Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Locally exact form $P\;dx+Q\;dy$ , and the property $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ This is a very known result, but I don't have some proof. Someone known or has some proof of it?
Let be $\omega = P\;dx + Q\;dy$ be a $C^1$ differential form on a domain $D$. If $$\frac{\partial P}{\par... | If you have a curl-free field $W = (W_1, W_2, W_3)$ in a neighborhood of the origin, it is the gradient of a function $f$ given by
$$ f(x,y,z) = \int_0^1 \; \left( \; x W_1(tx, ty,tz) + y W_2(tx, ty,tz) + z W_3(tx, ty,tz) \; \right) dt.$$
In your case, take $W_3 = 0$ and drop the dependence on $z$ from $f, \; W_... | {
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Proving that crossing number for a graph is the lowest possible How would one go about proving that the crossing number for a graph is the lowest possible?
To be more specific, given a specific representation of a particular cubic graph $G$, how do I prove that the crossing number can not be lowered any further?
This... | If the graph is small enough and you were willing to prove it by hand, you could do a case analysis similar to how students show a graph is non-planar ($cr(G) \geq 1$) by hand.
Take a long cycle (hopefully Hamiltonian), and place it evenly spaced on a circle, then start adding in edges. You're done if you can show by c... | {
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Checking for Meeting Clashes I've been sent here from StackOverflow with my mathematical / algorithm question.
I am currently working with an organisation developing a web system, one area attempting to resolve in-house training clashes.
An example (as best as I can describe is):
What the company is attempting to do is... | If you intend to estimate the expected number of clashes (not necessarily the unique or best measure, but perhaps the more easy to compute) you need a probabilistic model: in particular, you need to know the size of the total population ($N$) and if there is some dependency among courses attendance (i.e. if given that ... | {
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What can be said given that $\Psi$ is a Homomorphism? Let G and H be nontrivial finite groups with relatively prime orders When $\Psi: G\to H$ be a homomorphism, what can be said about $\Psi$ ?
| If $\Psi : G \to H$ is a homomorphism then by the (first) isomorphism theorem you know $G / \ker \Psi \cong \Psi (G)$. This means that $\frac{|G|}{|\ker \Psi|} = |\Psi(G)|$ so you know that $\Psi(G)$ divides $|G|$.
Next you know that $\Psi (G) $ is a subgroup of $H$ and hence by the Lagrange theorem it divides the orde... | {
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The letters ABCDEFGH are to be used to form strings of length four
The letters ABCDEFGH are to be used to form strings of length four.
How many strings contain the letter A if repetitions are not allowed?
The answer that I have is :
$$ \frac{n!}{(n-r)!} - \frac{(n-1)!}{(n-r)!} = \frac{8!}{4!} - \frac{7!}{4!} = 8 \t... | I presume I am correct. Here is a detailed proof.
First exclude 'A' and permute the rest (7P3). Which can be done in $\frac{7!}{4!}$ ways.
Then, include 'A' back into those permuted cases. $|X_1|X_2|X_3|$ and as indicated by the vertical lines can be in 4 locations. So, the answer is
$$\frac{7!}{4!} \times 4 = 840$$... | {
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reference for "compactness" coming from topology of convergence in measure I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here)
On page 2, I quote: "If one passes to the case of non-re... | So that this question has an answer: t.b.'s comment suggests that the quotes passage relates to the paper's Theorem 1.3, which states:
Theorem. Given a bounded sequence $(f_n)_{n \ge 1} \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ then there are convex combinations
$$g_n \in \operatorname{conv}(f_n, f_{n+1}, \dots)$$
... | {
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Does $\mathbb{R}^\mathbb{R}$ have a basis? I'm studying linear algebra on my own and saw basis for $\mathbb{R}, \mathbb{R}[X], ...$ but there is no example of $\mathbb{R}^\mathbb{R}$ (even though it is used for many examples). What is a basis for it? Thank you
| The dimension of $\mathbb R^\mathbb R$ over $\mathbb R$ is $2^{\frak c}$. It is not even the size of the continuum. As Jeroen says, this space is not finitely generated. Not even as an algebra.
Even as an algebra it is not finitely generated. What does that mean? Algebra is a vector space which has a multiplication ope... | {
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What are some "natural" interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $? (This is a spin-off of a recent question here)
In fiddling with the answer to that question I came to the set of sequences
$\qquad \small \begin{array} {llll}
A(1)=1,A(2)=1+2a,A(3)=1+2a+3a^2,A(4)=1+2a+3a^2+4a^... | I'll build from Michael's work (thanks for doing the heavy lifting!) and start with
$$A_n(r)=\frac1{n!}\frac{\mathrm d^n}{\mathrm da^n} \frac{a^{r+n}-1}{a-1}$$
Let's switch back to the series representation, and swap summation and differentiation:
$$A_n(r)=\frac1{n!}\sum_{k=0}^{r+n-1} \frac{\mathrm d^na^k}{\mathrm d a^... | {
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Continuous functions are Riemann-Stieltjes integrable with respect to a monotone function
Let $g:[a,b] \to \mathbb{R}$ be a monotone function. Could you help me prove that $\mathcal{C}([a,b])\subseteq\mathcal{R}([a,b],g)$?
(Here $\mathcal{R}([a,b],g)$ is the set of all functions that are Riemann-Stieltjes integrable... | Assume that $g$ is increasing. I suppose that you know that $f\in\mathcal{R}([a,b],g)$ iff $f$ satisfies the Riemann's condition. The Riemann's condition says:
$f$ satisfies the Riemann's condition respect to $g$ in $[a,b]$ if for every $\epsilon\gt 0$, there exist a partition $P_\epsilon$ of $[a,b]$ such that if $P$ i... | {
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How do I show that $\mathbb{Q}(\sqrt[4]{7},\sqrt{-1})$ is Galois? How do I show that $\mathbb{Q}(\sqrt[4]{7},\sqrt{-1})$ is Galois?
At first I thought it was the splitting field of $x^4-7$, but I was only able to prove that it was a subfield of the splitting field. Any ideas?
I'm trying to find all the intermediate fie... | The polynomial $f(x)=x^4-7$ factors as $(x-7^{1/4})(x+7^{1/4})(x-i7^{1/4})(x+i7^{1/4})$, and all these irreducible factors are distinct. Hence, $x^4-7$ is separable. Moreover, the field $L= \mathbb{Q}(7^{1/4},i)$ contains all its roots and is the minimal field where $x^4-7$ factors completely in. Hence, it is the minim... | {
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Chaos and ergodicity in hamiltonian systems EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be globally ergodic.
So the new formulation of my question is now this :
Do we call ... | *
*Yep: Alfredo M. Ozorio de Almeida wrote about this:
http://books.google.co.uk/books?id=nNeNSEJUEHUC&pg=PA60&lpg=PA60&dq=hamiltonian+chaos+liouville+measure&source=bl&ots=63Wnmn-xvT&sig=Z0eRtIQxmdQvgWUcLBab7ZJ9y-U&hl=en&ei=0EXfTuvzJcOG8gP5mZjaBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAw#v=onepage&q=ham... | {
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Proving the existence of point $z$ s.t. $f^{(n)}(z) = 0$
Suppose that $f$ is $n$ times differentiable on an interval $I$ and there are $n + 1$ points $x_0, x_1, \ldots, x_n \in I, x_0 < x_1 < \cdots < x_n$, such that $f(x_0) = f(x_1) = \cdots = f(x_n) = 0$. Prove that there exists a point $z \in I$ such that $f^{(n)}(... | Here’s a fairly broad hint:
You know from Rolle’s theorem that it’s true when $n=1$. Try it for $n=2$; Rolle’s theorem gives you points $y_0$ and $y_1$ such that $x_0<y_0<x_1<y_1<x_2$ and $f\;'(y_0)=f\;'(y_1)=0$. Can you now apply Rolle’s theorem to $f\;'$ on some interval to get something useful?
