Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Can a Unique Factorisation Domain be non-commutative? The definition that our lecturer gave us for Unique Factorisation Domains is:
An integral domain $R$ is called a Unique Factorisation Domain (UFD) if every non-zero non-unit element of $R$ can be written as a product of irreducible elements and this product is uniq... | In practice it's always safe to assume "integral domains" like UFD's are commutative, unless explicitly mentioned otherwise. (Commutative is the right word, not abelian. Abelian has other uses in ring theory.)
Of course, one can always go about trying to find an acceptable adaptation to the noncommutative case. One ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3592987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Prove $2^{29}$ has exactly 9 distinct digits There is a problem that say: The number $2^{29}$ has exactly 9 distinct digits. Which digit is missing?
It is an Olympiad problem and I see it solved by using remainder modulo 9.
$2^{29}=536870912$ interesting!
My Question: Can we find any mathematically way to show $2^{29}... | That $2^{29}$ has exactly nine distinct digits is a coincidence, and a fairly appropriate one. One (a very good one) mathematical proof that $81619^2$ contains only two distinct digits, is calculating it : $81619^2 = 6661661161$. Even though sometimes these do yield patterns that are interesting, my personal take is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3593173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For $\left|\frac{\alpha^n-\beta^n}{\alpha-\beta}\right| \leq \frac{|\alpha|^n-|\beta|^n}{|\alpha|-|\beta|}$, what will happen when $|\alpha|=|\beta|$? For the complex inequality
$$
\left|\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}\right| \leq \frac{|\alpha|^{n}-|\beta|^{n}}{|\alpha|-|\beta|},\quad|\alpha|\neq|\beta|,\qua... | Indeed the inequality can be rewritten as (and follows from the more obvious)
$$ |\alpha^{n-1}+\alpha^{n-2}\beta+\cdots+\alpha\beta^{n-2}+\beta^{n-1}|\le |\alpha|^{n-1}+|\alpha|^{n-2}|\beta|+\cdots+|\alpha||\beta|^{n-2}+|\beta|^{n-1} $$
This is simply an application of the triangle inequality to multiple summands, comb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3593488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Approximate $L^2$ function by convolving with mollifiers Let $\eta_\delta$ be a mollifier (i.e. positive real-valued function on $\mathbb{R}^2$, supported on the ball of radius $\delta$ centered at the origin, whose integral is 1), and $f$ is a compactly-supported $L^2$-function. How can we prove that
$$ || f - f*\eta_... | Minkowski’s inequality should be used to get $$\|f-f*\eta_\delta\|_2 = \left\| \int \eta_\delta (y)(f- f(\bullet -y)) dy \right\|_2 \le \int \| \eta_\delta (y)(f- f(\bullet -y)) \|_2 dy $$
Then you use the continuity of translations in $L^p$: only one factor of $\eta_\delta$ appears.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3593821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Any errors? My proof that if $\left(A - B\right) \cup \left(B - A\right) = A \cup B$, then $A \cap B = \emptyset$ If $A\cup B=(A-B)\cup(B-A)$, then $A\cap B=\emptyset$.
Proof by Contrapositive. If $A\cap B\ne\emptyset$, then $A\cup B\ne(A-B)\cup(B-A)$. Suppose that there exists a member of $A\cap B$, $x$. Then, $x\noti... | $(A - B)\cup(B - A) = (A \cap B^c) \cup (B \cap A^c) =$
$(A \cup B) \cap (A^c \cup B^c) = (A \cup B) - (A \cap B)$
$= A \cup B$
provides a direct proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3594013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Matrix representation of idempotent operator $V$ is a finite dimensional vector space over $F$, and $: → $ be an idempotent operator, i.e. $P^2=P$. It can be proved that $( − )$ is also an idempotent operator and $ker(P^m) = im((I - P)^n)$ for all $, ≥ 1$.
The question is to show that under some choice of basis $B$ ... | Hints: Proceed as follows:
*
*Show that for all $v \in im(P)$, we have $P(v) = v$.
*Prove that $ker(P) \cap im(P) = \{0\}$ and conclude that $V = im(P) \oplus ker(P)$.
*Take bases of $ker(P)$ and $im(P)$ö. This gives you a basis of $V$ (why?). Then calculate the matrix of $P$ with respect to this basis.
| {
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"timestamp": "2023-03-29T00:00:00",
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How can the chain rule be explained more rigorously? The 'proof' for the chain rule that is often used at school is unsatisfying for me because it treats derivatives as fractions:
$$
\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}
$$
However, the more rigorous proofs that are used in University are unfathomable to me be... | The chain rule helps us differentiate the composition of functions. It is often the case that the variables $y$ and $x$ can be linked by an intermediate variable $u$. If $y=f(u)$, where $u=g(x)$, then $y=f(g(x))$, meaning that $y$ and $x$ are linked through the composite function $f \circ g$. Here is an informal argume... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find the maximum of $\boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}$ subject to $\boldsymbol{q}^T \boldsymbol{x}=1$? I want to solve the following problem in $\boldsymbol{x} \in \mathbb R^{n}$
$$\begin{array}{ll} \text{maximize} & \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}\\ \text{subject to} & \boldsymbol{q... | Since $\mathbf q > \mathbf 0$, the feasible region $\{\mathbf x \in \mathbb R^n : \mathbf q^{\mathsf T} \mathbf x = 1, \mathbf x \ge \mathbf 0\}$ is bounded (we have $x_i \in [0, \frac1{q_i}]$ for each $i$). It's also closed, so the maximum of $f(\mathbf x) = \mathbf x^{\mathsf T} \!A \mathbf x$ must be achieved somewh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3594469",
"timestamp": "2023-03-29T00:00:00",
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$(a_k)$ is a sequence with $|a_k|\le M$ for all k . Show that with $|x|\lt 1$ the series $f(x)=\sum_{k=1}^{\infty}a_{k}x^{k}$ converges.
Question
Let $(a_n)_{n\in\mathbb N}$ be a sequence of real numbers with $|a_k|\le M$ for all $k \in\mathbb N$. Show that for each $x\in\mathbb R$ with $|x|\lt 1$ the series $f(x)=\su... | You may want to try the root test instead for then
$\sqrt[n]{|a_n x^n|}\leq \sqrt[n]{M}|x|\xrightarrow{n\rightarrow\infty}|x|$
and so, convergence occurs for all |x|<1.
Check the wikipedia links to root and ration test. Also, you nee to know that for any number $a>0$, $\sqrt[n]{a}\rightarrow1$ as $n\rightarrow\infty$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving Composition of Limits Prove that if $\lim_{x \to c}g(x)=b$ and $\lim_{x \to b}f(x)=L$ and there exists a sequence $a_n$ converging to the limit $c$ such that $g(a_n)=b$ then prove that $$\lim_{x \to c}f(g(x))$$ does not exists given that $f(b) \neq L$
$$$$If we consider the difference $|f(g(x))-L|$ and if we ch... | Your proof is mostly good, but not correct. But then, neither is the statement you are trying to prove. Here is a counter-example:
*
*$c = 0, g(x) = 0$ for all $x$, and $f(x) = \begin{cases}1, &x \ne 0\\0,& x = 0\end{cases}$. And $a_n = \frac 1n$ for all $n$. Then
$$\lim_{x\to 0}g(x) = 0\\\lim_{x \to 0} f(x) = 1\\f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3594766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that a function is continuous on a normed space So I have learned topology and we haven't touched normed space there. However, from my linear algebra class I know that every normed space can naturally induce a metric, so I wonder that, to show $f:X\rightarrow X$ is continuous on the normed space $X$, can we treat ... | You you can do that, knowing that the distance is defined by $d(x, y) = \Vert x - y \Vert$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3594886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Explaining derivative being used to derive tensor properties I'm doing some self-study on tensor calculus for physics, and I've come across a derivative that I can't quite wrap my head around. It looks like this:
$\frac{d}{dt}(\frac{\partial x'^i}{\partial x^j})=\frac{\partial^2 x'^i}{\partial x^j\partial x^k}\frac{dx^... | Your interpretation is correct. You could also see this as an operator equality:
$$\frac{d}{dt} = \frac{\partial x^k}{\partial t}\frac{\partial}{\partial x^k}$$
(which is just another statement of the chain rule) and then use this to replace the $\frac{d}{dt}$ in your equation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why do we add probabilities? I know that there is the addition rule of probability, but I want to understand the intuition behind it. Specifically, why does OR signifies addition in probability theory?
| Think about it like this
Lets say there is a $6$ sided dice.
