IFRAME SYNC
February 2020
Twitter Facebook

22:28

I'm trying to help my son with his homework and I'm out of my depth :(

As the question states, it look to me that I can concentrate on 5 and 1. so it will look li this:

[5,1,X,X,X,X] => 4!
[5,X,1,X,X,X] => 4!
[5,X,X,1,X,X] => 4!
[5,X,X,X,1,X] => 4!
[5,X,X,X,X,1] => 4!

[X,5,1,X,X,X] => 4!
[X,5,X,1,X,X] => 4!
[X,5,X,X,1,X] => 4!
[X,5,X,X,X,1] => 4!

[X,X,5,1,X,X] => 4!
[X,X,5,X,1,X] => 4!
[X,X,5,X,X,1] => 4!

[X,X,X,5,1,X] => 4!
[X,X,X,5,X,1] => 4!

[X,X,X,X,5,1] => 4!

so we end up with 4! * 15

Is this correct?

This is just me counting on my fingers ;), What would be the "Combinatorial" way of thinking and solving this kind of questions?

Thank you.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

19:28

I'm currently learning calculus (high school senior), and I am not comfortable with the idea that the limit of the sums of rectangles actually converges to the area under the curve. I know it looks like it does, but how do we know for sure? Couldn't the tiny errors beneath/over the curve accumulate as we add more and more rectangles? What's troubling me is the whole Pi = 4 thing with the staircase approximating a circle pointwise, and how it's wrong and the perimeter of the staircase shape does not approach the circumference of the circle, even though pointwise it does approach a circle. So how are the increasingly many, increasingly small errors in Riemann sums any different? How do we know the error in each step decreases faster than the number of errors increases? I would really like to see a proof of this.

Thanks so much!



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

13:18

Although this looks like elementary, I have trouble understanding the proof of Theorem 3.1 at page 7 of this paper by Dominic Orchard (Univ. of Cambridge). As hypotheses we are given a comonad $D$, a monad $T$ and an adjunction $D \dashv T$. But then in the course of the proof (at the top of page 8), the author constructs the monad $T$ from the comonad $D$. Why is the constructed monad identical to the one in hypotheses?



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

13:04

Hello, new to this community, please redirect me if there is a better subreddit. I am 37 and planning on going back to school for engineering. It has been about 18 years give or take since I have been to school. I am a little nervous that I am not very good at mathematics and have started taking some classes on edX to refresh myself. I am actually enjoying these, but feel like I need to review the basics before moving into anything more advanced. Are there any workbooks or resources that you could recommend? I think anything high school level and up would be really helpful. Edit: the current course I am taking on edX is basic geometry.

submitted by /u/GoddessOfChamomile
[link] [comments]

from math https://ift.tt/3ck1WxS
Twitter Facebook

12:48

How can we prove this identity: $$\sum _{k=1}^{n-1} \binom{n-1}{k} k^{k-1} (n-k)^{n-k-1}=n^{n-1}-n^{n-2}$$ A friend of mine gave me this problem several days ago. At first sight, the form of the summand may imply binomial theorem, but I haven't figured out the right way. Another possible way is to transform the sum into integration, however, the term $k^k$ does not seem to suit some famous integral representations. Therefore I'm kind of stuck and would like you to give some suggestions. Thank you.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

12:34

I found out later that this is already out there, so many of you must have seen it, but still felt like sharing.

If we assume √ab > a+b/2 squaring both sides, ab > (a+b)²/4 or 4ab > (a+b)². On expanding, we get 2ab > a² + b² on applying this to (a-b)², we end up with the result (a-b)² < 0, which is a contradiction, thus proving √ab ≤ a+b/2

P.S I posted the same thing on r/mathematics. Why are there two subreddits for this?

submitted by /u/BOIkratos1234
[link] [comments]

from math https://ift.tt/38cYVvK
Twitter Facebook

04:18

Show that if $N$ is large enough, then $x^5-Nx+1$ and $x^5-Nx^2+1$ are irreducible over $\mathbb{Q}$.

There is a hint for $x^5-Nx+1$: prove first that four of the roots in $\Bbb{C}$ has absolute values larger than $1$. I think that it follows from Rouche's theorem, but I don't know how to proceed.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

01:18

Define

$$I_n=\int _0^{\frac{\pi}{2}} \sqrt{1+\sin^nx}\, dx$$

I have to show this sequence is convergent and find its limit. I proved it is decreasing: $\sin^{n+1} x \le \sin^n x \implies I_{n+1} \le I_n$. Also, it is bounded because:

$$0 \le \sin^n x \le 1 \implies \frac\pi{2} \le \int _0^{\frac{\pi}{2}} \sqrt{1+\sin^n x}\, dx\le \frac{\pi\sqrt{2}}{2}$$

so it is convergent. I'm stuck at finding the limit. I think it should $\frac{\pi}{2}$ but I'm not sure.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

16:18

Three different situations:

  1. Take a group $G$. If you have a normal subgroup $H$, you can form a quotient $G/H$.
  2. (Can be considered as a corollary of 1.) Take an $R$-module $M$. If you have a submodule $N$, you can form a quotient $M/N$.
  3. Take a topological space $X$. If you have a subspace $U$, you can form a quotient space $X/U$.

However, if you take a (commutative and unital) ring, you need to quotient by an ideal, that (except for the trivial case in which it contains 1) is not a subring.

In general the construction of a quotient object in a given category requires the notion of a congruence, that is very different from a subobject. My question is

How typical is that we can go from a subobject (maybe requiring an additional property) to a congruence?

or more precisely

Is there a categorical notion encompassing this behaviour?

Edit:

(Based on comments by Rob Arthan and GEdgar). First two examples are effective congruences (Definition 1.5 in the link above). In other words we take two identical maps $f\colon A\to B$ and find a pullback $K$ with two maps $K\to A$. It can be treated as a subobject of $A\times A$.

But we want $K$ to be a subobject of $A$, so we could "replace $a\sim b$ by $ab^{-1}\in K$".



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

09:18

How can we prove that: $$\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy= -4 \pi ^2 \left(\frac{\pi }{\sqrt{3}}+\log (2)-\frac{\psi ^{(1)}\left(\frac{1}{6}\right)}{2 \sqrt{3} \pi }\right)$$ Where $\psi^{(1)}$ denotes trigamma function. It's J. Borwein's review on experimental mathematics that offers this interesting identity (I've verified it numerically). The literature refer this formula to V.Adamchik, but I haven't find any related source dealing with this kind of integrals. In fact, I've asked this question on this site a year before but no answer was given. Since I have still no solution, I'd like you to give some suggestions again. Any help will be appreciated.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

08:48

I passed my probability exam this morning, there was a question that I didn't solve it completely! I have been thinking about it for two hours after the exam but I don't know how it could be solved.

