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Ten Problems in Experimental Mathematics

Description: This article was stimulated by the recent SIAM ''100 DigitChallenge'' of Nick Trefethen, beautifully described in a recent book. Indeed, these ten numeric challenge problems are also listed in a recent book by two of present authors, where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent was to present ten problems that are characteristic of the sorts of problems that commonly arise in ''experimental mathematics''. The challenge in each case is to obtain a high precision numeric evaluation of the quantity, and then, if possible, to obtain a symbolic answer, ideally one with proof. Our goal in this article is to provide solutions to these ten problems, and in the process present a concise account of how one combines symbolic and numeric computation, which may be termed ''hybrid computation'', in the process of mathematical discovery.
Date: September 30, 2004
Creator: Bailey, David H.; Borwein, Jonathan M.; Kapoor, Vishaal & Weisstein, Eric
Partner: UNT Libraries Government Documents Department
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Experimental Mathemataics: Examples, Methods andImplications

Description: Recent years have seen the flowering of ''experimental'' mathematics, namely the utilization of modern computer technology as an active tool in mathematical research. This development is not limited to a handful of researchers, nor to a handful of universities, nor is it limited to one particular field of mathematics. Instead, it involves hundreds of individuals, at many different institutions, who have turned to the remarkable new computational tools now available to assist in their research, whether it be in number theory, algebra, analysis, geometry or even topology. These tools are being used to work out specific examples, generate plots, perform various algebraic and calculus manipulations, test conjectures, and explore routes to formal proof. Using computer tools to test conjectures is by itself a major time saver for mathematicians, as it permits them to quickly rule out false notions.
Date: January 31, 2005
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department
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A Survey of the Twentieth Century American Trends in Secondary Mathematics Education

Description: This investigation of twentieth century trends in mathematics education includes the survey of existing literature and questionnaires conducted with retired and active Texas teachers. Historical events, trends in curriculum, instruction, learning theories, and contradictions of twenty-year periods are delineated. Questionnaire responses are tabulated along the same periods and vignettes of typical classrooms are drawn from the data. Results of the survey show the impact of societal forces on mathematics curricula, a continued downward expansion of content into lower grades and expanding knowledge of learning processes. A unified mathematics curriculum, classroom-related learning theory research, and further development of team-teaching are postulated as future trends. Recommendations include further examination of trends through isolation of other variables such as region and ethnicity.
Date: May 1980
Creator: Maloney, Letty Lynn
Partner: UNT Libraries
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Doctoral Programs in Mathematics and Education as Related to Instructional Needs of Junior Colleges and Four Year Colleges

Description: The problem of this study was to analyze doctoral programs in mathematics and education for the preparation of teachers of undergraduate mathematics. The purpose of the study was to determine (1) the need for such programs, (2) the attitude of college and university officials toward them, (3) the composition of present offerings and (4) recommendations to the future course their development should take.
Date: June 1967
Creator: Hamilton, William Wingo
Partner: UNT Libraries
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A Strong Hot Spot Theorem

Description: A real number alpha is said to be b-normal if every m-long string of digits appears in the base-b expansion of alpha with limiting frequency b-m. We prove that alpha is b-normal if and only if it possesses no base-b ''hot spot''. In other words, alpha is b-normal if and only if there is no real number y such that smaller and smaller neighborhoods of y are visited by the successive shifts of the base-b expansion of alpha with larger and larger frequencies, relative to the lengths of these neighborhoods
Date: December 31, 2005
Creator: Bailey, David H. & Misiurewicz, Michal
Partner: UNT Libraries Government Documents Department
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An algebraic characterization of stability groups

Description: The goal of this paper is to establish necessary and sufficient conditions for a subgroup of the full homeomorphism group of a manifold to be the stability group of a point in the underlying space. Such subgroups are useful in identifying the underlying space in terms of its homeomorphism group even in cases in which this space is not necessarily a manifold. Thus, stability groups are useful in classifying various spaces.
Date: August 1991
Creator: Wright, William G. (William Glenn)
Partner: UNT Libraries
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The Relation Between Regression Coefficients in a Multiple-Regression Analysis

Description: In a multiple-regression analysis it is often necessary to assign a value, different from the least-squares value, to one or more of the regression coefficients. A simple method is described which eliminates the matrix algebra which, in such cases, is usually necessary in order to re-evaluate the remaining coeffcients, together with the associated variances and covariancesThe method is generalized to give an equally simple procedure for testing possible linear relations among the regression coefficients. (auth)
Date: September 1, 1959
Creator: Monahan, J. E.
Partner: UNT Libraries Government Documents Department
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Sixth Copper Mountain Conference on Multigrid Methods. Final report

Description: During the 5-day meeting, 112 half-hour talks on current research topics were presented. Session topics included: fluids, domain decomposition, iterative methods, Basics I and II, adaptive methods, nonlinear filtering, CFD I, II, and III, applications, transport, algebraic solvers, supercomputing, and student paper winners.
Date: July 1, 1994
Partner: UNT Libraries Government Documents Department
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Future Prospects for Computer-Assisted Mathematics

