Mathematics Commons^™

Multiscale Image Registration, Dana C. Paquin, Doron Levy, Eduard Schreibmann, Lei Xing

Mathematics

A multiscale image registration technique is presented for the registration of medical images that contain significant levels of noise. An overview of the medical image registration problem is presented, and various registration techniques are discussed. Experiments using mean squares, normalized correlation, and mutual information optimal linear registration are presented that determine the noise levels at which registration using these techniques fails. Further experiments in which classical denoising algorithms are applied prior to registration are presented, and it is shown that registration fails in this case for significantly high levels of noise, as well. The hierarchical multiscale image decomposition of E. …

Spectral Analysis Of The Supreme Court, B Lawson, M Orrison, David Uminsky

Mathematics

Should you decide that spectral analysis is worth looking into (as we hope to convince you), then you will be happy to know that there are efficient algorithms for doing spectral analysis. Perhaps more interestingly, at least from a mathematical perspective, these algorithms involve an intriguing mixture of ideas and techniques from linear algebra, abstract algebra, numerical analysis, and graph theory.

The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small …

On The Reliability Of An N-Component System, Don Rawlings, Lawrence Sze

Mathematics

Under assumptions compatible with the theory of Markov chains, we use a property of Vandermonde matrices to examine the reliability of an n-component system of production or service.

On The Existence Of Infinitely Many Closed Geodesics On Orbifolds Of Revolution, Joseph Borzellino, Christopher R. Jordan-Squire, Gregory C. Petrics, D. Mark Sullivan

Mathematics

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert’s theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert’s result …

Existence And Uniqueness Of V-Asymptotic Expansions And Colombeau’S Generalized Numbers, Todor D. Todorov

Mathematics

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau’s generalized number has a v-asymptotic expansion. A similar …

Image Registration With Auto-Mapped Control Volumes, Dana Paquin, Eduard Schreibmann, Lei Xing

Mathematics

Abstract of paper presented at the 47th Annual Meeting of the American Society for Therapeutic Radiology and Oncology.

Elemental T.G. Principles Of Relativistic T-Topos (Presheafification Of Matter, Space, And Time), Goro Kato

Mathematics

We would like to solve the following problem: find a mathematical model formulating I) quantum entanglement, II) particle-wave duality, III) universal objects (ur-sub-Planck objects): to be defined in terms of direct or inverse limits (defined by universal mapping properties) giving microcosm behaviors of space-time so as to give the smooth macrocosm space-time, and IV) the “curved” space-time associated with particles with mass in microcosm consistent with the notion of a light cone in macrocosm. Problems I) and II) are treated in Kato G., Europhys. Lett., 68 (2004) 467. In this paper, we will focus on III) and IV). As …

On The Metamorphosis Of Vandermonde's Identity, Don Rawlings, Lawrence Sze

Mathematics

No abstract provided.

The Expected Variation Of Random Bounded Integer Sequences Of Finite Length, Rudolfo Angeles, Don Rawlings, Lawrence Sze, Mark Tiefenbruck

Mathematics

From the enumerative generating function of an abstract adjacency statistic, we deduce the mean and variance of the variation on random permutations, rearrangements, compositions, and bounded integer sequences of finite length.

Elemental Principles Of T-Topos, Goro Kato

Mathematics

In this paper, a sheaf-theoretic approach toward fundamental problems in quantum physics is made. For example, the particle-wave duality depends upon whether or not a presheaf is evaluated at a specified object. The t-topos theoretic interpretations of double-slit interference, uncertainty principle(s), and the EPR-type non-locality are given. As will be explained, there are more than one type of uncertainty principle: the absolute uncertainty principle coming from the direct limit object corresponding to the refinements of coverings, the uncertainty coming from a micromorphism of shortest observable states, and the uncertainty of the observation image. A sheaf theoretic approach …

Shallow Water Modeling Of Antarctic Bottom Water Crossing The Equator, Paul F. Choboter, Gordon E. Swaters

Mathematics

The dynamics of abyssal equator-crossing flows are examined by studying simplified models of the flow in the equatorial region in the context of reduced-gravity shallow water theory. A simple “frictional geostrophic” model for one-layer cross-equatorial flow is described, in which geostrophy is replaced at the equator by frictional flow down the pressure gradient. This model is compared via numerical simulations to the one-layer reduced-gravity shallow water model for flow over realistic equatorial Atlantic Ocean bottom topography. It is argued that nonlinear advection is important at key locations where it permits the current to flow against a pressure gradient, a mechanism …

Radius Of Convergence Of A Power Series, Todor D. Todorov

Mathematics

We derive two simple and memorizable formulas for the radius of convergence of a power series which seem to be appropriate for teaching in an introductory calculus course.

On The Nonembeddability And Crossing Numbers Of Some Kleinical Polyhedral Maps On The Torus, Adrian Riskin, Judy L. Klein

Mathematics

We designed and constructed a sundial for the purpose of observing the declination of the sun and thus marking solar seasonal variation. The 122 × 122 cm vertical sundial on the south-facing wall of our library has two unusual features: a nodus on the gnomon that casts a shadow of a point for marking the height of the sun and a large blank working space for students to mark the shadow of the nodus at different hours of the day and to connect the marks of 1 day in a line of declination. We discuss the design of a dial …

The Derangement Problem Relative To The Mahonian Process, Jessica Delfert, Hillary Einziger, Don Rawlings

Mathematics

We survey n! plus q-derangement problems. Solutions to four Mahonian statistics arising in connection with cycle placement rules are presented. A few conjectures are also made.

