Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite? Let $R$ be a ring such that the only right ideals of $R$ are $(0)$ and $R$. Prove that either $R$ is a division ring or that $R$ is a ring with a prime number ... | Hint- any infinite group has infinitely many subgroups. If the ring operation is trivial, what are the ideals?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Non-trivial example of algebraically closed fields I'm beginning an introductory course on Galois Theory and we've just started to talk about algebraic closed fields and extensions.
The typical example of algebraically closed fields is $\mathbb{C}$ and the typical non-examples are $\mathbb{R}, \mathbb{Q}$ and arbitrary... | One interesting thing to note is that the first-order theory of algebraically closed fields of characteristic $0$ is $\kappa$-categorical for $\kappa > \aleph_0$. That means that there other algebraically closed field of characteristic $0$ of the same cardinality as $\mathbb{C}$! Maybe not the most helpful response, bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1858570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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"answer_id": 3
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Find the ratio of diagonals in Trapezoid
Given $ABCD$ a rectangular trapezoid, $\angle A=90^\circ$, $AB\parallel DC$, $2AB = CD$ and $AC \perp BD$.
What is the value of $AC/BD$ ?
Attempts so far:
I have tried using the ratio of the areas of triangles $AOB$ and $DOC$, which is $\frac14$ (where $O$ is the intersectio... | Put $AB=1$, $DC=2$, $AD=x$, and set the trapezoid with $A$ in the origin of a cartesian plane. Take the vector $AC=(x,2)$ and the vector $DB=(-x,1)$ and impose that they be orthogonal, i.e. their dot product be null, i.e. $x^2=2$. Now you have all the data to continue.
| {
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How linear map transform the unit ball? Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ maps the unit ball into an ellipsoid ?
Or how can we prove spectral theorem geo... | $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$An invertible linear operator $f$ on $\Reals^{n}$ maps the unit ball to an ellipsoid whether or not $f$ is symmetric (or even diagonalizable): One strategy is to write the unit ball as the locus of a quadratic inequality and perform a l... | {
"language": "en",
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"source": "stackexchange",
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Does every finite dimensional real nil algebra admit a multiplicative basis? We say that a finite dimensional real commutative and associative algebra $\mathcal{A}$ is nil if every element $a \in \mathcal{A}$ is nilpotent.
By multiplicative basis, I mean a basis $\{ v_1, \dots , v_n \}$ for $\mathcal{A}$ as a real vec... | No. The following constructs a counterexample.
Let $R$ be the graded ring $\mathbb{R}[x_1, \ldots, x_5]$ and $I = (x_1, \ldots, x_5)$. In any homgeneous degree $d$, define the "pure" polynomials to be the the products of $d$ linear polynomials, and let the rank of a homogeneous polynomial $f$ be the minimum number of t... | {
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How many integers $\leq N$ are divisible by $2,3$ but not divisible by their powers? How many integers in the range $\leq N$ are divisible by both $2$ and $3$ but are not divisible by whole powers $>1$ of $2$ and $3$ i.e. not divisible by $2^2,3^2, 2^3,3^3, \ldots ?$
I hope by using the inclusion–exclusion principle ... | The rules permit all numbers divisible by $6$, but excluding those also divisble by $4$ or $9$.
This is given by:
$$\lfloor\frac{N}{6}\rfloor-\lfloor\frac{N}{12}\rfloor-\lfloor\frac{N}{18}\rfloor+\lfloor\frac{N}{36}\rfloor$$
Firstly - Start by enumerating number of numbers divisible by 6.
Next term: Remove numbers divi... | {
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Find the probability of getting two sixes in $5$ throws of a die.
In an experiment, a fair die is rolled until two sixes are obtained in succession. What is the probability that the experiment will end in the fifth trial?
My work:
The probability of not getting a $6$ in the first roll is $\frac{5}{6}$
Similarly for ... | the odds of getting two sixes, in any order, within 6 dice (or is it die??) can be easily determined by using the binomial distribution
the odds would be 1/6 * 1/6 * 5/6 ^ 4
that opening statement very simply finds the chances of getting two sixes, but there is more than one way to get two sixes, so you need to multipl... | {
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nature of the series $\sum \tfrac{(-1)^{n}\ln(n)}{\sqrt{n+2}}$
I would like to prove the following series convergent
$$\dfrac{(-1)^{n}\ln(n)}{\sqrt{n+2}},\quad \dfrac{(-1)^{n}\ln(2)}{\sqrt{n+2}}$$
using Alternating series test:
*
*$u_n=\dfrac{(-1)^{n}\ln(n)}{\sqrt{n+2}}$, so
$|u_n|=\dfrac{\ln(n)}{\sqrt{n+2}}$. W... | Note that the first derivative is actually:
$$\frac{2x + 4 - x \ln x}{x (x+2)^{3/2}}$$
Consider $g(x) = (2 - \ln x)x + 4$; $g'(x) = 1 - \ln x < 0$ for $x \ge e$, and (say) $g(e^3) = 4-e^3 < 0$, so for $n \ge \lceil e^3 \rceil = 21$, $u_n$ is decreasing and we can apply the AST
| {
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"source": "stackexchange",
"question_score": "1",
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Fibonacci using Matrix Representation. Fibonacci using matrix representation is of the form :
Fibonacci Matrix. This claims to be of O(log n).However, isn't computing matrix multiplication of order O(n^3) or using Strassen's algorithm O(n^2.81)? How can this be solved in O(log n)?
| Yes, using Fibonacci Matrix $\begin{pmatrix} 1&1\\1&0\end{pmatrix}$ is the way to calculate the nth fibonacci number in $O(log(n))$ time. You can apply this to the matrix, and the solution is reduced to $O(log(n))$.
I put an example code.
long long fibonacci(int n)
{
long long fib[2][2]= {{1,1},{1,0}},ret[2][2]= {... | {
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Real Projective $n$ space $\mathbb{R}P^{n}$ In example 0.4 of Hatcher, he says that $\mathbb{R}P^{n}$ is just the quotient space of the sphere $S^{n}$ with antipodal points identified. He then says that this is equivalent to the quotient of a hemisphere $D^{n}$ with the antipodal points of the boundary identified.
I d... | So, a "point" in $\mathbb{RP}^n$ is secretly the same thing as a pair of antipodal points. But, if you look at two antipodal points in $S^n$, one of two things occurs:
*
*One of the points lies in the open hemisphere $\mathrm{Int}(D^n)$, and the other is in the opposite open hemisphere.
*Both points are on the equa... | {
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$\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$ means that $x$ is close to an integer
Suppose $x>30$ is a number satisfying $\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$. Prove that $\{x\}<\frac{1}{2700}$, where $\{x\}$ is the fractional part of $x$.
My heuristic is that $x$ needs ... | Let $\lfloor x \rfloor =y$ and $\{x\}=b$ Then
$\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor
=y\lfloor y^2+2by+b^2 \rfloor=
\lfloor y^3+3y^2b+3yb^2+b^3\rfloor$
One way this can happen is that $b$ is small enough that all the terms including $b$ are less than $1$, which makes both sides $y^3$. This... | {
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$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$ Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$.
*
*Does this module have a special name?
*Does a basis exist for every $n$? And if so, is there an algorithm to find a basis given an $n$?
I was just playing around w... | The ring $\mathbb Z[\zeta]$ is called the ring of cyclotomic integers.
| {
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A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$ Please look at this expression:
$$1^{2}-2^{2}+3^{2}-4^{2} + \cdots + (-1)^{n-1} n^{2}$$
I found this expression in a math book. It asks us to find a general formula for calculate it with $n$.
The formula that book suggests is this:
$$-\frac{1}{2}... | We wish to show that
$$
1^{2}-2^{2}+3^{2}-4^{2} + \dotsb + (-1)^{n-1} n^{2}=
(-1)^{n+1}\frac{n(n+1)}{2}\tag{1}
$$
To do so, induct on $n$.
The base case $n=1$ is simple to verify.
