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An impulsive SIR epidemic model characterized by an incidence rate governed by a mean-reverting Ornstein–Uhlenbeck process is investigated in this paper. First, the existence and uniqueness of the global positive solution to the impulsive SIR epidemic model are established by constructing an equivalent system without impulses. Second, a criterion for assessing the extinction of the epidemic is derived by introducing a periodic function. Then, the existence and global attraction of the boundary periodic solution are demonstrated using the comparison theorem, and a sufficient condition for the existence of a positive periodic solution is obtained with the aid of a Lyapunov function. Finally, the theoretical results are validated via a series of numerical simulations. [ABSTRACT FROM AUTHOR]

The powerful strikes generated by the smasher mantis shrimp require it to possess a robust protection mechanism to withstand the resultant forces. Although recent studies have suggested that phononic bandgaps complement the mantis shrimp’s defensive suite, direct experimental evidence for this mechanism has remained elusive. In this work, we explored the phononic properties of the mantis shrimp’s dactyl club using laser ultrasonic techniques and numerical simulations. Our results demonstrate that the dactyl club’s periodic region functions as a dispersive, high-quality graded system, exhibiting Bloch harmonics, flat dispersion branches, ultraslow wave modes, and wide Bragg bandgaps in the lower megahertz range. These features effectively shield the shrimp from harmful high-frequency stress waves generated by cavitation bubble collapse events during impact. [ABSTRACT FROM AUTHOR]

We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order n ≥ 3 , by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also p -almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation. [ABSTRACT FROM AUTHOR]

Flapping flight is an important mode of insect flight, and its unique flapping motion pattern enables it to fly efficiently in complex environments. This paper takes a biomimetic moth flapping-wing aircraft as the research object and proposes a periodic function composed of two sine functions with different frequencies as the flapping function. This paper explores the effect of this flapping function on the flight performance of flapping-wing aircraft and verifies whether it can be applied to the flight control of flapping-wing aircraft. Firstly, through the study of biomimetic mechanisms, the basic structure of the flapping-wing aircraft is roughly designed; then, the flapping motion is simplified, a rigid wing flapping motion model is established, and the key parameters affecting the average lift are determined. Next, a virtual wind tunnel simulation platform is built, and the key parameters of the flapping function that affect lift generation are simulated and calculated. Finally, an experimental prototype of a biomimetic moth flapping-wing aircraft is designed and manufactured. Through flight experiments, the effects of flapping amplitude, flapping frequency, and mid-position angle in the flapping function on the flight performance of the biomimetic flapping-wing aircraft are verified. The key control parameters are clarified, the control strategy of the flapping-wing aircraft is optimized, and the maneuverability and controllability of the aircraft are improved, providing a theoretical basis and practical support for the development of control methods for biomimetic flapping-wing aircraft. [ABSTRACT FROM AUTHOR]

The periodic standing waves of a nearly parallel vortex filaments model are studied. By the transformation of variables, the vortex filaments model can be reduced to a planar Hamiltonian system whose Hamiltonian function includes a logarithm term and a negative power term. We successfully handle the logarithm term in the study of the monotonicity of the period function of periodic solutions. It is proved that the period function is increasing in energy on a certain interval. Moreover, the asymptotic behaviour of the period function is revealed. The numerical simulation is made and the result is consistent with the theoretical analysis. Our results extend the studies on the dynamic behaviour of some periodic solutions. [ABSTRACT FROM AUTHOR]

We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the connectedness of certain sublevel sets, the dimension of the subspace generated by the imaginary part of the coefficients, and Diophantine conditions. In addition, we show that these properties are also linked to the regularity of the solutions. The results extend previous ones in Gevrey classes. [ABSTRACT FROM AUTHOR]

