Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove $x^n-p$ is irreducible over $Z[i]$ where $p$ is an odd prime. Prove $x^n-p$ is irreducible over $Z[i]$ where $p$ is an odd prime.
By gausses lemma this is equivalent to irreducability over $\mathbb{Q}(i)$. Using field extensions this is easy. $[\mathbb{Q}(i,\sqrt[n]{p}):\mathbb{Q}(i)][\mathbb{Q}(i):\mathbb{Q}]=[\... | To prove this via Eisenstein's criterion, use the fact that $\mathbb Z[i]$ is a principal ideal domain. In fact, it is Euclidean. Also, for odd primes $p$ in $\mathbb Z$, $p$ remains prime in $\mathbb Z[i]$ for $p=3\mod 4$, and $p$ factors as a product of two distinct primes $p=p_1p_2$ in $\mathbb Z[i]$ for $p=1\mod 4$... | {
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Is $f(x)=\frac{x^{2}-1}{x-1}$ continuous at $x=1$? Given $f(x)=\frac{x^{2}-1}{x-1}$. The function is said to be discontinuous at $x=1$ but since we can simplify it and rewrite $f(x)=x+1$, this removes the discontinuity. So is the function continuous or discontinuous at $x=1$
How do the two forms of $f(x)$ differ as bot... | $\lim_{x\rightarrow 1} f(x) = 2$, so the singularity is removable.
Let $$g(x) = \left\{ \begin{array}{cc} f(x), & x \ne 1\\ 2, & x=1 \end{array} \right.$$ The function $g(x)$ is continuous.
So, yes, $f(x)$ is discontinous but this discontinuity is easily repaired. If you were to graph $y=f(x)$, it would be the stra... | {
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Relation between diagonal entries of $A^{-1}$ and inverse values of $a_{ii}$ for positive definite $A$. I'd like to expand upon this question. Namely, it says that if $A$, $A=A^T$, is a positive definite matrix, then it holds that \begin{equation}\tag{*}(A^{-1})_{ii}\ge \frac1{A_{ii}}.\end{equation}
Can we prove the co... | No, that's even wrong for numbers, e.g. $A = -1$.
| {
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2 Cross Products? Usually, if we want to find the cross product of 2 vectors $\vec{b}$ and $\vec{c}$, we want to find the vector which is perpendicular to both of them. Let's say the cross product of $\vec{b}$ and $\vec{c}$ is $\vec{d}$. Isn't $-\vec{d}$ then also perpendicular to $\vec{b}$ and $\vec{c}$? Does that mea... | The cross product of $\vec b$ and $\vec c$ is defined as the vector with the following properties:
*
*The length of the product is equal to $|\vec b|\cdot|\vec c|\cdot\sin(\alpha)$, where $\alpha$ is the angle between the two vectors.
*The product is perpendicular to both $\vec b$ and $\vec c$.
*The direction of th... | {
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Let $p(x)$ be a polynomial with integer coefficients. Show that if $p(2)=3$ and $p(3)=5$ then $p(n)\ne0$ for all integers $n$. Let $p(x)$ be a polynomial with integer coefficients. Show that if $p(2)=3$ and $p(3)=5$ then $p(n) \neq 0$ for all integers $n$.
I did manage to solve it using the fact that $a-b | p(a)-p(b)$ ... | If $n$ is even, then $p(n)\equiv p(2)\bmod 2$ and likewise for $n$ odd. In both cases, we have $p(n)\equiv 1\bmod 2$, so $p$ is odd at every integer.
| {
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Inverse Laplace Transform, need help with this $\frac{s^2}{s^2+\sqrt{2}s+1}$ Inverse Laplace Transform of $\frac{s^2}{s^2+\sqrt{2}s+1}$
I transformed the denominator as $(s+\frac{\sqrt{2}}{2})^{2}$ + $\frac{1}{2}$
$\frac{s^2}{(s+\frac{\sqrt{2}}{2})^{2} + \frac{1}{2}}$ and I have no idea how to move forward because part... | Hint:
$$\frac{s^2}{s^2+\sqrt2\,s+1}=1-\frac{\sqrt2\left(s+\frac{\sqrt2}2\right)}{\left(s+\frac{\sqrt2}2\right)^2+\frac12}$$
| {
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Convergence of Lebesgue measurable sets I've been working on the following result:
Let $f$ be Lebesgue measurable on $[0,1]$ with $f(x)>0$ almost everywhere on $[0,1]$. Assume there are measurable sets $E_k \in [0,1]$ with $\int_{E_k} f(x)\to 0$ as $k \to \infty$. Then $m(E_k) \to 0$ as $k \to \infty$.
I've been attemp... | There exist $k_1,k_2<...$ such that $\int_{E_{k_j}} f(x)dx <\frac 1 {2^{j}}$. Hence $\int \sum_j 1_{E_{k_j}} f(x)dx <\infty$. This implies that $\sum_j 1_{E_{k_j}} f(x) <\infty$ almost everywhere. Since $f(x) >0$ a.e. this gives $\sum_j 1_{E_{k_j}} <\infty$ almost everywhere. Hence $\lim \sup E_{k_j}$ has measure $0$... | {
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Prove that $(a_1 − 1)(a_2 − 2)...(a_9 − 9)$ is always an even number. $a_1, a_2,..., a_9$ is an arbitrary permutation of positive integers from 1 to 9. Prove that $(a_1 − 1)(a_2 − 2)...(a_9 − 9)$ is always an even number.
So I don't understand what the question is asking by "arbitrary permutation of positive integers".... | The only way the product $(a_1 − 1)(a_2 − 2) \dots (a_9 − 9)$ could be odd is if $a_1, a_3, a_5, a_7$, and $a_9$ are all even numbers. That would take $5$ even numbers ...
| {
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How many $4$-digit numbers of the form $1a2b$ are divisible by $3$? How many $4$-digit numbers of the form $\overline{1a2b}$ are divisible by $3?$
Hello I am new here so I don’t really know how this works. I know that for something to be divisible by 3, you add the digits and see if they are divisible by $3$. So that m... | Giving you a hint :-
You got $3 + a + b = 6,9,12,15,18$ or $21$, which implies that $a + b = 3,6,9,12,15$ or $18.$ Now do Case-Work and find all possible $a,b$ which can satisfy these . This may take a bit of work.
$($For e.g. when $a + b = 3$ we have $(a,b) = (0,3),(1,2)(2,1)(3,0))$
Note that you forgot the case when... | {
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How to evaluate the volume of tetrahedron bounded between coordinate planes and tangent plane? Find the volume of the tetrahedron in $\mathbb{R}^3$ bounded by the coordinate planes $x =0, y=0, z=0$, and the
tangent plane at the point $(4,5,5)$ to the sphere $(x -3)^2 +(y -3)^2 +(z -3)^2 = 9$.
My attempt: I started with... | The plane intercept the axes in the points
\begin{align}
A&=(24,0,0), \\
B&=(0,12,0), \\
C&=(0,0,12)
\end{align}
so the volume is
$$
V=\frac{1}{6}\cdot24\cdot12\cdot12=576
$$
| {
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Open source software for calculation of eigenvalues of symbolic matrix I have following matrix
\begin{bmatrix}
-\alpha & 0 & \beta & \gamma\cdot\omega_m \\
0 & -\alpha & -\gamma\cdot\omega_m & \beta \\
R_r\frac{L_h}{L_r} & 0 & -\frac{R_r}{L_r} & -\omega_m \\
0 & R_r\frac{L_h}{L_r} & \omega_m & -\frac{R_r}{L_r}
\end{bm... | This response is (perhaps) barely appropriate as an answer rather than a comment. However, it may well be the best that the OP can do.
First of all, consider trying to programmatically identify the general roots of a quartic equation. Although the general formula is somewhat unwieldy, writing a computer program (e.g.... | {
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What is the intuition behind pushouts and pullbacks in category theory? What is the intuition behind pullbacks and pushouts? For example I know that for terminal objects kind of end a category, they are kind of last is some sense, and that a product is a kind of pair, but what about pullbacks and pushouts what are the ... | Pullbacks are fibred-products, i.e., a product with some compatibility restrictions. The terminology came from differential geometry when you really pull differential forms or their bundle on $B$ back to differential forms or their bundle on $A$ along immersion $A\to B$. Product $A\times B$ is just a special case whe... | {
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Evalution of a function where $t = x + \frac{1}{x}$ Consider a function $$y=(x^3+\frac{1}{x^3})-6(x^2+\frac{1}{x^2})+3(x+\frac{1}{x})$$ defined for real $x>0$. Letting $t=x+\frac{1}{x}$ gives: $$y=t^3-6t^2+12$$
Here it holds that $$t=x+\frac{1}{x}\geq2$$
My question is: how do I know that $t=x+\frac{1}{x}\geq2$ ?
