This is a follow-up question to Is there a reduced ring with exactly $3$ idempotents?, to which the answer was "no."
Note: In this question, 'ring' means ring with unity, but not necessarily commutative
In fact, in a (non-trivial) reduced ring, the number of idempotents is either even or $\infty$. The reason is that the idempotents come in pairs $e,1-e$. And $e \neq1-e$, otherwise $ee=e-ee$ and $e^2=0$, implying (since the ring is reduced) that $e=0$, which can't happen if $e=1-e$.
My next question is, does there exist a reduced ring whose number of idempotents is a multiple of $3$? (For example, can we find a reduced ring with $6$ idempotent elements? $12$? $18$? $3000$?)
What about rings in general? (i.e. not necessarily reduced)
Attempting the easiest case first, assume $R$ is a reduced ring and the idempotent elements are $\{0,1,a,(1-a),b,(1-b)\}$ (all distinct). I see that the product of two idempotents must be idempotent (since the idempotents commute with everything). Also, I see that the square of the difference of two idempotents must also be idempotent. So $ab \in \{0,1,a,(1-a),b,(1-b)\}$ . (I suspect that there might be a way to derive a contradiction from this, although I don't see how to do so yet.)
The professor teaching a class I am taking wants me to find the eigenvalues and the eigenvectors for the following matrix below.
$$\begin{bmatrix}-5 & 5\\4 & 3\end{bmatrix}$$
I have succeeded in getting the eigenvalues, which are $\lambda= \{ 5,-7 \}$. When finding the eigenvector for $\lambda= 5$, I get $\begin{bmatrix}1/2\\1 \end{bmatrix}$. However, the correct answer is $\begin{bmatrix}1\\2 \end{bmatrix}$ .
I have tried doing this question using multiple online matrix calculators. One of which gives me $\begin{bmatrix}1/2\\1 \end{bmatrix}$, and the other gives me $\begin{bmatrix}1\\2 \end{bmatrix}$.
The online calculator that gave me $\begin{bmatrix}1\\2 \end{bmatrix}$ explains, that y=2, hence $\begin{bmatrix}1/2*2\\1=>2 \end{bmatrix}$ =>$\begin{bmatrix}1\\2 \end{bmatrix}$.
What I do not understand is, why is y must equal to 2?Is it because there cannot be a fraction in an eigenvector?
There must be an error in my "proof" since it is evident that the sum of two irrational numbers may be rational, but I am struggling to spot it. A hint would be appreciated.
The "proof" is by contradiction:
Assume that the sum of two irrational numbers a and b is rational. Then we can write
$$ a + b = \frac{x}{y} $$
$$ \implies a + b + a - a = \frac{x}{y} $$
$$ \implies 2a + (b - a) = \frac{x}{y} $$
$$ \implies 2a = \frac{x}{y} + (-1)(b + (-1)(a)) $$
-> from our assumption that the sum of two irrational numbers is rational, it follows that $(b + (-1)(a))$ is rational
-> therefore, the right side is rational, being the sum of two rational numbers
-> but the left side, $2a$, is irrational, because the product of a rational and irrational number is irrational
-> this is a contradiction; since assuming that the sum of two irrational numbers is rational leads to a contradiction, the sum of two irrational numbers must be irrational.
Consider a group $(G,\cdot)$ with the property that $\exists a\in G, a\neq e$, such that $G\setminus \{a\}$ is a subgroup of $G$. Prove that $(G,\cdot) \cong (\mathbb{Z}/2\mathbb Z,+)$.
We know that if $H$ is a subgroup of $G$ then $\forall x \in H, y\in G\setminus H$ we have that $xy \in G \setminus H$.
In our case, $\forall x\neq a$, $xa \in G \setminus (G\setminus \{a\})=\{a\}$ and this implies that $\forall x\neq a$, $x=e$.
As a result, $G=\{e,a\}$ and it is well-known and easy to prove that any group of order $2$ is isomorphic to $(\mathbb{Z}/2\mathbb Z,+)$.
I would like to know if my proof is correct.
As the title implies: what is bigger $\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$. Specifically I am interested in working this out without actually calculating the values. So far I have tried applying order preserving operations on both and seeing if the comparison will become clearer but this has so far been unyieldy because I am stuck at the following point:
$\sqrt2^{\sqrt3^\sqrt3}$ or $\sqrt3^{\sqrt2^\sqrt2}$
$e^{\sqrt3^\sqrt3\ln\sqrt2}$ or $e^{\sqrt2^\sqrt2\ln\sqrt3}$
${\sqrt3^\sqrt3\ln\sqrt2}$ or ${\sqrt2^\sqrt2\ln\sqrt3}$
${\sqrt3^\sqrt3\ln2}$ or ${\sqrt2^\sqrt2\ln3}$
And at this point I have explored a few options but nothing has made it clear. Have I been pursuing the correct root (if you pardon the pun) and how should I proceed.
Update:
$\ln({\sqrt3^\sqrt3\ln2})$ or $\ln({\sqrt2^\sqrt2\ln3})$
$\frac{\sqrt3}{2}\ln3 +\ln({\ln2})$ or $\frac{\sqrt2}{2}\ln2 +\ln({\ln3})$
The William Lowell Putnam Mathematical Competition (see https://www.maa.org/math-competitions/putnam-competition for more details), commonly known as 'The Putnam', is a famously difficult mathematics competition taking place in the United States and Canada on Dec. 7 of this year. The median score on the Putnam is typically a 0 or 1 out of 120. The contest has two parts, each consisting of 6 questions worth 10 points each.
