The condition onto pair (F, G) of function Banach spaces under which there exists a integral operator T : F → G with analytic kernel such that the inverse mapping T −1 :imT → F does not belong to arbitrary a priori given Borel (or Baire) class is found. We begin with recalling some definitions [4, §31.IX]. Let X and Y be metric spaces. By definition, the family of analytically representable map...

For a meromorphic function $f$ in the complex plane, we shall introduce the definition of five-value rich line of $f$, and study the uniqueness of meromorphic functions of finite order in an angular domain by involving the five-value rich line and Borel directions. Finally, the relationship between a five-value rich line and a Borel direction is discussed, that is, every Borel direction of $f$ ...

Given a set of positive reals, we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.

In this section we want to understand what does the notation ∫ t 0 XsdWs mean. 1.1 Stochastic Process Xt Given a probability space {Ω,F,P} and time index set T , a stochastic process (Xs)s∈T is a measurable mapping from Ω×T to some space. In our most interested setting, the value space is R or Rd, while the time index set is T = [0,∞) with Borel σ-algebra. Another common option for T is the int...

This paper shows how a new approach to theorem proving by analogy is applicable to real maths problems. This approach works at the level of proof-plans and employs reformulation that goes beyond symbol mapping. The Heine-Borel theorem is a widely known result in mathematics. It is usually stated in R 1 and similar versions are also true in R 2 , in topology, and metric spaces. Its analogical tr...

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material about general linear algebraic groups before one c...

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal he...

We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin. We consider some topological and recursion-theoretic questions motivated by the following measure-theoretic result o...

We give several refinements of known theorems on Borel uniformizations of sets with ”large sections”. In particular, we show that a set B ⊂ [0, 1] × [0, 1] which belongs to Σ0α, α ≥ 2, and which has all ”vertical” sections of positive Lebesgue measure, has a Π0α uniformization which is the graph of a Σ0α-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollar...