Binary central relations and submaximal clones determined by nontrivial equivalence relations

Let k ≥ 3, θ a nontrivial equivalence relation on E _k = {0, . . . ,k – 1}, and ρ a binary central relation on E _k (a reflexive graph with a vertex having E _k as its neighborhood). It is known that the clones Pol θ and Pol ρ (of operations on E _k preserving θ and ρ, respectively) are maximal clones; i.e., covered by the largest clone in the inclusion-ordered lattice of clones on E _k . In this paper, we give the classification of all binary central relations ρ on E _k such that the clone Pol θ ∩ Pol ρ is maximal in Pol θ.     

Exact asymptotic bias for estimators of the Ornstein–Uhlenbeck process

We study the bias and the bias derivative for a family \({\mathcal{F}}\) of asymptotically efficient estimators of the Ornstein–Uhlenbeck process. That family contains the maximum likelihood, the conditional maximum likelihood and the empirical estimators. We show that, if g(θ _T ) is an estimator of g(θ), where θ is the parameter and \({\theta_{T} \in \mathcal{F}}\), then, under mild conditions,

$T\,E\left[g(\theta_{T})-g(\theta)\right]\xrightarrow[T\rightarrow\infty]{}c_{\theta}g^{\prime}(\theta)+\theta{g}^{\prime\prime}(\theta),$

where c _θ is an explicit constant that only depends on the choice of θ _T . In particular, if θ _T is one of the three previous estimators, one has

$T\,E_{\theta}(\theta_{T}-\theta)\xrightarrow[T\rightarrow\infty]\,2.$

     

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmoρ(ω) and establish the duality between h_ρ~1(ω) and bmo _ρ(ω),where ω∈A_1~(ρ, ∞)(R~n) and ω's locally behave as Muckenhoupt's weights but actually include them. We also give the Fefferman-Stein type decomposition of bmoρ(ω) with respect to Riesz transforms associated to Schrdinger operator L, where L =-? + V is a Schrdinger operator on R~n(n≥3) and V is a non-negative function satisfying the reverse Hlder inequality.     

Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

Packing,counting and covering Hamilton cycles in random directed graphs

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L ^p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L ^p (R ⁿ , (φ _iρ (x))² dx) where φ _iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.     

Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let (M, θ) be a pseudo-Hermitian space of real dimension 2n + 1, that is M is a CR-manifold of dimension 2n + 1 and θ is a contact form on M giving the Levi distribution \({HT(M) \subset TM}\). Let \({M^\theta \subset T^* M}\) be the canonical symplectization of (M, θ) and let M be identified with the zero section of M ^θ . Then M ^θ is a manifold of real dimension 2(n + 1) which admits a canonical foliation by surfaces parametrized by \({\mathbb{C} \ni t+i\sigma\mapsto \phi^{\theta}_{p}(t+i\sigma)=\sigma\theta_{g_t(p)}}\), where \({p \in M}\) is arbitrary and g _t is the flow generated by the Reeb vector field associated to the contact form θ. Let J be an (integrable) complex structure defined in a neighbourhood U of M in M ^θ . We say that the pair (U, J) is an adapted complex tube on M ^θ if all the parametrizations \({\phi^{\theta}_{p}(t+i\sigma)}\) defined above are holomorphic on \({(\phi^{\theta}_{p})^{-1}(U)}\). In this paper we prove that if (U, J) is an adapted complex tube on M ^θ , then the real function E on \({M^\theta\subset T^*M}\) defined by the condition \({\alpha=E (\alpha)\theta_{\pi(\alpha)}}\), for each \({\alpha \in M^\theta}\), is a canonical defining function for M which satisfies the homogeneous Monge–Ampère equation (dd ^c E)ⁿ⁺¹ = 0. We also prove that if M and θ are real analytic then the symplectization M ^θ admits an unique maximal adapted complex tube.     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

Asymptotic behavior of zeros of Mittag-Leffler functions of order 1/2

Let z _n denote the sequence of zeros of the Mittag-Leffler function E _ρ (z; μ), ρ > 0, μ ∈ ?, which is an entire function of order ρ. With the exception of the case ρ = 1/2, Re μ = 3 an asymptotic behavior of the sequence z _n ^ρ was known earlier up to infinitesimals o(1) having a sharply defined rate of decrease. In this paper the behavior of the sequence z _n ^1/2 is studied just in this exceptional case. Furthermore, for ρ = 1/2, μ > 3 we give the form of a curvilinear half-plane which is free of the points z _n .     

