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August 2019
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22:35

Consider a finite sequence of $n$ integers denoted $x_1,x_2,...,x_n$ where $x_1=x_n=0$ and $x_{n+1}$ is either equal to $x_n+1$, $x_n-1$ or $2x_n$. Is there a good way to count how many such sequences there are, with either an exact or asymptotic formula?

Phrased differently, if you start with the number $0$ and are allowed to add one to your number, subtract one from your number, or double your number $n$ times, how many ways can you end up back at $0$? Is there a good approximate/asymptotic formula for this in terms of $n$?

I have no idea how to start solving this. It would be nice to find some sort of generating function, but I’m not sure how to set one up.



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21:01

I've found an interesting infinite product formula without a proof: $$e=2 \sqrt{\frac{4}{3}} \sqrt[4]{\frac{6\ 8}{5\ 7}} \sqrt[8]{\frac{10\ 12\ 14\ 16}{9\ 11\ 13\ 15}}...$$ How can we prove it? Fow now I have no idea and would like you to help. Thanks!



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05:55

I'm a high school student.I love mathematics and I read a lot of maths books.I always like to discuss some mathematical problems with others.

But unfortunately, there aren't anyone in my school tending to truly love mathematics, they just care about the exams and they often ignore me when I'm talking mathematics with them.

This maybe because they can't understand what I'm talking about. So I often feel bloody lonely, it's painful. Could anyone help me? I wanna find a pen pal to communicate with each other.



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05:55

I have got this task at high-school math-contest seminar.

The theme is graphs.

Let us have $n \in \mathbb{N}$ and the cube $ n \times n \times n$. An ant can go over a diagonal of little cubes, but he can't turn at the intersection of two little diagonals.

Is it true that if in is odd then an ant can't visit every little facet (all of $6 n \times n$) exactly once? I didn't find the path for $n = 1,3$

I don't see a particular method to solve this task, because you can't treat facets as vertices. You can't treat them as edges also. It isn't an Euler or Hamilton cycle. I don't see an invariant that should be saved as well. The graph is planar.



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05:26

Click on this image to see it larger in a new window.

The official Atlantic hurricane season started on June 1 and doesn’t end until November 30th. Now Hurricane Dorian is approaching the East Coast.

Encourage your students to join the conversation, knowledgeably, by using our activity.  Students read about hurricane classfications and calculate the possible increases in destruction from various scaled events.  Estimation, data analysis with graphs to study trends, damage expense, average frequency, and histogram are all part of this lesson.

The activity: hurricane2019-update.pdf

CCSS: 6.SP.3, 6.SP.4, 6.SP.5, S-ID.1

For members we have an editable Word docx, our Excel data and charts, and solutions.

hurricane2019-update.docx     HurricaneData.xlsx    hurricane2019-update-solutions.pdf



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04:35

Question: Given a homomorphism $f:A \to B$, where $A$ and $B$ are algebras in any variety (in the sense of universal algebra), is it true that if $f(S)$ is isomorphic to $S$ for all subalgebras $S$ of $A$ (in symbols: $\forall S \le A \ f(S) \cong S$), then $f$ is necessarily injective?

For some specific varieties, the answer can easily be shown to be "yes":

  • Sets: Consider the 2-element subsets of $A$.
  • Groups (or modules or vector spaces or rngs (possibly non-unital rings)): Consider the kernel of $f$.
  • Lattices (not necessarily with top or bottom elements): For any $a_1, a_2 \in A$, consider the sublattice $\{a_1, a_2, a_1 \land a_2, a_1 \lor a_2\}$ of $A$.


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04:10

$$\lim_{n\to\infty}\Bigl(\frac{2+\sin n}{3}\Bigr)^n $$ where $n$ is a natural number.

The problem is if the numerator is less than 3. I think it is because $\sin n$ is never $1$ otherwise $\pi$ would be rational. But i am not sure if the limit exists because $\sin$ oscillates. How to solve this ?



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00:25

29 Aug 2019

In this Newsletter:

1. New on IntMath: CSS matrix math
2. Resources: Humble Pi, AnswerThePublic
3. Math in the news: Proof
4. Math movies: Parker, van Gogh
5. Math puzzle: Mystery object
6. Final thought: Dry leaves

1. New on IntMath: CSS matrix math

I recently gave a talk to a local meetup group on the mathematics behind CSS matrix. CSS stands for "cascading style sheets", and is the system where Web designers can set font sizes, colors, and also set sizes and vary shapes for objects like images and videos.

Matrices are used to transform geometric objects (scale, skew, rotate, translate and so on.) Computer games make extensive use of matrices to simulate depth, 3D objects and so on.

CSS Matrix - a mathematical explanation

Here is the content of the talk. It explores how CSS transform is the result of matrix multiplication. Even if you're not interested in Web design, it's interesting to see another real-life application of matrices.

See CSS Matrix - a mathematical explanation

I also developed the following interactive graph applet that demonstrates the concepts in the talk.

CSS matrix interactive applet

This is an interactive graph where you can vary sliders to see how CSS matrix changes the size, shape and location of an object.

