Mathematics Commons^™

Letter From The Editor, Issue 2, 1988, Alvin White

Humanistic Mathematics Network Journal

No abstract provided.

Please Carry Your Coals To Where They Are Needed, Professor Stein, Melvin Henriksen

Humanistic Mathematics Network Journal

No abstract provided.

Mathematics And Philosophy*, D. Bushaw

Humanistic Mathematics Network Journal

No abstract provided.

The Basis For The Success Of The Potsdam Program, Rick Luttmann

Humanistic Mathematics Network Journal

No abstract provided.

Two Poems, Miroslav Holub

Humanistic Mathematics Network Journal

No abstract provided.

A Reply To The Question "Why Math?", Louis A. Talman

Humanistic Mathematics Network Journal

No abstract provided.

Stability Of Steady Cross-Waves: Theory And Experiment, Seth Lichter, Andrew J. Bernoff

All HMC Faculty Publications and Research

A bifurcation analysis is performed in the neighborhood of neutral stability for cross waves as a function of forcing, detuning, and viscous damping. A transition is seen from a subcritical to a supercritical bifurcation at a critical value of the detuning. The predicted hysteretic behavior is observed experimentally. A similarity scaling in the inviscid limit is also predicted. The experimentally observed bifurcation curves agree with this scaling.

A Semilinear Wave Equation With Nonmonotone Nonlinearity, Alfonso Castro, Sumalee Unsurangsie

All HMC Faculty Publications and Research

We prove that a semilinear wave equation in which the range of the derivative of the nonlinearity includes an eigenvalue of infinite multiplicity has a solution. The solution is obtained through an iteration scheme which provides a priori estimates.

Fault Tolerance In Networks Of Bounded Degree, Cynthia Dwork, David Peleg, Nicholas Pippenger, Eli Upfal

All HMC Faculty Publications and Research

Achieving processor cooperation in the presence of faults is a major problem in distributed systems. Popular paradigms such as Byzantine agreement have been studied principally in the context of a complete network. Indeed, Dolev [J. Algorithms, 3 (1982), pp. 14–30] and Hadzilacos [Issues of Fault Tolerance in Concurrent Computations, Ph.D. thesis, Harvard University, Cambridge, MA, 1984] have shown that Ω(t) connectivity is necessary if the requirement is that all nonfaulty processors decide unanimously, where t is the number of faults to be tolerated. We believe that in forseeable technologies the number of faults will grow with the size of the …

Wide-Sense Nonblocking Networks, Paul Feldman, Joel Friedman, Nicholas Pippenger

All HMC Faculty Publications and Research

A new method for constructing wide-sense nonblocking networks is presented. Application of this method yields (among other things) wide-sense nonblocking generalized connectors with n inputs and outputs and size O( n log n ), and with depth k and size O( n^{1 + 1/k} ( log n )^{1 - 1/k} ).

Nonnegative Solutions For A Class Of Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

In the recent past many results have been established on non-negative solutions to boundary value problems of the form

-u''(x) = λf(u(x)); 0 < x < 1,

u(0) = 0 = u(1)

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.

Reliable Computation By Formulas In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

It is shown that if formulas are used to compute Boolean functions in the presence of randomly occurring failures then: (1) there is a limit strictly less than 1/2 to the failure probability per gate that can be tolerated, and (2) formulas that tolerate failures must be deeper (and, therefore, compute more slowly) than those that do not. The heart of the proof is an information-theoretic argument that deals with computation and errors in very general terms. The strength of this argument is that it applies with equal ease no matter what types of gate are available. Its weaknesses is …

The Bisection Method: Which Root?, Arthur T. Benjamin

All HMC Faculty Publications and Research

No abstract provided in this article.

Quasi F-Covers Of Tychonoff Spaces, Melvin Henriksen, J. Vermeer, R. G. Woods

All HMC Faculty Publications and Research

A Tychonoff topological space is called a quasi F-space if each dense cozero-set of X is C*-embedded in X. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi F-cover" QF(X) of a compact space X as an inverse limit space, and identify the ring C(QF(X)) as the order-Cauchy completion of the ring C*(X). In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi F-cover of an arbitrary Tychonoff space. In this paper the minimal quasi F-cover of a compact space X is constructed as the space of ultrafilters …

Ordered Products Of Topological Groups, Melvin Henriksen, Ralph Kopperman, Frank A. Smith

All HMC Faculty Publications and Research

The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

Infinitely Many Radially Symmetric Solutions To A Superlinear Dirichlet Problem In A Ball, Alfonso Castro, Alexandra Kurepa

All HMC Faculty Publications and Research

In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

Teaching With A Humanist, David Meredith

Humanistic Mathematics Network Journal

This paper is a report about a team-taught course at San Francisco State University, and what the author learned having taught it over 11 years with faculty in the humanities. As part of an interdisciplinary curriculum called NEXA, which explored the boundaries between science and the humanities, the course revealed to the author several ways mathematics and humanities pedagogies can reciprocally learn from each other and emphasize similar goals.