I w... | Because by AM-GM $$x+\frac{1}{x}\geq2\sqrt{x\cdot\frac{1}{x}}=2.$$
Your calculation of $y$ is right:
$$y=t^3-3t-6(t^2-2)+3t=t^3-6t^2+12$$ and you got it without using $t\geq2$.
| {
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Checking Presentations in GAP If I have the following presentation for $A_5$ $$\langle x,y,z\mid x^3 = y^3= z^3 =(xy)^2=(xz)^2= (yz)^2= 1\rangle$$ with subgroup $$ H = \left\langle {x,y} \right\rangle$$ and let GAP apply coset enumeration to my generators and relations, as with the code below, is there a command I can... | Yes.
Use IdGroup(G);. If G is indeed (a presentation for a group isomorphic to) $A_5$, the output of this command is [60, 5].
| {
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Under what conditions does $ \ (a+b)^{n}=a^{n}+b^{n}$ for a natural number $ n \geq 2$? Under what conditions does $ \ (a+b)^{n}=a^{n}+b^{n}$ holds for a natural number $ n \geq 2$?
My attempt at solving:
Using $(a+b)^2=a^2+2ab+b^2$; if $(a+b)^2=a^2+b^2$, $2ab=0$ therefore $a$ and/or $b$ must be $0$.
If $a$ and/or $b$ ... | Suppose $a\neq0$. Then
$$(a+b)^n=a^n\Big(1+\tfrac{b}{a}\Big)^n$$
Then, to is enough to consider the question for which values $x$ is $(1+x)^n=1+x^n$. For then $(a+b)^n=a^n+b^n$ with $b=ax$.
This leads to finding all the roots of
$$
p_n(x):=\sum^{n-1}_{k=1}\binom{n}{k}x^k=0
$$
When $n=2$, $p_2(x)=2x=0$ and so $x=0$
when... | {
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Uniform integrability and stochastic dominance
Let $(X_n)$ be a sequence of random variables, and $Y$ a integrable random variable with
$$\sup P(|X_n| \ge a) \le P(Y \ge a),$$
for all $a \in \mathbb{R}$. Show that $(X_n)$ is uniformly ntegrable.
This may be a stupid question, but I am having doubts if my solution is ... | Not quite. The best way to go about this is using the Darth Vader Rule.
Applied here, it gives us that
$$E(|X_n|1_{|X_n| \geq a})= \int_a^\infty P(|X_n|\geq a)dx \leq \int_a^\infty P(|Y|\geq a)dx = E(|Y|1_{|Y|\geq a}).$$
| {
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Every root of $x^n-1$ is simple in $ \mathbb{Z}_p[x]$ Let $p$ be a prime number s.t $p$ doesn't divide $n$. Show that every roots of $x^n-\overline{1}$ is simple in $\mathbb{Z}_p$
If $\overline {a} \in \mathbb{Z}_p$ is a root of $x^n - \overline{1}$ then $a^n \equiv 1\pmod p$ and $\gcd(a,p)=1$. By Fermat's little theor... | Hint:
If $f(a)=0$, and $a$ is not a simple root, what can you say about $f'(a)$? (The formal derivative)
Define $f(x)=x^n-1\in\Bbb Z_p[x]$, and let the group-homomorphism $$\frac{d}{dx}:\Bbb Z_p[x]\rightarrow\Bbb Z_p[x]$$ be the formal derivative.
Detecting multiplicity, and why it works
Let $f(x)\in R[x]$ be a polyn... | {
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Fundamental Group of Klein Bottle acts on $\mathbb{R}$ It is well know that the Fundamental Group of the Klein Bottle can be defined (up to isomorphism) as the group with two generator and one relation
$$BS(1,-1)=\langle a,b: bab^{-1}=a^{-1}\rangle $$
In Algebraic Topology this fundamental group is defined as the group... | Any one-dimensional representation of this group, will, as for any group, factor through its abelianization, which is $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.
Concretely, we can have $b$ act by multiplication by any non-zero scalar, and $a$ act by multiplication by $-1$ (we could also have $a$ acting trivially), and... | {
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Wronskian of two linearly independent differential functions. Show $c$ in [a,b] such that $g(c) = 0$ exists Let $f,g: [a,b] → R$, two differential functions and suppose $f(a) = f(b) = 0$.
If $W(f,g): [a,b] → R$ and $W(f,g)(x) = f(x)g'(x) - g(x)f'(x)$ doesn't equal 0 for all $x$ in $[a,b]$, show that a $c$ in $[a,b]$ mu... | If possible, take $g\neq 0 $ for all $x\in [a,b] $
Then we can easily define the differentiable function $\frac{f}{g} $ on $[a,b]$.
Clearly, $(\frac{f}{g})(a) = 0 = (\frac{f}{g})(b) $.
Then by Rolle's theorem, there exist at least one $c\in (a,b) $ such that $(\frac{f}{g})' (c) = \frac{gf'-fg'}{g^2}=\frac{-W(f,g)}{g^2}... | {
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Probability involved in information theory. I was reading Information, Entropy, and the Motivation for Source Codes chapter 2 MIT 6.02 DRAFT Lecture Notes(https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/readings/MIT6_02F12_chap... | It is quite immediate :
Suppose that $C$ is a random variable with values in a set $\mathcal{C}$ having a cardinality $N$. Suppose that all possible values of $C$ have the same probability.
Consider a subset $\mathcal{C}'$ of $\mathcal{C}$ that contains $M$ elements.
Then, if you consider the event $E$ : $C \in \math... | {
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Indefinite integral of $\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$ $$\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$$
My approach:
Since it is easy to evaluate $\int{\sec^2x}$ , integration by parts seems like a viable option.
Let $$I_n=\int{\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}}$$
$$I_n=\frac{\tan x}{(\sec x+\tan... | You missed a simplification after second step:
$$I=\frac{\tan x}{(\sec x+\tan x)}\frac{9}{2}+\frac{9}{2}\int \frac{\sec x \tan x}{(\sec x+\tan x)^{\frac92}}dx$$
Now, take the original expression for $I$
$$ I = \int \frac{\sec^2 x}{(\sec x + \tan x)^{\frac92}}$$
Add this $ \frac{9}{2} $ times this to previous expressio... | {
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prove $\left(3, 1+\sqrt{-5}\right)$ is prime ideal of $\mathbb{Z}\left[\sqrt{-5}\right]$ How to prove that $(3, 1+\sqrt{-5})$ is prime ideal of $\mathbb{Z}[\sqrt{-5}]$?
attempt 1: use definition
Consider $a, b, c, d, k_1, k_2 \in \mathbb{Z}$ s.t. $$ac-5bd=3k_1+k_2,\, \, ad+bc=k_2.$$ To prove $\exists j_1, j_2 \in \math... | Hint: The square of norm function $(a^2-5b^2)$ is a multiplicative function in the ring $\mathbb{Z}[\sqrt{-5}]$ for a number $a+b\sqrt {-5}$. Use this to prove the primality by proving one of the factors of the norm is $1$. After showing that the numbers are primes, it is correct that the ideal you describe is prime, b... | {
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How to efficiently sample edges from a graph in relation to its spanning tree Consider a connected, unweighted, undirected graph $G$. Let $m$ be the number of edges and $n$ be the number of nodes.
Now consider the following random process. First sample a uniformly random spanning tree of $G$ and then pick an edge from ... | While the other answer is correct, it requires the computation of $|E| + 1$ many determinants. There is a faster route when $|E|$ is large. The first thing to note is Kirchoff's theorem which states that if $T$ is a uniform spanning tree then
$$P(e \in T) = \mathscr{R}(e_- \leftrightarrow e_+)$$
where $e = \{e_-, e_+... | {
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Ideas for this integral: $\int \frac{\sqrt{\tan{x}}}{\sin{x}} dx$ $$\int \frac{\sqrt{\tan{x}}}{\sin{x}} \mathrm{d}x$$
So I was wondering if this correctly by converting $\sqrt{\tan{x}}$ into $\frac{\sqrt{\sin{x}}}{\sqrt{\cos{x}}}$ therefore I can divide it with $\sin{x}$ and that will give me $\frac{\sqrt{\cos{x}}}{\sq... | $$I=\int \frac{\sqrt{\tan x}}{\sin x} dx$$
Let $\tan x =t^2 \implies \sec^2 x dx=2t dt$
Then $$I=\int \frac{2}{\sqrt{1+t^4}} dt$$
This can be found interms of Elliptic functions.
| {
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Heine Borel Theorem statement (a) I have been following Prof Winston Ou's course on analysis on Youtube.
In the lecture on Heine Borel theorem, he mentioned that a set $E$ in $\mathbb R$ is closed and bounded implies that $E$ is a k-cell (hence $E$ is compact).
I don't understand how he came to this conclusion. For ins... | He means $D$ will be contained in a $k$-cell, and as $k$-cells are compact and $E$ is still a closed subset of it, it will also be compact.
The boundedness in Euclidean space/metric forces the set inside a product of compact intervals..
| {
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Show that the transformation $w=\frac{2z+3}{z-4}$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ Question:
Show that the transformation $w=\frac{2z+3}{z-4}$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$.
