We explain the economical meaning of all kinds of parameters,and analyze the extremum condition of the long-run cost function under constant factor production elasticity and factor price.

In this paper, the theorem of Rayleigh quotient is introduced at first, then by use of it, a new rigorous proof for an unconstrained extremum condition is given.

With the method, sub-pixel position of low-light-level strips central point is extracted by Hessian matrix algorithm according to second-order derivative extremum condition of strips image gray scale, which is used for considering two- dimensional variation of stripe intensity profile. Noises during image processing will be eliminated by three feature constraints such as continuity constraint, direction constraint and stripes spacing constraint.

Iterative functionals are constructed based on variation methods combined with Lagrangian multiplier methods,and iterative formulae for solving vibration equations are given after the Lagrangian multiplier is determined by functional extremum condition.

Based on ideal deformation theory, the authors developed a finite element inverse approach for sheet metal forming process simulation. The formulation of finite element equation is derived by the relative extremum condition of the global plastic work of the deforming body.

For the unmodified and chill-modified eutectic microstructures it seems probable that the basic equation is applicable but that the extremum condition is not.

The present measurements fit the extremum condition of the Hunt and Jackson theory of eutectic growth.

We reject the extremum condition with respect to one of the order parameters in spin glass models.

We show that it is possible to introduce an expansion at fixed magnetization amplitude in the inverse of lattice coordination number if the direction is selected by an extremum condition.

When the extremum condition in the thermodynamical potential is imposed, a family of Bogoliubov transformations that give us a planckian spectrum is found, even in "pathological" cases such as the minimally coupled scalar field.

Recently, there present many works to discuss the phase transition-like behaviour of the polaron in polar crystals. However, all these works didn't consider the incorporate effect of the temperature and electron-phonon Frohlich Coupling constant r. In this paper, starting from the Bogolyubov inequality for the free energy, we discuss this incorporate effect on the phase transition-like behaviour of the polaron.The Hamiltonian of a system consisting of the electrons and phonons is shown as (2.1) in the text....

Recently, there present many works to discuss the phase transition-like behaviour of the polaron in polar crystals. However, all these works didn't consider the incorporate effect of the temperature and electron-phonon Frohlich Coupling constant r. In this paper, starting from the Bogolyubov inequality for the free energy, we discuss this incorporate effect on the phase transition-like behaviour of the polaron.The Hamiltonian of a system consisting of the electrons and phonons is shown as (2.1) in the text. And assume that the Bogolyubov ineqality between the real free energy F(H) and the model free energy Fmod(H) also holds for the polaron. The Bogolyubov ineqality has the form (2.4) in the text. In this representation, H can be regarded as the sum of H0 and a perturbed term (H-H0). H0 is represented [by (2.8) which is called the model Hamiltonian. In the course of calculation, we introduce two unitary transformations 171 and U2. Then, applying these transformations to (2.4), we finally obtain the expression (2.27) from which the free energy can be determinded. Lagrange multiplier u in (2.27) has the meaning of the translation velocity of the polaron. The effective mass of the polaron and the mean numbers of phonons N in the cloud around the electron are, respectively, represented by (3.5) and (3.6). The expectation value of the polaron energy with respect to the ground state |0> at the different temperature is expressed by (3.4).From the extremum condition we can futher de-termine the parameters A, ε and λ. Then, inserting these parameters just determined into (3.4), (3.5) and (3.6), we can obtain the ground state energy, the mean phonon numbers and the effective mass. Fig.1 shows the α dependence of the ground state energy when the temperature is smaller than a certain critical value Tc. From this figure, one can see that the ground state energy is linearly dependent on α until the point αc, and then, it is approximately proportional to α2. There are two branches for each curves. One corresponds to small α and has the same rules as that obtained by Lee-Low-Pines approximation. The other begins at the point αc and corresponds to the results of the strong coupling approximation.The crroes-point αc of the two branches is the phase transition point. At the critical point the polaron state changes from a nearly-free-type state to a self-trapped one. In addition, there is also another branch which is represented in the dotted line. This branch corresponds to unstable solution. Fig. 2 and Fig.3 show how the mean phonon numbers and the effective mass vary with the coupling constant α. From these two figures the characteristic feature of the phase transition is most conspicuous. Both the mean phonon numbers and the effective mass have an abrupt change at the point αc.Reversely, when the temperature is larger than Tc, as shown in Fig.5, the rule of the ground state energy variating with α is the same for different temperatures. When α<αc, the ground state energy is equal to zero. As soon, as α arrives at αc, the ground state energy abruptly jumps to a certain value, then, it varies approximately with α2. This means that there only exists free electron state for α<αc, and polaron state for α>αc.αc is the phase transition point. For the different temperature, the phase transition point αc is different. If we plot the phase transition point αc as a function of the temperature T, as shown in Fjg. 8, the curve's shape is very intresting. There is a λ-like point on this (kT, αc) diagram. The temperature corresponding to this Y point is called the critical temperature Tc. The calculation indicates that this λ point is approximately (0.98 hω0, 6). Above the criticaltemperature Tc, a approximately varies with T1/2. on the contrary, when T

The unconstrained extremum conditions are the important parts of optimization theory, and have certain significance in the theory and practice. The optimality conditions for constrained problems become a logical extension of the conditions for unconstrained problems. One strategy for solving a constrained problem is to solve a sequence of unconstrained problems. In this paper, the theorem of Rayleigh quotient is introduced at first, then by use of it, a new rigorous proof for an unconstrained extremum condition...

The unconstrained extremum conditions are the important parts of optimization theory, and have certain significance in the theory and practice. The optimality conditions for constrained problems become a logical extension of the conditions for unconstrained problems. One strategy for solving a constrained problem is to solve a sequence of unconstrained problems. In this paper, the theorem of Rayleigh quotient is introduced at first, then by use of it, a new rigorous proof for an unconstrained extremum condition is given. This proof is more concise than the old one, and it has considerably practical value in learning and grasping the optimization theory for those readers who are not familiar with the theory of limit of sequence.

Based on ideal deformation theory, the authors developed a finite element inverse approach for sheet metal forming process simulation. The formulation of finite element equation is derived by the relative extremum condition of the global plastic work of the deforming body. Iterative algorithms are proposed to determine the initial guess of the inverse approach and the shape of the work piece relevant to given blank shape. Applying the FE inverse approach, the authors predict the blank shape of sheet metal...

Based on ideal deformation theory, the authors developed a finite element inverse approach for sheet metal forming process simulation. The formulation of finite element equation is derived by the relative extremum condition of the global plastic work of the deforming body. Iterative algorithms are proposed to determine the initial guess of the inverse approach and the shape of the work piece relevant to given blank shape. Applying the FE inverse approach, the authors predict the blank shape of sheet metal forming process, calculate the final shape and the strain distribution for a given blank shape. Numerical simulation shows that FE inverse approach can be used for rapid evaluation of sheet metal forming process design and for the optimization of the process parameters.