1,786 research outputs found
On the generic triangle group
We introduce the concept of a generic Euclidean triangle and study the
group generated by the reflection across the edges of . In
particular, we prove that the subgroup of all translations in
is free abelian of infinite rank, while the index 2 subgroup of all
orientation preserving transformations in is free metabelian of rank
2, with as the commutator subgroup. As a consequence, the group
cannot be finitely presented and we provide explicit minimal infinite
presentations of both and . This answers in the affirmative
the problem of the existence of a minimal presentation for the free metabelian
group of rank 2. Moreover, we discuss some examples of non-trivial relations in
holding for given non-generic triangles .Comment: 21 pages, 6 figure
Math Active Learning Lab: Math 107 Precalculus Notebook
This course notebook has been designed for students of Math 107 (Precalculus) at the University of North Dakota. It has been designed to help you get the most out of the ALEKS resources and your time. Topics in the Notebook are organized by weekly learning module. Space for notes from ALEKS learning pages, e-book and videos directs you to essential concepts. Examples and “You Try It” problems have been carefully chosen to help you focus on these essential concepts. Completed Notebook is an invaluable tool when studying for exams.https://commons.und.edu/oers/1023/thumbnail.jp
Odd Wheels Are Not Odd-Distance Graphs
An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane so that the lengths of the edges are odd integers
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Integer Solutions to Optimization Problems and Modular Sequences of Nexus Numbers
In this thesis, we examine the use of integers through two ideas. As mathematics teachers, we prefer students not use calculators on assessments. In order to require this, students compute the problems by hand. We take a look at the classic Calculus I optimization box problem while restricting values to integers. In addition, sticking with the integer theme, we take a new look at the nexus numbers. Nexus numbers are extensions of the hex and rhombic dodecahedral numbers. We put these numbers into a sequence, and through a few computations of modular arithmetic, we analyze the sequences and their patterns based upon the different moduli. These patterns are specific to whether the power is even or odd. Within each power, there are other properties to this set of sequences. Depending on modulus, there are some sequences that stand out more than others
Riemann surfaces and Schrodinger potentials of gauged supergravity
Supersymmetric domain-wall solutions of maximal gauged supergravity are
classified in 4, 5 and 7 dimensions in the presence of non-trivial scalar
fields taking values in the coset SL(N, R)/SO(N) for N=8, 6 and 5 respectively.
We use an algebro-geometric method based on the Christoffel-Schwarz
transformation, which allows for the characterization of the solutions in terms
of Riemann surfaces whose genus depends on the isometry group. The
uniformization of the curves can be carried out explicitly for models of low
genus and results into trigonometric and elliptic solutions for the scalar
fields and the conformal factor of the metric. The Schrodinger potentials for
the quantum fluctuations of the graviton and scalar fields are derived on these
backgrounds and enjoy all properties of supersymmetric quantum mechanics.
Special attention is given to a class of elliptic models whose quantum
fluctuations are commonly described by the generalized Lame potential
\mu(\mu+1)P(z) + \nu(\nu+1)P(z+\omega_1)+ \kappa(\kappa+1)P(z+\omega_2) +
\lambda(\lambda+1)P(z+\omega_1 +\omega_2) for the Weierstrass function P(z) of
the underlying Riemann surfaces with periods 2\omega_1 and 2\omega_2, for
different half-integer values of the coupling constants \mu, \nu, \kappa,
\lambda.Comment: 13 pages, latex; contribution to the proceedings of the TMR meeting
"Quantum Aspects of Gauge Theories, Supersymmetry and Unification" held in
Paris in September 199
Sines, Cosines, and Conjugates
This thesis is an investigation of angles whose sine and cosine are algebraic conjugates over the field of rational numbers. That is to say, sin(0) and cos(0) are roots of the same irreducible polynomial with integer coefficients. These interesting families are explored. First, it is shown that for n\u3e2, the angles have this property. Second, all angles which are conjugate in this sense and which have a quadratic minimum polynomial are identified. The relationship between these two families is explored, and a family of conjugate angles with 4^^ degree minimum polynomials is explored as well. Questions for further investigation are proposed, including an intriguing connection to chaos theory
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