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.     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball of and the radial derivative of h. In this paper we study the boundedness and compactness of the following integral operator:, from iterated logarithmic Bloch spaces to Zygmund-type spaces.     

Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

Weakly sequentially continuous differentiable mappings

It is well known that a (linear) operator between Banach spaces is completely continuous if and only if its adjoint takes bounded subsets of Y^* into uniformly completely continuous subsets, often called (L)-subsets, of X^*. We give similar results for differentiable mappings. More precisely, if UX is an open convex subset, let be a differentiable mapping whose derivative is uniformly continuous on U-bounded subsets. We prove that f takes weak Cauchy U-bounded sequences into convergent sequences if and only if f^′ takes Rosenthal U-bounded subsets of U into uniformly completely continuous subsets of . As a consequence, we extend a result of P. Hájek and answer a question raised by R. Deville and E. Matheron. We derive differentiable characterizations of Banach spaces not containing ℓ₁ and of Banach spaces without the Schur property containing a copy of ℓ₁. Analogous results are given for differentiable mappings taking weakly convergent U-bounded sequences into convergent sequences. Finally, we show that if X has the hereditary Dunford–Pettis property, then every differentiable function as above is locally weakly sequentially continuous.     

Convex and star-shaped sets associated with multivariate stable distributions, I: Moments and densities

It is known that each symmetric stable distribution in is related to a norm on that makes embeddable in L_p([0,1]). In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.     

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.     

Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the form

where , , f is a nonnegative function and satisfies the Carathéodory conditions on . Here, f can have a time singularity at t=0 and/or t=T and a space singularity at x=0 and/or y=0. We present conditions for the existence of solutions positive on [0,T) and having continuous first derivatives on [0,T].     

Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

On fractional maximal function and fractional integral on the Laguerre hypergroup

Let be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces to the spaces and from the spaces to the weak spaces .     

Weak and point-wise convergence in for filter convergence

We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study.     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

Spaces of holomorphic functions in regular domains

Let Ω be a regular domain in the complex plane , . Let be the linear space over of the holomorphic functions f in Ω such that f⁽ⁿ⁾ is bounded in Ω and is continuously extendible to the closure of Ω, n=0,1,2,… . We endow , in a natural manner, with a structure of Fréchet space and we obtain dense subspaces F of , with good topological linear properties, also satisfying that each function f of F, distinct from zero, does not extend holomorphically outside Ω.     

A new characterization of Bergman–Schatten spaces and a duality result

Let denote the space of all upper triangular matrices A such that lim_r→1⁻(1−r²)(A*C(r))^′_B(ℓ²)=0. We also denote by the closed Banach subspace of consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between and the Bergman–Schatten spaces . We also give a characterization of the more general Bergman–Schatten spaces , 1p<∞, in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L^p-behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.     

There's something about the diameter

We study diameter preserving linear bijections from onto where X, Y are compact Hausdorff spaces and V, Z are Banach spaces. For instance, we obtain that if X has at least four points, Z is linearly isometric to V and either Z is a space or Z^* is strictly convex or smooth, then there is a diameter preserving linear bijection from onto if and only if X is homeomorphic to Y. We also consider the case when X and Y are not compact but locally compact spaces.     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

Iterated function systems on multifunctions and inverse problems

In this paper, we first consider the problem of defining IFS operators on the space of non-empty compact and convex subsets of . After defining a complete metric on , we construct an IFS operator and show some properties. A notable feature is the definition of a type of weak inner product on . We then define a family of complete metrics on the space of all measurable set-valued functions (with values in ), and extend the weak inner product to this space. Following this, we construct IFS operators on these spaces. We close with a brief discussion of the inverse problem of approximating an arbitrary multifunction by the attractor of an IFS.     

Strong uniform continuity

Let be an ideal of subsets of a metric space X,d. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions.     

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 .     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,