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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]

In this study, by making use of the direct integral method and the complete discrimination system for the polynomial method, all the travelling wave solutions to the two-component Dullin–Gottwald–Holm (DGH2) system are obtained, including solitary wave solutions, singular periodic solutions and Jacobian elliptic function double periodic solutions. Some of them are initially given. Moreover, concrete examples are presented to make sure that several solutions can be realised, and the corresponding figures are also given to show their nature. This means every solution in the paper may reflect the corresponding natural phenomenon, such as tidal waves and tsunami waves. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) have shown remarkable achievements in solving partial differential equations (PDEs). However, their performance is limited when encountering oscillatory part in the solutions of PDEs. Therefore, this paper proposes a multi-scale deep neural network with periodic activation function to achieve high-frequency to low-frequency conversion, which can capture the oscillation part of the solution of PDEs. Moreover, the use of adaptive sampling method can adaptively change the location and distribution of residual points, improving the performance of the network. Additionally, the gradient-enhanced strategy is also utilized to embed the gradient information of the PDEs into the loss function of the neural network, which further improves the accuracy of PINNs. Through the numerical experiments verification, it is found that our method is better than PINNs in terms of accuracy and efficiency. • A MscaleDNNs structure with periodic activation functions is proposed. • The proposed neural network has better stability and robustness. • The effectiveness of the method was verified through numerical experiments. [ABSTRACT FROM AUTHOR]

The periodic sinc function interpolation offers a compelling solution to address the issue of noise in the analysis of thermogravimetric analysis (TGA) data, thereby enhancing the outcomes of differential techniques such as the Friedman isoconversional method. In this study, we introduce a novel approach that leverages the periodic sinc function interpolation to directly obtain smooth reaction rates from TGA data, eliminating the reliance on numerical differentiation methods. The efficacy of this method has been confirmed through its application to noisy experimental data derived from the thermal decomposition of various polymers, showcasing its robustness. Readers are provided with the corresponding code for Gnu Octave, serving as a free alternative to MATLAB. Additionally, the activation energies calculated from the experimental data using both the Friedman method and periodic sinc function interpolation closely align with those determined by the integral Vyazovkin method, emphasizing the validity and reliability of this new approach. [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]

The interlayer twist angle endows a new degree of freedom to manipulate the spatially separated interlayer excitons in van der Waals (vdWs) heterostructures. Herein, we find that the band-edge Γ-Γ interlayer excitation directly forms interlayer exciton in InSe/Sb heterostructure, different from that of transition metal dichalcogenides (TMDs) heterostructures in two-step processes by intralayer excitation and transfer. By tuning the interlayer coupling and breathing vibrational modes associated with the Γ-Γ photoexcitation, the interlayer twist can significantly adjust the excitation peak position and lifetime of recombination. The interlayer excitation peak in InSe/Sb heterostructure can shift ~400 meV, and the interlayer exciton lifetime varies in hundreds of nanoseconds as a periodic function of the twist angle (0°–60°). This work enriches the understanding of interlayer exciton formation and facilitates the artificial excitonic engineering of vdWs heterostructures. [ABSTRACT FROM AUTHOR]

The increasing availability of learning activity data in Massive Open Online Courses (MOOCs) enables us to conduct a large-scale analysis of learners' learning behavior. In this paper, we analyze a dataset of 351 million learning activities from 0.8 million unique learners enrolled in over 1.6 thousand courses within two years. Specifically, we mine and identify the learning patterns of the crowd from both temporal and course enrollment perspectives leveraging mutual information theory and sequential pattern mining methods. From the temporal perspective, we find that the time intervals between consecutive learning activities of learners exhibit a mix of power-law and periodic cosine function distribution. By qualifying the relationship between course pairs, we observe that the most frequently co-enrolled courses usually fall in the same category or the same university. We demonstrate these findings can facilitate manifold applications including recommendation tasks on courses. A simple recommendation model utilizing the course enrollment patterns is competitive to the baselines with 200× faster training time. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to find a solution of the Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple derivation of the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of four times continuously differentiable periodic infinite-gap functions is proven. The sum of the uniformly convergent functional series constructed by solving the infinite system of Dubrovin equations and the first trace formula are proven to satisfy the nonlinear Korteweg–de Vries equation. In addition, it is proven that if the number π n is the period of the initial function, then the number π n is the period for the solution of the Cauchy problem with respect to the variable x. Here n ≥ 2 is a natural number. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized sine-Gordon equation ψ t x = (1 + a ∂ x 2) sin ψ and the sinh-Poisson equation u x x + u y y + σ sinh u = 0 , where a is a real parameter, and σ is a positive parameter. Under different conditions, e.g., a = 0 , a ≠ 0 , and σ > 0 , the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions. [ABSTRACT FROM AUTHOR]

