instruction stringlengths 12 30k |
|---|
How to calculate volume of tetrahedron given lengths of all it's edges? |
So, what's unnatural about the complex differential equation...
f : C -> C satisfying f'(z) = f(z) and f(0)=1 ?
|
***Some Assumptions***
I will assume that you are ok with power series being used, just not Taylor's theorem. I will also assume you will allow us to observe a solution to a DE since you used it in your derivation.
**Defn** A series of the form $\sum_{n=0}^{\infty}c_n\left(z-z_0\right)^n$ for $c_n,z,z_0\in\mathb... |
***Some Assumptions***
I will assume that you are ok with power series being used, just not Taylor's theorem. I will also assume you will allow us to observe a solution to a DE since you used it in your derivation.
**Defn** A series of the form $\sum_{n=0}^{\infty}c_n\left(z-z_0\right)^n$ for $c_n,z,z_0\in\mathb... |
Let f and g be two periodic functions over R with the following property: If T is a period of f, and S is a period of g, then T/S is irrational.
**Conjecture**: f+g is *not* periodic.
Could you give a proof or a counter example? It is easier if we assume continuity. But is it true for arbitrary real valued fu... |
Suppose a finite group has the property that for every $x, y$, it follows that $(xy)^3 = x^3 y^3$.
How do you prove that it is abelian?
----------
Edit: I recall that the correct exercise needed in addition that the order of the group is not divisible by 3.
|
Why are noncommutative nonassociative Hopf algebras are called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
|
Why are noncommutative "algebraic" groups called quantum groups? |
Discrete valuations <-> points on a curve
For a nonsingular projective curve over an algebraically closed field, there is a one-one correspondence between the points on it, and the discrete valuations of the function field (i.e. all the meromorphic functions of the curve). The correspondence is point P -> the valuat... |
What are the differences between a (discrete) cosine transform and a (discrete) Fourier transform? I know the former is used in JPEG encoding while the latter plays a big part is signal and image processing- how related are they? |
Let A and B be two matrices which can be multiplied.
Then **rank(AB) <= min(rank(A), rank(B))**
I proved **rank(AB) <= rank(B)** interpreting AB as a composition of linear maps, observing that ker(B) \subseteq ker(AB) and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpre... |
Here is a simpler example. I claim that the function $h(x) = \sin x + \sin \pi x$ cannot possibly be periodic. Why? Suppose an equation of the form
$\sin x + \sin \pi x = \sin (x+T) + \sin \pi (x+T)$
held for all $x$ and some $T > 0$. Take the second derivative of both sides with respect to $x$ to get
$\si... |
Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you're doing number theory then you want to think of the prime numbers as acting "independently": knowing that a number is divisible by... |
Let $S$ be a set of size $n$. There is an easy way to count the number of subsets with an even number of elements. Algebraically, it comes from the fact that
$\displaystyle \sum_{k=0}^{n} {n \choose k} = (1 + 1)^n$
while
$\displaystyle \sum_{k=0}^{n} (-1)^k {n \choose k} = (1 - 1)^n$.
It follows that
... |
Disclaimer: I am not a finitist --- but as a theoretical computer scientist, I have a certain sympathy for finitism. The following is the result of me openly speculating what an "official" finitist response would be, based on grounds of computability.
The short version is this: **(a)** It depends on what you me... |
Disclaimer: I am not a finitist --- but as a theoretical computer scientist, I have a certain sympathy for finitism. The following is the result of me openly speculating what an "official" finitist response would be, based on grounds of computability.
The short version is this: **(a)** It depends on what you me... |
The [Burnside Lemma][1] looks like it should have an intuitive explanation. Does anyone have one?
[1]: http://en.wikipedia.org/wiki/Burnside%27s_lemma |
I ran into a problem dividing by imaginary numbers recently. I was trying to simplify:
$2 \over i$
I came up with two methods, which produced different results:
Method 1: ${2 \over i} = {2i \over i^2} = {2i \over -1} = -2i$
Method 2: ${2 \over i} = {2 \over \sqrt{-1}} = {\sqrt{4} \over \sqrt{-1}} = \sqrt{4 ... |
What calculation shortcuts exist to help or speed-up mental (or paper) calculations? |
What's the most effective ways of teaching kids - times tables? |
Art Benjamin is your man! He has many tricks to speed up mental calculation and other fun mathemagical tricks. He also wrote two books on the subject!
