Topological properties of the solution set of a class of nonlinear evolutions inclusions

In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field F(t, x), we are able to show that the solution set is in fact an R -set. Finally some applications to infinite dimensional control systems are also presented.     

Spaces in Which Countable Closed Sets Have Countable Character

It is proved that in a T ₃ space countable closed sets have countable character if and only if the set of limit point of the space is a countable compact set and every compact set is of countable character. Also, it is shown that spaces where countable sets have countable character are WN-spaces and are very close to M-spaces. Finally, some questions of Dai and Lia are discussed and some questions are proposed.     

Periodic problems and problems with discontinuities for nonlinear parabolic equations

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact R -set in (CT, L ²(Z)).     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

G-spaces and Fixed Point Sets

For a compact connected group G, in a compact free G-space M every closed subset is the fixed point set of an autohomeomorphism if the weight of M is not bigger than that of G. In a compact group with nonsingleton components, every closed subset is the fixed point set of an autohomeomorphism if and only if the weight of the whole group is not bigger than that of the component of the identity.     

DE LA VALLÉE POUSSIN'S THEOREM AND WEAKLY COMPACT SETS IN ORLICZ SPACES

Abstract

The classical theorem of Dunford and Pettis identifies the bounded, uniformly integrable subsets of L¹(μ) with the relatively weakly compact sets. Another characterization of uniform integrability is given in a theorem of De La Vallée Poussin which states that a subset K of L¹ (μ) is bounded and uniformly integrable if and only if there is an N-function F so that sup{f F(f)dμ: f ε K} < ∞. De La Vallée Poussin's theorem is the focal point of the fmt part of this paper as well as the driving force for the results in the second part. We refine and improve this theorem in several directions. The theorem of De La Vallée Poussin does not, for instance, specify just how well the function F can be chosen. It gives little additional information in case the set in question is relatively norm compact in L¹ (μ). Finally it gives no information on the structure of the set in the corresponding Band space of F-integrable functions. More specifically we establish the fact that a subset K of L¹ is relatively compact if and only if there is an N-function F ε δ' so that K is relatively compact in L*_F. Furthermore we prove that a subset K of L¹ is relatively weakly compact if and only if there is an N-function F ε δ' so that K is relatively weakly compact in L*_F. We then go on to show that a large class of non-reflexive Orlicz spaces has the weak Band-Saks property, by establishing a result for these spaces, very similar to the Dunford-Pettis theorem for L¹.     

Transfer Function Optimization Procedure for the H 2/H ∞ Problem

In this paper, the H ₂/H problem is considered in a transfer-function setting, i.e., without a priori chosen bounds on the controller order. An optimization procedure is described which is based on a parametrization of all feasible descending directions stemming from a given point of the feasible transfer-function set. A search direction at each such point can be obtained on the basis of the solution of a convex finite-dimensional problem which can be converted into a LMI problem. Moving along the chosen direction in each step, the procedure in question generates a sequence of feasible points whose cost functional values converge to the optimal value of the H ₂/H problem. Moreover, this sequence of feasible points is shown to converge in the sense of a weighted H ₂ norm; and it does so to the solution of the H ₂/H problem whenever such a solution exists.     

Predictor-corrector algorithm for solvingP *(κ)-matrix LCP from arbitrary positive starting points

A new predictor-corrector algorithm is proposed for solvingP _*(κ)-matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x ⁰,s ⁰). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has -iteration complexity, whereρ ₀ is the ratio of the smallest and average coordinate ofX ⁰ s ⁰. With appropriate initialization, a modified version of the algorithm terminates in O((1+κ)²(n/ρ ₀)L) steps either by finding a solution or by determining that the problem has no solution in a predetermined, arbitrarily large, region. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also propose an extension of a recent algorithm of Mizuno toP _*(κ)-matrix linear complementarity problems such that it can start from arbitrary positive points and has superlinear convergence without a strictly complementary condition. The work of this author was supported in part by NSF, Grant DMS 9305760 and by an Oberman fellowship from the University of Iowa Center for Advanced Studies.     

On Polynomials Sharing Preimages of Compact Sets, and Related Questions

In this paper we give a solution of the following problem: under what conditions on infinite compact sets and polynomials f ₁, f ₂ do the preimages f ₁⁻¹{K ₁} and f ₂⁻¹{K ₂} coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets. Received: May 2006, Accepted: June 2006     

Second Order Nonlinear Evolution Inclusions Ⅱ： Structure of the Solution Set

We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

Rigidity of the critical point equation

On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S³ if ker s^′*_g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization

Li and Sun [D. Li and X.L. Sun, Existence of a saddle point in nonconvex constrained optimization, J. Global Optim. 21 (2001), pp. 39--50; D. Li and X.L. Sun, Convexification and existence of saddle point in a p-th-power reformulation for nonconvex constrained optimization, Nonlinear Anal. 47 (2001), pp. 5611--5622], present the existence of a global saddle point of the p-th power Lagrangian functions for constrained nonconvex optimization, under second-order sufficiency conditions and additional conditions that the feasible set is compact and the global solution of the primal problem is unique. In this article, it is shown that the same results can be obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the primal problem.     

Minimum L1-Distance Projection onto the Boundary of a Convex Set: Simple Characterization

We show that the minimum distance projection in the L ₁-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec (Ref. 1).     

Linearly ordered compact sets and co-Namioka spaces

We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G _δ-set A ⊆ X dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 1001–1004, July, 2007.     

Optimization on directionally convex sets

Directional convexity generalizes the concept of classical convexity. We investigate OC-convexity generated by the intersections of C-semispaces that efficiently approximates directional convexity. We consider the following optimization problem in case of the direction set of OC-convexity being infinite. Given a compact OC-convex set A, maximize a linear form L subject to A. We prove that there exists an OC-extreme solution of the problem. We introduce the notion of OC-quasiconvex function. Ii is shown that if O is finite then the constrained maximum of an OC-quasiconvex function on the set A is attained at an OC-extreme point of A. We show that the OC-convex hull of a finite point set represents the union of a finite set of polytopes in case of the direction set being finite.