My try:
$$\begin{align}\\
&x^2+y^2-4x=0\\
&\implies (x-2)^2+y^2=4\\
&\implies |z-... | Ak19 answered what I would have answered before I got to it, so here's another possible way of how to do it. These transformations map generalized circles to generalized circles; a generalized circle is either a circle or a line. This particular transformation maps the point $z=0$ on the given circle to $w=-\frac34$,... | {
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$n$ is prime iff $\binom{n^2}{n} \equiv n \pmod{n^4}$? Can you prove or disprove the following claim:
Let $n$ be a natural number greater than two , then $$n \text{ is prime iff } \binom{n^2}{n} \equiv n \pmod{n^4}$$
You can run this test here. I have verified this claim for all $n$ up to $100000$ .
| Note that $\displaystyle\binom{n^2}{n} = \frac{1}{(n - 1)!} \frac{n^2 (n^2 - 1) ... (n^2 - (n - 1))}{n} = \frac{1}{(n - 1)!} n (n^2 - 1) ... (n^2 - (n - 1))$
Consider a prime $p > 2$. Then $1, 2, ..., p - 1$ are all invertible modulo $p^4$; thus, so is $(p - 1)!$.
Now consider $\displaystyle\binom{p^2}{p} = \frac{1}{(p... | {
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"url": "https://math.stackexchange.com/questions/3781690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 2
} |
Book recommendation : Olympiad Combinatorics book Can anyone recommend me an olympiad style combinatorics book which is suitable for a high schooler ? I know only some basics like Pigeon hole principle and stars and bars .
I hope to find a book which contains problems which purely test our originality ( the problems w... | One possibility is Problem-Solving Methods in Combinatorics: An Approach to Olympiad Problems by Pablo Soberon. As the title says, it's intended to prepare the student for Olympiad problems, and the author won a gold medal in the International Mathematical Olympiad. Some of the exercises in the book are drawn from re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3781790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 0
} |
Other absolute value definitions in $\mathbb R$ I know these definitions for the absolute value (or module): given a real number $x$, then
$$\bbox[yellow]
{|x|=\begin{cases}x & \text{if } x\geq 0\\ -x& \text{if } x< 0\end{cases}}$$
or
$$\bbox[yellow]
{|x|=\max\{x,-x\}}$$
Are there other definitions in $\mathbb R$ (for ... | Here's some I could think of:
*
*$|x|$ can be defined as the (unsigned) distance of $x$ from the origin.
*It's the even extension of $f:[0,\infty)\to\mathbb R$ where $f(x):=x$.
*$|x|$ is the unique norm on $\mathbb R$ with
$|1|=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3781861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Inverse Laplace Transform via Circuit Analysis [HELP] Inverse Laplace Transform $\frac{1}{s^2 + \sqrt{2}s + 1}$
so what I did it changed the denominator to complete the square format which is $\left(s+\frac{\sqrt{2}}{2}\right)^2 + \frac{1}{2}$, then I can solve for $s$, it will make it as
$$
\left(\left(s+ \frac{\sqrt{... | Once we complete the square, we can use the sine formula and Frequency Shift Theorem to evaluate the inverse transform:
If we accept that
$$\mathcal{L}(\sin(at)) = \frac{a}{s^2+a^2}$$
and
$$\mathcal{L}(e^{ct}f(t)) = F(s-c)$$
where $F(s) = \mathcal{L}(f(t))$, we can take our original fraction:
$\begin{align}
\mathcal{L}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3781963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
$\triangle ABC$ with a point $D$ inside has $\angle BAD=114^\circ$, $\angle DAC=6^\circ$, $\angle ACD=12^\circ$, and $\angle DCB=18^\circ$.
Let $ABC$ be a triangle with a point $D$ inside. Suppose that $\angle BAD=114^\circ$, $\angle DAC=6^\circ$, $\angle ACD=12^\circ$ and $\angle DCB=18^\circ$. Show that $$\frac{B... | Let $\omega$, $O$ be the circumcircle and circumcenter of $\triangle ABC$, respectively. Let $P,Q,R,S$ be four points on the shorter arc $AC$ of $\omega$ dividing this arc into five equal parts.
First, we shall prove that $\triangle RSD$ is equilateral. Let $D'$ be a point inside $\omega$ such that $\triangle RSD'$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3782069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 3,
"answer_id": 0
} |
Simplification of an algebraic determinant After studying some analytic geometry, I came across this step in a solution, however, I am not how they managed to simplify the determinant in this way.
When I tried to evaluate this, I got:
$\frac{bc-ad}{2}+\frac{ad-bc}{2b-2d}$, but didn’t see how this got to the desired fo... | We obtain
$$\frac12 b\frac{ad-bc}{d-b}-\frac12 d\frac{ad-bc}{d-b}+\frac12d=\frac12 (b-d)\frac{ad-bc}{d-b}+\frac12d=\frac12(bc-ad+d)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3782321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Range of Convergence of $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$ $$\sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \ 3^n (x-5)^n}$$
I am trying to use the alternating series test to find a range of $x$ for which $(1) b_n > b_{n+1}$ and $ (2) \lim_{n \to \infty} \frac{1}{n \ 3^n (x-5)^n} = 0$. If $... | By the root test, the series converses for all $x$ such that
$$
\limsup_n\sqrt[n]{\frac{1}{n3^n|x-5|^n}}=\frac{1}{3|x-5|}\lim_n\frac{1}{\sqrt[n]{n}}=\frac{1}{3|x-5|}<1
$$
Thus, the series converges for all $x$ such that $|x-5|>\frac{1}{3}$, i.e., all $x$ in $(-\infty,\tfrac{14}{3})\cup(\tfrac{15}{3},\infty)$.
At the po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3782557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Calculate $\lim_{h\to 0} \frac{\cos(x-2h)-\cos(x+h)}{\sin(x+3h)-\sin(x-h)}$ Calculate $$\lim_{h\to 0} \frac{\cos(x-2h)-\cos(x+h)}{\sin(x+3h)-\sin(x-h)}$$
If I take the limit it results in undefined value.
I try to change the formation using identity $\sin A + \sin B$
$$\lim_{h\to 0} \frac{-2\sin\frac{(2x-h)}2\sin(-3h/2... | Direct evaluation gives $0/0$ so apply L'Hospital's rule:
\begin{align}\lim_{h\to 0}\frac{\frac{d}{dh}\left(\cos(x-2h)-\cos(x+h)\right)}{\frac{d}{dh}\left(\sin(x+3h)-\sin(x-h)\right)}&=\lim_{h\to 0}\frac{2\sin(x-2h)+\sin(x+h)}{3\cos(x+3h)+\cos(x-h)}\\&=\frac{2\sin(x)+\sin(x)}{3\cos(x)+\cos(x)}\\&=\frac{3}{4}\tan(x)
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3782670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Proving a limit using the $\epsilon$ - $\delta$ definition of limit. Given $\lim _{x\to a}\left(f\left(x\right)\right)=\infty$ and $\lim _{x\to a}\left(g\left(x\right)\right)=c$ where $c \in R$, prove $\lim _{x\to a}\left[f\left(x\right)+g\left(x\right)\right]=\infty$.
My attempt:
Let for every $M>0$ exists $\delta_1$ ... | It's useful to start with stating what you want to prove. In this case:
For every $N>0$ there is $\delta>0$ such that
$$ 0<|x-a|<\delta \Rightarrow f(x) + g(x) > N.$$
So, if you choose your $M$ and $\epsilon$ so that $$ M+c-\epsilon \geq N,$$
you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3783040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
If $\lim_{\alpha \to \infty}\alpha P[X > \alpha] = 0$ then $E[X] < \infty$? Let $X$ be a positive random variable. Suppose that $\lim_{\alpha \to \infty}\alpha P[X > \alpha] = 0$ Does this implies that $X$ has finite expectation? that is $E[X] < \infty $
I know that if $E[X] < \infty$ $\Rightarrow$ $\lim_{\alpha \to \i... | Here is an another counter example based on a problem solved here
Consider the probability space $((0,1),\mathscr{B}((0,1)),\lambda)$ where $\lambda$ is Lebesgue's measure restricted to the unit interval, and consider the function $X:(0,1)\rightarrow\mathbb{R}$ defined by
$$X(t):=\frac{1}{t|\log t|}\mathbb{1}_{(0,e^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3783304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Find function $ f(x) $ to ensure the limit has certain value If $\lim_{x\to1}$ $\frac{f(x)}{(x-1)(x-2)} = -3$ , then provide a possible function $y = f(x)$
*I don't understand what the question is asking me and how I should solve it. Can a possible function be $y = f(1)$? Or must I do something else to figure out the a... | Suppose you wanted to get rid of the discontinuity at $x=1$. Create a function $f(x)$ that removes this discontinuity, that is, $f(x)=(x-1)g(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3783455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
If $abc=1$ where $a,b,c>0$, then show that $(a-1+b^{-1})(b-1+c^{-1})(c-1+a^{-1}) \leq 1$. I tried writing everything in terms of $a$ and $c$, but got stuck at $(a-1+ac)(a+1-ac)(1-a+ac) \leq a^2c$ where I thought of trying to show for all $x,y,z>0$ that $(x-y+z)(x+y-z)(-x+y+z) \leq xyz $ and substitute $x=1, y=a$ and $... | This is a well-known inequality from some years of IMO I think. After your substitution, use the following:
$$(x-y+z)(x+y-z)\leq\dfrac{(x-y+z+x+y-z)^2}{4}=x^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3783655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If $x_0=1$ and $x_n=\frac {1}{1+x_{(n-1)}}$, find: $\lim_{x\to\infty} x_n$
If $x_0=1$ and $x_n=\dfrac {1}{1+x_{(n-1)}}$, find $\displaystyle\lim_{x\to\infty} x_n$.