See CSS matrix interactive applet

2. Resources

(a) Humble Pi: A Comedy of Maths Errors

When I was teaching a group of engineering students some years ago, I went to my boss with an idea. I suggested we incorporate examples of cases where things went wrong, so students would learn the importance of accuracy and the safety issues that can arise out of sloppy and inaccurate mathematics.

He wasn't enthusiastic and squashed the idea, saying it would scare off students from choosing the degree. I felt it was a lost opportunity.

So I was interested when I came across the book "Humble Pi - A Comedy of Maths Errors" by Matt Parker.

Humble Pi - A Comedy of Maths Errors
Matt Parker's Humble Pi

This readable book was exactly what I had in mind when I approached my boss. Let's learn from cases where people made math errors, and see what the consequences were – not to apportion blame, but to learn what can go wrong.

We all make math errors, but usually the worst outcome is a drop in grade, or momentary embarrassment. The people working in science and engineering fields should be aware of why their math teachers insisted on accuracy.

So I suggested the book for the local library, and was pleasantly surprised how long I had to wait before I could read it (it turned out to be quite popular).

I recommend this book for any math student or teacher.

One of the videos below features Parker, covering some of the same errors detailed in the book.

Disclaimer: I have no connection with Matt Parker (other than through Twitter) and receive no commission.

(b) Teachers: Address the questions students are really asking

Some teachers see their job as simply giving out information, but there is no "value add" in that approach, especially since students can easily access such information in abundance.

One thing we can do better while teaching a topic is to actually address the questions students really have about that topic. One approach is to simply ask students what their questions are, and there are a lot of apps and sites that facilitate this process (e.g. Google Forms and Survey Monkey are both easy to use).

Another thing to consider is to look at the questions students are likely to ask, before you even start planning the lessons. AnswerThePublic is a good resource for this.

Answer the Public

AnswerThePublic is a database of common questions that people ask about topics. It provides a rich source of ideas on how we might go about introducing a topic, and pre-empting the stumbling blocks.

See Answer the Public

Topics to try are:

  • Algebra (you'll see e.g. "How is algebra used in real life?", "Who invented algebra?", etc)
  • Calculus (e.g. "When does a limit exist?", "Where to start?", "How is it used in computer science?", etc)
  • Matrix (e.g. "When is a matrix orthogonal?", "Matrix when a^2 = a?", etc)

You can choose the "Data" tab at the top of each visualization to get easier-to-read lists of questions, and download the lot as a CSV (for Excel).

Sometimes in this resource you see questions that may seem quite odd, like "Does calculus cause kidney stones?", but "calculus" means "stone" and in medicine, it refers to a build-up of hard substances in the body. I get it on my teeth.

3. Math in the news

Story of the Gaussian correlation inequality proof

The title "Gaussian correlation inequality" sounds scary, but it goes something like this.

I have a dart board sitting on a rectangular shape, and assume I'm a good dart player. When I throw a lot of darts at it, I expect the accuracy of my throws to form a somewhat bell-shaped curve distribution. That is, most of the darts land somewhere close to the middle, and there are less dart holes as I go out from the middle.

The greater the circle overlaps the rectangle, the probability of striking both goes up.

A Long-Sought Proof, Found and Almost Lost

The probability that a dart lands on both the circle and the rectangle is greater than or equal to the product of the individual probabilities.

At first, they didn't believe a German retiree had actually proved it.

See A Long-Sought Proof, Found and Almost Lost

4. Math Movies

(a) What Happens When Maths Goes Wrong?

What Happens When Maths Goes Wrong?

This is a one-hour presentation by Matt Parker at the Royal Institution, London. It covers some interesting examples that are worth considering.

See What Happens When Maths Goes Wrong?

Parker is the Public Engagement in Mathematics Fellow at Queen Mary University of London.

(b) The unexpected math behind Van Gogh's "Starry Night"

Turbulence is one of the most tricky phenomena to model using mathematics. It is complicated and chaotic.

The unexpected math behind Van Gogh's Starry Night

This video by Natalya St. Clair explores how Van Gogh incorporated turbulence in his art to give the impression of movement.

See: The unexpected math behind Van Gogh's "Starry Night"

5. Math puzzles

The puzzle in the last IntMath Newsletter asked about radii of mutually tangent circles. In fact, it turned out to be a 3x3 system of equations - it wasn't really a geometry question.

Correct answers with sufficient reasons were submitted by Russell, Nicola, Tomas and Thomas.

New math puzzle: Mystery object

This time, some investigation may be involved.

Mystery object
Mystery object

The above object was used to achieve a particular mathematical outcome – one that is still vitally important to this day. What is the object, where was it used, and what was the mathematical outcome?

(If you can't actually find it or figure it out, your speculation will prove interesting!)

You can leave your response here.

6. Final thought - what it could be like

Equatorial Singapore, where I live, is normally lush and green, and doesn't experience leaf falls as is normally the case for most places in Autumn.

However, this year we've had the driest and second hottest July on record, and practically no rain so far in August causing my local park to look like this:

Dry leaves Singapore
Dry leaves and dead grass in Singapore.

Such dry and hot conditions are caused by a positive Indian Ocean dipole, the situation where the Western Indian Ocean is hotter than the Eastern part, causing hot droughts over Australia and most SE Asian countries.

These make ideal conditions for farmers in Indonesia to set off forest fires in order to plant more palm oil, so there's been many fire hot spots reported there.

