Answer by Fixed Point for infinity, p2 and p1 Norm and its associated unit...
The comment was a little sloppy. Let $N$ be a norm in $\mathbb{R}^2$. Then the unit ball is the set of all points in $\mathbb{R}^2$ whose norm is one. So for example if we consider the Euclidean norm,...
View ArticleCommon Math Mistakes Made by Scientists
I am a mathematician by training working with a physicist. I have been invited to give an hour-long tutorial/presentation to incoming graduate students. These students are all coming in with physical...
View ArticleAnswer by Fixed Point for Prove that integral is independant of its parameter
Add the two fractions together and you get$$J_1+J_2 = \int_1^\infty {\left(\frac{1}{1+x^\alpha}+\frac{x^\alpha}{1+x^\alpha}\right) \frac...
View ArticleHow to fill in these steps to evaluate this Gaussian integral?
As a part of a much bigger problem, I came across this integral$$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$which represents the...
View ArticleTransforming a nonlinear system to a linear system
Suppose I have two points in $\mathbb{R}^2$ and I wish to find values of parameters $a$ and $b$ such that I obtain the power law $y=ax^b$ which goes through the two given points. I can solve the...
View ArticleAnswer by Fixed Point for What are some applications of elementary linear...
One of the examples my students have absolutely loved in the past is Hill cipher. It is a "real" application although outdated but the students do love playing with it. Using some sort of a encoding,...
View ArticleFind the last ten digits of this exponential tower.
I came across this challenge.Find the last ten (least significant) decimal digits of $x$ = 2^(3^(4^(5^(6^(7^(8^9)))))).First some notation. Let$x_n = n^{(n+1)^{(n+2)^{.^{.^{.^9}}}}}$denote the tower...
View ArticleAnswer by Fixed Point for If the field of a vector space weren't...
Two problems that I have myself come across are:The invertibility of a matrix can change. For example,the integer matrix $$\left[\begin{matrix}4 & 1\\2 & 2\end{matrix}\right]$$ is invertible...
View ArticleCases when the Intermediate Value Theorem is true - Part 2
I previously asked this question and was told that an answer is certainly possible but I am still looking for an example.The question was for cases when the intermediate value theorem is true and a...
View ArticleAnswer by Fixed Point for Understandable questions which are hard for...
Example 1:This question never fails to confuse non-mathematicians. It is pretty good at confusing even beginner math students, people who were "good" at math in high school.You have a summer vacation...
View ArticleHow to understand the exponential operator geometrically?
Consider the geometric interpretation of an orthogonal matrix, a projection matrix, a (Householder) reflector, or even just matrix-vector multiply in general.A matrix takes a vector from a vector space...
View ArticleAnswer by Fixed Point for Numerical approximation of a complex integral with...
Good answer by AD, it would have been a nice complement to my idea. I was going to suggest (now that you have two real integrals on an infinite domain) you can just use a transformation of variables, a...
View ArticleUnits of a log of a physical quantity
So I have never actually found a good answer or even a good resource which discusses this so I appeal to experts here at stack exchange because this problem came up again today. What happens to the...
View ArticleAnswer by Fixed Point for Real world application of Fourier series
It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if I was to record your voice for one second saying something, I can find its fourier series...
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Intersections of circles in a flat plane, when all radii are perturbed by a...
Consider a flat Euclidean plane and three distinct circles on it with one common intersection point shared by all three circles. If the three radii are perturbed by a constant, call it $\delta$, under...
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