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March 2020
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23:41

Integrate e^(-x) cos^(n) x. Hello friends, today I’ll show how to integrate e^(-x) cos^(n) x. Have a look!! If interested, read more on similar topics like: How to integrate e^(-x) sin^(n) x How to integrate cos^(n) x How to integrate Sec^(n) x How to integrate Tan^(n) x Example 1 – Integrate e^(-x) cos^(n) x   So...

The post Integrate e^(-x) cos^(n) x – how to do that? appeared first on Engineering math blog.



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20:49

The integral $$ \int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8}, $$ is given as equation $(17)$ in M.L. Glasser, Some integrals of the Dedekind $\eta$-function.

More general integral $$ \int_0^\infty \frac{\sin x\sinh (x/a)}{\cos (2 x)+\cosh \left(2x/a\right)}\frac{dx}{x}=\frac{\tan^{-1} a}{2},\tag{1} $$ can be deduced as a limiting case of formula $4.123.6$ in Gradsteyn and Ryzhik.

I have been looking for finite elementary analogs of integral $(1)$ and have proved that \begin{align}\label{} \int_0^{1}\frac{\sin \bigl(n \sin^{-1}t\bigr)\sinh \bigl(n \sinh^{-1}(t/a)\bigr)}{\cos \bigl( 2 n \sin^{-1}t\bigr)+\cosh \bigl(2 n \sinh^{-1}(t/a)\bigr)}\frac{dt}{t \sqrt{1-t^2} \sqrt{1+{t^2}/{a^2}}}=\frac{\tan^{-1} a}{2},\tag{1a} \end{align} for an odd integer $n$.

When $n\to\infty$ equation $(1a)$ will give equation $(1)$. This is easy to see because when $n$ is large then the main contribution to $(1a)$ comes from a small neighborhood around $0$.

Q: Can you explain why this integral has such a simple closed form and in particular why it has the same value for all odd $n$?

I want to stress that I have a proof which is based on partial fractions expansion for odd $n$ \begin{align} &\frac{\sin \bigl(n \sin^{-1}t\bigr)\sinh \bigl(n \sinh^{-1}(t/a)\bigr)}{\cos \bigl( 2 n \sin^{-1}t\bigr)+\cosh \bigl(2 n \sinh^{-1}(t/a)\bigr)}\frac{2n}{t^2}\\&=\sum _{j=1}^n\frac{i(-1)^{j-1} }{\sin\frac{\pi (2 j-1)}{2 n}}\cdot \frac{\left(a\cos\frac{\pi (2 j-1)}{2 n}+i\right) \left(a+i \cos\frac{\pi (2 j-1)}{2 n}\right)}{t^2 \left(a^2-1+2 ia \cos\frac{\pi (2 j-1)}{2 n}\right)-a^2 \sin ^2\frac{\pi (2 j-1)}{2 n}}, \end{align} the elementary integral \begin{align} \int_0^1 \frac{t}{t^2 \left(a^2-1+2 ia \cos\frac{\pi (2 j-1)}{2 n}\right)-a^2 \sin ^2\frac{\pi (2 j-1)}{2 n}}\frac{dt}{\sqrt{1-t^2} \sqrt{1+{t^2}/{a^2}}}\\=\frac{\tan^{-1}a+i\tanh^{-1}\cos\frac{\pi (2 j-1)}{2 n}}{i\left(a\cos\frac{\pi (2 j-1)}{2 n}+i\right) \left(a+i \cos\frac{\pi (2 j-1)}{2 n}\right)}, \end{align} and summation formula which can be deduced from the partial fractions above $$ \sum _{j=1}^n \frac{(-1)^{j-1}}{\sin \frac{\pi (2 j-1)}{2 n}}=n. $$

But despite this prove I don't understand why all these cancellations occur to give such a simple result at the end. I suspect there is a very short and transparent proof which explains why the integral is $\frac{\tan^{-1} a}{2}$ for all odd $n$. Maybe Glasser's master theorem or some contour integration can explain this formula? Motivation for this question is desire to understand this integration formula.

Any alternative proof is welcome if it is not just a detailed version of the proof above. Any ideas and comments are welcome. Thanks.



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18:49

How can we prove any of 3 following (numerically verified) identities? $$\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}=-2 \pi \Im(\text{Li}_3(1+i))-\frac{21 \text{Li}_4\left(\frac{1}{2}\right)}{2}-\frac{57}{8} \zeta (3) \log (2)+\frac{83 \pi ^4}{480}-\frac{23}{48} \log ^4(2)+\frac{7}{12} \pi ^2 \log ^2(2)$$ $$\int_0^1 \frac{x \text{Li}_2(x) \log (1-x)}{x^2+1} \, dx=\frac{C^2}{2}-\frac{1}{8} \pi C \log (2)+\frac{15 \text{Li}_4\left(\frac{1}{2}\right)}{16}-\frac{511 \pi ^4}{46080}+\frac{5 \log ^4(2)}{128}-\frac{7}{384} \pi ^2 \log ^2(2)$$ $$\int_0^1 \frac{\text{Li}_2(x) \log \left(x^2-2 x+2\right)}{x} \, dx=\frac{1}{2} \pi \Im(\text{Li}_3(1+i))+\frac{5 \text{Li}_4\left(\frac{1}{2}\right)}{8}+\frac{35}{64} \zeta (3) \log (2)-\frac{577 \pi ^4}{23040}+\frac{5 \log ^4(2)}{192}-\frac{1}{96} \pi ^2 \log ^2(2)$$ $$$$ Background: The first identity is a conjecture in one of J.M.Borwein's book on experimental mathematics. It was originally written in terms of multiple-polylogarithm $L_{3,1}$ and I've simplified it to the form above, using identities of polylogarithms. He claimed that it was found by an integer relation algorithm but not rigorously proved.

I proposed the first identity half a year ago but received no answers. During this period of time, I've successfully solved a class of quadratic log integrals (see section 4-2 and appendices of the article here if needed) and noticed that this series shares exactly the same structure with them. Based on my experience, I transformed the series into 2 equivalent integral forms, that is, the second and third equality. However, they belong to the class of quadratic polylog integrals which I haven't managed to solve completely.

I'd like to share them with you and see that whether they can be proved directly. As I've deduced the equivalence between three identities, we only need to prove one of them (although it is still a difficult task). Any kind of help will be appreciated.



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17:19

I'm going into college for applied mathematics in the next semester and was wondering what would be recommended in terms of a computer/tablet. I have been looking into the Macbook Air and the iPad Pro but I really can not decide which would work best for my major, and there seems to be no major-specific information on this topic.

