Physical Sciences and Mathematics Commons^™

Course In Computational Number Theory, David Bressoud, Stan Wagon

Stan Wagon, Retired

No abstract provided.

Paths In Graphs, Béla Bollobás, Amites Sarkar

Mathematics Faculty Publications

We prove that if 10 ≦ (^k₂) ≦ m < (^k+1₂) then the number of paths of length three in a graph G of size m is at most 2m(m – k)(k - 2)/k. Equality is attained if G is the union of K_k and isolated vertices. We also give asymptotically best possible bounds for the maximal number of paths of length s, for arbitrary s, in graphs of size m. Lastly, we discuss the more general problem of maximizing the number of subgraphs …

Ramanujan-Like Congreuences Of The Distinct Partition Function, Ian Blumenfeld, Christi Carlstead, Mimi Cukier, Wesley Terway

Mathematical Sciences Technical Reports (MSTR)

In his work with the partition function, Ramanujan observed several congruences of the form p(An + B) = 0 (mod m). We adapt this form to several congruences of the distinct partition function, p2(n). We show that one can determine all ordered pairs of integers (A;B) for which p₂(An + B)=0 (mod 2) and show families of congruences modulo 4. Finally, we offer a proof of a congruence modulo 5 satisfied by the distinct partition function.

Optimal Solvability For The Dirichlet And Neumann Problem In Dimension Two, Atanas Stefanov, Gregory C. Verchota

Mathematics - All Scholarship

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p₁,(partial D)(H^p (partial D)), p>2/3-E, where D C R² and E is a (small) number depending on the Lipschitz nature of D. This in turn implies that solutions to the Dirichlet problem with data in the Holder class C^1/2+E(partial D) are themselves in C^1/2+E(D). Both of these results are sharp. In fact, we prove a more general statement regarding the H^p solvability …

Constant Mean Curvature Surfaces With Three Ends, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan

Robert Kusner

We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

Fractional Derivatives, John M. Beach

Theses and Dissertations

In this thesis, the reader will not find a study of any kind; there is no methodology, questionnaire, interview, test, or data analysis. This thesis is simply a research paper on fractional derivatives, a topic that I have found to be fascinating. The reader should be delighted by a short history of the topic in Chapter 1, where he/she will read about the contributions made by some of the great mathematicians from the last three centuries.

In Chapter 2 the reader will find an intuitive approach for finding the general fractional derivative for functions such as e^ax, x …

Classification Of Cwatsets Through Order 23, Ben Goodwin, Dennis Lin

Mathematical Sciences Technical Reports (MSTR)

A cwatset of order n can be represented by a transitive subgroup of S_n. Previous work has shown that each conjugacy class of rep­resentation groups corresponds to an isomorphism class of cwatsets. We present a technique for determining whether a particular transitive subgroup of Sn can appear as the representation group for a cwatset of order n. Using this method, we provide a full classification of cwatset isomorphism classes through order 23.

Stochastic Dynamics Of Systems With Memory (Stochastic Analysis Seminar, University Of Oxford), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

Computational Geometry Column 40, Joseph O'Rourke

Computer Science: Faculty Publications

It has recently been established by Below, De Loera, and Richter-Gebert that finding a minimum size (or even just a small) triangulation of a convex polyhedron is NP-complete. Their 3SAT-reduction proof is discussed.

Commuting Self-Adjoint Extensions Of Symmetric Operators Defined From The Partial Derivatives, Palle Jorgensen, Steen Pedersen

Mathematics and Statistics Faculty Publications

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(∂/∂x_j):j=1,...,d} with domain C^∞_c(Ω) where the self-adjointness is defined relative to L²(Ω), and Ω is a given open subset of R^d.

Normal Forms, Canonical Forms, And Invariants Of Single Input Nonlinear Systems Under Feedback, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analysing, step by step, the action of homogeneous transformations on the homogeneous part of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, and canonical form.

The Averaging Lemma, Ronald A. Devore, Guergana Petrova

Faculty Publications

No abstract provided.

Stability And Largeness Of The Core Of Cooperative Games., Amit K. Biswas Dr.

