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Assume that u, v are conjugate harmonic functions on the unit disc of C, normalized so that u(0)=v(0)=0. Let u∗, |v|∗ stand for the one- and two-sided Brownian maxima of u and v, respectively. The paper contains the proof of the sharp weak-type estimate... [formula]. Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on...
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.
We consider some variational principles in the spaces C*(X) of bounded continuous functions on metrizable spaces X, introduced by M. M. Choban, P. S. Kenderov and J. P. Revalski. In particular we give an answer (consistent with ZFC) to a question stated by these authors.
Let G be a group acting on Ω and F a G-invariant algebra of subsets of Ω. A full conditional probability on F is a function P:F×(F∖{∅})→[0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB)=P(A|B) for all g∈G and A,B∈F, and strongly G-invariant provided that P(gA|B)=P(A|B) whenever g∈G and A∪gA⊆B. Armstrong (1989) claimed that weak and strong...
We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.
A space X is star-Hurewicz if for each sequence (Un:n∈N) of open covers of X there exists a sequence (Vn:n∈N) such that for each n, Vnis a finite subset of Un, and for each x∈X, x∈St(⋃Vn,Un) for all but finitely many n. We investigate the relationship between star-Hurewicz spaces and related spaces, and also study topological properties of star-Hurewicz spaces.
We describe the fields of rational constants of generic four-variable Lotka–Volterra derivations. Thus, we determine all rational first integrals of the corresponding systems of differential equations. Such systems play a role in population biology, laser physics and plasma physics. They are also an important part of derivation theory, since they are factorizable derivations. Moreover, we determine...
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