Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

Approximate Newton Methods for Nonsmooth Equations

We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic difference approximation method and the -generalized Jacobian method. The former can be applied to problems with specific structures, while the latter is expected to work well for general problems. Numerical tests show that the two methods are efficient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is briefly discussed.     

On Periodic Solutions of Nonlinear Functional Differential Equations

Sufficient conditions are established for the existence and uniqueness of an -periodic solution of the functional differential equation where f is a continuous operator acting from the space of n-dimensional -periodic continuous vector functions into the space of n-dimensional -periodic and summable on [0,] vector functions.     

On a One-Parameter Generalization of the Brownian Bridge and Associated Quadratic Functionals

A one-parameter generalization of the Brownian bridge is studied. These processes are then used to compute the laws of some quadratic functionals of Brownian motion, and to obtain identities in law involving local time of modified Bessel processes up to their first hitting time.     

Asymptotic Curvature of Θ-Metrics

We give an explicit and reasonably simple expression for the curvature tensor of a -metric at boundary points, in terms of the metric tensor and invariants of the -structure. We examine the behavior of the induced metric on level sets of a defining function near the boundary and describe the asymptotic behavior of its curvature tensor. Some applications of these results are given.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

Strongly Regular Graphs and Designs with Three Intersection Numbers

We investigate BIBDs with three intersection numbers,x, y, and z, such that the relation on the block set given byA B iff the cardinality of the intersectionof A and B is not equal to x is an equivalence relation. Withsuch a design, we associate a family of strongly regular graphswith the same parameters. Two constructions producing infinitefamilies of such designs are given.     

Multiple Stable Stochastic Integrals: Series Representation and Absolute Continuity of Their Law

We study multiple stable stochastic integrals. We begin with a definition of these integrals using a generalization of the LePage representation as Samorodnitsky et al. in Ref. 15 but with a simplified probabilistic background and for general stable measure when 0<<1 and for symmetric stable measure when 1<2. This representation allows to study the law of multiple stable integrals. We prove they are absolutely continuous using a stratification method on the Skorokhod space on which we have previously taken back the problem.     

Self-bounded controlled invariants versus stabilizability

Self-bounded controlled and self-hidden conditioned invariant subspaces, recently introduced by the authors for a more direct and neat handling of some fundamental concepts of the geometric approach, such as controllability subspaces, are proved in this paper to be very useful tools also in dealing with synthesis problems with stability requirements.Definitions concerning stability of invariants and stabilizability of controlled invariants, simple and self-bounded, are first presented and discussed. In particular, it is shown that a more straightforward definition for controlled invariant stabilizability allows a simpler development of the theory, Then, some fundamental results relating self-boundedness to stabilizability are derived. For the sake of completeness, all statements are dualized to conditioned invariants, simple and self-hidden.     

A New Class of Improved Convex Underestimators for Twice Continuously Differentiable Constrained NLPs

We present a new class of convex underestimators for arbitrarily nonconvex and twice continuously differentiable functions. The underestimators are derived by augmenting the original nonconvex function by a nonlinear relaxation function. The relaxation function is a separable convex function, that involves the sum of univariate parametric exponential functions. An efficient procedure that finds the appropriate values for those parameters is developed. This procedure uses interval arithmetic extensively in order to verify whether the new underestimator is convex. For arbitrarily nonconvex functions it is shown that these convex underestimators are tighter than those generated by the BB method. Computational studies complemented with geometrical interpretations demonstrate the potential benefits of the proposed improved convex underestimators.     

Semi-groupes de Riesz d'ordre α sur le demi-espace

Nous donnons une forme équivalente de la densité du semi-groupe de Riesz d'ordre sur le demi-espace. La nouvelle forme nous permet d'établir des critères de comparaison du semi-groupe de Riesz d'ordre et de la résolvante associée avec leurs perturbés par une mesure de Radon positive. Abstract. We give an equivalent form of the density of the Riesz semigroup on the half space. The new form allows us to establish criterions of comparison of the Riesz semigroup and the associated resolvent with their perturbed by a nonnegative Radon measure.     

Computational complexity of the integration problem for anisotropic classes

We determine the exact order of -complexity of the numerical integration problem for the anisotropic class W^r(I^d) and H^r(I^d) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions W_p(I^d) and H_p(I^d). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration. Project supported by the fund of Personnel Division of Nankai University and the Program of One Hundred Distinguished Chinese Scientists of the Chinese Academy of Sciences.     

A r (Ω)-Weighted Imbedding Inequalities for A-Harmonic Tensors

We prove the basic A _r ()-weighted imbedding inequalities for A-harmonic tensors. These results can be used to estimate the integrals for A-harmonic tensors and to study the integrability of A-harmonic tensors and the properties of the homotopy operator T: C (D, ^l )C (D, ^l–1).     

On some classes of modules

The aim of this paper is to investigate quasi-corational, comonoform, copolyform and -(co)atomic modules. It is proved that for an ordinal a right R-module M is -atomic if and only if it is -coatomic. And it is also shown that an -atomic module M is quasi-projective if and only if M is quasi-corationally complete. Some other results are developed.     

Stability for Quadratic K1

The general quadratic group GQ _2n and its elementary subgroup EQ _2n are analogs in the theory of quadratic forms of the general linear group GL _n and its elementary subgroup E _n . This article proves that the stabilization map GQ _2n /EQ _2n GQ _2(n+1) /EQ _2(n+1) is an isomorphism whenever n +1 and S denotes the -stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass–Serre dimension d, the stabilization map is an isomorphism whenever n d + 2.     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

Some Properties of Isologism of Groups and Baer-Invariants

Let be a variety of groups defined by the set of laws V. In this paper we study the concept of -isologism of groups in terms of -extensions and their connections with the Baer-invariant of groups are also discussed.AMS Subject Classification (2000): primary 20F14, 20F19, secondary 20E10     

Distribution of Increasing ℓ-sequences in a Random Permutation

This paper examines the distribution of the number, k, of increasing -sequences in a random permutation of . A new solution is determined based on the compositions of n which requires, at most, summands. This solution easily yields existing results for the special case and provides an alternate form for the case . The expected number of increasing -sequences in a random permutation is determined and it is shown that the limiting distribution is degenerate about 0 for 2$$ " align="middle" border="0"> . An alternate algorithm to determine the exact distribution is presented, based on the partitions of n, which is easy to implement and efficient for small n. Applications in non-parametric statistics and graph theory are discussed.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

Dual-lattice theorems in the geometric approach

A one-to-one correspondence is shown to exist between the lattice of all self-bounded (A, )-controlled invariants contained in and the lattice of all self-hidden (A, )-conditioned invariants containing . This correspondence, stated herein as the main dual-lattice theorem, allows a straightforward derivation of the universal bounds of the lattices, particularly when additional constraints are imposed, such as to contain a given subspace for the elements of the former lattice and to be contained in a given subspace for the elements of the latter. Then, two further minor dual-lattice theorems, dual to each other, are presented, and some connections and applications of the new theory to standard control and observation problems are briefly discussed.