Now, suppose that $(1)$ holds. Then
\begin{align*}
1^{2}-2^{2}+3^{2}-4^{2} + \dotsb + (-1)^{n} (n+1)^{2}
&= (-1)^{n+1}\frac{n(n+1)}{2}+(-1)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the maximum of $U (x,y) = x^\alpha y^\beta$ subject to $I = px + qy$
Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$.
I have tried to use the Lagrangian function to find the solution for the problem, with the equation
$$\nabla\mathsc... | The solution
The answer can be been found on the internet in any number of places. The function $U$ is a Cobb-Douglas utility function. The Cobb-Douglas function is one of the most commonly used utility functions in economics.
The demand functions you should get are:
$$x(p,I)=\frac{\alpha I}{(\alpha+\beta)p}\qquad y(p... | {
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Positive integer solution $pm = qn+1$
Let $m,n$ be relatively prime positive integers. Prove that there exist positive integers $p,q$ such that $pm = qn+1$.
We know Bézout's identity that there exist integers $p,q$ such that $pm+qn = 1$, but how do we know we can get positive integers $p,q$ with $pm = qn+1$?
| (1). Given integers $p',q'$ with $p'm+q'n=1,$ the set of all $(p'',q'')$ such that $p''m+q''n=1$ is $\{(p'+xn, q'-xm): x\in Z\}.$
If $p'>0$ then $q'<0,$ so let $p=p'$ and $q=-q'.$
If $p'\leq 0$ take $x\in Z^+$ where $x$ is large enough that $p'+xn>0 $ and $ xm-q'>0 . $ Let $p=p'+xn$ and $q=-q'+xm.$
(2). A one-step... | {
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The Thirty-one Game: Winning Strategy for the First Player I am going through UCLA's Game Theory, Part I. Below is an exercise on page 6:
The Thirty-one Game. (Geoffrey Mott-Smith (1954)) From a deck of cards, take the Ace, 2,3,4,5, and 6 of each suit. These 24 cards are laid out face up on a table. The players altern... | The main thing to note here is that this is analogous to the game where one has as many of each card as desired, rather than just four. In particular, it is easy to see that, in this modified game, the winning positions are exactly the positions where the sum is of the form $31-7n$ for some $n$. This is presumably what... | {
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Prove That If $(a + b)^2 + (b + c)^2 + (c + d)^2 = 4(ab + bc + cd)$ Then $a=b=c=d$ If the following equation holds
$$(a + b)^2 + (b + c)^2 + (c + d)^2 = 4(ab + bc + cd)$$
Prove that $a$,$b$,$c$,$d$ are all the same.
What I did is I let $a$,$b$,$c$,$d$ all equal one number. Then I substituted and expanded. I'm sort of p... | Consider the following steps
$$\begin{align}
(a + b)^2 + (b + c)^2 + (c + d)^2 &= 4(ab + bc + cd) \\
\left[ (a + b)^2-4ab \right] + \left[ (b + c)^2-4bc \right] + \left[ (c + d)^2-4cd \right] &=0 \\
\left[ a^2+b^2+2ab-4ab \right] + \left[ b^2+c^2+2bc-4bc \right] + \left[ c^2+d^2+2cd-4cd \right] &=0 \\
\left[ a^2+b^2-2a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ I stumbled on the following inequality: For all $n\geq 1,$
$$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}.$$
However I cannot find the proof of this anywhere.
Any ideas how to proceed?
Edit: I posted a... | \begin{align*}
2\sqrt{n+1}-2\sqrt{n} &= 2\frac{(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})}{(\sqrt{n+1}+\sqrt{n})} \\
&= 2\frac{1}{(\sqrt{n+1}+\sqrt{n})} \\
&< \frac{2}{2\sqrt{n}} \text{ since } \sqrt{n+1} > \sqrt{n}\\
&=\frac{1}{\sqrt{n}}
\end{align*}
Similar proof for the other inequality.
| {
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If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective? If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a ... | For the map $\Phi \colon H \to U'$, $x \mapsto \langle \cdot, x \rangle$ we have $\ker \Phi = U^\perp$.
Hence $\Phi$ is injective if and only if $U^\perp = 0$.
Because $H = U^\perp \oplus \overline{U}$ we have $U^\perp = 0$ if and only if $\overline{U} = H$, i.e. if $U$ is dense.
| {
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"timestamp": "2023-03-29T00:00:00",
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Switching limits of integration The solution in my textbook wrote $$\int_{\alpha \epsilon}^{\alpha N} \frac{f(u)}{u} \, du-\int_{\beta \epsilon}^{\beta N} \frac{f(u)}{u} \, du = \int_{\alpha \epsilon}^{\beta \epsilon} \frac{f(u)}{u} \, du-\int_{\alpha N}^{\beta N} \frac{f(u)}{u} \, du.$$ How can the limits of integrati... | If you write the equation with sums instead of differences, it reads:
$$
\int_{\alpha \epsilon}^{\alpha N} \dfrac{f(u)}{u}du
+\int_{\alpha N}^{\beta N} \dfrac{f(u)}{u}du
= \int_{\alpha \epsilon}^{\beta \epsilon} \dfrac{f(u)}{u}du
+\int_{\beta \epsilon}^{\beta N} \dfrac{f(u)}{u}du
$$
Now if you use the additive ... | {
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Probability problem with a die I've been practicing probability problems lately and I came to this problem
A number is formed in the following way. You throw a six-sided
die until you get a 6 or until you have thrown it three times at the
most. A sequence of dice throws form either one, two or three-digit
number... | For case 3: Not $6$, Not $6$, Any value. (Then use the multiplication principle).
| {
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A Riemannian manifold with constant sectional curvature is Einstein. A Riemannian manifold with constant sectional curvature is Einstein. Why?
It's true the inverse?
| By definition, a Riemannian manifold has constant sectional curvature if the sectional curvature $K$ is a constant that is independent of the point and $2$-plane chosen. If $R$ denotes the covariant curvature tensor and $g$ is the metric then, as a consequence of the definition of $K$, the components satisfy the relati... | {
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"source": "stackexchange",
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Why must $|z|\gt 1$ be the necessary condition
Question:- If $\left|z+\dfrac{1}{z} \right|=a$ where $z$ is a complex number and $a\gt 0$, find the greatest value of $|z|$.
My solution:- From triangle inequality we have
$$|z|-\left|\dfrac{1}{z}\right|\le\left|z+\dfrac{1}{z} \right|\le|z|+\left|\dfrac{1}{z}\right|
\... |
why the need of the specific condition $|z|\gt 1$
$$a=\left|z+\frac 1z\right|\ge|z|-\frac{1}{|z|}\tag1$$
If $0\lt |z|\le 1$, then $-\frac{1}{|z|}\le -1$, so
$$|z|-\frac{1}{|z|}\le 1-1=0\tag2$$
From $(1)$, we have
$$a=\left|z+\frac 1z\right|\ge |z|-\frac{1}{|z|}=(\text{non-positive})$$
which is true since $a\gt 0$, so... | {
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Does every locally compact group $G$ have a nontrivial homomorphism into $\mathbb{R}$? Does every locally compact group (second countable and Hausdorff) topological group $G$ that is not compact have a nontrivial continuous homomorphism into $\mathbb{R}$?
Obviously for compact groups it is not possible since continuous... | For a connected example one can take $G=\mathrm{SL}_2(\mathbb{R})$ (or any connected simple Lie group). If $f:G\to \mathbb{R}$ is a continuous homomorphism then $f(G)$ is a connected simple subgroup of $\mathbb{R}$, hence trivial.
