On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.     

Constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem with continuous time

Estimates for the rate of convergence in ergodic theorems are necessarily spectral. We find the equivalence constants relating the polynomial rate of convergence in von Neumann’s mean ergodic theorem with continuous time and the polynomial singularity at the origin of the spectral measure of the function averaged over the corresponding dynamical system. We also estimate the same rate of convergence with respect to the decrease rate of the correlation function. All results of this article have obvious exact analogs for the stochastic processes stationary in the wide sense.     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems

In the L _p spaces, 1 < _p < ∞, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff’s theorem in the presence of bounds on the convergence rate in von Neumann’s ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in L _p are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.     

Graham’s pebbling conjecture on product of complete bipartite graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs G and H, f( G x H) ⩽ f( G) f( H). We show that Graham’s conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are complete bipartite graphs.     

Lazard’s theorem for S-pure flat modules

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.     

Uniform version of Weyl–von Neumann theorem

We prove a “quantified” version of the Weyl–von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in Voiculescu’s theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C ₀(X), for a large class of spaces X.     

Chevalley��s theorem with restricted variables

First, a generalization of Chevalley’s classical theorem from 1936 on polynomial equations f(x ₁,...,x _N ) = 0 over a finite field K is given, where the variables x _i are restricted to arbitrary subsets A _i ⊆ K. The proof uses Alon’s Nullstellensatz. Next, a theorem on integer polynomial congruences f(x ₁,...,x _N ) ≡ 0 (mod p ^v ) with restricted variables is proved, which generalizes a more recent result of Schanuel. Finally, an extension of Olson’s theorem on zero-sum sequences in finite Abelian p-groups is derived as a corollary.     

Maps preserving product X*Y + YX* on factor von Neumann algebras

Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? satisfies Φ(X*?Y?+?YX*)?=Φ(X)*Φ(Y)?+?Φ(Y)Φ(X)* (?X,?Y?∈?𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.     

Khinchin��s theorem for approximation by integral points on quadratic varieties

We prove an analogue of the Khinchin??s theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often.     

Schur’s theorem for a block Hadamard product

We define a generalized Kronecker product for block matrices, mention some of its properties, and apply it to the study of a block Hadamard product of positive semidefinite matrices, which was defined by Horn, Mathias, and Nakamura. Under strong commutation assumptions we obtain generalizations of Schur’s theorem and of Oppenheim’s inequality.     

A generalization of the original Jordan–von Neumann theorem

The problem of representability of quadratic functionals (acting on modules over unital complex ∗-algebras), by sesquilinear forms, is generalized by weakening the homogeneity equation. The corresponding representation theorem can be considered as a generalization of (the original form of) the classical Jordan–von Neumann characterization of complex inner product spaces.     

Liouville’s theorem and the restricted biharmonic mean value property on the plane

In [3, 4], under some conditions we proved that a bounded Lebesgue measurable function satisfying the restricted biharmonic mean value property in ? ⁿ , where n ≥ 3 or n = 1, is constant. In the present paper, we study the case n = 2.     

An ergodic theorem of a parabolic Anderson model driven by L��vy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j)) _i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure ν _h starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang.