Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is the difference between a Definition and a Theorem? This may get into a discussion, but I have a homework problem and it tells me there is a difference between a definition and a theorem. I don't know how to differentiate the two in this question:
Consider the domain of all quadrilaterals. Let
A(x) = "x has fo... | A definition describes the exact meaning of something, whereas a theorem proves something.
| {
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"timestamp": "2023-03-29T00:00:00",
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Unsolved problems in number theory What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?
| I think nearly everything one does in number theory is understandable if you followed the story far enough. Even the most difficult to understand theorem usually makes sense in special cases if you understood the great theorems that came before it.
But yeah there is a wealth of immediately understandable theorems in nu... | {
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Limit of a Sum Involving Binomial Coefficients I would like to prove that $$\dfrac{\sqrt{n}}{{{2n \choose n}^2}} \cdot \sum_{j=0}^n {n \choose j}^4$$ converges to $\sqrt{\dfrac{2}{\pi}}$ as $n \to \infty$.
Evaluating the sum in Matematica for large values of $n$ suggests that $\sqrt{\dfrac{2}{\pi}}$ is indeed the corre... | It is enough to use the Central Limit Theorem. For large values of $n$, the binomial distribution converges to a normal distribution, and by Stirling's approximation
$$ \binom{2n}{n}\sim\frac{4^n}{\pi n}$$
holds, hence your limit is just the reciprocal of:
$$ \int_{-\infty}^{+\infty}e^{-2x^2}\,dx = \sqrt{\frac{\pi}{2}}... | {
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Strong Induction Proof: Fibonacci number even if and only if 3 divides index
The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$
Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green}{... | Since the period of $2$ in base $\phi^2$ is three places long = $0.10\phi\; 10\phi \dots$, and the fibonacci numbers represent the repunits of base $\phi^2$, then it follows that $2$ divides every third fibonacci number, in the same way that $37$ divides every third repunit in decimal (ie $111$, $111111$, $111111111$, ... | {
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Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$ I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title.
It is not hard to notice (given that one instinctively wants to complete the square in $P(2x - x^2)$ ) that $P(x) = 1-x$ is a solution; and the ... | Here’s an answer that has the same content as @Did‘s, but organized differently:
Let $f(x)=2x-x^2$ and also let $u(x)=1-x$, thought of as a transformation of the (real) line. Let $g=u^{-1}\circ f\circ u$, which you compute to be $g(x)=x^2$. Now let $Q=P\circ u$, so that $P=Q\circ u^{-1}$ and the requirement $P^2=P\circ... | {
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Dice probability problem collection I tag these questions as homework, as they are older exam questions and every year
Can you try to solve/explain how to solve with a method some of these?
If something is answered in an old post please post the link.
1. We toss 2 dice. Using probability-generating functions compute t... | 1:
The probability generating function for the total of a single die is
$$
G(z) = \frac16 (z+z^2+\cdots+z^6)
$$
The probability generating function of a sum is just the product of the two probability generating functions. So, the probability generating function for the sum of two dice will be
$$
G_2(z)=G(z)\cdot G(z)=\... | {
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Trying to understand an integral algebraically and conceptually $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$ It is defined that $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$ is the rate of population.
It is also given that the population at $t=0$ is given as 6.
Our goal is to find the time $t$ where the population hits $30$.
I want to ... | Well,
$$f(t)-f(0) = A \int_0^t dt' \frac{e^{t'}}{(0.02 A+e^{t'})^2}$$
To evaluate the integral, substitute $u=e^{t'}$ and get
$$f(t)-6 = A \int_1^{e^{t}} \frac{du}{(u+0.02 A)^2} = A \left [\frac{1}{1+0.02 A}-\frac{1}{e^t+0.02 A} \right ]$$
Now you gave us
$$\lim_{t \to \infty} f(t)=30$$
so that
$$30-6=24=\frac{A}{1+0.... | {
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Notation of exponential function? What is the difference between this notation of the exponential function
$$(1+\frac{1}{n})^n \rightarrow e \\ \mathbf{as} \\ n \rightarrow \infty$$
and this notation:
$$\lim_{n\rightarrow \infty} (1+\frac{x}{n})^n$$
Why is there a variable $x$ in the second equation, and a $1$... | Indeed the limits are not the same. The second one is $e^x$. Only if you choose $x=1$, you get the first limit. The second one is more general.
| {
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when does $f(a)^{f(b)}=f(a^b)$? First $\text{f}\left( 1 \right)=1$ beacause
$\text{f}\left( a \right)^{\text{f}\left( 1 \right)}=\text{f}\left( a \cdot 1 \right)$, and
$\log_{\text{f}\left( a \right)} \text{f}\left( a \right)^{\text{f}\left( 1 \right)} =\text{f}\left( 1 \right) \log_{\text{f}\left( a \right)} \text{f}... | Assume $a,b>0$ so that the RHS makes sense. Let $\mathcal{P}=(0,1),\;\mathcal{Q}=(1,\infty)$, noting that $f(1)=1$.
If $f(s)=1$ for some $s\in\mathcal{P}$, then for all $y,\;f(s^y)=1 \Rightarrow f(r)=1$ over $\mathcal{P}$. However, then for $y\in \mathcal{P}:\;f(x^y)=f(x) \Rightarrow f=\mathcal{C}$, and hence $f=1$ ov... | {
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Compositeness of $n^4+4^n$ My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number.
So we memorized this for future use in math competition. But I don't understand why is it?
| You can work $\bmod 5$:
As Jossie said, if $n$ is even, then both numbers are even. If $n$ is odd, set $n = 5k + r$;
If not, you can repeatedly use the fact that for $p$ a prime and $(a, p) = 1, a^{p - 1} = 1 \pmod p$ and so $a^p = a \pmod p$;
in this case, $(a, 5) = 1$ , then $a^{4n} = 1 \pmod 5$
$0 \leq r <5$ . Then ... | {
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Formula for summation of integer division series Consider '\' to be the integer division operator, i.e.,
$a$ \ $b = \lfloor a / b\rfloor$
Is there a formula to compute the following summation:
N\1 + N\2 + N\3 + ... + N\N
| This is not a closed form, but an alternate characterization of this sum is
$$
\sum_{k=1}^n\lfloor n/k\rfloor=\sum_{k=1}^nd(k)\tag{1}
$$
where $d(k)$ is the number of divisors of $k$. This can be seen by noticing that $\lfloor n/k\rfloor$ increases by $1$ when $k\mid n$:
$$
\begin{array}{c|cc}
\lfloor n/k\rfloor&1&2&3&... | {
"language": "en",
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Identity in Thom spaces. Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that
$$ T(\epsilon \oplus E) \simeq \Sigma T E\,\,\,\, ?$$
| Try and prove:
1) The Thom space of the trivial line bundle is $S^1$
2) For 'nice' spaces the one point compactification satisfies $(X \times Y)_+ = X_+ \wedge Y_+$
| {
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Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $? Is the equality below true?
$$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
| Since it wasn't specified, I'm assuming that the codomain of $f$ is something in which the symbols used make sense.
The equality $\displaystyle \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum_{i=1}^{n} f(i)$ holds by definition, if $\displaystyle \lim \limits_{n \to \infty} \sum \limits_{i=1}^{n} f(i)$ exi... | {
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Minimum number of colors I just read an old book today and it was stated that mathematicians are still unable to answer "What is the minimum number of colours needed to paint a map such that adjacent countries will not have the same colour" , so the mathematicians now know the answer? or is it still unknown !
| The 4-color theorem has been proven. In my "Graphs and Digraphs" book by Chartrand and Lesniak (4ed 2005), the story is told that in 1890 Heawood proved the 5-color theorem as a result of spotting an error in a flawed 4-color theorem by Kempe a decade earlier. After 1890 we had the 5-color theorem, and the 4-color conj... | {
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Constructing irreducible polynomials over the Polynomial Ring Well, my question is essentially:
Let $R$ be a Factorial Ring (UFD, basically) and let $p$ be a prime element in $R$.
