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 Notices of the AMS, Accepted March
"... the “third leg ” of science, complementing theory and experiment. That metaphor is outdated. Computing now pervades all of science. Massive computation is often required to reduce and analyse data; simulations are employed in fields as diverse as climate modeling and astrophysics. Unfortunately, sci ..."
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the “third leg ” of science, complementing theory and experiment. That metaphor is outdated. Computing now pervades all of science. Massive computation is often required to reduce and analyse data; simulations are employed in fields as diverse as climate modeling and astrophysics. Unfortunately, scientific computing culture has not kept pace. Experimental researchers are taught early to keep notebooks or computer logs of every work detail—design, procedures, equipment, raw results, processing techniques, statistical methods of analysis, etc. In contrast, few computational experiments are performed with such care. Typically, there is no record of workflow, computer hardware and software configuration, or parameter settings. Often source code is lost. While crippling reproducibility of results, these practices ultimately impede the researchers ’ own productivity.
Two Catalantype Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind
"... Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalantype Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2)(2x) n and (1/2)(4x) n ..."
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Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalantype Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2)(2x) n and (1/2)(4x) n in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representations of the central binomial coefficients and Catalan numbers. 1
Publishing Standards for Computational Science: “Setting the Default to Reproducible”
"... A group of computational scientists has developed a set of standards to guide the dissemination of reproducible research. Computation is now central to the scientific enterprise, and the emergence of powerful computational hardware combined with a vast array of computational software, presents novel ..."
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A group of computational scientists has developed a set of standards to guide the dissemination of reproducible research. Computation is now central to the scientific enterprise, and the emergence of powerful computational hardware combined with a vast array of computational software, presents novel opportunities for researchers. Unfortunately the scientific culture surrounding computational work has evolved in ways that make it difficult to verify findings, efficiently build on past research, or even to apply the basic tenets of the scientific method to computational procedures. As a result computational science is facing a credibility crisis [14]. The enormous scale of stateoftheart scientific computations, using tens or hundreds of thousands of processors, presents unprecedented challenges. Numerical reproducibility is a major issue, as is hardware reliability. For some applications, even rare interactions of circuitry with stray subatomic particles matter. In December of 2012, more than 70 computational scientists and stakeholders such as journal editors and funding agency officials gathered at Brown University for the ICERM Workshop on Reproducibility in Computational and Experimental Mathematics. This
The Life of Pi: From Archimedes to Eniac and Beyond 1
"... The desire to understand π, the challenge, and originally the need, to calculate ever more accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged mathematicians–great and less great—for many centuries. It has also, especially recently, provided compelling ex ..."
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The desire to understand π, the challenge, and originally the need, to calculate ever more accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged mathematicians–great and less great—for many centuries. It has also, especially recently, provided compelling examples of computational mathematics. π, uniquely in mathematics, is pervasive in popular culture and the popular imagination. I shall intersperse this largely chronological account of π’s mathematical status with examples of its ubiquity. More details will be found in the selected references at the end of the chapter—especially in Pi: a Source Book [9]. In [9] all material not otherwise referenced may be followed up upon, as may much other material, both serious and fanciful. Other interesting material is to be found in [21], which includes attractive discussions of topics such as continued fractions and elliptic integrals. Fascination with π is evidenced by the many recent popular books, television shows, and movies—even perfume—that have mentioned π. In the 1967 Star Trek episode “Wolf in the Fold, ” Kirk asks “Aren’t there some mathematical problems that simply can’t be solved? ” And Spock ‘fries the brains ’ of a rogue computer by telling it: “Compute to the last digit the value of Pi. ” (Figure 1 illustrates how much more is now known, see also http://carma.newcastle.edu.au/piwalk.shtml.) The May 6, 1993 episode of The Simpsons has the character Apu boast “I can recite pi to 40,000 places. The last digit is one.” In November 1996, MSNBC aired a Thanksgiving Day segment about π, including that scene from Star
Fluid Pinchoff
"... This 4608 2 image of a combustion simulation result was rendered by a hybridparallel (MPI+pthreads) raycasting volume rendering implementation running on 216,000 cores of the JaguarPF supercomputer. Combustion simulation data courtesy of J. Bell and M. Day ..."
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This 4608 2 image of a combustion simulation result was rendered by a hybridparallel (MPI+pthreads) raycasting volume rendering implementation running on 216,000 cores of the JaguarPF supercomputer. Combustion simulation data courtesy of J. Bell and M. Day
References
"... use of computers is revolutionizing doing mathematics. The most obvious and trivial use, that even “pure ” mathematicians (like Amanda Folsom and Ken Ono) should be able to do, is the empirical checking of formulas and statements. Unfortunately, they don’t. Frank Garvan (see [AGL]) discovered that t ..."
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use of computers is revolutionizing doing mathematics. The most obvious and trivial use, that even “pure ” mathematicians (like Amanda Folsom and Ken Ono) should be able to do, is the empirical checking of formulas and statements. Unfortunately, they don’t. Frank Garvan (see [AGL]) discovered that the statement of Theorem 1.2 in [FO] is false as stated. You can’t trust humans, they are such screwups! This prompted me to find out for myself. Once the short Maple code (written in a few minutes by D. Zeilberger):