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October 9

Pappus's hexagon theorem

I found a proof by analytic geometry for the Pappus's hexagon theorem here. Does anyone know if it could be described any simpler? יהודה שמחה ולדמן (talk) 22:46, 9 October 2017 (UTC)

It's basically a special/limiting case of Pascal's theorem, so you can adapt any of the proofs from that article; I'm especially fond of the proof using Bezout's theorem. This is a theorem on projective geometry, so things will go much more smoothly with projective coordinates rather than Cartesian coordinates. --RDBury (talk) 02:55, 10 October 2017 (UTC)

I understand "regular hexagon", I am happy with a non-regular hexagon. Can you really call six lines intersecting with each other a hexagon? Seems wrong somehow. So why is AbCaBc considered to be a hexagon? -- SGBailey (talk) 06:38, 10 October 2017 (UTC)

@SGBailey:, a polygon is defined as "a plane figure that is bounded by a finite chain of straight line segments closing in a loop". This includes self-intersecting polygons. Rojomoke (talk) 06:57, 10 October 2017 (UTC)

But I think the statement Regular specifically implies the concave version. Otherwise a regular pentagram would be considered one of two types of regular pentagon.Naraht (talk) 20:38, 10 October 2017 (UTC)

October 10

Going back to univeristy

I've suffered from a condition for a long time which affected me during school and at uni. I underachieved during A levels and did not finish my degree in Computer Science. Now I've become very interested in mathematics and have been thinking about whether I could work towards getting into a good uni, maybe even Cambridge, to study maths by retaking A levels.

Problem is I have a good idea of what maths is like at uni and I'm not sure I'd be cut out for it. It seems like you can't just memories stuff you have to have a natural ability for solving problems you've never encountered before. Unlike at school where you, for example, are shown a rule to differentiate equations and that's all you're expected to do.

It might just be that I'm not feeling very confident due to my past and the fact that I've been out of education for quite a while.

Any advice? — Preceding unsigned comment added by Polyknot (talk • contribs) 01:13, 10 October 2017 (UTC)

I certainly wouldn't advise you to major in a university subject you don't feel confident in. Sounds like you are in the British education system, but do they have something equivalent to a US community college, where you can get your skills up, figure out what you want to do, and not spend much money in the process, with the idea of eventually transferring back to university when you have it all sorted out ? (I see the UK does have something called a community college, but not sure how similar it is.)

As for an applied math field where you can just memorize formulas and apply them, maybe something more like statistics, demographics, or accounting ? StuRat (talk) 01:20, 10 October 2017 (UTC)

My plan is to first retake my A levels (though I may have some problems to get around as I want to mostly self teach and some of my subjects may involve practicals and thus need an exam center to accommodate to my needs). Two of these being maths and further maths. This I am confident I could get good grades in. However when it comes to maths at uni, having never experienced it, I can only go off of what I know which isn't much. Just that it seems you need to be a natural problem solver. --Polyknot (talk) 01:48, 10 October 2017 (UTC)

How are you at doing proofs ? That seems to be the part of math that requires the most creativity. Also, it might help if you reveal your condition, as this might affect our recommendations. You might also consider some form of engineering, as there's lots of applying existing formulas there. Some types of engineering also require a great deal of creativity, but others do not. For example, calculating the strength of an existing bridge design. Of course, a good deal of this type of "plug and chug" math can be done by computer now, but we still need qualified engineers to review the data and certify it as correct. StuRat (talk) 02:05, 10 October 2017 (UTC)

My condition hopefully shouldn't be a factor anymore so for now I'd rather not say. Proofs, it's hard to say I haven't had a go at them really since i probably need to brush up on my A level stuff first. I have however been going through questions at https://www.ukmt.org.uk/ . What level if any might suggest that I'd be comfortable with a maths degree?

I guess I could look into other degrees too. It's just that maths has really caught my interest lately. I love the idea of tackling problems you haven't been taught how to do among other things. --Polyknot (talk) 02:15, 10 October 2017 (UTC)

OK, if you don't want to reveal your condition I'll respect that. Can you tell us why you didn't do well in computer science ? Because using math for "tackling problems you haven't been taught how to do" does sound a lot like CS. StuRat (talk) 02:54, 10 October 2017 (UTC)

I didn't do well due to my condition, depression and maybe even some anxiety; nothing to do with the course itself being to hard for me. I actually loved the course and one of the things that has drawn me to Maths is how it underpins so much of CS. Are you suggesting that if I'm fine with CS I'd be fine with maths. Or just that maybe I should go back to CS? --Polyknot (talk) 03:13, 10 October 2017 (UTC)