My attempt:
$x_1=1+\dfrac 1 2=\dfrac 3 2$
$x_2=1+\dfrac {1}{1+\frac 3 2}=\dfrac2 5$
Which gives following series:
$$1, \frac32, \frac35,\frac 58, \frac 8 ... | You miscalculated the two first initial terms of the sequence.
$x_{1}=\frac{1}{1+x_{0}}=\frac{1}{1+1}=\frac{1}{2}$ and $x_{2}=\frac{1}{1+x_{1}}=\frac{1}{1+\frac{1}{2}}=\frac{2}{3}$.
Let's try to apply the Fixed Point Theorem. Consider $I=[\frac{1}{2},1]$ and $f(x)=\frac{1}{1+x}$ defined in $I$.
*
*$f$ is continuous i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3783796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Is $g(x) = \frac{x^3+9}{x^2}$ one-to-one? Let $g(x) =\frac{x^3+9}{x^2}$ restricted to $D(g) = (-\infty,0)$. Is it 1-1?
My approach
$g(x) \text{ 1-1 }: g(x_1)=g(x_2) \iff x_1 = x_2$
Let $x_1, x_2 \in D(g)$ then,
$$\frac{x_1^3+9}{x_1^2} = \frac{x_2^3+9}{x_2^2} \iff x_2^2 \cdot x_1^3 + 9x_2^2 - x_1^2 \cdot x_2^3 + 9x_1^... | I would use a calculus approach:
$$g(x) = \frac{x^3+9}{x^2}=x+\frac{9}{x^2}$$
Differentiating gives us $$1-\frac{9}{x^3}>0$$ which means it is a monotonic function over the negative domain.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Nonlinear differential equation with sine function If $y\in C^1(\mathbb{R})$ and $y'(x) =\sin(y(x) +x^2)$ for every $x\in\mathbb{R}$ with $y(0)=0$ I proved that $y$ is smooth and that $y'(0)=y''(0)=0$ and that $y'''(0)>0$ but how can I prove that $y>0$ in $(0,\sqrt{\pi}) $ and $y<0$ in $(-\sqrt{\pi}, 0)$?
| Hint: The constant function $z(x) = 0$ satisfies $z'(x) < \sin (z(x) + x^2)$ for every $x\in (-\sqrt{\pi}, 0) \cup (0, \sqrt{\pi})$, hence it is a sub-solution on those intervals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How is the Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ generated by the one element $(E_{11} - E_{22})$? In the article Classification of simple complex Lie algebras, 2nd paragraph of chapter 7 (p.15, bottom), the author considers the basis
$$
x = E_{21} := \left[
\begin{array}{ll}
0 & 0\\
1 & 0\\
\end{array}
\right], ... | It's not true. Each element generates a $1$-dimensional abelian Lie algebra. However, if you read the proof, he clearly means the ideal generated by $h$, not the subalgebra. Indeed, he is trying to prove that $\mathfrak{sl}_2(\mathbb{C})$ is simple, so he wants ideals.
Also, $[h,h]=0$, not $[h,h]=I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What does consistency of propositional logic means? I know a few proofs of consistency of propositional logic, and all of them are based on very similar things.
We are showing our axioms are tautologies and our inference rules are preserving truth, so we can only prove the tautologies. Since $\left(A\wedge\lnot A\right... | The question is: "What does consistency of propositional logic means?"
My reply .
There are two classical notions of consistency: consistency in the traditional sense and consistency in Post's sense (or consistency in the absolute sense) .
According to the definitions of these notions:
consistency of propositi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why does $A_s = k[U,T,S]/(UT-S) \otimes_{k[S]} k[S]/(S-s)$ simplify to $ A_s = k[U,T] / (UT-s)$? When reading algebraic geometry (on the technique of base change) in the book Algebraic Geometry 1 - Schemes by Ulrich Gortz, et.al, I came up with the following tensor product:
$$
A_s = k[U,T,S]/(UT-S) \otimes_{k[S]} k[S]/... | An arbitrary element of $A_s$ is a $k$-linear combination of elements of the form $p(U,T)S^i\otimes 1$, where $p$ is a polynomial over $k$, in two variables. As we are tensoring over $k[S]$, we have the equality:
\begin{eqnarray*}p(U,T)S^i\otimes 1&=&p(U,T)\otimes S^i\\&=&p(U,T)\otimes s^i\\&=&p(U,T)s^i\otimes 1\end{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Confusion in Applying Gauss Divergence Theorem. Evaluate $\displaystyle \iint \vec{F}\cdot ds$ where $\vec{F}= 3x\hat{i} + 2y\hat{j} -5z\hat{k}$ and $S$ is the portion of the $y = x^2 + z^2$ that lies behind $y =1$ oriented in the direction of positive y -axis
I am trying to solve this with the help of Gauss theorem.
S... | When you closed it off with $S_2$, the $S_1$ with its normal pointing in the positive $y$-direction is the inward normal. So there is an extra negative sign.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3784755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do a set of vertices uniquely determine a polytope? The question in title arises because I am trying to prove that a polytope is the convex hull of its vertices, i.e., $\mathcal{P}=conv(V)$. Here is how far I have got.
Convex hull of a finite set of vectors is a polytope. So for $v_1,...,v_k$, $\mathcal{Q}=conv(v_1,...... | Yes, in general this is how one defines a convex polytope. In general, a convex polytope is defined as the convex hull of a finite set (of vertices), page 14:
Def: A convex polytope $P\subset \mathbb{R}^n$, or symply a polytope, is defined as the convex hull of a non empty finite set $\{x_1,\dots,x_q\}$
So maybe you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is gradient co-variant ? (intuitively ) At 7:47 in this video, the professor defines a function F(x,y) on regular cartesian grid, then later defines the same function on a scaled cartesian grid (where x' = 2x and y'=2y) , now, after this he takes gradient on the function defined on the new grid. And for some reason... | For a single variable note that if $f(x)=x/2$ then $df/dx=1/2 i_x$ (writing the direction of coordinates would be useful here) and after re-scaling 2 times the coordinates we have:
$$f(y)=y/2,\quad y=2x$$
and $$df/dy=1/2 i_y=1i_x$$
and this is the derivative with respect to $y$ and if we convert it to $x$ using chain r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Definition of the Space $\mathbb{R}^\infty$ I'm trying to understand the definition of the topological space $\mathbb{R}^{\infty}$, which Hatcher defines as $\cup_n \mathbb{R}^n$ in his book Algebraic Topology. I'm having trouble making sense of this union since $\mathbb{R}^m$ is not literally a subset of $\mathbb{R}^n... | I think this is just a common shorthand in algebraic topology. More rigorously, I think he's defining $\mathbb{R}^\infty$ as the colimit of the diagram $\mathbb{R}^1\hookrightarrow\mathbb{R}^2\hookrightarrow\dots$ I believe the topology this colimit inherits from each Euclidean space will be the one you're thinking of,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Understanding Difference Between Cauchy-Goursat and Related Theorem I am reading Brown and Churchill's Introductory complex analysis book, which states the Cauchy-Goursat theorem as follows:
If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$ \int_{C} f(z) dz =0 $$ I unde... | I think the main issue here is that the `interior' of a closed curve is not necessarily as easy to define as you might think, especially when you start to consider some very pathological curves in the plane. That's why most authors focus on simple closed curves, for which the Jordan curve theorem gives a nice character... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Quasilinear PDE $u_t + (u^2)_x = 0$ cauchy problem The problem I am trying to solve is:
\begin{equation}\label{eq:3.1}
\begin{cases}
\partial_t u + \partial_x(u^2)=0 & x\in \mathbb{R}, t \in (0,\infty]\\
u(x,0)=
\begin{cases}
0 & x\leq 0\\
x & 0<x\leq 1\\
1 & x>1
\end{cases}
\end{cases}
\... | This PDE is very similar to Burgers equation, and the solution $u(x,t)$ deduced from the method of characteristics reads $u = f(x-2u t)$ in implicit form, where $f = u(\cdot, t=0)$. Following the steps in the linked post (see also the comments section), we find
$$
u(x,t) = \left\lbrace
\begin{aligned}
&0 & & x\leq 0\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why this simple relation between two complicated sums? I have the two following sums:
$$A_N =\sum_{n=0}^N\sum_{\substack{m=0 \\ m\neq n}}^N 1/\sqrt{n+m-2\sqrt{nm}}$$
$$B_{N,p} =\sum_{n=0}^N\sum_{\substack{m=0 \\ m\neq n}}^N 1/\sqrt{n+m-2\sqrt{nm}\cos{(2\pi(n-m)/p)}}$$
with $p$ a positive integer. Numerically, I find th... | Yes, the "conjecture" holds (in the form of $\color{blue}{B_{N,p}/A_N\to1/p}$ as $N\to\infty$). The basic idea is simple: the main contribution to $B_{N,p}$ is given by the terms with $n\equiv m\pmod p$. The next thing we need is $$\lim_{N\to\infty}\frac{1}{N^{3/2}\log N}\sum_{0<n<m<N}\frac{1}{\sqrt{m}-\sqrt{n}}=\frac4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Calculation involving determinant of a matrix Suppose I have the following Toeplitz symmetric matrix
\begin{align}
M=\begin{bmatrix}
1 & c & c & x \\
c & 1 & c & c \\
c & c & 1 & c \\
x & c & c & 1
\end{bmatrix}
\end{align}
I want to write an algorithm that takes $c$ as input and calculates the range of $x$ for which ... | For your specific example, done with pen and paper,
$$M_4=\left(
\begin{array}{cccc}
1 & c & c & x \\
c & 1 & c & c \\
c & c & 1 & c \\
x & c & c & 1
\end{array}
\right)$$
$$\Delta_4=\left(4 c^3-5 c^2+1\right)+\left(4 c^2-4 c^3\right) x+\left(c^2-1\right)
x^2$$ With a computer
$$M_5=\left(
\begin{array}{ccccc}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
find the matrix of $D$ a relative to this basis $ (-\cos x, \sin x).$ let $D: V\to V$ be the differentiation operator. find the matrix of $D$ a relative to this basis $ (-\cos x, \sin x).$
My attempt:
$D(-\cos x)=\sin x=0. \cos x + 1.\sin x $
$D(\sin x)=\cos x = 1.\cos x + 0 .\sin x$
Therefore the matrix representin... | $D(\sin x) = \cos x = -(-\cos x)$, as the basis element is $-\cos x$ not $\cos x$. This does not affect the first computation. So the matrix is then $\begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3785811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conjecture about graph with points and arrows with weights Postulates
*
*Each point has arrow(s), and each arrow is pointing to another point.
