Institut Fourier

Abstract We consider a finite analytic morphism $$\varphi =(f,g)$$ φ = ( f , g ) defined from a complex analytic normal surface ( Z , z ) to $$\mathbb {C}^2$$ C 2 . We describe the topology of the image by $$\varphi $$ φ of a reduced curve on ( Z , z ) by means of iterated pencils defined recursively for each branch of the curve from the initial one $$\langle f,g \rangle $$ ⟨ f , g ⟩ . This result generalizes the one obtained in a previous paper for the case in which ( Z , z ) is smooth and the curve irreducible. The methods we use also permit us to describe the topological type of the discriminant curve of $$\varphi $$ φ , in particular, the topological type of each branch of the discriminant can be obtained from the map without previous knowledge of the critical locus.

L’article a pour but de présenter quelques concepts liés à la science informatique ainsi que les plus-values qu’ils pourraient apporter aux mathématiques grâce à cette approche interdisciplinaire. Une séquence d’enseignement est ensuite proposée. Basée sur les mathématiques dans un premier temps, elle permettra, ensuite, de décliner et travailler différents objectifs du Plan d’Études Romand (PER) numérique.

We study the spectral gap behavior of an operator obtained by summing a random permutation M and a deterministic bistochastic matrix Q. We are interested in the asymptotic in terms of dimension. In the case where (M,Q) are asymptotically free with amalgamation over the diagonal, we can compute limit operators (u,q) which give the limit distribution. Therefore we introduce free with amalgamation operators that are suitable for computing the spectral gap limit of our operator in high dimensions. We then approximate the spectral radius of the corresponding limit operator and finally give an upper bound for the spectral radius of the finite-dimensional operator. In particular, we show that if the deterministic matrix underlying graph is an expander, then the underlying Markov chain associated to the sum with a random permutation is again an expander.

et gomtrie qui s'est tenu durant l'anne universitaire 2021-2021.Durant cette priode, le sminaire a t organis par Andrea Seppi.Les textes publis ne sont pas soumis un processus de rfr au sens strict ; certains (en particulier les exposs de synthse), constituent des contributions originales.Le prsent volume, ainsi que les sommaires des volumes prcdents, sont disponibles en ligne sur le site ddi https://tsg.centre-mersenne.org/

We present and test a protocol to learn the matrix-product operator (MPO) representation of an experimentally prepared quantum state. The protocol takes as an input classical shadows corresponding to local randomized measurements, and outputs the tensors of a MPO which maximizes a suitably-defined fidelity with the experimental state. The tensor optimization is carried out sequentially, similarly to the well-known density matrix renormalization group algorithm. Our approach is provably efficient under certain technical conditions which are expected to be met in short-range correlated states and in typical noisy experimental settings. Under the same conditions, we also provide an efficient scheme to estimate fidelities between the learned and the experimental states. We experimentally demonstrate our protocol by learning entangled quantum states of up to $N = 96$ qubits in a superconducting quantum processor. Our method upgrades classical shadows to large-scale quantum computation and simulation experiments.

This article deals the optimal design of a region [[EQUATION]] of the boundary [[EQUATION]] of a given domain [[EQUATION]], supporting a particular type of boundary conditions of a physical boundary value problem. We develop suitable notions of shape and topological derivatives, accounting for the sensitivity of a quantity [[EQUATION]] depending on a region [[EQUATION]]: on the one hand, we evaluate the sensitivity of [[EQUATION]] with respect to ``small'' perturbations of the boundary of [[EQUATION]] within [[EQUATION]]. On the other hand, we appraise the sensitivity of [[EQUATION]] with respect to the addition of a ``small'' surface disk to the region [[EQUATION]]. The calculation of both derivatives raises original difficulties, due to the weakly singularities caused by the change in boundary conditions in a boundary value problem. These aspects are detailed in a simple setting based on the conductivity equation. We notably propose formal arguments to calculate our derivatives, and show how they can be generalized to more intricate contexts, such as those of the Helmholtz equation and the linear elasticity system. Our derivatives are incorporated into a recent numerical framework for tracking arbitrarily dramatic motions of a region [[EQUATION]] within a fixed ambient surface. Finally, various 3d numerical examples are presented.