Asymptotic Expansion of the Bergman Kernel via Perturbation of the Bargmann–Fock Model

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated with the kth tensor powers of a positive line bundle L in a \(\frac{1}{\sqrt{k}}\)-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential \(k\varphi \) in a \(\frac{1}{\sqrt{k}}\)-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann–Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann–Fock Bergman kernel.     

Self-Dual Vortices in Chern–Simons Hydrodynamics

The classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie–Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant for the deformed strength of the quantum potential and to the pair of diffusion–antidiffusion equations for the strength . Specifying the gauge field as the Abelian Chern–Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw–Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson–Sabatier type to the phase of the vortex wave function, we show that deformation parameter , the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.     

Analytical Issues in the Construction of Self-dual Chern–Simons Vortices

In this note we discuss the solvability of Liouville-type systems in presence of singular sources, which arise from the study of non-abelian Chern Simons vortices in Gauge Field Theory and their asymptotic behaviour (for limiting values of the physical parameters). This investigation has contributed towards the understanding of singular PDE ’s in Mean Field form, in connection to surfaces with conical singularities, sharp Moser–Trudinger and log(HLS)-inequalities, bubbling phenomena and point-wise profile estimates in terms of Harnack type inequalities. We shall emphasise mostly the physical impact of the rigorous mathematical results established and mention several of the remaining open problems.     

On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

Journal of Nonlinear Science - A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a...     

On the Krein–Rechtman Theorem in the Presence of Multiple Points

Mathematical Notes -     

Carleson Conditions in Bergman–Orlicz Spaces

We give characterizations of (forward and reverse) Carleson conditions in terms of inequalities that involve functions in Bergman–Orlicz spaces.     

Separation of Variables in the Anisotropic Shottky–Frahm Model

We construct separated coordinates for the completely anisotropic Shottky–Frahm model on an arbitrary coadjoint orbit of SO(4). We find explicit reconstruction formulas expressing dynamical variables in terms of the separation coordinates and write the equations of motion in the Abel-type form.     

The Mass Shell in the Semi-Relativistic Pauli–Fierz Model

We consider the semi-relativistic Pauli–Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo’s iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain renormalized fiber Hamiltonian. The ground state energy is an exactly twofold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius ${\mathfrak{p} > 0}$ , provided that the coupling constant is sufficiently small depending on ${\mathfrak{p}}$ and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.     

Vortices in a 2d rotating Bose–Einstein condensate

We investigate the physical model for a two dimensional rotating Bose–Einstein condensate. We minimize a Gross–Pitaevskii functional defined in ${\mathbb{R}}^2$ under the unit mass constraint. We estimate the critical rotational speeds ${\Omega }_d$ for having d vortices in the condensate and we determine the location of the vortices. This relies on an asymptotic expansion of the energy. To cite this article: R. Ignat, V. Millot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Bilevel “Defender–Attacker” Model with Multiple Attack Scenarios

We consider a bilevel “defender-attacker” model built on the basis of the Stackelberg game. In this model, given is a set of the objects providing social services for a known set of customers and presenting potential targets for a possible attack. At the first step, the Leader (defender) makes a decision on the protection of some of the objects on the basis of his/her limited resources. Some Follower (attacker), who is also limited in resources, decides then to attack unprotected objects, knowing the decision of the Leader. It is assumed that the Follower can evaluate the importance of each object and makes a rational decision trying to maximize the total importance of the objects attacked. The Leader does not know the attack scenario (the Follower’s priorities for selecting targets for the attack). But, the Leader can consider several possible scenarios that cover the Follower’s plans. The Leader’s problem is then to select the set of objects for protection so that, given the set of possible attack scenarios and assuming the rational behavior of the Follower, to minimize the total costs of protecting the objects and eliminating the consequences of the attack associated with the reassignment of the facilities for customer service. The proposed model may be presented as a bilevelmixed-integer programming problem that includes an upper-level problem (the Leader problem) and a lower-level problem (the Follower problem). The main efforts in this article are aimed at reformulation of the problem as some one-level mathematical programming problems. These formulations are constructed using the properties of the optimal solution of the Follower’s problem, which makes it possible to formulate necessary and sufficient optimality conditions in the form of linear relations.     

Dispersion Relations in the Noncommutative \phi^3 and Wess–Zumino Model in the Yang–Feldman Formalism

We study dispersion relations in the noncommutative and Wess–Zumino model in the Yang–Feldman formalism at one-loop order. Nonplanar graphs lead to a distortion of the dispersion relation. We find that the strength of this effect is moderate if the scale of noncommutativity is identified with the Planck scale and parameters typical for a Higgs field are employed. The contribution of the nonplanar graphs is calculated rigorously using the framework of oscillatory integrals. Submitted: October 13, 2007., Accepted: August 11, 2008.     

A Model for Vortex Nucleation in the Ginzburg–Landau Equations

This paper studies questions related to the dynamic transition between local and global minimizers in the Ginzburg–Landau theory of superconductivity. We derive a heuristic equation governing the dynamics of vortices that are close to the boundary, and of dipoles with small inter-vortex separation. We consider a small random perturbation of this equation and study the asymptotic regime under which vortices nucleate.     

Exact Solutions for Linearized Gravity in the Randall–Sundrum Model

We find exact solutions of the equations of motion for linearized gravity in the Randall–Sundrum model with matter on the branes and calculate the Newtonian limit of the model. The results established include the contributions of the radion and the massive gravitons, which essentially modify Newton's law at short distances. We consider the effects produced by shadow matter situated on the other brane and compare them with the effect of the usual matter for branes with positive and negative tension. We also calculate the deflection of light and Newton's law in the zero-mode approximation and explicitly isolate the contribution of the radion field.     

Bifurcation Diagram of the Two Vortices in a Bose–Einstein Condensate with Intensities of the Same Signs

Doklady Mathematics - This paper deals with the problem of motion of a system of two point vortices in a Bose–Einstein condensate enclosed in a cylindrical trap. Bifurcation diagram is...     

On the Dimension of the Space of Dark States in the Tavis–Cummings Model

Mathematical Notes - The space of minimal energy of a qubit system is the dark subspace of quantum states of a system of two-level atoms in the finite-dimensional Tavis–Cummings (TC) model of...     

The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs.     

Dualities for the Domany–Kinzel Model

We study the Domany–Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p ₁,p ₂)[0,1]². When p ₁= and p ₂=(2– ²) with (,)[0,1]², the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities of a site being open and of a bond being open. This paper treats dualities for the Domany–Kinzel model _t ^A and the DKdual _t ^A starting from A. We prove that , as long as one of A,B is finite and p ₂p ₁.