Mathematics Commons^™

Experimentally Observed Dark Matter Confinement Clarifies A Discrepancy In Estimating The Universe's Expansion Speed, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known the our Universe is expanding. In principle, we can estimate the expansion speed either directly, by observing the current state of the Universe, or indirectly, by analyzing the cosmic background radiation. Surprisingly, these two estimates lead to somewhat different expansion speeds. This discrepancy is a important challenge for cosmologists. Another challenge comes from recent experiments that show that, contrary to the original idea that dark matter and regular (baryonic) matter practically do not interact, dark matter actually "shadows" the normal matter.

In this paper, we show that this "dark matter confinement" can explain the discrepancy between …

Hereditary Properties Of Co-Kähler Manifolds, Giovanni Bazzoni, Gregory Lupton, John F. Oprea

Mathematics and Statistics Faculty Publications

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: dim(H^∗(M; Q)) ≥2^r, for any r-torus T^r which acts almost freely on M.

Incorporating A Spatial Prior Into Nonlinear D-Bar Eit Imaging For Complex Admittivities, Sarah J. Hamilton, Jennifer L. Mueller, Melody Alsaker

Mathematics, Statistics and Computer Science Faculty Research and Publications

Electrical Impedance Tomography (EIT) aims to recover the internal conductivity and permittivity distributions of a body from electrical measurements taken on electrodes on the surface of the body. The reconstruction task is a severely ill-posed nonlinear inverse problem that is highly sensitive to measurement noise and modeling errors. Regularized D-bar methods have shown great promise in producing noise-robust algorithms by employing a low-pass filtering of nonlinear (nonphysical) Fourier transform data specific to the EIT problem. Including prior data with the approximate locations of major organ boundaries in the scattering transform provides a means of extending the radius of the low-pass …

The Mathematics Of Superoscillations, Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Jeff Tollaksen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the …

The Maximum Number Of Complete Subgraphs Of Fixed Size In A Graph With Given Maximum Degree, Jonathan Cutler, A. J. Radcliffe

Department of Mathematics Faculty Scholarship and Creative Works

In this article, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and Δ(G) ≤ r, which has the most complete subgraphs of size t, for t≥3. The conjectured extremal graph is aKr+1 ∪ Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when a ≤ 1, and also reduced the general conjecture to the case t = 3. We prove …

Gorenstein Projective Precovers, Sergio Estrada, Alina Iacob, Katelyn A. Coggins

Mathematical Sciences: Faculty Publications

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.

Ghost Series And A Motivated Proof Of The Andrews–Bressoud Identities, Shashank Kanade, James Lepowsky, Matthew C. Russell, Andrew Sills

Mathematical Sciences: Faculty Publications

We present what we call a “motivated proof” of the Andrews–Bressoud partition identities for even moduli. A “motivated proof” of the Rogers–Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a “motivated proof” of the somewhat analogous Göllnitz–Gordon–Andrews identities has been found. In the present work, we introduce “shelves” of formal series incorporating what we call “ghost series,” which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that …

A Refined Approach For Non-Negative Entire Solutions Of Δ U + Up = 0 With Subcritical Sobolev Growth, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

Let N≥2and 1uis a non-negative classical solution of the Lane–Emden equation, thenu≡0. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of Serrin and Zou, originally used for the Lane–Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.

The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson

Dissertations, Theses, and Capstone Projects

The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.

In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of …

On Gorenstein Fiber Products And Applications, Saeed Nasseh, Ryo Takahashi, Keller Vandebogert

Mathematical Sciences: Faculty Publications

We show that a non-trivial fiber product S×kT of commutative noetherian local rings S,T with a common residue field k is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both S and T are regular rings of dimension 1. We also give some applications of this result.

Domain Coloring And The Argument Principle, Frank A. Farris

Mathematics and Computer Science

The domain-coloring algorithm allows us to visualize complex-valued functions on the plane in a single image—an alternative to before-and-after mapping diagrams. It helps us see when a function is analytic and aids in understanding contour integrals. The culmination of this article is a visual discovery and subsequent proof of the Argument Principle, which relates the count of poles and zeros of a meromorphic function inside a contour to the accumulated change in argument of the function around the contour. Throughout, I offer connections to standard learning goals of courses in complex variables.