In order to generaliz... | {
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Encyclopedic dictionary of Mathematics I'm looking for a complete dictionary about Mathematics, after searching a lot I found only this one http://www.amazon.com/Encyclopedic-Dictionary-Mathematics-Second-VOLUMES/dp/0262090260/ref=sr_1_1?ie=UTF8&qid=1323066833&sr=8-1 .
I'm looking for a book that can give me a big pict... | You might first try the updated version of the book you mention, the Encyclopedia of Mathematics which is freely available via the quoted link. Then, since this encyclopedia is very weak on applied mathematics you could have a try with Engquist (ed): Encyclopedia of Applied and Computational Mathematics. This will give... | {
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Conditional probability with union evidence In the problem of cancer (C) and tests (t1, t2), or any other example,
How can I calculate: $P(C^+|(t1^+ \text{ or } t2^+)$
I think this would be the same as finding: $$P(t1^+ \text{ or } t2^+|C^+) P(C^+)\over P(t1^+ \text{ or } t2^+).$$
But is $$P(t1^+ \text{ or } t2... | The answer for your third (last) question is "yes"; this is just the definition of conditional probability. (I answer this first, since it is used later, here).
Your initial instinct is right. For any two events $A$ and $C$:
$$
P(A|C)={P(C\cap A)\over P(C)}={P(A\cap C)\over P(C)}={P(A)P(C| A)\over P(C)}.
$$
The answ... | {
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Rank of a degenerate conic This question comes from projective geometry. A degenerate conic $C$ is defined as $$C=lm^T+ml^T,$$
where $l$ and $m$ are different lines. It can be easily shown, that all points on $l$ and m lie on the $C$. Because, for example, if $x\in l$, then by definition $l^Tx=0$ and plugging it into c... | The rank of $lm^T$ is one. The same goes for $ml^T$. In most cases, the rank of the symmetric matrix $C$ as you define it will be 2. This corresponds to a conic degenerating into two distinct lines. If the lines $l$ and $m$ should coincide, though, the rank of $C$ will be 1.
If you need a proof, you can show this assum... | {
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Can you give an example of a complex math problem that is easy to solve? I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surfac... | What positive integers can be written as the sum of two squares? Sum of three squares? Sum of four?
For two squares, it's all positive integers of the form $a^2b$, where $b$ isn't divisible by any prime of the form $4k+3$, and the proof is easy.
For four squares, it's all positive integers, and the proof is moderately ... | {
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Trouble forming a limit equation Here's the question:
The immigration rate to the Czech republic is currently $77000$ peeople per year. Because of a low fertility rate, the population is shrinking at a continuous rate of $0.1$% per year. The current Czech population is ten million.
Assume the immigrants immediately ad... | We will proceed in two steps: First, assuming that the limit $\lim \limits_{n \to \infty} x_n$ exists, we will find it. Of course, we need to justify our assumption. So we will come back and show the existence of the limit.
Finding the limit. Suppose $x = \lim \limits_{n \to \infty} x_n$. Then allowing $n$ to go to i... | {
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Function which has no fixed points Problem:
Can anyone come up with an explicit function $f \colon \mathbb R \to \mathbb R$ such that $| f(x) - f(y)| < |x-y|$ for all $x,y\in \mathbb R$ and $f$ has no fixed point?
I could prove that such a function exists like a hyperpolic function which is below the $y=x$ axis and do... | $f(x)=2$ when $x\leq 1$, $f(x)=x+\frac{1}{x}$ when $x\geq 1$.
Another example:
Let $f(x)=\log(1+e^x)$. Then $f(x)>x$ for all $x$, and since $0<f'(x)<1$ for all $x$, it follows from the mean value theorem that $|f(x)-f(y)|<|x-y|$ for all $x$ and $y$.
| {
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Why does $\int \limits_{0}^{1} \frac{1}{e^t}dt $ converge? I'd like your help with see why does $$\int^1_0 \frac{1}{e^t} \; dt $$ converge?
As I can see this it is suppose to be:
$$\int^1_0 \frac{1}{e^t}\;dt=|_{0}^{1}\frac{e^{-t+1}}{-t+1}=-\frac{e^0}{0}+\frac{e}{1}=-\infty+e=-\infty$$
Thanks a lot?
| I assume you are integrating over the $t$ variable. $1/e^t$ is a continuous function, and you are integrating over a bounded interval, so the integral is well defined. An antiderivative of $1/e^t=e^{-t}$ is equal to $-e^{-t}$. So the integral equals $1-1/e$
| {
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Correspondences between Borel algebras and topological spaces Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...
Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given top... | I think that I can answer the second question. For each point $p \in \mathbb{R}$, let $\tau_p$ be the topology on $\mathbb{R}$ consisting of $\varnothing$ together with all the standard open neighbourhoods of $p$. Unless I've made some mistake, the Borel sigma-algebra generated by $\tau_p$ is the standard one. However... | {
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How to prove $k^n \equiv 1 \pmod {k-1}$ (by induction)? How to prove $k^n \equiv 1 \pmod {k-1}$ (by induction)?
| Well, I'll leave the case of $n=1$ to you.
So, for a fixed $k$, suppose that $k^n\equiv 1 \mod(k-1)$ for some $n\in \mathbb{N}$.
We want to show that $k^{n+1} \equiv 1 \mod(k-1)$. Well, $k^{n+1}=k^n k$, and we know that $k^n\equiv 1 \mod(k-1)$ (since this is the induction hypothesis). So, what is $k^{n+1}$ congruent ... | {
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Arithmetic Mean Linear? Is a function that finds the arithmetic mean of a set of real numbers a linear function?
So is $\left(X_1 + X_2 + \cdots + X_n\right)/n$ linear or not?
I'm not sure because so long as the set stays the same size $n$ could be defined as a constant.
| To talk about linearity, the domain and range of a function must be vector spaces, in this case over the real numbers. So your first question should be, what vector space do you take the arithmetic mean to be defined on? It turns out you must fix the value of $n$ to get any reasonable vector space with the arithmetic m... | {
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probability a credit-card number has no repeated digits It seems as though every Visa or Mastercard account number I've ever had (in the United States) has had at least two consecutive digits identical. I was wondering what the probability is that a particular account number will have at least two consecutive digits id... | It looks fine to me. What you’re using is the fact that if the random variables $X$ and $Y$ are uniformly distributed in $\{0,1,\dots,n-1\}$, then so is the reduced sum $Z=(X+Y)\bmod n$, where $\bmod$ here denotes the binary operation. (In your case $n$ is of course $10$.)
To see this, you can observe that $X+Y$ itself... | {
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Consequence of Cauchy Integral Formula for Several Complex Variables in Gunning's book I am reading Gunning's book Introduction to Holomorphic Functions of Several Variables, Vol. I, and I am stuck in the proof of Maximum modulus theorem: if $f$ is holomorphic in a connected open subset $D \subset \mathbb{C}^{n}$ and i... | It is an integrated form of the Cauchy formula. The single complex variable case illustrates what's going on. For example,
$$
f(0) = {1\over 2\pi i}\int_{|z|=1} {f(z)\over z}\;dz
= {1\over 2\pi} \int_0^{2\pi} {f(re^{i\theta})\over r\,e^{i\theta}}\,d(re^{i\theta})
= {1\over 2\pi} \int_0^{2\pi} f(re^{i\theta})\;i\,d\t... | {
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An application of Gronwall's lemma I've come across a creative use of Gronwall's lemma which I would like to submit to the community. I suspect that the argument, while leading to a correct conclusion, is somewhat flawed.
We have a continuous mapping $g \colon \mathbb{R}\to \mathbb{R}$ such that
$$\tag{1} \forall \va... | As you presented it, this is completely bogus: it is an example of the logical fallacy called "begging the question".
| {
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Prove equations in modular arithmetic Prove or disprove the following statement in modular arithmetic.
*
*If $a\equiv b \mod m$, then $ a^2\equiv b^2 \mod m$
*If $a\equiv b \mod m$, then $a^2\equiv b^2 \mod m^2$
*If $a^2\equiv b^2\mod m^2$, then $a\equiv b\mod m$
My proofs.