Consider:

$*$ A square $S = [0,1] \times [0,1]$

$*$ A disc $D$ of radius $1/2$, centered in $(1/2,1/2)$.

$*$ A sequence $X_1, \dots, X_n, \dots$ of a random vector of $\mathbb{R}^2$ independent and identical from the uniform distribution in the square $S$.

Clarify how it is possible to approximate $\pi$ using the the number of points $X_n$ fall inside the disc $D$, then use the central limit theorem to calculate the minimum number of samples needed so that the probability of deviating from $\pi$ by more than $0.01$ is less than $0.1 \%$

The only thing I wrote: "We are using here the Box-Muller method for sampling" but I don't know how this could be helpful for approximating $\pi$.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

08:35

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!

submitted by /u/AutoModerator
[link] [comments]

from math https://ift.tt/2Vvm6Pq
Twitter Facebook

07:04

Obligatory apologies if this subject is too "homework-y" for the mainline math sub.

  • I tend to get exceedingly long-winded when I type out reddit posts so I will do my best to keep this short. Skip to the quoted material if you don't eff with mostly-unnecessary context.

  • I am an ex math major who had my higher-education cut short for medical reasons. I have always been fond of recreational maths. Dual numbers in particular have always tickled my brain in a particular way. This question specifically I have been coming back to for over a year with little-to-no forward progress. I have done my best to read the appropriate reference material, but (as most technical writing online) it is rather dense. Here's hoping the answer to this leans-trivial or at least can be condensed to a handful of wikipedia articles rather than the 30 I've been going back to.

  • I am less familiar with all of this now than I once was; its been a while I've been back to this. Sorry. Hopefully I haven't retreaded some of my previous mistakes.

  • Also I've never used latex and cannot find a reliable plugin/script for firefox and thus have no way of knowing if this will display properly. At the very least it is formatted poorly. Sorry. I will be including a link to an image of the rendered text in case I wiff the formatting and/or for mobile users

  • I also realize I include a significant amount of math that's likely unnecessary. I get I could've condensed this to like three lines; most papers I read on grassman numbers didn't bother to touch this algebraic stuff. Or if they did they wrote it in lanquage i couldn't understand. It is included here to give context to my thought process.

[; \varepsiloni * \varepsilon_j = -\varepsilon_j * \varepsilon_i :, \ Let: \varepsilon_i = \varepsilon_j:; \varepsilon_i * \varepsilon_i = 0 = -\varepsilon_i * \varepsilon_i \ \dot{.\hspace{.095in}.}\hspace{.5in} \varepsilon_i2 = 0\ \forall\ \varepsilon \ \dot{.\hspace{.095in}.}\hspace{.5in} \varepsilon : \textrm{are } \emph{nilpotent} \ \ (\varepsilon_i + \varepsilon_j)2 = \varepsilon_i2 + \varepsilon_i \varepsilon_j + \varepsilon_j \varepsilon_i + \varepsilon_j2 = 0 + \varepsilon_i \varepsilon_j - \varepsilon_i \varepsilon_j + 0 = 0 \ (\sum{k=0}{n} \varepsilon{i_k})2 = \sum{k=0}{n} \varepsilon{i_k}2 + \sum{k=0}{n}(\varepsilon{i_k}\sum{j=k+1}{n}\varepsilon_{i_j}) - \sum{k=0}{n}(\varepsilon{ik}\sum{j=k+1}{n}\varepsilon_{i_j}) = 0 \ \dot{.\hspace{.095in}.}\hspace{.5in} \textrm{sums of } \varepsilon: \textrm{are } \emph{nilpotent} \ \ ex = 1 + x + \frac{x2}{2!} + ... + \frac{xn}{n!} + ... = \sum{i=0}{n} \frac{xn}{n!} \ e\varepsilon = 1 + \varepsilon :, : (\textrm{for } \emph{all} \textrm{ nilpotents of degree 2}) \ e{\varepsilon_i+\varepsilon_j} = 1 + \varepsilon_i + \varepsilon_j : (\textrm{as } \varepsilon_i + \varepsilon_j \textrm{ is nilpotent}) \ \dot{.\hspace{.095in}.}\hspace{.5in} e{\sum{k=0}{n} \varepsilon{i_k}} = 1 + \sum{k=0}{n} \varepsilon{i_k} \ \ \prod{k=0}{n}(1 + \varepsilon{i_k}) \ = 1 + \varepsilon{i0} + \varepsilon{i1} + ... + \varepsilon{in} \ + \varepsilon{i0}\varepsilon{i1} + \varepsilon{i0} \varepsilon{i2} + ... + \varepsilon{i0} \varepsilon{in} \ + \varepsilon{i1}\varepsilon{i2} + \varepsilon{i1}\varepsilon{i3} + ... + \varepsilon{i1}\varepsilon{in} \ + ... \ + \varepsilon{i{n-1}}\varepsilon{in} \ + \varepsilon{i0}(\varepsilon{i1}\varepsilon{i2} + \varepsilon{i1}\varepsilon{i3} + ... + \varepsilon{i1}\varepsilon{in}) \ + \varepsilon{i1}(\varepsilon{i2}\varepsilon{i3} + \varepsilon{i2}\varepsilon{i4} + ... + \varepsilon{i1}\varepsilon{in} + ... + \varepsilon{i{n-1}}\varepsilon{in}) \ + ... \ + \varepsilon{i{n-2}}\varepsilon{i{n-1}}\varepsilon{i{n}} \ + ... \ + \varepsilon{i{0}}\varepsilon{i{1}}...:\varepsilon{i{n-1}}\varepsilon{i{n}} \ \ =\prod{k=0}{n}e{\varepsilon_{i_k}} \ \ = e{\sum{k=0}{n}\varepsilon{i_k}} \ \ = 1 + \sum{k=0}{n}\varepsilon{ik} = 1 + \varepsilon{i0}+\varepsilon{i1}+ ... + \varepsilon{i{n-1}} + \varepsilon{i_n} ;]

SUMMARY AND WHY IT MATTERS:

