instruction stringlengths 12 30k |
|---|
You may want to take a look at *When is a Function that Satisfies the Cauchy-Riemann Equations Analytic?* by J. D. Gray and S. A. Morris, The American Mathematical Monthly, Vol. 85, No. 4, (Apr., 1978), pp. 246-256.
Yes, in a sense, we want open domains so limits can be computed, but we have a notion of directional... |
After learning about the binary number system(only 2 symbols i.e. 0 and 1), I just thought why did we adopt the decimal number system (10 symbols) after all?
I mean if you go to see, it's rather inefficient when compared to the octal(8 symbols) and the hexadecimal(16 symbols)?
Please tag this appropriately.
|
While reading about general topology, I started looking at the metric topology. At one point, the interior of a topological space $X$ is defined as such:
$${E}^{\circ}=\\{p\in E\ | \ B_p(\epsilon)\subseteq E, \epsilon>0\\}$$
Here, $B_p(\epsilon)$ is a ball with center $p$ and radius $\epsilon$. I tried verifying ... |
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.
If I have two independent random variables, X1 and X2, then I define two other random variables Y1 and Y2, where Y1 = f1(X1) and Y2 = f2(X2).
Intuitively, Y1 and Y2 should be independent, and I can't ... |
Are functions of independent variables also independent? |
Expanding on the comment by J.M., let me quote from the (highly recommended) book by Georges Ifrah [*The Universal History of Numbers*][1] (Wiley, 2000, pp. 21-22):
> Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" a... |
Expanding on the comment by J.M., let me quote from the (highly recommended) book by Georges Ifrah [*The Universal History of Numbers*][1] (Wiley, 2000, pp. 21-22):
> Traces of the anthropomorphic origin of counting systems can be found in many languages. In the Ali language (Central Africa), for example, "five" and... |
Consider a cube, with standard topological faces: 6 faces, 12 edges and 8 vertices.
Define a *cube part* as set of faces (possible of different dimension) so that no 2 set elements share same vertex.
E.g. there is no cube part consisting of 3 2-dimensional faces, because there have to be 2 faces that share a vertex.
... |
Hints:
**Prelude.**
a. If $\alpha+\beta+\gamma=\pi$, then $\gamma=\pi-\alpha-\beta$.
b. $\tan(\pi-\theta)=-\tan(\theta)$
c. $\tan(\alpha+\beta)=\frac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}$
d. $\tan\left(\frac{\pi}{2}-\theta\right)=\frac1{\tan(\theta)}$
**Act I.**
$$\tan(\alpha)+\tan(... |
If you want an intuition that explains *why* Farka's Lemma should be true, you will have to use the geometric interpretation; there's no way around that.
If you want an intuition that shows *what* Farka's Lemma achieves (but not why it should be true), there is indeed an algebraic explanation.
Namely, Farka's Lem... |
The [Reuleaux polygons][1] are analogs of the regular polygons, except that the "sides" are composed of circle arcs instead of lines. It is known that for an odd number of sides, e.g. the Reuleaux triangle, the polygon has constant width.
After reading the paper [*Roads and Wheels*][2] by Stan Wagon and Leon Hall, I... |
Four Color Theorem is equivalent to the statement: "Every cubic planar bridgeless graphs is 3-edge colorable". There is computer assisted proof given by Appel and Haken. Dick Lipton in of his [beautiful blogs](http://rjlipton.wordpress.com/2009/04/24/the-four-color-theorem/)
<a href="http://rjlipton.wordpress.com/2009... |
Human checkable proof of the Four Color Theorem? |
As a two-by-two matrix, applying [Jacobi][1] in fact gives the answer at once!
It is known that given a two-by-two matrix $\mathbf{A}$, one can construct an orthogonal matrix $\mathbf{V}=\bigl(\begin{smallmatrix}c&s\\\\-s&c\end{smallmatrix}\bigr)$ such that $\mathbf{V}^T\mathbf{A}\mathbf{V}$ is diagonal, where the t... |
Let $$\nu(x) = \frac{1}{2} \log{2\pi x} - x + \int\limits_{1}^{x} \frac{\lfloor{t\rfloor}}{t} \ dt$$ then what is $\displaystyle \lim_{x \to \infty} x \cdot \nu(x)$?
I really don't know to proceed. Tried something using the expansions of $\log$ and seeing how the integral behaves, but couldn't figure out.
|
What are the three cube roots of -1?
