SINGULAR PERTURBATION OF THE FOURTH ORDER ELLIPTIC EQUATION WHEN THE LIMIT EQUATION IS ELLIPTIC-PARABOLIC

This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operator, and sufficient smoothness. Under appropriate condition the sufficient condition of solvability is derived, the existence of solution is proved and a uniformly valid asymptotic solution of arbitrary order is given.

THE ASYMPTOTIC EXPRESSION OF THE SOLUTION OF THE CAUCHY’S PROBLEM FOR A HIGHER ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION WHEN THE LIMIT EQUATION HAS SINGULARITY

In this paper we consider the asymptotic expression of the solution of the Cauchy’sproblem for a higher order equation when the limit equation has singularity. In orderto construct the asymptotic expression of the solution, the region is divided into threesub-areas. In every small region, the solution of the differential equation is different.

ON THE EXISTENCE OF LIMIT CYCLES OF LIENARD EQUATION

In this paper, we have proved several theorems which guarantee that the Lienard equation has at least one or n limit cycles without using the traditional assumption G Thus some results in [3-5] are extended. The limit cycles can be located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or 'nth order compatible with each other' or 'nth order contained in each other'.

GIS-based 3D limit equilibrium analysis for design optimization of a 600 m high slope in an open pit mine

Combining the GIS （geographic information systems） grid-based data with four proposed column-based 3D slope stability analysis models, a comprehensive solution of a high-steep open-pit slope has been obtained. For six searching ranges, 19 critical slip surfaces of different sizes have been studied, in which the minimum 3D safety factor is 1.33. Comparison of 3D safety factors of designed and proposed slope plans shows for all the critical slip surfaces for the proposed plan, the smallest 3D safety factor is 1.33 under the most unfavorable condition. This means that the proposed plan of the high slopes, about 600 m, of an open pit （2-5° steeper than designed plan） is feasible.

Towards establishing practical multi-hazard bridge design limit states

In the U.S., the current Load and Resistance Factor Design （LRFD） Specifications for highway bridges is a reliability-based formulation that considers failure probabilities of bridge components due to the actions of typical dead load and frequent vehicular loads. Various extreme load effects, such as earthquake and vessel collision, are on the same reliability-based platform. Since these extreme loads are time variables, combining them with not considered frequent. non- extreme loads is a significant challenge. The number of design limit state equations based on these failure probabilities can be unrealistically large and unnecessary from the view point of practical applications. Based on the opinion of AASHTO State Bridge Engineers, many load combinations are insignificant in their states. This paper describes the formulation of a criterion to include only the necessary load combinations to establish the design limit states. This criterion is established by examining the total failure probabilities for all possible time-invariant and time varying load combinations and breaking them down into partial terms. Then, important load combinations can be readily determined quantitatively,

Meteoroid and Space Debris Risk Assessment for Satellites Orbiting the Earth/Moon

Interplanetary meteoroids and space debris can impact satellites orbiting the Earth or spacecraft traveling to the Moon.Targeting China Space Station(CSS),7 satellites selected from the constellation of Beidou Navigation Satellite System Phase III(BDS-3),and 3 spacecraft orbiting the Moon,we have adopted in the paper the Meteoroid Engineering Model 3,Divine-Staubach meteoroid environment model,and Jenniskens-McBride meteoroid steam model to analyze the meteoroid environment with the mass range of 10–6~10 g.Orbital Debris Engineering Model 3.1 space debris model is used to analyze the orbital debris environment faced by these satellites.The flux of space debris with a size larger than 100μm is compared with that of the meteoroids.The results show that the space debris flux encountered by China Space Station is much higher than that of the meteoroids with sizes in the above range.And quite the opposite,the meteoroids flux impacting the 7 satellites from the BDS-3 is higher.Upon adopting the double-layer Whipple protection measure,the catastrophic collision flux of these satellites encountering meteoroids is about 10–6 times of that without protection,or even less,implying that the Whipple protection effectively guarantees the safety of the satellites in orbit.Besides,it is also found that the flux of the high-density meteoroid population encountered by each satellite is greater than that of the low-density population,whereas the impact velocity is lower for each satellite.These results can aid the orbit selection and the protection design for satellites and spacecraft.

The periodic oscillations in the ENSO recharge–discharge oscillator model

A class of recharge–discharge oscillator model for the El Ni？o/Southern Oscillation （ENSO） is considered. A stable limit cycle is obtained by transforming the ENSO model into the van der Pol-Duffing equation. We proved that there exists periodic oscillations in the ENSO recharge–discharge oscillator model.

