Supersymmetry, variational method and the Lennard-Jones (12,6) potential

The formalism of supersymmetric quantum mechanics is used to determine trial functions in order to obtain eigenvalues for the Lennard-Jones (12, 6) potential from variational method. The superpotential obtained provides an effective potential which can be directly comparable to the original one.

Inequalities in access to education and healthcare

The burden of disease is borne by those who suffer as patients but also by society at large, including health service providers. That burden is felt most severely in parts of the world where there is no infrastructure, or foreseeable prospects of any, to change the status quo without external support. Poverty, disease and inequality pervade all the activities of daily living in low-income regions and are inextricably linked. External interventions may not be the most appropriate way to impact on this positively in all circumstances, but targeted programmes to build social capital, within and by countries, are more likely to be sustainable. By these means, basic oral healthcare, underpinned by the primary healthcare approach, can be delivered to more equitably address needs and demands. Education is fundamental to building knowledge-based economies but is often lacking in such regions even at primary and secondary level. Provision of private education at tertiary level may also introduce its own inequities. Access to distance learning and community-based practice opens opportunities and is more likely to encourage graduates to work in similar areas. Recruitment of faculty from minority groups provides role models for students from similar backgrounds but all faculty staff must be involved in supporting and mentoring students from marginalized groups to ensure their retention. The developed world has to act responsibly in two crucial areas: first, not to exacerbate the shortage of skilled educators and healthcare workers in emerging economies by recruiting their staff; second, they must offer educational opportunities at an economic rate. Governments need to lead on developing initiatives to attract, support and retain a competent workforce.

q-deformed variational study of the Lipkin-Meshkov-Glick model via coherent states

We show that the ground-state energy of the q-deformed Lipkin-Meshkov-Glick Hamiltonian can be estimated by q-deformed coherent states. We also use these coherent states to analyse qualitatively the suppression of the second order ground-state energy phase transition of this model. © 1993.

Variational iterative method for scattering problems

A parameter-free variational iterative method is proposed for scattering problems. The present method yields results that are far better, in convergence, stability and precision, than any other momentum space method. Accurate result is obtained for the atomic exponential (Yukawa) potential with an estimated error of less than 1 in 1015 (1010) after some 13 (10) iterations.

Higher order turän inequalities

The celebrated Turân inequalities P 2 n(x)-P n-x(x)P n+1(x) ≥ 0, x ε[-1,1], n ≥ 1, where P n(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ 2 n - γ n-1γ n+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the Laguerre-Pölya class, ψ(x) = ∑ ∞ n=0 γ nx n / n!. ©1998 American Mathematical Society.

Bell inequalities for entangled pairs of neutral kaons

We extend the use of Bell inequalities to Φ → K0K̄0 decays by exploiting analogies and differences to the well-known and experimentally verified singlet-spin case. Contrasting with other analyses, our Bell inequalities are violated by quantum mechanics and can strictly be derived from local realistic theories. In principle, quantum mechanics could then be tested using unstable, oscillating states governed by a CP-violating Hamiltonian. © 1999 The American Physical Society.

Morse potential energy spectra through the variational method and supersymmetry

The Variational Method is applied within the context of Supersymmetric Quantum Mechanics to provide information about the energy and eigenfunction of the lowest levels of a Hamiltonian. The approach is illustrated by the case of the Morse potential applied to several diatomic molecules and the results are compared with stabilished results. (C) 2000 Elsevier Science B.V.

Induced variational method from supersymmetric quantum mechanics and the screened Coulomb potential

The formalism of supersymmetric quantum mechanics supplies a trial wave function to be used in the variational method. The screened Coulomb potential is analyzed within this approach. Numerical and exact results for energy eigenvalues are compared.

