Mathematics Commons^™

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

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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.

Results On Periodic Solutions Of Parabolic Equations Suggested By Elliptic Theory, Alfonso Castro, A. C. Lazer

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No abstract provided.

Rearrangeable Networks With Limited Depth, Nicholas Pippenger, Andrew C.-C. Yao

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Rearrangeable networks are switching systems capable of establishing simultaneous independent communication paths in accordance with any one-to-one correspondence between their n inputs and n outputs. Classical results show that Ω( n log n ) switches are necessary and that O( n log n ) switches are sufficient for such networks. We are interested in the minimum possible number of switches in rearrangeable networks in which the depth (the length of the longest path from an input to an output) is at most k, where k is fixed as n increases. We show that Ω( n^{1 + 1/k} ) switches …

Some Properties Of Positive Derivations On F-Rings, Melvin Henriksen, Frank A. Smith

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Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set …

Algebraic Complexity Theory, Nicholas Pippenger

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Algebraic complexity theory, the study of the minimum number of operations sufficient to perform algebraic computations, is surveyed with emphasis on the general theory of bilinear forms and two of its applications: polynomial multiplication and matrix multiplication. Though by no means exhausting algebraic complexity theory, these topics illustrate well its development and its methods, and provide examples of its most striking successes.

Bounds On The Performance Of Protocols For A Multiple-Access Broadcast Channel, Nicholas Pippenger

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A general model is presented for synchronous protocols that resolve conflicts among message transmissions to a multiple-access broadcast channel. An information-theoretic method is used now to show that if only finitely many types of conflicts can be distinguished by the protocol, utilization of the channel at rates approaching capacity is impossible. A random-coding argument is used to show that if the number of conflicting transmissions can be determined (which requires distinguishing infinitely many types of conflicts) then utilization of the channel at rates arbitrarily close to capacity can be achieved.

On Periodic Solutions Of Weakly Coupled Systems Of Differential Equations, Alfonso Castro, A. C. Lazer

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No abstract provided.

Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro

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In this paper, we study the existence of weak solutions of the problem

□u + ∇G(u) = f(t,x) ; (t,x) є Ω ≡ (0,π)x(0,π)

u(t,x) = 0 ; (t,x) є ∂Ω

where □ is the wave operator ∂²/∂t² - ∂²/∂x², G: Rⁿ→R is a function of class C² such that ∇G(0) = 0 and f:Ώ→R^n is a continuous function having first derivative with respect to t in (L₂,(Ω))ⁿ and satisfying

f(0,x) = f(π,x) = 0

for all x є [0,π].

A New Lower Bound For The Number Of Switches In Rearrangeable Networks, Nicholas Pippenger

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For the commonest model of rearrangeable networks with $n$ inputs and $n$ outputs, it is shown that such a network must contain at least $6n \log _6 n + O( n )$ switches. Similar lower bounds for other models are also presented.

Two Point Boundary Value Problem With Jumping Nonlinearities, Alfonso Castro

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We prove that a certain two point BVP with jumping nonlinearities has a solution. Our result generalizes that of [2]. We use variational methods which permit giving a minimax characterization of the solution. Our proof exposes the similarities between the variational behavior of this problem and that of other semilinear problems with noninvertible linear part (see [5]).

On The Evaluation Of Powers And Monomials, Nicholas Pippenger

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Let $y_1 , \cdots ,y_p $ be monomials over the indeterminates $x_1 , \cdots ,x_q $. For every $y = (y_1 , \cdots ,y_p )$ there is some minimum number $L(y)$ of multiplications sufficient to compute $y_1 , \cdots ,y_p $ from $x_1 , \cdots ,x_q $ and the identity 1. Let $L(p,q,N)$ denote the maximum of $L(y)$ over all $y$ for which the exponent of any indeterminate in any monomial is at most $N$. We show that if $p = (N + 1^{o(q)} )$ and $q = (N + 1^{o(p)} )$, then $L(p,q,N) = \min \{ p,q\} \log N …

Critical Point Theory And The Number Of Solutions Of A Nonlinear Dirichlet Problem, Alfonso Castro, A. C. Lazer

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No abstract provided.

Optimal 2,3-Trees, Nicholas J. Pippenger, Raymond E. Miller, Arnold L. Rosenberg, Lawrence Snyder

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The 2,3-trees that are optimal in the sense of having minimal expected number of nodes visited per access are characterized in terms of their “profiles”. The characterization leads directly to a linear-time algorithm for constructing a K-key optimal 2,3-tree for a sorted list of K keys. A number of results are derived that demonstrate how different in structure these optimal 2,3-trees are from their “average” cousins.

A Semilinear Dirichlet Problem, Alfonso Castro

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Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:

Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω

u(x) = 0 ; x є ∂Ω

where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rⁿ⁺¹ → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

On The Application Of Coding Theory To Hashing, Nicholas Pippenger

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Quick proofs are given for the characterization (due to Schay, Raver, Hanan, and Palermo) of the collision distance of a linear hashing function and for a dual function (called the restriction distance), which relates to the accessibility of addresses by sets of keys and the uniform distribution of sets of keys over addresses.

