Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Euclidean distance and dot product I've been reading that the Euclidean distance between two points, and the dot product of the two points, are related. Specifically, the Euclidean distance is equal to the square root of the dot product. But this doesn't work for me in practice.
For example, let's say the points are $(... | distance=$D=\sqrt{((|A|^2-(A\cdot B)^2)+(|B|-(A\cdot B))^2)}$
I found this geometrically like this:
sorry for the lousy formatting, this is my first time posting here. I needed to find D in terms of norms and dot products for my linear algebra homework, as I assume most people asking this question are, and the other a... | {
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"timestamp": "2023-03-29T00:00:00",
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Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it. I've been stuck on this question for a long time now and was wondering if anyone could show me how it's done. So far I have done the following:
Since $\lim_{x\to \infty}f'(x)... | Here is an easier way out. By Mean Value theorem, given any $x$, there exists $y \in [x,x+1]$ such that
$$f(x+1) - f(x) = \dfrac{f(x+1)-f(x)}{(x+1)-x} = f'(y)$$
Now as $x \to \infty$, since $y \in [x,x+1]$, we have $y \to \infty$. Hence, we obtain that
$$\lim_{x \to \infty} \left(f(x+1) - f(x)\right) = \lim_{y \to \inf... | {
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Express $\ln(3+x)$ and $\frac{1+x}{1-x}$ as Maclaurin series its probably a lot to ask.. but how can I obtain the Maclaurin series for the two functions $f(x)=\ln(3+x)$ and $g(x)=\frac{1+x}{1-x}$ ? as far as I know I cant use any commonly known series to help me with this one ? so finding the derivatives at 0:
$f(0) = ... | for $(x+1)/(1-x)$ write it as
$$
x \frac{1}{1-x} + \frac{1}{1-x}
$$
substitute the series for $1/(1-x)$ and proceed. To get a single series, add the terms one by one.
Also
$$
\log(3 -x) = \log(3) + \log(1 - x/3)
$$
so substitute $x/3$ in the series for $\log(1-y)$ which can be found be integrating the series for $1/(... | {
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Why aren't these negative numbers solutions for radical equations? I was working on radical equations and I came across a few problems where I got answers that worked when I checked, but were not listed as solutions. My teacher's only explanation was, "just because." Here is one problem where the only solution is $1$.
... | The answer, as already the others has highlighted, is that $\sqrt{4} \neq -2$, but to aid you in the actual expression evaluation, consider that $\sqrt{x^2} \neq x$, but $\sqrt{x^2} = \left| x \right|$.
| {
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Identifying translations and rotations as compositions. I am having trouble understanding the below which are the ones underline in red and blue.
For the red: Why is that $R_{A,90}(A)=A$ and that $\tau_{AB}(A)=B$
As for the blue: Why is that $R_{A,90}(B)=A'$
| The transformation $R_{A,90^\circ}$ is a counterclockwise rotation of the plane through an angle of $90^\circ$ around the point $A$. Since $A$ is the centre of rotation, it doesn't move, and therefore $R_{A,90^\circ}(A)=A$. When $B$ is rotated $90^\circ$ counterclockwise around the point $A$, it ends of at $A'$: it sta... | {
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$5 \nmid 2^{n}-1$ when $n$ is odd I want to prove that $$5 \nmid 2^{n}-1$$ where $n$ is odd.
I used Fermat's little theorem, which says $2^4 \equiv 1 \pmod 5$, because $n$ is odd then $4 \nmid n$ , so it is done.
can you check it and say that my proof is right or wrong.
thanks.
| ${\rm mod}\ 5\!:\,\ 2^{1+2j}\equiv 2(4^j)\equiv 2(-1)^j\equiv \pm2 \not\equiv 1$
| {
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Caden has 4/3 kg of sand which fills 2/3 of his bucket. How many buckets will 1kg sand fill? I have already finished Calculus II but I go back and practice the basics on Khan Academy. This problem confuses me conceptually every time. I know what the answer is, but I am having a hard time rationalizing the steps. Wha... | As the bucket is filled by $2/3$, a half of its actual content is needed to get it filled completely, that is a half of $4/3$ kg.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the following statement true on $L^0$ spaces? Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X,Y\in L^0(\Omega;\mathbb{R})$ two random variables taking values in $\mathbb{R}$. Is it true that:
$$\int_{A} f(X(\omega)) P(d\omega) = \int_{A} f(Y(\omega)) dP(\omega)$$
for all $A\in \mathcal{F}$ and all Lipsch... | Under the further hypothesis that $(\Omega, \mathcal{F},P)$ is a standard probability space, I believe that this is true even if we restrict to only one $f$, namely $f(x) = x$.
Recall the Lebesgue Differentiation Theorem. To apply that here, assume that $\Omega =[0, 1]$, $\mathcal{F}$ is the completed $\sigma$-algebra,... | {
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Solving integral $\int \frac{3x-1}{\left(x^2+16\right)^3}$ I need to solve this one integral.
$$\int \frac{3x-1}{\left(x^2+16\right)^3}$$
You need to use the method of undetermined coefficients. That's what I get:
$$(3x-1) = (Ax + B)(x^{2}+16)^{2} + (Cx + D)(x^{2}+16) + (Ex + F)$$
$$1: 256B + 16D + F = -1$$
$$x: 256... | Hint: Alternately, evaluate $I(a)~=~\displaystyle\int\frac{3x-1}{x^2+a}~dx$ assuming $a>0$, and then try to express your integral in terms of $I''(16)$.
| {
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Proving $ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}$ in a geometric context
Prove or disprove
$$
\frac{1}{c} = \frac{1}{a} + \frac{1}{b}.
$$
I have no idea where to start, but it must be a simple proof.
Trivia. This fact was used for determination of resistance of two parallel resistors in some circumstances long time a... | In general, if the vertex angle is $2\theta$ and $OC$ is the angle bisector, since
$$\text{Area of triangle }OAB = \text{Area of triangle }OAC + \text{Area of triangle }OBC$$
we have
$$\dfrac{OA \cdot OB \cdot \sin(2\theta)}2 = \dfrac{OA \cdot OC \cdot \sin(\theta)}2 + \dfrac{OC \cdot OB \cdot \sin(\theta)}2$$
$$ab\sin... | {
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Integrating a jacobian to find the volume. I want to solve the following:
Prove that
$$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$
where $ R=[0,2\pi] \times [0,\pi]^{n-2}$. Hint: Compute $ \int_{\mathbb R^n}e^{-|x|^2}dx... | Hint all the variables $\phi_i$ can be separted in the integral, so you can transform the integral into:
$$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2}=\int_{0}^{2\pi}d\theta \prod_{i=1}^{n-2}\int_{0}^{\pi}\sin^i\phi_i d\phi_i$$
| {
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How to evaluate $\log x$ to high precision "by hand" I want to prove
$$\log 2<\frac{253}{365}.$$
This evaluates to $0.693147\ldots<0.693151\ldots$, so it checks out. (The source of this otherwise obscure numerical problem is in the verification of the Birthday problem.) If you had to do the calculation by hand, what me... | You could express this as $\log(1/2) > -\dfrac{253}{365}$. The series
$$\log(1/2) = \log(1-1/2) = -\sum_{n=1}^\infty \dfrac{1}{n 2^n}$$ converges quickly, and has nice bounds:
$$ \log(1/2) \ge - \sum_{n=1}^{N-1} \dfrac{1}{n 2^n} - \sum_{n=N}^\infty \dfrac{1}{N 2^n} = - \sum_{n=1}^{N-1} \dfrac{1}{n 2^n} - \dfrac{1}{N... | {
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Let $x \in \mathbb Q \setminus \{0\}$ and $y \in \mathbb R\setminus \mathbb Q$. Prove that $\frac{x}{y} \in \mathbb R \setminus \mathbb Q$ Let $x \in \mathbb Q\setminus \{0\}$ and $y \in \mathbb R\setminus \mathbb Q.$ Prove that $\frac{x}{y} \in \mathbb R \setminus \mathbb Q$
I saw this question in a basic analysis te... | From the hypothesis we have that
$$\exists n,m \in \mathbb{Z} \setminus \{ 0 \} : x = \frac{m}{n}$$
and that
$$ y \in \mathbb{R} \setminus \mathbb{Q}$$
Now assume that
$$ \frac{x}{y} \in \mathbb{Q}$$
Now we
$$ \exists p,q \in \mathbb{Z} : \frac{x}{y} = \frac{p}{q}$$
Well we can write this as
$$ \frac{1}{y} \frac{m}{n} ... | {
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"timestamp": "2023-03-29T00:00:00",
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Hamiltonian Graphs with Vertices and Edges I have this question:
Alice and Bob are discussing a graph that has $17$ vertices and $129$ edges. Bob argues that the graph is Hamiltonian, while Alice says that he’s wrong. Without knowing anything more about the graph, must one of them be right? If so, who and why, and if n... | The complete $K_{17}$ would have ${17\choose 2}=136$ edges. So we are missing only $7$ edges. Thus every vertex has degree at least $16-7=9>\frac{17}2$.
