Jonathan Shock – Page 7

Mathematical Foundations of Quantum Mechanics – By John Von Neumann, edited by Nicholas A Wheeler, a review

NB. I was sent this book as a review copy.

I have to admit that I was rather embarrassed to encounter this book, as I had never heard of it, and given the topic, and the author, it seemed that it must be one of the canonical texts in the field. However, it turns out that although Von Neumann wrote this book in 1932 (full German text here), it was not translated until 1955 (by Robert Beyer), and this edition aged quickly, particularly with the limitations of typesetting the equations. It wasn’t until now that a modern edition has been put together, by Nicholas Wheeler, and the result is lovely.

The book is really a collection and expansion of Von Neumann’s previously published works, attempting to put quantum mechanics on a firm mathematical footing. The first chapter is dedicated to the equivalence of Matrix Quantum Mechanics, and Schrodinger’s Wave Mechanics.…

By Jonathan Shock| May 6th, 2018|Book reviews, Reviews|1 Comment

An Introduction to analysis – By Robert G Gunning, a review

NB. I was sent this book as a review copy.

From Princeton University Press

While this book is called An Introduction to Analysis, it contains far more than one might expect from a book with such a title. Not only does it include extremely clear introductions to algebra, linear algebra, intregro-differential calculus of many variables, as well as the foundations of real analysis and beyond, building from their topological foundations, the explanations are wonderfully clear, and the way formal mathematical writing is shown will give the reader a perfect guide to the clear thinking and exposition needed to go on to further areas of mathematical study and research. I think that for an undergraduate student, taking a year to really get to grips with the content of this book would be absolutely doable and an extremely valuable investment of their time. While a very keen student would, I think, be able to go through this book by themselves, as it truly is wonderfully self-contained, if it were used as part of a one year course introducing mathematics in a formal way, I think that this really would be the ideal textbook to cover the foundations of mathematics.…

By Jonathan Shock| May 5th, 2018|Book reviews, Reviews|2 Comments

Reverse Mathematics – By John Stillwell, a review

NB. I was sent this book as a review copy.

From Princeton University Press

I’m not sure I’ve read a mathematics book which was so hard to review, not because of the quality of the book (which is superb), but because the way of thinking is in some senses so different to the way we normally think about mathematics. This, indeed, is also the book’s best feature. This book gets you thinking about mathematics in ways which I have never explored before, and which have definitely given me a new, and I think, improved perspective on formal mathematics.

In general in mathematics we start with a set of assumptions (axioms), and explore the consequences of them. Within Euclidean geometry we start with ideas about lines, and points, and circles, and then see what other theorems can be proved from these. Within set theory too, we start with a set of ideas about equalities of sets, existence, pairings, unions etc which we hold to be true and then see what can be said of other properties of sets, which are not straightforwardly stated in the axioms.…

By Jonathan Shock| March 18th, 2018|Uncategorized|2 Comments

Singalakha’s guide to plotting rational functions

To sketch the graph of a function $k(x)=\frac{f(x)}{g(x)}$ :

Find the intercepts:
1. X-intercepts, set y=0 (there can be multiple)
2. Y-intercept, set x=0 (there can be only one)
Factorise the numerator and denominator if possible:
1. Sign table: determine where the function is negative and where it is positive
Find the Vertical asymptotes:
1. This occur if the function in the denominator is equal to zero, i.e $g(x) = 0$ , AND that in the numerator must not be zero, i.e $f(x)\ne 0$ .
Find any Horizontal asymptote:
1. If the degree of the function in the numerator, i.e $f(x)$ , is less than the degree of the function in the denominator, i.e $g(x)$ , then the horizontal asymptote is the line $y = 0$ .
2. If the degree of the function in the numerator, i.e $f(x)$ , is equal to the degree of the function in the denominator, i.e $g(x)$ , say for example, the degree of $f(x)$ and $g(x)$ is $n$ for some non-negative $n$ element of integers, then there is a horizontal asymptote.

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By Jonathan Shock| March 15th, 2018|Uncategorized|1 Comment

My vlogging channel

Hi all, I’m not sure if it counts as vlogging, or making maths videos regularly fits into a slightly more niche category, but anyway, I wanted to advertise some videos that I’ve been putting up recently. I’m doing this in an attempt to find a different communication channel with my first year maths class, and so far the videos are getting reasonable feedback. I have a long way to go in terms of making them slick, and I goof up from time to time, but it’s an interesting experience. If you have specific questions that you would like me to discuss in a video. Let me know.

In this video I talk about a method for solving inequalities involving absolute values:

How clear is this post?

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By Jonathan Shock| March 13th, 2018|Uncategorized|2 Comments

A quick introduction to writing mathematics in WordPress using LaTeX

Here are a couple of very useful links about writing mathematics, for new authors of this blog:

LaTeX for wordpress
Equations in LaTeX in wordpress (and links in that same page, which are very useful).

I will update this as I find more useful material.

Generally I like to use the Visual Tab on the editor here rather than the Text Tab, unless there is some sort of strange formatting in which case I will go in and alter the Text.
I usually like to put formulas centrally justified on their own on a line with blank lines above and below.
Add Media to upload pictures or gifs and use the Fusion Shortcodes button (to the left of the yellow star in the blue box), to embed Youtube content.

Please let me know if, as an author, there is anything which is unclear about posting here and I will update accordingly.…

By Jonathan Shock| February 28th, 2018|Uncategorized|0 Comments

Can you find a simple proof for this statement?

I thought more about the last question I added into the addendum of the Numberphile, Graph theory and Mathematica post

It can be succinctly stated as:

$(\forall m\in\mathbb{Z}, m\ge 19) (\exists p,q\in\mathbb{Z}, 1\le p,q<m, p\ne q)$ such that $\sqrt{p+m}\in\mathbb{Z}$ and $\sqrt{q+m}\in\mathbb{Z}$ .

In words:

For all integers m, greater than 19, there are two other distinct positive integers less than m such that the sum of each with m, when square rooted is an integer.

What is the shortest proof you can find for this statement?

How clear is this post?

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By Jonathan Shock| January 17th, 2018|Uncategorized|2 Comments

Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.

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