A Summary Of Results On Order-Cauchy Completions Of Rings And Vector Lattices Of Continuous Functions, Melvin Henriksen

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This paper is a summary of joint research by F. Dashiell, A. Hager and the present author. Proofs are largely omitted. A complete version will appear in the Canadian Journal of Mathematics. It is devoted to a study of sequential order-Cauchy convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified for compact spaces X in …

Hammerstein Integral Equations With Indefinite Kernel, Alfonso Castro

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This paper deals with the problem of finding solutions of the Hammerstein integral equation. It is shown that this problem can be reduced to the study of the critical points of certain functional defined on L₂(Ω). Existence of a solution of the Hammersteln integral equation is proved. Some other related results of interest are obtained.

Generalized Connectors, Nicholas Pippenger

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An $n$-connector is an acyclic directed graph having $n$ inputs and $n$ outputs and satisfying the following condition: given any one-to-one correspondence between inputs and distinct outputs, there exists a set of vertex-disjoint paths that join each input to the corresponding output. It is known that the minimum possible number of edges in an $n$-connector lies between lower and upper bounds that are asymptotic to $3n\log _3 n$ and $6n\log _3 n$ respectively. A generalized $n$-connector satisfies the following stronger condition: given any one-to-many correspondence between inputs and disjoint sets of outputs, there exists a set of vertex-disjoint trees that …

Applying Mathematics Without A License, Melvin Henriksen

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A recent educational experience made me realize the extent to which the mathematical community has become fragmented and how this has served to inhibit communication both with others and ourselves.

An Algebraic Characterization Of The Freudenthal Compactification For A Class Of Rimcompact Spaces, Melvin Henriksen

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Throughout C(X) will denote the ring of all continuous real-valued functions on a Tychonoff space X, and C*(X) will denote the subring of bounded elements of C(X). The real line is denoted by R, and N denotes the (discrete) subspace of positive integers. A subset S of X such that the map f → f|_s is an epimorphism of C(X) (resp. C*(X)) is said to be C-embedded (resp. C*-embedded) in X. As is well-known, every f Є C*(X) has a unique continuous extension βf over its Stone-Čech compactification βX [GJ, Chapter 6]. That is, X is …

Some Sufficient Conditions For The Jacobson Radical Of A Commutative Ring With Identity To Contain A Prime Ideal, Melvin Henriksen

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Throughout, the word "ring" will abbreviate the phrase "commutative ring with identity element 1" unless the contrary is stated explicitly. An ideal I of a ring R is called pseudoprime if ab = 0 implies a or b is in I. This term was introduced by C. Kohls and L. Gillman who observed that if I contains a prime ideal, then I is pseudoprime, but, in general, the converse need not hold. In [9 p. 233], M. Larsen, W. Lewis, and R. Shores ask if whenever the Jacobson radical J(R) of an arithmetical ring is pseudoprime, it follows that J(R) …

Tychonoff Spaces That Have A Compactification With Countable Remainder, Melvin Henriksen

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In this paper, an attempt is made to characterize spaces that are Zippin or strongly Zippin. We succeed in this goal only in small part, but we do obtain a number of conditions on a space that are either necessary or sufficient for such compactifications to exist.

Multiplicatively Periodic Rings, Ted Chinburg, Melvin Henriksen

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We prove a generalization of Luh's result without using Dirichlet's Theorem. We then use Theorem 1 to show that the J-subrings of a periodic ring form a lattice with respect to join and intersection (the join of two subrings is the smallest subring containing both of them). After noting that every J-ring has nonzero characteristic, we determine for which positive integers n and m there exist J-rings of period n and characteristic m. This generalizes a problem posed by G. Wene.

On The Juror Utilization Problem, Melvin Henriksen, George H. Orland

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One of the authors, after hearing complaints night after night from his wife who was on jury duty, and finding it too much to bear, agreed to do something about the situation in return for peace. And so this study for the more efficient use of jurors was born. The aspects and magnitude of this problem have been discussed in many places. Basically we are concerned with achieving a better match between the number of jurors in a courthouse on a given day and those used in the judicial process.

On Graphs Which Contain All Small Trees, Ii, F. R. K. Chang, R. L. Graham, Nicholas Pippenger

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No abstract provided.

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

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In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Sums Of Kth Powers In The Ring Of Polynomials With Integer Coefficients, Ted Chinburg, Melvin Henriksen

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A working through of two theorems.

Suppose R is a ring with identity element and k is a positive integer. Let J(k, R) denote the subring of R generated by its kth powers. If Z denotes the ring of integers, then G(k, R) = {a ∈ Z: aR ⊂ J(k, R)} is an ideal of Z.

Fields Medals And Nevanlinna Prize Presented At Icm-94 In Zurich, Nicholas Pippenger, J. Lindenstrauss, L.C. Evans, A. Douady, A. Shalev

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The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner.

The Global Optimization Of Incoherent-Phase Signals, Henry A. Krieger, Charles Albert Schaffner

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No abstract provided.

Toeplitz Operators On Locally Compact Abelian Groups, Henry A. Krieger, C.A. Schaffner

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The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for $N = 2$, and the optimal signal sets are determined for $M = 2,3,4,6$ and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ${\bf s}_j $, $j = 1,2, \cdots ,M$, on the unit sphere $S_1 $ in $C^N $. If $W_{jk} $ is the half space determined by ${\bf s}_j …