Yes, if you like CS and think you would have done well at it, were it not for your condition, and are worried that pure math might be outside your abilities, then yes, going back to CS sounds like a better plan, to me. Also, as a practical matter, whatever classes you did complete previously are more likely to be able to be applied to a CS degree than a maths degree. StuRat (talk) 03:37, 10 October 2017 (UTC)

Since we appear to be reaching a horrifying situation in which someone might be taking life advice from StuRat, let me strongly suggest to the OP that getting advice from strangers on the internet who know nothing about you is a terrible way to make important life decisions. You studied CS before -- probably there are people you have met in your life who know or remember you, and with whom you might actually be able to discuss the details of your situation in a meaningful way. If there's a particular institution that you are thinking of applying to, they may have a faculty member whose job includes discussions of this sort with potential applicants. A good answer to your question involves too many details of your personal situation that none of us is privy too. --JBL (talk) 19:39, 10 October 2017 (UTC)

JBL, thank you for your reply. I understand this is not the best place to be looking for life advice. I will get round to speaking to such people as you mention but for now I just wanted to get some views from mathematicians and mathematically minded people. I see you are a mathematician. May I ask how you found the jump in mathematics from school to university and also whether you consider yourself talented at math or was it something you worked at? Thank you. --Polyknot (talk) 22:14, 10 October 2017 (UTC)

I took a college-level math class as a high school student and struggled a lot because I hadn't understood that the trade-off of having relatively little time spent in class is that one has to read independently and learn a lot of the material on one's own time. Once I actually reached university I implicitly understood this (at least in the context of mathematics) and did not have this problem again. Two or three times in my studies in university and graduate school I've taken a class in which I felt like I reached an "abstraction barrier" where I could understand the proofs in class line-by-line but I completely lost my sense of the big picture, had difficulty solving problems on my own, and struggled. (This experience is very common: for many people it happens on their first contact with algebra or calculus.) Some of the material I struggled with I have come to learn later, through incidental contact via my research. I have always (for as long as I can remember) loved doing mathematics, but even among people who get PhDs in mathematics this is not necessarily typical. (I also had a number of very good mathematics teachers, beginning in 2nd grade (age 7), which didn't hurt.) I am generally optimistic about the possibility of engaged, committed people to learn a substantial body of mathematics given adequate support, but all evidence suggests that it is much easier for some people than for others. Students coming back to school after a lapse often have a lot of "rust" to work off, both in terms of not having thought about certain ideas in a long time and also in terms of certain activities (studying, exam-taking) for which practice helps. Sometimes this can be very frustrating, but I also am generally optimistic about students overcoming it, if not hit with too many simultaneous obstacles. Discouragement is definitely a potential issue to be aware of. This is all a bit vague or general for obvious reasons, but hopefully some of it is helpful to you. Best of luck! --JBL (talk) 22:12, 11 October 2017 (UTC)

I studied maths at Cambridge nearly 20 years ago. The difference between A-level and university is significant for the reasons you identify. There is a mixture of “book work” which means regurgitating learnt proofs and then unseen problems where you have to apply those proofs in various ways. The finals of the maths tripos were the hardest thing I’ve ever done. Theoretically you don’t need further maths a-level in order to study maths at Cambridge. But I remember covering the entire further maths syallabus in the first week of lectures. Lectures then continued at the same pace for the next three years. 50% of the maths students in my year at my college dropped out or changed subjects. Best thing to do would be to go to a maths department open day and talk to some current students and fellows to discuss whether this is worth pursuing. Also, read “A Mathematician’s Apology” by GH Hardy and Littlewood’s “Miscellany”. Darkhorse06 (talk) 22:34, 10 October 2017 (UTC)

Thanks for the reply Darkhorse. Really given me something to think about and I will inquire in the future. I can't believe the drop out rate was that high at Cambridge. I posted a link above to ukmt.org.uk and I believe the closest thing there to university style questions are the BMO ones, would you agree? --Polyknot (talk) 23:03, 10 October 2017 (UTC)

Getting funding for university can be difficult after dropping out. A possibility is to do an Open University course. I'd look on it more as something to do for interest and enjoyment. It is quite hard work and unfortunately the fees have just gone up rather steeply so you really do need to be interested. Dmcq (talk) 08:33, 11 October 2017 (UTC)

