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Two notes on notation
 American Mathematical Monthly
, 1992
"... Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretic ..."
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Cited by 78 (2 self)
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Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretical emphases. Our mathematical language continues to improve, just as “the dism of Leibniz overtook the dotage of Newton ” in past centuries [4, Chapter 4]. In 1970 I began teaching a class at Stanford University entitled Concrete Mathematics. The students and I studied how to manipulate formulas in continuous and discrete mathematics, and the problems we investigated were often inspired by new developments in computer science. As the years went by we began to see that a few changes in notational traditions would greatly facilitate our work. The notes from that class have recently been published in a book [15], and as I wrote the final drafts of that book I learned to my surprise that two of the notations we had been using were considerably more useful than I had previously realized. The ideas “clicked ” so well, in fact, that I’ve decided to write this article, blatantly attempting to promote these notations among the mathematicians who have no use for [15]. I hope that within five years everybody will be able to use these notations in published papers without needing to explain what they mean.
A new look at Newton’s inequalities
 2000), Article 17. [ONLINE: http://jipam. vu.edu.au/v1n2/014_99.html
"... Communicated by A. Lupa¸s ABSTRACT. New families of inequalities involving the elementary symmetric functions are built as a consequence that all zeros of certain real polynomials are real numbers. ..."
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Cited by 22 (1 self)
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Communicated by A. Lupa¸s ABSTRACT. New families of inequalities involving the elementary symmetric functions are built as a consequence that all zeros of certain real polynomials are real numbers.
ASplines: Local Interpolation and Approximation using C^kContinuous Piecewise Real Algebraic Curves
, 1992
"... We present a concise characterization of the BernsteinBezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise C k continuous chain of such re ..."
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Cited by 17 (14 self)
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We present a concise characterization of the BernsteinBezier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise C k continuous chain of such real algebraic curve segments in BBform as an Aspline (short for algebraic spline). We show how to construct cubic and quartic Asplines to locally interpolate and/or approximate the vertices of an arbitrary planar polygon with up to C 3 and C 5 continuity, respectively. Quadratic Asplines are always locally convex. We also prove that our cubic Asplines are also always locally convex. Additionally, we derive evaluation formulas for the efficient display of all these Asplines and present examples of their use in geometric modeling. 1 Introduction Designing fonts with piecewise smooth curves or fitting curves to scattered data for image reconstruction are just two of the diverse applic...
Data Fitting with Cubic ASplines
 Proceedings of Computer Graphics International, CGI'94
, 1996
"... We present algorithms for constructing isocontours from image data or fitting scattered point data C 1 , C 2 or C 3 piecewise smooth chains of single sheeted real cubic algebraic curve segments called cubic Asplines (short for cubic algebraic splines). Using cubic Asplines we achieve data ..."
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Cited by 11 (6 self)
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We present algorithms for constructing isocontours from image data or fitting scattered point data C 1 , C 2 or C 3 piecewise smooth chains of single sheeted real cubic algebraic curve segments called cubic Asplines (short for cubic algebraic splines). Using cubic Asplines we achieve data fitting with either a higher order of continuity or greater local flexibility for fixed continuity, than numerous prior schemes. Key Words. isocontours, scattered points, curve fitting, algebraic splines, cubic 1 Introduction Generating contours in image data, reconstructing digitized signals, and designing scalable fonts are only some of the several applications of spline curve fitting techniques. In this paper, we generalize past fitting schemes with conic splines [4, 16, 17, 18] and even rational parametric splines [7, 13, 19], We exhibit efficient techniques to deal with cubic algebraic splines (Asplines) achieving fits with small number of pieces yet higher order of smoothness/contin...
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Cited by 10 (5 self)
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Motivations for MathLang
, 2005
"... FOMCAF13 What do we want? Open borders for productive collaboration or that we each stick to our borders without including and benefiting from other input? Do we want war+destruction or solid foundations for wisdom and prosperity? • Do we believe in the chosen framework? Should all the world believe ..."
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Cited by 3 (0 self)
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FOMCAF13 What do we want? Open borders for productive collaboration or that we each stick to our borders without including and benefiting from other input? Do we want war+destruction or solid foundations for wisdom and prosperity? • Do we believe in the chosen framework? Should all the world believe in the same framework? Does one framework fit all? Can such a framework exist? • Think of Capitalism, Communism, dictatorship, nationalism, etc... Which one worked in history? • But then, if we are committed to pluralism, are we in danger of being wiped out because being inclusive may well lead to contradictions? • Oscar Wilde: I used to think I was indecisive, but now I’m not sure anymore. FOMCAF13 1Things are not as somber: There is no perfect framework, but some can be invaluable • De Bruijn used to proudly announce: I did it my way. • I quote Dirk van Dalen: The Germans have their 3 B’s, but we Dutch too have our 3 B’s: Beth, Brouwer and de Bruijn. FOMCAF13 2There is a fourth B:
Quasiarithmetic means of covariance functions with potential applications to spacetime data
, 2006
"... data ..."
Upper and Lower Bounds for Text Indexing Data Structures
"... c○Alexander Golynski 2007I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. (Alexander Golynski) The main go ..."
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Cited by 2 (1 self)
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c○Alexander Golynski 2007I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. (Alexander Golynski) The main goal of this thesis is to investigate the complexity of a variety of problems related to text indexing and text searching. We present new data structures that can be used as building blocks for fulltext indices which occupies minute space (FMindexes) and wavelet trees. These data structures also can be used to represent labeled trees and posting lists. Labeled trees are applied in XML documents, and posting lists in search engines. The main emphasis of this thesis is on lower bounds for timespace tradeoffs for the following problems: the rank/select problem, the problem of representing a string of balanced parentheses, the text retrieval problem, the problem of computing a permutation and its inverse, and the problem of representing a binary relation. These results are divided in two groups: lower bounds in the cell probe model and lower bounds in the indexing model.
Bridging the gap between argumentation theory and the philosophy of mathematics
"... Abstract. We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work ..."
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Abstract. We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, which uses work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods.
Optimized product quantization for approximate nearest neighbor search. CVPR
, 2013
"... Product quantization is an effective vector quantization approach to compactly encode highdimensional vectors for fast approximate nearest neighbor (ANN) search. The essence of product quantization is to decompose the original highdimensional space into the Cartesian product of a finite number of ..."
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Cited by 2 (1 self)
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Product quantization is an effective vector quantization approach to compactly encode highdimensional vectors for fast approximate nearest neighbor (ANN) search. The essence of product quantization is to decompose the original highdimensional space into the Cartesian product of a finite number of lowdimensional subspaces that are then quantized separately. Optimal space decomposition is important for the performance of ANN search, but still remains unaddressed. In this paper, we optimize product quantization by minimizing quantization distortions w.r.t. the space decomposition and the quantization codebooks. We present two novel methods for optimization: a nonparametric method that alternatively solves two smaller subproblems, and a parametric method that is guaranteed to achieve the optimal solution if the input data follows some Gaussian distribution. We show by experiments that our optimized approach substantially improves the accuracy of product quantization for ANN search. 1.