We investigate the effect of three-dimensionality on the synchronisation characteristics of the wake behind an oscillating circular cylinder at ${\textit {Re}} = 300$. Cylinder oscillations in rotation, transverse translation and streamwise translation are considered. We utilise phase-reduction analysis, which quantifies the phase-sensitivity function of periodic flows, to examine the synchronisation properties. Here, we present an ensemble-based framework for phase-reduction analysis to handle three-dimensional wakes that are not perfectly time-periodic. Based on the phase-sensitivity functions, synchronisability to three types of cylinder oscillations is evaluated. In spite of similar trends, we find that phase-sensitivity functions involving three-dimensional wakes are lower in magnitude compared with those of two-dimensional wakes, which leads to narrower conditions for synchronisation to weak cylinder oscillations. We unveil that the difference between the phase-sensitivity functions of two- and three-dimensional flows is strongly correlated to the amplitude variation of the three-dimensional flow by the cylinder motions. This finding reveals that the cylinder motion modifies the three-dimensionality of the wake as well as the phase of vortex shedding, which leads to reduced phase modulation. The synchronisation conditions of three-dimensional wakes, predicted by phase-reduction analysis, agree with the identification by parametric studies using direct numerical simulations for forced oscillations with small amplitudes. This study presents the potential capability of phase-reduction to study synchronisation characteristics of complex flows. [ABSTRACT FROM AUTHOR]

This work aims to look into the dynamic research of coupled NLS-type equations with three components. The optical solitons, including the periodic function, trigonometric function, exponential function, solitary wave, and elliptic function solutions are built using the Jacobi elliptic function (JEF) method. The investigations will aid in improving comprehension of the soliton dynamics system's overall illustration. Using Mathematica software, we visually represent some solutions found in 3D, contour, and 2D graphs for tangible demonstration and visual presentation. These results are helpful in optical fiber, signal processing and data transmission. [ABSTRACT FROM AUTHOR]

In this paper, we establish and study a stochastic mosquito population suppression model incorporating the release of Wolbachia -infected males and time switching, where stochastic noises are given by independent standard Brownian motions. By combining the actual mosquito control strategy in Guangzhou, we assume that the waiting release period T between two consecutive releases of Wolbachia -infected males is less than the sexually active lifespan T ‾ of them. The existence and uniqueness of global positive solutions and stochastically ultimate boundedness for the stochastic model are obtained. Some sufficient conditions for the extinction and the existence of stochastic non-trivial periodic solutions are established. Furthermore, we assume that the release function is a general periodic function and some stochastic dynamical behaviors are obtained. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze various classes of multi-dimensional ρ-almost periodic type functions $ F : I \times X \rightarrow Y $ F : I × X → Y and multi-dimensional $ (\omega,\rho) $ (ω , ρ) -almost periodic type functions $ F : I \times X \rightarrow Y $ F : I × X → Y , where $ n\in {\mathbb N} $ n ∈ N , $ \emptyset \neq I \subseteq {\mathbb R}^{n} $ ∅ ≠ I ⊆ R n , X and Y are complex Banach spaces and ρ is a binary relation on Y. The proposed notion is new even in the one-dimensional setting, for the functions of the form $ F : I \rightarrow Y $ F : I → Y. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations, as well. [ABSTRACT FROM AUTHOR]

Due to the complex high - dimensional dynamics and highly dynamic characteristics of bipedal robots, achieving omnidirectional gait is a difficult problem. In order to achieve omnidirectional walking of bipedal robots, this study proposes a gait training method of biped robot based on deep reinforcement learning. Based on expert experience and the periodicity of bipedal walking, periodic symmetric functions that can achieve different gait styles are designed for imitation learning. In order to make the bipedal robot capable of omnidirectional walking, the foot-step planner in ROS (Robot Operating System) is used to generate target foothold points for imitation learning. The proposed method is validated on a self designed bipedal robot. The experimental results show that the proposed method can realize four gait modes of biped robot including forward, side, diagonal and turn, and realize omnidirectional gait of biped robot, and can realize different styles of cycles. [ABSTRACT FROM AUTHOR]

The method of inverse spectral problem is used to integrate a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions. The evolution of the spectral data for the periodic Dirac operator is introduced in which the coefficient of the Dirac operator is a solution of the nonlinear sine-Gordon–Liouville-type equation. The solvability of the Cauchy problems is proved for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin differential equations and the first-trace formula satisfies the sine-Gordon–Liouville-type equation. [ABSTRACT FROM AUTHOR]

