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.     

Values of the Euler phi function not divisible by a given odd prime

An asymptotic formula is given for the number of integers n≤x for which φ(n) is not divisible by a given odd prime.     

Covering the edges of a graph by three odd subgraphs

We prove that any finite simple graph can be covered by three of its odd subgraphs, and we construct an infinite sequence of graphs where an edge‐disjoint covering by three odd subgraphs is not possible. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 77–82, 2006     

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Bounds for the Local Distance Between Arithmetical Distributions

We obtain upper and lower bounds for the local distance ρ(ν _x ,P _x ). Here ν _x is the distribution of a set of strongly additive functions f _x with respect to the usual frequency on the set of positive integers, and P _x is the distribution of the sum of suitably chosen independent random variables. We consider the case where f _x (p)∈{0,1,2,…,C} for all prime numbers p with some fixed positive integer C.      

Quadratic decomposition of the symmetric semi-classical polynomial sequences of odd class: some examples from the class three

In this present work, we deal with the quadratic decomposition of symmetric semi-classical polynomial sequences of odd class. Some examples from class three are settled. The recurrence coefficients, the moments and the integral representations of the corresponding regular forms are established.     

Identities of symmetry for higher-order Euler polynomials in three variables (II)

We derive twenty five basic identities of symmetry in three variables related to higher-order Euler polynomials and alternating power sums. This demonstrates that there are abundant identities of symmetry in three-variable case, in contrast to two-variable case, where there are only a few. These are all new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the higher-order Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.     

Groups with handles of order different from three

We obtain a test for the unsimplicity of an infinite group.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1030–1042, August, 2004.     

Arithmetical thinking in children attending special schools for the intellectually disabled

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children's own ways of operating, which implies a triadic teaching process.     

On the representation of large odd integer as a sum of three almost equal primes

The well-known Goldback-Vinogradov theorem states that every large odd integer is a sum of three primes. In the present paper it is further proved that every large odd integerN can be represented as

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it has been proposed by Jameson, Schmidt, and Turkel to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed here, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which can move all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping can be effective in accelerating convergence to steady state.     

Extraction of orthogonal polynomials from generating function for reciprocal of odd numbers

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both even and odd order convergents of a regular C-fraction of a power series with coefficients as reciprocal of odd numbers are described. The two sequences of convergents are nothing but diagonal and upper diagonal Pade approximants for the power series. The two orthogonal polynomials extracted from denominators are shown to be classical orthogonal polynomials and two orthogonal polynomials extracted from numerators are shown to be non-classical orthogonal polynomials..     

Euler多项式的若干对称恒等式

Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.     

Precise regularity results for the Euler equations

It has already been proved, under various assumptions, that no singularity can appear in an initially regular perfect fluid flow, if the L^∞ norm of the velocity's curl does not blow up. Here that result is proved for flows in smooth bounded domains of (d?2) when the regularity is expressed in terms of Besov (or Triebel-Lizorkin) spaces.     

Exterior Problem for the Three-dimensional Euler Equations

A priori estimates for the exterior initial boundary value problems of the Euler equations are considered. The existence and uniqueness of a local solution is proved.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

Binary codes and partial permutation decoding sets from the odd graphs

For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω^{k}, the set of all k-subsets of Ω = {1, 2, …, 2k +1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k ⁴ is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement $\overline {O(k)} $ , is investigated.