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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]

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 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]

This paper introduces a fully discrete soft thresholding trigonometric polynomial approximation on [ − π , π ] , named Lasso trigonometric interpolation. This approximation is an ℓ 1 -regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is a sparse scheme which is efficient in dealing with noisy data. We theoretically analyze Lasso trigonometric interpolation quality for continuous periodic function. The L 2 error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. The performance of Lasso trigonometric interpolation for several testing functions (sin wave, triangular wave, sawtooth wave, square wave), is illustrated with numerical examples, with or without the presence of data errors. [ABSTRACT FROM AUTHOR]

We investigate the estimators obtained from reversing the three-step estimation procedure of Vogt and Linton (2014) for a regression model including a smooth trend function, a periodic component and a stochastic error term. The asymptotic properties of the estimators are given: (a) we provide the uniform weak convergence rates of the estimators of the trend function and periodic sequence; (b) we establish the asymptotic normality for the trend function estimator; and (c) we show that the period estimator is consistent. Plug-in type bandwidths are proposed for the trend estimator. A simulation study and an application to sea level data are employed in order to complement our theoretical results. • Asymptotic properties of estimators for a trend-periodic model are studied. • The order of the estimation procedure is theoretically relevant. • Plug-in bandwidth selectors were proposed for the estimation. • Choices on bandwidth and estimation order were important only for small samples. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear wave solutions for Gross–Pitaevskii equation on the periodic wave background are investigated by Darboux-Bäcklund transformation, from which the soliton and breather wave solutions on the Jacobi elliptic cn and dn functions backgrounds are derived. The corresponding evolutions and dynamical properties of nonlinear wave solutions under different parameters are discussed. These results reported in this paper may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, oceanography and so on. [ABSTRACT FROM AUTHOR]

Designing anisotropic structured materials by reducing symmetry results in unique behaviors, such as shearing under uniaxial compression. While rank-deficient materials such as hierarchical laminates have been shown to exhibit extreme elastic anisotropy, there is limited knowledge about the fully anisotropic elasticity tensors achievable with single-scale fabrication techniques. No established upper and lower bounds on anisotropic moduli exist. In this paper, we estimate the range of anisotropic stiffness tensors achieved by single-scale two-dimensional structured materials. We first develop a database of periodic anisotropic single-scale unit cell geometries using linear combinations of periodic cosine functions. The database covers a wide range of anisotropic elasticity tensors, which are then compared with the elasticity tensors of hierarchical laminates. We identify the regions in the property space where hierarchical design is necessary to achieve extremal properties. We demonstrate a method to construct various 2D functionally graded structures using this cosine function representation. These graded structures seamlessly interpolate between unit cells with distinct patterns, allowing for independent control of several functional gradients, such as porosity, anisotropic moduli, and symmetry. When designed with unit cells positioned at extreme parts of the property space, these graded structures exhibit unique mechanical behaviors such as selective strain energy localization, compressive strains under tension, and localized rotations.   • Estimating the bounds on the anisotropic elastic moduli of single-scale structured materials. • Identifying the regions in the elastic property space that require hierarchical designs. • Design of functionally graded anisotropic structures with seamless transition between unit cells with distinct patterns. • Exploiting functional gradients to demonstrate atypical mechanical behavior such as localized rotations. [ABSTRACT FROM AUTHOR]

• The deformation of unstable slopes in the reservoir area has distinct differences. • Displacement process of four reservoir landslide types was studied quantitatively. • Water level fluctuation and rainfall accelerate reservoir landslide deformation. Understanding the deformation patterns of different types of landslides around reservoirs is crucial for precise prevention and control of related hazards. However, under the complex coupling effect of extreme rainfall and reservoir level fluctuations, the response between various types of landslides and triggering factors is still unclear. Our study employed the Multi-Temporal Interferometric Synthetic Aperture Radar (MT-InSAR) method and wavelet analysis to undertake an early identification and quantitative analysis of deformation patterns of unstable slopes in the Baihetan Reservoir region, China, and explored the responses of different types of landslide deformation to rainfall and reservoir water level fluctuations. The findings reveal distinct differences in the deformation characteristics of unstable slopes around the reservoir. The quantitative analysis of normalized displacement indicated that the convergent-type landslide exhibits an initial rapid movement that gradually tends to be stable, following an exponential function. The steady-type landslide moves at a consistent speed along a linear trendline. The exponential-type landslide starts slowly and then progressively accelerates. The periodic-type landslide displays periodic fluctuations in displacement, which conforms to the Fourier periodic function. Furthermore, comparisons with antecedent rainfall and reservoir water levels reveal that the convergent-type landslide is significantly influenced by water level fluctuations, and both steady-type and exponential-type landslides are subject to the dual impact of rainfall and water level fluctuations, while the periodic-type landslide exhibits a notable response to rainfall. Therefore, this research aims to provide a scientific basis for the prevention and mitigation of geohazards in reservoir areas. [ABSTRACT FROM AUTHOR]

