ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.

Tuning Second Chern Number in a Four-Dimensional Topological Insulator by High-Frequency Time-Periodic Driving

Floquet engineering has attracted considerable attention as a promising approach for tuning topological phase transitions.We investigate the effects of high-frequency time-periodic driving in a four-dimensional(4D)topological insulator,focusing on topological phase transitions at the off-resonant quasienergy gap.The 4D topological insulator hosts gapless three-dimensional boundary states,characterized by the second Chern number C_(2).We demonstrate that the second Chern number of 4D topological insulators can be modulated by tuning the amplitude of time-periodic driving.This includes transitions from a topological phase with C_(2)=±3 to another topological phase with C_(2)=±1,or to a topological phase with an even second Chern number C_(2)=±2,which is absent in the 4D static system.Finally,the approximation theory in the high-frequency limit further confirms the numerical conclusions.

Solitons and Their Biperiodic Pulsation in Ultrafast Fiber Lasers Based on CB/GO

The carbon black(CB)is introduced to manufacture CB/graphene oxide(GO)composite material to mitigate limitations of GO as a saturable absorber with the excellent performance in ultrafast fiber lasers.At a central wavelength of 1555.5 nm,the stable mode-locked pulse with width of 656 fs,repetition rate of 20.16 MHz,and high signal-to-noise ratio of 82.07 dB is experimentally obtained.Additionally,experimental observations for pulsation phenomena of vector biperiodic solitons combining period-1 and period-17,period-2 and period-32,period-3 and period-36 are verified via simulations.

Interference-induced suppression of particle emission from a Bose-Einstein condensate in lattice with time-periodic modulations

Emission of matter-wave jets from a parametrically driven condensate has attracted significant experimental and theoretical attention due to the appealing visual effects and potential metrological applications.In this work,we investigate the collective particle emission from a Bose-Einstein condensate confined in a one-dimensional lattice with periodically modulated interparticle interactions.We give the regimes for discrete modes,and find that the emission can be distinctly suppressed.The configuration induces a broad band,but few particles are ejected due to the interference of the matter waves.We further qualitatively model the emission process and demonstrate the short-time behaviors.This engineering provides a way to manipulate the propagation of particles and the corresponding dynamics of condensates in lattices,and may find application in the dynamical excitation control of other nonequilibrium problems with time-periodic driving.

A Brief Introduction to the English Periodical of Contemporary Social Sciences

The English periodical of Contemporary Social Sciences is an English periodical founded by Sichuan Academy of Social Sciences and is published every two months.It was approved by the National Radio and Television Administration of the People’s Republic of China(formerly the State Administration of Press,Publication,Radio,Film and Television of the People’s Republic of China)in March 2016.

Periodic signal extraction of GNSS height time series based on adaptive singular spectrum analysis

Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection of appropriate embedding window size and principal components makes this method cumbersome and inefficient.To improve the efficiency and accuracy of singular spectrum analysis,this paper proposes an adaptive singular spectrum analysis method by combining spectrum analysis with a new trace matrix.The running time and correlation analysis indicate that the proposed method can adaptively set the embedding window size to extract the time-varying periodic signals from GNSS time series,and the extraction efficiency of a single time series is six times that of singular spectrum analysis.The method is also accurate and more suitable for time-varying periodic signal analysis of global GNSS sites.

Optimal design for rubber concrete layered periodic foundations based on the analytical approximations of band gaps and mapping relations

The seismic performance of rubber concrete-layered periodic foundations are significantly influenced by their design,in which the band gaps play a paramount role.Aiming at providing better designs for these foundations,this study first proposes and validates the analytical formulas to approximate the bounds of the first few band gaps.In addition,the mapping relations linking the frequencies of different band gaps are presented.Furthermore,an optimal design method for these foundations is developed,which is validated through an engineering example.It is demonstrated that ensuring the superstructure’s resonance zones are completely covered by the corresponding periodic foundation’s band gaps can achieve satisfactory vibration attenuation effects,which is a good strategy for the design of rubber concrete layered periodic foundations.

