Jewish Problems

This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value.Comment: 21 pages, 14 figure

AD in Fortran, Part 1: Design

We propose extensions to Fortran which integrate forward and reverse Automatic Differentiation (AD) directly into the programming model. Irrespective of implementation technology, embedding AD constructs directly into the language extends the reach and convenience of AD while allowing abstraction of concepts of interest to scientific-computing practice, such as root finding, optimization, and finding equilibria of continuous games. Multiple different subprograms for these tasks can share common interfaces, regardless of whether and how they use AD internally. A programmer can maximize a function F by calling a library maximizer, XSTAR=ARGMAX(F,X0), which internally constructs derivatives of F by AD, without having to learn how to use any particular AD tool. We illustrate the utility of these extensions by example: programs become much more concise and closer to traditional mathematical notation. A companion paper describes how these extensions can be implemented by a program that generates input to existing Fortran-based AD tools

AD in Fortran, Part 2: Implementation via Prepreprocessor

We describe an implementation of the Farfel Fortran AD extensions. These extensions integrate forward and reverse AD directly into the programming model, with attendant benefits to flexibility, modularity, and ease of use. The implementation we describe is a "prepreprocessor" that generates input to existing Fortran-based AD tools. In essence, blocks of code which are targeted for AD by Farfel constructs are put into subprograms which capture their lexical variable context, and these are closure-converted into top-level subprograms and specialized to eliminate EXTERNAL arguments, rendering them amenable to existing AD preprocessors, which are then invoked, possibly repeatedly if the AD is nested

The Art of the Propagator

We develop a programming model built on the idea that the basic computational elements are autonomous machines interconnected by shared cells through which they communicate. Each machine continuously examines the cells it is interested in, and adds information to some based on deductions it can make from information from the others. This model makes it easy to smoothly combine expression-oriented and constraint-based programming; it also easily accommodates implicit incremental distributed search in ordinary programs. This work builds on the original research of Guy Lewis Steele Jr. and was developed more recently with the help of Chris Hanson

Flexible and expressive substrate for computation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 167-174).In this dissertation I propose a shift in the foundations of computation. Modem programming systems are not expressive enough. The traditional image of a single computer that has global effects on a large memory is too restrictive. The propagation paradigm replaces this with computing by networks of local, independent, stateless machines interconnected with stateful storage cells. In so doing, it offers great flexibility and expressive power, and has therefore been much studied, but has not yet been tamed for general-purpose computation. The novel insight that should finally permit computing with general-purpose propagation is that a cell should not be seen as storing a value, but as accumulating information about a value. Various forms of the general idea of propagation have been used with great success for various special purposes; perhaps the most immediate example is constraint propagation in constraint satisfaction systems. This success is evidence both that traditional linear computation is not expressive enough, and that propagation is more expressive. These special-purpose systems, however, are all complex and all different, and neither compose well, nor interoperate well, nor generalize well. A foundational layer is missing. I present in this dissertation the design and implementation of a prototype general-purpose propagation system. I argue that the structure of the prototype follows from the overarching principle of computing by propagation and of storage by accumulating information-there are no important arbitrary decisions. I illustrate on several worked examples how the resulting organization supports arbitrary computation; recovers the expressivity benefits that have been derived from special-purpose propagation systems in a single general-purpose framework, allowing them to compose and interoperate; and offers further expressive power beyond what we have known in the past. I reflect on the new light the propagation perspective sheds on the deep nature of computation.by Alexey Andreyevich Radul.Ph.D

The Symmetriad: A Journey of Discovery Through the Land of the Polychora

MEng thesisI devised and implemented a method for constructing regular andsemiregular geometric objects in n-dimensional Euclidean space.Given a finite reflection group (a Coxeter group) G, there is a standard way to give G a group action on n-space.Reflecting a point through this group action yieldsan object that exhibits the symmetries specified by G. If the pointis chosen well, the object is guaranteed to be regular or semiregular,and many interesting regular and semiregular objectsarise this way. By starting with the symmetry group, I can use thegroup structure both to simplify the actual graphics involved withdisplaying the object, and to illustrate various aspects of itsstructure. For example, subgroups of the symmetry group (and theircosets) correspond to substructures of the object. Conversely, bydisplaying such symmetric objects and their various substructures, Ifind that I can elucidate the structure of the symmetry group thatgives rise to them.I have written The Symmetriad, the computer system whose name thisdocument has inherited, and used it to explore 3- and 4-dimensionalsymmetric objects and their symmetry groups. The 3-dimensionalobjects are already well understood, but they serve to illustrate thetechniques used on the 4-dimensional objects and make them morecomprehensible. Four dimensions offers a treasure trove of intriguingstructures, many of which have no ready 3D analogue. These are what Iwill show you here

Propagation Networks: A Flexible and Expressive Substrate for Computation

PhD thesisI propose a shift in the foundations of computation. Practically all ideas of general-purpose computation today are founded either on execution of sequences of atomic instructions, i.e., assembly languages, or on evaluation of tree-structured expressions, i.e., most higher level programming languages. Both have served us well in the past, but it is increasingly clear that we need something more. I suggest that we can build general-purpose computation on propagation of information through networks of stateful cells interconnected with stateless autonomous asynchronous computing elements. Various forms of this general idea have been used with great success for various special purposes; perhaps the most immediate example is constraint propagation in constraint satisfaction systems. These special-purpose systems, however, are all complex and all different, and neither compose well, nor interoperate well, nor generalize well. A foundational layer is missing. The key insight in this work is that a cell should not be seen as storing a value, but as accumulating information about a value. The cells should never forget information -- such monotonicity prevents race conditions in the behavior of the network. Monotonicity of information need not be a severe restriction: for example, carrying reasons for believing each thing makes it possible to explore but thenpossibly reject tentative hypotheses, thus appearing to undo something, while maintaining monotonicity. Accumulating information is a broad enough design principle to encompass arbitrary computation. The object of this dissertation is therefore to architect a general-purpose computing system based on propagation networks; to subsume expression evaluation under propagation just as instruction execution is subsumed under expression evaluation; to demonstrate that a general-purpose propagation system can recover all the benefits that have been derived from special-purpose propagation systems, allow them to compose andinteroperate, and offer further expressive power beyond what we have known in the past; and finally to contemplate the lessons that such a fundamental shift can teach us about the deep nature of computation.My graduate career in general, and this work in particular, have been sponsored in part by a National Science Foundation Graduate Research Fellowship, by the Disruptive Technology Office as part of the AQUAINT Phase 3 research program, by the Massachusetts Institute of Technology, by Google, Inc., and by the National Science Foundation Cybertrust (05-518) program.Doctor of Philosoph