This paper considers the same class of zero-relative-degree nonlinear systems for which a saturated Repetitive Learning Control (RLC) has been recently shown to ensure exponential – and not just asymptotical – convergence to zero of both the output and the input tracking errors in the case of periodic output reference signals with a known single period. The explicit role of the nonlinear unstructured uncertainties is here investigated within the more general scenario in which the output reference signal is multi-periodic. Special emphasis is provided to the generation of the input reference whose effect has to be exponentially nullified by the RLC. In particular, the following question is answered. Restrict the design to the sub-class of periodic output reference signals that admit a Periodic Signal Decomposition (PSD) (namely, the ones that can be written as a finite sum of periodic functions) with pairwise-commensurable periods (so that the knowledge of the multiple periods characterizing the input reference is preserved once the common multiple among the single periods is additionally included). Then, which robustness and convergence properties can be still achieved by the same output-feedback (definitely saturated) RLC, once it is intuitively generalized to transiently include an additional bank of learning estimation schemes corresponding to the single periods? Interestingly, the theoretical tools of this paper can be also used to successfully address identifiability and convergence issues regarding the identification of periodic components of general multi-periodic signals. [ABSTRACT FROM AUTHOR]

A mathematical model featuring the motion of a multilink wheeled vehicle is developed using a nonholonomic model. A detailed analysis of the inertial motion is made. Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. In addition, an analysis is made of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time. The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found. • A detailed analysis of the inertial motion of a multilink wheeled vehicle is made. • Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. • For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. • Analysis of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time is made. • The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found. [ABSTRACT FROM AUTHOR]

In this paper we consider a classical model of gasless combustion in a one dimensional formulation under the assumption of ignition temperature kinetics. We study the propagation of flame fronts in this model when the initial distribution of the solid fuel is a spatially periodic function that varies on a large scale. It is shown that in certain parametric regimes the model supports periodic traveling fronts. An accurate asymptotic formula for the velocity of the flame front is derived and studied. The stability of periodic fronts is also explored, and a critical condition in terms of parameters of the problem is derived. It is also shown that the instability of periodic fronts, in certain parametric regimes, results in a propagation-extinction-conduction-reignition pattern which is studied numerically. Novelty and significance statement: This work provides a closed form asymptotic description of periodic traveling fronts in a gasless combustion model with step-wise ignition temperature kinetics with a slowly varying concentration field. The stability analysis is performed, and the range of applicability of asymptotic formulas is given. A new propagation-extinction-conduction-reignition regime is identified. This regime emerges exclusively due to periodicity of the concentration field. [ABSTRACT FROM AUTHOR]

[Display omitted] • We modified the ZrO 2 thin films with three different POMs and we carried out experimental and DFT studies of adsorption of Methylene Blue on the these materials. • The adsorption rate values (K 2) of the pseudo-second order model follow these trend ZrO 2 /CrPOM > ZrO 2 /CoPOM > ZrO 2 /CuPOM > ZrO 2. • The ZrO 2 thin films showed the lowest photocatalytic degradation rate k ap value (k ap = 1.5 × 10−3 min−1), while the best result was obtained for ZrO 2 /CrPOM thin films (k ap = 5.7 × 10−3 min−1) under UV irradiation. • DFT study revealed to ZrO 2 /CrPOM its HOMO is composed by the d z 2 orbital of the metallic atom, whereas in the other two compounds the HOMO is composed by other d orbitals, that show less reactive d orbital composition. In this work, several ZrO 2 thin films modified with Anderson-type polyoxomolybdates (POMs) with general formula (NH 4) 6-n [XMo 6 O 24 H 6 ]-6+n where X = Co3+, Cr3+ and, Cu2+ were prepared. Thin films were characterized through SEM and EDX assay, UV–Vis diffuse reflectance and Fourier Transform Infrared (FTIR) assay. The optical bandgap of ZrO 2 thin films was determined to be 3.25 eV, while the modified thin films showed a red shift in the optical activity compared with bare ZrO 2 thin films. Methylene Blue (MB) adsorption studies showed that Freundlich isotherm describes properly the experimental data for modified-ZrO 2 thin films. Besides, the kinetic results showed the MB adsorption of modified-ZrO 2 thin films was superior to bare ZrO 2 thin film. The adsorption rate values (K 2) of the pseudo-second order model follow these trend ZrO 2 /CrPOM > ZrO 2 /CoPOM > ZrO 2 /CuPOM > ZrO 2. The photocatalytic activity of the thin films for MB decomposition under UV and Visible irradiation was studied. Among all the catalysts, the ZrO 2 thin films showed the lowest photocatalytic degradation rate k ap value (k ap = 1.5 × 10−3 min−1), while the best result was obtained for ZrO 2 /CrPOM thin films (k ap = 5.7 × 10−3 min−1) under UV irradiation. Besides, this was the only catalyst efficiently active in MB degradation under visible irradiation, these materials reach 10.4 % after 100 min under visible irradiation. Finally, chemical calculations supported the observed results, by means of TDDFT, EDA analysis, Fukui function and periodic DFT calculations. [ABSTRACT FROM AUTHOR]

