Welcome to the world of scale symmetry, the last elementary symmetry and the least explored!Find out how this long-neglected element transforms the traditional geometry of lines and planes into a rich landscape of trees, craggy mountains and rolling oceans.Enjoy a visual exploration through the intricate and elaborate structures of scale-symmetric geometry. See unique fractals, Mandelboxes, and automata and physical behaviors. Take part in the author's forage into the lesser-trodden regions of this landscape, and discover unusual and attractive specimens!You will also be provided with all the tools needed to recreate the structures yourself.Every example is new and developed by the author, and is chosen because it pushes the field of scale-symmetric geometry into a scarcely explored region. The results are complex and intricate but the method of generation is often simple, which allows it to be presented graphically without depending on too much mathematical syntax. If you are interested in the mathematics, science and art of scale symmetry, then read on!This is also a book for programmers and for hobbyists: those of us who like to dabble with procedural imagery and see where it leads.

Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models--the first examples of the musical spaces of the book's title--depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory. Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.

The Architecture of Music Volume 1.0 is the most complete combined chord, scale, and mode encyclopedia for the guitar and piano ever created, an effort almost two decades in the making. The chord encyclopedia includes more than 300 unique chords, and depicts every possible way to play them on the guitar and piano. The scale and mode encyclopedia contains 5 scales (33 modes) in every key, as well as complete lists of chords included in the chord encyclopedia that you are able to play with every note, in every scale, in every key. The book's two new innovative diagrams, the interval diagram and linear circle of fifths, expose the architecture behind chords, scales, and modes, and explain basic music theory without the use of standard musical notation. The information contained within is meant to be used as a comprehensive reference guide and tool for exploring and analyzing chords, scales, and modes through musical improvisation and composition.

"Kids love to move. But how do we harness all that kinetic energy effectively for math learning? In Math on the Move, Malke Rosenfeld shows how pairing math concepts and whole body movement creates opportunities for students to make sense of math in entirely new ways. Malke shares her experience creating dynamic learning environments by: exploring the use of the body as a thinking tool, highlighting mathematical ideas that are usefully explored with a moving body, providing a range of entry points for learning to facilitate a moving math classroom. ..."--Publisher description.

This is an expanded version of the report by the Electroweak Symmetry Breaking and Beyond the Standard Model Working Group which was contributed to Particle Physics — Perspectives and Opportunities, a report of the Division of Particles and Fields Committee for Long Term Planning. One of the Working Group's primary goals was to study the phenomenology of electroweak symmetry breaking and attempt to quantify the “physics reach” of present and future colliders. Their investigations encompassed the Standard Model — with one doublet of Higgs scalars — and approaches to physics beyond the Standard Model. These include models of low-energy supersymmetry, dynamical electroweak symmetry breaking, and a variety of extensions of the Standard Model with new particles and interactions. The Working Group also considered signals of new physics in precision measurements arising from virtual processes and examined experimental issues associated with the study of electroweak symmetry breaking and the search for new physics at present and future hadron and lepton colliders.This volume represents an important contribution to the efforts being made to advance the frontiers of particle physics.

In The Hidden Symmetry of 43 Octatonic Scales and 43 Tetrachords, Creamer provides an extensive explanation and analysis of his system of octatonic (eight-note) harmonizations and melodic organization as well as a series of exercises, complete musical examples and original compositions utilizing the system. While the book includes the diminished scale and all of the eight-note bebop scales, Creamer goes well beyond their traditional use in a jazz context and introduces a vast new musical language where all eight notes are utilized as scale tones creating thousands of chord combinations, tonal colors and melodic possibilities that can be used by improvisers and composers in any musical context for generating new ideas and expanding traditional harmonic and melodic approaches. Guitarists will also benefit from the inherit symmetrical fingerings of the system (eight-notes-per-two-strings) as well as full tablature for all examples have been provided. Much more than a series of possible mathematical combinations, the book is presented as a complete system, and while a thorough theoretical framework is presented for contextual understanding, the music is first and foremost the focus of the work as articulated in the book’s Foreward by Tuck Andress. Perhaps not since George Russell’s book, Lydian Chromatic Concept of Tonal Organization, has a book had such tremendous potential for modern composers and musicians.

