Mathematics Commons^™

How Do Neurons Work Together? Lessons From Auditory Cortex, Kenneth D. Harris, Peter Bartho, Paul Chadderton, Carina Curto, Jaime De La Rocha, Liad Hollender, Vladimir Itskov, Artur Luczak, Stephan Marguet, Alfonso Renart, Shuzo Sakata

Department of Mathematics: Faculty Publications

Recordings of single neurons have yielded great insights into the way acoustic stimuli are represented in auditory cortex. However, any one neuron functions as part of a population whose combined activity underlies cortical information processing. Here we review some results obtained by recording simultaneously from auditory cortical populations and individual morphologically identified neurons, in urethane-anesthetized and unanesthetized passively listening rats. Auditory cortical populations produced structured activity patterns both in response to acoustic stimuli, and spontaneously without sensory input. Population spike time patterns were broadly conserved across multiple sensory stimuli and spontaneous events, exhibiting a generally conserved sequential organization lasting approximately …

Bipartite Density Of Generalized Petersen Graphs, Lisa Jordan Ewell

Electronic Theses and Dissertations

The bipartite density b(G) of a graph G with m edges is the maximum ratio [special characters omitted] where m0 is the number of edges in a bipartitesubgraph of G. In this study we determine the bipartite density of several classes of Generalized Petersen Graphs. These graphs are denoted by P(n, k), where n ≥ 3 and 1 ≤ k < n with n ≠ 2k. The Generalized Petersen Graph P(n, k) has vertices [special characters omitted] and edges [special characters omitted] where subscript addition is modulo n. We define subgraphs P'(n, k) of P( n, k) by deleting the edge vn –1v0 and the edges w iwi+k for n – k ≤ i ≤ n – 1. For P'(n, k) and many classes of P(n, k), we determine the exact number of edges which must be removed from P( n, k) to reduce it to a bipartite subgraph. In many classes of Generalized Petersen Graphs the exact bipartite density is derived. For example: b(P(n, k)) = 1 for n even, k odd; b(P(n, k)) = 1 – [special characters omitted] for n and k odd and n > k²; b(P( n, k)) is asymptotically 1 – [special characters omitted] for n odd, k even.

Homology With Respect To A Kernel Transformation, Seyed Naser Hosseini, Mohammad Zaher Kazemi Baneh

Turkish Journal of Mathematics

In this article we first give the relations between commonly used images of a morphism in a category. We then investigate d-homology in a category with certain properties, for a kernel transformation d. In particular, we show that, in an abelian category, d-homology, where d is induced by the subtraction operation, is the standard homology and that in more general categories the d-homology for a trivial d is zero. We also compute through examples the d-homology for certain kernel transformations d in such categories as R-modules, abelian groups and short exact sequences of R-modules. Finally, we characterize kernel transformations in …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

Complexity Over Finite-Dimensional Algebras, Marju Purin

Mathematics - Dissertations

In this thesis we study two types of complexity of modules over finite-dimensional algebras.

In the first part, we examine the Ω-complexity of a family of self-injective k-algebras where k is an algebraically closed field and Ω is the syzygy operator. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H. As part of the proof, we show that a stable equivalence between self-injective algebras preserves …

Potential Theory On Compact Sets, Tony Perkins

Mathematics - Dissertations

The primary goal of this work is to extend the notions of potential theory to compact sets. There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in [the set of real numbers]ⁿ. One may let H(K) be the uniform closure of all functions in C(K) which are restrictions of harmonic functions on a neighborhood of K, or take H(K) as the subspace of C(K) consisting of functions which are finely harmonic on the fine interior …

The Jordan Curve Theorem Is Non-Trivial, Fiona Ross, William T. Ross

Department of Math & Statistics Faculty Publications

The formal mathematical definition of a Jordan curve (a non-self-intersecting continuous loop in the plane) is so simple that one is often lead to the unimaginative view that a Jordan curve is nothing more than a circle or an ellipse. In this paper, we pursue the theme that a Jordan curve can be quite fantastical in the sense that there are some bizarre properties such a curve might have (jagged at every point, space filling, etc.) or that such a curve can have a difficult to discover inside and outside as promised by the celebrated Jordan Curve Theorem (JCT). We …

Adjusted Empirical Likelihood Models With Estimating Equations For Accelerated Life Tests, Ni Wang, Jye-Chyi Lu, Di Chen, Paul H. Kvam

Department of Math & Statistics Faculty Publications

This article proposes an adjusted empirical likelihood estimation (AMELE) method to model and analyze accelerated life testing data. This approach flexibly and rigorously incorporates distribution assumptions and regression structures by estimating equations within a semiparametric estimation framework. An efficient method is provided to compute the empirical likelihood estimates, and asymptotic properties are studied. Real-life examples and numerical studies demonstrate the advantage of the proposed methodology.

