1,720,978 research outputs found
On Iterates of Moebius transformations on fields
No full textThis article has been published in the October 1997 issue of Mathematics of Computation.Let p be a quadratic polynomial over a splitting field K, and S be the set of zeros of p. We define an associative and commutative binary relation on G ≡ K ∪ {∞ } -S so that every Moebius transformation with fixed point set S is of the form x plus" c for some c. This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If K is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Moebius transformation m to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of m are distinct and the iterates of m converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.SUNY Plattsburg
Complex Descartes Circle Theorem
Get PDFThis article has been published in the December 2014 issue of American Mathematical Monthly, published by Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: http://www.jstor.org/stable/10.4169/amer.math.monthly.121.10.927We present a short proof of Descartes Circle Theorem on the curvature-centers of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We also prove an analogue for spheres.SUNY Plattsburg
On two types of exotic addition
No full textThis article has been published in Aequationes Mathematicae.We classify, under reasonable assumptions, all differentiable functions f for which the 'secant method' [xf(y)- yf(x)]/[f(y)- f(x)] is continuous and associative. Further, we classify all differentiable functions for which the similar type of addition xf(y) + yf(x) is associative
Not mixing is just as cool
Get PDFThis article has been published in the October 2004 issue of Mathematics Magazine.Newton's law of cooling, a staple of the Calculus curriculum, is an empirical law not meant for mathematical proof. However, we show it is mathematically equivalent to the intuitively appealing principle that the average temperature of two cooling objects is equal to the temperature of a single object with initial temperature the average of the other two.SUNY Plattsburg
On the spectrum and Martin boundary of homogeneous spaces
Get PDFThis article has been published in the March 1995 issue of Statistics and Probability Letters.Given a conservative, spatially homogeneous Markov process X on an homogeneous spaces χ, we show that if the bottom of the spectrum of the generator of X is zero then the Martin boundary of contains a unique point fixed by the isometry group of χ.SUNY Plattsburg
Sums across Pascal's triangle modulo 2
No full textThis article has been published in Congressus Numerantium.We consider sums of the binomial coefficients C(i + j, i) modulo 2 over lines ai + bj = n. Many interesting sequences (old and new) arise this way
Square Roots of 2x2 Matrices
No full textThis article has been published in the 2010 issue of Contemporary Mathematics.This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of two-by-two matrices. Each method gives rise to applications and examples. Topics touched upon include solutions to Abel's functional equation, Fibonacci numbers, Mobius transformations, systems of differential equations, Newton's method applied to matrices (including surprising pictures and open questions), continued fraction representations of matrices, quadratic number fields, and quadratic forms
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Square Roots of 2x2 Matrices
Get PDFThis article has been published in the 2010 issue of Contemporary Mathematics.This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of two-by-two matrices. Each method gives rise to applications and examples. Topics touched upon include solutions to Abel's functional equation, Fibonacci numbers, Mobius transformations, systems of differential equations, Newton's method applied to matrices (including surprising pictures and open questions), continued fraction representations of matrices, quadratic number fields, and quadratic forms.SUNY Plattsburg
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