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245A, prologue: The problem of measure
One of the most fundamental concepts in Euclidean geometry is that of the measure of a solid body in one or more dimensions. In one, two, and three dimensions, we refer to this measure as the length, area, or volume of respectively. In the classical approach to geometry, the measure of a body was often [...]
Automatic even/odd splitting
Statement of the Problem Suppose you have a real valued function on the reals, say f. We can split it into the sum of an even and odd part: f(x) = f odd (x)+f even (x) where f odd (x) = (f(x)-f(-x))/2, f even (x) = (f(x)+f(-x))/2 If f odd has a power series around zero, then all of its terms must have odd powers in x. Show More Summary
Philosophy is not a Mickey Mouse degree
In response to my previous post, someone brought my attention that BA in Combined studies in my own School of Mathematics also includes a number of seemingly bizarre combinations: This honours programme leads to the degree of BA, although the Mathematics component can make up about two thirds of the overall course. Other subjects will usually [...]
Wastes Management and Mathematics
Unbelievable report in the Telegraph about bizarre degree courses on offer in British univesities (duly linked to from the United Kingdom & Ireland Waste Management & Recycling News): Northampton University initially had 250 places available...Show More Summary
The Law of Small Numbers
This states: There aren’t enough small numbers to meet all the demands made of them. While that quote was good enough to post on its own, read the whole paper.
Guugu Yimithirr go mainstream
A wonderful article Does Your Language Shape How You Think? in NYT, by Guy Deutscher of the School of Languages, Linguistics and Cultures at the University of Manchester (my university, I am proud to use this occasion to say that). As I attempt to point out in my book Shadows of the Truth, the way [...]
LMS Elections: my statement
I have been serving as a Trustee and Member-at-Large of the LMS Council since my election at the AGM in 2006. I served on the Programme Committee and Research Meetings Committee (and had a chance to witness the scope and quality of British mathematicians’ research — as well as the remarkable efficiency of LMS grant [...]
Some Continuous Duals
I really wish I could just say in post titles. Anyway, I want to investigate the continuous dual of for. That is, we’re excluding the case where either (but not its Hölder conjugate ) is infinite. And I say that when is -finite, the space of bounded linear functionals on is isomorphic to. [...]
The Nature of Contemporary Core Mathematics
A very interesting and ambitious paper by Frank Quinn. However, I am not on the same tune with the author — I continue to be interested in a microscopic view of mathematics, not a panoramic landscape.
Data Encryption Standard: Part 2
In the previous exercise we examined the working of the DES block cipher for a single 64-bit block. In today’s exercise we extend encryption to an entire file, following the procedures of FIPS 81. In the descriptions that follow, Pi is the ith plain-text block, Ci is the ith cipher-text block, Ek is the encryption [...]
MathJax vs MathML?
This is a question that I placed at TeX and LaTeX question site: My organisation needs to update its website, which, in particular, will host a number of blogs and wikis on mathematics-related themes. We need to use some way of rendering of LaTeX on our web pages. Given a choice of MathJax and MathML, [...]
How to encourage TeX as a homework medium
A very useful thread in a new collaboratively edited question and answer site for people who love TeX, LaTeX and other related typesetting systems. It’s 100% free, no registration required.
Geometry @ Barriers
Bill Gasarch's summary of the Barriers II workshop omitted discussion of the geometry session, and Piotr Indyk kindly agreed to write a brief summary for your entertainment. Here it is, lightly edited and with links added. If there are...Show More Summary
Bounded Linear Transformations
In the context of normed vector spaces we have a topology on our spaces and so it makes sense to ask that maps between them be continuous. In the finite-dimensional case, all linear functions are continuous, so this hasn’t really come up before in our study of linear algebra. But for functional analysis, it becomes [...]
geometry sort of
So what I have is an open ball of unit radius centered at a point in R^2. At every point x in this ball lies another ball of radius d_x completely contained in this ball. What I need to do is to construct a mapping from each point in this ball to a direction (i.e. Show More Summary
Esperantism expands, but not so quickly
Today, as it has been for a long time, it can only be a fantastic dream to know and understand all of mathematics, and virtuous mathematicians must perforce look for alternatives. One of the best is to find some analogy between different areas — a brilliant instance, being Vojta’s rapprochement between questions of diophantine [...]
Function T-Algebras on ?-Stacks
Herman Stel's master thesis on Function Algebras on Infinity-Stacks.
P vs NP: What’s the problem?
As promised (better late than never), I’m going to begin explaining the (in)famous P vs NP question (see the previous post for a bit more context). As a start, here’s a super-concise, 30,000-foot version of the question: Are there problems whose solutions can be efficiently verified, but which cannot be efficiently solved? Of course, this [...]
The Extremal Case of Hölder’s Inequality
We will soon need to know that Hölder’s inequality is in a sense the best we can do, at least for finite. That is, not only do we know that for any and we have, but for any there is some for which we actually have equality. We will actually prove that That [...]
Bimonoids from Biproducts
Every object in a category with finite biproducts is a bimonoid in a unique way.