Let $d$ be an integer larger than 2, and let
$f(t) = t^d + c_{d-1} t^{d-1} + ... + c_0$ be a polynomial belonging to $R[t]$. Let $n \ge 1$ b... | As $p$ is a unit in $K$, the irreducibility of $g(t)$ is equivalent to the irreduciblity of $h(t):=p^{nd}g(t)$. Write
$$h(t)=(p^nt)^{d}+c_{d-1}p^n (p^nt)^{d-1}+\cdots + c_1p^{(d-1)n}(p^nt)+p(1+p^{nd-1}c_0).$$
Now apply Eisenstein criterion to the polynomial
$$ H(T)=T^d+c_{d-1}p^n T^{d-1}+\cdots + c_1p^{(d-1)n}T+p(1+p^... | {
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How to compute the integral $ \int\frac{1}{x\sqrt{x^2 +3x}}dx$ Given a problem :
$$ \int\frac{1}{x\sqrt{x^2 +3x}}dx, $$
what is the best solution for this?
I am thinking about solving this problem by using :
$$ u = x+3;\qquad x = u-3; $$
So that we get :
$ \int\frac{1}{x\sqrt{x}\sqrt{x+3}} dx$, then $ \int\frac{1}{(u-3... | I have not done an integral this cumbersome in quite a while. The technique that immediately stands out to me is trigonometric substitution. While I have performed the integration correctly on paper, I would appreciate someone being on the hunt for typesetting errors. Buckle seatbelt...
We have
$$\int \frac{1}{x\sqrt{... | {
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How to Make an Introductory Class in Set Theory and Logic Exciting I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first ... | maybe start with some riddles from http://en.wikipedia.org/wiki/Charles_Lutwidge_Dodgson or http://en.wikipedia.org/wiki/Raymond_Smullyan both have made books full with logical riddles
| {
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Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution? One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is,
Every symmetric positive semi-definite matrix is a covariance marix of some mul... | The wikipedia article on covariance matrices answers that (the excerpt below is taken verbatim from that article):
From the identity just above, let $\mathbf{b}$ be a $(p \times 1)$ real-valued vector, then:
$$\operatorname{var}(\mathbf{b}^{\rm T}\mathbf{X}) = \mathbf{b}^{\rm T} \operatorname{var}(\mathbf{X}) \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/489632",
"timestamp": "2023-03-29T00:00:00",
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How to evaluate this limit: $\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x} = \frac12$? How do I evaluate the limit of $$\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x} = \frac{1}{2}$$? As $x$ approaches $0$, I know the answer is $\frac{1}{2}$, but I got this question wrong. I think you have to multiply by the conjugate of the numerator? C... | Method 1 (basic)
$$\frac{\sqrt{x+1} - 1}{x} \stackrel{\sqrt{x+1}^2 = |x+1|}{=} \frac{|x + 1| - 1}{x (\sqrt{x+1} + 1)} \stackrel{x+1 \geq 0, \text{ for well-def.}}{=} \frac{1}{\sqrt{x + 1} + 1} \to \frac{1}{1+1} = \frac{1}{2}$$
as $x\to 0$, because $x\mapsto \sqrt{x}$ is continuous (and so the limit can be "used as inpu... | {
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Proof: $a = x^2 - y^2, x - y > 1 \implies a$ is not a prime number I am having trouble doing a proof on the subject:
$a = x^2 - y^2, x - y > 1 \implies a$ is not a prime number.
$a, x, y \in Z$
The book has not yet introduced the theory of congruences, so it should be a 'fairly' simple proof i guess ?
Thanks for your t... | $x^2-y^2 = (x-y)(x+y)$
if $x-y$ is not zero and $x+y$ is not zero and $(x-y)>1$ so is $(x+y)$
then $x^2-y^2$ is divisible by both $(x-y)$ and $(x+y)$ so it is not prime number
my concern is he doesn't mention if both $x$ and $y$ are natural numbers if they are not then this claim would be not correct
| {
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Infinite amount of additions, finite sum? I suggest it's a popular question, so if it was asked already (I couldn't find it anyway), close this question instead of downvoting, thanks!
Let's check this addition: $\sum_{n=0}^{\infty}\frac{1}{2^n}=2$
It looks like $1 + \frac12 + \frac14 + \frac18 + \frac1{16} + ... $ for ... | An infinite number of summands is not really a problem, if the summands decrease quickly enough. For example
$$\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6}<\infty$$
But
$$\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} = \infty$$
So for the convergence of series (which is what you are puzzled about), there... | {
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Show that there are exactly two values in $\{0, 1, ..., N - 1\}$ satisfying $x^{2} \equiv a \pmod{N}$.
Fix a positive integer $N > 1$. We say that $a$ is a quadratic residue modulo $N$ if there exists $x$ such that $a \equiv x^{2} \pmod{N}$.
Let $N$ be an odd prime and $a$ be a non-zero quadratic residue modulo $N$. S... | Few remarks just to remind you some theorems about congruences when the modulus is prime:
1- If $p$ is a prime number and $P(x)$ is a polynomial of degree $k$, then $P(x) \equiv 0 \pmod{p}$ has at most $k$ number of solutions.
2- when $p$ is a prime number, $\mathbb{Z}/p\mathbb{Z}$ is an integral domain. That means whe... | {
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Linear Algebra- Proof of trace property How can I prove that $\text{tr}(A)=\text{tr}(A^T)$ using the fact that $\text{trace}(A)= a_{11} +a_{22} +... +a_{nn}$?
| If $A$ has coefficients $(A)_{ij}=a_{ij}$ then by definition $A^t$ has as coefficients $(A^t)_{ij}=a_{ji}$. But in particular when $i=j$; $$(A)_{ii}=a_{ii}=(A^t)_{ii}$$
For example, if $$A=\begin{pmatrix}1&6&2\\7&-2&-9\\-1&0&2\end{pmatrix}$$ then $$A^t=\begin{pmatrix}1&7&-1\\6&-2&0\\2&-9&2\end{pmatrix}$$
| {
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perimeter of square inscribed in the triagle In the figure given below, PQR is a triangle with sides PQ=10, PR=17, QR=21. ABCD is a square inscribed in the triangle. I want to find perimeter of square ABCD that is to find the length of side AB. But by using of basic high school geometry concepts, not by trigonometry.
I... | Area of Triangle $PQR = 84$ by using semiperimeter formula. From this we get height of Triangle as 8.
Say, side of square $= x$, hence $AB=BC=CD=AD=x$
Now take small triangle $APB$, in this height will be $8-x$. Area of Triangle = $\frac{x\cdot(8-x)}{2}$ -----(1)
Now, Area of Trapezoid $ABRQ = \frac{(21+x)\cdot x}{2... | {
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Lorentz force and Newton's second law Here's the question.
Consider a particle of mass $m$ that carries a charge $q$. Suppose that the particle is under the influence of both and electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ so that the particle's trajectory is described by the path $\mathbf{x}(t)$ for... | The problem is that not only the force produced by the electric field acts on the particle, if this were so then the particle would accelerate (it would be subject to a non-zero force, this is N. Second law), so there is a force countering that of the electric field ${\bf F} = -q{\bf E}$, only then can ${\bf v}$ be con... | {
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Solve for $z$ in $z^5=32$ This was the last question on my Year 11 Complex Numbers/Matrices Exam
Name all 5 possible values for $z$ in the equation $z^5=32$
I could only figure out $2$. How would I go about figuring this on paper?
| We have $z^5=2^5$ or $$\left(\frac{z}{2}\right)^5=1$$ then we have the solutions are $$z=2*\large{(e^\frac{2ki\pi}{5})}$$ where $k=0,1,2,3,4.$
| {
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Fundamental group of product of schemes Is the \'etale fundamental group of the product of two schemes $X_1$ and $X_2$ related to the \'etale fundamental groups of $X_1$ and $X_2$ individually?
| The following is proven SGA 1, X.1.7:
Let $k$ be an algebraically closed field, and $X,Y$ connected $k$-schemes such that $X$ is proper and $Y$ is locally noetherian. Let $x,y$ be geometric points of $X,Y$ with values in the same algebraically closed field. Then the canonical homomorphism of profinite groups $\pi_1^{... | {
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what happens to rank of matrices when singular matrix multiply by non-singular matrix?? I need to find the prove for any rectangular matrix $A$ and any non-singular matrices $P$, $Q$ of appropriate sizes,
rank($PAQ$)= rank($A$)
I know that when singular matrix multiply by non-singular the result will be singular matrix... | Hint: multiplying matrices amounts to compose linear maps. In other words, suppose $A$ is an $n\times m$ matrix, i.e. a linear map $f_A:\mathbb R^m\to\mathbb R^n$. Then, to perform the product $PAQ$ is the same as to give a factorization of $f_{PAQ}$ as $$\mathbb R^m\overset{f_Q}{\longrightarrow}\mathbb R^m\overset{f_A... | {
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The power set of the intersection of two sets equals the intersection of the power sets of each set It would be great if someone could verify this proof.
