Reading “A variant of the Hales–Jewett theorem” on its anniversary

In a recent paper “A variant of the Hales–Jewett theorem”, M. Beiglböck provides a version of the classic coloring result in which an instance of the variable in a word giving rise to a monochromatic combinatorial line can be moved around in a finite structure of specified type (for example, an arithmetic progression). We give an elementary proof and infinitary extensions.     

Mathematical foundations of a cultural project or Ramchandra's treatise “through the unsentimentalised light of mathematics”

The nineteenth century witnessed a number of projects of cultural rapprochement between the knowledge traditions of the East and West. This paper discusses the attempt to render elementary calculus amenable to an Indian audience in the indigenous mathematical idiom, undertaken by an Indian polymath, Ramchandra. The exercise is specifically located in his book A Treatise on the Problems of Maxima and Minima. The paper goes on to discuss the “vocation of failure” of the book within the context of encounter and the pedagogy of mathematics.     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

The b-transform

The b-transform is used to convert entire functions into “primary b-functions” by replacing the powers and factorials in the Taylor series of the entire function with corresponding “generalized powers” (which arise from a polynomial function with combinatorial applications) and “generalized factorials.” The b-transform of the exponential function turns out to be a generalization of the Euler partition generating function, and partition generating functions play a key role in obtaining results for the b-transforms of the elementary entire transcendental functions. A variety of normal-looking results arise, including generalizations of Euler's formula and De Moivre's theorem. Applications to discrete probability and applied mathematics (i.e., damped harmonic motion) are indicated. Also, generalized derivatives are obtained by extending the concept of a b-transform.     

Extended obstruction tensors and renormalized volume coefficients

The behavior is investigated of the renormalized volume coefficients associated to a pseudo-Riemannian metric under conformal change of the metric. It is shown that the renormalized volume coefficients define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of “extended obstruction tensors”. These together with the Schouten tensor constitute building blocks for the coefficients in the ambient metric expansion. The renormalized volume coefficients have recently been considered by Chang and Fang motivated by comparison with the elementary symmetric functions of the eigenvalues of the Schouten tensor.     

Elementary theories of complete torsion-free Abelian groups with p-ary topology

The complete torsion-free Abelian groups are investigated as dibasic RPC models whose second basic set comprises subgroups forming a chain with factors of order p. Two theorems about the elementary theories of classes of such models are proved.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 201–208, August, 1973.     

Generic pairs of SU-rank 1 structures

For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T (generic T-pair). We show that the theory T^* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T^* coincides with the theory of infinite dimensional pairs, which was used in (S. Buechler, Pseudoprojective strongly minimal sets are locally projective, J. Symbolic Logic 56(4) (1991) 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T^* for the same purpose. In particular, we obtain a characterization of linearity for SU-rank 1 structures by giving several equivalent conditions on T^*, find a “weak” version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to “recover” projective geometries over division rings from non-trivial linear SU-rank 1 structures.     

Algebraic theories, clones and their segments

Isomorphism and elementary equivalence of segments of clones of objects in concrete categories are investigated. A survey of results about the finitary case is presented and a new theorem about the infinitary case is proved.Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 and of the Grant Agency of the Charles University under the grant GAUK 349 is gratefully acknowledged.     

An Oblivious Routing Algorithm for Two-Dimensional Meshes of Constant Queue-Size

This paper shows an important exception to the common perception that three-dimensional meshes are more powerful than two-dimensional ones. Let N be the total number of processors. Then permutation routing over three-dimensional mesh computers needs Θ(N^2/3) steps while it takes Θ(N^1/2) steps over two-dimensional ones under the following conditions: (1) The path of each packet must be determined solely by its initial position and destination, i.e., the algorithm must be oblivious. (2) Each path must be “elementary,” i.e., it must be shortest and as straight as possible. Thus the conditions, under which, somewhat surprisingly, three-dimensional meshes are significantly less powerful than two-dimensional ones for the fundamental network operation, are quite reasonable in practice.     

Case of a generating family of quasi-periodic solutions in the theory of small parameter : PMM vol. 37, n6, 1973, pp. 990–998

We consider the problem of the existence and the stability in-the-small of periodic solutions of systems of ordinary differential equations with a small parameter μ, which in the generating approximation (μ = 0) admit of a family of quasi-periodic solutions (we are concerned only with the solutions belonging to the indicated family when μ = 0). The case to be investigated is in a specific sense a more general case of the unisolated generating solution in the small parameter theory and, therefore, includes everything previously treated by Malkin [1], Blekhman [2], and others. The main difficulty in the investigation is the presence of a multiple zero root in the characteristic determinant of the problem's generating system, to which both simple as well as quadratic elementary divisors [3] correspond. This fact predestines the presence of three groups of stability criteria for the solution being examined. The method for constructing these criteria, proposed here, assumes, in contrast to a previous one [1], the preliminary determination of not only the generating approximation but also the first one to the desired periodic solution. Particular aspects of the general “mixed” problem treated here were studied earlier in [4, 5].     