Meanwhile, the dry season in the Amazon has been the excuse, along with President Bolsanaro's encouragement, for farmers there to burn vast amounts of the Earth's lungs for ever-expanding methane-producing cattle farms.

In the Arctic, fires across the tundra continue to burn, spewing even more carbon into the atmosphere.

These events are giving us an insight into how things will be if governments, companies and all of us fail to address our land use, our consumption, and our "economic growth at all costs" mentality.

We can stop it, but will we?

Until next time, enjoy whatever you learn.

Related posts:

  1. Updated matrix interactive Here's an applet where you can investigate how matrix...
  2. IntMath Newsletter: Resources, primes, matrix app and role models In this Newsletter: 1. Resources for the new school year...
  3. Is a 1×1 matrix a scalar? A 1×1 matrix is often regarded as a scalar...
  4. IntMath Newsletter: Piano trig applet, continued fractions In this Newsletter: 1. New applet: Piano note frequencies interactive...


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17:10

There seems to be a lot of conflicting information online about this. Some people say a Math Major is useless, others say it opens doors to so many fields.

Some people say you can only go into Academia or become a Software Engineer or Finance Analyst, and even then it's harder to land those positions because you're competing with students who actually majored in CS/Software or Finance. Other people (like the Bureau of Labor Statistics) say that Mathematician occupations are projected to grow 33% through 2026.

Who's right? For simplicity, let's assume a Master's degree is achieved.

--------------------------------------------------------------------------------------------------------------

​

Personal (Please read if you have the time):

I initially decided to go into Computer Science, but it required a ton of math prerequisites. I was decent at math in high school, understood it no problem, but I always disliked it. When I got to Community College I decided to start with College Algebra, just so I wouldn't overwhelm myself. Made it through Pre-Calculus and Trigonometry (two different courses at my school) and I fell in love with Math. I was thoroughly enjoying those classes more than my intro CS ones, and more than anything else I've ever studied. This was largely due to a wonderful professor who actually explained concepts, which was completely different from the mechanical, rote, repetitive style of math education I experienced in high school.

I went on to take Calculus 1 & 2, and am currently taking Calculus 3 right now. I'll be transferring in a couple months to a four-year university, and at this point, part of me just wants to keep studying Math because of the reward I feel every time I understand a concept, or the chills I get from every eerie coincidence between Math and our natural world. I just want more of it, and it seems that the more I learn, the less I actually know about Mathematics as a whole.

So that's why I decided to ask this question today. I am aware that a lot of similar posts do exist, however it seems that the replies vary from post to post, and from year to year.

I'm not interested in being rich, I just want 2 things:

1) An interesting, fulfilling, and impactful career I'm emotionally invested in, and

2) Just enough money for living expenses and modest savings (in case of emergency, gifts, travel, etc.)

I realize neither are guaranteed, and that thing 1 is a rarity these days. I'm just scared that if I decide to major in Math (hoping to complete a Master's Degree) I won't be able to find a relevant job. Any advice is much appreciated.

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18:35

Can we evaluate $\displaystyle\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$ ?

where $H_n=\sum_{k=1}^n\frac1n$ is the harmonic number.

A related integral is $\displaystyle\int_0^1\frac{\ln^2(1-x)\operatorname{Li}_2\left(\frac x2\right)}{x}dx$.

where $\operatorname{Li}_2(x)=\sum_{n=1}^\infty\frac{x^n}{n^2}$ is the dilogarithmic function.



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17:20

I am a math major and have this nasty habit of being very competitive in every class -- I don't talk to people, go to office hours, and attempt be the best student in every math class I am in. I am taking 4 math-based courses. To be honest I don't find this idea of perfection too difficult to attain, but just extremely stressful and gives me a headache every day. Does anybody else have this problem of trying to be to best at your university? I realize if I was at a very large university I would humbled by prodigies, but since I am not the fact that this idea of perfection is feasible makes it irresistible to try. Mathematics has become more of a competition for me it seems and it makes me sad. Is there a way to get over this? Anyone have similar experiences?

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17:18

Gradshteyn&Ryzhik 3.554.5 states that: $$\int_0^{\infty } \frac1x \biggl( 2qe^{-x}-\frac{\sinh (q x)}{\sinh \left(\frac{x}{2}\right)} \biggr) \, dx=\log\bigl(\cos (\pi q) \bigr)+2 \log \left(\Gamma \bigl(q+\frac12 \bigr)\right)-\log (\pi )~, \quad q^2<\frac{1}{2}$$ It looks like to have relation with the Binet representation of log-gamma function, but I haven't figure out how to solve this, and would like you to help. Any kind of approach is appreciated.$$$$ Note that a related problem in 3.559 is: $$\int_0^{\infty } \frac{ e^{-x} }{x} \left(e^{ (2-a) x} \frac{(1-a x) (1-e^{-x}) - x e^{-x}}{4 \sinh ^2\left(\frac{x}{2}\right)}+a-\frac{1}{2}\right) \, dx \\ =a+\log (\Gamma (a))-\frac{1}{2}-\frac{1}{2} \log (2 \pi )~, \qquad a>0$$ Maybe this can be solved together.