Patterns Of Emotion Within Mathematics Problem-Solving, Frances A. Rosamond

Humanistic Mathematics Network Journal

There is often a divide between the experience of positive emotions toward math on the part of mathematical educators and negative emotions toward math on the part of students. This paper utilizes psychologist Richard Lazarus's work on the effects of positive emotions in order to highlight their benefits for mathematical pedagogy, to explain the author's experiment applying Lazarus's theory, and to suggest ways this application might support and foster positive emotions in students.

Gresham's Law: Algorithm Drives Out Thought, Sherman K. Stein

Humanistic Mathematics Network Journal

Gresham's law in economics states, "Bad money drives good money out of circulation." An application of this law in mathematical pedagogy states that "Algorithm drives out thought." While universities are ideally places where classes are meant to develop students' independence and critical thinking skills, often mathematics courses reflect this altered version of Gresham's law. This paper demonstrates the ways traditional mathematical pedagogy has held up Gresham's law and presents several suggestions for ways to change this approach to mathematical education to focus more on critical thinking without sacrificing the necessity of algorithm.

Letter From The Editor, Issue 1, 1987, Alvin White

Humanistic Mathematics Network Journal

No abstract provided.

Applied Mathematics As Social Contract, Philip J. Davis

Humanistic Mathematics Network Journal

The author takes the position that mathematical education must redefine its goals so as to create a citizenry with sufficient knowledge to provide social backpressure on future mathematizations. This can be accomplished by increasing the part of mathematical education that is devoted to the description and interpretation of the processes of mathematization and by allowing the technicalities of the formal operations within mathematics itself to be deemphasized or automated out by computer.

Appendix

Humanistic Mathematics Network Journal

No abstract provided.

Sorting And Selecting In Rounds, Nicholas Pippenger

All HMC Faculty Publications and Research

We present upper bounds for sorting and selecting the median in a fixed number of rounds. These bounds match the known lower bounds to within logarithmic factors. They also have the merit of being “explicit modulo expansion”; that is, probabilistic arguments are used only to obtain expanding graphs, and when explicit constructions for such graphs are found, explicit algorithms for sorting and selecting will follow. Using the best currently available explicit constructions for expanding graphs, we present the best currently known explicit algorithms for sorting and selecting in rounds.

The Complexity Of Computations By Networks, Nicholas Pippenger

All HMC Faculty Publications and Research

We survey the current state of knowledge concerning the computation of Boolean functions by networks, with particular emphasis on the addition and multiplication of binary numbers.

Rigid And Nonrigid Achirality, Erica Flapan

Pomona Faculty Publications and Research

In order to completely characterize a molecule it is useful to understand the symmetries of its molecular bond graph in 3-space. For many purposes the most important type of symmetry that a molecule can exhibit is mirror image symmetry. However, the question of whether a molecular graph is equivalent to its mirror image has different interpretations depending on what assumptions are made about the rigidity of the molecular structure. If there is a deformation of 3-space taking a molecular bond graph to its mirror image then the molecule is said to be topologically achiral. If a molecular graph can be …

A Prime Strongly Positive Amphicheiral Knot Which Is Not Slice, Erica Flapan

Pomona Faculty Publications and Research

We begin by giving several definitions. A knot K in S³ is said to be amphicheiral if there is an orientation-reversing diffeomorphism h of S³ which leaves K setwise invariant. Suppose, in addition, that K is given an orientation. ThenK is said to be positive amphicheiral if h preserves the orientation of K. If, in addition, the diffeomorphism h is an involution then K is strongly positive amphicheiral. Finally, we say a knot is slice if it bounds a smooth disc in B⁴. In this note we shall give a smooth example of …

Reliable Computation In The Presence Of Noise, Nicholas Pippenger

All HMC Faculty Publications and Research

This talk concerns computation by systems whose components exhibit noise (that is, errors committed at random according to certain probabilistic laws). If we aspire to construct a theory of computation in the presence of noise, we must possess at the outset a satisfactory theory of computation in the absence of noise.

A theory that has received considerable attention in this context is that of the computation of Boolean functions by networks (with perhaps the strongest competition coming from the theory of cellular automata; see [G] and [GR]). The theory of computation by networks associates with any two sets Q and …

On The Steepest Descent For Nonpotential Locally Lipschitzian Vector Fields, Alfonso Castro

All HMC Faculty Publications and Research

In [4] J. Neuberger gave various sufficient conditions for the solvability of nonvariational operator equations via a variant of the steepest descent method. In this note we give versions of these conditions under weaker assumptions on the smoothness of the operators.

Necessary And Sufficient Conditions For Certain Homology 3-Spheres To Have Smooth Zp-Actions, Erica Flapan

Pomona Faculty Publications and Research

We derive necessary and sufficient conditions for a broad class of homology 3-spheres, obtained as the gluing of two knot complements, to have Zp-actions.

We explore when a homology sphere, obtained as the gluing of two knot complements has any smooth periodic diffeomorphisms.

Infinitely Periodic Knots, Erica Flapan

Pomona Faculty Publications and Research

One aspect of the study of 3-manifolds is to determine what infinite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action?