Here is a video of him in action: http://www.youtube.com/watch?v=M4vqr3_ROIk
Here is his new book: http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calcula... |
**To square a number ending in 5:**
Remove the ending 5. Let the resulting number be n, and compute n(n+1). Append 25 to the end of n(n+1) and that's your answer.
**Example:**
85<sup>2</sup>. Here, we drop the last digit to get 8, compute 8*9 = 72, so 85<sup>2</sup> = 7225. Similarly, we can compute
115<sup>2<... |
Are there 2 subsets, say, $A$ and $B$, of the naturals such that
$$\sum_{n\in A} f(n) = \sum_{n\in B} f(n)$$
where $f(n)=1/n^2$?
If $f(n)=1/n$ then there are many counterexamples, which is probably a consequence of the fact that the harmonic series diverges:
$$\frac23 = \frac12 + \frac16 = \frac14... |
The Wikipedia article on <a href="http://en.wikipedia.org/wiki/Honeycomb_(geometry)#Space-filling_polyhedra.5B2.5D">honeycombs</a> has several examples of 3d tilings by a single polyhedron which aren't vertex-transitive. I am not positive if they are vertex-uniform, though, since (again) I don't really understand your... |
Are there 2 subsets, say, $A$ and $B$, of the naturals such that
$$\sum_{n\in A} f(n) = \sum_{n\in B} f(n)$$
where $f(n)=1/n^2$?
If $f(n)=1/n$ then there are many counterexamples, which is probably a consequence of the fact that the harmonic series diverges:
$$\frac23 = \frac12 + \frac16 = \frac14... |
I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
> Problem:
> -----
> For any natural number n , n<sup>3</sup> + 2n is divisible by 3.
> `This makes sense`
>
> Proof:
> --
> **Basis Step:** If n = 0, then n<sup>3</sup> + 2n = 0<sup>3</sup> +
> 2*0 = 0. ... |
Are there 2 subsets, say, $A$ and $B$, of the naturals such that
$$\sum_{n\in A} f(n) = \sum_{n\in B} f(n)$$
where $f(n)=1/n^2$?
If $f(n)=1/n$ then there are many counterexamples, which is probably a consequence of the fact that the harmonic series diverges:
$$\frac23 = \frac12 + \frac16 = \frac14... |
Why don't you just test the validity of this using modular arithmetic? I.e. take n congruent to 1 mod 3 and we easily get 3=0. If you try n congruent to 2 mod 3, you get 8+4 =12 =0, so you're done. No ugly inductions. |
Why don't you just test the validity of this using modular arithmetic? I.e. take n congruent to 1 mod 3 and we easily get 3=0. If you try n congruent to 2 mod 3, you get 8+4 =12 =0, so you're done. No ugly inductions (although this particular case is not so dirty).
A useful idea when thinking of induction is to thin... |
I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
> Problem:
> -----
> For any natural number n , n<sup>3</sup> + 2n is divisible by 3.
> `This makes sense`
>
> Proof:
> --
> **Basis Step:** If n = 0, then n<sup>3</sup> + 2n = 0<sup>3</sup> +
> 2*0 = 0. ... |
I know that a superalgebra is a Z/2Z graded algebra and that it behaves nicely. I know very little physics though, so even though I know *that* the super- prefix is related to supersymmetry, I don't know what that means; is there a compelling *mathematical* reason to consider superalgebras? |
Why are superalgebras so important? |
I just came up with this problem yesterday.
**Problem**:
Assume there is an important segment of straight line `AB` that needs to be watched at all time. A watchdog can see in one direction in front of itself and must walk at a constant *non-zero* speed at all time. (All watchdogs don't need to have the same speed.... |
I can summarize one really basic reason, which is actually the reason I originally got interested in the definition. Take a finite-dimensional vector space $V$ of dimension $n$, and let $\left( {n \choose k} \right) = {n+k-1 \choose k}$ denote the number of multisets of size $k$ on a set of size $n$. (Multisets are l... |
Roots behave strangely over [complex numbers][1]. Given this, how do non-integer powers behave over negative numbers? More specifically:
- Can we define fractional powers such as (-2)^-1.5?
- Can we define irrational powers (-2)^pi?
[1]: http://math.stackexchange.com/questions/1183/division-by-imaginary-numb... |
Can we find f(x) given that 1-f(x) = f(-x) for all real x?
I start by rearranging to: f(-x) + f(x) = 1. I can find an example such as f(x) = abs(x) that works for some values of x, but not all. Is there a method here? Is this possible? |
According to [Wikipedia][1], a characteristic function completely determines the properties of a probability distribution. This means it must be unique. However, the definition give is:
`Char of X (t)=E[e^itX]`
Now e^ix repeats for every 2 pi increase in x. So how can it be unique?