Now, when asked to find $P(3 \mathbf {or} 6)$, consider what this statement actually means. The probability that either $3$ occurs or $6$ occurs. This is equivalent to adding the chances of $3$ and $6$. Therefore $\mathbf{or}$ represents addition.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3595616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Question about finite analog of $\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8}$ The integral
$$
\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8},
$$
is given as equation $(17)$ in M.L. Glasser, Some integrals of the Dedeki... | $$I_n\left(a\right)=\int_{0}^{1}{\frac{\sin{\left(n\sin^{-1}\left(t\right)\right)}\sinh{\left(n\sinh^{-1}{\left(\frac{t}{a}\right)}\right)}}{\cos{\left(2n\sin^{-1}\left(t\right)\right)}+\cosh{\left(2n\sinh^{-1}{\left(\frac{t}{a}\right)}\right)}}\frac{dt}{t\sqrt{1-t^2}\sqrt{1+\left(\frac{t}{a}\right)^2}}\ } $$
$$t\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3595770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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Integrability of $\int_{\mathbb{R}} 1/(1+|x|)^p d\mathcal{L}(x)$ Let $p>0$. For which $p$ is the integral $\int_{\mathbb{R}} \frac{1}{(1+|x|)^p} d\mathcal{L}(x)$ finite?
I assume $f(x)= \frac{1}{(1+|x|)^p}$ is integrable for $p>1$, but how exactly can I show this?
| You can do the following: split the integral al $2$ then
\begin{align*}\int_{\mathbb{R}}\frac{1}{(1+|x|)^p}\,d\mathcal{L}&=2\left( \int_{0}^2\frac{1}{(1+x)^p}\,d\mathcal{L}+ \int_{2}^\infty\frac{1}{(1+x)^p}\,d\mathcal{L}\right)\\
&=2\left( \int_{0}^2\frac{1}{(1+x)^p}\,d\mathcal{L}+ \int_{\color{red}3}^\infty\frac{1}{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3595891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Non-Strict Saddle Point vs Local Minima While going through Escaping Saddle Points Efficiently, I came across the definition of Strict Saddle Point. They define a stationary point to be a strict saddle if at least one of the eigenvalue of the Hessian Matrix is negative. This implies a non-strict saddle point will have ... | For a function $f\colon\mathbb{R}^n\to\mathbb{R}$, the condition $\nabla^2f(x^*)\succeq 0$ is necessary for $x^*$ to be an unconstrained local minimum of $f$, but it is in general not sufficient. Consider $f\colon\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^3$. Then $\nabla f(x) = 3x^2$ and $\nabla^2 f(x) = 6x$. Therefo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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3 square in a circle (geometric question) There is three square in a circle like the picture attached.
How can find radius of circle ?
I know that's not a hard problem,but I was away from geometry for years. If possible give me hint or idea to start to solving. Thanks in advance.
the smallest side is $6$
and med... | The circumradius of a triangle is
$$
R=\frac{abc}{4A}
$$
where $a,b,c$ are the sides of the triangle and $A$ is its area.
The sides are $18$, $\sqrt{36^2+6^2}=6\sqrt{37}$, and $\sqrt{36^2+12^2}=12\sqrt{10}$ and the area is $\frac12\,18\cdot36=324$. Therefore, the circumradius is
$$
R=\frac{18\cdot6\sqrt{37}\cdot12\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\sum_{k=0}^n \sum_{j=0}^k { {n} \choose {j}} { {n-j} \choose {k-j}} \left( \frac{x}{1-x}\right)^k = \left( \frac{1+x}{1-x} \right)^n$ How to prove the following formula?
$$\sum_{k=0}^n \sum_{j=0}^k { {n} \choose {j}} { {n-j} \choose {k-j}} \left( \frac{x}{1-x}\right)^k = \left( \frac{1+x}{1-x} \right)^n$$
I have tried... | Let $ n $ be a positive integer.
\begin{aligned} \sum_{k=0}^{n}{\sum_{j=0}^{k}{\binom{n}{j}\binom{n-j}{k-j}\left(\frac{x}{1-x}\right)^{k}}}&=\sum_{j=0}^{n}{\sum_{k=j}^{n}{\binom{n}{j}\binom{n-j}{k-j}\left(\frac{x}{1-x}\right)^{k}}}\\ &=\sum_{j=0}^{n}{\binom{n}{j}\left(\frac{x}{1-x}\right)^{j}\sum_{k=0}^{n-j}{\binom{n-j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Use $\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$ and $\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$ to bound $\vert s_n - s\vert$ Let $s, s_n\in\mathbb{R}$ and $\hat{s}_n$ be a random variable.
I have two concentration inequalities:
$$\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$$ for all $n\geq1$ an... | Assuming that $\hat{s}_n$ is integrable,
\begin{align}
|s_n-s|&\le \mathsf{E}|s_n-\hat{s}_n|+\mathsf{E}|s-\hat{s}_n| \\
&=\int_0^{\infty}\mathsf{P}(|s_n-\hat{s}_n|>x)\,dx+\int_0^{\infty}\mathsf{P}(|s-\hat{s}_n|>x)\,dx \\
&\le \int_0^{\infty}[a(n,x)+b(n,x)]\,dx.
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Compute $\int_{0}^{\infty} \frac{\arctan{x}}{1+x} \frac{dx}{\sqrt[4]{x}}$
Evaluate the following integral$$\int_{0}^{\infty} \frac{\tan^{-1}{x}}{1+x} \frac{dx}{\sqrt[4]{x}}$$
I was not able to find an antiderivative of this function, so I believe we must use properties of definite integrals to solve this integral. If... | \begin{aligned}\int_{0}^{+\infty}{\frac{\arctan{x}}{\sqrt[4]{x}\left(1+x\right)}\,\mathrm{d}x}&=\int_{0}^{+\infty}{\int_{0}^{1}{\frac{x}{\sqrt[4]{x}\left(1+x\right)\left(1+x^{2}y^{2}\right)}\,\mathrm{d}y}\,\mathrm{d}x}\\ &=\int_{0}^{1}{\int_{0}^{+\infty}{\frac{x}{\sqrt[4]{x}\left(1+x\right)\left(1+x^{2}y^{2}\right)}\,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For which values $x,\alpha$ does $\sum_{n=1}^{\infty}\frac{n^{nx}}{(n!)^ \alpha}$ converge? I want to know for which values of $x$ this series converges:
$$\sum_{n=1}^\infty \frac{n^{nx}}{(n!)^ \alpha};$$ here $\alpha \in\mathbb R$ is a constant
This series is defined $ \forall x \in \mathbb R$.
$$a_n=\frac{n^{nx}}{(n!... | For these kind of sums,
I use $n! \sim (n/e)^n$ so
$\sum_{n=1}^\infty \frac{n^{nx}}{(n!)^ a}
\sim \sum_{n=1}^\infty \frac{n^{nx}}{(n/e)^{an}}\\
=\sum_{n=1}^\infty (\frac{e^an^{x}}{n^a})^n\\
=\sum_{n=1}^\infty (e^an^{x-a})^n\\
$
By the n-th root test, we want
$|e^an^{x-a}| < 1$
and this requires
$x < a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Differential Equation Disease Modelling
The rate at which people are infected per week after the outbreak of a disease is $${dP\over dt} = 0.8P \left(1 - {P\over 350}\right)$$ At the outbreak for the disease, $7$ people were infected. What is the initial condition problem for $P$?