  • These epsilon operators are defined to anti-commute with one another. As a result each operator is nilpotent (epsilon2 = 0)
  • Sums and products of these operators are also nilpotent. ( i think? maybe this is as simple as me messing up the non-commutative multiplication.)
  • enilpotent = 1 + nilpotent
  • Every combination of n coefficient-less operators can be generated with the provided product formula
  • the product can be rewritten as the product of exponential functions which can be condensed into a single exponential function
  • this exponential can be rewritten as a simple sum of single operators

My problem is that this sum is included in the original product (which it is ostensibly equal to), but the product goes on to include more items. A few questions arise:

  • Where did I mess up this math?
  • when using one of these numbers is it simply identical to leave out the remainder of the original product? I can see intuitively why this may be the case but do not know how to "prove" or otherwise explore this exhaustively.
  • If there is nothing wrong this result, does it continue to hold with an infinite amount of operators? (i am interested in exploring the properties of these for myself)
  • Can a similar result be found when considering the "full" sum where each variable has unique coefficients? (i am interested in exploring the properties of these variations myself; this is what got me looking into this in the first place)
  • if not, do there exist coefficients where the equality still holds true?
  • why are these "supernumbers" always formatted even differently than this? example (it is particularly strange to me that, in the case of the infinite, I am hardpressed to find an example that does not include the converging factorials) *in Dewitt's paper on the subject (afaik the premier source on the topic) why does he say "the soul need not be nilpotant"? is this related? incomplete google archive. (Found directly under 1.1.4)

I would appreciate any insight on this topic. I find it hard to maintain my interest in mathematics for very long when I inevitably run into large roadblocks like these. I am not opposed to doing the research; the roadblock for me is knowing what exactly to research. I endeavor to get to a point where I can stay out of the hospital for long enough to continue taking classes; i can practically feel my synapses rerouting around the rudimentary base I had built up five years ago.

Please do not feel the need to speculate on my knowledge-base or use that to temper your response. I have more than enough time to actually look into this--the right way. I finished a significant amount of my undergraduate mathematics in high school and have a handful of semesters under my belt filling out that base. I suspect I am a course or two away from this being a "well duh"-type problem.

EDIT: sorry in advance if i do not reply to your comments. I promise they are no less appreciated. I cannot use my hands for long periods of time and I am afraid i may have already overdid it with all this. My access to people who are generous to notate for me is limited and i am working on a more permanent accessibility solution

submitted by /u/SigmaGod
[link] [comments]

from math https://ift.tt/3ab6zs2
Twitter Facebook

05:48

What I need to prove is a consequence of the following theorem.

Theorem A. Let $G$ be a finite $p$-group and suppose that its derived subgroup $G'$ is generated by 2 elements. Then there exists $x\in G$ s.t. $$ K_x(G):=\{[x,g]\mid g\in G \}=G'. $$

Now let $G$ be a pro-$p$ group whose derived subgroup is topologically generated by 2 elements. Define for any open normal subgroup $N$ of $G$ the set $$X_N:=\{x\in G\mid K_{xN}(G/N)=(G/N)'\}. $$ Notice that since $G$ is pro-$p$, every open normal subgroup of $G$ has finite index, therefore we can apply Theorem A to $G/N$ and conclude that $X_N\ne \emptyset$ for all $N\trianglelefteq_o G$.

What I need to prove is that the family $\{X_N\}_{N\trianglelefteq_o G}$ has the finite intersection property.

My guess is that we can take two any open normal subgroups $N$, $M$ of $G$, consider their product $NM$ which is still normal and open (since their product is equal to the union of cosets of an open subgroup) and prove that $$X_N\cap X_M \supseteq X_{NM}.$$ Since $NM\trianglelefteq_o G$, $X_{NM}$ would be non-empty and the statement would be proved.

All of this comes from here (at page 2 you can find Theorem A, as well as Theorem B whose proof, at page 11, is what I'm interested in). Notice that the author says that it's clear that $\{X_N\}_{N\trianglelefteq_o G}$ has the finite intersection property, so there could be an easier way to prove that.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

05:03

Charlotte Scott Centre for Algebra

On Wednesday the 26th of February 2020, Dr Daniel Woodhouse (University of Oxford) visited the Charlotte Scott Centre for Algebra and gave a talk “Leighton’s graph covering theorem”.

Abstract: Leighton’s theorem for graphs states that any pair of finite graphs with common universal covers have a common finite cover. Equivalently, this says that any pair of uniform lattices in the automorphism groups of a tree can be commensurated. I will discuss the significance of this theorem, recent generalizations, and how it can be applied in geometric group theory.

View original post



from Maths & Physics News
Twitter Facebook

01:18

While reading Borwein's review on experimental mathematics, I found a curious identity which is similar to the well-known Ahmed integral. $$\int_0^{\infty } \frac{\tan ^{-1}\left(\sqrt{a^2+x^2}\right)}{\left(x^2+1\right)\sqrt{a^2+x^2}} \, dx=\frac{\pi \left(2 \tan ^{-1}\left(\sqrt{a^2-1}\right)-\tan^{-1}\left(\sqrt{a^4-1}\right)\right)}{2 \sqrt{a^2-1}}, \ a>1$$ However, the original solution of Ahmed integral (depending on symmetry) does not seem to work here. Therefore I'd like to post it here to seek for suggestions. Any help will be appreciated!



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

01:18

I want to show the non-naturality of the splitting in the universal coefficient formula for homology. The s.e.s. is $$0\to H_q(X,X';R)\otimes_R N\to H_q(X,X';N)\to Tor^R_1(H_{q-1}(X,X';R),N)\to 0$$ where $R$ is a PID and $N$ an $R-$module.

I found a counter example with the map $\mathbb{RP}^2\to S^2$ that collapses the 1-cell of $\mathbb{RP}^2$ to a point, but I don't get why this map works.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

09:09

In each episode of the Inside Our MIND podcast, we take a look at issues and challenges facing education that we are working to address through research, technology and strategic initiatives.

In our latest episode, Brian welcomes Senior User Experience (UX) Researcher Alesha Arp back to the show for a discussion on what it truly means to be data-driven. They talk about the overuse of the term “data-driven,” and what it means for data to be actionable. Alesha shares insights from her research and the conversations she’s had with ST Math users that helped inform the redesign of ST Math.