Re BMO questions, someone once described these to me as "the daily Telegraph crossword of mathematics". They are not really what maths is all about - some people are very good at them and they will usually be very good mathematicians, but the converse is not always true. For myself I found the BMO fairly hard work and never did all that well, but flourished in both school and university maths. I'm sure the dropout rate for maths at Cambridge is not as high as 50% - that was just my experience with 12 students starting the course in my college. But the experience of the average Cambridge maths student including me is that at school they were top of the class and never found maths difficult, and then they arrive at Cambridge and find that everyone is like that. People who don't have a growth mindset really struggle with that - it can be quite bewildering. The guy who was Senior Wrangler in my year spent three years attending Anglo Saxon Norse and Celtic lectures instead of maths lectures as he knew it all anyway. If you compare yourself to people like that you will quickly be discouraged and give up. But the vast majority of maths students are in the same boat - they have to work very hard to keep up. Darkhorse06 (talk) 10:13, 11 October 2017 (UTC)

I got all A's in high school math (up to pre-calc) and computer classes, but found college math more difficult, while the computer classes were not any worse. Perhaps just my experience, though, since I'm a visual thinker, and I could do math well as long as I had an area under a curve or a tangent angle to visualize, but got lost when it all became purely theoretical. StuRat (talk) 19:33, 11 October 2017 (UTC)

Polyknot, just to supplement the helpful comments from Darkhorse06, I'd like to pass on an anecdote from a friend who also took Maths at Cambridge. He said that maths is very easy up to a certain level and then becomes impossibly hard and he reached the certain level in the middle of his second year! He works very hard, but apparently his third year did not go well. Also, it's quite difficult to get into Cambridge, but the UK has many other world-class unis. Why not try for one of those? You should also note that some of the newer universities are often much better at supporting mature students or students who need mental health support, so you shouldn't rule them out. And there's always the Open University. Matt's talk 13:15, 14 October 2017 (UTC)

• Statistics and demographics are fields "where you can just memorize formulas and apply them"? Someone please remind me why we still have a reference desk. (Accounting I can't speak for.) EEng 16:36, 15 October 2017 (UTC)

Are students required to do proofs in those fields or derive new formulas ? StuRat (talk) 17:10, 15 October 2017 (UTC)

The short answer is Yes (a random example: [1]). The long answer is that you seem to labor under the delusion referred to by Gian-Carlo Rota when he said We often hear that mathematics consists mainly of "proving theorems". Is a writer's job mainly that of "writing sentences"? EEng 18:02, 15 October 2017 (UTC)

Minute Calculation

If ‘'875kb” is used per min, how many minutes will I receive from ”750Mb”? — Preceding unsigned comment added by 202.134.11.154 (talk) 15:32, 10 October 2017 (UTC)

• Above, it says that we will not do your homework, and it looks awfully like a homework question. But you can have a look at cross-multiplication and megabyte. Tigraan^{Click here to contact me} 15:55, 10 October 2017 (UTC)

O, for God's sake. How will I acknowledge the articles you provided if I can't do the simple math?

This is not a homework question. I'm have to use Whatsapp and its rated as the 'second' app that consumes most kbs.

Further information: Please view this titled webpage somehow "Which voice calling app uses the most data per minute? We tested the top 10". And [2], my certifications that this is not a homework question.

202.134.9.136 (talk) 16:26, 11 October 2017 (UTC)

750Mb = 750,000Kb (possibly 2.4% off if using binary versus decimal, but that's not significant). So, divide 750,000 by 875 to get your answer. However, if that 875Kb is the peak usage per minute, not the average, then you will get more time when you divide by the average. StuRat (talk) 22:57, 11 October 2017 (UTC)

Rolling a marble in a box grid

Imagine a grid of length m and width n. A ball begins at (1,1) and always moves diagonally through the grid, beginning by moving to (2,2). When it encounters the side of the grid it is deflected, so that e.g. if it moves from (x-1,n-1) to (x,n) where 0<x<m then it is deflected to (x+1,n-1). At (m,n) it is always deflected to (m-1,n-1).

I am interested in classifying the different paths of the ball through the grid. Eventually the ball must hit a corner, at which point it will reverse course and retrace its steps back to (1,1). Therefore you can classify the paths according to the corner that the ball hits, which will always be one of the three corners other than (1,1).

Suppose you number the corners of the grid clockwise from (1,1) as 1,2,3,4. I would like to determine the first corner that the ball will hit for any m,n. I have begun by modelling this on an Excel spreadsheet where each cell (m,n) contains the number of the corner first hit in a grid of size m,n. A screen shot is below. A nice pattern emerges, with various repetitions of period 2,4,8, 16 and 32 in particular.