This paper presents an analysis on the convergence to a steady state for the traffic dynamics in a network of signalized intersections. The links carrying the traffic from one intersection to another in the network are partitioned into a series of cells whose dynamics is modelled as a nonlinear cell transmission model. A microscopic convergence analysis is presented for the network where the traffic lights at the intersections are controlled by pre-timed control in which the average flow rate along each link is assumed to be known accurately. A macroscopic convergence analysis is applied to the traffic network where the traffic lights are controlled by coordinated model predictive control. In coordinated model predictive control, each local agent optimizes the switching times of traffic light at one intersection online with sampled local traffic state around this intersection and exchanged information from neighbouring intersections. A unique periodic function for the microscopic steady state is proved to exist under pre-timed control while a unique macroscopic steady state is achieved under coordinated model predictive control for the traffic lights. This paper validates by simulations that the ability of coordinated model predictive control to stabilize the traffic demand is not worse than pre-timed control even though the average flow rate along each link is not assumed to be known and further no explicit cycle time for the traffic light is suggested. Moreover, the incurred delay from the steady state is lower on the average under coordinated model predictive control compared to pre-timed control. • Presenting microscopic convergence analysis on pre-timed control • Proposing distributed model predictive control for traffic signals. • Presenting macroscopic convergence analysis on distributed control. • Validating claims on convergence of traffic signal control by simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Boussinesq equation containing weak backward diffusion, delay in the convection term, dissipation and Marangoni effect. By applying geometric singular perturbation theory, a locally invariant manifold diffeomorphic to the critical manifold is established. For Boussinesq equation with delay and weak backward diffusion, the monotonicity of ratio of Abelian integrals is analyzed by utilizing the Picard–Fuchs equation. The conditions on existence of a unique periodic wave and solitary waves are obtained as well as the bound of wave speed. For Boussinesq equation with weak backward diffusion, dissipation and Marangoni effect, the corresponding Melnikov function containing three generic elements is given. The parametric conditions on existence of a unique and two periodic waves are derived respectively. Furthermore, the existence of a unique solitary wave is proved under some parametric conditions. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to a variational method for the construction of optimal quadrature formulas in the sense of Sard in the Hilbert space W ˜ 2 (m , m − 1) of complex-valued and periodic functions. In this, the coefficients of the optimal quadrature formula are found separately in the case ω h is integer and non-integer cases. In addition, using the constructed optimal quadrature formula, the numerical results of exponentially weighted integrals of certain functions in the case m = 2 is presented. The numerical results show that the order of convergence of the optimal quadrature formula is O 1 N + | ω | 2 in the space W ˜ 2 (2 , 1) . [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (called of central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then at least six limit cycles can bifurcate from the periodic annulus by linear perturbations. Four passing through the three zones and two passing through two zones. Now, if the central subsystem has a virtual center, then at leas four limit cycles can bifurcate from the periodic annulus by linear perturbations, three passing through the three zones and one passing through two zones. For this, we obtain a normal form for these piecewise Hamiltonian systems and study the number of zeros of its Melnikov functions defined in two and three zones. [ABSTRACT FROM AUTHOR]

By the use of the transfer matrix (TM) approach and effective non-Hermitian Hamiltonian (EnHH) approach we study transport and scattering properties of PT − symmetric tight-binding one-dimensional (1D) chains. In the TM approach we derive an exact expression for the transmission T through 1D chains that incorporates all the scattering-system parameters: incident-wave energy E , chain length L , gain/loss amplitude γ , and coupling strength to perfect leads χ. We show that the behavior of T vs. L strongly depends on the value of γ with respect to the critical value γ cr , defined through γ cr 2 = 4 − E 2 . For γ < γ cr , the transmission T is a periodic function of L , however, for γ > γ cr , T decreases exponentially with L. We also predict the unforeseen dependence T ∝ L − 2 at γ = γ cr . By the use of the EnHH approach to the same model which is now connected to continuum through left and write ends, we focus on the onset of the superradiance transition as a function of E , L , γ , and χ. The main interest here is in the competition between the balanced gain/loss terms in the bulk of the chain and the coupling to continuum through the ends of the chain. Our results demonstrate that the presence of the gain/loss in the bulk strongly modifies the width of non-superradiant states, however, did not change the width of the superradiant ones. The analytical predictions are supported by detailed numerical simulations. • Derivation of exact expression for the transmission T through 1D chains. • Transmission behavior largely depends on critical value of the gain/loss amplitude. • Superradiant transition is observed when increasing the coupling parameter χ. • The presence of the gain/loss strongly modifies the width of non-superradiant states. [ABSTRACT FROM AUTHOR]

• The validated turbulence model was implemented. • The aerodynamic loss was reduced by contoured endwall and the heat transfer level was lower than flat endwall. • The effectiveness of the contoured endwall was investigated on a typical gas turbine with cylindrical and laidback fan-shaped holes (7–7–7). • The outlet area of the film cooling hole was altered by the contoured endwall. • A higher film cooling level is achieved for the half-period trigonometric function of the non-axisymmetric endwal with laidback fan-shaped holes (7–7–7). Flow fields near the turbine vane endwall are complicated due to the endwall cross flows. The use of a non-axisymmetric endwall is regarded as an efficient technique to reduce the lateral pressure difference, decreasing the endwall cross flow. Numerical analysis was performed to determine how the non-axisymmetric endwall affected the vortex structure and heat transfer level. The cooling performance was investigated with cylindrical and laidback fan-shaped holes (7–7–7), which were arranged in rows aligned in the axial direction. The results showed that the non-axisymmetric endwall could significantly reduce the circumferential pressure difference and suppress the growth of the passage vortex, and the area-averaged heat transfer coefficient was reduced by 3.34%. The outlet area of the film hole was altered by the non-axisymmetric endwall, and the over-cooled regions may have appeared as a result of the excessive area increase. The influence of the non-axisymmetric endwall was concentrated at 0.4 < Z / C ax < 1.0 for the cylindrical hole. With the increase in M , the film cooling effectiveness of the non-axisymmetric endwall attained a higher level than that of the flat endwall. For the laidback fan-shaped hole, the effect of the non-axisymmetric endwall was confined within 0.25 < Z / C ax < 1.0. The half-period trigonometric function of the non-axisymmetric endwall (HTFN) achieved the optimal cooling performance for three blowing ratios. However, the periodic trigonometric function of the non-axisymmetric endwall (PTFN) only outperformed the flat endwall when M = 1.5. [ABSTRACT FROM AUTHOR]

