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Pierre-Olivier Parisé

Publications and source records attributed to Pierre-Olivier Parisé.

14 recordsLinked to original sources
arXiv2026

An Invariant Tangent-Angle Descriptor and a Band U-Net for 2D Fragment Adjacency Prediction

This paper addresses the prediction of adjacency between pairs of 2D fragments based on their contours. We improved the two-stage architecture proposed in Beaulac's thesis, in which a rotation-equivariant Siamese convolutional neural network scores pairs of local image windows along the two contours of two fragments. The scores are gathered in an adjacency matrix in which a ResNet detects the partial anti-diagonal band that reveals the adjacency of two fragments. In the current work, we keep the pipeline and replace the local score by a comparison of tangent-angle profiles of contour windows, making it, by construction, invariant to fragment rotation and agnostic to the selected contour-starting point. These adaptations may be either a training-free likelihood ratio or a small one-dimensional convolutional model trained on corresponding points. We also replaced the final classifier by a band U-Net that segments the band and classifies the pair, so that the shared arc is obtained along with the decision. In the synthetic data set of the original thesis, the tangent descriptor performs as well as or better than the image-window approach in all tested configurations. The proposed pipeline reaches an accuracy of 98%, vs 93% to 95% for the original approach once its evaluation is corrected. We tested our pipeline, with models trained only on synthetic data, on the PairingNet benchmark, and obtained an AUC of 0.93. Furthermore, under the PairingNet pair-searching protocol conditions, our learned descriptor obtains a Recall@10 of 0.82 on the real set against 0.56 from the best model of the original paper.

cs.CV↗
arXiv2026

Classes of Holomorphic Multicomplex-Valued Functions Generated by Elliptic-Admissible Involutions

We classify and count the real-algebra involutions of the multicomplex algebra $\mathbb{M}\mathbb{C} (n)$ that map each element of its canonical monomial basis to a signed monomial, using a matrix model over $\mathbb F_2$. An \emph{elliptic-admissible pair} consists of such an involution $σ$ and a monomial unit $\mathbf{i}\in\mathbb{I}(n)$ satisfying $\mathbf{i^2}=-1$ and $σ(\mathbf{i})=-\mathbf{i}$. The fixed algebra of $σ$ is a real form for the complex structure $\mathcal{L}_{\mathbf{i}}$ defined by multiplication by $\mathbf{i}$, and the pair yields a Cauchy--Riemann system. We prove that its solution class depends only on $\mathbf{i}$, not on $σ$, and is exactly the class of mappings holomorphic with respect to $\mathcal{L}_{\mathbf{i}}$. Multicomplex holomorphy is recovered as the intersection of the classes associated with the elementary generators. We obtain the analogous characterization for anti-holomorphic classes, describe twisted systems intertwining two such complex structures, and prove that every $\mathbf{i}{}$-holomorphic or $\mathbf{i}$-anti-holomorphic mapping is componentwise harmonic (solutions to Laplace's equation).

math.RA↗
arXiv2026

Weighted Hardy Inequalities for Nested Averages

We study a family of Hardy-type inequalities for weighted averages over nested subsets of a measure space. Given a partition of a measure space and a weight function $m$, we consider operators of the form \[ f \mapsto \frac{1}{M_n}\int_{X^{(n)}} m(x)f(x)\,\mathrm{d}μ(x), \] with additional weights on the resulting sequence of averages. In particular, we generalize an inequality obtained by Vincent and Sohani in \cite{VincentSohani2025} and characterize the boundedness in terms of the finiteness of a single testing quantity $β$. We also provide two-sided estimates for the best constant $C_{\mathrm{opt}}$, namely \[ β\leq C_{\mathrm{opt}} \leq p^{1/q} (p')^{1/p'}β\leq 2β. \] Thus the characterization is never off by more than a factor of 2. We also develop a second approach, inspired by Broadbent's proof of Hardy's inequality, which gives a local sufficient condition that often provides sharper constants and recovers several important cases, including the classical weighted Hardy inequality.

