Disjoint Supports Research Articles

On nonlinear effects in multiphase WKB analysis for the nonlinear Schrödinger equation *

We consider the Schrödinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that like in the linear case, a superposition principle holds on some time interval independent of the semiclassical parameter, in several régimes in term of the size of initial data with respect to the semiclassical parameter. When nonlinear effects are strong in terms of the semiclassical parameter, we invoke properties of compressible Euler equations. For weaker nonlinear effects, we show that there may be no nonlinear interferences on some time interval independent of the semiclassical parameter, and interferences for later time, thanks to explicit computations available for particular phases.

Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov

AbstractWe construct measures of maximal u-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.

Extreme generators of shock induced copulas

In a recent paper, extreme points (in the Krein-Milman sense) of the class of semilinear copulas were introduced, motivated by the lack of known extreme copulas such as shuffles of M. We propose an extension of this concept to the class of all bivariate shock induced copulas, the most well-known part of them being the Marshall-Olkin copulas. This class properly contains semilinear copulas. Our technique coincides with the existing notion on them and has some advantages. First, it is defined on a wider family of copulas, which is helpful in finding more examples of extreme copulas. Second, we show that they are dense in each class they belong to (including the class of semilinear copulas) in a stronger sense than in the Krein-Milman approach; actually, they are dense in a similar way as shuffles of M are dense in the set of all copulas. Third, this definition enables practitioners to give stochastic interpretation of extremality. Roughly speaking, a shock induced copula is extreme whenever the inducing shocks have pairwise disjoint supports.

Sparse PCA on fixed-rank matrices

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if the rank of the covariance matrix is a fixed value, then there is an algorithm that solves sparse PCA to global optimality, whose running time is polynomial in the number of features. We also prove a similar result for the version of sparse PCA which requires the principal components to have disjoint supports.

Signal Useful Information Recovery by Overlapping Supports of Time-Frequency Representations

This paper presents a novel method for automatic extraction of useful information from time-frequency distributions of noisy signals. Signals' features are examined through the combined analysis of their time-frequency energy distributions and inverse complexity maps. The inverse complexity approach gives a new entropy-based insight into the signal structure. These two approaches result in mostly disjoint time-frequency supports, overlapping only in the proximity of the signal components. Locations of the signal components (i.e., useful information) are thus identified by low complexity and high amplitude occurring together in the time-frequency plane. Compared to existing methods using only the time-frequency energy distribution, useful information extraction by the proposed method exhibits a significantly reduced error rate.

Disjointness-preserving operators and isospectral Laplacians

The most commonly considered counterexamples to Kac’s famous question “can one hear the shape of a drum?” – i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent? – consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping “local” isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann, and Robin Laplacians on spaces of continuous functions and on L^2 -spaces.

A combining method for the approximate solution of spatial segregation limit of reaction-diffusion systems

In this paper, we concern ourselves with the study of a class of stationary states for reaction-diffusion systems with densities having disjoint supports. Major contribution of this work is computing the numerical solution of problem as the rate of interaction between two different species tend to infinity. The main difficulty is the nonlinearity nature of problem. To do so, an efficient iterative method is proposed by hybrid of the radial basis function (RBF) collocation and finite difference (FD) methods to approximate the solution. Numerical results with good accuracies are achieved where the shape parameter is carefully selected. Finally, some numerical examples are given to illustrate the good performance of the method.

Hierarchically refined isogeometric analysis of trimmed shells

This work focuses on the study of several computational challenges arising when trimmed surfaces are directly employed for the isogeometric analysis of Kirchhoff–Love shells. To cope with these issues and to resolve mechanical and/or geometrical features of interest, we exploit the local refinement capabilities of hierarchical B-splines. In particular, we show numerically that local refinement is suited to effectively impose Dirichlet-type boundary conditions in a weak sense, where this easily allows to overcome the issue of over-constraining of trimmed elements. Moreover, we highlight how refinement can alleviate the spurious coupling stemming from disjoint supports of basis functions in the presence of “small” trimmed geometrical features such as thin holes. These phenomena are particularly pronounced in surface models defined by complex trimming patterns and with details at different scales. In this contribution we focus our effort on the analysis of single-patch geometries, where we show through several numerical examples the benefits and computational efficiency of the proposed methodology.