Calcium Transient Assays For Compound Screening With Human Ipsc-Derived Cardiomyocytes: Evaluating New Tools, Neil J. Daily, Radleigh Santos, Joseph Vecchi, Pinar Kemanli, Tetsuro Wakatsuki

Mathematics Faculty Articles

Calcium (Ca2+) plays a central role in regulating many biological processes in the cell from muscle contraction to neurotransmitter release. The need for reliable fluorescent calcium indicator dyes is of vast importance for studying many aspects of cell biology as well as screening compounds using phenotypic high throughput assays. We have assessed two of the latest generation of calcium indicator dyes, FLIPR Calcium 6 and Cal-520 AM for studying calcium transients (CaTs) in induced pluripotent stem cell (iPSC) -derived human cardiomyocytes. FLIPR Calcium 6 and Cal-520 dyes both displayed robust CaTs with a high signal-to-noise ratio (SNR) and were non-toxic …

Global Stability Of Nonlinear Stochastic Sei Epidemic Model With Fluctuations In Transmission Rate Of Disease, Olusegun Michael Otunuga

Mathematics Faculty Research

We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (R₀) and stochastic (R₀) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R₀ < 1), epidemic can still grow initially (if R₀ > 1) …

On The Three Dimensional Interaction Between Flexible Fibers And Fluid Flow, Bogdan Nita, Ryan Allaire

Department of Mathematics Faculty Scholarship and Creative Works

In this paper we discuss the deformation of a flexible fiber clamped to a spherical body and immersed in a flow of fluid moving with a speed ranging between 0 and 50 cm/s by means of three dimensional numerical simulation developed in COMSOL . The effects of flow speed and initial configuration angle of the fiber relative to the flow are analyzed. A rigorous analysis of the numerical procedure is performed and our code is benchmarked against well established cases. The flow velocity and pressure are used to compute drag forces upon the fiber. Of particular interest is the behavior …

Signed Lozenge Tilings, D. Cook Ii, Uwe Nagel

Mathematics Faculty Publications

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a …

Spectral Theory Of Operators On Manifolds, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Differential operators that are defined on a differentiable manifold can be used to study various properties of manifolds. The spectrum and eigenfunctions play a very significant role in this process. The objective of this chapter is to develop the heat equation method and to describe how it can be used to prove the Hodge Theorem. The Minakshisundaram-Pleijel parametrix and asymptotic expansion are then derived. The heat equation asymptotics can be used to give a development of the Gauss-Bonnet theorem for two-dimensional manifolds.

Removable Singularities In C*-Algebras Of Real Rank Zero, Lawrence A. Harris

Mathematics Faculty Publications

Let 𝕬 be a C*-algebra with identity and real rank zero. Suppose a complex- valued function is holomorphic and bounded on the intersection of the open unit ball of 𝕬 and the identity component of the set of invertible elements of 𝕬. We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of 𝕬. The author previously deduced this from a more general fact about Banach algebras.

Totally Acyclic Complexes, Sergio Estrada, Xianhui Fu, Alina Iacob

Mathematical Sciences: Faculty Publications

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. …

Numerical Solution Of Master Equation Corresponding To Schumann Waves, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Following a hypothesis by Marciak-Kozlowska, 2011, we consider one-dimensional Schumann wave transfer phenomena. Numerical solution of that equation was obtained by the help of Mathematica.

Discovering Liouville-Type Problems For P-Energy Minimizing Maps In Closed Half-Ellipsoids By Calculus Variation Method, Lina Wu

Publications and Research

The goal of this project is to investigate constant properties (called the Liouville-type Problem) for a p-stable map as a local or global minimum of a p-energy functional where the domain is a Euclidean space and the target space is a closed half-ellipsoid. The First and Second Variation Formulas for a p-energy functional has been applied in the Calculus Variation Method as computation techniques. Stokes’ Theorem, Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and the Bochner Formula as estimation techniques have been used to estimate the lower bound and the upper bound of the derived p-Harmonic Stability Inequality. One challenging point in …

Mathematics For Human Flourishing, Francis Su

All HMC Faculty Publications and Research

Why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?

