Any maths buffs?

Good question - I assume so. I think they are actuarial formulae and having given it a google, a appears to be the accumulated amount function where a(t)=(1+i)^t (which is basically compound interest I think?), except that doesn't lead to a"(t)= ln(1+i)... unless I'm being silly. I have asked.

@hamrack Ah! I had tried a substitution but have never come across the Product Log function before. That is super interesting and v useful. Pretty sure numerical only would not be an issue here. Thanks!

It's kind of close to: t = ln (A(t)/A(0))/ln(1 + i) which is the formula for time for some compounding investment to hit a value of A(t), starting value A(0), with interest i. But the numerator of that fraction doesn't make a lot of sense here.

In other words the question might be "how long is it before my investment is worth e^(1 - n/(1 + i)) times what I started with?" Unusual.

Oh and a''(t) = ln(1+i) doesn't seem to be something that would occur in the real world either, I agree

From some random actuarial lecture notes on the internet - looks like the same v.

1.2 Present Value

Definition: Let X be the amount that must be invested at the start of a
time period to accumulate $1 at the end of the period at effective annual
interest rate i. Then X satisfies the equation X(1 + i) = 1, or equivalently,
X = 1/(1 + i). The quantity v = 1/(1 + i) is called the present value of
an amount of $1 due in one time period (or present value factor or
discount factor).

Example 1.5: A person invests $X in a risky fund, which average annual
rate of return is 19.5%. What amount must the person invest in order to
accumulate $1,000,000 in 25 years?

Solution: The amount X solves the equation X(1 + i)^25 = $1,000,000.
Thus, X = $1,000,000 v^25 = $1,000,000(1.195)^−25 = $11,635.96.

Bad news, a double dot was a typo. Should have been lower case delta. It is not therefore a derivative. Even worse news, the vn in the x formula should have been v^n. Pasting into their email was not the best idea.

Lessons have been learnt:

• double check the formulae you are trying to rearrange. Find out what the variables actually mean rather than treating as an algebraic challenge.
  
• the Lambert W function / product log function is new to me and is rather lovely
  
• wolfram alpha is boss for working out if this sort of thing is possible
  
• using the correct formula, I now really don’t think it’s possible!
  
  

Appreciate all the replies on the topic!

I think you can get a form with the W function again, but that's it

Maths folkses.

At what stage of learning would Jacobean matrices be a thing? I'm thinking undergrad?

Yeah, first year undergrad I reckon

Quite sure we did at a level.

Actually, could it be a further maths thing? Defintely did differential calculus and matrices, and seem to vaguely remember doing both.

I don't think we called it Jacobean anything though.

Anyway. Sounds like I should know it then.

Fucking kalman filters, man.

Yeah. Dim memories of it in further maths. Didn't cover them in the maths parts of my Comp Sci degree (we did 20/120 credits of Maths in the first year). Don't think I covered them at all in my OU Maths degree but then it could have been a 2nd/3rd year option that I ran away from.

Yeah, definitely not in A Level Further Maths as is currently taught.

As an aside for anyone who enjoys a bit of integration, I have really been enjoying the challenge of Integral of the day recently.

I went through my entire physics undergrad without (explicitly) being introduced to the Jacobian, only to find I knew it as a different thing. I'd say its determinant naturally comes up with the introduction of multiple integrals and then the full Jacobian matrixis maybe introduced alongside vector calculus.

FWIW we don't explicitly cover it as part of our Y1 or Y2 maths courses in a STEM UG programme. Definitely not an A-level topic, even at further maths.

As an aside for anyone who enjoys a bit of integration, I have really been enjoying the challenge of Integral of the day recently.

Ooo, now that looks fun! #nerd

I thought we did it as a level.
We for sure did it first year engineering undergrad.

I had a friend who did do it in Y1 UG, I'm not suggesting you didn't either. It threw me when they mentioned it, then I realised what they meant (the volume element) and it made sense again. I think it's down to who teaches that particular topic as to whether they get diverted by the Jacobian.
Personally, if I was teaching it and was just going to use the Jacobian determinant for the volume element then I wouldn't introduce the terminology. I don't think it would add depth, just jargon and confusion.

We did 1D in maths IB. Definitely any course on vector calculus (generally first or second year [scotland] in a maths degree) will need to introduce Jacobians (and not just det J for volume element scaling) since grad transforms with J^{-T}.

Oh I didn't think you were, just had to recall my memory. Maths wasn't my strong point so I tried to forget, and now it's mostly gone!

Ah, like me and biology

I think its essential for a full treatment of vector calculus, hence showing up in a maths UG programme, but in non-maths STEM it seems ripe for confusion. That's probably why it is sometimes (often?) skipped/glossed over.

Back to the integral of the day, today's is a sneaky little sleight of hand isn't it!

Hi mathsfolks.

I'm writing formulaeses.

Does this make sense?

I have a series of numbers p, that occur every t seconds, where t is a continuous series (i.e. 1, 2, 3, 4, 5 ..... to the end of the series at t = n).

Is there a better way to write this?

Might just be simpler to do:

p = { p₁, p₂, ... , p_t }

As that defines both p and t in one go.

I'm rusty though.

That does make sense though.

I have corrected it so that P = {p1, p2 ...}

The second bit is to make it clear that t must be 1, 2, 3 . . . t with no missing numbers, (otherwise things break later).

Mathematicians are happy with ambiguity. If you write p = { p_0, p_1, ..., p_t }, everyone will assume that the index is contiguous (you can probably elide the p_1 even)

Specifying it as { p₁, p₂, p₃, ... , p_t } kind of removes any ambiguity, most people will infer that it starts with p₁, that the subscript increments by 1 each time, and ends at t also implying there are t terms in total.