Mathematics Commons^™

A Hyperbolic Two-Step Model Based Finite Difference Method For Studying Thermal Deformation In Three-Dimensional Micro Spheres Exposed To Ultrashort-Pulsed Lasers, Pan Wang

Doctoral Dissertations

Ultrashort-pulsed lasers with pulse durations of the order of sub-picoseconds to femtoseconds possess the capabilities in limiting the undesirable spread of the thermal process zone in a heated sample. Because of this, ultrashort-pulsed lasers have been attracting worldwide interest in science and engineering. The success of ultrashort-pulsed lasers in real application relies on: (1) well characterized pulse width, intensity and experimental techniques; (2) reliable microscale heat transfer models; and (3) prevention of thermal damage. Laser damage induced by ultrashort-pulsed lasers occurs after the heating pulse is over, since the pulse duration time is extremely short and the heat flux is …

Factoring Families Of Positive Knots On Lorenz-Like Templates, Michael C. Sullivan

Articles and Preprints

We show that for m and n positive, composite closed orbits realized on the Lorenz-like template L(m, n) have two prime factors, each a torus knot; and that composite closed orbits on L(−1,−1) have either two for three prime factors, two of which are torus knots.

Cell Groups Reveal Structure Of Stimulus Space, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to ‘‘decode’’ what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an …

Supplementary Text To Accompany “Cell Groups Reveal Structure Of Stimulus Space”, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

Here we present a brief exposition of some material from algebraic topology that we use in our methods. We include it for completeness, as it may not be familiar for many readers. In particular, we define simplicial complexes, simplicial homology groups, and state the theorem cited in the Results section. See [Bott and Tu, 1982, Ewald, 1996, Hatcher, 2002] for more details.

Permutation Representations On Schubert Varieties, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over ℂ and over ℂ[t1, t2, . . . , tn]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators of Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.

Biomimetic Subwavelength Antireflective Gratings On Gaas, Chih-Hung Sun, Brian J. Ho, Bin Jiang, Peng Jiang

Mathematics and Statistics Faculty Publications and Presentations

We have developed a simple and scalable bottom-up approach for fabricating moth-eye antireflective coatings on GaAs substrates. Monolayer, non-close-packed silica colloidal crystals are created on crystalline GaAs wafers by a spin-coating-based single-layer reduction technique. These colloidal monolayers can be used as etching masks during a BCl_3 dry-etch process to generate subwavelength-structured antireflective gratings directly on GaAs substrates. The gratings exhibit excellent broadband antireflective properties, and the specular reflection matches with the theoretical prediction using a rigorous coupled-wave analysis model. These bioinspired antireflection coatings have important technological applications ranging from efficient solar cells to IR detectors

2008 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2008 Fall Colloquium.

The Matrices R And G Of Matrix Analytic Methods And The Time-Inhomogeneous Periodic Quasi-Birth-And-Death Process, Barbara H. Margolius

Mathematics and Statistics Faculty Publications

We solve for the asymptotic periodic distribution of the continuous time quasi-birth-and-death process with time-varying periodic rates in terms of R^ and G^ matrix functions which are analogues of the R and G matrices of matrix analytic methods. We evaluate these QBDs numerically by solving for R^ numerically.

Coenzyme Q1 Redox Metabolism During Passage Through The Rat Pulmonary Circulation And The Effect Of Hyperoxia, Said H. Audi, Marilyn P. Merker, Gary S. Krenz, Taniya Ahuja, David L. Roerig, Robert D. Bongard

Mathematics, Statistics and Computer Science Faculty Research and Publications

The objective was to evaluate the pulmonary disposition of the ubiquinone homolog coenzyme Q₁ (CoQ₁) on passage through lungs of normoxic (exposed to room air) and hyperoxic (exposed to 85% O₂ for 48 h) rats. CoQ₁ or its hydroquinone (CoQ₁H₂) was infused into the arterial inflow of isolated, perfused lungs, and the venous efflux rates of CoQ₁H₂ and CoQ₁ were measured. CoQ₁H₂ appeared in the venous effluent when CoQ₁ was infused, and CoQ₁ appeared when CoQ1H2 was infused. In normoxic lungs, CoQ …

Applications Of Combinatorial Designs In Key Pre-Distribution In Sensor Networks., Dibyendu Chakrabarti Dr.

Doctoral Theses

Key pre-distribution is an important area of research in Distributed Sensor Networks (DSN). Some improved techniques over the existing schemes (employing combinatorial designs) have been proposed in this thesis and detailed mathematical analysis of the schemes has been presented. At first, combinatorial design followed by randomized merging strategy is applied to key pre-distribution in sensor nodes. Our main target is to get more than one pair of common keys between any pair of nodes to provide a robust network in terms of security under adversarial conditions where some nodes may get compromised. A transversal design is used to construct a …

Biggers, Covella (Houchens), 1921-2022 (Sc 1610), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Finding aid only for Manuscripts Small Collection 1610. Three handwritten sermons by Episcopalian clergyman John Wesley Venable, and one early nineteenth-century ciphering book (unbound) executed by Burrell Adams.