We discuss the fractional Leibniz rule for periodic functions on the d-dimensional torus, including the endpoint cases. As an application, we present a product estimate, involving distributions of negative regularities. [ABSTRACT FROM AUTHOR]

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams. [ABSTRACT FROM AUTHOR]

We start with a brief survey on the Höffding kernels, its properties, related spectral decompositions, and discuss marginal distributions of Höffding measures. In the second part of this note, one dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. Höffding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations. [ABSTRACT FROM AUTHOR]

We consider an elliptic system of Hamiltonian type in a strip, satisfying the periodic boundary condition. In the superlinear case with critical growth we prove the existence of bubbling solution for the system under suitable conditions on the coefficient function. As a result, we obtain the existence of periodic solution for the system in R N , if the coefficient function is periodic in its k variables. An non-existence result is also obtained. The novelty of the paper is based on the construction arguments and obtains the periodic solution for the system in R N directly, which we believe that the idea and the method can be applied to other related problems with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

We determine the solutions on various intervals in [ 0 , ∞ [ to the functional equation f (x m) = r f (x) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set S of all solutions. The formulas for r < 0 are more complicated. An approach to S with the help of the axiom of choice is also given. A special attention is laid on solutions that are continuous on [ 0 , ∞ [ or on various open subintervals. We also describe solutions satisfying some asymptotic properties at the boundary of these intervals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration. [ABSTRACT FROM AUTHOR]

The diffusion mechanism in Hamiltonian systems, close to completely integrable, is usually connected with the existence of the so-called "transition chains". In this case slow diffusion occurs in a neighborhood of intersecting separatrices of hyperbolic periodic solutions (or, more generally, lower-dimensional invariant tori) of the perturbed system. In this note we discuss another diffusion mechanism that uses destruction of invariant tori of the unperturbed system with an almost resonant set of frequencies. We demonstrate this mechanism on a particular isoenergetically nondegenerate Hamiltonian system with three degrees of freedom. The same phenomenon also occurs for general higher-dimensional Hamiltonian systems. Drift of slow variables is shown using analysis of integrals of quasi-periodic functions of the time variable (possibly unbounded) with zero mean value. In addition, the proof uses the conditions of topological transitivity for cylindrical cascades. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of periodic integer-valued autoregressive BL-PINAR(1) models with Bell innovations distribution based on the binomial thinning operator. The basic probabilistic and statistical properties of this class are studied. Indeed, the first and the second moment periodically stationary conditions are established. The closed forms of these moments are, under the obtained conditions, derived. Furthermore, the periodic autocovariance structure is also considered while providing the closed form of the periodic autocorrelation function. The conditional least squares (CLS), Yule–Walker (YW), weighted conditional least squares (WCLS), and conditional maximum likelihood (CML) methods are applied to estimate the underlying parameters. The asymptotic properties of the CLS and the YW estimators are obtained. The performances of these methods are compared through a simulation study. An application on a real data set is provided. [ABSTRACT FROM AUTHOR]

We establish exact order estimates for the approximations of the Sobolev classes W p , α r T d of periodic functions of numerous variables with bounded dominating mixed derivative. The approximations are performed by using trigonometric polynomials with spectra in step hyperbolic crosses, and the errors are estimated in the metric of the space B q , 1 T d , 1 ≤ p, q < ∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions. [ABSTRACT FROM AUTHOR]

The article "CORRIGENDUM AND CONTRIBUTION TO 'WEAKLY ALMOST PERIODIC FUNCTIONS INVARIANT MEANS AND FIXED POINT PROPERTIES IN LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES" published in Topological Methods in Nonlinear Analysis discusses corrections and contributions to a previous study on weakly almost periodic functions in locally convex topological vector spaces. The article presents mathematical proofs and theorems related to fixed point properties in semitopological semigroups, emphasizing the existence of common fixed points for actions on compact convex subsets. The author, Khadime Salame, explores the implications of left invariant means and theorems related to weakly compact subsets and invariant means. [Extracted from the article]