I got everything to the left side wi... | I think that your last formula is not correct.
You correctly wrote
$$\frac{5 }{4}\log (P)-\frac{5}{4} \log (350-P)=t+c$$ which rewrite
$$\frac{5 }{4}\log \left(\frac{P}{350-P}\right)=t+c_1\implies \log \left(\frac{P}{350-P}\right)=\frac{4 }{5}t+c_2$$ Now, exponentiate both sides
$$\frac{P}{350-P}= \exp\left(\frac{4 }{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3596871",
"timestamp": "2023-03-29T00:00:00",
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Prove that the Pell's Equation $x^2 −Dy^2 = 1$ always has a solution where $y$ is a multiple of $41$ $D$ is a positive integer that is not a perfect square
Recently I am taking a introductory number theory course and I met this question right after we learned Pell's equation and Diophantine Approximation. However, I ca... | There is an old parametric solution for Pell equation that says, if x, y and D are certain functions of a parameter such as $m$ , there can be infinite solutions"
We rewrite equation as:
$x^2-1=Dy^2$
$1$ is odd and number of terms on LHS is even so one of terms must be odd. Suppose $x^2$ is odd and we have:
$x=2m^2+1$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3597027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Negation of there exists quantifier? The problem states to find the negation of the following statement:
There exists a number which is equal to its square.
Original Answer: There does not exist a number which is equal to its square.
My answer: There exists a number which is not equal to its square
Are both the answe... | This will be helpful to understand negation:
Statement with universal quantifier: $\forall x, p(x)\implies q(x)$.
It's negation is $\exists x, p(x)\land \lnot q(x)$
Statement with existential quantifier: for some $x, p(x)\land q(x)$.
It's negation is $\forall x, \lnot p(x)\lor \lnot q(x)$ which is further equivalent to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3597209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Finitely generated subsemigroups of $\mathbb{N}^k$ It is well-know that every subsemigroup of $(\mathbb{N},+)$ is finitely generated. I am wondering if there is (any) similar characterization of subsemigroups of $\mathbb{N}^k$ for $k>1$? I am looking also for some examples of not finitely generated subsemigroups of $\m... | There are indeed some non finitely generated subsemigroups of $\mathbb{N}^k$. Consider for instance the subsemigroup $S$ of $\mathbb{N}^2$ defined by
$$
S = \{(m,n) \mid m > 0 \text{ and } n > 0\}
$$
Then every set of generators of $S$ necessarily contains all the elements of the form $(1, n)$ with $n \geqslant 2$. Ind... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $(X,\leq)$ is a set with a total order, how can I show that there is $Y\supset X$ s.t. $Y$ has the supremum property? In my course, they defined $\mathbb R$ as the smallest set that contain $\mathbb Q$ and that has the supremum property, i.e. that all upper-bounded set has a supremum.
1) My problem, it's that I don... | You are right, one has to construct this set. Call a pair $(L, R)$ of proper subsets of $\mathbb{Q}$ a Dedekind cut if :
*
*$L \cup R = \mathbb{Q}$
*$\forall x \in L, \ \forall y \in R, \ x < y$
One can then define $\mathbb{R}$ as the union of $\mathbb{Q}$ with the set $D$ of all Dedekind cuts. One can then define ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3597519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $S^2$ cannot have a smooth vector field with two zeros that are either both sources or both sinks Backround
I have just learned the Poincaré-Hopf Index Theorem which says that if $\overrightarrow{v}$ is a smooth vector field on a compact,
oriented manifold $X$ with only finitely many zeros, then the global sum of ... | Consider the following vector field $\vec v$ on $S^2$. Using standard spherical coordinates $\theta$, $\phi$, and let $\hat u_\theta$ and $\hat u_\phi$ be the unit vectors in the $\theta$ and $\phi$ directions, $\vec v$ is given by:
$$ \vec v =
\begin{cases}
v_0\sin(2\theta)\hat u_\theta +v_0\sin(\theta)\hat u_\phi &\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Construction of lines through a given point, cutting two given circles in congruent chords
Given a point $P$ and two circles $\Gamma$ and $\Gamma'$ with centers $O$ and $O'$. I'm searching for a method to construct lines passing through $P$ and defining on each circle chords of same length.
A geometer professor made ... | Suppose first that the two circles intersect each other. Let $F$ be the midpoint of the line of centres $O_1O_2$, and let $F'$ be the reflection of $F$ in the radical axis $\mathcal{R}$ of the two circles. Let $\mathcal{P}$ be the parabola with focus $F$ and directrix the line through $F'$ parallel to the radical axis ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Question about damped vibrations in ODEs I am currently studying ODEs and I came across this problem. For the first 2 parts, I just want to know if I am understanding this correctly. My main question is for part (c).
Given the spring-mass system represented by the equation $y'' + 4y' + ky = 0$,
a) for what value of k ... | The problem becomes a bit easier to manage if you rewrite the solution as $y=e^{-2t}\left(Ae^{it\sqrt{k-4}}+Be^{-it\sqrt{k-4}}\right)$.
Then the pair of equations become: $$Ae^{2i\sqrt{k-4}}+Be^{-2i\sqrt{k-4}}=0$$$$Ae^{3i\sqrt{k-4}}+Be^{-3i\sqrt{k-4}}=0$$
From these, we get: $$e^{4i\sqrt{k-4}}=\frac{-B}{A}=e^{6i\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving linearity of the differential operator $a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$. I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic fa... | Yes this proof is correct. One thing to note is that the vector spaces in question are very much not finite dimensional. Luckily, that doesn't impact your proof in any meaningful way.
Part (3) can also be shortened a lot by just observing that sums and multiples of linear maps are always linear. That removes a lot of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Combinatorics counting problem
There are $3n$ male students and $3n$ female students.
How many ways can they be divided into two groups of three, in such way that in each group has at least one male student and one female student.
I'd like to know if the following is correct:
First I organize them in lines of $2n$,... | I think that the answer should be
$$\frac{((3n)!)^2\binom{2n}{n}}{(2n)!2^{2n}}=\left(\frac{(3n)!}{2^{n}n!}\right)^{2}.$$
Explanation: we have two lines of $3n$ persons, one with males and another for females. Then we form $2n$ groups of $3$ persons each by taking $1$ male and $1$ female from the respective lines and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is $\lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right)$? So recently a friend asked me to compute this limit:
$$\lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right)$$
Question : Does the limit exist... | Brute force, but from the Stirling formula
$$
n! = \left( {\frac{n}{e}} \right)^n \sqrt {2\pi n} \left( {1 + \mathcal{O}\!\left( {\frac{1}{n}} \right)} \right),
$$
one has
$$
\sqrt[n]{{n!}} = \frac{n}{e}\exp \left( {\frac{1}{2n}\log (2\pi n)} \right)\left( {1 + \mathcal{O}\!\left( {\frac{1}{{n^2 }}} \right)} \right) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3598972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$ So the task is to find a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$
I was wondering if there is a more intelligible and less exhausting strategy in finding the coefficient, other than saying that $(x^2+x^7+x^9)^{20}=((x^2+x^7)+x^9)... | Rewrite as $x^{40}((1+x^5)+x^7)^{20}$, then the k-th term of the binomial expansion (ignoring the $x^{40}$) is $\binom{20}kx^{7(20-k)}\sum_{i=0}^k \binom kix^{5i}$.