We consider the problem of stability of nonzero integral manifolds of a nonlinear finitedimensional system of ordinary differential equations whose right-hand side is a periodic vector-valued function of the independent variable containing a parameter. We assume that the system has a trivial integral manifold for all values of the parameter and the corresponding linear subsystem does not possess the property of exponential dichotomy. The aim of this work is to find sufficient conditions for stability, instability, and asymptotic stability of a local nonzero integral manifold. For this purpose, we use the method of Lyapunov functions modified to the problem considered and singularities of the right-hand sides of the system. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, it is one-to-one, linear, and continuous with respect to Δ -convergence as well as Δ -convergence. Subsequently, we introduce a discrete variant of the OLCST. Ultimately, we broaden the application of the presented work by examining the OLCST within the domain of almost periodic functions. [ABSTRACT FROM AUTHOR]

This note is concerned with the linear BBM equation on the half-line. Its nonlinear counterpart originally arose as a model for surface water waves in a channel. This model was later shown to have considerable predictive power in the context of waves generated by a periodically moving wavemaker at one end of a long channel. Theoretical studies followed that dealt with qualitative properties of solutions in the idealized situation of periodic Dirichlet boundary conditions imposed at one end of an infinitely long channel. One notable outcome of these works is the property that solutions become asymptotically periodic as a function of time at any fixed point x in the channel, a property that was suggested by the experimental outcomes. The earlier theory is here generalized using complex-variable methods. The approach is based on the rigorous implementation of the Fokas unified transform method. Exact solutions of the forced linear problem are written in terms of contour integrals and analyzed for more general boundary conditions. For C ∞ -data satifisying a single compatibility condition, global solutions obtain. For Dirichlet and Neumann boundary conditions, asymptotic periodicity still holds. However, for Robin boundary conditions, we find not only that solutions lack asymptotic periodicity, but they in fact display instability, growing in amplitude exponentially in time. [ABSTRACT FROM AUTHOR]

AbstractThis 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]

Background: Measurement of Computed Tomography (CT) beam width is required by accrediting and regulating bodies for routine physics evaluations due to its direct correlation to patient dose. Current methods for performing CT beam width measurement require special hardware, software, and/or consumable films. Today, most 100‐mm pencil chambers with a digital interface used to evaluate Computed Tomography Dose Index (CTDIvol) have a sufficiently high sampling rate to reconstruct a high‐resolution dose profile for any acquisition mode. Purpose: The goal of this study is to measure the CT beam width from the sampled dose profile under a single helical acquisition with the 100‐mm pencil chamber used for CTDIvol measurements. Methods: The dose profiles for different scanners were measured for helical scans with varying collimation settings using a 100‐mm pencil chamber placed at the isocenter and co‐moving with the patient table. The measured dose profiles from the 100‐mm pencil chamber were corrected for table attenuation by extracting a periodic correction function (PCF) to eliminate table interference. The corrected dose profiles were then deconvolved with the response function of the chamber to compute the beam profile. The beam width was defined by the full width half maximum (FWHM) of the resulting beam profile. Reference dose profiles were also measured using Gafchromic film for comparison. Results: The beam widths, estimated using the innovative deconvolution method from the 100‐mm pencil chamber, exhibit an average percentage difference of 1.6 ± 1.8 when compared with measurements obtained through Gafchromic film for beam width assessment. Conclusion: The proposed approach to deconvolve the pencil chamber response demonstrates the potential of obtaining the CT beam width at high accuracy without the need of special hardware, software, or consumable films. This technique can improve workflow for routine performance evaluation of CT systems. [ABSTRACT FROM AUTHOR]