Not sure if this is a trick question, But I have been asked this. one of the ansers is -1, what are the other 2? |
What are the three cube roots of -1? |
I'm working to understand some noise on some of our analog-to-digital converter signals. I've analyzed a log of 2^18 samples in MATLAB, and can make the following observations:
- the noise consists of large single-sample spikes on top of my expected signal
- the FFT spectrum of the noise looks white (hard to say si... |
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix: $\begin{bmatrix} 0&0&0&-4 \\ 1&0&0&3 \\ 0&1&0&-9 \\ 0&0&1&0&1 \end{bmatrix}$
[[ 0, 0,... |
The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. Using analytic continuation, one can derive the fact that $\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}$ where $B_{s+1}$ ... |
Typically, one uses the [QR algorithm][1] on a matrix with a preliminary reduction to upper Hessenberg form (all zeroes below the subdiagonal) for determining the eigenvalues. However, since the Frobenius companion matrix is already Hessenberg to begin with, it can easily be fed into an eigenroutine without preliminary... |
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix:
$$\begin{bmatrix} 0&0&0&-4 \\\\ 1&0&0&3 \\\\ 0&1&0&-9 \\\\ 0&0&1&0&1 \end{bmatrix}$$
No... |
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix:
$$\begin{bmatrix} 0&0&0&-1 \\\\ 1&0&0&\frac{3}{4} \\\\ 0&1&0&-\frac{9}{4} \\\\ 0&0&1&\frac... |
I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix.
How do I do that?
For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix:
$$\begin{bmatrix} 0&0&\frac{3}{4} \\\\ 1&0&-\frac{9}{4} \\\\ 0&1&\frac{1}{4} \end{bmatrix}... |
You haven't given any information about the cost of turning A into B or turning C or D into B. Can we assume these are zero? In any case, it's not too hard to compute the probability distribution of costs of A by just working out similarly to what follows:
Suppose you go with the strategy "buy C and attempt a tran... |
Start with the identity
$\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$.
Differentiate the left-most term with respect to $i$ to get
$\frac{d}{di} \sum_{i=1}^n i^3 = 3 \sum_{i = 1}^n i^2$.
Differentiate the right-most term with respect to $n$ to get
$\frac{d}{dn} ... |
Prove that $f_n\to f$ in measure on $E$ if and only if given $\varepsilon>0$, there exists $K$ such that |{$x\in E : |f(x)-f_k(x)|>\varepsilon$}|$<\varepsilon$ for $k\ge K$.
The "if" direction of this is immediate from the definition of convergence in measure, but the other direction is less obvious to me.
Co... |
Prove that $f_n\to f$ in measure on $E$ if and only if given $\varepsilon>0$, there exists $K$ such that |{$x\in E : |f(x)-f_k(x)|>\varepsilon$}|$<\varepsilon$ for $k\ge K$.
The "only if" direction of this is immediate from the definition of convergence in measure, but the other direction is less obvious to me.
... |
I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series.
> The Hadamard Product
The Hadamard product is defined on generating functions $f(x)$ and $f(y)$ as $f(z)$ below:
$f(x) = \sum_{i=0}^\infty{c_i x^i}$
$f(y) = \sum_{i=0}^\infty{d_i y^i}$
$f(z) = \s... |
Is an easier formulation possible? |
Ryan already gave a very good answer. I just want to give a few clarifications on how one should think about PDEs in general. Most importantly: you should *not* think a differential operator as defined by "a formula". What do I mean by that? In your question, you wrote that the Laplacian in Euclidean coordinates is $\f... |
Let $B \subset [0,2\pi]$ be a Lebesgue measurable set. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{B} cos(nx) dx = 0$
OK I did this assuming B is an open interval, this is pretty easy using the fact that the sine function is bounded by 1. Now I'm stuck in the general case, I'm somewhat confused by "Lebes... |
**Motivation:** In odd dimensions, solutions to the wave equation: $u_{tt}(x,t)=\nabla u(x,t)$, $u_t(x,0)=0$, $u(x,0)=f(x)$, ($t\geq 0, x\in \mathbb{R}^n$) have the nice property that the value of $u(x,t)$ only depends on the values $f(y)$ with $|y-x|=t$. For even dimensions, the value $u(x,t)$ depend on all the values... |
Let $B \subset [0,2\pi]$ be a Lebesgue measurable set. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{B} \cos(nx) dx = 0$
OK I did this assuming B is an open interval, this is pretty easy using the fact that the sine function is bounded by 1. Now I'm stuck in the general case, I'm somewhat confused by "Lebe... |
The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory if they are <i>stably equivalent</i>:
there exists a trivial bundle $G$ with
$$
(E \oplus G) \cong (F \oplus G).