You can listen to the episode in the player below:

Topics Covered in the Podcast:

0:45 Intro
4:00 What does data-driven mean?
6:00 Focusing on how you get there
8:00 The danger of focusing on a single metric
12:00 Implementation challenges that informed new ST Math design
16:30 Providing the right metrics and making data actionable
28:00 Empowering students to set their own goals

Thanks for listening to the podcast! Please leave us a review on iTunes, Google Podcasts, Spotify, Spreaker or wherever you are listening to the show. Subscribe to get future episodes as soon as they are released! 
Get_it_on_iTunes_Badge_US_1114.svg Listen on Google Play Music Inside Our MIND Podcast RSS Feed

Additional Resources:

Designing Data to be Actionable
Podcast: How User Experience Research Informs Edtech Design
Fidelity Through Thoughtful and Strategic Implementation
Podcast: Talking About the New ST Math

from MIND Research Institute Blog https://ift.tt/2PviOrH
Twitter Facebook

08:18

I am curious about the nature of the primes in

$$ \frac{m(m+1)(m+2)...(m+k-1)}{k!} $$

when $k < m$ ( means they dont overlap).

We can show that it is always an integer using n choose k formulation or pascals rule or by prime factor counting.

But this last proof got me thinking. How can it be shown that there will be other primes as factors than those in $k!$ ?

Edit: I want to show that there is at least one prime greater than k as factor in that in every case.

I am looking for a non-handwavy yet insightful proof.

Edit2: I want to show that there is at least one ( specific form not needed ) prime greater than k that divides the product of any k consecutive natural numbers. just existence proof. In other words, I want to show that there is always a prime greater than k between k and n/2 or between n-k and n, whenever n is strictly greater than 2k.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

07:48

This is perhaps a soft question.

Let $X=\mathbb{S}^1 \times \mathbb{S}^1$. Let $\mathbb{Z}_2$ act on $X$ by setting $(-1) \cdot (\theta,\psi)=(\theta+\pi,\psi+\pi)$. Consider the quotient space $X/ \mathbb{Z}_2$ which is obtained after identifying $ (\theta,\psi) \sim(\theta+\pi,\psi+\pi)$.

Is there a succinct description of $X/ \mathbb{Z}_2$ as some product or twisted/fibered product or something like that?

Are there other "simple" descriptions of this space? Is it related to some projective space?

I feel like there should be a "right" terminology to describe it, or a way to recognize it as some familiar space, but I fail to see it.

I understand that identifying antipodal points on the $2$-torus embedded in $\mathbb{R}^3$ results in a Klein bottle- but this is not the same identification we are doing here:

Here we identify $(\theta,\psi)=(\theta+\pi,\psi+\pi)$, and in the embedded description we identify $(\theta,\psi)=(\theta+\pi,-\psi)$.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

04:18

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

The Iditarod Dog Sled Race will begin this year on Saturday, March 7th in Anchorage, Alaska. It is often called The Last Great Race on Earth because it is so long and grueling. Students do some rate, distance, and hardship comparisons to decide if they agree that the Iditarod is really the Last Great Race on Earth.

You could begin this activity with Dallas Seavey's Iditarod introduction movie below.  Dallas has won the Iditarod in 2012, 2014, 2015, and 2016.  His dad, Mitch Seavey, won in 2004, 2013 and 2017.

The activity: Iditarod2020.pdf

For members we have an editable Word docx and solutions.

Iditarod2020.docx    Iditarod2020-solution.pdf

CCSS: 5.NBT, 5.NF.B, 6.RP.A, 6.NS,7.RP.A



from Yummy Math
Twitter Facebook

03:04

Greetings Redditors. I wanted to share with you my latest creation. This is a video explaining how to do Geometric Two Column Proofs. It is an in depth method NOT TAUGHT ANYWHERE ELSE. The strategies shown here work for all textbooks and yet are not found in any textbook. This is my personally derived method for explaining to students the thought process behind a Two Column Proof. Try not to skip around in the video as it is progressively taught, with previously explained concepts underpinning future techniques. You will not find this method of thinking anywhere else online to my knowledge or any textbook or video. If it is found elsewhere then they independently had the same thought I did about how to explain Proofs to students. I am really proud of this technique and feel it can revolutionize your understanding of what exactly is happening in a Proof and how to approach the thinking behind them. Part 2 will be linked after I upload it for a total of one introduction and 5 examples. If you would like more examples, please request via comment on one of the two Proof videos. Please also leave feedback if you find this technique to be as useful as I think you will. If you have ANY questions please leave a comment and I try to respond as quick as possible.

I consider it one of the most important things you can learn in all of Mathematics, especially if you are not going to pursue a career in a STEM field, as it is the very foundation of logical thinking in Middle School and High School Curriculum.

All feedback is very welcome! Questions are ok as well. The video can be found here: https://youtu.be/wGozA2RSSrU

Part 2 with 3 more examples is linked at the 3nd of the first video. Thank you for your time and feedback!

submitted by /u/IceMatrix13
[link] [comments]

from math https://ift.tt/2VpvvYI
Twitter Facebook

01:17

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

We spoke to Dan Aspel, communications officer at the Isaac Newton Institute in Cambridge, about the Institute’s in-house podcast.

Podcast title: Isaac Newton Institute Podcast
Website: newton.ac.uk/news/podcast
Links: Apple – Podtail
Average episode length: 10-20 minutes
Recommended episode: This Interview with Matt Parker at the TMiP conference 2019

What is the Isaac Newton Institute?

Founded in 1992, the Isaac Newton Institute is the UK’s premier mathematical research centre. Based on the University of Cambridge’s mathematics campus, it runs research programmes with myriad applications over a wide range of cutting-edge science and technology. Selected by a panel of leading mathematicians, these programmes are chosen for their scientific merit and presence at the forefront of current developments where a significant scientific breakthrough can be expected. They are expected to be interdisciplinary and to have the highest quality leadership and participants invited from across the globe – with the result being that a third will typically come from the UK, a third from Europe as a whole and a third from the rest of the world. This mixture of minds, as well as the time and space given to them, allows the Institute to transcend the boundaries of departmental structures and truly move disciplines.

What kind of things happen at the INI?

Recent programmes have focused on subjects as diverse as: the mathematics of energy systems, the science of the human heart, the Wiener-Hopf technique, complex analysis, uncertainty quantification, computer vision and quantum field theory. Notable past participants have included more than 27 Fields Medallists and winners of the Abel Prize. INI also has a pivotal role in helping to champion diversity within the mathematical sciences. Numerous initiatives, including the Kirk Distinguished Visiting Fellowship and a supportive and open infrastructure for those travelling with families and young children, are helping the Institute in this regard as well as the availability of funding packages for visitors and participants from developing countries.