We use the inverse spectral problem method for integrating the nonlinear complex modified Korteweg-de Vries equation (cmKdV) with additional terms in the class of periodic infinite-gap functions. Also, we deduce the evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution to cmKdV. We prove that the Cauchy problem is solvable for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions. Moreover, we establish the solvability of the Cauchy problem for cmKdV with additional terms in the class of six times continuously differentiable periodic infinite-gap functions. [ABSTRACT FROM AUTHOR]

The pecten is a fold‐structured projection at the ocular fundus in bird eyes, showing morphological diversity between the diurnal and nocturnal species. However, its biological functions remain unclear. This study investigated the morphological and histological characteristics of pectens in wild birds. Additionally, the expression of non‐visual opsin genes was studied in chicken pectens. These genes, identified in the chicken retina and brain, perceive light periodicity regardless of visual communication. Similar pleat numbers have been detected among bird taxa; however, pecten size ratios in the ocular fundus showed noticeable differences between diurnal and nocturnal birds. The pectens in nocturnal brown hawk owl show extremely poor vessel distribution and diameters compared with that of diurnal species. RT‐PCR analysis confirmed the expression of Opn5L3, Opn4x, Rrh and Rgr genes. In situ hybridization analysis revealed the distribution of Rgr‐positive reactions in non‐melanotic cells around the pecten vessels. This study suggests a novel hypothesis that pectens develop dominantly in diurnal birds as light acceptors and contribute to continuous visual function or the onset of periodic behaviour. [ABSTRACT FROM AUTHOR]

The aim of our paper is to investigate the properties of the classical phase-dispersion minimization (PDM), analysis of variance (AOV), string-length (SL), and Lomb–Scargle (LS) power statistics from a statistician's perspective. We confirm that when the data are perturbations of a constant function, i.e. under the null hypothesis of no period in the data, a scaled version of the PDM statistic follows a beta distribution, the AOV statistic follows an F distribution, and the LS power follows a chi-squared distribution with two degrees of freedom. However, the SL statistic does not have a closed-form distribution. We further verify these theoretical distributions through simulations and demonstrate that the extreme values of these statistics (over a range of trial periods), often used for period estimation and determination of the false alarm probability (FAP), follow different distributions than those derived for a single period. We emphasize that multiple-testing considerations are needed to correctly derive FAP bounds. Though, in fact, multiple-testing controls are built into the FAP bound for these extreme-value statistics, e.g. the FAP bound derived specifically for the maximum LS power statistic over a range of trial periods. Additionally, we find that all of these methods are robust to heteroscedastic noise aimed to mimic the degradation or miscalibration of an instrument over time. Finally, we examine the ability of these statistics to detect a non-constant periodic function via simulating data that mimics a well-detached binary system, and we find that the AOV statistic has the most power to detect the correct period, which agrees with what has been observed in practice. [ABSTRACT FROM AUTHOR]

This study thoroughly investigates the dynamical properties, analytical solutions and chaotic behavior of the high-order nonlinear Schrödinger equation with conformal fractional derivative. After obtaining the dynamic properties by a third-order polynomial complete discriminant system, the study further utilizes a fourth-order polynomial complete discrimination system to derive all traveling wave solutions, including rational solutions, solitary wave solutions, triangular function periodic solutions, and Jacobian elliptic function periodic solutions. In addition, based on different phase diagrams and largest Lyapunov exponents, the chaotic behavior is numerically observed by considering different external perturbation terms and perturbation functions. The innovation of this work lies in using the same method to qualitatively and quantitatively analyze the proposed equation. The type of equation solution is predicted using the results of qualitative analysis, and the accuracy of the prediction can be verified through specific parameter values. [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]

