Mathematics Commons^™

Nested Canalyzing Depth And Network Stability, Lori Layne, Elena Dimitrova, Matthew Macauley

Publications

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input …

Copula Density Estimation By Total Variation Penalized Likelihood With Linear Equality Constraints, Leming Qu, Wotao Yin

Mathematics Faculty Publications and Presentations

A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on a maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by log-barrier method for second order cone program. A …

Free Torus Actions And Two-Stage Spaces, Barry Jessup, Gregory Lupton

Mathematics and Statistics Faculty Publications

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

Physiologic Noise Regression, Motion Regression, And Toast Dynamic Field Correction In Complex-Valued Fmri Time Series, Andrew D. Hahn, Daniel B. Rowe

Mathematics, Statistics and Computer Science Faculty Research and Publications

As more evidence is presented suggesting that the phase, as well as the magnitude, of functional MRI (fMRI) time series may contain important information and that there are theoretical drawbacks to modeling functional response in the magnitude alone, removing noise in the phase is becoming more important. Previous studies have shown that retrospective correction of noise from physiologic sources can remove significant phase variance and that dynamic main magnetic field correction and regression of estimated motion parameters also remove significant phase fluctuations. In this work, we investigate the performance of physiologic noise regression in a framework along with correction for …

Generalized Eigenproblem And Nonlinear Elliptic Equations With Nonlinear Boundary Conditions, Nsoki Mavinga

Mathematics & Statistics Faculty Works

We are concerned with the solvability of nonlinear second-order elliptic partial differential equations with nonlinear boundary conditions. We study the generalized Steklov-Robin eigenproblem (with possibly singular weights) in which the spectral parameter is both in the differential equation and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates.

Clinical And Genetic Association Of Serum Ceruloplasmin With Cardiovascular Risk, W.H. Wilson Tang, Yuping Wu, Jaana Hartiala, Yiying Fan, Alexandre F.R. Stewart, Robert Roberts, Ruth Mcpherson, Paul L. Fox, Hooman Allayee, Stanley L. Hazen

Mathematics and Statistics Faculty Publications

Objective—Ceruloplasmin (Cp) is an acute-phase reactant that is increased in inflammatory diseases and in acute coronary syndromes. Cp has recently been shown to possess nitric oxide (NO) oxidase catalytic activity, but its impact on long-term cardiovascular outcomes in stable cardiac patients has not been explored. Methods and Results—We examined serum Cp levels and their relationship with incident major adverse cardiovascular events (MACE; death, myocardial infarction [MI], stroke) over 3-year follow-up in 4177 patients undergoing elective coronary angiography. We also carried out a genome-wide association study to identify the genetic determinants of serum Cp levels and evaluate their relationship to prevalent …

Discerning Reported Suicide Attempts Within A Youthful Offender Population, Christopher Mallett, Lea A. Derigne, Linda Quinn, Paricia Stoddard Dare

Mathematics and Statistics Faculty Publications

With suicide being the third leading cause of death among young people, early identification of risk is critical, particularly for those involved with the juvenile courts. In this study of court-involved youth (N = 433) in two Midwest counties, logistic regression analysis identified some expected and unexpected findings of important demographic, educational, mental health, child welfare, and juvenile court-related variables that were linked to reported suicide attempts. Some of the expected suicide attempt risk factors for these youth included prior psychiatric hospitalization and related mental health services, residential placement, and diagnoses of depression and alcohol dependence. However, the most unexpected …

Subnormality Of Aluthge Transforms Of Weighted Shifts, Sang Hoon Lee, Woo Young Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

In this note we study the k-hyponormality and the subnormality of Aluthge transforms of weighted shifts. It is shown that Aluthge transforms of weighted shifts need not preserve the k-hyponormality. Moreover, we show that if W α is a subnormal weighted shift with 2-atomic Berger measure then its Aluthge transform W̃ α is subnormal if and only if at least one of two atoms is zero. © 2011 Springer Basel AG.

The Kolmogorov And Stechkin Problems For Classes Of Functions Whose Second Derivative Belongs To The Orlicz Space, Yuliya Babenko, Dmitry Skorokhodov

Faculty Articles

For any t ∈ [0, 1], we obtain the exact value of the modulus of continuity , where L* is the dual Orlicz space with Luxemburg norm and D is the operator of differentition at the point t. As an application, we state necessary and sufficient conditions in the Kolmogorov problem for three numbers. Also we solve the Stechkin problem, i.e., the problem of approximating an unbounded operator of differentition D by bounded linear operators for the class of functions x such that $\left\| {x''} \right\|_{L_N^* [0,1]} \leqslant 1$

Convergence Analysis Of A Multigrid Algorithm For The Acoustic Single Layer Equation, Simon Gemmrich, Jay Gopalakrishnan, Nilima Nigam

Mathematics and Statistics Faculty Publications and Presentations

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in Bramble et al. (1994) . A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator.

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.

Logarithmic Spirals And Projective Geometry In M.C. Escher's "Path Of Life Iii", Heidi Burgiel, Matthew Salomone

Journal of Humanistic Mathematics

M.C. Escher's use of dilation symmetry in Path of Life III gives rise to a pattern of logarithmic spirals and an oddly ambiguous sense of depth.

Some Aspects Of Toric Topology., Soumen Sarkar Dr.