The moiré effect is typically observed in regular periodic structures and sometimes in random (aperiodic) structures. However, currently, only regular graphical objects are used in measurements. We propose using graphical objects that are not regular but not entirely random and that resemble rows, such as grids of dotted lines or matrixes of dots. The moiré effect in such objects may become similar to the moiré effect in regular graphical objects if a relatively simple modification of the image processing algorithm is applied. We demonstrated that the results of measurements with five different graphical objects arranged in rows (including text) are similar. Using such objects can be helpful for practical moiré measurements. [ABSTRACT FROM AUTHOR]

In this article, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method, we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analyzing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ 1 , μ 2) -exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ 1 , μ 2) -exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research. [ABSTRACT FROM AUTHOR]

Background: Transanal irrigation (TAI) effectively addresses fecal incontinence and improves quality of life in individuals with spina bifida. Given the scarcity of follow-up studies lasting > 5 years and reports of numerous TAI discontinuations, we assessed the enduring effectiveness and impact of TAI > 10 years after its initiation on the quality of life in individuals with spina bifida. Methods: We recruited individuals with spina bifida enrolled in a bowel management program who initiated TAI in 2010 and participated in 4-month and 3-year follow-up studies at a spina bifida clinic. Raw data on bowel-related characteristics at baseline and after 4 months and 3 years of TAI were collected, and new survey-based demographic information, bowel-related characteristics, and the Fecal Incontinence Quality of Life scale scores were analyzed alongside extant datasets. Results: Among 34 participants (age, mean [standard deviation] 17.7 [3.2] years), the mean follow-up was 11.8 (0.3) years; 21 participants persistently used TAI (persistent users), 12 discontinued TAI (discontinued users), and 1 used TAI and antegrade continence enema at the time of analysis. The fecal incontinence rate among persistent users decreased from 76.2% at baseline to 14.3% at the time of analysis; 11 (91.7%) discontinued users had fecal incontinence before TAI initiation, and the majority of discontinued users (66.7%) discontinued TAI because of improved bowel function. The fecal incontinence rate and quality of life did not differ significantly between discontinued users and persistent users. Conclusions: TAI effectively alleviated fecal incontinence among persistent users. One-third of users discontinued TAI but had improved fecal continence. We recommend periodic bowel function evaluation in TAI users and to reevaluate the necessity for TAI maintenance. [ABSTRACT FROM AUTHOR]

Understanding the ecological and evolutionary dynamics of populations is critical for both basic and applied purposes in a variety of biological contexts. Although several modeling frameworks have been developed to simulate eco-evolutionary dynamics, many fewer address how to model structured populations. In a prior paper, we put forth the first modeling approach to simulate eco-evolutionary dynamics in structured populations under the G function modeling framework. However, this approach does not allow for accurate simulation under fluctuating environmental conditions. To address this limitation, we draw on the study of periodic differential equations to propose a modified approach that uses a different definition of fitness more suitable for fluctuating environments. We illustrate this method with a simple toy model of life history trade-offs. The generality of this approach allows it to be used in a variety of biological contexts. [ABSTRACT FROM AUTHOR]

The author has obtained the results that make it possible to consider the theory of dynamic game problems as an environment for constructing important mathematical objects. Namely, the triharmonic equation in the Cartesian coordinates with specially selected boundary conditions has been integrated. A triharmonic Poisson integral for the upper half-plane, which belongs to the class of positive operators, has been constructed. The functional dependence of the triharmonic operator on periodic functions has been considered, and an integral with the delta-shaped kernel has been obtained, which can be decomposed into three fractions of constant sign. The analysis of the asymptotic behavior of the triharmonic kernel shows the consistency of the obtained results with the already known results. [ABSTRACT FROM AUTHOR]