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23:50

Intense debate at work place around the solution for this:

Let $M \in C[a,b]$ be a continuous function on the closed interval $[a,b]$ that satisfies $$\int_{a}^{b}M(x)\eta^{\prime\prime}dx = 0,$$ for all $\eta \in C^{2}\left[a,b\right]$ satisfying $\eta(a) = \eta(b) = \eta^{\prime}(a) = \eta^{\prime}(b) = 0$. Prove that $M(x) = c_{0}+c_{1}x$ for suitable $c_{0}$ $c_{1}$. What can you say about $c_{0}$, $c_{1}$?

I tried to use integration by parts, and use the fundamental lemma of the calculus of variations, and the lemma of Du Bois and Reynolds to prove it, but that requires $M$ to be $C^1([a, b])$.



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06:39

Yaokun Wu wrote me about the China Three Gorges University logo:

CTGU logo

Another understanding of the logo of TGU is some sailing boats and books, meaning sailing in the sea (river) of books.

As evidence for this, here is a picture of Three Gorges, presumably as it was before the dam was built:

Three Gorges

You can see that the sails of the boats in the foreground do resemble the CTGU logo. And here is the real thing:

Boats in Three Gorges

The last picture was taken on the conference excursion, which took us to several beauty spots on the Xiling Gorge, below the dam. I should probably say that “beauty spots” might be more accurately translated as “tourist spots”: the number of tourists is so great that, rather than allow them to ramble as they choose through the rugged and dangerous terrain, they concentrate them in a few places where suitable entertainment can be provided. The boats are at Longjin, where they feature in a story which was never quite clear to me, something about a girl being married off against her will, I think.

On a walk up the small side stream to a waterfall, I passed a sign which said, “”Watch your head observe and pass”. This sounds to me like an advanced meditation technique, but it was really a warning about an overhanging rock.

We also visited a reconstructed village of the Ba people, prehistoric inhabitants of the region. This time the story was even less clear to me; the musical accompaniment was pleasant enough, but the music played before the performance was much too loud and mechanical to be listenable. The audience sat on benches under three large and ancient trees; pleasant on a very hot day until I observed that every leaf on the trees was made of plastic.

The scenery was spectacular, but unfortunately the muddy water of the river and the polluted air made it impossible to take any pictures worthy of a tourist brochure. This will have to suffice:

Longjin

The conference organisers, who delighted in satisfying our every desire, put on another ad hoc excursion at the end of the meeting, a half-day trip involving a visit to the famous dam followed by a cruise down the river. Here is a picture of the dam, the largest in the world, and some of the spectacular scenery we saw from the boat.

Three Gorges dam and Xiling Gorge



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02:57

I am confused of a question that needs to know the greatest common divisor of $2^m+1$ and $2^n+1$ ($m,n$ are positive integers), but I don't really know. I am pretty sure that the greatest common divisor of $2^m-1$ and $2^n-1$ ($m,n$ are positive integers) is $2^{\gcd\left(m,n\right)}-1$, even I can prove it by the Euclidean algorithm. However, it is hard to use it in this problem, so I want you guys to help me. Thanks!

P.S.

I created an excel and I observed the answer (maybe?) from it, but I can't prove or disprove it. Here is my conclusion from the excel: $$\gcd\left(2^m+1,2^n+1\right)=\begin{cases} 2^{\gcd\left(m,n\right)}+1 \\ 1 \end{cases}\begin{matrix} \text{when }m,n\text{ contain the exact same power of }2 \\ \text{otherwise} \end{matrix}$$ Hope it will help me and you guys solving this quesion :D

The link of The excel



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21:07

I was learning for the first time about partial fraction decomposition. Whoever explains it, emphasises that the fraction should be proper in order to be able to decompose the fraction. I was curious about knowing what happens if I try to decompose an improper fraction, So I tried to do one:

$\frac{x^2 - 4}{(x + 5)(x - 3)}$

I got the equation: $\frac{(Ax + B)(x - 3) + (Cx + D)(x + 5)}{(x + 5)(x - 3)} = \frac{x^2 - 4}{(x + 5)(x - 3)}$. I have 4 unknowns: A, B, C and D.

$\therefore (Ax + B)(x - 3) + (Cx + D)(x + 5) = x^2 - 4$

After expanding and regrouping the coefficients:

$(A + C) x^2 + (-3A + B + 5C + D)x + (-3B + 5D) = x^2 - 4$

Here the coefficient of the term $x^2$ is 1 therefore:

$(A+C) = 1$

similarly:

$(-3A + B + 5C + D) = 0$

$(-3B + 5D) = -4$

I still have to get one more equation to be able to solve this system so I substituted 1 for x and I got this equation:

$-2A - 2B + 6C + 6D = -3$

After getting four equations I used this site to solve the system of equations. Unfortinetly I got no soultion. Tried another site and also the same result.

I've tried to use different values for x and got another equaitons like:

for x = 2 : $-2A - B + 24C + 7D$

for x = -1 : $4A - 4B - 4C + 4D$

for x = -2 : $10A - 5B - 6C + 3D$

But also that didn't work. Always the system of equations have an infinite solutions.