Can anyone take this further and generalise to a formula for any m,n?

Screenshot

--Darkhorse06 (talk) 21:29, 10 October 2017 (UTC)

How about corner = 3 + (n mod 2) - (m mod 2)? (haven't tested that) [posting before thinking there] --catslash (talk) 22:39, 10 October 2017 (UTC)

One more wild guess:

$\mathrm {corner} =3-\left({\frac {m-1}{\gcd(m-1,n-1)}}{\bmod {2}}\right)+\left({\frac {n-1}{\gcd(m-1,n-1)}}{\bmod {2}}\right)$ --catslash (talk) 23:29, 10 October 2017 (UTC)

If the ball has traveled s steps and then hits a corner, we have m|s and n|s. So the ball first hits a corner when s = lcm(n,m) = nm/gcd(n,m). If 2n|s, then the ball is on the left side of the grid; otherwise, it's on the right. If 2m|s, then the ball is on the top of the grid; otherwise it's on the bottom. So the corner is $3+\left({\frac {m}{\gcd(m,n)}}{\bmod {2}}\right)-\left({\frac {n}{\gcd(m,n)}}{\bmod {2}}\right)$

By the way, the details of this calculation are made more complicated by the fact that you've chosen to start counting at 1 instead of 0.2601:245:C500:A5D6:CCBA:F5B1:D1B3:EC06 (talk) 00:15, 11 October 2017 (UTC)

circles to circles

Stereographic projection is conformal and takes circles to circles. Mercator projection is conformal but does not take circles to circles. Can a mapping take circles to circles without being conformal? —Tamfang (talk) 23:13, 10 October 2017 (UTC)

A mapping that always takes circles on a sphere to circles on the plane would be conformal. One can see this by considering what happens to two intersecting circles. Their intersection points can be used to generate more circles and intersection points till one gets as close to any point as one wants. The only variable bit is which way round the circles go and which point corresponds to which in the intersections. One can't have the circles going around twice as that would do something nasty to small circles on the circle. Therefore the faate of two intersecting circles determines everything and they can be mapped in the plane using inversions and reflections. A stereographic projection can be used between the sphere and one mapping on the plane, that is conformal, and every mapping on the plane would be conformal so any mapping from the sphere to the plane preserving circles would be conformal. Basically it is too rigid a condition to allow anything else. This is assuming the mapping is continuous - maybe something strange could be done using the axiom of choice. Dmcq (talk) 08:17, 11 October 2017 (UTC)

October 12

Sum of angles of a triangle

The sum of the angles of a triangle add up to 180 degrees. Where does this come from? How is it proven that this is always the case. --Polyknot (talk) 03:21, 12 October 2017 (UTC)

I found this proof http://www.apronus.com/geometry/triangle.htm . I have some more questions though:

• Who came up with this proof?
• What's the proof for <BAC = < B'CA
• As a mathematician when do you give up trying to find such proofs for yourself (assuming you fail to) and just look up the answer? --Polyknot (talk) 03:35, 12 October 2017 (UTC)

I have no idea who was the first to come up with the proof, but the first known to publish the theorem was Euclid – it was Proposition 32 in the Book I of his Elements. --CiaPan (talk) 06:55, 12 October 2017 (UTC)

• As to the question about proving <BAC = < B'CA, this is Euclid's Proposition 29 of book I of the Elements. The proof depends on the Parallel Postulate. --69.159.60.147 (talk)

References

Smith set and Smith-losing set

In elections where the Smith set doesn't contain all candidates, can it contain a candidate who'd still be in it if all preference rankings were reversed? NeonMerlin 06:30, 12 October 2017 (UTC)

• Huh... no? If A is a candidate from outside the Smith set (A exists per the original assumption), and we reverse all preferences, then A beats any candidate from the old Smith set in the new preference order, and thus none of the "old Smiths" are in the new Smith set. Tigraan^{Click here to contact me} 08:57, 12 October 2017 (UTC)

Tigraan, I think there's a non sequitur in your final clause ("and thus..."). Just because A defeats B which was in the old Smith set, that doesn't mean that B is not in the new Smith set. If the new Smith set contains at least two elements, then assuming no ties, it contains an element that is defeated by another element. Loraof (talk) 18:58, 12 October 2017 (UTC)

Hmm, I missed that. But I am still right , because new Smiths cannot be old Smiths.