Theorem:
$\mathcal{P}(A \cap B) = \mathcal{P}(A) \cap \mathcal{P}(B)$
Proof:
First I prove that $\mathcal{P}(A \cap B) \subseteq \mathcal{P}(A) \cap \mathcal{P}(B)$. ... | In the fullest generality, consider a nonempty set $I$ and a family $A$ of sets indexed by $I$ (such that we will write $A_i$ for the set corresponding to index $i \in I$).
We then have:
$$X \in \bigcap_{i \in I} \mathscr{P}(A_i) \Longleftrightarrow (\forall i)(i \in I \Rightarrow X \subseteq A_i) \Longleftrightarrow X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find largest $k$ such that the diophantine equation $ax+by=k$ does not have nonnegative solution. It is given that $a$ and $b$ are coprime positive integers. My question is, what is the largest integer $k$ such that the diophantine equation $ax+by=k$ does not have any solution where $x$ and $y$ are nonnegative integers... | Let $a,b$ be positive integers with $\gcd(a,b)=1$. Let $S=\langle a,b \rangle=\{ax+by: x, y \in {\mathbb Z}_{\ge 0}$ denote the numerical semigroup generated by $a,b$. Then the gap set of $S$,
$$ G(S) = {\mathbb Z}_{\ge 0} \setminus S $$
is a finite set. The largest element in $G(S)$ is denoted by $F(S)$, and called th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that an uncountable set X is equivalent to X\Y where Y is a denumerable subset of X How do I prove this?
The problem contains the following "hint": Prove that X/Y contains a denumerable family of pairwise disjoint denumerable subsets.
I am not sure how that proves cardinal equivalence. My first instinct was to t... | Let $g\colon \omega\to Y$ be a bijection.
Find an injection $f\colon \omega\to X\setminus Y$ recursively as follows: Assume we have already found an injective map $f_n\colon n\to X\setminus Y$ (the case $n=0$ being trivial). Then
$$ x\mapsto\begin{cases}f_n(x)&\text{if }x<n,\\g(x-n)&\text{if }x\ge n\end{cases}$$
defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What should be added to $x^4 + 2x^3 - 2x^2 + x - 1$ to make it exactly divisible by $x^2 + 2x - 3$? I'm a ninth grader so please try to explain the answer in simple terms .
I cant fully understand the explanation in my book .
It just assumes that the expression that should be added has a degree of 1.
I apologize if t... | You can get to a quartic divisible by $x^2+2x-3$ by writing
$$\begin{align}
x^4+2x^3-2x^2+x-1+\text{something}&=(x^2+2x-3)(x^2+ax+b)\cr
&=x^4+(a+2)x^3+(2a+b-3)x^2+(2b-3a)x-3b\cr
\end{align}$$
which leads to
$$\text{something} = ax^3+(2a+b-1)x^2+(2b-3a-1)x+(1-3b)$$
for any coefficients $a$ and $b$ (of the quotient) that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/490744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 5
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Proving that all sets of size n have $2^n$ subsets I've tried to construct a proof for this using recursion. My knowledge of recursion and set theory in general is quite limited so I'd appreciate some feedback!
The claim in symbolic logic: $\forall n \in \mathbb N, \exists u \in U, S(u) \wedge |u| = n \Rightarrow \math... | you have proved it for sets of size 0
now you have to prove:
if a set of size n has $ 2^n $ different subsets
then a set of size n+1 has $ 2^{n+1} $ different subsets
and then you can use induction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/490946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If Q-Cauchy sequences must limit to a rational, how can they construct the reals? I'm currently in a graduate Math for Economists course, and we spent last week learning how to construct the reals from the rationals using Cauchy sequences and their equivalence classes. I know that I'm missing something, because from th... | I'm not sure I understand your question properly but here goes..
It is not true that all Cauchy sequences in $\mathbb{Q}$ converge to a limit in $\mathbb Q$. For example consider $a_{n} = F_{n+1} / F_n$, where $F_n$ is the $n$th Fibonacci number. It is well known that this converges to the golden ratio $\phi = \dfrac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Teaching the concept of a function. I am doing a class for at risk high school math students on the concept of a function. I have seen all the Internet lesson plans and different differentiated instruction plans. The idea of a function as a machine has always sat well with me, so I was thinking of playing off that. ... | Some everyday concepts could help. Such as
In a restaurant menu (f=food item, p=price of item):
Is f a function of p? Is p a function of f?
On back of a mailed envelop (s=street address, z=5-digit zip code):
Is s a function of z? Is z a function of s?
In a teacher's grade book (n=name of student who took a test, g= gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Calculus: Area Between Two Curves Find the area between the two curves $y = x^2 + ax$ and $y = 2x$ between $x=1$ and $x=2$, given that $a>2$
I found the antiderivative of $x^2+ax-2x$ and included the area between $x=1$ and $x=2$ which is $\dfrac{3a-4}{6}-\dfrac{6a-4}{3}$. I don't understand what $(a>2)$ means in the pr... | Hint: the area between two curves $f(x)$ and $g(x)$ can be found by the formula
$$A=\int_c^d{[f(x)-g(x)]dx}$$
It seems the reason it's asking you for $a>2$ is it will cancel the $2x$ if $a=2$. Perhaps you should plot the graphs for different values of $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/491224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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increasing union of finitely generated submodules of M need not be finitely generated Show by an example that an increasing union of finitely generated submodules of M need not be finitely generated.
I was thinking about $R[x_1,x_2,x_3,....]$. Then if we consider the ideal $<x_1>$ , does it form a submodule?
$f(x_1,x_2... | You don't quite say what $M$ is. I'm assuming it's a module over a commutative ring $R$. IF $M$ is required to be finitely generated, then in order to get any non-(finitely generated) submodule of $M$ we need $R$ to be non-Noetherian, and your example seems as simple as any.
On the other hand, you don't say that you ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $ (n+1)(n+2)\cdots(2n)$ is divisible by $ 2^n$, but not by $ 2^{n+1}$ darij grinberg's note:
I. Niven, H. S. Zuckerman, H. L. Montgomery, An introduction to the theory of numbers, 5th edition 1991, §1.4, problem 4 (b)
Let $n$ be a nonnegative integer. Show that $ (n+1)(n+2)\cdots(2n)$ is divisible by $ 2^n... | You can do it by induction. The base case is easy. For the induction step, suppose the result is true for $n=k$. So we assume that we know that
$$(k+1)(k+2)\cdots(2k)\tag{1}$$
is divisible by $2^k$ but not by $2^{k+1}$.
Now the product when $n=k+1$ is
$$(k+2)(k+3)\cdots(2k)(2k+1)(2k+2).\tag{2}$$
To get from the produc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$".
| For such abstract questions, it is important that you stick to the definitions of the involved notions.
By definition, $A$ is a subset of $B$ if every element contained in $A$ is also contained in $B$.
