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arXiv2004

Tangential dimensions II. Measures

Notions of (pointwise) tangential dimension are considered, for measures of R^n. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to the given measure at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in math.FA/0305091, and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.

math.FA↗
arXiv2004

On convexified packing and entropy duality

A 1972 duality conjecture due to Pietsch asserts that the entropy numbers of a compact operator acting between two Banach spaces and those of its adjoint are (in an appropriate sense) equivalent. This is equivalent to a dimension free inequality relating covering (or packing) numbers for convex bodies to those of their polars. The duality conjecture has been recently proved (see math.FA/0407236) in the central case when one of the Banach spaces is Hilbertian, which - in the geometric setting - corresponds to a duality result for symmetric convex bodies in Euclidean spaces. In the present paper we define a new notion of "convexified packing," show a duality theorem for that notion, and use it to prove the duality conjecture under much milder conditions on the spaces involved (namely, that one of them is K-convex).

math.FA↗
arXiv2005

Quasiconvexity versus group invariance

The lower invariance under a given arbitrary group of diffeomorphisms extends the notion of quasiconvexity. The non-commutativity of the group operation (the function composition) modifies the classical equivalence between lower semicontinuity and quasiconvexity. In this context null lagrangians are particular cases of integral invariants of the group. Developments of parts of this paper can be found in math.FA/0105097 . Further informations at http://irmi.epfl.ch/cag/buliga_necv.html .

math.AP↗
arXiv2007

Construction of bipotentials and a minimax theorem of Fan

In Mechanics, the theory of standard materials is a well-known application of Convex Analysis. However, the so-called non-associated constitutive laws cannot be cast in the mould of the standard materials. From the mathematical viewpoint, a non associated constitutive law is a multivalued operator which is not supposed to be monotone. A possible way to study non-associated constitutive laws by using Convex Analysis, proposed first in [12], consists in constructing a "bipotential" function of two variables, which physically represents the dissipation. This is a second paper on the mathematics of the bipotentials, following math.FA/0608424 . We prove here another reconstruction theorem for a bipotential from a convex lagrangian cover, this time using a convexity notion related to a minimax theorem of Fan.

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arXiv2006

Flat cyclic Fréchet modules, amenable Fréchet algebras, and approximate identities

Let A be a locally m-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet A-module X=A_+/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that X is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally m-convex Fréchet algebra A is amenable if and only if A is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally m-convex Fréchet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable C^*-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fréchet algebra case. We also show that a quasinormable locally m-convex Fréchet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally m-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i. Some of the results of this paper were announced in ArXiv preprint math.FA/0511132.

math.FA↗
arXiv1993

Prevalence: an addendum

The authors mention work of Christensen, Mycielski, Tsujii, and others which is closely related to a survey article by the first author [math.FA/9210220].

math.FA↗
arXiv2019

Embeddability of $\ell_{p}$ and bases in Lipschitz free $p$-spaces for $0<p\leq 1$

Our goal in this paper is to continue the study initiated by the authors in [Lipschitz free $p$-spaces for $0<p<1$; arXiv:1811.01265 [math.FA]] of the geometry of the Lipschitz free $p$-spaces over quasimetric spaces for $0<p\le1$, denoted $\mathcal F_{p}(\mathcal M)$. Here we develop new techniques to show that, by analogy with the case $p=1$, the space $\ell_{p}$ embeds isomorphically in $\mathcal F_{p}(\mathcal M)$ for $0<p<1$. Going further we see that despite the fact that, unlike the case $p=1$, this embedding need not be complemented in general, complementability of $\ell_{p}$ in a Lipschitz free $p$-space can still be attained by imposing certain natural restrictions to $\mathcal M$. As a by-product of our discussion on basis in $\mathcal F_{p}([0,1])$, we obtain the first-known examples of $p$-Banach spaces for $p<1$ that possess a basis but fail to have an unconditional basis.

math.FA↗
arXiv2002

Lower order terms in Szego type limit theorems on Zoll manifolds

This is a detailed version of the paper math.FA/0212273. The main motivation for this work was to find an explicit formula for a "Szego-regularized" determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold. The idea of the Szego-regularization was suggested by V. Guillemin and K. Okikiolu. They have computed the second term in a Szego type expansion on a Zoll manifold of an arbitrary dimension. In the present work we compute the third asymptotic term in any dimension. In the case of dimension 2, our formula gives the above mentioned expression for the Szego-redularized determinant of a zeroth order PsDO. The proof uses a new combinatorial identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This identity is related to the distribution of the maximum of a random walk with i.i.d. steps on the real line. The proof of this combinatorial identity together with historical remarks and a discussion of probabilistic and algebraic connections has been published separately.