| {
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"url": "https://math.stackexchange.com/questions/1237774",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Prove $(\alpha_1 + ........ + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ....... + \alpha_n^2)$ For any real numbers $\alpha_1, \alpha_2, . . . . ., \alpha_n$,
$$(\alpha_1 + ...... + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ..... + \alpha_n^2)$$
And when is the inequality strict?
| This is just an application of Jensen's inequality. As $f(x) = x^2$ is a convex function, we can apply it here.
Reference : Jensen's inequality
| {
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Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y).
Eventually I managed to pull the (presumably) correct function:
$$f(x,y) = \ln \left( (1+2x)^{\frac{5}{4}y} - ... | My opinion is that you complicate things too much. Just try $f(x,y)=\ln (x^{0.25} + y^{0.02} -1)$ around $(1,1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1238034",
"timestamp": "2023-03-29T00:00:00",
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What did I do wrong in the permutations question. I was given the following question:
A hardware store sells numerals for house numbers. It has large quantities of the numerals 3, 5, and 8 but no other numerals. How many different house numbers, with no more than three digits can be made from these numbers?
So, I tried... | You've worked out the number of house numbers with exactly $3$ digits, but the question says `no more than' $3$ digits.
See if you can work out the number of house numbers with only $2$ or $1 $ digit!
| {
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"timestamp": "2023-03-29T00:00:00",
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$3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$ This is an exercise in my textbook in a chapter about the Hasse-Minkowski-theorem:
Show that the polynomial $3 X^3 + 4 Y^3 + 5 Z^3$ has a non-trivial root in $\mathbb{R}$ and all $\mathbb{Q}_p$. Show that it has only the t... | This is a classical counterexample to Hasse-Minkowski, due to Selmer. The solution of the second part is indeed difficult. For a detailed proof see, for example, Chapter $7$ of the thesis of Arnélie Schinck.
However, the first part is quite easy, and goes as follows: we can explicitly list a solution for each $\mathbb... | {
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Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ?
The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that means.
| I'm going to use the basic definitions of scalar and vector triple products to prove this. Let $\mathbf u= a_1 \mathbf i + a_2 \mathbf j + a_3 \mathbf k, \mathbf v= b_1 \mathbf i + b_2 \mathbf j + b_3 \mathbf k, \mathbf w= c_1 \mathbf i + c_2 \mathbf j + c_3 \mathbf k$,
Then $(\mathbf{v} \times \mathbf{w})= (b_2c_3-b_3... | {
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If $4=5$, then $6=8\,$ (yes or no?) I had an argument with a friend about the statement in the title. I asserted that if $4=5$, then $6=8$, as you can derive any conclusion from a false statement. However, he does not agree, and claims that you cannot know that $6$ would equal $8$ if $4$ were to equal $5$. Is the state... | $$4=5$$
$$4-1=5-1$$
$$2(4-1)=2(5-1)$$
| {
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Proof for $0a = 0$ Is this a valid proof for $0a =0$? I am using only Hilbert's axioms of the real numbers (for simplicity).
$(a+0)(a+0) = a^2 + 0a + 0a + 0^2 = (a)(a) = a^2$
Assume that $0a$ does not equal zero.
Then from $a^2 + 0a + 0a + 0^2$ we get $(a+0)(a+0) > a^2$ or $(a+0)(a+0) < a^2$ which contradicts the prove... | $0+0=0.\\\text {Multiplying on both sides by a from the right,}\\ [0+0].a=0.a.\\or, 0.a+0.a=0.a.\\\text {Adding -0.a on both sides,}\\-(0.a)+[0.a+0.a]=-(0.a)+0.a\\or, [-(0.a)+-0.a]+0.a=-(0.a)+0.a.\\or, 0+0.a=0.\\or, 0.a=0. $
| {
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Proving $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 I am trying to prove that $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 for all $k \in Z$.
I am not even sure what tags to use because I am not sure of right methods to use to solve this problem.
I don't see ... | Let $k = a_0 + 10a_1 + 10^2a_2 + ... + 10^na_n$, this is a decimal representation where $a_i$ represents an individual digit.
Consider $k \pmod 3$.
Since $10^i \equiv 1^i = 1\pmod 3$,
$k \equiv a_0 + a_1 + ... + a_n \pmod 3$, and you're done.
Using induction is more unwieldy, and you'll still have to use modular arithm... | {
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$a^2b + abc + a^2c + ac^2 + b^2a + b^2c + abc +bc^2$ factorisation
I came across this from a university mathematics resource page but they do not provide answer to this.
What I did was this:
$(a^2+b^2+c^2)(a+b+c) - (a^3 + b^3 + c^3) + 2abc$
But I don't think this is the correct solution.
How should I spot how to facto... | From What I see
$$ab(a+c)+ac(a+c)+b^2(a+c)+bc(a+c)=(a+c)(ab+ac+b^2+bc)$$
$$=(a+c)(a(b+c)+b(b+c))=(a+b)(b+c)(a+c)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Rearranging into $y=mx+c$ format and finding unknowns $a$ and $b$
Two quantities $x$ and $y$ are connected by a law $y = \frac{a}{1-bx^2}$ where $a$ and $b$ are constants. Experimental values of $x$ and $y$ are given in the table below:
$$
\begin{array}{|l|l|l|l|l|l|}
\hline
x & 6 &8 & 10 & 11 & 12\\
\hline
y & 5.50... | When you are asked questions like this one, basically the problem is to find the change of variables which transform the equation into the equation of a straight line. So, starting with $$y = \frac{a}{1-bx^2}$$ rewrite $$\frac 1y=\frac{1-bx^2}{a}=\frac 1a-\frac ba x^2$$
I am sure that you can take from here.
| {
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Equivalent definition of cograph A cograph is simple graph defined by the criteria
*
*$K_1$ is a cograph,
*If X is a cograph, then so is its graph complement, and
*If X and Y are cographs, then so is their graph union X union Y
and also it is said that a graph is called cograph if G does not contain the path graph... | To see that the first definition implies the second, you can use induction on $n$, the number of vertices.
The base case is trivial. For the induction step, the cograph $G$ can be obtained either by the union of two smaller cographs $G_1$ and $G_2$, or by complementing such a graph.
In the first case, apply induction ... | {
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Is Category Theory similar to Graph Theory? The following author noted:
Roughly speaking, category theory is graph theory with additional structure to represent composition.
My question is: Is Category Theory similar to Graph Theory?
| Category theory and graph theory are similar in the sense that both are visualized by arrows between dots.
After this the similarities quite much stop, and both have different reason for their existence.