Seeing as the competition is coming up, I thought it might be interesting to have a little discussion about it. What sort of tips and tricks do people like to employ on the Putnam (or other, similar contests I suppose)? What sort of strategies do people use for an approach; what are some unconventional theorems or lemmas that people have employed effectively? Does anyone have any particular favourite questions from past years, or stories about themselves writing the Putnam that they'd like to share? Let's chat!
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. We define the largest of among all the prime factor of these composite as the maximal prime factor of the gap.
Claim: Every prime is a maximal prime factor for some prime gap.
Experimental data for $p \le 10^5$ supports this claim. I am looking for a proof or disproof.
#tmwyk is a Twitter hashtag which stands for some approximation of “Talking math(s) with your/young kids”. It is used to share mathematical interactions with children. It is also the subject of my MathsJam talk this weekend.
For me, I tend to use #tmwyk to share playful interactions with my son, following his interests and the mathematics that we find in the world around him. I’m not trying to teach anything in particular, nor am I trying to limit his interests to what might come up at school.
“Algebra?” said Madam Frout … “But that’s far too difficult for seven-year-olds.”
Thief of Time, Terry Pratchett.
“Yes but I didn’t tell them that, and so far they haven’t found out,” said Susan.
So it covers some formal things like place-value, should this take his interest.
My 4yo & I are playing with tens and units. pic.twitter.com/rQgHXa7HQg
— Peter Rowlett (@peterrowlett) June 9, 2019
It also includes some recreational maths classics.
A giraffe, a flower/lion’s head, two rockets, some cargo and just constructing a funny sort of bone. So described by my 4yo. #tmwyk pic.twitter.com/X859i9LOFc
— Peter Rowlett (@peterrowlett) July 7, 2019
Further adventures in puzzling with my 3yo. We have a fox, a goose, a bag of beans and a boat. Let’s see if we can get them across this river! pic.twitter.com/SnOGDgfd9C
— Peter Rowlett (@peterrowlett) April 7, 2019
Also some items that might be less curriculum-relevant. His Six and Eight Bridges of Königsberg solutions were superb, as was his Two Utilities Problem.
My 4yo has enjoyed solving the Six and Eight Bridges of Königsberg, and the Two Utilities Puzzle. What other fun, achievable puzzles would you recommend we play with? (I haven’t told him he’s too young for graph theory.) #tmwyk pic.twitter.com/0n6sdJNwK9
— Peter Rowlett (@peterrowlett) October 13, 2019
But it also is just sometimes whatever mundane object has taken his interest.
“Wide, thin, wide, thin, …” he mutters to himself.
— Peter Rowlett (@peterrowlett) October 6, 2019
“What’s that you’re saying?” I ask.
“I’m just reading out the mat. There are 8 wide and 7 thin, so wides win!”#tmwyk pic.twitter.com/c6Sm8uWKUQ
We’re exploring self-similarity. #fractals #tmwyk pic.twitter.com/9TVsMNbtVu
— Peter Rowlett (@peterrowlett) September 7, 2019
“Do I have more than one thousand pieces of Lego?”
— Peter Rowlett (@peterrowlett) October 27, 2019
(We did some counting.)
“Yes!”
As a bonus, we noticed 1000 is a square number.#tmwyk pic.twitter.com/QSO1t61YEd
Object graphs are good fun.
Object graphs with my 4yo #tmwyk (HT @honeypisquared.) pic.twitter.com/zCaZNmJmTc
— Peter Rowlett (@peterrowlett) July 13, 2019
Object graphs, a game my son calls “sorting things”. Same objects, different classification for cars and dinosaurs. #tmwyk pic.twitter.com/rxyUZJeVNW
— Peter Rowlett (@peterrowlett) November 3, 2019
Number blocks are amazing, and lead to all sorts of mathematical play and interesting insight.
Inspired by @kyledevans’ book, a kind gift by @stecks, we’re trying to arrange blocks into squares, rectangles and lines.
— Peter Rowlett (@peterrowlett) July 27, 2019
My 4yo: “Twelve is a super-rectangle!” #tmwyk pic.twitter.com/Q7ESJqCQut
My son makes conjectures, which he then proves or disproves using number blocks.
At his insistence, we are testing my son’s conjecture that 21 is prime. #tmwyk pic.twitter.com/eHQvM7n8LJ
— Peter Rowlett (@peterrowlett) July 27, 2019
Looking at the pile of eleven nines, he conjectured that we could break 99 into “some sixes and one three”, so we’re trying that. #tmwyk pic.twitter.com/meXyGNI4kc
— Peter Rowlett (@peterrowlett) August 3, 2019
My 4yo’s conjecture this morning: all squares are even. Disproven by counterexample. (His lifetime 4th ever conjecture.) #tmwyk pic.twitter.com/f05rAZGkYC
— Peter Rowlett (@peterrowlett) August 18, 2019
His best invention may be the concept of the sillion. This is in the line of million, billion, trillion, etc. except it’s silly big, and behaves silly. For example, there are no bigger numbers than a sillion; if you adds lots of sillions the total get smaller.
