Trace formula of the integro-differential operator on a quantum graph

In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.

Study on Technology and Formula of Walnut-Roxburgh Rose Beverage

[Objective] The study aimed to explore technology and formula of walnut- roxburgh rose beverage. [Method] With walnut and roxburgh rose as the main raw material, technology and formula of walnut-roxburgh rose beverage was study by or- thogonal test. In addition, effects of emulsion stabilizers were investigated. [Result] The great emulsifying effect could be achieved by composite emulsion stabilizer at 0.40%. The orthogonal test showed that the optimum formula of walnut-roxburgh rose beverage was composed of 24% walnut milk, 16% cili juice and 12% sugar, with pH of 4.0. The product with stable organization state, pleasant sweetness and sourness may be prepared. [Conclusion] The study laid the foundation for production of walnut-roxburgh rose beverage.

Optimal Formula of Angelica sinensis Ameliorates Memory Deficits in β-amyloid Protein-induced Alzheimer's Disease Rat Model

Objective:Angelica(A.)sinensis is used as a traditional medical herb for the treatment of neurodegeneration,aging,and inflammation in Asia.A.sinensis optimal formula(AOF)is the best combination in A.sinensis that has been screened to rescue the cognitive ability in P-amyloid peptide(Ap25-35)-treated Alzheimer’s disease(AD)rats.The objective of this study was to investigate the effect of AOF on the learning and memory of AD rats as well as to explore the underlying mechanisms.

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

A Tentative Study on the Formula of English Compliment

Praising or compliment is a communicative behavior shared by people of different languages. However, due to the influence of unique culture and different society, the language of compliment differs greatly. The author of this essay is trying to analyze the formula of English compliment shared by all the English speaking nations, esp. in the United States.

An available formula of the sandy beach state induced by plunging waves

Laboratory experiments are performed to explore the response rule of a sandy beach profile under plunging wave on a non-uniform sediment-bed slope. The initial beach slope of combination of 1/10 and 1/20 is exposed to regular waves and cnoidal waves respectively. The free surface elevation, process of wave propagation, wave breaking, uprush and backwash and the change of a cross-shore beach profile are measured and recorded. The beach profile under the regular waves action exhibits two parts： a sandbar profile and a beach berm profile, and only one typical profile transformation under the cnoidal waves action is obtained, which is the beach berm profile. In the laboratory experiments, it is found that the beach states under wave action related to the previous factors. In addition, they are related to the characteristic of breaking waves such as the breaking intensity of the plunging wave. A concept about the characteristic angle of the plunging wave has been put forward through the observation and analysis of the phenomenon of the laboratory experiment. A qualitative analysis about the sediment transport carrying by currents generated from the plunging wave and the state of beach profile under the wave action has been done. The quantitative analysis about the relationship between the characteristic angle and Irribarren number has been done. An available formula of equilibrium states for the sandy beach induced by the plunging wave has been established based on the relationship between Irribarren number and the beach profile. By fitting these experimental results and others＇ experimental results to three lines, the three fitting coefficients can be calculated in their formula respectively. The recommended empirical formulas can divide three states of a beach morphology profile obviously, which include a depositive beach, an erosive beach and an intermediate beach.

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.

THE ASYMPTOTIC FORMULA OF THE ORTHOGONAL RATIONAL FUNCTION ON THE UNIT CIRCLE

In this paper we obtained the asymptotic formula of the orthogonal rational function on the unit circle with respect to the weight function μ(z) with preasigned poles, which are in the exterior of the unit disk.

Energy average formula of photon gas rederived by using the generalised Hermann-Feynman theorem

By virtue of the generalised Hermann-Feynmam theorem we re-derive the energy average formula of photon gas. This is another useful application of the theorem.

A New Loop Algebra and Corresponding Computing Formula of Constant γ in Quadratic-Form Identity

A new loop algebra containing four arbitrary constants is presented, -whose commutation operation is concise, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to computing formula of constant γ in the trace identity. As application, a new Liouville integrable hierarchy, which can be reduced to AKNS hierarchy is derived.

