Fatou's Lemma and the Lebesgue's Convergence Theorem
In this article we prove the Fatou's Lemma and Lebesgue's Convergence Theorem [10].MML identifier: MESFUN10, version: 7.9.01 4.101.1015
f-closure algebras
Fermat’s Little Theorem via Divisibility of Newton’s Binomial
Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in...
Fiberwise properties of definable sets and functions in o-minimal structures.
Fibrations and recursivity
Fields of surreal numbers and exponentiation
We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...
Fields with analytic structure
Filter descriptive classes of Borel functions
We first prove that given any analytic filter ℱ on ω the set of all functions f on which can be represented as the pointwise limit relative to ℱ of some sequence of continuous functions (), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of ℱ. We discuss several structural properties of this rank. For example, we prove that any free Π⁰₄ filter is of rank 1.
Filter games on ω and the dual ideal
We continue the efforts to characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorial or structural properties of the given filter. Previous results in the literature included those games where player II responded with natural numbers, or finite subsets of natural numbers. In this paper we concentrate on games where player II responds with members of the dual ideal. We also give a summary of known results on filter games.
Filter theory and covering law
Filters and annihilators in implication algebras
Filters and sequences
We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a filter is itself and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.
Filters of -algebras.
Filtres: mesurabilité, rapidité, propriété de Baire forte
Finitary axiomatizations of the true relational equations
Finitary fibrations
Finite alogorithmic procedures and computation theories.
Finite alogorithmic procedures and inductive definability.
Finite and infinite in intuitionistic mathematics
Finite dimensional intuitionistic fuzzy normed linear space.