Adjusted Hazard Rate Estimator Based On A Known Censoring Probability, Ülkü Gürler, Paul H. Kvam

Department of Math & Statistics Faculty Publications

In most reliability studies involving censoring, one assumes that censoring probabilities are unknown. We derive a nonparametric estimator for the survival function when information regarding censoring frequency is available. The estimator is constructed by adjusting the Nelson–Aalen estimator to incorporate censoring information. Our results indicate significant improvements can be achieved if available information regarding censoring is used. We compare this model to the Koziol–Green model, which is also based on a form of proportional hazards for the lifetime and censoring distributions. Two examples of survival data help to illustrate the differences in the estimation techniques.

Multi-Cause Degradation Path Model: A Case Study On Rubidium Lamp Degradation, Sun Quan, Paul H. Kvam

Department of Math & Statistics Faculty Publications

At the core of satellite rubidium standard clocks is the rubidium lamp, which is a critical piece of equipment in a satellite navigation system. There are many challenges in understanding and improving the reliability of the rubidium lamp, including the extensive lifetime requirement and the dearth of samples available for destructive life tests. Experimenters rely on degradation experiments to assess the lifetime distribution of highly reliable products that seem unlikely to fail under the normal stress conditions, because degradation data can provide extra information about product reliability. Based on recent research on the rubidium lamp, this article presents a multi‐cause …

Inverse Limits With Set Valued Functions, Van C. Nall

Department of Math & Statistics Faculty Publications

We begin to answer the question of which continua can be homeomorphic to an inverse limit with a single upper semi-continuous bonding map from [O, 1) to 2^(O,l). Several continua including (0, 1) x (0, 1) and all compact manifolds with dimension greater than one cannot be homeomorphic to such an inverse limit. It is also shown that if the upper semi-continuous bonding maps have only zero dimensional point values, then the dimension of the inverse limit does not exceed the dimension of the factor spaces.

A Topological Constructions For All Two-Row Springer Varieties, Heather M. Russell

Department of Math & Statistics Faculty Publications

Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification, Khovanov provides a topological construction of (m,m) Springer varieties. Here we extend his construction to all two-row Springer varieties. Using the combinatorial and diagrammatic properties of this construction we provide a particularly useful homology basis and construct the Springer representation using this basis. We also provide a skein-theoretic formulation of the representation in this case.

On The First-Order Expressibility Of Lattice Properties To Unicoherence In Continua, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

Many properties of compacta have “textbook” definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to the existence of “holes” in a continuum and in certain of its subcontinua.

Detecting Temporal Patterns Using Reconstructed Phase Space And Support Vector Machine In The Dynamic Data System, Wenjing Zhang, Xin Feng

Mathematics, Statistics and Computer Science Faculty Research and Publications

In this paper we present a method for detecting dynamic temporal patterns that are characteristic and predictive of significant events in a dynamic data system. We employ the Gaussian Mixture Model (GMM) to cluster the data sequence into three categories of signals, e.g. normal, patterns and events. The data sequence is then embedded into a Reconstructed Phase Space (RPS) which is topologically equivalent to the dynamics of the original system. We apply a hybrid method using Support Vector Machines (SVM) and Maximum a Posterior (MAP) to classify temporal pattern signals based on the event function. We performed two experimental applications …

Impact Of Curriculum Reform: Evidence Of Change In Classroom Instruction In The United States, John Moyer, Jinfa Cai, Ning Wang, Bikai Nie

Mathematics, Statistics and Computer Science Faculty Research and Publications

The purpose of the study reported in this article is to examine the impact of curriculum on instruction. Over a three-year period, we observed 579 algebra-related lessons in grades 6–8. Approximately half the lessons were taught in schools that had adopted a Standards-based mathematics curriculum called the Connected Mathematics Program (CMP), and the remainder of the lessons were taught in schools that used more traditional curricula (non-CMP). We found many significant differences between the CMP and non-CMP lessons. The CMP lessons, emphasized the conceptual aspects of instruction to a greater extent than the non-CMP lessons and the non-CMP lessons …

The Dynamics Of Integrate-And-Fire: Mean Vs. Variance Modulations And Dependence On Baseline Parameters, Joanna R. Wares, Todd W. Troyer

Department of Math & Statistics Faculty Publications

The leaky integrate-and-fire (LIF) is the simplest neuron model that captures the essential properties of neuronal signaling. Yet common intuitions are inadequate to explain basic properties of LIF responses to sinusoidal modulations of the input. Here we examine responses to low - and moderate-frequency modulations of both the mean and variance of the input current and quantify how these responses depend on baseline parameters. Across parameters, responses to modulations in the mean current are low pass, approaching zero in the limit of high frequencies. For very low baseline firing rates, the response cutoff frequency matches that expected from membrane integration. …

On The Distributions Of Certain Spacings, Gholamhossein Hamedani, Hans Volkmer

Mathematics, Statistics and Computer Science Faculty Research and Publications

A characterization of the uniform distribution based on distributions of spacings is presented which extends the existing result in this direction. Also, a result on the distribution of spacings for distributions close to the uniform one is discussed.

