In this paper we introduce a parameter ω∞ in the space of functions L0. The parameter measures the lack of equimeasurability using a sequence of functions which controls the oscillations of every function of a given subset of L0. We estimate the Hausdorff measure of noncompactness in terms of ω∞ and the parameter σ (see [3]) and characterize the totally bounded subsets of L0. A criterion of com...

For an n-element set X and a proper coloring ∆: [X] −→ {0, 1, . . .} where each color class is a matching with cardinality bounded by u, we show that there exists a totally multicolored subset Y ⊆ X with |Y | ≥ max { c1 · ( n/u ) 1 2k−1 , c2 · ( n/u ) 1 2k−1 · ( ln ( u/ √ n )) 1 2k−1 } This bound is tight up to constant factors for u = ω(n ) for any > 0. Moreover, for fixed k, we give a polynom...

for any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. the condition is that every minimal ideal of the ring consist entirely of bounded elements. it is not too stringent, and is satisfied, for instance, by rings of conti...

Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce notion totally $T$-adic function: that is algebraic and whose minimal polynomial splits completely completion $\mathbb{F}_q(\!(T)\!)$. give two proofs height nonconstant function bounded away from zero, each which provides sharp lower bound. spend majority paper providing explicit construc...