Now we look at $A = \{\emptyset\}$ and $B = \{\{\emptyset\}\}$.
$A$ has exactly one element, namely $\emptyset$.
This ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 5,
"answer_id": 3
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Inverse Function of $ f(x) = \frac{1-x^3}{x^3} $ I need to show that these function has a continuous inverse function and find this inverse function.
$$ f(x) = \frac{1-x^3}{x^3} $$
Defined on $ (1,\infty) $
I think I need to check for bijectivity. Don't know how.
I tried to solve the function to $x$ then. But somehow I... | The original function $f$ takes an $x$ and spits out $y=f(x)$. The inverse is a function $g$ that takes any $y$ from the range of $f$ and spits out $x=g(y)$ such that $g(f(x))=x$ for any $x$ in the domain. So you want to solve for $y$ - you already almost got the correct answer. Once you get $x=g(y)$ for some function ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find an example of non-locally finite collection I got stuck on this problem and can't find any hint to solve this. Hope some one can help me. I really appreciate.
Give an example of a collection of sets $A$ that is not locally finite, such that the collection $B = \{\bar{X} | X \in A\}$ is locally finite.
Note: Ev... | Let Q be the rationals in R, and A be the collection of sets of the form {a+√p, a ∈ Q}, p ∈ N.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Taylor series of fraction addition of common function What is the typical trick for finding the taylor series of a common function that is in the denominator when adding a constant.
eg:
$$f(x)=\frac{1}{e^x-c}$$
I know you can write $f(x)=\frac{e^x}{e^{2x}-ce^x}$ and then invoke
$$Taylor(\frac{f(x)}{g(x)})=\frac{Taylo... | If the problem for you is that the function is in the denominator, why not just take it as the numerator, i.e. re-write
$f(x) = \frac{1}{e^{x} - 2}$
as
$f(x) = (e^{x} - 2)^{-1}$
and proceed as
$f'(x) = -(e^{x} -2)^{-2}\cdot e^{x} \\
f''(x) = 2(e^{x} -2)^{-3}\cdot e^{2x} - (e^{x} -2)^{-2}\cdot e^{x} \\
f'''(x) = -6(e^{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\log_{2}(7)$ is irrational
Prove that $\log_{2}(7)$ is an irrational number.
My Attempt:
Assume that $\log_{2}(7)$ is a rational number. Then we can express $\log_{2}(7)$ as $\frac{p}{q}$, where $p,q\in \mathbb{Z}$ and $q\neq 0$. This implies that $7^q = 2^p$, where either $p,q>0$ or $p,q<0$.
My question ... | Consider the case $\log_27=p/q$ where $p, q<0$. You'll still have
$$
7^q=2^p
$$
and thus you'll have
$$
\frac{1}{7^{-q}}=\frac{1}{2^{-p}}
$$
which is the same as saying
$$
7^{-q}=2^{-p}
$$
and since $-q$ and $-p$ are both positive, you're back in the case where both exponents are positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/491884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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$e^{\ln(-2)} = -2$ but $\ln(-2) = \ln 2+i\pi$. How does this work? I'm messing with exponential growth functions. I noticed that I can write
$y(t)=y(0)\alpha^t$ as
$y(t)=y(0)e^{\ln(\alpha)t}$
(and then I can go ahead and replace $\ln(\alpha)$ with $\lambda$.)
But how do I handle when the $\alpha$ in $\ln(\alpha)$ is... | With Euler's identity, $e^{i\theta}=\cos \theta+\sin\theta$. This is just a nice identity, proven to work for complex numbers.
Any complex number can be written then as :$$z=re^{i\theta}$$
where $r$ is its modulus.
Now take $\ln$ of that:
$$\ln z=\ln (r e^{i\theta})=\ln r+\ln e^{i\theta}=\ln r+i\theta $$
Now, because $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/491952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Difference between Deformation Retraction and Retraction I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as follows:
A deformation retraction of a space $X$ onto a subspac... | The difference between a retraction and a deformation retraction does have to do with the "notion of time" as you suggest.
Here's a strong difference between the two:
1) For any $x_0 \in X$, $\{x_0\} \subset X$ has a retract. Choose $r : X \to \{x_0\}$ to be the unique map to the one-point set. Then, certainly, $r(x_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 2,
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what is the proper contour for $\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz$ what is the proper contour for $$\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz:2\leq n$$
I tried with rectangle contour but the problem which I faced how to make the contour contain all branches point because $1+e^{nz}=0$ for every $z=\frac{... | Set $w=nz$, then your integral is equal to $\displaystyle\frac{1}{n}\int_{\mathbb R}\frac{e^{w/n}}{1+e^w}\,dw=\frac{I}{n}$. Now, for $M>0$ big, consider the following rectangle contour: $$t_1\in[-M,M],\\ M+t_2i,\,t_2\in[0,2\pi],\\t_3+2\pi i,\,t_3\in[M,-M],\\-M+t_4i,t_4\in[2\pi,0].$$ On the second line, $$|I|\leq\int_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Finding the sum of two numbers given the difference of LCM and GCD.
l-h=57
Let, the two numbers be a and b.
Then,
ab=lh
How to proceed further? Please give me just hints? They have asked the minimum value of the sum so, I think that I have to take different possibilities.
| You want to minimize $lh$ subject to the condition $l-h=57$. So $l=57+h$, and the formula you want to minimize is $lh=(57+h)h$. Since we are looking at natural numbers, this will be the smallest possible when $h=1$, and then $lh=(57+1)\times1=58$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/492185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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what's the maximum volume of a cone with a 10 cm generator i have no idea how to solve this i tried writing down the volume expression depending on the angle $ v= \frac{1}{3}\pi r^2h $ knowing that $ r= 10\sin\theta $ and $h= 10\cos\theta $ then $ V= \frac{1000}{3}\pi(\sin^2\theta \cos\theta) $ and that way the volume ... | The angle seems fine to me as a variable, but since it the formula contains $r^2$ the function should be $v(\theta)=\cfrac {1000\pi}3\sin^2 \theta \cos \theta$. Then you need to take care over the differentiation.
Now $\sin^2 \theta \cos \theta=(1-\cos^2\theta) \cos \theta=\cos \theta -\cos^3\theta$
So $\cfrac {dv}{d \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What do the curvature and torsion measure? Consider a smooth surface for simplicity. What does its curvature measure? What does its Gaussian/Riemannian curvature measure? What does its torsion measure?
What does the Ricci curvature measure?
| If you search the questions on MSE and/or MO, I think you will,find some pretty good insights into these topics; if I recall correctly, these topics have come up more than once; many times, in fact. For example, this one might be a good place to start:
Geometric interpretation of connection forms, torsion forms, curva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use Laplace transform to solve the following initial–value problems. Use Laplace transform to solve the following initial–value problem.
$y′′′′ + 2y′′ + y = 0, y(0) = 1, y′(0) = −1, y′′(0) = 0, y′′′(0) = 2$
Answer
$s^4 L(s) - s^3y(0) -s^2 y'(0) - s y''(0) - y'''(0) +2[s^2L(s)-sy(0)-y'(0)] +L(s) \\\\$
I get the partial ... | I guess it is the last fraction, $\frac{1+s}{(s^2+1)^2}$, that causes you problems.
For the (one sided) Laplace transform we have the transform pairs
$$\sin{t}\stackrel{\mathcal{L}}{\longmapsto}\frac{1}{s^2+1},$$
$$tf(t)\stackrel{\mathcal{L}}{\longmapsto}-F'(s),$$
and
$$f'(t)\stackrel{\mathcal{L}}{\longmapsto}sF(s).$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Group theory with analysis I've studied group theory upto isomorphism.
Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems.