Porous structures are intricate solid materials with numerous small pores, extensively used in fields like medicine, chemical engineering, and aerospace. However, the design of such structures using computer-aided tools is a time-consuming and tedious process. In this study, we propose a novel representation method and design approach for porous units that can be infinitely spliced to form a porous structure. We use periodic B-spline functions to represent periodic or symmetric porous units. Starting from a voxel representation of a porous sample, the discrete distance field is computed. To fit the discrete distance field with a periodic B-spline, we introduce the constrained least squares progressive-iterative approximation algorithm, which results in an implicit porous unit. This unit can be subject to optimization to enhance connectivity and utilized for topology optimization, thereby improving the model's stiffness while maintaining periodicity or symmetry. The experimental results demonstrate the potential of the designed complex porous units in enhancing the mechanical performance of the model. Consequently, this study has the potential to incorporate remarkable structures derived from artificial design or nature into the design of high-performing models, showing the promise for biomimetic applications. • Periodic B-spline functions are used as new representations of porous structures. • The CON-LSPIA algorithm is proposed to solve constrained fitting problems. • The periodicity of the porous unit is maintained during connectivity optimization. • Implicit units can improve model stiffness through topology optimization. [Display omitted] [ABSTRACT FROM AUTHOR]

In this work, we are interested in a class of Bi-directional associative memory neural networks (BAMNNs) with hybrid time-varying delays and D operator. We establish some sufficient conditions to guarantee the existence and global exponential stability of measure-pseudo almost periodic solutions of the considered BAMNNs, using contraction mapping principle, exponential dichotomy theory, differential inequality techniques, and the properties of the measure-pseudo almost periodic functions. A numerical example along with a graphical illustration are presented to support the effectiveness and feasibility of the obtained results. The results in this paper are original and complement the previous outcomes. [ABSTRACT FROM AUTHOR]

The integration of wave energy devices and coastal structures may be an innovative and sustainable way to achieve energy production purposes with a secondary benefit of coastal protection, which can increase accessibility and reduce the costs of wave energy technology. In this paper, a 3-D theoretical model was developed to investigate the hydrodynamic efficiency and breakwater function of a periodic array of oscillating buoys embedded in a caisson breakwater. The generalized radiation problem was solved to derive generalized radiation force. The theoretical model was validated using Haskind's relation and energy flux conservation law. The influences of wave/geometrical parameters and PTO damping were revealed. In particular, hydrodynamic phenomenon of multiple orders reflected and transmitted propagating waves and their influence on wave power extraction and coastal protection was examined. Results show that a satisfactory hydrodynamic efficiency and coastal defense are realized simultaneously under oblique waves for this proposed system. A decline of hydrodynamic efficiency is found beyond a critical wavenumber, accompanied by the occurrence of the strong reflection phenomenon. The findings of this paper contribute towards the preliminary design of the hybrid breakwater-WEC system for the synergy effect between the wave energy devices and breakwaters. [ABSTRACT FROM AUTHOR]

• Axons are lined by a submembrane periodic scaffold of actin and spectrin. • Circumferential actin rings are regularly spaced every 190 nm by spectrin tetramers. • Super-resolution microscopy has revealed the periodic scaffold and its organization. • Electron microscopy was recently used to resolve its ultrastructure. • Open questions include the assembly mechanisms and cellular functions of this scaffold. Neurons rely on a unique organization of their cytoskeleton to build, maintain and transform their extraordinarily intricate shapes. After decades of research on the neuronal cytoskeleton, it is exciting that novel assemblies are still discovered thanks to progress in cellular imaging methods. Indeed, super-resolution microscopy has revealed that axons are lined with a periodic scaffold of actin rings, spaced every 190 nm by spectrins. Determining the architecture, composition, dynamics, and functions of this membrane-associated periodic scaffold is a current conceptual and technical challenge, as well as a very active area of research. This short review aims at summarizing the latest research on the axonal periodic scaffold, highlighting recent progress and open questions. [ABSTRACT FROM AUTHOR]

