Exponential objects and Cartesian closedness in the constructPrtop

We give an internal characterization of the exponential objects in the constructPrtop and investigate Cartesian closedness for coreflective or topological full subconstructs ofPrtop. If $ is the set {0} {1/n;n 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct ofPrtop containing $ and which is Cartesian closed. With regard to topological full subconstructs ofPrtop we give an example of a Cartesian closed one that is large enough to contain all topological Fréchet spaces and allT ₁ pretopological Fréchet spaces.Aspirant NFWO     

On concretely both reflective and coreflective subconstructs of Chang fuzzy topological spaces

In this paper, the concretely both reflective and coreflective subconstructs of the construct CFTS of Chang fuzzy topological spaces are characterized and their relationship to such subconstructs of stratified fuzzy topological spaces is investigated. Properties of a particular such subconstruct of CFTS, the construct of weak fuzzy neighborhood spaces, are discussed in detail. And it is proved that there exists exactly c such subconstructs between the subconstruct of weakly induced spaces and CFTS.     

Classes of pretopological spaces closed under the formation of final structures

We investigate coreflective subconstructs of the construct Prtop of pretopological spaces and continuous maps and in particular the inclusion “order” between these subconstructs. We describe the smallest, second and third coreflective class and then all minimal elements that are strictly larger. Using these minimal elements we obtain a “partition” of the whole conglomerate of coreflective subconstructs of Prtop. The results dealing with classes in one member of this partition have an immediate interpretation in the framework of reflexive relations.     

On the largest coreflective Cartesian closed subconstruct of Prtop

We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subconstruct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

Simultaneously reflective and coreflective full subconstructs of stratified L-topological spaces are concretely reflective and coreflective

Let L be a completely distributive lattice. A stratified L-topology on a set X is a subfamily of L-subsets of X which is closed with respect to arbitrary suprema and finite infinima, and contains all the constants. In this paper, it is shown that every simultaneously reflective and coreflective full subconstruct of stratified L-topological spaces is necessarily concretely reflective and coreflective. In other words, every such subconstruct is necessarily both initially and finally closed. As an application, it is demonstrated that the construct of bitopological spaces has exactly 4 simultaneously reflective and coreflective full subconstructs.     

Convergence of Greedy Algorithms in Banach Spaces

We study the convergence of greedy algorithms in Banach spaces. We construct an example of a smooth Banach space, where the X-greedy algorithm converges not for all dictionaries and initial vectors. We also study the R-greedy algorithm, which, along with the X-greedy algorithm, is a generalization of the simple greedy algorithm in Hilbert space. We prove its convergence for a certain class of Banach spaces. In particular, this class contains, the spaces ^p,p 2.     

Exponential integrability of sub-Gaussian vectors

Summary In this paper we define two classes of Banach space (B, ·)-valued random vectors called sub-Gaussian vectors and -sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is, for some >0, in the case whereB=L _p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a -sub-Gaussian vector in an arbitrary separable Banach space.These two definitions of sub-Gaussian vectors and -sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.     

On σ-discrete borel mappings via quasi-metrics

Let X and Y be metrizable spaces. We show that, for a mapping f : X Y, there exists a quasi-metric X inducing the topology of X such that f regarded as a mapping from (X, max{, ^–1}) to Y is continuous if and only if f in the original topology of X is a -discrete map of Borel class 1. Further, we prove that, for every -discrete mapping f: X Y of Borel class + 1, there exists a compatible quasi-metric on X such that f : (X, max{, ^–1}) Y is of Borel class . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Biclosed Binary Relations and Galois Connections

Given two closure spaces (E,) and (E,), a relation RE×E is said biclosed if every row of its matrix representation corresponds to a closed subset of E, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational applications are derived from this result.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

Universal correlations in finite posets

Posets A, BX×X, with X finite, are said to be universally correlated (AB) if, for all posets R over X, (i.e., all posets RY×Y with XY), we have P(RA) P(RB)P(RAB) P(R). Here P(RA), for instance, is the probability that a randomly chosen bijection from Y to the totally ordered set with |Y| elements is a linear extension of RA. We show that AB iff, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)).Winkler proved a theorem giving a necessary and sufficient condition for AB. We suggest an alteration to his proof, and give another condition equivalent to AB.Daykin defined the pair (A, B) to be universally negatively correlated (A B) if, for all posets R over X, P(RA) P(RB)P(RAB) P(R(AB)). He suggested a condition for AB. We give a counterexample to that conjecture, and establish the correct condition. We write AB if, for all posets R over X, P(RA) P(RB)P(RAB) P(R). We give a necessary and sufficient condition for AB.We also give constructive techniques for listing all pairs (A, B) satisfying each of the relations AB, AB, and AB.     

Scattered spaces not having scattered compactifications

Some examples of scattered spaces not having scattered compactifications are given, which solves a problem of Semadeni. Thus, let S be any extremally disconnected dense-in-itself subspace of N/N. Then for every point S the subspacen {} does not have any scattered compactification.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 127–136, January, 1978.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

A global and stochastic analysis approach to bosonic strings and associated quantum fields

We construct a probability measure giving a mathematical realization of Polyakov's heuristic measure for bosonic strings in space-time dimensions 3d13, having as world sheet compact Riemann surfaces of arbitrary genus. The measure involves the path space measures for scalar fields with exponential interaction on and a measure on Teichmüller space.Deceased 24 January 1988     

On the Completeness and Decidability of the Horn‐Like Fragment of the First‐Order Linear Temporal Logic

We prove the completeness and decidability of the Hornlike sequents, specifically, the socalled D₂sequents (of the firstorder linear temporal logic) considered in the author's paper [Lith. Math. J., 41(3), 266–281 (2001)]. In this paper, with the help of the infinitary calculus G_L, grounded by the author in his earlier papers, for D₂sequents we construct a D₂Sat calculus of the socalled saturated type consisting of decidable deductive procedures replacing the omegarule for the always operator. In the present paper, in order to prove the completeness and decidability of the calculus D₂Sat, we construct the socalled invariant decidable calculus D₂IN. We prove the equivalence of the calculi D₂IN, D₂Sat, and G _L ^** for the socalled saturated D₂sequents. From this equivalence, by reducing an arbitrary D₂sequent to a saturated D₂sequent, and also from the completeness of the G _L ^** calculus and decidability of the invariant calculus D₂IN, we deduce the completeness and decidability of the calculus D₂Sat in the class of D₂sequents.