EDIT: I see now someone already mentioned this in the comments.
| {
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"timestamp": "2023-03-29T00:00:00",
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Closure of an operator I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$
$$A_0= f^{(4)}-f^{(6)}$$
$$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(1)=f^{(5)}(1)=0\big\}$$
| The closure of the domain in $L^2$ is simply $L^2$: Obviously it holds $C_0^\infty(0,1)\subset D(A_0)$. The set of smooth function is dense in $L^2(0,1)$, hence its closure is $L^2(0,1)$. This implies that the closure of $D(A_0)$ is $L^2(0,1)$ as well.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Consider the function $f(x) = x^2 + 4/x^2$ a) Find$f ^\prime(x)$ b) Find the values of $x$ at which the tangent to the curve is horizontal. So far I have this...
a) $f^\prime(x) = 2x + (0)(x^2)-(4)\dfrac{2x}{(x^2)^2}$
$= 2x - \dfrac{8x}{x^4}$
$= \dfrac{2x^5 - 8x}{x^4}$
$= \dfrac{2(x^4 - 4)}{x^3}$
I believe I derived ... | Your derivative is correct. You could have saved yourself some work by using the power rule.
\begin{align*}
f(x) & = x^2 + \frac{4}{x^2}\\
& = x^2 + 4x^{-2}
\end{align*}
Using the power rule yields
\begin{align*}
f'(x) & = 2x^1 - 2 \cdot 4x^{-3}\\
& = 2x - 8x^{-3}\\
& = 2x - \frac{8}{x^3}
\end{align*}
... | {
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Bending a line segment REVISED QUESTION
With the help of the existing answers I have been able to put together this clearer animation, and I asked this question to discover the shape is called a cochleoid. What I am really trying to find out at this point is the following:
It seems like this curve should be perfectly ... | The angle in radians subtended by an arc of length $s$ on the circumference of a circle of radius $r$ is given by
\begin{equation}
\theta=\dfrac{s}{r}
\end{equation}
In this instance $s=1$ and $r\ge\tfrac{1}{\pi}$. The circle has center $(r,0)$ and radius $r$. The arc $s$ of length $1$ extends upward along the circumfe... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can $\frac{\sqrt{3}}{\sin20^{\circ}}-\frac{1}{\cos20^{\circ}}$ have two values? I would like to confirm a solution. The question goes as:
Show that $$\frac{\sqrt{3}}{\sin20^{\circ}}-\frac{1}{\cos20^{\circ}}=4$$
Firstly I combined the two terms to form something like: $$\dfrac{\sqrt{3}\cos20^{\circ} - \sin20^{\circ}}{\s... | When $R=-2,-2\sin\theta=\sqrt3\iff\sin\theta=?,-2\cos\theta=1\iff\cos\theta=?$
Observe that $\theta$ lies in the third quadrant $\theta=(2n+1)180^\circ+60^\circ$ where $n$ is any integer
$$\sin\{20^\circ-(2n+1)180^\circ+60^\circ)\}=-\sin220^\circ$$
and $$\sin220^\circ=\sin(180+40)^\circ=-\sin40^\circ$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Equivalence of surjectivity and injectivity for linear operators on finite dimensional vector spaces I'd like to show that for a linear operator $T$ and finite-dimensional vector space $V$ such that $T:V\rightarrow V$, $T$'s injectivity is equivalent to its surjectivity. I started by trying to show $T$'s surjectivity i... | No such statement can be true for infinite-dimensional vector spaces. For example, let $V$ be a vector space with a countable basis $\left\{e_n\right\}_{n\in{\mathbb N}}$, then
$$Te_i=e_{i+1}\ \forall i\in{\mathbb N}$$
defines an injective but not surjective operator, and
$$Te_0=e_0, Te_i=e_{i-1}\ \forall i\ge 1$$
defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that there is a subsequence of $(f_n)_n$ that converges to $f$ almost everywhere. Let $(X,\mathcal{B}, \mu)$ be a measure space and assume the sequence $(f_n)_n$ converges to $f$ in $L^p(\mu)$, where $1\leq p<\infty$. Show that there is a subsequence of $(f_n)_n$ that converges to $f$ almost everywhere.
Isn't it t... | The standard counterexample to your claim that the pointwise convergence holds for every subsequence is the following. Set
$$ A_{n,m}:=[(n-1)/m, n/m] $$
Then
$$1_{A_{1,1}}, 1_{A_{1,2}}, 1_{A_{2,2}}, 1_{A_{1,3}}, \dots$$
converges in $L_p[0,1]$ to the zero function. But it does not converge pointwise to the zero functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such that $A^TA = I_n$
We identify $\mathbb R^{n \times n}$ with $\mathbb R^{n^2}$ and define $f:\mathbb R^{n^2} \to \mathbb R^{n^2}, A \mapsto A^TA$.
Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such that $A^TA = I_n$.
The solution by Om... | First, let's compute the derivative as it is defined here, noting that this coincides with the usual definition except that $Df(A)$, rather than producing an explicit $n^2 \times n^2$ matrix, produces a linear map from $\Bbb R^{n \times n}$ to $\Bbb R^{n \times n}$.
In particular, we compute that
$$
(A + H)^T(A + H) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding the rank of an endomorphism Recently I tried to prove a statement I know should be easy, but for some reason I just can't prove it. The statement is: given a $9 \times 9$ matrix $N$ sucht that $N^3 = 0$ and that $rk(N^2)$ = 3, proof that $rk(N)=6$.
I tried to proof this by using the dimension kernel formula, bu... | That the rank of $N^2$ is $3$ means that the image of $N^2$ is three dimensional. That $N^3$ is $0$ means that $N$ restricted to the image of $N^2$ is $0$. Thus the kernel of $N$ restricted to the image of $N^2$ has dimension $3$.
This implies that the kernel of $N$ has dimension at least $3$, it also implies that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Minimum value of algebraic expression.
If $0\leq x_{i}\leq 1\;\forall i\in \left\{1,2,3,4,5,6,7,8,9,10\right\},$ and $\displaystyle \sum^{10}_{i=1} x^2_{i}=9$
Then $\max$ and $\min$ value of $\displaystyle \sum^{10}_{i=1} x_{i}$
$\bf{My\; Try::}$
Using Cauchy-Schwarz Inequality
$$\left(x^2_{1}+x^2_{2}+.......+x^2_{10... | You have already got the right inequality for the maximum, all you need to add is that equality can be achieved when $x_i = 3/\sqrt{10}$.
For the minimum, note $x_i\in [0,1]\implies x_i^2\leqslant x_i\implies 9=\sum x_i^2\leqslant \sum x_i$ Equality is possible here when one of the $x_i$ is $0$ and all others $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Does subtracting a positive semi-definite diagonal matrix from a Hurwitz matrix keep it Hurwitz? I am having a linear algebra problem here. I will be grateful if someone can help me.
Let $A\in \mathbb{R}^{n\times n}$ be Hurwitz and diagonizable, and let $B$ be a diagonal matrix whose diagonal elements are non-negative... | If $A$ is not only Hurwitz, but also symmetric, then it is negative definite and, thus, $-A$ is positive definite. Let
$$D := \mbox{diag} (d_1, d_2, \dots, d_n)$$
where $d_i \geq 0$, be a positive semidefinite diagonal matrix. Hence, $-(A-D) = -A + D \succ 0$ and, thus, $A - D \prec 0$. As $A-D$ is negative definite, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Russell's paradox from Cantor's I learnt how Russell's paradox can be derived from Cantor's theorem here, but also from S C Kleene's Introduction to Metamathematics, page 38.
In his book, Kleene says that if $M$ is set of all sets, then $\mathcal P(M)=M$ but since this implies $\mathcal P(M)$ has same cardinality as $... | Cantor's theorem shows that for any set $X$ and any function $f:X\to \mathcal{P}(X)$, there is some subset $T\subseteq X$ that is not in the image of $f$. Specifically, $T=\{x\in X:x\not\in f(x)\}$. Kleene is saying that if you apply this theorem to the identity function $f:M\to\mathcal{P}(M)$, the counterexample $T$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Intuition behind proof of bounded convergence theorem in Stein-Shakarchi
Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ a.e. $x$ as $n \to \infty$. Then $f$ is measur... | To remember the proof, maybe it is best to keep a particular example in mind.