• OSB-PD model for granular fracture considering arbitrary grain orientations is proposed. • Effects of grain orientations on mechanical and fracture behaviors are studied. • Transgranular and intergranular cracks considering randomness of microstructure are captured. Granular fracture holds significant implications in material mechanics. However, the previous studies in ordinary state-based peridynamic (OSB-PD) framework often neglect the internal crystalline structure of particles or only consider limited crystal orientation. To address this gap, a novel OSB-PD model for granular fracture within polycrystalline materials is proposed, in which the periodic functions are incorporated in the PD strain energy density, taking into account the inherent random orientation in cubic crystals. By comparing energy density from PD and the classical continuum mechanics, four PD material parameters are defined. Moreover, the corresponding surface correction method in the global coordinate system is also proposed. Several numerical examples including fracture analysis of polycrystalline materials are conducted to validate the effectiveness of the proposed method. The proposed ordinary state-based peridynamic model offers a fresh perspective for investigating granular fracture behaviors within polycrystalline materials. [ABSTRACT FROM AUTHOR]

The Integer Quantum Hall Effect transverse conductivity is derived assuming the electrons form a lattice described by the doubly periodic states of Rodrigues and Doria (2021). We consider the simplest scenario of one electron per Landau level in the unit cell to find complex electronic patterns revealing that besides the counter clockwise circulation set by the magnetic field, there is also clockwise circulation contrary to it. These circulations describe vortices and anti-vortices, thus there are novel topological excitations whose spatial distributions and individual vorticities are numerically calculated for the first eleven Landau levels. Although we work in a particular gauge, the Landau gauge, our results are gauge invariant as they hold for gauge invariant quantities, such as the wave function probability and the current density. • The topological properties of the IQHE are fixed by the integer N associated with the transverse conductivity plateaus, N e 2 / h , N is the number of filled Landau levels (n = 0 , ... , N − 1). Here we find that N does not describe all the topological properties of the IQHE thanks to the use of the doubly periodic functions obtained by Rodrigues and Doria (2021). • The treatment developed in details in the present paper, is for the case of one electron per Landau level in the unit cell (p = 1), where the number of trapped magnetic flux, and also the vorticity in the unit cell, is one. • Novel topological properties of the IQHE are shown here, as besides the counter clockwise circulation set by the magnetic field, there is also clockwise circulation contrary to it in each Landau level above the fundamental one. • We explicitly obtain the spatial distributions of vortices and anti-vortices and their individual vorticities in the unit cell for the first time. • The present approach also provides a straightforward description of the IQHE solely based on bulk properties without any appeal to the use of boundaries, and so, proves that edge states are not fundamental to the description of the IQHE. [ABSTRACT FROM AUTHOR]

We define a zeta function of a finite graph derived from the time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph. [ABSTRACT FROM AUTHOR]

In this paper, we provide a criterion to guarantee the existence of a globally attracting positive periodic solution for the nonautonomous Ricker system whose coefficients are all periodic functions of a common period, that is, under certain circumstances, all species will coexist in a limit cycle fashion. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the spreading dynamics of a nonlocal diffusion KPP model with free boundaries which is firstly explored in time almost periodic media. As the spreading occurs, the long-run dynamics are obtained. Especially, when the threshold condition for the kernel function is satisfied, applying the novel positive time almost periodic function, we accurately express the unique asymptotic spreading speed of the free boundary problem. [ABSTRACT FROM AUTHOR]

In this work, we use recent results on pseudo almost periodicity to study a class of non-autonomous cellular neural networks with mixed delays. Sufficient conditions are obtained in context of general measure theory (d μ (x) = ρ (x) dx + d μ 1 (x)), for existence, uniqueness and global exponentially stability of μ -pseudo almost periodic solutions of the considered model. As a consequence, we establish the exponential attractiveness of μ -pseudo almost periodic solutions of the addressed model as t ⟶ + ∞. Some examples and numerical simulations are given to illustrate our results. [ABSTRACT FROM AUTHOR]

• Two nonlinear transversely vibrating beams are considered. • Parametric and closed-form solutions are obtained using trigonometric function and Jacobi elliptic function respectively. • Our solutions can recover some of other's exact solutions. Nonlinear transversely vibrating beams, including a uniform beam carrying a lumped mass and a transversely vibrating quintic nonlinear beam, are considered in this paper. Firstly, using trigonometric function, analytical solutions to their Cauchy initial problems are constructed in parametric and closed-form. Secondly, it is found that one has the freedom to choose any periodic function to simulate the periodic vibration theoretically. As an example, we also construct parametric and closed-form solution expressed by Jacobi elliptic function. Thirdly, by comparing the two kinds of derived solutions, it is shown that the Jacobi elliptic function solution can degenerate to the corresponding trigonometric function solution when the modulus tends to zero. Comparison are also made between our derived Jacobi elliptic function solution and other's exact solution, which indicates that the presented parametric solution method is more general. [ABSTRACT FROM AUTHOR]

A multifield asymptotic homogenization technique for periodic thermo-diffusive elastic materials is provided in the present study. Field equations for the first-order equivalent medium are derived and overall constitutive tensors are obtained in closed form. These lasts depend upon the micro constitutive properties of the different phases composing the composite material and upon periodic perturbation functions, which allow taking into account the effects of microstructural heterogeneities. Perturbation functions are determined as solutions of recursive non homogeneous cell problems emanated from the substitution of asymptotic expansions of the micro fields in powers of the microstructural characteristic size into local balance equations. Average field equations of infinite order are also provided, whose formal solution can be obtained through asymptotic expansions of the macrofields. With the aim of investigating dispersion properties of waves propagating inside the medium, proper integral transforms are applied to governing field equations of the homogenized medium. A quadratic generalized eigenvalue problem is thus obtained, whose solution characterizes the complex valued frequency spectrum of the first-order equivalent material. The validity of the proposed technique has been confirmed by the very good matching obtained between dispersion curves of the homogenized medium and the lowest frequency ones relative to the heterogeneous material. These lasts are computed from the resolution of a quadratic generalized eigenvalue problem over the periodic cell subjected to Floquet–Bloch boundary conditions. An illustrative benchmark is conducted referring to a Solid Oxide Fuel Cell (SOFC)-like material, whose microstructure can be modeled through the spatial tessellation of the domain with a periodic cell subjected to thermo-diffusive phenomena. [ABSTRACT FROM AUTHOR]