*An arrow of a point cannot point to itself.
*Arrows can only point to any direction on the right. That means $\uparrow$, $\downarrow$ are not allowed, and of course $\lefta... | This is not true. For example, consider a network with four nodes $A,B,C,D$ on the left and four nodes $E,F,G,H$ on the right. Give $A$ two arcs of weight $0.5$ to $E,F$. Give $D$ two arcs of weight $0.5$ to $G,H$. Give each of $B,C$ four arcs of weight $0.25$ to $E,F,G,H$. Then every node on the left has $1$ unit goin... | {
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partition and equivalence relation's classes My question concerns the definitions more than semantics.
That is, a family of sets $P$ is a partition of $X$ if the following conditions hold:
*
*$P$ doesn't contain the empty set;
*union of all $P$'s sets gives $X$;
*elements of $P$ are pairwise disjoint.
Now for an e... | When you talk about the equivalence classes of a relation on a set:
*
*if you violate (1), i.e. $\emptyset \in P$, you have an invalid relation
*if you violate (2), either the union of class elements has something which is not in the original set (which means you have an invalid relation since it is defined on thing... | {
"language": "en",
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Solving quintic equations of the form $x^5-x+A=0$ I was on Wolfram Alpha exploring quintic equations that were unsolvable using radicals. Specifically, I was looking at quintics of the form $x^5-x+A=0$ for nonzero integers $A$. I noticed that the roots were always expressible as sums of generalized hypergeometric fun... | The answer to your third question is yes! The method uses Bring radicals, whose explicit form in terms of generalized hypergeometric functions can be found using the Lagrange inversion theorem. (In fact since any quintic can be reduced to this form, in principle this method can be used to solve any quintic.) I can answ... | {
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Visualising the sum of the first $n$ positive odd integers Using the fact that $1+2+\cdots+n=\frac{n(n+1)}{2}$, we can deduce that sum of first $n$ positive odd integers is $n^2$. However, is there a way of finding the sum of $1+3+5+\cdots+(2n-1)$ visually?
| Here is a ‘proof’ I once found in a book for young children. It is not a real proof in the mathematical sense, but rather a convincing example that any mathematician feels could be transformed into a rigourous proof:
Imagine wooden cubes stacked in rows, with the basis containing, say, $7$ cubes, the row above, $5$ cu... | {
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How do you prove $\pi =\sqrt{12}\sum_{n\ge 0}\frac{(-1)^n}{3^n(2n+1)}$? In the book Pi: A Source Book I found the following:
Extract the square root of twelve times the diameter squared. This is the first term. Dividing the first term repeatedly by 3, obtain other terms: the second after one division by 3, the third a... | Consider that
$$\sum_{n=0}^\infty (-1)^n\frac{ x^{2 n+1}}{2 n+1}=\tan ^{-1}(x)$$
$$\sum_{n=0}^\infty (-1)^n\frac{ x^{2 n}}{2 n+1}=\frac{\tan ^{-1}(x)}x$$ Make $x=\frac 1{\sqrt 3}$ and the rhs is $\frac{\pi }{2 \sqrt{3}}$.
Multiply by $\sqrt{12}$ to get $\pi$ as desired.
| {
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What does it mean when a number is subscripted with a truth statement? I have seen the following in several papers: $1_{\lvert r\rvert>1}$. What does this mean? Does this evaluate to 1 if $\lvert r\rvert>1$ and 0 otherwise? What would this evaluate to if instead of 1 we had a variable like $x$?
Thanks.
| Yes, it is an indicator function: $1$ when the condition is true, $0$ when it is not. I have not seen it used with $x$ instead of $1$.
| {
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Probability number comes up and then comes up again before another number I was trying to follow the logic in a similar question (Probability number comes up before another), but I can't seem to get it to work out.
Some craps games have a Repeater bet. You can bet on rolling aces twice before rolling a 7, rolling 3 th... | Let $E_1$ denote the event that you roll snake eyes before a 7.
Let $E_2$ denote the event that you roll snake eyes before a 7, given that event E_1 has already occurred.
In fact, the chance of $E_2$ is the same as the chance of $E_1$.
I simply separated the events for clarity.
The key formula here is that if
$A$ and ... | {
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Closed form of $\int\limits_0^{2\pi} \prod\limits_{j=1}^n \cos(jx)dx$ and combinatorial link I have been trying to find a closed form for this integral:
$$I_n = \int\limits_0^{2\pi} \prod_{j=1}^n \cos(jx)dx$$
The first values are: $I_1=I_2=0,I_3=\frac{\pi}{2}, I_4=\frac{\pi}{4}, I_5=I_6=0, I_7=\frac{\pi}{8}, I_8=\frac{... | As pointed out in the comments, the result is
$$ I_n = a_n \frac{2\pi}{2^{n}} $$
where $a_n$ is the numbers of solutions of $\sum_{j=1}^n s_n \,j =0$ where $s_j \in \{1,-1\}$ (or number of ways of marking a subset of $\{ 1,2, \cdots n\}$ such that the sum of the marked subset equals the sum of the unmarked subset). Thi... | {
"language": "en",
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Integration of $\sqrt {\tan x}$ I have tried many ways to integrate $\sqrt {\tan x}$ including integration by parts but didn't get to any final result.
I also assumed,
$$ \tan x = t^2 $$
$$ \int \sqrt {\tan x} \,dx $$
$$⇒\int \frac{2t^2}{1+t^4}dt$$
but it's getting a bit complicated further, kindly help. Also, are the... | You can factor the denominator as the product of two quadratics (finding the [necessarily complex] roots can help there), then use partial fractions. It gets ugly, but Wofram Alpha gives the same ugly answer as you get by doing this, so I assume there isn't a simpler way.
BTW, are you sure about the numerator? I have v... | {
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Can a function be differentiable at its endpoints? If not, are these endpoints critical values?
The reader is asked to identify the function's critical values. Point $x=c$ is a critical value because $f'(c)=0$. Also, points $x=b,d,e$ are critical values because $f'$ is undefined at those points.