In category theory, we may have a huge amount of dots, and these dots do often represent some abstract algebraic st... | {
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Proving $6^n - 1$ is always divisible by $5$ by induction I'm trying to prove the following, but can't seem to understand it. Can somebody help?
Prove $6^n - 1$ is always divisible by $5$ for $n \geq 1$.
What I've done:
Base Case:
$n = 1$: $6^1 - 1 = 5$, which is divisible by $5$ so TRUE.
Assume true for $n = k$, whe... | $6$ has a nice property that when raised to any positive integer power, the result will have $6$ as its last digit. Therefore, that number minus $1$ is going to have $5$ as its last digit and thus be divisible by $5$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\bigcup \alpha$ where $\alpha$ is a finite ordinal. Given a finite ordinal, is it correct in saying $\bigcup \alpha = \alpha - 1$?
As an illustrative example consider $3 = \{\emptyset , \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$.
I believe $\bigcup 3 = \{\emptyset , \{\emptyset\}\} = 2$.
Is this accurate?
| Assume $\alpha=\beta+1=\beta\cup\{\beta\}$ where $\beta$ is an ordinal (note that this is applicable for all nonzero finite ordinals, but also for many infinite ones). Then $x\in\bigcup\alpha$ iff $x\in y$ for some $y\in \alpha$. And this is equivalent to $x\in\beta$ or $x\in y$ for some $y\in \beta$. Since $x\in y\in ... | {
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To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $. To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $.
Thus there exists an ideal $J$ of $ \Bbb Z \times \Bbb Z $ such ... | Yes, it will. It is clearly between $I$ and the full ring and is an ideal.
Alternatively, without exhibiting $J$, you can calculate $(1,0)\cdot(0,1)=(0,0)$, which shows that the product of two elements $\notin I$ can be $\in I$; hence $I$ is not even prime, let alone maximal.
| {
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4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose. Question: $4$ cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You ... | Probablity of losing is not 1/3. It should be 2/3.
As Below,
He wins if,
*
*First three cards are Water/Earth/Wind = 3!
*First two cards are Water/Earth and third is Fire = 2
so the total = 6 + 2 = 6
Probability of winning = 8/24 = 1/3 , so loosing 2/3.
| {
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Dimension of garden to minimize cost Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while on the other three sides it cost only $\$2$ per foot, find the dimension that will... | Let $A=5000$, length of the side along driveway$=x$ and width $=y$.
Then $A=xy=5000$. Total cost $\displaystyle =C=6x+2(x+2y)=8x+\frac{4\times 5000}{x}$. Now we have to find $x$ that minimizes $C$. So differentiating $C$ w.r.t. $x$ and equating it to $0$,
$\displaystyle \frac{dC}{dx}=8-\frac{20000}{x^2}=0\Rightarrow x... | {
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How to Prove Unboundedness Suppose I have a submartingale $X_k$, what results/theorems can be useful if I want to show that $X_k$ is unbounded in the limit. There are results (basically bounding $\mathbb{E}X_k$) for convergence of submartinagles. But, I wanted to show that a submartingale goes to infinity in the limit ... | If the submartingale is of the form $$X_k = \sum_{j=1}^k \xi_j$$ where $\xi_j \in L^1$ are independent identically distributed random variables, then, by the strong law of large numbers,
$$\frac{X_k}{k} \to \mathbb{E}\xi_1 \qquad \text{almost surely as $k \to \infty$.}$$
This means that $$\lim_{k \to \infty} X_k = \inf... | {
"language": "en",
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Proof of Mean-value for Laplace's equation. $\textbf{ Statement of Theorem:}$ If $u \in C^2(U)$ is harmonic, then $$u(x) = \frac{1}{m(\partial B(x,r))}\int_{\partial B(x,r)} u dS = \frac{1}{m(B(x,r))}\int_{B(x,r)} u dy$$ for each $B(x,r) \subset U$.
$\textbf{Proof:}$ Let $$ \phi(r) := \frac{1}{m(\partial B(x,r))} u(y... | Divergence Theorem : $$ \int_U {\rm div}\ \nabla f =\int_{\partial U} \nabla f\cdot \nu $$ where $\nu$ is an unit outnormal to $\partial U$
| {
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How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$ I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has:
1 element of order 1 (AKA the identity)
3 elements of order 2
4 elements of order 4
I'm considering the set of all automorphisms on this group, denoted $\operatorname{Aut}(\math... | Note that $G$ is generated by two elements of order $4$.
To count the possible automorphisms we can focus on the maps to these two generators. To the first generator we can map any of the $4$ elements of order $4$, giving $4$ possibilities.
To the second generator we don't want to map the same element of order $4$ as... | {
"language": "en",
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Why does the Gaussian-Jordan elimination work when finding the inverse matrix? In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix
$$(A \mid I)$$
to obtain
$$(I \mid C),$$
where $C$ is indeed $A^{-1}$. However, I fail to see why this actually works, and readin... | You want to find a matrix $B$ such that $BA = I$. $B$ can be written as a product of elementary matrices iff it is invertible. Hence we attempt to obtain $B$ by left-multiplying $A$ by elementary matrices until it becomes $I$. All we have to do is to keep track of the product of those elementary matrices. But that is e... | {
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An integral related to the derivative of Legendre polynomials I want to calculate the integral
$$
I=\int_{-1}^{1}
\Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big)
\Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big)
\mathrm{d}t
$$
where $P_n(t)$ is Legendre polynomials. By virtue of the identity
$$
\frac{\mathrm{d}P_{... | Legendre polynomials are orthogonal.here So it may work to calculate that integral by using integrating by parts twices.
| {
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Solve $x + y + z = xyz$ such that $x , y , z \neq0$ I came across the equation $x+y+z=xyz$ such that $x , y , z \neq 0$.
I set $x=1, y=2, z=3$ but how can i reach formal mathematical solution without " guessing " the answer ? Thank you
| you can write $y+z=x(yz-1)$ and if $yz\ne1$ you will get $$x=\frac{y+z}{yz-1}$$
if $yz=1$ we get $y+z=0$ or $y^2+1=0$ which gives us complex solutions.
| {
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is 1 greater than i? I'm not sure this question even makes sense because complex numbers are a plane instead of a line. The magnitudes are obviously the same because i is a unit vector, but is there any inequality you can write about an imaginary number and a real number without just using their magnitudes?
| When we go from real numbers to complex numbers, we lose ordering of values.
You cannot compare between two non-real values or between a real and non-real value. So, you cannot compare $i$ and $1$ since $i$ is non-real while $1$ is real.
The only thing you can compare is their absolute values.
$$|i|=|1|=1$$
But that do... | {
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Error in my proof? What is wrong in this proof. It seems correct to me but still doesn't make proper sense.
$$\sqrt{\cdots\sqrt{\sqrt{\sqrt{5}}}}=5^{1/\infty}=5^0=1$$
EDIT
So does this mean that
$5^{1/\infty} = 1$
$(5^{1/\infty})^\infty = 1^\infty$
But according to me, $1^\infty$ is indeterminate.
| $$\lim_{n\to\infty}x^{1/2^n}=1$$ because the exponent approaches zero
| {
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"timestamp": "2023-03-29T00:00:00",
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Too strong assumption in the Uniqueness Theorem of Rudin's Real and Complex Analysis? In Rudin's Real and Complex Analysis, there is the following result about Fourier transforms.
The Uniqueness Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}(t)=0$ for all $t\in\mathbb{R}$, then $f(x)=0$ almost everywhere.
Isn't the ass... | As Ian pointed out, the Fourier transform of an $L^1$ function is continuous. For continuous functions, being zero a.e. and being zero everywhere are equivalent; so it makes more sense to use the shorter version, without a.e.
| {
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Looking for an intuitive approach of ODE I started reading up on the topics of differential equations and tried to solve certain problems to get used to those kind of equations, in particular I tried to understand every "$=$" and "$\Rightarrow$".