[1]: http://en.wikipedia.o... |
I'm trying to read a document that applies Reimann-Roch left, right and center. I don't know this theorem or the theory it comes from so I need to build up a bit more background before I can tackle this.
Can you please recommend good books or (online) lecture notes which cover "multi-valued functions", Reimann Surfa... |
I'm trying to read a document that applies Riemann-Roch left, right and center. I don't know this theorem or the theory it comes from so I need to build up a bit more background before I can tackle this.
Can you please recommend good books or (online) lecture notes which cover "multi-valued functions", Riemann Surfa... |
Consider instead the functions g that satisfy the identity -g(x)=g(-x) for all x. If (x,g(x)) is a point of the function g, then (-x,-g(x)) is also a point (since g(-x)=-g(x)). Therefore, every function g is symmetric when rotated by 180 degrees about the point (0,0).
How do things change for the identity 1-f(x)=f... |
I am assuming that you need the error function only for real values. For complex arguments there are other approaches, more complicated than what I will be suggesting.
If you're going the Taylor series route, the best series to use is formula [7.1.6][1] in Abramowitz and Stegun. It is not as prone to subtractive can... |
It's just a Fourier transform. E(x) is an integral over the probability distribution. The function is unique; if you focus on the value inside the expectation integral, that's not, but so what? |
I don't have enough mojo to comment on Greg's answer.
1. Greg made a silly calculational mistake: The transfer function $A(\omega)$ should be $c/(1-(1-c)e^{-i\omega})$.
2. What you want is the modulus of $A(\omega)$. Note that $\sin \omega n$ is precisely the imaginary part of $e^{i\omega n}$. Because the relatio... |
This was really inspired by Solitaire, but a few people reacted with ``oh, it's like the towers of Hanoi, isn't it?'' so I'll try to pose the problem in terms of discs here.
Let's start. On the floor in front of you are n wooden disks of sizes 1, 2, 3, ..., and n. They are placed in a line and the distances between ... |
Is this version of the Hanoi towers problem NP-complete? |
How have experts estimated the amount of oil that was shooting out of that pipe in the Gulf? I bet there's some neat math or physics involved here, and some interesting assumptions considering how little concrete data are available. |
How do you estimate the flow rate of one fluid into another like the Deep Horizon Oil leak? |
A fun question I ask students or interviewees (in engineering) is:
> **This is not my question, this is an example:**
> Using only what you know now, how many
> cans of soda would you estimate are
> produced per day (on average) in the
> United States?
For this question, the result doesn't matter so much a... |
What is your favorite estimation exercise? |
This was really inspired by Solitaire, but a few people reacted with ``oh, it's like the towers of Hanoi, isn't it?'' so I'll try to pose the problem in terms of discs here.
Let's start. On the floor in front of you are n wooden disks of sizes 1, 2, 3, ..., and n. They are placed in a line and the distances between ... |
"How many estimation questions are asked in interviews across the world during a typical 24h period?" |
The generalized mean (power mean) with exponent $p$ of $n$ numbers $x_1, x_2, \ldots, x_n$ is defined as
$$ \bar x = (\frac{1}{x} \sum x_i^p)^{1/p}. $$
This is equivalent to the harmonic mean, arithmetic mean, and root mean square for $p = -1$, $p = 1$, and $p = 2$, respectively. Also its limit at $p = 0$ is equa... |
Which average to use? (RMS vs. AM vs. GM vs. HM) |
WolframAlpha provides a solution to this (and many other) recurrence equations:<br>
http://www.wolframalpha.com/input/?i=1-f(x)+%3D+f(-x) |
I am in the process of trying to learn algebraic geometry via schemes and am wondering if there are simple motivating examples of why you would want to consider these structures.
I think my biggest issue is the following: I understand (and really like) the idea of passing from a space to functions on a space. In ... |
To start off, the [discussion][1] at Sbseminar has comments from lots of people who actually know algebraic geometry, and if anything I say contradicts something they say, please trust them and not me.