... |
Why is stable equivalence necessary in topological K-theory? |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory if they are <i>stably equivalent</i>:
there exists a trivial bundle $G$ with
$$
(E \oplus G) \cong (F \oplus G).
... |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
It comes down to whether or not you want $K(X)$ to be a group. If all you care about is isomorphism classes of vector bundles, that's a perfectly good monoid. People wouldn't call the monoid $K$-theory because $K$-theory of stuff generally means you've performed some kind of group-completion of a monoid or a category... |
I have a semigroup $G$ (a set with associative binary operation) such that for all $a,b\in G$ there exists $x,y\in G$ such that $ax=ya=b$. Is this property enough to show that $G$ is a group, and if so, how? |
Semigroup with "transitive" operation is a group? |
A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).
1. What is the special feature of Euclidean spaces that makes them interesting?
1. Is there a field that studies spaces that are neither globally nor locally Euclidean spaces?
1. If such a field ... |
Why the interest in locally Euclidean spaces? |
Américo Tavares pointed out in his answer to [this question](http://math.stackexchange.com/questions/8476/rearrange-the-formula-for-the-sum-of-a-geometric-series-to-find-r) that finding the ratio of a geometric progression only from knowledge of the sum of its first $n+1$ terms $S = 1+x+x^2+\cdots+x^n$ amounts to solvi... |
Why does the polynomial equation $1 + x + x^2 + \cdots + x^n = S$ have at most two solutions in $x$? |
Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?
|
I don't believe you understand the notion of efficiency in terms of encoding. Informally speaking, you have to keep it mind there are two factors involved: (i) cost of having different symbols (in case of base 10 there as 10 different symbols, in case of base 16 there are 16 different symbols etc) and the length of the... |
I am working in the field of machine learning, and I have come across a few papers that show relationships between Gröbner bases and discrete probability. So I come here for help.
Can you please explain how can Gröbner bases used to describe discrete probability?
I have looked at Gröbner bases and I understand th... |
How can Gröbner bases used to describe discrete probability ? |
My question is:
Basktball player shoots basket for 100 times. On the first throw the ball is always in, on the second - always out. For all consequent shoots the probability of hit is equal to precentage of all scored shoots diveded by total number of shoots done.
What is the probability the he will score total 5... |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial I}{\partial x}\frac{\partial I}{\partial y} \\\
\frac{\partial I}{\partial y}\frac{\partial I}{\partial x} & \frac{\partial I}{... |
I recall seeing a hand-drawn picture of spec of a ring (maybe of Z?) that had been passed around in the early days of the Zariski topology. Does anyone know where I can find a copy? |
Picture of spec? |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial I}{\partial x}\frac{\partial I}{\partial y} \\\
\frac{\partial I}{\partial y}\frac{\partial I}{\partial x} & \frac{\partial I}{... |
Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a sprinter or a marathon runner in mathematics? |
Is speed an important quality in a mathematician? |
As a two-by-two matrix, applying [Jacobi][1] in fact gives the answer at once!
It is known that given a two-by-two matrix $\mathbf{A}$, one can construct an orthogonal matrix $\mathbf{V}=\bigl(\begin{smallmatrix}c&s\\\\-s&c\end{smallmatrix}\bigr)$ such that $\mathbf{V}^T\mathbf{A}\mathbf{V}$ is diagonal, where the t... |
To some degree speed matters. If you are going to sit there thinking "hmmmm ... what *is* a subspace?" every single time you see the word "subspace" you may never get through even the simplest of linear algebra proofs. So, some knowledge must be there automatically, that's why we all practice so much. That's why we don... |
I have a branching process of the form p0=0.1, p1 = 0.6, p2 = 0.3. (any other pn = 0). Z0, the original population is 1. Z1 is the population after 1 timestep, Z2 is the population after 2 timesteps, etc.
How can I compute the probability mass function of Z3? |
Given probabilities for a branching process, how do I compute the probability mass function at a time? |
I have a branching process of the form p0=0.1, p1 = 0.6, p2 = 0.3. (any other pn = 0). Z0, the original population is 1. Z1 is the population after 1 timestep, Z2 is the population after 2 timesteps, etc.