Tell us more about your podcast.

Launched in March 2019, the INI podcast series aims to highlight these diverse people and explore the many interconnected topics linked to the Institute’s activities. Interviewees may range from visiting academics and lecturers to mathematicians, other scientists, and prominent figures within the University of Cambridge and beyond. The podcast typically involves mathematical themes, but is specifically aimed at a general audience. The focus is on the subjects being interviewed and the social stories they have to tell, not just on the significance and details of the research they may be undertaking.

What’s special about your podcast?

With 2,500 visitors passing through the Institute each year, there is a rich pool of interviewees from which to choose. We hope that 15+ episodes already available and the many more to come will inspire, interest and entertain.



from The Aperiodical https://ift.tt/2HWaDAi
Twitter Facebook

22:18

It's known that Fibonacci numbers satisfy the following relation:

$$F_mF_{n+1}-F_{m+1}F_n=(-1)^nF_{m-n}$$

Which is called d'Ocagne's identity.

This identity with the following identities are well-known:

$$F_{n-1}F_{n+1}-F_{n}^2=(-1)^n\tag{Cassini's identity}$$ $$F_{n}^2-F_{n-r}F_{n+r}=(-1)^{n-r}F_r^2\tag{Catalan's identity }$$ $$F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_iF_j\tag{Vajda's identity }$$ $$F_{k−1}F_n + F_kF_{n+1} = F_{n+k} \tag{Honsberger identity}$$

Cassini's identity is a special case of Catalan's identity and can be derived with $r=1$.

The usual way for proving these identities is using $2×2$ matrix , another way would be induction,I know how to prove Catalan's identity using induction but still I have not seen any proof of d'Ocagne's identity,I'm asking if someone know a proof of that (induction preferred)?

Also is their any combinatorial poof for d'Ocagne's identity? if yes, so it would be really nice to see the proof.

My try:

  • Define: $$a:=\frac{1+\sqrt{5}}{2}\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\; b:=\frac{1-\sqrt{5}}{2}$$ Then using this follows: $$F_mF_{n+1}-F_{m+1}F_n$$ $$=\left(\frac{a^{m}-b^{m}}{\sqrt{5}}\right)\left(\frac{a^{\left(n+1\right)}-b^{\left(n+1\right)}}{\sqrt{5}}\right)-\left(\frac{a^{\left(m+1\right)}-b^{\left(m+1\right)}}{\sqrt{5}}\right)\left(\frac{a^{n}-b^{n}}{\sqrt{5}}\right)$$

$$=\frac{\color{red}{a^{\left(m+n+1\right)}}-a^{m}b^{\left(n+1\right)}-a^{\left(n+1\right)}b^{m}+\color{blue}{b^{\left(m+n+1\right)}}}{5}-\frac{\color{red}{a^{\left(m+n+1\right)}}-a^{\left(m+1\right)}b^{n}-a^{n}b^{\left(m+1\right)}+\color{blue}{b^{\left(m+n+1\right)}}}{5}$$ $$=\frac{-a^{m}b^{\left(n+1\right)}-a^{\left(n+1\right)}b^{m}+a^{\left(m+1\right)}b^{n}+a^{n}b^{\left(m+1\right)}}{5}$$$$=\frac{a^{m}b^{n}\left(a-b\right)+a^{n}b^{m}\left(b-a\right)}{5}=\frac{\left(a-b\right)\left(a^{m}b^{n}-a^{n}b^{m}\right)}{5}$$$$=\left(a-b\right)\frac{\left(a^{\left(m-n\right)}-b^{\left(m-n\right)}\right)}{\sqrt{5}}\frac{a^{n}b^{n}}{\sqrt{5}}$$$$=\left(a-b\right)\frac{a^{n}b^{n}}{\sqrt{5}}F_{m-n}$$$$=\bbox[5px,border:2px solid #00A000]{\left(-1\right)^{n}F_{m-n}}$$

Which is the claim.

  • Another way for proving this identity would be setting $i \mapsto m-n$ and $j \mapsto 1$ in Vajda's identity.


from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

04:01

Revisited the Monty Hall Problem recently and realised it is interesting probability question and gets you thinking. There might be many such paradoxes/questions(not just related to probability but any topic) which are intriguing and counterintuitive in Mathematics. Is there any book where there is a collection of such cool stuff

submitted by /u/thehaddi
[link] [comments]

from math https://ift.tt/3a9LRZM
Twitter Facebook

02:28

I have a problem in which I am trying to prove over GF(2) that a binary symmetric matrix (A) with a diagonal of ones has a rank always equal to the rank of its augmented matrix with a ones vector (C) $$ C=\left[\begin{array} \\ 1 \\ \vdots \\ 1 \end{array}\right] $$

To clarify, such matrix is constructed like so: $$ A=\left[\begin{array}{rrrr} 1 & a_{1,1} & a_{1,2} & \dots & a_{1,n} \\ a_{1,1} & 1 & a_{2,1} & \ddots & \vdots \\ a_{1,2} & a_{2,1} & \ddots & a_{n-1,n-1} & a_{n-1,n} \\ \vdots & \ddots & a_{n-1,n-1} & 1 & a_{n,n} \\ a_{1,n} & \dots & a_{n-1,n} & a_{n,n} & 1 \end{array}\right] $$

For example, a 3 by 3 matrix like this has a rank of 2: $$ A=\left[\begin{array}{rrr} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ When we augment it with a ones vector, we get this matrix which also has a rank of 2: $$ A|C=\left[\begin{array}{rrr|r} 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{array}\right] $$ Cleary rank(A) = rank(A|C) over GF(2).

Why is this always true for such type of matrices?

If you have a proof, an idea, or a suggestion on how to proceed, please let me know. Any help is appreciated.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

01:28

It is a celebrated equation that $$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\ddots}}}}$$

However, there are two other conjectured equations that I found which, if true (they seem to be), might reveal a pattern.

$$\frac{\pi^2}{12}=\cfrac{1}{1+\cfrac{1^4}{3+\cfrac{2^4}{5+\cfrac{3^4}{7+\ddots}}}}$$

$$\frac{\pi^3}{36}=\cfrac{1}{1+\cfrac{1^6}{3+\cfrac{2^6}{5+\cfrac{3^6}{7+\ddots}}}}$$

Conjectured General Formula: For natural $n\geqslant 1$, $$\frac{\pi^n}{4\cdot 3^{n-1}}=\cfrac{1}{1+\cfrac{1^{2n}}{3+\cfrac{2^{2n}}{5+\cfrac{3^{2n}}{7+\ddots}}}}$$

Can these be numerically verified? I have not the skill to by-hand prove/disprove these, and have only been using Wolfram Alpha to arrive at these conjectures.

