Experimentation in mathematics computational paths to discovery pdf

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Aiene Realuyo. Brandon Lim. Anonymous WtjVcZCg. Saifullah Suhairi. Michelle Gonzales Caliuag. Kaliswary Kathan. Rishi Balan. Elaiprisezra Rosmi. These expectations have actually been checked by computer simulations.

Hint: Reduce the first integral to a three dimensional one, and use the binomial theorem on both. Each has a distinguished history. We will show that computational techniques can ac- celerate both solution and understanding of these problems. Formula 5. We return to the analytic behavior of this series below. Many geometric representations exist. See Figure 5. Such techniques provide alternate ways to prove results such as the number of partitions of n with all parts odd is the number of partitions of n into distinct parts, see Volume 2, Chapter 4, Exercise 1.

Now, however, we can use the most naive approach: Computing terms of the series for the inverse product in 5. A recent discussion of this formula is given by Almkvist and Wilf in [2]. It is interesting to speculate how much corresponding beautiful mathematics is not done when computation becomes too easy—both Maple and Mathematica have good built-in partition functions.

Fur- ther details of what follows are given fully in [15]. He asked: How many pairs of rabbits can be produced from a single pair in a year if every month each pair begets a new pair which from the second month on becomes productive?

Lest one thinks the problem is imprecise, Fibonacci describes the solution in the text and in the margin. Moreover, it has a straightforward proof, as did the preceding. Whereupon I leave it to you for homework. The Fibonacci sequence occurs in many contexts both serious and quirky.

For example, is the only Fibonacci square. It is easy to check that the sequence in 5. What happens for negative integers is more interesting. The proof of the first formula is a consequence of the result on theta functions in Volume 1, 4. Moreover, since both the initial sums and especially the theta functions are easy to compute numerically, we can hunt for other such identities using integer relation methods. If we compute the corresponding continued fractions of the two sums, we obtain the quite different results [1, 1, 4, 1, 2, 3, 6, 2, 1, 3, 1, , 1, 3, 12] and [1, 8, 2, 8, 2, 8, 2, 8, 2, 8] in partial confirmation.

A combinatorial determinant problem. Taken from [32]. Solution: The pattern is clear from the first few cases on simplifying in Maple or Mathematica.

A sum-of-powers determinant. Solution: The first few instances of this sequence are 1, 4, , , , , which can be quickly identified as q! Hint: 4 3 Observe numerically,"then prove by induction, that An has determinant 1 an bn and is of the form. A polygon problem. Hint: In each case the sequence starts 1, 2, 5, 14, 42, , , , Fibonacci and Lucas numbers in terms of hyperbolic functions. Also, if the process of computing the consequences is indefinite, then with a little skill any experimental result can be made to look like the expected consequences.

Richard Feynman, , from Gary Taubes, The Political Science of Salt, In this chapter, we will examine in more detail some additional underlying com- putational techniques that are useful in experimental mathematics. In partic- ular, we shall briefly examine techniques for theorem proving, prime number computations, polynomial root finding, numerical quadrature, and infinte se- ries summation.

As in the first volume, we focus here on practical algorithms and techniques. In some cases, there are known techniques that have superior efficiencies or other characteristics, but for various reasons are not considered suitable for practical implementation.

We acknowledge the existence of such algorithms but do not, in most cases, devote space to them. To briefly reprise one example, we were inspired by a recent problem in the American Mathematical Monthly [1].

The third of these results P is the result from the Monthly. However, these packages do have some limitations, and in many cases much faster performance can be achieved with custom-written programs. And in general it is beneficial to have some understanding of quadrature techniques, even if you rely on software packages to perform the actual computation. We describe here three state-of-the-art, highly efficient techniques for numer- ical quadrature. While error function quadrature is not as efficient as Gaussian quadrature for continuous, bounded, well-behaved functions on finite intervals, it often produces highly accurate results even for functions with integrable singularities or vertical derivatives at one or both endpoints of the interval.

