Quantum Integrals of Motion for the Heisenberg Spin Chain

An explicit expression for all the quantum integrals of motion for the isotropic Heisenberg $s=1/2$ spin chain is presented. The conserved quantities are expressed in terms of a sum over simple polynomials in spin variables. This construction is direct and independent of the transfer matrix formalism. Continuum limits of these integrals in both ferrromagnetic and antiferromagnetic sectors are briefly discussed.Comment: 10 pages Report #: LAVAL-PHY-94-2

Local electromigration model for crystal surfaces

We analyze the dynamics of crystal surfaces in the presence of electromigration. From a phase field model with a migration force which depends on the local geometry, we derive a step model with additional contributions in the kinetic boundary conditions. These contributions trigger various surface instabilities, such as step meandering, bunching and pairing on vicinal surfaces. Experiments are discussed

Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank

Gravity, strings, modular and quasimodular forms

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ... The latter often appear as gravitational instantons i.e. as special solutions of Einstein's equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by requiring (conformal) self-duality and Bianchi IX homogeneity. In this case, a vast range of configurations exist, which exhibit interesting modular properties. Examples of other Einstein spaces, without Bianchi IX symmetry, but with similar features are also given. Finally we discuss the emergence and the role of Eisenstein series in the framework of field and string theory perturbative expansions, and motivate the need for unravelling novel modular structures.Comment: 45 pages. To appear in the proceedings of the Besse Summer School on Quasimodular Forms - 201

Higher Order Corrections to the Asymptotic Perturbative Solution of a Schwinger-Dyson Equation

Building on our previous works on perturbative solutions to a Schwinger-Dyson for the massless Wess-Zumino model, we show how to compute 1/n corrections to its asymptotic behavior. The coefficients are analytically determined through a sum on all the poles of the Mellin transform of the one loop diagram. We present results up to the fourth order in 1/n as well as a comparison with numerical results. Unexpected cancellations of zetas are observed in the solution, so that no even zetas appear and the weight of the coefficients is lower than expected, which suggests the existence of more structure in the theory.Comment: 16 pages, 2 figures. Some points clarified, typos corrected, matches the version to be published in Lett. Math. Phy

Experimental study of depolarization and antenna correlation in tunnels in the 1.3 GHz band

Measurements have been carried out in a low-traffic road tunnel to investigate the influence of the polarization of the transmitting and receiving antennas on the channel characteristics. A real-time channel sounder working in a frequency band around 1.3 GHz has been used, the elements of the transmitting and receiving arrays being dual-polarized patch antennas. Special emphasis is made on cross-polarization discrimination factor and on the spatial correlation between array elements which has a great influence on the performances of transmit/receive diversity schemes. Various polarizations both at the transmitter and the receiver have been tested to minimize this spatial correlation while keeping the size of the array as small as possible