*
*$$ a\equiv b \mod m \implies (... | HINT $\: $ for $\rm (3),\ \ m^2\ |\ a^2 - b^2\ \Rightarrow\ m\ |\ a-b\ $ fails if $\rm\: m > 1 = a - b\:.\:$ Then $\rm\:a^2-b^2 = 2\:b+1\:$ so any odd number with a square factor $\rm\:m^2 \ne 1\:$ yields a counterexample.
| {
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Algorithm wanted: Enumerate all subsets of a set in order of increasing sums I'm looking for an algorithm but I don't quite know how to implement it. More importantly, I don't know what to google for. Even worse, I'm not sure it can be done in polynomial time.
Given a set of numbers (say, {1, 4, 5, 9}), I want to enume... | Here's an algorithm. The basic idea is that each number in the original set iterates through the list of subsets you've already found, trying to see if adding that number to the subset it's currently considering results in the smallest subset sum not yet found.
The algorithm uses four arrays (all of which are indexed... | {
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Foreign undergraduate study possibilities for a student in Southeastern Europe In the (non-EU) country I live in, the main problem with undergraduate education is that it's awfully constrained. I have only a minimal choice in choosing my courses, I cannot take graduate courses, and I have to take many applied and compu... | Hungary also has a very strong mathematical tradition, especially in discrete math, and has relatively cheap living standards. Many great mathematicians have studied at Eötvös Loránd University (ELTE) in Budapest. You can try looking there as well.
| {
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Trick to find multiples mentally We all know how to recognize numbers that are multiple of $2, 3, 4, 5$ (and other). Some other divisors are a bit more difficult to spot. I am thinking about $7$.
A few months ago, I heard a simple and elegant way to find multiples of $7$:
Cut the digits into pairs from the end, multipl... | One needn't memorize motley exotic divisibility tests. There is a universal test that is simpler and much easier recalled, viz. evaluate a radix polynomial in nested Horner form, using modular arithmetic. For example, consider evaluating a $3$ digit radix $10$ number modulo $7$. In Horner form $\rm\ d_2\ d_1\ d_0 \ $ i... | {
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"timestamp": "2023-03-29T00:00:00",
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Sparsest cut is solvable on trees The problem is to prove that Sparsest cut is solvable on trees in polynomial time.
A short review, a sparsest cut is linear program
$$\min \frac{c(S,\overline{S})}{D(S,\overline{S})}$$
where $c(S,\overline{S})$ - sum of edge weights for every edge that crosses the cut $S,\overline{S}$... | Actually, you understand more than you think :). The proof indeed goes by contradiction, in that if the optimal cut induced disconnected components, then one of the components would give a better cut value. The rest of the proof follows from the fact that you can now parametrize the set of candidate optimal solutions b... | {
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Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$
And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$
In both cases, $x_1,x_2,x_3,x_4,x_5$ must be nonnegative integers.
Is there a general form... | $\infty$, if you have no constraint on $x_2$ and $x_5$ other than that they are integers: note that you can always add $1$ to one of these and subtract $1$ from the other. Or did you mean nonnegative (or positive) integers?
| {
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Rules for algebraically manipulating pi-notation? I'm a bit of a novice at maths and want to learn more about algebraically manipulating likelihoods in statistics.
There are a lot of equations that involve taking the product of a set of values given a model.
I know a few rules for manipulating sigma-notation (e.g., her... | This might be inappropriate for an answer but I believe you tried yourself too hard at here. The $\prod $ sign just means multiplying some elements together, with a label in the bottom to denote the beginning element and a label on the top to denote the end element. The files you provided are very good and you should g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Curve defined by 3 equations Suppose that $X$ is a curve in $\mathbb{A}^3$ (in the AG sense, let's say over an algebraically closed field $k$) that contains no lines perpendicular to the $xy$ plane, and that there exist two polynomials $f,g\in k[x,y,z]$ such that $\{f=0\}\cap\{g=0\}=X\cup l_1\cup\cdots\cup l_n$, where ... | Yes. I believe this is from a Shafarevich problem?
For instance, suppose $\ell_1$ intersects the $x-y$ plane at $(x,y) = (a,b)$. Consider the homomorphism $k[x,y,z] \rightarrow k[z]$ sending $x\mapsto a$ and $y \mapsto b$. The image of $I(X)$ is some prime ideal of $k[z]$, which is principal. Now look at the pullback o... | {
"language": "en",
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Find all analytic functions such that... Here is the problem: find all functions that are everywhere analytic, have a zero of order two in $z=0$, satisfy the condition $|f'(z)|\leq 6|z|$ and such that $f(i)=-2$. Any hint is welcomed.
| Here is a hint: consider $f'(z)/z$.
Since $f(z)$ has a zero of order two at $z=0$, the derivative $f'(z)$ is also holomorphic, and $f'(0)=0$. Thus, you may write $f'(z)$ as $z\cdot g(z)$, with $g(z)$ holomorphic. Then, the bound in the statement tells you that $|g(z)|$ is bounded.
| {
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A problem related to basic martingale theory In our probability theory class, we are supposed to solve the following problem:
Let $X_n$, $n \geq 1 $ be a sequence of independent random variables such that
$ \mathbb{E}[X_n] = 0, \mathbb{Var}(X_n) = \sigma_n^2 < + \infty $ and
$ | X_n | \leq K, $ for some constan... | Set $S_n:=\sum_{i=1}^n X_i$ and $S_0=0$. $S$ is a martingale (wrt the natural filtration), so $S^2$ is a sub-martingale.
Using the Doob's decomposition we can write $S^2=M+A$ where M is a martingale and A is predictable (increasing) process, both null at 0.
It turns out that $A_n=\sum_{i=1}^n \sigma_i^2$.
Define the st... | {
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The expected value of magnitude of winning and losing when playing a game Suppose we play a game at a casino. There is a \$5 stake and three possible outcomes: with probability $1/3$ you lose your stake, with with probability $1/3$ the bank returns your stake plus \$5, and with probability $1/3$ the bank simply returns... | If I understand correctly, you are interested in the expected value of the magnitude of the win (or) loss (and not the magnitude of the expected value of the win (or) loss). Hence, you are interested in computing $\mathbb{E}(S_n)$ of the underlying random variable where $$S_n = \left| X_1 + X_2 + \cdots + X_n \right|.$... | {
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Properties about a certain martingale I asked this question here. Unfortunately there was not a satisfying answer. So I hope here is someone who could help me.
I'm solving some exercises and I have a question about this one:
Let $(X_i)$ be a sequence of random variables in $ L^2 $ and a filtration $ (\mathcal{F}_i)$ su... | For 3, compute $E[M_n^2]$. Having done this conclude that $E[M_n^2]\le M_\infty$ for all $n$. This means that $\{M_n\}$ is an $L^2$-bounded martingale, to which the martingale convergence theorem may be applied.
For 4, the $i$th term in the sum defining $M_\infty$ is equal to
$E[(X_i-E[X_i|\mathcal{F}_{i-1}])^2]$, whic... | {
"language": "en",
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A primitive and irreducible matrix is positive for some power $k.$ Prove it's positive for any power $k+i,$ where $i=1,\dots.$
Let $P = [p_{ij}]_{1 \leqslant i,j \leqslant m} \geqslant 0$ a primitive and irreducible matrix. And $P^k > 0$ for some $k.$ Prove that $ P^{k+i} > 0, i =1,2, \dots.$
I have used a hint sug... | Hint Prove first that any row of $P$ has a non-zero element. Then
$P^{k+i+1}=PP^{k+i}$
| {
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Need help finding limit $\lim \limits_{x\to \infty}\left(\frac{x}{x-1}\right)^{2x+1}$ Facing difficulty finding limit
$$\lim \limits_{x\to \infty}\left(\frac{x}{x-1}\right)^{2x+1}$$
For starters I have trouble simplifying it
Which method would help in finding this limit?
| $$
\begin{eqnarray}
\lim \limits_{x\to \infty}\left(\frac{x}{x-1}\right)^{2x+1}=\lim \limits_{x\to \infty}\left(\frac{x-1+1}{x-1}\right)^{2x+1}
=\lim \limits_{x\to \infty}\left(1+\frac{1}{x-1}\right)^{2x+1}\\= \lim \limits_{x\to \infty}\left(1+\frac{1}{x-1}\right)^{(x-1)\cdot\frac{2x+1}{x-1}}
=\lim \limits_{x\to \inft... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple Combinations with Repetition Question
In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar and 5 (IDENTICAL) Dollars ?