The Periodic Table of Primes

Over millennia, nobody has been able to predict where prime numbers sprout or how they spread. This study establishes the Periodic Table of Primes (PTP) using four prime numbers 2, 3, 5, and 7. We identify 48 integers out of a period 2×3×5×7=210 to be the roots of all primes as well as composites without factors of 2, 3, 5, and 7. Each prime, twin primes, or composite without factors of 2, 3, 5, and 7 is an offspring of the 48 integers uniquely allocated on the PTP. Three major establishments made in the article are the Formula of Primes, the Periodic Table of Primes, and the Counting Functions of Primes and Twin Primes.

Effective Elimination of Hazardous Chromium (VI) Using Periodic Elements and Contemporary Adsorption Methods by Using Magnesium Ferrite Nanoparticle: A Review

A well-known hazardous metal and top contaminant in wastewater is hexavalent chromium. The two forms of most commonly found chromium are chromate ( CrO 4 2− ) and dichromate ( Cr 2 O 7 2− ). Leather tanning, cooling tower blow-down, plating, electroplating, rinse water sources, anodizing baths etc. are the main sources of Cr (VI) contamination. The Cr (VI) is not only non-biodegradable in the environment but also carcinogenic to living population. It is still difficult to treat Cr contaminated waste water effectively, safely, eco-friendly, and economically. As a result, many techniques have been used to treat Cr (VI)-polluted wastewater, including adsorption, chemical precipitation, coagulation, ion-exchange, and filtration. Among these practices, the most practical method is adsorption for the removal of Cr (VI) from aqueous solutions, which has gained widespread acceptance due to the ease of use and affordability of the equipment and adsorbent. It has been revealed that Fe-based adsorbents’ oxides and hydroxides have high adsorptive potential to lower Cr (VI) content below the advised threshold. Fe-based adsorbents were also discovered to be relatively cheap and toxic-free in Cr (VI) treatment. Fe-based adsorbents are commonly utilized in industry. It has been discovered that nanoparticles of Fe-, Ti-, and Cu-based adsorbents have a better capacity to remove Cr (VI). Cr (VI) was effectively removed from contaminated water using mixed element-based adsorbents (Fe-Mn, Fe-Ti, Fe-Cu, Fe-Zr, Fe-Cu-Y, Fe-Mg, etc.). Initial findings suggest that Cr (VI) removal from wastewater may be accomplished by using magnesium ferrite nanomaterials as an efficient adsorbent.

Elastic wave insulation and propagation control based on the programmable curved-beam periodic structure

Curved-beams can be used to design modular multistable metamaterials(MMMs)with reprogrammable material properties,i.e.,programmable curved-beam periodic structure(PCBPS),which is promising for controlling the elastic wave propagation.The PCBPS is theoretically equivalent to a spring-oscillator system to investigate the mechanism of bandgap,analyze the wave propagation mechanisms,and further form its geometrical and physical criteria for tuning the elastic wave propagation.With the equivalent model,we calculate the analytical solutions of the dispersion relations to demonstrate its adjustability,and investigate the wave propagation characteristics through the PCBPS.To validate the equivalent system,the finite element method(FEM)is employed.It is revealed that the bandgaps of the PCBPS can be turned on-and-off and shifted by varying its physical and geometrical characteristics.The findings are highly promising for advancing the practical application of periodic structures in wave insulation and propagation control.

Self-propelled Leidenfrost droplets on femtosecond-laser-induced surface with periodic hydrophobicity gradient

The controllable transfer of droplets on the surface of objects has a wide application prospect in the fields of microfluidic devices,fog collection and so on.The Leidenfrost effect can be utilized to significantly reduce motion resistance.However,the use of 3D structures limits the widespread application of self-propulsion based on Leidenfrost droplets in microelectromechanical system.To manipulate Leidenfrost droplets,it is necessary to create 2D or quasi-2D geometries.In this study,femtosecond laser is applied to fabricate a surface with periodic hydrophobicity gradient(SPHG),enabling directional self-propulsion of Leidenfrost droplets.Flow field analysis within the Leidenfrost droplets reveals that the vapor layer between the droplets and the hot surface can be modulated by the SPHG,resulting in directional propulsion of the inner gas.The viscous force between the gas and liquid then drives the droplet to move.