math.CA↗
arXiv2025

Rational lemniscates and the matching problem

The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of the curve, which are continuous up to the curve and complex conjugate to each other on the curve. We prove that there exist Jordan curves of any Hausdorff dimension between $1$ and $2$ for which the matching problem has a solution. This answers a question of Ebenfelt--Khavinson--Shapiro and provides the first examples of solutions to the matching problem other than rational lemniscates. Our approach relies on conformal welding and harmonic measure. We also obtain new examples of Jordan curves for which the matching problem has no solution, and give a characterization of the subsets of the Riemann sphere that are rational lemniscates in terms of harmonic measure.

math.CV↗
arXiv2025

Classification of Principle 3D Slices of Filled-in Julia Sets in Multicomplex Spaces

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals sets on a computer. We therefore introduce an equivalence relation between 3D representations and show that, for the filled-in Julia sets associated to the polynomial $z^p + c$, there are nine 3D slices when $p$ is an odd integer and four when $p$ is even. These results differs from the recent characterization obtained by Brouillette and Rochon in 2019 and the proofs require different arguments in the context of the filled-in Julia sets.

math.DS↗
arXiv2023

Kernel-Summability Methods and the Silverman-Toeplitz Theorem

We introduce kernel-summability methods in Banach spaces using the vector-valued integrals and prove an analogue of the Silverman-Toeplitz Theorem for regular kernel-summability methods. We also show that if $X$ is a Banach space and one kernel-summability method is included in another kernel-summability method for scalar-valued functions, then the first method is included in the second method, for $X$-valued functions. This extends a previous result from Javad Mashreghi, Thomas Ransford and the author. We then apply these abstract results to the summability of Taylor series of functions in a Banach space of holomorphic functions on the unit disk.

math.FA↗
arXiv2023

On the Divergence of Taylor Series in de Branges-Rovnyak Spaces

It is known that there exist functions in certain de Branges--Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Cesàro means.

math.CV↗
arXiv2022

Involutions of Bicomplex Numbers

An involution of a real commutative algebra $A$ is a real-linear homomorphism $f : A \rightarrow A$ such that $f^2 = \mathrm{Id}$. We show that there are six involutions of the algebra of bicomplex numbers, contrary to the actual number of four stated in the literature. We also characterize $n$-involutions satisfying the additional property $f^n = \mathrm{Id}$ for some integer $n \geq 2$. We show there are eight $n$-involutions and they occur only for $n = 2$ and $n= 4$. We use our result to give a new characterization of the invertible elements of the algebra of bicomplex numbers.

math.RA↗
arXiv2021

Power-series summability methods in de Branges-Rovnyak spaces

We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Cesàro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.

math.FA↗
arXiv2020

Cesàro summability of Taylor series in weighted Dirichlet spaces

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.

math.CV↗
arXiv2020

Failure of Approximation of Odd Functions by Odd Polynomials

We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$. As a consequence, there exists a function $f$ in $H$ that lies outside the closed linear span of its Taylor partial sums $s_n(f)$, so it cannot be approximated by any triangular summability method applied to the $s_n(f)$. We also show that there exists a function $f$ in $H$ that lies outside the closed linear span of its radial dilates $f_r, ~r<1$.

math.FA↗
arXiv2017

Tricomplex dynamical systems generated by polynomials of odd degree

In this article, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial $z^p+c$ where $z$ and $c$ are complex numbers and $p > 2$ is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial $η^p+c$ where $p > 2$ is an odd integer.

math.GM↗
arXiv2017

Tricomplex dynamical systems generated by polynomials of even degree

In this article, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial $z^p+c$ where $z$ and $c$ are complex numbers and $p \geq 2$ is an even integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the tricomplex polynomial $η^p+c$ where $p \geq 2$ is an even integer.

math.GM↗