In contrast, Gaussian quadrature typically performs very poorly in such instances. The error function quadrature scheme and the tanh-sinh scheme to be de- scribed in the next section are based on the Euler-Maclaurin summation formula, which can be stated as follows [4, pg.

In the circumstance where the function f x and all of its derivatives are zero at the endpoints a and b, the second and third terms of the Euler-Maclaurin formula are zero. Thus the error in a simple step-function approximation to the integral, with interval h, is simply E. In such cases we have that the error in the above approximation decreases faster than any power of h. We summarize this scheme with the following algorithm statement.

Algorithm 4 Error function complement [erfc] evaluation. As with the Gaussian scheme, m levels or phases of abscissas and weights are precomputed in the error function scheme. Then we perform the computation, increasing the level by one each of which approximately doubles the computation, compared to the previous level , until an acceptable level of estimated accuracy is obtained see Volume 2, Section 7. Algorithm 5 Error function quadrature.

Evaluation of integrals. These examples are taken from Gradsteyn and Ryzhik [31]. We recognize that many of these can be eval- uated analytically using symbolic computing software depending on the available versions. The intent here is to provide exercises for numerical quadrature and constant recognition facilities. Julia sets. Figure 6. The filamentary structure shown is a Julia set, a set of measure zero separating disconnected regions. Chaitin on randomness.

They have absolute certainty and all the rest of us have doubts. Even the best physics is uncer- tain, it is tentative.

Newtonian science was replaced by relativ- ity theory, and then—wrong! But mathematicians like to think that mathematics is forever, that it is eternal. Well, there is an element of that.

Certainly a mathematical proof gives more certainty than an argument in physics or than experimental ev- idence, but mathematics is not certain. But my theory just measures mathematical information. Once you mea- sure mathematical information you see that any mathematical theory can only have a finite amount of information.

But the world of mathematics has an infinite amount of information. Therefore it is natural that any given mathematical theory is limited, the same way that as physics progresses you need new laws of physics. Mathematicians like to think that they know all the laws. My work suggests that mathematicians also have to add new axioms, simply because there is an infinite amount of mathematical information.

This is very controversial. I think mathematicians, in general, hate my ideas. Physicists love my ideas because I am saying that mathematics has some of the un- certainties and some of the characteristics of physics. Another aspect of my work is that I found randomness in the foundations of mathematics. Bibliography [1] Zafar Ahmed. Definitely An Integral. American Mathematical Monthly, —, Journal of Number Theory, —, Pi Unleashed. Springer-Verlag, Heidelberg, An Introduction to Numerical Analysis.

A Fortran Based Multiprecision System. Integer Relation Detection. Computing in Science and Engineering, 2 1 —28, Bailey, Peter B. Borwein, and Simon Plouffe. Mathematics of Computation, —, Bailey and David J. A History of Pi. A Rational Approach to Pi. Nieuw Archief voor Wiskunde, —, Borwein and J. Proceedings of the American Mathematical Society, —, Borwein, J. Borwein, and R. Explicit Evaluation of Euler Sums.

Proceedings of the Edinburgh Mathematical Society, —, Pi, Euler Numbers and Asymptotic Expansions. Borwein, David Borwein, and William F. Borwein and Peter B. Mathematical Association of America, Washington, Zeitschrift fur Physik, C—, Broadhurst and Dirk Kreimer. Physics Letters, B—, Arithmetical Functions. Topics in Advanced Scientific Computation. New Representations for the Madelung Constant. Background Citations.

Methods Citations. Results Citations. Topics from this paper. Numerical analysis Computer experiment. Floor and ceiling functions Polynomial. Book Numerical method. Citation Type. Has PDF. Publication Type. More Filters. Experimental Mathematics: Examples, Methods and Implications. View 1 excerpt, cites background.

The mathematical community appropriately defined faces a great challenge to re-evaluate the role of proof in light of the power of current computer systems, the sophistication of modern … Expand.