More on properties of a sillion. My son tells me if you add up lots of sillions, the total gets smaller. Because, you know, it’s silly. #tmwyk https://t.co/1iwB24nLqL
— Peter Rowlett (@peterrowlett) October 8, 2019
He has the idea that numbers go on forever – in both positive and negative directions.
Trying to explain, I said 1 is special because it’s the first number and he loudly declared: “But there’s no first number! It goes on and on forever backwards and on and on forever forwards.” Heavy sigh. “We’ll never know what the last number is.” #tmwyk
— Peter Rowlett (@peterrowlett) November 19, 2019
All this thinking about numbers can get pretty serious. For example, I’m entirely happy with this as a four-year-old’s proof.
Normally if I #tmwyk it’s just playing around or his query, but on the walk to school I thought I’d test his intuition. I asked if I lined them up, would there be more odds or evens?
— Peter Rowlett (@peterrowlett) September 23, 2019
“The same.”
How do you know?
“There’s evens between odds.”
Think I’m happy with that!#4yoproofs
I’m not at all qualified to speak about early years or primary education, but I am fascinated in how much good his playing with mathematical ideas seems to do. I’m interested more broadly in the use of play in mathematics education, though. I teach a final year undergraduate module called Game Theory and Recreational Mathematics. If you are interested to learn more about using recreational maths in education and the design and operation of our module, you can read our article about this: ‘The potential of recreational mathematics to support the development of mathematical learning‘.
Every day I put 1 or 2 dollars in the piggy bank with probability $1/2$. What's is expected value number of 2-dollar bills when in the piggy bank will be for the first time at least 100 dollars ?
I know what is going on. Let $X$- number of 2-dollar bills.
1) $X=1$ :
1 dollar- 98 bills and 2 dollars- 1 bill or 1 dollar - 99 bills and 2-dollars- 1 bill
2) $X=2$
1 dollar- 96 bills and 2 dollars- 2 bills or 1 dollar- 97 bills and 2 dollars- 2 bills
etc.
Unfortunately, I can't think of a quick way.
Casella & Berger (2001) write:
the Axiom of Countable Additivity, is not universally accepted among statisticians. ... [It] is rejected by a school of statisticians led by deFinetti (1972), who chooses to replace this axiom with the Axiom of Finite Additivity.
What might possibly be wrong or objectionable about the Axiom of Countable Additivity?
(Simple examples would be helpful!)
The Axiom of Countable Additivity is the 3rd condition below:
Take a (kind of) arrowhead real-symmetric matrix of the general form
$$ M = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{12} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{13} & a_{23} & a_{33} & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & a_{44} & 0 & 0 \\ a_{15} & a_{25} & 0 & 0 & a_{55} & 0 \\ a_{16} & a_{26} & 0 & 0 & 0 & a_{66} \\ \end{bmatrix} $$
where the size of the blocks may vary, however in general, the diagonal submatrix will be of dimension close to that of the entire matrix. Is there a method to diagonalise this matrix which takes advantage of this largely diagonal structure? My desire is computational efficiency, i.e. compared to dgemm.
I require all of the eigenvalues and eigenvectors of this matrix, i.e. $V^{-1}MV = W$ where $V$ are the eigenvectors of $M$, and $W$ a diagonal matrix containing the eigenvalues.
Does $\displaystyle \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$ converge uniformly on $(-\infty,\infty)$?
My attempt:
No. Consider the case where $x=0$, then $\displaystyle \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n} = 0$.
For $x \neq 0$, observe $\displaystyle 0 \lt \frac{1}{(1+x^2)^n} \lt 1$, so by geometric series formula
$\displaystyle \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$ $\displaystyle = \frac{x^2}{1 - \frac{1}{1+x^2}} = 1+x^2$
(1) So clearly the series doesn't even converge for all $x$, let alone converge uniformly.
Now, my question is about the case where $x \neq 0$. Does it converge uniformly to $1 + x^2$?
(2) I think, "yes". By Dini's theorem for series the convergence of the series to $1 + x^2$ must be uniform since $1+x^2$ is continuous and $(-\infty,0) \cup (0,\infty)$ is compact.
Is my reasoning for (1) and (2) correct?
On Thursday the 5th of December 2019, Elena Di Domenico (University of Trento and University of the Basque Country) will be giving a talk on “Branch structures of some GGS-groups”. The talk will be at 4pm in INB 3305. This seminar is part of the Lincoln Algebra Research Afternoons (LARA) talks.
Let $p, q \in \mathbb Q$, $n \in \mathbb Z^+$ and label $a = \sqrt[n]p, b=\sqrt[n]q$.
Conjecture: If $a + b$ is a non-zero rational, then both $a$ and $b$ are rational.
(Preliminary question: is this a known result I'm not aware of?)
I believe I have found a partial proof of the above. Namely,
28 Nov 2019
In this Newsletter:
1. New on IntMath: Witch of Agnesi
2. Resources: Hypertools, tech
3. Math in the news: Light, Etalumis
4. Math movies: Mod, culture
5. Math puzzle: Mean, mode
6. Final thought: 20
I've started a series of articles on historical math curves.
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The first one is the "Witch of Agnesi", which is not a witch at all, and was included in a math handbook written by a rarity of the 18th century: a female mathematics academic. See: Witch of Agnesi |
The page has some background to the curve and some animations explaining how it is constructed. There's even a connection between this curve and calculus.
Data will be the most valuable commodity in the 21st century, and those able to make sense of it will do well. This tool could help.
