Mathematics Commons^™

Cell-Like Images And UvM Groups, Gerard A. Venema

University Faculty Publications and Creative Works

Let X and A be compact metric spaces. The main problem studied in this paper is that of finding conditions under which a map f{hook}:A→X can be lifted to a cell-like space; i.e., conditions are sought under which there exist a cell-like continuum Z and continuous maps g:Z→X and f{hook}′:A→Z such that g{ring operator}f{hook}′=f{hook}. A theorem is proved which spells out technical conditions on the embedding of A into X and on the homotopy pro-groups of X under which such a lifting exists. The main corollary asserts the following: If X is UVk+1, dim A≤k, and f{hook}′:A→X is continuous, then …

Ext In The Nineties, Robert R. Bruner

Mathematics Faculty Research Publications

We describe a package of programs to calculate minimal resolutions, chain maps, and null homotopies in the category of modules over a connected algebra overe Z_2 and in the category of unstable modules over the mod 2 Steenrod algebra. They are available for free distribution and intended for use as an Adams spectral sequence 'pocket calculator'. We provide a sample of the results obtained from them.

Introduction To Fractal Geometry: Definition, Concept, And Applications, Mary Bond

Presidential Scholars Theses (1990 – 2006)

It has become evident that fractals are not to be tied down to one compact, Webster-style, paragraph definition. The foremost qualities of fractals include self-similarity and dimensionality. One cannot help but appreciate the aesthetic beauty of computer generated fractal art. Beyond these characteristics, when trying to grasp the idea of fractal geometry, it is helpful to learn about its many applications. Fractal geometry is opening new doors for study and understanding in diverse areas such as science, art, and music. All of these facets of fractal geometry unite to provide an intriguing, and alluring, wardrobe for mathematics to wear, so …

K Dimension Continued Fractions And K Dimension Golden Ratios, Tascha Gwyn Yoder

Presidential Scholars Theses (1990 – 2006)

The following is an investigation dealing with continued fractions based on research conducted by Professor John C. Longnecker at the University of Northern Iowa.

Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this paper we are concerned with the splitting of the identity component G₀ in an LCA group G. As Pontrjagin duality shows, this splitting is to the splitting of the torsion part tA in a discrete abelian group A, if G is assumed to be compact.

Characterization Of Knot Complements In The 4-Sphere, Vo Thanh Liem, Gerard A. Venema

University Faculty Publications and Creative Works

Knot complements in S4 are characterized as follows: A connected open set W ⊂ S4 is homeomorphic to the complement of some locally flat 2-sphere in S4 if and only if H1(W) is infinite cyclic, W has one end, and the fundamental group of that end is infinite cyclic. Applications include a characterization of weakly flat 2-spheres in S4 and a complement theorem for 2-spheres in S4.

Some Complex Grassmannian Manifolds That Do Not Fibre Nontrivially, John Ferdinands

University Faculty Publications and Creative Works

A finite CW complex X is said to be prime if, given a Hurewicz fibration F→E→B with E homotopy equivalent to X, and B and F homotopy equivalent to finite CW complexes, either B or F is contractible. We show that certain 3- and 4-plane complex Grassmanian manifolds are prime. © 1991.

Integer Triangles With Rational Medians, Bart Goddard, Dale Mesner

Mathematical Sciences Technical Reports (MSTR)

A characterization of all integer-sided triangles with a rational median is given, similar to the categorization of Pythagorean triangles. An infinite family of integer-sided triangles with two rational medians is given, along with several examples of three rational medians. All examples come from solutions to systems of quadratic Diophantine equations.

Shadow Casting Phenomena At Newgrange, Frank Prendergast

Articles

A digital model of the Newgrange passage tomb and surrounding ring of monoliths known as the Great Circle is used to investigate sunrise shadow casting phenomena at the monument. Diurnal variation in shadow directions and lengths are analysed for their potential use in the Bronze Age to indicate the passage of seasonal time. Computer-aided simulations are developed from a photogrammetric survey to accurately show how three of the largest monoliths, located closest to the tomb entrance and archaeologically coded GC1, GC-1 and GC-2, cast their shadows onto the vertical face of the entrance kerbstone, coded K1. The phenomena occur at …

Mat 751 Algebraic Topology I - Fall '89, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course Mat 751, Algebraic Topology I, prepared by Professor David Handel of the Wayne State University Mathematics Department. The notes include examples, exercises, and additional lecture notes on related concepts.

Changing Modes Of Thought: Non-Euclidean Geometry And The Liberal Arts, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Dupin Submanifolds In Lie Sphere Geometry, Thomas E. Cecil, Shiing-Shen Chern

Mathematics and Computer Science Department Faculty Scholarship

No abstract provided.

Corrigendum, John A. Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Neighborhoods Of Compacta In 4-Manifolds, Gerard A. Venema

University Faculty Publications and Creative Works

In this paper we investigate conditions under which a compact set in a piecewise linear 4-manifold has close neighborhoods with 1-dimensional spines. We prove that such neighborhoods exist in case the compactum satisfies the inessential loops condition and either has fundamental dimension 0 or has the shape of S1. Corollaries include two complement theorems and a weak flatness theorem for compacta in S4.

Equations Of Variation For Ordinary Differential Equations On Manifolds, J. B. Bennett

Journal of the Arkansas Academy of Science

No abstract provided.