After tring to figure out why this is happening, I've managed to prove logically that this equation:

$(Ax + B)(x - 3) + (Cx + D)(x + 5) = x^2 - 4$

has an infinite solutions and my approach was as follows:

After doing polynomial long division and decomposing the fraction using the traditional way, the result should be:

$\frac{5}{8(x - 3)} - \frac{21}{8(x - 5)} + 1$

Now I can add the last term (the one) to the first term and get the follows:

$\frac{8x-19}{8(x-3)} - \frac{21}{8(x+5)}$

From that solution I can see that $A = 0, B = \frac{-21}{8}, C = 1, D = \frac{-19}{8}$. After all these are just the coefficients of the terms. and this solution worked fine.

Alternatively I can add the one to the second term instead and get:

$\frac{5}{8(x-3)} + \frac{8x + 19}{8(x+5)}$

Now $A = 1, B = \frac{19}{8}, C = 0, D = \frac{5}{8}$

Generally, after adding the one to any of the terms, I can add any number to one of the terms and add its negative to the other term and the equation will remain the same, But the value of the 4 constants (A, B, C, and D) will change. And from that I got convinced that there are an infinite solutions to this equation.

But Algebraically? I'm not able to prove that it has an infinite solutions algebraically. And my questions is how to prove algebraically that this equation has an infinite solutions? Or generally how to know whether the equation has just one solution or an infinite?



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03:42

I am trying to show that $\forall N\in\mathbb{N}$,

$$\sum\limits_{n=0}^{N}\sum\limits_{k=0}^{N}\frac{\left(-1\right)^{n+k}}{n+k+1}{N\choose n}{N\choose k}{N+n\choose n}{N+k\choose k}=\frac{1}{2N+1}$$

It's backed by numerical verifications, but I can't come up with a proof.

So far, I tried using the generating function of $\left(\frac{1}{2N+1}\right)_{N\in\mathbb{N}}$, which is $\frac{\arctan\left(\sqrt{x}\right)}{\sqrt{x}}$, by showing that the LHS has the same generating function, but this calculation doesn't seem to lead me anywhere...

Any suggestion ?



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00:57

I am a 2nd year student doing an honors program in math and statistics.

Everything that I have been learning has been formulas, theorems, and mathematical concepts that other people have discovered/invented/created.

Some very simplistic formulas I am able to modify to meet the needs of what I am trying to accomplish, but still I am using someone else discoveries as a basis for what I am modifying.

Friends and family say I am becoming a mathematician, but I dont feel that way as I do not have the ability to invent/create/discover new math, I am simply regurgitating what others have discovered.

At what point in your mathematical education are you able to invent new math?

For example, the linear regression formula. How did Francis Galton know that the formula he created would accomplish what he wanted to accomplish?

Note: Sorry to the editors as I could not find a relevant tag for this question.



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21:27

From this page I found an interesting equality:$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$ It's known as the average distance between two random points in the unit square, and higher dimensional results are also obtained in that page. My question is: is there any elementary method to obtain this result? Thank you. $$$$Update: In comments one user offered a link which solves the problem by basic methods. Still, alternative methods for this problem (as well as 3 dimension's analogue, see the link above) are welcomed.



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20:27

I like sequences. Specifically, I like making my own sequences. I want to start generating my own large sets of sequences and graphing them, but I have zero experience with computer science. Where should I start learning the basics of the languages/programs that would help me do this, which I can also hopefully extend to study in other areas of mathematics?

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19:07

One common definition of arclength is to just define it as a supremum of the set of lengths obtained by approximating your curve as a union of line segments. The natural analogue of this to the surface area of a surface in 3 space fails quite spectacularly thanks to constructions such as the Schwarz lantern, which shows we can approximate a cylinder by polyhedra whose surface areas approach infinity!

Is there an intuitive reason that polygonal approximation works so well for curves but fails so spectacularly for surfaces?



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18:47

Given a polynomial with integer roots, is it possible to add an integer to the polynomial so the roots of the new polynomial are also integers.

Apparently it is possible for some polynomials.

For example $$ x^2-6x+5$$ and $$x^2-6x+8$$ satisfy our conditions.

Third order polynomials such as $$x^3-12x^2+41x-30$$ and $$x^3-12x+41x-42$$ satisfy the conditions as well.

My question is what is the highest possible degree of polynomials satisfying the said conditions.



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04:52

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\cap(0,1).$

Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this.

I believe it would follow from Schanuel's conjecture,

$\mathbb{Q}(\ln x,\mathrm{e}^{1/\ln x})$

has transcendence degree $2$ whenever $x$ is algebraic.

I've tried very hard to solve this problem over the course of several months using the little abstract algebra I know. My first abstract algebra class starts on Monday so I hope to gain better tools to attack this problem.

My main point of attack has been to analyze various classes of algebraic equations that satisfy the condition $f(0)=0$ and $f(1)=1.$ This is because when you don't know how to think of a single object, sometimes it's better to study a class of similar objects.

Then I've reflected the entire class about $x=1/2$ to generate points in $(0,1)^2,$ and taken the collection of these points as the elements in the space. I'm specifically looking to see if I can generate points in this way, that satisfy closure w.r.t. the binary operation:$(\times).$ For example, if I have points that have algebraic numbers as coordinate pairs $P_{\in(0,1)^2}(\Bbb A,\Bbb A)$ then possibly by multiplying two points component wise, closure could be accomplished.

