Recent zbMATH articles in MSC 97https://www.zbmath.org/atom/cc/972021-05-28T16:06:00+00:00WerkzeugThe last two days in elementary linear algebra.https://www.zbmath.org/1459.970082021-05-28T16:06:00+00:00"Muench, Donald L."https://www.zbmath.org/authors/?q=ai:muench.donald-l(no abstract)Unimodular roots of quadrinomials.https://www.zbmath.org/1459.970012021-05-28T16:06:00+00:00"Brilleslyper, Michael A."https://www.zbmath.org/authors/?q=ai:brilleslyper.michael-aSummary: We show finding all the unimodular roots (roots of modulus 1) of the quadrinomial \(p(z)=zn+zk+zj-1\), with integer exponents \(n>k>j\geq 1\), requires checking a finite number of conditions on a set of integer equations.Proof without words: magic of tangential polygons.https://www.zbmath.org/1459.970062021-05-28T16:06:00+00:00"Laudano, Francesco"https://www.zbmath.org/authors/?q=ai:laudano.francescoSummary: Starting from the two tangent theorem, we show an elegant properties of tangential polygons with an even number of sides that generalize the well-known \textit{Pitot} theorem for quadrilaterals. Moreover, we extend it to the tangential polygons with an odd number of sides, including triangles.Fold-over regions in nonlinear first order PDEs.https://www.zbmath.org/1459.970112021-05-28T16:06:00+00:00"Maritz, Milton F."https://www.zbmath.org/authors/?q=ai:maritz.milton-f"Cloete, Marèt"https://www.zbmath.org/authors/?q=ai:cloete.maret(no abstract)Idempotent factorizations in the cryptography classroom.https://www.zbmath.org/1459.970192021-05-28T16:06:00+00:00"Fagin, Barry S."https://www.zbmath.org/authors/?q=ai:fagin.barry-sSummary: When students first learn RSA cryptography and probabilistic primality testing, students may ask about the use of composite numbers. What happens if RSA is employed with \(p\) and/or \(q\) that are not prime? Although it will not work correctly most of the time, there are composite integers that ``fool'' RSA and produce correct \((e,d)\) encryption and decryption keys. We explain how to find and construct such \(\overline{p},\overline{q}\) and use them in the classroom. This paper expands on work first presented in [\textit{B. Fagin}, ``Idempotent factorizations: a new addition to the cryptography classroom'', in: Proceedings of the 2019 ACM conference on innovation and technology in computer science education (ITiCSE '19), Aberdeen, GB-SCT, July 15--17 2019. New York, USA: Association for Computing Machinery. p. 303 (2019; \url{doi:10.1145/3304221.3325557})].The dynamics of the greenhouse effect.https://www.zbmath.org/1459.970172021-05-28T16:06:00+00:00"Kiers, Claire"https://www.zbmath.org/authors/?q=ai:kiers.claireSummary: The greenhouse effect is crucial for regulating temperature on the Earth. To explore how it works, we introduce some simple dynamical systems layer models of radiation transfer. We begin by finding solutions to a 1-dimensional bare rock model to see what would happen if the Earth had no atmosphere. Then, we analyze solutions to a 2-dimensional model to see how the temperatures of the Earth and atmosphere would affect each other over time if the atmosphere were one homogeneous layer. In particular, we show using forward invariant boxes that if there is one equilibrium it is globally attracting. Finally, we describe an \(n\)-layer model in which the atmosphere is divided up into horizontal layers. We compare equilibrium values of the models for different numbers of layers to see how adding more layers to the atmosphere affects the temperature of the Earth.Chance encounters with large polynomials.https://www.zbmath.org/1459.970142021-05-28T16:06:00+00:00"Jones, Brian D."https://www.zbmath.org/authors/?q=ai:jones.brian-dSummary: An activity on the first day of probability class has the students plotting the co-efficients that result after a small polynomial with unit coefficients is raised to a large power. Surprisingly, this plot reveals the shape of a normal density curve. Presented here is a mathematical explanation of the normal curve's appearance and how it can be used to estimate the coefficients. The argument brings together important connections among algebra, combinatorics, and probability -- connections that draw from nearly every topic in a first probability course. Probability instructors and students will find the activity and these connections useful as material for class discussions and assignments, and as a path to deeper understanding.Counting connected sets of squares.https://www.zbmath.org/1459.970132021-05-28T16:06:00+00:00"Wagon, Stan"https://www.zbmath.org/authors/?q=ai:wagon.stan(no abstract)The proportion of comets in the card game SET.https://www.zbmath.org/1459.000082021-05-28T16:06:00+00:00"May, Dan"https://www.zbmath.org/authors/?q=ai:may.dan"Swenson, Dan"https://www.zbmath.org/authors/?q=ai:swenson.danSummary: We use recurrence relations to answer a question that was posed in this \textit{Journal}: in the game of SET, what is the probability that nine randomly chosen cards form what is called a ``comet''? We then generalize our results to count subsets of an arbitrary finite vector space which sum to the zero vector.The most beautiful equation ever. From basic mathematical knowledge to Euler's identity.https://www.zbmath.org/1459.260022021-05-28T16:06:00+00:00"Rinkens, Hans-Dieter"https://www.zbmath.org/authors/?q=ai:rinkens.hans-dieter.1"Krüger, Katja"https://www.zbmath.org/authors/?q=ai:kruger.katjaThe equation in question is $e^{i\pi} + 1 = 0$. In this formula you find the circle constant $\pi$, the imaginary unity $i$, the Euler number $e$, as well as $0$ and $1$. The first three chapters of the book are devoted to $\pi$, the imaginary unity $i$, and the base $e$: their history, definitions, their applications, approximations, extensions and so on. In Chapter 4, the reader finds further aspects concerning complex $e$-functions and the extension of powers, formulas including $
\pi$, as well as $e$ and $\pi$ in the area of the real numbers. An appendix contains foundations of elementary mathematics, in geometry, arithmetic and algebra and analysis. This is a wonderful book with many illustrations and pictures, the print in different colours, easy to understand and very interesting. Both authors are specialists in the didactics of mathematics. This book is primarily intended for teachers and students who want to become teacher, but also for all people who are interested in mathematics and history of mathematics. The book can be highly recommended.