I actually pieced together a ridiculously long proof (which I will leave below in case the revised one has a fault). However: the complementary of the old Smith set is a dominant set (each element beats each non-element) in the new order (and it is non-empty by assumption). It may not be minimal, but it proves the intersection between new and old Smiths is empty, which patches replaces the above demo. Tigraan^{Click here to contact me} 12:40, 13 October 2017 (UTC)

Ridiculously long proof of a useless lemma - do not read

Lemma: if an element from the old Smith set is in the new Smith set, then the new Smith set contains the whole set. Let S be the whole set of candidates. Let S_1 be the Smith set in the original order, i.e. the minimal set such that $\forall a\in S_{1},\forall b\notin S_{1},a>b$. Let S2 be the new Smith set, and $A=S_{1}\cap S_{2}$. Assume that A is nonempty, and let a be one of its elements. In that case, $\forall b\notin S_{1},b\in S_{2}$ because each of those b beats a in the new order. Then there are two cases depending on how much of S1 is included in S2: If A=S1, then it means S1 is included in S2. Since by the above its complementary is also included, $S_{2}=S$. Otherwise, $S_{1}-A$ is nonempty. It must have at least one element x that is beaten by at least one element of A in the old order, otherwise, it contradicts the minimality of S1 as a Smith set (since A would do). Thus, in the new order, x beats an element of the Smith set (since A is included in S2), and thus x is in S2. This is a contradiction per the construction of A. Hence, this case does not happen. QED

Calculator problem

If you use a traditional, non-scientific calculator and type

a + + b, then keep pressing =, the values will consistently add b.

But if you type:

a * * b, then keep pressing =, the values will consistently multiply by a.

Any reason for this inconsistency?? Please try both of these with different calculators. Georgia guy (talk) 15:18, 12 October 2017 (UTC)

I just tried it, and can confirm it on two different models of (cheapo, dollar store) calculators. 3+4 = = = = returns the pattern 7, 11, 15, 19... 3x4 = = = = returns the pattern 12, 36, 108, 324. I have absolutely no idea why. --Jayron32 16:02, 12 October 2017 (UTC)

Georgia guy, on your calculator is it necessary to press "+" and "*" twice (as you typed), or does the same thing happen if you only press them once? I see some suggestion online that on some Casio calculators, pressing the operation key twice in succession is necessary to enter "K" mode, to avoid an inadvertent error from accidentally pressing only the "=" twice. -- ToE 18:55, 12 October 2017 (UTC)

The popup calculator that comes with Windows 7 does these repeated calculation without a double press of the operation key, and unlike your and Jayron's calculator, applies b repeatedly for both addition and multiplication. Thus 3+4=== gives 7, 11, 15 and 3*4=== gives 12, 48, 192. -- ToE 19:17, 12 October 2017 (UTC)

This YouTube video from Casio suggests that their K or Constant mode, entered when you press the operation key twice, repeatedly applies a. That is 3++4=== will give 7, 10, 13 and 3**4=== will give 12, 36, 108. This works with the inverse operations also, so 3--4=== gives 1, -2, -5 and 3÷÷4=== gives 1.3333, 0.4444, 0.1481. Thus 3--4= and 3÷÷4= will give different values than 3-4= and 3÷4=. -- ToE 19:34, 12 October 2017 (UTC)

Yes, that's how Casio scientific calculators have always worked. It's very useful for "subtracted from" and "divided into", and for VAT calculations, especially when it was 17.5% (1.175** n= then subsequent amounts= etc.) Dbfirs 15:36, 14 October 2017 (UTC)

On my calculator, double and single operator gives the same result and doesn't reverse the operands. 8+2=== gives 10, 12, 14. 8-2=== gives 6, 4, 2. 8÷2 gives 4, 2, 1. But 8*2=== gives 16, 128, 1024. I suspect it's chosen because a*b is more often viewed as "multiplying a by something" than "multiplying something by b", while "a+b" is viewed as "adding b to something" and not "adding a to something". PrimeHunter (talk) 23:42, 12 October 2017 (UTC)

October 13

Differential equations

I had a few problems that I was stuck on:

1. Suppose a cell is suspended in a solution containing a solute of constant concentration C_s. Suppose further that the cell has constant volume V and that the area of its permeable membrane is the constant A. By Fick's law the rate of change of its mass m is directly proportional to the area A and the difference C_s − C(t), where C(t) is the concentration of the solute inside the cell at time t. Find C(t) if m = V · C(t) and C(0) = C₀. Use k > 0 as the proportionality constant.

