Physical Sciences and Mathematics Commons^™

Teaching Symmetry In The Elementary Curriculum, Christy Knuchel

The Mathematics Enthusiast

Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry every day but often don’t realize it. People use concepts of symmetry, including translations, rotations, reflections, and tessellations as part of their careers. Examples of careers that incorporate these ideas are artists, craftspeople, musicians, choreographers, and not to mention, mathematicians.

It is important for students to grasp the concepts of geometry and symmetry while at the elementary level as a means of exposing them to things they see everyday that aren’t obviously related to mathematics but have …

Radius, Diameter, Circumference, Pi, Geometer's Sketchpad, And You!, Scott Edge

The Mathematics Enthusiast

I truly believe learning mathematics can be a fun experience for children of all ages. It is up to us, the teachers, to present math as an interesting application. The addition of computers into our ever-changing world has given us an important tool, which can assist us on our journey to teach math in new fun and interesting ways. The Program Geometer’s Sketchpad© is one of many mathematic programs we as teachers can use to better help kids understand different geometric concepts. I would like to use Geometer’s Sketchpad© in my classroom to help teach my students about circles and …

Tme Volume 1, Number 1

The Mathematics Enthusiast

No abstract provided.

Fermat's Last Theorem For Rational Exponents, Curtis D. Bennett, A. M. W. Glass, Gábor J. Székely

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

Two New Criteria For Comparison In The Bruhat Order, Brian Drake, Sean Gerrish, Mark Skandera

Dartmouth Scholarship

We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type $A$). One criterion utilizes totally nonnegative polynomials and the other utilizes Schur functions.

Transcendence Measures And Algebraic Growth Of Entire Functions, Dan Coman, Evgeny A. Poletsky

Mathematics - All Scholarship

In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in C2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z, f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {nj} of degrees of polynomials. But for special classes of functions, including the Riemann zeta-function, they hold …

A Forbidden Subgraph Characterization Problem And A Minimal-Element Subset Of Universal Graph Classes, Michael D. Barrus

Theses and Dissertations

The direct sum of a finite number of graph classes H_1, ..., H_k is defined as the set of all graphs formed by taking the union of graphs from each of the H_i. The join of these graph classes is similarly defined as the set of all graphs formed by taking the join of graphs from each of the H_i. In this paper we show that if each H_i has a forbidden subgraph characterization then the direct sum and join of these H_i also have forbidden subgraph characterizations. We provide various results which in many cases allow us to exactly …

Problems Related To The Zermelo And Extended Zermelo Model, Benjamin Zachary Webb

Theses and Dissertations

In this thesis we consider a few results related to the Zermelo and Extended Zermelo Model as well as outline some partial results and open problems related thereto. First we will analyze a discrete dynamical system considering under what conditions the convergence of this dynamical system predicts the outcome of the Extended Zermelo Model. In the following chapter we will focus on the Zermelo Model by giving a method for simplifying the derivation of Zermelo ratings for tournaments in terms of specific types of strongly connected components. Following this, the idea of stability of a tournament will be discussed and …

Lattices And Their Applications To Rational Elliptic Surfaces, Gretchen Rimmasch

Theses and Dissertations

This thesis discusses some of the invariants of rational elliptic surfaces, namely the Mordell-Weil Group, Mordell-Weil Lattice, and another lattice which will be called the Shioda Lattice. It will begin with a brief overview of rational elliptic surfaces, followed by a discussion of lattices, root systems and Dynkin diagrams. Known results of several authors will then be applied to determine the groups and lattices associated with a given rational elliptic surface, along with a discussion of the uses of these groups and lattices in classifying surfaces.

