Mathematics Commons^™

Mathematical Beauty, Mike Janssen

Faculty Work Comprehensive List

"With all the brokenness in the world, is the study of the mathematical aspects of Creation worthwhile?"

Posting about ­­­­­­­­the beauty of mathematics from In All Things - an online hub committed to the claim that the life, death, and resurrection of Jesus Christ has implications for the entire world.

http://inallthings.org/mathematical-beauty/

Classification Of Rectifying Space-Like Submanifolds In Pseudo-Euclidean Spaces, Bang Yen Chan, Yun Myung Oh

Faculty Publications

The notions of rectifying subspaces and of rectifying submanifolds were introduced in [B.-Y. Chen, Int. Electron. J. Geom 9 (2016), no. 2, 1–8]. More precisely, a submanifold in a Euclidean m-space Em is called a rectifying submanifold if its position vector field always lies in its rectifying subspace. Several fundamental properties and classification of rectifying submanifolds in Euclidean space were obtained in [B.-Y. Chen, op. cit.]. In this present article, we extend the results in [B.-Y. Chen, op. cit.] to rectifying space- like submanifolds in a pseudo-Euclidean space with arbitrary codimension. In particular, we completely classify all rectifying space-like submanifolds …

Best Sandbox, Tom Clark

Faculty Work Comprehensive List

"Mathematics ought to be for everyone, not just for mathematicians, scientists, and engineers. Mathematics is not only about application; it is part of the created world."

Posting about ­­­­­­­­learning about math through creative play from In All Things - an online hub committed to the claim that the life, death, and resurrection of Jesus Christ has implications for the entire world.

http://inallthings.org/the-best-sandbox/

Algorithmically Complex Residually Finite Groups, Olga Kharlampovich, Alexei Myanikov, Mark Sapir

Publications and Research

We construct the first examples of algorithmically complex finitely presented residually finite groups and the first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn functions, and arbitrarily large depth functions. The groups are solvable of class 3.

Applications Of The Heine And Bauer-Muir Transformations To Rogers-Ramanujan Type Continued Fractions, Jongsil Lee, James Mclaughlin, Jaebum Sohn

Mathematics Faculty Publications

In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq, b, λq)/G(a, b, λ) may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine’s continued fraction for a ratio of 2φ1 functions, or other similar continued fraction expansions of ratios of 2φ1 functions. Further, by employing essentially the same methods, a new continued fraction for …

Computational Algorithms For Improved Representation Of The Model Error Covariance In Weak-Constraint 4d-Var, Jeremy A. Shaw

Dissertations and Theses

Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and …

Superdegenerate Hypoelliptic Differential Operators., Denis Bell

Showcase of Faculty Scholarly & Creative Activity

A proof of a Hormander theorem applicable to sum of squares operators with degeneracies of exponential order.

Application Of Inverse Problems In Imaging, Xiaoyue Luo

Post-Grant Reports

In this project, we studied how to enhance image quality by denoising and deblurring a given image mathematically. We compared some existing state-of-the-art methods for image denoising and deblurring. We implemented the algorithms numerically using Matlab.

We studied the possibility of combining statistical analysis with the traditional image restoration methods including using wavelets and framelets and we derived some encouraging preliminary results.

My research student Alleta Maier gave a sequence of talks on the project including the Pacific Northwest Mathematical Association of America conference at Oregon State University in April, 2016; Linfield College Taylor Series in March, 2016, and Linfield …

Sexual Dimorphism In Digit Ratios Derived From Dorsal Digit Length Among Adults And Children, Sanjay Kumar, Martin Voracek, Maharaj Singh

Mathematics, Statistics and Computer Science Faculty Research and Publications

Sexual dimorphism in ventrally measured digit ratios (2D:4D and other) has been related to prenatal sex-hormone levels. In the present series of three studies, we measured all digit lengths (excluding the thumb) on the dorsal, rather than the ventral, side of left and right hands and investigated the sexual dimorphism in digit ratios in three independent samples, two of them comprising adults (Study I, N = 104; Study II, N = 154), and one further, comprising kindergarten children (Study III, N = 64). Results show that men have lower digit-ratio values compared to women in digit ratios that include digit …

Session A-3: Three-Act Math Tasks, Lindsey Herlehy

Professional Learning Day

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

Hurwitz Number Fields, David P. Roberts

Mathematics Publications

The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The scheme-theoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics.

3d Printing For Mathematics Education, Cecile M. Piret

TechTalks

3D printing is particularly well-suited for teaching concepts of mathematics. We (Drs. Piret and Ellis) have started using 3D printers in a calculus 3 classroom. Our goals are to analyze the impact of the printers on students' understanding and motivation towards calculus, and to further incorporate this new technology in our teaching of mathematics. In this talk, we will show a few examples of projects that were presented to the students in our classes, and we will try to motivate you to also use the printers in your classrooms.

Why Linear Interpolation?, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Linear interpolation is the computationally simplest of all possible interpolation techniques. Interestingly, it works reasonably well in many practical situations, even in situations when the corresponding computational models are rather complex. In this paper, we explain this empirical fact by showing that linear interpolation is the only interpolation procedure that satisfies several reasonable properties such as consistency and scale-invariance.

Derivation Of Gross-Pitaevskii Version Of Nonlinear Schroedinger Equation From Scale Invariance, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions D. It turns out that while for D ≥ 3 we still get only the Schroedinger's equation, for D = 2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for D = 1, we also get a new nonlinear …

Towards Decision Making Under General Uncertainty, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.

Why Stable Teams Are More Efficient In Education, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that study groups speed up learning. Recent studies have shown that stable study groups are more efficient than shifting-membership groups. In this paper, we provide a theoretical explanation for this empirical observation.

Classification Of Book Representations Of K6, Dana Rowland

Mathematics Faculty Publications

A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and …

Can We Detect Crisp Sets Based Only On The Subsethood Ordering Of Fuzzy Sets? Fuzzy Sets And/Or Crisp Sets Based On Subsethood Of Interval-Valued Fuzzy Sets?, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy sets are naturally ordered by the subsethood relation. If we only know which set which fuzzy set is a subset of which -- and have no access to the actual values of the corresponding membership functions -- can we detect which fuzzy sets are crisp? In this paper, we show that this is indeed possible. We also show that if we start with interval-valued fuzzy sets, then we can similarly detect type-1 fuzzy sets and crisp sets.

Convergence Rates In Periodic Homogenization Of Systems Of Elasticity, Zhongwei Shen, Jinping Zhuge

Mathematics Faculty Publications

This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L² for the mixed boundary value problems with bounded measurable coefficients.

Nash Twist And Gaussian Noise Measure For Isometric C1 Maps, Amites Dasgupta, Mahuya Datta

Communications on Stochastic Analysis

No abstract provided.