math.FA↗
arXiv2024

Holomorphic functions on the quantum polydisk and on the quantum ball

We introduce and study noncommutative (or ``quantized'') versions of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$. Specifically, for each $q\in\mathbb C\setminus\{ 0\}$ we construct Fréchet algebras $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk $\mathbb D^n$ and on the open ball $\mathbb B^n$, respectively. In the case where $0<q<1$, we establish a relation between our holomorphic quantum ball algebra $\mathcal O_q(\mathbb B^n)$ and L. L. Vaksman's algebra $C_q(\bar{\mathbb B}^n)$ of continuous functions on the closed quantum ball. Finally, we show that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are not isomorphic provided that $|q|=1$ and $n\ge 2$. This result can be interpreted as a $q$-analog of Poincaré's theorem, which asserts that $\mathbb D^n$ and $\mathbb B^n$ are not biholomorphically equivalent unless $n=1$. This paper replaces the first part of Version 1: arXiv:1508.05768v1 [math.FA].

math.FA↗
arXiv2021

Lipschitz Homotopy Groups of Contact 3-Manifolds

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb{H}^n$ with its \cc metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young (arXiv:1210.6943 [math.GT]) on the Lipschitz homotopy groups of $\mathbb{H}^1$ to an arbitrary contact 3-manifold endowed with a \cc metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(π,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

math.GT↗
arXiv2026

Comments on Symplectic bipotentials arXiv:2410.23122

This is a reaction to the article Symplectic bipotentials, in published form [2] Harakeh M, Ban M, de Saxce G. Symplectic bipotentials. Mathematics and Mechanics of Solids. 2026;0(0) doi:10.1177/10812865251413554, and in preprint form [1] arXiv:2410.23122v1. We give evidence that most of the content of the article [2] is already covered in previous works, partially cited like [7] arXiv:0810.1419 [math.FA], or uncited, like [10] arXiv:1902.04598 [math-ph], [3] arXiv:2304.14158 [math-ph], which already introduced and studied symplectic bipotentials. These comments also apply to the conference paper version of [1] arXiv:2410.23122v1, namely to the article Harakeh, M., Ban, M., de Saxce, G. (2026). Symplectic Bipotentials for the Dynamics of Dissipative Systems with Non Associated Constitutive Laws. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2025. Lecture Notes in Computer Science, vol 16034. Springer, Cham. doi:10.1007/978-3-032-03921-7_31 .

math.SG↗
arXiv2007

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.

math.NT↗
arXiv2010

A model for semantic integration of business components

Today, reusable components are available in several repositories. These last are certainly conceived for the reusing However, this re-use is not immediate; it requires, in the fact, to pass through some essential conceptual operations, among them in particular, research, integration, adaptation, and composition. We are interested in the present work to the problem of semantic integration of heterogeneous Business Components. This problem is often put in syntactical terms, while the real stake is of semantic order. Our contribution concerns a model proposal for Business components integration as well as resolution method of semantic naming conflicts, met during the integration of Business Components.

cs.SE↗
arXiv2015

Fourier series in weighted Lorentz spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces.

math.FA↗
arXiv2023

Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part I

We study the approximation of functions by tensor networks (TNs). We show that Lebesgue $L^p$-spaces in one dimension can be identified with tensor product spaces of arbitrary order through tensorization. We use this tensor product structure to define subsets of $L^p$ of rank-structured functions of finite representation complexity. These subsets are then used to define different approximation classes of tensor networks, associated with different measures of complexity. These approximation classes are shown to be quasi-normed linear spaces. We study some elementary properties and relationships of said spaces. In part II of this work, we will show that classical smoothness (Besov) spaces are continuously embedded into these approximation classes. We will also show that functions in these approximation classes do not possess any Besov smoothness, unless one restricts the depth of the tensor networks. The results of this work are both an analysis of the approximation spaces of TNs and a study of the expressivity of a particular type of neural networks (NN) -- namely feed-forward sum-product networks with sparse architecture. The input variables of this network result from the tensorization step, interpreted as a particular featuring step which can also be implemented with a neural network with a specific architecture. We point out interesting parallels to recent results on the expressivity of rectified linear unit (ReLU) networks -- currently one of the most popular type of NNs.

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