How can I compute the probability mass function of Z3? It would suffice to have an expression I could expand on ... |
Given that X_i is a partition of the probability space, and Y,Z are events in the probability space such that P[Y intersect X_i] > 0 for all i, how can I prove that P[Z|Y] = sum(from i=1 to infinity) P[Z|Y intersect X_i] * P[X_i | Y] ?
|
How can I prove this law of "generalized" total probability? |
The classic text of Deligne and Rapoport giving a model for the mod p reduction of the modular curve is found in a "Lecture Notes in Mathematics" volume.
Has it been translated from the original french and/or LaTeX-ed so that it can be searched more easily? |
Has Deligne-Rapoport been translated? |
My class had a good time redeveloping the unit circle under the [taxicab metric][1]. Now some of them want to do it again with another similar metric. I may give it to some of my "honors" high-school* calculus I students to do independently ... so it can't be too difficult or require too many advanced concepts.
**Wh... |
My class had a good time redeveloping the unit circle under the [taxicab metric][1]. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" high-school* calculus I students to work on independently. As such, it can't be too difficult or require too many advanced con... |
This is somewhat of a follow-up question to [‘Elliptic integrals with parameter outside 0<m<1’](http://math.stackexchange.com/q/8426/2961).
I have an equation that I'm attempting to simplify that looks something like this:
$
\Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) -b \mathrm K(1-c) +2 d \\, \Pi(1-... |
Two funny distances on $\mathbb{R}^2$:
**The elevator distance.**
$$
d((x_1,x_2),(y_1,y_2)) =
\begin{cases}
\vert y_1 - y_2 \vert & \text{if}\ x_1 = x_2 \\\
\vert y_1 \vert + \vert x_1 - x_2 \vert + \vert y_2\vert & \text{otherwise}
\end{cases}
$$
**The mail office distance.**
$$
d... |
Two funny distances on $\mathbb{R}^2$:
**The elevator distance.**
$$
d((x_1,y_1),(x_2,y_2)) =
\begin{cases}
\vert y_1 - y_2 \vert & \text{if}\ x_1 = x_2 \\\
\vert y_1 \vert + \vert x_1 - x_2 \vert + \vert y_2\vert & \text{otherwise}
\end{cases}
$$
**The mail office distance.**
$$
d... |
This is somewhat of a follow-up question to [‘Elliptic integrals with parameter outside 0<m<1’](http://math.stackexchange.com/q/8426/2961).
I have an equation that I'm attempting to simplify that looks something like this:
$
\Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) -b \mathrm K(1-c) +2 d \\, \Pi(1-... |
This is somewhat of a follow-up question to [‘Elliptic integrals with parameter outside 0<m<1’](http://math.stackexchange.com/q/8426/2961).
I have an equation that I'm attempting to simplify that looks something like this:
$
\Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) -b \mathrm K(1-c) +2 d \\, \Pi(1-... |
Two funny distances on $\mathbb{R}^2$:
**The elevator distance.**
$$
d((x_1,y_1),(x_2,y_2)) =
\begin{cases}
\vert y_1 - y_2 \vert & \text{if}\ x_1 = x_2 \\\
\vert y_1 \vert + \vert x_1 - x_2 \vert + \vert y_2\vert & \text{otherwise}
\end{cases}
$$
**The mail office distance.**
$$
d... |
I'm a bit curious as to how "real" mathematicians would solve this problem.
"Find the least prime number greater than 2000."
Of course, I can always go brute force:
IF N IS PRIME, OUTPUT N
ELSE INCREASE N BY 1
But that's no fun. (Especially when the number is really high.)
Are there any algorithms/t... |
This is somewhat of a follow-up question to [‘Elliptic integrals with parameter outside 0<m<1’](http://math.stackexchange.com/q/8426/2961).
I have an equation that I'm attempting to simplify that has terms that look something like this:
$
\Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) \bigr)
$
where ... |
I expect there are more powerful tools available; but for a simple approach I would apply the Prime Number Theorem (PNT) and a sieve (as Yuval has already suggested).
One useful consequence of the PNT is that around a number *N*, approximately one out of every *log(N)* numbers is prime. (By '*log*,' number theori... |
Two funny distances on $\mathbb{R}^2$:
**The elevator distance.**
$$
d((x_1,y_1),(x_2,y_2)) =
\begin{cases}
\vert y_1 - y_2 \vert & \text{if}\ x_1 = x_2 \\\
\vert y_1 \vert + \vert x_1 - x_2 \vert + \vert y_2\vert & \text{otherwise}
\end{cases}
$$
**The mail office distance.**
$$
d... |
What do YOU think?