We now actually want the exponent to be 17. Check $i=0, 1, 2, 3$ to see the only possibility is $i=2$, which gives $k=19$.
Hence the coefficient is $\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3599094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Constructing a Graph from Squared Adjacency Matrix I have a homework problem that has got me stumped:
Draw a graph whose adjacency matrix A is such that:
\begin{bmatrix}3&1&0&1&2\\1&2&1&1&1\\0&1&2&2&0\\1&1&2&3&0\\2&1&0&0&2\end{bmatrix}
is equal to A2
This seems almost impossible to do without either square rootin... | Not a full answer, but potentially helpful.
I used the following MiniZinc model to get a suitable matrix:
int: n = 5;
set of int: N = 1..n;
array[N,N] of int: A2 =
array2d(N, N, [3, 1, 0, 1, 2,
1, 2, 1, 1, 1,
0, 1, 2, 2, 0,
1, 1, 2, 3, 0,
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition: cofinal functor between triangulated categories In Def 1.1 this paper it says about a functor between triangulated categories $i:C \rightarrow D$ is cofinal.
This does not seem to be the usual one for categories. Am I correct in understanding that it means
$i$ is cofinal iff For all $d \in D$ exists some ... | As the definition says, $i$ must be fully faithful and every object of $D$ must be a summand of an object in the image of $i$. You are right that this is essentially unrelated to the generally category theory notion of cofinality.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative of $\sqrt{x}$ using geometry I am having trouble with a problem given in a video by 3Blue1Brown, to which I have already found a response, here.
My is issue is understanding why the equation $$\mathrm dx = 2(\sqrt{x})\left(\mathrm d\sqrt{x}\right)+\left(\mathrm d\sqrt{x}\right)\left(\mathrm d\sqrt{x}\right) ... | Consider the initial example in the video - how to find the derivative of $x^2$. The area of the sqaure he has constructed is $x^2$, with the sides as x. Hence, your f is $x^2$. The change in area you find is df or d($x^2$).
3B1B wants you take a slightly different approach here and create a square with sides as $\sqr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Two different roots for $P(x) = x^4+ax^2+bx+c$
Let $a, b, c \in \mathbb{R}$ and $a > 0$. Also let $P: \mathbb{R} \to \mathbb{R}$, $P(x) = x^4+ax^2+bx+c$. Show that the function has at most two different roots.
My assumption was to use Bolzano's theorem, but I couldn't figure out how to use it here. Also I'm curious i... | We have
$$P'(x)=4x^3+2ax+b$$
and $$P''(x)=12x^2+2a>0$$
this means $P$ is a convex function over $\mathbb{R}$, and has at most two roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3600016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Residue of $\dfrac{z^{1/4}}{z+1}$ at $ z = -1$. This question is from Churchill and Brown's Complex Variables and Applications 8th edition, page 248:
Find $Res ({f},-1) $ for $f = \dfrac{z^{1/4}}{z+1}$ given $|z| > 0, 0 < \arg z< 2\pi$
Attempt:
Since the denominator has a simple zero, the residue is $(-1)^{1/4}$. ... | The domain of $f$ is four copies of the complex plane. They are arranged like a four-level carpark. Every time you go around the origin you end up on the next copy. $\theta$ increases by $2\pi$ so $z^{1/4}$ is multiplied by $i$.
So there are four functions $f$, each one is consistent and continuous as long as you st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3600127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Area of largest triangle under $y=e^{-x}$ I came across this question and I'm unsure how to solve it. The question wasn't originally in English, and English isn't my first language so please excuse any terminology/grammar mistakes.
A tangent is drawn through the point P on the curve $y=e^{-x}$. The tangent together w... | Let $P(\alpha, e^{-\alpha})$ be the point then the tangent
$y-e^{-\alpha}=-e^{-\alpha}(x-\alpha)$
meets Y-axis at $A(0,e^{-\alpha}(\alpha +1))$. The horizontal line through $P$ meets Y-axis at $B(0,e^{-\alpha})$.
The area of triangle $PAB$ is
$S(\alpha)=\frac{1}{2} (\alpha)^2 e^{-\alpha}$
Use derivatives to see that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3600249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the least number of balls that must be taken from the bag.. I'm building onto my previous question, but I will include it also in here. I'm again struggling with the probability questions we've been given.
Six balls numbered 1,2,2,3,3,3 are placed in a bag. Balls are taken one at a time from the bag at random and ... | a) Let $X_i$ denote the label of the $i^{th}$ drawn ball and let $N=\inf\{i>0:X_i=2\}$. Then $N\sim Geo(2/6)$, since there are $6$ balls and $2$ of them are numbered $2$.
$$ \mathbb P(N=n) = \bigg(1-\frac{1}{3} \bigg)^{n-1}\frac13 \quad n\geq 1. $$
Let $A_n$ denote the event that among $n$ draws there was at least 1 b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $\sqrt{1+x}<1+\frac{x}{2}$ for all $x>0$ I am a little stuck on this question and would appreciate some help. The question asks me to prove that $\sqrt{1+x}<1+\frac{x}{2}$ for all $x>0$.
I squared both sides of the question to get $1+x<\frac{x^2}{4}+x+1$ for all $x>0$. Then, I multiplied both sides by $4$ to... | You did well. Let's finish it:
we have $x^2>0$ thus $\dfrac{x^2}{4}>0$. Adding $x+1$ to both sides, we have
$$\dfrac{x^2}{4}+x+1> x+1$$
or $$(\frac{x}{2}+1)^2>x+1$$
or
$$\frac{x}{2}+1>\sqrt{x+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3600695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Suppose you have functions $f$ and $g$ such that $f: Y \rightarrow T$ and $g: X \rightarrow Y$. I having a tough time understanding this? I've learned about onto and 1 to 1 functions from a basic perspective, but I'm having a hart time understanding where to even start with this.
(a) Suppose (f o g) is onto. Claim ... | If (f o g) is onto, that means that for any x value that enters into g, there is a resulting value of g AND that each of these g values has a corresponding f output. This means that f's outputs are the ones that must have be fulfilled by all of the given inputs. This means that between (a) and (b), the claim that f is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3601030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does it mean for an equation to contain another equation within it? I'm reading a paper, specifically "Forecasting correlated time series with exponential
smoothing models" by Corberan-Vallet et al, and I came across this little beauty: A lower triangular $n\times n$ Matrix $L$ whose entries below the main diagona... | (Typing an answer because I can't comment yet.) That equality should be a congruence. My best bet is that this is a typo,
$$ L_{ij}=\alpha+\alpha\beta(i-j)+\gamma \ \ (i \equiv j\text{ mod } s) $$
It shows that $i$ is always congruent to $j \mod s$. $ j < i$ would follow from $ s < i$ if that was ever implied before.... | {
"language": "en",
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What does the 5-dimensional representation of $\mathfrak{so}(3)$ explicitly look like? I found three $5 \times 5$ matrices that fulfil the defining Lie algebra relation of $\mathfrak{so}(3)$:
However, these matrices are not antisymmetric, which implies that when we put them into the exponential map, the corresponding ... | There is nothing wrong with the representation you found. The eigenvalues of your matrices are, indeed, ±2i,±i,0, the ones required for the "spin-2" quintet of physics.
You may check directly that the quadratic Casimir is an invariant,
$$
T_x^2+ T_y^2+T_z^2= -6 ~ 1\!\! 1,
$$
as expected,
so that it must be equival... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Compute $\lim \limits_{n\to \infty} \int_3^4 (-x^2+6x-8)^\frac{n}{2} dx$ Compute $$\lim \limits_{n\to \infty} \int_3^4 (-x^2+6x-8)^\frac{n}{2}dx.$$
I am interested in a method to compute this as simply as possible. I know that by DCT this is $0$, but I am not allowed to use it. With the substitution $t=x-3$ I got that ... | May be too complex.
