Jacobi polynomials and its special forms are all fundamental orthogonal polynomials,these polynomial all have important application in the mathematics-physics question.

It' s proved that the four forms of chemical potential are identical according ot the properties of Jacobi determinant and thermodynamic principle.

Jacobian Determinant and Its Application to Thermodynamics;

With applying Jacobian determinant,the equilibrium and stability conditions CV>0,{P/V}T<0 and other forms in the isolated homogeneous system have been deduced by entropy criterion and internal energy criterion.

The Maxwell relations between the thermodynamic functions were derived with the matrix analysis according to the characteristics of Jacobian determinant in present paper.

The generation and utilization of extrinsic information is deeply analyzed during the iterative decoding process,and some simplification for branch metric calculation is presented in detail with the trait of Jacobian logarithm formula.

A theoretical explanation is given by analyzing the Jacobian logarithm formula and iterative decoding algorithm, and based on the explanation a new T-TCM decoding scheme without SNR estimation is further proposed, with which the T-TCM systems have no performance loss and can be more conveniently applied and easily implemented.

The second order Jacobian J_2 is calculated to determine the delay time T of two-dimensional reconstruction about ordinary differential equation according to the first extreme value of J_2.