However I soon stumbled over this equation:
$ {1 \over y} * { dx \over d... | $x$ depends on $y$.
So :
$$\frac{d}{dy} \left( \frac{x}{y} \right)=\frac{d}{dy} \left( x \cdot \frac{1}{y} \right)=\frac{dx}{dy} \cdot \frac{1}{y}+x \frac{d}{dy}\left(\frac{1}{y} \right)=\frac{dx}{dy} \cdot \frac{1}{y}+x \left(-\frac{1}{y^2} \right)$$
| {
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"source": "stackexchange",
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Convergence in the weak operator topology implies uniform boundedness in the norm topology? If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology?
I know how to apply the uniform bo... | Hint: Try to prove the following result
$\textbf{Result}:$ If for a sequence $(x_n)$ in a Banach space $X$, the sequence $(f(x_n))$ is bounded for all $f\in X^*$, then the sequence $(\|x_n\|)$ is bounded.
Now suppose that you have proved the result. Suppose $(T_n)$ converges to $T$ in weak operator topology, i.e.,
\b... | {
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Change the matrix by multiplying one column by a number. Consider a positive definite matrix A. We can think of the matrix as a linear transformation. Now supopse we get matrix B by multiplying only one column of A by a number. Is there a geometric relation between A and B?
| We have that all Matrix $A\in M_{n\times m}(\mathbb{R})$, $m\leq n$ has a vectorial subspace of $\mathbb{R}^n$ asociated. if $V$ is the vectorial subspace generated for the columns of $A$. Then we can see that each element of $V$ has the form:
$$v=Ax, x\in \mathbb{R}^m.$$
Now if $A=(a_{1},\dots, a_{m})$, where $a_{i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1240821",
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Green's Theorem and limits on y for flux I'm working through understanding the example provided in the book for the divergence integral. The theorem (Green's):
$$
\oint_C = \mathbf{F}\cdot \mathbf{T}ds = \oint_CMdy-Ndx=\int\int_R(\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y} )dxdy
$$
The example uses the... | For Green's Theorem
$$\oint_C (Mdx+Ndy)=\int \int_R \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) dxdy$$
Here, $M=(x-y)$ and $N=x$ such that
$$\begin{align}\oint_C (Mdx+Ndy)&=2\int \int_R dxdy\\\\
&=2\int_{-1}^{1} \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}dxdy\\\\
&=4\int_{-1}^{1} \sqrt{1-y^2}dy\\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Algebra verbal find the amount of sold items Hey I am having an exam tomorrow, so I looked up at some verbal algebra questions, and found one that I could not solve, because I don't really understand how would I do this.
The question is like this:
An IKEA agent bought beds for 60,000 dollars. 1/4 of the beds that he bo... | Let $B$ be the number of beds. Let $P$ be the price he paid per bed.
$$60000=BP$$
$$\begin{align}9500&=\frac{B}4\cdot0.8\cdot P + \left(\frac{3B}{4}-4\right)\cdot(-0.1)\cdot P\\
&=\frac{BP}{4}\left(0.8+3\cdot(-0.1)\right)-4\cdot(-0.1)\cdot P\end{align}$$
| {
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Show $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$
Let $n \geq 2$ be an integer and $i, j \in \{1, 2, ..., n\} $ be distinct elements. Let $\sigma \in S_n$, Show that $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$
let $\tau=\sigma^{-1}(ij)\sigma$, then $((i)\sigma) \tau=((i)\sigma)\sigma^{-1} (i j)\sigma=i(i... | $$((i)\sigma)\sigma^{-1}(ij)\sigma=(i)(ij)\sigma=(j)\sigma$$
and
$$((j)\sigma)\sigma^{-1}(ij)\sigma=(j)(ij)\sigma=(j)\sigma$$
So, $(i)\sigma\mapsto(j)\sigma$ and $(j)\sigma\mapsto(i)\sigma$.
| {
"language": "en",
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Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $. I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$):
$$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$
However, I am not certain how one would go about findin... | One reason that it is hard to describe a systematic procedure that will let you find that solution to $A^2=-I$ is that there are so many other solutions to this equation. A solution cannot have real eigenvalues, so the image of the first standard basis vector $e_1$ (which give the firs column of your matrix) cannot be ... | {
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expectation calculation problem I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here.
A system made up of 7 components with independent, identically distributed lifetimes will operate until any of 1 of the system's components fails. If t... | You want the expected time until the earliest component failure, of seven i.i.d. components. This is the seventh least order statistic.
$$\begin{align}
\mathsf E[X_{(7)}]
& = \binom{7}{1}\int_1^\infty x\cdot f_{X}(x)\cdot (1-F_X(x))^6 \operatorname d x
\\ & = 7\int_1^\infty x \cdot\frac {3}{x^4}\cdot \left(\int_x^\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finite conjugate subgroup In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached-
What is finite conjugate subgroup of a group? It is not clear to me what is author referring here.
| I had never heard of this before, but after a bit of searching, I found the definition $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ or, in other words, the elements of $G$ whose conjugacy classes in $G$ are finite. It is easy to see that $\Delta(G)$ is a normal (in fact characteristic) subgroup of $G$. Apparently ... | {
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Verbal question problem help I had this question today and I got confused on how to construct my solution for this. At the first view of the question, I decided to use $X$ and $Y$ and no other option:
Maria purchased $X$ books for 300 dollars. After a day, the shop sold the books for 2 dollars cheaper. So Ana purchased... | As I mentioned in my comment, the first step should be $X=300/Y$. But that will make it hard to solve. A better way is to look at your second equation. If you multiply it out, there is a term $XY$ which can be replaced with $300$ according to your first equation. You might then go on to do your substitution with this s... | {
"language": "en",
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Determining half life without logs, given only reduction undergone and total time taken I have a half-life question that I can't solve. There's very limited information given. Even the half-life formula has not been taught yet.
The mass of a radioactive substance in a certain sample has decreased 32 times in 10 years.... | The solution $2$ assumes the following interpretation of the problem statement:
Suppose you have a radioactive sample of mass $m_0$ which after $10$ years is reduced to mass $m_{10} = \frac{m_0}{32}$. What is the half-life of the substance?
Recall that the half-life of a radioactive substance is the time after which ... | {
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Giving integer images (bis) Prove the statement below (the restriction $x < y < z$ is to avoid apparent uncertainties but the property is valid for all $x, y, z$ really).
$$F(x,y,z) = \frac{(y+z)x^n}{(z-x)(y-x)} +\frac{(z+x)y^n}{(z-y)(x-y)}+\frac{(x+y)z^n}{(x-z)(y-z)} \in \mathbb Z.$$
| If $n = 1$, then the whole expression is equal to $-1$. If $n=2$, it is equal to $0$. Let us therefore assume that $n > 2$.
We have:
$$F(x,y,z) = \frac{y^nz^2 - y^2z^n + x^n(y-z)(y+z) + x^2(z^n-y^n)}{(x-y)(x-z)(y-z)} = \frac{f(x,y,z)}{(x-y)(x-z)(y-z)}.$$
What we want to know is if the numerator is divisible by $x-y$, $... | {
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Prove that a symmetric distribution has zero skewness Prove that a symmetric distribution has zero skewness.
Okay so the question states : First prove that a distribution symmetric about a point a, has mean a.
I found an answer on how to prove this here: Proof of $E(X)=a$ when $a$ is a point of symmetry
Of course I use... | I have a simpler proof. I hope this is ok. Let $Y = X - a$ be a random variable. Now note that due to symmetricity $Y$ and $-Y$ have the same distribution. That implies
$$E[Y^3] = E[(-Y)^3]$$
This implies $E[Y^3] = 0$.
| {
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How many non prime factors are in the number $N=2^5 \cdot 3^7 \cdot 9^2 \cdot 11^4 \cdot 13^3$. to find non prime factors in the number
$N=2^5 \cdot 3^7 \cdot 9^2 \cdot 11^4 \cdot 13^3$.
