Neighboring extremals of dynamic optimization problems with path equality constraints

Neighboring extremals of dynamic optimization problems with path equality constraints and with an unknown parameter vector are considered in this paper. With some simplifications, the problem is reduced to solving a linear, time-varying two-point boundary-value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Two example problems are solved to illustrate the validity and usefulness of the solution technique. This research was supported in part by the National Aeronautics and Space Administration under NASA Grant No. NCC-2-106. The author is indebted to Professor A. E. Bryson, Jr., Department of Aeronautics and Astronautics, Stanford University, for many stimulating discussions.     

Volume variation of nuclear Grüneisen parameter for equation of state studies

The method of first principle pseudopotential is used to simulate the volume variation of nuclear Grüneisen parameter. The elements Al and Be, for which Neal’s experimental data exist are investigated. The nuclear Grüneisen obtained from the details of phonon frequencies are in better agreement with the experimental data as compared to those obtained from approximate Slater and Dugdale-MacDonald methods, often used in shock wave studies.     

Collisions and Fracture: a 1-D Theory. How to Tear off a Chandelier from the Ceiling

When a plate falls on the ground, it breaks. We study this phenomenon at the macroscopic level. We restrict ourselves to 1-D problems and illustrate the theory with a chandelier to which a falling stone is tied. The collisions are assumed instantaneous. Percussions are introduced at the unknown fracture points. Equations of motion and constitutive laws give a set of differential equations, whose corresponding variational problem may be solved in SBV (special functions of bounded variation). The example shows how the theory applies and gives realistic results.     

Two-Parametric Compound Binomial Approximations

We consider two-parametric compound binomial approximation of the generalized Poisson binomial distribution. We show that the accuracy of approximation essentially depends on the symmetry or shifting of distributions and construct asymptotic expansions. For the proofs, we combine the properties of norms with the results for convolutions of symmetric and shifted distributions. In the lattice case, we use the characteristic function method. In the case of almost binomial approximation, we apply Steins method.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 443–466, October–December, 2004.     

Degree of approximation to function of bounded variation by Bézier variant of MKZ operators

Guo (Approx. Theory Appl. 4 (1988) 9-18) introduced the integral modification of Meyer-Konig and Zeller operators and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Bézier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364-376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106-107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng.     

On the invariance of classes under composition

The necessary and sufficient condition for to be in the class for every of that class whose range is in the domain of is that be in .

     

ON THE RATE OF CONVERGENCE OF BEZIER VARIANT OF SZASZ—DURRMEYER OPERATORS

In the present paper, we introduce Szasz-Durrmeyer-Bezier operators M.,.(f,x) , which generalize the Szasz-Durrmeyer operators. Here we obtain an estimate on the rate of convergence of Mn,a(f,x) for functions of bounded variation. Our result extends and improves that of Sahai and Prasad and Gupta and Pant.     

Non-contemporaneous variations and Hölder’s principle

In the process of deducing the Hölder principle, a key step is to use the concept of non-contemporaneous variations. In this paper, whether starting from analytic method or from graphic solution method, the authors prove that the expression formula of non-contemporaneous variations is incorrect when the variable functions have zero-order nearness degree, and obtain a new expression. From the view of calculus of variations and differential calculus, the non-contemporaneous variations are studied. The study result shows that the concept of non-contemporaneous variations is a combination of the concept of variations and the concept of differentiation. The authors prove that the new expression is correct and obtain an equivalent expression of it. By means of this equivalent expression, this paper proves that the above expression formula of non-contemporaneous variations is correct when the variable functions have one-order nearness degree. Further study shows that, in the process of deducing Hölder’s principle, there is an implicit expression. Whether starting from analytic method or from graphic solution method, the authors discovered that the implicit expression of non-contemporaneous variations is incorrect when the variable functions have zero-order nearness degree and have one-order nearness degree. This paper proves that the implicit expression of non-contemporaneous variations is correct when the variable functions have two-order nearness degree. Further study shows that Hölder’s principle is tenable when the variable functions have two-order nearness degree.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Second-Order Subexponential Sequences and the Asymptotic Behavior of Their De Pril Transform

Suppose that { f(n), n N ₀ } is a sequence of positive real numbers and suppose that the sequence { a(n), n N ₀ } is given by a(0) = 0, and, for n 1, by the convolution equation nf(n) = a* f(n). The resulting sequence is denoted by a(n) = _f (n) and is called the De Pril transform of { f(n), n N ₀ } . In this paper, we consider first- and second-order asymptotic behavior of { _{f (n), n N 0} } for a large class of subexponential sequences { f(n), n N ₀ } . We also discuss some applications.