Infinitely Many Solutions and a Ground-State Solution for Klein-Gordon Equation Coupled with Born-Infeld Theory

In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION

In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRODINGER SYSTEMS

In this paper,we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H^(1)(ℝ^(2)).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in[L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].

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.

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL

In this paper,we consider the nonlinear Kirchhoff type equation with a steep potential well−(a+b∫_(R)^(3)|∇u|^(2 )dx)Δu+λV(x)u=f(u)in R^(3),where a,b>0 are constants,λ is a positive parameter,V∈C(R3,R)is a steep potential well and the nonlinearity f∈C(R,R)satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing(PS)C sequence,we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when λ is large enough,and find that its energy is strictly larger than twice that of the ground state solutions.In addition,we also prove the concentration of ground state sign-changing solutions.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound

In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R2 and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.

STEADY-STATE SOLUTIONS FOR A ONE-DIMENSIONAL NONISENTROPIC HYDRODYNAMIC MODEL WITH NON-CONSTANT LATTICE TEMPERATURE

A one-dimensional stationary nonisentropic hydrodynamic model for semiconductor devices with non-constant lattice temperature is studied. This model consists of the equations for the electron density, the electron current density and electron temperature, coupled with the Poisson equation of the electrostatic potential in a bounded interval supplemented with proper boundary conditions. The existence and uniqueness of a strong subsonic steady-state solution with positive particle density and positive temperature is established. The proof is based on the fixed-point arguments, the Stampacchia truncation methods, and the basic energy estimates.

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.

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.

High-Intensity Interval Training v/s Steady-State Cardio in Rehabilitation of Neurological Patients

Neuropathy is nerve damage that can cause chronic neuropathic pain, which is challenging to cure and has a significant financial burden. Exercise therapies, including High-Intensity Interval Training (HIIT) and steady-state cardio, are being explored as potential treatments for neuropathic pain. This systematic review compares the effectiveness of HIIT and steady-state cardio for improving function in neurological patients. This article provides an overview of the systematic review conducted on the effects of exercise on neuropathic patients, with a focus on high-intensity interval training (HIIT) and steady-state cardio. The authors conducted a comprehensive search of various databases, identified relevant studies based on predetermined inclusion criteria, and used the EPPI automation application to process the data. The final selection of studies was based on validity and relevance, with redundant articles removed. The article reviews four studies that compare high-intensity interval training (HIIT) to moderate-intensity continuous training (MICT) on various health outcomes. The studies found that HIIT can improve aerobic fitness, cerebral blood flow, and brain function in stroke patients;lower diastolic blood pressure more than MICT and improve insulin sensitivity and skeletal muscle mitochondrial content in obese individuals, potentially helping with the prevention and management of type 2 diabetes. In people with multiple sclerosis, acute exercise can decrease the plasma neurofilament light chain while increasing the flow of the kynurenine pathway. The available clinical and preclinical data suggest that further study on high-intensity interval training (HIIT) and its potential to alleviate neuropathic pain is justified. Randomized controlled trials are needed to investigate the type, intensity, frequency, and duration of exercise, which could lead to consensus and specific HIIT-based advice for patients with neuropathies.

A CONDITION OF THE EXISTENCE OF STABLE POSITIVE STEADY-STATE SOLUTIONS FOR A ONE PREDATOR TWO PREY SYSTEM

One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system. Under this condition we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem, discuss the positive stable solution problem bifurcated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.

Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.

Interaction of 4-aminosalicylic Acid and Surfactants in Aqueous Solutions Using UV-Vis Spectra and Steady-state Fluorescence Spectroscopy

The interactions of 4-aminosalicylic acid （4-ASA） and surfactants in aqueous solutions were investigated by using UV-Vis spectra and steady-state fluorescence spectroscopy.The results showed that the strongest peak at UV-vis spectra of 4-ASA aqueous solution in the presence of cationic surfactant and cetyltrimethyl ammonium bromide （CTAB） appeared at 206 nm and took a red shift from 206 nm to 221 nm with the increase of 4-ASA concentrations from 0.8×10-5 to 4.4×10-4 mol/L.Similarly,the strongest peak at UV-vis spectra of 4-ASA aqueous solution in the presence of nonionic surfactant and polyvinylpyrrolidone （PVP） appeared at 206 nm and took a red shift from 206 nm to 219 nm with the increase of 4-ASA concentrations from 0.8×10-5 to 4.4×10-4 mol/L.However,the similar phenomena did not appeared in the presence of anion surfactant,sodium dodecyl sulfate （SDS）,the UV-vis spectra of 4-ASA aqueous solution remained the same peak position and the peak value increased with the 4-ASA concentration increase.The results could be attributed to the electrostatic attraction between 4-ASA and CTAB or PVP,as well as the electrostatic repulsion between 4-ASA and SDS.Furthermore,the value of critical micelle concentration （CMC） of surfactants in the presence of 4-ASA was determined with Fluorescence method.The first and second CMC of CTAB was 1.2×10-4 M and 2.4×10-4 M,respectively.The first and second CMC of PVP was 1.2×10-4 M and 2.8×10-4 M.SDS realized the multiple micellizations to form multiple CMC.