First I tried finding all the factors by adding 1 to each of the exponents and then multiplying them
and then finding the prime facto... | Write it as $2^5 * 3^{11} * 11^4 * 13^3$. Thus, as you said, adding one to each exponent and multiplying, we get the total number of divisors, which is $6* 12 * 5* 4 = 1440$. Subtracting the four prime factors ($2,3,11,13$) leaves us with 1436.
| {
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How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ I know there are many methods to scale a number from range $[A,B]$ to a range $[C,D]$, and I've searched over and over the web. I've seen this math.SE thread.
I need to scale a big number (signed 32-bit integer) between $-2147483648$ and $... | This works in special cases:
Let $[a,b]$ be an interval and consider a uniform distribution on the intervals in this interval. Let $k$ be the number of integers in $[a,b]$.
Let $[c,d]$ be another interval and suppose that the number of integers in this interval is $l$.
BIG ASSUMPTION: Suppose that $l\mid k$.
Then, any... | {
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Proving the sequence $f_{x_n}(x)= (n+1)x^n$ converges in distribution I am preparing for a final exam and just working on sample problems.
Let $X_1,X_2,\dots$ be an infinite sequence of continuous random variables such that $f_{x_n}(x)= (n+1)x^n$ for $0<x<1$ and $0$ otherwise.
Show that $\{X_n\}$ converges in distribu... | Hint: A sequence of rvs converge in distribution is the sequence of corresponding characteristic functions converge for points where the function sequence is continuous. This is Levy's continuity theorem.
So find the characteristic function sequence and show its convergence.
| {
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Vector times symmetric matrix For most of you here, this is probably quite basic.
As for a symmetric matrix $A$ the first row equals the first column, multiplying the matrix with a column vector $b$ equals multiplying the transposed vector $b'$ with the symmetric matrix, i.e. if $A=A'$ then
$$Ab=b'A$$
Could you please ... | Except in dimension $1$, your claim is not correct. However, because transposition swaps factors we have
$$ (Ab)^T=b^TA^T=b^TA$$
that is multiplying with a column vector from the right equals the transpose of multiplying the transposed vector from the left.
| {
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Show that the Wronskian of solutions of $y''+p(x)y'+q(x)y=0$ satisfies $\frac{dW}{dx}+pW=0$ So I am given: $\{y_1(x),y_2(x)\}$ is a fundamental solution set of the ODE:
$$y''+p(x)y'+q(x)y=0$$ I need to show that the Wronskian $W(y_1,y_2)$ satisfies the ODE $\frac{dW}{dx}+pW=0$ and hence, $W(x)=C \cdot \exp(-\int p(x) ... | you have $$y_1'' + py_1' + y_2 = 0\tag 1$$ $$y_2'' + py_2' + qy_2 = 0 \tag 2 $$ multiplying $(1)$ by $y_2,(2)$ by $y_1$ and subtracting gives you,
$$y_1''y_2 - y_2y_1'' + p(y_1'y_2 - y_2'y_1) = 0 \to W' + pW = 0$$ the solution is $$W = Ce^{\int_0^x p\, dt} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of Newton Girard formula symmetric polynomials Newton Girard formula states that for $k>2$:
\begin{equation}
p_k=p_{k-1}e_1-p_{k-2}e_2+\cdots +(-1)^{k}p_1e_{k-1}+(-1)^{k+1}ke_{k}
\end{equation}
where $e_i$ are elementary symmetric functions and $p_0=n$ with $p_k=x_1^k+\cdots+x_n^k$.
I am using induction to prove ... | By way of enrichment here is an alternate formulation using
cycle indices.
Recall that the OGF of the cycle index $Z(P_n)$ of the unlabeled set
operator $\mathfrak{P}_{=n}$ is given by
$$G(w) = \sum_{n\ge 0} Z(P_n) w^n =
\exp\left(\sum_{q\ge 1} (-1)^{q+1} a_q \frac{w^q}{q}\right).$$
Differentiating we obt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$?
If the set $B=\{f(x) : x\in A\}$ has supremum and $C=\{k+f(x): x\in A\}$, then what is $\sup C$?
Since $C$ is not an empty set and $f(x)\le \sup(B) ⇒ k+f(x)\le k+\sup(B)$.
So $C$ is bounded above. Thus, $\sup(C)\le k+\su... | *
*$\sup(C) \geq k + \sup (B)$ is equivalent with $\sup (C) - k \geq \sup (B)$.
*Given that $\sup (C)$ is an (in fact, the smallest) upper bound on $C$, can you show that $\sup (C) - k$ is an upper bound on $B$?
*Since $\sup(B)$ is its smallest upper bound, you'd automatically have the inequality $\sup (C) - k \geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Absolute Value Algebra with inverses I noticed the following equality in some material regarding limits and infinite series.
$$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$
And I'm honestly stumped (and slightly ashamed) on how to algebraically go from the lefthand side to the righthand side. An... | It's just a little bit of algebra to get there.
\begin{align*}
\left|\frac{x}{x+1} -1\right| &= \left|\frac{x}{x+1} - \frac{x+1}{x+1}\right| \\ &= \left|\frac{x-x-1}{x+1}\right| \\ &= \left|\frac{-1}{x+1}\right|
\end{align*}
I hope that helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1242868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Seeking after notation for two objects equal up to a constant Sometimes we want to express that two objects are equal up to a constant but there is no need to keep writing out the constant or constants. For example, often times the constant or constants involved in the derivation of a primitive of a function plays or p... | Apparently there is no standard notation. I have seen $\propto$ used in the additive context as well as the multiplicative, leaving it to the reader to overcome the abuse of notation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Clockwise rotation of $3\times3$ matrix? I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me.
Find the $3\times3$ matrix which represents a rotation clockwise through $43°$ about the point $(\frac{1}{2},1+\frac{8}{10})$
For example: if the rotation ... | The center of rotation is the point $C=(c_1,c_2)=(1/2,9/5)$ and the angle is $\theta= -43°$.
You can represent this transformation whith a $3 \times 3$ matrix using homogeneous coordinates in the affine plane.
Note that you can perform your transformation in three steps:
1) translate the origin to the point $C$
2) rot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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$\lim_{x\to\infty}{f(x)}=\lim_{x\to\infty}{g(x)}\Rightarrow\lim_{x\rightarrow\infty}{\frac{f(x)}{2^x}}=\lim_{x\rightarrow\infty}{\frac{g(x)}{2^x}}$? Is this statement true? Why?
$$\lim_{x \rightarrow \infty}{f(x)} =\lim_{x \rightarrow \infty}{g(x)} \quad \Rightarrow\quad \lim_{x \rightarrow \infty}{\frac{f(x)}{2^x}}=\... | The answer is definitely no. Think of 2 functions which tends to $\infty$ as $x\to\infty$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove the inclusion-exclusion formula We just touched upon the inclusion-exclusion formula and I am confused on how to prove this:
$|A ∪ B ∪ C| =|A| + |B| + |C|
− |A ∩ B| − |A ∩ C| − |B ∩ C|
+ |A ∩ B ∩ C|$
We are given this hint: To do the proof, let’s denote $X = A ∪ B$, then
$|(A ∪ B) ∪ C| = |X ∪ C|$,
and we can appl... | $|(A\cup B)\cup C|=|A\cup B|+|C|-|(A\cup B)\cap C|$
Now, $|A\cup B|=|A|+|B|-|A\cap B|$
$$\text{and }|(A\cup B)\cap C|=|(A\cap C)\cup (B\cap C)|=|(A\cap C)|+|(B\cap C)|-|(A\cap C)\cap (B\cap C)|=|(A\cap C)|+|(B\cap C)|-|A\cap B\cap C|$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Real Numbers are Roots $r, s$.