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23:19

Although one cannot find an elementary antiderivative of $f(x)=x^x$, we can still give a series representation for $\int_0^1 x^x dx$, namely:

$$I_1=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^n}=0.78343\ldots$$

One can even find an expression for the complete antiderivative in terms of infinite sums and the incomplete gamma function $\Gamma(a,x)$:

$$\int x^x dx =\sum_{n=1}^\infty \left(\frac{(-1)^{n+1}\Gamma(-n\ln(x),n)}{n^n \Gamma(n)}\right)+C$$

Considering special, non-elementary function, series, infinite products, etc. , is this also possible for $\int_0^1 x^{x^x} dx$?

Thank you in advance!



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21:19

The problem is as follows (verbatim) : Prove that it is impossible to write $x = f(x)g(x)$ where $f$ and $g$ are differentiable and $f(0)=g(0)=0$.

I rephrased the question as I realized it wasn't optimally worded : Prove that it is impossible to write $x = f(x)g(x)$ for all $x \in \mathbb R$, where $f$ are $g$ are differentiable on $\mathbb R$ and $f(0)=g(0)=0$.

I proved this by differentiating both sides of $x = f(x)g(x)$, which gives $1 = f'(x)g(x) + f(x)g'(x)$; thus when $x=0, f'(0)g(0) + f(0)g'(0) = 0 \neq 1$. However, this proof doesn't give me much insight on why this particular exercise is true. I am really interested in seeing a more intuitive explanation, but here are some of my attempts/hypotheses :

  1. $f'(x)g(x) + f(x)g'(x)$ is constant when $f$ or $g$ is constant. However, WLOG, if $f(x) = 0$, then we will have to divide by zero in order to get a function $g$ such that $x = f(x)g(x)$.
  2. If we define $f(x) = \frac{x}{g(x)},$ then clearly if $g(x) = 0$ then $f$ is not even continuous at $x=0$, so $f$ would not be differentiable on $\mathbb R$.

I am more or less looking for a graphical(?), intuitive explanation. Thank you for your help!



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20:49

Let $f :\mathbb{R}\to \mathbb{R}$ be a continuous function with period $1$ and $$\lim_{n\to\infty}\int_0^1\sin^2(\pi x)f(nx)dx= \frac{1}{k}\int_0^1f(x)dx.$$ Find $k$.

My approach till now:

Applying half angle formula $2\sin^2(x) = 1-\cos(2x)$ I got : $$\frac{1}{2}\int_0^1f(nx)dx- \frac{1}{2}\int_0^1\cos(2\pi x)f(nx)dx.$$

I can't think of a way forward from here without applying integration by parts but i don't know if its right to apply it as we don't know about the differentiability of the function $f$.

Please help me with this problem.



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20:19

For $(x,y,z)\in S^2=\{\vec{x}\in\mathbb{R}^3: \lVert\vec{x}\rVert=1\}$, I am given the 3D-system $$ \begin{align*} x'&=x(-x+f(x,y,z))\\ y'&=y(x-y+f(x,y,z))\\ z'&=z(y-z+f(x,y,z)) \end{align*} $$ with $f(x,y,z)=x^3-xy^2+y^3-yz^2+z^3$.

One equilibrium (among others) is given by $$ E=(a,2a,3a),\quad a=\sqrt{1/14}. $$

I would like to show that $E$ is globally stable within the set $$ M:=\{(x,y,z)\in S^2: x,y,z>0\} $$ (In fact, $E$ is the only equilibrium in $M$.)

Is there a Lyapunov-function $V$?

———

I tried the function
$$ V(x,y,z)=\alpha(x-a)^2+\beta(y-2a)^2+\gamma(z-3a)^2, $$ where $\alpha,\beta,\gamma>0$.

At least, one gets that $V$ is positive definite on $M$, i.e. $V(E)=0$ and $V(x,y,z)>0$ for all $(x,y,z)\in M\setminus\{E\}$.

The derivative is $$ \begin{align*} V'&=2\alpha(x-a)x'+2\beta(y-2a)y'+2\gamma(z-3a)z'\\ &=2\alpha(x-a)(-x^2+x f)+2\beta(y-2a)(xy-y^2+yf)+2\gamma(z-3a)(yz-z^2+zf). \end{align*} $$

Unfortunately, I do not see whether $V'$ is negative definite on $M$: It is clear that $V'(E)=0$ but not whether $V'<0$ on $M\setminus\{E\}$. Actually, I tend to think that the sign of $V’$ changes on $M$ and thus my $V$ is not the right choice.

Do you have an idea?



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03:04

Question: Define $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx$ for every $n\in\mathbb{N}$. Prove that $$\lim_{n\to\infty}nI_n=\frac{1}{\sqrt 2}$$.

My approach: Given that $I_n=\int_0^1\frac{x^n}{\sqrt{x^2+1}}dx, \forall n\in\mathbb{N}.$ Let us make the substitution $x^n=t$, then $$nI_n=\int_0^1\frac{dt}{\sqrt{1+t^{-2/n}}}.$$

Now since $0\le t\le 1\implies \frac{1}{t}\ge 1\implies \left(\frac{1}{t}\right)^{2/n}\ge 1 \implies 1+\left(\frac{1}{t}\right)^{2/n}\ge 2\implies \sqrt{1+\left(\frac{1}{t}\right)^{2/n}}\ge \sqrt 2.$

This implies that $$\frac{1}{\sqrt{1+\left(\frac{1}{t}\right)^{2/n}}}\le\frac{1}{\sqrt 2}\\ \implies \int_0^1 \frac{dt}{\sqrt{1+\left(\frac{1}{t}\right)^{2/n}}}\le \int_0^1\frac{dt}{\sqrt 2}=\frac{1}{\sqrt 2}.$$

So, as you can see, I am trying to solve the question using Sandwich theorem.

Can someone help me to proceed after this?

Also, in $$\lim_{n\to\infty}nI_n=\lim_{n\to\infty}\int_0^1\frac{dt}{\sqrt{1+t^{-2/n}}},$$ the limit and integral interchangeable?



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01:34

Earlier, a very interesting proof of an inequality has been proposed at MSE:How prove this inequality $\tan{(\sin{x})}>\sin{(\tan{x})}$

Here the question is: How to prove that $$\sin(\tanh x) \ge \tanh(\sin x), ~~ for~~ x \in [0,\pi/2].$$ Interestingly, the first three terms of the Mclaurin series are identical for both the functions.



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01:03

Hey everybody,

When you look on the internet for the definition of "Calculus", for example Wikipedia says the following: (https://en.wikipedia.org/wiki/Calculus)

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

I find this definition of odd, as it does not really explain what calculus is. In German, there a word that is often used called "Kalkül", which the dictionary directly translates to "calculus". The definition of "Kalkül" seems to be (please corret me if I'm wrong) a system of axioms or postulates from which theorems are derived.

