Search results for "Mathematical physics"
showing 10 items of 2687 documents
Back to the Amitsur-Levitzki theorem: a super version for the orthosymplectic Lie superalgebra osp(1, 2n)
2003
We prove an Amitsur-Levitzki type theorem for the Lie superalgebras osp(1,2n) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras gl(p,q) cannot satisfy an Amitsur-Levitzki type super identity if p, q are non zero and conjecture that neither can any other classical simple Lie superalgebra with the exception of osp(1,2n).
Optimal Control of Dissipative Quantum Systems
2008
We study the control of finite dimensional quantum systems by external laser fields. After examining the concrete example of the diatomic molecular alignment in dissipative media, we are interested in the problem of optimal control, where the objective is to bring the system from an initial state into a given final state while minimizing a cost functional. The Pontryagin maximum principle (PMP) provides necessary conditions for optimality, by establishing that any optimal trajectory is the extremal solution of an extended problem of Hamiltonian structure. In this context, we perform the analysis of two particular systems. The first one is a dissipative 2-level system, for which we determine…
The J-invariant, Tits algebras and Triality
2012
In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group $G$ and the degree one parameters of its motivic $J$-invariant. Our main technical tool are the second Chern class map and Grothendieck's $\gamma$-filtration. As an application we recover some known results on the $J$-invariant of quadratic forms of small dimension; we describe all possible values of the $J$-invariant of an algebra with orthogonal involution up to degree 8 and give explicit examples; we establish several relations between the $J$-invariant of an algebra $A$ with orthogonal involution and the $J$-invariant of the corresponding quadratic form over the functi…
Algebraic groups as difference Galois groups of linear differential equations
2019
We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.
From $1$ to $6$: a finer analysis of perturbed branching Brownian motion
2020
The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\alpha}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<\alpha<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depen…
Iterative approach to the exponential representation of the time–displacement operator
2005
An iterative method due to Voslamber is reconsidered. It provides successive approximations for the logarithm of the time–displacement operator in quantum mechanics. The procedure may be interpreted, a posteriori, as an infinite re-summation of terms in the so-called Magnus expansion. A recursive generator for higher terms is obtained. From two illustrative examples, a detailed comparative study is carried out between the results of the iterative method and those of the Magnus expansion.
A second-order differential equation for the two-loop sunrise graph with arbitrary masses
2011
We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.
Vanishing of certain cuts or residues of loop integrals with higher powers of the propagators
2019
Starting from two-loops, there are Feynman integrals with higher powers of the propagators. They arise from self-energy insertions on internal lines. Within the loop-tree duality approach or within methods based on numerical unitarity one needs (among other things) the residue when a raised propagator goes on-shell. We show that for renormalised quantities in the on-shell scheme these residues can be made to vanish already at the integrand level.
MULTI-SCALE ANALYSIS OF LUNG COMPUTED TOMOGRAPHY IMAGES
2007
A computer-aided detection (CAD) system for the identification of lung internal nodules in low-dose multi-detector helical Computed Tomography (CT) images was developed in the framework of the MAGIC-5 project. The three modules of our lung CAD system, a segmentation algorithm for lung internal region identification, a multi-scale dot-enhancement filter for nodule candidate selection and a multi-scale neural technique for false positive finding reduction, are described. The results obtained on a dataset of low-dose and thin-slice CT scans are shown in terms of free response receiver operating characteristic (FROC) curves and discussed.
Invariant varieties of discontinuous vector fields
2004
We study the geometric qualitative behaviour of a class of discontinuous vector fields in four dimensions around typical singularities. We are mainly interested in giving the conditions under which there exist one-parameter families of periodic orbits (a result that can be seen as one analogous to the Lyapunov centre theorem). The focus is on certain discontinuous systems having some symmetric properties. We also present an algorithm which detects and computes periodic orbits.