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Dartmouth College has released the open-source package HyperTools, which can transform data into visualizable shapes or animations for data comparisons and testing theories. See: HyperTools transforms complex data into visualizable shapes |
Most of you will know I'm a fan of technological approaches to teaching and learning of mathematics.
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Here are some resources (some free, some paid) that can help with this. |
Image processing is at the core of augmented reality, autonomous driving and facial recognition. But because it's currently achieved using electronic circuits, it's too slow.
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A new optical computing and imaging technique operates at the speed of light and the mathematical operation itself consumes no energy. |
Simulations are developed to try to understand and predict the future of such things as climate change, disease transmission and cosmic events. But how do we interpret and make best use of new data?
A new system, called Etalumis ("simulate" spelled backwards) developed by a group of scientists from the University of Oxford and others, makes use of Bayes' Theorem (which is a way to calculate probabilities based on probabilities for events that have already occurred) to solve such problems.
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Etalumis uses Bayes inference to improve existing simulators via machine learning. |
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This video (by Brilliant) explains modular arithmetic. It moves fairly quickly, but that's the beauty of videos - you can pause and go over parts. |
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This is an interesting way to visualize the spread of culture throughout history. The animation distils hundreds of years of culture into just five minutes. See: Charting Culture |
The comments rightly point out some of the cultural biases in the video, but it probably can't be helped when compressing such a highly complex data set into just 5 minutes.
Clarification: The puzzle in the September IntMath Newsletter asked about the probability involved in a given octagon. There were conflicting answers given, and I provided a solution. However, it turned out my attempt had a problem, and Tomas' answer was correct. You can see the discussion here: Octagon discussion
The puzzle in the last (October) IntMath Newsletter asked about "square ages".
There were two parts to the puzzle, and Thomas was the only one who got both parts correct. (Special mention to Don who got the first part, and the range of years correct for the second part.)
The table below shows the daily earnings for two workers for 5 days and the mean and the median salary.
| Person A | Person B | |
|---|---|---|
| Day 1 | $100 | $72 |
| Day 2 | $87 | $97 |
| Day 3 | $90 | $70 |
| Day 4 | $10 | $71 |
| Day 5 | $91 | $100 |
| Mean | $75.60 | $82 |
| Median | $90 | $72 |
(a) Which statistic, the mean or the median, would be best to describe the typical daily salary for the 5 days for Person A? How about for person B?
You can leave your responses here.
Jean-Francois Rischard, ex-World Bank vice president wrote the book High Noon: 20 Global Problems, 20 Years To Solve Them in 2003. Sadly, the scorecard for his 20 Problems is rather abysmal as we approach the 20th year of the century.
I wrote a summary review of the book in 2008. I concluded by saying:
Mathematics is involved in the solutions for all these problems. So instead of doing algebra with no purpose other than to pass an examination, let's put our mathematical efforts into coming up with ways to solve (at least some of) these problems.
Governments should be placing many of the 20 Global Problems in the center of their policy frameworks, but they're not, and too many governments are actively doing the opposite.
As a counterpoint to the destruction of the Amazon, here's what one Brazilian photographer achieved in 18 years, something that should be happening worldwide:
Please be careful who you vote for...
Until next time, enjoy whatever you learn.
Related posts:
Consider a divergent series that tends to infinity such as $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$. The limit of this series is unbounded, and I have often seen people say that the sum 'equals infinity' as a shorthand for this. However, is it acceptable to write $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \infty$ in formal mathematics, or is it better to denote that the limit is equal to infinity. If so, how does one do this?
I had just learned in measure theory class about the monotone convergence theorem in this version:
for every monotonically increasing series of functions $f_n$ from measurable space $X$ to $[0, \infty]$,
$$ \text{if}\quad \lim_{n\to \infty}f_n = f, \quad\text{then}\quad \lim_{n\to \infty}\int f_n \, \mathrm{d}\mu = \int f \,\mathrm{d}\mu . $$
I tried to find out why this theorem apply only for Lebesgue integral, but I didn't find counter example for Riemann integrals, so I would appreciate your help.
(I guess that $f$ might not be integrable in some cases but I want a concrete example)
This is my favorite puzzle in the last issue of the Emissary, proposed by Peter Winkler.
Puzzle. Alice and Bob each have 100 dollars and a biased coin that flips heads with probability 51%. At a signal, each begins flipping his or her coin once per minute, and betting 1 dollar (at even odds) on each flip. Alice bets on heads; poor Bob, on tails. As it happens, however, both eventually go broke. Who is more likely to have gone broke first?
Follow-up question: As above, but this time Alice and Bob are flipping the same coin (biased 51% toward heads). Again, assume both eventually go broke. Who is more likely to have gone broke first?
this is the CUDA code for Path tracing Thurston's sphere eversion in CUDA | 49k triangles, 200 trillion intersections. The sphere is represented as an array of triangles. The triangles are represented as one vertex plus two edges. The edges are computed by subtracting vertices. The vertices are generated by (essentially) Nathaniel Thurston's evert. The triangle mesh path tracer is based on Sam Lapere's. It uses no acceleration data structures (so it takes forever to run)
also here's a little video with more animations & stuff
I've tried to "simplify" Nathaniel Thurston's original implementation by removing all 2nd-order and 3rd-order partial derivatives, which weren't being used (I think). Thurston's code features something called "jets)", which I think are only there to implement automatic differentiation (the latest part of the eversion uses partial derivatives for... something)
This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on over the week/weekend. This can be anything from math-related arts and crafts, what you've been learning in class, books/papers you're reading, to preparing for a conference. All types and levels of mathematics are welcomed!
