Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that if $ \ A\cup B \subseteq C \cup D,\ A \cap B =$ ∅ $\land \ C \subseteq A \implies B \subseteq D.$ Question:
Prove that if $ \ A\cup B \subseteq C \cup D,\ A \cap B =$ ∅ $\land \ C \subseteq A \implies B \subseteq D$.
My attempt:
Let $ \ x\in B \implies x \in A \cup B \implies x \in C \cup D \because A\cup B... | It is correct. Well done.
Perhaps a bit of suggested modification in the "But that's not possible part" to make it clearer.
Since we started with $x \in B$, if $x \in A$, then $x \in A \cap B$ which is not possible, as $A \cap B = \phi$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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Prove or disprove $ \ (A \times A) - (B \times B) = (A-B) \times (A-B)$ Question:
Let $ A,B$ be sets.
Prove or disprove: $ \ (A \times A) - (B \times B) = (A-B) \times (A-B)$
My attempt:
Let $ \ (x,y) \in (A \times A) - (B \times B) \implies (x,y) \in (A \times A)$ and $ \ (x,y) \notin (B \times B) \implies x \in A $ ... | This is incorrect.
The set $A \times A$, is the sets where both coordinates belong in $A$.
The set $B \times B$ is the set where both coordinates belong in $B$.
So, $A \times A-B \times B$ are the points where both coordinates belong in $A$ except the ones that both coordinates belong in $B$. In other words, the first ... | {
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"question_score": "5",
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"answer_id": 5
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Is there any framework similar to Bayesian Preposterior Analysis for Value of Information? I am trying to use the Value of Information concept using Bayesian Preposterior Analysis as proposed by Raiffa and Schlaifer 1961. However, due to certain limitations, mainly associated with the decision model, I am looking for a... | You're probably looking for "Entropy" or "Information Gain," also called mutual information. It's the expected value of the Kullback–Leibler divergence from the conditional distribution.
https://courses.cs.washington.edu/courses/cse455/10au/notes/InfoGain.pdf
You can also look for articles on info-gain ratios and relat... | {
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How is induction on $p-q > 0$ used correctly here? I'm self learning Rotman's Algebraic topology and I've come across this theorem and proof. I have two questions:
$(1.)$ How did the author arrive at $H_n(X^p, X^{q+1}) = 0$ if $q+1 \ge n$?
$(2.)$ How is induction used properly here? Normally induction is done on a... | You have
$$(p',q') = (p,q+1)$$
where
$$p-q >0,\;\;\text{and either}\;\;n > p\;\;\text{or}\;\;q \ge n$$
To apply the inductive hypothesis, you need to have
\begin{align*}
&{\small{\bullet}}\;\;p'-q' \ge 0\\[4pt]
&{\small{\bullet}}\;\;p'-q' < p-q\\[4pt]
&{\small{\bullet}}\;\;\text{either}\;\;n > p'\;\;\text{or}\;\;q' \g... | {
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Choosing branch cuts for complex integration When calculating integrals like $$\int_0^\infty \frac{x^\alpha}{1 + x^2}dx$$ for $\alpha \in (-1,1 )$, it is convenient to take the branch cut of the integrand along the positive real axis and then use the keyhole contour.
I was wondering is there a way to use the principal ... | Using the principal branch, we can write the integral $\oint_C \frac{z^a}{z^2+1}\,dz$, where $C$ is comprised of (i) the real line segment from $-R$ to $R$ and (ii) the semi-circle in the upper-half plane, centered at the origin and with radius $R$, as
$$\begin{align}
\oint_C \frac{z^a}{z^2+1}\,dz&=\int_{-R}^0 \frac{x^... | {
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Multi-variable function Integration I want to calculate this integral.
$$ \int\int_B x^2y \space dx dy$$ where $$B = \{ (x,y) \in \mathbb{R}^2 |
y \leq x \leq y^2+1, 0 \leq y \leq 1 \}$$
Now from the definition we know that the "$y$"-integral needs to be from $0$ to $1$.
I have tried to calculate it like this
$$ \in... | If
$$B = \{ (x,y) \in \mathbb{R}^2 : y \leq x \leq y^2, 0 \leq y \leq 1 \},
$$
then you should be integrating:
$$
\int_0^1 \int_{y}^{y^2} x^2y \ dx dy
= - \int_0^1 \int_{y^2}^{y} x^2y \ dx dy = -\frac{1}{40}.
$$
But if
$$B = \{ (x,y) \in \mathbb{R}^2 : y \leq x \leq y^2\color{blue}{+1}, 0 \leq y \leq 1 \}... | {
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Any neat proof that $0$ is the unique solution of the equation $4^x+9^x+25^x=6^x+10^x+15^x$? It is obvious that both $f(x)= 4^x+9^x+25^x$ and $g(x)=6^x+10^x+15^x$ are strictly monotonic increasing functions. It is also easy to check that $0$ is a solution of the equation.
Also I chart the functions, and it looks that f... | HINT: $$a^2+b^2+c^2\geq ab+bc+ca$$
| {
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Are the elements of a set within a set also the elements of the latter? It is my understanding that an event is a subset of the set of all possible outcomes (sample space). If however the sample space consists of elements which are sets, can an event be defined as one the elements from these "inner" sets?
Ex. A coin is... | No.
In the first place, if $x\in y$ and $y\in z,$ then in most cases it is not true that $x\in z.$
For example, consider the set $\{\ \{1,2,3\},\ \{2,3,4\}\ \}.$ This set has only two members. If $1,2,3,4$ were members of it, then it would have at least four members.
In the second place, in the set $\{\,(H,T),(T,H),... | {
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Show that $\lim_{n \to \infty} \int_{0}^{1}|f_n(x)| \, dx= 0$ Let $\{f_n\}$ be a sequence of Lebesgue integrable functions and $g: [0,\infty)\to [0,\infty)$ be an increasing and continuous function such that $\displaystyle\lim_{x\to\infty}g(x) = \infty$.
We also have that
(i) $\int_0^1|f_n(x)|g(|f_n(x)|)\,dx < 100$
(ii... | $\forall \varepsilon>0$
$\exists M>0,\ s.t.\ 100/g(M)<\varepsilon/3$.
Due to Egoroff. $\exists E(measurable)\subset[0, 1]\ \&\ m([0, 1]-E)<\varepsilon/(3M)$. And $f_n\xrightarrow{u.}0$ on E.
Find an enough large N such that $\forall n>N$, $|f_n|<\varepsilon/3$ on E.
Then we have
\begin{eqnarray}
\int_0^1|f_n|dx&=&\int_... | {
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Prove that $\frac{\tan x}{x}>\frac{x}{\sin x}, x\in(0,\pi/2)$
Prove that $$\frac{\tan x}{x}>\frac{x}{\sin x},\;\;\; x\in(0,\pi/2).$$
My work
I formulated $$f(x)=\tan x \sin x - x^2$$ in hope that if $f'(x)>0$ i.e. monotonic then I can conclude for $x>0, f(x)>f(0)$ and hence, prove the statement.
However, I got $$f'(x... | I believe the simplest proof is through the Cauchy-Schwarz inequality:
$$\tan(x)\sin(x)=\int_{0}^{x}\frac{d\theta}{\cos^2\theta}\int_{0}^{x}\cos(\theta)\,d\theta\geq\left(\int_{0}^{x}\frac{d\theta}{\sqrt{\cos\theta}}\right)^2\geq\left(\int_{0}^{x}d\theta\right)^2=x^2. $$
In a similar fashion, for any $x\in\left(0,\frac... | {
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Showing a solution of a PDE is bounded. Let $x \in \mathbb{R}^3$, $t \in [1,\infty)$ and $ u(x,t)$ be a solution of the PDE
$$\partial_{t}^{2}u - \Delta u = 0 \\ u(x,0) = 0 \\ \partial_{t} u(x,0) = v(x)$$
where $v$ and $\partial_{x_i}v$ are both integrable on $\mathbb{R}^3$ for all $1 \leq i \leq 3$.
Show that there... | $\DeclareMathOperator{\p}{\partial}
$Recall Kirchoff's formula for the solution to the wave equation with initial conditions $u(x,0)=g$ and $\p_tu(x,0)=v(x)$:
$$u(x,t)=\frac{1}{4\pi t^2}\int_{\p B(x,t)}g(y) + \nabla g(y)\cdot(y-x) + tv(y)~d\sigma(y).$$
In your case $g=0$ and so the representation formula simplifie... | {
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How many numbers less than $100$ can be expressed as a sum of distinct factorials? How many numbers less than $100$ can be expressed as a sum of distinct factorials?
Example:
a) $4 = 2! + 2!$
b) $3 = 2! + 1!$
| Lemma(I): For every positive integer $n$ we have:
$$
1! + 2! + ... + (n-1)! < n!
\ \
$$
There are
$
\color{Green}{15} =
\color{Green}{16}
\color{Red}{-1} =
\color{Green}{2^4}
\color{Red}{-1}$.
$$n=
\varepsilon_1 (1!)
