Computing the Abel map

We present the next step in an ongoing research program to allow for the black-box computation of the so-called finite-genus solutions of integrable differential equations. This next step consists of the black-box computation of the Abel map from a Riemann surface to its Jacobian. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of this Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way. This generality is necessary in order for these algorithms to be relevant for the computation of the finite-genus solutions of any integrable equation.     

Analytical expressions for partial wave two-body Coulomb transition matrices at ground-state energy

Leaning upon the Fock method of the stereographic projection of the three-dimensional momentum space onto the four-dimensional unit sphere the possibility of the analytical solving of the Lippmann–Schwinger integral equation for the partial wave two-body Coulomb transition matrix at the ground bound state energy has been studied. In this case new expressions for the partial p$p$-, d$d$- and f$f$-wave two-body Coulomb transition matrices have been obtained in the simple analytical form. The developed approach can also be extended to determine analytically the partial wave Coulomb transition matrices at the energies of excited bound states.     

Recent Results on Recurrent Solutions and Limit Distributions of SDEs

The limit distribution for homogeneous Markov processes is studied extensively and well understood, but it is not the case for inhomogeneous Markov processes. In this paper, we review some recent results on inhomogeneous Markov processes generated by non-autonomous stochastic (partial) differential equations (SDE in short). Under some suitable conditions, we show that the distribution of recurrent solutions of SDEs constitutes the limit distribution of the corresponding inhomogeneous Markov processes.     

The minimal excludant and Schmidt's partition theorem

In this work, we study Schmidt's partition theorem in a combinatorial manner, and find a strong refinement which connects the minimal excludant of ordinary partitions to the length of Schmidt's partitions. As a byproduct, we obtain a bivariate form of an identity recorded in Ramanujan's lost notebook.     

Global existence inL 1 for the modified nonlinear Enskog equation in ℝ3

A global existence theorem with large initial data inL ¹ is given for the modified Enskog equation in ³. The method, which is based on the existence of a Liapunov functional (analog of theH-Boltzmann theorem), utilizes a weak compactness argument inL ¹ in a similar way to the DiPerna-Lions proof for the Boltzmann equation. The existence theorem is obtained under certain condition on the behavior of the geometric factorY. The condition onY amounts to the fact that theL ¹ norm of the collision term grows linearly when the local density tends to infinity.     

Phase space density representations in fluid dynamics

Phase space density representations of inviscid fluid dynamics were recently discussed by Abarbanel and Rouhi. Here it is shown that such representations may be simply derived and interpreted by means of the Liouville equation corresponding to the dynamical system of ordinary differential equations that describes fluid particle trajectories. The Hamiltonian and Poisson bracket for the phase space density then emerge as immediate consequences of the corresponding structure of the dynamics. For barotropic fluids, this approach leads by direct construction to the formulation presented by Abarbanel and Rouhi. Extensions of this formulation to inhomogeneous incompressible fluids and to fluids in which the state equation involves an additional transported scalar variable are constructed by augmenting the single-particle dynamics and phase space to include the relevant additional variable.     

Membrane lipid domains and dynamics as detected by Laurdan fluorescence

2-Dimethylamino-6-lauroylnaphthalene (Laurdan) is a membrane probe of recent characterization, which shows high sensitivity to the polarity of its environment. Steady-state Laurdan excitation and emission spectra have different maxima and shape in the two phospholipid phases, due to differences in the polarity and in the amount of dipolar relaxation. In bilayers composed of a mixture of gel and liquid-crystalline phases, the properties of Laurdan excitation and emission spectra are intermediate between those obtained in the pure phases. These spectral properties are analyzed using the generalized polarization (GP). TheGP value can be used for the quantitation of each phase. The wavelength dependence of theGP value is used to ascertain the coexistence of different phase domains in the bilayer. Moreover, by following the evolution of Laurdan emission vs. time after excitation, the kinetics of phase fluctuation in phospholipid vesicles composed of coexisting gel and liquid-crystalline phases was determined.GP measurements performed in several cell lines did not give indications of coexistence of phase domains in their membranes. In natural membranes, Laurdan parameters indicate a homogeneously fluid environment, with restricted molecular motion in comparison with the phospholipid liquid-crystalline phase. The influence of cholesterol on the phase properties of the two phospholipid phases is proposed to be the cause of the phase behavior observed in natural membranes. In bilayers composed of different phospholipids and various cholesterol concentrations, Laurdan response is very similar to that arising from cell membranes. In the absence of cholesterol, from the steady-state and time-resolved measurements of Laurdan in phospholipid vesicles, the condition for the occurrence of separate coexisting domains in the bilayer has been determined: the molecular ratio between the two phases must be in the range between 30% and 70%. Below and above this range, a single homogeneous phase is observed, with the properties of the more concentrated phase, slightly modified by the presence of the other. Moreover, in this concentration range, the calculated dimension of the domains is very small, between 20 and 50 Å.     

S-wave baryons in an equally mixed scalar-vector square root potential model of independent quarks

A relativistic model of independent quarks based on Dirac equation with an equally mixed scalar-vector square root confining potential is used to compute the quark core contributions to the static properties like magnetic moments, charge radii and axial vector coupling constant ratios of the baryon octet. The results obtained with appropriate corrections due to centre-of-mass motion agree fairly well with experimental values. The model is also extended to the study of magnetic moments of the quark core of baryons in the charmed andb-flavoured sectors and the overall predictions so obtained compare well with other model predictions.