I'm currently a master's student of mathematics applying for Pure Mathematics PhD programs. I was diagnosed with ADHD after graduating from undergrad and currently wonder whether I should mention it in personal statements or not. I wouldn't bother if my grades throughout undergrad were perfect, but they were not. My undergrad GPA is just shy of a 3.4 with math GPA just below a 3.5. It's safe to say that I began my mathematics career with poor time management skills and no idea how to study math, but my diagnosis retroactively explains why it took so long to complete problems, why I had trouble focusing during lectures despite interest in the material, etc.
I began taking medication before starting my first semester of graduate coursework and I'm currently on track to get As in all three classes. I attribute the improvement partially to the diagnosis and treatment, so on one hand, I can point to it and say "Hey, the person who fucked up Calculus had an undiagnosed disorder, look at how he's doing now that he's begun treatment." This has explanatory power for academic record, could be used to talk about overcoming obstacles (I worked hard and improved my performance with proof-based classes even before seeking treatment), etc. I can think of two strong reasons to not mention it.
My issue with 1 is that I have to offer some explanation as to why my undergraduate math GPA is on the lower end, and any explanation could be seen as an excuse. The second I think could be resolved by simply noting that I'm doing well now after receiving treatment.
My friend recently gave me this system of functional equations, asking me if I could find holomorphic $f,g: \mathbb{C} \to \mathbb{C}$ satisfying:
To which I promptly said, “no.” Thoughts? Hints?
Let $[q]:=\{1,2,3, \ldots,q\}$, where $q$ is a positive integer. Consider a vector $\underline{\alpha}=(\alpha_A)_{A\subseteq [q], A\neq \emptyset}$, where each $\alpha_A \in \mathbb{R}$. Note that such a vector $\underline{\alpha}$ has $2^q-1$ entries.
Given such an $\underline{\alpha}$, we define the following:
$$ F(\underline{\alpha}):=\sum_{A\neq \emptyset} \alpha_A \log(|A|), \quad v(\underline{\alpha})=\sum_{A\neq \emptyset} \alpha_A, \quad E(\underline{\alpha})=\sum_{\{ \{A,B\}: A,B \subseteq [q], A\cap B=\emptyset \}} \alpha_A \alpha_B, $$
where the sum in the definition of $E(\underline{\alpha})$ is taken over all unordered pairs of disjoint subsets of $[3]$.
Define $T(1/4)=\{\underline{\alpha}: \alpha_A \geq 0 \text{ for all nonempty A },\, v(\underline{\alpha})=1, \, E(\underline{\alpha})\geq 1/4 \}$.
I believe that the following is true: $$ \max_{\underline{\alpha} \in T(1/4) } F(\underline{\alpha})=\frac{\log(\lfloor q/2 \rfloor \cdot \lceil q/2 \rceil)}{2}. $$
I know that it is true when $q$ is even (it was proven), but I want to show that it also holds for odd $q$. I have verified that this is true for $q=3,5,7,9$ using SageMath, but would like to prove it by hand for general odd $q$.
I think that this problem can be solved using the language of probability. It is natural to do so given that the sum $v(\underline{\alpha})=\sum \alpha_A=1$ in the set $T(1/4)$.
Consider a random variable $X$ on $2^{q}\setminus \{\emptyset\}$ (the power set of $[q]$, excluding the empty set) such that $\mathbb{P}[X=A]=\alpha_A$, where $A \subseteq [q]$.
Note that $2\cdot E(\underline{\alpha})$ can be interpreted as the probability that from the sets in $2^{[q]}\setminus \{\emptyset\}$ one selects two disjoint sets $A$ and $B$. Since by definition of $T(1/4)$, $2\cdot E(\underline{\alpha}) \geq 1/2$, we see that we are more likely to select two disjoint sets rather than two sets which have a nonempty intersection.
One can verify that the set $T(1/4)$ is compact, so there exists some $\underline{\alpha}^*\in T(1/4)$ for which $F(\underline{\alpha}^*)=\max_{\underline{\alpha} \in T(1/4) } F(\underline{\alpha})$. I conjecture that this vector $\underline{\alpha}^*$ has exactly 2 nonzero entries $\alpha^*_{A_1}=1/2$ and $\alpha^*_{A_2}=1/2$, where $|A_1|=\lfloor q/2 \rfloor$, $|A_2|=\lceil q/2 \rceil$, $A_1\cap A_2 =\emptyset$, and $A_1\cup A_2=[q]$. That is, the sets $A_1$ and $A_2$ form a partition of the first $q$ positive integers and their sizes are as equal as possible.
If it is difficult to prove this for general odd $q$, how would I prove it for, say, $q=5$? I would like to avoid using Lagrange multipliers with so many variables.
