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Counterexamples to Witness Conjectures
, 2003
"... this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures ..."
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this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures
What’s experimental about experimental mathematics?
, 2008
"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."
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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computerbased tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging subdiscipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the socalled PSLQ algorithm.
Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm
"... From a philosophical viewpoint, mathematics has traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the ..."
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From a philosophical viewpoint, mathematics has traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the
ComputerMediated Thinking
, 2004
"... This paper discusses computermediated thinking and some of its possible implications for curriculum design in mathematics education. We begin with a discussion of today’s context and of ideas related to computermediated thinking. We continue with examples of the use of computermediated thinking in ..."
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This paper discusses computermediated thinking and some of its possible implications for curriculum design in mathematics education. We begin with a discussion of today’s context and of ideas related to computermediated thinking. We continue with examples of the use of computermediated thinking in modern applied mathematics. We then extract some suggestions for a curriculum in mathematics centred at the calculus level. We include specific suggestions for removing material from the current syllabus. We end with a discussion of the unintentional power of the calculus. 1
Now
"... ¨ Consider a population whose members are of N different types (e.g. colors). ¨ For 1 ≤ j ≤ N we denote by pj the probability that a member of the population is of type j. ¨ The members of the population are sampled independently with replacement and their types are recorded. ¨Our main object of stu ..."
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¨ Consider a population whose members are of N different types (e.g. colors). ¨ For 1 ≤ j ≤ N we denote by pj the probability that a member of the population is of type j. ¨ The members of the population are sampled independently with replacement and their types are recorded. ¨Our main object of study is the number TN of trials it takes until all N types are detected (at least once). ¨ Of course, P {TN ≥ k} = 1, if 1 ≤ k ≤ N. 2 It is convenient to introduce the events Akj, 1 ≤ j ≤ N, that the type j is not detected until trial k (included). Then P {TN ≥ k} = P Ak−11 ∪ · · · ∪ Ak−1N, k = 1, 2,.... By invoking the inclusionexclusion principle one gets P {TN ≥ k} = P Ak−11
Counterexamples to Witness Conjectures Joris van der Hoeven
"... Consider the class of explog constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let z be such an explog constant and let n be its size as an expression. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which n ..."
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Consider the class of explog constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let z be such an explog constant and let n be its size as an expression. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which need to be evaluated in order to test whether z equals zero. For this purpose, it is convenient to assume that exponentials are only applied to arguments with absolute values bounded by 1. In that context, several witness conjectures have appeared in the literature and the strongest one states that it is possible to choose $(n) = O(n). In this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures. 1.