Volume 67, Issue 6

We establish some basic properties of quotients of projective Fraïssé limits and exhibit some classes of compact metric spaces that are the quotient of a projective Fraïssé limit of a projective Fraïssé family in a finite language. We prove the result for the arcs directly, and by applying some closure properties we obtain all hypercubes and graphs as well.

We consider the Horn-Tarski condition for the extension of (signed) measures (resp., non-negative measures) in the setup of field-valued assignments. For a finite collection 𝓒 of subsets of Ω, we find that the extension from 𝓒 over the collection exp Ω of all subsets of Ω is implied by, and indeed equivalent to, a certain type of Frobenius theorem (resp. a certain type of Farkas lemma). This links classical notions of linear algebra with a generalized version of Horn-Tarski condition on extensions of measures. We also observe that for a general (infinite) 𝓒 the Horn-Tarski condition guarantees the extension of signed measures (here the standard Zorn lemma applies). However, we find out that the extensions for non-negative ordered-field-valued measures are generally not available.

A d 0 -algebra, which is a generalization of a D-lattice, is an algebraic structure with one operation, called difference, and a distinguished element denoted by 0. In this paper we investigate those uniformities on a d 0 -algebra which make the difference uniformly continuous. We prove that such uniformities are univocally determined the neighbourhoods of 0, thus extending the corresponding result known for D-lattices.

In this paper we will consider 𝓙-density topology connected with a sequence 𝓙 of closed intervals tending to 0 and a 𝓙-approximately continuous function associated with that kind of density points. It will be the continuation of the investigations started in “𝓙- approximately continuous functions ” by J. Hejduk, A. Loranty and R. Wiertelak published in Tatra Mountains Mathematical Publications. In particular, we will show that topology generated by the sequence 𝓙 from the special family 𝔍 α is always completely regular.

The comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property. We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.

We improve a Duda’s theorem concerning metric and w * -Gâteaux differentiability of Lipschitz mappings, by replacing the σ -ideal 𝓐 of Aronszajn null sets [ARONSZAJN, N.: Differentiability of Lipschitzian mappings between Banach spaces , Studia Math. 57 (1976), 147–190], with the smaller σ -ideal 𝓐 of Preiss-Zajíček null sets [PREISS, D.—ZAJÍČEK, L.: Directional derivatives of Lipschitz functions , Israel J. Math. 125 (2001), 1–27]. We also prove the inclusion C̃ o ⊂ 𝓐, where C̃ o is the σ -ideal of Preiss null sets [PREISS, D.: Gâteaux differentiability of cone-monotone and pointwise Lipschitz functions , Israel J. Math. 203 (2014), 501–534].

In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable.

We prove linearity theorems for modular measures on pseudo-D-lattices (= lattice ordered pseudo-effect algebras) and we study the consequences for the core of measure games.

We investigate some properties of (uniformly) absolutely continuous, singular and non-concentrated k -subadditive lattice group-valued capacities, and we examine some relations between them. Furthermore, we pose some open problems.

L p spaces are investigated for vector lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the vector lattice context, and some properties of the Brownian motion and the Brownian bridge are studied, to solve some stochastic differential equations.

Fuzzy measures and Choquet asymmetric integral are considered here. As an application to economics some Core-Walras results are given.

We give another proof for Kluvanek and Knowles’ characterization of Liapounoff measures [KLUVANEK, I.—KNOWLES, G.: Vector Measures and Control Systems . North-Holland Mathematics Studies 20, Amsterdam, 1976] and of the fact that the range of an exhaustive measure with values in a complete locally convex space is relatively weakly compact.

Let 𝓙 be an ideal on ℕ which is analytic or coanalytic. Assume that ( f n ) is a sequence of functions with the Baire property from a Polish space X into a Polish space Z , which is divergent on a comeager set. We investigate the Baire category of 𝓙-convergent subsequences and rearrangements of ( f n ). Our result generalizes a theorem of Kallman. A similar theorem for subsequences is obtained if ( X , μ ) is a σ -finite complete measure space and a sequence ( f n ) of measurable functions from X to Z is 𝓙-divergent μ -almost everywhere. Then the set of subsequences of ( f n ), 𝓙-divergent μ -almost everywhere, is of full product measure on {0,1} ℕ . Here we assume additionally that 𝓙 has property (G).

The theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform and L p norms, for continuous and L p functions, respectively. The above family of operators, are based upon kernels generated by sigmoidal functions. Multivariate approximation by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of sigmoidal functions for which the above theory holds have been presented.

