On some arithmetic properties of Mahler functions

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ? ≥ 2 such that f(x), f(x^?),..., \(f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})\) are linearly dependent over ?(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(x^?) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.     

Level Hitting Probabilities and Extremal Indexes for Some Particular Dynamical Systems

We establish an exact formula for the distribution of the partial maximum sequence generated by the stationary process obtained by iterations of the Rényi map x → βx mod 1, β = 2, 3, .... We thus obtain a simple proof of some asymptotic behaviour of the extremes and the values of the extremal index. A numerical application is presented.     

Commuting Krichever–Novikov differential operators with polynomial coefficients

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w² = z³+c₂z2+c₁z+c₀ with arbitrary coefficients c_i, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.     

Critical exponents and the pseudo-<Emphasis Type="Italic">є</Emphasis>-expansion

We present the pseudo-?-expansions (τ-series) for the critical exponents of a λ?⁴-type three-dimensional O(n)-symmetric model obtained on the basis of six-loop renormalization-group expansions. We present numerical results in the physically interesting cases n = 1, n = 2, n = 3, and n = 0 and also for 4 ≤ n ≤ 32 to clarify the general properties of the obtained series. The pseudo-?-expansions or the exponents γ and α have coefficients that are small in absolute value and decrease rapidly, and direct summation of the τ -series therefore yields quite acceptable numerical estimates, while applying the Padé approximants allows obtaining high-precision results. In contrast, the coefficients of the pseudo-?-expansion of the scaling correction exponent ω do not exhibit any tendency to decrease at physical values of n. But the corresponding series are sign-alternating, and to obtain reliable numerical estimates, it also suffices to use simple Padé approximants in this case. The pseudo-?-expansion technique can therefore be regarded as a distinctive resummation method converting divergent renormalization-group series into expansions that are computationally convenient.     

Lebesgue Constants for Some Interpolating <Emphasis Type="Italic">L</Emphasis>-Splines

We find exact values for the uniform Lebesgue constants of interpolating L-splines that are bounded on the real axis, have equidistant knots, and correspond to the linear thirdorder differential operator L₃(D) = D(D₂ + α₂) with constant real coefficients, where α > 0. We compare the obtained result with the Lebesgue constants of other L-splines.     

On the Asymptotic Locations of the Largest and Smallest Extremes of a Stationary Sequence

This paper deals with the asymptotic independence of the normalized kth upper- and rth lower-order statistics and their locations, defined on some strictly stationary sequences \(\left\{ X_n\right\} _{n\ge 1}\) admitting clusters of both high and low values. The main result is the asymptotic independence of the joint locations of the k-largest extremes and the joint locations of the r-smallest extremes of \(\left\{ X_{n}\right\} _{n\ge 1}\), which allows us to censor a sample, by ensuring that the set of observations that we selected contains the k-largest and r-smallest order statistics of the stationary sequence \(\left\{ X_{n}\right\} _{n\ge 1}\) with a predetermined probability.     

On the Nörlund means of Vilenkin-Fourier series

We prove and discuss some new (H _p ,L _p )-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q _k : k ? 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results.     

On some mean values for the divisor function and the Riemann zeta-function

Let Δ(x) and E(x) denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for ζ(s) on the critical line. We consider some general mean values for Δ(x) and E(x) and discover interesting differences between these two functions. In particular, this yields evidence that E(x) is more negative than Δ(x).     

Characterization theorems for <Emphasis Type="Italic">Q</Emphasis>-independent random variables with values in a locally compact Abelian group

Let X be a locally compact Abelian group, Y be its character group. Following A. Kagan and G. Székely we introduce a notion of Q-independence for random variables with values in X. We prove group analogues of the Cramér, Kac–Bernstein, Skitovich–Darmois and Heyde theorems for Q-independent random variables with values in X. The proofs of these theorems are reduced to solving some functional equations on the group Y.     

Calculation of Bezout’s coefficients for the <Emphasis Type="Italic">k</Emphasis>-ary algorithm of finding GCD

Bezout’s equation is a representation of the greatest common divisor d of integers A and B as a linear combination Ax + By = d, where x and y are integers called Bezout’s coefficients. The task of finding Bezout’s coefficients has numerous applications in the number theory and cryptography, for example, for calculation of multiplicative inverse elements in modular arithmetic. Usually Bezout’s coefficients are caclulated using the extended version of the classical Euclidian algorithm.We elaborate a new algorithm for calculating Bezout’s coefficients based on the k-ary GCD algorithm.     

Estimate of solution stability in a two-dimensional inverse problem for elasticity equations

The problem of determining the density of the medium and one of its elasticity moduli is considered. Properties of the elastic medium and external forces are assumed to be independent of the coordinate x ₃. In this case, the third component of the displacement vector satisfies a scalar equation of the second order, which contains the density ρ of the medium and elasticity modulus μ as coefficients. The parameters ρ and μ are known to be positive and constant everywhere outside some compact domain D ? ?², but they are unknown inside D. The problem of determining these coefficients in D via information, given on the boundary of the domain D for some finite time interval, about a solution of two direct problems is considered. An estimate of the conditional stability of a solution of the inverse problem under consideration is established.     

On the discontinuity of lyapunov exponents of an almost periodic linear differential system affinely depending on a parameter

We construct a linear differential system \(\dot x\) = (A(t) + μB(t))x, x ∈ ?², t ≥ 0, with almost periodic coefficients which is almost reducible for all μ ∈ ? except for an at most countable set and whose singular and higher characteristic exponents treated as functions of the parameter μ are discontinuous at some point.     

A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.     

Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Hüsler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.     

Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local L-splines with uniform knots (such splines were constructed in the authors’ earlier papers) corresponding to a linear differential operator L of order r with constant coefficients and real pairwise different roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which an L-spline locally inherits the property of the generalized k-monotonicity (k ≤ r ? 1) of the input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the L-spline. The parameters of an L-spline that is exact on the kernel of the operator L are written explicitly.     

On Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with periodic coefficients and Eisenstein series

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of \({SL_2(\mathbb{Z})}\) and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.     

Lower semicontinuity of mappings with bounded (<Emphasis Type="Italic">θ</Emphasis>, 1)-weighted (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-distortion

We prove that, under some extra conditions, the locally uniform limit of mappings with bounded (θ, 1)-weighted (p, q)-distortion is a mapping of bounded (θ, 1)-weighted (p, q)-distortion too. Moreover, we obtain the lower semicontinuity of the distortion coefficients.     

Boundary displacement control at one end of a string in the presence of a nonlocal boundary condition of one of four types,and its optimization

In the present paper, we exhaustively solve the problem of boundary control by the displacement u(0, t) = µ(t) at the end x = 0 of the string in the presence of a model nonlocal boundary condition of one of four types relating the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point \(\mathop x\limits^ \circ\).     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

On the Conservativeness of Some Markov Processes

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L ²(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the “increasingness of the volume of some sets(balls)” and “that of the coefficients on the sets” of the Markov processes.