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1] ⁿ transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρ _μ and ρ _ν . It is shown that if there exist positive constants ? and M such that ? < ρ _ν < M, ? < ρ _ν < M, there exist numbers α, β > 1 such that p = αβ(n ? 1)^?1 (α + β)^?1 > 1 and ρ _μ ∈ W ^1,α (Θ), ρ _ν ∈ W ^1,β ) (Θ), where W ^1,q denotes a Sobolev class, then the mapping T belongs to the class W ^1,p (Θ).     

A realization theorem in the context of the Schur-Szegő composition

Every real polynomial of degree n in one variable with root ?1 can be represented as the Schur-Szeg? composition of n ? 1 polynomials of the form (x + 1) ^n?1(x + a _i ), where the numbers a _i are uniquely determined up to permutation. Some a _i are real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ? ρ, r ? [n/2], there exists a polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a _i .     

Rigidity theorem for Willmore surfaces in a sphere

Let M² be a compact Willmore surface in the (2 + p)-dimensional unit sphere S^{2 + p}. Denote by H and S the mean curvature and the squared length of the second fundamental form of M², respectively. Set ρ² = S?2H². In this note, we proved that there exists a universal positive constant C, such that if ∥ρ²∥₂<C, then ρ²=0 and M² is a totally umbilical sphere.     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

On decompositions of trigonometric polynomials

Let R _t [θ] be the ring generated over R by cosθ and sinθ, and R _t (θ) be its quotient field. In this paper we study the ways in which an element p of R _t [θ] can be decomposed into a composition of functions of the form p = R ? q, where R ∈ R(x) and q ∈ R _t (θ). In particular, we describe all possible solutions of the functional equation R ₁ ? q ₁ = R ₂ ? q ₂, where R ₁,R ₂ ∈ R[x] and q ₁, q ₂ ∈ R _t [θ].     

A moving object and observers in ℝ<Superscript>2</Superscript> with a piecewise smooth shading set

We consider the motion of an object t in the space ?², where a bodily bounded set G with piecewise smooth boundary hinders the motion and visibility. In a neighborhood of convex parts of the boundary, there are observers, which can hide from t in a shading set s(t) ? ?² G in the case of danger from t. We find characteristic properties of a trajectory T of the object that maximizes the value min{ρ(t, s(t)): t ∈ T}.     

On a problem of P. Hall for Engel words

Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ ^*(G), respectively. The following problems are generally attributed to P. Hall:

1. (I)
   If π is a set of primes and |G : θ ^*(G)| is a finite π-group, is θ(G) also a finite π-group?
    
2. (II)
   If θ(G) is finite and G satisfies maximal condition on its subgroups, is |G : θ ^*(G)| finite?
    
3. (III)
   If the set \({\{\theta(g_1,\ldots,g_n) \;|\; g_1,\ldots,g_n\in G\}}\) is finite, does it follow that θ(G) is finite?
    

We investigate the case in which θ is the n-Engel word e _n  = [x, _n y] for \({n\in\{2,3,4\}}\) .

     

(0, 2) Pál-type interpolation on a circle in the complex plane involving Möbius transforms

We study the regularity of certain (0,?2) Pál-type interpolation problems involving the Möbius transforms of the zeros of z ²ⁿ ?ρ ²ⁿ to the circle ∣z∣?=?ρ′.     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

Global strong solutions of the Cauchy problem for 1D compressible Navier-Stokes equations with density-dependent viscosity

We consider the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = Aρ ^α , where α > 0 and A > 0. The global existence of strong solutions is obtained, which improves the previous results by enlarging the interval of α. Moreover, our result shows that no vacuum is developed in a finite time provided the initial data does not contain vacuum.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.