Essential extension type maps for general classes of maps

This paper discusses admissible (and more general) maps between topological spaces. We show that if F is H-essential and F ≅ G then G has a fixed point.     

On Beck’s coloring of posets

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x,y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide. Also, we give a condition under which posets are not finitely colorable.     

On stationary zeros of solutions of linear elliptic equations

For a nontrivial solution of a linear homogeneous elliptic equation, we study the dimension of the set of zeros whose multiplicity is not less than the order of the equation. In the case of a linear homogeneous differential operator P = P(D) with constant coefficients and three variables, we show that if, for a solution of the equation Pu = 0, a point x ⁰ is a zero of multiplicity not less than the order of the equation, then the intersection of a sufficiently small neighborhood of the point x ⁰ with the set of all other zeros of this kind is a finite set of segments with common endpoint x ⁰.     

A curious behavior of capillary surfaces

It is known that for a given base domainΩ and «contact angle»γ, there will be a critical γ₀, 0 ?γ₀, ?π/2., such that a solution overΩ of the scalar capillarity equation in the absence of external (e.g. gravity) force field will exist if γ₀, 0 ?γ, ?π/2, and will fail to exist if 0 ?γ ?γ₀. In the particular case for whichΩ is a circular disk, a solution exists for everyγ, and is unique up to an additive constant. For a piecewise smoothΩ that is not circular, there will be a boundary pointP of maximal inward directed curvatureK _M (at a protruding corner we defineK _M=∞). We suppose that a solution exists for such a domain, and ask whether a solution continues to exist if the domain is made closer to circular by smoothing with an inscribed interior arc of constant curvatureK<K _M. That is so in many cases, for example it is true ifΩ is a rectangle, and in fact in that case smoothing decreases γ₀. In the present note, we show that the answer can also be negative. To that effect, we give an example of a convex domain Ω at which the valueK _M is achieved only at a single isolated point, and for which smoothing at that point changes existence to non-existence.     

Decomposition of weighted pseudo-almost periodic functions

First, we show by constructing two counterexamples that the decomposition of weighted pseudo-almost periodic functions is not unique in general. Then we prove that the decomposition of such functions is unique if PAP₀(X,ρ) is translation invariant, but not necessarily unique without the assumption. Moreover, we give an example to show that the mean value under a certain weight ρ may not exist for all almost periodic functions. With these results, we answer some fundamental questions on weighted pseudo-almost periodic functions.     

Comparability invariance of the fixed point property

We show that the fixed point property is comparability invariant for finite ordered sets; that is, if P and Q are finite ordered sets with isomorphic comparability graphs, then P has the fixed point property if and only if Q does. In the process we give a characterization of comparability invariants which can also be used to give shorter proofs of some known results.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb(K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of , then the set of all strongly norm attaining elements in is dense. In particular, the set of all points at which the norm of is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.     

Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V(G) = P, that the outer cycle of G coincides with the convex hull Conv(P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv(P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv(P) is even and that any two consecutive points on Conv(P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv(P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of \({\frac{n}{2}}\) red and \({\frac{n}{2}}\) blue points in general position, then adding Steiner points Q with \({|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}\) , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove \({|Q| \leq \frac{7n+34m-48}{18}}\) , where |P| = n and the number of points of P in Conv(P) is 2m.     

Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖Tx_n−x_n‖→0 due to Chidume and Zegeye in 2004 [14] where (x_n) is a certain perturbed Krasnoselski-Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.     

General Existence Theorem of Zero Points

Let X be a nonempty, compact, convex set in and let be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of . It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under with the normal cone of X at the point is either empty or contains the origin 0 ⁿ , then must have a zero point on X; i.e., there exists a point in X such that 0 ⁿ lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, the mapping must have a zero point. In this paper, we extend all these existence results by giving a general zero-point existence theorem, of which the previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping . Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.     

Two point sets with additional properties

A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some σ-ideal, being (completely) nonmeasurable with respect to different σ-ideals, being a κ-covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations of two point sets, namely: partial two point sets and n point sets for n = 3, 4, …, ?₀, ?₁. We obtain consistent results connecting partial two point sets and some combinatorial properties (e.g. being an m.a.d. family).     

Stationary point conditions for the FB merit function associated with symmetric cones

For the symmetric cone complementarity problem, we show that each stationary point of the unconstrained minimization reformulation based on the Fischer-Burmeister merit function is a solution to the problem, provided that the gradient operators of the mappings involved in the problem satisfy column monotonicity or have the Cartesian P₀-property. These results answer the open question proposed in the article that appeared in Journal of Mathematical Analysis and Applications 355 (2009) 195-215.     

Preserving asphericity of virtually nilpotent spaces under P-localization

For a set of prime numbers P, we study when the P-localization of an Eilenberg-Mac Lane space with virtually nilpotent fundamental group is again aspherical. While investigating this problem, we devote special attention to infra-nilmanifolds.We prove that the P-localization of an orientable infra-nilmanifold is aspherical if and only if its holonomy group is P-torsion. The same holds for non-orientable infra-nilmanifolds if 2 is in P. We also develop computational techniques to check preservation of asphericity. These are explicitly applied to show that the P-localization of a non-orientable infra-nilmanifold of dimension ?3 is always aspherical. We point out that this is no longer true from dimension 4 onwards.     

Density of positive rank fibers in elliptic fibrations

A question of Mazur asks whether for any non-constant elliptic fibration {Er}_r∈Q, the set {r∈Q:rank(Er(Q))>0}, if infinite, is dense in R (with respect to the Euclidean topology). This has been proved to be true for the family of quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. Here we settle Mazur's question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when Q is replaced by any real number field.     

Geoffrion type characterization of higher-order properly efficient points in vector optimization

We consider the constrained vector optimization problem minCf(x), x∈A, where X and Y are normed spaces, A⊂X₀⊂X are given sets, C⊂Y, C≠Y, is a closed convex cone, and is a given function. We recall the notion of a properly efficient point (p-minimizer) for the considered problem and in terms of the so-called oriented distance we define also the notion of a properly efficient point of order n (p-minimizers of order n). We show that the p-minimizers of higher order generalize the usual notion of a properly efficient point. The main result is the characterization of the p-minimizers of higher order in terms of “trade-offs.” In such a way we generalize the result of A.M. Geoffrion [A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (3) (1968) 618-630] in two directions, namely for properly efficient points of higher order in infinite dimensional spaces, and for arbitrary closed convex ordering cones.     

On the integrality of an extreme solution to pluperfect graph and balanced systems

Let A be a nonnegative integer matrix, and let e denote the vector all of whose components are equal to 1. The pluperfect graph theorem states that if for all integer vectors b the optimal objective value of the linear program minse′xvbAx ? b, x ? 0 s is integer, then those linear programs possess optimal integer solutions. We strengthen this theorem and show that any lexicomaximal optimal solution to the above linear program (under any arbitrary ordering of the variables) is integral and an extreme point of sxvbAx ? b, x ? 0 s. We note that this extremality property of integer solutions is also shared by covering as well as packing problems defined by a balanced matrix A.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.