For my answer, I used the logic, how many dollars are there in the 5 we choose?
I added the case for 5 ... | Decide for each small coin whether you select that or not. Then top up with dollars until you have selected 5 coins in total.
The topping-up step does not involve any choice, so you have 5 choices to make, each with 2 options, giving $2^5=32$ combinations in all.
| {
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Interesting Property of Numbers in English I was playing with the letters in numbers written in English and I found something quite funny. I found that if you count the number of letters in the number and write this as a number and then count the number of letters in this new number and keep repeating the process, you ... | Define $f: \mathbb{N} \to \mathbb{N}$ as the number of letters in a given natural number spelled out.
Four is the only fixed point under $f$, and it's not too difficult to see that $f$ is almost always strictly decreasing with the only exceptions being one, two, three and four. So the $n^{th}$ iterate of $f$ must event... | {
"language": "en",
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Isometries of $\mathbb{R}^3$ So I'm attempting a proof that isometries of $\mathbb{R}^3$ are the product of at most 4 reflections. Preliminarily, I needed to prove that any point in $\mathbb{R}^3$ is uniquely determined by its distances from 4 non-coplanar points, and then that an isometry sends non-coplanar points to... | Would this work? Construct the altitude from $D$ to the plane containing $A$, $B$, and $C$. Call the foot of this altitude $E$ (the point where the altitude meets the plane). Triangles $ADE$, $BDE$, and $CDE$ all have right angles at $E$ and you know that the isometry preserves angles, so triangles $A'D'E'$, $B'D'E'... | {
"language": "en",
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Why is the Kendall tau distance a metric? So I am trying to see how the Kendall $\tau$ distance is considered a metric; i.e. that it satisfies the triangle inequality.
The Kendall $\tau$ distance is defined as follows:
$$K(\tau_1,\tau_2) = |(i,j): i < j, ( \tau_1(i) < \tau_1(j) \land \tau_2(i) > \tau_2(j) ) \lor ( \tau... | Kendall tau rank distance is a metric only if you compare ranking of the elements.
If you perform Kendall function comparing elements you will find cases where the triangular inequality does not work.
Example:
0 0 0 10 10 10
and
5 5 5 0 0 0
scores 9 (using Kendall comparing elements)
While
0 ... | {
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Can this be simplified any further I've been working on a formula, which I have managed to simplify to the following expression, but I wonder if anyone can spot a way to simplify it further?
$$2^{1 -\frac{1}{2}\sum_i \log_2 \frac{(a_i + c_i)^{(a_i + c_i)}}{a_i^{a_i}c_i^{c_i}}}$$
| Does this look simpler to you?
$$
2 \left( \prod_i \frac{a_i^{a_i}c_i^{c_i}}{(a_i + c_i)^{(a_i + c_i)}} \right)^\frac{1}{2}
$$
| {
"language": "en",
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"source": "stackexchange",
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Probability of getting different results when tossing a coin Here's a question I got for homework:
In every single time unit, Jack and John are tossing two different
coins with P1 and P2 chances for heads. They
keep doing so until they get different results. Let X be the number of
tosses. Find the pmf of X (in d... | You can work out the probability that they get different results on the first toss, namely $p_1 (1-p_2)+ (1-p_1)p_2 = p_1+p_2 - 2p_1 p_2$.
If they have not had different results up to the $n$th toss, then the conditional probability they get different results on the next toss is the same; this is the memoryless prope... | {
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Prove that exist $z_0 \in \mathbb C$ that satisfy $f(z_0)=0$. I would be glad to get some help with this question:
Let $f(z)$ be an entire function. Assume that there exists a
monotonous increasing and unbounded sequence $\{r_n\}$ such that $\lim\limits_{n \to \infty} \min\limits_{|z|=r_n} |f(z)|=\infty$. I want to s... | Assume $f(z)\ne0$ for all $z\in\mathbb{C}$. Then $h(z)=1/f(z)$ is also an entire function. Apply the maximum modulus principle to $h$.
| {
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What function does $\sum \limits_{n=1}^{\infty}\frac{1}{n3^n}$ represent, evaluated at some number $x$? I need to know what the function $$\sum \limits_{n=1}^{\infty}\frac{1}{n3^n}$$ represents evaluated at a particular point.
For example if the series given was $$\sum \limits_{n=0}^{\infty}\frac{3^n}{n!}$$ the answer ... | Take $f(x) = \displaystyle \sum_{n=1}^\infty \frac{x^n}{n}$.
Then,
$f^\prime (x) = \displaystyle \sum_{n=1}^\infty \frac{n x^{n-1}}{n} = \sum_{n=0}^\infty x^n$.
The last expression is a geometric series and, as long as $x < 1$, it can be expressed as
$f^\prime (x) = \displaystyle \frac{1}{1-x}$.
Therefore,
$f(x) = - \l... | {
"language": "en",
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A group of order $195$ has an element of order $5$ in its center Let $G$ be a group of order $195=3\cdot5\cdot13$. Show that the center of $G$ has an element of order $5$.
There are a few theorems we can use here, but I don't seem to be able to put them together quite right. I want to show that the center of $G$ is di... | Hint: There are unique, hence normal, $5$- and $13$-Sylows. Their internal direct product is thus normal and has complementary subgroup equal to one of the $3$-Sylows, so $G$ is a semidirect product of $H_5 H_{13}$ and $H_3$, where $H_p$ denotes a $p$-Sylow (not necessarily unique). What can you say about $\varphi: H_3... | {
"language": "en",
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Maxima of bivariate function [1] Is there an easy way to formally prove that,
$$
2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y\ge -x^{4} -y^{4} +2x^{3} +2y^{3} -2x^{2} -2y^{2} +x+y$$
$${0<x,y<1}$$
without resorting to checking partial derivatives of the quotient formed by the two sides, and finding local maxima?
[2] Similar... | Your first question:
With a little manipulation you get that it is equivalent to
$$x^2((1-x)^2+1)+y^2((1-y)^2+1) \ge 2xy[(1-x)(1-y)+1].$$
This can be obtained from addition of two inequalities
$$x^2(1-x)^2+y^2(1-y)^2 \ge 2xy(1-x)(1-y)$$
$$x^2+y^2\ge 2xy.$$
Both of them are special cases of $a^2+b^2\ge 2ab$, which follo... | {
"language": "en",
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formal proof challenge I am desperately trying to figure out the formal proof for this argument.
$$\begin{array}{r}
A\lor B\\
A\lor C\\
\hline
A\lor (B \land C)
\end{array}$$
I am trying to apply the backwards method here. I am trying to infer A, in order to use vIntro in the last step and introduce the final dis... |
I am trying to apply the backwards method here. I am trying to infer A, in order to use vIntro in the last step and introduce the final disjunction. But I got stuck finding sufficient proof for A.
You don't prove it; you assume it -- To be precise: you assume both cases aiming to derive the same conclusion from each.... | {
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Is my trig result unique? I recently determined that for all integers $a$ and $b$ such that $a\neq b$ and $b\neq 0$,
$$
\arctan\left(\frac{a}{b}\right) + \frac{\pi}{4} = \arctan\left(\frac{b+a}{b-a}\right)
$$
This implies that 45 degrees away from any angle with a rational value for tangent lies another angle with a ... | If you differentiate the function $$f(t)=\arctan t - \arctan\frac{1 + t}{1 - t},$$ you get zero, so the function is constant in each of the two intervals $(-\infty,1)$ and $(1,+\infty)$ on which it is defined.
