Mathematics Commons^™

The Cross Curvature Flow Of 3-Manifolds With Negative Sectional Curvature, Bennett Chow, Richard S. Hamilton

Turkish Journal of Mathematics

We consider the cross curvature flow, an evolution equation of metrics on 3-manifolds. We establish short time existence when the sectional curvature has a sign. In the case of negative sectional curvature, we obtain some monotonicity formulas which support the conjecture that after normalization, for initial metrics on closed 3-manifolds with negative sectional curvature, the solution exists for all time and converges to a hyperbolic metric. This conjecture is still open at the present time.

Surgery Diagrams For Contact 3-Manifolds, Fan Ding, Hansjörg Geiges, Andras I. Stipsicz

Turkish Journal of Mathematics

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact (\pm 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S^3 by a sequence of such contact (\pm 1)-surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into (\pm 1)-surgeries. We use this to give explicit surgery …

Quasipositivity Problem For 3-Braids, Stepan Yu. Orevkov

Turkish Journal of Mathematics

A braid is called quasipositive if it is a product of conjugates of standard generators of the braid group. We present an algorithm deciding if a given braid with three strings is quasipositive or not. The complexity (the time of work) of our algorithm is O(n^{k+1}) where n is the length of the word in standard generators representing the braid and k is the algebraic length of the braid. The algorithm is based on the Garside normal form. The problem of quasipositivity in braid groups is motivated by the topology of plane real algebraic curves (16th Hilbert's problem). In particular, …

Solution Of The Word Problem In The Singular Braid Group, Stepan Yu. Orevkov

Turkish Journal of Mathematics

Singular braids are isotopy classes of smooth strings which are allowed to cross each other pairwise with distinct tangents. Under the usual multiplication of braids, they form a monoid. The singular braid group was introduced by Fenn-Keyman-Rourke as the quotient group of the singular braid monoid. We give a solution of the word problem for this group. It is obtained as a combination of the results by Fenn-Keyman-Rourke and some simple geometric considerations based on the mapping class interpretation of braids. Combined with Corran's normal form for the singular braid monoid, our algorithm provides a computable normal form for the …

Faà Di Bruno’S Formula And Nonhyperbolic Fixed Points Of One-Dimensional Maps, Vadim Ponomarenko

Mathematics Faculty Research

Fixed-point theory of one-dimensional maps of R does not completely address the issue of non-hyperbolic fixed points. This paper generalizes the existing tests to completely classify all such fixed points. To do this, a family of operators are exhibited that are analogous to generalizations of the Schwarzian derivative. In addition, a family of functions f are exhibited such that the MacLaurin series of f(f(x)) and x are identical.

Effective Asymptotics For Some Nonlinear Recurrences And Almost Doubly-Exponential Sequences, Eugen J. Ionascu

Faculty Bibliography

We develop a technique to compute asymptotic expansions for recurrent sequences of the form an+1 = f(an), where f(x) = x − axα + bxβ + o(x β) as x → 0, for some real numbers α, β, a, and b satisfying a > 0, 1 < α < β. We prove a result which summarizes the present stage of our investigation, generalizing the expansions in [Amer. Math Monthly, Problem E 3034[1984, 58], Solution [1986, 739]]. One can apply our technique, for instance, to obtain the formula: an = √ 3 √ n − 3 √ 3 10 ln n n √ n + 9 √ 3 50 ln n n2√ n + o ln n n5/2 , where an+1 = sin(an), a1 ∈ IR. Moreover, we consider the recurrences an+1 = a 2 n + gn, and we prove that under some technical assumptions, an is almost doubly-exponential, namely an = bk 2 n c, an = bk 2 n c + 1, an = bk 2 n − 1 2 c, or an = bk 2 n + 5 2 c for some real number k, generalizing a result of Aho and Sloane [Fibonacci Quart. 11 (1973), 429–437].

Using Humanistic Content And Teaching Methods To Motivate Students And Counteract Negative Perceptions Of Mathematics, Roger Haglund

Humanistic Mathematics Network Journal

This paper examines the following questions: How is math commonly taught, why is it taught this way, and what are the outcomes? Who are some of the voices calling for change and what are they saying? Can a humanistic approach produce positive results in students who have learned to dislike math and have not been successful in a traditional classroom?

Man's Cards And God's Dice: A Conceptual Analysis Of Probability For The Advanced Student, Elie Feder

Humanistic Mathematics Network Journal

No abstract provided.

What Are Mathematical Problems?, Emam Hoosain

Humanistic Mathematics Network Journal

No abstract provided.

A Linear Perspective To Art, Sarah Littler

Humanistic Mathematics Network Journal

No abstract provided.

Humanistic Mathematics As Mathematics For All, Michael N. Fried

Humanistic Mathematics Network Journal

No abstract provided.

Base And Subbase In A Number System, Walter S. Sizer

Humanistic Mathematics Network Journal

No abstract provided.

Conversations Among Women In Mathematics (Workshop Information), University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

Frobenius Classes In Alternating Groups, David P. Roberts

Mathematics Publications

We present a method, based on an old idea of Serre, for completely computing Frobenius classes in alternating groups. We contrast this method with other approaches in examples involving the alternating groups A₃ and A₉. The method can be useful for proper subgroups of alternating groups as well, and we present examples involving the 168-element group PSL₂(7) = GL₃(2) and the Mathieu group M₂₄.

Nonic 3-Adic Fields, John W. Jones, David P. Roberts

Mathematics Publications

We compute all nonic extensions of Q₃ and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields.

An Abc Construction Of Number Fields, David P. Roberts

Mathematics Publications

We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation ax^p + by^q + cz^r = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2^a3^b .

