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### Abstract Korovkin-type theorems in modular spaces and applications

Open Mathematics

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the...

### Dieudonné-type theorems for lattice group-valued $k$-triangular set functions

Kybernetika

Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for $k$-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems.

### Extending the ideal of nowhere dense subsets of rationals to a P-ideal

Commentationes Mathematicae Universitatis Carolinae

We show that the ideal of nowhere dense subsets of rationals cannot be extended to an analytic P-ideal, ${F}_{\sigma }$ ideal nor maximal P-ideal. We also consider a problem of extendability to a non-meager P-ideals (in particular, to maximal P-ideals).

### F-limit points in dynamical systems defined on the interval

Open Mathematics

Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider...

### Generalized vector-valued sequence spaces defined by modulus functions.

Journal of Inequalities and Applications [electronic only]

### Ideal version of Ramsey's theorem

Czechoslovak Mathematical Journal

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

### Lacunary weak statistical convergence

Mathematica Bohemica

The aim of this work is to generalize lacunary statistical convergence to weak lacunary statistical convergence and $ℐ$-convergence to weak $ℐ$-convergence. We start by defining weak lacunary statistically convergent and weak lacunary Cauchy sequence. We find a connection between weak lacunary statistical convergence and weak statistical convergence.

### On double statistical $P$-convergence of fuzzy numbers.

Journal of Inequalities and Applications [electronic only]

### On I-convergence of double sequences in the topology induced by random 2-norms

Matematički Vesnik

### On ideal equal convergence

Open Mathematics

We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów...

### On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function.

Journal of Inequalities and Applications [electronic only]

### On the ideal convergence of sequences of quasi-continuous functions

Fundamenta Mathematicae

For any Borel ideal ℐ we describe the ℐ-Baire system generated by the family of quasi-continuous real-valued functions. We characterize the Borel ideals ℐ for which the ideal and ordinary Baire systems coincide.

### Probabilistic norms and statistical convergence of random variables.

Surveys in Mathematics and its Applications

### Some generalizations of Olivier's theorem

Mathematica Bohemica

Let $\sum _{n=1}^{\infty }{a}_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $\left({a}_{n}\right)$ is non-increasing, then $\underset{n\to \infty }{lim}n{a}_{n}=0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\underset{n\to \infty }{lim}n{a}_{n}=0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$-convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$. Again, Olivier’s theorem is a consequence of...

### Some statistically convergent difference sequence spaces defined over real 2-normed linear spaces.

APPS. Applied Sciences

### Statistical convergence of a sequence of random variables and limit theorems

Applications of Mathematics

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order $r$ and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

### Statistical convergence of sequences of functions with values in semi-uniform spaces

Commentationes Mathematicae Universitatis Carolinae

We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzelà, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kočinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kočinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett....

### Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers

Open Mathematics

In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively....

### The classes of strongly ${V}_{\lambda }^{F}$(A,p)-summable sequences of fuzzy numbers.

The New York Journal of Mathematics [electronic only]

### The reaping and splitting numbers of nice ideals

Colloquium Mathematicae

We examine the splitting number (B) and the reaping number (B) of quotient Boolean algebras B = (ω)/ℐ where ℐ is an ${F}_{\sigma }$ ideal or an analytic P-ideal. For instance we prove that under Martin’s Axiom ((ω)/ℐ) = for all ${F}_{\sigma }$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin’s Axiom ((ω)/ℐ) = for all ${F}_{\sigma }$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications of these characteristics...

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