Nowadays, fast, reliable and accurate optimization methods are extensively used to realize edge intelligence in industrial applications of internet of things in order to better processing of big data collected by different environmental sensors on the ground and under the sea. Sometimes, the collected data is a one-dimensional signal with periodic/semi-periodic pattern (in terms of time, distance, etc.) which shows important features of the sensed data. However, finding the general extremum (global maximum or minimum) may not be easy in some measurements of such observed signals. The global points are critical to be found very accurately because of their importance to find optimal velocity, effective distance (or optimum range), the moment of optimality, and so on (related to physical measurements that need to be optimized). In this study, an analysis to find global optimal points of generic trigonometric functions with relatively complicated periodic patterns is carried out based on a modified form of grid search (GS) technique while there is a possibility of repetitive global points. As it is shown, some marine signals behave in such a way that can make the optimization process more complicated with losing optimal points while using the basic GS method. The basic method cannot find all repetitive maximum/minimum points in some signals. There is therefore a challenge in the use of the basic method in practice, because some global optimum points may be classified as local optimum points. We used some trigonometric functions to model the signals, and apply the modified GS method to them to find all repetitive points. Our results confirm that the modified solution can find all the repetitive points of the functions under a normally determined accuracy (not very high). Thus, we have proposed this new version of grid search to safely find all global points without a high accuracy parameter setting for sensitive data. This is indeed toward less complexity of the optimization algorithm. [ABSTRACT FROM AUTHOR]

Fourier expansions employing polyharmonic–Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic–Neumann eigenfunctions and exhibit similar benefits. Furthermore, we enhance the convergence of these interpolations by incorporating the concept of quasi-periodicity, wherein the basis functions are periodic over a slightly extended interval. We demonstrate that those interpolations achieve significantly better convergence rates away from the endpoints of the approximation interval and offer increased accuracy over the entire interval. We establish these properties for a specific case of polyharmonic–Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide supporting evidence through numerical experiments. While the latter methods display superior convergence rates, we demonstrate that interpolations using the modified Fourier basis offer distinct advantages. Firstly, they permit explicit representations via the inverses of certain Vandermonde matrices, whereas other interpolation methods require approximate computations of the eigenvalues and eigenfunctions involved. Secondly, these matrix inverses can be efficiently computed for numerical applications. Thirdly, the introduction of quasi-periodicity improves the convergence rates, making them comparable to those of other interpolation techniques. [ABSTRACT FROM AUTHOR]

Let f be a bounded and uniformly continuous function from \mathbb {R} to a Banach space X and \mathbf {sp}(f) be its Carleman spectrum. A classical Tauberian theorem states that f is constant if and only if \mathbf {sp}(f) \subset \{ 0 \}, and f is \omega-periodic if and only if \mathbf {sp}(f) \subset \frac {2\pi }{\omega } \mathbb {Z} for some \omega >0. However, one cannot expect analogous results on \mathbb {R}^+ since there is a counterexample showing that the case of \mathbb {R}^+ contrasts dramatically with the case of \mathbb {R}. In this paper, we succeed in extending the above classical Tauberian theorem to \mathbb {R}^+ and obtain an extension of the well-known Ingham theorem. We also apply our Tauberian theorems to abstract Cauchy problems and improve a result in [Russian Math. 58 (2014), pp. 1–10]. Moreover, as an application, we present an extension of a Katznelson-Tzafriri theorem in [J. Funct. Anal. 103 (1992), pp. 74–84] with weaker assumptions. In addition, it is interesting to note that several of our results and examples show that \mathcal {S}-asymptotically \omega-periodic functions on \mathbb {R}^{+} is just the "natural" analogue of periodic functions on \mathbb {R}. [ABSTRACT FROM AUTHOR]

Given a field K, a rational function \phi \in K(x), and a point b \in \mathbb {P}^1(K), we study the extension K(\phi ^{-\infty }(b)) generated by the union over n of all solutions to \phi ^n(x) = b, where \phi ^n is the nth iterate of \phi. We ask when a finite extension of K(\phi ^{-\infty }(b)) can contain all m-power roots of unity for some m \geq 2, and prove that this occurs for several families of rational functions. A motivating application is to understand the higher ramification filtration when K is a finite extension of \mathbb {Q}_p and p divides the degree of \phi, especially when \phi is post-critically finite (PCF). We show that all higher ramification groups are infinite for new families of iterated extensions, for example those given by bicritical rational functions with periodic critical points. We also give new examples of iterated extensions with subextensions satisfying an even stronger ramification-theoretic condition called arithmetic profiniteness. We conjecture that every iterated extension arising from a PCF map should have a subextension with this stronger property, which would give a dynamical analogue of Sen's theorem for PCF maps. [ABSTRACT FROM AUTHOR]