Variational studies and replica symmetry breaking in the generalization problem of the binary perceptron

We analyze the average performance of a general class of learning algorithms for the nondeterministic polynomial time complete problem of rule extraction by a binary perceptron. The examples are generated by a rule implemented by a teacher network of similar architecture. A variational approach is used in trying to identify the potential energy that leads to the largest generalization in the thermodynamic limit. We restrict our search to algorithms that always satisfy the binary constraints. A replica symmetric ansatz leads to a learning algorithm which presents a phase transition in violation of an information theoretical bound. Stability analysis shows that this is due to a failure of the replica symmetric ansatz and the first step of replica symmetry breaking (RSB) is studied. The variational method does not determine a unique potential but it allows construction of a class with a unique minimum within each first order valley. Members of this class improve on the performance of Gibbs algorithm but fail to reach the Bayesian limit in the low generalization phase. They even fail to reach the performance of the best binary, an optimal clipping of the barycenter of version space. We find a trade-off between a good low performance and early onset of perfect generalization. Although the RSB may be locally stable we discuss the possibility that it fails to be the correct saddle point globally. ©2000 The American Physical Society.

Variational calculation of positronium-helium-atom scattering length

A basis-set calculation scheme for S-waves Ps-He elastic scattering below the lowest inelastic threshold was formulated using a variational expression for the transition matrix. The scheme was illustrated numerically by calculating the scattering length in the electronic doublet state: a=1.0±0.1 a.u.

Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: A variational approach

The soliton propagation in a medium with Kerr nonlinearity and resonant impurities was studied by a variational approach. The existence of a solitary wave was shown within the framework of a combined nonintegrable system composed of one nonlinear Schrödinger and a pair of Bloch equations. The analytical solution which was obtained, was tested through numerical simulations confirming its solitary wave nature.

Variational method for excited states from supersymmetric techniques

We suggest a method for constructing trial eigenfunctions for excited states to be used in the variational method. This method is a generalization of the one that uses a superpotential to obtain the trial functions for the ground state. The construction of an effective hierarchy of Hamiltonians is used to determine excited variational energies. The first four eigenvalues for a quartic double-well potential are calculated for several values of the potential parameter. The results are in very good agreement with the eigenvalues obtained by numerical integration.

Confined systems through variational supersymmetric quantum mechanics

The energy states of the confined harmonic oscillator and the Hulthén potentials are evaluated using the Variational Method associated to Supersymmetric Quantum Mechanics.

Landau and Kolmogoroff type polynomial inequalities II

Let 0 < j < m ≤ n. Kolmogoroff type inequalities of the form ∥f(j)∥2 ≤ A∥f(m)∥ 2 + B∥f∥2 which hold for algebraic polynomials of degree n are established. Here the norm is defined by ∫ f2(x)dμ(x), where dμ(x) is any distribution associated with the Jacobi, Laguerre or Bessel orthogonal polynomials. In particular we characterize completely the positive constants A and B, for which the Landau weighted polynomial inequalities ∥f′∥ 2 ≤ A∥f″∥2 + B∥f∥ 2 hold. © Dynamic Publishers, Inc.

Robust control to parametric uncertainties in smart structures using linear matrix inequalities

The study of algorithms for active vibrations control in flexible structures became an area of enormous interest, mainly due to the countless demands of an optimal performance of mechanical systems as aircraft and aerospace structures. Smart structures, formed by a structure base, coupled with piezoelectric actuators and sensor are capable to guarantee the conditions demanded through the application of several types of controllers. This article shows some steps that should be followed in the design of a smart structure. It is discussed: the optimal placement of actuators, the model reduction and the controller design through techniques involving linear matrix inequalities (LMI). It is considered as constraints in LMI: the decay rate, voltage input limitation in the actuators and bounded output peak (output energy). Two controllers robust to parametric variation are designed: the first one considers the actuator in non-optimal location and the second one the actuator is put in an optimal placement. The performance are compared and discussed. The simulations to illustrate the methodology are made with a cantilever beam with bonded piezoelectric actuators.