Let $E = [0,1]$, the closed unit interval on the line. Let $f_n(x) = x^n$, which is bounded by $M=1$. Then $f_n \rightarrow 0$ almost everywhere on $E$ but not uniformly.
But we can exclude the bits where uniform convergence fails (this is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1861987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution Find the values of 'b' for which the equation
$$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution.
=$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$
My try:
After removing the logarithmic... | You have
$$x^2+(4+b)x+16=0\tag1$$
This is correct.
However, note that when we solve
$$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$
we have to have
$$bx+28\gt 0\quad\text{and}\quad 12-4x-x^2\gt 0,$$
i.e.
$$bx\gt -28\quad\text{and}\quad -6\lt x\lt 2\tag2$$
Now, from $(1)$, we have to have $(4+b)^2-4\cdot 16\geqslant ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as... | Given a noetherian integrally closed domain $A$ we have the equivalence $$ A \operatorname { is a } UFD \iff Cl(A)=0$$where $Cl(A)=0$ is the class group of $A$.
If your case $\operatorname {Spec}(R)$ is smooth so that $Cl(R)=Pic(R)$, the Picard group of $R$.
So the problem boils down to proving that the algebraic Picar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral of the product $x^n e^x$ I would be very pleased if you could give me your opinion about this way of integrating the following expression. I think that it has no issues, but just wanted to confirm:
$$ \int x^n e^x dx $$
$$ e^x = u $$
$$ e^x dx = du $$
$$ \int x^n e^x dx = \int \ln(u)^n du = nu(\ln(u) - 1) + C ... | Here is one way to proceed. Note that
$$\begin{align}
\int x^ne^x\,dx&=\left.\left(\frac{d^n}{da^n}\int e^{ax}\,dx\right)\right|_{a=1}\\\\
&=\left.\left(\frac{d^n}{da^n}\frac{e^{ax}}{a}\right)\right|_{a=1}\\\\
&=\left.\left(\sum_{k=0}^n\binom{n}{k}\frac{d^k a^{-1}}{da^k}\frac{d^{n-k} e^{ax}}{da^{n-k}}\right)\right|_{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
} |
Analytic continuation of $\sum (z/a)^n$ I'm having trouble continuing this function beyond its convergence radius, $R=a$.
$$f(z)=\sum (z/a)^n$$
Given the context (a textbook in complex analysis) I suspect it should have a simple closed-form expression. I've tried differentiating and trying to relate it to the geomtric ... | For $|z/a|<1$ the sum is $\frac{1}{1-(z/a)} = \frac{a}{a-z}$. That is the continuation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
A series involving digamma function I am trying to solve the series
$$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$
The best I got is
$$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$
I am not able to simplify it more.
Maybe there is another approach to solve the series. Any idea how ?
You can assume that $n$ is an int... | In fact, your result is correct and it agrees to Eq.(6.3.17) of Abramovic & Stegun, where you'll find a nice zeta series representation for it. See link http://people.math.sfu.ca/~cbm/aands/page_259.htm
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Fundamental group of $S^{1}$ unioned with its two diameters Is my solution correct?
Call $X$ the riflescope space (I made this name up). I let $p$ be the point of intersection of the two diameters, and $q$ be the right point of intersection of the horizontal diameter with the circle. I let $A=X-{p}$ and $B=X-q$. Then $... | If your space looks like $\bigoplus$ then it is homotopy equivalent to a wedge of four circles (collapse the diameters to a point) so Van Kampen gives its fundamental group is the free group on four generators (not three).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Diameter of set in metric space I do agree with the statement that
$$d(A) = \sup{\{d(x, y):x, y \in A\}}$$
But why can't we use maximum because according to me its max will also give diameter.
I know it should not be correct, so please give me the correct explanation with example where the $\max$ formulation will be w... | It is possible for the supremum of a set to be a value that is not in the set, and as a result the set has no maximum value (since the maximum of a set is always taken to be the largest element in the set).
For a simple example, consider the open interval $A = (0, 1) = \{x \in \mathbb{R}: 0 < x < 1\}$. What's the large... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ is generator of $G$.
Am I interpreting this action correctly. I am thinking ... | Thinking first of $\mathbb{C}^2$, the action of the group $G = \mathbb{Z}/3\mathbb{Z}$ is generated by the linear transformation
$$\begin{pmatrix}0 & 1 \\ -1 & 1 \end{pmatrix} \cdot \begin{pmatrix}w \\ z \end{pmatrix} = \begin{pmatrix}z \\ -w+z \end{pmatrix}
$$
Under the projection map $\mathbb{C}^2 - \{0\} \to P^1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1862896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The relation between axes of 3D rotations Let's suppose we have two rotations about two different axes represented by vectors $v_1$ and $v_2$:
$R_1(v_1, \theta_1)$, $R_2(v_2,\theta_2)$.
It's relatively easy to prove that composition of these two rotations gives rotation about axis $v_3$ distinct from axes $v_1$ and $v... | This is easily seen if we assume familiarity with the use of unit quaternions in representing rotations. A rotation $R$ about the axis given $\vec{v}=v_1\bf{i}+v_2\bf{j}+v_3\bf{k}$ by the angle $\theta$ is represented by the quaternion
$$
q=\cos\frac\theta2+\sin\frac\theta2\vec{v}.
$$
Here it is essential that $\vec{v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Building volume using lagrange multipliers A rectangular building with a square front is to be constructed of materials that costs 20 dollars per square foot for the flat roof, 20 dollars per square foot for the sides and the back, and 14 dollars per square foot for the glass front. We will ignore the bottom of the bui... | Since the building has a square front its dimensions are $x \times x \times y$ where $x$ is both the height and width of the building and $y$ is its length. The areas of the front and back are $x^2$, and the areas of the sides and roof are $xy$. The cost of the materials to construct the building is given by $$C(x,y) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given $P(C)$ and $P(A\mid C)$, what is $P(C\mid A)$? I am wondering if there's a way to find the solution if we know:
$P(C) = 0.01$
$P(A\mid C) = 0.7$
what is $P(C\mid A)$?
I think we need to know $P(A)$ to answer this question right? There is no other way around it?
Thank you!
| Considering the given data, without knowing anything more, one may just write
$$
P_A(C)=\frac{P(A \cap C)}{P(A)}=\frac{P(A \cap C)}{P(C)}\cdot \frac{P(C)}{P(A)}=0.7\frac{0.01}{P(A)}=\color{red}{\frac{0.007}{P(A)}}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Applying distortion to Bézier surface I am trying to simulate the image warp effect, that is used in Adobe Photoshop.
The rectangular image is warped according to a cubic Bézier surface (in 2D, all Z components are 0). Having any Bézier surface, vertical distortion $d \in[0,1]$ can be applied to it.
Left: input bézier ... | Clearly the $y$-coordinates of the Bezier patch control points are being left unchanged, and the $x$-values are being "tapered". When I say "tapered", I mean that the upper edge of the patch is being shrunk inwards, the lower edge is being expanded outwards, and the mid-height curve is being left unchanged.
I don't rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does it follow that $\mu$ is a measure? Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. Does it follow that $\mu$ is a measure?
| This is overkill since I use the
non trivial result that if the sequences $ x_n, x$ are in $l_1$, then $x_n \to x$ in norm iff $x_n(k) \to x(k)$ for all $k$.
It is straightforward to verify that $\mu \emptyset = 0$ and $\mu A \ge 0$ for any $A \in \cal A$.
Suppose $A_k \in \cal A$ are disjoint, let $x_n(k) = \mu_n A... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve functional equation $f(x^4+y)=x^3f(x)+f(y)$ I need help solving this equation, please:
$$f(x^4+y)=x^3f(x)+f(y),$$ such that $ f:\Bbb{R}\rightarrow \Bbb{R},$ and differentiable
I've found that $f(0)=0$ and $f(y+1)=f(1)+f(y)$, but I couldn't continue, I think the solution is $f(x)=ax$.
Thanks for your help
| To help clarify or expand the hint by @xidgel and avoid any possible confusion.
