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Measures of EEG Dynamic Properties

Paul L. Nunez and Ramesh Srinivasan

in Electric Fields of the Brain: The neurophysics of EEG

This chapter suggests a general working framework for experimental study of large-scale dynamics of EEG, including interpretation of data recorded at different spatial and temporal scales. It views ... More

This chapter suggests a general working framework for experimental study of large-scale dynamics of EEG, including interpretation of data recorded at different spatial and temporal scales. It views source dynamics as a stochastic (random) process having statistical properties that change with changes in behavior and cognition. It avoids any attempts to choose the “best” methods of EEG or SSVEP data analysis because any such optimization requires prior knowledge of the underlying source dynamics. Fourier-based methods are emphasized to estimate power, phase, coherence, and closely related dynamic measures in distinct frequency bands. Additional methods include estimates of phase velocity across the scalp and frequency-wavenumber spectral analysis. The latter is expressed in terms of spherical harmonics, the natural spatial functions for dynamics on a spherical surface. This approach is used to pick out individual Schumann resonances as an example, in preparation for a similar application to EEG and SSVEP presented in Chapter 10.Less

Rotating Stars and Revolving Planets: Bayesian Exploration of the Pulsating Sky *

Thomas J. Loredo

in Bayesian Statistics 9

I describe ongoing work on the development of Bayesian methods for exploring periodically varying phenomena in astronomy, addressing two classes of sources: pulsars, and extrasolar planets ... More

I describe ongoing work on the development of Bayesian methods for exploring periodically varying phenomena in astronomy, addressing two classes of sources: pulsars, and extrasolar planets (exoplanets). For pulsars, the methods aim to detect and measure periodically varying signals in data consisting of photon arrival times, modeled as non‐homogeneous Poisson point processes. For exoplanets, the methods address detection and estimation of planetary orbits using observations of the reflex motion “wobble” of a host star, including adaptive scheduling of observations to optimize inferences.Less

Toward an Analysis of the Music of Prokofiev’s Middle Period

Konrad Harley

in Rethinking Prokofiev

Prokofiev’s music is so individual that it is not uncommon for analysts to come up with new terms and concepts to describe it. Many of his devices have their origin in eighteenth- and ... More

Prokofiev’s music is so individual that it is not uncommon for analysts to come up with new terms and concepts to describe it. Many of his devices have their origin in eighteenth- and nineteenth-century practices, but he reinvents them in a personal way. This chapter focuses on four topics of particular relevance to the analysis of music from Prokofiev’s middle period: first, the integration of set-theoretic relationships in tonality and the importance of listening to harmonic function even in works that have simplistically been described as atonal; second, the question of how Prokofiev’s harmonic material begins to change throughout the 1930s in his search for “new simplicity”; third, the idea of multiple meanings in harmony and the related notions of branching paths and decisive moments; and finally, the tonal significance of chromatic displacement, a technique that proves central to at least a few works in each period of his life.Less

Pedagogy of Fundamentals and Diatonic Harmony

Jennifer Snodgrass

in Teaching Music Theory

The earliest levels of the undergraduate music theory core might be some of the more challenging courses to teach. Because students enter the undergraduate theory core with a variety of backgrounds, ... More

The earliest levels of the undergraduate music theory core might be some of the more challenging courses to teach. Because students enter the undergraduate theory core with a variety of backgrounds, experiences, and knowledge, instructors face the challenge of inspiring some students with new material while keeping the more experienced students involved. How can educators make this material both relevant and engaging for all students? Teaching the lower levels of written theory is more than just memorization of patterns and rules; it is an opportunity to engage students in creative music making from the very first day with an introduction that helps them understand why a certain element of music works. By participating in engaging and creative methods of learning scales, key signatures, intervals, triads, harmonic function, and voice leading, students are immersed in a music experience that is more than just printed notes on the page.Less

Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in ^R3, and Newton Polyhedra Detlef Müller

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a ... More

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.Less

Square Functions and Maximal Operators Associated with Radial Fourier Multipliers

Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and ... More

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harmonic analysis and many important variants and generalizations in various monographs. The chapter proves new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers. The majority of the chapter is devoted to these proofs, such as for convolutions with spherical measures.Less

On Oscillatory Integral Operators in Higher Dimensions

Jean Bourgain

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier ... More

This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier restriction operators to hypersurfaces in Rn and their variable coefficient generalizations are intimately related to questions of a combinatorial nature. Over recent years there has been quite a bit of research around these underlying issues. In some way, it became interdisciplinary with connections towards geometric measure theory, the theory of finite fields, incidence geometry, and mathematical computer science. While the central original problems remain unsolved, this line of research has produced many new results of independent interest, though the chapter focuses primarily on developments around the theory of oscillatory integrals.Less

Threshold Phenomena and Influence: Perspectives from Mathematics, Computer Science, and Economics