ON THE MINIMUM COVERING NUMBER FORMULA OF GRAPHS

Throughout this paper,D=（d1,d2,...,dn）denote a sequence of nonnegative inte-gers.We let（?）（D）denote the class of all graphs with degree sequence D,or equally,theclass of all symmetric（0,1）--matrices with trace 0 and row sum vector D.The structure matrix S=S（D） of D is a matrix of order n+1,whose entries

AN ASYMPTOTIC EXPANSION FORMULA OF KERNEL FUNCTION FOR QUASI FOURIER-LEGENDRE SERIES AND ITS APPLICATION

Is this paper we shall give cm asymptotic expansion formula of the kernel functim for the Quasi Faurier-Legendre series on an ellipse, whose error is 0(1/n2) and then applying it we shall sham an analogue of an exact result in trigonometric series.

Corrected Formula of Bed Resistance Coefficient for Plane Numerical Simulation of Tidal Current

By means of a logarithm law for the velocity profile, a corrected formula of bed resistance coefficient, which involves many factors such as gradient of still water depth, variation of surface elevation, flow direction, and so on, is derived from the 3D governing equations of tidal current by averaging over the whole water depth. Theoretical analysis and application have shown that the 2D plane tidal current numerical model would be more reasonable and could be applied to steep bottom topography when the corrected bed resistance coefficient is used, therefore the results of reproduction simulation and engineering calculation would be more scientific and reasonable.

Simple Formula of β^+-Decay Half-Lives of Nuclei Far From β-Stable Line

We generalize the formula of β^+-decay half-lives [Phys.Rev.C 78 (2006) 014305] to include all availabledata of nuclei far from the β-stable line. Good agreement with experimental data is obtained by taking into accountthe even-odd (e-o) effect.It is shown that the e-o effect on β^+-deeay half-life is apparent for the allowed β^+-transition,whereas it is not apparent for the first and second forbidden transitions.This demonstrates that the allowed β^+-transitionis more sensitive to the nuclear even-odd effect than the forbidden β^+-transition.The underlying physics is discussed.This formula can be used to predict the β^+-decay half-lives of the nuclei far from β-stability.It is useful to experimentalphysicists for analyzing the data of β^+-decay.

Unifying the theory of integration within normal-,Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators （which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases）. The s-ordered operator expansion （denoted by s…s ） formula of density operators is derived, which isρ=2/1-s∫d^2β/π〈-β｜ρ｜β〉sexp{2/s-1（s｜β｜^2-β＊α＋βa-αα）}s The s-parameterized quantization scheme is thus completely established.

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C^（1） smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.

Formula of Overtopping Discharge of Vertical Seawall with A Form Similar to Solitary Wave Function

An empirical formula to predict overtopping discharge of vertical wall is presented,in which an expression similar to the solitary wave function is proposed to describe the rule of the influence of relative water depth.The formula is derived from performing an investigation to the well-known overtopping graphs of Goda,and for the sake of interest,the process of the derivation is detailed.To make clear the formula’s performance,relevant test datasets in the CLASH database are extracted to examine the error levels of the formula.As a result,an overall good agreement has been found between the predictions of the formula and the extracted datasets in a wide range outside the extreme shallow region,i.e.,the range of relative water depth smaller than 0.6,in which the water depth near the wall is so shallow that nearly all the incoming waves have been broken before reaching the wall.

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 Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda＇s Method and Its Limitation

Since the roll damping of ships has significant effects of viscosity, it is difficult to calculate it theoretically. Therefore, experimental results or some prediction methods are used to get the roll damping in design stage of ships. Among some prediction methods, Ikeda＇s one is widely used in many ship motion computer programs. Using the method, the roll damping of various ship hulls with various bilge keels can be calculated to investigate its characteristics. To calculate the roll damping of each ship, detailed data of the ship are needed to input. Therefore, a simpler prediction method is expected in primary design stage. Such a simple method must be useful to validate the results obtained by a computer code to predict it on the basis of Ikeda＇s method, too. On the basis of the predicted roll damping by Ikeda＇s method for various ships, a very simple prediction formula of the roll damping of ships is deduced in the present paper. Ship hull forms are systematically changed by changing length, beam, draft, mid-ship sectional coefficient and prismatic coefficient. It is found, however, that this simple formula can not be used for ships that have high position of the center of gravity. A modified method to improve accuracy for such ships is proposed.