On Super-KMS functionals and entire cyclic cohomology

We formulate the super-KMS condition suggested by Connes and Kastler, in the context of entire cyclic cohomology of quantum algebras. We show that the Chern character of Jaffe, Lesniewski, and Osterwalder — associated by Kastler to a super-KMS functional — satisfies the entire growth condition. Hence, a super-KMS functional defines a cocycle for the entire cyclic cohomology of quantum algebras.Supported in part by the National Science Foundation.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

Remarks on the angle property and solid cones

This paper is devoted to the investigation of the relations between cones satisfying the angle property and solid cones. The investigation shows that the two classes of cones are dual in some sense. As an application of our results, we improve some related results due to Cesari, Suryanarayna, Sterna-Karwat, and Yu.This research was supported by the National Natural Science Foundation of China. It was completed while the author was a PhD Student at the Department of Mathematics, Shandong University, Jinan, China. The author would like to thank Professor D. Guo for encouragement. The author is also grateful to the referees for comments and suggestions.     

Spaces with Vanishing ℓ2-Homology and their Fundamental Groups (after Farber and Weinberger)

We characterize those groups which can occur as the fundamental groups of finite CW-complexes with vanishing ²-homology (the first examples of such groups were obtained by Farber and Weinberger).     

Robustness of minimax controllers to nonlinear perturbations

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, >0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH -controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.This research was supported in part by the US Department of Energy, Grant DE-FG-02-88-ER-13939 and in part by the National Science Foundation, Grant ECS-91-13153.An abridged version was presented at the 32nd IEEE Conference on Decision and Control, San Antonio, Texas, December 15–17, 1993, and it appeared in the Conference Proceedings.     

Three theorems on ρ* random fields

This paper generalizes results by Bradley.⁽³⁾ Suppose that for 1=1,2,...X _k ¹ :k ^d is a centered, weakly stationary ^*-mixing random field, and suppose lim_l Cov(X ₀ ¹ ,x _k ¹ ) exists, anyk ^d . Then the successive spectral densities converge uniformly to a continuous function. For a sequence of strictly stationary random fields that are uniformly ^*-mixing and satisfy a indeberg condition, a CLT is proved for sequences of sums from the fields. This result is then applied: given a centered strictly stationary ^*-mixing random field whose probability density and joint densities are continuous, then a kernel estimator for the probability density obeys the CLT.     

Semicontinuous solutions for Hamilton-Jacobi equations and theL ∞control problem

We prove uniqueness for extended real-valued lower semicontinuous viscosity solutions of the Bellman equation forL -control problems. This result is then used to prove uniqueness for lsc solutions of Hamilton-Jacobi equations of the form –u _t +H(t, x, u, –Du)=0, whereH(t, x, r, p) is convex inp. The remaining assumptions onH in the variablesr andp extend the currently known results.Supported in part by Grant DMS-9300805 from the National Science Foundation.     

SOLUTIONS CONTAINING DELTA－WAVES OF CAUCHY PROBLEMS FOR A NONSTRICTLY HYPERBOLIC SYSTEM

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Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field X defined on some open set M ², we characterize the prolongations X _n ^* of X to the nth jet space M ⁽ⁿ⁾, n1, such that a complete system of invariants for X _n ^* can be obtained by derivation of lower-order invariants. This leads to characterizations of C -symmetries and to new procedures for reducing the order of an ordinary differential equation.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Novel approach to accelerating Newton's method for sup-norm optimization arising inH ∞-control

We present algorithms for solving general sup-norm minimization problems over spaces of analytic functions, such as those arising inH control. We also give an analysis and some theory of these algorithms. Part of this is specific to analytic optimization, while part holds for general sup-norm optimization. In particular, we are proposing a type of Newton-type algorithm which actually uses very high-order terms. The novel feature is that higher-order terms can be chosen in many ways while still maintaining a second-order convergence rate. Then, a clever choice of higher-order terms greatly reduces computation time. Conceivably this technique can be modified to accelerate Newton algorithms in some other circumstances. Estimates of order of convergence as well as results of numerical tests are also presented.This work was partially supported by the Air Force Office of Scientific Research and the National Science Foundation.     

