Mathematics Commons^™

The Composition Of Operator-Valued Measurable Functions Is Measurable, Albert Badrikian, G. W. Johnson, Il Yoo

Department of Mathematics: Faculty Publications

Given separable Frechet spaces, E, F , and G , let L(E, F), L(F, G), and L(E, G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of point-wise convergence and the topology of compact convergence. We will show that if (X, F) is any measurable space and both A: X → L(E, F) and B: X → L(F, G) are Borelian, then the operator composition BA: X → L(E, G) is …

Generalized Geynman Integrals: The 𝓛(L2, L2) Theory, Chull Park, David Skough

Department of Mathematics: Faculty Publications

In this paper we develop an 𝓛(L₂(R), L₂(R)) theory for the Feynman integral of functionals of general stochastic processes.

Perturbations Of Certain Reflexive Algebras, David R. Pitts

Department of Mathematics: Faculty Publications

In this note we use cohomological techniques to prove that if there is a linear map between two CSL algebras which is close to the identity, then the two CSL algebras are similar. We use our result to show that if 2' is a purely atomic, hyperreflexive CSL with uniform infinite multiplicity which satisfies the 4-cycle interpolation condition, then there are constants d, C > 0 such that whenever L is another CSL such that d(Alg2' , AlgL) < d, then there is an invertible operator S such that S Alg2'S-1 = AlgL and IISII liS-III < 1 + Cd(AIg2' , AIgL).

The Proportion Of Fixed-Point-Free Elements Of A Transitive Permutation Group, Nigel Boston, Walter Dabrowski, Tuval Foguel, Paul J. Gies, Judy Leavitt Walker, David T. Ose, David A. Jackson

Department of Mathematics: Faculty Publications

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1/n is such a bound.

Lenstra’s problem arose from his work on the number field sieve. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial …

Rewriteability In Finite Groups, Judy Leavitt Walker, G. J. Sherman, Mark E. Walker

Department of Mathematics: Faculty Publications

What's the probability that two elements in a finite group commute? A formal answer,

Pr₂(G) = {(x, y) [element of] G² |xy = yx}| / |G|²

begs our next question. How many ordered pairs of elements of a finite group commute?

A Note On The Connectedness Problem For Nest Algebras, David R. Pitts

Department of Mathematics: Faculty Publications

It has been conjectured that a certain operator T belonging to the group ff of invertible elements of the algebra Alg Z of doubly infinite uppertriangular bounded matrices lies outside the connected component of the identity in W. In this note we show that T actually lies inside the connected component of the identity of & .

Homoclinic Bifurcations With Nonhyperbolic Equilibria, Bo Deng

Department of Mathematics: Faculty Publications

A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilbrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong A-lemma, and Lyapnunov- Schmidt reduction for the Poincare maps under Sil'nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.

An Illustrative Model Describing The Refraction Of Long Water Waves By A Circular Island, Gregory A. Kriegsmann

Department of Mathematics: Faculty Publications

The refraction of small shallow water waves by an idealized island is studied in this paper. The island's shoal is modeled by a quartic polynomial in the radial variable. This particular model allows the explicit construction of the rays (wave orthogonals) and the determination of several important features of the wave motion. The various shortcomings of the particular profile are discussed.

The Differentiability Of AX, J. A. Eidswick

Department of Mathematics: Faculty Publications

A "from scratch" proof of the differentiability of a^x, a > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define a^x as e^x log a its differentiability and other properties following from that of the functions e^x and log x. Unfortunately, the usual definitions of e^x and log x involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of e, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is …

Measurable Choice And The Invariant Subspace Problem, Edward Azoff, Frank Gilfeather

Department of Mathematics: Faculty Publications

In [1], J. Dyer, A. Pedersen and P. Porcelli announced that an affirmative answer to the invariant subspace problem would imply that every reductive operator is normal. Their argument, outlined in [1], provides a striking application of direct integral theory. Moreover, this method leads to a general decomposition theory for reductive algebras which in turn illuminates the close relationship between the transitive and reductive algebra problems.
The main purpose of the present note is to provide a short proof of the technical portion of [1] : that invariant subspaces for the direct integrands of a decomposable operator can be assembled …

Decompositions Of Modules And Matrices, Thomas Shores, Roger Wiegand

Department of Mathematics: Faculty Publications

A canonical form for a module M over a commutative ring R is a decomposition M ≈ R/I₁ Ο … Ο R/I_n, where the I_j are ideals of R and 1₁ ≤ . . . ≤ I_n. A complete structure theory is developed for those rings for which every finitely generated module has a canonical form. The (possibly larger) class of rings, for which every finitely generated module is a direct sum of cyclics, is also considered, and partial results are obtained for rings with fewer than 2^{c …}