Suppose I have a sequence of positive integers $\{a_n\}$. Let us denote $b_n=\max_{1\le i\le n} a_i$. Suppose $$\frac{b_n}{\sum\limits_{i=1}^n a_i} \to 0$$ then show that $$\frac{b_n^2}{\sum\limits_{i=1}^n a_i^2} \to 0$$
I am not sure if it is true. But I didn't find any Counterexample. I was trying to get a reasonable lower bound for the denominator. I could not find any. Bounds like $$\sum_{i=1}^n a_i^2 \ge \sum_{i=1}^n a_i$$ won't help though. Note that the converse is true. As you can easily get an upper bound using: $$\sum_{i=1} a_i^2 \le b_n\sum_{i=1} a_i$$
Any help/suggestions?
Edit: Note that $a_n$'s are positive integers, that's why $\sum a_i^2 \ge \sum a_i$ is true.
This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.
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Let $R = (A,+,\cdot)$ be a ring. If there exists a positive integer $n$ such that any element $x$ of a ring $R$ satisfies $x^{4^n+2} = x$, then every element $x$ in $R$ is idempotent.
I have studied some group theory but I don't think it's that helpful here. Basically by multiplying both sides a bunch of times with $x^{4^n + 1}$ you can show that $$x^{k(4^n + 1) + 1} = x \hspace{5px} \forall k \in \mathbb{N}.$$
My intuition goes along the lines of "if $x^a = x$ and $x^b = x$ then $x^{(a,b)} = x$" but in the absence of multiplicative inverses that wouldn't work (the reason I thought this might have been helpful is because the exponent of $2$ in $4^n + 2$ is always $1$, so by finding an appropriate $k$ we could make the gcd $2$).
How should I proceed?
I want to prove that $$8^x+4^x\geq 6^x+5^x$$ for all $x\geq 0$. How can I do this?
My attempt:
I try by AM-GM: $$8^x+4^x\geq 2\sqrt{8^x4^x}=2(\sqrt{32})^x.$$
However, $\sqrt{32}\approx 5.5$ so I am not sure if $$2(\sqrt{32})^x\geq 5^x+6^x$$ is true.
Also, I try to compute derivatives but this doesn't simplify the problem. What can I do?
$$\underset{n\rightarrow\infty}\lim{\frac{n}{a^{n+1}}\left(a+\frac{a^2}{2}+\frac{a^3}{3}+\cdots+\frac{a^n}{n}\right)}=?, \;\;a>1$$
In Shaum's Mathematical handbook of formulas and tables I've seen: $$\;\;\;\;\;\;\;\;\;\;\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots\;,x\in\langle-1,1]\;\;\;\;\;\;\;$$
$$\frac{1}{2}\ln{\Bigg(\frac{1+x}{1-x}\Bigg)}=1+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots\;\;\;,x\in\langle-1,1\rangle$$ The term in parentheses reminded me of the harmonic series. I thought of using the Taylor series. Is that a good idea? It says $a>0$ so I probably can't use these two formulas. On the other hand: $$e^x=x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots\;\;\;\;\;\;,$$ but there are no factorials in the denominators.
Inspired by this question, I decided to ask about $$I = \int_{0}^1 \ln \left\lfloor \frac{1}{x} \right\rfloor dx$$
This can be easily converted to an infinite summation by considering each segment: the $n$th segment, starting at $n = 1$ from the right and going to the left, has length $\frac{1}{n}-\frac{1}{n+1}$ and height equal to $\ln(n)$. This then means that the integral is equal to $$S_1 = \sum_{n=1}^{\infty} \left( \frac{1}{n}-\frac{1}{n+1} \right) \ln(n)$$
Through telescoping, this can be rewritten as $$S_2 = \sum_{n=2}^{\infty} \frac{\ln(n)-\ln(n-1)}{n}$$
I don't know how $S_2$ or either of the other representations can be solved, but I am fairly certain that it converges since numerical approximations have given me an answer around $0.788$.
Any help in solving the integral would be appreciated.
Edit: By using $$\frac{\ln(n)-\ln(n-1)}{n} = \sum_{m=2}^{\infty} \frac{1}{(m-1)n^m}$$ the series can be rewritten as $$S_3 = \sum_{m=2}^{\infty} \sum_{m=2}^{\infty} \frac{1}{(m-1)n^m} = \sum_{m=2}^{\infty} \frac{\zeta(m) - 1}{m-1}$$
I thought I would buy some Christmas presents for my 8-year-old niece but I'm having a hard time figuring out what to buy her.
Do you have any ideas? Say, I'd use a budget of around $200.
I thought of giving her books to get ahead in mathematics and physics which will be useful for her in the future but I think would like to give her something that could be more accessible.
What would you give an 8-year-old girl?
Repunits are numbers whose digits are all $1$. In general, finding the full prime factorization of a repunit is nontrivial.
Sequence A067063 in the OEIS gives the smallest prime factor of repunits. There are no $7$'s (just the number $7$, not as a digit in a different prime) that I can see in the sequence.
$7$ does divide many repunits, but always does when $3$ is also a factor, so $7$ never shows up as the smallest prime factor. The table of the first $508$ smallest prime factors of repunits shows that $7$ is not the smallest prime factor of any of those repunits.
My question is can $7$ ever be the smallest prime factor of a repunit?
I tried to find a repunit with $7$ as the smallest factor but searched very far and $3$ is always a factor whenever $7$ is.
I tried to prove that it can't be but I have no idea how.
I was recently asked to describe a result in mathematics that profoundly surprised me, and I thought it would be worth posting here for those interested. It's a rather advanced topic, so I'll provide some soft background so that it may be conceptually accessible to a broader audience.