The definition given by Wikipedia does kinda sorta imply this definition, I feel. Is there a specific reason other than "mainly it's used in this context", that would justify this definition?

Additionally, why this question came up: Doesn't Hilberts "Grundlagen der Geometrie" do this exact thing, building a "calculus"? I've haven't seen that word in that context yet.

It's very likely that some of my points might be nonsensical. Please tell me where, so I can understand what's up :)

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00:03

I'm an undergrad freshman in an American college and I recently became fascinated with mathematics. I took Calculus II last semester, and I know most people in the sub are way beyond my level, but there was one class that used taylor series to prove Euler's Identity that just blew my mind and I decided to at least double-major in math.

The main problem is, I went to my school convinced I was gonna study the humanities. My school has a big brand name and has top notch humanities programs, but a lot of things STEM related are average at best. We don't even have an engineering school. The math courses they offer, especially excluding statistics, are very limited. While the professors we do have are all good, I feel like there's a lot of missing areas and I might get a subpar math education if I ever want to pursue math at a graduate level. My current math professor has literally admitted that the course offerings are "pretty bad" in a lecture, and courses we do have move pretty slowly sometimes.

Right now, I'm taking Multivariable Calculus, which corresponds to calc 3 in most colleges, that was split into two. Apparently the second half of the "standard" calc 3 became Vector Analysis/Differential Geometry. I feel like this class moves way too slowly as we've only finished covering pretty basic vector knowledge and vector valued functions stuff in half a semester. Since I'm getting this coronavirus break time, I want to do some self-learning and get a headstart on topics like analysis, linear algebra (my next course), and number theory. What are some of the books you recommend? Are there any free or even paid course/online resources that I can use to get something useful out of? And for my fellow Americans, what was your undergrad experience like and what general advice would you give me?

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23:33

I am looking for video lectures.I searched on google and found so many but problem is that i don't know where to start and as all of you know probability and statistics is tough part of mathematics, and i don't want to take any risk.And you people are more experienced then me so please kindly suggest me some video lectures or resources to to kick start and begin my journey.

Your Recommendations will help me a lot. Thanks

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02:34

The usual definition of a category states: a category $\mathbf{C}$ consists of:

  1. A collection $\text{ob}(\mathbf{C})$ of objects
  2. A collection $\text{arr}(\mathbf{C})$ of arrows
  3. Some rules on the behaviour of these two types of objects

Leaving aside what collection here means, I realized that I have never seen a clear definition of what an arrow in a category is. Surely, in the typical categories like $\textbf{Set}$, $\textbf{Top}$ or $\textbf{Grp}$ arrows are functions that ... But there are categories whose arrows are not "functions".

If you have a poset $(P,\le)$ then you can get a category whose objects are elements of $P$ and such that there is an arrow $x\to y$ if and only if $x\le y$ in $P$. Okay but what kind of "entity" is the arrow there?, is there a precise, rigorous definition or do just have to accept that there are objects and arrows and asume that they follow the required rules, just like we asume there some things called "numbers" that follow the rules of arithmetic?

I hope I made myself clear

Thanks!



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22:34

For every cardinal $\kappa$, is there a cardinal $\lambda$ such that for all groups $G$ with $|G| > \lambda$, we have $|\mathrm{Aut}(G)| > \kappa$? I believe a similar result holds for finite groups (see Groups with given automorphism groups), but I'm wondering about the infinite case. If this fails, how badly does it fail? Are there arbitrarily large groups with finite automorphism group?



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21:34

I found a theorem from a book 'Diophantine equations', L. J. Mordell, which says

The equation $y^2 = Dx^4+1$ where $D>0$ and is not a perfect square, has at most two solutions in positive integers.

But I can't find any proof in this book, and I tried to find its proof but I failed. Is there anybody knows its proof?



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22:34

I have this integral: $$\int \frac {dx}{x \sqrt {x^2-1}}.$$ I solved it like this: $$ \sqrt {x^2-1} = t \to x = \sqrt {t^2+1} \to $$ $$dx = \frac {t}{\sqrt {t^2+1}}\,dt .$$ Solving for t: $$ \int \frac {dt}{t^2+1} = \arctan(t) + c .$$ In the solution I have, however, it is solved in another way: $$ x = \sec(t), \ \sqrt{x^2-1}=\tan(t), $$ $$dx=\sec(t)\tan(t)\,dt \to $$ $$\int \frac {\sec(t)\tan(t)} {\sec(t)\tan(t)}\,dt= \int dt = t +c= \operatorname{arcsec}(x) + c.$$ Which solution is correct and why?



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20:33

Do you want to discover, share, and discuss the theorems, definitions, axioms, and conjectures used in mathematics, both big and small?

(Re)Introducing mathlore.org.

Previously I was calling this Mathpendium. However, based on the great feedback I got on Reddit, I have renamed it to mathlore.org.

In addition, I've added the ability to comment on and discuss any math result in mathlore.org.

What are your thoughts? Your feedback is always greatly appreciated.

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19:23

I need some help in untangling and solving the following exercise:

Let the curve $c:[a,b] \to \mathbb{R}^2, t \mapsto (t, y(t))$ be a solution for the ODE $$ y'(x) = f(x, y(x)). $$ Justify the "Physicist's method" (no offense intended) of rearranging the equation $\frac{dy}{dx} = f(x,y)$ through formal multiplication of $dx$ to $$ dy = f(x,y)dx, $$ by showing that both differential forms agree in every point $c(t)$ on the tangent space.

As far as I understand the situation, we are considering the two differential forms $$\begin{equation} dy: \mathbb{R}^2 \to {\bigwedge}^1(T_p\mathbb{R}^2)\\ f(x,y)dx: \mathbb{R}^2 \to {\bigwedge}^1(T_p\mathbb{R}^2)\\ \end{equation} $$ Here $dy$ is a constant differential form, in the sense that $dy(p)(x,y) = y$ independent of $p=(v,w) \in \mathbb{R}^2$. However, in general $f(x,y)dx$ is not a constant differential form. Now, if we choose a point on the curve $c$, say $p = c(t_0) \in c([a,b])$ then we have $$ f(p)dx = f(c(t_0))dx = y'(t_0)dx, $$ since $c$ is a solution to the ODE given above. I don't know how to proceed from this point. How can we argue that this equals $dy$?



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18:23

So, I am supposed to show that to each representation $\rho$ of a group $G$ on $\mathbb{C}^n $ corresponds a representation $\tilde{\rho}$, of the same group $G$, on $\mathbb{R}^{2n} $.