What you did is good. You end with
$$I_n=\int\limits_0^1 (1-t^2)^\frac{n}{2}\,dt$$ Now, make $t=\sin(u)$ to work with
$$I_n=\int_0^\frac \pi 2 \cos^{n+1}(u)\,du=\frac{\sqrt{\pi }}2 \frac{ \Gamma \left(\frac{n+2}{2}\right)}{ \Gamma
\left(\frac{n+3}{2}\right)}$$ Now, take logarithms, use Stirling a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3601427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability of being picked more than once in a simple random sample. A school has $500$ girls and $500$ boys. A simple random sample is obtained by selecting names from a box (with replacement) to a get a sample of $10$.
Find the probability of someone being picked more than once.
My working is:
\begin{align}
P(\tex... | This is a variation on the Birthday Problem, with names instead of birthdays and drawings from the hat instead of people in a room.
The answer is $1$ minus the probability that all ten names are different,
which is the product of the probabilities that the $n$th name is different from all previous names, given that all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3601552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Canonical form of a Quadratic form. I've been given the following quadratic form to find the canonical form of:
$$
Q(\bf{z})= z_1z_2 + 2z_2z_3 − 3z_3z_4
$$
through the method of forming perfect squares.
The method I've been taugh/show is to look at terms consisting of a specific variable, say $z_1$ and form the perfect... | Gantmacher's version of Lagrange's method leads to
$$ \left( \frac{1}{2} x_1 + \frac{1}{2} x_2 + x_3 \right)^2 - \left( -\frac{1}{2} x_1 + \frac{1}{2} x_2 - x_3 \right)^2 + \left( \frac{1}{2} x_2 + \frac{1}{2} x_3 -\frac{3}{2} x_4 \right)^2 - \left( -\frac{1}{2} x_2 + \frac{1}{2} x_3 +\frac{3}{2} x_4 \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3601734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Verifying $P(\lim_{n \to \infty}\inf A_n) \leq \lim_{n \to \infty}\inf P(A_n) \leq \lim_{n \to \infty}\sup P(A_n) \leq P(\lim_{n \to \infty}\sup A_n)$ From Probability Through Problems By Marek Capinski,Tomasz Jerzy Zastawnaik
Verify that
$P(\lim_{n \to \infty}\inf A_n) \leq \lim_{n \to \infty}\inf P(A_n) \leq \lim_{... | I think there may be some mistake in the book.We can write proof as
$B_n=\cap_{k=n}^{\infty}A_k$
for all $k \geq n$,$B_n \subset A_k$
so we can say $P(B_n) \leq P(A_k)$ for all $k \geq n$
So,we can say $P(B_n) \leq \inf_{k \geq n} P(A_k)$....(1)
So we proceed as
\begin{eqnarray*}P(\lim_{n \to \infty}\inf A_n) &=& P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3601937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Understanding Why a Limit (w/ Factorials) Approaches 0 I am given the following limit:
$\lim_{x \to \infty} \frac{a^{x+1}}{(x+1)!} $
I am aware that this limit approaches $0$ after plugging large enough numbers on the calculator, but I am looking for a more mathematical approach. Do I need to use L'Hôpital's rule for ... | Assuming $x\in\mathbb{N}$ in your question and w.l.o.g. $a\geq0$ (else just look at the absolute values).
Let $f:\mathbb{N}\to\mathbb{R},\; n\mapsto a^{n+1}/(n+1)!$. There exists an $n_0$ s.t. for all $n\geq n_0$
$$\frac{f(n+1)}{f(n)}=\frac{a^{n+2}}{a^{n+1}}\frac{(n+1)!}{(n+2)!}=\frac{a}{n+2}<1.$$
Hence, $0\leq f(n+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3602175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the velocity of an orthogonal matrix? Consider an orthogonal $n\times n$ matrix $\boldsymbol{A}$ so that$\boldsymbol{A}^T \boldsymbol{A} = \boldsymbol{A} \boldsymbol{A}^T = \boldsymbol{I}$. Suppose this matrix can vary with time, hence denoting it as $\boldsymbol{A}(t)$. I would like to obtain the velocity matr... | Having reflected more on my comments, I think I can formulate this as an answer: No.
Let $\mathcal{A}$ be the set of smooth curves through the manifold of $O(n)$ of of orthogonal $n\times n$ matrices (i.e. time dependent orthogonal matrices, as in your question) such that for all $A \in \mathcal{A},$ $A(0) = I$. It can... | {
"language": "en",
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A question on the product of positive definite functions Show that if $f$ and $g$ are continuous positive definite functions in $\mathbf{R}^1$, then $f(x)g(y)$ is positive definite on $\mathbf{R}^2$.
I just wanted to check if the approach I'm using is correct,
I defined the matrix $A_{j,k}=f(x_j-x_k)$ and $B_{j,k}=g(... | Bochner theorem states that $f$ is positive definite iff $f = \widehat{\mu}$, with $\mu$ a positive Radon measure. If $f_1$, $f_1$ are positive definite then:
$$
f_1 \cdot f_2 = \widehat{\mu_1} \cdot \widehat{\mu_2} = \big(\mu_1 \ast \mu_2 \big)^\wedge,
$$
and the convolution of two positive measures is positive.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Where do the constants in this formula come from? I do not exactly know what they are. I am reading about inverse operators and the book is going over this one.
After proving some stuff about this operator it finally says if $$P(D)y=(a_nD^n+...+a_1D+a_0)y=bx^k$$
then
$$y_p=\frac{1}{P(D)}(bx^k)$$
$$=\frac{1}{a_0(1+\frac... | There are two key points here.
*
*First, the computation is expanding the formal function (in $D$)
$$ \frac{1}{1 + c_1 D + c_2 D^2 + \cdots + c_n D^n} $$
using its Maclaurin series, which would look like
$$ 1 + b_1 D + b_2 D^2 + b_3 D^3 + \ldots $$
*When looking at the action on $x^k$, the Maclaurin series can be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3602714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why to simplify $\sin x - 1 =\cos x$ you have to multiply both side by $(\frac{\sqrt 2}{2}) $? Why do you have to multiply both side of $$\sin x - 1 = \cos x$$
by:
$$(\frac{\sqrt 2}{2}) $$ to simplify it?
I am doing Shaum's pre-calculus and the final answer to this question is: π but I have no clue where the idea of mu... | It is because $\dfrac{\sqrt{2}}{2}=\sin\dfrac{\pi}{4}=\cos\dfrac{\pi}{4}$. Then we get
$$\dfrac{\sqrt{2}}{2}\sin x-\dfrac{\sqrt{2}}{2}\cos x=\cos\dfrac{\pi}{4}\sin x-\sin\dfrac{\pi}{4}\cos x=\sin(x-\dfrac{\pi}{4}).$$
Generally, for expressions like $$a\cos x+b\sin x=c$$
we do $$\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find an analytical function if initial value and the imaginary part is given. So I must find $f(z) = u(x,y) + v(x,y), \ \ f(0)=1$ if $v(x,y) = 2x^3 - 6xy^2-4xy+2x$
$$\frac{\partial v}{\partial x} = 6x^2-6y^2-4y +2 \\ \frac{\partial v}{ \partial y} = -12xy -4x \\ f'(z) = -12xy-4x + i(6x^2-6y^2-4y+2) \\ \int-4z+i(6z^2+2)... | You got the right answer: $f(z)$ is $-2z^2+2iz^3+2iz+1$ indeed. And\begin{multline}2(x+yi)^2+2i(x+yi)^3+2i(x+yi)+1=\\=-6 x^2 y-2 x^2+2 y^3+2 y^2-2 y+1+i(2x^3-6xy^2-4xy+2x).\end{multline}So, again, yes, you got the right answer.
| {
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"source": "stackexchange",
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Projectivization of the tangent bundle to $S^2$ is nontrivial I want to prove that the projectivization of the tangent bundle to $S^2$ is nontrivial. It looks very similar to the hairy ball theorem.
