Mathematics Commons^™

Optimal Token Allocations In Solitaire Knock 'M Down, Arthur Benjamin, Matthew T. Fluet, Mark L. Huber

All HMC Faculty Publications and Research

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: …

Enumeration Of Equicolourable Trees, Nicholas Pippenger

All HMC Faculty Publications and Research

A tree, being a connected acyclic graph, can be bicolored in two ways, which differ from each other by exchange of the colors. We shall say that a tree is equicolorable if these bicolorings assign the two colors to equal numbers of vertices. Labelled equicolored trees have been enumerated several times in the literature, and from this result it is easy to enumerate labelled equicolorable trees. The result is that the probability that a randomly chosen n-vertex labelled tree is equicolorable is asymptotically just twice the probability that its vertices would be equicolored if they were assigned colors by …

A Mathematical Tumor Model With Immune Resistance And Drug Therapy: An Optimal Control Approach, Lisette G. De Pillis, Ami E. Radunskaya

All HMC Faculty Publications and Research

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy …

Existence Of Many Positive Nonradial Solutions For A Superlinear Dirichlet Problem On Thin Annuli, Alfonso Castro, Marcel B. Finan

All HMC Faculty Publications and Research

We study the existence of many positive nonradial solutions of a superlinear Dirichlet problem in an annulus in R^N. Our strategy consists of finding the minimizer of the energy functional restricted to the Nehrai manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a desired solution. Then we show that the minimal energy solutions in different symmetric classes have mutually different energies. The same approach has been used to prove the existence of many sign-changing nonradial solutions (see [5]).

The School/Home Communication Project: A Study Of The Effect Of More Frequent Grade Reporting On The Achievement Of High School Mathematics Students, Robert Dean Rogers

Humanistic Mathematics Network Journal

No abstract provided.

Ode To The Square Root: A Historical Journey, Dorothy W. Goldberg

Humanistic Mathematics Network Journal

The author gives a personal history of experiences in finding the square root of a number by the “do it thus” method—from algorithm to table to calculator. Why each procedure works is elucidated, making liberal use of the history of mathematics.

Book Review: Experiencing School Mathematics By Jo Boaler, Stephen Sproule

Humanistic Mathematics Network Journal

No abstract provided.

Qualitative Quantities, Susan Parman

Humanistic Mathematics Network Journal

No abstract provided.

Skolem’S Paradox And Contradictory Popular Songs, Maurice Machover

Humanistic Mathematics Network Journal

No abstract provided.

Number, Infinity And Truth: Reflections On The Spiritual In Mathematics, James V. Rauff

Humanistic Mathematics Network Journal

No abstract provided.

Will You Still Be Teaching In The Twenty-First Century?, Theodore Panitz

Humanistic Mathematics Network Journal

No abstract provided.

The Story Of A Service-Learning Project: Mathematics In The Park, Joyce O'Halloran

Humanistic Mathematics Network Journal

No abstract provided.

Training Elementary Teachers For The New Millennium, Dixie Metheny, David Davison

Humanistic Mathematics Network Journal

No abstract provided.

True Prime, Pat Mower

Humanistic Mathematics Network Journal

No abstract provided.

Book Review: Knowing And Teaching Elementary Mathematics By Liping Ma, Roger Howe

Humanistic Mathematics Network Journal

No abstract provided.

Senior Seminar: A Capstone Course In The Computer And Mathematical Sciences, Ken Oberhoff, Ron Barnes

Humanistic Mathematics Network Journal

This paper describes the evolution of a course developed to tie together many strands of activity encountered by students in the computer and mathematical sciences (CMS). The senior level course is required of all majors in our computer science, applied mathematics and statistics undergraduate degree programs. One of the primary purposes of the course is to refine writing and presentation skills needed for those who will later pursue individual research projects. Writing projects are organized around the theme of “Ethical Decision Making in the Computer and Mathematical Sciences”. Numerous case studies are investigated. Additional topics in the course include designing …

Weizmann Day’S “Math Night” Brings Parents To Their Knees

Humanistic Mathematics Network Journal

No abstract provided.

Book Review: Geometry From Africa: Mathematical And Educational Explorations By Paulus Gerdes, Claudia Zaslavsky

Humanistic Mathematics Network Journal

No abstract provided.

Optimal Token Allocations In Solitaire Knock'm Down, Arthur T. Benjamin, Mark L. Huber, Matthew T. Fluet '99

All HMC Faculty Publications and Research

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.

Phased Tilings And Generalized Fibonacci Identities, Arthur T. Benjamin, Jennifer J. Quinn, Francis E. Su

All HMC Faculty Publications and Research

Fibonacci numbers arise in the solution of many combinatorial problems. They count the number of binary sequences with no consecutive zeros, the number of sequences of 1's and 2's which sum to a given number, and the number of independent sets of a path graph. Similar interpretations exist for Lucas numbers. Using these interpretations, it is possible to provide combinatorial proofs that shed light on many interesting Fibonacci and Lucas identities (see [1], [3]). In this paper we extend the combinatorial approach to understand relationships among generalized Fibonacci numbers.

Given G₀ and G₁ a generalized Fibonacci sequence G …

Random Approaches To Fibonacci Identities, Arthur T. Benjamin, Gregory M. Levin, Karl Mahlburg '01, Jennifer J. Quinn

All HMC Faculty Publications and Research

No abstract provided in this article.

Evolution Of Positive Solution Curves In Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We study the existence, multiplicity, and stability of positive solutions to -u''(x) = λf(u(x)) for x є (-1,1), u(-1) = 0 = u(1), where λ > 0 and f:[0,∞)→R is monotonically increasing and concave with f(0) < 0 (semipositone). We establish that f should be appropriately concave (by establishing conditions on f) to allow multiple positive solutions. For any λ > 0, we obtain the exact number of positive solutions as a function of f(t)/t. We follow several families of nonlinearities f for which f(∞) := lim t→∞ f(t) > 0 and study how the positive solution curves to the above problem evolve. Also, we give examples where our results apply. This work extends the work of A. Castro and R. Shivaji (1988, Proc. Roy. Soc. Edinburgh …

What We Say, What Our Students Hear: A Case For Active Listening, Dorothy Buerk

Humanistic Mathematics Network Journal

No abstract provided.

The Word Problem And The Child, Kenneth J. Preskenis

Humanistic Mathematics Network Journal

No abstract provided.

A Tribute To Ramanujan, Mahesh Dube

Humanistic Mathematics Network Journal

No abstract provided.

Math Induction, Lawrence Mark Lesser

Humanistic Mathematics Network Journal

No abstract provided.

Written To Me Upon Getting A B In Linear Algebra, Sandra Z. Keith

Humanistic Mathematics Network Journal

No abstract provided.

The Need For Interviews In The Mathematics Classroom, Emam Hoosain

Humanistic Mathematics Network Journal

No abstract provided.

A Glorious Constant, Ze'ev Barel

Humanistic Mathematics Network Journal

No abstract provided.

Using Environmental News To Help Teach Mathematics, Barry Schiller

Humanistic Mathematics Network Journal

No abstract provided.