*
*Its value at zero is $\pi/2$, so that $f(t)=-\pi/4$ for all $t<1$, so
$$ \arctan t + \frac\pi4 = \arcta... | {
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Prove that $\mathbb{Z}_p^{\times}/(\mathbb{Z}_p^{\times})^2$ is isomorphic to $\{\pm1\}$. Prove that $\mathbb{Z}_p^{\times}/(\mathbb{Z}_p^{\times})^2$ is isomorphic to $\{\pm1\}$, where $p$ is a prime integer.
| I take it that you mean to prove that $\mathbb{F}_p^\times/(\mathbb{F}_p^\times)^2 \cong \{\pm 1\}$, where $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
If so, use the fact that the map $(\mathbb{Z}/p\mathbb{Z})^\times \to \{\pm 1\}$ given by $a\bmod p\mapsto (\frac{a}{p})$ is a homomorphism of groups, where $(\frac{a}{p})$... | {
"language": "en",
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Proving the Cantor Pairing Function Bijective How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$
How would you show that for a function like the Cantor pairing function?
| I will denote the pairing function by $f$. We will show that pairs $(x,y)$ with a particular value of the sum $x+y$ is mapped bijectively to a certain interval, and then that the intervals for different value of the sum do not overlap, and that their union is everything.
Let $m$ be a natural number and suppose $m=x+y$... | {
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Characteristic equation of a recurrence relation Yesterday there was a question here based on solving a recurrence relation and then when I tried to google for methods of solving recurrence relations, I found this, which gave different ways of solving simple recurrence relations.
My question is how do you justify wri... | The characteristic equation is the one that a number $\lambda$ should satisfy in order for the geometric series $(\lambda^n)_{n\in\mathbf N}$ to be a solution of the recurrence relation. Another interpretation is that if you interpret the indeterminate $s$ as a left-shift of the sequence (dropping the initial term and ... | {
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How to find all rational points on the elliptic curves like $y^2=x^3-2$ Reading the book by Diophantus, one may be led to consider the curves like:
$y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$,
the first two of which are easy (after calculating some eight curves to be solved under some certain conditions, one can directly der... | Given your interest in Mordell's equation, you really ought to buy or borrow Diophantine Equations by Mordell, then the second edition of A Course in Number Theory by H. E. Rose, see AMAZON
Rose discusses the equation starting on page 286, then gives a table of $k$ with
$ -50 \leq k \leq 50$ for which there are integr... | {
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Generalization of manifold Is there a generalization of the concept of manifold that captures the following idea:
Consider a sphere that instead of being made of a smooth material is actually made up of a mesh of thin wire. Now for certain beings living on the sphere the world appears flat and 2D, unware that they are ... | One thing to look at is foliations (and laminations), which are decompositions of manifolds into lower-dimension manifolds. While there is no "mesh" because each lower-dimension manifold has another lower-dimension manifold in any neighborhood, there is still a lower-dimensionality that is something like what you seek.... | {
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Find the ordinary generating function $h(z)$ for a Gambler's Ruin variation. Assume we have a random walk starting at 1 with probability of moving left one space $q$, moving right one space $p$, and staying in the same place $r=1-p-q$. Let $T$ be the number of steps to reach 0. Find $h(z)$, the ordinary generating func... | A classical way to determine $h(z)$ is to compute $h_n(z)=\mathrm E_n(z^T)$ for every $n\geqslant0$, where $\mathrm E_n$ denotes the expectation starting from $n$, hence $h(z)=h_1(z)$.
Then $h_0(z)=1$ and, considering the first step of the random walk, one gets, for every $n\geqslant1$,
$$
h_n(z)=rzh_n(z)+pzh_{n+1}(z... | {
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} |
Free products of cyclic groups Given $G$, $H$, $G'$, and $H'$ are cyclic groups of orders $m$, $n$, $m'$, and $n'$ respectively.
If $G*H$ is isomorphic to $G'* H'$, I would like to show that either $m = m'$ and $n = n'$ or else $m = n'$ and $n = m'$ holds. Where * denotes the free product.
My approach:
$G*H$ has an el... | Let me give you an alternative argument for the claim $m=m'$ in Arturo Magidin's answer.
Take the abelianizations of the groups $G*H$ and $G'*H'$, since $G,G,H,H'$ are abelian, their abelianizations are $G\oplus H$ and $G'\oplus H'$ respectively. Then you get $G\oplus H\cong G'\oplus H'$ and in particular, their order... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Proving $\mathbb{N}^k$ is countable Prove that $\mathbb{N}^k$ is countable for every $k \in \mathbb{N}$.
I am told that we can go about this inductively.
Let $P(n)$ be the statement: “$\mathbb{N}^n$ is countable” $\forall n \in \mathbb{N}$.
Base Case: $\mathbb{N}^1 = \mathbb{N}$ is countable by definition, so $\checkm... | The function $f:\mathbb{N^K}\to \mathbb{N^K\times \{m \}}$ defined by $$f(a_1,a_2,\cdots, a_k)=(a_1,a_2,\cdots, a_k,m)$$ is clearly a bijection for fixed $m\in \mathbb{N}$ and we can write $\mathbb{N^{K+1}}$ as $$\mathbb{N^{K+1}}=\bigcup_{m=1}^{\infty}\{\mathbb{N^K\times \{m \}}\}$$ and this being a countable union of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
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Define when $y$ is a function of $x$ Hello guys I want to make sure myself in determming when is $y$ as function of $x$,so for this, let us consider following question. If the equation of circle is given by this
$$x^2+y^2=25$$
and question is find the equation of tangent of circle at point $(3,4)$,then it is c... | The equation $$
\tag{1}x^3+y^3=6xy
$$does define $y$ as a function of $x$ locally (or, rather, it defines $y$ as a function of $x$ implicitly). Here, it is difficult to write the defining equation as $y$ in terms of $x$. But, you don't have to do that to evaluate the value of the derivative of $y$.
[edit] The point ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Sum of irrational numbers Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an answer, but I just proved $\sqrt[100]{\sqrt3 + \sqrt2} \notin \mathbb{Q}$ and $\sqrt[100]{\sqrt3 - ... | Here is a useful trick, though it requires a tiny bit of field theory to understand: If $\alpha + \beta$ is a rational number, then $\mathbb{Q}(\alpha) = \mathbb{Q}(\beta)$ as fields. In particular, if $\alpha$ and $\beta$ are algebraic, then the degrees of their minimal polynomials are equal.
So, for example, we can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Heads or tails probability I'm working on a maths exercise and came across this question.
The probability of a "heads" when throwing a coin twice is 2 / 3. This could be explained by the following:
• The first time is "heads". The second throw is unnecessary. The result is H;
• The first time is "tails" and twice "head... | The reason your result, as Shitikanth has already pointed out, is wrong, is that you've applied the principle of indifference where it doesn't apply. You can only assume that events will all be equally likely if they're all qualitatively the same and there's nothing (other than names and labels) to distinguish them fro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding angles in a parallelogram without trigonometry
I'm wondering whether it's possible to solve for $x^{\circ}$ in terms of $a^{\circ}$ and $b^{\circ}$ given that $ABCD$ is a parallelogram. In particular, I'm wondering if it's possible to solve it using only "elementary geometry". I'm not sure what "elementary ge... | The example by alex.jordan does finish the matter, and similar ones may be constructed. We have an angle
$$ \theta = \arctan \left( \frac{1}{\sqrt{12}} \right) $$
and we wish to know whether $ x = \frac{\theta}{\pi} $ is the root of an equation with rational coefficients.
Well,
$$ e^{i \theta} = \sqrt{\frac{12}{13}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 1
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Adding a different constant to numerator and denominator Suppose that $a$ is less than $b$ , $c$ is less than $d$.