Additionally: Show that, if the representation $(\tilde{\rho},\mathbb{R}^{2n}) $ is irreducible, then the representation $(\rho,\mathbb{C}^{n} )$ is also irreducible.

My thinking:

For the first part:

Suppose we have the following map: $\phi(z):\mathbb{C}^{n}\rightarrow \mathbb{R}^{2n}, \space \phi(z)= \begin{bmatrix} Re(z) \\ Im(z) \\ \end{bmatrix}$. This is obviously a bijection, additionally it follows that $\phi(z_1+z_2)=\phi(z_1)+\phi(z_2)$. My idea then was to find a bijection $\eta$ between the sets $\rho(g)$ and $\tilde{\rho}(g)$, for $g \in G$ .

And this is what I came up with (I'll denote $\rho(g)$ as $\rho_g$):

$$\eta:\mathbb{R}^{2n}\rightarrow \mathbb{C}^{n}, \space \eta:\space \tilde{\rho}_g(.)\rightarrow \phi^{-1}(\tilde{\rho}_g(\phi(.)))=\rho_g(.)$$

The map $\rho_g (.)=\phi^{-1}(\tilde{\rho}_g(\phi(.)))$ is clearly $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$, it is also a bijection and it follows that $\rho_g(z_1z_2)=\rho_g(z_1)\rho_g(z_2)$, all together meaning that $\rho_g $ is a representation of $G$ on $\mathbb{C}^{n}$. I have therefore shown that to each representation $(\tilde{\rho},\mathbb{R}^{2n})$ corresponds a representation $(\rho,\mathbb{C}^{n})$, via $\eta$.

For the second part: (proof by contradiction)

Irreducible means that for $\tilde{\rho}$ there is no (non-trivial) invariant subspace in $\mathbb{R}^{2n}$. Lets take a (non-trivial) subset $R\subset \mathbb{R}^{2n}$, so that for a subset $W \subset \mathbb{C}^{n}: \space$ $\phi(W)=R$ and $\phi^{-1}(R)=W$. We suppose that the opposite holds ($\rho$ is reducible): that $W$ is some non-trivial invariant subset for $(\rho,\mathbb{C}^{n} )$. This means that $\rho_g(W) \subset W$ for each $g \in G$. But since $\phi^{-1}(\tilde{\rho}_g(\phi(W)))=\rho_g(W) \subset W$, it follows that $\tilde{\rho}(\phi(W))\subset R $ and using $\phi(W)=R$, we get that $\tilde{\rho}(R)\subset R$. So we found a (non-trivial) invariant subspace for $\tilde{\rho}$, but this contradicts the above statements that $\tilde{\rho}$ is irreducible, therefore $\rho$ must be irreducible.

My question:

Is my thinking correct? Could you please suggest some improvements or maybe show a different, more compact way to prove the above statements.



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03:53

Let $X_1,\dots,X_n$ be $n$ iid non negative rv's and let $f:\mathbf{R}_+\times \mathbf{R}_+\to \mathbf{R}_+$ be such that $f(t,x)-f(t,x')=(x'-x)t$. Define for all $x$ $$x \mapsto \frac{1}{n}\sum_{i=1}^n f(X_i,x)\mathbb{1}(f(X_i,x)\geq 0)$$ and assume it has a unique fixed point $\overline{x_0}$. Now fix $\overline{x_0}$ and define $$x \mapsto \mathbb{E}_X[f(X,x)\mathbb{1}(f(X,\overline{x_0})\geq 0)]$$ and assume it has a unique fixed point $x_0$.

I would like confirmation that $\mathbb{E}_{X_1,\dots,X_n}[\overline{x_0}] = \mathbb{E}_{X_1,\dots,X_n}[x_0]$.

I have $\mathbb{E}_{X_1,\dots,X_n}[\overline{x_0}-x_0] = \mathbb{E}_{X_1,\dots,X_n}[(x_0-\overline{x_0})X\mathbb{1}(f(X,\overline{x_0})\geq 0)]$ and I'm not sure how to conclude.



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03:05

For those who do not know, an oxford comma is "a comma used after the penultimate item in a list of three or more items, before ‘and’ or ‘or’ ".

I've often noticed in various theorems and mathematical papers how significant this comma is to group together various conditions. Do you have any interesting examples of when this comma has saved you many days trying to decipher an un-comma'd condition?

Sorry for my english, it's not very good.

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02:59

Integrate Cot^n (x). Hello friends, today I’ll talk about how to integrate Cot^n (x). Have a look!!   Read more on similar topics like: Reduction formulas for algebraic functions   Integrate Cot^n (x)   Solved example of how to integrate Cot^n (x) Disclaimer: This example is not mine. I have chosen it from some book....

The post Integrate Cot^n (x) | How to integrate a function of the form Cot^n (x) appeared first on Engineering math blog.



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02:57

This day is known for pranks and puzzles.  
Try one of these 4 activities on April Fools Day.

Ask students to analyze where I made my mistake

       PerplexingProblem.pdf

For members we have solutions. PerplexingProblem-solution.pdf

         CCSS: MP1, MP3, 4.OA, 5.G, 8.EE, 8.SP

Or baffle 'em with this Ghost Whisperer lesson.  They will think you are pulling one over on them until they figure out the math!

CCSS:  4.OA.4, 5.OA.1, HSA.CED.A.2, MP1, MP3, MP4, MP7, MP8

Or "Who got the extra buck?" or "Where did that extra dollar go?"

ExtraBuck.pdf

For members we have a solution.   ExtraBuck-solution.pdf

       CCSS: MP1, MP3, 4.OA, 5.G, 8.EE, 8.SP

Another great April Fool's day activity is Robert Kaplinsky's Foil Prank analysis.

CCSS: 6.G.4, 6.RP.2, 6.RP.3, 7.G.6, HSG.MG.1



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22:53

Let $C$ be a small category, let $D$ be a locally small category. Given a functor $F:C\to D$, the image of $F$ may not be a category. Now following the nLab, let's instead call the image of $F$ the subcategory of $D$ generated by the images of the objects and morphisms of $C$ under $F$, i.e. close it under composition.

With this notion of image, by construction we have a subcategory of $D$. Is this subcategory again small?



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17:35

I feel like discussions about foundations of maths pop up pretty regularly on this subreddit, and also on pop math places like Youtube. Yet my own university doesn't dedicate much time to FoM and, while looking for universities to go to on exchange, the same seems to be true for many universities who don't go far beyond basic logic. There also doesn't seem to be much research in the area but I'm not the best person to be the judge of that

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17:35

Say I have the proof of Andrew Wiles Fermats Last Theorem, which is 100 pages long and involves numerous branches of difficult mathematics.

