Maximal Reduced Form Research Articles

Analytic transition matrix for the Manning-Rosen potential in all partial waves

The off-shell Jost and physical solutions of the Manning-Rosen potential are derived by adapting two different approaches to the problem. In contrast to the semi-classical approaches to the problem we construct exact analytical expression of the off-shell transition matrix for the Manning-Rosen potential in all partial waves by adapting pure quantum mechanical method. Our constructed expressions are in their maximal reduced form and consistent with respect to their limiting behaviors. The off-shell transition matrix satisfies the unitarity relation for positive energy. To authors’ knowledge, these results for the off-shell quantities have not yet been discussed in the literature.

THE ANALYTIC T-MATRIX FOR THE HULTHÉN POTENTIAL IN ALL PARTIAL WAVES

Analytical expression for the Hulthén off-shell T -matrix in all partial waves is constructed in its maximal reduced form. As a basic requirement the off-shell solutions and Jost function are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics . The off-shell T -matrix is found to be in order with respect to its limiting behaviours and off-shell unitarity.

Nucleon–nucleus inelastic scattering by Manning–Rosen distorted nonlocal potential

Within the framework of non-relativistic quantum scattering theory, we treat the charged hadron scattering by replacing the nuclear interaction by a separable nonlocal one and the electromagnetic part by the Manning–Rosen potential. The off-energy-shell scattering is rigorously studied by this additive interaction by including the effect of electromagnetic interaction. The exact analytical expressions for the off-shell solutions and half-shell T-matrix are obtained in maximal reduced form. The half-shell T-matrix for the proton–oxygen system is computed and the resultant phase shifts are found in order.

The Hankel Transform of the Hulthén Green’s Function

It is shown that the Hankel transform of the s-wave Hulthen physical Green’s function satisfies a second-order differential equation. This equation is solved by applying the proper boundary conditions in association with the properties of the special functions of mathematics to get a closed form expression for the same. The Hankel transform of the physical Green’s function is exploited to extract off-shell solutions and Half- and off-shell T-matrices in the maximal reduced form. The check on our expressions with particular emphasis on their limiting behaviour is made and is found in order. The bound state spectrums of n–p and $$\hbox {n-C}^{12}$$ systems are computed by exploiting the associated Jost function and found excellent agreement with experimental results.

Off-shell Jost function for the Hulthén potential in all partial waves

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.

On the Alpha-Carbon-12 Elastic Scattering

A closed form expression for the Fredholm determinant is constructed for motion in Coulomb-distorted nonlocal separable potential and expressed it in its maximal reduced form. We demonstrate the usefulness of our constructed expression by means of a model calculation.

Off-Shell Jost Solution for the Hulthén Potential

A closed form expression for the off-shell Jost solution is constructed for motion in the Hulthen potential. This is derived via two different approaches to the problem namely; differential equation approach and the integral transform method. The off-shell quantities are expressed in their maximal reduced form.

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrodinger equation. Using the particular integral of the inhomogeneous Schrodinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.

Integral transform of the Coulomb Green's function by the Hankel function and off-shell scattering

In the representation space approach a useful analytical expression for the integral transform of the Coulomb Green's function by the Hankel function is constructed via Sturmian representation of the bound-state Coulomb Green's function. This integral transform is exploited to construct off-shell Jost solutions for motion in Coulomb and Coulomb plus separable interactions in the maximal reduced form. The expressions for the corresponding off-shell $T$ matrices are also constructed by using a modified relation between the off-shell physical solution and the $T$ matrix that does not involve the potential explicitly. Finally, off-shell $T$ matrices are computed to examine the role of the Coulomb interaction in proton-proton scattering in the ${}^{1}{S}_{0}$ channel.