Also, I would like to understand if the projectivization of the tangent bundle to $X$ is nontrivial for arbitrary closed ... | Lemma. Let $M$ be a smooth compact connected manifold. Then $PT(M)$ admits a section if and only if $M$ has a nonzero vector field.
Proof. Suppose that $PT(M)$ admits a section $s$. This section yields a real line bundle $L\to M$ over $M$. This bundle may or may not be orientable. If it is orientable then we obtain a... | {
"language": "en",
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"source": "stackexchange",
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Equivalence of exterior algebra definitions For a finite-dimensional vector space $V$, let $A$ be the subspace of $V\otimes V$ generated by all elements of the form $v\otimes v$ for some $v\in V$ and let $B$ be the subspace generated by all elements of the form $v\otimes w+w\otimes v$. I then learned that I can define
... | You're right that the problem arises when you try to show $A\subseteq B$.
To show $B\subseteq A$ we note that for arbitrary $v$ and $w$, the tensors $v\otimes v$,$w\otimes w$ and $(v+w)\otimes (v+w)$ are all in $A$. Since $A$ is a subspace, it also contains
$$(v+w)\otimes (v+w) -v\otimes v - w\otimes w=v\otimes w+w\oti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inequality of Expectation of Indicator function Is the following succession correct? Would it not make more sense if the last inequality was reversed?
$|E[X I_{X > t}]| = E[X] - E[XI_{X < t}] \geq E[X] - t$?
Here $I_{X>t}$ is the indicator function ($1$ if $X>t$ else $0$) and $E[X] > 0$ is given.
thank you in advanc... | I think the answer is no.
let $P(X=1)=P(X=2)=P(X=3)=\frac{1}{3}$
$$E(X1_{X>2})=3*P(X=3)=1$$
$$E(X1_{X<2})=1*P(X=1)=\frac{1}{3}$$
$$E(X)=2$$
Maybe following inequalities help you: (I think they are hold)
$$E(X) \leq
E(X 1_{\{ X> t \}} ) + t P(X\leq t) \hspace{.5cm} (1)$$
$$E(X) \geq
t P(X\geq t)+ E(X 1_{\{ X< t \}} ) \... | {
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"source": "stackexchange",
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The Functional-Calculus Version of the Spectral Theorem In the book Analysis Now by Pedersen, the Spectral Theorem is that, for a normal operator $T$ acting on a Hilbert space $H$, there is an isometric star-isomorphism between $C(\text{sp}(T))$ and the $C^*$-algebra that is generated by $I$ and $T$. This star-isomorph... | The first spectral theorem was von Neumann's representation of a self-adjoint operator $A$
$$
Ax = \int_{-\infty}^{\infty} \lambda dE(\lambda)x
$$
where $E(\lambda)$ is a non-decreasing orthogonal projection-valued function of $t$ on $\mathbb{R}$ and $x\in\mathcal{D}(A)$.
Earlier specialized versions were ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can this congruence rule be generalized? Let n be a positive integer which representation in base 10 is $a_ka_{k-1}a_{k-2}...a_2a_1a_0$.
It's not particularly hard to prove that:
$n\equiv a_0\pmod2$
$n\equiv2a_1+a_0 \pmod 4$
$n\equiv 4a_2+2a_1+a_0\pmod8$
However this doesnt seem to hold for $\pmod{16}$, what I mean is:... | The actual rule for $n = a_k \ldots a_0$ mod $2^d$ is
$$n \equiv a_0 + 10^1 a_1 + 10^2 a_2 + \ldots + 10^{d-1} a_{d-1} \mod 2^d$$
So for $16$ that would be
$$n \equiv a_0 + 10 a_1 + 4 a_2 + 8 a_3 \mod 16$$
for $32$ it would be
$$n \equiv a_0 + 10 a_1 + 4 a_2 + 8 a_3 + 16 a_4 \mod 32$$
and for $64$ it would be
$$ n \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion about normal bundles This is an elementary question about some basic confusion I have.
Consider the Segre embedding $S: \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{3}$. Then the image of a line $\mathbb{P}^{1} \times \{p_{0}\}$ is a line $ L \subset \mathbb{P}^{3}$. Denote the quadric $S(\mat... | The exact sequence of normal bundles is:
$$0\to \mathcal{O}\xrightarrow{i} \mathcal{O}(1)\oplus \mathcal{O}(1)\to \mathcal{O}(2)\to 0.$$
To understand the inclusion $i$, let's twist it by $\mathcal{O}(-1)$:
$$0\to \mathcal{O}(-1)\xrightarrow{i} \mathcal{O}\oplus \mathcal{O}\xrightarrow{\pi} \mathcal{O}(1)\to 0.$$
Doe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the points on the curve $x^4+y^4+3xy=2$ closest and farthest to the origin I need to find the points on the curve $x^4+y^4+3xy=2$ that are closest and farthest to the origin.
I believe this might be a Lagrange multiplier problem, but I am not sure. I was thinking that maybe minimizing/maximizing the function woul... | You've already got adequate answers, but this is just another way to do it -- especially fitted for this particular curve, as we'll find out.
The problem, reformulated, is to find the smallest circle intersecting the curve so that no point on the curve lies within the circle; on the other hand we need the largest circl... | {
"language": "en",
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Analytic branch of log z Let $g(z)$ denote an analytic branch of $log z$ in a domain $D \subset\mathbb{C}$. Show that $g'(z) = 1/z$.
Show also that if $h(z)$ is another analytic branch of $log z$ in $D$, then the function
$$\frac{g(z)-h(z)}{\pi i}$$
is constant in $D$ and equal to an even integer.
Here is how I approac... | Note that\begin{align}\exp\bigl(g(z)-h(z)\bigr)&=\frac{\exp\bigl(g(z)\bigr)}{\exp\bigl(h(z)\bigr)}\\&=\frac zz\\&=1.\end{align}So, for each $z\in D$, $g(z)-h(z)=2\pi in$, for som integer $n$. Since $D$ is connected, the range of $g-h$ must be connected too, and therefore there is some $n\in\mathbb Z$ such that, for eac... | {
"language": "en",
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Solve $3\cosh(x)+2\sinh(2x) =0$ When trying to solve $$3\cosh(x)+2\sinh(2x)=0$$
I have subbed in the definitions of the cosh and sinh functions:
$${\cosh x=\frac{e^{x}+e^{-x}}{2}}$$
$${\sinh x=\frac{e^{x}-e^{-x}}{2}}$$
Which has given me:
$$\frac{3e^{x}+3e^{-x}}{2} +e^{2x}-e^{-2x}=0$$
I can recognise that $(e^x)^2=e^{2... | Hint to continue $\textbf{your work}$: substitute $e^x=t$ to get $$3(t+\frac{1}{t})+2(t^2+\frac{1}{t^2})=0.$$ Then substitute $u=t+\frac{1}{t}$. Can you continue this?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about Rene Schilling's proof of the submartingale maximal inequality I am reading the proof on the submartingale maximal inequality from Rene Schilling's Measures, Integrals and Martingales. The proof below uses an equivalence theorem on submartingales. Namely, look at the second $\le$ in the last line. I don'... | One could consider the stopping time $\tau = N \mathcal{X}_A + (N+1) \mathcal{X}_{A^c}$, where $\mathcal{X}_A$ is the characteristic function of $A = \{ \sigma \leq N \}$. Clearly, we have that $\sigma \leq \tau$ a.s. and we have $$
\int_A u_\sigma d\mu \leq \int_A u_\tau d\mu
$$ but on $A$, $\tau \equiv N$, giving us ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show $\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+ b^2\cos^2 x)^{2}}=\frac {\pi^2 (a^2+b^2)}{4a^3b^3}$ Show that
$$\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+ b^2\cos^2 x)^{2}}=\frac {\pi^2 (a^2+b^2)}{4a^3b^3}$$
My Attempt:
Let $$I=\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+b^2 \cos^2 x)^2} $$
Using $\int_{a}^{b} f(x) dx=\int... | Continue with the substitution $t=\tan x$,
$$\begin{align}
& \pi \int_{0}^{\frac {\pi}{2}} \frac {dx}{(a^2\sin^2 x+b^2 \cos^2 x)^2} \\
& = \pi \int_0^\infty \frac{1+t^2}{(b^2+a^2t^2)^2}dt \\
&=\frac{\pi(a^2-b^2)}{2a^2b^2} \frac t {b^2+a^2t^2}\bigg|_ 0^\infty
+ \frac{\pi(a^2+b^2)}{2a^2b^2} \int_0^\infty \frac {dt}{b^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is $\lim_{n\to \infty}f(x_n)=f(\lim_{n\to \infty}x_n)$ always true? I was studying about sequences of numbers and their limits. My book states the standard rules for algebra of limits involving sums, differences, products and quotients of convergent sequences.