Doctoral Theses

The main goal of this thesis is to study the topology of torus actions on manifolds and orbifolds. In algebraic geometry actions of the torus (C * ) n on algebraic varieties with nice properties produce bridges between geometry and combinatorics see [Dan78], [Oda88] and [Ful93]. We see a similar bridge called moment map for Hamiltonian action of compact torus on symplectic manifolds see [Aud91] and [Gui94]. In particular whenever the manifold is compact the image of moment map is a simple polytope, the orbit space of the action. A topological counterpart called quasitoric manifolds, a class of topological manifolds …

Dispersive Estimates In R^3 With Threshold Resonances, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

We prove dispersive estimates in R^3 for the Schroedinger evolution generated by the Hamiltonian H = -\Delta+V, under optimal decay conditions on V, in the presence of zero energy eigenfunctions and resonances.

Subnormality For Arbitrary Powers Of 2-Variable Weighted Shifts Whose Restrictions To A Large Invariant Subspace Are Tensor Products, Raúl E. Curto, Sang Hoon Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. We study LPCS within the class of commuting 2-variable weighted shifts T≡(T1, T2) with subnormal components T1 and T2, acting on the Hilbert space ℓ2(Z+2) with canonical orthonormal basis {e(k1,k2)}k1,k2≥0. The core of a commuting 2-variable weighted shift T, c(T), is the restriction of T to the invariant subspace generated by all vectors e(k1,k2) with k1, k2≥1; we say that c(T) is of tensor form if it is unitarily equivalent to a shift …

Existence Of A Minimizer For The Quasi-Relativistic Kohn-Sham Model, Carlos Argáez García, Michael Melgaard

Articles

We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator (see article) . For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge (see article).

Incorporating Quantitative Reasoning In Common Core Courses: Mathematics For The Ghost Map, John R. Jungck

Numeracy

How can mathematics be integrated into multi-section interdisciplinary courses to enhance thematic understandings and shared common readings? As an example, four forms of quantitative reasoning are used to understand and critique one such common reading: Steven Berlin Johnson’s "The Ghost Map: The Story of London's Most Terrifying Epidemic - and How it Changed Science, Cities and the Modern World" (Riverhead Books, 2006). Geometry, statistics, modeling, and networks are featured in this essay as the means of depicting, understanding, elaborating, and critiquing the public health issues raised in Johnson’s book. Specific pedagogical examples and resources are included to illustrate applications and …

Parts Of The Whole: Learn More, Learn Better, Dorothy Wallace

Numeracy

Building on previous columns in Numeracy, this column analyzes various teaching techniques in terms of their ability to build cognitive schema, extend existing schema, reinforce learning, move mean understanding of a group of students, and reduce variance in understanding of a group. We offer a pedagogical cycle as an example of how to address multiple learning goals using common teaching methods.

If Only Math Majors Could Write..., Bernard L. Madison

Numeracy

This text of the opening plenary address to the 2011 Summit of the Appalachian College Association and the meeting of the National Numeracy Network makes an argument that quantitative reasoning and writing should be taught together. The argument is set up by noting that humanists have historically banished quantitative issues from their study of the liberal arts and that science, engineering, and mathematics education suffers from lack of approaches to learning that promote complex, deeper understanding, most notably integrative and reflective learning. Therefore, everyone would profit from combining writing and quantitative reasoning. Five more specific reasons are discussed, drawing evidence …

Period Life Tables: A Resource For Quantitative Literacy, Thomas J. Pfaff, Stanley Seltzer

Numeracy

A period life table provides an estimate of the probability that a person will die at a particular age. Using data available online, we examine tables of expected years to live for males and females against age for three populations: the United States in 2007, the U.S. at the turn of the twentieth century, and the Roman Empire. Scatter plots of males and females for each population show how life expectancy increases with age (e.g., U.S. 2007: 50 year-old female > 40 year-old female > 45 year-old male). The three data sets allow historical comparisons (e.g., of gender disparity, larger now; of …

Motivation For Achievement And Attitudes Toward Mathematics Instruction In A Required Calculus Course At The Norwegian University Of Science And Technology, Donna Sundre, Carol Barry, Vidar Gynnild, Erin Tangen Ostgard

Numeracy

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students.

Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and …

Quantitative Literacy Interventions At University Of Cape Town: Effects Of Separation From Academic Disciplines, Vera Frith

Numeracy

The aim of the Numeracy Centre at the University of Cape Town is to develop students’ quantitative literacy (QL) in a manner consistent with their programmes of study and intended roles in the community. Our theoretical perspective on the nature of QL is in line with that of the New Literacies Studies and sees academic QL as practices in different academic disciplinary contexts. This means that for us the ideal curriculum structure for developing QL would fully integrate it into the teaching of the disciplines. This is in practice not achievable in most cases, especially since many students do not …

Numeracy, Financial Literacy, And Financial Decision-Making, Annamaria Lusardi

Numeracy

Financial decisions, be they related to asset building or debt management, require the capacity to do calculations, including some complex ones. But how numerate are individuals, in particular when it comes to calculations related to financial decisions? Studies and surveys implemented in both the United States and in other countries that are described in this paper show the level of numeracy among the population to be very low. Moreover, lack of numeracy is not only widespread but is particularly severe among some demographic groups, such as women, the elderly, and those with low educational attainment. This has potential consequences for …

The Joy Of Quantitative Reasoning, Caren Diefenderfer

Numeracy

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 …

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.