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-usFri, 18 Jul 2014 17:38:05 PDT3600Methods for Estimating Peak Physiological Performance and Correlating Performance Measures
http://scholarship.claremont.edu/hmc_fac_pub/1052
http://scholarship.claremont.edu/hmc_fac_pub/1052Thu, 10 Jul 2014 15:20:59 PDT
Estimates of animal performance often use the maximum of a small number of laboratory trials, a method which has several statistical disadvantages. Sample maxima always underestimate the true maximum performance, and the degree of the bias depends on sample size. Here, we suggest an alternative approach that involves estimating a specific performance quantile (e.g., the 0.90 quantile). We use the information on within-individual variation in performance to obtain a sampling distribution for the residual performance measures; we use this distribution to estimate a desired performance quantile for each individual. We illustrate our approach using simulations and with data on sprint speed in lizards. The quantile method has several advantages over the sample maximum: it reduces or eliminates bias, it uses all of the data from each individual, and its accuracy is independent of sample size. Additionally, we address the estimation of correlations between two different performance measures, such as sample maxima, quantiles, or means. In particular, because of sampling variability, we propose that the correlation of sample means does a better job estimating the correlation of population maxima than the estimator which is the correlation of sample maxima.
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Austen W. Head et al.Tiling Proofs of Recent Sum Identities Involving Pell Numbers
http://scholarship.claremont.edu/hmc_fac_pub/1051
http://scholarship.claremont.edu/hmc_fac_pub/1051Mon, 30 Jun 2014 17:04:07 PDT
In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.
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Arthur T. Benjamin et al.Combinatorial Interpretations of Spanning Tree Identities
http://scholarship.claremont.edu/hmc_fac_pub/1050
http://scholarship.claremont.edu/hmc_fac_pub/1050Mon, 30 Jun 2014 15:10:24 PDT
We present a combinatorial proof that the wheel graph Wn has L2n − 2 spanning trees, where Ln is the nth Lucas number, and that the number of spanning trees of a related graph is a Fibonacci number. Our proofs avoid the use of induction, determinants, or the matrix tree theorem.
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Arthur T. Benjamin et al.Fibonacci Determinants — A Combinatorial Approach
http://scholarship.claremont.edu/hmc_fac_pub/1049
http://scholarship.claremont.edu/hmc_fac_pub/1049Mon, 30 Jun 2014 15:10:19 PDT
In this paper, we provide combinatorial interpretations for some determinantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to generalize and discover new ones.
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Arthur T. Benjamin et al.Double Birthday Magic Square
http://scholarship.claremont.edu/hmc_fac_pub/1048
http://scholarship.claremont.edu/hmc_fac_pub/1048Mon, 30 Jun 2014 15:10:14 PDTArthur T. BenjaminThe Probability of Relatively Prime Polynomials
http://scholarship.claremont.edu/hmc_fac_pub/1047
http://scholarship.claremont.edu/hmc_fac_pub/1047Mon, 30 Jun 2014 15:10:09 PDT
Euclid does integers
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Arthur T. Benjamin et al.Optimal Token Allocations in Solitaire Knock 'M Down
http://scholarship.claremont.edu/hmc_fac_pub/1046
http://scholarship.claremont.edu/hmc_fac_pub/1046Fri, 27 Jun 2014 17:44:00 PDT
In the game Knock ’m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a ﬁxed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution. MR Subject Classiﬁcations: primary: 91A60
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Arthur Benjamin et al.Estimating Winning Probabilities in Backgammon Races
http://scholarship.claremont.edu/hmc_fac_pub/1045
http://scholarship.claremont.edu/hmc_fac_pub/1045Fri, 27 Jun 2014 17:15:48 PDT
In modern backgammon, it is advantageous to know the chances each player has of winning, and to be able to compute the chances without the aid of calculators or pencil and paper. A simple model of backgammon is used to approximate those chances, and a readily computable and sufficiently accurate approximation of that is developed. From there, the model is compared to simulated backgammon games, and the previous approximation is modified to fit the real data.