​

How would one make that proof as rigourous as possible? Which of course means converting every step into set theory.

​

Next. Here is a very good essay about errors in mathematical proofs.

​

https://www.gwern.net/The-Existential-Risk-of-Mathematical-Error

​

More or less. How are theorems confirmed in an absolutely rigorous mathematical framework? Or as rigorous as Bayes Theorem and the Dark Lords will allow?

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05:53

Let us define, $$r(b)=\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k})$$ After playing around with the $r(b)$ function for sometime I noticed that $r(b)$ appreared to be more even than odd. So to see the difference between the number of even and odd terms of $r(b)$, I defined a function, $$z(x)=\sum_{n=1}^x(-1)^{r(n)}$$ When user Peter ran a program for computing values of $z(x)$ in PARI, I observed that for $x\le 10^{10}$, $z(x)\gt 0$. This suggests that there are always more even terms of $r(n)$ than odd terms for any $x$.

This leads to my two questions:

  1. Is $z(x)$ always positive? If so, then how do we prove this?
  2. Is $|z(x)|$ bounded by some maximum value? If so, then what is this maximum value? Till now the maximum value of $|z(x)|$ found was $49$ for $x = 5424027859$. I find it odd that $|z(x)|$ goes to these large values and then returns back to small values as small as $1$.


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05:23

I want to solve the following problem in $\boldsymbol{x} \in \mathbb R^{n}$

$$\begin{array}{ll} \text{maximize} & \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}\\ \text{subject to} & \boldsymbol{q}^T \boldsymbol{x} = 1\\ & x_i \geq 0\end{array}$$

where matrix $\boldsymbol{A}$ is positive definite matrix and $x_i$ denotes the $i$-th entry of $\boldsymbol{x}$.

Actually, I have tried to use Lagrangian multiplier. I directly transformed the objective function to $-\boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x} + \lambda ( \boldsymbol{q}^T \boldsymbol{x} - 1 )$ and take its first derivative and set that to zero.

However, the solution obtained did not maximize the objective function, it just makes $\boldsymbol{x}^T\boldsymbol{A} \boldsymbol{x}$ smaller and smaller. Then I found that the solution of $\min_{\boldsymbol{x}} \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}$ with the same constraints is the same with that of $\max_{\boldsymbol{x}} \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}$.

Any comments would be appreciated!

Update As comments suggested, I changed the situation to $x_i \geq 0, \forall i$. Thus for example, when $\boldsymbol{A}= \left[\begin{matrix} {2 \; 0\\ 0 \;1 }\end{matrix} \right]$ and $\boldsymbol{q} = [1,1]^T$. The problem has a solution $\boldsymbol{x} = [1 ,0]^T$ that maximize the objective function. Can this extend to more general case?



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01:53

Let $F_n$ be the free group on $n$ letters. Let $g_1,...,g_{2m} \in F_n$, can the group $$F_n / \langle\langle[g_1,g_2],...,[g_{2m-1},g_{2m}]\rangle\rangle$$ ever have torsion elements?

The double angle brackets means "normal subgroup generated by" and $[a,b] = aba^{-1}b^{-1}$.

This problem arose when I saw a question in an old paper of Yanagawa ("On Ribbon 2-knots, II") that mentioned that it was unknown if a complement of a ribbon 2-knot could have torsion. These groups have presentations that are special cases of what I asked about in the question so I was guessing there was a simple counterexample to my question, but I had no inspiration to find it. Just FYI - the main result of that paper is a well known open problem (are ribbon disk complements aspherical?), so the proof of the main result is flawed... Although I didn't go looking for the error.

Edit : I just wanted to clarify that the elements $g_i$ are arbitrary elements (not necessarily the generators). So user1729's nice answer answers the question in the case where the $g_i$ are generators, but I am still interested in the question for general $g_i$. Also, for those interested in the topological origin of this problem, I actually messed up with the relations that arise in the context of the aforementioned paper. The groups that arise as ribbon group complements are of the form $F_n / << x_1 = x_2^{g_1}, x_2 = x_3^{g_2},...,x_{n_1} = x_n^{g_n} >>$ where here the $x_i$ are the generators of $F_n$ and again the $g_i$ are arbitrary elements of $F_n$. I would like to know if these groups can have any torsion as well.



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01:23

Let $T:V\to V$ be a bounded linear operator on a finite vector space $V$. If the sequence $\frac{1}{n}T^n$ converges, can we prove that its limit is the zero operator?

I think that the answer is yes, but I am struggling a bit with the proof. One approach could be to prove that $\|T\|\leq 1$ but I don't know how to proceed. One could also play around with the sequence terms by setting $S_n=\frac{1}{n}T^n$, $S_0:=\lim_{n\to+\infty}S_n$ and observing that $S_{n+1}=\frac{n}{n+1}TS_n$ which gives $S_0=TS_0$ but I don't know if it is helpful.



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22:53

Let $A$ be a (not necessarily commutative) algebra over a field $k$. Suppose that for all $a,b\in A$, we have $kab=kba$, i.e. commutativity up to scalar. Show that then $A$ is commutative.

In the assumption, it is important that it holds for all $a,b\in A$, otherwise it would be false. This is a step in Exercise 2.4.8 of Radford's book "Hopf algebras".



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20:05

Hi r/math, I had a question on whether I should take a graduate course in algebra next semester. For reference, I’m currently a second year undergraduate math major, and I’m finishing courses using Rudin’s PMA, Dummit and Foote (up to chapter 13), and Stein’s Fourier Analysis book. Next semester I’ll be taking courses on measure theory and complex analysis. I really enjoyed my analysis classes but I can’t say I enjoyed the algebra course very much. I was wondering I should take a course next semester using Hungerford’s Algebra. The course syllabus indicates that the focus of the first semester will mainly be on introducing category theory and Galois Theory. Although I understand Algebra is important for all branches of mathematics, I have trepidation about taking the course since I feel my interests lie strongly in analysis. For reference, I will be applying to graduate schools next year and was wondering if they like to see breadth in coursework. Thanks.

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18:23

Below, "logic" means "regular logic containing first-order logic" in the sense of Ebbinghaus/Flum/Thomas.