The Bose–Einstein condensates (BECs) are seen during the studies in atomic optics, cavity opto-mechanics, cavity quantum electrodynamics, black-hole astrophysics, laser optics and atomtronics. Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential which describes the dynamics of a (2+1)-dimensional BEC. Based on the Kadomtsev–Petviashvili hierarchy reduction, we construct the bilinear forms and the N th order rogue-wave solutions in terms of the Gramian. With the help of the analytic and graphic analysis, we exhibit the first- and second-order rogue waves under the influence of the strength of the interatomic interaction, α (t) , and of Ω (t) = ω R ̃ ∕ ω Z , where t is the scaled time, ω R ̃ and ω Z are the confinement frequencies in the radial and axial directions: When Ω (t) = 0 , the first-order rogue wave exhibits as an eye-shaped distribution; When Ω (t) is a periodic function, the rogue wave periodically raises; When Ω (t) is an exponential function, the rogue wave appears on the exponentially increasing background; When α (t) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when Ω (t) is a constant. With Ω (t) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with t increasing. [ABSTRACT FROM AUTHOR]

This paper is motivated by an open problem around deep networks, namely, the apparent absence of over-fitting despite large over-parametrization which allows perfect fitting of the training data. In this paper, we analyze this phenomenon in the case of regression problems when each unit evaluates a periodic activation function. We argue that the minimal expected value of the square loss is inappropriate to measure the generalization error in approximation of compositional functions in order to take full advantage of the compositional structure. Instead, we measure the generalization error in the sense of maximum loss, and sometimes, as a pointwise error. We give estimates on exactly how many parameters ensure both zero training error as well as a good generalization error. We prove that a solution of a regularization problem is guaranteed to yield a good training error as well as a good generalization error and estimate how much error to expect at which test data. [ABSTRACT FROM AUTHOR]

In this work, we have developed a one-dimensional tight-binding model to investigate the conductivity of tunneling through a quasiperiodic or slowly-varying potential. Our study focuses on the hopping tunneling properties of a tight-binding model in which the on-site potential is replaced by both quasiperiodic and periodic functions. By comparing the hopping transport in scenarios involving a periodic potential and an slowly-varying model, the latter being the only known case of a 1D potential with a mobility edge and a metal-insulator transition, we gain valuable insights. In our analysis, we consider a proposed device situated between two quantum reservoirs in both quasiperiodic and periodic cases. Remarkably, we find that the tunnel transport through the lattice completely diminishes at the band center. This intriguing phenomenon is likely to hold fundamental importance in the interpretation of experimental data. • We have developed a one-dimensional tight-binding model to investigate tunneling through a quasiperiodic potential. • We consider a proposed device situated between two quantum reservoirs in both quasiperiodic and periodic cases. • We find that the tunnel transport through the lattice completely diminishes at the band center. [ABSTRACT FROM AUTHOR]

We study approximation by arbitrary linear combinations of n translates of a single function of periodic functions. We construct some linear methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of the L p -approximation convergence rate by these methods, when n → ∞, for 1 < p < ∞. We also prove a lower bound of the quantity of best approximation of this class by arbitrary linear combinations of n translates of arbitrary function, for the particular case p = 2. [ABSTRACT FROM AUTHOR]

In this paper, we establish an embedding theorem for the fuzzy multidimensional space and introduce the new types of multiplications of fuzzy vectors determined by the determinant algorithm. Then some fundamental results of calculus of fuzzy vector-valued functions on time scales are established. Based on these results, we introduce the concept of shift almost periodic fuzzy vector-valued functions and develop a theory of almost periodic fuzzy functions on time scales. This function theory can be used to study almost periodic fuzzy dynamic equations on time scales including a new type of fuzzy dynamic equations called fuzzy q -dynamic equation (i.e., quantum fuzzy dynamic equation). [ABSTRACT FROM AUTHOR]

• This paper analyses some important measures responsible for the performance of ZCZ in quasi synchronous direct sequence (QS-DS) CDMA. • The BER and the MUI interference of binary and ternary spreading codes were investigated as functions of the codes' even and odd periodic correlation functions. • Numerical results based on availability of ZCZ codes in the existing literature is provided. Zero correlation zone (ZCZ) codes have been recently adopted for many communication systems based on code division multiple access (CDMA). In this paper, we focus on the use of ZCZ codes in quasi-synchronous direct-sequence (QS-DS) CDMA. To ensure Interference-free communication, spreading ZCZ codes must be carefully designed. Indeed, numerous ZCZ code constructions with various correlation properties have been proposed and studied in literature. However, most of them are not adequately designed for communication systems. Here, we aim at determining specific criteria in order to choose the suitable ZCZ code family. A new bit error rate (BER) for QS-DS-CDMA as a function of the code's even and odd periodic correlation functions (PCFs) is derived. It is shown that the adequate spreading codes are those with zero even and odd PCFs. [ABSTRACT FROM AUTHOR]

