All HMC Faculty Publications and ResearchCopyright (c) 2014 Claremont Colleges All rights reserved.
http://scholarship.claremont.edu/hmc_fac_pub
Recent documents in All HMC Faculty Publications and Researchen-usMon, 21 Apr 2014 17:27:03 PDT3600Aftermath: Every Math Major Should Take a Public-Speaking Course
http://scholarship.claremont.edu/hmc_fac_pub/1022
http://scholarship.claremont.edu/hmc_fac_pub/1022Thu, 17 Apr 2014 14:53:56 PDT
Rachel Levy argues that all mathematics majors should learn the art of public speaking.
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Rachel LevyThe Scientist–Reporter Collaboration: A Guide to Working with the Press
http://scholarship.claremont.edu/hmc_fac_pub/1021
http://scholarship.claremont.edu/hmc_fac_pub/1021Thu, 10 Apr 2014 15:39:48 PDT
Science, technology, engineering, and mathematics (STEM) to the public can be challenging. Often, the language that researchers use among themselves is technical and difficult for non-experts to decipher. But as you probably know, communicating your research to non-experts is becoming mandatory. In a direct sense, funding agencies often require outreach for grant fulfillment. There are indirect benefits as well: Conveying the joy of discovery and the relevance of scientific results builds scientific literacy among the public---which of course includes both students who will eventually do research of their own and people who elect the policy makers who allocate funding. How many people know that what scientists do can be fun and interesting?
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Rachel Levy et al.Engineering Design at Harvey Mudd College: Innovation Institutionalized, Lessons Learned
http://scholarship.claremont.edu/hmc_fac_pub/1020
http://scholarship.claremont.edu/hmc_fac_pub/1020Mon, 24 Feb 2014 16:59:11 PST
This paper outlines the development of Harvey Mudd College’s Engineering program, describes its resulting form, and articulates how some truly innovative ideas in engineering design education, now widely accepted as “best practices,” were developed and implemented. The paper also describes the lessons learned from Harvey Mudd College’s Engineering curriculum, as well as some of the efforts made to disseminate these lessons.
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Clive L. Dym et al.Buckling of Supported Arches Under Three Pressure Distributions
http://scholarship.claremont.edu/hmc_fac_pub/1019
http://scholarship.claremont.edu/hmc_fac_pub/1019Mon, 24 Feb 2014 16:59:09 PST
The literature on the buckling of arches is extensive. In one of the most authoritative papers, Chwalla and Kollbrunner [1]² obtained exact solutions for the inextensible buckling of arches subjected to hydrostatic, centrally directed, and constant directional ("dead") pressure distributions.
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Clive L. DymOn Approximations of the Buckling Stresses of Axially Compressed Cylinders
http://scholarship.claremont.edu/hmc_fac_pub/1018
http://scholarship.claremont.edu/hmc_fac_pub/1018Mon, 24 Feb 2014 16:59:08 PST
An uncoupled eighth-order equation for the buckling of axially compressed cylinders, within the framework of the Koiter-Budiansky shell equations, is derived for the first time herein. A solution of this equation is then used to generate some different types of approximations for the buckling stresses of a circular cylinder. Comparisons to previous solutions are given, as are some remarks on the nature of the approximations and their relationships to the limiting cases of infinitely short and infinitely long shells. Also given is a summary of the uncoupled system of equations of Sanders’ linear shell theory.
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Clive L. DymOn the Buckling of Cylinders in Axial Compression
http://scholarship.claremont.edu/hmc_fac_pub/1017
http://scholarship.claremont.edu/hmc_fac_pub/1017Mon, 24 Feb 2014 16:59:06 PST
The paper presents for the first time buckling results for cylinders in axial compression based upon shell equations advanced by Koiter and Budiansky. The numerical results are compared to those of Flügge and of Donnell, and limiting results for the infinitely long shell are also given. The principal conclusion is that the Flügge buckling results are in excellent agreement with those obtained from the highly refined Koiter-Budiansky equations.
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Clive L. DymConsistent Derivations of Spring Rates for Helical Springs
http://scholarship.claremont.edu/hmc_fac_pub/1016
http://scholarship.claremont.edu/hmc_fac_pub/1016Mon, 24 Feb 2014 16:59:05 PST
The spring rates of a coiled helical spring under an axial force and an axially directed torque are derived by a consistent application of Castigliano’s second theorem, and it is shown that the coupling between the two loads may not always be neglected. The spring rate of an extensional spring is derived for the first time through the use of the displacement based principle of minimum total potential energy. The present results are also compared with available derivations of and expressions for the stiffness of a coiled spring.