C(t) =

2a. According to Stefan's law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature T_m is given by ⁠dT/dt⁠ = k(T⁴ - T_m⁴), where k is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling. Solve the differential equation.

T(t) =

2b. Using the binomial series, expand the right side of the following equation. (Only write the first three terms of the expansion.)

⁠dT/dt⁠ = k(T⁴ - T_m⁴)

⁠dT/dt⁠ = k[T_m + (T - T_m)]⁴ - T_m⁴

⁠dT/dt⁠ = kT_m⁴[1 + (⁠T/T_m⁠))⁴ - 1]

⁠dT/dt⁠ = kT_m⁴[(____________________) - 1]

3. A classical problem in the calculus of variations is to find the shape of a curve C such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x₁, y₁) in the least time. It can be shown that a nonlinear differential equation for the shape y(x) of the path is y(1 + (⁠dy/dx⁠)²) = k, where k is a constant. First solve for dx in terms of y and dy. Then use the substitution y = k sin²(θ) to obtain a parametric form of the solution. The curve C turns out to be a cycloid.

x(θ) =

For 1, I wrote C(t) = ⁠m/V⁠, but didn't know how to proceed from there.

For 2a, I tried separation of variables then factoring then partial fractions:

Work

⁠dT/dt⁠ = k(T⁴ - T_m⁴)

⁠dT/T⁴ - T_m⁴⁠ = k dt

⁠dT/(T² + T_m²)(T + T_m)(T - T_m)⁠ = k dt

⁠A/(T² + T_m²)⁠ + ⁠B/(T + T_m)⁠ + ⁠C/(T - T_m)⁠ = 1, where A, B, C ∈ ℝ

⁠A(T + T_m)(T - T_m)/(T² + T_m²)(T + T_m)(T - T_m)⁠ + ⁠B(T² + T_m²)(T - T_m)/(T² + T_m²)(T + T_m)(T - T_m)⁠ + ⁠C(T² + T_m²)(T + T_m)/(T² + T_m²)(T + T_m)(T - T_m)⁠ = 1

⁠A(T + T_m)(T - T_m) + B(T² + T_m²)(T - T_m) + C(T² + T_m²)(T + T_m)/T⁴ - T_m⁴⁠ = 1

⁠B(T² + T_m²)(T - T_m)/(T² + T_m²)(T + T_m)(T - T_m)⁠ = 1, setting T = -T_m

B(2T_m²)(-2T_m) = 1

B = ⁠-1/4⁠T_m³

⁠C(T² + T_m²)(T + T_m)/(T² + T_m²)(T + T_m)(T - T_m)⁠ = 1, setting T = T_m

C(2T_m²)(2T_m) = 1

C = ⁠1/4⁠T_m³

⁠A(T + T_m)(T - T_m) + B(T² + T_m²)(T - T_m) + C(T² + T_m²)(T + T_m)/T⁴ - T_m⁴⁠ = 1

A(T + T_m)(T - T_m) + (⁠-1/4⁠T_m³)(T² + T_m²)(T - T_m) + (⁠1/4⁠T_m³)(T² + T_m²)(T + T_m) = T⁴ - T_m⁴

A(T² - T_m²) + ⁠-1/4⁠T_m³(T³ - T²T_m + TT_m² - T_m³) + ⁠1/4⁠T_m³(T³ + T²T_m + TT_m² + T_m³) = T⁴ - T_m⁴

A(T² - T_m²) + ⁠-1/4⁠T_m³(- T²T_m - T_m³) + ⁠1/4⁠T_m³(T²T_m + T_m³) = T⁴ - T_m⁴

A(T² - T_m²) + ⁠1/2⁠T_m³(T²T_m + T_m³) = T⁴ - T_m⁴

A(T² - T_m²) = T⁴ - T_m⁴ - ⁠1/2⁠T_m³(T²T_m + T_m³)

A = ⁠T⁴ - T_m⁴ - ⁠1/2⁠T²T_m⁴ - ⁠1/2⁠T_m⁶/T² - T_m²⁠

A = ⁠- T_m⁴ - ⁠1/2⁠T_m⁶/- T_m²⁠, setting T = 0

A = T_m² + ⁠1/2⁠T_m⁴

A = T_m² + ⁠1/2⁠T_m⁴, B = ⁠-1/4⁠T_m³, C = ⁠1/4⁠T_m³

⁠T_m² + ⁠1/2⁠T_m⁴/T² + T_m²⁠ - ⁠⁠1/4⁠T_m³/T + T_m⁠ + ⁠⁠1/4⁠T_m³/T - T_m⁠ = k dt

∫
⁠T_m² + ⁠1/2⁠T_m⁴/T² + T_m²⁠ dT - ∫
⁠⁠1/4⁠T_m³/T + T_m⁠ dT + ∫
⁠⁠1/4⁠T_m³/T - T_m⁠ dT = k dt

