<p>The aim of this book is to make accessible to mathematicians, physicists and other scientists interested in quantum theory, the mathematically beautiful but difficult subjects of the Feynman integral and Feynman's operational calculus. Some advantages of the four approaches to the Feynman integral which are given detailed treatment in this book are the following: the existence of the Feynman integral is established for very general potentials in all four cases; under more restrictive but still broad conditions, three of these Feynman integrals agree with one another and with the unitary group from the usual approach to quantum dynamics; these same three Feynman integrals possess pleasant stability properties. Much of the material covered here was previously only in the research literature, and the book also contains some new results. The background material in mathematics and physics that motivates the study of the Feynman integral and Feynman's operational calculus is discussed and detailed proofs are provided for the central results.</p>

The Feynman Integral and Feynman's Operational Calculus (Oxford Mathematical Monographs)

Science & Math

Lapidus, Michel L.

Johnson, Gerald W.

Mathematics

Clarendon Press

<p>This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book.<br><br>The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections with certain analytic Feynman integrals are noted. <br><br>This volume is essentially self-contained and we only assume that the reader has a reasonable, graduate level, background in analysis, measure theory and functional analysis or operator theory. Much of the necessary remaining background is supplied in the text itself.<br></p>

Feynman's Operational Calculus and Beyond: Noncommutativity and Time-Ordering (Oxford Mathematical Monographs)

Nielsen, Lance

Oxford University Press

<p>Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.</p>

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)

van Frankenhuijsen, Machiel

Springer

<p>This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings"). </p>
<p>In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable "gauge functions" other than power functions. (This content is used to measure the irregularity (or "fractality") of the boundary of an open set in $R^n$ by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural "gauge function", this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.</p>

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function (Memoirs of the American Mathematical Society)

He, Christina Q.

Amer Mathematical Society

<p>A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.</p>

Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Health, Fitness & Dieting

Psychology & Counseling

Birkhauser

<p>Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary.</p>
<p>Throughout <i>Geometry, Complex Dimensions and Zeta Functions, </i>Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.</p>

Reference

In its Second Edition, this in-depth study of the vibrations of fractal strings interlinks number theory, spectral geometry and fractal geometry. Includes a geometric reformulation of the Riemann hypothesis and a new final chapter on recent topics and results.

most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t > O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x,O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt,E* ®E), locally given by 00 K(x,y; t) = L>-IAk(~k ® 'Pk)(X,y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g. , the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.

Progress in Inverse Spectral Geometry (Trends in Mathematics)

Crafts, Hobbies & Home

Andersson, Stig I.

Gardening & Landscape Design

<p>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.  <br>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.</p>

Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions (Springer Monographs in Mathematics)

Radunović, Goran

Žubrinić, Darko

<p><div>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. &nbsp;</div> <div><br /></div> <div>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.</div></p>

See the best price to sell, buy, or rent books by Lapidus, Michel L.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Lapidus, Michel L.

See the best price to sell, buy, or rent books by Lapidus, Michel L.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

The Feynman Integral and Feynman's Operational Calculus (Oxford Mathematical Monographs)

Feynman's Operational Calculus and Beyond: Noncommutativity and Time-Ordering (Oxford Mathematical Monographs)

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function (Memoirs of the American Mathematical Society)

Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)

Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)

Progress in Inverse Spectral Geometry (Trends in Mathematics)

Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions (Springer Monographs in Mathematics)

Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions (Springer Monographs in Mathematics)