An embedding of the Bannai–Ito algebra in $$\mathscr {U}(\mathfrak {osp}(1,2))$$ and $$-1$$-1 polynomials

An embedding of the Bannai–Ito algebra in the universal enveloping algebra of \(\mathfrak {osp}(1,2)\) is provided. A connection with the characterization of the little \(-1\) Jacobi polynomials is found in the holomorphic realization of \(\mathfrak {osp}(1,2)\). An integral expression for the Bannai–Ito polynomials is derived as a corollary.     

Correlations between surface and interface energies with respect to crystal nucleation

The energetic chemical and structural properties of interfaces between solid and liquid metals are of great interest in numerous applications. However, solid-liquid interfacial energies of metals are often determined by nucleation experiments, requiring particular attention to the measurements so obtained. The purpose of this paper is to conduct an analysis of the level of liquid undercooling of 3d-, 4d-, and 5d-transition metals, as well as of other common undercooled elements, through a critical survey of thermophysical data and experimental results. As already pointed out for the liquid-vapor surface energy sigma(LV), the solid-liquid interface energy sigma(LS) determined from the maximum amount of liquid undercooling is well connected to the position of the element in the periodic table, leading to the individualization of each behavior, including the size of the critical nucleus. The beta ratio of sigma(LS)/sigma(LV) is demonstrated to be an interesting dimensionless number by which to classify the elements into distinctive groups. No significant unexplained anomaly is identified, except for Co, supporting the belief that Turnbull's classical theory furnishes a robust support to describe the crystal nucleation in pure elements.     

Hidden symmetries and accidental degeneracy for a spin-1/2 particle in the field of a dyon

We show that the recently discovered hidden symmetries for the motion of a spin-1/2 particle in the field of a dyon completely account for the accidental degeneracy of the spectrum.     

On the Discretization of the Coupled Integrable Dispersionless Equations

We study the integrable discretization of the coupled integrable dispersionless equations. Two semi-discrete version and one full-discrete version of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.     

How to prepare and stabilize very small nanoemulsions

Practical and theoretical considerations that apply when aiming to formulate by ultrasonication very small nanoemulsions (particle diameter up to 150 nm) with very high stability are presented and discussed. The droplet size evolution during sonication can be described by a monoexponential function of the sonication time, the characteristic time scale depending essentially on the applied power. A unique master curve is obtained when plotting the mean diameter size evolution as a function of sonication energy. We then show that Ostwald ripening remains the main destabilization mechanism whereas coalescence can be easily prevented due to the nanometric size of droplets. The incorporation of "trapped species" within the droplet interior is able to counteract Ostwald ripening, and this concept can be extended to the membrane compartment. We finally clarify that nanoemulsions are not thermodynamically stable systems, even in the case where their composition lies very close to the demixing line of a thermodynamically stable microemulsion domain. However, as exemplified in the present work, nanoemulsion systems can present very long-term kinetic stability.     

Dynamic invariants and the berry phase for generalized driven harmonic oscillators

We present quadratic dynamic invariants and evaluate the Berry phase for the time-dependent Schrödinger equation with the most general variable quadratic Hamiltonian.     

An algebraic interpretation of the <Emphasis Type="Italic">q</Emphasis>-Meixner polynomials

An algebraic interpretation of the q-Meixner polynomials is obtained. It is based on representations of \({\mathscr {U}}_q({\mathfrak {su}}(1,1))\) on q-oscillator states with the polynomials appearing as matrix elements of unitary q-pseudorotation operators. These operators are built from q-exponentials of the \({\mathscr {U}}_q({\mathfrak {su}}(1,1))\) generators. The orthogonality, recurrence relation, difference equation, and other properties of the q-Meixner polynomials are systematically obtained in the proposed framework.     

Resetting and annihilation of reentrant activity in a model of a one-dimensional loop of ventricular tissue

Resetting and annihilation of reentrant activity by a single stimulus pulse (S1) or a pair (S1-S2) of coupled pulses are studied in a model of one-dimensional loop of cardiac tissue using a Beeler-Reuter-type ionic model. Different modes of reentry termination are described. The classical mode of termination by unidirectional block, in which a stimulus produces only a retrograde front that collides with the activation front of the reentry, can be obtained for both S1 and S1-S2 applied over a small vulnerable window. We demonstrate that another scenario of termination-that we term collision block-can also be induced by the S1-S2 protocol. This scenario is obtained over a much wider range of S1-S2 coupling intervals than the one leading to a unidirectional block. In the collision block, S1 produces a retrograde front, colliding with the activation front of the pre-existing reentry, and an antegrade front propagating in the same direction as the initial reentry. Then, S2 also produces an antegrade and a retrograde front. However, the propagation of these fronts in the spatial profile of repolarization left by S1 leads to a termination of the reentrant activity. More complex behaviors also occur in which the antegrade fronts produced by S1 and S2 both persist for several turns, displaying a growing alternation in action potential duration ("alternans amplification") that may lead to the termination of the reentrant activity. The hypothesis that both collision block and alternans amplification depend on the interaction between the action potential duration restitution curve and the recovery curve of conduction velocity is supported by the fact that the dynamical behaviors were reproduced using an integro-delay equation based on these two properties. We thus describe two new mechanisms (collision block and alternans amplification) whereby electrical stimulation can terminate reentrant activity. (c) 2002 American Institute of Physics.