On the BCH-formula in so(3)

We find local closed-form expression for the Baker—Campbell—Hausdorff formula in the Lie algebra so(3), and interpret the formula geometrically in terms of rotation vectors in ³.This revised version was published online in October 2005 with corrections to the Cover Date.     

Analysis on adjoint non-recurrent property of nonlinear time series in random environment domain

By introducing a random interference into the typical of nonlinear time series model, this paper establishes a RENLAR model: . The author introduces the definition of adjoint non-recurrence, and utilizing general state space Markov chain theorem, we obtain some criteria for non-recurrence and adjoint non-recurrence of nonlinear time series models in random environment domain and analyze adjoint non-recurrence of some models by using these criteria. Research supported Science Foundation of China (10171009).     

Partial Results in Δ 3 0 -Categoricity in Linear Orderings and Boolean Algebras

The notion of ₃ ⁰ -categoricity in linear orderings and Boolean algebras is examined. We provide a proof for the fact that there are uncountably many relatively ₃ ⁰ -categorical linear orderings, and furnish a proof of another fact which suggests that the (unrelatively) ₃ ⁰ -categorical linear orderings may be very difficult to classify. In stark contrast to these results for linear orderings, a complete classification of the relatively ₃ ⁰ -categorical Boolean algebras is given.     

Largest <Emphasis Type="Italic">E</Emphasis>-thin, <Emphasis Type="Italic">E</Emphasis>-invariant sets below Δ<Superscript>1</Superscript><Subscript>3</Subscript>

In this paper, we construct, by effective induction, a ¹₁ equivalence relation E on for which there is no largest E-thin E-invariant ¹₁ subset of by effective induction. This result answers a problem asked by Kechris in [1]Mathematics Subject Classification: 03E15Revised version: 13 January 2000     

Rates of convergence of α-stable random motions

The rate-of-convergence problem in the central limit theorem is considered for -stable distributions on the noncommutative group of motions in ^d. The method used here is based on the theory of probability metrics.     

‘Twice’ Equivariant C*-Index Theorem and the Index Theorem for Families

We prove the index theorem for elliptic operators acting on sections of bundles where the fiber is equal to a projective module over a C ^*-algebra in the situation of the action of a compact Lie group on this algebra as well as on the total space commuting with a symbol. For this purpose, we prove in particular the corresponding Thom isomorphism theorem. As an application, the equivariant family index theorem for a direct product of a base by the space of parameters is obtained.     

Local structure of Rogers semilattices of Σ<Subscript>n</Subscript><Superscript>0</Superscript>-computable numberings

We deal in specific features of the algebraic structure of Rogers semilattices of _n ⁰-computable numberings, for n 2. It is proved that any Lachlan semilattice is embeddable (as an ideal) in such every semilattice, and that over an arbitrary non 0-principal element of such a lattice, any Lachlan semilattice is embeddable (as an interval) in it.Supported by INTAS grant No. 00-499, by FP Universities of Russia grant UR.04.01.013, and by the Grant Center for Fundamental Research (GCFR), project PD02-1.1-475.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 148–172, March–April, 2005.     

Well-conditioned computation for H ∞ controller near the optimum

The present paper gives a procedure for determining a H optimal controller in the assumption that the game Riccati equations have stabilizing positive definite solutions at the optimum value. A specific feature of the construction is its first step consisting in balancing with respect to the positive definite stabilizing solutions of the Riccati equations. The justification is based on singular perturbations reduction.     

Positive Numberings of Families of Sets in the Ershov Hierarchy

It is proved that there exist infinitely many positive undecidable _n ^-1 -computable numberings of every infinite family that admits at least one _n ^-1 -computable numbering and contains either the empty set, for even n, or N for odd n.