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Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the MeanField Theory for Probabilists
 Bernoulli
, 1997
"... Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by ..."
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Cited by 142 (13 self)
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Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by the Smoluchowski coagulation equations, have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x; y) = 1 and K(x; y) = xy. We attempt a wideranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS9622859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...
Tragic Loss or Good Riddance? The Impending Demise of Traditional Scholarly Journals
 INTERNATIONAL JOURNAL OF HUMANCOMPUTER STUDIES
, 1995
"... Traditional printed journals are a familiar and comfortable aspect of scholarly work. They have been the primary means of communicating research results, and as such have performed an invaluable service. However, they are. ..."
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Cited by 72 (11 self)
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Traditional printed journals are a familiar and comfortable aspect of scholarly work. They have been the primary means of communicating research results, and as such have performed an invaluable service. However, they are.
Estimating Covariances of Locally Stationary Processes: Rates of Convergence of Best Basis Methods
, 1996
"... Mallat, Papanicolaou and Zhang [MPZ98] recently proposed a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in a specially constructed CoifmanMeyer basis of cosine packets. In this paper we extend this approach to estimating the covariance ..."
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Cited by 20 (10 self)
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Mallat, Papanicolaou and Zhang [MPZ98] recently proposed a method for approximating the covariance of a locally stationary process by a covariance which is diagonal in a specially constructed CoifmanMeyer basis of cosine packets. In this paper we extend this approach to estimating the covariance from sampled data. Our method combines both wavelet shrinkage and cosinepacket bestbasis selection in a simple and natural way. The resulting algorithm is fast and automatic. The method has an interpretation as a nonlinear, adaptive form of anisotropic timefrequency smoothing. We introduce a new class of locally stationary processes which exhibits a form of inhomogeneous nonstationarity; our processes have covariances which typically change little from row to row, but might occasionally change abruptly. We study performance in an asymptotic setting involving triangular arrays of processes which are becoming increasingly stationary, and are able to prove rates of convergence results for our...
Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 8 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Cited by 7 (7 self)
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
CRYSTALLINE CONCEPTS IN LONGTERM MATHEMATICAL INVENTION AND DISCOVERY
"... In this paper [1], I formulate a cognitive concept that offers a simple unifying foundation for the most sophisticated level of development in the growth of mathematical thinking while also having implications for mathematics at all levels. It relates to my formulation of three distinct worlds of ma ..."
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Cited by 3 (2 self)
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In this paper [1], I formulate a cognitive concept that offers a simple unifying foundation for the most sophisticated level of development in the growth of mathematical thinking while also having implications for mathematics at all levels. It relates to my formulation of three distinct worlds of mathematical thinking (Tall, 2004), which in turn links naturally to a wide range of theories of cognitive development [2], some of which I discuss in this article. The idea offers common ground for all these theoretical frameworks, relating the development of human thinking to the structure of mathematical concepts. It enables us to respond to deep questions about the nature of mathematics itself in an essentially simple, structured way. For example, “Is mathematics discovered or is it invented?” “Does it exist out there in some pure platonic world, or is it
Mathematical Intuition vs. Mathematical Monsters
, 1998
"... Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of ..."
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Cited by 3 (1 self)
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Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of intuition. We examine several famous geometrical, topological and settheoretical examples of such monsters in order to see to what extent, if at all, intuition is undermined in its everyday roles.
Exploratory Experimentation: Digitallyassisted Discovery and Proof
 Chapter in ICMI Study 19: On Proof and Proving in Mathematics Education. In press
, 2010
"... Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate the role of proof in light of the power of current computer systems, the sophistication of modern mathematical computing packages, and the growing capacity to datamine on the internet. Added to those ..."
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Cited by 2 (2 self)
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Abstract The mathematical community (appropriately defined) faces a great challenge to reevaluate the role of proof in light of the power of current computer systems, the sophistication of modern mathematical computing packages, and the growing capacity to datamine on the internet. Added to those are the enormous complexity of many modern mathematical results such as the Poincaré conjecture, Fermat’s last theorem, and the classification of finite simple groups. With great challenges come great opportunities. Here, I survey the current challenges and opportunities for the learning and doing of mathematics. As the prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished. Much of this material was presented as a plenary talk in May