Almost every "object" in modern mathematics boils down to a set equipped with some extra structure (a notion of distance, operations on the set like addition/multiplication, linearity, etc.). The objects you deal with in early mathematics courses, typically open subsets of Rn , have a particularly rich structure to them. We can reintepret Rn as being a field, a vector space, a manifold, and nearly everything in-between. They have almost any property you could want, which makes sense seeing as though Rn is often the basis for considering these properties in the first place.
Differentiable manifolds arise from asking "how similar must an object be to Rn for us to retain a meaningful notion of calculus?" Or rather, what type of structure should a set be equipped with to disucss calculus. The answer is not as easily seen as the question, but the crux of it is that the object must locally resemble Rn . For example, if you were to zoom in on a circle, you would see it getting flatter and flatter. In the limit, it looks like a line--R.
To get to the realm of differentiable manifolds, however, there's a hierarchy of structures that you must equip to some underlying set that (in the context of geometry) goes:
Set ---> topology ---> topological manifold ---> differentiable manifold
The topological structure allows one to talk about the notion of continuity within the set. The topological manifold structure is just a super nice topological structure that allows us to omit some of the weirdness that you can get in topology. Specifically, a topological manifold is a topological space that, in some sense, looks sufficiently close to euclidean space. The differentiable structure is a level beyond this--it's a topological space that looks locally linear, so that we can discuss the idea of differentiation and tangent spaces.
An interesting question to ask is "given a topological manifold, how many different differentiable structures can you add to it?" Where different essentially means that the spaces have a fundamentally different notion of 'calculus'. Even more practically, having different differentiable structures means that calculations involving calculus on one manifold cannot be used to determine calculation involving calculus on the other (despite them being equivalent as sets, topological spaces, and topological manifolds).
The answer is quite surprising, and is partitioned by the dimension of the manifold, where this dimension is given by the dimension of euclidean space (i.e. the 'n' in Rn ) that the manifold locally resembles.
Manifolds with dimensions 1, 2, and 3 have a unique differentiable structure. That is, the underlying topological manifold admits a natural choice in calculus.
In dimensions 5 and above, the differentiable structure is not generally unique, but there are only finitely many different differentiable structures you can have. In principle, this means that we could classify all of the different types of calculus up to diffeomorphism.
In dimension 4, there are uncountably infinite different differentiable structures you can add. In effect meaning that the notions of a topological manifold structure and a differentiable manifold structure are the most separated from each other in dimension 4.
Feel free to discuss this in the comments or post your own experience with an extremely counterintuitive result in mathematics. Cheers!
I’m grateful to Jemma Sherwood and Rob Low for reading an early draft of this and for their comments thereon. All opinions are, of course, my own.
This post is inspired by something that I see crop up now and again in discussions with other Maths teachers. It usually manifests itself as a rallying cry to use ≡ in place of = in identities and reserve = for equations. My standard response is to mutter something about identities being equations and leave it at that. But in the latest round, Jemma Sherwood challenged me, in the nicest possible way, to explain a bit further. This is that explanation.
Although I’m going to state my case here, I’m well aware that there are different opinions. In matters of opinion, such as this, agreement and disagreement is less important than that all sides think. So if what I write seems to you wrong, that’s fine so long as it makes you think about why you think that it is wrong.
I’m actually going to give two answers to the question “Should we use ≡ for identities?”. Both are “No”, but for different reasons:
The second answer is the one that I usually mutter about when I come across this idea of using ≡ but it’s the first that is the important one.
In preparation for writing this I posted a poll on Twitter with four mathematical statements and asked which of them were identities. The four statements were:
You may wish to ponder what your answer would be before continuing.
From the discussion that ensues whenever anyone posts about ≡, the rationale for insisting on it would seem to be that students find it difficult to distinguish between identities and equations so using notation to clarify the difference would be a good idea.
Seems reasonable. But to my mind, it’s trying to solve the wrong problem.
In the comments around my twitter poll, someone linked to the Wikipedia entry on Mathematical Identity which starts (emphasis mine):
In mathematics an identity is an equality relation $A = B$, such that $A$ and $B$ contain some variables and $A$ and $B$ produce the same value as each other regardless of what values (usually numbers) are substituted for the variables.
Another person gave a similar criterion for an identity which involved, as I understood it, putting “$\forall x$” at the start (or whatever unbound variables existed in the expressions).
The poll wasn’t long published before someone made a comment that slightly let the cat out of the bag. They queried the $\sin (180 n) = 0$ and said that it would be okay if $n$ was an integer but that I hadn’t made that clear. (Actually, they also queried the fact that I’d written $180$ rather than $180^{\circ }$; I must confess that one was due to me not being bothered to hunt down a unicode degree symbol but it really just underlines my point.) After that, some others remarked that they wanted to change their vote as they hadn’t noticed that.
So just putting $\forall x$ or $\forall n$ in front of an expression and seeing if it is still true isn’t a valid test of anything. We have to provide a context for the variables, and that allows me the freedom to make any of my equations into an identity or not.
To be a valid mathematical sentence, an identity requires a context. My contention is that the real problem behind the equation vs identity debate is that students are filling in the missing context for themselves and often getting it wrong. And once the context is made explicit, we no longer think of the identity as anything special and no longer need special notation for it.
I would also contend that the distinction between a double and triple line is not sufficient. If someone is having difficulty with the difference between an equation and an identity then an extra horizontal line will not make it clear.