• Unsteady-state exergy analysis was conducted for a homogenous solid. • Complete forms of unsteady-state exergy equations were solved numerically. • Exergetic behavior of the solid was examined under periodic boundary conditions. • A zonal classification method was proposed to better understand exergetic behavior. • Subsystem unsteady-state exergetic behavior was analyzed by zonal classification. Thermal exergy analysis is typically performed under a steady-state assumption. This assumption is reasonable when the temperature difference between the environment and the system of interest is very large; however, when the temperature difference is small, the exergetic behavior of the system becomes highly sensitive to the environmental temperature. Additionally, when the system has a large thermal capacity, the storage effect cannot be ignored; as such, the steady-state assumption is not suitable. Under such conditions, unsteady-state exergy analysis should be performed to improve understanding of the system behavior. In this study, we conducted numerical unsteady-state exergy analyses for heat conduction based on the complete forms of energy, entropy, and exergy equations. The system of interest was a one-dimensional homogeneous solid with a thermal diffusivity of 0.5 × 10−6 J/(m3∙K). We analyzed two cases by assigning two different time-varying temperature boundary conditions using a 24 h periodic sinusoidal function with a constant amplitude of 10 °C but different midlines of 20 °C and 10 °C. This study was focused on the complex exergetic behavior inside the solid based on the concept of exergy balance, how the exergy flows in and out of a subsystem, and how it is stored and consumed. For intuitive interpretation of the unsteady-state results, a zonal classification method was proposed. This method was used to interpret the state of exergy flow (warm/cool) and its direction (inflow/outflow) that varies depending on the temperature relationships. Additionally, the zonal classification method was utilized to interpret the state (warm/cool) and temporal increase/decrease of the exergy storage rate, which uniquely appears in unsteady-state analyses. The application of the developed method to various transient thermal problems will lead to more complete understanding of and greater insight into various thermodynamic phenomena. [ABSTRACT FROM AUTHOR]

• A constructive approach for random 3D composites is developed. • The question of conditionally divergent series for random composites is resolved. • The famous Jeffreys formula for the effective conductivity of suspensions is revised and extended. • A theory of 3D Eisenstein functions is developed. Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f. Jeffrey (1973) suggested an analytical formula valid up to O (f 3) for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the f 2-term. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed up to O (f 10 3). The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f 2-term, and in the f 3-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter. [ABSTRACT FROM AUTHOR]

In this work, we formulate a minimum-time optimal control problem to steer a FWMAV dynamical system from a hovering condition to forward flight with a prescribed forward speed using time-periodic and averaged dynamics formulations. For the averaged dynamics representation, we optimize the back and forth flapping angle and the up and down-stroke angles of attack of the wing. We represent the flapping angle via a generic periodic function with some parameters that determine the waveform of the flapping angle over the cycle. We formulate the optimal control problem such that the cost functional is the final time, and the slowly time-varying parameters of the flapping angle wave form and the up and down-stroke angles of attack are considered inputs to the averaged dynamics. On the other hand, the instantaneous the flapping speed and wing pitching angle are considered direct inputs to the time periodic system. The problem is then to steer the averaged dynamics from the hovering fixed point (origin) to a prescribed average forward speed, and the time periodic dynamics from the hovering periodic orbit to the orbit of the forward flight condition. We show that the averaging is not suitable for the steering between hovering and forward flight and that time-periodic dynamics are required for the controller to achieve proper transition. Also, we investigated the effect of using the time-averaged stability derivatives obtained using a computational fluid dynamics simulation versus using the time-varying hovering derivatives. [ABSTRACT FROM AUTHOR]

This paper retrieves, bright, dark and singular highly dispersive optical solitons with quadratic-cubic law of nonlinearity. The F -expansion scheme is the integration algorithm adopted in the paper. There are other solutions to the model that are also recovered such as doubly periodic functions, Weierstrass elliptic functions and periodic solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the n -width problem for the Sobolev space of periodic functions, H per r (0 , 2 π). Building on a theorem of Pinkus we show that it admits optimal even-dimensional spline spaces of all degrees ≥ r − 1. Then, by using a theorem of Karlovitz, we show that it does not admit any optimal spline space of odd dimension > 1. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the problem of periodic solutions is studied for Liénard equations with anindefinite singularity x ′ ′ (t) + f (x (t)) x ′ (t) + φ (t) x m (t) − α (t) x μ (t) = 0 , where f : (0 , + ∞) → R is a continuous function which may have a singularity at the origin, the signs of φ and α are allowed to change, m is a non-negative constant, and μ is a positive constant. The approach is based on a continuation theorem of Manásevich and Mawhin with techniques of a priori estimates. The main results partly answer the open problem proposed by R. Hakl, P.J. Torres and M. Zamora in the known literature. [ABSTRACT FROM AUTHOR]

