Mathematics Commons^™

The Joy Of Quantitative Reasoning, Caren Diefenderfer

Mathematics & Statistics Faculty Scholarship

One of the advantages of focusing on quantitative reasoning is that it spans a wide variety of topics. As incoming president of the National Numeracy Network, I would like to take the opportunity of this editorial to tell my story of intellectual reward from finding common purpose in quantitative reasoning with colleagues from disciplines outside of mathematics. The story starts with an NSF-funded faculty development project (DUE-9952807) to further a QR across-the-curriculum program and the finding from that program that merging authentic context with mathematics brings interaction and collaboration. That joy in learning from and working with colleagues in other …

Nodal Geometry Of Eigenfunctions On Smooth Manifolds And Hardy-Littlewood-Sobolev Inequalities On The Heisenberg Group, Xiaolong Han

Wayne State University Dissertations

Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M:

-Δu=Λu,

in which Δ is the Laplace-Beltrami operator. That is, u is an eigenfunction with eigenvalue Λ. We analyze the asymptotic behavior of eigenfunctions as Λ go to ∞ (i.e., limit of high energy states) in terms of the following aspects.

(1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation.

(2) Write the nodal set of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that …

Fixed Points And Excedances In Restricted Permutations, Sergi Elizalde

Dartmouth Scholarship

Using an unprecedented technique involving diagonals of non-rational generating functions, we prove that among the permutations of length $n$ with $i$ fixed points and $j$ excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for any given $i,j$. Our theorem generalizes a result of Robertson, Saracino and Zeilberger. Even though bijective proofs have later been found by the author jointly with Pak and with Deutsch, this paper contains the original analytic proof that was presented at FPSAC 2003.

Problems And Solutions., Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Regular Octahedrons In {0, 1, K, N} 3, Eugen J. Ionascu

Faculty Bibliography

In this paper we describe a procedure for calculating the number of regular octahedra, RO(n), which have vertices with coordinates in the set {0, 1, ..., n}. As a result, we introduce a new sequence in The Online Encyclopedia of Integer Sequences (A178797) and list the first one hundred terms of it. We improve the method appeared in [12] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0, 1, ..., n}. A new fact proved here helps increasing considerably the speed of all programs used before. The procedure is put together in …

Solution Of Problem 955, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Solution Of Problem 947, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Weight: Does It Really Matter?, Jennifer Brown

Faculty Bibliography

Differential weighting promises to improve the validity of a measure. This study examines whether similar results would be found using weighted, unweighted and standardized z scores from the All Stars Core survey. It was concluded that the weighted systems were developed to equate the questions within the scales and to ease the process for customers without access to data analysis programs; however, the standardized scores were the more appropriate method for equating the test items.

A Generalized Birkhoff-Rott Equation For Two-Dimensional Active Scalar Problems, Hui Sun, David Uminsky, Andrea L. Bertozzi

Mathematics

In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves. These equations are a generalization of the Birkhoff–Rott equation when vorticity is the active scalar. The formulation is Lagrangian and it is valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, …

Network-Based Criterion For The Success Of Cooperation In An Evolutionary Prisoner's Dilemma, Stephen Devlin, T Treloar

Mathematics

We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.

Convergence And Energy Landscape For Cheeger Cut Clustering, X Bresson, T Laurent, David Uminsky, J Von Brecht

Mathematics

This paper provides both theoretical and algorithmic results for the l ₁-relaxation of the Cheeger cut problem. The l₂- relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The l₁-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The l₁-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for …

Model Reduction Of Linear Pde Systems: A Continuous Time Eigensystem Realization Algorithm, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

The Eigensystem Realization Algorithm (ERA) is a well known system identification and model reduction algorithm for discrete time systems. Recently, Ma, Ahuja, and Rowley (Theoret. Comput. Fluid Dyn. 25(1) : 233-247, 2011) showed that ERA is theoretically equivalent to the balanced POD algorithm for model reduction of discrete time systems. We propose an ERA for model reduction of continuous time linear partial differential equation systems. The algorithm differs from other existing approaches as it is based on a direct approximation of the Hankel integral operator of the system. We show that the algorithm produces accurate balanced reduced order models for …

