# Volume I, Issue 2

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Remarks: From the Editors | From the Dean | From the CMS Student Committee

Remarks: From the Editors | From the Dean | From the CMS Student Committee

# A Combinatorial Approach to Finding Dirichlet Generating Function Identities

—This paper explores an integer partitions-based method for obtaining Dirichlet generating function identities. In the process we shall generalize a previous result, obtain previously unknown formulae for the Möbius and Liouville Dirichlet generating functions, and obtain a formula on unit fractions.

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*January 23, 2012: Corrected mis-print of identities in the introduction.*Download the article | Cite this article (BibTeX) | This article cites (BibTeX)

# An Introduction to Calibration Estimators

—In survey sampling, the use of auxiliary information can greatly improve the precision of estimates of population total and/or means. In this
paper, we explain the basic theory and use of calibration estimators proposed by Deville and Särndal, which incorporate the use of
auxiliary data. Results of a simulation study conducted using real data from the 2008 Survey of Household Spending by Statistics Canada are presented,
comparing the performance of two calibration estimators against the Horvitz-Thompson estimator. Limitations of
calibration estimators and recent extensions made by other leading statisticians in this topic are also discussed.

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# Eigenvalues and Eigenfunctions of the Laplacian

—The problem of determining the eigenvalues and eigenvectors for linear operators acting on finite dimensional vector spaces is a problem known to every student of linear algebra. This problem has a wide range of applications and is one of the main tools for dealing with such linear operators. Some of the results concerning these eigenvalues and eigenvectors can be extended to infinite dimensional vector spaces. In this article we will consider the eigenvalue problem for the Laplace operator acting on the

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*L*^{2}space of functions on a bounded domain in**R**^{n}. We prove that the eigenfunctions form an orthonormal basis for this space of functions and that the eigenvalues of these functions grow without bound.Download the article | Cite this article (BibTeX) | This article cites (BibTeX)

# A Systematic Construction of Almost Integers

—Motivated by the search for "almost integers", we describe the algebraic integers known as Pisot numbers, and explain how they can be used to easily find irrational values that can be arbitrarily close to whole numbers. Some properties of the set of Pisot numbers are briefly discussed, as well as some applications of these numbers to other areas of mathematics.

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# Relativistic Fluid Dynamics

—Understanding the evolution of a many bodied system is still a very important problem in modern physics. Fluid mechanics provides a mechanism to determine the macroscopic motion of the system. These equations are additionally complicated when we consider a fluid moving in a curved spacetime. The following paper discusses the derivation of the relativistic equations of motion, uses numerical methods to provide solutions to these equations and describes how the curvature of spacetime is modified by the fluid.

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Volume I, Issue 2 of The Waterloo Mathematics Review is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Canada License.