Since critical values ... | None of the claimed equalities are true because they simply do not make any sense. Limits are not defined by taking limits from the left and right sides. Rather, limits are based on taking limits from any direction within the domain. If the domain includes a left and right side of a point, then the limit is defined fro... | {
"language": "en",
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Fourier series of $f(x) = |x|^3$ and evaluating series I found the Fourier serie for the function $$f: [-\pi, \pi], \quad f(x) = |x|^3$$
Coefficents:
$$
a_0 = \frac{\pi^3}{2}
$$
$$
a_n = \frac{6 \pi}{n^2} \cos(n\pi) - \frac{12}{n^4\pi} \cos(n\pi) + \frac{6}{n^4}
$$
$$
b_n = 0
$$
So, the Fourier serie is given by
$$
f(x... | I think the fourier coefficients are
$$n\neq1,\;\;a_n=\frac{6(2-n^2\pi^2)}{\pi n^4}(-1)^n-\frac{12}{\pi n^4}$$
So when we susntitute $\;x=0\;$ , we get (Dirichlet's convergence theorem)
$$0=\frac{\pi^3}4+6\sum_{n=1}^\infty\left(\frac{(2-n^2\pi^2)}{\pi n^4}(-1)^n-\frac{2}{\pi n^4}\right)\implies-\frac{\pi^3}{24}=\sum_{n... | {
"language": "en",
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Poisson process into a series of Bernouilli trial problem The ants of a colony arrives at a location with two food sources according to a poisson process $N(t),t\geq 0$ at the rate of $\lambda$. Once there, each ant will independently choose to eat form one of the sources $A$ or $B$ with respective probabilities $p, (1... | Clearly the long term arrival rate splits into $\lambda = p \lambda + (1-p)\lambda$.
If the number of ants at the origin is kept (supposed to be) constant then you can think to split the procession right at the origin and the two are clearly independent and Poisson distributed.
However the actual process looks to be th... | {
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How to multiply out brackets when they contain vectors Little confused on the rules here, obviously if I treat it as a vector and its transpose I can compute this if I knew each vector entry but I am keen to know the general rule for any vector:
$(\mathbf x - \mathbf y)(\mathbf x - \mathbf y)^T$
How does it relate to $... | To see why removing brackets is a bit different, you can remove them step by step:
$$(\mathbf x - \mathbf y)(\mathbf x - \mathbf y)^T = \mathbf x (\mathbf x - \mathbf y)^T - \mathbf y(\mathbf x - \mathbf y)^T = \mathbf x \mathbf x^T - \mathbf x\mathbf y^T - \mathbf y\mathbf x^T - \mathbf y \mathbf y^T$$
The result cann... | {
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How to prove $\frac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}$? Give $a,b,c>0$. Prove that: $$\dfrac{a^{n+1}+b^{n+1}+c^{n+1}}{a^n+b^n+c^n} \ge \sqrt[3]{abc}.$$
My direction: (we have the equation if and only if $a=b=c$)
$a^{n+1}+a^nb+a^nc \ge 3a^n\sqrt[3]{abc}$
$b^{n+1}+b^na+b^nc \ge 3b^n\sqrt[3]{abc}$
$c... | Let $A_p:=\left(\frac{1}{N}\sum_{i=1}^Na_i^p\right)^{1/p}$ be the $p$th mean of $(a_i)$.
By the extended AM-GM inequality, $GM\le A_n\le A_{n+1}$. Hence $$GM\times A_n^n\le A_{n+1}\times A_{n+1}^n=A_{n+1}^{n+1}$$ or $$\sqrt[3]{abc}\times\frac{a^n+b^n+c^n}{3}\le\frac{a^{n+1}+b^{n+1}+c^{n+1}}{3}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Understanding the map about the classification of all abelian extensions with Galois groups with a fixed exponent (Kummer Theory) Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$. Also, let $E/F$ be a finite Galois extension with Galois group $G$.
Now I am trying to understand the following... | As you are aware, this is the main theorem of Kummer theory. I keep you notations and assumptions, but beware that you forgot the hypothesis that the characteristic of $F$ does not divide $n$. Besides, it will be more convenient to replace a subgroup $B$ containing ${F^\times}^n$ as a subgroup of finite index, by the q... | {
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Asymptotic Estimate of Vector Function I would like to compute the asymptotic limit of the of the following function
$$f(x,\omega) = \frac{x - \omega\sqrt{1+|x|^2}}{x\cdot \omega - \sqrt{1+|x|^2}}$$
Where $x\in \mathbb{R}^3$ and $\omega \in \mathbb{S}^2 = \{y\in\mathbb{R}^3 \ | \ |y| = 1\}$ is a point on the unit spher... | Let $x=|x|\hat{x}$ and $\hat{x}=(\hat{x}\cdot\omega)\omega+\alpha\omega^\perp$, then
$$
f(x,\omega)=\frac{x-\omega\sqrt{1+|x|^2}}{x\cdot\omega-\sqrt{1+|x|^2}}=\omega+\frac{\alpha}{\hat{x}\cdot\omega-\sqrt{(1+|x|^2)/|x|^2}}\omega^\perp$$
Hence for $\xi=\sqrt{(1+|x|^2)/|x|^2}$, \begin{align*}|f(x,\omega)|^2&=1+\frac{\si... | {
"language": "en",
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$\lim_{n\to\infty} f_{n}(x) = g(x) \implies \lim_{n\to\infty} f_{n}^{'}(x) = g^{'}(x) $ Let $f_{n} :\mathbb R \to\mathbb R$ be differentiable for each $n \in\mathbb N$ with $|f_{n}^{'}(x)| ≤ 1$ for all $n$ and $x$.
Assume $\lim_{n\to\infty} f_{n}(x) = g(x) $ for all $x$. Is $\lim_{n\to\infty} f_{n}^{'}(x) = g^{'}(x) $... | There's no reason for $f_n'(x)$ to even converge. Consider, for instance, $$f_n(x) = \dfrac {\sin(nx)}n$$
Clearly $f_n \to 0$ pointwise on $\mathbb{R}$. However the sequence of derivatives $f^{'}_{n} = \cos(nx)$ does not converge pointwise on $\mathbb{R}$:
$$f^{'}_n(\pi) = (-1)^{n}$$
| {
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Evaluate $\frac{1+3}{3}+\frac{1+3+5}{3^2}+\frac{1+3+5+7}{3^3}+\cdots$ It can be rewritten as
$$S = \frac{2^2}{3}+\frac{3^2}{3^2}+\frac{4^2}{3^3}+\cdots$$
When $k$ approaches infinity, the term $\frac{(k+1)^2}{3^k}$ approaches zero. But, i wonder if it can be used to determine the value of $S$. Any idea?
Note: By using ... | Calculus is not required to evaluate the sum. Let $$f(z) = \sum_{k=1}^\infty (k+1)^2 z^k.$$ Then $$z f(z) = \sum_{k=1}^\infty (k+1)^2 z^{k+1} = -z + \sum_{k=1}^\infty k^2 z^k$$ hence $$f(z) - zf(z) = \sum_{k=1}^\infty (k+1)^2 z^k + z - \sum_{k=1}^\infty k^2 z^k = z + \sum_{k=1}^\infty (2k+1) z^k.$$ Now let $$g(z) = ... | {
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Evaluate $\lim _{n\to \infty }\int _{0}^{1}nx^ne^{x^2}dx$
Evaluate $\lim _{n\to \infty }\int _{0}^{1}nx^ne^{x^2}dx.$
I applied the mean value thorem of integral to $\int _{0}^{1}nx^ne^{x^2}dx.$ We get $c\in (0,1):$
$$\int _{0}^{1}nx^ne^{x^2}dx=(1-0)nc^ne^{c^2}.$$ Taking limit ($\lim_{n\to \infty}$)on the both side,
W... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\new... | {
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Definition of finite field with fixed characteristic In this article Discrete logarithms in quasi-polynomial time in finite fields of fixed characteristic the term finite fields of fixed characteristic is not defined and I couldn't find it on the literature, too.
*
*What is the definition of finite fields of fixed ch... | The term quasi-polynomial time means quasi-polynomial in...which parameters? What fixed characteristic says is that the problem is quasi-polynomial in the field size as the field size varies, as long as we keep the characteristic fixed.
So, for instance, if their algorithm takes $\le C2^pk^r$ steps over a field of char... | {
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Partial reciprocal sum How can I show that $$\sum_{k=1}^{n}\frac{1}{n+k}\leq\frac{3}{4}$$ for every integer $n \geq 1$?
I tried induction, estimates with logarithms and trying to bound the sum focusing on the larger terms or things like $\frac{1}{n+1}+\frac{1}{n+2}\leq\frac{2}{n+1}$ but nothing seems to work. Do you ha... | First, note that $f(n)=\sum_{k=1}^n\frac{1}{k+n}$. When $n$ is a positive integer greater than 1, this is a special case of $g(n)=\int_{n+1}^{2n+1}\frac{1}{\lfloor x\rfloor}dx$
$$\frac{dg}{dn}=\frac{d}{dn}\int_{n+1}^{2n+1}\frac{1}{\lfloor x\rfloor}dx=2\frac{1}{\lfloor 2n+1\rfloor}-\frac{1}{\lfloor n+1\rfloor}=\frac{2\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find the winding number and number of zeros of certain function about $|z|=2$. I have the function $f(z)=z^3+\frac{1}{(z-1)^2}$ and I am asked to find the winding number about $C:=\{|z|=2\}$ and then the number of zeros inside $C$.
