On the -fuzzy topological spaces

As a natural generalization of fuzzy metric spaces due to George and Veeramani [George A, Veeramani P. On some result in fuzzy metric space. Fuzzy Sets Syst 1994;64:395–9], the present author defined the notion of -fuzzy metric spaces. In this paper we prove some known results of metric spaces including Uniform continuity theorem and Ascoli–Arzela theorem for -fuzzy metric spaces. We also prove that every -fuzzy metric space has a countably locally finite basis and use this result to conclude that every -fuzzy metric space is metrizable.     

Parseval frame wavelets with -dilations

We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

Coboundaries in

Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|S_n(f)|>n) are of order n^−p roughly corresponds to the existence of an approximation in L^∞ of f by functions (coboundaries) g−g○T, gL^p. Similarly, the probabilities μ(|S_n(f)|>n) are exponentially small iff f can be approximated by coboundaries g−g○T where g have finite exponential moments.

Résumé

Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|S_n(f)|>n) d'ordre n^−p et l'existence de l'approximation dans L^∞ de la fonction f par des cobords g−g○T où g est “presque” dans L^p. De manière similaire, les probabilités μ(|S_n(f)|>n) sont d'ordre e^−cn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L^∞ par des cobords g−g○T avec g ayant des moments exponentiels.     

Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

Global well-posedness and scattering for the defocusing -subcritical Hartree equation in

We prove the global well-posedness and scattering for the defocusing -subcritical (that is, 2<γ<3) Hartree equation with low regularity data in , d3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space with s>4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s>max(1/2,4(γ−2)/(3γ−4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ)ξ^p. As a byproduct of our proof, we obtain that the H^s norm of the solution obeys the uniform-in-time bounds.     

The operator equation

The solution of the linear operator equation:A^n-1X+A^n-2XB++AXB^n-2+XB^n-1=Y is given by if the spectra of A and B are in the sector {z:z≠0,-π/n<argz<π/n}.     

On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.     

Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

Generalized rough approximations in

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call -inner and -outer approximations respectively ( being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic by means of step functions from the hypercube [0,1]^k and taking value in an arbitrary (finite) subset of . Then, we will show that if a fuzzy set is definable as truth table of a formula of , then both its -inner and -outer approximation are definable as truth table of formulas of . Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.     

The implicitization problem for

We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in , based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions.Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of , and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.     

On the recursive sequence

Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation

where B, C and α, β, γ are positive, k {1, 2, 3, … }, and the initial conditions x_−2k+1, … , x₋₁, x₀ are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.