Stochastic and Integral Geometry de Rolf Schneider

Stochastic and Integral Geometry

Autor Rolf Schneider, Wolfgang Weil

en Limba Engleză Paperback – 25 noi 2010

EDIȚIA: Publicat în seria Probability and Its Applications, volumul Stochastic and Integral Geometry din 2008 reprezintă o sinteză riguroasă a fundamentelor matematice necesare pentru modelarea structurilor geometrice aleatorii. Față de abordările introductive, această ediție consolidează legătura dintre geometria integrală și modelele stochastice, introducând capitole esențiale despre modelele non-staționare și transformările geometrice integrale.

Subliniem faptul că autorii Rolf Schneider și Wolfgang Weil au structurat materialul pentru a servi cercetătorilor, punând accent pe modelele staționare — seturi aleatorii, procese punctuale și mozaicuri. Reținem organizarea logică a volumului, care debutează cu bazele geometriei stochastice și progresează spre concepte extinse de geometrie integrală, incluzând măsuri invariante și fapte din geometria convexă. Cititorii familiarizați cu Stochastic Geometry de A. Baddeley vor aprecia aici profunzimea tratamentului teoretic al mediilor geometrice în raport cu măsurile invariante, oferind o bază matematică mult mai densă pentru aplicațiile în științele naturii sau inginerie.

În contextul operei lui Rolf Schneider, acest tratat reprezintă punctul culminant al cercetărilor sale în geometria convexă și stochastică, fiind o lucrare de referință net diferită de incursiunile sale în traduceri literare precum König Lear. Volumul reușește să demonstreze interdependența dintre probabilitățile geometrice și geometria integrală, oferind note detaliate la finalul fiecărei secțiuni pentru a ghida cititorul prin literatura de specialitate vastă.

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Specificații

ISBN-13: 9783642097669
ISBN-10: 3642097669
Pagini: 708
Ilustrații: XII, 694 p.
Dimensiuni: 155 x 235 x 38 mm
Greutate: 1.05 kg
Ediția:Softcover reprint of hardcover 1st ed. 2008
Editura: Springer
Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

De ce să citești această carte

Recomandăm acest volum cercetătorilor care au nevoie de un fundament matematic solid pentru analiza structurilor spațiale aleatorii. Spre deosebire de manualele aplicate, această lucrare oferă demonstrații riguroase și o acoperire sistematică a mozaicurilor și proceselor punctuale. Este o resursă esențială pentru a înțelege cum geometria integrală devine un instrument de calcul în probabilitățile moderne.

Despre autor

Rolf Schneider este Profesor Emeritus la Facultatea de Matematică și Fizică din cadrul Universității Albert-Ludwigs din Freiburg, Germania. Este recunoscut la nivel internațional pentru contribuțiile sale fundamentale în geometria convexă și geometria stochastică. Wolfgang Weil, co-autor al acestui tratat, a colaborat extensiv cu Schneider pentru a defini standardele moderne în studiul măsurilor invariante și al modelelor geometrice aleatorii, poziționând școala germană de matematică în avangarda acestui domeniu interdisciplinar.

Descriere scurtă

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry – random sets, point processes, random mosaics – and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

Cuprins

Foundations of Stochastic Geometry.- Prolog.- Random Closed Sets.- Point Processes.- Geometric Models.- Integral Geometry.- Averaging with Invariant Measures.- Extended Concepts of Integral Geometry.- Integral Geometric Transformations.- Selected Topics from Stochastic Geometry.- Some Geometric Probability Problems.- Mean Values for Random Sets.- Random Mosaics.- Non-stationary Models.- Facts from General Topology.- Invariant Measures.- Facts from Convex Geometry.

Recenzii

From the reviews:
“In ‘Stochastic and Integral Geometry,’ R. Schneider and W. Weil give priority to the basic concepts in stochastic geometry … while keeping from integral geometry only what is relevant for applications in stochastic geometry. … Each chapter section is concluded by notes in which the main references are cited and numerous possible extensions are discussed. … Stochastic and Integral Geometry is a profound work by two eminent specialists which is essential reading for those willing to learn deep theory.” (Pierre Calka, Mathematical Geosciences, Vol. 45, 2013)
“This book … provides the systematic and exhaustive account of mathematical foundations of stochastic geometry with particular emphasis on tools from convex geometry. … The thorough and up-to-date presentation in this text makes it an invaluable source for researchers pursuing studies not only in stochastic geometry, but also in convex geometry and various applications … . an absolutely indispensable part of all mathematical libraries. … also beneficial for personal collections of all mathematicians who ever deal with probability measures on spaces of geometric objects.” (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1175, 2010)
“The book presents a number of results that are otherwise scattered among an immense number of research papers and mostly provides full proofs for them. … The most remarkable aspect of the book is the reader-friendly structure and the style in which it has been written. The book is also worth owning not only for those working in stochastic geometry and immediately related fields of theoretical and applied probability and spatial statistics. … This book … will be an essential part of every mathematical library.” (V. K. Oganyan, Mathematical Reviews, Issue 2010 g)

Notă biografică

Rolf Schneider: Born 1940, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1964, PhD 1967 (Frankfurt), Habilitation 1969 (Bochum), 1970 Professor TU Berlin, 1974 Professor Univ. Freiburg, 2003 Dr. h.c. Univ. Salzburg, 2005 Emeritus
Wolfgang Weil: Born 1945, Studies of Mathematics and Physics in Frankfurt/M, Diploma 1968, PhD 1971 (Frankfurt), Habilitation 1976 (Freiburg), 1978 Akademischer Rat Univ. Freiburg, 1980 Professor Univ. Karlsruhe

Textul de pe ultima copertă

Stochastic geometry has in recent years experienced considerable progress, both in its applications to other sciences and engineering, and in its theoretical foundations and mathematical expansion. This book, by two eminent specialists of the subject, provides a solid mathematical treatment of the basic models of stochastic geometry -- random sets, point processes of geometric objects (particles, flats), and random mosaics. It develops, in a measure-theoretic setting, the integral geometry for the motion and the translation group, as needed for the investigation of these models under the usual invariance assumptions. A characteristic of the book is the interplay between stochastic and geometric arguments, leading to various major results. Its main theme, once the foundations have been laid, is the quantitative investigation of the basic models. This comprises the introduction of suitable parameters, in the form of functional densities, relations between them, and approaches to their estimation. Much additional information on stochastic geometry is collected in the section notes.
As a combination of probability theory and geometry, the volume is intended for readers from either field. Probabilists with interest in random spatial structures, or motivated by the prospect of applications, will find an in-depth presentation of the geometric background. Geometers can see integral geometry "at work" and may be surprised to learn how classical results from convex geometry have elegant applications in a stochastic setting.

Caracteristici

First book since Santalo's classic 1976 to combine stochastic geometry and integral geometry. It presents rigorous foundations of the models of stochastic geometry as well as of the tools from integral geometry, and supplies with clear, complete, and comprehensible proofs of the major results. Includes supplementary material: sn.pub/extras