By applying some novel inequality techniques, the properties of almost periodic function and the fluctuation lemma, this manuscript establishes the criteria for the existence and globally exponential stability of positive almost periodic solutions of a non-autonomous delayed Nicholson's blowflies model under weaker conditions, which refine and complement some existing literature. In particular, when the influence of delay disappears and the addressed model is autonomous, the adopted assumption in the obtained result is a sharp condition ensuring the global exponential stability of positive equilibrium points. Moreover, some numerical examples are provided to illustrate the effectiveness and feasibility of the theoretical findings. [ABSTRACT FROM AUTHOR]

Monitoring equipment wear is an important problem that requires constant attention, since this process can lead to a decrease in its efficiency, accidents or breakdown. The problem of replacing equipment is a production problem, in which it is necessary to consider many factors that influence the efficiency of the enterprise itself. This paper examines various formulations of the equipment replacement problem: in the classical formulation of dynamic programming, in the stochastic formulation and fuzzy graph representation. The idea to solve the problem using periodic fuzzy graphs is proposed. We also consider the age of equipment when determining the wear coefficient, determined by the degree of belonging to a particular class of parameters defined in the formulation of the problem. Periodic fuzzy graphs allow considering the problem of replacing equipment not for one type of equipment, but for a complex of machines, which ensures the scalability of the classical dynamic programming problem under uncertain initial data. When determining the membership function, it is possible to consider those factors that may influence the solution of the optimization production problem. Setting the problem in a fuzzy form makes it possible to forecast and plan the activities of an enterprise for future periods. [ABSTRACT FROM AUTHOR]

In this paper, exact wave propagation patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional nonlinear Schrödinger equation are studied. The topological structure of the dynamic system of the equation is studied by the complete discrimination system for the cubic polynomial method, in which the existence conditions of the soliton solutions and periodic solutions are obtained. Then, by the trial equation method, thirteen exact solutions are obtained, including solitary wave solutions, triangular function solutions, rational solutions and the elliptic function double periodic solutions, especially the elliptic function double periodic solutions. Finally, it is found that the system has chaotic behaviors when given the appropriate perturbations. [ABSTRACT FROM AUTHOR]

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Let sk, 1 ≤ k ≤ m, m ≥ 2, be disjoint segments lying in a parallelogram Q. Denote by ℘(z) a doubly periodic Weierstrass function with the fundamental parallelogram Q. Let fk : sk → ℂ be functions, and let f k ′ ∈ L p k (sk), 1 ≤ k ≤ m, 1 < pk < ∞. Consider the Green function G(z) of the domain C \ ⋃ k = 1 m s k with the pole at infinity and define L h = def ζ : ζ ∈ C \ ⋃ k = 1 m s k , G ζ = log 1 + h , h > 0 ; ρ h ζ = def dist ζ , L h . Theorem. There exist polynomials Pn(u, v), deg Pn ≤ n, n = 1, 2, · · · , such that ∑ k = 1 m ∫ s k f k ζ - P n ℘ ζ , ℘ ′ ζ ρ 1 n ζ p k d ζ ≤ c. [ABSTRACT FROM AUTHOR]

Let P z be a doubly periodic Weierstrass function with periods 2ω1, 2ω2, and let Q be the parallelogram of periods, Q = {z ∈ C : z = 2α1ω1+2α2ω2, α1, α2 ∈ [0, 1)}. We consider a simply connected domain D, D ¯ ⊂ Q, such that its boundary ∂D contains cusps, and a function f that is analytic in D and continuous on ∂D. We assume that the modulus of continuity ω(t) satisfies the relation ∫ x 0 ω t t d t + x ∫ ∞ x ω t t 2 d t ≤ c ω x. Let Φ map conformally the domain C \ D onto C \ D with the normalization Φ(∞) = ∞,Φ′(∞) > 0. We put L1+t = {z ∈ C \ D : |Φ(z)| = 1+t}, δn(z) = dist(z, L 1 + 1 n ), z ∈ ∂D. The main result of the paper is the following statement. Theorem 1. Assume that there exists a sequence of polynomials Pn(u, v), deg Pn ≤ n, such that f z - P n P z , P ′ z ≤ C δ n r z ω δ n z , z ∈ ∂ D , C is independent of n and z. Then f ∈ Hr+ω(D). [ABSTRACT FROM AUTHOR]