We consider the mean curvature flow V = H in a cylinder Ω ≔ B × R , where B is the unit ball in R N , V is the normal velocity of a moving surface Γ t : x N + 1 = u ( x , t ) ( x ∈ B ) and H is the mean curvature of Γ t . Assume that the surface is radially symmetric and it contacts the boundary of Ω with a prescribed angle θ = θ ( t , u ) . In case θ tends to two (spatially or temporally) periodic functions as u → ± ∞ , we show that the problem has a unique entire solution U , which propagates from − ∞ to ∞ and connects two (spatially or temporally) periodic traveling waves at t = − ∞ and at t = ∞ . [ABSTRACT FROM AUTHOR]

Abstract Plants are essential to life. They are a source of food, medicine, clothing, and are important to a healthy environment. Unfortunately, plants can become infected with a disease. A viral infection is one way that a plant may become diseased. Often times, plants die from this infection. These viruses hurt the agriculture industry as billions of dollars are lost due to crop loss every year. An insect vector is typically the cause for the virus propagation. The vectors exhibit seasonal behavior as they are active in the warm months, but not as much so in the cooler months. To defend against the vectors, pesticides have been used. While the pesticides might be effective in controlling the vectors, they can have harmful side effects on the environment. An alternative solution is to introduce a predator, or just increase the number of a naturally present one, to prey on the insects. In this paper, we use a mathematical model of ordinary differential equation to model the dynamics of this biological process. We first present an autonomous system, then two nonautonomous systems, accounting for the periodic nature of the insects. To analyze the models, the basic reproductive number is used. We demonstrate a couple of approaches for determining this number: a time average approach and a linear operator approach. Afterwards, numerical simulations are used to demonstrate the results. Finally, comparisons are made between the models and the approaches. [ABSTRACT FROM AUTHOR]

Highlights • Ag islands in 2D periodic pattern on GaP nanocones for SERS are prepared. • Two different techniques for photoresist deposition on 3D surface are compared. • Detectable enhancement in Raman signal for Rhodamine 6G is demonstrated. Abstract This contribution brings novel concept of Ag islands formation in 2D periodic arrangement with square symmetry on special 3D surfaces – gallium phosphide (GaP) nanocones, for surface-enhanced Raman spectroscopy application. 2D periodic structure in square symmetry with period of app. 470 nm is patterned in photoresist layer by interference lithography using double exposure of a two-beam interference optical field. Two different techniques for photoresist deposition on 3D surface are compared: spin-coating and photoresist deposition from water level. After Ag sputtering and lift-off, GaP nanocones are decorated with Ag islands in 2D periodical arrangement. Detectable enhancement in Raman signal for Rhodamine 6G adsorbed on prepared samples is demonstrated. [ABSTRACT FROM AUTHOR]

Highlights • The problem of time average estimation is addressed in the fraction-of-time (FOT) probability framework. • The observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. • A central limit theorem for the normalized estimation error of the time average is derived. • Results on the distribution of the estimation error are derived for the almost-periodic functions. • The estimation of the cyclic autocorrelation function of a non-zero mean cyclostationary signal is considered. Abstract The problem of time average estimation is addressed in the fraction-of-time probability framework. In this approach, the observed signal is modeled as a single function of time rather than as a sample path of a stochastic process. Under mild regularity assumptions on temporal cumulants of the signal, a central limit theorem (CLT) is proved for the normalized error of the time average estimate, where the normalizing factor is the square root of the observation interval length. This rate of convergence is the same as that obtained in the classical stochastic approach, but is derived here without resorting to mixing assumptions. That is, no ergodicity hypothesis is needed. Examples of functions of interest in communications are provided whose time average estimate satisfies the CLT. For the class of the almost-periodic functions, a non normal limit distribution for the normalized error of the time average estimate can be obtained when the normalizing factor equals the observation interval length. The limit distribution, nevertheless, depends on the sequence adopted to perform the limit, a result that cannot be obtained in the classical stochastic approach. Numerical examples illustrate the theoretical results and an application to test the presence of a nonzero-mean cyclostationary signal is presented. [ABSTRACT FROM AUTHOR]

We study in this paper the existence of periodically modulated in one variable and localized in another variable solutions to the cubic Swift–Hohenberg equation on the plane R 2. In the first part we try to apply the method by Kirschgässner–Mielke to reduce the problem to the search of finite dimensional submanifolds with periodic orbits on them in some formal infinite-dimensional dynamical system generated by the stationary SH equation. It turns out that it cannot be done immediately due to properties of the spectrum for the linearized system at the localized roll (a one-dimensional pulse). In the second part we change roles of variables and formulate the problem as funding homoclinic orbits to an equilibrium of the formal infinite-dimensional system in the space of periodic functions in variable y. Staying apart the proof of the exact theorem on the existence of related center manifold, we exploit the Bubnov–Galerkin method to derive the Hamiltonian finite-dimensional ODEs with four or six degrees of freedom having the saddle type equilibrium whose homoclinic orbits correspond to approximate solution on needed type. The search for homoclinic orbits is performed by means of numerical methods. • SH equation solutions localized in one variable and periodic in another are sought. • Kirchgassner–Mielke approach is justified if temporal variable is taken localized. • The related finite-dimensional approximations confirm numerical existence. • Theory of dynamical systems proves to be useful to solve the problem. [ABSTRACT FROM AUTHOR]