Fuzzy \Beta-Compactness And Fuzzy \Beta-Closed Spaces, I. M. Hanafy

Turkish Journal of Mathematics

The concepts of \beta-compactness and \beta-closed spaces in the fuzzy setting are defined and investigated. Fuzzy filterbases are used to characterize these concepts. A comparison between these types and some different forms of compactness in fuzzy topology is established.

The Trace Formula For A Differential Operator Of Fourth Order With Bounded Operator Coefficients And Two Terms, Erdal Gül

Turkish Journal of Mathematics

L Yıldız Teknik Üniversitesi Fen-Edebiyat Fakültesi Matematik Bölümü Davutpaşa Kampüsü İstanbul-TURKEY e-mail: [email protected] We investigate the spectrum of a differential operator of fourth order with bounded operator coefficients and find a formula for the trace of this operator.

Splitting Of Sharply 2-Transitive Groups Of Characteristic 3, Seyfi̇ Türkelli̇

Turkish Journal of Mathematics

We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3.

A New Characteristic Of Möbius Transformations By Use Of Apollonius Points Of Pentagons, Serap Bulut, Ni̇hal Yilmaz Özgür

Turkish Journal of Mathematics

In this paper, we give a new characterization of Möbius transformations. To this end, a new concept of ''Apollonius points of pentagons'' is used.

Rate Of Convergence Of Durrmeyer Type Baskakov-Bezier Operators For Locally Bounded Functions, Vijay Gupta

Turkish Journal of Mathematics

In the present paper, we introduce the Durrmeyer variant of Baskakov-Bezier operators B_{n,\alpha} (f,x), which is the modified form of Baskakov-Beta operators. Here we obtain an estimate on the rate of convergence of B_{n,\alpha} (f,x) for functions of bounded variation in terms of Chanturiya's modulus of variation. In the end we also propose an open problem for the readers.

I Don't Care If I Ever Get Back: Marathons Lasting 20 Or More Innings, Phil Lowry, Darren B. Glass

Math Faculty Publications

This article looks at marathon games of baseball. For purposes of this article, a marathon is defined as a game lasting 20 or more innings. In my research I have discovered 341 marathons. These games are hard to find. Leagues either keep no records, or keep track only of their longest game; only the Texas League keeps records on all marathons. Nobody has ever before explored such questions as: What is the probability a game will go x number of innings? How often should we expect a marathon of 20 or more innings, or 40 or more innings? What is …

The Use Of Filters In Topology, Abdellatif Dasser

Electronic Theses and Dissertations

Sequences are sufficient to describe topological properties in metric spaces or, more generally, topological spaces having a countable base for the topology. However, filters or nets are needed in more abstract spaces. Nets are more natural extension of sequences but are generally less friendly to work with since quite often two nets have distinct directed sets for domains. Operations involving filters are set theoretic and generally certain to filters on the same set. The concept of a filter was introduced by H. Cartan in 1937 and an excellent treatment of the subject can be found in N. Bourbaki (1940).

Examples Of Cayley 4-Manifolds, Weiqing Gu, Christopher Pries '03

All HMC Faculty Publications and Research

We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.

We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We …

Cr Approximation On A Nonrigid Hypersurface Graph In Cⁿ, Al Boggess, Roman Dwilewicz

Mathematics and Statistics Faculty Research & Creative Works

Let M be a hypersurface in Cⁿ that is the graph over a (2n - 1)-linear real space. the main result of the paper is that any CR function on M can be uniformly approximated on compact subsets by entire functions on Cⁿ.

Asymptotic Solutions Of A Discrete Schrödinger Equation Arising From A Dirac Equation With Random Mass, Bernd Aulbach, Saber Elaydi, Klaus Ziegler

Mathematics Faculty Research

For a Dirac particle in one dimension with random mass, the time evolution for the average wavefunction is considered. Using the supersymmetric representation of the average Green’s function, we derive a fourth order linear difference equation for the low-energy asymptotics of the average wavefunction. This equation is of Poincar´e type, though highly critical and therefore not amenable to standard methods. In this paper we show that, nevertheless, asymptotic expansions of its solutions can be obtained.

Radiotherapy Treatment Design And Linear Programming, Allen G. Holder

Mathematics Faculty Research

Intensity modulated radiotherapy treatment (IMRT) design is the process of choosing how beams of radiation will travel through a cancer patient to treat the disease, and although optimization techniques have been suggested since the 1960s, they are still not widely used. Instead, the vast majority of treatment plans are designed by clinicians through trial-and-error. Modern treatment facilities have the technology to treat patients with extremely complicated plans, and designing plans that take full advantage of the technology is tedious. The increased technology found in modern treatment facilities makes the use of optimization paramount in the design of successful treatment plans. …

Taxicab Geometry As A Vehicle For The Journey Toward Enlightenment, Neil Greenspan

Humanistic Mathematics Network Journal

No abstract provided.

Small Zeros Of Quadratic Forms With Linear Conditions, Lenny Fukshansky

CMC Faculty Publications and Research

Given a quadratic form and M linear forms in N + 1 variables with coefficients in a number field K, suppose that there exists a point in KN+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.

Pokémon® Cards And The Shortest Common Superstring, Mark Stamp, Austin Stamp

Faculty Publications, Computer Science

Evidence is presented that certain sequences of Pokémon cards are determined by selecting consecutive elements from a longer sequence. We then consider the problem of recovering the shortest common superstring (SCS), i.e., the shortest string that contains each of the Pokémon card sequences as a consecutive substring. The SCS problem arises in many applications, most notably in DNA sequencing.