The Fibonacci word f=010010100100101⋯ is one of the most well-studied words in area combinatorics on words. It not periodic, but nevertheless contains many highly periodic factors (contiguous subwords). For example, it cubes (i.e., non-empty form xxx). We study prefixes that end with a cube. Using computer prover Walnut, we obtain an exact description positions at which cube ends. This gives ce...

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(n− 1, xn−k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p gcd(p,k) periodic...

Let A and M be closed linear operators defined on a complex Banach space X. Using operator-valued Fourier multipliers theorems, we obtain necessary and sufficient conditions to guarantee existence and uniqueness of periodic solutions to the equation d dt (Mu(t)) = Au(t) + f(t), in terms of either boundedness or R-boundedness of the modified resolvent operator determined by the equation. Our res...

This article concerns the stability and periodicity of solutions to the delay dynamic system x4(t) = A(t)x(t) + F (t, x(t), x(g(t))) + C(t) on a time scale. By the inequality technique for vectors, we obtain some stability criteria for the above system. Then, using the Horn fixed point theorem, we present some conditions under which our system is asymptotically periodic and its periodic solutio...

We obtain optimal trigonometric polynomials of a given degree N that majorize, minorize and approximate in L(R/Z) the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann ([9] and [10]) that generalize a paper of J. Vaaler ([14]). As applications we provide the corresponding Erdös-Turán-type inequalities, approximations to other periodic functions and bound...

This paper studies the existence of periodic solutions to the delay differential equation ẋ(t) = f(x(t− μτ(t)), ) . The analysis is based on a perturbation method previously used for retarded differential equations with constant delay. By transforming the studied equation into a perturbed non-autonomous ordinary equation and using a bifurcation result and the Poincaré procedure for this last eq...

We prove that a group of injection-locked oscillators, each modelled using a nonlinear phase macromodel, responds as a single oscillator to small external perturbations. More precisely, we show that any group of injection-locked oscillators has a single effective PRC [1] or PPV [2], [3] that characterises its phase/timing response to small external perturbations. This result constitutes a found...

Let K be a field of characteristic 0, f : N → K be a multiplicative function, and F (z) = P n≥1 f (n)z n ∈ K[[z]] be algebraic over K(z). Then either there is a natural number k and a periodic multiplicative function χ(n) such that f (n) = n k χ(n) for all n, or f (n) is eventually zero. In particular , the generating function of a multiplicative function f : N → K is either transcendental or r...

Let T be a periodic time scale. We use Krasnoselskii’s fixed point theorem to show that the neutral functional differential equation with impulses x(t) = −A(t)x(t) + g(t, x(t− h(t))) + f(t, x(t), x(t− h(t))), t 6= tj , t ∈ T, x(t+j ) = x(t − j ) + Ij(x(tj)), j ∈ Z + has a periodic solution. Under a slightly more stringent conditions we show that the periodic solution is unique using the contrac...

Let $f(t)$ be a smooth and periodic function of one real variable. Then the planar curves $t\mapsto \big(f'(t),f(t)\big)$ \big(f''(t)-f(t),f'(t)\big)$ both have non-negative rotation number around every point not on curve. These are two simplest cases beautiful Theorem by C. Loewner. This article is expository, we prove statements elementary means following work Bol [3]. After that, present Loe...