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.     

Universal Associated Legendre Polynomials and Some Useful Definite Integrals

We first introduce the universal associated Legendre polynomials, which are occurred in studying the non-central fields such as the single ring-shaped potential and then present definite integrals I_A^±(a, τ)=∫_-1⁺¹x^a[P_l'^m'(x)]²/(1±x)^τdx, a=0, 1, 2, 3, 4, 5, 6, τ=1, 2, 3, I_B(b, σ)=∫_-1⁺¹x^b[P_l'^m'(x)]²/(1- x²)^σdx, b=0, 2, 4, 6, 8, σ=1, 2, 3, and I_C^±(c, κ)=∫_-1⁺¹x^c[P_l'^m'(x)]²/[(1-x²)^κ(1±x)]dx, c=0, 1, 2, 3, 4, 5, 6, 7, 8, κ=1, 2. The superindices “±” in I_A^±(a, τ) and I_C^±(c, κ) correspond to those of the factor (1±x) involved in weight functions. The formulas obtained in this work and also those for integer quantum numbers l' and m' are very useful and unavailable in classic handbooks.     

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

A few important integrals involving the product of two universal associated Legendre polynomials P_(l′)~(m′)(x),P_k~n′~′(x)and x~(2a)(1-x~2)~(-p-1),x~b(1±x)~(-p-1)and x~c(1-x~2)~(-p-1)(1±x)are 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 also given.We also verify the correctness of those integral formulas numerically.     

A Note on the Rarita-Schwinger Equations

Assuming that the background spacetime is a solution of the Einstein vacuum equations without cosmological constant, we analyze how the Rarita-Schwinger equations can be obtained via a particular generalization of the usual spin-3/2 massless free field equations. On the basis of this analysis we speculate on the possibility of finding other generalizations of the Rarita-Schwinger equations.     

A Note on the Quantization of Constrained Generalized Dynamics

It is proved that the canonical quatization starting from a singular first-order Lagrangian yields the same physical results as a second-order Lagrangian which differs from the original singular Lagrangian by a total-time derivative term for a system with finite degrees of freedom. A typical example is given. For the case in the field theories a brief discussion is given and an example is illustrated.     

On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1-x2) and Their Byproducts

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials P_l'^m'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1-x²)^-p-1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.     

A Note on Similarity Solutions of Navier-Stokes Equations

This note looks at the two similarity solutions of the Navier Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.     

A Note on Stress-Tensors,Conservation and Equations of Motion

Some unusual relations between stress tensors, conservation and equations of motion are briefly reviewed. This work was supported by the National Science Foundation under grants PHY99-73935:04-01667.     

On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree

Formulas for calculating Legendre functions and associated Legendre functions of the first kind of complex degree using an Ermakov—Lewis invariant are presented. These formulas are straight-forward to implement numerically and are motivated by the lack of computational routines in standard university tools like those of MatLab and Maple. Angular waves propagating in opposite directions are also obtained. The results are particularly useful in complex angular momentum theories and nearside/farside analysis of spin-dependent angular scattering from central potentials.     

On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree

Formulas for calculating Legendre functions and associated Legendre functions of the first kind of complex degree using an Ermakov–Lewis invariant are presented. These formulas are straight-forward to implement numerically and are motivated by the lack of computational routines in standard university tools like those of Mat Lab and Maple.Angular waves propagating in opposite directions are also obtained. The results are particularly useful in complex angular momentum theories and nearside/farside analysis of spin-dependent angular scattering from central potentials.     

A Note on the Relationship Between Genuinely Coherence and Generalized Entanglement Monotones

International Journal of Theoretical Physics - We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for...     

Generalized Lie Symmetries and the Integrability of Generalized Emden-Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden-Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden-Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.     

A New Hopf Algebra and the Universal R-Matrix Associated with It

We present a new Hopf algebra and following Drinfeld's quantum double construction, we obtain an explicit formula for the universal R-matrix associated with it.     

A Note on the Equivalence of Methods to finding Nonclassical Determining Equations

In this note we prove that the method of Bîlã and Niesen to determine nonclassical determining equations is equivalent to that of Nucci’s method with heir-equations and thus in general is equivalent to using an appropriate form of generalised conditional symmetry.     

A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.     

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

It is known that the solution to a Cauchy problem of linear differential equations: $$x'(t)=A(t)x(t), \quad {with}\quad x(t_0)=x_0,$$ can be presented by the matrix exponential as $\exp({\int_{t_0}^tA(s)\,ds})x_0,$ if the commutativity condition for the coefficient matrix $A(t)$ holds: $$\Big[\int_{t_0}^tA(s)\,ds,A(t)\Big]=0.$$ A natural question is whether this is true without the commutativity condition. To give a definite answer to this question, we present two classes of illustrative examples of coefficient matrices, which satisfy the chain rule $$ \frac d {dt}\, \exp({\int_{t_0}^t A(s)\, ds})=A(t)\,\exp({\int_{t_0}^t A(s)\, ds}),$$ but do not possess the commutativity condition. The presented matrices consist of finite-times continuously differentiable entries or smooth entries.     

Note on the Time Reversal Symmetry of Equations of Motion

Rohrlich's recent claim that the equation of motion for a point charge is symmetric under time reversal is shown to be the result of an inappropriate definition. The equation of motion for a charged sphere of finite size, which is claimed, in contrast, to be asymmetric because of the finite propagation time of its (retarded) self-forces, is shown to possess the same asymmetry (or the same symmetry, depending on the definition) as that for a point charge. The arguments apply similarly to other equations of motion.     

Einstein's Equations and Associated Linear Systems

The relationship between Einstein's field equations and classical higher spin field equations is investigated using two-component spinor valued differential forms. Linear systems of equations associated to both the vacuum and coupled gravitational matter field equations are constructed. The latter equations are shown to be the integrability conditions of the linear systems.     

A Generalized Representation Formula for Systems of Tensor Wave Equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433, 2007) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic Equ. 4(3):401–433, 2007) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in (J. Amer. Math. Soc. 23(2):345–382, 2009) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.