Descriptions of equivalence of O-connectivity

We first study some properties of the subspace, and investigate into the relationship of separation between a fuzzy topological space (fts) and its subspace. Then we obtain the equivalence conditions for O-connectivity. The results on O-connectivity and separation are very similar to those in general topology. Finally we discuss the relationship of connectivity between an O-connected set A in the fts (X, ω ($\text{T}$)) induced by the crisp topological space (X, $\text{T}$) and the crisp set A₀ (=supp A) in (X, $\text{T}$).     

H-spectral spaces

Let X be a T₀-space, we say that X is H-spectral if its T₀-compactification is spectral. This paper deal with topological properties of H-spectral spaces. In the case of T₁-spaces the T₀-compactification coincides with the Wallman compactification. We give necessary and sufficient condition on the T₁-space X in order to get its Wallman compactification spectral.     

Characterizingω 1 and the long line by their topological elementary reflections

Given a topological space 〈X, T〉 ∈M, an elementary submodel of set theory, we defineX _Mto beX ∩M with the topology generated by {U ∩M : U ∈T ∩M}. We prove that it is undecidable whetherX _Mhomeomorphic toω ₁ impliesX =X _M,yet it is true in ZFC that ifX _Mis homeomorphic to the long line, thenX =X _M.The former result generalizes to other cardinals of uncountable confinality while the latter generalizes to connected, locally compact, locally hereditarily LindelöfT ₂ spaces.     

Characterization of Hermitian and skew-Hermitian maps between matrix algebras

Let D be a division ring with an involution J such that D is finite-dimensional over its center Z and char D≠2. Let T:M_m(D)→M_n(D) be a Z-linear map between matrix rings over D. We show that T satisfies [T(X)]¹=T(X¹) if and only if T(X)=∑±A¹_kXA_k. Similarly, T satisfies [T(X)]¹ = ? T(X¹) if and only if T(X = ∑(A¹_kXB_k ? B¹_kXA_k). The first of these results generalizes and extends a theorem of R.D. Hill [2] on Hermitian-preserving transformations.     

Lelong-Skoda Transform for Compact Kähler Manifolds and Self-Intersection Inequalities

Let X be a compact Kähler manifold of dimension k and T be a positive closed current on X of bidimension (p,p) (1≤p<k?1). We construct a continuous linear transform ? _p (T) of T which is a positive closed current on X of bidegree (1,1) which has the same Lelong numbers as T. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

On similarity between topologies

Let T ₁ and T ₂ be topologies defined on the same set X and let us say that (X, T ₁) and (X, T ₂) are similar if the families of sets which have nonempty interior with respect to T ₁ and T ₂ coincide. The aim of the paper is to study how similar topologies are related with each other.     

On the cancellation problem for the affine space \mathbb{A}^{3} in characteristic p

We show that the Cancellation Conjecture does not hold for the affine space $\mathbb{A}^{3}_{k}$ over any field k of positive characteristic. We prove that an example of T. Asanuma provides a three-dimensional k-algebra A for which A is not isomorphic to k[X ₁,X ₂,X ₃] although A[T] is isomorphic to k[X ₁,X ₂,X ₃,X ₄].     

On matrix mappings into some strong Cesàro sequence spaces

We determine the classes (X, Y_T) of matrix transformations from X into Y_T where X is one of the classical sequence spaces c₀, c, ?_∞ and ?₁ of all null, convergent and bounded complex sequences and all absolutely convergent complex series, T is a triangle, Y_T is the matrix domain of T in Y and Y is any of the sets of all sequences that are summable, summable to zero or bounded by the strong Cesàro method of order 1, with index 1 ? p < ∞. Furthermore, we determine the representations of the general bounded linear operators from c into Y. We also establish estimates for the norms of the operators in each case.     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

Minmax subtree cover problem on cacti

Let G=(V,E) be a connected graph such that edges and vertices are weighted by nonnegative reals. Let p be a positive integer. The minmax subtree cover problem (MSC) asks to find a pair (X,T) of a partition X={X₁,X₂,…,X_p} of V and a set T of p subtrees T₁,T₂,…,T_p, each T_i containing X_i so as to minimize the maximum cost of the subtrees, where the cost of T_i is defined to be the sum of the weights of edges in T_i and the weights of vertices in X_i. In this paper, we propose an O(p²n) time (4-4/(p+1))-approximation algorithm for the MSC when G is a cactus.     

On a family of IFSs whose attractors are not connected

For an infinite-dimensional Banach space X, S and T bounded linear operators from X to X such that ‖S‖,‖T‖<1 and w∈X, let us consider the IFS Sw=(X,f₁,f₂), where f₁,f₂:X→X are given by f₁(x)=S(x) and f₂(x)=T(x)+w, for all x∈X. We prove that if the operator S is finite-dimensional, then the set {w∈X|the attractor of Sw is not connected} is open and dense in X.     

A renorming in some Banach spaces with applications to fixed point theory

We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk(⋅)} which satisfy some special conditions. We define an equivalent norm ?⋅? on X such that if C is a convex bounded closed subset of (X,?⋅?) which is τ-relatively sequentially compact, then every nonexpansive mapping T:C→C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier-Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G=T, we recover P.K. Lin's renorming in the sequence space ?₁. Moreover, we give new norms in ?₁ with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L₁(μ) can be renormed to have the FPP.     

Lévy random bridges and the modelling of financial information

The information-based asset-pricing framework of Brody-Hughston-Macrina (BHM) is extended to include a wider class of models for market information. To model the information flow, we introduce a class of processes called Lévy random bridges (LRBs), generalising the Brownian bridge and gamma bridge information processes of BHM. Given its terminal value at T, an LRB has the law of a Lévy bridge. We consider an asset that generates a cash-flow X_T at T. The information about X_T is modelled by an LRB with terminal value X_T. The price process of the asset is worked out, along with the prices of options.     

Fixed points and iterations of mean-type mappings

Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T ⁿ ) _n∈? of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µ _T leads to the invariance equation µ _T ○ T = µ _T , which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I ^p , where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M ₁,...,M _p ) of I ^p . In this paper we give the explicit forms of µ_M for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.     

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ _± such that the stopping time T=inf?{t>0:X _t ∈{?φ _?(L _t ),φ ₊(L _t )}} induces X _T ～μ, where (L _t ) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup. Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.     

The pseudoweights of a space

Let X be a T_i-space, i ⩽ 2. We define the T_i-pseudoweight of X, ψ _i(X), to be the least weightof a coarser T_i topology on X. Reed and Zenor have shown that if X is a Moore space, and |X| ⩽ 2^ω, then ψ₁(X) = ω, but there is a Moore space, X, such that ψ₂(X) = w(X) = |X| = ω₁.Theorem 1: If X is metric, ψ₀(X) = log w(X), where log κ = min{λ:2^λ ⩾ κ}. Theorem 2: If X is compact and T₂, then ψ₁(X) = ψ₂(X) = w(X) (but it is possible to have ψ₀(X) = log w (X)< w(X)). Theorem 3: If X is a GO-space, then ψ₁(X) = ψ₂(X) (but it is possible to have ψ₀(X) =log ψ₁(X) < ψ₁(X) < w(X) even if X is a LOTS). Finally, Hart has shown that if X is an infinite LOTS, then w(X) = c (X) · ψ₁(X). Theorem 4: If X is an infinite LOTS, then w(X) =c(X) · ψ₀ (X).