I know that the winding number is:
$$
n(f,C)=\frac{1}{2\pi i}\int_C \frac{f'(z)}{f(z)}dz... | The zeroes of $f$ are the zeroes of the polynomial $P(z)=z^3(z-1)^2+1$ and for $|z| \ge 2$, one has $|P(z)| \ge 7$ by trivial majorizations so all the 5 zeroes of $P$ are inside $C$, hence $f$ has indeed $5$ zeroes there.
To compute the integral one uses the above observation that all the zeroes of the denominator are ... | {
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Does the existence of a minimal cover for a subset of reals need some form of choice? In Kanamori's book, "The Higher Infinite" p. 376, he defines a minimal cover of some $A \subseteq \omega^\omega$, to be any $B \subseteq \omega^\omega$, such that $A\subseteq B$ and that $B$ is Lebesgue measurable and if $Z \subseteq ... | This is in fact a theorem of ZF. The Caratheodory construction of Lebesgue measure works in ZF, though without choice it need not be $\sigma$-additive. Every Borel codable set of reals is measurable but not necessarily every Borel set. See the paper cited below.
Let $U_i$ enumerate the basic open sets of $\omega^{\omeg... | {
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Problem with evaluating $\lim_{n \rightarrow \infty} e^{-n-t\sqrt{n}}\cdot \left(e^{e^{\frac{t}{\sqrt{n}}}\cdot n}-1\right)$ As above, I have a problem with evaluating
$$\lim_{n \rightarrow \infty} e^{-n-t\sqrt{n}}\cdot \left(e^{e^{\frac{t}{\sqrt{n}}}\cdot n}-1\right).$$
I checked the result in the limit calculator an... | Hint use Taylor expansion of $\displaystyle e^{\frac{t}{\sqrt{n}}}$.
$$\exp\left(\frac{t}{\sqrt{n}}\right) \approx 1 + \frac{t}{\sqrt{n}} + \frac{t^2}{2n} + O\left(\frac{t^3}{n^{3/2}}\right)$$
| {
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What is the smallest real number $m$ such that $n < m^n$ for all $n \geq 1$? I have taken a short look at this problem and found it to be much harder than expected to solve. Per the title, I am looking to find the smallest number $m\in\mathbb{R}$ such that the inequality $$n < m^n$$ is true for all $n\in\mathbb{R}$, $n... | As was pointed out in the comments, there is no smallest such $m$. To see why $m = e^{1/e}$ doesn't work, notice that with $n=e$ we get $m^n = (e^{1/e})^e = e = n$.
However, if you're willing to ask instead about the smallest positive $m$ for which $n \leq m^n$ for all $n \geq 1$, then $m = e^{1/e}$ is indeed correct.... | {
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Prove there exists $v$ such that $f^3(v) = f(f(f(v))) \neq 0$
Let $V$ be a vector space of dimension $n$ and $f: V \to V$. If $\dim Im f \geq 2n/3$, then prove there exists $v$ such that $f^3(v) = f(f(f(v))) \neq 0$
The only thing I can deduce is that $\dim \ker f \leq n/3$.
| $dimImf =dim(Kerf_{\mid Im f}+dim(Im f^2)$ since $dim(Ker f)<n/3$ and $dim(Imf)\geq {{2n}\over 3}$, we deduce that $dim(Imf^2)>n/3$,
$dimImf^2=dim(Ker f_{\mid Imf^2}+dim(Imf^3)$ since $dim(Ker f)<n/3$ and $dim(Imf^2)>n/3$, we deduce that $dimImf^3>0$
| {
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The Quaternions are the smallest.... I've been reading https://github.com/GleasSpty/MATH-104-----Introduction-to-Analysis, and the author formulates the integers as the smallest (by inclusion under isomorphism) nontrivial totally ordered cring that contains the natural numbers, the rationals as the smallest totally ord... | By Frobenius' theorem, the quaternions $\Bbb{H}$ can be characterized as the smallest noncommutative division ring that contains $\Bbb{C}$.
| {
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"source": "stackexchange",
"question_score": "4",
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How many different ways can you go about completing a course/ class at university? How many different combinations of results in assignments are possible in a University course?
I am interesting in calculating the number of unique ways I can finish this course that I am doing. To make things easier, there are no partia... | Eureka
By writing a small block of code I was able to find out 134431 unique combinations of grades!! This surpised me, mostly because I don't understand entirely why this is the case.
I am still curious as to how you would solve this mathematically instead of programmatically... don't hesitate to correct me or prove o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3789216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$? Can anyone give some hints about the following question?
How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$?
Normally this kind of question uses stars and bars but there are $2x_1$, which I do... | One idea is to deal with $x_1$ separately in order to use stars and bars on $x_2,x_3,x_4$. For example you fix $x_1=0$ and then you have $x_2+x_3+x_4=n$ or you fix $x_1=1$ and then get $x_2+x_3+x_4=n-2$ and so on and so forth. This then generates the summations
$$x_2+x_3+x_4=n-2i$$
which have ${n-2i+2 \choose 2}$ solut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3789347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Calculating the acceleration vector of elliptical curve. Satisfying Kepler's first law but not second. I am trying to solve problem 16 in section 1.6 of David Bressoud book Second Year Calculus. He gives a hint at the end of the book which says. If $r$ and $\theta$ are related by $\frac{r^2\cos^2(\theta)}{a^2}+\frac{r^... | we have $\frac{r^2 \cos^2(\theta)}{a^2}+\frac{r^2 \sin^2 (\theta)}{b^2} = 1$
Differentiating this equation with respect to $t$, gives $$r^2\sin(2\theta)\frac{d\theta}{dt}\left(\frac{1}{b^2}-\frac{1}{a^2}\right)+2\frac{1}{r}\frac{dr}{dt}=0$$
Now replacing $\frac{d\theta}{dt} = \frac{k}{r^2}$, gives
$$\sin(2\theta)\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3789553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $\{x_1,x_2,\cdots,x_n\}$ is a basis, is $\{x_1+x_2,x_2+x_3,\cdots,x_n+x_1\}$ a basis too? Let's say we have a vector space $V$ with a basis $\{x_1,x_2,\cdots,x_n\}$ then is $\{x_1+x_2,x_2+x_3,\cdots,x_{n-1}+x_n,x_n+x_1\}$ a basis too?
My Answer:
For n=2 clearly this is false because of the following counter example:... | Let there exist scalars $c_j$'s such that the following linear combination is equal to $0.$$c_1(x_1+x_2)+c_2(x_2+x_3)+\ldots+c_{n-1}(x_{n-1}+x_n)+c_n(x_n+x_1)=0$
Case 1: $n$ is even:
Let $n=2m,\ m\in\Bbb N$.
\begin{aligned}&c_1(x_1+x_2)+c_2(x_2+x_3)+\ldots+c_{2m-1}(x_{2m-1}+x_{2m})+c_{2m}(x_{2m}+x_1)=0\\\implies& (c_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3789976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
$m^*(f(E))\leq\int_E|g'(x)|dx$ for absolutely continuous function $f$ Suppose $f$ is an absolutely continuous function on $[0,1]$, and suppose $E\subset (0,1)$ is any measurable set. I'd like to show that $m^*(f(E))\leq\int_E|f'(x)|dx$.
I know that since $f$ is AC on $[0,1]$, we can write $f(x)=\int_0^xf'(t)dt+f(0)$. ... | See Measure Theory, Vol I by Bogachev, Proposition 5.5.4, p. 348 for the following:
If $f$ is differentiable at each point of a measurable set $E$ then $m^{*}(f(E)) \leq \int_E |f'(x)|dx$.
Your result follows follows from this since absolute continuity of $f$ implies differentiabilty of $f$ at almost all points and al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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I've Hit a Major Snag While Writing a Paper on Deriving the Cubic Formula! So I've writing a paper for school on deriving the cubic formula. As of now I have written the cubic formula as a system of two equations in terms of original coefficients $a$, $b$, $c$, and $d$. The system is below:
$$z=\sqrt[3]{\frac{9abc-2b^3... | You start something of the form:
$z = \sqrt [3] {A \pm \sqrt {A^2+B^3}}\\
x = z - \frac {B}{z} -\frac {b}{3a}$
Lets choose $z = \sqrt [3] {A + \sqrt {A^2+B^3}}$ and let $\bar z = \sqrt [3] {A - \sqrt {A^2+B^3}} $ represent the conjugate (option with the negative sign).
Then
$z-\frac {B}{z} = z-\frac {B}{\sqrt [3] {A +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Show that the matrix $I-uu^T$ has rank $n-1$ where $u$ is a unit vector in $R^n$ I've tried a few examples and I know that $I-uu^T$ is symmetric but I'm stuck here. Any help will be appreciated!
| Note that $\rm P_u := u u^\top$ is the (rank-$1$) projection matrix that projects onto the line spanned by vector $\rm u$. Hence, ${\rm I}_n - {\rm P_u}$ is the projection matrix that projects onto the $(n-1)$-dimensional orthogonal complement of the line. Since the rank of a projection matrix is equal to its trace,
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Find probability of certain number of cards being dealt from remaining cards I did find similar questions to this but I didn't understand the complex answers, so here goes:
I need to find the probability of being dealt a specified amount of cards from the remaining cards in the deck, for example:
I have being dealt 2 c... | If I understand well then $5$ cards are drawn from a deck of $50$ cards of which exactly $11$ are clubs.