POSITIVE STEADY STATES AND DYNAMICS FOR A DIFFUSIVE PREDATOR-PREY SYSTEM WITH A DEGENERACY

In this article, we consider positive steady state solutions and dynamics for a spatially heterogeneous predator-prey system with modified Leslie-Gower and Holling-Type II schemes. The heterogeneity here is created by the degeneracy of the intra-specific pressures for the prey. By the bifurcation method, the degree theory, and a priori estimates, we discuss the existence and multiplicity of positive steady states. Moreover, by the comparison argument, we also discuss the dynamical behavior for the diffusive predator-prey system.

Rheological behavior of semi-solid 7075 aluminum alloy at steady state

The further application of semi-solid processing lies in the in-depth fundamental study like rheological behavior. In this research, the apparent viscosity of the semi-solid slurry of 7075 alloy was measured using a Couette type viscometer. The effects of solid fraction and shearing rate on the apparent viscosity of this alloy were investigated under different processing conditions. It can be seen that the apparent viscosity increases with an increase in the solid fraction from 10% to 50%(temperature 620 oC to 630 oC) at steady state. When the solid fraction was fi xed, the apparent viscosity can be decreased by altering the shearing rate from 61.235 s-1 to 489.88 s-1 at steady state. An empirical equation that shows the effects of solid fraction and shearing rate on the apparent viscosity is fi tted:. The microstructure of quenched samples was examined to understand the alloy's rheological behavior.

A novel kind of multiple steady states characteristics in the dividing wall column

In this article, one kind of multiple steady states(MSS) phenomenon was investigated for a dividing wall column(DWC). The four-section model constructed in Aspen Plus was employed to simulate two DWC cases: mixture of n-hexane, n-heptane and n-octane;system of methanol, ethanol and n-propanol. It can be seen that there is a range of vapor split ratio in which multiple solutions of reflux ratio exist for fixed DWC configuration with the same feed and product streams. The width and the curve shapes of the MSS region, and the number of solutions change with the liquid split ratio. This MSS phenomenon was further explained using the component recovery around the prefractionator and the component recycling flow inside the DWC. This MSS phenomenon is helpful for DWC design by knowing the probable existence of multiple solutions in advance.

Predicting of Power Quality Steady State Index Based on Chaotic Theory Using Least Squares Support Vector Machine

An effective power quality prediction for regional power grid can provide valuable references and contribute to the discovering and solving of power quality problems. So a predicting model for power quality steady state index based on chaotic theory and least squares support vector machine (LSSVM) is proposed in this paper. At first, the phase space reconstruction of original power quality data is performed to form a new data space containing the attractor. The new data space is used as training samples for the LSSVM. Then in order to predict power quality steady state index accurately, the particle swarm algorithm is adopted to optimize parameters of the LSSVM model. According to the simulation results based on power quality data measured in a certain distribution network, the model applies to several indexes with higher forecasting accuracy and strong practicability.

On the Fifth-Order Stokes Solution for Steady Water Waves

This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear （1961）, confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.

40 Hz Auditory Steady State Response to Linguistic Features of Stimuli during Auditory Hallucinations

The auditory steady state response （ASSR） may reflect activity from different regions of the brain, depending on the modulation frequency used. In general, responses induced by low rates （_〈40 Hz） emanate mostly from central structures of the brain, and responses from high rates （〉80 Hz） emanate mostly from the peripheral auditory nerve or brainstem structures. Besides, it was reported that the gamma band ASSR （30-90 Hz） played an important role in working memory, speech understanding and recognition. This paper investigated the 40 Hz ASSR evoked by modulated speech and reversed speech. The speech was Chinese phrase voice, and the noise-like reversed speech was obtained by temporally reversing the speech. Both auditory stimuli were modulated with a frequency of 40 Hz. Ten healthy subjects and 5 patients with hallucination symptom participated in the experiment. Results showed re- duction in left auditory cortex response when healthy subjects listened to the reversed speech compared with the speech. In contrast, when the patients who experienced auditory hallucinations listened to the reversed speech, the auditory cortex of left hemispheric responded more actively. The ASSR results were consistent with the behavior results of patients. Therefore, the gamma band ASSR is expected to be helpful for rapid and objective diagnosis of hallucination in clinic.