On The Classification Of Gradient Ricci Solitons, Peter Petersen, William Wylie

Mathematics - All Scholarship

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor.

On Gradient Ricci Solitons With Symmetry, Peter Petersen, William Wylie

Mathematics - All Scholarship

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper "Rigidity of gradient Ricci solitons" to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.

Constant-Sign Solutions Of A System Of Volterra Integral Equations In Orlicz Spaces, Ravi P. Agarwal, Donal O'Regan, Patricia J.Y. Wong

Mathematics and System Engineering Faculty Publications

We consider the following system of Volterra intergral equations uu1(t) = gi(t, s)fi(s, u1(s), u2(s), · · ·, un(s))ds, a.e. t ∈ [0,T], 1 ≤ i ≤ n. Criteria are offered for the existence of one and more constantsign solutions u = (u1, u2, · · ·, un) of the system in Lp and the Orlicz spaces. We say u is of constant sign if for each 1 ≤ i ≤ n, Θiui(t) ≥ 0 for a.e. t ∈ [0,T], where Θi ∈ {1,-1} is fixed. © 2008 Rocky Mountain Mathematics Consortium.

Confidence Intervals For Long Memory Regressions, Kyungduk Ko, Jaechoul Lee, Robert Lund

Mathematics Faculty Publications and Presentations

This paper proposes an accurate confidence interval for the trend parameter in a linear regression model with long memory errors. The interval is based upon an equivalent sum of squares method and is shown to perform comparably to a weighted least squares interval. The advantages of the proposed interval lies in its relative ease of computation and should be attractive to practitioners.

Testing Diffusion Processes For Non-Stationarity, Jeff Hamrick, Murad S. Taqqu

Master of Science in Analytics (MSAN) Faculty Research

Financial data are often assumed to be generated by diffusions. Using recent results of Fan et al. (J Am Stat Assoc, 102:618–631, 2007; J Financ Econometer, 5:321–357, 2007) and a multiple comparisons procedure created by Benjamini and Hochberg (J R Stat Soc Ser B, 59:289–300, 1995), we develop a test for non-stationarity of a one-dimensional diffusion based on the time inhomogeneity of the diffusion function. The procedure uses a single sample path of the diffusion and involves two estimators, one temporal and one spatial. We first apply the test to simulated data generated from a variety of one-dimensional diffusions. We …

A Parallel Computational Method For Simulating Two-Phase Gel Dynamics, Jian Du, Aaron L. Fogelson, Grady Wright

Mathematics Faculty Publications and Presentations

We develop a parallel computational algorithm for simulating models of gel dynamics where the gel is described by two phases, a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volumeaveraged incompressibility constraint. Multigrid with Vanka-type box relaxation scheme is used as preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Through numerical experiments of a model problem, the efficiency, robustness and scalability of the algorithm are illustrated.

Enumeration Of Rational Plane Curves Tangent To A Smooth Cubic, C. Cadman, Linda Chen

Mathematics & Statistics Faculty Works

We compute the number of rational degree d plane curves having prescribed fixed and moving contacts to a smooth plane cubic E . We use twisted stable maps to the stack View the MathML sourceP²_E,r for r large, where View the MathML sourceP²_E,r is the r th root of P² along E. We prove that certain Gromov–Witten invariants of this stack are enumerative, and establish recursive formulas for these numbers.

Carman, William, 1790-1841 (Mss 52), Manuscripts & Folklife Archives

Manuscript Collection Finding Aids

Ciphering books (2) of William Carman, a teacher in Mayfield, Kentucky. One book is dated 1815-1816; the other one is undated, but evidently of the same period. The ciphering used is the British pound system. The title page of one volume is by Gavin G. Craig of WKU's Penmanship Department.

Limiting Subgradients Of Minimal Time Functions In Banach Spaces, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory: and Hamilton-Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Frechet type epsilon-subgradients in arbitrary Banach spaces.

