There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

The Visual Display of Quantitative Information

Environment and Planning B: Planning and Design

Examples are discussed to show the differences among discriminant analysis, logistic regression, and multiple regression. Chapter 6, “Multivariate Analysis of Variance,” presents advantages of multivariate analysis of variance (MANOVA) over univariate analysis of variance (ANOVA), discusses assumptions of MANOVA, and assesses validations of MANOVA assumptions and model estimation. The authors also discuss post hoc tests of MANOVA and multivariate analysis of covariance. Chapter 7, “Conjoint Analysis,” explains what conjoint analysis does and how it is different from other multivariate techniques. Guidelines of selecting attributes, models, and methods of data collection are presented. Chapter 8, “Cluster Analysis,” studies objectives, roles, and limitations of cluster analysis. Two basic concepts: similarity and distance are discussed. The authors also discuss details of five most popular hierarchical algorithms (singlelinkage, complete-linkage, average-linkage, centroid method, Ward’s method) and three nonhierarchical algorithms (the sequential threshold method, the parallel threshold method, and the optimizing procedure). Profiles of clusters and guidelines for cluster validation are studied as well. Chapter 9, “Multidimensional Scaling and Correspondence Analysis,” introduces two interdependence techniques to display the relationships in the data. The book describes clearly and intuitively the differences between the two techniques and how these two techniques are performed. Chapters 10–12 cover topics in SEM. Chapter 10, “Structural Equation Modeling: An Introduction,” introduces SEM and related concepts such as exogenous, endogenous constructs, and so on, points out the differences between SEM and other multivariate techniques, overviews the decision process of SEM. Chapter 11, “Confirmatory Factor Analysis,” explains the differences between exploratory and confirmatory factor analysis, discusses how to construct, validate, and assess the goodness of fit of a measurement model in SEM by confirmatory factor analysis. Chapter 12, “Testing a Structural Model,” presents some methods of SEM in examining the relationships between latent constructs. The book is an excellent book for people in management and marketing. For the Technometrics audience, this book does not have much flavor of physical, chemical, and engineering sciences. For example, partial least squares, a very popular method in Chemometrics, is discussed but not as detailed as other techniques in the book. Furthermore, due to the amount of materials covered in the book, it might be inappropriate for someone who is new to multivariate analysis.

The Grammar of Graphics

Visualization applications nowadays not only face increasingly larger datasets, but have to solve increasingly complex research questions. They often require more than a single algorithm and consequently a software solution will exceed the possibilities of simple research prototypes. Well-established systems intended for such complex visual analysis purposes have usually been designed for classical, mesh-based graphics approaches. For particle-based data, however, existing visualization frameworks are too generic—e.g. lacking possibilities for consistent low-level GPU optimization for high-performance graphics—and at the same time are too limited—e.g. by enforcing the use of structures suboptimal for some computations. Thus, we developed the system software MegaMol for visualization research on particle-based data. On the one hand, flexible data structures and functional module design allow for easy adaption to changing research questions, e.g. studying vapors in thermodynamics, solid material in physics, or complex functional macromolecules like proteins in biochemistry. Therefore, MegaMol is designed as a development framework. On the other hand, common functionality for data handling and advanced rendering implementations are available and beneficial for all applications. We present several case studies of work implemented using our system as well as a comparison to other freely available or open source systems.

http://doi.ieeecomputersociety.org/10.1109/TVCG.2014.2350479

MegaMol—A Prototyping Framework for Particle-Based Visualization

We present a novel visualization method for mono- and dipolar molecular simulations from thermodynamics that takes advantage of modern graphics hardware to interactively render specifically tailored glyphs Our approach allows domain experts to visualize the results of molecular dynamics simulations with a higher number of particles than before and furthermore offers much better visual quality We achieve this by transferring only visualization parameters to the GPU and by generating implicit surfaces directly in the fragment program As a result, we can render up to 500000 glyphs with about 10 fps displaying all the simulation results as geometrical properties that resemble the classical abstract representation used in this research area Thus we enable researchers to visually assess the results of simulations of greater scale than before We believe that the proposed method can be generalized to create other kinds of parametrized surfaces directly on graphics hardware to overcome the bandwidth bottleneck that exists between CPU and GPU

