Numerical treatment of delay differential equations by Hermite Interpolation

Summary A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.     

An algorithmic approach to Hermite-Birkhoff-interpolation

Summary This paper deals with an algorithmic approach to the Hermite-Birkhoff-(HB)interpolation problem. More precisely, we will show that Newton's classical formula for interpolation by algebraic polynomials naturally extends to HB-interpolation. Hence almost all reasons which make Newton's method superior to just solving the system of linear equations associated with the interpolation problem may be repeated. Let us emphasize just two: Newton's formula being a biorthogonal expansion has a well known permanence property when the system of interpolation conditions grows. From Newton's formula by an elementary argument due to Cauchy an important representation of the interpolation error can be derived. All of the above extends to HB-interpolation with respect to canonical complete ebyev-systems and naturally associated differential operators [7]. A numerical example is given.     

Discretization by finite elements of a model parameter dependent problem

The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.     

A variable metric algorithm for unconstrained minimization without evaluation of derivatives

Summary This paper presents a modification of the BFGS-method for unconstrained minimization that avoids computation of derivatives. The gradients are approximated by the aid of differences of function values. These approximations are calculated in such a way that a complete convergence proof can be given. The presented algorithm is implementable, no exact line search is required. It is shown that, if the objective function is convex and some usually required conditions hold, the algorithm converges to a solution. If the Hessian matrix of the objective function is positive definite and satisfies a Lipschitz-condition in a neighbourhood of the solution, then the rate of convergence is superlinear.     

Backward differentiation type formulas for Volterra integral equations of the second kind

Summary Numerical integration formulas are discussed which are obtained by differentiation of the Volterra integral equation and by applying backward differentiation formulas to the resulting integro-differential equation. In particular, the stability of the method is investigated for a class of convolution kernels. The accuracy and stability behaviour of the method proposed in this paper is compared with that of (i) a block-implicit Runge-Kutta scheme, and (ii) the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas. The present method is particularly advantageous in the case of stiff Volterra integral equations.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Cellularity in polyhedra

A compact subset X of a polyhedron P is cellular in P if there is a pseudoisotropy of P shrinking precisely X to a point. A proper surjection between polyhedra f:P→Q is cellular if each point inverse of f is cellular in P. It is shown that if f:P→Q is a cellular map and either P or Q is a generalized n-manifold, n≠4, then f is approximable by homeomorphisms. Also, if P or Q is an n-manifold with boundary, n≠4, 5, then a cellular map f:P→Q is approximable by homeomorphisms. A cellularity criterion for a special class of cell-like sets in polyhedra is established.     

Solution of a diffusion equation of smoke molecules due to constant point source in gravitational field

From both the Langevin equation, including a gravitational term, the Fokker-Planck equation based on the dynamical behavior of Brownian particles, and equation of smoke molecular diffusion due to a constant point source is introduced and is solved by applying the Laplace transformation with the convolution theorem.The solution is expressed by the complementary error function with a mean pathway of smoke molecules affected by gravity and is proved to be reduced to conventional forms by a certain restriction neglecting gravity without any forced term.     

Algebraic fermion bosonization

By exploiting the construction of charged field algebras as canonical extensions of CCR current algebras in 1+1 dimensions and nonregular representations of extended algebras, we provide an algebraic construction of local Fermi fields as ultrastrong limits of bosonic variables in all representations which are locally Fock with respect to the ground-state representation of the massless scalar field.     

夫勒烯二氮杂桥衍生物对吡啶荧光猝灭机制以及它的电化学的研究

用荧光光谱研究了富勒烯二氮杂桥衍生物(eddy2)对吡啶荧光的猝灭机制,发现吡啶在426.27 nm处有荧光发射峰,当向吡啶中分别加入C₆₀和eddy2时,原有的发射峰强度明显减弱,且向短波长稍有移位,C₆₀和eddy2对吡啶荧光有猝灭作用。吡啶-C60和吡啶-eddy2体系的猝灭过程是分子之间结合形成了复合物引起的静态猝灭。eddy2对吡啶的离解常数为K_D＝2.28×10-6 (mol·L-1),结合常数K_S＝4.39×10-5 (L·mol-1)。同时研究了eddy2的电化学行为,发现eddy2的前3个还原峰电位比母体C₆₀的峰电位正移,表明eddy2的电化学还原比C₆₀的电化学还原容易,更容易接收电子。