I don't know if this method alone will lead to a definitive answer as to whether $z$ is algebraic but it might give a helpful perspective on the road to a solution.



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04:32

Here's the sketch:

enter image description here

From the inner point P of a triangle ABC the three connecting lines to the corner points are drawn. In addition, the lines PE, PD and PF are each drawn parallel to a median of ABC. Show that the grey areas cover half of the triangular area.

I thought about using ceva's theorem, but I don't really know how to start... any ideas?



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03:02

How to show that the integral $$\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx$$ equals to $\frac{3 \pi ^2}{160}$? I've already verified this numerically but failed to prove it. The problem came from a Chinese blog (closed), however the blogger didn't show the method to arrive at the result. Does it has some relation with the Ahmed integral? Basic methods, residue cauculus as well as other methods are all welcomed. Thank you!



from Hot Weekly Questions - Mathematics Stack Exchange
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01:07

Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$

I tried studying this exam problem but I need help in understanding it.

I found $f_{x}=4x^3-4x \\ f_y=4y^3$

From this, I get the stationary points $A(0,0)$, $B(1,0)$ and $C(-1,0)$

$f_{xx}=12x^2-4 \\ f_{yy}=12y^2 \\ f_{xy}=0$

Now after computing $D(x,y)$ I got $D(A)=D(B)=D(C)=0$ which is inconclusive.

In the graph, it shows that the point $B$ and $C$ are local minimum.

I would really appreciate some good explanation because I have an exam soon and I would like some help.



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00:42

I was playing around with this demo of Project Euler Problem 208 which allows you to take steps which are "left" or "right" arcs of $1/5$ of a unit circle.

Here's an example walk, which starts at the blue dot pointing vertically up, and which consists of steps

RLLLRLLLRRLLLRLLLRRLRLLLR

Question

Which points in $\mathbb R^2$ can be reached in a finite number of steps, starting at the origin and pointing in the positive $y$-direction. Is this set of points dense in $\mathbb R^2$? If not, what's the greatest number of points that can land in $[0,1] \times [0,1] \subset \mathbb R^2$?

enter image description here



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00:42

For natural numbers $n\ge m$, let $n\underset{m \text{ times}}{\underbrace{!!!\dots!}}=n(n-m)(n-2m)(n-3m)\dots$ where all factors are natural numbers (we exclude $0$ and negative factors).

Question:

What is the units digit of $1!+2!+3!+4!!+5!!+\dots+k\underset{\left \lfloor \sqrt{k} \right \rfloor \text{ times}}{\underbrace{!!!\dots!}}+\dots+1992\underset{44 \text{ times}}{\underbrace{!!!\dots!}}$? ($\left \lfloor \cdot \right \rfloor$ denotes the floor funtion).

My Attempt (Is wrong as Peter Foreman commented below):

Consider the first $9$ terms:

$1!+2!+3!+4!!+5!!+6!!+7!!+8!!+9!!!$

$=1+2+6+8+15+48+105+384+162=731$

Each of the remaining terms includes at least on factor that ends with $0$. Therefore, the each term ends with $0$.

Hence the units digit of the given expression is equal to the units digit of the sum of the first $9$ terms. So, $1$ is the units digit of the given expression.

Peter Foreman said: "$17!!!!=9945$". This showed me that my attempt is wrong. Thanks Peter Foreman.

Any help would be appreciated. THANKS.



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23:42

So over the past couple of years, I've been studying what I'd like to call "olympiad math"(so not subjects that require a lot of theory such as linear algebra, calculus, real analysis, etc but instead, those which require little theory but use a lot of "tricks" and "manipulation" such as number theory, elementary geometry, combinatorics, etc. Basically, the sort of problems that come up in math olympiads) to prepare for the IMO board exam which, as you might know, is in a few months.

But after that, I don't think that's something I'd like to continue and even up to this point, this was more of a hobby for me than anything. Instead, I'd like to start studying what I call school math; basically things such as calculus, linear algebra, etc but I've no idea where to even begin so I was wondering what books would you recommend?

A couple things to note:

$1.$ I'm a CS student so I'm mostly interested in calculus and linear algebra but please note that although I already know a little bit of calculus and linear algebra, I'm by no means good at them.

$2.$ I'm currently in grade 9 but I do grade 12 CS and grade 10 math so unfortunately, going up another grade for me is not an option.

$3.$ Other than "olympiad math", I know up to(and including) grade 10 math.



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22:37

A group $G$ is called pleasant if it is abelian and every non-identity element $g$ in $G$ has prime order. Up to isomorphism, how many pleasant groups are there of order at most $25?$

Options: $0, 9, 16, 25, 31$, or infinitely many.

I got $15.$ Any help would be greatly appreciated!

Thanks,

I got:

  • the trivial one,

  • cyclic groups of prime order up to $25$ (there are $9$) and also

  • $C_2 \times C_2$,

  • $C_2 \times C_2 \times C_2$,

  • $C_2 \times C_2 \times C_2 \times C_2$,

  • $C_3 \times C_3$ and

  • $C_5 \times C_5$.

Thank you again!



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20:57

Let $(T,\mu,\eta)$ be a nontrivial monad on $\mathbf{Set}$. By nontrivial here I mean there is $X$ with $|T(X)|>1$.