+
\varepsilon_2 (2!)
+
\varepsilon_3 (3!)
+
\varepsilon_4 (4!)
;
$$
where $\var... | {
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$\ x^3\:+\:a\left(a+1\right)x^2\:+\:ax\:-\:a\left(a+b\right)\:-\:1\:=\:0 $ $$\ x^3\:+\:a\left(a+1\right)x^2\:+\:ax\:-\:a\left(a+b\right)\:-\:1\:=\:0 $$
For what values of$\ b$ does the equation have a root which is independent of a?
Tried the Horner's Method, but doesn't seem to work with this. Could I have some hints ... | Put x = 1
You will find that for b = 2, irrespective of the value of a, you will find a root. It is that simple. But not scientific.
$1+a(a+1) + a - a(a+b) -1 = 2a-ab = 0$
In the above expresssion, if you put b=2, the root does not have to be dependent on a
| {
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$r=\pm1$ are the only rationals with $\,r+1/r\in \Bbb Z$ (sum with its reciprocal is an integer)
Can sum of a rational number and its reciprocal be an integer?
My brother asked me this question and I was unable to answer it.
The only trivial solutions which I could think of are $1$ and $-1$.
As to what I tried, I am... | Key Idea $\ r\ \&\ 1/r\,$ have integer sum & product so by RRT both are integers, so $\,r =\pm1.\,$
For convenience we reproduce the proof below, slightly generalized to $\,r\ \&\ c/r,\,$ for $\,c\in\Bbb Z$.
Lemma $ $ If $\ r\in \Bbb Q,\,c\in\Bbb Z\ $ then $\ r + c/r = b\in\Bbb Z \iff r,\, c/r \in \Bbb Z\,\ $ [OP is ... | {
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Let $a_n$ be a sequence of real numbers such that $\lim(a_n\sum_{k = 1}^n a_k^2) = 1$ Let $a_n$ be a sequence of real numbers such that $\lim(a_n\sum_{k = 1}^n a_k^2) = 1$ . Prove that $\lim((3n)^{\frac{1}{3}}a_n)=1$
I'm more concerned with how I can derive the prove of this question
| Let $S_n=a_1^2+\cdots+a_n^2$. The sequence $(S_n)_{n\ge1}$ is nondecreasing. If it is convergent then $\lim_{n\to\infty}a_n=0$ and this contradicts the hypotgesis $\lim_{n\to\infty}a_nS_n=1$. Thus,
$\lim_{n\to\infty}S_n=+\infty$ and $\lim_{n\to\infty}a_n=
\lim_{n\to\infty}\frac{a_nS_n}{S_n}=0$. Now
$$S_{n}^3-S_{n-1}^3... | {
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When is interchange of quantifiers allowed? Ex: $\forall w \in \bigcup A_n$ There is a myriad of question for the interchange of different quantifiers, mainly between $\exists$ and $\forall$.
I'm interested in knowing when both can be interchanged.
The motivation came from this:
$\forall w \in \bigcup_n A_n \Leftrighta... | Example
Suppose that for every man, the there exists a women that is his mother. Symbolically, we can state this relationship as follows:
$$\forall x:[x \in Men \implies \exists y: [y\in Women \land Mother(y,x)]]$$
Or equivalently:
$$\forall x\in Men: \exists y\in Women: Mother(y,x)$$
It should be obvious that we cann... | {
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Roots of an equation over the finite field $\operatorname{GF}(p^q)$ Consider the following equation over the finite field $\operatorname{GF}(p^q)$
such that $r \mid p^q-1$:
\begin{align}
x^r=y^r \tag{1} \\
\end{align}
The solutions of $(1)$ over $\operatorname{GF}(p^q)$ are:
\begin{align}
x=\gamma^i\, y \quad , \quad... | I would just use the factorization
$$f(x)=x^r-1=\prod_{i=0}^{r-1}(x-\gamma^i).$$
It implies that
$$
x^r-y^r=y^rf(x/y)=y^r\prod_{i=0}^{r-1}(x/y-\gamma^i)=\prod_{i=0}^{r-1}(x-y\gamma^i).
$$
You can then cancel the factor $x-y$ corresponding to $i=0$.
The trick is known as homogenization. It adds one more variable to a po... | {
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Calculation of Lower Box and Box Dimension I am new to this site, so sorry if the question is stupid. I am learning fractals and my teacher gave me the following exercise.
Let $$E=\{0,0\}\cup\left\{\bigcup_{n=1}^\infty (x,1/\sqrt{n}\,):0\leq x\leq 1/\sqrt{n}\right\}$$ Find $\dim_\text{lower box}(E)$ and $\dim_\text{bo... | Since it is probably a homework problem, I'll just get you started.
Suppose the width of the box is $\epsilon$. For each line segment $(0,1/\sqrt n)\times\{1/\sqrt n\}$ you need at least $1/(\sqrt n \epsilon)$ boxes, and these boxes won't cover more than one line segment if $\frac1{\sqrt n}-\frac1{\sqrt{n+1}} > \epsil... | {
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If a bird fly to height of $h$ ,What's the area that it can see? Suppose a bird fly to height $h$ from earth . The bird can see area under by it's eyes ,name as $S$ ,What's $max \{S\}$ ?
Is it possible to solve ?
my first trial was to assume a cone by $height =h$ and $S=\pi R^2$ as area like the figure I attached with... | I am assuming that there is no limitation on the angle of vision of the bird.
The only known values I am assuming is the half apex angle $\theta$.
By property of tangent, the half angle subtended at the center will be $90-\theta$.
By using the solid angle formula,
$$
A=\phi r^2=2\pi(1-\cos{(90-\theta}) = 2\pi (1-\sin{\... | {
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How does $t \rightarrow \infty$ then $t[1-F(t)+F(-t)] \rightarrow 0$ relate to the weak law of large numbers? Refering to the notes here http://www.stat.umn.edu/geyer/8112/notes/weaklaw.pdf
In Theorem 1, I understand how (i) $\iff$ (iii). I also understand the second part of (ii) where $\lim_{t \to\infty} \int ^t _{-t... |
$1-F(t)+F(-t) \rightarrow 0$, and therefore $t(1-F(t)+F(-t)) \rightarrow 0$.
If for a function $h\colon\mathbb R\to\mathbb R$, we have $h(t)\to 0$, it does not mean that $t\cdot h(t)$ goes to zero (for example if $h(t)=\sqrt{1+\left\lvert t\right\rvert}$).
Here, we have
$$1-F(t)+F(-t)=\mathbb P\left\{X_1 \gt t \righ... | {
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Given a sequence {bn}, with two subsequence which both converge, proof that {bn} not have to be convergent Given a sequence $\{bn\}$, with two subsequences which both converge, prove that $b_n$ need to converge.
Given $bn$ and subsequences $b_{2n}$ and $b_{2n+1},$
where $b_{2n}$ and $b_{2n+1}$ are both covergent,
Sho... | Take $b_n=(-1)^n$.
$\{b_{2n}\}$ converges and $\{b_{2n-1}\}$ converges, but $\{b_n\}$ does not.
Let $A=\lim\limits_{n\rightarrow+\infty}b_n$ and $\epsilon=\frac{1}{2}.$
Thus, for all $N>0$ there is $n>N$ for which $|b_n-A|\geq\epsilon$.
| {
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Derivative of matrix-valued function with respect to matrix input I have the expression
$$\bf \phi = \bf X W$$
where $\bf X$ is a $20 \times 10$ matrix, $\bf W$ is a $10 \times 5$ matrix.
How can I calculate $\frac{d\phi}{d\bf W}$? What is the dimension of the result?
| There is a similar question.
Also, you could define it
$$C = \frac{\partial \phi}{\partial W} $$
where C is a 4D matrix (or tensor) with
$$ C_{a,b,c,d} = \frac{\partial \phi_{a,b}}{\partial W_{c,d}} $$
Actually, when derivatives are expressed as matrices, for example, $f=x^TAx$ where $x\in R^{n\times1}, A\in R^{n\times... | {
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Binomial distribution to approximate 90 % successes A student is taking a multiple choice test where each question has four options for an answer. The student mastered mastered 70% of the material. Assume this means that the student has a 0.7 chance of knowing the correct answer to a random test question. On the other ... | At least $90$ per cent correct means that the number of correct answers could be $27$, $28$, $29$ or $30$. So to be exact, you're looking for $P(\ge90 \% \, correct)$, which is the sum of probabilities of the four cases computed the way you did for $27$.
| {
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Simplify $\sqrt{6-\sqrt{20}}$ My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$
Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$
But I sti... | In this particular problem, you can pretty much guess the answer.
$$\sqrt{6-\sqrt{20}}=\sqrt{6-2\sqrt{5}}$$
Now, suppose that the $-2\sqrt{5}$ was the middle term of a perfect square trinomial, where $x = \sqrt{5}$. In other words, that middle term is $-2x$.
What would the first and last term look like? Obviously it wo... | {
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An IMO graph theory problem In the second 1990 IMO problem, I saw a solution (probably of the proposer) as follows, of which I have some questions.
