Luca Brandolini's Books

Hormander Operators

By: Marco Bramanti,Luca Brandolini

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Hörmander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hörmander and is intimately connected to the geometry of vector fields.Motivations for the study of Hörmander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hörmander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.

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Page Count

722

Publisher

World Scientific

Published Date

ISBN 10

9811261709

ISBN 13

9789811261701

Fundamental Solutions and Local Solvability for Nonsmooth Hormander's Operators

By: Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni

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Fundamental Solutions and Local Solvability for Nonsmooth Hormander's Operators

By: Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni

The authors consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that also possesses second derivatives, and they deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . The authors also prove estimates on this solution.

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American Mathematical Soc.

Published Date

ISBN 10

1470425599

ISBN 13

9781470425593

Borders and the Politics of Space in Late Medieval Italy

Milan, Venice, and their Territories

By: Luca Zenobi

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Borders and the Politics of Space in Late Medieval Italy

Milan, Venice, and their Territories

By: Luca Zenobi

Space matters. It situates our history, structures our daily lives, and often determines what we can and cannot do. Borders are central to this reality. Tools and symbols of separation, power, and identity, they bring people together as much as they set them apart. This book explores how borders were understood, made, and encountered at the end of the Middle Ages, and what they can tell us about the spatial fabric of society at the threshold of modernity. It shows that pre-modern borders were nothing like the fuzzy lines they are typically made out to be, that border-making was rarely a top-down process and should instead be studied as an interactive endeavour, and that space was shaped by communities far more than states in this period. At its core, Borders and the Politics of Space in Late Medieval Italy is the account of a frontier which would mark the Italian peninsula for centuries, that between the territories of the Duchy of Milan and those of the Republic of Venice. But it is also a study of how rulers and subjects alike defined spaces they could call their own. Luca Zenobi combines methods from several disciplines and applies them to a range of evidence from twenty different libraries and archives, including theoretical treatises and pragmatic records, written chronicles and cartographic visualisations, private documents and official correspondence. The cast of characters is equally eclectic, featuring influential thinkers and pragmatic statesmen, zealous factions and clumsy bureaucrats, hopeless beggars and ambitious princes. On the border, their stories intersect and reveal their part in a shared history.

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Page Count

282

Publisher

Oxford University Press

Published Date

ISBN 10

0198876882

ISBN 13

9780198876885

A Visual Atlas for Soil Micromorphologists

By: Eric P. Verrecchia,Luca Trombino

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A Visual Atlas for Soil Micromorphologists

By: Eric P. Verrecchia,Luca Trombino

This open access atlas is an up-to-date visual resource on the features and structures observed in soil thin sections, i.e. soil micromorphology. The book addresses the growing interest in soil micromorphology in the fields of soil science, earth science, archaeology and forensic science, and serves as a reference tool for researchers and students for fast learning and intuitive feature and structure recognition. The book is divided into six parts and contains hundreds of images and photomicrographs. Part one is devoted to the way to sample properly soils, the method of preparation of thin sections, the main tool of soil micromorphology (the microscope), and the approach of soil micromorphology as a scientific method. Part two focuses on the organisation of soil fragments and presents the concept of fabric. Part three addresses the basic components, e.g. rocks, minerals, organic compounds and anthropogenic features. Part four lists all the various types of pedogenic features observed in a soil, i.e. the imprint of pedogenesis. Part five gives interpretations of features associated with the main processes at work in soils and paleosols. Part six presents a view of what the future of soil micromorphology could be. Finally, the last part consists of the index and annexes, including the list of mineral formulas. This atlas will be of interest to researchers, academics, and students, who will find it a convenient tool for the self-teaching of soil micromorphology by using comparative photographs.

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Page Count

184

Publisher

Springer Nature

Published Date

ISBN 10

3030678067

ISBN 13

9783030678067

Hormander Operators

By: Marco Bramanti,Luca Brandolini

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Hormander Operators

By: Marco Bramanti,Luca Brandolini

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Page Count

722

Publisher

World Scientific

Published Date

ISBN 10

9811261709

ISBN 13

9789811261701

Fundamental Solutions and Local Solvability for Nonsmooth Hörmander's Operators

By: Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni

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Fundamental Solutions and Local Solvability for Nonsmooth Hörmander's Operators

By: Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni

The authors consider operators of the form L=\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \mathbb{R}^{p} where X_{0},X_{1},\ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \gamma for L and provide growth estimates for \gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \gamma also possesses second derivatives, and the.

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ISBN 10

1470441314

ISBN 13

9781470441319

Book's by Luca Brandolini

Hormander Operators

Hormander Operators

Fundamental Solutions and Local Solvability for Nonsmooth Hormander's Operators

Fundamental Solutions and Local Solvability for Nonsmooth Hormander's Operators

Borders and the Politics of Space in Late Medieval Italy

Borders and the Politics of Space in Late Medieval Italy

A Visual Atlas for Soil Micromorphologists

A Visual Atlas for Soil Micromorphologists

Hormander Operators

Hormander Operators

Fundamental Solutions and Local Solvability for Nonsmooth Hörmander's Operators

Fundamental Solutions and Local Solvability for Nonsmooth Hörmander's Operators

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