What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to $\dfrac{a+c}{b+d}$?
| One nice thing to notice is that
$$
\frac{a}{b}=\frac{c}{d} \Leftrightarrow \frac{a}{b}=\frac{a+c}{b+d}
$$
no matter the values of $a$, $b$, $c$ and $d$. The $(\Rightarrow)$ is because $c=xa, d=xb$ for some $x$, so $\frac{a+c}{b+d}=\frac{a+xa}{b+xb}=\frac{a(1+x)}{b(1+x)}=\frac{a}{b}$. The other direction is similar.
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Tricky Factorization How do I factor this expression: $$ 0.09e^{2t} + 0.24e^{-t} + 0.34 + 0.24e^t + 0.09e^{-2t} ? $$
By trial and error I got $$ \left(0.3e^t + 0.4 + 0.3 e^{-t}\right)^2$$ but I'd like to know how to formally arrive at it.
Thanks.
| The most striking thing about the given expression is the symmetry. For anything with that kind of symmetric structure, there is a systematic approach which is definitely not trial and error.
Let
$$z=e^t+e^{-t}.$$
Square. We obtain
$$z^2=e^{2t}+2+e^{-2t},$$
and therefore $e^{2t}+e^{-2t}=z^2-2$.
Substitute in our exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ Trying to solve
$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.
I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$
Would appreciate your help!
| Suppose $$\tag{1}\lim\limits_{x\rightarrow\infty}f(x)\ne 0.$$
Then we may, and do, select an $\alpha>0$ and a sequence $\{x_n\}$ so that for any $n$, $$\tag{2}x_n\ge x_{n-1}+1$$
and
$$\tag{3}|f(x_n)|>\alpha.$$
Now, since $f$ is uniformly continuous, there is a $1>\delta>0$ so that
$$\tag{4}|f(x)-f(y)|<\alpha/2,\q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
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Simplifying trig expression I was working through some trig exercises when I stumbled upon the following problem:
Prove that: $ \cos(A+B) \cdot \cos(A-B)=\cos^2A- \sin^2B$.
I started out by expanding it such that
$$ \cos(A+B) \cdot \cos(A-B)=(\cos A \cos B-\sin A \sin B) \cdot (\cos A \cos B+ \sin A \sin B),$$
which s... | The identities
$$\cos(\theta) = \frac{e^{i \theta}+e^{- i \theta}}{2}$$
$$\sin(\theta) = \frac{e^{i \theta}-e^{- i \theta}}{2i}$$
can reduce a trigonometric identity to a identity of polynomials. Let's see how this works in your example:
$$\cos(A+B) \cos(A-B)=\cos(A)^2-\sin(B)^2$$
is rewritten into:
$$\frac{e^{i (A+B)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Quadratic forms and prime numbers in the sieve of Atkin I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point.
For example, in the paper linked above, theorem 6.2 on page 1028 says that if $n$ is prime then the cardinality of the set which contains all ... | The main thing is that the norm of $s + t \omega$ is $s^2 + s t + t^2,$ which is a binary form that represents exactly the same numbers as $3x^2 + y^2.$
It is always true that, for an integer $k,$ the form $s^2 + s t + k t^2$ represents a superset of the numbers represented by $x^2 + (4k-1)y^2.$ For instance, with $k=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the limit of a sequence $\lim _{n\to \infty} \sqrt [3]{n^2} \left( \sqrt [3]{n+1}- \sqrt [3]{n} \right)$ If there were a regular square root I would multiply the top by its adjacent and divide, but I've tried that with this problem and it doesn't work. Not sure what else to do have been stuck on it.
$$ \lim _{... | $$
\begin{align*}
\lim _{n\to \infty } \sqrt [3]{n^2} \left( \sqrt [3]{n+1}-
\sqrt [3]{n} \right)
&= \lim _{n\to \infty } \sqrt [3]{n^2} \cdot \sqrt[3]{n} \left( \sqrt [3]{1+ \frac{1}{n}}-
1 \right)
\\ &= \lim _{n\to \infty } n \left( \sqrt [3]{1+ \frac{1}{n}}-
1 \right)
\\ &= \lim _{n\to \infty } \frac{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Expected value of $XYZ$, $E(XYZ)$, is not always a $E(X)E(Y)E(Z)$, even if $X$, $Y$, $Z$ are not correlated in pairs Could you prompt me, please, is it true?
Expected value of $XYZ$, $E(XYZ)$, is not always $E(X)E(Y)E(Z)$, even if $X$, $Y$, $Z$ are not correlated in pairs, because if $X$, $Y$, $Z$ are not correlated in... | Suppose
$$
(X,Y,Z) = \begin{cases}
(1,1,0) & \text{with probability }1/4 \\
(1,0,1) & \text{with probability }1/4 \\
(0,1,1) & \text{with probability }1/4 \\
(0,0,0) & \text{with probability }1/4
\end{cases}
$$
Then $X,Y,Z$ are pairwise independent, and $E(X)E(Y)E(Z)=1/8\ne 0 = E(XYZ)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proofs for an equality I was working on a little problem and came up with a nice little equality which I am not sure if it is well-known (or) easy to prove (It might end up to be a very trivial one!). I am curious about other ways to prove the equality and hence I thought I would ask here to see if anybody knows any or... | For whatever it is worth, below is an explanation on why I was interested in this equality. Consider a rectangle of size $x \times 1$, where $x < 1$. I was interested in covering this rectangle with squares of maximum size whenever possible (i.e. in a greedy sense).
To start off, we can have $\displaystyle \left \lfloo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Deriving SDE(s) and Expectation from Given PDE We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the expectation of some function of the path $X_t, Y_t$.
Attempt: I tried to... | What do you think about the system of SDEs :
$$dX_t=\sqrt{X_t^2+Y_t^2}dW_t$$
$$dY_t=(X_t-Y_t^2)dt$$
And finally :
$$u(X_t,Y_t,t)=\mathbb{E}[V(X_T,Y_T).e^{-\int_t^TrY_s.ds}|X_t,Y_t]$$
You can check that $u$ is satisfying your PDE, but as always check my calculations as I am used to making errors.
The way I found this is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Given a matrix, is there always another matrix which commutes with it? Given a matrix $A$ over a field $F$, does there always exist a matrix $B$ such that $AB = BA$? (except the trivial case and the polynomial ring?)
| Another example is the adjoint of $A$:
$$
A \operatorname{adj}(A)= \operatorname{adj}(A) A = \det(A)I
$$
(but for invertible matrices it is equal to the scalar $\det(A)$ multipliying the inverse of $A$, so is trivial that commutes. with $A$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/92480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 5,
"answer_id": 4
} |
A solvable Lie-algebra of derived length 2 and nilpotency class $n$ Given a natural $n>2$, I want to show that there exists a lie algebra $g$ which is solvable of derived length 2, but nilpotent of degree $n$.
I have seen a parallel idea in groups, but i can't see how i can implement it for Lie-algebras.
Thanks!
| The so called standard graded filiform nilpotent Lie algebra $\mathfrak{f}_{n+1}$ of dimension $n+1$ has nilpotency class $n$, and derived length $2$.
The non-trivial brackets are $[e_1,e_i]= e_{i+1}$ for $i=2,\ldots ,n$. We have
$[\mathfrak{f}_{n+1}, \mathfrak{f}_{n+1}]=\langle e_3,\ldots ,e_{n+1}\rangle$ and
$[[\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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For which value(s) of parameter m is there a solution for this system Imagine a system with one parameter $m$:
\begin{cases}
mx + y = m\\
mx + 2y = 1\\
2x + my = m + 1
\end{cases}
Now the question is: when does this system of equations have a solution?
I know how to do it with the Gaussian method, but how can I do this... | Compute the values of $x$ and $y$ dependent on $m$ for the following system, then solve $2x + my = m + 1$ (the last equation) to find the values of parameter $m$ for $x$ and $y$:
\begin{cases}
mx + y = m\\
mx + 2y = 1\\
\end{cases}
So,
\begin{cases}
2mx + 2y =2 m\\
mx + 2y = 1\\
\end{cases}
Subtracting two equations,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A problem about stochastic convergence (I think) I am trying to prove the convergence of the function $f_n = I_{[n,n+1]}$ to $f=0$, but first of all I don't in which way it converges, either in $\mathcal{L}_p$-measure or stochastically, or maybe some other form of convergence often used in measure-theory.