Problem 2: Suppose $n\geq 3$, and let $S$ be a set of $2n-1$ distinct points on a circle. Assume that exactly $k$ points of $S$ are colored black. A coloring of $S$ is cal... | If you label the points with $\mathbb Z/(2n-1)\mathbb Z$ in order, then node $x$ is adjoined to nodes $x+n+1$ and $x-(n+1)$. So the nature of the graph depends on the sequence:
$$0,n+1,2(n+1),\dots,$$
Two nodes $a,b$ in the graph are in the same loop if $a-b=(n+1)A$ for some $A$. But the number of distinct multiples of... | {
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Simple probability question, with faulty screws. I have translated the problem as follows:
A factory produces screws, the probability of them being faulty is 0.01 independently. The factory makes a box with 10 screws and recalls the boxes containing 2 or more faulty screws. What is the percentage of boxes that the fac... | My solution is:
The percentage of faulty boxes is equal to the probability of faulty boxes.
In order to make easy my calculation, I calculate the probability of the boxes which have no faulty screws or have at least one faulty screw. Then I can find the probability I am searching for as follows:
$$S_{FaultyBox} = 1 - S... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Find the period of $f(x) =\{x\}+\{x+1/3\}+\{x+2/3\}$ is equal to what?({.} denotes fractional part of function) I tried the basic way of solving this question $f(x+T)=f(x)$ and writing $3x$ as $x+x+x$ but I don't think it can be solved directly like that.
| Hint: $\{x+1\} = \{x\}\,$ so $f(x+1/3) = \{x+1/3\}+\{x+2/3\}+\{x+1\} = f(x)$.
Alt. hint: write $\{x\}=x - \lfloor x \rfloor$ and use Hermite's identity $\;\lfloor x \rfloor + \lfloor x+1/3 \rfloor + \lfloor x+2/3 \rfloor = \lfloor 3x \rfloor\,$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2400754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help with solving word problems pertaining to limits of a function. This is my first post, and I apologise if I make any mistakes in setting this out, but I am just getting used to formatting.
So In my current math unit, we are being assigned problems to do with maxima and minima of a function. Specifically, the one I... | I'm going to suppose that you're working with a rectangle, hopefully you can generalise to other shapes as necessary.
For an $l \times b$ rectangle, the area is $y = lb$ and the perimeter is $x = 2(l + b)$. Since we have a fixed amount of material, $x$ is a known value, so $l$ and $b$ are related by $l = \frac{x}{2} - ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is nᵐ>mⁿ if m>n? I remember playing with my calculator when I was young. I really liked big numbers so I'd punch big numbers like $20^{30}$ to see how big it really is.
On such a quest, I did observe that $20^{30}$ is greater than the value of $30^{20}$. In fact, in many cases, I found that $n^m>m^n$ if $m>n$.
Is this ... | In simple terms, for integers you can start with smallest no. i.e. (1,2), (2,3), (2,4).
*
*$1^2 < 2^1$,
*$2^3 < 3^2$ and
*$2^4 = 4^2$.
In all the above cases $n^m > m^n$ was false for all m>n.
By observing the pattern for all n>=2 and m>4 we have $n^m > m^n$ true. Consider
*
*(2,5) $\implies$ 32 > 25 or
*(3,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2400996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 5,
"answer_id": 3
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Prime ideals lying above a prime number in $\mathbb{Q}(\zeta_n)$ If we assume we have a prime number $p$ such that $p\nmid n$, and $\mathscr{P}$ is a prime ideal lying above $p$ in the field extension $\mathbb{Q}(\zeta_n)$, is it always true that $\mathscr{P}\nmid n$?
| Yes. If $\mathscr{P} \mid k$, then $\mathrm{Nm}(\mathscr{P}) \mid \mathrm{Nm}(k)$. Since $\mathrm{Nm}(\mathscr{P})$ is a power of $p$ and $\mathrm{Nm}(k)$ is a power of $k$, so this can only happen if $p \mid k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2401089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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how to prove "monoid object in the category of monoids is a commutative monoid"? I've read about Eckmann–Hilton theorem (a * b) . (c * d) = (a . c) * (b . d). But category of monoids is not a 2-category, why are there two binary operators here?
| If $(M, \cdot, 1)$ is a monoid, and $f: M \times M \rightarrow M$ a monoid homomorphism with $f(x,1) = x = f(1,x)$, then:
$x\cdot y = f(x,y)$
Since
$x \cdot y = f(x,1) \cdot f(1,y) = f((x,1) \cdot (1,y)) = f(x,y)$
Therefore
$x \cdot y = f(1,x) \cdot f(y,1) = f((1,x) \cdot (y,1)) = f(y,x) = y \cdot x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2401181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Compute the determinant The following problem is taken from here exercise $2:$
Question: Evaluate the determinant:
\begin{vmatrix}
0 & x & x & \dots & x \\
y & 0 & x & \dots & x \\
y & y & 0 & \dots & x \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
y & y & y & \dots & 0
\end{vmatrix}
My attempt:
I tried to use... | Let $A$ be a $n\times n$ matrix of the form described above. You can easily compute the determinant by hand for the case up to $n=4$, which suggests the following relation:
$$\det(A_n) = (-1)^{n+1}\sum_{i=1}^{n-1}x^iy^{n-i}$$
This can be proved by induction, by using the calculated small cases, followed by using the fa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Which of the following is bigger (logarithms) I need to compare those two expressions and decide which is bigger.
$2 \sqrt2$ or $\log_2(3)+\log_3(4) $.
So I tried to simplify so the log expression so I know
and so
$$ \log_2(4) \times (\log_4(3) + \log_3(2)) ?? 2 \times \sqrt2$$
and then
$$2 \times \log_2(2)\times(\lo... | $\log_3 4 = \dfrac{\log_2 4}{\log_2 3} = \dfrac{2}{\log_2 3}$
$A := {\log_2 3}+ \log_3 4 = {\log_2 3} + \dfrac{2}{\log_2 3}$
Dividing by$A$ by $\sqrt 2$, observe $ \dfrac{\log_2 3}{\sqrt 2} + \dfrac{\sqrt 2}{\log_2 3} > 2$ by AM-GM inequality (since ${\log_2 3 \ne \sqrt 2}$)
Thus $A>2\sqrt 2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2401638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding map from Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups. I want to find a map from the Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups. Since the fundamental group of $RP^2$ is $Z_2$, intuitively it feels like a good start would be to look for any non-trivial m... | Let $X$ and $Y$ be two connected manifolds of dimension $n \geq 2$.
There is a map $\varphi : X\# Y \to X$ given by mapping $Y$ to a disc $D$. By the Seifert van Kampen theorem, $\pi_1(X\# Y) \cong \pi_1(X^{\circ})*_{\pi_1(S^{n-1})}\pi_1(Y^{\circ})$ where $X^{\circ}$ denotes $X$ with an embedded open disc removed; lik... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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"Note that connectedness is not defined for closed sets" explanation I'm learning Complex Analysis, and we are given the following definitions:
Definition. Suppose that
$\Omega \subseteq$ C and that
$\Omega$ is open.
(1) The set $\Omega$
is connected if any two points of $\Omega$
can be joined by a polygonal
path ... | Let $X$ be a topological space, e.g. a subset of $\mathbb{C}$. Your definition of what it means for $X$ to be connected is different from the usual definition of connectedness. However, the two definitions coincide when $X$ is open. That is why they didn't want to apply their definition for nonopen sets.
The usual d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Squares of positive semidefinite matrices Suppose $L_1 \succeq L_2$, where $L_1,L_2$ are positive semidefinite matrices (actually combinatorial Laplacians). Is the following inequality true, and if no, under which conditions?
$$L_1^2 \succeq L_2^2$$
| It's not always true. Counterexample:
$$
\begin{align*}
&L_1=\pmatrix{1&-1&0\\ -1&2&-1\\ 0&-1&1},
\ L_2=\pmatrix{1&-1&0\\ -1&1&0\\ 0&0&0},\\
&L_1^2-L_2^2=\pmatrix{0&-1&1\\ -1&4&-3\\ 1&-3&2}.
\end{align*}
$$
I'm not sure if there is any good (non-restrictive) sufficient condition for the inequality to hold.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I show that for any natural number n, there exists a natural number m such that $4^{2n+1} + 3^{n+2} = 13m$?
Show that for any natural number $n$, there exists a natural number $m$ for which:
$$4^{2n+1} + 3^{n+2} = 13m$$
I don't know where to start. I tried to use Mathematical Induction, denoting the top stat... | The statement $P(n)$ is:
There exists $m\in\mathbb N$ such that $$4^{2n+1}+3^{n+2}=13m$$
Step $1$: Proving that $P(0)$ is true should be easy, you just have to calculate what $$4^{2\cdot 0+1}+3^{0+2}$$ is equal to.
