FORMULA OF GLOBAL SMOOTH SOLUTION FOR NON-HOMOGENEOUS M-D CONSERVATION LAW WITH UNBOUNDED INITIAL VALUE

In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.

A Formula of General Solution for a Class of Homogeneous Trinomial Recurrence of Variable Coefficients with Two Indices

In practice, it is very difficult to find the solution of recurrence relation by using the characteristic roots. By applying iteration and induction we present an explicit formula of general solution for a class of homogeneous trinomial recurrence of variable coefficients with two indices. It provides a concrete and applicable model to solve the relevant problem with computer.

A Formula of Solution for a Class of Linear Recurence with Two Indices

It is very difficult, sometimes impossible, to get a formula solution of recurrence relation, even for the case of homogeneous recurrence with one indite. In this paper, according to the principle of soluting algebraic equation, we present the formula of solution for a class of recurrnce relations with two indices by appling iteration and induction. It provides a concrete model to solve the concerning problems with modern computing tools.

Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators

This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.

Recursion-transform method and potential formulae of the m×n cobweb and fan networks

In this paper, we made a new breakthrough, which proposes a new recursion–transform（RT） method with potential parameters to evaluate the nodal potential in arbitrary resistor networks. For the first time, we found the exact potential formulae of arbitrary m × n cobweb and fan networks by the RT method, and the potential formulae of infinite and semi-infinite networks are derived. As applications, a series of interesting corollaries of potential formulae are given by using the general formula, the equivalent resistance formula is deduced by using the potential formula, and we find a new trigonometric identity by comparing two equivalence results with different forms.

Adjusting the nutrient solution formula based on growth stages to promote the yield and quality of strawberry in greenhouse

To optimize the nutrient supply of strawberry,fertilization was reduced as much as possible on the premise of satisfying the elements needed for strawberry growth.This study put forward a modified nutrient solution formula(NM)by adjusting the ratios of N/K,K/Ca,NO_(3)^(-)-N/NH_(4)^(+)-N in vegetative growth,flowering and fruiting,and fruit expanding stages based on Yamasaki strawberry nutrient solution formula(NY).Taking the NY as the control group,the NM was verified with elevated-grown‘Ssanta’strawberry using three different substrates(Vulcanics-added substrate,Horticultural substrate,Comme rcial substrate)in greenhouse.The results indicated that under the NM treatment,strawberry plants had better photosynthesis in the vegetative growth stage and flowering and fruiting stage,and the first crop yield and total crops yield per plant also increased by 20%.Although the vegetative growth stage was longer,it entered the full fruiting stage earlier so that the maturity stage was advanced one week.Compared to strawberries grown in horticultural substrate irrigated with NY,the fruit yield per plant of‘Ssanta’strawberry grown in horticultural substrate irrigated with NM increased by 26%,and the sugar-acid ratio and vitamin C content of primary fruits reached 10.6 and 74.1 mg/100g,improving by 41%and 34%respectively.The NM based on the growth stage can optimize the element needs of strawberry in all growth and development stages and maintain the substrate’s pH stability,improving the yield and quality of‘Ssanta’strawberry under substrate cultivation.Therefore,it is recommended to use the horticultural substrate irrigated with the modified nutrient solution for the‘Ssanta’strawberries cultivation elevated in greenhouse to improve the economic benefit of strawberry cultivation.

Smooth Solution of Multi-dimensional Nonhomogeneous Conservation Law: Its Formula, and Necessary and Sufficient Blowup Criterion

In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES ON AN ELASTIC FOUNDATION

On the basis of von Karnmnequations, the axisymmetric buckling and post-buckling of annular plates on anelastic foundation is so wematically discussed byusing shooting

A General Solution for Trinomial Linear Recurrence with Two Indices

In this paper, we present the explicit formula of general solution for a class of homogeneous trinomial recurrence of variable coefficients with two indices by applying iteration and induction. It provides a concrete and applicable model to solve the relevant problem with computer.

A General Solution for a Class of Non-Homogeneous Recurrence Relations with Two Indices

In this paper,we present the formula of a solution for a class of recurrence relations with two indices by applying iteration and induction according to the principle of solving algebraic equa-tions. It provides a concrete model to solve the concerned prob-lems with modern computing tools.