Setup

For a logic $\mathcal{L}$, let $\mathcal{PA}(\mathcal{L})$ be the set of $\mathcal{L}$-sentences in the language of arithmetic consisting of:

  • the ordered semiring axioms, and

  • for each $\mathcal{L}$-formula $\varphi(x,y_1,...,y_n)$ the induction instance $$\forall y_1,...,y_n[[\varphi(0,y_1,...,y_n)\wedge\forall x(\varphi(x,y_1,...,y_n)\rightarrow\varphi(x+1,y_1,...,y_n))]\rightarrow\forall x\varphi(x,y_1,...,y_n)].$$

(Note that even if the new logic $\mathcal{L}$ has additional types of variable - e.g. "set" variables - the corresponding induction instances will only allow "number" variables $y_1,...,y_n$ for the parameters.)

For example, $\mathcal{PA}(FOL)$ is just the usual (first-order) PA, and $\mathcal{PA}(SOL)$ characterizes the standard model $\mathbb{N}$ up to isomorphism. Also, we always have $\mathbb{N}\models_\mathcal{L}\mathcal{PA}(\mathcal{L})$.

  • Note that it's not quite true that $\mathcal{PA}(SOL)$ "is" second-order PA as usually phrased - induction-wise, we still have a scheme as opposed to a single sentence. However, induction applied to "has finitely many predecessors" gets the job done. (If we run the analogous construction with ZFC in place of PA, though, things seem more interesting ....)

Question

Say that a logic $\mathcal{L}$ is PA-intermediate if we have $PA<\mathcal{PA}(\mathcal{L})<Th_{FOL}(\mathbb{N})$ in the following sense:

  • There is a first-order sentence $\varphi$ such that $PA\not\models\varphi$ but $\mathcal{PA}(\mathcal{L})\models\varphi$.

  • There is a first-order sentence $\theta$ such that $\mathbb{N}\models\theta$ but $\mathcal{PA}(\mathcal{L})\not\models\theta$.

Is there a "natural" PA-intermediate logic?

(This is obviously fuzzy; for precision, I'll interpret "natural" as "has appeared in at least two different papers whose respective authorsets are incomparable.")

Partial progress

  • A couple non-examples:

    • Infinitary logic $\mathcal{L}_{\omega_1,\omega}$ is not PA-intermediate, since in fact $\mathcal{PA}(\mathcal{L}_{\omega_1,\omega})$ pins down $\mathbb{N}$ up to isomorphism (consider $\forall x(\bigvee_{i\in\omega}x=1+...+1\mbox{ ($i$ times)}$). So both second-order and infinitary logic bring in too much extra power.

    • Adding an equicardinality quantifier ($Ix(\varphi(x);\psi(x))$ = "As many $x$ satisfy $\varphi$ as $\psi$") also results in pinning down $\mathbb{N}$ up to isomorphism (consider "$\neg$ As many $x$ are $<k$ as are $<k+1$"). If we add the weak equicardinality quantifer $Qx\varphi(x) = Ix(\varphi(x);\neg\varphi(x))$, on the other hand, we wind up with a conservative extension of $PA$. That said, the latter is stronger semantically: no countable nonstandard model of $PA$ satisfies $\mathcal{PA}(FOL[Q])$ (to prove conservativity over $PA$ we look at the $\omega_1$-like models).

  • The proof of Lindstrom's theorem yields a weak negative result: if $\mathcal{L}$ is a logic strictly stronger than first-order logic with the downward Lowenheim-Skolem property, then we can whip up an $\mathcal{L}$-sentence $\varphi$ and an appropriate tuple of formulas $\Theta$ such that $\varphi$ is satisfiable and $\Theta^M\cong\mathbb{N}$ in every $M\models\varphi$. So an example for the above question would have to either allow nasty implicit definability shenanigans or lack the downward Lowenheim-Skolem property. This rules out a whole additional slew of candidates. That said, there is a surprising amount of variety among logics without the downward Lowenheim-Skolem property even in relatively concrete contexts - e.g. there is a compact logic strictly stronger than FOL on countable structures.



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02:08

In this series of posts, we’ll be featuring mathematical podcasts from all over the internet, by speaking to the creators of the podcast and asking them about what they do.

We spoke to Evelyn Lamb and Kevin Knudson, who interview mathematicians for their podcast, My Favorite Theorem.

Evelyn Lamb and Kevin Knudson, looking all podcast-hosty

Podcast title: My Favorite Theorems
Website: kpknudson.com/my-favorite-theorem (@myfavethm on Twitter)
Links: Apple – Player.fm – RSS
Average episode length: 30 minutes
Recommended episode: obviously, the one with Katie Steckles in

What is your podcast about, and when/why did it start?

As the name suggests, we talk with mathematicians about theorems! Of course, the podcast is also about who those mathematicians are as people, the interesting things that they do, and how they think about math. One of our favorite parts of the show is that we ask mathematicians to “pair” their theorems with something, like a food and wine pairing. The pairings have been everything from food and drink to music and literature and sports, so we leave it wide open. Getting mathematicians to talk a bit impressionistically about why something from everyday life reminds them of this bit of math is really fun.

Kevin had the idea in early 2017 to have a podcast focused on talking with mathematicians about theorems, and he invited Evelyn on. (They had interacted a bit in the math Twitter/blog world.) Evelyn didn’t want to sign on initially and suggested some other people he should talk to, but luckily he didn’t. As she thought about the idea more, she got more enthusiastic, and when she came up with the idea for theorem pairings, she really wanted to do it. We recorded our first episode March 23 (Emmy Noether’s birthday, so it’s easy to remember) and recorded a few more before our launch in July of that year.

How is your podcast published?

We are entirely independent. Kevin hosts the episodes (including audio, show notes, and transcripts) on his website, and we have Twitter and Facebook pages. Evelyn writes blog posts on her Scientific American blog, Roots of Unity, about each episode to get them a little more exposure from that direction. We are fortunate that we don’t need the podcast to be a money-making venture, at least right now, and we relish the freedom to set our own schedule, talk with exactly the guests we want in exactly the way we want to, and not read Casper or HelloFresh ads. (Not sure if those are universal references, but in the US it seems like every podcast is sponsored by mattresses and meal delivery kits.) Although we do not earn money from the podcast, it has definitely fed into Evelyn’s career in other ways. (She is a freelancer, and Kevin has a steady paycheck from an academic job.)

Who do you hope will listen to your podcast?

We’d like to think the episodes are entertaining for a broad audience regardless of math background, but realistically speaking, it seems like math-interested undergraduates and math graduate students/faculty make up most of our audience. The accessibility level varies a bit from episode to episode because our guests all want to talk about something different, and that’s something that we love about the podcast, but it also makes it a bit difficult to say exactly who the podcast is for. (Another reason we love being independent!)

What is a typical episode like?