Gil Kalai and Shmuel Safra

in Computational Complexity and Statistical Physics

Threshold phenomena refer to settings in which the probability for an event to occur changes rapidly as some underlying parameter varies. Threshold ... More

Threshold phenomena refer to settings in which the probability for an event to occur changes rapidly as some underlying parameter varies. Threshold phenomena play an important role in probability theory and statistics, physics, and computer science, and are related to issues studied in economics and political science. Quite a few questions that come up naturally in those fields translate to proving that some event indeed exhibits a threshold phenomenon, and then finding the location of the transition and how rapid the change is. The notions of sharp thresholds and phase transitions originated in physics, and many of the mathematical ideas for their study came from mathematical physics. In this chapter, however, we will mainly discuss connections to other fields. A simple yet illuminating example that demonstrates the sharp threshold phenomenon is Condorcet's jury theorem, which can be described as follows. Say one is running an election process, where the results are determined by simple majority, between two candidates, Alice and Bob. If every voter votes for Alice with probability p > 1/2 and for Bob with probability 1 — p, and if the probabilities for each voter to vote either way are independent of the other votes, then as the number of voters tends to infinity the probability of Alice getting elected tends to 1. The probability of Alice getting elected is a monotone function of p, and when there are many voters it rapidly changes from being very close to 0 when p < 1/2 to being very close to 1 when p > 1/2. The reason usually given for the interest of Condorcet's jury theorem to economics and political science [535] is that it can be interpreted as saying that even if agents receive very poor (yet independent) signals, indicating which of two choices is correct, majority voting nevertheless results in the correct decision being taken with high probability, as long as there are enough agents, and the agents vote according to their signal. This is referred to in economics as asymptotically complete aggregation of information. Less

Introduction To Stochastic Processes

Abraham Nitzan

in Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not ... More

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(t) are made at discrete time 0 < t1 < t2, . . . ,< tn < T. In this case the sequence {z(ti)} is a discrete sample of the stochastic process z(t). Let us start with an example. Consider a stretch of highway between two intersections, and let the variable of interest be the number of cars within this road segment at any given time, N(t). This number is obviously a random function of time whose properties can be deduced from observation and also from experience and intuition. First, this function takes positive integer values but this is of no significance: we could redefine N → N − (N) and the new variable will assume both positive and negative values. Second and more significantly, this function is characterized by several timescales: 1. Let τ1 is the average time it takes a car to go through this road segment, for example 1 min, and compare N(t) and N(t +∆ t) for ∆ t << τ1 and ∆ t τ1. Obviously N(t) ≈ N(t + ∆ t) for ∆ t << τ1 while in the opposite case the random numbers N(t) and N(t + ∆ t) will be almost uncorrelated. Figure 7.1 shows a typical result of one observation of this kind. The apparent lack of correlations between successive points in this data set expresses the fact that numbers sampled at intervals equal to or longer than the time it takes to traverse the given distance are not correlated. 2. The apparent lack of systematic component in the time series displayed here reflects only a relatively short-time behavior. Less

Mathematical Models and Quantitative Methods

David R. Legates and Sucharita Gopal

in Geography in America at the Dawn of the 21st Century

Although the use of mathematical models and quantitative methods in geography accelerated in earnest with the development of quantitative geography and ... More

Although the use of mathematical models and quantitative methods in geography accelerated in earnest with the development of quantitative geography and regional science in the late 1950s, such techniques had already made their way into the mainstream of physical geography much earlier. Today, mathematical models and quantitative methods are used in a number of subfields in geography with their proliferation being aided, in part, by the widespread use of remote sensing, geographic information systems (GIS), and computer-based technology. As a consequence, geography as a whole has witnessed a new growth in the development of models and quantitative methods over the last decade, and it is this growth that we seek to elucidate here. Highlighting the advances in the use of models and methods in geography is a difficult undertaking. Such techniques are so widely used in GIS and remote sensing that many developments in these areas also could be considered in this chapter. Moreover, modeling and quantitative techniques are so strongly integrated within some geographic subfields (e.g. climatology and geomorphology, economic and urban geography, regional science) that it is often difficult to separate technique development from application. This is illustrated by the fact that many members of the Association of American Geographers who frequently use and develop quantitative techniques and models are not active participants in the Mathematical Models and Quantitative Methods Specialty Group, choosing instead to favor specialty groups with a more topical, rather than methodological, focus. In a very real sense, the quantitative revolution has been completed in many subfields of geography, with the goals and aims of the revolutionaries having long since passed into the mainstream. Furthermore, geographers who are involved with quantitative methods and mathematical models are extremely diverse in their interests and applications— they contribute to an extremely wide variety of disciplines. While they excel at spreading the geographic word to other disciplines, summarizing their multifarious contributions is nearly impossible. The rather trite statement, “Geography is what geographers do,” seems to apply strongly here. Geographers are largely a collection of individuals who, although united by their interest in spatial models and methods, are unique in the ways that they make contributions to various fields. Less