper,some special θ-graphs are determined by their Laplacian spectra.

Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations

Two new convection-dominated are derived under the approximate solutions least-squares mixed finite element procedures are formulated for solving Sobolev equations. Optimal H（div;Ω）×H1（Ω） norms error estimates standard mixed finite spaces. Moreover, these two schemes provide the with first-order and second-order accuracy in time increment, respectively.

Estimation of the Hausdorff Measure of a Kind of Sierpinski Carpet

Suppose F0 is an arbitrary triangle and F is a kind of Sierpinski carpet generated by F0.We construct a projection mapping to obtain the lower bound of the Hausdorff measure of F ;meanwhile the upper bound of the Hausdorff measure of F is calculated by the general covering.

Unified Approach to the Expression for the Moore-Penrose Inverse of a- xy

Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).

Maximum Genus of the Generalized Permutation Graph

In this paper we prove that the generalized permutation graph G（n, k） is upper embeddable if it has at most two odd subcycles, and that the maximum genus of G（n, k） is more than 「β（G（n,k））/3」 in most cases.