Search results for "proof"
showing 10 items of 187 documents
Abstract and concrete tangent modules on Lipschitz differentiability spaces
2020
We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli'…
The Fatou coordinate for parabolic Dulac germs
2017
We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.
Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures
2008
In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.
Entropy, Lyapunov exponents, and rigidity of group actions
2018
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems…
Rectifiability of RCD(K,N) spaces via δ-splitting maps
2021
In this note we give simplified proofs of rectifiability of RCD(K,N) spaces as metric measure spaces and lower semicontinuity of the essential dimension, via -splitting maps. The arguments are inspired by the Cheeger-Colding theory for Ricci limits and rely on the second order differential calculus developed by Gigli and on the convergence and stability results by Ambrosio-Honda. peerReviewed
Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type
2021
The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.
Energy-Efficiency and Coverage Quality Management for Reliable Diagnostics in Wireless Sensor Networks
2020
International audience; The processing of data and signals provided by sensors aims at extracting rnrelevant features which can be used to assess and diagnose the health state rnof the monitored targets. Nevertheless, Wireless Sensor Networks (WSNs) present rna number of shortcomings that have an impact on the quality of the gathered rndata at the sink level, leading to imprecise diagnostics rnof the observed targets. To improve data accuracy, two main critical and related issues, namely the energy consumption and coverage quality, need to be considered. The goal is to maximize the network lifetime while guaranteeing the complete coverage of all the targets. Unfortunately, these performance…
RFID passive gas sensor integrating carbon nanotubes
2011
Carbon nanotube (CNT) composites are sensitive to the presence of gases due to their high surface-to-volume ratio and hollow structure that are well suited for gas molecule absorption and storage. Such sensing capability is here integrated with UHF RF identification (RFID) technology to achieve passive and low-cost sensors, remotely readable. CNT film (buckypaper) is used as a localized variable resistive load integrated into a tag antenna, which becomes able to transduce the presence of hazardous gas in the environment, ammonia in this case, into a change of its electromagnetic features. The dynamic range and the hysteresis of the radio sensor are investigated by simulations, equivalent ci…
Razonamientos no rigurosos y demostraciones asistidas por ordenador
2007
RESUMENPresentamos la contribución de Th. Tymoczko a la filosofía de la matemática y analizamos y evaluamos las demostraciones asistidas por ordenador y los razonamientos no rigurosos en la matemática experimental, con particular referencia al Teorema de los Cuatro Colores.PALABRAS CLAVETYMOCZKO – CUASI-EMPIRISMO – MATEMÁTICA EXPERIMENTAL – RAZONAMIENTO NO RIGUROSO – DEMOSTRACIONES ASISTIDAS POR ORDENADORABSTRACTWe present Th. Tymoczko’s contribution to the philosophy of mathematics, and we analyze and evaluate the computer-assisted proofs and the non-rigorous reasonings in the experimental mathematics, particularly in reference to the Four-Colour Theorem.KEYWORDTYMOCZKO – QUASI-EMPIRICISM …
Heuristics and Memory Strategies Used by Mathematicians
1996
The study of the cognitive processes involved in learning and acquisition of technically complex material is a main focus of interest for basic and applied research. Our research program tries to identh memory aids and heuristic training strategies useful for improving mathematics performance. Part of the effectiveness of a course, designed by taking into account knowledge about the cognitive system, has to do with the development of an adequate relationship with the belief system of the learner. As a first step in that direction, we present a survey of the opinions of a group of mathematicians about the dd€iculty of their subjecr matter, the strategies they use spontaneously to overcome di…