Then the probability that exactly $3$ clubs are drawn equals:$$\frac{\binom{11}3\binom{39}2}{\binom{50}5}$$
If you are looking for the probability that at least $3$ clubs are drawn then see the answer of Jfischer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Arrangement of $12$ people in a row such that neither of $2$ particular persons sit on either of $2$ ends of the row
If $12$ persons are arranged in a row such that neither of two particular persons can sit on either end of the row, is
My attempt:
Total ways $=$ Sitting $12$ persons in a row $-$ Sitting $2$ particula... | The $10$ normal persons can be seated in $10!$ ways. There are $9$ slots in between them for the special persons. When the first special person is seated there are $10$ slots for the second special person. Therefore there are
$$10!\cdot 9\cdot 10=326\,592\,000$$
admissible seatings for the $12$ persons.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Why $\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$ is equal to 1? $\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$
By maths calculator it results 1.
I calculate and results $\sqrt{-\frac{1}{2}}$.
$\sqrt{\left(\frac{-\sqrt3}2\right)^2+{(\frac12)}^2}$
$\sqrt{\frac{-{(3)}^{{\displaystyle\frac12}\times2}}{2^2}... | Hint:$(-\sqrt{3})^2=(-1)^2(3)^{{\frac{1}{2}}2}=3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Are there any identities for the determinant of almost upper triangular matrices of the following form? I've encountered a problem in which I need to compute the determinant of an almost upper triangular matrix of the following form:
$$ A = \begin{pmatrix} 1 & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5} & \dots \\
1 & a_{2,2... | Note that we can write this matrix in the form $A = B + uv^T$, where
$$
B = \begin{pmatrix} 1 & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5} & \dots \\
0 & a_{2,2} & a_{2,3} & a_{2,4} & a_{2,5} & \dots \\
0 & 0 & a_{3,3} & a_{3,4} & a_{3,5} & \dots \\
0 & 0 & 0 & a_{4,4} & a_{4,5} & \dots \\
0 & 0 & 0 & 0 & a_{5,5} & \\
\vdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3790845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $\beta$ a basis of vector space $V$? Let $V=\{p\in\mathbb{R}[X]:\deg(p)\leq n\}$, knowing that $\{1,X,\dots,X^n\}$ is a basis of $V$, determine whether $\beta=\{1,X,X^2+1,X^3+X,\dots,X^n+X^{n-2}\}$ is a basis of $V$.
Consider: $\quad c_0+c_1X+c_2(X^2+1)+\dots+c_n(X^n+X^{n-2})=0$
$\implies (c_0+c_2)+(c_1+c_3)X+(c_2+... | More simply, you may consider the determinant of $\beta$ in the standard basis:
$$\det\beta=\begin{vmatrix}
1 & 0 & 1 & 0 & \dots\dots & 0 \\
0 & 1 & 0 & 1 & \dots\dots & 0 \\
0 & 0 & 1 & 0 & \dots\dots & 0 \\
0 & 0 & 0 & 1 & \dots\dots & 0 \\
\vdots & & & & \ddots& \vdots \\
0 & 0 & 0 & 0 & \dots\dots & 1
\end{v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding $|K^\times/\ker(s)|$ and isomorphism $K^\times/S\cong\mathbb{Z}/2\mathbb{Z}$ for finite field K Let $K$ be a finite field with $q$ elements and $K^\times := K\setminus\{0\}$ be the multiplicative group. Assume that the characteristic of $K$ is not $2$, and let $s:K^\times\to K^\times$ given by $x\mapsto x^2$ be... | The elements of the kernel are exactly the roots of the polynomial $x^2-1=(x-1)(x+1)\in K[x]$. Clearly its roots are $1$ and $-1$, and since the characteristic of $K$ is not $2$ they are two different elements. Thus $|Ker(s)|=2$.
As for the second question, all groups of order $2$ are isomorphic to each other. Each gro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many ways are there to place $15$ pieces of size $1 \times 2$ into a $3 \times 10$ rectangle? How many ways are there to place $15$ pieces of size $1 \times 2$ into a $3 \times 10$ rectangle? (rotating and flipping are considered different ways)
I think this question might be solved by recursion.
I tried to split e... | Recursion: we're tiling left to right. Let's say we have already tiled some part of the rectangle by somehow including leftmost untiled square, then we're left 7 possibilities of how the leftmost untiled squares (gray) look like:
1 2 3 4 5 6 7
Let's denote $f_k(n)$ the number of w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why $8^{\frac{1}{3}}$ is $1$, $\frac{2\pi}{3}$, and $\frac{4\pi}{3}$ The question is:
Use DeMoivre’s theorem to find $8^{\frac{1}{3}}$. Express your answer in complex form.
Select one:
a. 2
b. 2, 2 cis (2$\pi$/3), 2 cis (4$\pi$/3)
c. 2, 2 cis ($\pi$/3)
d. 2 cis ($\pi$/3), 2 cis ($\pi$/3)
e. None of these
I think that ... | $8^{\frac{1}{3}}$=$2(1)^{\frac{1}{3}}=2,2\omega,2{\omega}^2$
here $\omega$ is cube root of unity
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Show $ \prod_{k=1}^{n} 4^k = 2^{n*(n+1)}$, where did I go wrong in my induction step? Can someone help me out with the induction step?
Show $ \prod_{k=1}^{n} 4^k = 2^{n*(n+1)}$
Base case n=1: $$4^1 = 2^{1*(1+1)} = 2^2$$
Induction step (to show: $2^{(n+1)*(n+2)} = 2^{n^2+3n+2}$ ) :
$$ \prod_{k=1}^{n+1} 4^k = 4^{n+1} *... | The sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$ and so
$\prod_{k=1}^n4^k=4^{\sum_{k=1}^nk}=4^{\frac{n(n+1)}{2}}=2^{n(n+1)}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating the limit of the quotient of two infinite sums How can I evaluate this limit?
$$\lim_{n\to\infty}\underbrace{\frac{\sum_{k=1}^n \frac 1k}{\sum_{k=1}^{n+1} \frac{1}{2k-1} }}_{=:a_n}$$
By WolframAlpha, the limit has to be 2 but how can I show this? I see it is monotonous increasing so when i could show $\sup_{... | Comparing term by term, we have
$$
\begin{align}
\sum_{k=1}^n\frac1k
&\le\sum_{k=1}^n\frac1{k-\frac12}\\
&=\sum_{k=1}^{n+1}\frac1{k-\frac12}-\frac1{n+\frac12}
\end{align}
$$
Similarly,
$$
\begin{align}
\sum_{k=1}^n\frac1k
&\ge\sum_{k=1}^n\frac1{k+\frac12}\\
&=\sum_{k=2}^{n+1}\frac1{k-\frac12}\\
&=\sum_{k=1}^{n+1}\frac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Why is this sequence not uniformly convergent? In this problem is explained that $f_n(x)$ is pointwise convergent, however not uniformly convergent. The explanation why is not unifromly convergent is also given. However I cannot understand it, when I use the theorem below I get that limit of $f_n - f = 0$
Could maybe ... | Since $\displaystyle(\forall n\in\Bbb N):\left|f_n\left(\frac1{2n}\right)\right|=\frac n4$, you have $\displaystyle\sup_{x\in[0,1]}\left|f_n(x)\right|\geqslant\frac n4$. In other words, $\displaystyle\|f-f_n\|_\infty\geqslant\frac n4$ and, in particular, it is not true that $\displaystyle\lim_{n\to\infty}\|f-f_n\|_\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3791893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Local-global test of algebraicity Let $\alpha \in \mathbb{C}$. Suppose for all primes $p$ and all isomorphisms $j : \mathbb{C} \rightarrow \mathbb{C}_p$, $j(\alpha) \in \bar{\mathbb{Q}}_p$. Is $\alpha \in \bar{\mathbb{Q}}$?
| Sure, if $\alpha$ is not algebraic take $\beta\in \Bbb{C}_p,\not \in \overline{\Bbb{Q}}_p$ and $\sigma\in Aut(\Bbb{C}), \sigma(\alpha)=j^{-1}(\beta)$.
(those things require axiom of choice)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove $\lim\limits_{n \to \infty }\sqrt[n]{a}=1$, if $a>0$ I tried solving by using $\log$ and got $\log(a)/n = \log (1)$ which after applying limit (of $n \to \infty$) gives $0= \log(1)$. Is this right?
| Write , $a^{\frac{1}{n}} = e^{\ln(a^{\frac{1}{n}} )}= e^{\frac{1}{n} \cdot \ln(a) } $
Now , just apply the limit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3792121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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