Reviewer: Karin Reich (Berlin)Fluid-structure interaction for the classroom: interpolation, hearts, and swimming!https://www.zbmath.org/1459.650172021-05-28T16:06:00+00:00"Battista, Nicholas A."https://www.zbmath.org/authors/?q=ai:battista.nicholas-aRotor coordinates, vector trigonometry and quadrilaterals, with applications to the four bar linkage.https://www.zbmath.org/1459.510122021-05-28T16:06:00+00:00"Wildberger, Norman J."https://www.zbmath.org/authors/?q=ai:wildberger.norman-johnThe author replaces the standard polar coordinate representation \((r, \theta)\) of vectors in the plane by what he calls rotor coordinates \(|r, h \rangle\). The new quantity \(h\) introduced herein is called \textit{half-slope} and can for instance be defined as \[h = \frac{r-x}{y}\] for any vector \(\mathbf{v}\) whose Cartesian coordinates are \(x\) and \(y\not= 0\). (In fact, one could express \(h\) by means of \(\theta\) as \(h = \tan(\frac{\theta}{2})\) (classical half-angle substitution) which is not mentioned in the paper as one of the author's primary concerns is to avoid trigonometric expressions.)
Subsequently, the author derives various formulas for this half-slope, for instance, the adapted cosine and sine laws and a formula for the composition of two rotations about the origin. He also presents some vector trigonometry equations for triangles and quadrilaterals in terms of the half-slope concept.
Finally, four bar linkages are studied in this new light. The relation between drive angle and output angle of such a mechanism can now be re-formulated as a relation between drive and output half-slope. Moreover, an expression for the coupler half-slope in terms of the drive and the output half-slopes is derived and parameterizations of coupler curves are presented.
Reviewer: Anton Gfrerrer (Graz)Half row sums in Pascal's triangle.https://www.zbmath.org/1459.970122021-05-28T16:06:00+00:00"Plaza, Ángel"https://www.zbmath.org/authors/?q=ai:plaza.angelSummary: We demonstrate visually that the sum of every other term in the \((n + 1)\)st row of Pascal's triangle is equal to the sum of all the terms in the previous row.An identity from Viète.https://www.zbmath.org/1459.970072021-05-28T16:06:00+00:00"Wu, Rex H."https://www.zbmath.org/authors/?q=ai:wu.rex-hSummary: We provide a visual proof for a sine identity used by Viète in the computation of his sine table. The figure also proves the counterpart for the cosine function.Yet another proof without words of the Pythagorean theorem.https://www.zbmath.org/1459.970052021-05-28T16:06:00+00:00"Cantone, Domenico"https://www.zbmath.org/authors/?q=ai:cantone.domenicoSummary: We present a visual, dissection proof for the Pythagorean theorem.True grit in real analysis.https://www.zbmath.org/1459.970092021-05-28T16:06:00+00:00"Bressoud, David M."https://www.zbmath.org/authors/?q=ai:bressoud.david-mSummary: For far too many students, Real Analysis is a dreaded course that proceeds from unmotivated definitions to formal and impenetrable theorems with little sense of why the course unfolds as it does. This article describes the experience of a radically different approach that drew on the history of mathematics to confront students with the uncertainties and confusion that real mathematicians encounter as they explore new mathematical territory. The argument is made that such an historical approach enables students to appreciate the precise definitions that have evolved and the power of the theorems that are built upon them.Squared primes modulo 24.https://www.zbmath.org/1459.970042021-05-28T16:06:00+00:00"Nelsen, Roger B."https://www.zbmath.org/authors/?q=ai:nelsen.roger-bSummary: We show visually that the square of a prime greater than or equal to 5 is congruent to 1 modulo 24.Exercises and problems in linear algebra.https://www.zbmath.org/1459.150012021-05-28T16:06:00+00:00"Erdman, John M."https://www.zbmath.org/authors/?q=ai:erdman.john-mPublisher's description: This book contains an extensive collection of exercises and problems that address relevant topics in linear algebra. Topics that the author finds missing or inadequately covered in most existing books are also included. The exercises will be both interesting and helpful to an average student. Some are fairly routine calculations, while others require serious thought.