We consider a multidimensional controlled system with a constant matrix, a significant nonlinearity of the two-position relay type with hysteresis as a control, and a continuous periodic perturbation function. The system matrix has simple real nonzero eigenvalues, among which one can be positive. Conditions for the system parameters, including the nonlinearity ones, are established under which there exists a single two-point oscillatory periodic solution with a period comparable to the period of the perturbation function in the case of a special type of the feedback vector. The asymptotic stability of the solution has been proven using the phase plane method. The results obtained are illustrated by examples for three-dimensional systems. [ABSTRACT FROM AUTHOR]

Enzyme catalysis in reactors for industrial applications usually require an external intervention of the species involved in the chemical reactions. We analyze the most elementary open enzyme catalysis with competitive inhibition where a time-dependent inflow of substrate and inhibitor supplies is modeled by almost periodic functions. We prove global stability of an almost periodic solution for the non-autonomous dynamical system arising from the mass-law action. This predicts a well behaved situation in which the reactor oscillates with global stability. This is a first case study in the path toward broader global stability results regarding intraspecific and monotone open reaction networks. [ABSTRACT FROM AUTHOR]

This work delves into the behavior of cold bosonic atoms within a zigzag optical lattice which offers a rich platform for studying quantum many‐body physics and has potential applications in various fields including quantum simulation, quantum computing, and precision measurement. The study aims to derive exact solutions and explore qualitative properties through dynamical analysis. Exact solutions of the nonlinear differential equation model can provide insights into the behavior of bosonic atoms in complex optical lattice configurations, allowing researchers to simulate and study phenomena such as quantum phase transitions, many‐body localization, and exotic states of matter. The invariant analysis has been performed for the first time on the specified equation, yielding a reduced system of equations with forms that are easier to handle. Novel techniques are applied to extract solutions from the reduced system of equations obtained via invariant analysis. The results reveal a rich set of solutions, including various traveling wave solutions and doubly periodic functions in the form of Jacobian elliptic functions. Bifurcation analysis, conducted through phase portraits, provides insights into the system's long‐term behavior under different parameter values. This work contributes to a deeper understanding of the dynamics of cold bosonic atoms in optical lattices via 2D and 3D plots of obtained solutions depicting the change in the behavior of soliton solutions with fractional derivative parameter. [ABSTRACT FROM AUTHOR]

This paper deals with some probabilistic and statistical properties of periodic generalized Poisson integer-valued autoregressive processes of order p, PG P INAR (p). Necessary and sufficient conditions for the periodic stationarity, both in mean and second order, are established. The closed-forms of the mean and the second moment are, under these conditions, obtained. Moreover, the Wold–Cramér expression of the underlying second-order periodically stationary process is then established. The autocovariance structure is studied, while providing the closed-form of the periodic autocorrelation function. The Yule–Walker (YW), the two-stage conditional least squares (CLS) and the conditional maximum likelihood (CML) estimation methods of the underlying parameters are obtained. An intensive simulation study and an application on a real count data consisting of the daily number of daytime road accidents in Schiphol area, in Netherlands are provided. [ABSTRACT FROM AUTHOR]