Analysis Of The Dpg Method For The Poisson Equation, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We give an error analysis of the recently developed DPG method applied to solve the Poisson equation and a convection-dffusion problem. We prove that the method is quasioptimal. Error estimates in terms of both the mesh size h and the polynomial degree p (for various element shapes) can be derived from our results. Results of extensive numerical experiments are also presented.

A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in Cockburn et al. (2010) A new elasticity element made for enforcing weak stress symmetry. Math. Comput., 79, 1331–1349 . In this sequel we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements is of one degree less than the first method. The analysis, while similar to …

Analysis Of Hdg Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, Francisco-Javier Sayas

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree $ k$ for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the …

Seeking More Than Nothing: Two Elementary Teachers Conceptions Of Zero, Gale Russell, Egan J. Chernoff

The Mathematics Enthusiast

Zero is a complex and important concept within mathematics, yet prior research has demonstrated that students, pre-service teachers, and teachers all have misconceptions about and/or lack of knowledge of zero. Using a hermeneutic approach based upon Gadamer’s philosophy, this study examined how two elementary mathematics teachers understand zero and how and when zero enters into their teaching of mathematics. The results of this study add new insights into the understandings of teachers and students related to zero and the origins, relationships between, and consequences of those understandings. Significant gaps and misconceptions within both teachers’ understandings of zero suggest the need …

The Education Of Mathematically Gifted Students: Some Complexities And Questions, Roza Leikin

The Mathematics Enthusiast

In this paper I analyze some complexities in the education of mathematically gifted students. The list of issues presented in this paper is not inclusive; however, all of them seem to be typical on the international scope. Among these issues are: (1) the gap between research in mathematics education and the research in gifted education; (2) the role of creativity in the education of the gifted and the theoretical perspective on the relationship between creativity and giftedness, and (3) teaching the gifted and the teachers of gifted, including relationships between the equity principle in mathematics education and views on the …

Historical Perspectives On A Program For Mathematically Talented Students, Harvey B. Keynes, Jonathan Rogness

The Mathematics Enthusiast

The University of Minnesota Talented Youth Mathematics Program (UMTYMP) is a highly accelerated program for students who are extremely talented in mathematics. This paper describes our experiences running UMTYMP since its inception thirty years ago, the challenges in implementing such a program, and how changes in the student body have necessitated changes in the program over three decades.

Independence Polynomials Of Molecular Graphs, Cameron Taylor Byrum

Electronic Theses and Dissertations

In the 1980's, it was noticed by molecular chemists that the stability and boiling point of certain molecules were related to the number of independent vertex sets in the molecular graphs of those chemicals. This led to the definition of the Merrifield-Simmons index of a graph G as the number of independent vertex sets in G. This parameter was extended by graph theorists, who counted independent sets of different sizes and defined the independence polynomial F_G(x) of a graph G to be \sum_k F_k(G)x^k where for each k, F_k(G) is the number of independent sets of k vertices. This thesis …

Analytic & Numerical Study Of A Vortex Motion Equation, Daniel Bueller

Electronic Theses and Dissertations

A nonlinear second order differential equation related to vortex motion is derived. This equation is analyzed using various numerical and analytical techniques including finding approximate solutions using a perturbative approach.

Algebraic Aspects Of (Bio) Nano-Chemical Reaction Networks And Bifurcations In Various Dynamical Systems, Teng Chen

Electronic Theses and Dissertations

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems.

Dsm Vector Spaces Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of DSm linear algebra of refined labels have been done by Florentin Smarandache, Jean Dezert, and Xinde Li.

In this book the authors introduce the notion of DSm vector spaces of refined labels. The reader is requested to refer the paper as the basic concepts are taken from that paper [35]. This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly …

Weighted Scores Method For Regression Models With Dependent Data, Aristidis K. Nikoloulopoulos, Harry Joe, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

There are copula-based statistical models in the literature for regression with dependent data such as clustered and longitudinal overdispersed counts, for which parameter estimation and inference are straightforward. For situations where the main interest is in the regression and other univariate parameters and not the dependence, we propose a "weighted scores method", which is based on weighting score functions of the univariate margins. The weight matrices are obtained initially fitting a discretized multivariate normal distribution, which admits a wide range of dependence. The general methodology is applied to negative binomial regression models. Asymptotic and small-sample efficiency calculations show that our …

Putting The X In Biology: A Review Of The Mathematics Of Life, John Adam

Mathematics & Statistics Faculty Publications

(First Paragraph) Charles Darwin's 1859 work On the Origin of the Species contained no equations. But that does not mean mathematics has no role to play in the science of life; in fact, the field of biomathematics is burgeoning and has been for several decades. Ian Stewart's new book does an admirable job of unfolding the mathematics undergirding so much of the research being carried out today in the many fields that comprise the subject of biology. Stewart sets the context by noting five great revolutions that have changed the way scientists think about life. These five revolutions are: (i) …

On The Betti Number Of Differential Modules, Justin Devries

Department of Mathematics: Dissertations, Theses, and Student Research

Let R = k[x₁, ..., x_n] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d² = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rank_R D ≥ 2ⁿ. This …