I too have done metric spaces and real analysis properly. Can you recommend any good topic to be presented in a short discussion. A good proof on an intere... | If you have covered elementary point set topology a possibility might be to discuss basics of topological groups. For example, show how having a (continuous) group structure on a topological space simplifies the coarsest separation axioms ($T_0$ implies Hausdorff). Not a cool theorem, but may be the first encounter wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Limit $\lim_{n\to\infty}\frac{n^3+2}{n^2+3}$ Find the limit and prove your answer is correct
$$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}$$
By divide everything by $n^3$ I got
$$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}=\frac10 $$
which is undefined. So I conclude that there is no limit for this sequence. However, I don't know ... | Hint:
$$\frac{n^3+2}{n^2+3}=\frac{n^3+3n-3n +2 }{n^2+3}= \frac{n(n^2+3)}{n^2+3} +\frac{2-3n}{n^2+3}=n +\frac{2-3n}{n^2+3} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/492633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Why $f(x) = \sqrt{x}$ is a function? Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a rule of corre-
spondence that assigns to each element in set A exactly one element i... | I will assume that in this portion of your textbook it is assumed that $x \in \mathbb{R}$, and with that condition $f(x)=\sqrt{x}$ is certainly a function. Specifically $f:[0,\infty) \rightarrow [0,\infty)$. It meets the formal definition of a function (not one to many).
Your confusion is due to an inappropriate extrap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 1
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Is this an analog of the mean value theorem for vector-valued functions? I was reading through various proofs of the multi-dimensional analogues of the mean value theorem. Suppose we have a $C^1$ function $f: \mathbb{R}^n \supseteq U \to \mathbb{R}^m$. I had thought there was a theorem that
*
*Given a ball $B\subset... | There is nothing like your proposed theorem (1). In one dimension, the MVT says that if you cover $d$ miles in $t$ hours, you must at some time have been traveling at $\frac dt$ miles per hour. But in more than one dimension this is false, because you can drive all over town and return home, and so have a net velocity... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Block Decomposition of a linear map on $\Lambda^2TM$ I'm trying a exercise from Peter Petersen's book, and I did the following:
Let $*$ be the Hodge star operator, I know that $\Lambda^2TM$ decompose into $+1$ and $-1$ eigenspaces $\Lambda^+TM$ and $\Lambda^-TM$ for $*$.
I know that, if $e_1,e_2,e_3,e_4$ is an oriente... | Presumably $M$ is a $4$-manifold here, since you are working with $\Lambda^2 TM$. If $M$ is $2n$-dimensional, the Hodge star gives rise to a decomposition $\Lambda^n TM = \Lambda^+ TM \oplus \Lambda^- TM$.
First, note that any $\omega \in \Lambda^2 TM$ can be written as
$$\omega = \tfrac{1}{2}(\omega + \ast \omega) + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Principal Component Analysis in Face Recognition - number of Eigenvalues? I'm at the beginning of learning about the PCA as it is applied in the field of face recognition (Eigenface algorithm) and I came about the following question:
"You're using a training set of 80 images (150x150 pixels). After visualizing the Eige... | Yes. That's correct. Although 40% of eigenvectors is not very meaningful. It's better to talk about how much variance is captured, i.e., the proportion of the sum of eigenvalues.
See Eckart–Young theorem to understand how eigenvalues related to reconstruction error.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/492883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How is a simple birth process is time-homogeneous? Why is it that a simple birth process is time-homogeneous?
The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this population size clearly depend on the time (since the population grows over ti... | The notion of time-homogeneity of a stochastic process $(X_t)$ can refer to the invariance of its distribution, that is, to the fact that the distribution of $X_t$ does not depend on $t$ and more generally to the fact that, for every set $T$ of nonnegative time indices, the distribution of $X_{t+T}=(X_{t+s})_{s\in T}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/492980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Placing Ts on the $x$-axis
A "T" consists of two perpendicular intervals $\{c\}\times[0,a]$ and $[b,d]\times \{a\}$ (with $b<c<d$) on the plane. We say that the T is placed on point $c$. Is it possible to place non-intersecting T's on all real numbers on the $[0,1]$ interval of the $x$-axis?
I believe the answer shou... | This may not actually address your question, but it may help you think about it in a different way:
For any set of finitely many $x_i \in [0,1]$, we know you can construct a set of open intervals $I_i$ such that $x_i \in I_i$ and $x_i \notin I_j \;\forall j < i$. This is (should be) fairly easy to understand why; a rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$... | HINT:
Putting $x=r\cos\theta,y=r\sin\theta$
$$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1,$$
$$r^2=\frac{a^2b^2}{b^2\cos^2\theta+a^2\sin^2\theta}=b^2\frac{\sec^2\theta}{\frac{b^2}{a^2}+\tan^2\theta}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
How to solve this ODE tT' +2T = 0.
I'm solving for T here, and I think that T' is actually T'(t), but I'm not sure because the above is how my instructor emailed it to us, but then I don't know why the independent variable is multiplying T' though.
I used the characteristic and got that T= ce^((-2/t)*t) which I'm very... | This is a separable DE. Rewrite it as $\dfrac{T'}{T}=-\dfrac{2}{t}$, and integrate.
We get
$$\ln |T|=-2\ln|t|+C.$$
Take the exponential of both sides. We get $|T|=\dfrac{K}{t^2}$. If you know an "initial" value (the value of $T$ at some given $t_0$, you can find $K$.
Remark: The $T$, as you expected, is supposed to be ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $B$ is nilpotent.
Let $A$ and $B$ be complex matrices with $AB^2-B^2A=B$. Prove that $B$ is nilpotent.
By the way: This problem is from American Mathematical Monthly, Problem 10339,and this solution post 1996 American Mathematical Monthly, page 907.
My question: This problem has other nice solutions... | Multiply both sides by $B^{n-1}$ on the right so the given condition is that $B^n = AB^{n+1} - B^2 A B^{n-1}.$ The trace of a product is invariant under cyclic permutations of the factors, so the trace of the right hand side is zero. The trace of $B^n$ is zero for all $n\geq 1$ so $B$ is nilpotent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Ideals generated by roots of polynomials Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in $\mathscr{O}_K$. I want to prove that this ideal is actually the entire ring $\maths... | Hint/steps for one route that caught my eye. There may be a shorter one out there. Call that ideal $I$. Observe that $\alpha \in {\cal O}_K$.
*
*Show that $\alpha+1\in I$.
*Show that $\alpha^2-1\in I$.
*Show that $1\in I$.
Remark: this is my second suggestion - the first suggestion had 5 steps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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A Modular Diophantine Equation $a = (N \bmod c)\bmod d$
$b = (N \bmod d)\bmod c$
That is $a$ and $c$ is remainder of $N$ when divided by $c$ and $d$ in different order.
What can we say about $N$ if $a,b,c,d$ are known and $N < cd$?
| Without loss of generality, assume that $c<d$. Moreover assume that $x \bmod n$ denotes the unique representative in $[0,n-1]$.
The first remark is that $a=N \bmod c$, because (since $c<d$) this value is already reduced modulo $d$. For the other reduction, we only know that $N \bmod d=b+tc$, where $$0\leq t \leq \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Factorizing a difference of two $n$-th powers How can be proved that
$$a^n-b^n=\displaystyle\prod_{j=1}^{n}(a-\omega^j b)$$
where $\omega=e^{\frac{2\pi i}{n}}$ is a primitive $n$-th root of $1$?
| Hint: $$a^n-b^n = b^n((a/b)^n-1).$$
Now you can represent a polynomial as a product of terms $(x-x_i)$ where $x_i$ are its roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the field F is a vector space over itself. How can I prove that a field F is a vector space over itself?
Intuitively, it seems obvious because the definition of a field is
nearly the same as that of a vector space, just with scalers instead
of vectors.