But the author, while solving an example problem implicitly... | Not true in general. For a simple example, let $f(x)=\begin{cases} x,\,x\ne1\\2,\,x=1\end{cases}$.
Now consider a sequence converging to $1$, like $x_n=1+1/n$.
There's a limit point definition of continuous functions that you may want to take a look at. In particular, it's true when $f$ is continuous.
| {
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"source": "stackexchange",
"question_score": "2",
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Evaluating Series Integral How do I show that $$\int_0^\infty\frac{(\ln x)^2dx}{1+x^2}$$=$$4(1-1/3^3+1/5^3-1/7^3...)$$
I expanded the integral to $$\int_0^\infty(\ln x)^2(1-x^2+x^4...)dx$$ using the power series for $$\frac{1}{1+x^2}$$ but I'm not sure how to continue from here.
| Let $f(x)=\frac{\ln^n(x)}{1+x^2}$
$$I_n=\int_0^\infty f(x)\ dx=\int_0^1f(x)\ dx+\underbrace{\int_1^\infty f(x)\ dx}_{1/x\to x}=\int_0^1f(x)\ dx+\int_0^1(-1)^nf(x)\ dx$$
Clearly, for odd $n$, $I_n=0$, so for even $n$ , we have
$$I_n=2\int_0^1\frac{\ln^n(x)}{1+x^2}\ dx=2\sum_{k=0}^\infty(-1)^k\int_0^1 x^{2k}\ln^n(x)\ dx... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\sum_{i=1}^{n}v_i^t \cdot v_i = I$ for orthonormal base Let
$$
\{v_1,..,v_n\}
$$
Be an orthonormal base in $R^n$ with the standard inner product.
I need to prove that:
$$
\sum_{i=1}^{n}v_i^t \cdot v_i = I
$$
Where $v_i$ is a row vector.
What i tried:
I tried to look at an example but still - i dont feel its... | Hint: Rewrite $\sum_{i=1}^n v_i^Tv_i$ as a matrix product.
| {
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Find a polynomial of degree 5 that is irreducible over $\mathbb{Z}_3$. Can someone please let me know if this looks ok? Thanks in advance!
An irreducible degree $5$ polynomial over $\mathbb{Z}_3$is one such that $f(0)\neq0,f(1)\neq0,f(2)\neq0$.
Take e.g. $p(x)=x^5+x^4+x^3+x^2+1$
$p(0)=1$
$p(1)=2$
$p(2)=1$
And, $$x^2,... | Your problem is to find an irreducible polynomial of degree $5$ over $\mathbb Z_3$. You haven't found one yet, because
$$x^5+x^4+x^3+x^2+1=(x^2+x+2)(x^3+2x+2).$$
Note that the irreducible (monic) polynomials of degree $2$ are $x^2+1$ and $x^2+x+2$ and $x^2+2x+2$. You need to check all three of them as possible factors.... | {
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Let $G$ and $H$ be finite groups and let $p$ be a positive prime number that divides the order of $G$ and that divides the order of $H.$ Let $S$ be a Sylow $p$-subgroup of $G$ and let $T$ be a Sylow $p$-subgroup of $H$. Prove that
$S×T$ is a Sylow $p$-subgroup of $G× H$.
Suppose $S \times T$ is a Sylow $p$-subgroup. Th... | Oops. It looks like you started off assuming what you wanted to prove, if i'm not mistaken.
Anyway, as I said, it suffices to prove that $S\times T$ has the right order.
We know $|S|=p^k$, where $|G|=p^kl$, and $|T|=p^r$, where $|H|=p^rs$ and $(p,l)=(p,s)=1$.
Now $|S\times T|=|S||T|=p^{k+r}$. But $|G\times H|=|G||H|=p... | {
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how many combination of coins add up to \$20? we have five coins:
*
*Coin 1: \$1.
*Coin 2: \$2.
*Coin 3: \$5.
*Coin 4: \$10.
*Coin 5: \$20.
In how many way can we get \$20 using those coins and combinations of them?
The only way I could do that was by counting all possibilities and it took forever.I started coun... | There are $41$ combinations in all. The following solution is essentially a twist on the usual approach using generating functions.
Start by noticing that if we want to make a total of 20 dollars, we can use any combination of the 2, 5, 10, and 20 dollar coins and make up the rest with 1 dollar coins. So we can sol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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There is a number, the second digit of which is smaller than its first digit by 4, and if the number There is a number, the second digit of which is smaller than its first digit by 4, and if the number was divided by the digit's sum, the remainder would be 7.
Actually I know the answer is 623
I found it by using comput... | You have $6$ possibilities for the two first digits: $a_1a_2=40,51,62,73,84,95$ and you can verify that the number cannot have two digits. Then you try with $3$ digits keeping in mind that the number minus $7$ is a multiple of $a_1+a_2+a_3$ so you have
$$\frac{a_1a_2a_3-7}{a_1+a_2+a_3}=entero$$
(1) $\dfrac{400+a_3-7}{... | {
"language": "en",
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How does a homotopy $f\simeq g$ induces a homotopy equivalence between their mapping toruses $M_f$ and $M_g$? By a mapping torus $M_f$ of a continuous map $f:X\to X$ we mean the space $$X\times I/\{(x,0)\sim (f(x),1)\}.$$
Now given $f,g:X\to X$ and $F:X\times I\to X$ with $F(x,0)=f(x)$ and $F(x,1)=g(x)$, namely $F$ a ... |
Lemma: Let $A\subseteq X$ be a cofibration and $f\simeq g:A\rightarrow Y$ homotopic maps. Then the adjunction spaces $X\cup_fY$ and $X\cup_gY$ are homotopy equivalent.
This is Proposition 0.18 on pg. 16 of Hatcher's Algebraic Topology. The statement is for a CW pair $(X,A)$, but you'll notice that the proof only requ... | {
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The existence of the derivative of a Banach space valued function In Evans' Partial Differential Equations $\S 7.1$, there is a motivation for definition of weak solution.
\begin{align*}
(1) \begin{cases} u_t + Lu = f \ &\text{in} \ U_T\\
u = 0 \ &\text{on} \ \partial U \times [0,T]\\
u = g \ &\text{on} \ U \times \... | In the context of the book, $\mathbf{u}'$ isn't the classical derivative as defined in your post.