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Andrew M. Ross et al.Faster Circuits and Shorter Formulas for Multiple Addition, Multiplication and Symmetric Boolean Functions
http://scholarship.claremont.edu/hmc_fac_pub/1044
http://scholarship.claremont.edu/hmc_fac_pub/1044Thu, 26 Jun 2014 17:09:49 PDT
A general theory is developed for constructing the shallowest possible circuits and the shortest possible formulas for the carry-save addition of n numbers using any given basic addition unit. More precisely, it is shown that if BA is a basic addition unit with occurrence matrix N, then the shortest multiple carry-save addition formulas that could be obtained by composing BA units are of size n1p+o(1)/, where p is the unique real number for which the Lp norm of the matrix N equals 1. An analogous result connects the delay matrix M of the basic addition unit BA and the minimal q such that multiple carry-save addition circuits of depth (q+o(1)) log n could be constructed by combining BA units. On the basis of these optimal constructions of multiple carry-save adders, the shallowest known multiplication circuits are constructed
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Michael Paterson et al.On Determinism Versus Non-Determinism And Related Problems
http://scholarship.claremont.edu/hmc_fac_pub/1043
http://scholarship.claremont.edu/hmc_fac_pub/1043Thu, 26 Jun 2014 17:01:44 PDT
We show that, for multi-tape Turing machines, non-deterministic linear time is more deterministic Turing machines (that receive their input on their work tape) require time Q(n2) to powerful than deterministic linear time. We also recognize non-palindromes of length n (it is easy to discuss the prospects for extending this result to see that time O(n log n) is. sufficient for a more general Turing machines. non-deterministic machine). 1. Introduction
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Wolfgang J. Paul et al.On Graphs Which Contain All Small Trees, II
http://scholarship.claremont.edu/hmc_fac_pub/1042
http://scholarship.claremont.edu/hmc_fac_pub/1042Thu, 26 Jun 2014 17:01:39 PDTF. R. K. Chang et al.Asymptotic Behavior of the Chromatic Index for Hypergraphs
http://scholarship.claremont.edu/hmc_fac_pub/1041
http://scholarship.claremont.edu/hmc_fac_pub/1041Wed, 25 Jun 2014 15:01:41 PDT
We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rödl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to , resolving a long-standing conjecture.
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Nicholas Pippenger et al.Polynomial Hash Functions Are Reliable
http://scholarship.claremont.edu/hmc_fac_pub/1040
http://scholarship.claremont.edu/hmc_fac_pub/1040Wed, 25 Jun 2014 15:01:36 PDT
Polynomial hash functions are well studied and widely used in various applications. They have gained popularity because of certain performances they exhibit. It has been shown that even linear hash functions are expected to have such performances. However, quite often we would like the hash functions to be reliable, meaning that they perform well with high probability; for some certain important properties even higher degree polynomials were not known to be reliable. We show that for certain important properties linear hash functions are not reliable. We give indication that quadratic hash functions might not be reliable. On the positive side, we prove that cubic hash functions are reliable. In a more general setting, we show that higher degree of the polynomial hash functions translates into higher reliability. We also introduce a new class of hash functions, which enables to reduce the universe size in an efficient and simple manner. The reliability results and the new class of hash functions are used for some fundamental applications: improved and simplified reliable algorithms for perfect hash functions and real-time dictionaries, which use significantly less random bits, and tighter upper bound for the program size of perfect hash functions.
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M. Dietzfelbinger et al.Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines
http://scholarship.claremont.edu/hmc_fac_pub/1039
http://scholarship.claremont.edu/hmc_fac_pub/1039Tue, 24 Jun 2014 16:53:27 PDT
It is shown that a probabilistic Turing acceptor or transducer running within space bound Scan be simulated by a time S^{2} parallel machine and therefore by a space S^{2} deterministic machine. (Previous simulations ran in space S^{6}.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.