Linearizable Feedforward Systems: A Special Class, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We address the problem of linearizability of systems in feedforward form. In a recent paper [22] we completely solved the linearizability for strict feedforward systems. We extend here those results to a special class of feedforward systems. We provide an algorithm, along with explicit transformations, that linearizes the system by change of coordinates when some easily checkable conditions are met. We also re-analyze type II class of linearizable strict feedforward systems provided by Krstic in [9] and we show that this class is the unique linearizable among the class of quasi-linear strict feedforward systems (see Definition III.1). Our results allow …

Borda Meets Pascal, Marie K. Jameson, Gregory Minton '08, Michael E. Orrison

All HMC Faculty Publications and Research

Every so often (especially in mathematics), unforeseen connections between different ideas arise and beg explanation. This happened to us when, in an effort to generalize the voting procedure known as the Borda count, we began to see vectors of the form (-1, 1), (1, -2, 1), (-1, 3, -3, 1), (1, -4, 6, -4, 1), and so on. As you might imagine, we were instantly intrigued by this unanticipated relationship with Pascal's triangle, and we quickly set out to find an explanation. This article describes some of our initial findings.

Some Rather Mechanical Reflections On Symmetry: In Art, Science, Engineering, Mathematics, Etc., Jim Mcgovern

Articles

This Inaugural Lecture consists of some of my rather mechanical, being an engineer, reflections on symmetry in diverse areas such as art, science, engineering, mathematics, etc. I explain what symmetry is to me, giving examples with lots of images and mentioning or at least barely referencing art, science, architecture, engineering, heritage, cosmology, bicycles, flight, invention, ingenuity, history, wallpaper, mathematics, typography, structures, regular shapes, coordinate systems, spacetime, thermodynamics and suchlike.

Threefold Flops Via Matrix Factorization, Carina Curto, David R. Morrison

Department of Mathematics: Faculty Publications

The structure of birational maps between algebraic varieties becomes increasingly complicated as the dimension of the varieties increases. There is no birational geometry to speak of in dimension one: if two complete algebraic curves are birationally isomorphic then they are biregularly isomorphic. In dimension two we encounter the phenomenon of the blowup of a point, and every birational isomorphism can be factored into a sequence of blowups and blowdowns. In dimension three, however, we first encounter birational maps which are biregular outside of a subvariety of codimension two (called the center of the birational map). When the center has a …

The Mixed Problem In L P For Some Two-Dimensional Lipschitz Domains, Loredana Lanzani, Luca Capogna, Russell M. Brown

Mathematics Sciences: Faculty Publications

We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution …

Environmental Limits On The Nonresonant Cosmic-Ray Current-Driven Instability, Brian Reville, John Kirk, Peter Duffy, Stephen O'Sullivan

Articles

We investigate the so-called nonresonant cosmic-ray streaming instability, first discussed by Bell (2004). The extent to which thermal damping and ion-neutral collisions reduce the growth of this instability is calculated. Limits on the growth of the nonresonant mode in SN1006 and RX J1713.7-3946 are presented.

What's In A Name? The Matrix As An Introduction To Mathematics, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

In lieu of an abstract, here is the article's first paragraph:

In my classes on the nature of scientific thought, I have often used the movie The Matrix to illustrate the nature of evidence and how it shapes the reality we perceive (or think we perceive). As a mathematician, I usually field questions related to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object?

Langer's Method For The Calculation Of Escape Rates And Its Application To Systems Of Ferromagnets, Gerard Duff

Doctoral

This work is a study of the application of a theory proposed by J. S. Langer (J.S. Langer, Statistical Theory of the Decay of Metastable States, Annals of Physics 54, 258-275 (1969)) for the calculation of the decay rate (relaxation rate) of a metastable state. The theory is set in the context of statistical mechanics, where the dynamics of a system with a large number of degrees of freedom (order 1023) are reduced to N degrees of freedom, where N is small, when a steady state or equilibrium position is maintained by the entire system. In this thesis N equals …

On The Rolling Motion Of Viscous Fluid On A Rigid Surface, Xinli Wang

Dissertations

This thesis considers two closely related problems. First, the influence of insoluble surfactant at a moving contact line is considered. This work is mostly motivated by the air entrainment during the coating process where there is a three-phase contact point (e.g., air, liquid and solid). For moving contact line problems, when the fluid is assumed to be an incompressible Newtonian fluid and a no-slip boundary condition is enforced at the solid boundary, the non-integrable stress singularity arises at the contact line, which is physically unrealistic. The contact angle of 180° is considered as a special case in which the singularity …

A Mathematical And Computational Exploration Of The Effect Of The A-Current In Determining The Activity Phase Of Follower Neurons, Yu Zhang

Dissertations

Bursting oscillations are prevalent in neurons of central pattern generators (CPGs) that produce rhythmic motor activity, and the activity phase plays an important role in determining the normal or dysfunctional network output. The activity phase is the delay time-with respect to some reference time in each cycle and normalized by the oscillation cycle period-of the onset of action potentials by a neuron. This dissertation investigates how the A-current, in conjunction with other intrinsic properties, sets the activity phase of a neuron driven by inhibition.

This dissertation is divided into two major components. In the first component, methods of dynamical systems …