Professor Leo Moser -- Reflections Of A Visit, Walter E. Mientka

Department of Mathematics: Faculty Publications

Professor Leo Moser' was known throughout the Mathematical Community as a significant researcher and excellent lecturer. I first met Leo during the Summer Research Institute in the Theory of Numbers held at the University of Colorado in 1959. After talking with him and hearing his lectures during the Institute, I felt that arrangements would have to be made in the near future for a visit to Nebraska. During the academic year 1962-63 while Professor Moser was on a lecture tour for the MAA, I invited him to present two research lectures to the Nebraska Section on May 3 and 4, …

Fourier Transforms Of Unbounded Measures, Loren Argabright, Jesus De La Lamadrid

Department of Mathematics: Faculty Publications

Let G be a locally compact abelian group. For a (generally unbounded) measure μ on G we shall say that μ is transformable if there is a measure μˆ on the character group Γ of G such that, for every ƒЄK(G), the space of continuous functions with compact support on G,ƒЄL₂(μ) and (1) ∫_Gƒ^**ƒ(x) dμ(x) = ∫_r|∫(γ^-1)|²dμ(γ). The resulting "Fourier transformation” μ→μˆ contains the classical theory and leads to generalizations of a variety of classical results, including the Plancherel theorem and the Poisson summation …

A Proof Of Newton's Power Sum Formulas, J. A. Eidswick

Department of Mathematics: Faculty Publications

For a polynomial P(z) = α₀ + α₁z + ... + α_nzⁿ = α_n (z – z₁) (z – z₂) ... (z – z_n), the power sums S_m = ∑ ⁿ _k=1 z^m _k, m = 1, 2, ... , can be calculated from the formulas ...

A Theorem On Repeating Decimals, William G. Leavitt

Department of Mathematics: Faculty Publications

It is well known that a real number is rational if and only if its decimal expansion is a repeating decimal. For example, 2/7 = .285714285714 . . . . Many students also know that if n/m is a rational number reduced to lowest terms (that is, n and m relatively prime), then the number of repeated digits (we call this the length of period) depends only on m. Thus all fractions with denominator 7 have length of period 6. A sharp-eyed student may also notice that when the period (that is, the repeating digits) for 2/7 is split …

A New Family Of Partially Balanced Incomplete Block Designs With Some Latin Square Design Properties, Dale M. Mesner

Department of Mathematics: Faculty Publications

An extensive family of two-class PBIB designs is defined in Section 1and given the title of NL designs. Several analogies of NL designs with Latin square type designs are discussed in Sections 1 and 2. Some of these designs have already appeared in scattered contexts. Section 3 gives a construction based on finite geometry for an infinite class of XL designs, and others can be constructed by methods to be taken up elsewhere. An extension to designs with more than two associate classes is indicated in Section 4.

A Note On The Parameters Of Pbib Association Schemes, Dale M. Mesner

Department of Mathematics: Faculty Publications

In an m-class partially balanced incomplete block (PBIB) design [3], any two distinct treatments are related as first, second, . . . ,or mth associates in accordance with certain rules, and the resulting classification of pairs of treatments is called an association scheme [1], [4]. Parameters, including v, n_i, pⁱ_jk, which depend only on the association relation between treatments and are common to all designs having a given association scheme, are called association scheme parameters. Other parameters, including b, r, k, λ_i, depend, in addition, on the arrangement of the treatments …

Modules Over Commutative Rings, William G. Leavitt

Department of Mathematics: Faculty Publications

The following is another short proof of the fact that for a commutative ring with unit R, any finitely based R-module is "dimensional" in the sense that all of its bases have the same number of elements.

THEOREM. Let R be a commutative ring with unit. If M is a unitary R-module with a basis of n elements, then all bases of M contain exactly n elements.

A Note On Hausdorff Separation, Edwin Halfar

Department of Mathematics: Faculty Publications

The examples usually given as instances of topological spaces that have T₁-separation but not T₂-separation (Hausdorff) also have the property that some compact subset is not closed. This with the classic result concerning closedness of compact subsets of a Hausdorff space suggests the question of the equivalence of Hausdorff separation and the condition that the class of compact subsets be a subclass of the class of the closed subsets of a given space. The following is a simple result of this type and may be of some use in an introductory course in point set topology. …

On Linear Associative Algebras Corresponding To Association Schemes Of Partially Balanced Designs, R. C. Bose, Dale M. Mesner

Department of Mathematics: Faculty Publications

Given v objects 1, 2, .. , v, a relation satisfying the following conditions is said to be an association scheme with m classes:
(a) Any two objects are either 1st, or 2nd, . . ,or mth associates, the relation of association being symmetrical, i.e., if the object α is the ith associate of the object β, then β is the ith associate of α.
(b) Each object a has n_i ith associates, the number n_i being independent of α.
(c) If any two objects α and β are ith associates, then …

On A Theorem Of Hölder, Donald W. Miller

Department of Mathematics: Faculty Publications

A well-known result, due to Hölder [1], is the following: The symmetric group S_n, has outer automorphisms if and only if n = 6. The classical proof of the existence of a class of outer automorphisms of S₆, as formulated by Burnside [2], rests in part on the theory of primitive groups and entails extensive computation. In this note we offer a direct method for constructing such automorphisms. The author is grateful to Professor R. H. Bruck for raising this problem and for subsequent helpful remarks.