It would also be much appreciated if one could suggest a program I could install in order to evaluate these continued fractions independently, as well as the code required. Will PARI/GP suffice?

Thanks.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

00:28

It's very common to solve partial differential equations via "separable solution", in the following way. Say we have the wave equation,

$$u_t=u_{xx}.$$

We often solve this by assuming a form $u(x,t)=X(x)T(t)$, which gives

$$\frac{T_t}{T}=\frac{X_{xx}}{X}=\lambda,$$

where $\lambda$ is the separation constant, and the final solution looks something like

$$u(x,t)=C\exp (\lambda t -i\sqrt{\lambda} x)$$

However, it seems to me we could just as easily have tried the ansatz $u(x,t)=X(x)+T(t)$. Then our PDE would look like

$$T_t=X_{xx}=\lambda$$

And we would find

$$X(x)=\frac{1}{2}\lambda x^2 + C_2 x + C_3,\qquad T(t)=\lambda t + C_1.$$

Basically, a polynomial solution instead of an exponential one.

Is there any good reason why we often present the first way instead of the second? I guess there are "niceness" properties that the exponentials have, but polynomial solutions are nice in some ways too.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

00:28

Let $A$ be a $N\times N$-matrix with elements $$ a_{ii}=1 \quad\text{and}\quad a_{ij} = \frac{1}{ij} \quad\text{for}~ i\neq j. $$ Then $A$ is positive-definite, as can be easily seen from $$ x^T A x = \sum_i x_i^2 + \sum_{i \neq j} \frac{x_i x_j}{ij} \geq \sum_i \frac{x_i^2}{i^2} + \sum_{i \neq j} \frac{x_i x_j}{ij} = \left(\sum_i \frac{x_i}{i}\right)^2 \geq 0. $$

Assume now that $A$ is a real symmetric $N\times N$-matrix with elements $$ a_{ii}=1 \quad\text{and}\quad |a_{ij}| \leq \frac{1}{ij} \quad\text{for}~ i\neq j. $$

Is it possible to show that $A$ is also positive-definite (or positive-semidefinite)?



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

19:01

I am close to graduating with a bachelor's degree in mathematics, and one of my last assignments is to write a paper and give several presentations over a topic in mathematics. It can't be strictly historical, and it can't be something I've already learned about. I have to pick a topic that's new to me that involves some kind of computation or proofing, and self study that topic to present to the department. My topic can not be too broad, as I have to cover it within 7 pages.

I was wondering if anybody had an interesting idea for this. I have taken topology and really enjoyed it, so I'm starting my search there. I have had abstract algebra 1, real analysis 1, differential equations, and probability, so I have a good understanding of the basic ideas in those fields. I'll be flipping to the later chapters in those books to see if something catches my eye.

submitted by /u/IFearTomatoes
[link] [comments]

from math https://ift.tt/2SZNrHK
Twitter Facebook

08:58

A theory is categorical if it has a unique model up to isomorphism. First-order Peano arithmetic is not categorical, but second-order Peano arithmetic is categorical, with the natural numbers as its unique model. The first-order theory of real closed fields is not categorical, but the second-order theory of Dedekind-complete ordered fields is categorical, with the real numbers as its unique model. ZFC is not categorical, but Morse-Kelley Set Theory with an appropriate axiom about inaccessible cardinals is categorical.

My question is, what theory of the complex numbers is categorical? The first order theory of algebraically closed fields of characteristic zero is not categorical, because both the field of algebraic complex numbers and the field of complex numbers satisfy it. So is there some second-order axiom we can add to this theory to make it categorical?



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

08:31

So I'm writting my master's thesis on dynamical systems comming from a biological background. Hence, I don't know some of the formalities of mathematics. Part of my project is based on boolean network models, so I need to explain a bit on that in the thesis. Writting about them I started wondering if a boolean network can have no equilibria, as happens with ODE's, or if you just end up with an n-period equilibrium which dissappears with alternative updating methods.

submitted by /u/_between3-20
[link] [comments]

from math https://ift.tt/380rrk6
Twitter Facebook

07:58

A well known interesting fact is that if two integers are picked "at random" (in an appropriate asymptotic sense), the chances they generate the integers is $6/\pi^2$. So, the integers can be generated by two randomly selected elements with non-vanishing probability.

I am wondering if there exists a (finitely generated) group that is almost certainly not generated by any finite number of "randomly chosen" elements. That is, is there a group $G$ with generators $\{g_1, ... , g_n\}$ such that for every $k$, our if we pick $k$ elements of $G$ at random, there is a vanishingly small probability that the selected elements generate $G$.

To clarify, select elements randomly from balls of a given radius in the group, using the word metric with given generators, and see if the probability $k$ elements chosen from the ball generate $G$ tends to $0$ as the radius of the balls grows. This definition is compatible with the above result about the integers.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

05:21

Cesare Tronci visited the School of Mathematics at the University of St Andrews last week and gave a talk on Friday 21st February in the Applied Mathematics Seminar Series. The title of his talk was “Modeling efforts in hybrid kinetic-MHD theories of magnetized plasmas” (link to the StA seminar page here). He also had research interaction with the Solar and Magnetosphere Theory Group. The photo below shows a part of the St Andrews campus.



from Surrey Mathematics Research Blog https://ift.tt/39VRWc2
Twitter Facebook

03:28

I have a question about Gödel-Gentzen negative translation. According to the Wikipedia article for negative translations, "a sentence $\phi$ may not imply its negative translation $\phi^{\rm N}$". I am not sure if I understand correctly this sentence. Does it mean that if $\phi$ is true in intuitionistic logic, $\phi^{\rm N}$ is not necessarily intuitionistically true? If that's the case, could anyone give me an illustrating example?

Thanks!



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

02:29

Texas gets to send a lot more delegates to the July National Democratic Convention than New Hampshire does.  How do they figure out what each state's fair number of delegates should be?  What are those weird abbreviations in the allocation formula above? Do the math!