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin–Voigt friction. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient σ(r,t) through blood vessels, and the tumor attracts blood vessels at a rate proportional to α(t). So, the boundary value condition ∂σ ∂r + α(t)(σ − ψ(t)) = 0,r = R(t),t > 0, holds on the boundary, where the function ψ(t) is the concentration of nutrient externally supplied to the tumor. Considering that the nutrients provided by the outside world are often periodic, the research in this paper assumes that ψ(t) is a periodic function. Sufficient conditions for the global stability of zero steady state are presented. Under certain conditions, we prove that there exists at least one periodic solution to the model. The results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

• The analytical solutions of the current and energy flux of the thermoelectric half-plane under periodic contact are obtained. • The influence of topography and material on the contact response is discussed. • The stress distribution of thermoelectric and elastic materials under partial and full contact conditions is revealed. • The influence of load level on contact behavior is revealed. The solution of the contact problem with multiple contact areas between a thermoelectric device and a surface is crucial for the design and optimization of thermoelectric devices. This study focuses on the contact problem of a thermoelectric half-plane and a rigid solid with periodic wavy surfaces. The interface profile is represented by a periodic function, which is likely more accurate when addressing periodic structures compared with an assumed flat or smooth interface. The exact contact areas between the thermoelectric device and the rigid solid are not predetermined but need to be calculated as part of the analysis. To tackle the nonlinear problem caused by waviness and thermoelectric effects, the integral transformation method is used to simplify the stated problem into three singular integral equations (SIEs) with the Hilbert kernels. The equations in terms of the interface normal current density and interface normal energy flux are transformed into the Cauchy SIEs of the first type through variable transformation, and solved analytically. Two special cases are checked with the thermoelectric load being zero or the periodic region being infinite, and the analytical solutions obtained are consistent with the classical results. The equation in terms of the interface normal contact stress is solved numerically by the collocation methods. The numerical results show that the interface stress of the material can be changed effectively by adjusting the thermoelectric load, thus reducing the thermoelectric-mechanical contact damage of the material. The present work may provide a benchmark for studying more complex multi-area surface/interface contact problems of thermoelectric materials and structures. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

For addressing the limitations of traditional elastic metamaterials in opening wide bandgaps below 100 Hz, a tunable pneumatic metamaterial plate with airbag local resonators is proposed. Utilizing the characteristics of airbags, such as small volume, large load-bearing capacity, easy stiffness adjustment, and the ability to provide multi-directional restoring forces, a structured low-stiffness local resonator with a certain load-bearing capacity is designed. By varying the gauge pressure of the airbag, the bandgap can be moved toward lower frequencies, thereby achieving a broad low-frequency vibration suppression capability for various wave propagations. The low-frequency vibration bandgap characteristics of the tunable pneumatic metamaterial are analyzed and verified by applying the finite element method. The results illustrate that this tunable pneumatic metamaterial can attenuate bending waves in the range of 22–121 Hz by adjusting the air pressure. Moreover, increasing the gauge pressure will not only shift the complete bandgap toward lower frequencies but also significantly expand the bandwidth of the complete bandgap. For instance, increasing the gauge pressure from 0 to 50 kPa reduces the opening frequency of the complete bandgap from 36 to 22 Hz while enhancing the relative bandwidth from 0.52 to 0.85. Extending from this, a parametric study was conducted to examine the impact of the structural parameters of airbag-type resonant units on bandgap evolution, summarizing the general principles for achieving wide low-frequency bandgaps. Finally, the bandgap characteristics of the tunable pneumatic metamaterial are confirmed through the frequency response function of a finite periodic structure. [ABSTRACT FROM AUTHOR]

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The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld's hypothesis for the He atom. [ABSTRACT FROM AUTHOR]

We introduce the continuous and discrete versions of multi-dimensional linear canonical wave packet transform with the non-separable kernel. At first, we establish a relationship between the non-separable linear canonical wave packet transform (NSLCWPT) and n-dimensional windowed Fourier transform (WFT) and then establish the fundamental results like reconstruction formula, orthogonality relation and some bounds for the NSLCWPT. Some examples were also provided for the enhancement of our proposed transform. We apply the discrete version of the NSLCWPT in the context of the almost periodic functions. At last, a simple conclusion is given, which provides the idea for future work. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the (μ 1 , μ 2) -pseudo-asymptotically τ -periodicity and its applications. Then, the problems of existence, uniqueness and global exponential stability of (μ 1 , μ 2) -pseudo-asymptotically τ -periodic solutions for neural networks with delay are considered. Finally, three examples are given to illustrate the validity of the theoretical results [ABSTRACT FROM AUTHOR]