Description: The recent rise of ''computer-assisted'' and ''experimental'' mathematics raises intriguing questions as to the future role of computation in mathematics. These results also draw into question the traditional distinctions that have been drawn between formal proof and computationally-assisted proof. This article explores these questions in the context of the growing consensus among computer technologists that Moore's Law is likely to continue unabated for quite some time into the future, producing hardware and software much more powerful than what is available today.
Date: October 26, 2005
Creator: Bailey, David H. & Borwein, Jonathan M.
Partner: UNT Libraries Government Documents Department
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Continua and Related Topics

Description: This paper is a study of continue and related metric spaces, Chapter I is an introductory chapter. Irreducible continua and noncut points are the main topics in Chapter II. The third chapter begins with a few results on locally connected spaces. These results are then used to prove results in locally connected continua. Decomposable and indecomposable continua are dealt with in Chapter IV. Totally disconnected metric spaces are studied in the beginning of Chapter V. Then we see that every compact metric space is a continuous image of the Cantor set. A continuous map from the Cantor set onto [0,1] is constructed. Also, a continuous map from [0,1] onto [0,1]x[0,1] is built, Then an order preserving homeomorphism is constructed from a metric arc onto [0,1],
Date: August 1982
Creator: Brucks, Karen M. (Karen Marie), 1957-
Partner: UNT Libraries
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Teachers' Use of Children's Literature, Mathematics Manipulatives, and Scaffolding to Improve Preschool Mathematics Achievement: Does It Work?

Description: The primary purpose of this study was to determine if the implementation of an intervention involving teachers' use of children's literature, related storybook manipulatives, and a scaffolding (LMS) approach to learning would improve preschool children's mathematics test scores. Additionally, the LMS approach was examined to determine whether teachers' perceptions of their effectiveness in mathematics instruction changed from the beginning to the end of the study. The subjects of the study included 60 preschool-aged children and six teachers from two child care centers. The preschool teachers participated in either a control or experimental condition (the LMS approach) in their daily mathematics instruction with their preschool children. The researcher tested the children using the Test of Early Mathematics Ability and an abbreviated version of the Stanford-Binet Intelligence Scale. The study was based on two main research questions. The first question asked if there was a difference in the Test of Early Mathematics Ability total posttest scores between children in the literature-manipulatives-scaffolding intervention group and children in the control group after assuring equivalency of the two groups. The second question addressed if preschool teachers believed they were more effective in their mathematics instruction after implementing the LMS approach with young children. The answer to the first research question was that there was no statistically significant difference in the Test of Early Mathematics Ability total posttest scores between children in the literature-manipulatives-scaffolding group and children in the control group. However, the answer to the second question was that preschool teachers believed they were more effective in their mathematics instruction after implementing the LMS approach with young children. Recommendations for future research on early childhood mathematics include the investigation of preschool children's ability, achievement, and interest in mathematics; teachers' use of mathematics scaffolding techniques; and longitudinal mathematics interventions beginning during the preschool years.
Date: December 2000
Creator: Bennett, Tisha L.
Partner: UNT Libraries
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An Application of Spencer's Method to the Stieltjes and Hamburger Moment Problems

Description: Given the moments of a function it is often desirable to obtain a numerical approximation of the function. A method for estimating the solutions of the Stieltjes and Hamburger moment problems is developed. Calculations indicate that the method is both feasible and accurate. (auth)
Date: July 1, 1959
Creator: Guilinger, W. H.
Partner: UNT Libraries Government Documents Department
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Periodized Wavelets

Description: The properties of periodized Daubechies wavelets on [0,1] are detailed and contrasted against their counterparts which form a basis for L{sup 2}(R). Numerical examples illustrate the analytical estimates for convergence and demonstrate by comparison with Fourier spectral methods the superiority of wavelet projection methods for approximations. The analytical solution to inner products of periodized wavelets and their derivatives, which are known as connection coefficients, is presented, and several tabulated values are included.
Date: December 1993
Creator: Schlossnagle, George; Restrepo, Juan Mario & Leaf, Gary K.
Partner: UNT Libraries Government Documents Department
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GRAPHS OF THE HYPERGEOMETRIC O.C. AND A.O.Q. FUNCTIONS FOR LOT SIZES 10 TO 225

Description: Graphs of the hypergeometric operating characteristic (O.C) and average outgoing quality (A.O.Q.) functions are given for lot sizes 10 to 225 and various sample sizes. Graphs for acceptance numbers O(1)2 and 0(1)3 are included. (auth)
Date: September 1, 1959
Creator: Clark, C.R. & Koopmans, L.H.
Partner: UNT Libraries Government Documents Department
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Final grant report to the U.S. Department of Energy (DOE): Olga Taussky Todd Celebration of Careers in Mathematics for Women.

Description: This grant was proposed to be supplemental funding for a conference which would be documented by publication of conference information in the AWM Newsletter, and then inclusion of the Proceedings and other information generated in a book. The conference was held and documented in the AWM Newsletter (Vol. 29, No. 6; Vol. 30, No. 1 (attached)). Work is in progress on the extensive and complicated related publication which involves 50 authors beyond those from the conference. The Celebration was held at the Mathematical Sciences Research Institute in Berkeley, Calif. It was attended by over 100 women and men mathematicians. Plenary talks, panels, and poster sessions were held.
Date: June 3, 2002
Creator: Case, Bettye Anne
Partner: UNT Libraries Government Documents Department
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