Doctoral Theses

This monograph deals with the area from game theory known as co operativo games. Except the last chapter on NTU games, it deals with transferable utility games. llere we will introiduce and diseus the involved game theuretie notions and set a mathematical hase for the chapters to come.In 1944 von Neamann and Morgenstern(15) introduced a theory of solutions for n-person games in characteristic function form in which cooperation and coalition formation is a crucial aspect. The primary mathematical concern regarding this model is the existence of solutions or stable set. In 1968 Lucas|19 described a tem person game which has …

An Excursion From Enumerative Geometry To Solving Systems Of Polynomial Equations With Macaulay 2, Frank Sottile

Mathematics and Statistics Department Faculty Publication Series

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions …

Topology And Metastability In The Lattice Skyrme Model, Alec Schramm, Benjamin Svetitsky

Alec J Schramm

We offer the Skyrme model on a lattice as an effective field theory—fully quantized—of baryon-meson interactions at temperatures below the chiral phase transition. We define a local topological density that involves the volumes of tetrahedra in the target space S3 and we make use of Coxeter’s formula for the Schläfli function to implement it. This permits us to calculate the mean-square radius of a Skyrmion in the three-dimensional lattice Skyrme model, which may be viewed as a Ginzburg-Landau effective theory for the full quantum theory at finite temperature. We find that, contrary to expectations, the Skyrmion shrinks as quantum and …

Sfde's As Dynamical Systems (Symposium 2000/2001, University Of Warwick), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

The Expected Wet Period Of Finite Dam With Exponential Inputs, Eui Yong Lee, Kimberly Kinateder

Mathematics and Statistics Faculty Publications

We use martingale methods to obtain an explicit formula for the expected wet period of the finite dam of capacity V, where the amounts of inputs are i.i.d exponential random variables and the output rate is one, when the reservoir is not empty. As a consequence, we obtain an explicit formula for the expected hitting time of either 0 or V and a new expression for the distribution of the number of overflows during the wet period, both without the use of complex analysis.

Transverse Asymmetry AT' From The Quasielastic 3He (E, E') Process And The Neutron Magnetic Form Factor, X. Wu, D. Dutta, F. Xiong, B. Anderson, L. Auberbach, T. Averett, W. Bertozzi, T. Black, J. Calarco, L. Cardman, G. D. Cates, Z. W. Chai, J. P. Chen, S. Choi, E. Chudakov, S. Churchwell, G. S. Corrado, C. Crawford, D. Dale, A. Deur, P. Djawotho, B. W. Filippone, J. M. Finn, H. Gao, R. Gilman, A. V. Glamazdin, C. Glashausser, W. Glockle, J. Golak, J. Gomez, V. G. Gorbenko, J. O. Hansen, F. W. Hersman, D. W. Higinbotham, R. Holmes, C. R. Howell, E. Hughes, B. Humensky, S. Incerti, C. W. De Jager, J. S. Jensen, X. Jiang, C. E. Jones, M. Jones, R. Kahl, H. Kamada, A. Kievsky, I. Kominis, W. Korsch, K. Kramer, G. Kumbartzki, M. Kuss, Enkeleida K. Lakuriqi, M. Liang, N. Liyanage, J. Lerose, S. Malov, D. J. Margaziotis, J. W. Martin, K. Mccormick, R. D. Mckeown, K. Mcilhany, Z. E. Meziani, R. Michaels, G. W. Miller, E. Pace, T. Pavlin, G. G. Petratos, R. I. Pomatsalyuk, D. Pripstein, D. Prout, R. D. Ransome, Y. Roblin, M. Rvachev, A. Saha, G. Salme, M. Schnee, T. Shin, K. Slifer, P. A. Souder, S. Strauch, R. Suleiman, M. Sutter, B. Tipton, L. Todor, M. Viviani, B. Vlahovic, J. Watson, C. F. Williamson, H. Witala, B. Wojtsekhowski, J. Yeh, P. Zolnierczuk

Enkeleida K. Lakuriqi

We have measured the transverse asymmetry AT' in 3He (e,e') quasielastic scattering in Hall A at Jefferson Laboratory with high precision for Q2 values from 0.1 to 0.6(GeV/c)2. The neutron magnetic form factor GnM was extracted based on Faddeev calculations for Q2=0.1 and 0.2(GeV/c)2 with an experimental uncertainty of less than 2%.

Dependence Among Spacings, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we study the dependence properties of spacings. It is proved that if X₁,..., X_n are exchangeable random variables which are TP₂ in pairs and their joint density is log-convex in each argument, then the spacings are MTP₂ dependent. Next, we consider the case of independent but nonhomogeneous exponential random variables. It is shown that in this case, in general, the spacings are not MTP₂ dependent. However, in the case of a single outlier when all except one parameters are equal, the spacings are shown to be MTP₂ dependent and, hence, …

Existence Of Many Positive Nonradial Solutions For A Superlinear Dirichlet Problem On Thin Annuli, Alfonso Castro, Marcel B. Finan

All HMC Faculty Publications and Research

We study the existence of many positive nonradial solutions of a superlinear Dirichlet problem in an annulus in R^N. Our strategy consists of finding the minimizer of the energy functional restricted to the Nehrai manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a desired solution. Then we show that the minimal energy solutions in different symmetric classes have mutually different energies. The same approach has been used to prove the existence of many sign-changing nonradial solutions (see [5]).

