Subsets ofX Research Articles

Bump Functions with Hölder Derivatives

AbstractWe study the range of the gradients of aC1,α-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case ofC1-smooth bump functions. Finally, we give a sufficient condition on a subset ofX* so that it is the set of the gradients of aC1,1-smooth bump function. In particular, ifXis an infinite dimensional Banach space with aC1,1-smooth bump function, then any convex open bounded subset ofX* containing 0 is the set of the gradients of aC1,1-smooth bump function.

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF σ subset ofX which is neither prevalent nor Haar-null.

Weak equivalence of cocycles and Mackey action in generic dynamics

LetR be an equivalence relation generated by a countable ergodic homeomorphism group of a perfect Polish spaceX. We consider cocycles taking values in Polish groups onR modulo meager subsets ofX. Two cocycles are called weakly equivalent if they are cohomologous up to an automorphism ofR. The notion of generic associated Mackey action is introduced, which is an invariant of weak equivalence for cocycles. Regular cocycles with values in an arbitrary Polish group and transient cocycles with values in an arbitrary countable group are completely classified up to weak equivalence.

A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers

Given a Banach spaceX, letc0(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc0(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc0(X) admits a center iffX is quasi uniformly rotund. We construct a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the condition (Q) andX is not quasi uniformly rotund. It follows that the Banach spaceE=c0(X) has the property from the title.

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * p , parametrized by a parameterp eX. As a result we obtain a family of join spaces (X, * p ). We show that: for everya,b eX the family {a*pb} peX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b eX.

Enumeration of Hypergraphs

In this paper, we will study enumeration of hypergraphs. Let Spbe a symmetric group acting on ap-setX. It induces the permutation group Sp*acting on the set of all subsets ofX. Our problem is reduced to finding a good formula for the cycle index ofSp* . A crucial point is to calculate the cycle inventory of the permutation induced by a single cycle. Their formulas are given inductively. We will give the cycle index ofSp* explicitly and concretely. We will find a relation between the cycle inventory of the induced permutation and the number of convexk-gons of a given regularp-gon and obtain the formula for its cycle inventory.

Embedding problems with local conditions

LetY→X be a connectedG-Galois cover of affine varieties in characteristicp, and supposeG=Γ/P for somep-groupP. We show that there is a connected Γ-Galois coverZ→X dominatingY→X, and thatZ→X can be chosen to have prescribed behavior over a given closed subset ofX. There are several versions of this result, depending on whether ramification is permitted, and whether adelic behavior is prescribed. The results are deduced from a general assertion about embedding problems, which is proven for profinite groups.

On the convexity of the weakly compact Chebyshev sets in Banach spaces

A sufficient condition for a Banach spaceX is given so that every weakly compact Chebyshev subset ofX is convex. For this purpose a class broader than that of smooth Banach spaces is defined. In this way a former result of A. Brondsted and A. L. Brown is partially extended in every finite dimensional normed linear space and a known result in Hilbert spaces is proved in a different way.

Local solvability of a constrainedgradient system of total variation

Suppose X is a real q‐uniformly smooth Banach space and F, K : X → X with D(K) = F(X) = X are accretive maps. Under various continuity assumptions on F and K such that 0 = u + KFu has a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is imposed on K and the operators K and F need not be defined on compact subsets of X. Our method of proof is of independent interest.

Symmetrization Inequalities for Difference Equations on Graphs

We prove symmetrization inequalities for positive solutions of (not necessarily linear) difference equations of the form−Δu=φu−c·u+λ,where Δ is a discrete Laplacian, φ is a convex decreasing function,cis a positive function and λ is a real function, on subsets ofX×Y, whereXis a graph andYis the line Z, the circle graph Zm, them-regular treeTm, the line graphL(Tm) ofTm, or the edge graph of the octahedron. Instead of Δ, we may also have some other operators, including the heat operator. The typical result says that we use a discrete version of Steiner symmetrization to symmetrize the domain on which the equation is defined, and if all the functions and boundary values involved are appropriately symmetrized as well, then the increasing convex means ofu(x,·) are increased for each fixedx∈X.

When isf(x,y) = u(x) + v(y)?

LetX andY be arbitrary non-empty sets and letS a non-empty subset ofX ×Y. We give necessary and sufficient conditions onS which ensure that every real valued function onS is the sum of a function onX and a function onY.

A Note on Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process

LetXbe a uniformly convex Banach space that satisfies Opial's condition or whose norm is Fréchet differentiable, letCbe a bounded closed convex subset ofX, and letT:C→Cbe a nonexpansive mapping. It is shown that for any initial datax0∈C, the Ishikawa iterates {xn}, defined byxn+1=tnT(snTxn+(1−sn)xn)+(1−tn)xn,n≥0, with the restrictions thatlimnsnis less than 1, and for any subsequence {nk}∞k=0of {n}∞n=0, ∑∞k=0tnk(1−tnk) diverges, converge to a fixed point ofTweakly. Thus, such a result complements Theorem 1 of Tan and Xu (J. Math. Anal. Appl.178(1993), 301–308) and generalizes, to a certain extent, Theorem 2 of Reich (J. Math. Anal. Appl.67(1979), 274–276).

Vector-Valued capacities

It is well-known that ifX is a compact metric space and ifI is a capacity onX then every analytic subset ofX isI-capacitable [2], [1]. We introduce the notion of vector-valued capacity in the case when the codomain is a Banach lattice and we prove an analogous theorem for analytic subsets of a Polish space. Finally, we show that every vectorvalued outer measure is a capacity and, in connection with the so-called “marginal problem”, we give an example of a capacity taking values in a reflexive Banach lattice.

Convexity of sublevel sets of plurisubharmonic extremal functions

LetX be a convex domain in C n and letE be a convex subset ofX. The relative extremal functionuE,X forE inX is the supremum of the class of plurisub- harmonic functionsv� 0 onX withv� 1 onE. We show that ifE is either open or compact, then the sublevel sets ofuE,X are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

Evolutions of Tubes under Operability Constraints

In this paper, we study the evolutions of tubes that can be regarded as set-valued maps with compact images from R+to a finite dimensional vector spaceXunder some constraints that we call operability constraints. We first recall some properties of the mutational calculus in the metric space (K(X),D) where K(X) is the set of nonempty compact subsets ofX, and D is the Hausdorff distance. The main result is the geometric characterization of the dynamics of the tubes that satisfy the operability constraint.

Closure of Rigid Semianalytic Sets

LetKbe an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. LetXbe aK-rigid analytic variety and Σ a semianalytic subset ofX. Then the closure of Σ inXwith respect to the canonical topology is again semianalytic. The proof uses embedded resolution of singularities.

A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetCbofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA−1(b) for which this canalways be done. Finally, in many cases, the best approximationPC(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2X is homeomorphic to Π2 ifX is a Baire space, and toQ∖Π1 ifX contains a dense absoluteGδ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π1, Π1⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes.

Some Remarks on Relative Chebyshev Centers

In this work we study a relative Chebyshev center ofKwith respect toY, whereKis a closed bounded convex subset of a Hilbert spaceX, andYis a closed convex subset ofX. Some results of Amir and Mach [J. Approx. Theory40, (1984), 364–374] are extended.

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.