Comment by Fixed Point on Influence of small constant term on roots of...
@StefanRickli I have done some more investigation and added my results to my post. I really want to investigate how the multroot package you pointed to behaves and how exactly does it fail but I don't...
View ArticleComment by Fixed Point on Riddles that can be solved by meta-assumptions
@CiaPan PieterB and JeremyP are both correct. I have added a picture in the post to show what the solutions are.
View ArticleComment by Fixed Point on Understanding Linear Algebra Geometrically -...
@Omnomnomnom I don't know too much Lie Group/Lie Algebras. Can you please add an answer clearly explaining the geometric connection with the exponential operator?
View ArticleComment by Fixed Point on Find the two missing angles in a quadrilateral
-1 I specifically said using high school geometry in the spirit of Euclid. Trigonometry, law of sines, and law of cosines are "not allowed". It is easy if we allow these.
View ArticleComment by Fixed Point on Find the two missing angles in a quadrilateral
@fleablood Because this was a student who came into an open math tutoring session and said that he was not allowed to use any trigonometry. Furthermore, I am actually inclined to believe the student...
View ArticleComment by Fixed Point on Find the two missing angles in a quadrilateral
While you are at it, can you also change the labels to match the problem I posted so that it is easier to follow the proof.
View ArticleComment by Fixed Point on General request for a book on mathematical history,...
Undergraduate/Graduate level books don't have to be patronizing. I am sure there are some mathematical fields in which he is not an expert but he has always wanted to learn more about. He might be very...
View ArticleComment by Fixed Point on How to solve simultaneous equations using...
@LuaiGhunim There are lots of ways Newton's method can fail. It could be that your function is non-smooth and doesn't have a well-defined Jacobian. It could be that the Jacobian is just difficult to...
View ArticleComment by Fixed Point on Why does separation of variable gives the general...
+1 At the end of part 4, we have a basis for two 1D operators. How do we know that multiplying these two bases (element wise) will give us a basis for the Laplacian in a 2D rectangle? Why don't we need...
View ArticleComment by Fixed Point on Constructing Very "Flat" Functions
@CalvinKhor Yes it is that trivial. Please see my addendum to my question above.
View ArticleAnswer by Fixed Point for Is there an impossible to solve equality in...
Sure, you can't solve equations like$$xy-yx=1$$in the complex plane. But if we extend the complex plane into quaternions then this equation does have a solution. The basis elements are usually written...
View ArticleMathematical Analysis of the Electoral College
Let us consider the electoral college voting system used to elect the American president. I have a few questions from the point of view of decision-making/gaming theory. My ultimate goal is to vote in...
View ArticleAnswer by Fixed Point for How to test a computer language random number...
There are many many tests. First these are all statistical tests. You first decide on a $p$-value, your cutoff, usually 5% for normal everyday use (gaming, scientific simulations stuff) and 1% for very...
View ArticleAnswer by Fixed Point for Does there exist a system such that the additive...
I am surprised no one mentioned elliptic curves with its "usual" elliptic-curve-addition. You pick two points $a$ and $b$ on an elliptic curve. Then $a+b$ is defined as the negative of the third point...
View ArticleAnswer by Fixed Point for Why can't a Series Converge to 1?
Because$$1+1+1+1+1+...=\sum_{n=0}^{\infty} 1$$doesn't converge. This sum is not a finite number. So if there is any hope of an infinite series converging, the terms MUST go to zero. It is not true the...
View ArticleAnswer by Fixed Point for Checksum of a set of real numbers
Some of the algorithms like check digit algorithms can be adapted to be used here. For example you can compute$$\frac{a_1}{2}+\frac{a_2}{4}+\frac{a_3}{8}+...+\frac{a_n}{2^n}.$$Depending on how you...
View ArticleAnswer by Fixed Point for An example of a function whose domain is the set of...
Zero maps to zero. All even numbers $n$ map to $n/2$ and all odd numbers $n$ map to $-(\frac{n-1}{2}+1).$ This map is one to one and onto and hence invertible.
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Answer by Fixed Point for What is wrong with this method for a rotated and...
This parabola cannot be written in $x=f(y)$ form (like a sideways parabola) or in $y=f(x)$ form (vertical parabola). You can see this by the mixed $xy$ term when you expand everything. It is a parabola...
View ArticleAnswer by Fixed Point for What is the sum of $1^4 + 2^4 + 3^4+ \dots + n^4$...
Here is my favorite method which works for any polynomial summand and you only need to remember two basic facts, one from calculus one and one about polynomials. First, since summations are analogous...
View ArticleAnswer 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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