We’ve got a short intro and usually get to introducing our guest pretty quickly. They tell us a little bit about themselves and then their favorite theorem. We try to ask somewhat relevant questions about the theorem and their relationship to it, and then we ask them for a pairing. At the end we let them plug anything they have to plug and then say bye. Evelyn thinks podcasts can have a tendency to go too long, so we try to keep them on the shorter side. Towards the beginning, they were usually 15-20 minutes. They’ve grown a bit since then, and now our average length is probably around 30 minutes.

Why should people listen to My Favorite Theorem?

We think you should listen if you want to hear a diverse lineup of mathematicians talking about a diverse range of specific topics in a fairly casual way. We don’t just mean race and gender diversity, though those are important. We have mathematicians who work in a variety of fields at a variety of institutions with a variety of mathematical life stories behind them. We do not think of it as an educational podcast per se, which may be a bit different from some other math podcasts. We think the audio format and the situations in which people tend to listen to podcasts (commuting, cooking, cleaning, etc.) don’t lend themselves to that goal. Instead, we hope you feel like you’ve eavesdropped on an interesting and entertaining conversation about math and can find out more about the topic later if it piques your interest.

What are some highlights of the podcast so far?

Well of course, we loved having the one and only Katie Steckles on the show! There’s a bit of a recency bias when we think about highlights, so we’ll just lean in to that. (Otherwise, we’d end up with a list of 51 favorite episodes!) The last two we published were a lot of fun. Carina Curto talked about linear algebra, and aBa Mbirika talked about the theorem that made him decide to be a mathematician.

Last year at the Joint Mathematics Meetings, we recorded short “flash favorite” theorems from 16 different mathematicians, who each talked about their favorite theorem for a couple minutes. It seems like our listeners really enjoyed the change of pace.

A couple of our early episodes that were really great for really different reasons were our episodes with Eriko Hironaka and Henry Fowler. Hironaka’s theorem is pretty technical, but the way she talked about its importance in her life felt incredibly relatable. Fowler talked about the Pythagorean theorem, perhaps the only theorem a majority of US-educated adults know the name of, and how it connected to his Navajo heritage. I think those two episodes, which happened to be published back to back, highlight what an interesting mix of different mathematicians and theorems we have on the show.

What exciting plans do you have for the future? 

For the most part, we’re just kind of trucking along. This is a side project for both of us, and we’re pretty happy with how it’s going, and so plan on keeping on with that. We’ve got some nice guests lined up for the next few months and hope our listeners have as much fun listening as we did recording the episodes.



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23:23

Apologies in advance if this was asked before.

Say we have $X$ is embedded into $Y$, my understanding is that there exists an injective ring homomorphism from $X$ to $Y$. I have the following questions about the idea of 'embedding': (as I want to get more familiar with this terminology)

$1)$ Is the idea of embedding, loosely speaking, saying that there is a subring in $Y$ that is isomorphic to $X$?

$2)$ I saw this question earlier today, where the answer says 'properly, you get an embedding $\mathbb N\to K$'. However though, can the naturals be embedded to an arbitrary field $K$ if $K$ is finite?

$3)$ Lastly, maybe just a check of my understanding, is an inclusion map always an embedding map?



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21:23

Suppose $A = (a_n) = (a_1, a_2, a_3, . . .)$ is an positive, increasing sequence of integers.

Define an $A$- expressible number $c$ if $c$ is the alternating sum of a finite subsequence of $A.$ To form such a sum, choose a finite subset of the sequence $A,$ list those numbers in increasing order (no repetitions allowed), and combine them with alternating plus and minus signs. We allow the trivial case of one-element subsequences, so that each an is $A-$expressible.

Definition. Sequence $A = (a_n)$ is an “alt-basis” if every positive integer is uniquely $A-$ expressible. That is, for every integer $m > 0,$ there is exactly one way to express $m$ as an alternating sum of a finite subsequence of $A.$

Examples. Sequence $B = (2^{n−1}) = (1, 2, 4, 8, 16, . . .)$ is not an alt-basis because some numbers are B-expressible in more than one way. For instance $3 = −1 + 4 = 1 − 2 + 4.$

Sequence $C = (3^{n−1}) = (1, 3, 9, 27, 81, . . .)$ is not an alt-basis because some numbers (like 4 and 5) are not C-expressible.

An example of an alt-basis is $\{2^n-1\}=\{1,3,7,15,31,\ldots\}$

Is there a fairly simple test to determine whether a given sequence is an alt basis?

I have attempted to solve this from a limited knowledge in sequences and have found out various kinds of sequences do not work but fail to see what it is that could make it work.



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06:48

I want to show that any group extension

$$1\to \mathbb C^*\to A\xrightarrow{\pi} \mathbb C\to 0$$

is trivial, where the group structure is multiplicative on $\mathbb C^*$ and additive on $\mathbb C$.

Here is my thought: From the point of view of topology, $A$ is a principal $\mathbb C^*$-bundle over the contractible base $\mathbb C$. This implies $A$ is trivial as principal $\mathbb C^*$-bundle, so there is a section $s:\mathbb C\to A$ (satisfying $\pi\circ s=Id$). However, the section need not preserve the group structure.

Can I modify the section $s$ so as preserve the group structure?



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03:48

I am trying to prove the following assertion (, which is an exercise from a section on tensor products in the book Algebra: Chapter 0 by P.Aluffi):

Let $F=k(\alpha)\supset k$ be a finite simple extension such that $F\otimes_kF\cong F^{[F:k]}$ as a ring. Then the extension is Galois.

A related question has been asked several times on this site, for example in here. Reading the answers to these questions, I learned that the above assertion may be proved as follows:

  1. There is a canonical isomorphism $F\otimes_k F\cong F[x]/(m(x))$ of $k$-algebras. Now $F^n$ is obviously reduced, so $F[x]/(m(x))$ is also reduced. It follows that $\alpha$ is separable over $k$. (This step requires a little more argument, but I am OK with this step.)
  2. Therefore, to prove that $F/k$ is Galois, it suffices to show that the $\alpha$ splits over $F$.
  3. Let $m(x)\in k[x]$ be the minimal polynomial of $\alpha$ over $k$. But $F[x]/(m(x))\cong F^n$ as a ring by hypothesis. By the Chinese remainder theorem, this implies that $m(x)$ factors into linear factors over $F$.

I do not understand the bold faced part in step 3. Sure, the Chinese remainder theorem implies that if $m(x)$ splits over $F$, then $F[x]/(m(x))\cong F^n$ as rings. However, we want to go in the other direction. How do we do this?

Thanks in advance.

In addition to the above question, I would greatly appreciate the answers to the following additional questions. (But you do not have to answer these unless you want to.)

