Toward the theory of ring Q-homeomophisms

We investigate classes of the so-called ring Q-homeomorphisms including, in particular, Q-homeomorphisms, various classes of homeomorphisms with finite length distortion, Sobolev’s classes etc. In terms of the majorant Q(x), we give a series of criteria for normality based on estimates of the distortion of the spherical distance under ring Q-homeomorphisms. In particular, it is shown that the class of all ring Q-homeomorphisms f of a domain D ⊂ ℝ ⁿ into , n ≥ 2, with , forms a normal family, if Q(x) has finite mean oscillation in D. We also prove normality of , for instance, if Q(x) has singularities of logarithmic type whose degrees are not greater than n − 1 at every point x ∈ D. The results are applicable, in particular, to mappings with finite length distortion and Sobolev’s classes.     

Similarity classes of3 × 3 matrices over a discrete valuation ring

Let Q be a complete discrete valuation ring. Let Π be a prime element in Q. Write P = ΠQ. For n = 1,2,…, letQ_n be the factor ring Q | Pⁿ. Let G = G1₃(Q_n. Denote by M?_n the G-module of 3 × 3 matrices over Q_n modulo scalar matrices. Canonical forms are found for every element in M?_n, and it is shown that there exist five sets of similarity classes. Some results about the general case of NxN matrices over Q also are proved.     

Intriguing Sets of Points of Q(2n, 2) \Q +(2n ? 1, 2)

Intriguing sets of vertices have been studied for several classes of strongly regular graphs. In the present paper, we study intriguing sets for the graphs Γ _n , n ≥ 2, which are defined as follows. Suppose Q(2n, 2), n?≥ 2, is a nonsingular parabolic quadric of PG(2n, 2) and Q ⁺(2n ? 1, 2) is a nonsingular hyperbolic quadric obtained by intersecting Q(2n, 2) with a suitable nontangent hyperplane. Then the collinearity relation of Q(2n, 2) defines a strongly regular graph Γ _n on the set Q(2n, 2) \ Q ⁺(2n ? 1, 2). We describe some classes of intriguing sets of Γ _n and classify all intriguing sets of Γ₂ and Γ₃.     

Equicontinuity of mean quasiconformal mappings

We establish the equicontinuity and normality of the families R ^Φ of ring Q(x)-homeomorphisms with integral-type restrictions ∫^Φ(Q(x))dm(x) < ∞ on a domain D ⊂ R ⁿ with n ≥ 2. The resulting conditions on Φ are not only sufficient but also necessary for the equicontinuity and normality of these families of mappings. We give some applications of these results to the Sobolev classes W _loc^1,n .     

On mappings in the Orlicz–Sobolev classes on Riemannian manifolds

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes W_loc^1,j W_{loc}^{1,\varphi } under a condition of the Calderon type for the function φ and, in particular, in the Sobolev classes W_loc^1,p W_{loc}^{1,p} for p > n − 1.     

On the lipschitz property of a class of mappings

Open discrete annular Q-mappings with respect to the p-modulus in ? ⁿ , n ≥ 2, are considered in this paper. It is established that such mappings are finite Lipschitz for n ? 1 < p < n if the integral mean value of the function Q(x) over all infinitesimal balls B(x ₀, ?) is finite everywhere.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

A Characterization of L-orthogonal Polynomials from?Three Term Recurrence Relations

We consider the sequence of polynomials {Q _n } satisfying the L-orthogonality ?[z ^?n+m Q _n (z)]=0, 0??m??n?1, with respect to a linear functional ? for which the moments ?[t ⁿ ]=?? _n are all complex. Under certain restriction on the moment functional these polynomials also satisfy a three term recurrence relation. We consider three special classes of such moment functionals and characterize them in terms of the coefficients of the associated three term recurrence relations. Relations between the polynomials {Q _n } associated with two of these special classes of moment functionals are also given. Examples are provided to justify this characterization.     

On extension of some generalizations of quasiconformal mappings to a boundary

This work is devoted to the investigation of ring Q-homeomorphisms. We formulate conditions for a function Q(x) and the boundary of a domain under which every ring Q-homeomorphism admits a homeomorphic extension to the boundary. For an arbitrary ring Q-homeomorphism f: D → D’ with Q ∈ L ¹(D); we study the problem of the extension of inverse mappings to the boundary. It is proved that an isolated singularity is removable for ring Q-homeomorphisms if Q has finite mean oscillation at a point.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems

The paper deals with the Sturm-Liouville operator $$ Ly = - y'' + q(x)y, x \in [0,1], $$ generated in the space L ₂ = L ₂[0, 1] by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to the Sobolev spaceW ₁ ^p [0, 1] for some integer p ≥ 0 and satisfy the conditions q ^(k)(0) = q ^(k)(1) = 0 for 0 ≤ k ≤ s ? 1, where s ≤ p. Let the functions Q and S be defined by the equalities $$ Q(x) = \int_0^x {q(t)dt, S(x) = Q^2 (x)} $$ and let q _n , Q _n , and S _n be the Fourier coefficients of q, Q, and S with respect to the trigonometric system $ \{ e^{2\pi inx} \} _{ - \infty }^\infty $ . Assume that the sequence q _2n ? S _2n + 2Q ₀ Q _2n decreases not faster than the powers n ^?s?2. Then the system of eigenfunctions and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L ₂[0, 1] (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n} - s_{2n} + 2Q_0 Q_{2n} \asymp q_{ - 2n} - s_{2n} + 2Q_0 Q_{ - 2n} , n > 1, $$ holds.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Boundary behavior of ring Q-homeomorphisms on Riemannian manifolds

We study the problems of continuous and homeomorphic extensions to the boundary for so-called ring Q-homeomorphisms between domains on the Riemannian manifolds and establish conditions for the function Q(x) and the boundaries of the domains under which every ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary. This theory can be applied, in particular, to the Sobolev classes.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Arithmetical progressions formed byk different Lehmer pseudoprimes

LetU _n=(αⁿ-β²)/(α-β) forn odd andU _n=(αⁿ-βⁿ)/(α²-β²) for evenn, where α and β are distinct roots of the trinomialf(z)=z ²-√Lz+Q andL>0 andQ are rational integers.U _n is then-th Lehmer number connected withf(z). A compositen is a Lehmer pseudoprime for the bases α and β ifU _n??(n)≡0 (modn), where?(n)=(LD/n) is the Jacobi symbol. IfD=L?4Q>0, U _n denotesn-th Lehmer number,p>3 and 2p?1 are primes,p(2p-1)+(α²-β²)², (α^2p-1±β^2p-1)/(α±β) are composite then the numbers (α^2p-1+β^2p-1)/(α+β), (α^2p-β^2p)/(α²-β²), (α^2p-1-β^2p-1)/(α-β) are lehmer pseudoprimes for the bases α and β and form an arithmetical progression. IfD>0 then from hypothesisH of A. Schinzel on polynomials it follows that for every positive integerk there exists infinitely many arithmetic progressions formed fromk different Lehmer pseudoprimes for the bases α and β.