Weighing Fog: Hands On Modeling For Day 1 Of Differential Equations, Tom Clark

Faculty Work Comprehensive List

The first day of many mathematics classes contains formalities and very little mathematics. Here an alternative is presented where modeling is placed as the centerpiece to orient students to the real work of differential equations. Namely, to capture as beautifully and compactly as possible through the process of conjecture and investigation, the deep and interesting aspects of the physical world. A demonstration of the sublimation of dry ice sits at the center of the lesson. Students collaborate in groups to design an experiment that could measure the change in mass of a piece of dry ice that is dropped into …

Functionality And Robustness Of Injured Connectomic Dynamics In C. Elegans: Linking Behavioral Deficits To Neural Circuit Damage, James M. Kunert, Pedro D. Maia, J. Nathan Kutz

Mathematics Faculty Publications - Archive

Using a model for the dynamics of the full somatic nervous system of the nematode C. elegans, we address how biological network architectures and their functionality are degraded in the presence of focal axonal swellings (FAS) arising from neurodegenerative disease and/or traumatic brain injury. Using biophysically measured FAS distributions and swelling sizes, we are able to simulate the effects of injuries on the neural dynamics of C. elegans, showing how damaging the network degrades its low-dimensional dynamical responses. We visualize these injured neural dynamics by mapping them onto the worm’s low-dimensional postures, i.e. eigenworm modes. We show that a diversity …

Teaching And Learning Mathematics In The Ar/Vr Environment, Alexander Vaninsky

Publications and Research

This presentation discusses teaching and learning mathematics in augmented (AR) or virtual (VR) reality created by a combination of goggles and earphones. It claims that interactive learning in such an environment is more attractive and efficient. It increases motivation and interest in the subject matter. The approach is underlain by the findings of educational neuroscience considering the learning process as the formation of domains in the brain forming mathematics knowledge centers. The teaching process provides sensory excitation and establishes connections among these and other domains. Hardware and software are available in the market. The suggested approach allows for practical implementation …

Specifications Grading In A First Course In Abstract Algebra, Mike Janssen

Faculty Work Comprehensive List

Specifications grading offers an alternative to more traditional, points-based grading and assessment structures. In place of partial credit, students are assessed pass/fail on whether or not they have achieved the learning outcomes being assessed on a given piece of work according to certain specifications, with limited opportunities for revision of non-passing work. This talk will describe the learning outcomes and specifications grading system I used in my Fall 2016 abstract algebra course, as well as student responses.

A Madison-Numeracy Citation Index (2008-2015): Implementing A Vision For A Quantitatively Literate World, Nathan D. Grawe, H. L. Vacher

Numeracy

This editorial recognizes the contributions made by Bernard Madison to the field of quantitative literacy with a bibliographic index of his papers, edited volumes, and works contained therein that were cited in the first eight volumes (2008-2015) of Numeracy. In total, 61 citing papers ("sources") cite 42 Madison works ("citations") a total of 218 times. The source and citation indexes provided in the appendix at the end of this editorial make it easy to see the direct contribution of Madison's work to the arguments and debates contained in the founding years of the journal. For those who are new …

Property Analysis Of Exfoliated Graphite Nanoplatelets Modified Asphalt Model Using Molecular Dynamics (Md) Method, Hui Yao, Qingli Dai, Zhanping You, Andreas Bick, Min Wang, Shuaicheng Guo

Michigan Tech Publications, Part 1

This Molecular Dynamics (MD) simulation paper presents a physical property comparison study between exfoliated graphite nanoplatelets (xGNP) modified and control asphalt models, including density, glass transition temperature, viscosity and thermal conductivity. The three-component control asphalt model consists of asphaltenes, aromatics, and saturates based on previous references. The xGNP asphalt model was built by incorporating an xGNP and control asphalt model and controlling mass ratios to represent the laboratory prepared samples. The Amber Cornell Extension Force Field (ACEFF) was used with assigned molecular electro-static potential (ESP) charge from NWChem analysis. After optimization and ensemble relaxation, the properties of the control and …

Rhombic Tilings And Primordia Fronts Of Phyllotaxis, Pau Atela, Christophe Golé

Mathematics Sciences: Faculty Publications

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically …

Dealing With Mathematical Anxiety: Should One Size Fit All?, Jon Warwick

The Mathematics Enthusiast

Many students who have to study mathematics as an enabling subject within higher education experience mathematical anxiety to a greater or lesser extent. This affliction can impact student learning and achievement in mathematics and so a number of strategies have been suggested for alleviating mathematical anxiety or at least moderating its effects. This paper reports on a comparison of the mathematical anxiety experienced by two groups of students each studying a different subject discipline. The results indicate that the groups have quite different levels of anxiety and the differing contributing factors between the groups suggest that approaches to remediation need …

The Mean Value Theorem, Dave Ruch

Analysis

No abstract provided.