Step $2$: Assume that $P(n)$ is true, and write
$$\begin{align}4^{2(n+1)+1}+3^{(n+1)+2} &= 4^{2n+1+2}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Trouble understanding ij-element of a matrix
Let $\;f:\mathbb R^n \rightarrow \mathbb R^m\;$ and $\;G:\mathbb R^m
\rightarrow \mathbb R_{+}\;$ and consider the $\;n\times n\;$ tensor
$\;\mathcal A=(a_{ij})_{1\le i,j \le n}\;$ where $\;a_{ij}=f_{x_i}
\cdot f_{x_j} -{\delta}_{ij}(\frac{1}{2} {\vert \nabla f
\vert}^... | Yes it is correct. This is what the author meant. Note that the matrix $\mathcal A$ can also be written in more compact form as
$$\mathcal A = ff^T - (\frac 12 \left |\nabla f\right| + G(f))I$$
where $I$ is the identity matrix
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2402203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x\left(x-1\right)\left(x-2\right)\left(x-3\right)=m$ has all roots real Given the equation:
$x\left(x-1\right)\left(x-2\right)\left(x-3\right)=m$
For what values of $m$ are all the roots real?
I've rewritten the equation as: $x^4-6x^3+11x^2-6x-m=0$
I'm quite sure this is done with Vieta's but didn't really figure out ... | Start from $x^4-6 x^3+11 x^2-6 x-m=0$ and substitute $x=z+\dfrac{3}{2}$
we get
$\left(z+\frac{3}{2}\right)^4-6 \left(z+\frac{3}{2}\right)^3+11 \left(z+\frac{3}{2}\right)^2-6 \left(z+\frac{3}{2}\right)-m=0$
Expand and reorder
$z^4-\frac{5 }{2}z^2+\frac{9}{16}-m=0$
substitute $z^2=w$
$w^2-\frac{5 }{2}w^2+\frac{9}{16}-m=0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Shortest Hamilton Path Planar Problem I think that the problem to obtain the shortest path visiting once time each point (it is not needed to come back to start point so it is a Hamilton path), in its planar euclidean and symmetric version is an NP-complete problem.
Wikipedia says:
"If the distance measure is a metric... | Cristos H. Papadimitriou says that "The Euclidean Travelling Salesman Problem Is NP-Complete" He based its demonstration reducing the Exact Cover Problem to it, wich is known NP-complete. In general the ETSP is NP-hard when the inputs are real coordenates, but restricting the input to integers such as the distances can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2402492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to minimize the integral of the functional of a function, with respect to that function? I need to obtain the function $f(x)$ for which the following integral has its minimum value:
$I=\int F(f(x))dx= \int [A (B^2-f(x)^2)^2-Cf(x)f''(x)]dx$
One special solution is $f(x)=constant=\pm B$, but I need the general solut... | The Euler Lagrange equation for $f$ to be an extremum of the integral $\int^b_a F(x,f,f',f'') dx$ is
$\frac{\partial F}{\partial f}-\frac{d}{dx}\frac{\partial F}{\partial f'}+\frac{d^2}{dx^2}\frac{\partial F}{\partial f''}$.
For the given $F$ this gives $\frac{\partial}{\partial f}[A(B^2-f^2)^2]-Cf''-C\frac{d^2}{dx^2} ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Given a decreasing function s.t. $\int_0^\infty f(x)\,dx<\infty,$ prove $\sum_{n=1}^\infty f(na)$ converges
Let $f\in C([0,\infty))$ be a decreasing function such that $\int_0^\infty f(x)\,dx$ converges.
Prove $\sum_{n=1}^\infty f(na)$ converges, $\forall a>0$
My attempt:
By the Cauchy criterion, there exists $M>0,$ ... | This is the integral test for convergence of series. Do a change of variable x --- to --- ax. The integral converges so does series g(n)=f(na).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2402659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Characterizing the kernel of a certain map For a given $A \in \mathrm{O}_n(\mathbb{R})$, consider the map
\begin{align*}
\phi_A: \mathfrak{o}_n(\mathbb{R}) & \to \mathrm{Mat}_{n \times n}(\mathbb{R}) \\
x &\mapsto Ax-xA^T.
\end{align*}
Recall that $\mathfrak{o}_n(\mathbb{R})$, the Lie algebra of $\mathrm{O}_n(\mathbb{... | Hint. By a change of orthonormal basis, we may assume that $A=I_p\oplus-I_q\oplus R_{\theta_1}\oplus\cdots\oplus R_{\theta_m}$, where each each $R_{\theta_k}$ denotes a $2\times2$ rotation matrix for an angle $\theta_k\in(0,\pi)$. It is not hard to see that $Ax=xA^T$ has a nonzero skew symmetric solution if and only if... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Simplex: duplicate constraints I'm trying to understand how the two-phase simplex algorithm works, this site explains it using a simple example: http://optlab.mcmaster.ca/feng/4O03/Two.Phase.Simplex.pdf
I've tried to come up with some edge cases myself, and I'm stuck on this one, what am I doing wrong?
Minimize $x$ wh... | The salient point is that the entries of the RHS have to be non-negative. You calculate the minimum of the fractions. Only the entries in the matrix has to be positive. The short notation is
$\min\bigg\{\frac{b_i}{a_{ij^*}}|a_{ij^*}>0\bigg\} $
where $b_{i}\geq 0$
In your case $\min\bigg\{\frac{b_2}{a_{22^*}}\bigg\}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2402841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of a series of translations of a Lebesgue integrable function
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue integrable function. Prove that $$\sum_{n=1}^{\infty} \frac{f(x-\sqrt{n})}{\sqrt{n}}$$ converges almost for every $x \in \mathbb{R}$.
My tactic here (which of course might lead me to nowh... | A brute-force method: Define $g(x):= \sum_0^{\infty} \frac{|f(x-\sqrt n)|}{\sqrt n}$, which a priori might be infinite at many points. Fix some $j \in \Bbb Z$. By Tonelli's theorem we may interchange sum and integral to compute \begin{align*}\int_j^{j+1} g(x)dx &=\sum_{n=1}^{\infty} \int_j^{j+1}\frac{|f(x-\sqrt n)|}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2402988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Prove that for all $k \geq 1$, $\mathbb{log}(n)^k \in o(n)$ without using limits Prove that for all $k \geq 1$, $[\mathbb{log}(n)]^k \in o(n)$ without using limits, i.e prove that for any $c > 0$, there is a $n_0 > 0$ such that for all $n \geq n_0$, $[\mathbb{log}(n)]^k \lt cn$
I first proved it using limits and L'hopi... | Let $n= x^k$, we need to prove $\log(x^k)^k = o(x^k)$, or $$k\log(x) = \log(x^k) = o(x),$$ which is what you just proved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show $\frac{2-x}{2+x} \le e^{-x}$ for $x\ge0$? Let $x\geq0$. Show that:
$$\frac{2-x}{2+x} \le e^{-x}.$$
I have some troubles to prove it. Would you give me any hint?
| We need to prove that
$$(2-x)e^x\leq2+x$$ or
$$x(e^x+1)\geq2(e^x-1)$$ or
$f(x)\geq0$, where
$$f(x)=x-\frac{2(e^x-1)}{e^x+1}$$ and calculate $f'(x)$:
$$f'(x)=1-\frac{4e^x}{(e^x+1)^2}=\frac{(e^x-1)^2}{(e^x+1)^2}\geq0,$$
which gives $f(x)\geq f(0)=0$ and we are done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Part of sum inequality to number of terms Suppose you have a series
$$ S=\sum_{i=1}^n a_i$$ and rearrange the terms such that, $$ a_1\geq a_2 \geq \cdots \geq a_l \geq a_i\geq 0 $$ for all $$ i > l$$, then it should be obvious that $$ \frac{a_1+a_2+\cdots+a_l}{S}\geq \frac{l}{n}$$ How would one go about proving this? ... | Hint: first rearrange your inequality to $\displaystyle \frac{S}{n} \leqslant \frac{a_1+a_2+\ldots+a_l}{l}$. Then note that
$$\frac{S}{n} = \frac{l}{n} \cdot \frac{a_1+a_2+\ldots+a_l}{l} + \left( 1-\frac{l}{n} \right) \cdot \frac{a_{l+1} + \ldots + a_n}{n-l}$$
which is a weighted arithmetic mean of two numbers, where t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the probability that the digit sum of a randomly chosen integer between 0000 and 9999 is divisible by 5? If I have a randomly selected integer between 0000 and 9999, what is the probability that the digit sum of that number is divisible by 5?
[E.g. 1234 = 1 + 2 + 3 + 4 = 10]
I've started off with knowing that ... | In this case we had it easy since the base 10 is a multiple of the divisor 5. What happens if we have $n$ digits in base $B$ and are interested in divisor $d$?
