Topological Expansion Research Articles

Topological expansion and boundary conditions

In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.

Topological expansion of mixed correlations in the Hermitian 2-matrix model and x–y symmetry of the Fg algebraic invariants

We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e. a generating function for counting bicolored discrete surfaces with non-uniform boundary conditions. As an application, we prove the x–y symmetry of Eynard and Orantin (2007 Invariants of algebraic curves and topological expansion Preprint math-ph/0702045).

An angle dependent site-renormalized theory for the conformations of n-butane in a simple fluid

The angular dependent site-renormalized integral equation theory is developed to compute the dihedral conformation distribution and intermolecular pair distributions of n-butane at infinite dilution in a Lennard-Jones solvent. The equations take advantage of the topological diagrammatic expansion of the full angular dependent molecular system by resumming the series in conjunction with the intramolecular degree of freedom. To first order in an angular basis set, the numerical results of these site-renormalized equations are a systematic quantitative improvement over previous methods. In particular, the thermodynamics and conformational distribution of the solute are essentially indistinguishable from simulation.

Combining topological sensitivity and genetic algorithms for identification inverse problems in anisotropic materials

The identification inverse problem is solved here for flaw detection in anisotropic materials by means of an innovative approach: the combination of Genetic Algorithm and the Topological Sensitivity in anisotropic elasticity. The Topological Sensitivity provides a measure of the susceptibility of a defect being at a given location. This is based on a linearized topological expansion, applying Boundary Integral Equations and using solely information of the non-damaged state. It is proved that the Topological Sensitivity provides an accurate tool for estimating the location and size of defects. First, it is shown that the minimum of the residual (cost function) topological sensitivity pinpoints the location and size of the actual flaws, and secondly, the minimization of the residual topological sensitivity is carried out using Genetic Algorithm. When the Genetic Algorithm is applied to the residual Topological Sensitivity instead of to the full residual, the applicability of this method is enhanced since the computational effort, which is the major drawback of this type of search methods, is drastically reduced. In this paper, the formulation for linearly anisotropic elastic media is composed for the case of circular flaws, although the procedure is extensible to other kinds of defects like elliptical cavities, elastic or rigid inclusions or cracks.

Topological asymptotic expansions for the generalized Poisson problem with small inclusions and applications in lubrication

The asymptotic expansion of the solution u to the equation ∇ ⋅ (α∇u) = −∇ ⋅ β + γ in , with respect to the size ϵ of an inclusion (at a point x0 of the domain Ω) in which the parameters α, β and γ are changed, is studied. Writing the difference with the unperturbed solution as u − uϵ = ϵnzϵ, it is shown that the sequence zϵ converges weakly, for all p < n/(n − 1) and ρ > 0, to a function z in Lp(Ω)∩H1(Ω ∖ Bρ(x0)), where Bρ(x0) is the ball of radius ρ around x0. This allows for the calculation of asymptotic expansions of cost functions of the form J(ϵ) = ∫ΩF(uϵ) dx, for example, rendering it useful for many applications. It also extends available estimates which hold uniformly on the boundary of Ω. In addition, a link is provided with the adjoint method of calculating topological expansions of cost functions. A specific application to lubricated devices is illustrated.

From Differential Calculus to 0‐1 Topological Optimization

The topological asymptotic expansion gives the variation of a cost function when a small hole is created in a domain. This approach leads to very powerful algorithms in topological optimization. Unfortunately, these asymptotic expansions are obtained with the use of complicated mathematical tools. The goal of this paper is to provide a straightforward way to derive a topological asymptotic expansion using a classical gradient. We will illustrate this general approach by some numerical experiments for the elasticity and the Stokes problems.

Methods for Reliable Topology Changes for Perimeter-Regularized Geometric Inverse Problems

This paper is devoted to the incorporation of topological derivativelike expansions into level set methods for perimeter-regularized geometric inverse problems. The expansions are done up to the second order with respect to the Lebesgue measure of the symmetric difference. They provide simpler shape functionals, still including the perimeter, and therefore allow the construction of steepest descent- and Newton-type algorithms to force topology changes during the level set evolution. Numerous numerical examples are provided that show the strong and also the weak points of the newly developed algorithms.

Free energy topological expansion for the 2-matrix model

We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/ N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.

Second order topological sensitivity analysis

The topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal holes. However, for practical applications, we need to insert holes of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we apply the topological-shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the Poisson’s equations, taking the total potential energy as cost function and the state equation as constraint. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). Finally, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features: it allows us to deal with hole of finite size and provides a better descent direction in optimization process.

An introduction to the topological asymptotic expansion with examples

To find an optimal domain is equivalent to look for Its characteristic function. At first sight this problem seems to be nondifferentiable. But it is possible to derive the variation of a cost function when we switch the characteristic function from 0 to 1 or from 1 to 0 a small area. Classical and two generalized adjoint approaches are considered in this paper. Their domain of validity is given and Illustrated by several examples. Using this gradient type Information, It is possible to build fast algorithms. Generally, only one Iteration Is needed to find the optimal shape. Trouver un domaine optimal est équivalent à la recherche de sa fonction caractéristique. A première vue, ce problème semble non différentiable, mais Il est possible de calculer la variation de la fonction coût lorsque la fonction caractéristique passe de 1 à 0 ou de 0 à 1 dans une région de petite taille. On s’appuiera sur une approche adjointe classique et deux généralisations de cette méthode. Le domaine de validité de ces différentes approches est donné et illustré par différents exemples. Cette Information de type gradient permet de construire des algorithmes très efficaces: en général, une seule Itération suffit pour trouver le domaine optimal.