First introduce $t(x,y) = x^4+y$, then we can write:
$$f(t) = x^3f(x)+f(y)$$
Now we can express differentiation with respect to x and y with the chain rule:
$$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial t} \cdot \frac{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 4
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Is there an intuitive meaning of $p - p^2$ If $p$ is the probability of an event occurring, does $p - p^2$ have an intuitive meaning?
| Since $p-p^2=p(1-p)$ it is the probability of the event occurring multiplied by the probability of it not occurring. For example if $p$ is the probability of a coin coming up heads. Then $p(1-p)$ is the probability it comes up head then tail after two throws.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.
Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying:
i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$.
ii) there is some $\theta \in (0,1)$ such that: $$\left|\int_I... | Hint: Suppose $\theta = 1/2$ just to scratch around a bit. For $h>0$ we have
$$|\int_a^{a+h} f\,\,|^p \le \frac{1}{2}\cdot h^{p-1}\int_a^{a+h} |f|^p \implies |\frac{1}{h} \int_a^{a+h} f|^p \le \frac{1}{2}\cdot \frac{1}{h}\int_a^{a+h} |f|^p.$$
In the last inequality, let $h\to 0^+$ and apply the Lebesgue differentiatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Domain of $f(x)=x^{\frac{1}{\log x}}$ What is the domain of $$f(x)=x^{\frac{1}{\log x}}$$
Since there is logarithm , the domain is $(0 \: \infty)$
But the book answer is $(0 \: \infty)-\{1\}$
but if $x=1$ $$f(x)=1^\infty=1$$
So is it necessary to exclude $1$
| Just as MathematicsStudent1122 answered, if $$f=x^{\frac{1}{\log (x)}}$$ $$\log(f)=\frac{1}{\log(x)}\times\log(x)=1 \implies f=e$$ provided $x>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The set of all real or complex invertible matrices is dense I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, and $\lambda\neq 0$, $A\in\Omega\implies \lambda A\in\Omega$, we... | No, it is all wrong. What you need to show is that given $C \in M_{n \times n}(F)$ and $\epsilon > 0$ there exists $A \in \Omega$ with $\|A - C\| < \epsilon$. But you are assuming that $A \in \Omega$ with $\|A - C\| < 1/\|A\|$. Then you make the absurd assertion that $\|\lambda A - C \| < \dfrac{1}{\lambda \|A\|}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1863916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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If $g>0$ is in $L\ln\ln L$, then $\#\{n: g(\theta x)+\cdots+g(\theta^nx)\le t\,g(\theta^nx)\}\le Ct$ when $t\to\infty$ Here are two theorems:
*
*For every dynamical system $(X, Σ, m, T )$ and function $f \in L \ln \ln L(X,m)$
(that is, such that $\int |f| \ln^+ \ln^+ |f|\, {\rm d}m$ is finite), $$N^∗f(x)=\sup... | By the pointwise ergodic theorem, we have (by ergodicity) that $$\frac 1n\sum_{j=1}^ng\circ \theta^k(x)\geqslant \frac{\mathbb E\left[g\right]}2$$
for each $n\geqslant n_0(x)$. Therefore, we have
$$\left\{n\geqslant n_0(x)\mid b_n\leqslant t\right\}
\subset \left\{n\geqslant n_0(x)\mid \frac{g\left(\theta^nx\right)}n
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1864005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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fourier transform of cos(kx) using the formula given. i want to find the fourier transform of $$ f(x) = \cos (kx)$$
using the fourier transformation formula $$f(k)={1\over \sqrt(2\pi)}\int _{-\infty}^\infty (f(x) e^{ikx}dk$$
how can i do that??
| As tired mentioned, a delta function, $\delta(x)$, is a distribution that satisfies
$$\int _{-\infty}^\infty\delta(x-a)f(x)=f(a)$$
Indeed the Fourier transform of the function $g(x)=1$ is a delta function.So we should Write
$$\delta(\omega)={1\over \sqrt{2\pi}}\int _{-\infty}^\infty e^{i\omega x}dx$$
$$\mathcal{F}[f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1864103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$. The main question is :
$z^x=x$, $z^y=y$, $y^y=x$
Find $z$, $y$, $x$.
My method :
I first attempted to get two equation for the unknowns $x$ and $y$.
We can happily write :
$z=x^{1/x}$ and $z=y^{1/y}$
Thus we get,
$x^{1/x}=y^{1/y}$
Which is,
... | $$z^{y}=y$$
$$(z^y)^y=y^y$$
$$z^{y^2}=x=z^x$$
therefore
$$x=y^2$$
on the other hand
$$y^y=x=y^2\implies y=2$$
thus
$$x=4\quad,\quad z=\sqrt{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1864161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Matrix decomposition into square positive integer matrices This is an attempt at an analogy with prime numbers. Let's consider only square matrices with positive integer entries. Which of them are 'prime' and how to decompose such a matrix in general?
To illustrate, there is a product of two general $2 \times 2$ matric... | It's a strange question... Let $A\in M(N)$ s.t. $A=PQ$ where $P,Q\in M(N)$ are random. I calculate "the" Smith normal decomposition of $A$: $A=UDV$ where $U,V\in GL(\mathbb{Z})$ and $D$ is a diagonal in $M(\mathbb{Z})$. During each Maple test, I consider the matrix $UD=[C_1,\cdots,C_n]$, where $(C_i)_i$ are its columns... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that $(p \to q) \to (\neg q \to \neg p)$ is a tautology using the law of logical equivalence I'm new to discrete maths and I have been trying to solve this:
Decide whether $$(p \to q) \to (\neg q \to \neg p)$$ is a tautology or not by using the law of logical equivalence
I have constructed the truth table and c... | The following line of reasoning may help:
$\qquad\begin{align}
(p\to q)\to(\neg q\to\neg p)&\equiv\neg(\neg p\lor q)\lor(q\lor\neg p)&&\text{material implication}\\[1em]
&\equiv\neg(\neg p\lor q)\lor(\neg p\lor q)&&\text{commutativity}\\[1em]
&\equiv \neg M\lor M&&{M:\neg p\lor q}\\[1em]
&\equiv \mathbf{T}&&\text{negat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1864346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Matrices that represent rotations So the question is
What 3 by 3 matrices represent the transformations that
a) rotate the x-y plane, then x-z, then y-z through 90?
I believe this is the matrix that rotates the xy plane
\begin{bmatrix}
0 &-1 &0 \\
1 &0 &0 \\
0 &0 &1 \\
\end{bmatrix}
But I couldn't think of a rotation ... | Hint:
since you are a beginner I give you a general, simple and powerful method that works well in many situations.
You can interpret the action of your matrix
\begin{bmatrix}
0 &-1 &0 \\
1 &0 &0 \\
0 &0 &1 \\
\end{bmatrix}
You can interpret the action of your matrix looking at the columns. The first column is the tran... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find the n'th number in this sequence The first numbers of the sequence are {2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2}.
I.e. the even in... | The n-th term is equal to the largest power of 2 which divides (n+1), plus 1.
In other words if $n+1=2^lm$ in which l is non-negative and m is odd, then $a_n=l+1$.
(For example the 11-th term is equal to 2+1,since $2^2$ divides 12 but not $2^3$, the 23-th term is equal to 3+1, because $2^3$ is the greatest power which ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$ I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also explained in the video... | As $n$ goes to infinity, $n^{1/n}$ approaches $1$ from above. In particular the fractional part of $n^{1/n}$ approaches $0$, so the $p^{1/p}$ are not equidistributed modulo $1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1864615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Laplace transform in ODE Use any method to find the laplace transform of coshbt
Looking to get help with this example for my exam review
| The laplace transformations of $coshbt$ is the following
$$\int_0^\infty cosh({bt})e^{-st} dt$$
$$= \int_0^\infty \frac{(e^bt + e^{-bt})e^{-st}}{2} dt$$
$$= \frac{1}{2} \int_0^\infty e^{-st + bt} + e^{-st - bt} dt $$
$$= \frac{1}{2} \int_0^\infty e^{(-s + b)t} + e^{(-s - b)t} dt$$
$$=\frac{1}{2} \begin{bmatrix} \frac {... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with Telescopic Series with 3 terms in denominator
All the examples i have done and seen only have 2 terms in the denominator so I am a bit stuck with this one. I have attached what I have done so far, not sure how to proceed with it.
