Precise bounds for the petersen time-varying real parameter uncertainty system stabilized via linear static controllers

Using a class of linear static controllers, we stabilize the Petersen open-loop two-dimensional linear system (Ref. 1), which consists of one time-varying uncertainty in the state matrixA and one timevarying uncertainty in the input matrixB. We show that the worst-case uncertainty strategy for the closed-loop system is a piecewise constant strategy of the angular state with three switches on the half-turn, –/2/2; it is unique with respect to a set of measure zero. Formulas are derived for the worst-case half-turn radius gainr _HT as a function of the parameters of the class of stabilizing linear static controllers. Using the class of scalar-quadratic Lyapunov functions, we show that a necessary and sufficient condition for the closed-loop system to be robustly stable against all time-varying admissible uncertainties is thatr _HT be less than unity. The bound on the time-varying real parameter uncertainties for the closed-loop system to be robustly stable is derived for the class of linear static feedback controllers. We obtain stabilizing linear static controllers such that the bound is as close to infinity as desired. The derived results are compared with numerical results obtained using commerical robust-control software.     

An invariant trace formula for the universal covering group of SL(2,ℝ)

Automorphic forms of arbitrary real weight can be considered as functions on the universal covering group of SL(2, ). In this situation, we prove an invariant form of the Selberg trace formula for Hecke operators. For this purpose, the Fourier transforms of weightet orbital integrals, obtained by J. Arthur, R. Herb and P. Sally, jr., are explicitly calculated. Our formula does not follow from Arthur's invariant trace formula, since the group has infinite centre, and vector-valued automorphic forms with respect to non-congruence lattices are considered.     

Density computations for real quadratic units

In order to study the density of the set of positive integers for which the negative Pell equation is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in as the factorization of is a natural part of the input for the values of we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm .

     

Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals

In this paper we study a particular class of primal-dual path-following methods which try to follow a trajectory of interior feasible solutions in primal-dual space toward an optimal solution to the primal and dual problem. The methods investigated are so-called first-order methods: each iteration consists of a long step along the tangent of the trajectory, followed by explicit recentering steps to get close to the trajectory again. It is shown that the complexity of these methods, which can be measured by the number of points close to the trajectory which have to be computed in order to achieve a desired gain in accuracy, is bounded by an integral along the trajectory. The integrand is a suitably weighted measure of the second derivative of the trajectory with respect to a distinguished path parameter, so the integral may be loosely called a curvature integral.     

Degenerate polynomial patches of degree 11 for almostGC 2 interpolation over triangles

We consider the problem of interpolating scattered data in ³ by analmost geometrically smoothGC ² surface, where almostGC ² meansGC ² except in a finite number of points (the vertices), where the surface isGC ¹. A local method is proposed, based on employing so-called degenerate triangular Bernstein-Bézier patches. We give an analysis of quintic patches forGC ¹ and patches of degree eleven for almostGC ² interpolation.     

Triplet states of humic acids studied by laser flash photolysis using different excitation wavelengths

The spectra and kinetics of short-lived intermediates formed from aqueous (0.1 N NaOH) solutions of the natural mixture of humic and fulvic acids (HFA) were studied by laser flash photolysis using excitation wavelengths of 337, 390, 470, and 520 nm. Laser photolysis of HFA with light of 520 and 470 nm results in the formation of triplet excited states (T_HFA) characterized by the broad absorption spectrum with a maximum near 630 nm and lifetimes of 0.15 ms in deoxygenated solutions. The formation of two types of T_HFA with lifetimes of 0.1 and 2 ms and absorption spectra with maxima at 570 nm is observed under photolysis with light of 337 and 390 nm. The estimation of quantum yields of T_HFA gives 1 and 0.3% under photolysis with excitation wavelengths of 337 and 520 nm, respectively. The rate constants of T_HFA quenching by molecular oxygen are equal to (7—8)·10⁸ L mol^–1 s^–1.     

Surface modification of proteins by covalent binding of acrylic polymers

Surface modification of enzymes for a potential use in therapy was obtained with a new type of tailor-made copolymers ofNacryloylmorpholine andN-acryloxysuccinimide. The first monomer was designed to confer solubility on the polymer, whereas the second was used to give it reactivity toward protein amino groups. The reactivity of polymers of different composition towards amino acid derivatives and model proteins, such as catalase and ribonuclease-A, is described. Water soluble and catalytically active enzyme derivatives were obained using copolymers prepared with a mixture of N-acryloxysuccinimide andn-acryloylmorpholine in a 1:99 molar ratio. At increasing molar ratio (3:97, 10:90) extensive crosslinking between polymer and enzymes takes place, yielding insoluble adducts.     

液体色谱,凝胶色谱分析低聚芳砜及其双烯大分子单体

本文用液相色谱和凝胶色谱对双酚A、双酚S型低聚芳砜及α,ω-双甲基丙烯酸聚芳砜酯大分子单体进行了组分分析,通过改合成条件、测定数均分子量等辅助手段确定液相色谱各峰的归属,计算其分子量及分子量分布指数,并对两种方法测定的结果进行了比较。用液相色谱观察低聚体每个组分的含量及其反应过程中的消长情况比凝胶色谱清晰。