05136nam 22005655 450 991025428070332120220407000254.03-319-44706-810.1007/978-3-319-44706-3(CKB)3710000001406102(DE-He213)978-3-319-44706-3(MiAaPQ)EBC4873442(PPN)202991288(EXLCZ)99371000000140610220170607d2017 u| 0engurnn#008mamaatxtrdacontentcrdamediacrrdacarrierFractal zeta functions and fractal drums higher-dimensional theory of complex dimensions /by Michel L. Lapidus, Goran Radunović, Darko Žubrinić1st ed. 2017.Cham :Springer International Publishing :Imprint: Springer,2017.1 online resource (XL, 655 p. 55 illus., 10 illus. in color.)Springer Monographs in Mathematics,1439-73823-319-44704-1 Includes bibliographical references at the end of each chapters and indexes.Overview -- Preface -- List of Figures -- Key Words -- Selected Key Results -- Glossary -- 1. Introduction -- 2 Distance and Tube Zeta Functions -- 3. Applications of Distance and Tube Zeta Functions -- 4. Relative Fractal Drums and Their Complex Dimensions -- 5.Fractal Tube Formulas and Complex Dimensions -- 6. Classification of Fractal Sets and Concluding Comments -- Appendix A. Tame Dirchlet-Type Integrals -- Appendix B. Local Distance Zeta Functions -- Appendix C. Distance Zeta Functions and Principal Complex Dimensions of RFDs -- Acknowledgements -- Bibliography -- Author Index -- Subject Index. .This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions. The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.Springer Monographs in Mathematics,1439-7382Number theoryMeasure theoryMathematical physicsNumber Theoryhttps://scigraph.springernature.com/ontologies/product-market-codes/M25001Measure and Integrationhttps://scigraph.springernature.com/ontologies/product-market-codes/M12120Mathematical Physicshttps://scigraph.springernature.com/ontologies/product-market-codes/M35000Number theory.Measure theory.Mathematical physics.Number Theory.Measure and Integration.Mathematical Physics.512.7Lapidus Michel Lauthttp://id.loc.gov/vocabulary/relators/aut47890Radunović Goranauthttp://id.loc.gov/vocabulary/relators/autŽubrinić Darkoauthttp://id.loc.gov/vocabulary/relators/autBOOK9910254280703321Fractal Zeta Functions and Fractal Drums2238299UNINA04768nam 22008053 450 991096075860332120231110232453.097814704701421470470144(MiAaPQ)EBC6852907(Au-PeEL)EBL6852907(CKB)20667664900041(RPAM)22493618(OCoLC)1292081275(EXLCZ)992066766490004120220117d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierThe Brunn-Minkowski Inequality and a Minkowski Problem for Nonlinear Capacity1st ed.Providence :American Mathematical Society,2022.©2022.1 online resource (128 pages)Memoirs of the American Mathematical Society ;v.275"Volume 275. January 2022."Print version: Akman, Murat The Brunn-Minkowski Inequality and a Minkowski Problem for Nonlinear Capacity Providence : American Mathematical Society,c2022 9781470450526 Includes bibliographical references.Notation and statement of results -- Basic estimates for A-harmonic functions -- Preliminary reductions for the proof of theorem A -- Proof of theorem A -- Final proof of theorem A -- Appendix -- Introduction and statement of results -- Boundary behavior of A-harmonic functions in Lipschitz domains -- Boundary Harnack inequalities -- Weak convergence of certain measures on Sn-1 -- The Hadamard variational formula for nonlinear capacity -- Proof of theorem B."In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, CapA, where A-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the p-Laplace equation and whose solutions in an open set are called A-harmonic"--Provided by publisher.Memoirs of the American Mathematical Society Minkowski geometryInequalities (Mathematics)Nonlinear theoriesElliptic functionsHarmonic functionsPartial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equationsmscPotential theory -- Higher-dimensional theory -- Potentials and capacities, extremal lengthmscDifference and functional equations -- Functional equations and inequalities -- Functional inequalities, including subadditivity, convexity, etc.mscConvex and discrete geometry -- General convexity -- Inequalities and extremum problemsmscPartial differential equations -- Elliptic equations and systems -- Variational methods for second-order elliptic equationsmscConvex and discrete geometry -- General convexity -- Convex sets in $n$ dimensions (including convex hypersurfaces)mscPartial differential equations -- Elliptic equations and systems -- Quasilinear elliptic equations with $p$-LaplacianmscMinkowski geometry.Inequalities (Mathematics)Nonlinear theories.Elliptic functions.Harmonic functions.Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations.Potential theory -- Higher-dimensional theory -- Potentials and capacities, extremal length.Difference and functional equations -- Functional equations and inequalities -- Functional inequalities, including subadditivity, convexity, etc..Convex and discrete geometry -- General convexity -- Inequalities and extremum problems.Partial differential equations -- Elliptic equations and systems -- Variational methods for second-order elliptic equations.Convex and discrete geometry -- General convexity -- Convex sets in $n$ dimensions (including convex hypersurfaces).Partial differential equations -- Elliptic equations and systems -- Quasilinear elliptic equations with $p$-Laplacian.515/.3533515.353335J6031B1539B6252A4035J2052A2035J92mscAkman Murat1802239Gong Jasun1802240Hineman Jay1802241MiAaPQMiAaPQMiAaPQBOOK9910960758603321The Brunn-Minkowski Inequality and a Minkowski Problem for Nonlinear Capacity4347814UNINA