Thank you for the hints they were useful, after working it out more I ended up ... | You are close to the end. Express what you got as
$$\left(\frac{1/2}{n}-\frac{1/2}{n+1}\right)-\left(\frac{1/2}{n+1}-\frac{1/2}{n+2}\right).$$
It looks a little better as
$$\frac{1/2}{n(n+1)}-\frac{1/2}{(n+1)(n+2)}.$$
Now add up, and (in either version) watch almost all the terms cancel.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1865001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fundamental theorem of calculus statement Let f be an integrable real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
F(x)=$\int _{a}^{x}\!f(t)\,dt$
Doesn't this make F(a) = $0$ ?
| Yes, yes it does. In this particular formation of the FTOC, you are defining a function F(x) such that it represents the area underneath the function f(t) as t ranges from 0 to x. When x = a, you are measuring an area that is f(a) high, and of zero width, so therefore it must be zero area.
EDIT To respond to your comme... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Give an approximation for $f(-1)$ with an error margin of less than $0.01$ $f$ is defined by the power series: $f(x) = \sum_{n=1}^{\infty}\frac{x^n}{3^n (n+2)}$
I need to find an approximation for $f(-1)$ such that the error margin will be less than $0.01$.
I know I need to use the Taylor remainder and the Laggrange t... | Just for your curiosity.
As @gammatester answered, you are looking fo $n$ such that $$\frac 1{3^n(n+2)}\lt \frac 1{100}$$ which can rewrite $$3^n(n+2)\gt 100$$ Just by inspection, $n=3$ is the samllest value for which the inequality holds.
In fact, there is an analytical solution to the equation $$x^n(n+k)=\frac 1 \eps... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$? This question is related to a previous question of mine.
I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the premises. S... | The same objection as before holds. If we consider
$$ f(z)=1-a_2 z+a_4 z^2-a_6 z^3 +\ldots $$
the fact that $\{a_{2n}\}_{n\geq 1}$ is a positive decreasing sequence do not give that Newton's inequalities are fulfilled. If Newton's inequalities are not fulfilled, $f(z)$ cannot have only real roots and the same applies t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about the term "depressed cubic" The depressed cubic equation is a cubic equation of the form $x^3+px+q=0$. This expression sounds strange especially for someone that English is not his mother tongue. Why this equation is called "depressed"? What is so depressing in it? Thanks!
|
Why this equation is called "depressed"?
It is from latin deprimitur : lowered.
It seems that the terminology was intoduced by François Viète (1540 – 1603) into his posthumous :
*
*Francisci Vietae Fontenaeensis ab aequationum recognitione et emendatione (1615), page 79:
Anastrophe [anastrophe] is the transfor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A series with logarithms Can we express in terms of known constants the sum:
$$\mathcal{S}=\sum_{n=1}^{\infty} \frac{\log (n+1)-\log n}{n}$$
First of all it converges , but not matter what I try or whatever technic I am about to apply it fails. In the mean time if we split it apart (let us take the partial sums) then:
... | We may exploit Frullani's theorem to get an integral representation of our series.
$$\begin{eqnarray*}S=\sum_{n\geq 1}\frac{\log(n+1)-\log(n)}{n}&=&\int_{0}^{+\infty}\sum_{n\geq 1}\frac{e^{-nx}-e^{-(n+1)x}}{nx}\,dx\\ &=&\int_{0}^{+\infty}\frac{1-e^{-x}}{x}\left(-\log(1-e^{-x})\right)\,dx\\&=&\int_{0}^{1}\frac{x\log x}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1865408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Arrangement of 12 boys and 2 girls in a row. 12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls.
My result:
Total number of arrangements = 14!
P1 = number of ways girls can sit together = $2!×13!$
Now I want to find P2 the number of ways in one boy sits bet... | The number of arrangements in which exactly one boy sits between the girls is
$$12 \cdot 2! \cdot 12!$$
since there are twelve ways to choose the boy who sits between the girls, two ways of choosing the girl who sits to his left, one way of choosing the girl who sits to his right, and $12!$ ways of arranging the block... | {
"language": "en",
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nilpotent linear transformation and invariant subspaces I'm trying to proof a biconditional statement about a nilpotent linear transformation, and I think I already proved it one way,but I'm stuck on the other way.
The statement is as follows:
Let $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, be a nilpotent linear tr... | Your first proof is correct. In other words, if $\phi^2 = 0$, then the dimension of the kernel is at least $2$.
On the other hand, if $\phi^2$ fails to be zero. Then, every invariant subspace of $\phi$ contains an eigenspace, but the only eigenvalue $\phi$ can have is $0$. So, every invariant subspace of $\phi$ cont... | {
"language": "en",
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Why can we change a limit's function/expression and claim that the limits are identical? Say you have limit as $x$ approaches $0$ of $x$. You could just write it as $\frac{1}{\frac{1}{x}}$ and then the expression would be undefined. So what are you really doing when you "rearrange" an expression or function so its limi... | Remark: sort of a long comment.
What you are rediscovering are so called removable singularities: Note that the functions
$$f(x)=x$$
and
$$\tilde{f}(x)=\frac{1}{\frac{1}{x}},$$
are not the same, as they domain of definition differs:
the first is defined for all real numbers, whereas the second is not defined at zero ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Non-negative, integrable random variables which converge in probability and whose expected values have a finite limit Suppose we have a sequence $X_1, X_2,...$ of non-negative, real random variables (not necessarily increasing) in $L^1$ which converge in probability to an integrable, non-negative random variable $X \in... | It seems that you know how to handle the case where the convergence in probability is replaced by almost sure convergence.
Let's do the general case. As David Mitra suggests, the key point is to extract an almost everywhere convergent subsequence.
Suppose that we do not have the convergence in $\mathbb L^1$. Then the... | {
"language": "en",
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What are the constraints on $\alpha$ so that $AX=B$ has a solution? I found the following problem and I'm a little confused.
Consider $$A= \left( \begin{array}{ccc}
3 & 2 & -1 & 5 \\
1 & -1 & 2 & 2\\
0 & 5 & 7 & \alpha \end{array} \right)$$ and $$B= \left( \begin{array}{ccc}
0 & 3 \\
0 & -1 \\
0 & 6 \end{array} \ri... | Ignoring the fourth column, notice that
$$\begin{pmatrix} 3 & 2 & -1 \\ 1 & -1 & 2\\ 0 & 5 & 7 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0 \end{pmatrix}$$
and
$$\begin{pmatrix} 3 & 2 & -1 \\ 1 & -1 & 2\\ 0 & 5 & 7 \end{pmatrix} \begin{pmatrix} \frac15 \\ \frac65 \\ 0 \end{pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1865900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why would we use the radius of a circle instead of the diameter when calculating circumference? Forgive me if this question is a little too strange or maybe even off. Mathematics has never been my strong point, but I definitely think it's the coolest...
Anyway, I was looking into tau, pi's up-and-coming sibling. I sta... | You can define a circle knowing the centre and the radius (distance $r$). A circle is the set of all points, on a 2D-plane, at distance $r$ from the centre.
That's a concise and elegant definition; try doing so using the diameter (distance $d$).
Then, having defined circles using the radius, it becomes convenient to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate $\int\sin^{7}x\cos^4{x}\,dx$
$$\int \sin^{7}x\cos^4{x}\,dx$$
\begin{align*}
\int \sin^{7}x\cos^4{x}\,dx&= \int(\sin^{2}x)^3 \cos^4{x}\sin x \,dx\\
&=\int(1-\cos^{2}x)^{3}\cos^4{x}\sin x\,dx,\quad u=\cos x, du=-\sin x\,dx\\
&=-\int(1-u^{2})^3u^4{x}\,du\\
&=-\int (1-3u^2+3u^4-u^6)u^4\,du\\
&=u^4-3u^6+3u^8-u^{1... |
is it correct?
