Aleksandrov surfaces and hyperbolicity

Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply-connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.

     

Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions

In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second order horizontal derivatives are Radon measures.     

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

On the Aleksandrov problem in linear n-normed spaces

The aim of this article is to generalize the Aleksandrov problem to the case of linear $n$-normed spaces.     

Kolmogorov Complexity and Noncomputability

We use a method suggested by Kolmogorov complexity to examine some relations between Kolmogorov complexity and noncomputability. In particular we show that the method consistently gives us more information than conventional ways of demonstrating noncomputability (e. g. by embedding in the halting problem). Also, many sets which are (at least) awkward to embed into the halting problem are easily shown noncomputable. We also prove a gap‐theorem for outputting consecutive integers and find, for a given length n, a statement of length n with maximal (shortest) proof length.     

Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x L ^x (r), namely,

Aleksandrov reflection and nonlinear evolution equations,I: The n-sphere and n-ball

We consider the (degenerate) parabolic equationu _t =G(u + ug, t) on then-sphereS ⁿ . This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationu _t =G(u +cu, ¦x¦,t) on then-ballB ⁿ , wherec ₂(B ⁿ ).     

Kolmogorov complexity and cryptography

We consider (in the framework of algorithmic information theory) questions of the following type: construct a message that contains different amounts of information for recipients that have (or do not have) certain a priori information. Assume, for example, that a recipient knows some string a and we want to send him some information that allows him to reconstruct some string b (using a). On the other hand, this information alone should not allow the eavesdropper (who does not know a) to reconstruct b. This is indeed possible (if the strings a and b are not too simple). Then we consider more complicated versions of this question. What if the eavesdropper knows some string c? How long should our message be? We provide some conditions that guarantee the existence of a polynomial-size message; we show then that without these conditions this is not always possible.     

The Aleksandrov problem in linear 2-normed spaces

We introduce the concept of 2-isometry which is suitable to represent the notion of area preserving mappings in linear 2-normed spaces. And then we obtain some results for the Aleksandrov problem in linear 2-normed spaces.     

Essential norms of weighted composition operators and Aleksandrov measures

We derive exact formulas for the essential and weak essential norms of weighted analytic composition operators acting on certain function spaces in the unit disc, extending and improving earlier results due to Sarason and to Kriete and Moorhouse. Differences of composition operators are also considered. The formulas involve the Aleksandrov measures associated to the symbol of the operator. The results are based on a variant of a general method due to Weis of constructing best compact and weakly compact approximants for linear operators on L¹ spaces.     

Hamiltonian Feynman Measures,Kolmogorov Integral,and Infinite-Dimensional Pseudodifferential Operators

Doklady Mathematics - Properties of infinite-dimensional pseudodifferential operators (PDO) are discussed. In particular, the connection between two definitions of PDO is considered: one given in...     

Kolmogorov Averages and Approximate Identities

We study the action of Kolmogorov-type nonlinear averaging operators of the form V ⁻¹ AV on smooth functions. Here, A runs through a family of convolution operators A _ε ^[K], ε>0, generated by a single kernel K∈L ¹(ℝ ⁿ ) in the usual way and forming an “approximate identity” as ε→0, while V is a superposition map given by Vf=v○f, with a monotone continuous function v. The main result characterizes the kernels K with the property that the natural estimate

Aleksandrov maximum principle and bony maximum principle for parabolic equations

The author simplifies the proof of Aleksandrov maximum principle for parabolic equations given by Krylov and obtains finer results. He further proves Bony maximum principle for parabolic equations by using the above results.     

Kolmogorov law and Kolmogorov and Taylor scales in anisotropic turbulence. Turbulence as a result of three-scale interaction

The interaction of waves of different scales in the development of turbulence is considered. A mechanism of occurrence of a wave whose frequency and amplitude satisfy Kolmogorov's law is described.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp. 368–374, March, 1993.