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Allan Borodin et al.A Mathematical Model of Immune Response to Tumor Invasion
http://scholarship.claremont.edu/hmc_fac_pub/1038
http://scholarship.claremont.edu/hmc_fac_pub/1038Mon, 23 Jun 2014 16:56:12 PDT
Recent experimental studies by Diefenbach et al. [1] have brought to light new information about how the immune system of the mouse responds to the presence of a tumor. In the Diefenbach studies, tumor cells are modiﬁed to express higher levels of immune stimulating NKG2D ligands. Experimental results show that sufﬁciently high levels of ligand expression create a signiﬁcant barrier to tumor establishment in the mouse. Additionally, ligand transduced tumor cells stimulate protective immunity to tumor rechallenge. Based on the results of the Diefenbach experiments, we have developed a mathematical model of tumor growth to address some of the questions that arise regarding the mechanisms involved in the immune response to a tumor challenge. The model focuses on the interaction of the NK and CD8+ T cells with various tumor cell lines using a system of differential equations. We propose new forms for the tumor-immune competition terms, and validate these forms through comparison with the experimental data of [1].
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Lisette de Pillis et al.Project-Based ORMS Education
http://scholarship.claremont.edu/hmc_fac_pub/1037
http://scholarship.claremont.edu/hmc_fac_pub/1037Mon, 23 Jun 2014 16:12:19 PDT
Operations research and management science (ORMS) is not simply the rote application of formulas and algorithms but also an involved process of modeling ill-posed real-world problems. Students must therefore learn both the mathematical foundations of the field as well as the craft of the practice of ORMS. In this article, the author surveys different approaches to project-based education, which attempts to teach the practice of ORMS. The author also provides best practices for the specific example of student field projects with clients.
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Susan E. MartonosiUndergraduate Operations Research Prize
http://scholarship.claremont.edu/hmc_fac_pub/1036
http://scholarship.claremont.edu/hmc_fac_pub/1036Mon, 23 Jun 2014 16:01:35 PDTSusan E. Martonosi et al.A Globalization of the Implicit Function Theorem with Applications to Nonlinear Elliptic Equations
http://scholarship.claremont.edu/hmc_fac_pub/1035
http://scholarship.claremont.edu/hmc_fac_pub/1035Mon, 23 Jun 2014 14:43:37 PDT
In this paper we introduce some basic concepts from nonlinear analysis through a discussion of a well-known degree theoretic globalization of the Implicit Function Theorem. Applications to the Liouville-Gelfand problem and a related problem for k-Hessian operators are considered.
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Jon T. JacobsenGlobal Bifurcation for Monge-Ampère Operators
http://scholarship.claremont.edu/hmc_fac_pub/1034
http://scholarship.claremont.edu/hmc_fac_pub/1034Mon, 23 Jun 2014 12:22:25 PDT
In this paper I wish to describe some recent results concerning bifurcation for nonlinear equations of Monge-Ampère type. Many of the results below may be found, with more detailed proofs, in [8]. I would like to thank Professor Klaus Schmitt under whose guidance this work was carried out.
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Jon T. JacobsenGlobal Bifurcation Problems Associated With k-Hessian Operators
http://scholarship.claremont.edu/hmc_fac_pub/1033
http://scholarship.claremont.edu/hmc_fac_pub/1033Mon, 23 Jun 2014 12:08:53 PDT
In this paper we study global bifurcation phenomena for a class of nonlinear elliptic equations governed by the h-Hessian operator. The bifurcation phenomena considered provide new methods for establishing existence results concerning fully nonlinear elliptic equations. Applications to the theory of critical exponents and the geometry of k-convex functions are considered. In addition, a related problem of Liouville–Gelfand type is analyzed.
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Jon T. Jacobsen