As defined in Section 5.9.2 (Spaces involving time), $\mathbf{u}'$ stands for the "weak" derivative of $\mathbf{u}$, that is, a function $\mathbf{v}$ such that
$$\int_0^T \varphi'(t)\mathbf{u}(t)\;dt=-\int_0^T \varphi(t)\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3607009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mobius transformations problem
The Mobius transformation $T_1(z)=\dfrac{z+b}{z+d}$ maps the line $\operatorname{Im}(z)=\operatorname{Re}(z)+3$ onto the unit circle in such a way that the region above the line is mapped to the interior of the circle and that $5i$ is mapped to the origin. Find the value of $d$.
The Mo... | Hint: Calculate $a,b,c,d$ where $T_3(z)=(az+b)/(cz+d)$.
This is possible because the Mobius transformations form a group, with composition as the operation.
I get $T_3(z)=1/2((1-i)z-1-3i)$. The dilation is by $|a|=1/\sqrt2$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is this symbol equivalent to taking a partial derivative? On the Wikipedia page for the Leibniz Rule for Integration, it displays this formula:
$$
\frac{d}{dx}\int f(x,t)\,dt = \int\partial_xf(x,t)\,dt
$$
Is the symbol $\partial_x$ equivalent to $\frac{\partial}{\partial x}?$ If so, is this just a preference of convent... | $\partial_xf(\cdot)$ and $\frac\partial{\partial x}f(\cdot)$ denote completely the same thing, and nobody will be confused. The former way is just a more concise formulation.
| {
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"url": "https://math.stackexchange.com/questions/3607431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Writing power set of $\{\emptyset,x,y,z\}$ That $\emptyset$ is confusing me, should we consider $\emptyset$ as an element or should we discard it?
So here is my work:
$\mathcal P(\{\emptyset,x,y,z\})=\{\emptyset,\{ \emptyset \},\{x \} ,\{ y \},\{ z \},\{ \emptyset ,x \},\{ \emptyset, y \},\{ \emptyset,z \},\{ x,y \},\... | The power set of an $n$-element set always has $2^n$ elements. The empty set is also always an element in the power set of any set.
So yes, your work is correct, and since $\varnothing$ is an element of the original set, it gets considered like any other element.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Diagonalisability of a certain rank 1 matrix
Let $E$ and $F$ be non-zero $n$-tuples, and set $P$ = $EF^H.$
(a) Find the rank of $P$.
(b) Determine when $P$ is diagonalisable, and in that case find an eigenbasis for $P$. (Hint: consider $Px = \lambda x.$)
We have
\begin{equation}
\begin{split}
P &= \left[ \begin{arr... | If $x$ is the eigenvector for $\lambda_1$, you have$$\tag1\lambda_1x=Pe=ef^*x=(f^*x)e.$$Since $\lambda\ne0$, you get that $x=\alpha e$ for some scalar $\alpha$. If you now substitute this into $(1)$, you get $$\lambda_1=f^*e.$$
The condition for diagonalizability is that $e,f$ are colinear. Indeed, if $f=\beta e$ and ... | {
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} |
Isomorphism from $Z[i]/(p)$ to $(p^n)/(p^{n+1})$ where $p$ is a irreducible element .
Let $p\in\Bbb{Z}[i]$ be an irreducible element. Prove that for any positive integer $n\ge0$ the ideal $(p^{n+1})$ is an ideal in $(p^{n})$, and prove that multiplication by $p^n$ induces a isomorphism (defined above) between $\Bbb{Z}... | The map defined by
$\sigma:x-> p^nx$ is an isomorphic map from $Z[i]/(p)->(p^n)/(p^{n+1})$.
We first try to see whether the map is homomorphic or not
$\sigma((a+ib + c+id)) = p^{n}(a+c) + p^{n}(b+d)i = \sigma(a+ib) + \sigma(c+id)$.
1)injectivity:
$\sigma(x)=\sigma(y).p^n(x)= p^n(y) .x=y$
2) surjectivity:
In $ (p^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3607909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Smooth table/smooth pulley question problems
I'm struggling with the above question.
As it is a smooth table, there should be no friction, so I came up with the two equations (using $F=ma$):
$T=5a$ and $T=(2*9.8)+2a$.
This solved out to give me the answer of $a = 6.5333333$; however, the back of the book states the an... | First you should treat the T as unknown and for g is the gravitational acceleration 2g-T=2*a then T=2g-2a using the last equation insert in the equation for the force on the mass on the table which is 5*a=T. Why you are treating the T as unknown initially is because you know the gravitational force on the hanged ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
eigenvalues of sum of a matrix and its conjugate transpose We know that the eigenvalues of a matrix and its conjugate transpose are conjugate to each other. My question is: If $\lambda$ is an eigenvalues of $A$, what is the eigenvalue of $A+A^*$ in terms of $\lambda$, where $A^*$ denotes the complex conjugate of $A$
| Your question can't possibly work / be meaningful.
Consider $A$ as the strictly upper triangular matrix with all ones above the diagonal. Then $A + A^*$ is real symmetric and hence diagonalizable (it in fact has eigenvalues of $n-1$ and $-1$, though that's outside the scope). Yet all eigenvalues of $A$ were zero. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all roots of the polynomial equation $p(p(x)) - x = 0$ Let $p(x)$ be a quadratic polynomial such that for distinct reals $\alpha$ and $\beta$, $$p(\alpha)=\alpha\ \&\ p(\beta)=\beta$$
Show that $\alpha$ and $\beta$ are the roots of the following equation $$p(p(x))-x=0$$
Also find the remaining roots.
The first par... | Write $q(x)= p(x)-x$, then given equation is equivalent to $$q(q(x)+x)+q(x)=0$$
Since $\alpha $ and $\beta $ are roots for $q$ we have $$q(x)=c(x-\alpha)(x-\beta)$$ where $c\ne 0$, so $$ c(q(x)+x-\alpha )(q(x)+x-\beta)+c(x-\alpha)(x-\beta)=0$$
so $$(x-\alpha)(x-\beta)\Big(\color{\red}{(cx-c\alpha+1)(cx-c\beta+1)+1}\Big... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Use of the mixed product for the Wikipedia description of Barycentric Co-ordinates In reading Wikipedia entry for Barycentric co-ordinates, the authors states without proof that for a vector $h \in \mathbb{R}^3$ and a basis $\{e,f,g \}$ that
$h = \frac{1}{(e,f,g)} \cdot [(h,f,g)e + (e,h,g)f+(e,f,h)g]$ where $(e,f,g) =... | This is just an application of Cramer’s rule for solving systems of linear equations.
Recall that the coordinates of a vector are the coefficients of the unique linear combination of basis vectors that produces the vector. In other words, the coordinates of $h$ are the solution to the equation $h_1e+h_2f+h_3g = h$, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $A^2 = A$ then $A$ is diagonalizable I've stumbled upon this question in my assignment:
Prove if $A_{nxn}(\mathbb C)$ with $A^2 = A$, then $A$ is diagonalizable
My first thought is to solve for $p(A)$ where $p(x) = x^2 - x$ and you get real roots.
Would that be sufficient in showing that $A$ is diagonalizable giv... | $A^2=A$ means that $\operatorname{col}A$ is a subset of the eigenspace of $1$. Moreover, $\ker A$ is the eigenspace of $0$. Therefore, $\operatorname{col} A\cap \ker A=\{0\}$, which means that $\operatorname{col}A+\ker A=\operatorname{col}A\oplus \ker A$. By rank-nullity, $\operatorname{col}A\oplus \ker A=\Bbb R^n=V_1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
If $0So the problem is asking me to supose $0<x<\ln\ 2$ and then prove that $x+1<e^x<2x+1$. However, per the problem, I need to use the MVT. So far, I have noticed that if you exponentiate by e your hypothesis you get $e^0=1<e^x<2=e^{\ln 2}$. This happens to coincide with the derivative of your result. I thought about ... | