On the regularity of the locally integrable solutions of the functional equations $$\sum\limits_{i = 1}^k {a_i \left( {x,t} \right)f\left( {x + \varphi _i \left( t \right)} \right) = b\left( {x,t} \right)} $$

Aequationes mathematicae -     

Qualitative Investigation of the Singular Cauchy Problem \sum\limits_{k = 1}^n {(a_{k1} t + a_{k2} x)(x')^k = b_1 t + b_2 x + f(t,x,x'),x(0) = 0}

We prove the existence of continuously differentiable solutions $x:(0,\rho ] \to {\mathbb{R}}$ with required asymptotic properties as t → +0 and determine the number of these solutions.     

Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.     

Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )}

Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.     

Exact formulae for periodic solutions of\dot x(t + 1) = \alpha x(t) + bx^3 (t))

An explicit formula for the symmetric period-4-solutions of the odd cubic differential delay equation is calculated together with a formula for its primary branch. The qualitative behavior of the primary branch is discussed.

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Zusammenfassung Die symmetrischen Periode-4-Lösungen der ungeraden kubischen verzögerten Differentialgleichung werden zusammen mit einer Formel für ihren Primärzweig exakt berechnet. Das qualitative Verhalten des Primärzweiges wird diskutiert.

     

On the diophantine equation F $$\left( {\left( {_n^x } \right)} \right) = b\left( {_m^y } \right)$$

As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions.     

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation \sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C

This paper is concerned with iterative solution to general Sylvester-conjugate matrix equation of the form $\sum_{i = 1}^{s} A_{i}V + \sum_{j = 1}^{t} B_{j}W = \sum_{l = 1}^{m} E_{l}\overline{V}F_{l} + C$ . An iterative algorithm is established to solve this matrix equation. When this matrix equation is consistent, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.     

A New Proof of Diophantine Equation $\Bigg( \begin{matrix} n \\ 2 \end{matrix} \Bigg) = \Bigg( \begin{matrix} m \\ 4 \end{matrix} \Bigg)$

By using algebraic number theory and $p$-adic analysis method, we give a new and simple proof of Diophantine equation $\Bigg( \begin{matrix} n \\ 2 \end{matrix} \Bigg) =\Bigg( \begin{matrix} m \\ 4 \end{matrix} \Bigg)$.     

On convergence of the averages\frac{1}{N}\sum\nolimits_{n = 1}^N {f_1 (R^n x)f_2 (S^n x)f_3 (T^n x)}

In this note, we will prove that for commuting ergodic measure preserving transformationsR, S andT, ifRT ^?1,ST ^?1 are also ergodic, then the limit $$\lim \frac{1}{N}\sum\nolimits_{n = 1}^N {f_1 (R^n x)f_2 (S^n x)f_3 (T^n x)} $$ exists inL ¹-norm. The method used in this note was developed byConze, Furstenberg, Lesigne andWeiss.