Let $\omega \neq 1$ be a $d$th root of unity, and consider
$$
P(\omega) = \left(\frac{1 + \omega + \omega^2 + \cdots + \omega^{B-1}}{B}\right)^n = \left(\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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How do I prove this method of determining the sign for acute or obtuse angle bisector in the angle bisector formula works? The formula for finding the angular bisectors of two lines $ax+by+c=0$ and $px+qy+r=0$ is $$\frac{ax+by+c}{\sqrt{a^2+b^2}} = \pm\frac{px+qy+r}{\sqrt{p^2+q^2}}$$
I understand the proof of this formu... | If you follow the method described by me here, we simply have to check the angle between normal after finding the bisector vectors:
Now, check the angle between $n_1$ and $n_2$, if it is greater than ninty degrees then it means $\chi$ is the obtuse bisector and if not, it $\chi$ is the acute bisector. To understand wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Set operations on connected sets in $R^2$ An exercise wants me to give an example of the following in $R^2$
*
*$A$ and $B$ are connected but $A \cap B$ is not.
*$A$ and $B$ are connected but $A \setminus B$ is not.
*$A$ and $B$ are not connected but $A \cup B$ is.
I think I found an example for 3. Take two curve... |
$1)$ Take two circles $A,B$ with same radii and different centers and intersect them using a translation.The intersection in of these circles woulb be a two point set namely $A=\{(a_1,a_2),(b_1.b_2)\}$.Then $A$ is not a connected set as a union of two singletons which are closed sets with respect to the usual topolog... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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prove that every non-empty open set contains an open sphere disjoint from $A$.
If $A$ is nowhere dense in $(X,d)$ then prove that every non-empty open set contains an open sphere disjoint from $A$.
Suppose that $A$ is n.w.d. in $X$ , and every non-empty open set say , $B$ contains all open sphere $S_r(x),x\in X,r>0$ ... | Let B be a non-empty open set. Suppose every open sphere inside B intersects A. Then every point of B belongs to the closure of A. But this contradicts the definition of a nowhere dense set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2403843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Pull back of universal cover is universal iff the map induces a fundamental group isomorphism I'm trying to solve the following task:
Let $f: Y \rightarrow X$ be a map and $p: \bar{X} \rightarrow X$ be the universal cover of $X$. The spaces Y and X are path-connected. The pull back of the cover $p$ by $f$ is a universa... | You want to be a little careful, since the pullback $Z = Y \times_X \tilde{X}$ need not be path-connected even if $X$ and $Y$ are. (For example, take $f: * \to S^1$.) However, each path component of $Z$ will be simply-connected.
I think your proof works, but here's another way to think about this problem. Associat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to determine probability of an outcome when the number of tries is variable. Given a known probability of an event being successful, how do you calculate the odds of at least one successful outcome when the number of times you can attempt is determined by a separate dice roll.
For example: I know the probability o... | Lets say the outcome of first dice roll is $k(1 \le k \ge 6$) each with probability $\frac{1}{6}$.
P(atleast 1 success)= 1-P(no success)
P(atleast 1 success) = P(k=1)(1-P(no success with 1 throws)) + P(k=2)(1-P(no success with 2 throws))+...6 terms
Since P(k=1)=P(k=2)=...=P(k=6)=1/6
Hence
P(atleast 1 success) = $\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2404013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate $\int_0^1 \mathrm e^{-x^2} \,\mathrm dx$ using power series? I'm trying to evaluate
$$\int_0^1 \mathrm e^{-x^2} \, \mathrm dx$$
using power series.
I know I can substitute $x^2$ for $x$ in the power series for $\mathrm e^x$:
$$1-x^2+ \frac{x^4}{2}-\frac{x^6}{6}+ \cdots$$
and when I calculate the ant... | $$\left[ x-\frac{x^3}{3}+ \frac{x^5}{5*2}-\frac{x^7}{7*6}+ \dots \right]_0^1 = \left( 1-\frac{1}{3} + \frac{1}{5 \cdot 2} - \frac{1}{7 \cdot 6} + \dots \right) - 0 = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!}$$
So $\int_{0}^{1}e^{-x^2}=\sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!}$, that is, the answer is whatever the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2404105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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$\square+\square+\square=30$, with boxes filled using $1, 3, 5, 7, 9, 11, 13, 15$, possibly repeated. How? From the days I started to learn Maths, I've have been taught that
Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
$$3 + 5 + 1 = 9$$
OK, but look at this question
This question was solve... |
you can also repeat the numbers
Wonder if that means $\,11,5+13,5+5=30\,$ (where the $\,,\,$ comma works as
decimal separator).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a$ is a positive integer that is not a perfect square then $[\Bbb Q(a^{\frac{1}{4}}):\Bbb Q]=4$ If $a$ were a square free number I am done by Eisenstein criterion but if $a$ is not a perfect square how do I use Eisenstein criterion. The book by Joseph Rotman gives an argument that that there exists a prime $p$ suc... | May be Rotman means the following.
Consider the prime factorization
$$
a=\prod_{i=1}^kp_i^{a_i}.
$$
Because
$$\Bbb{Q}(a^{1/4})=\Bbb{Q}([a/p^4]^{1/4})\tag{*}$$ for any prime $p$, we can without loss of generality assume that $a_i<4$ for all $i$. As we assumed that $a$ is not a perfect square we also know that at least... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2404269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Expected stopping time Let $X_1,X_2,\ldots,$ be iid uniform in $[0,1]$ and define $S_n:=\sum_{i=1}^n X_i$. Let $\tau$ be the smallest $n$ for which $S_n>1$. It is known that $E[\tau]=e\approx 2.7183$, and this can be easily shown using the
Irwin-Hall https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution
distri... | For $x \in [0, 1)$, let $\tau(x)$ denote the smallest $n$ for which $S_n > x$. Let $g(x) := E[\tau(x)]$. We're interested in $g(1)$.
Using its definition, $g$ satisfies an integral recurrence
$$ g(x) \ = \ \int_{y = 0}^{x} [1 + g(x - y)] \ dy \ + \ \int_{y = x}^{1} 1 \ dy $$
which if I'm not mistaken, is satisfi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complex number quadratic Is the following equation
$$z^2 + z^* + \frac14 = 0$$
where $z$ is a complex number and $z^*$ is its conjugate
completely separate from ordinary quadratic equations? i.e. can I use the discriminant, quadratic formula etc. If not what, what type of equation is this? Can z* be treated independe... | Here is a more geometric approach. We have
$$z^{2}+z^{*} = -\frac{1}{4} \in \mathbb{R}$$
so $z^{2}+z^{*}$ is real. Write this in polar form
$$r^{2}e^{2i\theta}+re^{-i\theta}=-\frac{1}{4}$$
to obtain the constraint
$$r^{2}\sin(2\theta)-r\sin(\theta)=0$$
so that either $r=0$ (so $z=0$), or $\sin(\theta)=0$ (so $z \in \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2404493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ax=b has no solution - version 3 I am struggling to prove the following theorem which is popped up in a linear optimization textbook,
Theorem : For $A \in \mathbb{R}^{m \times n}, b\in \mathbb{R}^{m}$, $ \quad Ax=b $ has no solution if and only if there exists a vector $y \in \mathbb{R}^m $ with $ A^Ty =0 $ and $ b^Ty... | Let me identify the matrix $A$ with the linear map $T_A \colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ defined using left multiplication ($T_A(x) = Ax$). There are two basic relations between the kernel and image of $A$ (or, more precisely, $T_A$) and the kernel and image of $A^T$ given by
$$ \ker(A) = \operatorname{im}... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Is a number meeting these conditions divisible by forty-nine? I am not a mathematician, I'm a linguistics PhD student. As part of my research I need to put various convoluted sentences through various syntactic transformations and see then check whether people think they are true or not. Mathematical statements (well, ... | Yes. The reason is that the only way a square number can be divisible by $7$ is if its square root is divisible by $7$. So your number is the result of squaring a multiple of $7$, and when you do that you get a multiple of $49$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2404833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Proof Silverman-Toeplitz theorem Proof Silverman-Toeplitz theorem: Let $A$ be an infinite matrix with entries $(a_{ij})$. Two sequences $\sigma $ and $s$ are related by this matrix as follows
$$\sigma_i =\sum_{j=0}^{\infty} a_{ij}s_i$$
Prove that for a convergent sequence $s$, $\sigma$ converges to the same value iff
... | Although this answer is very late, I still think it might be useful to others. Let $A_n$ be the $n$th row of $A$ and let $a_k$ denote its $k$th element. If $\sum_k |a_k|$ does not converge we can choose an index sequence $k_r$ such that $k_0 = 0$, and
$$ \sum_{k = k_{r-1}}^{k_r-1} |a_k| > r \quad \text{for $r \geq 1.$}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the quadratic equation from its given roots.