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Clive L. DymArtificial Intelligence and Geometric Reasoning in Manufacturing Technology
http://scholarship.claremont.edu/hmc_fac_pub/1015
http://scholarship.claremont.edu/hmc_fac_pub/1015Mon, 24 Feb 2014 16:59:04 PST
This article presents a brief review of the current literature on the applications of artificial intelligence (AI) technologies, and especially expert (knowledge-based) systems, to manufacturing. Emphasis is placed on geometric representation and reasoning in design as an aid to manufacturing. Also discussed are applications of AI to process planning and design, process control, assembly, and other phases of manufacturing.
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John R. Dixon et al.Extending Castigliano’s Theorems to Model the Behavior of Coupled Systems
http://scholarship.claremont.edu/hmc_fac_pub/1014
http://scholarship.claremont.edu/hmc_fac_pub/1014Mon, 24 Feb 2014 16:59:02 PST
Extensions of the Castigliano theorems are developed in the context of modeling the behavior of both discrete coupled linear systems and various coupled beams. It is shown that the minimization of the displacement of a parallel system determined from Castigliano’s second theorem can be used to formally define the apportionment of loads among the system elements, while the minimization of the load in a series system determined from Castigliano’s first theorem can be used to formally define the apportionment of the displacement among the system elements. These extensions provide a means for apportioning loads in coupled continuous systems, as will be shown for the cases of coupled cantilever Timoshenko beams supporting discrete and continuous loads.
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Clive L. DymPerturbation Solutions for the Buckling Problems of Axially Compressed Thin Cylindrical Shells of Infinite or Finite Length
http://scholarship.claremont.edu/hmc_fac_pub/1013
http://scholarship.claremont.edu/hmc_fac_pub/1013Mon, 24 Feb 2014 16:59:01 PST
A study is presented of the effect of initial deviations on the load carrying capacity of thin circular cylindrical shells under uniform axial compression. A perturbation expansion is used to reduce the nonlinear equations of von Karman and Donnell to an infinite set of linear equations, of which only the first few need be solved to obtain a reasonably accurate solution. The results for both infinite shells and shells of finite length indicate that a small imperfection can sharply reduce the maximum load that a thin-walled cylinder will sustain. In addition, for a particular set of boundary conditions, it is shown that the effect of the length of a finite shell is small as far as the load carrying capacity is concerned, but significant when the number of waves around the circumference has to be determined. A further result of the study is that axisymmetric initial deviations reduce the load carrying capacity only slightly more than deviations characterized by a product of trigonometric functions of the axial and circumferential coordinates if the wave lengths are properly chosen.
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Clive L. Dym et al.Sums of z-Ideals and Semiprime Ideals
http://scholarship.claremont.edu/hmc_fac_pub/1012
http://scholarship.claremont.edu/hmc_fac_pub/1012Mon, 24 Feb 2014 16:58:59 PST
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.
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Melvin Henriksen et al.Proof Without Words: The Pigeonhole Principle
http://scholarship.claremont.edu/hmc_fac_pub/1011
http://scholarship.claremont.edu/hmc_fac_pub/1011Mon, 17 Feb 2014 11:48:32 PST
This article contains one image and no text.
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Ran Libeskind-HadasAn Isomorphism Theorem for Real-Closed Fields
http://scholarship.claremont.edu/hmc_fac_pub/1010
http://scholarship.claremont.edu/hmc_fac_pub/1010Mon, 17 Feb 2014 11:48:30 PST
A classical theorem of Steinitz states that the characteristic of an algebraically closed fields, together with its absolute degree of transcendency, uniquely determine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the absolute transcendence degree are needed in order characterize a real-closed field.
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P. Erdös et al.On Difficulties in Embedding Lattice-Ordered Integral Domains In Lattice-Ordered Fields
http://scholarship.claremont.edu/hmc_fac_pub/1009
http://scholarship.claremont.edu/hmc_fac_pub/1009Mon, 17 Feb 2014 11:48:29 PST
A lattice ordered ring (or l-ring) A = A(+, •, v, ʌ) is an abstract algebra closed under four binary operations +, •, v, ʌ such that A(+, •) is a ring, A(v, ʌ) is a lattice, and if 0 is the identity element of A(+), then