For now I'm a... | The sequence $\{f_n\}$ doesn't converge in $\mathcal L^p$ norm, since for all $n$ $$\lVert f_{n+1}-f_n\rVert_{L^p}^p=\int_{\mathbb R}|\mathbf 1_{[n+1,n+2]}-\mathbf 1_{[n,n+1]}|^p =\int_{[n,n+2]}1d\mu =2.$$
This sequence cannot converge in measure since $\mu(\{|f_{n+1}-f_n|\geq \frac 12\})\geq \mu([n,n+1))=1$, but conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Representing the $q$-binomial coefficient as a polynomial with coefficients in $\mathbb{Q}(q)$? Trying a bit of combinatorics this winter break, and I don't understand a certain claim.
The claim is that for each $k$ there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the fiel... | The $q$-binomial coefficient satisfies the recurrence
$$
\binom{n}{k}_q = q^k \binom{n-1}{k}_q + \binom{n-1}{k-1}_q,
$$
which follows easily from the definition. We can assume inductively that each term on the right is a polynomial and therefore the LHS is a polynomial.
Edit: Unfortunately this does not seem to yie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92772",
"timestamp": "2023-03-29T00:00:00",
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Showing $\tau(n)/\phi(n)\to 0$ as $n\to \infty$ I was wondering how to show that $\tau(n)/\phi(n)\to 0$, as $n\to \infty$. Here $\tau(n)$ denotes the number of positive divisors of n, and $\phi(n)$ is Euler's phi function.
| Here's a hint: let $Q(n)$ denote the largest prime power that divides $n$. Then prove:
*
*$\displaystyle \frac{\tau(n)}{\phi(n)} \le 2 \frac{\tau(Q(n))}{\phi(Q(n))} \le \frac4{\log2} \frac{\log Q(n)}{Q(n)}$;
*$Q(n) \to \infty$ as $n\to \infty$.
For #1, you'll want to use the fact that $\tau(n)/\phi(n)$ is multiplic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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Find limit of polynomials Suppose we want to find limit of the following polynomial
$$\lim_{x\to-\infty}(x^4+x^5).$$
If we directly put here $-\infty$, we get "$-\infty +\infty$" which is definitely undefined form, but otherwise if factor out $x^5$, our polynomial will be of the form $x^5(1/x+1)$.
$\lim_{x\to-\in... | Your factoring method is fine.
In general
given a polynomial, $$P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1 x+a_0,\quad a_n\ne0,$$
you can factor out the leading term when $x\ne0$:
$$
P(x)= x^n\Bigl(\,a_n+{ a_{n-1}\over x}+ \cdots +{a_1\over x^{n-1}} +{a_0\over x^n} \,\Bigr),\quad x\ne0.
$$
When taking the limit as $x$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Software to display 3D surfaces What are some examples of software or online services that can display surfaces that are defined implicitly (for example, the sphere $x^2 + y^2 + z^2 = 1$)? Please add an example of usage (if not obvious).
Also, I'm looking for the following (if any):
*
*a possibility to draw man... | Try these for algebraic surfaces:
*
*surf generates excellent images.
*surfer
*surfex
from http://www.algebraicsurface.net/.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/92963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 1
} |
Why is $\lim\limits_{x \space \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$? As part of this problem, after substitution I need to calculate the new limits.
However, I do not understand why this is so:
$$\lim_{x \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$$
I tried drawing the unit circle to see what happens with $... | Here's a slightly different way of seeing that $\lim\limits_{\theta\rightarrow {\infty}}\arctan\theta={\pi\over2}$.
Thinking of the unit circle, $\tan \theta ={y\over x}$, where $(x,y)$ are the coordinates of the point on the unit circle with reference angle $\theta$, what happens as $\theta\rightarrow\pi/2$? In partic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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How to prove that geometric distributions converge to an exponential distribution?
How to prove that geometric distributions converge to an exponential distribution?
To solve this, I am trying to define an indexing $n$/$m$ and to send $m$ to infinity, but I get zero, not some relevant distribution. What is the techni... | The waiting time $T$ until the first success in a sequence of independent Bernoulli trials with probability $p$ of success in each one has a geometric distribution with parameter $p$: its probability mass function is $P(x) = p (1-p)^{x-1}$
and cumulative distribution function $F(x) = 1 - (1-p)^x$ for positive integer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 5,
"answer_id": 2
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Evaluating a definite integral by changing variables. How can I evalute this integral?
$$\psi(z)=\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-a)^2+(y-b)^2+z^2]^{-3\over 2}f(x,y)\,\,\,dxdy\;.$$
I think we can treat $z$ as a constant and take it out of the integral or something. Maybe changing variables l... | There's no general way to evaluate this integral, for if there were, you could integrate any function $g(x,y)$ by calculating this integral for $f(x,y)=g(x,y)[(x-a)^2+(y-b)^2+z^2]^{\frac32}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Is $22/7$ equal to the $\pi$ constant?
Possible Duplicate:
Simple numerical methods for calculating the digits of Pi
How the letter 'pi' came in mathematics?
When I calculate the value of $22/7$ on a calculator, I get a number that is different from the constant $\pi$.
Question: How is the $\pi$ constant calculated... | In answer to your second question, NOVA has an interactive exhibit that uses something like Archimedes method for approximating $\pi$. Archimedes method predates calculus, but uses many of its concepts.
Note that "simple" and "calculus" are not disjoint concepts.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
A question on transcendental numbers Transcendental numbers are numbers that are not the solution to any algebraic equation.
But what about $x-\pi=0$? I am guessing that it's not algebraic but I don't know why not. Polynomials are over a field, so I am guessing that $\mathbb{R}$ is implied when not specified. And since... | To quote Wikipedia "In mathematics, a transcendental number is a number (possibly a complex number) that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients." so the field is $\mathbb{Q}$ and $\pi$ is not included.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating Integral $\int e^{x}(1-e^x)(1+e^x)^{10} dx$ I have this integral to evaluate: $$\int e^{x}(1-e^x)(1+e^x)^{10} dx$$
I figured to use u substitution for the part that is raised to the tenth power. After doing this the $e^x$ is canceled out.
I am not sure where to go from here however due to the $(1-e^x)$.
Is ... | let $x=\ln(u)$
$dx=du/u$
$I=\int e^{x}(1-e^x)(1+e^x)^{10} dx$ = $\int ((u(1-u)(1+u)^{10})/u)du$=$\int (1-u)(1+u)^{10}du$
You may want to take it from here...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
A question on Taylor Series and polynomial Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$.
For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial.
Is true that $f(x)$ is a polynomial?
I can show it is true if for every $c\in [a,b]$, there exists a neig... | As I confirmed here, if for every $c\in[a,b] $, the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial, then for every $c\in[a,b]$ there exists a $k_c$ such that $f^{(n)}(c)=0$ for $n>k_c$.
If $\max(k_c)$ is finite, we're done: $f(x)$ is a polynomial of degree $\le\max(k_c)$.
If $\max(k_c)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 2,
"answer_id": 0
} |
For which $n\in\mathbf{N}$ do we have $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$? Put $z_{n} = e^{2\pi i /n}$. I am searching for $n \in \mathbf{N}$ so that $\mathbf{Q}(z_{5},z_{7}) = \mathbf{Q}(z_{n})$.
I know that : $z_{5} = \cos(\frac{2\pi}{5})+i\sin(\frac{2\pi}{5}) $ and $z_{7} =\cos(\frac{2\pi}{7})+i\sin(\frac{... | That's one too many hints in the comments, but the OP still seems in doubt, so here is a proof that $\mathbb{Q}(\zeta_5,\zeta_7)=\mathbb{Q}(\zeta_{35})$, where $\zeta_n=e^{2\pi i/n}$ is a primitive $n$th root of unity.