We know very little about the time-evolution of many-particle dynamical systems, the subject of our book. Even the 3-body problem has no explicit solution (we cannot solve the corresponding system of differential equations, and computer simulation indicates hopelessly chaotic behaviour). For example, what can we say about the typical time evolution of a large system starting from a stage far from equilibrium? What happens in a realistic time scale? The reader's first reaction is probably: What about the famous Second Law (of thermodynamics)?Unfortunately, there are plenty of notorious mathematical problems surrounding the Second Law. (1) How to rigorously define entropy? How to convert the well known intuitions (like 'disorder' and 'energy spreading') into precise mathematical definitions? (2) How to express the Second Law in forms of a rigorous mathematical theorem? (3) The Second Law is a 'soft' qualitative statement about entropy increase, but does not say anything about the necessary time to reach equilibrium.The object of this book is to answer questions (1)-(2)-(3). We rigorously prove a Time-Quantitative Second Law that works on a realistic time scale. As a by product, we clarify the Loschmidt-paradox and the related reversibility/irreversibility paradox.

This book is based on a series of lectures at the Mathematics Department of the University of Jena, developed in the period from 1995 up to 2015. It is completed by additional material and extensions of some basic results from the literature to more general metric spaces.This book provides a clear introduction to classical fields of fractal geometry, which provide some background for modern topics of research and applications. Some basic knowledge on general measure theory and on topological notions in metric spaces is presumed.

Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics.Originally, M L Lapidus and M van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M L Lapidus and H Maier on inverse spectral problems for fractal strings and the Riemann hypothesis.One of the main themes of the book is to provide a rigorous framework within which the corresponding question 'Can one hear the shape of a fractal string?' or, equivalently, 'Can one obtain information about the geometry of a fractal string, given its spectrum?' can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator.The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space.It is shown that the quasi-invertibility of the spectral operator is intimately connected to the existence of critical zeros of the Riemann zeta function, leading to a new spectral and operator-theoretic reformulation of the Riemann hypothesis. Accordingly, the spectral operator is quasi-invertible for all values of the dimensional parameter c in the critical interval (0,1) (other than in the midfractal case when c =1/2) if and only if the Riemann hypothesis (RH) is true. A related, but seemingly quite different, reformulation of RH, due to the second author and referred to as an 'asymmetric criterion for RH', is also discussed in some detail: namely, the spectral operator is invertible for all values of c in the left-critical interval (0,1/2) if and only if RH is true.These spectral reformulations of RH also led to the discovery of several 'mathematical phase transitions' in this context, for the shape of the spectrum, the invertibility, the boundedness or the unboundedness of the spectral operator, and occurring either in the midfractal case or in the most fractal case when the underlying fractal dimension is equal to ½ or 1, respectively. In particular, the midfractal dimension c=1/2 is playing the role of a critical parameter in quantum statistical physics and the theory of phase transitions and critical phenomena.Furthermore, the authors provide a 'quantum analog' of Voronin's classical theorem about the universality of the Riemann zeta function. Moreover, they obtain and study quantized counterparts of the Dirichlet series and of the Euler product for the Riemann zeta function, which are shown to converge (in a suitable sense) even inside the critical strip.For pedagogical reasons, most of the book is devoted to the study of the quantized Riemann zeta function. However, the results obtained in this monograph are expected to lead to a quantization of most classic arithmetic zeta functions, hence, further 'naturally quantizing' various aspects of analytic number theory and arithmetic geometry.The book should be accessible to experts and non-experts alike, including mathematics and physics graduate students and postdoctoral researchers, interested in fractal geometry, number theory, operator theory and functional analysis, differential equations, complex analysis, spectral theory, as well as mathematical and theoretical physics. Whenever necessary, suitable background about the different subjects involved is provided and the new work is placed in its proper historical context. Several appendices supplementing the main text are also included.

Economic Risk in Hydrocarbon Exploration provides a total framework for assessing the uncertainties associated with exploration risk from beginning to end. Numerous examples with accompanying microcomputer algorithms illustrate how to quantitatively approach economic risk. The text compares detailed assumptions and models of economic risk, and presents numerical examples throughout to facilitate hands-on calculations using popular spread-sheet packages on personal computers. - Covers economic risk from exploration through production models - Brings methods to a level where all can be done on a PC - Analyzes numerical examples from the real world - Removes "mystery" from how economics is done - Addresses assumptions in models and shows how they influence projections