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (− Δ Φ) s u + V (x) a (| u |) u = f (x , u) ,   x ∈ R N , where (− Δ Φ) s (s ∈ (0 , 1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ -Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V (x) . The proof is based on a variant version of the mountain pass theorem and a Lions' type result in fractional Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic properties for the drift parameter estimators in the fractional Ornstein-Uhlenbeck process with periodic mean function and long range dependence. The Cremér-type moderate deviations, as well as the moderation deviation principle with explicit rate function can be obtained. [ABSTRACT FROM AUTHOR]

We investigate the following two types of nonlinear differential-difference equations L (z , f) + H (z , f) = ∑ k = 1 r α k (z) e β k z ; L (z , f) + H (z , f) = ∑ k = 1 r F k (z) , where α 1 , … , α r are meromorphic functions of order < 1 , and F 1 , … , F r are periodic transcendental entire functions, and L, H are defined by L (z , f) = ∑ k = 1 p a k (z) f (m k) (z + τ k) ≢ 0 , H (z , f) = ∑ k = 1 q b k (z) [ f (n k) (z + ζ k) ] s k with small meromorphic coefficients a i , b j. By introducing a new method, we obtain the exact forms of the solutions of these two equations under certain growth conditions. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the rigorous numerical analysis of a Multiscale Finite Element Method (MsFEM) for the Stokes system, when dealing with highly heterogeneous media, as proposed in B.P. Muljadi et al., Non-conforming multiscale finite Element method for Stokes flows in heterogeneous media. Part Ⅰ: Methodologies and numerical experiments, SIAM MMS (2015), 13(4) 1146-–1172. The method is in the vein of the classical Crouzeix-Raviart approach. It is generalized here to arbitrary sets of weighting functions used to enforce continuity across the mesh edges. We provide error bounds for a particular set of weighting functions in a periodic setting, using an accurate estimate of the homogenization error. Numerical experiments demonstrate an improved accuracy of the present variant with respect to that of Part Ⅰ, both in the periodic case and in a broader setting. [ABSTRACT FROM AUTHOR]

Consider the Navier-Stokes equations in a half space. It is shown that this system admits a unique, $ T $-periodic mild solution in some suitable Lorentz space provided the outer force is $ T $-periodic and small enough. This result is then extended to the case of almost periodic functions. Moreover, it is shown that these solutions are polynomially stable. [ABSTRACT FROM AUTHOR]

A describing function method with balancing in time domain (DFBT) is used to efficiently solve 2-DOF airfoil system with nonlinear energy sink (NES). This method uses a set of basis functions to describe the periodic motion, discretes the dynamic equation in time domain and lists the equilibrium equation point by point to solve the basis function and obtain the harmonic coefficient. The point-by-point equilibrium equation obtained by discretization can provide overdetermined algebraic equations for solving harmonic coefficients, and the iterative convergence can be easily combined with minimum optimization.The method does not need to switch the time-domain and frequency-domain responses of the nonlinear system frequently, which brings convenience to the solution. The results of the solution agree well with those of the numerical method, which verifies the effectiveness of the method. At the same time, due to the semi-analytic form of DFBT results, the unsteady solution of the flutter system and the complete subcritical bifurcation diagram can be obtained. Therefore, DFBT is suitable for solving the steady-state response of multi-degree-of-freedom flutter systems with complex nonlinearity. [ABSTRACT FROM AUTHOR]

We construct a Parseval wavelet frame with compact support for an arbitrary continuous -periodic function , , satisfying the inequality . The frame refinement mask uniformly approximates . The refining function has stable integer shifts. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the (recently introduced by Alavrez et al.) concept of the so-called asymptotically (ω,c) periodic functions. We investigate further properties of this class of functions which complement the currently known ones. These properties allowed us to prove the existence and uniqueness of asymptotically (ω,c) - periodic mild solution of an integro-differential equation in a Banach space, a result which generalizes some recent works in the class of S-asymptotically - periodic functions by Brindle and N'Guérékata. [ABSTRACT FROM AUTHOR]