  • The question which I cited above actually assumed that the isomorphism $F\otimes_k F\cong F^{[F:k]}$ to be an $F$-algebra isomorphism. In my book, this isomorphism is assumed to be merely a ring isomorphism. Maybe the author took it for granted that the isomorphism to be $F$-linear? Or maybe we can do without the $F$-linearity.
  • In general, which quotient of $F[x]$ isomorphic to the direct product of $F$ as a ring (or as an $F$-algebra)?


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03:48

Suppose I write down the Manhattan distance from the origin to a point (x,y) in terms of a series of n steps of length x/n in the x direction, alternated with m steps of length y/m in the y direction: $$d_{Manhattan} = \sum_{i=1}^n \frac{x}{n} + \sum_{i=1}^m \frac{y}{m}$$

Imagine walking an approximately diagonal line towards (x,y), zig-zagging parallel to the x and y axes.

Now, if we take the limits as $n\rightarrow \infty$ and $m\rightarrow \infty$ our path should approach the straight line connecting the origin to (x,y), suggesting that in the limit the Manhattan distance should equal $\sqrt{x^2+y^2}$. Why is this not the case? Is there a way to correctly arrive at Pythagoras by taking a limit using infinitesimal steps along the axis directions?



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03:18

Let $a>2$ be a real number and consider the following integral $$ I(a)=\int_0^\pi\int_0^\pi \frac{\sin^2(x)\sin^2(y)}{a+\cos(x)+\cos(y)} \mathrm{d}x\,\mathrm{d}y $$

My question. Does there exist a closed-form expression of $I(a)$?

Some comments. Since $a-2<a+\cos(x)+\cos(y)<a+2$ and $\int_0^\pi \int_0^\pi \sin^2(x)\sin^2(y)\ \mathrm{d}x\, \mathrm{d}y=\frac{\pi^2}{4}$, we have the following bounds $$ \frac{\pi^2}{4(a+2)} < I(a) < \frac{\pi^2}{4(a-2)}, $$ however I didn't manage to find an exact expression for $I(a)$. Any help is welcome!



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02:53

I am a Maths teacher. Teaching of Mathematics in Senior High School.

My school has been closed down. I am trying to figure out what is the best way to teach students long distance.

Is anyone familiar with the different options available and can recommend me one?

I was thinking of setting up a way where I could write on a screen and it was recorded, with my voice over, explaining things and streaming/making the video available to students. Just wondering what the best way to do it is and if there are better ways of basically "giving the classes you were supposed to be giving if you could go to the school class".

Apologies if this has been answered and/or is too much of a "noob" question.

Thank you

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22:53

I was just accepted to a masters degree program in applied mathematics. My advisor recommended that I do a concentration in dynamical systems given my background and interests.

I currently have a BS in mechanical engineering but I have been unsatisfied with my career prospects in that field. The only kind of jobs I’ve had have involved paper-pushing, project management, and forwarding emails. I’m interested in jumping into fast growing tech fields like software development, AI, algorithms, data analytics. I’m curious to know if there are any applications to those fields with dynamical systems. What kind of careers am I looking at with this concentration?

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22:18

Maybe, we can make use of Tannery's Theorem, or dominated convergence theorem, to exchange the order of the limit and summation:

\begin{align*} \lim_{n \to \infty}\sum_{k=1}^n\left(\frac{k}{n}\right)^k&=\lim_{n \to \infty}\sum_{k=0}^{n-1}\left[\left(1-\frac{k}{n}\right)^{n}\right]^{\frac{n-k}{n}}=\sum_{k=0}^{\infty}\lim_{n \to \infty}\left[\left(1-\frac{k}{n}\right)^{n}\right]^{\frac{n-k}{n}}=\sum_{k=0}^{\infty} e^{-k}=\frac{e}{e-1} \end{align*}

This is correct? How to verify that it satisfy the conditons of the theorem?



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22:18

I read very often and i highlight definitions in orange and theorems in purple, but sometimes I cant really distinguish a definition from a theorem, for example:

Let $A$ be a finite set and $B$ a nonempty set. $|A|≥|B|$ if and only if there exists a function that maps A onto B.

$|A|$ and $|B|$ represent the cardinality of $A$ and $B$ respectively.

I don't understand if this is a theorem (or a rule in general), or a definition, how do you distinguish definitions from theorems since both usually use the sentence "if and only if"?



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21:48

I tried to get an answer to this question (which was hastily closed) but couldn't find a proof, so I decided to ask it again, adding some of my efforts.

Suppose we have a finite sequence of $n$ circles ($n\ge10$, see figure below) whose centres lie on the major axis of an ellipse. All circles are internally tangent to the ellipse and each circle is also externally tangent to the preceding and following circle (if they exist). If $r_1$, $r_2$, ..., $r_n$ are the radii of these circles, prove that: $$ r_7(r_1 + r_7) = r_4(r_4 + r_{10}). $$

enter image description here

If $x_0$, $x_1$, ..., $x_n$ are the abscissae of the intersection points between the circles and the major axis (taking as origin the ellipse centre, see figure above), then it is not difficult to find a recursive relation for $x_k$. Let $a$, $b$ be the semi-major axis and semi-minor axis of the ellipse, $A$ and $B$ its foci, $O$ its centre and $c=AO=BO=\sqrt{a^2-b^2}$. If $C_k$ is the centre of $k$-th circle and $P_k$ one of its tangency points with the ellipse, then radius $P_kC_k$ is the normal to the ellipse at $P_k$ and thus the bisector of $\angle AP_kB$. It follows from the length of bisector formula that

$$ P_kC_k={b^2\over a^2}\sqrt{AP_k\cdot BP_k}={b^2\over c^2}\sqrt{AC_k\cdot BC_k}= {b^2\over c^2}\sqrt{c^2-c_k^2}, $$ where $c_k$ is the abscissa of centre $C_k$. Inserting here $P_kC_k=(x_{k}-x_{k-1})/2$ and $c_k=(x_{k}+x_{k-1})/2$, then squaring both sides and rearranging, one finds: $$ x_k^2+x_{k-1}^2-2(2e^2-1)x_kx_{k-1}=4e^2b^2, $$ where $e=c/a$ is the eccentricity of the ellipse. From the above recursive equation one can find, once $x_0$ is given, all $x_k$ and thus compute $r_k=(x_{k}-x_{k-1})/2$ for all values of $k$. I used these results with GeoGebra to draw the first figure, and could numerically check that the formula to prove holds for any value of $x_0$.

Nonetheless, I couldn't obtain a real proof of that formula using algebra, hence I believe I'm missing a simpler way to find those radii. Any idea to prove the statement is welcome.

enter image description here



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