The activity: How-to-earn-delegates.pdf

Extra info: The Green Papers.pdf

CCSS: 6.RP, 6.EE.A, 6.EE.B, 7.RP, 7.EE, HSA.SEE

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

How-to-earn-delegates.docx   DelegateVotes.xlsx   How-to-earn-delegates-solution.pdf



from Yummy Math
Twitter Facebook

01:01

Hello, Informatics student here. I had a course on numerical methods this semester (going to the finals just now actually) and I kept learning methods named after Gauss (like Gaussian quadrature or approximating the solution for large systems of linear equations with Gauss-Seidel) and other (old) mathematicians over and over again. A lot of what we deal with is approximations. My question is that naively I thought of numerics as a field of Mathematics that was relevant in the era of computers, so I didn't expect a lot of methods to be specified that long ago. Why am I wrong? What was the historical context in which these findings came to be?

Side note: any book recommendations for the history of numerics would be appreciated

submitted by /u/TheActualMc47
[link] [comments]

from math https://ift.tt/2TaO1Bi
Twitter Facebook

00:01

I know that there’s a lot of mathematics that goes into how we count votes, but I’m not familiar with any of it. As the primaries are rolling around I’m thinking a lot about how votes are counted, reported, etc, and wondering why we don’t award the presidency to the person with the most votes. Does anyone have insight on this, or on voting theory in general?

submitted by /u/appleoftheorangetree
[link] [comments]

from math https://ift.tt/2w4ONYA
Twitter Facebook

21:01

Hello,

Just looking for some insight on how to approach this toy project that a friend and I are interested in approaching. Our tools to solve problems are minimal in terms of the grand scheme of things because we are undergraduates, but we would like to expand our horizon.

Our goal is to take a large set of data points and create a function for these data points. We were wondering about implementing different methods of approximation using Taylor's polynomial or regression analysis. Essentially, we want to create an approximated function from a large set of data points. After understanding and duplicating the process on a small sample size. We want to create a program that will approximate any large set of data points. From this program, we want to input data points from a variety of industries into which we can make predictions deriving these functions and analysing the patterns that may arise.

What are the best methods to build a model from a large set of data points?

submitted by /u/Certitudes
[link] [comments]

from math https://ift.tt/2Poag5H
Twitter Facebook

16:31

I'm a 4th year undergrad about to graduate. I've applied to graduate schools and already gotten a few acceptances, which makes this all 10x more scary and frustrating for me.

What do I mean by fail? Well, I opened the exam book, answered the first question which was 1/3rd of the test, and completely blanked on the other two questions. Like total blank out. All the theorems I knew, all the problems I did, my way of mathematical reasoning just dissapeared in a flash. I kept trying different things and none of them worked. This coupled with the fact that I probably didn't get the first questions perfect means I likely got in the 20%'s on this test. The test is worth 30% of my total grade. Meaning, assuming I got 20% and taking into account my assignment grades thus far, the maximum grade I can get ASSUMING I DO PERFECT FROM HERE ON is a 74%. That sucks.

I did all the homework assignments. I'd say I was spending approximately 10-12 hours a week on them, on top of going to lectures. So that's 13-15 hours weekly spent on this course. Could I have done more? Sure. It's definitely my fault. I just feel so, so frustrated with myself considering I've spent way less time on courses that are considered "just as hard" and done very well.

I don't know what this really is. I guess I'm just super frustrated with myself and ranting here because nobody else understands/cares in real life. I have two good grad school acceptances in my back pocket which I feel like I'm about to fucking blow after doing so poorly on this test. My other courses aren't doing so spectacular either. I guess the worst part is that I decided to give myself such a challenging courseload in my last semester to "prepare" myself for grad school even though I didn't have to. Altogether this is making me seriously reconsider and wonder if I can even handle mathematics at the graduate level to begin with, if I'm struggling with basic fundamental undergrad stuff.

submitted by /u/The-Drapery-Falls
[link] [comments]

from math https://ift.tt/2HSlw6e
Twitter Facebook

12:58

Prove that:

$$e^\pi+\frac{1}{\pi}< \pi^{e}+1$$

Using Wolfram Alpha $\pi e^{\pi}+1 \approx 73.698\ldots$ and $\pi(\pi^{e}+1) \approx 73.699\ldots$

Can this inequality be proven without estimations? I've just seen this question and I remembered I've seen the question from this thread in an older paper, but I don't remember the details.

Note that this is sharper because it can be written as:

$$e^{\pi}-\pi^e<1-\frac{1}{\pi}<1$$

I've tried, but none of the methods in the linked question (which study the function $x^\frac{1}{x}$) can be applied here.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

09:58

I am working on my Senior Thesis for my Bachelor's Degree in Mathematics. My project involves Japanese San Gaku problems, and moving said problems from Euclidean Geometry to Spherical and Hyperbolic Geometry.

I've been working on a particular problem for weeks now. The problem is stated as follows:

Problem 1.2.5: A circle $O(r)$ has its center on a line $m$, and has a tangent line $\ell$. The circles $O_1(r_1)$ and $O_2(r_2)$ both touch $O(r)$ externally and also the lines $\ell$ and $m$. Show that \begin{align*} 4r=r_1+6\sqrt{r_1r_2}+r_2. \end{align*}

enter image description here

I have worked very hard on this and have come up with a lot of stuff. One very useful result is the following:

Useful Result. Given tangent circles $O_1(r_1)$ and $O_2(r_2)$, and a line $AB$ tangent to $O_1$ at $A$ and to $O_2$ at $B$, it follows that $$|AB| = 2\sqrt{r_1 r_2}$$ See Useful Result.

Additionally, I've managed to construct the figure in Geogebra. This is quite a difficult task unless you know what you're doing (which I did not at first!). Here is what the figure looks like after construction.

By equating $AC$ and $A'C'$ in this image, I've been able to solve the problem, but not by hand. After using the Useful Result to rewrite $AC$, and a clever usage of the Pythagorean Theorem to write $A'C'$ in terms of $r$, $r_1$, and $r_2$, I get an algebraic nightmare. Mathematica can solve it for $r$ and provide us with the desired result, but that's a little unsatisfying.

This problem comes from the book Japanese Temple Geometry Problems: San Gaku by H. Fukagwa and D. Pede. I can't find the problem anywhere online, and the "solution" in the back only says "Written on a surviving tablet in the Yagamata prefecture in 1823." If some incredibly smart individual out there could help me come up with a better way of solving this problem, I would be forever in your favor!



