Related papers: Comparison theorems and some of their applications
We provide new simple proofs of the Kolmogorov extension theorem and Prokhorovs' theorem. The proof of the Kolmogorov extension theorem is based on the simple observation that $\mathbb{R}$ and the product measurable space…
This is an old article of 2000. Its aim is to illustrate how a Lie-theoretic result of Zelmanov enables one to treat various problems in group theory.
We obtain a necessary and sufficient condition for the convergence of independent products on Lie groups, as a natural extension of Kolmogorov's three-series theorem. Application to independent random matrices is discussed.
We improve on Gonek-Montgomery's quantitative version of Kronecker's approximation theorem.
A new generalization of the classical separate algebraicity theorem is suggested and proved.
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
After introducing some cohomology classes as obstructions to orientation and spin structures etc., we explain some applications of cohomology to physical problems, in special to reduced holonomy in M- and F-theory.
Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.
We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.
We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete…
We consider generalized Kolmogorov system. We prove the existence of heteroclinic trajectory. We apply the results to astrophysical models for self-gravitating particles and predator-prey systems.
In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases.
This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.
An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.
The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.
An analog of Picard's little theorem for entire functions of matrices is proved.
Using Morse theory and a new relative homological linking of pairs, we prove a ``homological linking principle'', thereby generalizing many well known results in critical point theory.
We study $\Sigma^1_2$ definable counterparts for some algebraic equivalent forms of the Continuum Hypothesis. All turn out to be equivalent to "all reals are constructible".