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.     

An Lp-view of the Bahadur-Kiefer Theorem

Summary Let ]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>\alpha_n$ and $\beta_n$ be respectively the uniform empirical and quantile processes, and define $R_n = \alpha_n + \beta_n$, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$ almost surely, as $n$ goes to infinity, where $\| f\| =\sup_{0\le t\le 1} |f(t)|$ is the $L^\infty$-norm. We prove that $\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4}$ almost surely, where $\| \, \cdot \, \|_p$ is the $L^p$-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that $n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}}$ converges almost surely to a finite positive constant whose value is explicitly known.     

Semi-parametric multivariate modelling when the marginals are the same

A model is developed for multivariate distributions which have nearly the same marginals, up to shift and scale. This model, based on “interpolation” of characteristic functions, gives a new notion of “correlation”. It allows straightforward nonparametric estimation of the common marginal distribution, which avoids the “curse of dimensionality” present when nonparametically estimating the full multivariate distribution. The method is illustrated with environmental monitoring network data, where multivariate modelling with common marginals is often appropriate.     

Old Math, New Math: Parents' Experiences with Standards-Based Reform

We focus on how African American parents in a low-income neighborhood experience, interpret, and respond to current reform efforts as implemented in their children's school. As part of a larger project on parent-child numeracy connections in an elementary school, we interviewed 10 parents and held 2 focus group meetings, during which parents shared their experiences with mathematics as students themselves and as parents of children using a Standards-based curriculum. Even though parents saw themselves as critical players in their children's learning, we found that the implementation of reform-oriented curriculum tended to disempower parents with respect to school mathematics. Parents had little understanding of the reform-based approaches, and thus limited access to the discourse of reform. Our findings call for examination of the effect that reforms have on parents, particularly when the current educational climate calls for increased parent participation and involvement.

If an 8 year old can do it, I know I can do it. I was like—wait a minute, he's the kid and I'm the parent, and he knows and I don't know. He had got upset one day and said, “Mom, you're going to make me get a bad grade. That's not right. That's not right. That's wrong.”—Shanice, mother of three     

A random walk proof of the Erdős-Taylor conjecture

Summary For the simple random walk in ]]>\mathbb{Z}^2$ we study those points which are visited an unusually large number of times, and provide a new proof of the Erdős-Taylor Conjecture describing the number of visits to the most visited point.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

On the decision problem for theories of finite models

An infinite extension of the elementary theory of Abelian groups is constructed, which is proved to be decidable, while the elementary theory of its finite models is shown to be undecidable. Tarski’s proof of undecidability for the elementary theory of Abelian cancellation semigroups is presented in detail. Szmielew’s proof of the decidability of the elementary theory of Abelian groups is used to prove the decidability of the elementary theory of finite Abelian groups, and an axiom system for this theory is exhibited. It follows that the elementary theory of Abelian cancellation semigroups, while undecidable, has a decidable theory of finite models.     

Passage to the limit in Proposition I, Book I of Newton's Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F_m is used for motive force.     

Construction of the value function in a pursuit-evasion game with three pursuers and one evader

A differential pursuit-evasion game is considered with three pursuers and one evader. It is assumed that all objects (players) have simple motions and that the game takes place in a plane. The control vectors satisfy geometrical constraints and the evader has a superiority in control resources. The game time is fixed. The value functional is the distance between the evader and the nearest pursuer at the end of the game. The problem of determining the value function of the game for any possible position is solved.

Three possible cases for the relative arrangement of the players at an arbitrary time are studied: “one-after-one”, “two-after-one”, “three-after-one-in-the-middle” and “three-after-one”. For each of the relative arrangements of the players a guaranteed result function is constructed. In the first three cases the function is expressed analytically. In the fourth case a piecewise-programmed construction is presented with one switchover, on the basis of which the value of the function is determined numerically. The guaranteed result function is shown to be identical with the game value function. When the initial pursuer positions are fixed in an arbitrary manner there are four game domains depending on their relative positions. The boundary between the “three-after-one-in-the-middle” domain and the “three-after-one” domain is found numerically, and the remaining boundaries are interior Nicomedean conchoids, lines and circles. Programs are written that construct singular manifolds and the value function level lines.     

Discrete time Galerkin approximations to the nonlinear filtering solution

This paper concerns discrete time Galerkin approximations to the solution of the filtering problem for diffusions. Two families of schemes approximating the unnormalized conditional density, respectively, in an “average” and in a “pathwise” sense, are presented. L² error estimates are derived and it is shown that the rate of convergence is linear in the time increment or linear in the modulus of continuity of the sample path.