No, it isn't. You have errors in the following part :
$$=-\int(1-u^{2})^3u^4{x}du=-\int (1-3u^2+3u^4-u^6)u^4du$$
$$=u^4-3u^6+3u^8-u^{10}=\frac{u^5}{5}-\frac{3u^7}{7}+\frac{3u^9}{9}-\frac{u^{11}}{11}+c$$
They should be
$$-\int(1-u^{2})^3u^4du=-\int (1-3u^2+3u^4-u^6)u^4du$$
$$=\int \left(-u^4+3u^6-3u... | {
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"timestamp": "2023-03-29T00:00:00",
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Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to ${2^n}{-1}$. Please explain in quotations!
"Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to $2^n-1$." In a very simple case, for $n = 3, 1 + 2 + 4 = 7 = 8 - 1$.
| If you’re familiar with binary, simply note that
$$2^n-1 = {1\underbrace{000\dots00}_{\text{$n$ zeros}}}\text{$_2$} - 1 = \underbrace{111\dots11}_{\text{$n$ ones}}\text{$_2$}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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related rates- rate a man's shadow changes as he walks past a lamp post (is a fixed distance away from it) A $186$ cm man walks past a light mounted $5$ m up on the wall of a building, walking at $2\ m/s$ parallel to the wall on a path that is $2$ m from the wall. At what rate is the length of his shadow changing when ... | The first thing to do with a problem like this is to draw a diagram:
The man (at $M$) is walking parrallel to the wall at $2\mbox{ m/s}$ and $2\mbox{ m}$ from the wall.
The plan distance from the wall below the lamp and the man's feet is:
$$
p=\sqrt{2^2+x^2}
$$
where $x$ is the distance of the man past his closest p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866319",
"timestamp": "2023-03-29T00:00:00",
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Derivation of the Euler Lagrange Equation I'm self studying a little bit of physics at the moment and for that I needed the derivation of the Euler Lagrange Equation. I understand everything but for a little step in the proof, maybe someone can help me. That's were I am:
$$
\frac{dJ(\varepsilon=0 )}{d\varepsilon } = \... | In the Euler-Lagrange equation, the function $\eta$ has by hypothesis the following properties:
*
*$\eta$ is continuously differentiable (for the derivation to be rigorous)
*$\eta$ satisfies the boundary conditions $\eta(a) = \eta(b) = 0$.
In addition, $F$ should have continuous partial derivatives.
This is why $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$ I want to prove that all "closed unit ball" of
$$
c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}
$$
do not have any extreme point. Would you please help me?
(Extreme Po... | Hint: if $x$ is in the unit ball of $c_0$, there is some $i$ such that $|x_i| < 1$. What happens if you increase or decrease $x_i$ a little bit?
| {
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How to solve this Sturm Liouville problem? $\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$
Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
| Hint:
Let $\phi=e^{ax^3}y$ ,
Then $\dfrac{d\phi}{dx}=e^{ax^3}\dfrac{dy}{dx}+3ax^2e^{ax^3}y$
$\dfrac{d^2\phi}{dx^2}=e^{ax^3}\dfrac{d^2y}{dx^2}+3ax^2e^{ax^3}\dfrac{dy}{dx}+3ax^2e^{ax^3}\dfrac{dy}{dx}+(9a^2x^4+6ax)e^{ax^3}y=e^{ax^3}\dfrac{d^2y}{dx^2}+6ax^2e^{ax^3}\dfrac{dy}{dx}+(9a^2x^4+6ax)e^{ax^3}y$
$\therefore e^{ax^3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integral of solid angle of closed surface from the exterior Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, demonstrating that the integral depends only on the charge enclos... | Jacob, you probably won't like this, but the easiest way I see to do the calculation rigorously is to use differential forms. (One reference that's somewhat accessible is my textbook Multivariable Mathematics ..., but you can find plenty of others.)
If $S$ is a closed (oriented) surface in $\Bbb R^3$ not containing th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Given a function, how can one tell if it doesn't have a limit at $x=a$ due to a discontinuity? For example, if you have the $$\lim_{x \to 2} \frac{1}{x-2},$$ the limits approaching from the positive and negative are different. You can tell because the $x-2$ becomes $0$ and the entire binomial is raised to an odd power.... | You have to look at the one-sided limits
$$
\lim_{x \to a^-} f(x) \quad \text{and} \quad \lim_{x \to a^+} f(x).
$$
The 2-sided limit exists iff both one-sided limits exist and are equal to each other, and $f(a)$ also has that value.
If 2-sided limits exist and are equal, but $f(a)$ has a different value, you have a poi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove by induction that $a^{4n+1}-a$ is divisible by 30 for any a and $n\ge1$ It is valid for n=1, and if I assume that $a^{4n+1}-a=30k$ for some n and continue from there with $a^{4n+5}-a=30k=>a^4a^{4n+1}-a$ then I try to write this in the form of $a^4(a^{4n+1}-a)-X$ so I could use my assumption but I can't find any $... | Here is how I would write up the main part of the induction proof (DeepSea and Bill handle the base case easily), in the event that you may find it useful:
\begin{align}
a^{4k+5}-a&= a^4(a^{4k+1}-a)+a^5-a\tag{rearrange}\\[1em]
&= a^4(30\eta)+a^5-a\tag{by ind. hyp.; $\eta\in\mathbb{Z}$}\\[1em]
&= a^4(30\eta)+30\ell\tag{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
I don't know what this symbol in root systems means (of coxeter groups) I'm reading Humphreys, Reflection groups and Coxeter groups. The section "Construction of root systems" and the books uses the symbol $ \mathop {\alpha}\limits^{\sim} $ to denote an special element. But I don't know what it is. I looked for it but ... | I believe it is just a formal symbol that represents the element. They wanted something that was a modified $\alpha$ for continuity reasons, and decided to go with that symbol for unknown reasons. It's the same as if they had written $\alpha'$ (or even $x$ for that matter)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1866955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$?
Let $E=\mathbb{Q}(2^{1/3})$. What is the normal closure of $E/E$?
My thought is the $A(2^{1/3})$ where $A$ is an algebraic closure of $\mathbb{Q}$. But I am not sure whether it is correct and why...
Thanks for your time.
| The normal closure of $E/E$ is $E$.
Note that a normal closure of a finite extension is always a finite extension, excluding your answer. Also note that $A[\sqrt[3]{2}]$ is in fact equal to $A$.
Recall that a characterization of normal is: "Every irreducible polynomial in K[X] that has one root in L, has all of its r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
counting number of steps using permutation-combination We need to climb 10 stairs. At each support, we can walk one stair or you can jump two stairs.
In what number alternative ways we'll climb ten stairs?
How to solve this problem easily using less calculation?
| Consider $f(n)$ as the number of ways to climb $n$ stairs. We note that $f(1) = 1$ and $f(2) = 2$. Now consider the case when $n>2$. The first action one can take is climb $1$ or $2$ stairs.
Suppose $1$ stair is climbed for the first step. Then there are $n-1$ stairs left and thus there are $f(n-1)$ alternate ways to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Problem including three circles which touch each other externally The circles $C_{1},C_{2},C_{3}$ with radii $1,2,3$ respectively,touch each other externally. The centres of $C_{1}$ and $C_{2}$ lie on the x-axis ,while $C_{3}$ touches them from the top. Find the ordinate of the centre of the circle that lies in the reg... | HINT...if $(a,b)$ is the centre of the circle and its radius is $r$ you can set up and solve a system of three simultaneous equations. So for example, for circle $C_1$ you have $$(a+1)^2+b^2=(r+1)^2$$ and likewise for the other two circles.