One reason is that you lose the functoriality of $Spec$ if you stick to $MaxSpec$: the inverse image of a maximal ... |
In my geometry class last year I remember putting down the statement in a column proof "That all isosceles are always and only similar to other isosceles". I do not remember what I was trying to prove. But, I do remember that I was stressed and that was the only thing I could think of and made a guess thinking I would ... |
Are isosceles always and only similar to other isosceles? |
See my answer [here](http://math.stackexchange.com/questions/942/meaning-of-closed-points-of-a-scheme/984#984) for a brief discussion of how points that are closed in one optic (rational solutions to a Diophantine equation, which are closed points on the variety over Q attached to the Diophantine equation) become non-c... |
There are many types of numbers, though the natural numbers, the integers, the rational, the decimal, the real, and the complex form a nice self-complete expository whole. Hopefully, the following block(s) of text aren't too poorly formatted.
1) The set of **natural numbers** consists of numbers with which we coun... |
I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime? |
Is there a known mathematical equation to find the nth prime? |
* $\forall x, \frac{0}{x} = 0 \Rightarrow \frac{0}{0} = 0?$<br/><br/>
* $\forall x, \frac{x}{x} = 1 \Rightarrow \frac{0}{0} = 1?$<br/><br/>
* Moreover, if one could say, for _any_ k, that `0/0 = k` we could then say `2 = 3` -- just divide both sides by 0 and get `k = k`.
Since there is no reasonable value `0/0` ca... |
Below is my geometric understanding of why irrational powers of negatives are difficult to define. As such it is probably not rigorous and may be wrong.
The way irrational powers of the real numbers are usually defined is by limits of fractional powers.
For complex numbers, the same is true, except the limiting ... |
The generalized mean (power mean) with exponent $p$ of $n$ numbers $x_1, x_2, \ldots, x_n$ is defined as
$$ \bar x = (\frac{1}{n} \sum x_i^p)^{1/p}. $$
This is equivalent to the harmonic mean, arithmetic mean, and root mean square for $p = -1$, $p = 1$, and $p = 2$, respectively. Also its limit at $p = 0$ is equa... |
Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.
This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent. Hence every element of $P$ is a zero-divisor. (As M... |
As other posters have indicated, the problem is that the complex logarithm isn't well-defined on $\mathbb{C}$. This is related to my comments in <a href="http://math.stackexchange.com/questions/1183/division-by-imaginary-number">a recent question</a> about the square root not being well-defined (since of course $\sqrt... |
As other posters have indicated, the problem is that the complex logarithm isn't well-defined on $\mathbb{C}$. This is related to my comments in <a href="http://math.stackexchange.com/questions/1183/division-by-imaginary-number">a recent question</a> about the square root not being well-defined (since of course $\sqrt... |
In part of my research, the following problem has come up.
Consider the system of equations (in complex numbers)
z^b w^c = 1
z^d w^e = 1.
I am interested in the solution set when we restrict both z and w to be ath roots of unity, for some positive integer a. Of course, one immediately sees that (z,w) = (1,... |
Your question: given that X and Z are independent, X is Gaussian (I'll use "normal"), and Y = X+Z, prove that Y is normal iff Z is normal. Right? As you observed, one direction is easy: *if* Z is normal, then so is Y=X+Z. So for the other direction, assume that Y is normal. We need to prove that is Z normal too.
Per... |
According to [Wikipedia][1], a characteristic function completely determines the properties of a probability distribution. This means it must be unique. However, the definition give is:
`Char of X (t)=E[e^itX]`
Now e^iz repeats for every 2 pi increase in z. So how can it be unique?
[1]: http://en.wikipedia.o... |
See my answer [here](http://math.stackexchange.com/questions/942/meaning-of-closed-points-of-a-scheme/984#984) for a brief discussion of how points that are closed in one optic (rational solutions to a Diophantine equation, which are closed points on the variety over Q attached to the Diophantine equation) become non-c... |
Is there a name for a shape that is like a capsule, but with different radii? |
Is there a name for a shape that is like a capsule, but with two different radii? |
What is the name for a shape that is like a capsule, but with two different radii? |
Here is a counterexample. Let $a, b, c \in \mathbb{Q}$ be linearly independent. Let $\text{span}(x, y, z, ...)$ be the $\mathbb{Q}$-vector space in $\mathbb{R}$ spanned by $x, y, z, ...$. Let $AB = \text{span}(a, b), BC = \text{span}(b, c), AC = \text{span}(a, c)$. And for a subset $S$ of $\mathbb{R}$, let $\chi_S$... |
Here is a counterexample. Let $a, b, c \in \mathbb{R}$ be linearly independent over $\mathbb{Q}$. Let $\text{span}(x, y, z, ...)$ be the $\mathbb{Q}$-vector space in $\mathbb{R}$ spanned by $x, y, z, ...$. Let $AB = \text{span}(a, b), BC = \text{span}(b, c), AC = \text{span}(a, c)$. And for a subset $S$ of $\mathbb... |
Here is a counterexample. Let $a, b, c \in \mathbb{R}$ be linearly independent over $\mathbb{Q}$. Let $\text{span}(x, y, z, ...)$ be the $\mathbb{Q}$-vector space in $\mathbb{R}$ spanned by $x, y, z, ...$. Let $AB = \text{span}(a, b), BC = \text{span}(b, c), AC = \text{span}(a, c)$. And for a subset $S$ of $\mathbb... |
I found this question a while ago on a SAT practice exam or something, can't quite remember. So given an acute triangle $ABC$ with $P$ a point inside it and $AP$, $BP$, and $CP$ meeting the opposite sides at $D$, $E$, and $F$ respectively:
![alt text][1]
How can you find the area of triangle $ABC$ given the areas... |
Here is an article at Mathworld on [Circle-Circle Tangents][1]. Perhaps "CirclesWithTangents"?