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
Using Vieta's Formulas,
$r+s+x_1$ $=0$ $\Rightarrow x_1$ $=-r-s$, where $x_1$ is the third root. Similarly, $x_2=-r-s-1$ $=x_1-1$, w... | Since $r,s$ are two roots of $x^3+ax+b=0$, we have
$$ r^3+ar+b=0,s^3+as+b=0 \tag{1}$$
and hence
$$ (r^3-s^3)+a(r-s)=0. $$
Assuming $r-s\neq0$, we have
$$ r^2+rs+s^2+a=0.\tag{2}$$
Similarly since $r+4,s-3$ are two roots of $x^3+ax+b+240=0$, we have
$$ (r+4)^3+a(r+4)+b+240=0,(s-3)^3+a(s-3)+b+240=0\tag{3}$$
and hence
$$ [... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $A,B \in {M_2}$ and $C=AB-BA$. Why is ${C^2} = \lambda I$ true? Let $A,B \in {M_2}$ and $C=AB-BA$. Why does ${C^2} = \lambda I$?
| It is a well-known fact that for any two matrices $A$ and $B$ with
$C = [A, B] = AB - BA, \tag{1}$
we have
$\text{Tr} (C) = 0, \tag{2}$
where
$\text{Tr}(X) = \text{trace}(X) = X_{11} + X_{22} \tag{3}$
for any $2 \times 2$ matrix $X = [X_{ij}]$, $1 \le i, j \le 2$.
It is also easy to verify (2) by direct calculation, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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a relation in logic Suppose $\prec$ is a relation defined in the set of well defined formulas such that
$\phi \prec \psi$ iff $\models \phi \rightarrow \psi$ and $ \nvDash \psi \rightarrow \phi$
I would like to prove the following:
*
*If $\phi \prec \psi$ then there exists $\chi$ such that $\phi\prec\chi\prec\psi$.... | The best way to explain this is going to be with the aid of an example. Consider two basic formulae $\phi$ and $\psi$ consisting of propositional variables $p$ and $q\,$. That is let $\phi :=p\,$, and $\psi:= p\lor q\,$. Now, since $\vDash p\rightarrow (p \lor q )\,$, and $\not\vDash (p \lor q )\rightarrow p\,$, it fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If each term in a sum converges, does the infinite sum converge too? Let $S(x) = \sum_{n=1}^\infty s_n(x)$ where the real valued terms satisfy
$s_n(x) \to s_n$ as $x \to \infty$ for each $n$.
Suppose that $S=\sum_{n=1}^\infty s_n< \infty$.
Does it follow that $S(x) \to S$ as $x \to \infty$?
I really do not recall any ... | No.
Let $s_n(x)=\frac 1x$, $s_n=0$. Then $S=\sum_{n=1}^\infty s_n$ trivially converges. However, $S(x)=\sum_{n=1}^\infty s_n(x)$ is clearly divergent, since the summand is a constant (with respect to $n$) greater than $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
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Determinants of triangular matrices Does a lower triangular matrix have a determinant that is equal to the product of the elements in the diagonal similar to an upper triangular matrix.
| The matrix looks like this: $$\begin{bmatrix}
a_{1,1}&0&\cdots&\cdots&0
\\b_{1,2}&a_{2,2}&\ddots&&\vdots
\\ \vdots&\ddots&\ddots&\ddots&\vdots
\\ \vdots&&\ddots&\ddots&0
\\ b_{1,n}&\cdots&\cdots&b_{n-1,n}&a_{n,n}
\end{bmatrix}$$
The determinant can be written as the sum of the product of the elements in the top row wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
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Parameterizing cliffs I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to 1, it approximates a sharp cliff, as in the figure below. The function needs not be defined for $... | The linear equation is given by
$$\frac{x - x_1}{\lvert x_2 - x_1\rvert} + \frac{y - y_2}{\lvert y_1 - y_2\rvert} = 1.$$
We can get successively 'sharper' curves by using
$$\left(\frac{x - x_1}{\lvert x_2 - x_1\rvert}\right)^{k} + \left(\frac{y - y_2}{\lvert y_1 - y_2\rvert}\right)^k = 1$$
for integer $k \geq 1$, using... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1243927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Collection of all partial functions is a set I'm studying real analysis from prof. Tao's book "Analysis 1" and I'm stuck on the following exercise:
"Let $ X $ , $ Y $ be sets. Define a partial function from $ X $ to $ Y $ to be any function $ f: X' -> Y' $ whose domain $ X' $ is a subset of $ X $, and whose range $ Y' ... | Completely revised.
I’m assuming that you’ve already shown (or assumed) that for any sets $S$ and $T$ the set $T^S$ is well-defined.
Note that if $f:S\to T\subseteq Y$, then $f:S\to Y$ as well. Thus, we really need only
$$\bigcup\left\{Y^S:S\in 2^X\right\}\;.$$
Let $P(x,y)$ hold iff $x\in 2^X$ and $y=Y^x$, or $x\noti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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I need help finishing this proof using the Intermediate Value Theorem? Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$.
Here's what I have so far:
Let $h$ be a continuous function where $h = f -g$
Since $h(b)=f(b)... | There exists some $y \in (a,b)$ such that $h(y) = 0$. So $f(y) = g(y)$. The other case $a=b$ is trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns? When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return ... | If you use the logarithmic values, then you will get the log of the geometric mean by calculating $$log(\mu_r)=\frac{1}{T} \cdot \sum_{t=1}^T r_t$$
The formula you have posted is only valid for returns, which have not been logarithmized.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show this is not a manifold with boundary Consider a curve $\alpha: \mathbb R \to \mathbb R^2$ defined by $t \mapsto (e^t \cos(t), e^t \sin(t))$. Show the closure of $\alpha(\mathbb R )$ is not a manifold with boundary.
Denote $\alpha(\mathbb R )$ by M. The closure of M is the spiral plus the limit point, the origin. ... | Suppose $M$ is a manifold with boundary. Then there is an open set $U$ in $\mathbb{R}^2$ and a diffeomorphism $F:U\to\mathbb{R}^2$ such that $F(M\cap U)$ is either
*
*the line $\{(x,0):x\in\mathbb{R}\}$, or
*the half-line $\{(x,0):x\ge 0\}$
Consider the $y$ component of $F$, call it $g$. It vanishes on $M$. Co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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What is my error in this matrix / least squares derivation? I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is.
let,
$y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny and full rank, and $x_{ls} = (A^T A)^{-1}A^Ty$ is the standard least squares ... | The issue is this: your problem is overdetermined and there is no solution vector $x$ such that
$$
\mathbf{A}x - b = 0.
$$
Instead of asking for an exact solution, we relax requirements and ask for the best solution. Picking the $2-$norm, the best solution is
$$
x_{LS} = \left\{ x\in\mathbb{C}^{n} \colon \lVert \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Proving or disproving $c-d|p(c)-p(d)$ where $p$ is a polynomial. I have $p(X)=\sum_{i=0}^{n}{a_iX^i}$, where $a_i\in\Bbb{Z}$. Let $c,d\in\Bbb{Z}$.
Prove or disprove: $c-d|p(c)-p(d)$.
I did some algebra but I can't think of a way to divide high power parts by $c-d$. I can't on the other hand find a counter example, an... | It is true because $c-d \mid c^k-d^k, k \in \mathbb {N}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Oriented Hypersurface admits unique unit normal vector field This question is the converse of this question and is taken from Lee's Smooth Manifolds, problem 15.7. Namely:
Suppose $M$ is an oriented Riemannian manifold and $S\subset M$ is an oriented smooth hypersurface. Show that there is a unique smooth unit normal ... | I know this is an old post, but here is a detailed solution, which I wrote partly as an exercise for myself to go through the details.