Of course, quoting Descartes' Theorem will be a short-cut to finding $r$ but yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
A question about the product functor on finite sets I am a beginner in Category Theory so please excuse me if this is a trivial question.
Let $\mathbf{FSet}$ denote the category of finite sets. The product functor $X\times -:\mathbf{FSet}\to \mathbf{FSet}$ has a right adjoint for every finite set $X$.
My question is, ... | No, it doesn't unless $X$ is a singleton. The very first condition to check for a functor to have a left adjoint is that it should preserve limits (such as products, equalizers...). But clearly in general, if $X$ has at least two element,
$$X \times (Y \times Z) \not\cong (X \times Y) \times (X \times Z),$$
and so the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Squeeze fractions with $a^n+b^n=c^n+d^n$ Let $0<x<y$ be real numbers. For which positive integers $n$ do there always exist positive integers $a,b,c,d$ such that $$x<\frac ab<\frac cd<y$$ and $a^n+b^n=c^n+d^n$?
For $n=1$ this is true. Pick any $a,b$ such that $x<\frac ab<y$ -- this always exists by the density of the r... | Partial answer I: if $x < 1 < y$, then we can find $a,b$ with $\frac{a}{b}, \frac{b}{a}$ arbitrarily close to one, satisfying the requirements for any $n$. Then it can be seen that that it suffices to prove the result for $y<1$ or $1<x$, since we have symmetry about 1 by inversion: $$x < \frac{a}{b} < \frac{c}{d} < y <... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding a tricky composition of two piecewise functions I have a question about finding the formula for a composition of two piecewise functions. The functions are defined as follows:
$$f(x) =
\begin{cases}
2x+1, & \text{if $x \le 0$} \\
x^2, & \text{if $x > 0$}
\end{cases}$$
$$g(x) =
\begin{cases}
-x, & \text{if $x ... | You're correct about the value of $g(f(x))$ when $x\leq 0$; since $f(x)$ will be at most $2\cdot0+1=1$, $g$ is only going to evaluate $f(x)$ according to the definition for $x<2$. Testing for cases here is a good approach, and you've just resolved the $x\leq0$ case. When $x>0$, consider the values of $f(x)$: when will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Gradient and Hessian of function on matrix domain Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and hessian of $f(X)$ with respect to $X$. Thank you!
| Let $f(X)=\log(|\det(I+XA)|)$; we calculate $Df_X$ in a point $X$ s.t. $I+XA$ is invertible, that is, $-1$ is not an eigenvalue of $XA$.
$Df_X:H\in M_{p,k}\rightarrow tr(HA(I+XA)^{-1})=tr((I+XA)^{-T}A^TH^T)$ or
$Df_X(H)=<(I+XA)^{-T}A^T,H>$ (the scalar product over the matrices). In other words, the gradient of $f$ is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Summing a series of integrals I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here.
I have a series of integrals I would like to sum, but I don't understand how I would begin to do that considering the structure of the integrals.
Questi... | This really isn't so bad.
$$\int_0^i\frac{i}{(i+x^2)^{3/2}}dx=\frac{i}{\sqrt{i(i+1)}}.$$
So you're summing:
$$\sum_{i=m}^n\frac{i}{i+1}.$$
The latter sum unfortunately doesn't have an explicit form, unless you're willing to use digamma functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Correlated brownian motions and Lévy's theorem $W^{(1)}_t$ and $W^{(2)}_t$ are two independent Brownian motions. How can I use Lévy's Theorem to show that
$$W_t:=\rho W^{(1)}_t+\sqrt{(1-\rho^2)} W^{(2)}_t,$$
is also a Brownian motion for a given constant $\rho\in(0,1)$.
Also, it is clear why $\rho$ in front of the firs... |
If $W_t:=\rho W^{(1)}_t+\sqrt{1-\rho^2} W^{(2)}_t$ then we can show that,
$W_t$ is a Brownian motion.
Proof
Let $(\Omega, \mathcal{F},\mathbb{P},\{\mathcal{F_t}\})$ be a probability space . Clearly, $W_t$ has continuous sample paths and $W_0=0$.
$$\mathbb{E}[W_t|\mathcal{F_s}]=\rho\,\mathbb{E}[W^{(1)}_t|\mathcal{F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1867936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is the way to show the following derivative problem? If $f$ is function twice differentiable with $|f''(x)|<1, x\in [0,1]$ and $f(0)=f(1)$, then $|f'(x)|<1$ for all $x\in [0,1]$
I have tried with Rolle's theorem, but fail
| Hint: For some $c\in[0,1]$, $f'(c)=0$ (can you see why?). Now apply the mean value theorem to $f'$ to bound $f'(x)$ for any other $x\in[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $A$ is infinite and bounded, the infimum of the difference set of $A$ is zero. Let $A$ be a non-empty subset of $\mathbb{R}$. Define the difference set to be
$A_d := \{b-a\;|\;a,b \in A \text{ and } a < b \}$
If $A$ is infinite and bounded then $\inf{A_d} = 0$.
Since $a < b$ we have $b - a > 0$. Thus zero is a low... | Your argument is ok for me. If you want to apply the Pigeon-Hole Principle: We have $A\subset [\inf A, \sup A]=[x,y]$ with $x<y$. For any $r>0$ take $n\in N$ such that $(y-x)/n<r.$ The set of $n$ intervals $S= \{[x+j(y-x)/n,x+(j+1)(y-x)/n] : 0\leq j<n\}$ covers $[x,y].$ Take any set $B$ of $n+1$ members of $A.$ At le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$ Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$.
I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded?
I have noted that $$x_{n+1}-x_n = (x_n-1)^2\geq 0$$ so $x_n$ is i... | Notice that $x_{n+1}=1-x_n(1-x_n)$. Thus $x_{n+1}\in[0,1]$ if and only if $x_n\in[0,1]$. Thus the sequence is bounded above by $1$ if and only if $\alpha\in[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
What does the notation $\overline{\mathbb R}$ mean in that context? In an old question, it can be read that "the finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$" is one of the "impressive finiteness results in mathematics".
I commented the question to know what was meant by the notation $\overline{\mathbf R}... | My reading of this quote is that
*
*$\overline{\bf R}$ refers to $\Bbb{C}$, and that
*"impressive finiteness result" refers to the (not that obvious) fact that $\Bbb{C}$ is algebraically closed.
Most of us hear about $\Bbb{C}$ being algebraically closed early in our studies, may be in the same course the complex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the ring of integers of $\Bbb Q(\sqrt[4]{2})$
I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt[4]{2})$ is $\Bbb Z[\sqrt[4]{2}]$ and I would like to prove it.
A related question is this one, but it doesn't answer mine.
I computed quickly the discriminant $\text{disc}(1,\sqrt[4]{2},\sqrt[4]{4},\sqrt[... | Following the approach of Keith Conrad, suppose that $$\alpha = a + b \sqrt[4]2+c\sqrt[4]4+d\sqrt[4]8,\quad a,b,c,d\in\mathbb Q$$
is an element of $\mathcal O_K$. We will show that $\alpha\in\mathbb Z[\sqrt[4]2]$. Calculating traces,
$$
\mathrm{Tr}_{K/\mathbb Q}(\alpha) = 4a\\
\mathrm{Tr}_{K/\mathbb Q}(\sqrt[4]2\alpha)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Getting characteristic polynomial from a small matrix Sorry I don't know how to format matrices, but if I have this matrix
$\pmatrix{1& 1& 0\\
0& 0& 1\\
1 &0& 1\\}$
How is the characteristic polynomial $λ^3 − 2λ^2 + λ − 1$? Is there some methodical approach to getting the characteristic polynomial from a matrix... | As mentioned in the comments, you just find $\det(A-\lambda I)$ (or $\det(\lambda I-A)$ if you want the leading term positive). Alternatively, if you find all of the (complex) eigenvalues $\lambda_1, \lambda_2, \lambda_3$, counted with multiplicity, then the characteristic polynomial will be $(\lambda-\lambda_1)(\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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