Let Λ be a sub-semigroup of the reals. We show that the Bass and topological stable ranks of the algebras APΛ = {f ∈ AP : σ(f) ⊆ Λ} of almost periodic functions on the real line and with Bohr spectrum in Λ are infinite whenever the algebraic dimension of the Q-vector space generated by Λ is infinite. This extends Suárez’s result for APR = AP. Also considered are general subalgebras of AP. Intro...

Let $${{\mathcal {F}}}_n$$ be the Farey series of order $$n\ge 1$$ . We obtain sharp effective estimates sums $$\begin{aligned} F_n(f)= \sum _{\frac{{\kappa }}{{\lambda }}\in {{\mathcal {F}}}_n} f\Big (\frac{{\kappa }}\Big ),{\qquad }F_{n,{\sigma }}(f)= \frac{1}{{\kappa }^{\sigma }{\lambda }}f\Big }0< {\sigma }\le 1, \end{aligned}$$ for 1-periodic functions f satisfying weak local regularity as...

where a = a(t), h = h(t), and τ = τ(t) are continuous T-periodic functions. We will also assume that T > 0, λ > 0, f = f (t) as well as h = h(t) are positive, ∫T 0 a(t)dt > 0. Functional differential equations with periodic delays appear in a number of ecological models. In particular, our equation can be interpreted as the standard Malthus population model y′ = −a(t)y subject to perturbation w...

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations, and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It was pointed out by many authors that the non-regular...

In the paper is given a method of finding periodical solutions differential equation form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions $t$. This reduces $n$-periodic soluble problem to system $n+1$ linear equations, $n=2,3$. Furthermore, by using well known properties in algebra, all existenc...

We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a " twisted " version of the recent generalisation ...

We show that there is no blowup solutions, for positive viscosity constant , to the equation f xxt f xxxx +ff xxx f x f xx = 0; x 2 (0; 1); t > 0 with (i) periodic boundary condition, or (ii) Dirichlet boundary condition f = f x = 0 or (iii) Neumann boundary condition f = f xx = 0 on the boundary x = 0; 1. Furthermore we show that every solution decays to the trivial steady state as t goes to i...

Recently, the Fokker-Planck dynamics of particles in periodic potentials ±V , have been investigated by using the matrix continued fraction method. It was found that the two periodic potentials, one being bistable and the other metastable give the same diffusion coefficient in the overdamped limit. We show that this result naturally follows from the fact that the considered potentials in the co...