Frequency-dependent viscoelasticity effects on the wave attenuation performance of multi-layered periodic foundations

In this paper,layered periodic foundations(LPFs)are numerically examined for their responses to longitudinal and transverse modes in the time and frequency domains.Three different unit-cells,i.e.,2-layer,4-layer,and 6-layer unit-cells,comprising concrete/rubber,concrete/rubber/steel/rubber,and concrete/rubber/steel/rubber/lead/rubber materials,respectively,are taken into account.Also,the viscoelasticity behavior of the rubber is modeled with two factors,i.e.,a frequency-independent(FI)loss factor and a linear frequency-dependent(FD)loss factor.Following the extraction of the complex dispersion curves and the identification of the band gaps(BGs),the simulations of wave transmission in the time and frequency domains are performed using the COMSOL software.Subsequent parametric studies evaluate the effects of the rubber viscoelasticity models on the dispersion curves and the wave transmission for the longitudinal and transverse modes.The results show that considering the rubber viscoelasticity enhances the wave attenuation performance.Moreover,the transverse-mode damping is more sensitive to the viscoelasticity model than its longitudinal counterpart.The 6-layer unit-cell LPF exhibits the lowest BG,ranging from 4.8 Hz to 6.5 Hz.

Conditional Generative Adversarial Network Enabled Localized Stress Recovery of Periodic Composites

Structural damage in heterogeneousmaterials typically originates frommicrostructures where stress concentration occurs.Therefore,evaluating the magnitude and location of localized stress distributions within microstructures under external loading is crucial.Repeating unit cells(RUCs)are commonly used to represent microstructural details and homogenize the effective response of composites.This work develops a machine learning-based micromechanics tool to accurately predict the stress distributions of extracted RUCs.The locally exact homogenization theory efficiently generates the microstructural stresses of RUCs with a wide range of parameters,including volume fraction,fiber/matrix property ratio,fiber shapes,and loading direction.Subsequently,the conditional generative adversarial network(cGAN)is employed and constructed as a surrogate model to establish the statistical correlation between these parameters and the corresponding localized stresses.The stresses predicted by cGAN are validated against the remaining true data not used for training,showing good agreement.This work demonstrates that the cGAN-based micromechanics tool effectively captures the local responses of composite RUCs.It can be used for predicting potential crack initiations starting from microstructures and evaluating the effective behavior of periodic composites.

Existence of a Hölder Continuous Extension on Embedded Balls of the 3-Torus for the Periodic Navier Stokes Equations

This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of cells of the 3-Torus. Satisfying a divergence-free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which have finite time singularities can occur starting with the first derivative and higher with respect to time. The existence of a subspace of the solution space where v3 is continuous and {C, y1, y12}, is linearly independent in the additive argument of the solution in terms of the Lambert W function, (y12=y2, C∈R) together with the condition v2=-2y1v1. On this subspace, the Biot Savart Law holds exactly [see Section 2 (Equation (13))]. Also on this subspace, an expression X (part of PNS equations) vanishes which contains all the expressions in derivatives of v1 and v2 and the forcing terms in the plane which are related as with the cancellation of all such terms in governing PDE. The y3 component forcing term is arbitrarily small in ε ball where Weierstrass P functions touch the center of the ball both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3 only governing PDE resulting. With viscosity present as v changes from zero to the fully viscous case at v =1 the solution for v3 reaches a peak in the third component y3. Consequently, there exists a dipole which is not centered at the center of the cell of the Lattice. Hence since the dipole by definition has an equal in magnitude positive and negative peak in y3, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere and where a given cell of dimensions [-1, 1]3 is circumscribed on a sphere of radius 1. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of v3 wrt to surface normal n is zero including at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. It is shown that v3 is spatially finite on the Riemann Sphere and the forcing is oscillatory in y3 component if the velocity v3 is. It is true that . A boundary condition on the sphere shows the rotation of a sphere of viscous fluid. Finally on the sphere a solution for v3 is obtained which is proven to be Hölder continuous and it is shown that it is possible to extend Hölder continuity on the sphere uniquely to all of the interior of the ball.