![Circle-Circle Tangents][2]
[1]: http://mathworld.wolfram.com/Circle-CircleTangents.html
[2]: http://mathworld.wolfram.com/images/eps-gif/CircleCircleTangentGeneral_1000.gif "Circle-Circle Tangents" |
One that I remember from some book (I think it was Innumeracy by John Allen Paulos) was "How fast does your hair grow, in miles per hour?" |
The "[sum and difference][1]" formulas often come in handy, but it's not immediately obvious that they would be true.
![sin(a +/- b) = ...][2]
![cos(a +/- b) = ...][3]
So what I want to know is,
1. How can I prove that these formulas are correct?
2. More importantly, how can I understand these formulas in... |
How can I understand and prove the "sum and difference formulas" in trigonometry? ($cos(a \pm b) = $..., etc.)? |
This was really inspired by Solitaire, but a few people reacted with ``oh, it's like the towers of Hanoi, isn't it?'' so I'll try to pose the problem in terms of discs here.
Let's start. There are n disks on the real line, one of size 1 at position $x_1$, one of size 2 at position $x_2$, ..., and one of size n at po... |
How to test if a point is inside the convex hull of two circles? |
Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $Spec(A)$ is a noetherian topological space, but $A$ is nonnoetherian. (Indeed, otherwise $A$ would be a discrete valuation ring.)
Is there a concrete example of such a ring $A$? |
I have seen and studied this class equation for a finite group acting on itself by conjugations. The only applications I know are Cauchys' theorm and Sylow's theorem. Are there more?
|
I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive integer, therefore there are the same number of rationals as integers. I've ignored sign-related issues, but these are eas... |
Are there more rational numbers that integers? |
Are there more rational numbers than integers? |
Mathematicians have very precise definitions for terms like "infinite" and "same size". The single unambiguous correct answer to this question is that using the standard mathematical definitions, the rationals have the "same size" as the integers.
First, here are the definitions:
1. Define "0" = emptyset, "1" =... |
Suppose you have a normal distribution with mean=0, and stdev=1. So the expected value is 0.
Now suppose you limit the outcomes, such that no values can be below 0. So 50% of values now equal 0, and rest of distribution is still normal. Running 1000000 trials, I come out with an expected value of .4
My que... |
How do I calculate expected return of partial normal distribution? |
In mathematics a set is called *infinite* if it can be put into a 1-1 correspondence with a proper subset of it, and *finite* it is not infinite. (I know it seems crazy to have the concept of infinite as primitive and finite as a derivate, but it's simpler to do this, since otherwise you must assume that the integers e... |
My favorite is [Elementary Number Theory][1] by Rosen, which combines computer programming with number theory, and is accessible at a high school level.
[1]: http://www.amazon.com/Elementary-Number-Theory-Kenneth-Rosen/dp/0321500318/ref=sr_1_1?s=books&ie=UTF8&qid=1280607379&sr=1-1 |
You may not be very satisfied with this answer, but I'll try to explain anyway.
**Countability.** We're not really talking about whether you can "count all of the rationals", using some finite process. Obviously, if there is an infinite number of elements, you cannot count them in a finite amount of time using any r... |
How do I calculate expected value of partial normal distribution? |
I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive integer, therefore there are the same number of rationals as integers. I've ignored sign-related issues, but these are eas... |
Is there an atlas of Algebraic Groups and corresponding Coordinate rings? |
What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more detail.
I'm unsure of what a primitive polynomial is, and why it is useful for these random number generators.
I'd find... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.