Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T ₁, T ₂ and T ₃: K → E be asymptotically nonexpansive mappings with {k _n }, {l _n } and {j _n }. [1, ∞) such that Σ _n=1 ^∞ (k _n − 1) < ∞, Σ _n=1 ^∞ (l _n − 1) < ∞ and Σ _n=1 ^∞ (j _n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T _1x = T _2x = T ₃ x} = x} denotes the common fixed points set of T ₁, _{T 2} and T ₃. Let {α _n }, {α′ _n } and {α″ _n } be real sequences in (0, 1) and ∈ ≤ {α _n }, {α′ _n }, {α″ _n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x ₁ ∈ K define the sequence {x _n } by

Geodesics in Graphs,an Extremal Set Problem,and Perfect Hash Families

 A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection 𝒮 of subsets of [n]={1,2,…,n} such that, for any two distinct elements x,y∈[n], there exists disjoint subsets A _x ,A _y ∈𝒮 such that x∈A _x and y∈A _y . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions. Received: May 19, 2000 Final version received: July 5, 2001     

The Möbius function on Abelian semigroups

Let X be an Abelian semigroup such that the following conditions hold: (i) if x × y = II (II is the identity element), then x = y = II; (ii) the set {{x, y}: x × y = a} is finite for any a ∈ X. Let Λ be any field, and let ℰ be the algebra of all Λ-valued functions on X. The convolution of u, υ ∈ ℰ is defined by

Ergodic Retraction Theorem and Weak Convergence Theorem for Reversible Semigroups of Non-Lipschitzian Mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt ： t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F（T） of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u（.） of T, ∩s∈G ^-CO{u（t） ： t ≥ s} ∩ F（T） consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx ： t ≥ s} ∩ F（T） is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F（T） such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx ： s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u（t） ： t ∈ G} to a common fixed point of T.     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

Visibility Graphs of Point Sets in the Plane

The visibility graph of a discrete point set X⊂ℝ² has vertex set X and an edge xy for every two points x,y∈X whenever there is no other point in X on the line segment between x and y. We show that for every graph G, there is a point set X∈ℝ², such that the subgraph of induced by X is isomorphic to G. As a consequence, we show that there are visibility graphs of arbitrary high chromatic number with clique number 6 settling a question by Kára, Pór and Wood. Supported by the DFG Research Center Matheon (FZT86).     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

On weak positive supercyclicity

A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT ⁿ x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ _p (T ^*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT ⁿ x: n ∈ ℂ, r ∈ ℝ₊} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions. Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225. Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225     

Transitive points via Furstenberg family

Let (X,T) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z₊ with hereditary upward property). A point x∈X is called an F-transitive one if {n∈Z₊:Tnx∈U}∈F for every non-empty open subset U of X; the system (X,T) is called F-point transitive if there exists some F-transitive point. In this paper, we aim to classify transitive systems by F-point transitivity. Among other things, it is shown that (X,T) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive).It is shown that every weakly mixing system is Fip-point transitive, while we construct an Fip-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ^?(Fwt)-transitive if and only if it is weakly disjoint from every P-system.     

Dependent sets of a family of relations of full measure on a probability space

For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {γr}γ∈Γis a countable family of relations of the full measure on a probability space (X,B,μ), i.e. for everyγ∈Γthere is a positive integer sγsuch that Rγ(?)Xsγwithμsγ(Rγ) = 1. In the present paper we show that if (X, B,μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K(?)X withμ*(K) = 1 such that (x1,...,xsγ)∈γr for anyγ∈Γand for any sγdistinct elements x1,..., xsγof K, whereμ* is the outer measure induced by the measureμ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.     

Limit T-spaces

Let F be a field of prime characteristic p and let V _p be the variety of associative algebras over F without unity defined by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 and by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 (here [x, y] = xy − yx). Let A/V _p be the free algebra of countable rank of the variety V _p and let S be the T-space in A/V _p generated by x ₁² x ₂² ⋯ x _k² + V ₂, where k ∈ ℕ if p = 2, and by {ie4170-01}, where k ∈ ℕ and α ₁, …, α _2k ∈ {0, p − 1} if p > 2. As is known, S is not finitely generated as a T-space. In the present paper, we prove that S is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative F-algebra without unity of countable rank. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 135–159, 2007.     

Piecewise Convex Maximization Problems: Piece Adding Technique

In this article, we provide a global search algorithm for maximizing a piecewise convex function F over a compact D. We propose to iteratively refine the function F at local solution y by a virtual cutting function p _y (⋅) and to solve max {min {F(x)−F(y),p _y (x)}∣x∈D} instead. We call this function either a patch, when it avoids returning back to the same local solutions, or a pseudo patch, when it possibly yields a better point. It is virtual in the sense that the role of cutting constraints is played by additional convex pieces in the objective function. We report some computational results, that represent an improvement on previous linearization based techniques.     

On Jakovlev spaces

A topological spaceX is called weakly first countable, if for every pointx there is a countable family {C _n ^x |n ∈ω} such thatx ∈C _n ^+1x ⊆C _n ^x and such thatU ⊂X is open iff for eachx ∈U someC _n ^x is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing. The second author was supported by the Ben-Gurion University Center for Advanced Studies in Mathematics, Be’er Sheva. The third author was supported by OTKA grant no. 37758 of Hungary.     

Estimates for the real taylor coefficients in one function class

Considering the class {ie48-01} of analytic functions {ie48-02} in the unit disk with a _m,n ε ℝ and the nonvanishing nth divided difference [F(z);z ₀, ⋯, z _n ] for all z ₀, ℝ, z _n ∈ E we establish that {ie48-03}, where {ie48-04}. If n is an odd number then {ie48-05}.