(T_m² + ⁠1/2⁠T_m⁴)∫
⁠1/T² + T_m²⁠ dT - ⁠1/4⁠T_m³∫
⁠1/T + T_m⁠ dT + ⁠1/4⁠T_m³∫
⁠1/T - T_m⁠ dT = k dt

(T_m² + ⁠1/2⁠T_m⁴)⁠arctan(⁠T/T_m⁠)/T_m⁠ - ⁠1/4⁠T_m³ln(T + T_m) + ⁠1/4⁠T_m³ln(T - T_m) = kt + D, where D ∈ ℝ

⁠T_m² + ⁠1/2⁠T_m⁴/T_m⁠arctan(⁠T/T_m⁠) + ⁠1/4⁠T_m³(ln(T - T_m) - ln(T + T_m)) = kt + D

⁠T_m² + ⁠1/2⁠T_m⁴/T_m⁠arctan(⁠T/T_m⁠) + ⁠1/4⁠T_m³ln(⁠T - T_m/T + T_m⁠) = kt + D

⁠T_m² + ⁠1/2⁠T_m⁴/T_m⁠arctan(⁠T/T_m⁠) + ln(⁠T - T_m/T + T_m⁠)^{⁠1/4⁠T_m3} = kt + D

however, this answer was verified to be incorrect. I think I might have the wrong values for A, B, and C.

For 2b, I tried ⁠T⁴/T_m⁴⁠ + ⁠4T³/T_m³⁠ + ⁠6T²/T_m²⁠ and ⁠T⁴/T_m⁴⁠ - ⁠4T³/T_m³⁠ + ⁠6T²/T_m²⁠, both of which were verified to be incorrect.

Any help would be appreciated. 147.126.10.148 (talk) 10:23, 13 October 2017 (UTC)

Differential equation 2a.

$-{\frac {dT}{dt}}=k(T^{4}-T_{m}^{4})$

Note the minus sign. The body is cooling when it is warmer than the surroundings. Choose your units of time and temperature such that the equation takes the form

$-{\frac {dT}{dt}}=T^{4}-1$.

Use perturbation.

$-{\frac {dT}{dt}}=T^{4}-\epsilon$

$T(t,\epsilon )\approx \sum _{n=0}^{\infty }T_{n}(t)\epsilon ^{n}$ (as $\epsilon \rightarrow 0$)

The first term in the perturbation series

$T_{0}(t)=\lim _{\epsilon \rightarrow 0}T(t,\epsilon )$

satisfies the differential equation

$-{\frac {dT_{0}}{dt}}=T_{0}^{4}$

$dt=-T_{0}^{-4}dT_{0}$

Integrate

$t=3^{-1}T_{0}^{-3}+t_{0}$

Let the beginning of time be $t_{0}=0$.

$t=3^{-1}T_{0}^{-3}$

$T_{0}=(3t)^{-3^{-1}}$

This is the cooling formula when the surrounding is at absolute zero.

Differentiate

$dT_{0}=d((3t)^{-3^{-1}})=3^{-3^{-1}}(-3^{-1})t^{-3^{-1}-1}dt=-3^{-3^{-1}4}t^{-3^{-1}4}dt$

$=-(3^{-3^{-1}}t^{-3^{-1}})^{4}dt=-T_{0}^{4}dt$

checking that the differential equation $dT_{0}=-T_{0}^{4}dt$ is satisfied.

Insert the next term in the perturbation series

$T\approx T_{0}+T_{1}\epsilon$ (as $\epsilon \rightarrow 0$)

into the differential equation

$-{\frac {d(T_{0}+T_{1}\epsilon )}{dt}}=(T_{0}+T_{1}\epsilon )^{4}-\epsilon$

and expand to the first power in $\epsilon$

$-{\frac {dT_{0}}{dt}}-\epsilon {\frac {dT_{1}}{dt}}=T_{0}^{4}+4T_{0}^{3}T_{1}\epsilon -\epsilon$

The constant terms vanish and the rest is divided by $\epsilon$

$-{\frac {dT_{1}}{dt}}=4T_{0}^{3}T_{1}-1$

Inserting $T_{0}^{3}=((3t)^{-3^{-1}})^{3}=(3t)^{-1}=3^{-1}t^{-1}$

$-{\frac {dT_{1}}{dt}}={\frac {4T_{1}}{3t}}-1$

or

$({\frac {d}{dt}}+{\frac {4}{3t}})T_{1}=1$

This is an inhomogenous Linear differential equation#First-order equation with variable coefficients.

