Non-random behavior in sums of modular symbols

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

(&omega,c)- PSEUDO ALMOST PERIODIC FUNCTIONS, (&omega,c)- PSEUDO ALMOST AUTOMORPHIC FUNCTIONS AND APPLICATIONS

In this paper, we introduce the classes of $(\omega, c)$-pseudo almost periodicfunctions and $(\omega, c)$-pseudo almost automorphicfunctions. These collections include $(\omega, c)$-pseudo periodicfunctions, pseudo almost periodic functions and their automorphic analogues.We present an application to the abstract semilinear first-order Cauchy inclusions in Banach spaces.

Bochner definition of Stepanov-like almost automorphic functions on time scales and an application to cellular neural networks with delays

The definition of Stepanov-like almost automorphic functions on time scales had been proposed in the literature, but at least one result was incorrect, which involved Bochner transform. In our work, we give the Bochner definition of Stepanov-like almost automorphic functions on time scales, and prove that a function is Stepanov-like almost automorphic if and only if it satisfies Bochner definition of Stepanov-like almost automorphic function on time scales. The Bochner definition of Stepanov-like almost automorphic functions on time scales corrects the faulty result, and perfects the definition of Stepanov-like almost automorphic functions. As applications, we discuss the almost automorphy of a certain dynamic equation and some cellular neural networks with delays on time scales.

(µ, η)-pseudo almost automorphic solutions of a new class of competitive Lotka-Volterra model with mixed delays

In the natural world, competition is an important phenomenon that can manifest in various generalized environments (economy, physics, ecology, biology,...). One of the famous models which is able to represent this concept is the Lotka-Volterra model. A new class of a competitive Lotka-Volterra model with mixed delays and oscillatory coefficients is investigated in this work. Thus, by using the (µ, η)-pseudo almost automorphic functions function class and the Leray-Schauder fixed-point theorem, it can be proven that solutions exist. In addition, in such situations, we have a number of species that coexist and all the rest will be extinct. Therefore, the study of permanence becomes unavoidable. Therefore, sufficient and new conditions are given in order to establish the permanence of species without using a comparison theorem. By the new Lyapunov function we prove the asymptotic stability for the considered model. Moreover, we investigate the globally exponential stability of the (µ, η)-pseudo almost automorphic solutions. In the end, an example is given to support theoretical result feasibility.

Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

Using the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.