The format of the questions makes them suitable for teachers to use in quizzes and assigned homework. Some of the problems may provide excellent topics for presentation and discussions. Furthermore, answers are given for all odd-numbered exercises which will be extremely useful for self-directed learners. In each chapter, there is a short background section which includes important definitions and statements of theorems to provide context for the following exercises and problems.Proof without words: sums of polygonal numbers.https://www.zbmath.org/1459.970022021-05-28T16:06:00+00:00"Caglayan, Günhan"https://www.zbmath.org/authors/?q=ai:caglayan.gunhan(no abstract)On the sum of powers of consecutive integers.https://www.zbmath.org/1459.970032021-05-28T16:06:00+00:00"Ho, Chungwu"https://www.zbmath.org/authors/?q=ai:ho.chungwu"Mellblom, Gregory"https://www.zbmath.org/authors/?q=ai:mellblom.gregory"Frodyma, Marc"https://www.zbmath.org/authors/?q=ai:frodyma.marcSummary: It is well known that the sum of cubes of three consecutive integers is always divisible by 9. Is this an isolated incident? We showed that this can be generalized to the fact that the sum of the \(mk\)th power of anyk consecutive integers is always divisible by the square of \(k\) if the exponent \(mk\) is an odd integer. If \(m\) is even, the situation is more complicated: the sum of the \(mk\)th powers of \(k\) consecutive integers is divisible by the square of \(k\) for some even integers \(m\) and not divisible by the square of \(k\) for other even integers. For the case when \(k\) is an odd prime, we have a complete characterization on the integer \(m\) for which the sum of the \(mk\)th power of any \(k\) consecutive integers is divisible by the square of \(k\).A tour of discrete probability guided by a problem in genomics.https://www.zbmath.org/1459.970162021-05-28T16:06:00+00:00"Hanin, Leonid"https://www.zbmath.org/authors/?q=ai:hanin.leonid-gSummary: The classic binomial, geometric, negative binomial, and hypergeometric distributions differ by their mathematical form and the nature of underlying random experiments. In this article, we discuss a unifying framework for these distributions that comes from an unlikely source: computational genomics. One important problem in genomics is to find all protein-coding genes. A mathematical/computational solution to this problem begins with identifying open reading frames (ORFs) that serve as natural gene candidates and testing the hypothesis that a given ORF belongs to the non-coding region of the genome modeled as a randomly and independently assembled segment of DNA. Rejection of this hypothesis with a high degree of certainty increases the likelihood that the ORF in question is an actual gene. To test the above hypothesis, one has to compute the distribution of the number and length of ORFs ina long sequence of non-coding DNA. This computation leads naturally to the above-mentioned discrete probability distributions or their analogs and reveals various relationships between them through conditioning and compounding. Computation of the ORF length is based onan important, yet rarely visited, negative hypergeometric distribution. Although conceptually related to the binomial and negative binomial distributions, it is structurally similar to the hypergeometric distribution.Derivatives are multipliers.https://www.zbmath.org/1459.970102021-05-28T16:06:00+00:00"Flath, Dan"https://www.zbmath.org/authors/?q=ai:flath.daniel-eSummary: Generations of calculus students have been taught to repeat the mantra ``a derivative is a rate of change.'' We suggest replacing it with ``a derivative is a multiplier.'' this formulation focuses attention on how a derivative is used and its role as a sensitivity parameter. It generalizes easily to higher dimensional contexts. The word ``multiplier'' also facilitates inclusion of relative derivatives and elasticities as standard topics in a calculus course. The phrase ``rate of change'' need not be dropped completely, but it is most meaningful for students when it's a time rate of change.Flattening the curve.https://www.zbmath.org/1459.970182021-05-28T16:06:00+00:00"Kennedy, Gary"https://www.zbmath.org/authors/?q=ai:kennedy.garySummary: The logistic differential equation provides a plausible model for the spread of an infection; realistic features naturally emerge from it. In particular, reducing the constant of proportionality in the model has the beneficial effect of ``flattening the curve,'' which both delays and reduces the peak rate of new infection.How to win at Tenzi.https://www.zbmath.org/1459.970152021-05-28T16:06:00+00:00"Bacinski, Steve"https://www.zbmath.org/authors/?q=ai:bacinski.steve-j"Pennings, Timothy"https://www.zbmath.org/authors/?q=ai:pennings.timothy-jSummary: Tenzi is a game which is won by being the first to roll ten dice to the same agreed-upon number. Natural questions arise: How many rolls are expected for a player to get Tenzi? How many rolls are expected for a win when \(n\) people are playing? How many of the losers will have almost won -- with nine out of ten? And, most interesting, what's the advantage of rollingfaster than your opponents? Exploring these questions -- using Markov processes, probability, analysis and technology -- demonstrates how college-level mathematics can be used to gain insight into interesting problems.