Hardware-accelerated glyphs for mono- and dipoles in molecular dynamics visualization

Molecular dynamics simulations are a principal tool for studying molecular systems. Such simulations are used to investigate molecular structure, dynamics, and thermodynamical properties, as well as a replacement for, or complement to, costly and dangerous experiments. With the increasing availability of computational power the resulting data sets are becoming increasingly larger, and benchmarks indicate that the interactive visualization on desktop computers poses a challenge when rendering substantially more than millions of glyphs. Trading visual quality for rendering performance is a common approach when interactivity has to be guaranteed. In this paper we address both problems and present a method for high-quality visualization of massive molecular dynamics data sets. We employ several optimization strategies on different levels of granularity, such as data quantization, data caching in video memory, and a two-level occlusion culling strategy: coarse culling via hardware occlusion queries and a vertex-level culling using maximum depth mipmaps. To ensure optimal image quality we employ GPU raycasting and deferred shading with smooth normal vector generation. We demonstrate that our method allows us to interactively render data sets containing tens of millions of high-quality glyphs.

Coherent culling and shading for large molecular dynamics visualization

Modern virtual reality display technologies engender spatial immersion by using a variety of depth cues such as perspective and head-tracked binocular presentation to create visually realistic 3D worlds. While 3D visualisations are common in scientific visualisation, they are much less common in information visualisation. In this chapter we explore whether immersive analytic applications should continue to use traditional 2D information visualisations or whether there are situations when 3D may offer benefits. We identify a number of potential applications of 3D depth cues for abstract data visualisation: using depth to show an additional data dimension, such as in 2.5D network layouts, views on non-flat surfaces and egocentric views in which the data is placed around the viewer, and visualising abstract data with a spatial embedding. Another important potential benefit is the ability to arrange multiple views in the 3D space around the user and to attach abstract visualisations to objects in the real world.

Immersive Analytics: Time to Reconsider the Value of 3D for Information Visualisation

A current research topic in molecular thermodynamics is the condensation of vapor to liquid and the investigation of this process at the molecular level. Condensation is found in many physical phenomena, e.g. the formation of atmospheric clouds or the processes inside steam turbines, where a detailed knowledge of the dynamics of condensation processes will help to optimize energy efficiency and avoid problems with droplets of macroscopic size. The key properties of these processes are the nucleation rate and the critical cluster size. For the calculation of these properties it is essential to make use of a meaningful definition of molecular clusters, which currently is a not completely resolved issue. In this paper a framework capable of interactively visualizing molecular datasets of such nucleation simulations is presented, with an emphasis on the detected molecular clusters. To check the quality of the results of the cluster detection, our framework introduces the concept of flow groups to highlight potential cluster evolution over time which is not detected by the employed algorithm. To confirm the findings of the visual analysis, we coupled the rendering view with a schematic view of the clusters' evolution. This allows to rapidly assess the quality of the molecular cluster detection algorithm and to identify locations in the simulation data in space as well as in time where the cluster detection fails. Thus, thermodynamics researchers can eliminate weaknesses in their cluster detection algorithms. Several examples for the effective and efficient usage of our tool are presented.

Guido Reina

Papers

MegaMol—A Prototyping Framework for Particle-Based Visualization

Hardware-accelerated glyphs for mono- and dipoles in molecular dynamics visualization

Coherent culling and shading for large molecular dynamics visualization

Immersive Analytics: Time to Reconsider the Value of 3D for Information Visualisation

Visual Verification and Analysis of Cluster Detection for Molecular Dynamics