On Reconfiguring Tree Linkages: Trees Can Lock, Therese Biedl, Erik D. Demaine, Martin L. Demaine, Sylvain Lazard, Anna Lubiw, Joseph O'Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, Sue Whitesides

Computer Science: Faculty Publications

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N-link tree can have 2^Ω(N) equivalence classes of configurations.

On Logistic And Some New Discrimination Rules:Charecterizations,Inference And Application., Supratik Roy Dr.

Doctoral Theses

Introduction and Summary Consider the problem of classification of an observation into one of two specified populations. Fisher's classification rale, just as several other rules commonly used in practice, depends only on the ratio of the individual densities fi(x), i = 1,2. This led Cox (1966),/27) to model the "posterior odds" by a simple function. Specifically,Cox's logistic discrimination (LGD) rule is then based on the statistic a + 'ßx. This has the advantage that individual densities f.(x) need not be known and we only need to estimate the parameters a and B.Another advantage, which is claimed , is that the …

Stochastic Dynamics Of Infinite-Dimensional Systems (Stochastic And Non-Linear Analysis Seminar, University Of Illinois), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

We describe an approach to the dynamics of non-linear stochastic differential systems with finite memory using multiplicative cocycles in Hilbert space. We introduce the notion of hyperbolicity for stationary solutions of stochastic systems with memory. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses ideas from infinite-dimensional multiplicative ergodic theory and interpolation arguments.

Quest For Tilings On Riemann Surfaces Of Genus Six And Seven, Robert Dirks, Maria Sloughter

Mathematical Sciences Technical Reports (MSTR)

The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, T, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.

Dynamical Properties Of Forced Shear Layers In An Annular Geometry, Evangelos A. Coutsias, Keith Bergeron, J.P. Lynov, A.H. Nielsen

Branch Mathematics and Statistics Faculty and Staff Publications

Results of numerical simulations of a forced shear flow in an annular geometry are presented. The particular geometry used in this work reduces the effects of centrifugal and Coriolis forces. However, there are still a large number of system parameters (shear width, shear profile, radius of curvature, initial conditions, etc.) to characterize. This set of variables is limited after the code has been validated with experimental results (Rabaud & Couder 1983; Chomaz et al. 1988) and with the associated linear stability analysis. As part of the linear stability characterization, the pseudo-spectrum for the associated Orr-Sommerfeld operator for plane, circular Couette …

Pushpush And Push-1 Are Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that two pushing-blocks puzzles are intractable in 2D. One of our constructions improves an earlier result that established intractability in 3D [OS99] for a puzzle inspired by the game PushPush. The second construction answers a question we raised in [DDO00] for a variant we call Push-1. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a …

Singularly Perturbed Control Systems Using Non-Commutative Computer Algebra, J. W. Helton, F. Dell Kronewitter, W. M. Mceneaney, Mark Stankus

Mathematics

Most algebraic calculations which one sees in linear systems theory, for example in IEEE TAC, involve block matrices and so are highly non-commutative. Thus conventional commutative computer algebra packages, as in Mathematica and Maple, do not address them. Here we investigate the usefulness of non-commutative computer algebra in a particular area of control theory − singularly perturbed dynamic systems − where working with the non-commutative polynomials involved is especially tedious. Our conclusion is that they have considerable potential for helping practitioners with such computations. Commutative Gröbner basis algorithms are powerful and make up the engines in symbolic algebra packages' …

On Singular Critical Points Of Positive Operators In Krein Spaces, Branko Ćurgus, Aurelian Gheondea, H. Langer

Mathematics Faculty Publications

We give an example of a positive operator B in a Krein space with the following properties: the nonzero spectrum of B consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of B are uniformly bounded and the point ∞ is a singular critical point of B.

The Story Of A Service-Learning Project: Mathematics In The Park, Joyce O'Halloran

Humanistic Mathematics Network Journal

No abstract provided.

Training Elementary Teachers For The New Millennium, Dixie Metheny, David Davison

Humanistic Mathematics Network Journal

No abstract provided.