Different methods are possible for estimating tidal deformations of Earth and telluric planets. On the one hand, the code ALMA 3 solves analytically the governing equations in considering symmetrical and incompressible bodies with homogeneous layers and different possible rheologies. Tidal deformations are considered with periodic excitation functions and the output of the model is frequency-dependent complex Love numbers. On the other hand, the 3-D finite element code CitcomSVE integrates numerically the governing equations with possibly lateral variations in viscoelastic structures on the regional and global scales. In this work, we present how tidal deformations have been implemented in CitcomSVE by the introduction of a periodic forcing potential. For validation and benchmarking, we realized comparisons between the ALMA 3 output for Moon tidal deformations and the numerical CitcomSVE in terms of frequency-dependent Love numbers k 2 and h 2 real and imaginary parts with 1-D viscoelastic structure. Considering two possible profiles for the Moon,we compared the frequency-dependent quality factor deduced from ALMA 3 with the one obtained with CitcomSVE. We found that with a sufficient numerical resolution for CitcomSVE (with an horizontal resolution of about 14 km and 10 - 5 for numerical accuracy), the results of the two methods for computing tidal deformations (i.e., k 2 and h 2 ) and quality factor Q are in good agreement for different periods including the monthly period (less than 0.025% for the real part of Love numbers and for the Q, about 1% for periods of excitation from 5 to 10 8 days). We also computed tidal dissipation energy from CitcomSVE and found it consistent with that expected from quality factor calculation. Our study demonstrates the potential for CitcomSVE to be applied for planetary tidal deformation calculations for a planet with a 3-D structure. [ABSTRACT FROM AUTHOR]

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency. [ABSTRACT FROM AUTHOR]

Two mismatched binary sequences are called a mismatched binary periodic complementary pair (MB-PCP) if the sum of their periodic cross-correlation functions is a delta function. Such a pair is a generalization of the well-known Golay sequence pair, and has applications in the scenario where the mismatched filter is used. Up to now, there are only MB-PCPs of length $ 3q $ with peak value 4 or $ 2q-6 $, where $ q \equiv 3\; (\bmod \,4) $ is a prime. The objective of this paper is to construct four families of MB-PCPs of length $ pq $ via the generalized cyclotomy and the Chinese Remainder Theorem, where $ p $ and $ q $ are two different odd primes. [ABSTRACT FROM AUTHOR]

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results. [ABSTRACT FROM AUTHOR]

Stochastic ultimate boundedness has always been a very important property, which plays an important role in the study of stochastic models. Thus, in this paper, we will study a stochastic periodic chemostat system, in which we assume that the nutrient input concentration and noise intensities are periodic. In order to make the stochastic periodic model have mathematical and biological significance, we will study a very important issue: the existence, uniqueness and ultimate boundedness of a global positive solution for a stochastic periodic chemostat system. [ABSTRACT FROM AUTHOR]

A fully coupled network of Mackey–Glass generators is considered. Each generator is described by a limit equation for the Mackey–Glass equation. The right parts are represented by a relay function obtained when the exponent in the denominator of the nonlinearity tends to infinity. Discrete traveling waves are sought in the system. These modes are such that all components are represented by the same periodic function with successive (multiple of the same value) shifts. Moreover, this periodic function has the smallest number of switchings, that is, points at which the right-hand sides change the analytical form. It is shown that discrete traveling waves coexist in the system. Moreover, their number is equal to the factorial of the number of generators. [ABSTRACT FROM AUTHOR]

Advances in additive manufacturing enable fabrication of architected materials composed of microstructures with extreme mechanical properties. In the design of such architected materials, the parameterization of microstructures determines not just the computational cost but also connectivity between adjacent microstructures. In this paper, we propose a periodic composite function (PCF)-based approach for designing microstructures. The shape of the microstructures is characterized by the value of the periodic composite functions. The proposed method can program microstructures with both positive and negative Poisson's ratios by a small number of parameters. Furthermore, due to its implicit representation, the proposed method allows for continuously tiling of microstructures with different mechanical properties. Explicit geometric features of the PCF-based microstructures are extracted, and the condition to maintain connectivity between adjacent microstructures is derived. Based on the proposed approach, multiple groups of 2D and 3D microstructures with Poisson's ratios ranging from negative to positive are presented. Combining with a deep neural network (DNN)-based surrogate model to predict macroscopic material properties of the microstructures, the proposed method is applied to the design of architected materials for elastic deformation control. Numerical examples on both microstructure representation and architected materials design are presented to demonstrate the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]