Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm

Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

Determining knots by optimizing the bending and stretching energies

For a given set of data points in the plane, a new method is presented for computing a parameter value（knot） for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.

Evaluate More General Integrals Involving Universal Associated Legendre Polynomials via Taylor's Theorem

Abstract A few important integrals involving the product of two universal associated Legendre polynomials Pl＇m＇, （x）,Pk＇n＇（x）and x2a（1-x2）-p-1,xb（1± x）-p-1and xc（1-x2）-p-1（1 ± x）axe evaluated using the operator form of Taylor＇s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e.l＇ ≠ k＇ and m＇≠ n＇ .Their selection rules are a/so given. We also verify the correctness of those integral formulas numerically.

Solutions of Laplace Equation in n-Dimensional Spaces

The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.

Model of the Nerve Impulse with Account of Mechanosensory Processes: Stationary Solutions.

Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.

A Note on the Generalized and Universal Associated Legendre Equations

A class of second-order differential equations commonly arising in physics applications are considered,and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated Legendre Equations are examined and established. The hypergeometric solutions, presented in this work,will promote future investigations of their mathematical properties and applications to problems in theoretical physics.

Dual Lie bialgebra structures of Poisson types

Let A = F [x, y] be the polynomial algebra on two variables x, y over an algebraically closed field F of characteristic zero. Under the Poisson bracket, A is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of A* induced from the multiplication of the associative commutative algebra A coincides with the maximal good subspace of A* induced from the Poisson bracket of the Poisson Lie algebra A. Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products,five classes of new infinite-dimensional Lie algebras are obtained.