If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c
=0$ , then form an equation whose roots are:
$\alpha+\dfrac{1}{\beta},\beta+\dfrac{1}{\alpha}$
Now, using Vieta's formula,
For new equation,
Product of roots ($P$) = $\dfrac{a^2+c^2+2ac}{ac}$
Su... | $$S=\alpha+\beta+\frac{1}{\alpha}+\frac{1}{\beta}=(\alpha+\beta)\left(1+\frac{1}{\alpha\beta}\right)$$
If I write this in terms of $a,b,c$ I get $$S=-\frac{b}{a}\left(1+\frac{a}{c}\right)=-\frac{b(a+c)}{ac}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to think about open sets and continuous functions on discrete metrics I'm working through practice problems for the Math Subject GRE (Which seems to me to be all Analysis and Algebra even though I'd heard it was mostly multivariate calculus). This problem came up:
Let $\mathbb{Z^+}$ be the set of positive integers ... | $1.$ Open balls in a metric help you determine just how close two points are. In a way, the more open balls around one that contain the other, the closer they are. The discrete metric says all points are equidistant from each other. So, there is a smallest closed ball containing them all and smallest open ball contai... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a problem using upper bounds Let S be a non empty subset of the real numbers which is bounded above. Let c be a real number and define T ={cx|x∈S}. Show that,if c > 0,then T is non empty, bounded above and that sup T = c sup S. Give an example of a set S as above, such that with c = −1, the set T is still bounded above... | Since $S$ is bounded above, $\overline s=\sup S<\infty .$ We show that $\sup T=c\overline s:$
$a). \ \overline s$ is an upper bound for $S$ by definition of $\sup.$ Then, by definition of $T,$ and the fact that $c>0,\ c\overline s$ is an upper bound for $T$.
$b).$ Let $t$ be $\textit {any}$ upper bound for $T.$ Then,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The closed unit ball to generate a linear normed space. Let $X$ be a linear normed space and let B denote the closed unit ball of $X$. Then
we can stretch the unit ball to get every vector in $X$: $$X=\bigcup_{n=1}^\infty nB.$$
Is this true? Why?
| Indeed, in a normed space every neighbourhood of $0$ is absorbing, as this property is also called. For, if $x \in X$, take $n = 1 + \lceil\|x\|\rceil \in \mathbb{N}$ so that $\|x\| < n$. Then $\|\frac{1}{n}\cdot x\| = \frac{1}{n}\|x\| <1$ So $x = n \cdot (\frac{1}{n}\cdot x) \in nB$, as required.
IIRC, a locally conv... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does independence necessarily mean uncorrelatedness? We all know that two independent events are uncorrelated, don't we?
Nonetheless, we can find events that are correlated, yet they are independent such as the examples found on the Suprious Correlations website$^*$.
Is there a problem here or is it just me who's missi... | You should keep stochastic independence distinct from causal independence.
Two random variables that are stochastically independent are uncorrelated by definition.
Two random variables that are causally independent ($A$ does not imply/causes $B$, nor vice versa) may be correlated.
It is also possible that some third... | {
"language": "en",
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Is it possible to derive a Ring from a Monoid? Multiplication is just repeated addition, so we can derive multiplication from addition.
Addition over integers is a monoid. Addition and multiplication over integers form a Ring, so there's one example where this is possible, but addition and multiplication have more pro... | It depends on what you want to do.
For any monoid $M$ there is alway the monoid ring $\Bbb Z[M]$. This is the left adjoint to the forgetful functor from rings to monoids.
If you want to take a commutative monoid as ground for the underlying group of the ring you probably want to take the of its Grothendieck group befor... | {
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Proving $\sqrt{3+\sqrt{13+4\sqrt{3}}} = 1+\sqrt{3}$. How would I show $\sqrt{3+\sqrt{13+4\sqrt{3}}} = 1+\sqrt{3}$?
I tried starting from the LHS, and rationalising and what-not but I can't get the result...
Also curious to how they got the LHS expression from considering the right.
| $$ \sqrt{3+\sqrt{13+4\sqrt3}}= \sqrt{3+\sqrt{1+2\times2\sqrt{3}+(2\sqrt{3})^2}}$$
$$=\sqrt{3+\sqrt{(1+2\sqrt{3})^2}}$$
$$=\sqrt{3+1+2\sqrt{3}}$$
$$=\sqrt{(1+\sqrt{3})^2}$$
$$=1+\sqrt3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2405667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equivalence between $\sigma$-additivity and a certain condition, on a certain set of equivalence classes I'm facing a certain part of a problem, and I don't know how to solve it. The background is the following:
In a probability space $(\Omega,\mathcal{B},\mathbb{P})$, two sets $A,B$ are equivalent if $\mathbb{P}(A\big... | Let $A_k$ be a sequence of pairwise disjoint sets. Set $B_n = \bigcup_{k=n}^\infty A_k$. Then $B_n \downarrow \emptyset$ so by assumption $d(B_n^\#, 0^\#) \to 0$ which is to say $\mathbb{P}(B_n) \to 0$. Now by finite additivity, for any $n$ you have
$$\mathbb{P}\left(\bigcup_{k=1}^\infty A_k\right) = \sum_{k=1}^{n-1... | {
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On polynomials over finite ring Let $R$ be a finite commutative ring with unity ; then does there exist a non-empty proper subset $A \subseteq R$ and $f(X) \in R[X]$ such that $f(r)=1 , \forall r \in A$ and $f(r)=0 , \forall r \notin A$ ?
| Such an $f$ exists iff $R$ is local.
First, suppose $R$ is local. Since $R$ is finite, its unique maximal ideal is the nilradical, so every element of $R$ is either nilpotent or a unit. There is then some $n$ such that $r^n=0$ for every nilpotent $n$ and $r^n=1$ for every unit $r$. You can then take $f(X)=X^n$.
Now ... | {
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Binary Relation composition is associative proof explanation and set memberships and set definitions request
What set or relation does y,x belong to?
What set or thing does w belong to?
What set or thing does z belong to?
I have a hard time keeping track what set w,z,y are members of, are part of whose domain and cod... | In general if $R\subseteq A\times B$ and $S\subseteq B\times C$ then: $$S\circ R:=\{\langle a,c\rangle\mid\exists b\in B[\langle a,b\rangle\in R\wedge\langle b,c\rangle\in S\}\subseteq A\times C$$
So starting with relations $R\subseteq A\times B$, $S\subseteq B\times C$ and $T\subseteq C\times D$ we have $S\circ R\subs... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $\int_{-\infty}^{+\infty}te^{-|t|} \, dt$ converge? Good evening
Does $\displaystyle \int_{-\infty}^{+\infty}te^{-|t|} \, dt$ converge?
I have got a series of exercices without corrections so I carry on with a new exercice.
My solution :
$te^{-|t|}=\dfrac{t}{e^{|t|}}= \dfrac{t}{e^{\frac{|t|}{2}}}\times\dfrac{1}{... | You can also integrate by parts to an upper bound $R$ and take limits as $R \to \infty$:
$$
\int_0^R t \, e^{-t} \, dt
= \left[ t \, \left( -e^{-t} \right) \right]_0^R - \int_0^R \left( -e^{-t} \right) \, dt
= -R \, e^{-R} + \int_0^R e^{-t} \, dt \\
= -R \, e^{-R} + \left[ -e^{-t} \right]_0^R
= -R \, e^{-R} - e^{-R} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2406214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compactness of the closed interval [0,1] In general topology, a topological space is said to be compact, if every one of its open cover has a finite subcover.
However, I cannot see the compactness of the close interval [0,1] from the above definition.
To be a little specific,let us consider the following open cover for... | So you get an open cover by retaining $[0,1/2)$ and $(2/3,1]$
but replacing $(1/3,3/4)$ by a bunch $P$ of open sets where no finite
collection covers $(1/3,3/4)$.
You can do this.
But it is still the case that this new covering $C'$ has a finite
subcovering. Don't forget that $[0,1/2)$ and $(2/3,1]$
are still available... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2406432",
"timestamp": "2023-03-29T00:00:00",
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Proof of $ r(I)^e\subset r(I^e)$ in ring theory with extension and radical
Let $f:A\to B$ be a ring homomorphism and $I$ be an ideal of $A$. Then prove that $\displaystyle r(I)^e\subset r(I^e)$, where $I^e$ denotes the extension of the ideal $I$ and $r(I)$ denotes the radical of the ideal $I$.
Let $x\in r(I)^e=Bf(r(I... | As user26857 pointed out, $x=yz$ is not right. Remember that $I^e=\langle f(I)\rangle$, so it should be instead $x=\sum_{i=0}^n b_if(u_i)$ for some $n\in \Bbb Z^+$, where $b_i\in B$ and $u_i\in \sqrt{I}$.
Now, as $u_i\in \sqrt{I}$, then there is $k_i\in \Bbb Z^+$ such that $u_i^{k_i}\in I$. Therefore, $$\bigl(b_if(u_i... | {
"language": "en",
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Show that for $x \in (-\pi, \pi)$ and $T(x) := \sum_{n=0}^\infty(\frac{x}{\pi})^n$ the derivation $T'(x) = \frac{\pi}{(x-\pi)^2}$ $\frac{x}{\pi} < 1$ for $x\in (-\pi, \pi)$, therefore we could use the formula for geometric series to get the limit.
$$\sum_{n = 0}^\infty(\frac{x}{n})^n = \frac{1}{1-\frac{x}{\pi}} = \frac... | You have already proved that, for $x \in (-\pi,\pi)$, by using a geometric series result,
$$
T(x)= \frac{\pi}{\pi-x}
$$ then to obtain $T'(x)$ just use
$$
\left(\frac 1u\right)'=-\frac{u'}{u^2}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2406640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Endomorphism rings of isogeneous elliptic curves Let $E$ and $E'$ be isogenous elliptic curves and $K=\text{end}(E) \otimes \mathbb(Q) $
Is it true that $\text{end}(E') $ is a subring of $K$?