Here's what I'm thinking:
Let V = { (a) | a in F } descr... | If in the axioms of vector spaces you assume that the vector space is the same as the field, and you identify vector addition and scalar multiplication respectively with addition and multiplication in the field, you will see that all axioms are contained in the set of axioms of a field. There is nothing more to check t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Need help with $\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx$ Let
$$I=\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx,\tag1$$
where
$$S(x)=-\frac12+\int_0^x\sin\frac{\pi t^2}2dt,\tag2$$
$$C(s)=-\frac12+\... | Notice the integrand can be rewritten as:
$$\pi x + \frac{S C' - C S'}{S^2 + C^2}
= \pi x + \frac{1}{2i}\left(\frac{C' - iS'}{C - iS} - \frac{C' + iS'}{C + iS}\right)
= \pi x + \frac{1}{2i} \log\left(\frac{C-iS}{C+iS}\right)'
$$
The integral is equal to
$$\lim_{x\to\infty} \left[\frac{\pi t^2}{2} + \frac{1}{2i}\log\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
"answer_id": 0
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Will the rules of calculus stay the same when a real-valued function is defined over infinite number of variables? So the question would be:
*
*Can we ever talk about a real-valued function that is defined over infinite number of variables?
*Will the rules of calculus remain the same for such functions described in... | There is a theory of "calculus on Banach spaces", which comes close to what you are asking for. See for example Wikipedia:Fréchet derivatives for some information about Fréchet and Gateaux derivatives for functions $f : U \to V$ where $U$ and $V$ are Banach spaces.
Some things remain the same in the infinite dimensiona... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there a triangle like this? This is my question that I posted at https://mathematica.stackexchange.com/questions/32338/is-there-a-triangle-like-this "I want to find the numbers $a$, $b$, $c$, $d$ of the function $y = \dfrac{a x + b}{c x + d}$ so that the triangle $ABC$ with three points $A$, $B$, $C$ have integer ... | My student found a function $y = \dfrac{93 x + 6}{x+2}$. Then, $A(-6,138)$, $B(8,75)$, $C(-62,96)$ and centroid $G(20,103)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How can I explain this formally? Let $f: \Bbb R \to \Bbb R$ and $x \in \Bbb R$. Suppose that $\lim_{y \to x+} f(y)$ exists as a real number. If there is an $r \in \Bbb R$ such that
$$\lim_{y \to x+} f(y) > r$$
then there exists $n \in \Bbb N$ (dependind on $x$ and $r$) such that
$$f(z)>r$$
whenever $x<z<x+1/n$.
I thi... | If $L$ is the limit, let $\epsilon = L - r$. Then by definition of a limit, there exists a $\delta > 0$ such that whenever $z \in (x, x + \delta$, we have $|f(z) - L| < \epsilon$. In particular, it follows that $f(z) > r$ for all such $z$.
Now just choose $n$ sufficiently large that $\frac{1}{n} < \delta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/493898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Matrix Multiplication Problem I'm working on the following problem and I can't seem to come up with the right answer.
$$ \text{Let}: A^{-1} =
\begin{bmatrix}
1 & 2 & 1 \\
0 & 3 & 1 \\
4 & 1 & 2 \\
\end{bmatrix}
$$
Find a matrix such that:
$$ ACA =
\begin{bmatrix}
2 & 1... | Let $$ B =
\begin{bmatrix}
2 & 1 & 3 \\
-1 & 2 & 2 \\
2 & 1 & 4 \\
\end{bmatrix}
$$
$ACA = B$ if and only if $A^{-1}ACA = A^{-1}B$ if and only if $A^{-1}ACAA^{-1} = A^{-1}BA^{-1}$. Now, multiplication between matrices is not commutative but it is associative! Hence you have:
$$A^{-1}ACA... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/493964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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The order of permutation groups and alternating groups The question was:
True or False: $\forall{n}\in{\mathbb{N}}$ the group $S_n$ and $A_n$ have different sizes.
My answer is False. That is since both $A_1 =(\text{id})$ and $S_1 =(\text{id})$.
Can any one confirm my answer please? Thank you very much.
| Of course $S_n$ in general is much larger than $A_n$. $A_n$ is the group of permutations with even parity only, while $S_n$ is the group of all permutations of $\{1,\cdots,n\}$. Not all permutations are of even parity, e.g., consider
$$(1,2)\circ(1,3)\circ(1,5)$$
the product of three transpositions that give rise to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/494047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Drawing 3 cards of same suit and 2 of a different suit What is the probability of having 3 cards of the same suit and $2$ cards of the same suit (but a different suit than the first three) in a $5$ card hand from a standard $52$ card deck?
What is the difference between solving this problem using approach (1) and (2).... | Select the suit with 2 cards (1 out of 4), select their values (2 out of 13), then select the suit with 3 cards (1 out of the remaining 3) and then the 3 values (3 out of 13) for a total of
$$
\binom{4}{1} \cdot \binom{13}{2} \cdot \binom{3}{1} \cdot \binom{13}{3} = 267696
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Binomial Expansion involving two terms? How would you find the 4th term in the expansion $(1+2x)^2 (1-6x)^{15}$?
Is there a simple way to do so?
Any help would be appreciated
| $$(1+2x)^2 (1-6x)^{15}=\sum_{i=0}^{2}\binom{2}{i}(2x)^i\sum_{j=0}^{15}\binom{15}{j}(-6x)^j=$$
using $i+j=3$ for fourth term we get
$$\sum_{i+j=3}\binom{2}{i}2^i\binom{15}{j}(-6)^jx^3=$$
$$=\left(\binom{15}{3}(-6)^3+4\binom{15}{2}(-6)^2+4\binom{15}{1}(-6)^1\right)x^3=$$
$$=(-98280+15120-360)x^3=-83520x^3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Sum of Fourier Series I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by
$$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$
in the interval $]-\pi,\pi[$ and give the sum of the series for $x=p\pi,p\in\mathbb{Z}$.
What I know:
If $f(x)=\sum_{n=-\infty}^{\infty}\a... | Here is how you advance
$$ \alpha_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx= \frac{1}{2\pi}\int_{-\pi}^{0}(0)e^{-inx}dx + \frac{1}{2\pi}\int_{0}^{\pi} \cos(x)e^{-inx}dx$$
$$ \implies \alpha_n = {\frac {in \left( {{\rm e}^{i\pi \,n}}+1 \right) }{2\pi({n}^{2}-1)}}.$$
The case $n=1$ can be obtained from the above ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $S$ is a subring of $R$ then $\operatorname{char}(S)\leq\operatorname{char}(R)$, when $S$ and $R$ have finite characteristic $\newcommand{\ch}{\operatorname{char}}$
a) Let $S$ be a subring of a ring $R$ and let $\ch (S)$ and $\ch (R)$ be finite. Then $\ch (S) \le \ch (R)$. Could someone give me a hint.
b) Prove t... | $\newcommand{\ch}{\operatorname{ch}}$Let us start with the second part. If a ring has a unity, then its characteristic is the additive period of the unity. So if $\mathbf{r}$ and its subring $\mathbf{s}$ have the same unity, then they have the same characteristic.
In a ring that may or may not have a unity, the charact... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/494371",
"timestamp": "2023-03-29T00:00:00",
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Result $a \equiv b \mod m$ or $c \equiv d \mod m$ is false $\Rightarrow ac \equiv cd \mod m$ is false? Root extraction $\mod m$ I know that $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$ implies $ac \equiv bd \pmod{m}$.
However, can one show that if $a \equiv b \pmod{m}$ is false, then:
$ac \equiv bd \pmod{m}$ is fals... | Isn't this as simple as noting that $1\cdot k \equiv k\cdot 1\pmod m$ for all $k$? So if we take $k$ such that $k\not\equiv 1\pmod m$ then we have a counter-example to your claim.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Is there a name for this particular linear fractional transformation? Is there a conventional name for this function?
$$
\begin{align}
g(t) & = \frac{1+it}{1-it} \\[15pt]
& = \frac{1-t^2}{1+t^2} + i\frac{2t}{1+t^2}.
\end{align}
$$
This function comes up from time to time.