First we define a rough unit vector field $N$ that determines the orientation on $S$. Let $p\in S$, and note that we can write $T_p M=T_pS\oplus (T_pS)^\perp$. Since $S$ is a hypersurfa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Where do I best learn informal set theory? I want to learn math for machine learning, and I want to start with informal set theory.
I was reading 'naive set theory' (1960) by halmos, and it didn't seem to contain modern set notations.
If anyone knows a good material for learning informal set theory, please leave a comm... | You can see the book "Book of Proof" of Richard Hammack; it have many diagrams and pics. The chapter about cardinals is very educational.
P.S.: Machine leaning is more about Linear Algebra and Probability Theory.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to prove that it is possible to make rhombuses with any number of interior points? I was given some square dot paper which can be found on this link: http://lrt.ednet.ns.ca/PD/BLM/pdf_files/dot_paper/sq_dot_1cm.pdf and was told to draw a few rhombuses with the vertices on the dots but no other dots on the edges.
H... | The square dot paper you describe can be thought of as something called the integer lattice, that is, the set of points $(m,n)$ in the plane where $m,n$ are integers. In order to make this identification, pick an arbitrary point on the square dot paper and call it $(0,0)$. If $m,n$ are positive integers, then $(m,n)$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 5,
"answer_id": 1
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My dilemma about $0^0$ We know that $0^0$ is indeterminate.
But if do this:
$$(1+x)^n=(0+(1+x))^n=C(n,0)\cdot ((0)^0)((1+x)^n) + \cdots$$
we get
$$(1+x)^n=(0^0)\cdot(1+x)^n$$
So, $0^0$ must be equal to $1$.
What is wrong in this?
Or am I mistaking $0^0$ as indeterminate?
Other threads are somewhat similar but not exact... | The binomial theorem states that:
$$(a+b)^n=\sum_k\binom nka^kb^{n-k}$$
(assuming I made no typo). What you noticed is basically that, when $a=0$, this only works if $0^0=1$.
More specifically: When $k=0$, you're supposed to evaluate:
$$a^0b^n$$
when $a=0$. Now, while $0^0$ is an indeterminate form, it makes sense to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Factorising ideals in the ring of integers of a quadratic field In an undergraduate algebraic number theory course, I was given the question "If $K = \mathbb Q(\sqrt{-33})$ Factorise the ideal $(1+\sqrt{-33})\subset \mathcal O_K$ into a product of prime ideals." I know that the norm is multiplicative and that the norm... | For your second question: in a Dedekind domain $(I+J)(I \cap J) = IJ$, so
$$
(2, 1 + \sqrt{-33})(17, 1 + \sqrt{-33}) = (2, 1 + \sqrt{-33}) \cap (17, 1 + \sqrt{-33}) \ni 1 + \sqrt{-33}
$$
because $(2, 1 + \sqrt{-33})$ and $(17, 1 + \sqrt{-33})$ are different maximal ideals.
As for the other one, I have been taught to ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1244946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Evaluate the integral $\int \sin(x)\cos(3x^2)dx$ I am looking for a solution for the following integral problem.
$$\int \sin(x)\cos(3x^2)dx$$
Passed over these integral things long time ago. I cannot see how to go for a solution.
|
$$\begin{align}I=\dfrac12~\sqrt{\dfrac\pi6}~\bigg\{\cos\dfrac1{12}\bigg[S\bigg(\dfrac{6x+1}{\sqrt{6\pi}}\bigg)-S\bigg(\dfrac{6x-1}{\sqrt{6\pi}}\bigg)\bigg]+\\\\+\sin\dfrac1{12}\bigg[C\bigg(\dfrac{6x-1}{\sqrt{6\pi}}\bigg)-C\bigg(\dfrac{6x+1}{\sqrt{6\pi}}\bigg)\bigg]\bigg\}\end{align}$$
where S and C are the two Fresne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Injection of the mapping cone of $z^2$ We define the mapping cone of $f:S^1\to S^1=:Y$, $f (z)=z^2$ as the quotient space of $S^1\times [0,1]\sqcup Y$ where $(z,0)$ and $(z',0)$ are identified and where $(z,1)$ and $f(z)$ are identified for all $z,z'\in S^1$. We call this space $C_f$.
Probably my question is very easy:... | The question is not that trivial. Though it's pretty obvious that the composition
$$Y\hookrightarrow S^1\times[0,1]\sqcup Y\xrightarrow q C_f$$ is injective, you still need to show that it's a homeomorphism onto its image, i.e. an embedding. To this end, let $C\subseteq Y$ be closed. Then $\bar C=f^{-1}(C)\times\{0\}\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given a system of differential equations, how can one tell if $\textbf{x}_c = (0,0)^T$ is a unique critical point? I have:
$$\frac{d\textbf{x}}{dt}=\begin{bmatrix}
-1 & 2 \\
-2 & -1 \\
\end{bmatrix}\textbf{x}(t)$$ with $\textbf{x}(0)=(1,-1)^T$.
I am asked whether the critical point $\textbf{x}_c=(... | The critical point(s) occur where
$\dfrac{d\mathbf x}{dt} = 0; \tag{1}$
since, as a vector field,
$\dfrac{d\textbf{x}}{dt}=\begin{bmatrix}
-1 & 2 \\
-2 & -1 \\
\end{bmatrix}\textbf{x}, \tag{2}$
the critical points occur wherever
$\begin{bmatrix}
-1 & 2 \\
-2 & -1 \\
\end{bmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. Setting
For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every fini... | Hint: each formula uses only finitely many symbols.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$. How many words with letters from the word ABRACADABRA if they must end in a consonant and $d$ must be after $r$.
What I did:
I have
$A:5$
$B:2$
$R:2$
$C:1$
$D:1$
If the words must end in a consonant and d mu... | Since D must appear after both R's, the last letter can be a B, C, or D.
Case 1: The last letter is D.
We have ten places to fill with five A's, two B's, two R's, and one C. We can fill five of the ten places with A's in $\binom{10}{5}$ ways. We can fill two of the remaining five places with B's in $\binom{5}{2}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does $\mathrm{Rank}(A{A^*} - {A^*}A) \ne 1$? Given $A \in M_n$, why does $\mathrm{Rank}(A{A^*} - {A^*}A) \neq 1$?
| Hint.
Observe that
$$
\mathrm{Trace}(AA^*)=\mathrm{Trace}(A^*A),
$$
and hence
$$
\mathrm{Trace}(AA^*-A^*A)=0.
$$
Next show that, if $B$ is diagonalizable and $\mathrm{Rank}(B)=1$, then $\mathrm{Trace}(B)\ne 0$.
Finally, observe that $AA^*-A^*A$ is hermitian and hence diagonalizable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Higher Order ODE with Differential Operators I am trying to solve an ODE problem involving higher order.
Let $p(s) = s(s^2-s+1)(s-1)$ and $D = d/dt$. Solve the initial value problem $$p(D)x = t + e^t,$$ $x'''(2) = 1$, $x''(2) = 1$, and $x'(2) = 1$, and $x(2) = 0$.
Attempt: I believe I need to solve the homogenous eq... | You want
$$
p(D)f=D(D^{2}-D+1)(D-1)f = t+e^{t}
$$
The operator $D^{2}$ annihilates $t$ and $(D-1)$ annihilates $e^{t}$. Therefore,
$$
D^{3}(D^{2}-D+1)(D-1)^{2}f = 0.
$$
Because $D^{2}-D+1=(D-1/2+i\sqrt{3}/2)(D-1/2-i\sqrt{3}/2)$,
That gives a solution
$$
f = A + Bt + Ct^{2}+Ee^{t/2}\cos(\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Cauchy-Riemann equations in polar form.