The only thing I thought is that the isogeny between $E$ and $E'$ and its dual give a map between the endomorphism rings, which... | I think it is not true at least over $\mathbb{C}$
Let us consider the elliptic curve $E$ whose lattice is given by $\langle 1,\frac{i}{2} \rangle$ which has not the CM
Now let us consider the isogeny given by the quotient for the translation of the point $\frac{1}{2}$ then you get an elliptic curve $E’$ whose lattice i... | {
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proof that the pattern exists for all n Explain the pattern
$$(\sqrt2-1)^1= \sqrt2 - \sqrt1$$
$$(\sqrt2-1)^2 = \sqrt9 - \sqrt8$$
$$ (\sqrt2-1)^3 = \sqrt{50} - \sqrt{49} $$
that is $(\sqrt2-1)^n$ is equal to difference of two consecutive numbers one of which are squares.
| This is related to the problem of finding which triangular numbers are a square, which leads to a Pell equation for $\sqrt 2$, and is related to the units of $\mathbb Z[\sqrt 2]$. Indeed, $(\sqrt2-1)^{-1}=\sqrt2+1$ is the fundamental unit.
The pattern is $(\sqrt2-1)^n = \sqrt{a_n+1} - \sqrt{a_n}$, where the $a_n$-th tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2406860",
"timestamp": "2023-03-29T00:00:00",
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Prime dividing repunit Let $ R(n) = \underbrace{111\ldots111}_{\text n\ ones}$. Prove that if a prime number $ p \neq 3 $ divides $ R(n) $ then $ n $ and $ p - 1 $ are not coprime.
So obviously $ R(n) = \frac{10^n - 1}{9}$. Now if $ p $ divides $ R(n) $ then
$$ \frac{10^n - 1}{9} \equiv 0 \pmod p $$
which implies
$$ 10... | Use little Fermat $10^{p-1} \equiv 1\bmod p$
| {
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"url": "https://math.stackexchange.com/questions/2406976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all positive integers $n > 1$ such that the polynomial $P(x)$ belongs to the ideal generated by the polynomial $x^2 +x +1$ in $\Bbb Z_n[x]$
Find all positive integers $n > 1$ such that the polynomial $x^4 + 3x^3 + x^2 + 6x + 10$ belong to the ideal generated by the polynomial $x^2 + x + 1$ in $\Bbb Z_n[x]$.
My a... | I think Bill Dubuque addressed the problem, but I would note that when you were looking at the remainder $r(x) = x^2 + x + 1$, you were trying to plug in values for $x$ to show that $r = 0$. This is not what you want to do.
A polynomial is zero if and only if all its coefficients are zero. Like in your example, you can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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For which values of $a\in\mathbb{Q}$ does integer solutions to $x^2+x+1=a(y^2+1)$ exist? I am unable to determine for which values of $a\in\mathbb{Q}$ does integer solutions to
$$x^2+x+1=a(y^2+1)$$
in the form $(x,y)$ exist.
My initial idea was to set $a=\frac{c}{d}$ to get $dx^2+dx+d=c(y^2+1)$ for $c,d\in\mathbb{Z}$ b... | Surely a zest of Galois theory could do no harm.
1) Consider first the equation with rational parameter and variables : $x^2+x+1=a(y^2 +1), a, x, y \in \mathbf Q^*$ (1). With $i^2=-1$ and $j^3=1$, (1) is equivalent to $a= N_3(x-j).N_1(y-i)^{-1}$, where $N_n$ denotes the norm map of $\mathbf Q(\sqrt -n)/\mathbf Q$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 2
} |
Power set of any set. Question:
Let $A$ be any set. Let $\mathbb{P}(A)$ be the power set of $A$. Then which one is true?
*
*$\mathbb{P}(A)=\emptyset$ for some $A$.
*$\mathbb{P}(A)$ is finite for some $A$.
*$\mathbb{P}(A)$ is countable for some $A$.
*$\mathbb{P}(A)$ is uncountable for some $A$.
Now 1. is not t... | Assuming countable means "countably infinite" (so finite sets are not considered countable), 3 is false. If $A$ is finite, then $\mathbb{P}(A)$ is finite.
If $A$ is infinite it contains a countably infinite subset $B$ (mild form of choice used here). And then $2^{\mathbb{N}} = |\mathbb{P}(B)| \ge |\mathbb{P}(A)|$ so th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Prove that $ \lfloor 2x \rfloor \leq 2\lfloor x \rfloor + 1$
Theorem. For $x \in \mathbb R$, $$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1.$$
I tried to prove this theorem by first proving a few helper theorems. I have proved the following Lemma.
Lemma. For $x \in \mathbb R$ and $n \in \m... | $$\begin{array}{rcl}
x &<& \lfloor x \rfloor + 1 \\
\lfloor 2x \rfloor &\le& 2x \\
\lfloor 2x \rfloor &<& 2\lfloor x \rfloor + 2 \\
\lfloor 2x \rfloor &\le& 2\lfloor x \rfloor + 1 \\
\end{array}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Simplify $\frac1{\sqrt{x^2+1}}-\frac{x^2}{(x^2+1)^{3/2}}$ I want to know why
$$\frac1{\sqrt{x^2+1}} - \frac{x^2}{(x^2+1)^{3/2}}$$
can be simplified into
$$\frac1{(x^2+1)^{3/2}}$$
I tried to simplify by rewriting radicals and fractions. I was hoping to see a clever trick (e.g. adding a clever zero, multiplying by a cle... | Factor from $\dfrac{1}{\sqrt{x^2+1}}$. You will have:
$$\frac{1}{\sqrt{x^2+1}} \bigg(1 - \frac{x^2}{x^2+1}\bigg) = \frac{1}{\sqrt{x^2+1}} \frac{1}{x^2+1} = \frac{1}{(x^2+1)^\frac{3}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Measure theory in practice I am trying to unite my knowledge of statistics and measure theory by considering the following example.
Suppose we have a measurable space $(\Omega_1,B_1)$ and a random variable (measurable function) on the space, call it $X$: $\Omega_1 \rightarrow R$.
Suppose we know the distribution funct... | In order to properly speak of a random variable's distribution, you need a measure space (with a measure), and specifically a probability space $(\Omega, \mathscr A, \mathbb P)$, where of course $\mathbb P (\Omega) = 1$.
The distribution of a random variable $X$ is the image measure $X(\mathbb P)$, i.e.
$$ X(\mathbb P)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
The logarithmic inequality $\ln^q (1+x) \le \frac{q}{p} x^p \quad (x \ge 0, \; 0 < p \le q)$ $$\ln^q (1+x) \le \frac{q}{p} x^p \quad (x \ge 0, \; 0 < p \le q)$$
For $p=q$ this reduces to the familiar $\ln(1+x) \le x$. Otherwise I haven't had much success in proving it. General suggestions would be appreciated.
| put $p=\frac{1}{\ln x}$ and when $x > e$ then $p <1$ and we arrive at
$\ln^q(x+1) \leq q \ln x * x^{p}$ which is $\ln^q (x+1) \leq q \ln x *e$
when $q \geq 2$ we get that $\ln^q(x+1) \leq e q \ln x$
Because $\ln^q(x+1)\geq \ln^q x \geq e q \ln x$ divide by $\ln x$
We get that $\ln^{q-1} x \geq e q $ and since $q \g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve this PDE with method of characteristics? I have this problem $$y \frac{\partial u}{\partial x}-x\frac{\partial u}{\partial y}=1,\\u(x,0)=0$$
Using the method of characteristics I have
$$\frac{dx}{dt}=y \\ \frac{dy}{dt}=-x \\ \frac{du}{dt}=1$$
Then $$\frac{dx}{y}=\frac{dy}{-x} \\ x^2+y^2=\eta $$
and
$$u=t... | You have done well so far. The system of ODE's is correct. Notice that $\xi=0$ due to your condition $u(x,0)=0$. However, your method to invert the $x$ and $y$ coordinates does not lead you anywhere. The following method to solve the system will be more useful.
$$\frac{d}{dt}\left(\frac{dx}{dt}=y\right)\implies\frac{d^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2407994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
For any $\epsilon$, if $\epsilon>0$ and $|x|<\epsilon$, then $x=0$ For any $\epsilon$, if $\epsilon>0$ and $|x|<\epsilon$, then $x=0$.
