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Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows
 APEIRON
, 2001
"... re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different set ..."
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re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different sets of adjacent prime number spacing intervals show that the power spectra follow the model predicted universal inverse powerlaw form of the statistical normal distribution. The prime number distribution therefore exhibits selforganized criticality, which is a signature of quantumlike chaos. (iii) The continuum real number field contains unique structures, namely, prime numbers, which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows. Keywords: quantumlike chaos in prime numbers, fractal structure of primes, quantification of prime number distribution, prime numbers and fluid flows 1. Introduction he continuum real number field (infinite numbe
Data, Shapes, Symbols: Achieving Balance in School Mathematics
"... Mathematics is our “invisible culture ” (Hammond 1978). Few people have any idea how much mathematics lies behind the artifacts and accoutrements of modern life. Nothing we use on a daily basis—houses, automobiles, bicycles, furniture, not to mention cell phones, computers, and Palm Pilots—would be ..."
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Mathematics is our “invisible culture ” (Hammond 1978). Few people have any idea how much mathematics lies behind the artifacts and accoutrements of modern life. Nothing we use on a daily basis—houses, automobiles, bicycles, furniture, not to mention cell phones, computers, and Palm Pilots—would be possible without mathematics. Neither would our economy nor our democracy: national defense, Social Security, disaster relief, as well as political campaigns and voting, all depend on mathematical models and quantitative habits of mind. Mathematics is certainly not invisible in education, however. Ten years of mathematics is required in every school and is part of every state graduation test. In the late 1980s, mathematics teachers led the national campaign for high, publicly visible standards in K12 education. Nonetheless, mathematics is the subject that parents most often recall with anxiety and frustration from their own school experiences. Indeed, mathematics is the subject most often responsible for students ’ failure to attain their educational goals. Recently, mathematics curricula have become the subject of ferocious debates in school districts across the country. My intention in writing this essay is to make visible to curious and uncommitted outsiders some of the forces that are currently shaping (and distorting) mathematics education. My focus is on the
Diophantus’ 20th problem and fermat’s last theorem for n=4  Formalization of . . .
, 2005
"... We present the proof of Diophantus’ 20th problem (book VI of Diophantus’ Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the ..."
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We present the proof of Diophantus’ 20th problem (book VI of Diophantus’ Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller nonnegative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are bounded by zero. We describe the formalization of this proof which has been carried out in the Coq proof assistant. Moreover, as a direct and no less historical application, we also provide the proof (by Fermat) of Fermat’s last theorem for n = 4, as well as the corresponding formalization made in Coq.
MARTIN SCHIRALLI 1 and NATHALIE SINCLAIR 2∗ A CONSTRUCTIVE RESPONSE TO ‘WHERE MATHEMATICS
"... researchers with a novel, and perhaps startling perspective on mathematical thinking. However, as evidenced by reviewers ’ criticisms (Gold, 2001; Goldin, 2001; Madden, 2001), their perspective – though liberating for many, with its humanistic emphases – remains controversial. Nonetheless, we believ ..."
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researchers with a novel, and perhaps startling perspective on mathematical thinking. However, as evidenced by reviewers ’ criticisms (Gold, 2001; Goldin, 2001; Madden, 2001), their perspective – though liberating for many, with its humanistic emphases – remains controversial. Nonetheless, we believe this perspective deserves further constructive response. In this paper, we propose that several of the book’s flaws can be addressed through a more rigorous establishment of conceptual distinctions as well as a more appropriate set of methodological approaches. In the past decade, several mathematics education researchers have emphasised the embodied nature of mathematical understanding, working toward displacing the more prevalent, conventional views found in both psychology and philosophy and studying implications for mathematics learning. These researchers have argued that sensorymotor action plays a crucial role in mathematical activity. A major struggle has been to explain how abstract, formal mathematical ideas can emerge from concrete sensorymotor experiences. This struggle has more recently found promising
Quantumlike Chaos in the Frequency Distributions of Bases A, C, G, T in Human Chromosome1 DNA
, 2004
"... Introduction DNA topology is of fundamental importance for a wide range of biological processes [1]. Since the topological state of genomic DNA is of importance for its replication, recombination and transcription, there is an immediate interest to obtain information about the supercoiled state fro ..."
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Introduction DNA topology is of fundamental importance for a wide range of biological processes [1]. Since the topological state of genomic DNA is of importance for its replication, recombination and transcription, there is an immediate interest to obtain information about the supercoiled state from sequence periodicities [2,3]. Identification of dominant periodicities in DNA sequence will help understand the important role of coherent structures in genome sequence organization [4,5]. Li [6] has discussed meaningful applications of spectral analyses in DNA sequence studies. Recent studies indicate that the DNA sequence of letters A, C, G and T exhibit the inverse power law form 1/f frequency spectrum where f is the frequency and a the exponent. It is possible, therefore, that the sequences have longrange order [714]. Inverse powerlaw form for power spectra of fractal spacetime fluctuations is generic to dynamical systems in nature and is identified as selforganized criticality
C. Roy Keys Inc.
"... this paper shows (Section 2) that Fibonacci series underlies fractal fluctuations on all spacetime scales ..."
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this paper shows (Section 2) that Fibonacci series underlies fractal fluctuations on all spacetime scales
THE NATURE OF CONTEMPORARY CORE MATHEMATICS
, 2010
"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."
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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.
Learning Goal: Students will represent and analyze mathematical patterns, relationships, and
"... functions to model and solve problems. With this learning goal in mind, Minnesota students will have the opportunity to pursue the following instructional components: • Recognize, describe, and generalize patterns and build mathematical models to make predictions. • Analyze the interaction between q ..."
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functions to model and solve problems. With this learning goal in mind, Minnesota students will have the opportunity to pursue the following instructional components: • Recognize, describe, and generalize patterns and build mathematical models to make predictions. • Analyze the interaction between quantities and/or variables to model patterns of change. • Use algebraic concepts and processes to represent and solve problems that involve variable quantities. As biology is the science of life and physics the science of energy and matter, so mathematics is
Financed by Federal Money: Dollar Amount of Federal Funds for Grant:
"... government sponsorship are encouraged to express freely their judgement in professional and technical matters. Points of view or opinions do not, therefore, necessarily represent ..."
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government sponsorship are encouraged to express freely their judgement in professional and technical matters. Points of view or opinions do not, therefore, necessarily represent
Redefining Literacy Pages 813 Numeracy: The New Literacy for a DataDrenched Society
"... To develop an informed citizenry and to support a democratic government, schools must graduate students who are numerate as well as literate. ..."
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To develop an informed citizenry and to support a democratic government, schools must graduate students who are numerate as well as literate.