Show that in polar coordinates, the Cauchy-Riemann equations take the form
$\dfrac{\partial u}{\partial r} = \dfrac{1}r \dfrac{\partial v}{\partial \theta}$ and $\dfrac{1}r \dfrac{\partial u}{\partial \theta} = −\dfrac{\partial v}{\partial r}$.
Use these equations to show that... | Proof of Polar C.R Let $f=u+iv$ be analytic, then the usual Cauchy-Riemann equations are satisfied
\begin{equation}
\frac{\partial u}{\partial x} =\frac{\partial v}{\partial y} \ \ \ \ \ \text{and} \ \ \ \ \frac{\partial u}{\partial y} =-\frac{\partial v}{\partial x} \ \ \ \ \ \ \ (C.R.E)
\end{equation}
Since $z=x+iy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 2,
"answer_id": 1
} |
Path of diffusion process with discontinuous drift Let $(B_t)$ be a standard Brownian motion on some probability space and let $X_t$ be the process defined by the SDE $dX_t = \mu_t dt + dB_t$, where $\mu_t$ is adapted, deterministic, and only takes the values $0$ or $1$. (For example, $\mu_t = 0$ on $(2n, 2n+1]$ for a... | Applying Girsanov's theorem is overkill. Note that, by definition,
$$X_t = X_0 + \int_0^t \mu_s \, ds+ B_t, \qquad t \geq 0.$$
We know that $t \mapsto B_t$ is continuous (almost surely); moreover, it is well-known that mappings of the form
$$t \mapsto I(t) := \int_0^t \mu_s \, ds$$
are continuous whenever the integral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1245942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$a,b,c,p$ are rational number and $p$ is not a perfect cube Given that $a,b,c,p$ are rational number and $p$ is not a perfect cube, if $a+bp^{1\over 3}+cp^{2\over 3}=0$ then we have to show $a=b=c=0$
I concluded that $a^3+b^3p+c^3p^2=3abcp$ but how can I go ahead? could you please help? Thanks
| Clearly $p^{1/3}$ is one of the roots of the Quadratic equation: $cx^2+bx+a = 0$
Now, sum of roots = $-b/c$. Since RHS is rational and one of the terms in LHS is irrational, it is logical to assume that the other root is: $-p^{1/3}$ (so that the LHS becomes rational). The sum of roots then becomes $0$. Hence, $b = 0$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
2010 unit circles $C$ is a unit circle with radius $r$. $C_1,C_2,\ldots, C_{2010}$ are unit circles along the circumference of $C$ touching $C$ externally. Also the pairs $C_1C_2;C_2C_3,\ldots;C_{2010}C_1$ touch. Then find $r$. Options:
1) $cosec(\pi/2010)$
2) $sec(\pi/2010)$
3) $cosec(\pi/2010)-1$
4) $sec(\pi/2010)-1$... |
Made it before I noticed its already done, but anyway here it is...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Prove an interesting function $f$ is semicontinuos.
Let $r_n$ be the sequence of all rational numbers and $$f(x)=\sum_{n,r_n<x}\frac{1}{2^n}.$$ Show that $f$ is lower semicontinuous in $R$.
Since $f$ is continuous on irrationals.
It is also an increasing function. So we only need to look at limit from the left side ... | Let $$ f_K (x) :=\sum_{n\leq K,\ r_n< x } \frac{1}{2^n}$$ Then
$f_K\rightarrow f $ uniformly since $\sum_{n>M} \frac{1}{2^n } <
\varepsilon$ for some $M$. Note that each $f_K$ is lower. So the
limit is lower.
(1) Lower Continuity of $f_K$ : Define $g_n$ :
$$
g_n(x) =\left\{
\begin{array}{ll}
\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute $f(2)$ if $\int_0^{x^2(1+x)} f(t) dt = x$ for all $x \geq 0$ This is exercise 22(d) of section 5.5 (p.209) of Apostol's Calculus, vol.I.
Compute $f(2)$ is $f$ is continuous and satisfies the given formula for all $x \geq 0$. $$\int_0^{x^2(1+x)} f(t) dt = x$$
I tried to differentiate both sides so that I have ... | here is another way to do this. we will use the fact that $$(1+0.1)^2(1+1+0.1)=(1+2\times 0.1 + \cdots)(2 + 0.1) = 2+5\times0.1+\cdots $$ putting $x = 1, 1+0.1$ in $\int_0^{x^2(1+x)} f(t) dt = x, $ we get $$\int_0^2f(t)\, dt = 1, \int_0^{2+5\times0.1+\cdots}f(t)\,dt=1+0.1$$ subtracting one from the other, $$\int_2^{2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Has the 3x3 magic square of all squares entries been solved? It is my understanding that it has not yet been determined if it is possible to construct a $3$x$3$ magic square where all the entries are squares of integers. Is this correct? Has any published work been done on this problem?
| The existence or not of a non-trivial integer 3x3 magic square of squares is STILL a unsolved problem.
The quoted reference to Kevin Brown's web pages only discusses an extremely special configuration of numbers, which does not exist. The page does NOT claim to prove non-existence for all possible magic squares.
If you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Vectors in a plane I have a question where i need to prove that three vectors are all on one plane.
I proved that the angle between ALL the three vectors is 120.
Is that enough to prove that it is on one plane? because together they are 360.
Basiclly, is there any other way to build three vectors with 120 angle betwee... | Assume WLOG that all the vectors are normalized. Then
$$u\cdot v=v\cdot w=u\cdot w=-\frac12$$
Now, consider the equation
$$au+bv+cu\times v=w$$
and dot-product with $u$:
$$a-\frac b2=-\frac12$$
now with $v$:
$$-\frac a2+b=-\frac12$$
and we get $a=b=-1$. Then
$$-u-v+cu\times v=w$$
and dot-product with $w$:
$$\frac12+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Uniqueness of prime ideals of $\mathbb F_p[x]/(x^2)$ What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ corresponds to a prime ideal $P$ of $\mathbb F_p[x]$ containing $(x^2)$. An... | Prime ideals do correspond under the correspondence theorem, so your argument suffices.
To see this,
Let $I\subset P\subset R$ be any prime in $R$ containing an ideal $I$, then
$R/P \cong \frac{R/I}{P/I}$ by the 3rd isomorphism theorem. Since $P$ is prime, $R/P$ is an integral domain, hence so is $\frac{R/I}{P/I}$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Map between free sheaf modules not arising from a "matrix" My question is about this example in the Stacks project.
For convenience and completeness, here is the relevant part.
Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$
of the real line all glued together at $0$; a fundamental
system of neighbourhoo... | I think I have found your misunderstanding:
The sum is not finite on $U_n$. Observe that $e_j$ maps to an infinite collection of functions $(g_{i,j})$ on $X$, where $g_{i,j}$ takes the value $f_j$ on $L_i$ and is zero on all the other lines. Since $U_n$ is the union of the copies of $(-1/n,1/n)$ in every line $L_i$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sum identity involving sin How one can prove that $$\sum_{k=1}^n(-1)^k\sin(2k\theta)=\cos(n\pi/2+\theta+n\theta)\sec\theta\sin(n\pi/2+n\theta)?$$
It looks difficult as there is sum on the other side and product of trigonometric functions on the other side. The book Which Way did the Bicycle Go gave a hint of induction ... | $$\begin{align*}
\sum_{k=1}^{n} (-1)^{k} \sin(2k\theta) &= {} \mathrm{Im}\Bigg[ \sum_{k=1}^{n} \big[ -\exp(2i\theta) \big]^{k} \Big) \Bigg] \\[2mm]
&= \mathrm{Im}\Bigg[ \big( -\exp(2i\theta) \big)\frac{1 - \Big( - \exp(2i\theta) \Big)^{n}}{1 + \exp(2i\theta)} \Bigg] \quad \mathrm{if} \; 1 + \exp(2i\theta) \neq 0 \\[2mm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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