On Matrices With Elements In A Principal Ideal Ring, William Leavitt, George Whaples

Department of Mathematics: Faculty Publications

We prove the following theorem.
THEOREM 1. Let D be any commutative principal ideal ring without divisors of zero, and A any matrix with elements in D whose characteristic equation factors into linear factors in D. Then there exists a unimodular matrix T, with elements in D, such that T^-1 AT has zeros below the main diagonal.

On Certain Arithmetical Functions Due To G. Humbert, M. A. Basoco

Department of Mathematics: Faculty Publications

G. Humbert has discussed, in a series of brief notes, a certain class of entire functions with interesting arithmetical properties. These functions are defined, in an essentially unique manner, by certain functional equations. The Fourier series representations of the solutions of these equations are similar in form to those for the elliptic functions snu, cnu, dnu, and so on. They differ from these, however, in that their domain of validity extends throughout the entire complex plane (Z_∞ excluded) and moreover, in that their arithmetized forms involve incomplete numerical functions of the divisors of an integer.
In the …

A Theorem On The Unit Groups Of Simple Algebras, Ralph Hull

Department of Mathematics: Faculty Publications

Let k be an algebraic number field of finite degree m and let A be a normal simple algebra of degree n, order n², over k. Our object is to prove the following theorem.
THEOREM. If A is an R-algebra, that is, if n>2 or at least one infinite prime place of k is unramified in A when n=2, then any two distinct maximal orders of A have distinct groups of units.

On The Fourier Developments Of A Certain Class Of Theta Quotients, M. A. Basoco

Department of Mathematics: Faculty Publications

In this paper we shall be concerned with the functions φ^k_a(z) defined by the relation
(1) φ^k_a(z) ≡{d/(dz)logℓ_α (z,q)}^k = {[ℓά(z,q)] ÷ [ℓα(z,q)]^k, α + 0, 1, 2, 3,
where ℓ_α(z, q) is a Jacobi theta function and k is a positive integer. In the first place, we shall derive the Fourier developments which represent these functions in a certain strip of the complex plane; it …

The Basic Analogue Of Kummer’S Theorem, J. A. Daum

Department of Mathematics: Faculty Publications

About one hundred years ago, E. E. Kummer proved the formula
(1) ₂F₁ [_{1 + a – b} ^{a, b; -1}] = [Γ(1 + a - b)Γ(1 + a/2) ÷ Γ(1+a)Γ(1+a/2 - b)
which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( - 1).
All the relations in the theory of hypergeometric series _rF₈ which have analogues in the theory of basic serie3 are those in which the argument is ( + 1 ) …

On Certain Basic Series, John Daum

Department of Mathematics: Faculty Publications

The identity
(1) Σ_n=1^∞ (qⁿ)/[(1-qⁿ)²] {(1÷(1-q)) + (1÷(1-q²)) + … + (1 ÷ (1 -qⁿ))} = Σ_n=1^∞ [(n²qⁿ)
was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation
(2) Σ_n=0^∞ [(1-q)(1-q²)…(1-qⁿ] ÷ [(1-z)(1-qz)…(1-qⁿz)] x [(zⁿ + 1) ÷ (1 - q^{n …}

Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster

Department of Mathematics: Faculty Publications

We are concerned with the following assertion:
THEOREM. If {φ_n(x)} and { φ_n’(x)} are orthogonal systems of polynomials, then {φ_n(x)} may be reduced to the classical polynomials of Jacobi, Laguerre, or Hermite by means of a linear transformation on x.

Note On The Greatest Integer Function, M. A. Basoco

Department of Mathematics: Faculty Publications

In this note we wish to record certain finite sums involving the greatest integer function E(x), which seem to be of some interest. Hermite* has shown that the generating function for E(x) has the simple form, (1) x^b ÷(1 -x)(1 -x^a) = Σ_(n)E[(n + a - b) ÷ a]xⁿ, where a, b are positive integers. To him is due, likewise, the development (2) ) x^b ÷(1 -x)(1 + x^a) =Σ_(n)E₁[( …

On The Summability Of A Certain Class Of Series Of Jacobi Polynomials, A. P. Cowgill

Department of Mathematics: Faculty Publications

The result obtained in this paper is as follows:
The series Σnⁱ[((p + 1)(p +3)…(p +2n -1)) ÷(2ⁿn!) X^((p-1)/2)_n (x), where X_n^(p-1)/2(x) (hereafter indicated simply by X_n) is a symmetric Jacobi polynomial p >-1, and i a positive integer, is summable (C, k),k>i—1/2, for the range -1 <x<1.