Determination Of The Topological Structure Of An Orbifold By Its Group Of Orbifold Diffeomerphisms, Joseph E. Borzellino, Victor Brunsden

Mathematics

We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let Diff^r_Orb (O) denote the C^r orbifold diffeomorphisms of an orbifold O. Suppose that Φ: Diff^r_Orb(O₁) → Diff^r_Orb (O₂) is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds O₁ and O₂. We show that Φ is induced by a homeomorphism h: X_O1 → X_O2 , where X_O denotes the underlying topological …

The Continuum Problem, John Stillwell

Mathematics

No abstract provided.

Sheaf Cohomology Of Conscious Entity, Goro Kato

Mathematics

Awareness of a conscious entity can exist without elements; therefore, the general notion of an object of a category is employed. One of the characterization of understanding is: for a given local infonnation (awareness) there exists a global information whose restriction is the given information. For such mental activities, category and sheaf theories are employed to formulate consciousness. We will show that the cohomology (more general precohomology) object, a subquotient object, better represents the essence of a conscious entity than an object itself. We will also give a definition of an observation to fonnulate the collapse of the wave and …

Category Theory And Consciousness, Goro Kato, Daniele C. Struppa

Mathematics

No abstract provided.

Orbifold Homeomorphism And Diffeomorphism Groups, Joseph E. Borzellino, Victor Brunsden

Mathematics

In this paper we outline results on orbifold diffeomorphism groups that were presented at the International Conference on Infinite Dimensional Lie Groups in Geometry and Representation Theory at Howard University, Washington DC on August 17-21, 2000. Specifically, we define the notion of reduced and unreduced orbifold diffeomorphism groups. For the reduced orbifold diffeomorphism group we state and sketch the proof of the following recognition result: Let O1 and O2 be two compact, locally smooth orbifolds. Fix r ≥ 0. Suppose that Φ : Diffr (O1) → Diffr (O2) is a group isomorphism. Then Φ is induced by redred a (topological) …

Large Sets Of Zero Analytic Capacity, John Garnett, Stan T. Yoshinobu

Mathematics

We prove that certain Cantor sets with non-sigma-finite one-dimensional Hausdorff measure have zero analytic capacity.

Sequential Searches: Proofreading, Russian Roulette, And The Incomplete Q-Eulerian Polynomials Revisited, Don Rawlings

Mathematics

No abstract provided.

Whose Limit Is It Anyway?, Joseph E. Borzellino

Mathematics

In a tongue-in-cheek manner, we investigate the notion of limit. We illustrate some of its shortcomings and show that addressing these shortcomings can often lead to unexpected consequences.

The Breakdown Of Synchronization In Systems Of Non-Identical Chaotic Oscillators: Theory And Experiment, Jennifer Chubb, Ernest Barreto, Paul So, Bruce J. Gluckman

Mathematics

The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persist as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of nonidentical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of the evolution of the unstable periodic orbit structure. Our results are illustrated with data from physical experiments.

Modular Miracles, John Stillwell

Mathematics

Over the last 20 years, the modular function has become widely known through its miraculous intervention in two great mathematical achievements: the proof of Fermat's last theorem and the "moonshine" of the monster simple group. In both cases, the modular function appears where no one expected it, and it bridges a chasm between seemingly unrelated fields. It is probably fair to say that, in these two cases, we do not yet fully understand how the modular magic works.

However, it can at least be said that these are not the first modular miracles. Ever since its discovery, in the early …

On The Nonembeddability And Crossing Numbers Of Some Toroidal Graphs On The Klein Bottle, Adrian Riskin

Mathematics

We show that toroidal polyhedral maps with four or more disjoint homotopic noncontractiblecircuits are not embeddable on the projective plane and that toroidal polyhedral maps with -veor more disjoint homotopic noncontractible circuits are not embeddable on the Klein bottle. Wealso show that the Klein bottle crossing numbers ofCm×Cn(m6n) form=3;4;5;6 are 1,2,4,and 6, respectively, and give upper bounds for all other values ofn. These crossing numbersdispla yat ypical behavior in that the value depends onl yonminstead of on bothmandnas isthe case for the plane and projective plane.

A Sequential Search Distribution: Proofreading, Russian Roulette, And The Incomplete Q-Eulerian Polynomials, Travis Herbranson, Don Rawlings

Mathematics

The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k = n and the find probability remains constant, relatively simple and efficient formulas are obtained. From our main theorem, we further (1) deduce the inverse absorption distribution and (2) determine the expected number of times the survivor pulls the trigger in an n-player game of Russian roulette.

Back To Classics: Teaching Limits Through Infinitesimals, Todor D. Todorov

Mathematics

The usual ϵ, δ-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a “candidate L” for the limit value. Thus, we have to start our first calculus course with “guessing” instead of “calculating”. In this paper we criticize the method of using calculators for the purpose of selecting candidates for L. We suggest an alternative: a working formula for calculating the limit value L of a real function in terms of infinitesimals. Our formula, if considered as a definition of limit, is equivalent to the usual ϵ, δ-definition but does not involve …

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' …

Modelling The Dynamics Of Abyssal Equator-Crossing Currents, P. F. Choboter, G. E. Swaters

Mathematics

No abstract provided.

Restricted Words By Adjacencies, Don Rawlings

Mathematics

A recurrence, a determinant formula, and generating functions are presented for enumerating words with restricted letters by adjacencies. The main theorem leads to refinements (with up to two additional parameters) of known results on compositions, polyominoes, and permutations. Among the examples considered are (1) the introduction of the ascent variation on compositions, (2) the enumeration of directed vertically convex polyominoes by upper descents, area, perimeter, relative height, and column number, (3) a tri-variate extension of MacMahon's determinant formula for permutations with prescribed descent set, and (4) a combinatorial setting for an entire sequence of bibasic Bessel functions.