I understand that supposing $\epsilon=\frac{x}{2}$ will lead to a contradiction, but let’s take a correct case:
Let $\epsilon=3$, then $x$ would have a whole set of values. Can you exp... | If it is true for any real $\epsilon>0$, it will be true in particular, for any rational $r>0$,
so we will have, for any natural $n $,
$$|x|<\frac {1}{n+1} $$
or
$$-\frac {1}{n+1}<x <\frac {1}{n+1} $$
and the squeeze theorem gives
$$0\le x\le 0$$
The downvoter is a zam.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What are some mathematically interesting computations involving matrices? I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices.
Preferably these examples would be ... | Rotation matrices are a typical example of useful matrices in computer graphics
$${\bf R_\theta} = \begin{bmatrix}\cos(\theta)&\sin(\theta)\\-\sin(\theta)&\cos(\theta)\end{bmatrix}$$
They rotate a vector around origo by the angle $\theta$.
If you want to make it more complicated you can make them in 3 dimensions. For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 26,
"answer_id": 17
} |
Show right half-plane with points of closed unit disk removed is not a star domain Consider the right half-plane $\{z\in {\mathbb {C}}:{\mbox{Re}}(z)>0\}$. We define a set $X$ by removing from the right half-plane the points of the closed unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| \leq 1 \}$.
I want to show that... | Suppose that $X$ is a star domain and let $z=(x,y)$ be the center of the star.
Claim: For $\epsilon>0$ sufficiently small, the line between $z$ and $(\epsilon,1+\epsilon)$ or the line between $z$ and $(\epsilon,-1-\epsilon)$ intersects the unit disk.
Sketch: If the imaginary part of $z$ is greater than $0$, i.e., $y>... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
vector normal to a plane There is this "shortcut" we learned that helps us find a vector perpendicular to a plane. Say, $ax+by+cz+d=0$ is the plane equation, then the vector $(a,b,c)$ is normal to this plane.
But why is this? Why does $d$ contribute nothing to the normal vector?
For example, let $(x,y,z)$ be a point on... | A normal vector $\textbf{N}$ to a plane $P$ is a vector such that for all $\textbf{v} \in P$, $\textbf{v} \perp \textbf{N} \Rightarrow \textbf{N} \cdot \textbf{v} = 0$. Given any point $p_0, \textbf{x} = (x,y,z) \in P$, we defined $\textbf{x} - p_0$ to be the vector which extends from $p_0$ to $\textbf{x}$. Hence, $\te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to solve the recurrence relation $a_n - 2 a_{n-1} = 3 \times 2^n, a_0 = 1$ How to solve the recurrence relation $a_n - 2 a_{n-1} = 3 \times 2^n, a_0 = 1$. By looking at the terms of the relation, it can be seen that it is linear in nature but it is not homogeneous. How to solve such a recurrence relation?
| Hint:
Let $a_m=am2^m+b_m$
$$3\cdot2^n=a_n-2a_{n-1}=an2^n+b_n-2\{a(n-1)2^{n-1}+b_{n-1}\}=a2^n+b_n-2_{n-1}$$
Set $a=3$ to find $b_n-2_{n-1}=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Is following option is correct or incorrect? Which of the following statements are true?
(a). Let $X$ be a set equipped with two topologies $\tau_1$ and $\tau_2$. Assume that any
given sequence in $X$ converges with respect to the topology $\tau_1$ if, and only
if, it also converges with respect to the topology $\tau_2... | In $b)$ the $" \Leftarrow"$ part is not always true unless $(X,\tau)$ is a first countable space.
Here is a counterexample:
sequentially continuous on a non first-countable
$c)$ is not always true.Here is a reference for counterexample. The space $[0,1]^{[0,1]}$ which is compact from Tychonov's theorem.
https://www.a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How much cash is in the wallet In the wallet we have $26$ banknotes. If we take, arbitrarily, $20$ of them we are sure that we have at least one of $\$5$, at least two of $\$10$ and at least five of $\$20$. How much cash is in the wallet?
I have tried determining the number of compositions of $20$ in three parts with r... | The problem is much simpler. If we take 20 of the 26 notes, and we have at least five $\$20$ notes, then there must be at least eleven $ \$20$ notes (since otherwise the guarantee does not work). For similar reasons, there must be at least seven $ \$5$ notes, and at least eight $ \$10$ notes. Because $11+7+8 = 26$, thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Express it in its reduced form : $\sum\limits_{k=1}^{n}{(C(n,k-1)*C(n,k))}$ As we know $C^{2}(n,0)$+$C^{2}(n,1)$+$C^{2}(n,2)$+....+$C^{2}(n,n)$ =
$C (2n,n)$
By deducing it from $ (1+x)^{n}$
So, how can I find the reduced form of $\sum\limits_{k=1}^{n}{(C(n,k-1)*C(n,k))}$
From $ (1+x)^{n}$
Please help me to solve thi... | Let $[x^k]: (1+x)^n$ denote the coefficient of $x^k$ for the function $(1+x)^n$ ; that is $\binom{n}{k}$. Now
\begin{eqnarray*}
\sum_{k=1}^{n} \binom{n}{k-1} \binom{n}{k} = \sum_{k=1}^{n} [x^k]: \binom{n}{k-1} (1+x)^n \\
= [x^n]: \sum_{k=1}^{n} \binom{n}{k-1} x^{n-k} (1+x)^n \\
= [x^n]: x^{n-1}(1+\frac{1}{x})^n (1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Difference between “ proof by reductio ad absurdum” and “proof by contradiction”? I always thought that both “proof by reductio ad absurdum” and “proof by contradiction” mean the same, but now my professor asked this question on my homework and I don't know.
I believe that in both cases you assume the negation of the c... | Regarding the rule of indirect proof:
"if from assumption $\lnot A$ a contradiction follows, we can infer $A$",
we can see:
*
*Jan von Plato, Elements of Logical Reasoning, Cambridge UP (2013), page 81:
Sometimes the nomenclature RAA is used; it stands for reductio ad absurdum,
the mediæval Latin name of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2408906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\mathcal{M}_{n}(\mathbb{C})$ as a Hilbert space Let $\mathcal{M}_{n}(\mathbb{C})$ be the set of $n\times n$ matrices over $\mathbb{C}.$ I know that for $A,B\in \mathcal{M}_{n}(\mathbb{C}),$ $$\langle A,B \rangle=\text{tr}(B^{*}A)$$ defines an inner product on $\mathcal{M}_{n}(\mathbb{C})$ and hence we can induce the n... | Since $\mathcal{M}_n(\mathbb{C})$ is finite-dimensional, it is complete. Whence the result.
In fact, $\mathcal{M}_n(\mathbb{C})$ equipped with the given Hermitian product is just $\mathbb{C}^{n^2}$ with its usual Hermitian structure.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Given $M=\{1,2,3,4\}$ find a topology on $M$ of minimum $3$ elements which makes $x=\{1,2,1,2,\dots\}$ converge. So, I've been having trouble with understanding this exercise:
It sounds like this:
Given $M=\{1,2,3,4\}$ find a topology on $M$ of minimum $3$ elements which makes $x=\{1,2,1,2,\dots\}$ converge.
I don't re... | Recall that a sequence $\{x_n\}$ converges to some $x$ in a topological space if for every open set $U$ with $x\in U$, there is some $N$ sufficiently large that $x_n \in U$ whenever $n>N$.
If the sequence $\{1,2,1,2,\dotsc\}$ is to converge to some $x\in M$, then every open set containing $x$ must contain both 1 and 2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
From normal distribution to the lognormal distrubtion; where does $1/x$ come from? So the normal distribution is given by $\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$; Now the lognormal distribution is related to this by $y = e^x$ so the distrbution should be $\frac{1}{\sqrt{2\pi\sigma^2... | Apply the following theorem, from Requirements for transformation functions in probability theory
Let $X$ be an absolutely continuous random variable with support $S$ and probability density function $f(x)$. Let $g: \mathbb{R} \to \mathbb{R}$ be one-to-one and differentiable on $S$. If
$$\frac{dg^{-1}(y)}{dy} \ne 0, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
There is an easier way to compute manually $e^{tA}$? I have this matrix
$$A:=\begin{bmatrix}0&-1&1\\0&0&1\\-1&0&1\end{bmatrix}$$
And I have tested that $A^3\neq I$ but $A^4=I$, and I want to find $e^{tA}$. Then what I did was
$$e^{tA}=\sum_{k=0}^\infty\frac{(tA)^k}{k!}=I\sum_{k=0}^\infty\frac{t^{4k}}{(4k)!}+A\sum_{k=0}... | Let$$P=\begin{pmatrix}1-i&1+i&0\\-i&i&1\\1&1&1\end{pmatrix}.$$The columns of $P$ are eigenvectors of $A$. Then$$P^{-1}=\frac14\begin{pmatrix}1+i & -1+i & 1-i \\ 1-i & -1-i & 1+i \\-2 & 2 & 2\end{pmatrix}$$and$$P^{-1}.A.P=\begin{pmatrix}i&0&0\\0&-i&0\\0&0&1\end{pmatrix}.$$Therefore$$P^{-1}.e^{tA}.P=\begin{pmatrix}e^{ti}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2409282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
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