a, b ≧ 0 imply that a + b ≧ 0 and a • b ≧ 0.

As usual, we say that a ≧ 0 if a v 0 = a, and a ≧ b if (a - b ) ≧ 0. Moreover, we let |a| = a v (- a).

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Melvin HenriksenSums of k-th powers in the Ring of Polynomials with Integer Coefficients
http://scholarship.claremont.edu/hmc_fac_pub/1008
http://scholarship.claremont.edu/hmc_fac_pub/1008Mon, 17 Feb 2014 11:48:26 PST
Suppose R is a ring with identity element 1 and k is a positive integer. Let H (k, R) denote the set of kth powers of elements of R, and let J(k, R) denote the additive subgroup of R generated by H(k, R). If Z denotes the ring of integers, then

G(k, R) = {a∊Z: aR ⊆ J(k, R)}

is an ideal of Z.

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Ted Chinburg '75 et al.Order-Cauchy Completions of Rings and Vector Lattices of Continuous Functions
http://scholarship.claremont.edu/hmc_fac_pub/1007
http://scholarship.claremont.edu/hmc_fac_pub/1007Mon, 17 Feb 2014 11:48:25 PST
This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is realted 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 [6]. This condition is that every dense cozero set S in X should be (^-embedded in X(that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the /^-spaces of [12]).
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F. Dashiell et al.F-Spaces and Substonean Spaces General Topology as a Tool in Functional Analysis
http://scholarship.claremont.edu/hmc_fac_pub/1006
http://scholarship.claremont.edu/hmc_fac_pub/1006Mon, 17 Feb 2014 11:48:23 PST
K. Grove and G. Pedersen define a substoneanspace to be a locally compact (Hausdorff) space in which disjoint σ-compact open subspaces have disjoint compact closures. It is routine to verity that a locally compact space X is substonean if and only if every continous f: S → K, where K is a compact, has a unique continuous extension f: Cℓ_xS→K whenever S is a σ-compact open subspace of X. Spaces with the property obtained by deleting "σ-compact" in the above are called stonean spaces and must b compact. If the only requirement is that open subspaces have open closures, such spaces are said to be extremally disconnected. Thus, a spacie is stonean if and only if is compact and extremally disconnected.
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Melvin Henriksen et al.Rings of Continuous Functions from an Algebraic Point of View
http://scholarship.claremont.edu/hmc_fac_pub/1005
http://scholarship.claremont.edu/hmc_fac_pub/1005Mon, 17 Feb 2014 11:48:21 PST
When George Martinez invited me, close to a year ago to give a talk on rings of continuous functions at a conference on ordered algebraic systems in an exotic location, I felt honored and elated. The accompanying feeling of euphoria stayed with me until I started to think seriously about what I could say to an audience containing people thoroughly familiar with the Gillman-Jerison text [GJ1] and many of the ensuing developments since 1960 — a lot of them due to mathematicians to whom I would be talking. I began to wonder if I would be doing the equivalent of carrying coals to Curaçao in August.
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Melvin HenriksenSemiprime f-Rings That Are Subdirect Products of Valuation Domains
http://scholarship.claremont.edu/hmc_fac_pub/1004
http://scholarship.claremont.edu/hmc_fac_pub/1004Mon, 17 Feb 2014 11:48:20 PST
Recall that an f-ring is a lattice-ordered ring in which a Λ b = 0 implies a Λ bc = a Λ cb = 0 whenever c ≥ 0. In [BKW], an f-ring is defined to be a lattice-ordered ring which is a subdirect product of totally ordered rings. These two definitions are equivalent if and only if the prime ideal theorem for Boolean Algebras is assumed; see [FH]. We regard these two definitions as equivalent henceforth. Our main concern is with f-rings that are semiprime; i.e., such that the intersection of the prime ideals is 0. A ring whose only nilpotent element is 0 is said to be reduced. (An f-ring is semiprime if and only if it is reduced; see [BKW, 8.5].) We will, however, maintain more generality when it does not take us too far afield. An ℓ-ideal I of an f-ring A is the kernel of a homomorphism of A into an f-ring. Equivalently, I is a ring ideal of A such that if a ∈ I, b ∈ A, and ∣b∣ < ∣a∣, then b ∈ I. A left ideal with this latter property is called a left ℓ-ideal, and a right ℓ-ideal is defined similarly. We let N(A) denote the set of nilpotent elements of the f-ring A.
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Melvin Henriksen et al.The Intermediate Value Theorem for Polynomials over Lattice-ordered Rings of Functions
http://scholarship.claremont.edu/hmc_fac_pub/1003
http://scholarship.claremont.edu/hmc_fac_pub/1003Mon, 17 Feb 2014 09:41:58 PST
The classical intermediate value theorem for polynomials with real coefficients is generalized to the case of polynomials with coefficients in a lattice-ordered ring that is a subdirect product of totally ordered rings. Several candidates for a generalization are investigated, and particular attention is paid to the case when the lattice-ordered ring is the algebra C(X) of continuous real-valued functions on a completely regular topological space X. For all but one of these generalizations, the intermediate value theorem holds only if X is an F-space in the sense of Gillman and Jerison. Surprisingly, for the most interesting of these generalizations, if X is compact, the intermediate value theorem holds only if X is an F-space and each component of X is an hereditarily indecomposable continuum. It is not known if there is an infinite compact connected space in which this version of the intermediate value theorem holds.
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Melvin Henriksen et al.