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Lucas Oliveira

Publications and source records attributed to Lucas Oliveira.

16 recordsLinked to original sources
arXiv2026

Modular Forms and Numerical Explorations of Rational Approximations to $\zeta(3)$

We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Ap\'ery approximations and satisfy the same denominator-growth estimate needed in Beukers' irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

math.NT↗
arXiv2024

David regularity of the Yoccoz extension

A central problem in the study of critical circle dynamics is understanding the regularity of Yoccoz conjugators - circle homeomorphisms that conjugate critical circle maps with irrational rotation numbers to their corresponding rigid rotations. One can approach this problem from a different angle by studying the regularity of extensions of these maps to the unit disk. Of particular interest is the question of when such a conjugator admits a David extension. Building on the work of Petersen and Zakeri, we classify the David regularity of a specific extension process known as the Yoccoz extension.

math.DS↗
arXiv2024

On the Uniqueness of the Norton-Sullivan Quasiconformal extension

We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.

math.CA↗
arXiv2023

Sharp embeddings between weighted Paley-Wiener spaces

In this paper we address the problem of estimating the operator norm of the embeddings between multidimensional weighted Paley-Wiener spaces. These can be equivalently thought as Fourier uncertainty principles for bandlimited functions. By means of radial symmetrization mechanisms, we show that such problems can all be shifted to dimension one. We provide precise asymptotics in the general case and, in some particular situations, we are able to identify the sharp constants and characterize the extremizers. The sharp constant study is actually a consequence of a more general result we prove in the setup of de Branges spaces of entire functions, addressing the operator given by multiplication by $z^k$, $k \in \mathbb{N}$. Applications to sharp higher order Poincar\'{e} inequalities and other related extremal problems are discussed.

math.CA↗
arXiv2022

On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schrödinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.

math.CA↗
arXiv2021

CORAA: a large corpus of spontaneous and prepared speech manually validated for speech recognition in Brazilian Portuguese

Automatic Speech recognition (ASR) is a complex and challenging task. In recent years, there have been significant advances in the area. In particular, for the Brazilian Portuguese (BP) language, there were about 376 hours public available for ASR task until the second half of 2020. With the release of new datasets in early 2021, this number increased to 574 hours. The existing resources, however, are composed of audios containing only read and prepared speech. There is a lack of datasets including spontaneous speech, which are essential in different ASR applications. This paper presents CORAA (Corpus of Annotated Audios) v1. with 290.77 hours, a publicly available dataset for ASR in BP containing validated pairs (audio-transcription). CORAA also contains European Portuguese audios (4.69 hours). We also present a public ASR model based on Wav2Vec 2.0 XLSR-53 and fine-tuned over CORAA. Our model achieved a Word Error Rate of 24.18% on CORAA test set and 20.08% on Common Voice test set. When measuring the Character Error Rate, we obtained 11.02% and 6.34% for CORAA and Common Voice, respectively. CORAA corpora were assembled to both improve ASR models in BP with phenomena from spontaneous speech and motivate young researchers to start their studies on ASR for Portuguese. All the corpora are publicly available at https://github.com/nilc-nlp/CORAA under the CC BY-NC-ND 4.0 license.

cs.CL↗
arXiv2020

Fold maps associated to geodesic random walks on non-positively curved manifolds

We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold. We show that for odd powers of the unit tangent sphere the mappings are fold maps. Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.

math.DG↗
arXiv2019

Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

For $ξ= (ξ_1, ξ_2, \ldots, ξ_d) \in \mathbb{R}^d$ let $Q(ξ) := \sum_{j=1}^d σ_j ξ_j^2$ be a quadratic form with signs $σ_j \in \{\pm1\}$ not all equal. Let $S \subset \mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = \big\{(ξ, τ) \in \mathbb{R}^{d}\times \mathbb{R} \ : \ τ= Q(ξ)\big\}$. In this note we prove that Gaussians never extremize an $L^p(\mathbb{R}^d) \to L^{q}(\mathbb{R}^{d+1})$ Fourier extension inequality associated to this surface.

math.CA↗
arXiv2018

Analysis of hyper-singular, fractional, and order-zero singular integral operators

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional derivatives, pseudodifferential operators, Calderón-Zygmund operators, and many others. The main results of this article are built around the notion of an operator calculus that connects operators with different kernel singularities via vanishing moment conditions and composition with fractional derivative operators. We also provide several boundedness results on weighted and unweighted distribution spaces, including homogeneous Sobolev, Besov, and Triebel-Lizorkin spaces, that are necessary and sufficient for the operator's vanishing moment properties, as well as certain behaviors for the operator under composition with fractional derivative and integral operators. As applications, we prove $T1$ type theorems for singular integral operators with different singularities, boundedness results for pseudodifferential operators belonging to the forbidden class $S_{1,1}^0$, fractional order and hyper-singular paraproduct boundedness, a smooth-oscillating decomposition for singular integrals, sparse domination estimates that quantify regularity and oscillation, and several operator calculus results. It is of particular interest that many of these results do not require $L^2$-boundedness of the operator, and furthermore, we apply our results to some operators that are known not to be $L^2$-bounded.

math.FA↗
arXiv2017

Design of an Integrated Analytics Platform for Healthcare Assessment Centered on the Episode of Care

Assessing care quality and performance is essential to improve healthcare processes and population health management. However, due to bad system design and lack of access to required data, this assessment is often delayed or not done at all. The goal of our research is to investigate an advanced analytics platform that enables healthcare quality and performance assessment. We used a user-centered design approach to identify the system requirements and have the concept of episode of care as the building block of information for a key performance indicator analytics system. We implemented architecture and interface prototypes, and performed a usability test with hospital users with managerial roles. The results show that by using user-centered design we created an analytical platform that provides a holistic and integrated view of the clinical, financial and operational aspects of the institution. Our encouraging results warrant further studies to understand other aspects of usability.

cs.CY↗
arXiv2017

Bilinear Spherical Maximal Function

We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of $L^p$ with $p<1$. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.

math.CA↗
arXiv2013

Weighted Multilinear Square Function Bounds

In this work we study boundedness of Littlewood-Paley-Stein square func- tions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolu- tion operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.

math.FA↗
arXiv2012

Multilinear local Tb for Square functions

In the present work we extend a local Tb theorem for square functions of Christ and Hofmann to the multilinear setting. We also present new BMO type interpolation result for square functions associated to multilinear operators. These square function bounds are applied to prove a multilinear local Tb theorem for singular integral operators.

math.CA↗
arXiv2009

A Note on Maximal Averages in the Plane

Using variants of the TT* method we give a self-contained proof of the result of Alfonseca, Soria and Vargas on maximal operators on arbitrary directions in $\rr^2$. We also give a sharp $L^2$ estimate for a maximal function extending a Theorem of Cordoba.

math.CA↗