Image restoration and edge detection by topological asymptotic expansion

A new method for edge detection and image restoration is proposed. This method is based on topological gradient approach. Experimental results obtained on noisy images illustrate the capabilities of this promising method in image processing. To cite this article: L.J. Belaid et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

We continue our study of the large N phase transition in q-deformed Yang-Mills theory on the sphere and its role in connecting topological strings to black hole entropy. We study in detail the chiral theory defined in terms of uncoupled single U(N) representations at large N and write down the resulting partition function by means of the topological vertex. The emergent toric geometry has three Kaehler parameters, one of which corresponds to the expected fibration over the sphere. By taking a suitable double-scaling limit we recover the chiral Gross-Taylor string expansion. To analyse the phase transition we construct a matrix model which describes the chiral gauge theory. It has three distinct phases, one of which should be described by the closed topological string expansion. We verify this expectation by explicit comparison between the matrix model and the chiral topological string free energies. We also show that the critical point in the pertinent phase of the matrix model corresponds to a divergence of the topological string perturbation series.

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

We solve the loop equations of the hermitian 2-matrix model to all orders in the topological $1/N^2$ expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.

Effective density terms in proper integral equations

Two complementary routes to a new integral equation theory for site-site molecular fluids are presented. First, a simple approximation to a subset of the atomic site bridge functions in the diagrammatically proper integral equation theory is presented. This in turn leads to a form analogous to the reactive fluid theory, in which the normalization of the intramolecular distribution function and the value of the off-diagonal elements in the density matrix of the proper integral equations are the means of propagating the bridge function approximation. Second, a derivation from a topological expansion of a model for the single-site activity followed by a topological reduction and low-order truncation is given. This leads to an approximate numerical value for the new density coefficient. The resulting equations give a substantial improvement over the standard construction as shown with a series of simple diatomic model calculations.

The topological asymptotic expansion for the Maxwell equations and some applications

The aim of the topological sensitivity analysis is to derive an asymptotic expansion of a design functional with respect to a topological perturbation of the domain. In this paper, such an expansion is obtained for the 3D Maxwell equations when we introduce a small dielectric object in the domain and when we insert a small metallic obstacle. Some numerical results are presented in the context of buried object detection and shape inversion of 3D objects in free space from time-domain scattered field data.

Topological expansion for the 1-hermitian matrix model correlation functions.

We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the topological 1/N2 expansion, as residues on an hyperelliptical curve. Those residues, can be represented as Feynman graphs of a cubic field theory on the curve.

The topological asymptotic expansion for the Quasi-Stokes problem

In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized lake.

PartialN=2 toN=1 supersymmetry breaking and gravity deformed chiral rings.

We present a derivation of the chiral ring relations, arising in ${\cal N}=1$ gauge theories in the presence of (anti-)self-dual background gravitational field $G_{\alpha\beta\gamma}$ and graviphoton field strength $F_{\alpha\beta}$. These were previously considered in the literature in order to prove the relation between gravitational F-terms in the gauge theory and coefficients of the topological expansion of the related matrix integral. We consider the spontaneous breaking of ${\cal N} =2$ to ${\cal N} =1$ supergravity coupled to vector- and hyper-multiplets, and take a rigid limit which keeps a non-trivial $G_{\alpha\beta\gamma}$ and $F_{\alpha\beta}$ with a finite supersymmetry breaking scale. We derive the resulting effective, global, ${\cal N}=1$ theory and show that the chiral ring relations are just a consequence of the standard ${\cal N}=2$ supergravity Bianchi identities . We can also obtain models with matter in different representations and in particular quiver theories. We also show that, in the presence of non-trivial $F_{\alpha\beta}$, consistency of the Konishi-anomaly loop equations with the chiral ring relations, demands that the gauge kinetic function and the superpotential, a priori unrelated for an ${\cal N}=1$ theory, should be derived from a prepotential, indicating an underlying ${\cal N}=2$ structure.

Supersymmetry relics in one-flavor QCD from a new 1/N expansion.

We suggest a new large-N(c) limit for multiflavor QCD. Since fundamental and two-index antisymmetric representations are equivalent in SU(3), we have the option to define SU(N(c)) QCD keeping quarks in the latter. We can then define a new 1/N(c) expansion (at a fixed number of flavors N(f)) that shares appealing properties with the topological (fixed N(f)/N(c)) expansion while being more suitable for theoretical analysis. In particular, for N(f)=1, our large-N(c) limit gives a theory that we recently proved to be equivalent, in the bosonic sector, to N=1 supersymmetric gluodynamics. Using known properties of the latter, we derive several qualitative and semiquantitative predictions for N(f)=1 massless QCD that can be easily tested in lattice simulations. Finally, we comment on possible applications for pure SU(3) Yang-Mills theory and real QCD.

LargeNexpansion of the 2-matrix model

We present a method, based on loop equations, to compute recursively all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model. We illustrate the method by computing the first subleading term, i.e. the free energy of a statistical physics model on a discretized torus.