Suppose that $TX \cong TY$ as $T$-algebras (both with the usual free $T$-algebra structure). Does it then follow that $X$ and $Y$ have the same cardinality ? (Which we could then define as the dimension of the free $T$-algebra, by analogy with vector spaces).

I think the motivation for this question is pretty clear (analogy with vector spaces, lots of examples), here are my thoughts :

  • all the examples I know satisfy this property : vector spaces over a fixed field $k$, or more generally $R$-modules, groups, boolean algebras, the ultrafilter monad, the powerset monad,...

  • If the monad $T$ is furthermore finitary, that is it preserves (as an endofunctor) filtered colimits, then the claim is true when either $X$ or $Y$ is infinite, this is not totally straightforward but not hard to show, I will add the details at the end of the question (*)

  • If the monad has at least one finite $T$-algebra with cardinality $>1$, then the claim holds when either $X$ or $Y$ is finite. This is easy, details at (**).

-If the monad has $T$-algebras of arbitrary infinite cardinalities, then the claim holds when either $X$ or $Y$ is infinite. This relies on some (easy) cardinal arithmetic and will be detailed at (***)

  • $T$ being nontrivial implies that it always reflects isomorphisms, which is weaker than what we want. This is standard and not too hard, but I think the post is already long enough, so unless specifically asked to I will not detail this.

-In the examples I gave one can notice that there are 2 extremely different behaviours : in the case of monoids, the ultrafilter monad, or the powerset monad, isomorphisms are "created", in the sense that an isomorphism of $T$-algebras between $TX$ and $TY$ always comes from a bijection $X\to Y$; whereas in other cases such as vector spaces or groups, there can be other isomorphisms, but there is a bijection between $X$ and $Y$.

For the first behaviour, one can see that this is because somehow the monad is "positive", one can not cancel the "atoms" (elements of $TX$ of the form $\eta_X(x)$), and they are the only ones with this property.

This leads me to think one could define a notion of "positive monad" for which the proof would be easy, but I have not yet found the right definition (the ones I find either don't work or seem ad hoc). The idea would be that elements of the form $\eta_X(x)$ are precisely the $z$'s such that if $\mu_X(a)=z$, then $a=\eta_{TX}(z)$. The thing is, any algebraic characterization of these elements will give this behaviour, because then we can ensure that the isomorphism comes from a morphism $X\to Y$ and then the previous point (about reflecting isomorphisms) solves the question.

For the second behaviour (vector spaces and "weird" isomorphisms), I don't really know how to attack the problem.

So my questions are, more specifically : is the result true in all generality, and if so, how can one prove it ? Could it be valid for nice categories other than $\mathbf{Set}$ (with an appropriate notion of nontriviality) ? If not, what are some nice conditions on $T$ that ensure that the result holds ? Are any of them similar to the vague idea of positivity that I tried to describe ?

EDIT : As was pointed out in the comments, it is not true for all monads, indeed there are some noncommutative rings without the IBP, and so it fails for finitely generated free modules. However, since $R$ modules always give a finitary monad, it still holds for infinitely generated free algebras, and so two questions remain : is it always true for infinitely generated free algebras ? (I now suspect that it is false) and what are some nice conditions on $T$ that yield a positive result ?

(*) this is essentially the same proof as for vector spaces : if $T$ is finitary, $X$ infinite and $f:TX\cong TY$ then for any $x\in X$, there is a finite $Y_x\subset Y$ with $f(\eta_X(x))\in TY_x$ and since $T$ is nontrivial, it reflects epimorphisms (this is standard and I will not detail it unless specifically asked to) so that $\bigcup_x Y_x = Y$. Therefore $|Y|\leq \aleph_0 |X|$, the converse holding as well. It is clear that by a similar argument, $Y$ is infinite so that these inequalities imply $|X|=|Y|$.

This can be adapted to $<\kappa$-ary monads and sets of cardinality $\geq \kappa$ in an obvious manner but it didn't seem worth it to mention it.

(**) Suppose $h: TA\to A$ is a finite $T$-algebra with $|A|\geq 2$, suppose $X$ is finite and $TX\cong TY$ as $T$-algebras. Then the universal property of free algebras yields that $A^X \cong A^Y$ as sets. But $X$ is finite and $|A|\geq 2$ so $Y$ is finite as well and $|X|=|Y|$, therefore there is a bijection.

(* * *) : We use the universal property of free algebras as for ( * *) and the fact that if two cardinals (here $|X|, |Y|$) have the same continuum function $\mu \mapsto \mu^\kappa$ on infinite cardinals $\mu$, then they are equal, cf the answer to my previous post here



from Hot Weekly Questions - Mathematics Stack Exchange
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11:22

Show that the following equation has a solution in $ \mathbb{Z}/n\mathbb{Z} $ for every $n > 1$:

\begin{align} (x^2-2)(x^2-17)(x^2-34)=0 \end{align}

I know you can cite Chinese Remainder Theorem so it suffices to find solutions for powers of primes. Then you can somehow choose to consider only solutions $\pmod{2^e}$ * or $\pmod{8}$?* and $\pmod{p}$ for odd $p$. But I don't know how to get there or farther to solve the problem. I know the Legendre symbol can be used since $34 = 2 \cdot 17$.



from Hot Weekly Questions - Mathematics Stack Exchange
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11:17

Dr Phil Sutton

On 31st July I visited Lincoln shortwave club to talk about some of the important discoveries made by the Cassini spacecraft by utilising the radio part of the electromagnetic spectrum. We started by looking at some of the many radar images taken of Titan, Saturn’s largest moon, before discussing some of the least known science about the internal structures of Saturn and its moons by measure the doppler shift in radio transmissions sent back to Earth.

