ON THE GROWTH OF SOLUTIONS OF HIGHER-ORDER ALGEBRAIC DIFFERENTIAL EQUATIONS

Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.

Insulin like growth factor-1 increases fatty liver preservation in IGL-1 solution

AIM: To investigate the benefits of insulin like growth factor-1 (IGF-1) supplementation to serum-free institut georges lopez-1 (IGL-1) solution to protect fatty liver against cold ischemia reperfusion injury. METHODS: Steatotic livers were preserved for 24 h in IGL-1  solution supplemented with or without IGF-1 and then perfused "ex vivo " for 2 h at 37℃. We examined the effects of IGF-1 on hepatic damage and function (transaminases, percentage of sulfobromophthalein clearance in bile and vascular resistance). We also studied other factors associated with the poor tolerance of fatty livers to cold ischemia reperfusion injury such as mitochondrial damage, oxidative stress, nitric oxide, tumor necrosis factor-α (TNF-α) and mitogen-activated protein kinases.RESULTS: Steatotic livers preserved in IGL-1 solutionsupplemented with IGF-1 showed lower transaminase levels, increased bile clearance and a reduction in vascular resistance when compared to those preserved in IGL-1solution alone. These benefits are mediated by activation of AKT and constitutive endothelial nitric oxide synthase (eNOS), as well as the inhibition of inflammatory cytokines such as TNF-α. Mitochondrial damage and oxidative stress were also prevented.CONCLUSION: IGL-1  enrichment with IGF-1 increasedfatty liver graft preservation through AKT and eNOS activation, and prevented TNF-α release during normothermic reperfusion.

A steady solution of the gasar eutectic growth in directional solidification

This paper presents the general mathematical model on gasar eutectic growth in directional solidification. Using multiple scale expansion and matching method, we obtain the global steady solution of gasar eutectic growth as the Peclet number ε≤1, where ε is defined as the ratio of half an inter-pore spacing and solutal diffusion length. We also give the interfacial shape and predict the porosity of gasar eutectic growth. Results show that porosity is mainly dependent on gas pressure above the metal melt, but independent of pulling velocity. Our predicted results are in agreement with experimental data.

Model of apparent crystal growth rate and kinetics of seeded precipitation from sodium aluminate solution

Based on the population balance equation in a batch crystallizer characteristic of seeded precipitation, a model to calculate the rate of apparent crystal growth of aluminum hydroxide from the size distribution was deve- loped. The simulation results indicate that the rate of apparent crystal growth during seeded precipitation exhibits a manifest dependence on the crystal size. In general, there is an obvious increase in the apparent crystal growth rate with the augment in crystal size. The apparent activation energy increases with the increase of characteristic crystal size, which indicates that the growth of small crystals is controlled by surface chemical reaction; it is gradually controlled by both the surface reaction and diffusion with the augment in crystal size.

THE GROWTH OF SOLUTIONS TO HIGHER ORDER DIFFERENTIAL EQUATIONS WITH EXPONENTIAL POLYNOMIALS AS ITS COEFFICIENTS

By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.

GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A FR ACTIONAL SCHRODINGER-POISSON SYSTEM WITH CRITICAL GROWTH

We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.

Adjusted Nutrition Solution Formula Improves the Growth and Yield of Soilless Tomato Culture in Southern Xinjiang

The tomato production facilities in southern Xinjiang have unique climatic conditions.However,due to the high salinity and degree of mineralization of the soil and the extensive management of irrigation water sources,the tomato production efficiency is low,and there is still much room for improvement.In this study,the formulas of tomato nutrient solutions were adjusted according to the local irrigation water quality characteristics,and tomato was grown in a greenhouse using sand cultivation.To select a nutrient solution formula suitable for the tomato cultivated in the local facilities,various parameters of tomato treated with different formulas were compared.The results showed the following:①Adjusting the nutrient solution formula could effectively improve the vegetative and reproductive growth indexes of tomato.②Properly decreasing the nutrient solution concentration could improve the growth indexes of tomato.③Decreases in Ca^(2+) and Mg^(2+) in the nutrient solution did not affect the growth and yield of tomato.The possible reason for the third result was that excessive cations in the water inhibited the absorption of Ca^(2+) and Mg^(2+) in the nutrient solution and had antagonistic effects.Therefore,a high nutrient solution concentration does not necessarily promote the growth of tomatoes.In summary,this study will improve the production conditions of protected tomato in the southern Xinjiang area by supporting tomato nutrient solution adjustments according to actual conditions in combination with reasonable irrigation systems and scientific management.