Impact of Alcian blue and periodic acid Schiff expression on the prognosis of gastric signet ring cell carcinoma

BACKGROUND The Alcian blue(AB)and periodic acid Schiff(PAS)stains are representative mucus markers in gastric signet ring cell carcinoma(SRCC).They are low-cost special staining methods used to detect acidic mucus and neutral mucus,respectively.However,the clinical importance of the special combined AB and PAS stain is unclear.AIM To investigate AB expression,PAS expression and the AB-to-PAS(A/P)ratio in gastric SRCC patients and to assess patient prognosis.METHODS Paraffin-embedded sections from 83 patients with gastric SRCC were stained with AB and PAS,and signet ring cell positivity was assessed quantitatively.Immuno-histochemical staining for Ki67,protein 53(P53)and human epidermal growth factor receptor 2(HER2)was performed simultaneously.The cancer-specific survival(CSS)rate was estimated via Kaplan-Meier analysis.Cox proportional hazards models were used for univariate and multivariate survival analyses.RESULTS Kaplan-Meier survival analysis revealed that the 3-year CSS rate was significantly greater in the high-PAS-expression subgroup than in the low-PAS-expression subgroup(P<0.001).The 3-year CSS rate in the A/P≤0.5 group was significantly greater than that in the A/P>0.5 group(P=0.042).Univariate Cox regression analysis revealed that the factors affecting prognosis included tumor diameter,lymph node metastasis,vessel carcinoma embolus,tumor stage,the A/P ratio and the expression of Ki67,P53 and the PAS.Cox multivariate regression analysis confirmed that low PAS expression[hazard ratio(HR)=3.809,95%confidence interval(CI):1.563-9.283,P=0.003]and large tumor diameter(HR=2.761,95%CI:1.086-7.020,P=0.033)were independent risk factors for poor prognosis.CONCLUSION A/P>0.5 is potentially a risk factor for prognosis,and low PAS expression is an independent risk factor in the prognosis of gastric SRCC.PAS expression and the A/P ratio could help in predicting the clinical prognosis of patients with SRCC.

Frictional contact analysis of a rigid solid with periodic surface sliding on the thermoelectric material

Understanding and characterizing rough contact and wavy surfaces are essential for developing effective strategies to mitigate wear,optimize lubrication,and enhance the overall performance and durability of mechanical systems.The sliding friction contact problem between a thermoelectric(TE)half-plane and a rigid solid with a periodic wavy surface is the focus of this investigation.To simplify the problem,we utilize mixed boundary conditions,leading to a set of singular integral equations(SIEs)with the Hilbert kernels.The analytical solutions for the energy flux and electric current density are obtained by the variable transform method in the context of the electric and temperature field.The contact problem for the elastic field is transformed into the second-kind SIE and solved by the Jacobi polynomials.Notably,the smoothness of the wavy contact surface ensures that there are no singularities in the surface contact stress,and ensures that it remains free at the contact edge.Based on the plane strain theory of elasticity,the analysis primarily examines the correlation between the applied load and the effective contact area.The distribution of the normal stress on the surface with or without TE loads is discussed in detail for various friction coefficients.Furthermore,the obtained results indicate that the in-plane stress decreases behind the trailing edge,while it increases ahead of the trailing edge when subjected to TE loads.

Calibration of CO and CO2 Monitors Used in Periodic Inspection of Vehicles at Fixed Stations for Environmental Control