For mathematics, intelligence is needed. And intelligence is instantaneous. And something that is instantaneous has no notion of speed; speed requires time.
Speed as a measure cannot be applied.
For arithmetic, speed can be applied.
But there is an almost universal confusion between arit... |
I came across an interesting problem but unable to see how to approach it. How do I use the LDCT convergence theorem, to show that first derivative of characteristic function of probability distribution at t = 0, $\phi^′(0)=iE[X]$? Any ideas? |
What does this expression mean - $\lim_{n\rightarrow\infty} E|X_n-X|=0$? $X_n$ is a sequence of random variables and $X$ is a random variable. What does this expression imply? Can I say that the sequence $X_n$ converges to $X$ in probability and almost sure convergence? |
I came across an interesting problem but unable to see how to approach it. How do I use the dominated convergence theorem (LDCT), to show that first derivative of characteristic function of probability distribution at t = 0, $\phi^′(0)=iE[X]$? Any ideas?
References:
http://en.wikipedia.org/wiki/Characteristic_fun... |
You need to show that if $(a_n)$ is a sequence of nonzero numbers
which tends to zero, that $(\phi(a_n)-\phi(0))/a_n\to iE[X]$. Now
$$\frac{\phi(a)-\phi(0)}{a}=E\left(\frac{e^{iaX}-1}{a}\right).$$
You need some hypothesis on $X$ for the result to work, for
instance that $|X|$ has finite expectation. Certainly $(e^{... |
Let $\mathbf{x} =$ \{$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result
\begin{eqnarray}
AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq HM(\mathbf{x}),
\end{eqnarray}
where $AM(\mathbf{x}) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ is the arithmetic mean, $GM(\mathbf{x}... |
I need to find the number of undirected trees on $n$ vertices such that the edges (and not the vertices) are labeled and exactly one label appears twice (i.e. there are $n-2$ possible labels and they all appear).
My reasoning goes like this: From Cayley's formula we know there are $n^{n-2}$ trees with labeled vertic... |
This one comes from my module without proper explanation.Please suggest how to approach this problem ?
PS:How do you write degree (in symbol) in math-mode ? |
In right triangle PQR, ∠Q = 90 and PR = 2PQ, then ∠R is equal to ? |
In right triangle PQR, ∠Q = $90 ^\circ$ and PR = 2PQ, then ∠R is equal to ? |
As Djaian pointed out use the fact that $$\sin\angle{R} = \frac{PQ}{PR}=\frac{PQ}{2PQ}=\frac{1}{2}$$ So $\angle{R} = ?$
Also i hope you are aware of the fact that in a right angled triangle $$\sin\theta =\frac{\text{Opposite Side}}{\text{Hypotenuse}}$$ |
Let A and B be nxn matrices over C. How to prove $(AB)^*=B^*A^*$? $A^*$ is complex conjugate transpose( if matrix $A$ is real then $A^* = A^t$. This is homework and is in the area of linear algebra. |
How to prove $(AB)*=B*A*$? |
Do you have the theorem that $(AB)^t=B^tA^t$ for real matrices? Can you follow that proof through for the complex case (using conjugates)? Without going back to look, I think it works. |
Let A and B be nxn matrices over C. How to prove $(AB)^{\ast}=B^{\ast}A^{\ast}$? $A^{\ast}$ is complex conjugate transpose( if matrix $A$ is real then $A^{\ast} = A^{t}$. This is homework and is in the area of linear algebra. |
How to prove $(AB)^{\ast}=B^{\ast}A^{\ast}$? |
Online tool for diagonalizing matrices ? |
Let M, K, and L be isomorphic subgroups of a group G. Suppose $M \cap K \leq M \cap L$. Can we conclude anything about the index of MK in G and the index of ML in G? I would like it to be the case that MK has smaller index than ML, but I'm having a hard time convincing myself either way. |
Can we say something about the index of these subgroups? |
How to prove polarization identity?
For any $u, v \in V$
$4 \langle T(u), v \rangle = \langle T(u+v), u+v \rangle - \langle T(u-v), u-v \rangle + i\langle T(u+iv), u+iv \rangle - i\langle T(u-iv), u-iv \rangle$ |
How to prove polarization identity? |
This is one of those results in calculus which were anticipated by
Archimedes. He gave a correct formula for the volume but it not known exactly
how Archimedes solved this problem. There is, however, a simple way to
obtain the answer without much calculus. Let me quote from late Gardner's [*The Unexpected Hanging a... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.