and it is part of his contribution the proof that this definition is equivalent to ( 1.17 ). All this for smooth, albeit possibly abstract, structures. On the other hand, there is as of now a quite well established theory of (non-smooth) metricmeasurespaces satisfying a curvature-dimension condition: that of RCD ∗ (K, N ) spaces introduced by Ambrosio- Gigli-Savar´e (see [ 9 ] and [ 24 ]) as a refinement of the original intuitions of Lott-Sturm- Villani ([ 35 ] and [ 40 , 41 ]) and Bacher-Sturm ([ 12 ]). In this setting, there is a very natural Laplacian and inequality ( 1.15 ) is known to be valid in the appropriate weak sense (see [ 9 ] and [ 21 ]) and one can therefore wonder if even in this low-regularity situation one can produce an effective notion of N-Ricci curvature. Part of the problem here is the a priori lack of vocabulary, so that for instance it is unclear what a vector field should be. In the recent paper [ 23 ], Gigli builds a differential structure on metricmeasurespaces suitable to handle the objects we are discussing (see the preliminary section of Chapter- 5
Keywords and phrases: statistical test, subsampling, metric-measurespaces, dis-
tance to a measure, (Gromov)-Wasserstein distances.
Very often data comes in the form of a set of points from a metric space. A natural question, given two such sets of data, is to decide whether they are similar. For example, do they come from the same distribution? Are their shapes similar? From the seminal two-samples tests
SUB-GAUSSIAN HEAT KERNEL ESTIMATES
Abstract. Hardy space theory has been studied on manifolds or metricmeasurespaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviour (locally Gaussian and at infinity sub-Gaussian) in which case the previous theory doesn’t apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the H p space corresponding to Gaussian estimates may
Spaces, volume 1. Princeton University Press, 1971.
A Useful Probabilistic Results
In this section we present two useful probability lemmas that we used several times throughout the paper. The first one is needed to avoid relying on conditional probabilities to obtain independence properties (allowing us to state results in non-separable metricspaces). The second one is the classic “freezing lemma”. Lemma 7. Let (Ω, F , P) be a probability space. Let (V, FV ) and (W, F W ) be two measurable spaces. Let U : Ω → [0, ∞], f : V×W → [0, ∞], V : Ω → V, W : Ω → W be four measurable functions. If (U, V ) and W are P-independent, then
is a resolving set if the distance vectors are pairwise distinct for every t ∈ V . The metric dimension of G, denoted by M D(G), is then the minimum number of vertices that must be probed simultaneously (in one step) to determine the location t of the target wherever it is. For instance, in the case of a path, probing one of its ends is sufficient to locate the target, i.e., M D(P ) = 1 for every path P . Another example (that we use throughout the paper) is the case of a star (tree with a universal vertex) with n leaves, denoted by S n , for which it is necessary
In this paper, we address a sequential variant of this problem, which we deal with through the following terminology. Let us consider a graph G = (V, E) where an unknown vertex t ∈ V hosts a hidden (invisible) and immobile target. Probing one vertex v ∈ V results in the knowledge of the distance between t and v, denoted by d G (v, t) , which is the length of a shortest path from t to v. Probing a set R ⊆ V of vertices results in the distance vector (d G (v, t)) v∈R and R is resolving if no two vertices of G get the same distance vector (by R). The metric dimension of G, denoted by MD(G), is then the minimum number of vertices that must be probed simultaneously to immediately (in one step) determine the location t of the target (wherever it is). For instance, in the case of a path, probing one of its ends is sucient to locate the target, i.e., MD(P ) = 1 for every path P . Another example is the case of a star (tree with a universal node) with n leaves, denoted by S n , for which it is necessary and sucient to probe every leaf but one, i.e., MD(S n ) = n − 1 .
Proposition 5.1. Let d be a partial dissimilarity with a 2-tree support graph G , and d T the tree function extending d obtained by the triangle method. Then, d T is a tree metric if and only if d is a partial metric.
Since 2-trees are chordal graphs, it is easy to verify that d, with a 2-tree support graph G = (X , E), is a partial metric if and only if all the triangles of G are metric. Assume that G is just 2-acyclic. We then can add new pairs to E until a 2-tree is obtained. To obtain a tree metric extension by the triangle method, we have to give to each new edge xy a length preserving the property of being a partial metric. For that purpose, a simple solution consists of taking the minimum length d m (x,y) of a path of G between x and y as d(x,y).
of approximating compact spaces by ﬁnite simplicial complexes (nerves of ﬁnite covers), so even if it not obvious that the 1931 result applies directly to an inverse limit of ﬁnite simpli- cial complexes of dimension at most d (i.e., whether such an inverse limit is of dimension at most d), the proofs still apply. And in any case, in 1937 H. Freudenthal [Fre37] proved that a compact metrizable space is of dimension at most d if and only if it is an inverse limit of ﬁnite simplicial complexes of dimension at most d. See Sections 1.11 and 1.13 of [Eng78] for more about the history, including later improvements.
analyse the nature of political bonds in authoritarian conditions, and show how the favelas are not political spaces but rather places of great social responsiveness.