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

08:31

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!

submitted by /u/AutoModerator
[link] [comments]

from math https://ift.tt/38VJcCv
Twitter Facebook

04:58

Ada Lovelace Day (which is much more than just a day) is fundraising.


Ada Lovelace Day is a global day of events celebrating and supporting women in STEM. It’s much more than this too, with the GoFundMe page explaining

Since ALD began, we’ve produced two anthologies of biographies of women in STEM , free women in STEM crochet patterns , a podcast , several careers posters , and a free education pack . We have run an Online Recruitment Fair for Women in STEM , and our Twitter campaigns have included a Christmas STEM advent calendar and the Twelve Days of STEMmas .
We recently launched the Finding Ada Network , a unique mentorship and knowledge sharing platform for women in STEM and the people who act as advocates for gender equality. 

All that activity costs money, and this has until now relied on sponsorship. But, the post explains

Ada Lovelace Day needs your support to survive! As a sponsor-funded event, we have been badly affected by Brexit as companies cut their marketing and CSR budgets and delay spending decisions. So we need you to help us out and ensure that we can continue our work supporting girls and women in STEM. 

There’s a FAQ list and more information at GoFundMe: Ada Lovelace Day: Supporting women & girls in STEM.



from The Aperiodical https://ift.tt/2T8veGY
Twitter Facebook

01:28

Let $g(x,y)\colon \mathbb{R}^2\to \mathbb{R}$ be defined as $$ g(x,y)=\frac{2\sin^2(2\pi x)\sin^2(2\pi y)}{2k -(\cos(2\pi x)+\cos(2\pi y))}, $$ where $k>1$ is a real number, and consider the following integral $$ I(z) = \int_{-1/2}^{1/2}\int_{-1/2}^{1/2}e^{2\pi i z (x+y)}\,g(x,y)\,\mathrm{d}x\,\mathrm{d}y, $$ where $z>0$ is a real number.

My question. Is it possible to find a closed form expression for $I(z)$? Or, at least, approximately characterize the behavior of $I(z)$ as a function of $z$?

Numerics suggest that $|I(z)|\sim \left(k-\sqrt{k^2-1}\right)^{2z}$, however, after several unsuccessful attempts, I have not yet been able to formally derive this fact.

This is not an homework question. Any help is welcome! Thanks in advance!

My attempt. I briefly sketch my attempt to solve the problem, hoping that someone may find this useful. Note that $g(x,y)$ is analytic in a neighborhood of $\mathbb{R}^2$. Thus, we can shift the integral in the complex direction $it$ as long as $g$ remains analytic. This is indeed the case for all $$ t < t_0 := \frac1{2\pi} \cosh^{-1} k = \frac1{2\pi} \ln \left(k + \sqrt{k^2-1}\right)= -\frac1{2\pi} \ln \left(k - \sqrt{k^2-1}\right). $$ Thus, by letting $t=t_0$, we get \begin{align*} I(z) &= e^{-4\pi z t_0}\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}e^{2\pi iz (x+y)}\,g(x+it_0,y+it_0)\,\mathrm{d}x\,\mathrm{d}y\\ &= \left(k-\sqrt{k^2-1}\right)^{2z} I'(z), \end{align*} where \begin{align*} I'(z) &= \int_{-1/2}^{1/2}\int_{-1/2}^{1/2}e^{2\pi iz (x+y)}\,g(x+it_0,y+it_0)\,\mathrm{d}x\,\mathrm{d}y\\ &=\int_{-1/2}^{1/2}\int_{-1/2}^{1/2}\frac{e^{2\pi i z (x+y)}}{K}\frac{(K^2(2-K^2e^{-4 \pi i x})-e^{-4 \pi ix})(K^2(2-K^2e^{-4 \pi i y})-e^{-4 \pi iy}) }{K(4k-K(e^{-2\pi i x}+e^{-2\pi i y}))-e^{2\pi i x}-e^{2\pi i y}}\,\mathrm{d}x\,\mathrm{d}y\\ \end{align*} where $K:=k+\sqrt{k^2-1}$. Although the integrand of $I'(z)$ blows up at $x=0$, $y=0$, $I'(z)$ still converges absolutely, and I think it should be possible to prove that $I'(z)$ decays slower than $\left(k-\sqrt{k^2-1}\right)^{2z}$. However I'm currently stuck here.



from Hot Weekly Questions - Mathematics Stack Exchange
Twitter Facebook

20:01

Hi. Hopefully it is OK to share this.

Sorry, I've just been wondering what I can do to help solving/containing (whatever that is helpful) this outbreak, more than following the general government measures or letting those epidemiologists do their thing.

I worked in graph theory and never have I felt so useless. I saw a post here using dynamical system which is amazing. I believe there is at least something..that math community or amateur graduate like me can do..not just those who are in the field, not about pure or applied stuff, but just mathematics in general.

This might be too vague and I might be exaggerating, but feel free to blame me for feeling useless (as I might not blame myself hard enough) 😭 I am just thinking that instead of searching the answer myself, maybe I can share this here.

Notes: - Negative answers are welcome too. - If the answer depends (on the field or anything), please use "if" instead. 😁

Edit: - The fact that I am involved a little bit in graph theory makes me even more sad as I know there should be at least something...

submitted by /u/21understanding
[link] [comments]

from math https://ift.tt/3c1qlYB
Twitter Facebook

18:01

I've recently been learning about the power of martingales for calculating various stochastic quantities. This is a bit of a meta question: is coming up with the right martingale for the quantity you want just complete black magic?

As an example, consider an unbiased random walk, where we start at position $x$ and want the probability we either hit $0$ or $L$. In this case, a good martingale to use for the calculation is $$Y_n=X_n^2-n$$ where $X_n$ is the random variable of the steps ($X_n=\pm 1$ with equal probability).

However, this is not a martingale when the walk is biased, i.e. $X_n=+1$ with probability $p$ and $X_n=-1$ with probability $q=1-p$. Then it turns out that we should use a different martingale for the calculation, namely $$Y_n=\left(\frac{q}{p}\right)^{\sum_{i=1}^n X_n}.$$

How does one come up with the appropriate martingale to calculate a given quantity in general?



from Hot Weekly Questions - Mathematics Stack Exchange
Mathematicianadda © 2016. All Rights Reserved.
Theme By BloggersStand

Contact Form

Name

Email *

Message *

copyrighted to mathematicianadda.com. Powered by Blogger.
Javascript DisablePlease Enable Javascript To See All Widget

Blog Archive