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Articles 361 - 390 of 395
Full-Text Articles in Mathematics
The Differentiability Of AX, J. A. Eidswick
The Differentiability Of AX, J. A. Eidswick
Department of Mathematics: Faculty Publications
A "from scratch" proof of the differentiability of ax, a > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define ax as ex log a its differentiability and other properties following from that of the functions ex and log x. Unfortunately, the usual definitions of ex 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
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
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/In, where the Ij are ideals of R and 11 ≤ . . . ≤ In. 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 2c …
Professor Leo Moser -- Reflections Of A Visit, Walter E. Mientka
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
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
A Proof Of Newton's Power Sum Formulas, J. A. Eidswick
Department of Mathematics: Faculty Publications
For a polynomial P(z) = α0 + α1z + ... + αnzn = αn (z – z1) (z – z2) ... (z – zn), the power sums Sm = ∑ n k=1 zm k, m = 1, 2, ... , can be calculated from the formulas ...
A Theorem On Repeating Decimals, William G. Leavitt
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
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
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, ni, pijk, 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, λi, depend, in addition, on the arrangement of the treatments …
Modules Over Commutative Rings, William G. Leavitt
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
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
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 ni ith associates, the number ni being independent of α.
(c) If any two objects α and β are ith associates, then …
On A Theorem Of Hölder, Donald W. Miller
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 Sn, 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
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.
A Theorem On The Unit Groups Of Simple Algebras, Ralph Hull
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 Certain Arithmetical Functions Due To G. Humbert, M. A. Basoco
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 …
On The Fourier Developments Of A Certain Class Of Theta Quotients, M. A. Basoco
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 φka(z) defined by the relation
(1) φka(z) ≡{d/(dz)logℓα (z,q)}k = {[ℓά(z,q)] ÷ [ℓα(z,q)]k, α + 0, 1, 2, 3,
where ℓα(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
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
On Certain Basic Series, John Daum
Department of Mathematics: Faculty Publications
The identity
(1) Σn=1∞ (qn)/[(1-qn)2] {(1÷(1-q)) + (1÷(1-q2)) + … + (1 ÷ (1 -qn))} = Σn=1∞ [(n2qn)
was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation
(2) Σn=0∞ [(1-q)(1-q2)…(1-qn] ÷ [(1-z)(1-qz)…(1-qnz)] x [(zn + 1) ÷ (1 - qn …
Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster
Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster
Department of Mathematics: Faculty Publications
We are concerned with the following assertion:
THEOREM. If {φn(x)} and { φn’(x)} are orthogonal systems of polynomials, then {φn(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
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) xb ÷(1 -x)(1 -xa) = Σ(n)E[(n + a - b) ÷ a]xn, where a, b are positive integers. To him is due, likewise, the development (2) ) xb ÷(1 -x)(1 + xa) =Σ(n)E1[( …
On The Summability Of A Certain Class Of Series Of Jacobi Polynomials, A. P. Cowgill
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 Xn(p-1)/2(x) (hereafter indicated simply by Xn) 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.
On The Summability And Generalized Sum Of A Series Of Legendre Polynomials, W. C. Brenke
On The Summability And Generalized Sum Of A Series Of Legendre Polynomials, W. C. Brenke
Department of Mathematics: Faculty Publications
The results obtained in this paper are as follows. (A) The series of Legendre polynomials ΣnpXn(x), where p is a positive integer, is summable (H, p) for -1< x < 1, and summable (H, p +1) for - 1 ≤ x < 1 .
On The Element Of Decomposition Of A Doubly Periodic Function Of The Second Kind, M. A. Basoco
On The Element Of Decomposition Of A Doubly Periodic Function Of The Second Kind, M. A. Basoco
Department of Mathematics: Faculty Publications
Hermite has shown that a meromorphic function which satisfies periodicity relations of the form (1) F (z + 2ω) = μF (z), F (z + 2ω’) = μ’F (z), where ω’/ω = a+ib, b>0, and μ, μ' are independent of z, maybe expressed in terms of the function v{z + X) (2) G(z) = σ(z + λ) ÷ σ(z) + σ(λ) eρz, and its derivatives, in which λ, ρ are suitably determined constants and σ(u) is the …
On The Trigonometric Developments Of Certain Doubly Periodic Functions Of The Second Kind, M. A. Basoco
On The Trigonometric Developments Of Certain Doubly Periodic Functions Of The Second Kind, M. A. Basoco
Department of Mathematics: Faculty Publications
The class of meromorphic functions which satisfy periodicity relations of the form (1) ƒ(z + 2ωl) = c1f(z), f(z + 2ω2) = c2f{z), where the multipliers c1 and c2 are independent of z, and ωl/ω2 is a complex number with non-vanishing imaginary part, has been named by Hermite doubly periodic of the second kind. It is possible to make the study of these functions depend on others of the same type, but such that one of the multipliers, say cl, is unity. In what follows …
The Practical Evaluation Of Resultants, T. A. Pierce
The Practical Evaluation Of Resultants, T. A. Pierce
Department of Mathematics: Faculty Publications
The purpose of the present note is to give a practical method of evaluating the resultant of two equations. The method is particularly effective when the degree of one of the equations is high while that of the other is low. Use will be made of certain results in the theory of matrices.
On The Trigonometric Expansion Of Elliptic Functions, M. A. Basoco
On The Trigonometric Expansion Of Elliptic Functions, M. A. Basoco
Department of Mathematics: Faculty Publications
The problem of expressing an elliptic function in terms of infinite sums of trigonometric functions has been treated by Hermite, Briot and Bouquet, A. C. Dixon and others. In the present paper we treat the same problem from the point of view of Cauchy's residue theorem in function theory, which is also Briot and Bouquet's starting point, but we differ from these authors in that the integrand we use leads to an expansion for an elliptic function which is valid in an arbitrarily wide, but finite, strip of the complex plane, and which contains certain classical results as special cases. …
A Certain Multiple-Parameter Expansion, H. P. Doole
A Certain Multiple-Parameter Expansion, H. P. Doole
Department of Mathematics: Faculty Publications
C. C. Camp has shown the convergence of the expansion of an arbitrary function in terms of the solutions of the systems of equations
X1’ (λa1 - Σi=2nμi)X1 = 0,
X1’ (λai + μi)Xi = 0, (j = 2, 3, …, n),
where the ai’s are functions of x, with the boundary conditions
Xi(-π) = Xi(π), (j = 1, 2, …, n).
In this paper it is intended to use a …
Parametric Solutions Of Certain Diophantine Equations, T. A. Pierce
Parametric Solutions Of Certain Diophantine Equations, T. A. Pierce
Department of Mathematics: Faculty Publications
In this note parametric solutions of certain diophantine equations are given. The method of obtaining the solutions is derived from an equation involving the determinants of certain matrices. It will be recognized that the method is a generalization of the method of Euler and Lagrange which depends on forms which repeat under multiplication. The matrices used in this paper must be such that their forms are retained under matric multiplication and addition. When integer values are assigned to the parameters of our solutions we obtain integer solutions of the particular equation under consideration; however not all integer solutions are necessarily …
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
Department of Mathematics: Faculty Publications
Certain well known polynomials have a number of common properties. They arise as coefficients of tn in the expansion of a generating function ; they may be obtained by means of orthogonalization of a set of functions xⁿg(x) when the function ρ(x) = g2(x) and the interval are properly chosen ; they may be regarded as polynomials which become orthogonal when multiplied by a proper factor g(x) ; they satisfy a certain type of difference equation ; they satisfy a certain type of differential equation. The results …