THE EXISTENCE AND UNIQUENESS OF TIME-PERIODIC SOLUTIONS TO THE NON-ISOTHERMAL MODEL FOR COMPRESSIBLE NEMATIC LIQUID CRYSTALS IN A PERIODIC DOMAIN

In this paper,we are concerned with a three-dimensional non-isothermal model for the compressible nematic liquid crystal flows in a periodic domain.Under some smallness and structural assumptions imposed on the time-periodic force,we establish the existence of the time-periodic solutions to the system by using a regularized approximation scheme and the topological degree theory.We also prove a uniqueness result via energy estimates.

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued,nonmonotone friction law.First,a variational formulation of the model is obtained;that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field.Then,an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated.Under mild assumptions,the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven.Furthermore,a uniqueness theorem for the abstract inequality is established by using a monotonicity argument.Finally,we employ the theoretical results to examine the nonstationary Oseen model.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS

We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.

Analytical three-periodic solutions of Korteweg-de Vries-type equations

Based on the direct method of calculating the periodic wave solution proposed by Nakamura,we give an approximate analytical three-periodic solutions of Korteweg-de Vries(KdV)-type equations by perturbation method for the first time.Limit methods have been used to establish the asymptotic relationships between the three-periodic solution separately and another three solutions,the soliton solution,the one-and the two-periodic solutions.Furthermore,it is found that the asymptotic three-soliton solution presents the same repulsive phenomenon as the asymptotic three-soliton solution during the interaction.

Diversity of Rogue Wave Solutions to the (1+1)-Dimensional Boussinesq Equation

A periodically homoclinic solution and some rogue wave solutions of (1+1)-dimensional Boussinesq equation are obtained via the limit behavior of parameters and different polynomial functions. Besides, the mathematics reasons for different spatiotemporal structures of rogue waves are analyzed using the extreme value theory of the two-variables function. The diversity of spatiotemporal structures not only depends on the disturbance parameter u0 but also has a relationship with the other parameters c0, α, β.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation

This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.

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.

THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE

In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.

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.

GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

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.

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.

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

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.