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In this paper we study a new class of functions, which we call (ω,c)\\(( ,c)\\)-pseudo periodic functions. This collection includes pseudo periodic, pseudo anti-periodic, pseudo Bloch-periodic, and unbounded functions. We prove that the set conformed by these functions is a Banach space with a suitable norm. Furthermore, we show several properties of this class of functions as the convolution invariance. We present some examples and a composition result. As an application, we prove the existence and uniqueness of (ω,c)\\(( ,c)\\)-pseudo periodic mild solutions to the first order abstract Cauchy problem on the real line. Also, we establish some sufficient conditions for the existence of positive (ω,c)\\(( ,c)\\)-pseudo periodic solutions to the Lasota–Wazewska equation with unbounded oscillating production of red cells.

Our paper is focused on spaces of generalized almost periodic functions which, as in classical Fourier analysis, are associated with a Fourier series with real frequencies. In fact, based on a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch’s sense, we refine the Bochner-type property by showing that the condition of almost periodicity of a function in any of these generalized spaces can be interpreted in the way that, with respect to the topology of each space, the closure of its set of translates coincides with its corresponding equivalence class.

We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

Classes of target functions containing a large number of approximately orthogonal elements are known to be hard to learn by the Statistical Query algorithms. Recently this classical fact re-emerged in a theory of gradient-based optimization of neural networks. In the novel framework, the hardness of a class is usually quantified by the variance of the gradient with respect to a random choice of a target function. A set of functions of the form x → a x mod p , where a is taken from Z p , has attracted some attention from deep learning theorists and cryptographers recently. This class can be understood as a subset of p -periodic functions on Z and is tightly connected with a class of high-frequency periodic functions on the real line. We present a mathematical analysis of limitations and challenges associated with using gradient-based learning techniques to train a high-frequency periodic function or modular multiplication from examples. We highlight that the variance of the gradient is negligibly small in both cases when either a frequency or the prime base p is large. This in turn prevents such a learning algorithm from being successful.

In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly:$$\\mu $$μ -a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of$$\\mu $$μ -a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of$$\\mu $$μ -a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of$$\\mu $$μ -a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have$$\\mu $$μ -a.p. solutions.

In this paper we introduce the class of (N,λ)\\((N, )\\)-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N,λ)\\((N, )\\)-periodic solutions to the following class of Volterra difference equations with infinite delay: u(n+1)=α∑j=−∞na(n−j)u(j)+f(n,u(n)),n∈Z,α∈C,\\[ u(n+1)= _j=-ınfty ^na(n-j)u(j)+f (n,u(n) ), n ın Z, ın C, \\] where the kernel a and the nonlinear term f satisfy suitable conditions.

As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c -almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c -almost periodic function in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) is also c m -almost periodic in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) for each non-zero integer number m . This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complex functions defined on vertical strips in the complex plane.

As a consequence of a general result, we prove that in the case of singular integrals the set of convergence consists only of the two functions 1 and cos . We prove also a multivariate version of this result and apply it to find the necessary and sufficient conditions for the convergence of the sequences of positive linear operators associated to the rectangular and triangular summation.

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by Dunkl (Am Math Soc 179:331–348, 1973), Jewett (Adv Math 18(1):1–101, 1975) and Spector (Apercu de la theorie des hypergroups, (French) Analyse harmonique sur les groupes de Lie (Sém. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups.

In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (tk) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.