A new invariant-based method for building biomechanical behavior laws - Application to an anisotropic hyperelastic material with two fiber families

2013

Abstract In this article, we present a general constructive and original approach that allows us to calculate the invariants associated with an anisotropic hyperelastic material made of two families of collagen fibers. This approach is based on mathematical techniques from the theory of invariants: • Definition of the material symmetry group. • Analytical calculation of a set of generators using the Noether’s theorem. • Analytical calculation of an integrity basis. • Comparison between the proposed invariants and the classical ones.

Automated detection of contextuality proofs with intermediate numbers of observables

2021

<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtQuantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality </span&gt</font&gt<span style="font-size: 13px; font-family: arial, helvetica;"&gtproofs, aka. observable-based proofs of the Kochen-Specker Theorem, we first formulate the</span&gt</div&gt<div style=""&gt<font face="arial, helvetica"&gt<span style="font-size: 13px;"&gtcontextuality property as the absence of solutions to a linear system. Then we explain why </span&gt</font&gt<span style="font-size: 13px; font-family: arial, helvetica…

Three-qutrit entanglement and simple singularities

2016

In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is $X\cap H_\varphi \subset X$). We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated singularities, has…

A constructive approach of invariants of behavior laws with respect to an infinite symmetry group – Application to a biological anisotropic hyperelas…

2014

Abstract In this paper, six new invariants associated with an anisotropic material made of one fiber family are calculated by presenting a systematic constructive and original approach. This approach is based on the development of mathematical techniques from the theory of invariants: • Definition of the material symmetry group. • Definition of the generalized Reynolds Operator. • Calculation of an integrity basis for invariant polynomials. • Comparison between the new (constructed) invariants and the classical ones.

On the projective geometry of entanglement and contextuality

2019

Classification of multipartite systems featuring only $|W\rangle$ and $|GHZ\rangle$ genuine entangled states

2015

In this paper we present several multipartite quantum systems featuring the same type of genuine (tripartite) entanglement. Based on a geometric interpretation of the so-called $|W\rangle$ and $|GHZ\rangle$ states we show that the classification of all multipartite systems featuring those and only those two classes of genuine entanglement can be deduced from earlier work of algebraic geometers. This classification corresponds in fact to classification of fundamental subadjoint varieties and establish a connection between those systems, well known in Quantum Information Theory and fundamental simple Lie algebras.

A new hyperelastic model for anisotropic hyperelastic materials with one fiber family

2016

International audience; The main goal of this study is to propose a practical application of a new family of transverse anisotropic invariants by designing a strain energy function (SEF) for incompressible fiber-reinforced materials. In order to validate the usability and creativeness of the proposed model, two different fiber-reinforced rubber materials under uniaxial and shear testing are considered. For each kind of material, numerical simulations based on the proposed model are consistent with experimental results and provide information about the effect of the new family of invariants in the construction of the SEF.

Computer-assisted enumeration and classification of multi-qubit doilies

2022

For N ≥ 2, an N-qubit doily is a doily living in the N-qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of [SdBHG21] that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any N &gt; 2. Then we present an effective algorithm for the generation of all N-qubit doilies. Using this algorithm for N = 4 and N = 5, we provide a classification of N-qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with.