Three-dimensional critical slip surface locating and slope stability assessment for lava lobe of Unzen volcano

Even Unzen volcano has been declared to be in a state of relative dormancy,the latest formed lava lobe No.11 now represents a potential slope failure mass based on the latest research.This paper concentrates on the stability of the lava lobe No.11 and its possible critical sliding mass.It proposes geographic information systems (GIS) based three-dimensional (3D) slope stability analysis models.It uses a 3D locating approach to identify the 3D critical slip surface and to analyze the 3D stability of the lava lobe No.11.At the same time,the new 3D approach shows the effectiveness in selecting the range of the Monte Carlo random variables and locating the critical slip surface in different parts of the lava lobe No.11.The results are very valuable for judging the stability of the lava lobe and assigning the monitoring equipments.

Recovering Navier–Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation

This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employing different time and space scalings.The paper shows that,depending on the magnitude of the parameters which define the scaling,the macroscopic quantities(number density,mean velocity and local temperature)are solutions of the acoustic equation,the linear incompressible Euler equation and the incompressible Navier–Stokes equation.The derivation is formally tackled by the recent moment method proposed by[C.Bardos,et al.,J.Stat.Phys.63(1991)323]and the results generalize the analysis performed in[C.Bianca,et al.,Commun.Nonlinear Sci.Numer.Simulat.29(2015)240].

Incompressible limit and stability of all-time solutions to 3-D full Navier-Stokes equations for perfect gases

This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.

Research of Pipeline Pressure Based on the Reliability Theory

The former gas pipeline operating pressure transmission determined mostly according to the determined formula into the corresponding pipeline parameters are obtained, and the parameters of the actual pipeline due to many reasons, such as measurement error, production batch, etc., is not a fixed value. This paper on pipeline integrity established limit state equation, using Monte Carlo method to calculate the gas pipeline in different pressures of reliability, according to the API （American Petroleum Institute） 579 recommended target reliability to determine the operating pressure of the different regions, and design coefficient method, the calculated results are compared, results of calculation reliability is more reasonable and improve the delivery pressure of the area, the pipeline safety management provides the basis.

Limit Boundary Value Problem for Nonlinear Differential Equation of the Second Order

1 .Introduetion Considered in thio PaPer 15 the exi tenoe of the solution of the limit boundaryvalue problem■ which occurs in the study of Physios

Fluctuation Limits of the Single Point Catalytic Super-stable Processes

We prove a fluctuating limit theorem of a sequence of super-stable processes overR with a single point catalyst.The weak convergence of the processes on the space of Schwartz distributions is established.The limiting process is an Ornstein–Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional stable process.

L^6 BOUND FOR BOLTZMANN DIFFUSIVE LIMIT

We consider diffusive limit of the Boltzmann equation in a periodic box.We establish L~6 estimate for the hydrodynamic part Pf of particle distribution function, which leads to uniform bounds global in time.

Estimating EC and ionic EC contribution percentage of nutrient solution based on ionic activity

The dynamic monitoring technology of inorganic ions using ion selective electrodes has some problems such as low precision,vulnerability to other ions,short service life,and high price.Due to the difficulty of dynamic control based on ionic concentration of nutrient solution,EC and pH values of nutrient solution are often used as feedback control indexes in hydroponic system.In this study,estimation algorisms of EC and ionic EC contribution percentage based on ionic activity were proposed to understand the quantitative relationship between ionic concentration and EC.With a view to predicting the EC accurately by mean ionic activities of specific salts in nutrient solution based on a specific formula,ionic concentration could also be calculated by the actual measurement of EC combined with ionic EC contribution percentage.With Japanese horticultural experimental nutrient formula and Yamasaki tomato nutrient formula,significant linear correlations between estimated EC and measured EC were found with determination coefficients over 0.99.Ionic EC contribution percentage was not affected by different relative concentrations of nutrient solutions.However,ionic EC contribution percentage changed significantly when adding specific salts with different concentrations,and different changes were found in each anions and cations of specific salt added.When the same K+concentration was added in different forms of KNO_(3),K_(2_SO_(4),KCl,and KH_(2)PO_(4),the changes of ionic EC contribution percentage of K^(+)were similar,but those of other anions in potassium salts varied greatly.The relative errors of estimated EC of nutrient solutions based on ionic activities were only 1.3%in horticultural experimental nutrient solution and 1.8%in Yamasaki tomato nutrient solution with different relative concentrations compared to measured EC.The relative errors of estimated EC of nutrient solutions with specific salt added were only 0.1%-0.5%compared to measured EC in two nutrient solution.Therefore,the dynamic feedback control of ionic concentration of nutrient solution could be realized by using EC measurement combined with ionic EC contribution percentage to improve the ionic quantitative control in nutrient solution.The EC control of nutrient solution in automatic irrigation system might be upgraded to ionic concentration control by using algorisms above of ionic EC contribution percentage and EC estimation to meet dynamic demands of hydroponic crops for ionic concentration in different growth stages.