The integrating factor is

$e^{\int {\frac {4dt}{3t}}}=e^{{\frac {4}{3}}\ln t}=t^{\frac {4}{3}}$

and the differential equation becomes

$d(T_{1}t^{\frac {4}{3}})=t^{\frac {4}{3}}dt$

integrate

$T_{1}t^{\frac {4}{3}}={\frac {3}{7}}t^{\frac {7}{3}}+C$

and divide

$T_{1}={\frac {3}{7}}t+Ct^{-{\frac {4}{3}}}$

Bo Jacoby (talk) 12:24, 13 October 2017 (UTC).

(Bo, I don't know what you're doing here, but it's probably well beyond the scope of what's expected.) The original poster's attempt at separating variables, and then integrating via partial fraction decomposition is most likely what's intended. I'll point out that in general, when you have a term like ${\frac {1}{T^{2}+T_{m}^{2}}},$ the numerator needs to be $AT+B,$ not just $B.$ In this case, it happens to work out because $A=0,$ but it won't always be like that. Otherwise, it's just a matter of being more careful with your algebra, because I think you've got the basic idea right (and I'm not going to try to wade through all your work, sorry ). --Deacon Vorbis (talk) 15:35, 14 October 2017 (UTC)

Thank you! I expect that the differential equation is not solvable in terms of elementary functions. Most differential equations are not. So I tried to find an asymptotic expression. I think I failed. As $t\rightarrow \infty$ you should get $T\rightarrow 1$ , and that is not what I got. I must have made mistakes. Bo Jacoby (talk) 18:51, 14 October 2017 (UTC).

Summary. The equation is $-{\frac {dT}{dt}}=k(T^{4}-T_{m}^{4})$. Substituting $T=T_{m}y$ gives $3dy=-3kT_{m}^{3}(y^{4}-1)dt$. Substituting $x=3kT_{m}^{3}t$ gives the dimensionless differential equation $3dy+y^{4}dx=dx$.

This is attacked by perturbation theory: $y\approx y_{0}+y_{1}\epsilon$ satisfies $3dy+y^{4}dx=\epsilon dx$ for $\epsilon \rightarrow 0$.

The unperturbated equation is $3dy_{0}+y_{0}^{4}dx=0$. The solution is $y_{0}=x^{-3^{-1}}$.

The perturbated equation $3d(y_{0}+y_{1}\epsilon )+(y_{0}+y_{1}\epsilon )^{4}dx=\epsilon dx$ gives $3dy_{1}+4^{1}3^{-1}x^{-1}y_{1}dx=dx$. The solutions are $y_{1}=7^{-1}x+Cx^{-3^{-1}4}$ where $C$ is a constant of integration.

The approximate solutions to the equation $3dy+y^{4}dx=dx$ are $y\approx x^{-3^{-1}}+7^{-1}x+Cx^{-3^{-1}4}$.

Bo Jacoby (talk) 06:42, 15 October 2017 (UTC).

October 14

Basic math, compound interest, multiply out

In the common compound interest formula: Capital C, interest rate i and periods p:

C * (1 + i)^p

multiplying out would lead to:

(C + iC)^p

which is a completely different result, so the parenthesis has to be solved first. But isn't algebraically valid transforming:

(ab+ac)^2

into

a(b+c)^2

or back? Shouldn't the rule rather be expressed as:

C * ((1 + i)^p)?

--Dikipewia (talk) 12:46, 14 October 2017 (UTC)

No. $(ab+ac)^{2}=(a(b+c))^{2}=a^{2}(b+c)^{2}\neq a(b+c)^{2}.$ And the extra set of parentheses aren't necessary (but aren't wrong either) because exponentiation is generally understood to have higher precedence than multiplication. --Deacon Vorbis (talk) 12:57, 14 October 2017 (UTC)

You cannot get from

C * (1 + i)^p

to

(C + iC)^p

because power (Exponentiation) comes first then multiplication. 110.22.20.252 (talk) 15:04, 14 October 2017 (UTC)

October 15

Rational trigonometry

The mathematician Norman J. Wildberger writes this lemma here (at 04:48), but does not prove it. Can anyone help? יהודה שמחה ולדמן (talk) 17:56, 15 October 2017 (UTC)