A Mathematical Model Of Tumor Growth By Diffusion, John A. Adam

Mathematics & Statistics Faculty Publications

A diffusion model of the prevascular stage of tumor growth is presented. The basic feature of such a model is the diffusion of growth inhibitor, which is produced at a spatially non-uniform rate within the tissue. Regimes of limited and unlimited tissue growth are determined, and the consistency of this and simpler models is discussed in the light of observational results.

Ce Equivalence And Shape Equivalence Of 1-Dimensional Compacta, R. J. Daverman, Gerard A. Venema

University Faculty Publications and Creative Works

In this paper the relationship between CE equivalence and shape equivalence for locally connected, 1-dimensional compacta is investigated. Two theorems are proved. The first asserts that every path connected planar continuum is CE equivalent either to a bouquet of circles or to the Hawaiian earring. The second asserts that for every locally connected, 1-dimensional continuum X there is a cell-like map of X onto a planar continuum. It follows that CE equivalence and shape equivalence are the same for the class of all locally connected, 1-dimensional compacta. In addition, an example of Ferry is generalized to show that for every …

Math 752 Algebraic Topology Ii - Winter '84, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course MAT 752, Algebraic Topology II, prepared by Professor David Handel of the Wayne State University Mathematics Department. This course builds on MAT 751, Algebraic Topology I, and the notes include examples, exercises, and suggestions for further reading.

Sums Of Z-Ideals And Semiprime Ideals, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

If B is a ring (or module), and K is an ideal (or submodule) of B, let B(K) = {(a,b) є B x B:a-b є K}. The relationship between ideals (or submodules) of B and those of B(K) is examined carefully, and this construction is used to find a lattice-ordered subring of the ring C(R) of all continuous real-valued functions on the real line R with two z-ideals whose sum is not even semiprime.

Two Generalizations Of The Adams Spectral Sequence, Robert R. Bruner

Mathematics Faculty Research Publications

No abstract provided.

An Approximation Theorem In Shape Theory, Gerard A. Venema

University Faculty Publications and Creative Works

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.

Transformational Geometry Unit, Elizabeth Ann O'Neill

All Graduate Projects

The study included the development and writing of a unit on transformational geometry which involved a holistic approach including the cognitive, psychomotor, and affective domains. This unit was taught to the eighth grade class in the Oakville School District in Oakville, Washington. The results showed support that the teaching of this unit was effective.

K-Theory And Steenrod Homology: Applications To The Brown-Douglas-Fillmore Theory Of Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.

Finite-Difference Approach To The Hodge Theory Of Harmonic Forms, Jozef Dodziuk

Publications and Research

No abstract provided.

Faces, Edges, Vertices Of Some Polyhedra, Charles H. Harbison

Journal of the Arkansas Academy of Science

A proof that: for any given polyhedron so shaped that every closed non-self intersecting broken line composed of edges of the polyhedron divides the surface of the polyhedron into precisely two disjoint regions each of which is bounded by the closed broken line, v - e + f = 2, where v is the number of vertices of the polyhedron, e the number of edges and f the number of faces.

Infinite Product Spaces Under The Tychonoff And Goofynoff Topologies, James A. Capps

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The only topology considered for the infinite product of topological spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topology of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not refer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of …

Radicals And Torsion Theories In Locally Compact Groups, Robert R. Bruner

Mathematics Faculty Research Publications

In this paper we will study the properties of locally compact Abelian Hausdorff topological groups (hereafter known as LCA groups) by means of their mapping properties. The results contained herein are an outgrowth of work done by Professor Armacost [Al] on "sufficiency classes" of LCA groups. The sufficiency class S\textunderscore(H) of an LCA group H is the class of all LCA groups G such that there are sufficiently many continuous homomorphisms from G to H to separate the points of G. This condition is easily seen to be equivalent to the requirement that ∩ker(f)=0, where f ranges over all elements …

The Fundamental Groups Of The Complements Of Some Solid Horned Spheres, Norman William Riebe

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M₂, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M₃ and M₄, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization.

The fundamental groups of the complements of M₂, M₃, and M₄ are derived, and representations of these groups onto the Alternating Group, A₅, are found. The form of the presentations …

An Investigation Of The Properties Of Join Geometry, Louis John Giegerich Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper presents a proof that the classical geometry as stated by Karol Borsuk [1] follows from the join geometry of Walter Prenowitz [2].

The approach taken is to assume the axioms of Prenowitz. Using these as the foundation, the theory of join geometry is then developed to include such ideas as 'convex set', 'linear set', the important concept of 'dimension', and finally the relation of 'betweenness'. The development is in the form of definitions with the important extensions given in the form of theorems.

With a firm foundation of theorems in the join geometry, the axioms of classical geometry …

On The Theory Of Head Waves, Patrick Heelan

Research Resources

When a combined longitudinal and transverse disturbance, diverging from a localized source, strikes a plane boundary between two solid elastic media, several systems of head waves and second order boundary waves are generated, each associated with grazing incidence of one or the other of the reflected or refracted waves. Associated with grazing incidence of P ₁ P₂, the refracted P-wave, is the head wave system comprising P₁P₂P₁ (the "refracted wave" of seismic prospectors), and P₁P₂S₁ (a transverse head wave) in the upper medium, and P₁P_{2 …}