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

We define and study the moduli d ( x , 𝓐, D ) and i ( x , 𝓐, D ) related to monotonicity of a given function x of the space L 0 (Ω) of real-valued “measurable” functions defined on a linearly ordered set Ω. We extend the definitions to subsets X of L 0 (Ω), and we use the obtained quantities, d ( X ) and i ( X ), to estimate the Hausdorff measure of noncompactness γ ( X ) of X . Compactness criteria, in special cases, are obtained.

A synaptic algebra A is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace V of A in regard to the question of when V is a vector lattice. Our main theorem states that if V contains the identity element of A and is closed under the formation of both the absolute value and the carrier of its elements, then V is a vector lattice if and only if the elements of V commute pairwise.

In this paper we consider classes of continuous functions with three kinds of topologies on the domain and/or range of the function: the natural topology, density topology and ψ -density topology. We focus on functions from the classes 𝓒 ψd and 𝓒 dψ .

Topologies of special type in a product of two topological spaces are introduced and considered here, in a general case, then in ℝ n to finish on the plane. In the last case, the graph of any function is closed in the considered topology, so it leads to interesting examples of open sets. Our purpose is to study properties of these topologies and connections between them. We prove also that every continuous real function from the plane equipped with the considered topology is in the first class of Baire.

The aim of the paper is to examine topological properties of disruptions of nonautonomous discrete dynamical system by other “close” systems. Our considerations will be connected with entropy of a system at a point and with stable points of a system.

Let S be a finite set, ( X n ) an exchangeable sequence of S -valued random variables, and μ n = (1/ n ) ∑ i=1n$\sum\limits_{i=1}^{n} $ δ X i the empirical measure. Then, μ n ( B )⟶ a.s.$\overset{a.s.}{\longrightarrow} $ μ ( B ) for all B ⊂ S and some (essentially unique) random probability measure μ . Denote by 𝓛( Z ) the probability distribution of any random variable Z . Under some assumptions on 𝓛( μ ), it is shown that an≤ ρ [L(μ n),L(μ )]≤ bnandρ [L(μ n),L(an)]≤ cnu$$\begin{array}{} \displaystyle \frac{a}{n}\le\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(\mu)\bigr]\le\frac{b}{n}\qquad\text{and}\qquad\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(a_n)\bigr]\le\frac{c}{n^u} \end{array} $$ where ρ is the bounded Lipschitz metric and a n (⋅) = P ( X n +1 ∈ ⋅ | X 1 , …, X n ) is the predictive measure. The constants a , b , c > 0 and u ∈ (12$\frac{1}{2} $, 1] depend on 𝓛( μ ) and card ( S ) only.

The additivity of considered measures or integrals resp. can be omitted in some problems of mathematical analysis and its applications. In the paper it is shown that similar situations are possible also in the probability theory. As an example is proved a generalized version of the central limit theorem about the convergence of arithmetical means of independent random variables to the Gauss distribution.

We consider skew tent maps T α , β ( x ) such that ( α , β )∈[0,1] 2 is the turning point of TT α , β , that is, T α , β = βα $\begin{array}{} \frac{{\beta}}{{\alpha}} \end{array} $ x for 0≤ x ≤ α and T α , β ( x ) = β1− α $\begin{array}{} \frac{{\beta}}{1- {\alpha}} \end{array} $ (1− x ) for α < x ≤ 1. We denote by M = K ( α , β ) the kneading sequence of TT α , β and by h ( α , β ) its topological entropy. For a given kneading squence M we consider equi-kneading, (or equi-topological entropy, or isentrope) curves ( α , φ M ( α )) such that K ( α , φ M ( α )) = M . To study the behavior of these curves an auxiliary function Θ M ( α , β ) is introduced. For this function Θ M ( α , φ M ( α )) = 0, but it may happen that for some kneading sequences Θ M ( α , β ) = 0 for some β < φ M ( α ) with ( α , β ) still in the dynamically interesting quarter of the unit square. Using Θ M we show that the curves ( α , φ M ( α )) hit the diagonal {( β , β ): 0.5 < β < 1} almost perpendicularly if ( β , β ) is close to (1,1). Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for M = RLLRC the curve ( α , φ M ( α )) is not orthogonal to the diagonal. On the other hand, for M = RLC it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J. C. Marcuard, M. Misiurewicz and E. Visinescu.

Let Ω be a Lebesgue-measurable set in ℝ n of finite positive Lebesgue measure. In this note we calculate the lack of equi-measurability of the set K c (Ω), c > 0, of all Lebesgue-measurable functions f : Ω → ℝ such that 0 ≤ f ≤ c , a.e. on Ω. From our result we repair a gap in the Example 2.3 of the paper [APPELL, J.—DE PASCALE, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili , Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515].