It enjoys these properties:
*
*(A restrictio... | It is a rational equivalence between the line and the circle. That's what an algebraic geometer would say in any case. More precisely, it is a birational map.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Laplace transform of product of $\sinh(t)$ and $\cos(t)$ If I have a function $f(t)=\sinh(t)\cos(t)$ how would I go about finding the Laplace transform? I tried putting it into the integral defining Laplace transformation:
$$
F(s)= \int_0^\infty \mathrm{e}^{-st}\sinh(t)\cos(t)\,\mathrm{d}t
$$
But this integral looks ve... | = L [ { (( e^(-t) - e^(t) ))/2 }.{ cost } ]
Taking Constant outside and separating Laplace to different terms,
= (1/2).{ L[ e^(-t).cost ] - L[ e^(t).cost ] }
Separating solving,
L[ e^(-t).cost ],
first solve ----> L[ cost ] = s/ (s^(2) + 1)
Therefore for L[ e^(-t).cost ],
Using shifting property,
L[ e^(-t).cost ] = (s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/494580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $\int \limits_0^{2a}f(x)dx=\int_0^a[f(x)+f(2a-x)]dx$ How do I attempt this? What is the aim of the proof?
I've been trying silly things with the LHS and the RHS but cannot produce anything of use.
Can someone offer a very slight hint on proceeding?
Thanks
I did this so far:
$\displaystyle \int \limits_0^{2... | $$\int_0^a f(2a-x)dx = \int_a^{2a} f(y)dy.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expectation conditioned on a sub sigma field Let $X$ and $Y$ be two integrable random variables defined on the same probability space $(\Omega,\mathcal F,\mathbf P)$
Let $\mathcal A$ be a sub-sigma-field such that X is $\mathcal A$-measureable.
*
*Show that $E(Y|A)=X$ implies $E(Y|X)=X$
*Show by counter-example tha... | Because $X$ is $A-$measurable, $\sigma(X)\subseteq A$ therefore:
$$
\mathbb{E}(Y|X)=\mathbb{E}[\mathbb{E}(Y|A)|X]=\mathbb{E}[X|X]=X.
$$
For the next part, suppose that $\sigma(X)\subset A$ therefor there is a set $G\notin \sigma(X)$ and $G\in A$ and we have $\mathbb{E}[\mathbb{1}_G|X]=\mathbb{P}(G|X)=p>0$. Consider $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/494727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the Determinant of a $3\times 3$ matrix. Show that:
$$
\begin{vmatrix}
x & a & b \\
x^2 & a^2 & b^2 \\
a+b & x+b & x+a \\
\end{vmatrix} = (b - a)(x-a)(x-b)(x+a+b)
$$
I tried expanding the whole matrix out, but it looks like a total mess. Does anyone have an idea how this could be simplified?
| you can expand the matrix determinant and look for "shortcuts" , my solution :
$$
\begin{vmatrix}
x & a & b \\
x^2 & a^2 & b^2 \\
a+b & x+b & x+a \\
\end{vmatrix} =
x \begin{vmatrix}
x & a\\
x^2 & a^2\\
\end{vmatrix}-a\begin{vmatrix}
x^2 & b^2\\
a+b & x+a\\
\end{vmatrix}+b\begin{vmatrix}
x^2 & a^2\\
a+b & x+b\\
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/494820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Why can't I take the square root of both sides when finding the roots of a quadratic equation? Is $x^2 - x - 12 = 0$ equivalent to $x = \sqrt{x + 12}$?
I started with
$x^2 - x - 12 = 0$
and made the following changes:
$x^2 - x - 12 = 0$
$x^2 = x + 12$
$x = \sqrt{x + 12}$
From here I can eyeball it and see that x = 4 an... | It's not equivalent for two reasons:
*
*We don't know that $x + 12$ is non-negative, so it might not be valid to take square roots
*$\sqrt{x + 12}$ is always non-negative by definition, provided it's defined. So $x = -3$ is not actually a solution to this new equation.
| {
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"url": "https://math.stackexchange.com/questions/494896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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When multiplying 2 positive integers, why when we round the larger number up and the smaller number down, the product will be lower and vice versa. I experimented with 2 digits multiplication and found out that when we rounded the larger number up and the smaller number down, the product will be lower. When we rounded ... | Consider $a+b=2k$ and assume WLOG $a\ge b$
Then $a=k+x$, $b=k-x$ for any $x\ge0$
Finally, $ab=k^2-x^2$, and since
$$x^2\ge0\implies k^2-x^2\ge k^2$$
And equality is achieved when $x=0$, that is, $a=b$. Finally, we note that we note that "rounding $a$ down and $b$ up" is precisely minimizing $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/494975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Why does the Method of Successive Approximations for a Differential Equation work? Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous outline of proof as to why this shoul... | Cleaning stuff up, here are your givens:
$$i\frac{\partial{|\psi(t)\rangle}}{\partial t}= {H(t)}{|\psi(t)\rangle},\quad{|\psi(t_0)\rangle}=|\psi\rangle$$
Now, I claim that this series is a solution:
$$|\psi(t)\rangle=\left(1+(-i)\int_{t_0}^tdt_1 {H(t_1)}+(-i)^2\int_{t_0}^tdt_1\int_{t_0}^{t_1}dt_2{H(t_1)H(t_2)}+...\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How do the terms “countable” and “uncountable” not assume the continuum hypothesis?
*
*Every countable set has cardinality $\aleph_0$.
*The next larger cardinality is $\aleph_1$.
*Every uncountable set has cardinality $\geq 2^{\aleph_0}$
Now, an infinite set can only be countable or uncountable, so how does this ... | When we say that a set is finite if there is a bijection between the set and a proper initial segment of the natural numbers. We say that a set is infinite if it is not finite.
Similarly, we say that a set is countable if there is an injection from that set into the set of natural numbers. We say that it is uncountabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
homomorphism from $S_3$ to $\mathbb Z/3\mathbb Z$
TRUE/FALSE TEST:
*
*There is a non-trivial group homomorphism from $S_3$ to $\mathbb Z/3\mathbb Z.$
My Attempt:
True: Choose $a,b\in S_3$ such that $|a|=3,|b|=2.$
Then $S_3=\{1,a,a^2,b,ba,ba^2\}.$
Define $f:S_3\to\mathbb Z/3\mathbb Z:b^ia^j\mapsto j+3\mathbb Z$
The... | HINTS:
*
*The image of a homomorphism is a subgroup of co-domain. Does $\mathbb{Z}_3$ has any subgroups except $\{\bar{0}\}$ and itself?
*Since $|S_3|=3!=6$ and $|\mathbb{Z}_3|=3$ then any function $\varphi: S_3 \to \mathbb{Z}_3$ will be not one-to-one. So, the kernel must be non-trivial.
*If $\varphi: S_3 \to \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Minimal Counterexample for False Prime-Generating Quadratic Polynomials (Chartrand Ex 7.66)
Factor the quadratic: $n^2 \pm n + 41 = n(n \pm 1) + 41 = n\left[(n \pm 1) + \cfrac{41}{n}\right]$.
So if we find at least one $n$ such that $\frac{41}{n}$ is an integer, or equivalently an $n$ such that $n \mid 41$,
then w... | *
*Nothing written so far actually proves $n=40$ is the minimal counterexample for $n^2+n+41.$ All that has been shown is that the factorization $n(n+1)+41$ makes it clear that $n=40$ is a counterexample, while the second factorization you wrote makes it clear that $n=41$ is also a counterexample.
*For a similar rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Product polynomial in $\mathbb{F}_7$ I need to compute the product polynomial $$(x^3+3x^2+3x+1)(x^4+4x^3+6x^2+4x+1)$$
when the coefficients are regarded as elements of the field $\mathbb{F}_7$.
I just want someone to explain to me what does it mean when a cofficient (let us take 3 for example) is in $\mathbb{F}_7$ ? I ... | We use $3$ as shorthand for the coset $$7\mathbb{Z}+3=\{\ldots,-11,-4,3,10,17,\ldots\}$$ in $\mathbb{Z}/7\mathbb{Z}$. Since $$\cdots=7\mathbb{Z}-11=7\mathbb{Z}-4=7\mathbb{Z}+3=3\mathbb{Z}+10=3\mathbb{Z}+17=\cdots,$$ when working in $\mathbb{Z}/7\mathbb{Z}$, we have $$\cdots=-11=-4=3=10=17=\cdots.$$
Practically, this m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Valid Sobolev Norm on $\mathbb{R}$? I have seen many questions along this line, but none quite answered my question as far as I could tell.