In her article, “Favelas: Towards a Fairer Space? Democratic Transition and Collective Mobilisation in Authoritarian Space ”, Justine Ninin offers a reflection on the processes whereby favelas are constructed as authoritarian spaces of exclusion. She therefore focuses on the way in which they are structurally constituted by state domination and on the reappropriation that they are experiencing under the impact of more private constraints. In particular, she shows how representations and spatial perceptions feed a specific local citizenship, institutionally organised by intermediate agents, who are particularly able to transform the contact zone between state and society into a space of mobilisation and to respond – even if only partially – to the claims for justice formulated in these spaces.
experienced extinctions. We also found that the functional distance between species (as measured by FDis, MPD and
Maire, E., Grenouillet, G., Brosse, S. & Vill eger, S. (2015). How many dimensions are needed to accurately assess functional diversity? A pragmatic approach for assessing the quality of functional spaces. Glob. Table 1 Changes in the functional and phylogenetic diversity of island bird assemblages
APPLICATIONS TO FIXED POINT THEORY AND ARZELA-ASCOLI TYPE THEOREM
MOHAMMED BACHIR, BRUNO NAZARET
Abstract. Schweizer, Sklar and Thorp proved in 1960 that a Menger space (G, D, T ) under a continuous t-norm T , induce a natural topology τ wich is metrizable. We extend this result to any probabilistic metric space (G, D, ⋆) provided that the triangle function ⋆ is continuous. We prove in this case, that the topological space (G, τ ) is uniformly homeomorphic to a (deterministic) metric space (G, σ D ) for some canonical metric σ D on G. As applications, we
has a nonnegative discriminant, that is
(3.10) |h∂ x v, z i| 2 ≥ 4(FR(∂ tx ϕ, ∂ x ϕ) − ε)kzk 2 L 2 (D) .
These two inequalities and the first equality on the kinetic energy ( 3.2 ) give the claimed equality between the Lagrangian and Eulerian functionals. We now prove that if the initial and final diffeomorphisms are smooth enough, the minimization on vector fields that are in the same smoothness category gives the same minimization result than in Theorem 1 . The result is based on the right- invariance of the metric which enables the construction of smooth approximating sequences. Let us describe the underlying strategy developed in the proof below. Recall the result of Proposition 6 : If a Lagrangian ϕ(t, x) has a non-empty jump set, an approximation of it can be defined by introducing an interval at each jump point on which we will define a minimal norm interpolation. By right-invariance of the metric, this solution, which is defined on a larger interval than [0, 1], can be mapped to [0, 1] while preserving the total energy. This new solution provides a continuous path which is an approximation of the initial flow. Starting from this candidate, we use standard smoothing arguments, the main point consists in dealing with the boundary conditions.
We address here the problem of discontinuities of total orders in a metric space and its implications for mathematical morphology. We first give a rigorous formulation of the problem. Then, a new approach is proposed to tackle the discontinuity issue by adapting the order to the image to be processed. Given an image and a total order we define a cost that evaluates the importance of the discontinuities for morpho- logical processing. The proposed order is then built as a minimization of this cost function. One of the strength of the proposed framework is its generality: the only ingredient required to build the total order is the graph of distances between values of the image. The adapted or- der can be computed for any image valued in a metric space where the distance is explicitly known. We present results for color images, dif- fusion tensor images (DTI) and images valued in the hyperbolic upper half-plane.
The goal of this paper is to contribute to the understanding of the geometry of Wasserstein spaces. Given a metric space X, the theory of optimal transport (with quadratic cost) gives birth to a new metric space, made of probability measures on X, often called its Wasserstein space and denoted here by W 2 (X) (precise definitions are recalled in the first part of this paper). One can use this theory to study X, for example by defining lower Ricci curvature bounds as in the celebrated works of Lott-Villani [LV09] and Sturm [Stu06]. Conversely, here we assume some understanding of X and try to use it to study geometric properties of W 2 (X). A similar philosophy underlines the works of Lott in [Lot08] and Takatsu and Yokota in [TY12].
We consider the problem of laying out routes that connect simultaneously given source- destination vertex pairs over a metric graph G 0 (V 0 , E 0 ). Vertices of the metric graph G 0
other than the sources and destinations may be used, but we are uncertain of their avail- ability, in that each such vertex is present with some probability independently of all other vertices. Sources and destinations are present with probability 1. Our objective is to take some a priori decisions regarding the layout of required routes, so as to be able to come up with feasible routes for every possibly materializable subgraph G 1 (V 1 , E 1 ), V 1 ⊆ V 0 , of G 0 ,
P(W p (µ, ˆ µ) > ) ≤ C exp (−KN d/p )
where C and K are constants that can be computed from moments of µ. This bound shows that the Wasser- stein distance suffers the dimensionality and as such a Wasserstein distance embedding for distribution learn- ing is not expected to be efficient especially in high- dimension problems. However, a recent work of Weed & Bach (2017) has also proved that under some hy- pothesis related to singularity of µ better convergence rate can be obtained (some being independent of d). Interestingly, we demonstrate in what follows that the estimated Wasserstein distance for Gaussians using Bu- res metric and plugin estimate of m and Σ has a better bound related to the dimension.
Financial risk management is fundamentally based on the comparison of risk measures across different assets, portfolios, or financial institutions. Examples include the compar- ison of total risk of two portfolios measured by their volatility, of tail risk measured by the Value-at-Risk (VaR) or the Expected Shortfall (ES), of systematic risk measured by the beta, or the comparison of systemic risk scores of two financial institutions and many others. Comparing unconditional risk measures can be done using a variety of paramet- ric or non-parametric tests. However, most risk measures are expressed conditionally on an information set and the corresponding forecasts are generally issued from a dynamic parametric or semi-parametric model. For instance, a (M-)GARCH model can be used to produce conditional VaR or ES forecasts, or a DCC can be used to estimate a dynamic conditional beta (Engle, 2012). As a consequence, the conditional distribution of the es- timated risk measure is generally unknown and depends on which estimation procedure is used.