Shallow Water Modeling Of Antarctic Bottom Water Crossing The Equator, Paul F. Choboter, Gordon E. Swaters

Mathematics

The dynamics of abyssal equator-crossing flows are examined by studying simplified models of the flow in the equatorial region in the context of reduced-gravity shallow water theory. A simple “frictional geostrophic” model for one-layer cross-equatorial flow is described, in which geostrophy is replaced at the equator by frictional flow down the pressure gradient. This model is compared via numerical simulations to the one-layer reduced-gravity shallow water model for flow over realistic equatorial Atlantic Ocean bottom topography. It is argued that nonlinear advection is important at key locations where it permits the current to flow against a pressure gradient, a mechanism …

Computing Isotypic Projections With The Lanczos Iteration, David K. Maslen, Michael E. Orrison, Daniel N. Rockmore

Dartmouth Scholarship

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric group.

Quasianalyticity And Pluripolarity, Dan Coman, Norman Levenberg, Evgeny A. Poletsky

Mathematics - All Scholarship

We show that the graph gamma f = {(z, f(z)) in C2 : z in S} in C2 of a function f on the unit circle S which is either continuous and quasianalytic in the sense of Bernstein or C1 and quasianalytic in the sense of Denjoy is pluripolar.

Smooth Submanifolds Intersecting Any Analytic Curve In A Discrete Set, Dan Coman, Norman Levenberg, Evgeny A. Poletsky

Mathematics - All Scholarship

We construct examples of C^inifinity smooth submanifolds in Cⁿ and Rⁿ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real n-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.

Intersection Properties Of Balls In Banach Spaces And Related Topics., Sudipta Dutta Dr.

Doctoral Theses

In the first part of this chapter, we explain in general terms the background and the main theme of this thesis and provide a chapter-wise summary of its principal results. In the second part, we introduce some notations and preliminaries that will be used in the subsequent chapters.As a prototype of the properties we will study in this thesis, let us call a closed linear subspace Y of a Banach space X a (P)-subspace of X if Y has a certain property P as a subspace of X. If a Banach space X, in its canonical embedding, is a (P)-subspace …

A Frame Bundle Generalization Of Multisymplectic Momentum Mappings, J Lawson

Mathematics Faculty Research

We construct momentum mappings for covariant Hamiltonian field theories using a generalization of symplectic geometry to the bundle L_Vϒ of vertically adapted linear frames over the bundle of field configurations ϒ. Field momentum observables are vector-valued momentum mappings generated from automorphisms of ϒ, using the (n + k)-symplectic geometry of L_Vϒ. These momentum observables on LVϒ generalize those in covariant multisymplectic geometry and produce conserved field quantities along flows. Three examples illustrate the utility of these momentum mappings: orthogonal symmetry of a Kaluza-Klein theory generates the conservation of field angular momentum, affine …

Self-Similarity And Symmetries Of Pascal’S Triangles And Simplices Mod P, Richard P. Kubelka

Faculty Publications

No abstract provided.

Self-Similarity And Symmetries Of Pascal’S Triangles And Simplices Mod P, Richard P. Kubelka

Richard P. Kubelka

No abstract provided.

Magical Miscellany, Francis Su

All HMC Faculty Publications and Research

What is a Math Fun Fact, you ask? A Math Fun Fact is any mathematical tidbit that can be presented or grasped quickly, is surprising or captivating, can be generally enjoyed by friends of mathematics, and is hopefully fun! Of course, part of the fun is thinking about why the Fun Fact is true--so we won't spoil the fun. Though, we may give you some hints and references

However, since there are infinitely many Math Fun Facts (prove this), we can only bring you a few each time... here are a few whose conclusions might be considered "magical".

Finite Horizon Riemann Structures And Ergodicity, Victor J. Donnay, Charles Pugh

Mathematics Faculty Research and Scholarship

In this paper we show that any surface in R-3 can be modified by gluing on small 'focusing caps' so that its geodesic flow becomes ergodic. A new concept, finite horizon cap geometry, is what makes the construction work.

Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin

Publications

We study a Hamiltonian system that has a superquadratic potential and is asymptotic to an autonomous system. In particular, we show the existence of a nontrivial solution homoclinic to zero. Many results of this type rely on a convexity condition on the nonlinearity, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. This paper replaces that condition with a different condition, which is automatically satisfied when the autonomous system is radially symmetric. Our proof employs variational and mountain-pass arguments. In some similar results requiring the convexity condition, solutions inhabit a submanifold homeomorphic to the …

On The Number Of Embeddings Of Minimally Rigid Graphs, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the …

Invariant Currents And Dynamical Lelong Numbers, Dan Coman, Vincent Guedj

Mathematics - All Scholarship

Let f be a polynomial automorphism of C^k of degree lamda, whose rational extension to P^k maps the hyperplane at infinity to a single point. Given any positive closed current S on Pk of bidegree (1,1), we show that the sequence lamda−ⁿ(fⁿ)*S converges in the sense of currents on P^k to a linear combination of the Green current T₊ of f and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for …

Measures Of Concordance Of Polynomial Type, Heather Edwards

Electronic Theses and Dissertations

A measure of concordance, $\kappa$, is of polynomial type if and only if $\kappa (tA+(1-t)B)$ is a polynomial in $t$ where $A$ and $B$ are 2-copulas. The degree of such a type of measure of concordance is simply the highest degree of the polynomial associated with $\kappa$. In previous work [2], [3], properties of measures of concordance preserving convex sums (equivalently measures of concordance of polynomial type degree one) were established; however, a characterization was not made. Here a characterization is made using approximations involving doubly stochastic matrices. Other representations are provided from this characterization leading naturally to two interpretations …

A Fixed Point Theorem For Analytic Functions, Valentin Matache

Mathematics Faculty Publications

We prove that each analytic self-map of the open unit disk which interpolates between certain n-tuples must have a fixed point.

Constructing Random Probability Distributions, Theodore P. Hill, David E.R. Sitton

Research Scholars in Residence

This article surveys several classes of iterative methods for constructing random probability distributions (or random convex functions, or random homeomorphisms), and includes illustrative applications in statistics, optimal-control theory, and game theory. Computer simulations of these methods are fast and easy to implement

Integral Transforms, Convolution Products, And First Variations, Bong Jin Kim, Byoung Soo Kim, David Skough

Department of Mathematics: Faculty Publications

We establish the various relationships that exist among the integral transform F_α,βF, the convolution product (F∗G)α, and the first variation δF for a class of functionals defined on K[0,T], the space of complex-valued continuous functions on [0,T] which vanish at zero.

Discrete Approximations And Necessary Optimality Conditions For Functional-Differential Inclusions Of Neutral Type, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper deals with necessary optimality conditions for optimal control systems governed by constrained functional-differential inclusions of neutral type. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functional-differential inclusions and obtain new necessary optimality conditions of both Euler-Lagrange and Hamiltonian types.

Norms Of Linear-Fractional Composition Operators, Paul S. Bourdon, E. E. Fry, Christopher Hammond, C. H. Spofford

Mathematics Faculty Publications

No abstract provided.

Fixed Point Theorems For Infinite Dimensional Holomorphic Functions, Lawrence A. Harris

Mathematics Faculty Publications

This talk discusses conditions on the numerical range of a holomorphic function defined on a bounded convex domain in a complex Banach space that imply that the function has a unique fixed point. In particular, extensions of the Earle-Hamilton Theorem are given for such domains. The theorems are applied to obtain a quantitative version of the inverse function theorem for holomorphic functions and a distortion form of Cartan's uniqueness theorem.

A Computational Model For Martensitic Thin Films With Compositional Fluctuation, Pavel Bělík, Mitchell Luskin

Faculty Authored Articles

We develop a computational model for the martensitic first-order structural phase transformation in a single crystal thin film, and we use this model to study the effect of spatial compositional fluctuation, spatial temporal noise, and the loss of stability of the metastable phase at temperatures sufficiently far from the transformation temperature.