On all of $\mathbb{R}$, is the Sobolev norm ever defined as follows
$$\|f\|_{W_2^k(\mathbb{R})} := \|f\|_{L_2(\mathbb{R})}+|f|_{W_2^k(\mathbb{R})},$$
where $|f|_{W_2^k(\mathbb{R})}... | Yes, this is a perfectly fine definition on $\mathbb R$. The purpose of the uniform (interior) cone condition is to make sure that all points of the domain are uniformly easy to approach "from the inside"; this enables us to control lower order derivatives globally, by integrating higher order derivatives (Poincaré's i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Strange delta function I don't know what to do when I see a delta-function of the following sort appear in an integral (3d-spherical here):
$$\delta^3(r\sin \theta - r_0).$$
E.g. the argument is a function of two of the variables.
I'm familiar with the standard properties of delta-fns. Are there some tricks I should kn... | In this answer, it is shown that when composing the dirac delta with $g(x)$, we get
$$
\int_{\mathbb{R}^n} f(x)\,\delta(g(x))\,\mathrm{d}x=\int_{\mathcal{S}}\frac{f(x)}{|\nabla g(x)|}\,\mathrm{d}\sigma(x)
$$
where $\mathcal{S}$ is the surface on which $g(x)=0$ and $\mathrm{d}\sigma(x)$ is standard surface measure on $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$? A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used sequences/convergent subsequences... | That's a nice idea for a proof. I think perhaps it works well to turn it inside out, so to speak:
Lemma. Assuming $X$ is a compact metric space, for each $\delta>0$ there is a finite upper bound to the number of points in $X$ with a pairwise distance $\ge\delta$. (Let us call such a set of points $\delta$-separated.)
P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
How to solve this geometry problem?
I have been given 2 squares here, and I supposed the ratio of (area of shaded region/ are of outer of square)
Obviously there are four right angled triangles here, once we have their size length we can calculate their are and get the final answer.
My book's answer says that all of t... | Well, the obvious answer is that they are all $3,4,5$ triangles, but how might we see that?
Let's look at the angles which meet at the corner between the marked $3$ and $4$. Let the angle in the triangle with $3$ be $\alpha$, then the angle in the triangle with $4$ which is $90^{\circ}-\alpha$ because the unshaded area... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
$\lambda : (x,y,x^2+\lambda y^2)$ contains straight lines For which $\lambda\in \mathbb{R}$ $$\varphi(x,y) = (x,y,x^2+\lambda y^2)$$ is a ruled surface, i.e. it can also be parametrized as
$$ \mathbb{R}^2 \ni (t,u) \mapsto g(t) + uw(t) $$
with $g$ a differentiable curve and $w$ a differentiable vector field?
| If $\lambda<0$, then $\lambda=-k^2$ and $x^2+\lambda y^2=(x-ky)(x+ky)$.
Set $u=x-ky$ and $v=x+ky$. Then $x=\frac{u+v}{2}$, $y=\frac{u-v}{2k}$ and $z=uv$.
Then:
$$
(u,v)\rightarrow \; (\frac{u+v}{2},\frac{u-v}{2k},uv)=(\frac{u}{2},\frac{u}{2k},0)+v\,(\frac{1}{2},\frac{-1}{2k},u)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/495826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Dedekind cuts: Showing that the set B has no smallest element Let $A$ and $B$ be subset of the rational numbers defined as follows:
$A = \{p \in \mathbb{Q} \mid p\leq0 \lor p^2<2\}$
$B = \{q \in \mathbb{Q} \mid q>0 \land q^2 >2\}$
*
*Show that the set $B$ has no smallest element.
*Show that the set $A$ fails to hav... | We try to discover something that might work.
Suppose we are given a positive rational $r$ such that $r^2\gt 2$. We want to produce a smaller positive rational $s$ such that $s^2\gt 2$.
We will produce $s$ by taking a little off $r$, say by using $s=r-\epsilon$, where $\epsilon$ is a small positive rational.
So we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K A similar question to mine was answered here on stackexchange:
Probability of winning the game "1-2-3"
However, I am unable to follow the formulas so perhaps someone could show the calculation and the way they arrived at it to answer this question.
My card game... | I think you'll find what you want in the paper Frustration solitaire by Doyle, Grinstead, and Snell at http://arxiv.org/pdf/math/0703900.pdf -- it looks like they get the answer
$$\begin{align}
{R_{13} \over 52!} &= {4610507544750288132457667562311567997623087869 \over 284025438982318025793544200005777916187500000000}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/495991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Gradient of a dot product The wikipedia formula for the gradient of a dot product is given as
$$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$
However, I also found the formula $$\nabla(a\cdot b) = (\nabla a)\cdot b + (\nabla b)\cdot a $$
So... what is going... | Since there are not many signs that one may easily use in mathematical notations, many of these symbols are overloaded. In particular, the dot "$\cdot$" is used in the first formula to denote the scalar product of two vector fields in $\mathbb R^3$ called $a$ and $b$, while in the second formula it denotes the usual pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "53",
"answer_count": 4,
"answer_id": 3
} |
Use every non-abelian group of order 6 has a non-normal subgroup of order 2 to classify groups of order 6.
Prove that every non-abelian group of order $6$ has a non-normal subgroup of order $2$. Use this to classify groups of order $6$.
I proved that every non-abelian group of order 6 has a nonnormal subgroup of orde... | Assuming $H$ is a non normal subgroup of order $2$.
Consider Action of $G$ on set of left cosets of $H$ by left multiplication.
let $\{g_iH :1\leq i\leq 3\}$ be cosets of $H$ in $G$.
(please convince yourself that there will be three distinct cosets)
we now consider the action $\eta : G\times\{g_iH :1\leq i\leq 3\} \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
Can an integer of the form $4n+3$ written as a sum of two squares? Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact?
| I'll write another argument with more group theoretic flavor in my opinion. Suppose that $p=4k+3$ is a prime number and you can write $p=x^2+y^2$. then $x^2+y^2 \equiv 0 \pmod{p} \iff x^2 \equiv -y^2 \pmod{p} \iff (xy^{-1})^2 \equiv -1 \pmod{p}$. Therefore $t=xy^{-1}$ is a solution of $x^2 \equiv -1 \pmod{p}$.
Now cons... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 0
} |
Showing that $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ contains multiplicative inverses
Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients -- contain multiplicative inverses?
I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ tak... | Let $K$ be an extension field of the field $F$ and let $0\ne a\in K$ be algebraic over $F$; then $F[a]$ contains the inverse of $a$.
Indeed, if $f(X)=c_0+c_1X+\dots+c_{n-1}X^{n-1}+X^n$ is the minimal polynomial of $a$ over $F$, then it's irreducible, so $c_0\ne 0$ and
$$
c_0+c_1a+\dots+c_{n-1}a^{n-1}+a^n=0.
$$
Multiply... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Find solution of PDE $x \frac {\partial z} {\partial x} +y \frac {\partial z} {\partial y}=z$
Problem:Find solution of Cauchy problem for the first order PDE $x \frac {\partial z} {\partial x} +y \frac {\partial z} {\partial y}=z$,on $ D= {(x,y,z): x^2 +y^2 \neq0,z>0} $ with initial condition $x^2+y^2=1,z=1$
Solution... | This is Euler's equation for homogeneous functions of degree $k=1$. Hence we know that $z( \lambda x, \lambda y)=\lambda z(x,y)$ for all $\lambda>0$. Since $z \equiv 1$ on the unit circle, we find that $$z(x,y)=z \left( \sqrt{x^2+y^2}\frac{(x,y)}{\sqrt{x^2+y^2}} \right)=\sqrt{x^2+y^2} \;z \left( \frac{(x,y)}{\sqrt{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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