Below: Maps of the surface of Titan show large lakes and seas of methane. Due to the thick atmosphere, one of the only moons in the solar system to have such an atmosphere, radar was very important to gain detailed surface maps. Image credit: NASA / JPL

98E3231A-AAD1-4967-9F22E55E2D38328A_source 

Below: The internal structure of Titan with Saturn and its rings in the background. by measure the doppler shift in the radio transmission sent back to earth…

View original post 53 more words



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09:17

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.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits: /r/GradSchool, /r/AskAcademia, /r/Jobs, /r/CareerGuidance

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09:04

What’s getting students so engaged that they’re explaining their mathematical thinking? According to our teachers, ST Math Chats provide experiences that spark rich math discussion!

Earlier this year, we announced ST Math Chats as a new feature coming to ST Math®. Teachers across the country have been sharing their excitement.

math-chats-classroom

Liz Blose, a 5th Grade Teacher at Fogelsville Elementary in the Parkland School District was among the teachers that took to Twitter to voice her support.

Teachers who have enjoyed early access to ST Math Chats are seeing the rich math discussion and powerful math experiences that result from implementing them in the classroom. These can be used to introduce new math concepts, dive deeper into current concepts, and review concepts to strengthen understanding.

ST Math Chats use the visual models and manipulatives of ST Math puzzles to deepen learning through whole-class discussion. ST Math Chats are 25-30 minute lessons that utilize 1:1 or 2:1 technology to create an interactive learning and sharing experience. 

Now that teachers, educators, and ST Math users have had the opportunity to experience ST Math Chats firsthand, we’ve gathered their feedback and want to share it with you.

What Teachers Are Saying About ST Math Chats

One teacher who implemented ST Math Chats once every other week, shared, “The kids really enjoyed it and it allowed me to see what a lot of their misconceptions were.” 

This was an overwhelming theme in the feedback from teachers. By including discourse and encouraging conversations around mathematical concepts, students are able to deepen their learning, introduce correct language, and comprehend these concepts better.

Teacher Learning Impacts Student Learning

Another teacher shared, “My most positive experience was having students explain concepts that they understand, but lack the words. Seeing how their flexible minds find answers that we didn't even think about at that time. I can do a quick teach of a difficult concept like area/perimeter and I can walk around checking each one for understanding, reteaching, and deeper thinking.”

Abbi Lievens, a second grade teacher, shared her excitement on Twitter.

Another powerful, common theme in ST Math Chat feedback was the increased engagement that teacher’s saw as a direct result of implementing the program in the classroom.

Sandra Consilio, Director of Learning for MIND Research Institute, shared her feedback from Melanie Cooper, a STREAM teacher at Freeport Elementary.

How To Incorporate ST Math Chats

ST Math Chats lessons are aligned to grades 3-5 standards, and are designed for easy implementation. Using a Play-Discuss-Apply model, these lessons designate time for students to play an ST Math game, participate in discussion, and apply what they’ve learned on a conclusion problem. Each lesson provides opportunities for whole class, small group, and partner discussion.

math-chats-play-discuss-apply-nobg-1

Why Incorporate ST Math Chats?

Teachers can easily launch ST Math Chat lessons directly from their ST Math console and jump into a guided experience that uses the visual models and manipulatives of ST Math puzzles to spark rich math discussion.

We asked teachers why they implemented ST Math Chat lessons and the feedback showed that their motivation differed with each teacher. Some of the teachers introduced them to share new content. Some teachers implemented ST Math Chats to review mathematical concepts.

Other reasons for implementation varied, including:

  • To enhance conceptual development.
  • For formative assessments.
  • As an innovative way to engage students.
  • To grow problem-solving and discussion.
  • To replace existing learning resources.

Meagan Erwin, a third grade educator from Columbus City Schools shared, "If a teacher is just using ST Math during their computer time or their lab time right now, this is a great, easy way to step into doing whole group, whole class lessons using this incredible tool.”

How Educators Are Using ST Math Chats

Schools collaborating with us to improve the program varied in the way they introduced ST Math Chats into their classrooms. Teachers and schools alike face constraints that affect their time, resources, and implementation practices.

We asked teachers using ST Math Chats what their implementation was like. We discovered that implementation looked different across state lines and in each classroom. Some teachers used ST Math Chats several times a week. Others, used it every other week. Some teachers chose to use it each time they introduced a new topic or concept to the class.

Across the board, regardless of the implementation method or what they were solving for, over 92.9% of teachers agreed that ST Math Chats helped to increase math discourse with students. Thank you to all of the teachers for proving their feedback and helping us improve this feature in beta for grades 3-5!

One teacher shared, “They have been fantastic for engaging my students and pushing their thinking this year. Thank you!”



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