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Tu Nguyen
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We answer in the affirmative a question posed by Landis and Oleinik on unique continuation of variable coefficients parabolic equations.
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We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepys Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
In this paper we study various aspects of the Ekedahl-Serre problem. We formulate questions of Ekedahl-Serre type and Coleman-Oort type for general weakly special subvarieties in the Siegel moduli space, propose a conjecture relating these two questions, and provide examples supporting these questions. The main new result is an upper bound of genera for curves over number fields whose Jacobians are isogeneous to products of elliptic curves satisfying the Sato-Tate equidistribution, and we also refine previous results showing that certain weakly special subvarieties only meet the open Torelli locus in at most finitely many points.
We make several improvements to methods for finding integer solutions to $x^3+y^3+z^3=k$ for small values of $k$. We implemented these improvements on Charity Engines global compute grid of 500,000 volunteer PCs and found new representations for several values of $k$, including $k=3$ and $k=42$. This completes the search begun by Miller and Woollett in 1954 and resolves a challenge posed by Mordell in 1953.
Answering questions related to art pieces (paintings) is a difficult task, as it implies the understanding of not only the visual information that is shown in the picture, but also the contextual knowledge that is acquired through the study of the history of art. In this work, we introduce our first attempt towards building a new dataset, coined AQUA (Art QUestion Answering). The question-answer (QA) pairs are automatically generated using state-of-the-art question generation methods based on paintings and comments provided in an existing art understanding dataset. The QA pairs are cleansed by crowdsourcing workers with respect to their grammatical correctness, answerability, and answers correctness. Our dataset inherently consists of visual (painting-based) and knowledge (comment-based) questions. We also present a two-branch model as baseline, where the visual and knowledge questions are handled independently. We extensively compare our baseline model against the state-of-the-art models for question answering, and we provide a comprehensive study about the challenges and potential future directions for visual question answering on art.
The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Omega)$ compactly embeds in $W^{-1,q}(Omega)$, for every $q < 2$ and for any open and bounded set $Omega$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities.
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