None other than the great Don Knuth once said that in a mathematical document it should be possible to replace all the bits of maths by “blah” and for it to still make grammatical sense. I strongly suspect that my students do the opposite and replace all non-maths by “blah”. For example, fill in the “blah”s in these two questions and consider how the different possibilities would lead you down different routes to an answer:
Then add in the fact that a novice learner is likely to overlook the fact that the second doesn’t have an “$= 0$” in it and try to “solve” that quadratic.
If we make the context clearer, we are lessening the work that the student has to do to understand what they are being asked to do. And this is not an artificial weakening: context becomes more and more important the deeper one goes into mathematics. In school, certainly pre-16, it is a safe assumption that the context is “numbers”. It is only later that students learn that the context could be vectors, functions, matrices, sets, objects, morphisms, groups, rings, fields, manifolds, sheaves, schemes, … if I missed your favourite, I apologise.
But even a context of “numbers” can be misconstrued. How many students look at an answer with extreme puzzlement when it turns out to be a fraction? They were expecting a whole number.
And wouldn’t it set up expectations for quadratics and trigonometry much better if we consistently said “Find all (real) numbers $x$ for which …” instead of just “Solve”? And “Show that for all real numbers $x$ …” instead of just “Show that”?
The language doesn’t even have to be that formal, we don’t need $\forall $ or $\exists $ in Y7, but it should make clear the context. It can even be something like “I’m thinking of a real number, call it $x$; it satisfies $2 x + 6 = 10$. What is it?”
I have very few memories of my own time at school, but one that I do recall very vividly is my A-level Chemistry teacher announcing at the start of the course that everything we’d been told up to then had been a lie. “Sodium,” he declared, “doesn’t want to lose an electron. It doesn’t want anything.”
It was dramatic, I’ll give him that, but it did make me lose a bit of faith in Chemistry. For all I knew, everything I was going to be told in A-level would also be a lie (spoiler: it was).
I try my utmost not to do the same in my own teaching.
Of course, I can’t tell my students the whole truth. When teaching about negative integers, for example, I don’t set up an equivalence relation on pairs of positive integers and prove that the operations of arithmetic descend through the relation. What I aim for is the following thought experiment: suppose that one of my students did go on to do a mathematics degree, possibly even further, and encountered some fancy part of mathematics that recast something that they’d learnt in school. What I would hope is that they would feel that the recasting fitted in with the story that they already knew. That if they ever came back to visit, they’d say, “Now I understand why you told the story that way.”
So when I consider something like identities, I think about how the concept is used later on and try to use that to inform how I talk about it in school.
And that’s a bit tricky with identities because, in my mathematical experience, they all but disappear. The Wikipedia page does rather give the game away when it says (emphasis mine):
In other words, $A = B$ is an identity if $A$ and $B$ define the same functions. This means that an identity is an equality between functions that are differently defined.
Thus once we are happy talking about functions, the need for the word identity disappears.
When I think of the word identity, the first concept that springs to mind is the identity function (or, rather, the identity functions since there are rather a lot of them), which might happen to be representable by the identity matrix. There’s also the identity element in a group or ring.
The closest I get to the concept of identity under discussion here is in a topic called universal algebra. Very briefly, this is the area of mathematics that studies operations like $+$ or $\times $ in the abstract. Such operations satisfy relations which are sometimes called identities. These are things like $x + y = y + x$. The catch is that in this area, the identities are imposed. They don’t occur by accident but by design.
This idea of imposing identities also chimes with where I see the ≡ sign used. I don’t think of it as “is identically equal to” but as “is equivalent to in this context”. The classic situation is in modular arithmetic, where I will happily write things like $4 \equiv 1 \mod 3$, by which I mean that in the context where I ignore multiples of $3$ then I can view $4$ as equivalent to $1$. In the wider context of integers then I know that $4$ and $1$ are different, but in the smaller context of modular arithmetic then I can consider them equivalent.
So I feel that I should exercise caution in using the term “identity” to refer to what is an equality of functions, and where the term is used differently later on. Particularly because, as I argue above, using ≡ is unlikely to solve the underlying issue of establishing context.
Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I necessarily done something wrong?
Often, an indefinite integral can be evaluated using different techniques. For example, an integrand might be simplified via partial fractions or other algebraic techniques before integration, or it might be amenable to a clever substitution. These techniques give different results. For example, looking over a few other questions on MSE:
From this question: evaluate $$ \int x(x^2+2)^4\,\mathrm{d}x. $$
Via the substitution $u = x^2+2$, this becomes $$ \int x(x^2+2)^4\,\mathrm{d}x = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + \frac{32}{5} + C. $$
However, multiplying out the polynomial and integrating using the power rule gives $$ \int x(x^2+2)^4\,\mathrm{d}x = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + C $$
From this question: evaluate $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x. $$
Simplifying the integrand using partial fractions then integrating gives $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x = \frac{2}{(x+1)} - \ln|x+1| + C. $$
Via integration by parts, we get $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x = \frac{x-1}{(x+1)} - \ln|x+1| + C. $$
From this question: evaluate $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x. $$
Using the substitution $u = \sec(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x = \frac {\sec^2(\pi x)}{4\pi} + C. $$
Using the substitution $u = \tan(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x = \frac {\tan^2(\pi x)}{4\pi} + C. $$