Extracting conflict-free information from multi-labeled trees

BackgroundA multi-labeled tree, or MUL-tree, is a phylogenetic tree where two or more leaves share a label, e.g., a species name. A MUL-tree can imply multiple conflicting phylogenetic relationships for the same set of taxa, but can also contain conflict-free information that is of interest and yet is not obvious.ResultsWe define the information content of a MUL-tree T as the set of all conflict-free quartet topologies implied by T, and define the maximal reduced form of T as the smallest tree that can be obtained from T by pruning leaves and contracting edges while retaining the same information content. We show that any two MUL-trees with the same information content exhibit the same reduced form. This introduces an equivalence relation among MUL-trees with potential applications to comparing MUL-trees. We present an efficient algorithm to reduce a MUL-tree to its maximally reduced form and evaluate its performance on empirical datasets in terms of both quality of the reduced tree and the degree of data reduction achieved.ConclusionsOur measure of conflict-free information content based on quartets is simple and topologically appealing. In the experiments, the maximally reduced form is often much smaller than the original tree, yet retains most of the taxa. The reduction algorithm is quadratic in the number of leaves and its complexity is unaffected by the multiplicity of leaf labels or the degree of the nodes.

Off-shell Jost solutions for Coulomb and Coulomb-like interactions in all partial waves

By exploiting the theory of ordinary differential equations, with judicious use of boundary conditions, interacting Green's functions and their integral transforms together with certain properties of higher transcendental functions, useful analytical expressions for the off-shell Jost solutions for motion in Coulomb and Coulomb-nuclear potentials are derived in maximal reduced form through different approaches to the problem in the representation space. The exact analytical expressions for the off-shell Jost solutions for Coulomb and Coulomb-like potentials are believed to be useful for the description of the charged particle scattering/reaction processes.

An integral transform of Green’s function, off-shell Jost solution and T-matrix for Coulomb-Yamaguchi potential in coordinate representation

By exploiting the theory of ordinary differential equations together with certain properties of higher transcendental functions, a useful analytical expression for the integral transform of the Green’s function for motion in Coulomb-Yamaguchi potential is derived via the r-space approach. This integral transform is applied to construct an analytical expression for off-shell Jost solution in its ‘maximal reduced form’ involving confluent and Gaussian hypergeometric functions. Corresponding Jost functions automatically follow from this solution. Finally, as another application of the off-shell Jost solution, the off-shell T-matrix is calculated by using a modified relation between off-shell physical wave function and T-matrix which does not involve the potential explicitly, thereby avoiding certain difficult integrals, and expressed it in terms of rational functions and simple hypergeometric functions which is in exact agreement with the results given previously by other authors.

The set of primes: Towards an optimized algorithm, prime generation and validation, and asymptotic consequences

Starting from the result that the prime distribution is deterministic, we show in its maximal reduced form, the set of prime numbers P. This set can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The asymptotic behaviour is considered too. We go head towards an optimized algorithm to generate primes, whose computational complexity is C ( n ) ∈ O ( n ) . In addition a pre-computed algorithm is also considered and its computational complexity C ( n ) ∈ O ( 1 ) .

On the integral representations of the Jost function and Coulomb off-shell Jost solution

The integral representations of the Jost function (on- and off-shell) are rederived by the judicious use of the transposed operator relation on the particular integrals for Jost solution and using one of these particular integrals an analytical expression for the Coulomb off-shell Jost solution is presented in the maximal reduced form.

Off-shell Jost solution for a Coulomb-like potential

The theory of ordinary differential equations together with judicious use of boundary conditions and certain properties of higher transcendental functions is exploited to derive a useful analytical expression for the Coulomb-Yamaguchi Jost solution through an $r$-space approach to the problem. Note that the off-shell Jost solution is expressed in its maximal reduced form involving confluent and Gaussian hypergeometric functions. As an application of the Jost solution the off-shell $T$ matrix is also expressed in terms of Gaussian hypergeometric functions.

Half-shellT matrix for Coulomb-modified Graz separable potential

We construct a closed form expression for the off-shell Jost function for scattering by the Coulomb-distorted Graz separable potential and express it in the ‘maximal reduced form’. Our result is particularly suitable for numerical computation. We present a case study in support of this and examine the role of Coulomb interaction in thep — p half-shell scattering in the1S0 channel.