Comments on the spectra of Pisot numbers

For q∈(1,2), Erd?s, Joó and Komornik studied the spectra of q, defined as

     

The structure of the spectra of Pisot numbers

For q∈(1,2), Erdös, Joó and Komornik studied the set:

     

A linear algorithm for nonhomogeneous spectra of numbers

Given a sequence of integers [a_i]_i=1ⁿ, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that a_i = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of $\text{d}$ and $\text{d}$ such that $\text{d < α <}\text{d}$ if and only if a_i = [iα + β] (1 ? i ? n) for suitable β = β(α).     

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of for Pisot numbers , where

Although the quantity is known for some Pisot numbers , there has been no general method for computing . This paper gives such an algorithm. With this algorithm, some properties of and its generalizations are investigated.

A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .

     

Nonhomogeneous waring equations

It is proved that for an arbitrary positive integer k the equation n=x²+y²+z³+u³+v⁴+w¹⁴+t^4k+1 has a positive integer solution for all sufficiently large n. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 65–68.     

Nonhomogeneous Rankin convolutions

The properties of the Rankin convolutions of two eigenfunctions of different parities from the discrete part of the spectrum of the Laplace operator are studied. Analytical continuability of these convolutions into the left half-plane is proved and a functional equation of Riemann type is obtained. Applications to arithmetical convolutions are given. In particular the asymptotics of such convolutions is obtained by using the nonhomogeneous Rankin convolutions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 37–51.     

Nonhomogeneous Markov processes

We present the foundations of the theory of nonhomogeneous Markov processes in general state spaces and we give a survey of the fundamental papers in this topic. We consider the following questions:

1. The existence of transition functions for a Markov process.
2. The construction of regularization of processes.
3. The properties of right and left processes: the strict Markov property, the behavior of excessive functions, etc.
4. The relation of right and left processes with dual homogeneous processes and the application of the results of the nonhomogeneous theory to dual homogeneous processes, etc.

     

Resonant-Superlinear Nonhomogeneous Dirichlet Problems

We consider a Dirichlet nonlinear equation driven by the(p, 2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.     

Nonhomogeneous G-spaces of compact Lie groups

A duality theorem for a compact G-manifold M of a compact Lie group G is proved. For the case when all the orbits in M are principal, it is proved that the quotient space of the complexification of M by the action of the associated algebraic group G^c is isomorphic to the quotient space of M by the action of G.Translated from Matematicheskie Zametki, Vol, 13, No. 4, pp. 523–529, April, 1973.This paper was completed under the guidance of A. L. Onishchik, to whom the author expresses his thanks.     

非齐次马氏链的禁止概率

基于齐次马氏链禁止概率理论,研究非齐次马氏链禁止概率.经推导得出其基本分解公式和若干结论,并在引入假设条件下,得出非齐次马氏链禁止概率与齐次马氏链禁止概率的联系.     

非齐次特征值问题扰动分析

本文的目的是想对Ax=λx+b,x~Tx=1解的性态和扰动作一些分析,讨论解的存在范围,给出一个解扰动的估计不等式,并针对A对称情形,作较具体的分析,对大||b||和小||b||两种情形,作专门讨论.     

Solutions of a Nonhomogeneous Burgers Equation

In this article, we construct solutions of a nonhomogeneous Burgers equation subject to certain unbounded initial profiles. In an interesting study, Kloosterziel [ 1 ] represented the solution of an initial value problem (IVP) for the heat equation, with initial data in , as a series of the self‐similar solutions of the heat equation. This approach quickly revealed the large time behavior for the solution of the IVP. Inspired by Kloosterziel [ 1 ]'s approach, we express the solution of the nonhomogeneous Burgers equation in terms of the self‐similar solutions of a linear partial differential equation with variable coefficients. Finally, we also obtain the large time behavior of the solution of the nonhomogeneous Burgers equation.     

Nonhomogeneous eigenvalue problem with indefinite weight

This paper, following the theory of partial differential equations on the Orlicz–Sobolev spaces, is mainly concerned with the nonhomogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. The results show that the spectrum of such problems contains a continuous family of eigenvalues.     

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

非齐次线性方程组无解条件的应用

给出了非齐次线性方程组无解的条件在向量组的线性相关性、矩阵的秩及线性方程组解的判定上的一些应用.