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Nov
12
comment Cardinality of infinite sets - Help with intuition
"why can't I do the exact same thing when dealing with the mapping between odds/integers?" Have you tried?
Nov
7
comment Function 'result arity'
In my many years of mathematics I've never come across such a term.
Aug
13
comment If $ P(A) = 0 $ is $ A $ a null event?
@user159813 The fact that the "event is a non empty set" does not make the event "non-null". The set of pairs of integers that sum to 13 is non-empty but it's impossible to roll 13 with two ordinary dice.
Aug
5
comment In calculus, which questions can the naive ask that the learned cannot answer?
First, you don't need table look-ups to compute elliptic integrals. There are a variety of methods for computing them. Second, in what sense can you not exactly calculate the arc-length of an ellipse but can exactly calculate the arc-length of a circle?
Aug
4
comment using the same symbol for dependent variable and function?
This is a problematic area. Despite functions playing a central role in much of mathematics there isn't a clear unambiguous notation that mathematicians agree on. I think a good modern view is that $y$ is a function and $y(t)$ is the value of the function evaluated at $t$ in which case it (usually) makes no sense to say $y=y(t)$. But others take the view that $y=y(t)$ is a special use of notation that emphasises that $y$ is a variable that depends on $t$ and so is perfectly acceptable.
Jul
24
comment Can universal instantiation be used more than once?
If technique X can only be used once in a proof, and I use it to prove A implies B, and use it to prove B implies C, then I wouldn't be able to combine these proofs to show A implies C as that would require two uses, and mathematics would pretty well come to a halt.
Jul
14
comment Why is the height of a heap defined as $\lg n$?
See mathworld.wolfram.com/Lg.html for some discussion of what the notation $\mathop{lg}$ (as opposed to $\log$ or $\ln$) means.
Jul
14
comment Are the real numbers really uncountable?
I'm not sure what the issue is @frogeyedpeas . When a mathematician says "there exists an x such that..." they're not saying "there exists an x, with definition Y, such that...". If you want to, you can study numbers with this property but that has no bearing on the existence of the other numbers.
Jul
9
comment How do I know if I have imaginary numbers when using Newton Raphson Method?
I guess you're wondering what happens if you're trying to solve an equation that contains complex values, or from a complex starting point. The answer is that it gets complicated. Seriously complicated: chiark.greenend.org.uk/~sgtatham/newton
Jul
7
comment Integral of the square of a function
The integral can easily be non-negative even if the integrand is sometimes not real. Consider a function $f$ that takes a small imaginary value for a small region and otherwise takes large real values. Overall the real parts will dominate the integral giving a non-negative result.
Jul
5
comment Axiom of Choice and finite sets
"How do I choose such an x?" is often a good question if you're writing a computer program. But in mathematics, if you want to use the existence of $x$ then tautologically you need only a proof that $x$ exists.
Jul
4
comment Understanding matrices.
That code isn't right. Maybe you mean newX = cos(angle)*oldX-....
Jun
30
comment Given a histogram, programatically, how do I find the normal distributions that comprise it?
See en.wikipedia.org/wiki/…
Jun
30
comment What's wrong with having the same variable in the integrand as in the limits?
@QiaochuYuan You point to the real issue. In principle $\int_0^x(\int_0^xx^3dx)x^4dx$ is unambiguous. But mathematicians like the abuse of notation that allows them to write what you just wrote and they find it worthwhile to sacrifice $\alpha$-equivalence to get it. (See en.wikipedia.org/wiki/Lambda_calculus#Alpha_equivalence .) I suspect that modern mathematics students with some computing experience may have slightly different ideas of what's reasonable.
Jun
27
comment Topologies on n-manifolds
Do you mean other topologies? Or other topologies on $n$-manifolds? If the former then you'll find you need to understand the compact-open topology if you intend to study homotopy theory. en.wikipedia.org/wiki/Compact-open_topology
Jun
25
comment Where does this unit vector come from?
There is a sphere here :-) The reason is that unit vectors are vectors of length 1, i.e. if you place the vector's tail at the origin its head lies on the unit sphere. So, in disguise, your question is "where does the vector $F_1$ meet a unit sphere?"
Jun
25
comment Is this mathematically correct?
Well $v_1/v_2=13/15$ but $r_1/r_2=1$ for row 1. But suppose there are errors up to 10% in your voltage measurements, so, eg. $v_1$ is actually 2.8 and $v_2$ is actually 2.8, etc., then the relationship might hold exactly.
Jun
25
comment Is this mathematically correct?
The relationship doesn't hold exactly. But if your voltage measurements are only accurate to 0.5V, say, then it is possible that the relationship holds. That's why experimenters give error bars when they publish results. Do you have some estimates of your uncertainty?
Jun
24
comment What should I learn first, Mathematica or MatLab?
Want to define a function? Stick an underscore after the argument names. Want to apply a function to some arguments? Use square brackets. Want to get the $i$th element of an array, use double square brackets. While one might quibble about what is or isn't natural, saying my claim that Mathematica notation "can look very unnatural" is outrageous is, well, interesting.
Jun
24
comment How to establish this inequality without using induction?
Seeing as you defined the $a_n$ by induction it's going to be completely impossible to prove any non-trivial property of them without using induction in some form or other. Maybe you're trying to rule out one particular induction proof.