The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach and the Backlund transformation, we search for soliton solutions of the fractional Zoomeron nonlinear equation. A number of solutions have been put forth, such as kink, anti-kink, cuspon kink, lump-type kink solitons, single solitons, and others defined in terms of pseudo almost periodic functions. The (2 + 1)-dimensional fractional Zoomeron equation given in a form undergoes precise dynamics. We use the computational software, Matlab 19, to express these solutions graphically by changing the value of various parameters involved. A detailed analysis of their dynamics allows us to obtain completely better insights necessarily with the elementary physical phenomena controlled by the fractional Zoomeron equation. [ABSTRACT FROM AUTHOR]

Recently, the authors introduced the notion of weighted semigroups which apply to sun-dual semigroups and especially to the translation semigroup on the space of left continuous functions with values in dual spaces. In this article, we will show that it is sufficient that we either assume geometry on the Banach, or an abelian structure on the minimal subgroup to prove almost periodicity. This yields a different approach to the almost periodicity of semigroups and integrals, by extending Basit generalized Kadets result to general groups, to obtain almost periodicity. [ABSTRACT FROM AUTHOR]

We consider a system of ordinary differential equations with a relay hysteresis and a harmonic perturbation. We propose an approach that allows one to decompose an n-dimensional system into one- and two-dimensional subsystems. The approach is illustrated by a numerical example for the system of dimension 3. As a result of the decomposition, a two-dimensional subsystem with non-trivial Jordan block in right-hand side is studied. For this subsystem we prove a theorem on the existence and uniqueness of an asymptotically stable solution with a period being multiple to period of the perturbation. Moreover, we show how to obtain this solution by tuning the parameters defining the relay. We also provide a supporting example in this regard. [ABSTRACT FROM AUTHOR]

The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper, we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to R of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions. [ABSTRACT FROM AUTHOR]

We propose a method for multiple-depth extraction in diffraction grating imaging. A diffraction grating can optically generate a diffraction image array (DIA) having parallax information about a three-dimensional (3D) object. The optically generated DIA has the characteristic of forming images periodically, and the period depends on the depth of the object, the wavelength of the light source, and the grating period of the diffraction grating. The depth image can be extracted through the convolution of the DIA and the periodic delta function array. Among the methods for extracting depth images through the convolution characteristics of a parallax image array (PIA) and delta function array, an advanced spatial filtering method for the controllable extract of multiple depths (CEMD) has been studied as one of the reconstruction methods. And that possibility was confirmed through a lens-array-based computational simulation. In this paper, we aim to perform multiple-depth extraction by applying the CEMD method to a DIA obtained optically through a diffraction grating. To demonstrate the application of the CEMD in diffraction grating imaging, a theoretical analysis is performed to apply the CEMD in diffraction grating imaging; the DIA is acquired optically, and the spatial filtering process is performed through computational methods and then compared with the conventional single-depth extraction method in diffraction grating imaging. The application of the CEMD to DIA enables the simultaneous reconstruction of images corresponding to multiple depths through a single spatial filtering process. To the best of our knowledge, this is the first research on the extraction of multiple-depth images in diffraction grating imaging. [ABSTRACT FROM AUTHOR]

Numerically validated analytical predictions for electro-osmosis over a charged surface decorated with a nanoscale groove pattern are developed for the situation when the electrical double layer thickness is comparable to the spatial period of the grooves. For the analytical predictions, the groove shape can be specified by any continuous periodic function, such as the triangular, trapezoidal, and sinusoidal waveforms, which are investigated as special cases. We discover that the classical Helmholtz–Smoluchowski expression for electrokinetic mobility, notwithstanding its widespread use in measurements, is rendered invalid by the presence of Debye-length-scale unevenness in the surface topography. Furthermore, we use the depth-resolved anisotropic response of oblique grooves to design and optimize a novel electro-microfluidic strategy for separating constituents of a nano-particulate mixture. [ABSTRACT FROM AUTHOR]