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Philippe Bouafia

Publications and source records attributed to Philippe Bouafia.

11 recordsLinked to original sources
arXiv2026

Magnitude and diversity of trees

We compute the magnitude (an isometric invariant of metric spaces) of compact $\mathbb{R}$-trees and show that it equals $1 + L/2$, where $L \in [0, \infty]$ denotes the total length. Although length is the only geometric invariant captured by magnitude, we show that diversity-maximizing measures on compact $\mathbb{R}$-trees are more sensitive to the branching structure as they tend to be more concentrated toward the leaves: their support contains no branch points. In the finite case, we further show that maximum diversity on a weighted tree can be computed in polynomial time.

math.MG↗
arXiv2026

Non-absolute integration and application to Young geometric integration

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on Young geometric integration, namely the integration of certain generalized differential forms over $m$-dimensional subsets of $\mathbb{R}^d$. This is achieved by introducing appropriate notions of chains and cochains, in the spirit of Whitney's geometric integration theory.

math.FA↗
arXiv2025

Fractional currents and Young geometric integration

We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by H\"older continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $\alpha$-fractional currents is in duality with a class of cochains, $\alpha$-fractional charges, that extend both Whitney's flat cochains and $\alpha$-H\"older continuous forms. We construct a partially defined wedge product between fractional charges, enabling a generalization of the Young integral to arbitrary dimensions and codimensions. This helps us identify $\alpha$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+\alpha)/(m+1)})$.

math.FA↗
arXiv2024

On the exterior product of H\"older differential forms

We introduce a complex of cochains, $\alpha$-fractional charges ($0 < \alpha \leq 1$), whose regularity is between that of De Pauw-Moonens-Pfeffer's charges and that of Whitney's flat cochains. We show that $\alpha$-H\"older differential forms and their exterior derivative can be realized as $\alpha$-fractional charges, and that it is possible to define the exterior product between an $\alpha$-fractional and a $\beta$-fractional charge, under the condition that $\alpha + \beta > 1$. This construction extends the Young integral in arbitrary dimension and codimension.

math.DG↗
arXiv2024

A regularity property of fractional Brownian sheets

A function $f$ defined on $[0, 1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1, \dots,x_d)$ equals the flux of $v$ through the rectangle $[0, x_1] \times \cdots \times [0, x_d]$ for all $(x_1, \dots, x_d) \in [0, 1]^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d \geq 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1, \dots, H_d)$ are shown to be almost surely strongly chargeable whenever \[ \frac{H_1 + \cdots + H_d}{d} > \frac{d - 1}{d}. \]

math.PR↗
arXiv2024

Young Integration with respect to H\"older Charges

We present a multidimensional Young integral that enables to integrate H\"older continuous functions with respect to a H\"older charge. It encompasses the integration of H\"older differential forms introduced by R. Z\"ust: if $f$, $g_1, \dots, g_d$ are merely H\"older continuous functions on the cube $[0, 1]^d$ whose H\"older exponents satisfy a certain condition, it is possible to interpret $\mathrm{d}g_1 \wedge \cdots \wedge \mathrm{d}g_d$ as a H\"older charge and thus to make sense of the integral \[ \int_B f \mathrm{d} g_1 \wedge \cdots \wedge \mathrm{d}g_d \] over a set $B \subset [0, 1]^d$ of finite perimeter.

math.FA↗
arXiv2021

A representation formula for members of SBV dual

We give an integral representation formula for members of the dual of $SBV(\mathbb{R}^n)$ in terms of functions that are defined on $\hat{\mathbb{R}}^n$, an appropriate fiber space that we introduce, consisting of pairs $(x,[E]_x)$ where $[E]_x$ is an approximate germ of an $(n-1)$-rectifiable set $E$ at $x$.

math.CA↗
arXiv2021

Localizable locally determined measurable spaces with negligibles

We study measurable spaces equipped with a $σ$-ideal of negligible sets. We find conditions under which they admit a localizable locally determined version -- a kind of fiber space that describes locally their directions -- defined by a universal property in an appropriate category that we introduce. These methods allow to promote each measure space $(X, \mathcal{A},μ)$ to a strictly localizable version $(\hat{X}, \hat{\mathcal{A}},\hatμ)$, so that the dual of $L_1(X, \mathcal{A}, μ)$ is $L_\infty(\hat{X},\hat{\mathcal{A}},\hatμ)$. Corresponding to this duality is a generalized Radon-Nikodým theorem. We also provide a characterization of the strictly localizable version in special cases that include integral geometric measures, when the negligibles are the purely unrectifiable sets in a given dimension.

math.CA↗