First, let us show that $\mathbb{Q}(\zeta_5,\zeta_7)\subseteq\mathbb{Q}(\zeta_{35})$. Notice that
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Fourier transform (logarithm) question Can we think, at least in the sense of distribution, about the Fourier transform of $\log(s+x^{2})$? Here '$s$' is a real and positive parameter
However $\int_{-\infty}^{\infty}dx\log(s+x^{2})\exp(iux)$ is not well defined.
Can the Fourier transform of logarithm be evaluated ??
| Throughout, it is assumed that $s>0$ and $u \in \mathbb{R}$.
Define:
$$
\mathcal{I}_\nu(u) = \int_{-\infty}^\infty \left(s+x^2\right)^{-\nu} \mathrm{e}^{i u x} \,\,\mathrm{d} x = \int_{-\infty}^\infty \left(s+x^2\right)^{-\nu} \cos\left(u x\right) \,\,\mathrm{d} x
$$
The integral above converges for $\nu > 0$. W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Determinant of a special kind of block matrix I have a $2\times2$ block matrix $M$ defined as follows:
$$\begin{pmatrix}X+|X| & X-|X| \\ Y-|Y| & Y+|Y|\end{pmatrix}$$
where $X$ and $Y$ are $n\times n$ matrices and $|X|$ denotes the modulus of the entire matrix $X$ that essentially comprises modulus of individual elemen... | I shall assume that $X+|X|$ is invertible, although a similar solution exists under the assumption that $Y+|Y|$ is. I shall use $A,B,C,D$ to denote the respective block matrices in your problem to avoid giant equations. The decomposition $$M = \begin{pmatrix}A & B \\ C & D\end{pmatrix} = \begin{pmatrix}A & 0\\ C & I\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Software for Galois Theory Background: While studying Group Theory ( Open University M208 ) I had a lot of benefit from the Mathematica Add-on package AbstractAlgebra and later from the GAP software. I am currently self-studying Galois Theory ( using Ian Stewart's Galois Theory ).
Question: Is there a program that ca... | Canonical answers are Sage, Pari, Magma. The first two are open source, the last one costs money but has an online calculator. Type for example
P<x>:=PolynomialRing(Rationals());
GaloisGroup(x^6+3);
in the online calculator and hit submit. See the online manual on how to interpret the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Non-algebraically closed field in which every polynomial of degree $My problem is to build, for every prime $p$, a field of characteristic $p$ in which every polynomial of degree $\leq n$ ($n$ a fixed natural number) has a root, but such that the field is not algebraically closed.
If I'm not wrong (please correct me if... | Let $k$ be a field, $\bar k$ an algebraic closure of $k$. Fix $n>1$ natural. Consider the family $\mathcal{K}_n$ of fields $K$, $k\subset K\subset \bar k$ with the property: there exists a family of intermediate fields
$$k = K_0 \subset K_1 \subset \ldots K_s= K$$
so that $[K_{i+1}\colon K_i]< n$ for all $1\le i \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 2
} |
What are the interesting applications of hyperbolic geometry? I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous parallel lines postulate, putting an end to centuries of uns... | It is my understanding that the principle application of hyperbolic geometry is in physics. Specifically in special relativity. The Lorentz group of Lorentz transformations is a non-compact hyperbolic manifold. But it also shows up in general relativity and astrophysics as the space surrounding black holes is hyperboli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 16,
"answer_id": 13
} |
Understanding bounds on factorions I am trying to understand the upper bound on factorions (in base $10$). The Wikipedia page says:
"If $n$ is a natural number of $d$ digits that is a factorion, then
$10^{d − 1} \le n \le 9!d$. This fails to hold for $d \ge 8$ thus $n$
has at most $7$ digits, and the first upper b... | By definition, $n$ is the sum of the factorials of its digits. Since each digit of $n$ is at most 9, this can be at most $9!\cdot d$, where $d$ is the number of digits of $n$:
If $n$ is a factorion:
$$
n=d_1d_2\cdots d_d\quad\Rightarrow\quad n= d_1!+\,d_2!\,+\cdots+ \,d_d!\le \underbrace{9!+\,9!+\,\cdots+\, 9!}_{d -\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Trying to figure out how an approximation of a logarithmic equation works The physics books I'm reading gives $$\triangle\tau=\frac{2}{c}\left(1-\frac{2m}{r_{1}}\right)^{1/2}\left(r_{1}-r_{2}+2m\ln\frac{r_{1}-2m}{r_{2}-2m}\right).$$
We are then told $2m/r$
is small for $r_{2}<r<r_{1}$
which gives the approximatio... | It actually seems to me they use
$$\frac{2}{c}\left(1-\frac{2m}{r_{1}}\right)^{1/2}\approx\frac{2}{c}\left(1-\frac{m}{r_{1}}\right)$$
and
$$2m\ln\frac{r_{1}-2m}{r_{2}-2m}\approx 2m\ln\left(\frac{r_{1}}{r_{2}}\right) \; .$$
EDIT: Just realized the following:
$$2m\ln\frac{r_{1}-2m}{r_{2}-2m}\approx 2m\ln\left(\frac{r_{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Why is $0$ excluded in the definition of the projective space for a vector space?
For a vector space $V$, $P(V)$ is defined to be $(V \setminus \{0 \}) / \sim$, where two non-zero vectors $v_1, v_2$ in $V$ are equivalent if they differ
by a non-zero scalar $λ$, i.e., $v_1 = \lambda v_2$.
I wonder why vector $0$ is ... | Projective space is supposed to parametrize lines through the origin. A line is determined by two points, so a line through the origin is determined by any nonzero vector.
As Nate's explains, you can certainly include 0, but you will get a different space. Is there a reason to care about it?
One reason we care about th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
The Dimension of the Symmetric $k$-tensors I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the dimension of covariant $k$-tensors is $n^k$ but how can I eliminate non-symmetric t... | A basis for symmetric tensors, say $\text{Sym}_r(V)$ with $\{v_1,...,v_n\}$ a basis for $V$, is given by the symmetrizations of $\{v_{i_1}\otimes ... \otimes v_{i_r} \ | \ 1\leq i_1\leq...\leq i_r\leq n\}$. You must count the number of non-decreasing sequences (repetitions allowed) of length $r$ with entries in $[1,n]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
Altitudes of a triangle are concurrent (using co-ordinate geometry)
I need to prove that the altitudes of a triangle intersect at a given point using co-ordinate geometry.
I am thinking of assuming that point to be $(x,y)$ and then using slope equations to prove that the point exists and I can think of another way t... | I'll prove this proposition by Vector algebra, (not to solve OP's problem, but essentially for other users who might find it useful):
Let $\Delta ABC$ be a triangle whose altitudes $AD$, $BE$ intersect at $O$. In order to prove that the altitudes are concurrent, we'll have to prove that $CO$ is perpendicular to $AB$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Proving that $G/N$ is an abelian group
Let $G$ be the group of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$ where $ad \neq 0$ under matrix multiplication. Let $N=\left\{A \in G \; \colon \; A = \begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix} \right\}$ be a subset of the group $G$.... | One way is using first isomorphism theorem.
To do this you should find a group homomorphism such that $\operatorname{Ker} \varphi=N$.
Let us try $\varphi: G\to \mathbb R^*\times \mathbb R^*$ given by
$$\begin{pmatrix} a & b \\ 0 & d\end{pmatrix} \mapsto (a,d).$$
(By $\mathbb R^*$ I denote the group $\mathbb R^*=\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 1
} |
Generalize the equality $\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{n\cdot(n+1)}=\frac{n}{n+1}$ I'm reading a book The Art and Craft of Problem Solving. I've tried to conjecture a more general formula for sums where denominators have products of three terms. I've "got my hands dirty", but don't see any regular... | I have the following recipe in mind, see how far it helps (leave pointers in this regards as comments):
Let $a_1, a_2, a_3, \cdots, a_n, \cdots$ be the terms of an $A.P$. Let $d$ be the common difference of the given $A.P$. We are interested to find the sum for some $r \in \mathbb{N}$. $$\sum_{k=1}^n \dfrac{1}{a_k a_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
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