Global efforts for environmental cleanliness through the control of gaseous emissions from vehicles are gaining momentum and attracting increasing attention. Calibration plays a crucial role in these efforts by ensuring the quantitative assessment of emissions for informed decisions on environmental treatments. This paper describes a method for the calibration of CO/CO2 monitors used for periodic inspections of vehicles in cites. The calibration was performed in the selected ranges: 900 - 12,000 µmol/mol for CO and 2000 - 20,000 µmol/mol for CO2. The traceability of the measurement results to the SI units was ensured by using certified reference materials from CO/N2 and CO2/N2 primary gas mixtures. The method performance was evaluated by assessing its linearity, accuracy, precision, bias, and uncertainty of the calibration results. The calibration data exhibited a strong linear trend with R² values close to 1, indicating an excellent fit between the measured values and the calibration lines. Precision, expressed as relative standard deviation (%RSD), ranged from 0.48 to 4.56% for CO and from 0.97 to 3.53% for CO2, staying well below the 5% threshold for reporting results at a 95% confidence level. Accuracy measured as percent recovery, was consistently high (≥ 99.1%) for CO and ranged from 84.90% to 101.54% across the calibration range for CO2. In addition, the method exhibited minimal bias for both CO and CO2 calibrations and thus provided a reliable and accurate approach for calibrating CO/CO2 monitors used in vehicle inspections. Thus, it ensures the effectiveness of exhaust emission control for better environment.

Modeling a Periodic Signal Using Fourier Series

This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a regular periodic signal, under Dirichlet conditions, via a linear superposition of trigonometric functions, thus Fourier polynomials are constructed. The Dirichlet conditions, are a set of mathematical conditions, providing a foundational framework for the validity of the Fourier series representation. By understanding and applying these conditions, we can accurately represent and process periodic signals, leading to advancements in various areas of signal processing. The resulting Fourier approximation allows complex periodic signals to be expressed as a sum of simpler sinusoidal functions, making it easier to analyze and manipulate such signals.

Ultrafast dynamics of femtosecond laser-induced high spatial frequency periodic structures on silicon surfaces

Femtosecond laser-induced periodic surface structures(LIPSS)have been extensively studied over the past few decades.In particular,the period and groove width of high-spatial-frequency LIPSS(HSFL)is much smaller than the diffraction limit,making it a useful method for efficient nanomanufacturing.However,compared with the low-spatial-frequency LIPSS(LSFL),the structure size of the HSFL is smaller,and it is more easily submerged.Therefore,the formation mechanism of HSFL is complex and has always been a research hotspot in this field.In this study,regular LSFL with a period of 760 nm was fabricated in advance on a silicon surface with two-beam interference using an 800 nm,50 fs femtosecond laser.The ultrafast dynamics of HSFL formation on the silicon surface of prefabricated LSFL under single femtosecond laser pulse irradiation were observed and analyzed for the first time using collinear pump-probe imaging method.In general,the evolution of the surface structure undergoes five sequential stages:the LSFL begins to split,becomes uniform HSFL,degenerates into an irregular LSFL,undergoes secondary splitting into a weakly uniform HSFL,and evolves into an irregular LSFL or is submerged.The results indicate that the local enhancement of the submerged nanocavity,or the nanoplasma,in the prefabricated LSFL ridge led to the splitting of the LSFL,and the thermodynamic effect drove the homogenization of the splitting LSFL,which evolved into HSFL.

3D Printing of Periodic Porous Metamaterials for Tunable Electromagnetic Shielding Across Broad Frequencies

The new-generation electronic components require a balance between electromagnetic interference shielding efficiency and open structure factors such as ventilation and heat dissipation.In addition,realizing the tunable shielding of porous shields over a wide range of wavelengths is even more challenging.In this study,the well-prepared thermoplastic polyurethane/carbon nanotubes composites were used to fabricate the novel periodic porous flexible metamaterials using fused deposition modeling 3D printing.Particularly,the investigation focuses on optimization of pore geometry,size,dislocation configuration and material thickness,thus establishing a clear correlation between structural parameters and shielding property.Both experimental and simulation results have validated the superior shielding performance of hexagon derived honeycomb structure over other designs,and proposed the failure shielding size(D_(f)≈λ/8-λ/5)and critical inclined angle(θf≈43°-48°),which could be used as new benchmarks for tunable electromagnetic shielding.In addition,the proper regulation of the material thickness could remarkably enhance the maximum shielding capability(85-95 dB)and absorption coefficient A(over 0.83).The final innovative design of the porous shielding box also exhibits good shielding effectiveness across a broad frequency range(over 2.4 GHz),opening up novel pathways for individualized and diversified shielding solutions.