Defect of compactness for Sobolev spaces on manifolds with bounded geometry

Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.

Distortion in groups of affine interval exchange transformations

In this paper, we study distortion in the group \mathcal A of affine interval exchange transformations (AIET). We prove that any distorted element f of \mathcal A has an iterate f^k that is conjugate by an element of \mathcal A to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences, we prove that no Baumslag–Solitar group, BS (m,n) with \vert m \vert \neq \vert n \vert , acts faithfully by elements of \mathcal A ; every finitely generated nilpotent group of \mathcal A is virtually abelian and there is no distortion element in \mathcal A_{\mathbb Q} , the subgroup of \mathcal A consisting of rational AIETs.

Explicit analytical formulae of ranking indices without the requirement of multiplicative compatibility

The ranking of fuzzy numbers is often given by a special class of utility functions, better known as ranking indices of the first kind. Recently, we have proved that the effective ranking of fuzzy numbers from a set containing all real numbers can be done by involving a very particular class of ranking indices. In this way, the searching of effective ranking methods is essentially simplified. In this paper, firstly we extend this result in a sense which allows us to find explicit formulae for ranking indices defined on other important subsets of fuzzy numbers, as for example the set of positive trapezoidal fuzzy numbers or the set of positive triangular fuzzy numbers. Another novelty is that, reducing all those ranking indices that generate the same ordering to a special member of their class, under some reasonable regularity assumptions, we find all the orderings on trapezoidal fuzzy numbers, or positive trapezoidal fuzzy numbers, respectively, which satisfy all the basic requirements of Wang and Kerre, except for the compatibility with scalar multiplication. This special member of the class is what we call a basic ranking index, that is, a ranking index which assigns to a fuzzy number a real number that belongs to its support. Actually, these basic ranking indices can be extended to linear functionals, so their form is very simple and ready to be used in applications. Similar results are obtained for the set of triangular fuzzy numbers and positive triangular fuzzy numbers, respectively. The key element in all these theoretical results, is that under some reasonable regularity properties, we can reduce a ranking index to this so called basic ranking index, which has a simpler form but generates the same order, that is, a binary relation with the same graph. As a first application, we review several ranking approaches got from ranking indices which are additive functions with respect to addition of fuzzy numbers and check their properties from the list of Wang and Kerre. Then, we do the same for a group of ranking indices none of them being additive functions with respect to the addition of fuzzy numbers. We can always do that, as long as we reduce a nonadditive ranking index to a basic ranking index, with both ranking indices generating the same ordering. For the newly obtained ranking index, we will just need to study additivity or scale invariance, depending on which type of property from the list of Wang and Kerre are we interested in. An interesting special case of nonadditive ranking index that cannot be reduced to a basic ranking index is the so called signed distance. We slightly modify this ranking index, so that it can be reduced to a basic ranking index. In this way, we never get contradictory results for fuzzy numbers with disjoint supports. Lastly, we propose a method to extend the ordering of positive trapezoidal fuzzy numbers to the ordering of arbitrary positive fuzzy numbers.

Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems

One of the most fascinating phenomenon observed in reaction diffusion systems is the emergence of segregated solutions, i.e., population densities with disjoint supports. We analyze such a reaction...

Variational Gram Functions: Convex Analysis and Optimization

We introduce a new class of convex penalty functions, called variational Gram functions (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, multitask learning, and estimating vectors with disjoint supports, among other applications. We study convexity of VGFs, and give characterizations for their convex conjugates, subdifferentials, proximal operators, and related quantities. We discuss efficient optimization algorithms for regularized loss minimization problems where the loss admits a common, yet simple, variational representation and the regularizer is a VGF. These algorithms enjoy a simple kernel trick, an efficient line search, as well as computational advantages over first order methods based on the subdifferential or proximal maps. We also establish a general representer theorem for such learning problems. Last, numerical experi...

Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓp summability of the (∥ψj∥L∞)j ≥ 1 implies the ℓp summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n− s of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L∞(U,V) or s = (1/p)−(1/2) in L2(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓp summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψj|)j ≥ 1. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, these results imply that for all 0 <p< 2, the ℓp summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψj∥L∞)j ≥ 1 is ℓq summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j ≥ 1yjψj where y = (yj) ∈ ℝNare i.i.d. standard scalar Gaussian variables and (ψj)j ≥ 1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(yj) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 <p ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥V)ν ∈ ℱ. Such results ensure convergence rates n− s with s = (1/p)−(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥ψj∥L∞)j ≥ 1 for some β>1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥V)ν ∈ ℱ. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

Lexicographic beliefs and assumption

Foundations for iterated admissibility (i.e., the iterated removal of weakly dominated strategies) need to confront a fundamental challenge. On the one hand, admissibility requires that a player consider every strategy of their opponents possible. On the other hand, reasoning that the opponents are rational requires ruling out certain strategies. Brandenburger, Friedenberg, Keisler's (BFK, Econometrica, 2008) foundations for iterated admissibility address this challenge with two ingredients: lexicographic beliefs and the concept of “assumption.” However, BFK restrict attention to lexicographic beliefs whose supports are essentially disjoint. This restriction does not have a compelling behavioral rationale, or a clear intuitive interpretation. At the same time, it plays a crucial role in BFK's foundations for iterated admissibility—specifically, in their analysis of assumption. We provide an alternate characterization of assumption, which applies to all lexicographic beliefs. We also characterize two variants of assumption, based on two extensions of ‘weak dominance’ to infinite state spaces. These notions of assumption coincide with BFK's notion when the state space is finite and lexicographic beliefs have disjoint support; but they are different in more general settings. Leveraging these characterization results, we show that disjoint supports do not play a role in the foundations for iterated admissibility.

Insensitizing exact controls for the scalar wave equation and exact controllability of $$2$$ -coupled cascade systems of PDE’s by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of \(2\)-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger \(2\)-coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).

Path Properties of Dilatively Stable Processes and Singularity of Their Distributions

First, we present some results about the Hölder continuity of the sample paths of so-called dilatively stable processes which are certain infinitely divisible processes having a more general scaling property than self-similarity. As a corollary, we obtain that the most important (H, δ)-dilatively stable limit processes (e.g., the LISOU and the LISCBI processes, see [4]) almost surely have a local Hölder exponent H. Next we prove that, under some slight regularity assumptions, any two dilatively stable processes with stationary increments are singular (in the sense that their distributions have disjoint supports) if their parameters H are different. We also study the more general case of not having stationary increments. Throughout the article, we specialize our results to some basic dilatively stable processes such as the above-mentioned limit processes and the fractional Lévy process.

A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth

We consider a tumor growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell population densities with contact inhibition. In one space dimension, it is known that global solutions exist and that they satisfy the so-called segregation property: if the two populations are initially segregated - in mathematical terms this translates into disjoint supports of their densities - this property remains true at all later times. We apply recent results on transport equations and regular Lagrangian flows to obtain similar results in the case of arbitrary space dimension.

The Optimal Partial Transport Problem

Given two densities f and g, we consider the problem of transporting a fraction \({m \in [0,\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}\) of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by |x − y|2, we will see that uniqueness of solutions holds for \({m \in [\|f\wedge g\|_{L^1},\min\{\|f\|_{L^1},\|g\|_{L^1}\}]}\) . This extends the result of Caffarelli and McCann in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active regions are shown to be (n − 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets \({{\Omega,\Lambda \subset \mathbb {R}^n}}\) , and bounded away from zero and infinity on their respective supports, \({C^{0,\alpha}_{\rm loc}}\) regularity of the optimal transport map and local C1 regularity of the free boundaries away from \({{\Omega\cap \Lambda}}\) are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.