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Mathematical Reflections - Two Lockdown Years

Titu Andreescu and Maxim Ignatiuc
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Russell Jay Hendel
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Although the authors primarily see this book, “encouraged by appreciative and constructive feedback from faithful readers, as a compilation and revision of the 2020 and 2021 volumes of the online journal, Mathematical Reflections", this reviewer would emphasize two other aspects of the book: i) a multi-national statement of human resilience in response to the pandemic, and ii) a collection of 16 beautiful articles succinctly summarizing current and interesting mathematical topics not easily accessible.
The first part of the book is organized as a collection of challenging mathematical problems and complete worked-out solutions. The problems are organized by level, not by technique: junior level, aimed at high-school students, senior and Olympiad level, aimed at students preparing for national and international contests such as the United States of America Mathematical Olympiad (USAMO) or the International Mathematical Olympiad (IMO), and undergraduate level, targeted to college students familiar with linear algebra, calculus, and graph theory. 
Thus the book represents material targeted to diverse nationalities and a variety of age groups. This problem solving activity, during the Covid years, provides a statement of human resilience in the face of the pandemic; it contrasts with the negative aspects of Covid such as increased isolation, decreased social interaction, consequent depression and reduced productivity.
The second part and an important feature of the book is a beautiful collection of 16 articles on a variety of interesting topics outside of the mainstream curriculum affording supplemental educational opportunities to instructors and students.
One chapter proves in a few pages the beautiful Marden Theorem which states that if the three roots of a cubic polynomial are plotted in the complex plane forming a triangle (unless they are collinear) and an ellipse is drawn inside this triangle tangent to its sides (the inellipse) then the focii of this ellipse are the roots of the derivative of the original cubic polynomial.
Another two-part article summarizes deep learning comprehensively and succinctly. The article presents the three factors that converged in 2012 to produce the deep learning big bang: i) datasets had tremendously improved; for example, pre-2009 the Pascal dataset had 20000 images and 20 categories; post-2009, the Imagenet dataset had 14 million hand-labeled images in 20,000+ categories. ii) Digital computers achieved a 100-fold efficiency in computing through the GPU, graphics processing unit, replacing the CPU (central processing unit). iii) Algorithm improvement as illustrated by the 2012 AlexNet convolutional network surpassing the 1998 LeNet-5 which was used in mail-sorting by the US postal service (for handwritten recognition and processing of envelopes). The article defines the major categories of artificial intelligence including machine learning, supervised learning, and (deep) neural networks. The article concludes by showing the basis of iterative solutions to the linear multiple regression problem which forms a basis for neural network layers.
The other 14 articles similarly deal with a wide variety of topics including skillful use of generating functions, inversions through a circle, tangential quadrilaterals, and how proper use of discriminants can facilitate problem solving.

Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, graph theory, actuarial science, general theory of pedagogy
including applications of technology, and biblical exegesis.

See the publisher's website