Rose-Hulman Institute of Technology: Rose-Hulman Scholar
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    Heat Conduction

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    Further Generalizations of Happy Numbers

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    A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven

    Divisibility Probabilities for Products of Randomly Chosen Integers

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    We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test

    A Characterization of Complex-Valued Random Variables With Rotationally-Invariant Moments

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    A complex-valued random variable Z is rotationally invariant if the moments of Z are the same as the moments of W=e^{i*theta}Z. In the first part of the article, we characterize such random variables, in terms of vanishing unbalanced moments, moment and cumulant generating functions, and polar decomposition. In the second part, we consider random variables whose moments are not necessarily finite, but which have a density. In this setting, we prove two characterizations that are equivalent to rotational invariance, one involving polar decomposition, and the other involving entropy. If a random variable has both a density and moments which determine it, all of these characterizations are equivalent

    Number of Regions Created by Random Chords in the Circle

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    In this paper we discuss the number of regions in a unit circle after drawing n i.i.d. random chords in the circle according to a particular family of distribution. We find that as n goes to infinity, the distribution of the number of regions, properly shifted and scaled, converges to the standard normal distribution and the error can be bounded by Stein\u27s method for proving Central Limit Theorem

    Elliptic triangles which are congruent to their polar triangles

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    We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines

    Vitreoscilla Globin Promoter Cloning and Testing in \u3ci\u3eEscherichia coli\u3c/i\u3e

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    On Solutions of First Order PDE with Two-Dimensional Dirac Delta Forcing Terms

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    We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided

    2023 Rose-Hulman Institute of Technology : ONE HUNDRED AND FORTY-FIFTH COMMENCEMENT MAY 27, 2023

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    https://scholar.rose-hulman.edu/commencementprograms/1043/thumbnail.jp

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    Rose-Hulman Institute of Technology: Rose-Hulman Scholar is based in United States
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