On boundaries and influences

We prove an inequality relating the size of the boundary of a monotone subset of {0,1} ⁿ and the influences of the coordinates upon this set. It shows that if the boundary (resp. the influences) are small, the influences (resp. the boundary) are large.Work partially supported by an NSF grant.     

, Vertex Isoperimetry and Concentration

. This approach refines results relating the spectral gap of a graph to the so-called magnification of a graph. A concentration result involving is also derived. Received July 22, 1998     

Quantitative relation between noise sensitivity and influences

A Boolean function f: {0,1} ⁿ → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ _p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand’s lemma, which easily generalizes in various directions, including non-monotone functions.     

Concentration of maps and group actions

In this paper we study actions of compact groups and of Lévy groups on a large class of metric spaces, such as \mathbbR{\mathbb{R}} -trees, doubling spaces, metric graphs, and Hadamard manifolds, from the viewpoint of the theory of concentration of maps.     

Concentration curve for truncated and censored data

Concentration curve is the inverse Lorenz curve. Together, they form the basis for most measures of distributional inequality. In this paper, we consider the empirical estimator of the concentration curve when the data are subjected to random left truncation and/or right censorship. Simultaneous strong Gaussian approximations for the associated Lorenz and normed concentration processes are established under appropriate assumptions. Functional laws of the iterated logarithm for the two processes are established as easy consequences. The construction provides a solid foundation for the study of functional statistics based on the two processes.     

Concentration of Multivariate Polynomials and Its Applications

are independent random variables which take values either 0 or 1, and Y is a multi-variable polynomial in 's with positive coefficients. We give a condition which guarantees that Y concentrates strongly around its mean even when several variables could have a large effect on Y. Some applications will be discussed. Received March 29, 1999     

Concentration and the variability of orthopaedic demand

Concentrating health services with centres providing selected, specialist care offers a number of potential advantages. The benefits may include the opportunity to improve the quality of care by providing more specialist services and greater expertise, more attractive working conditions with a larger pool of specialists providing the on-call rota and an enhanced opportunity for training. Concentration will produce greater volumes of patients in the selected specialties with the possibility of various economies of scale. A series of simulation experiments explored the potential for efficiencies associated with the increasing volume of non-elective patients in an orthopaedic specialty. As the annual volume of patients increases so the relative variability of the demand for operating theatre time declines: concentrating non-elective orthopaedic activity could offer considerable savings in the theatre time allocated to trauma patients. However, the impact on the wards is much less significant, with concentration having a negligible effect on the requirement for beds.     

Concentration through large advection

In this paper we extend the elegant results of Chen, Lam and Lou [6, Section 2], where a concentration phenomenon was established as the advection blows up, to a general class of adventive–diffusive generalized logistic equations of degenerate type. Our improvements are really sharp as we allow the carrying capacity of the species to vanish in some subdomain with non-empty interior. The main technical devices used in the derivation of the concentration phenomenon are Proposition 3.2 of Cano-Casanova and López-Gómez [5], Theorem 2.4 of Amann and López-Gómez [1] and the classical Harnack inequality. By the relevance of these results in spatial ecology, complete technical details seem imperative, because the proof of Theorem 2.2 of [6] contains some gaps originated by an “optimistic” use of Proposition 3.2 of [5]. Some of the general assumptions of [6] are substantially relaxed.     

Concentration of symmetric eigenfunctions

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.     

Concentration of norms and eigenvalues of random matrices

We prove concentration results for ?_pⁿ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.     

Concentration inequalities and limit theorems for randomized sums

Concentration properties and an asymptotic behaviour of distributions of normalized and self-normalized sums are studied in the randomized model where the observation times are selected from prescribed consecutive integer intervals. Research supported in part by NSF Gr. No. 0405587     

Velocity and Concentration Boundary Layers on Spinning Fibers

In this paper a mathematical model for the viscose wet spinning process is presented: We consider a single fibre, which is produced by pressing a basic solution of viscose into a bath containing sulphuric acid. H₂SO₄ diffuses into the viscose solution and reacts with the natrium hydroxide so that a solidifying fibre is formed which is pulled through the bath by drives. Due to the movement of the fibre and of diffusive transport of sulphuric acid into the fibre velocity and concentration boundary layers develop. Starting from the laminar boundary layer equations we investigate the flow and concentration fields in the bath induced by the fibre utilizing the Local Non‐Similarity method. Mass transfer in the fibre is modelled by transport equations for sulphuric acid and natrium hydroxide taking into account the neutralization reaction. The model of the fibre is coupled to the bath phase model by appropriate boundary conditions for the mass flow density and the chemical potential of sulphuric acid. The non‐constant diameter of the fibre is taken into account by a perturbation approach.     

Multiple Sets Exponential Concentration and Higher Order Eigenvalues

Potential Analysis - On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that...     

Concentration functions and a class of non-compact groups

Mathematische Annalen -     

Concentration inequality for evolutionary trees

Maximum likelihood inferred topologies are commonly used to draw conclusions in evolutionary biology and molecular evolution. Considering the sampling error when estimating the topology is a critical issue. Bootstrap-based methods are the most popular tools to assess the robustness of clades, i.e. the stability of a tree and subtrees. Unfortunately, there is no analytical result to connect the bootstrap values to the sampling variability, or at least to the number of sites and species in the study. Using concentration measure tools, we first bound the variations of the computed likelihood around its true value and then bound the sampling variability of likelihood as measured by bootstrap. In particular and unlike most bootstrap-based methods, these bounds are explicitly sensitive to both the number of species and of nucleotides.     

Functional Inequalities for Two-Level Concentration

Potential Analysis - Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and...