Onto Orthogonal Projections in the Space of Polynomials Pn[x]

In this article, I consider projection groups on function spaces, more specifically the space of polynomials Pn[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of Pn[x] where the function h ∈Pn[x] represents the closets function to f ∈Pn[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in Rn+1 using the metric tensor on Pn[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in Rn+1. This can be also achieved with the Kronecker Product as detailed in this paper.

DIRECT MANIPULATION OF B-SPLINE SURFACES

Engineering design and geometric modeling often require the ability to modifythe shape of parametric curves and surfaces so that their shape satisfy some given geometricconstraints, including point, normal vector, curve and surface. Two approaches are presented todirectly manipulate the shape of B-spline surface. The former is based on the least-square, whereasthe latter is based on minimizing the bending energy of surface. For each method, since unified andexplicit formulae are derived to compute new control points of modified surface, these methods aresimple, fast and applicable for CAD systems. Algebraic technique is used to simplify the computationof B-spline composition and multiplication. Comparisons and examples are also given.

Matrix description of differential relations of moment functions in structural reliability sensitivity analysis

In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessary. Based on that, the structural system reliability is analyzed with a fourth-order moment method. The reliability sensitivity is required to conduct the differential operation of the numerical characteristic functions. A reliability sensitivity analysis formula is then derived in combination with the relation of the differential operation. Based on the matrix theory and Kronecker algebra, this paper systematically derives a matrix expression of the first four moments of the state functions, and establishes the matrix relation between the first four moments of the state functions and those of the basic random variables. On this basis, a differential operation formula of the first four moments of the state functions is further derived against the first four moments of the basic random variables. The vector relation between the state functions and the multidimensional basic random variables is described by means of the matrix operation to extend the operation method. Finally, a concise and intuitive formula is obtained to explore the inherent essential relation between the numerical characteristics of the state functions and those of the basic random variables, leading to a universal equation for the two kinds of numerical characteristics.

CONFIGURATION FROM TRUTH VECTOR TO XOR FUNCTION

Kamaugh maps are widely used in the logic synthesis. However, the number of the variable it can deal with is limited. In this paper, two kinds of function shrinking techniques are proposed, and a fast algorithm to configure a truth vector into a XOR function is realized. There is no variable number limitation for this algorithm.

Parameterized Solution to a Class of Sylvester Matrix Equations

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.

Some Results on Graph Products Determined by Their Spectra

In this paper, we prove that some Kronecker products of G and K2 are determined by their spectra where the graph G is also determined by its spectrum. And a problem for further researches is proposed.

A Necessary and Sufficient Condition for Products of Quasi-Positive Definite Matrices and Generalization of Schur's Theorem

A quasi positive definite matrix is the generalization of a positive definite matrix. A necessary and sufficient condition of quasi positive definite matrix is obtained in this paper for the Kronecker product and Hadamard product of two quasi positive definite matrices, and Schur's achievements in Hadamard product of the positive definite matrix is generalized to quasi positive definite matrix theory.

Kronecker Product of Two-dimensional Arrays

Kronecker sequences constructed from short sequences are good sequences for spread spectrum communication systems. In this paper we study a similar problem for two-dimensional arrays, and we determine the linear complexity of the Kronecker product of two arrays, Our result shows that similar good property on linear complexity holds for Kronecker product of arrays.

TWO INTEGRABLE COUPLINGS FOR A LATTICE HIERARCHY BY THE KRONECKER PRODUCT

It is shown that the Kronecker product can be applied to construct integrable couplings for discrete systems. In this paper, using this method, we derive two integrable couplings for a lattice hierarchy.

Finite-Time Consensus of a Leader-Following Multi-Agent Network with Non-Identical Nonlinear Dynamics and Time-Varying Topologies

In this paper, the finite-time consensus of a leader-following multi-agent network with non-identical nonlinear dynamics and time-varying topologies is investigated. All the agents, especially the leaders, have non-identical and nonlinear dynamics. According to the algebraic graph theory, Lyapunov stability theory and Kronecker product, a control strategy strategy is established to guarantee the finite-time consensus of multi-agent network with multiple leaders. Furthermore, several numerical simulations illustrate the effectiveness and feasibility of the proposed method.

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

In this paper the problem−div(a(x,y)∇u)=f with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for u and∇u using a new scheme called“hermitian box-scheme”.The design of the scheme is based on a“hermitian box”,combining the approximation of the gradient by the fourth order hermitian derivative,with a conservative discrete formulation on boxes of length 2h.The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh.The problem is suitably scaled before iteration.The numerical results obtained show the efficiency of the numerical scheme.This work is the extension to strongly elliptic problems of the hermitian box-scheme presented by Abbas and Croisille(J.Sci.Comput.,49(2011),pp.239–267).

Near theoretical upper bound new loosely synchronous complementary spreading sequences

A new class of loosely synchronous(LS)spreading sequences with zero correlation window(ZCW)was presented.It was constructed by making Kronecker product of orthogonal matrix and ZCW complementary sequences.This new LS sequence increases the number of perfect complementary pairs and extends ZCW within the same group.Moreover,both auto-correlation and cross-correlation of ZCW in the same group remain identical.The minimum ZCW among different groups is the same as that of basic LS sequences.The method for constructing these new LS sequences is presented and ZCW properties are also verified.The number of these new LS sequences is only smaller than theoretical upper bound by one.