Existence and Uniqueness of Almost Periodic Solution for a Mathematical Model of Tumor Growth

This article is concerned with a mathematical model of tumor growth governed by 2nd order diffusion equation . The source of mitotic inhibitor is almost periodic and time-dependent within the tissue. The system is set up with the initial condition C(r, 0) = C0(r) and Robin type inhomogeneous boundary condition . Under certain conditions we show that there exists a unique solution for this model which is almost periodic.

A PRIORI ESTIMATES TO THE MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH ANISOTROPIC GROWTH CONDITIONS

In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.

Kinetics of crystal growth on seeded precipitation of sodium aluminate solutions with new device

A new device was designed,which can effectively avoid the undesired nucleation and agglomeration of fine particles on the experimental results during the sead ed precipitation of sodium aluminate solution,and moreover,the experimental co nditions are nearly kept constant during the experiment. With the new device,it is proven that a good result can be obtained on the kinetics study of the cryst al growth in seeded precipitation of sodium aluminate solution.Experiments were carried out with the concentration of Na2O （Nk）170 g/L,the mole ratio of Na2O to Al2O3 （αk） all between 1.52 to 2.01,at 65,70,75 ℃,respectively. And the kinetics equation of crystal growth of gibbsite was deduc ed.

ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C. YANG

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less

Solute distribution in KNbO 3 melt-solution and its effect on dendrite growth during rapid solidification

This paper reports that the rapid solidification of mixed Li2B4O7 and KNbO3 melted in a Pt loop heater has been performed experimentally by the method of quenching, and various morphologies of KNbO3 crystals have been observed in different regions of the quenched melt-solution. Dendrites were formed in the central region where mass transfer is performed by diffusion, whereas polygonal crystals with smooth surface grew in the marginal region where convection dominates mass transport. Based on measurement of KNbO3 concentration along crystal interface by electronic probe analysis, it finds the variety of crystal morphologies, which is the result of different solute distributions： in the central region the inhomogeneity of solute concentration is much sharper and morphological instability is easier to take place; nevertheless in the marginal region the concentration homogeneity has been greatly enhanced by convection which prevents the occurrence of morphological instability. Additional solute distribution in the melt along the primary dendrite trunk axis as well as that in mushy zones has also been determined. Results show that the solute concentration in the liquid increases linearly with distance from the trunk tip and more solutes were found to be concentrated in mushy zones. The closer the mushy zone is to trunk tip, the lower the solute concentration will be there.

GROWTH OF MEROMORPHIC SOLUTIONS OF SOME ALGEBRAIC DIFFERENTIAL EQUATIONS

In this paper, by means of the normal family theory, we estimate the growth order of meromorphic solutions of some algebraic differential equations and improve the related result of Barsegian et al. [6]. We also give some examples to show that our results occur in some special cases.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH CRITICAL GROWTH

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF PARABOLIC EQUATIONS WITH NATURAL GROWTH TERMS AND L^1 DATA

We study a class of nonlinear parabolic equations of the type：δb（u）/δt-div（a（x,t,u）△u）＋y（u）｜△u｜^2=f,where the right hand side belongs to L^1（Q）, b is a strictly increasing C^1-function and -div（a（x, t, u）△u） is a Leray-Lions operator. The function g is just assumed to be continuous on R and to satisfy a sign condition. Without any additional growth assumption on u, we prove the existence of a renormalized solution.

SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR SCHRODINGER-POISSON SYSTEM WITH CRITICAL GROWTH

In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.

Stability of Crystal Growth Face and Dissolution Face in Crystallization from Solution under Microgravity

The stability of the shapes of crystal growth face and dissolution face in a two-dimensional mathematical model of crystal growth from solution under microgravity is studied. It is proved that the stable shapes of crystal growth face and dissolution face do exist, which are suitably shaped curves with their upper parts inclined backward properly.The stable shapes of crystal growth faces and dissolution faces are calculated for various values of parameters, Ra, Pr and Sc. It is shown that the stronger the convection relative to the diffusion in solution is, the more backward the upperparts of the stable crystal growth face and dissolution face are inclined. The orientation and the shape of dissolution face hardly affect the stable shape of crystal growth face and vice versa.

The Growth Orders of Meromorphic Solutions of Differential Equations

By meas of the Nevanlinna theory of the value distribution of meromorphic functions, this paper discusses the orders of growth of meromorphic solutions of differential equation and proves that the form of the solution is determined if the order are sufficiently large.

Behavior of the Solutions for the Chemotaxis Equations with Saturation Growth

In this paper,we study the behavior of the solutions to a kind of chemotaxis equations that

A Property of Solutions for Elliptic Equation with Nonstandard Growth Conditions in Unbounded Domain

In this paper the author proves that the Phragmen Lindelof principle holds for solutions of elliptic equation (1) with nonstandard growth conditions.