Mathematical Modeling And Simulation Of Biologically Inspired Hair Receptor Arrays In Laminar Unsteady Flow Separation, John R. Singler, Belinda A. Batten, Benjamin T. Dickinson

Mathematics and Statistics Faculty Research & Creative Works

Bats possess arrays of distributed flow-sensitive hair-like mechanoreceptors on their dorsal and ventral wing surfaces. Bat wing hair receptors are known to play a significant role in flight maneuverability and are directionally most sensitive to reversed flow over the wing. in this work, we consider the mechanics of flexible hair-like structures for the time accurate detection and visualization of hydrodynamic images associated with unsteady near surface flow phenomena. a nonlinear viscoelastic model of a hair-like structure coupled to an unsteady nonuniform flow is proposed. Writing the hair model in nondimensional form, we identify five dimensionless groups that govern hair behavior. …

A Linear Energy Stable Scheme For A Thin Film Model Without Slope Selection, Wenbin Chen, Sidafa Conde, Cheng Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

We present a linear numerical scheme for a model of epitaxial thin film growth without slope selection. the PDE, which is a nonlinear, fourth-order parabolic equation, is the L2 gradient flow of the energy ∫Ω(-1/2 ln(1 + |ø|2) + ε2 2 |Ø(x)|2) dx. the idea of convex-concave decomposition of the energy functional is applied, which results in a numerical scheme that is unconditionally energy stable, i.e., energy dissipative. the particular decomposition used here places the nonlinear term in the concave part of the energy, in contrast to a previous convexity splitting scheme. as a result, the numerical scheme is fully …

Productivity Formulae Of An Infinite-Conductivity Hydraulically Fractured Well Producing At Constant Wellbore Pressure Based On Numerical Solutions Of A Weakly Singular Integral Equation Of The First Kind, Chaolang Hu, Jing Lu, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these …

Almost Oscillatory Three Dimensional Dynamic Systems, Elvan Akin, Zuzana Dosla, Bonita Lawrence

Mathematics and Statistics Faculty Research & Creative Works

In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems.

S-Index: A Comprehensive Scholar Impact Index, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

Limitations of impact indices to compare scholars across disciplines and time based only the number of publications and citations are discussed. The S-index, based on more comprehensive scholar impact factors, is proposed.

A Note On Hopfian And Co-Hopfian Abelian Groups, Brendan Goldsmith, Ketao Gong

Conference papers

The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we exploit some unpublished ideas of Corner to answer questions relating to such groups. In particular, we extend an answer given by Corner to a problem of Beaumont and Pierce and show how the properties may be lifted from subgroups to the whole group in certain situations.

Cubes In {0,1, ...N}³, Eugen J. Ionascu, Rodrigo Obando

Faculty Bibliography

The main aim of this paper is to describe a procedure for calculating the number of cubes that have coordinates in the set {0, 1, . . . , n}. For this purpose we continue and, at the same time, revise some of the work begun in a sequence of papers about equilateral triangles and regular tetrahedra all having integer coordinates for their vertices. We adapt the code that was included in a paper by the first author and was used to calculate the number of regular tetrahedra with vertices in {0, 1, . . . , n}3. The idea …

On The Logic Of Reverse Mathematics, Alaeddine Saadaoui

Theses, Dissertations and Capstones

The goal of reverse mathematics is to study the implication and non-implication relationships between theorems. These relationships have their own internal logic, allowing some implications and non-implications to be derived directly from others. The goal of this thesis is to characterize this logic in order to capture the relationships between specific mathematical works. The results of our study are a finite set of rules for this logic and the corresponding soundness and completeness theorems. We also compare our logic with modal logic and strict implication logic. In addition, we explain two applications of S-logic in topology and second order arithmetic.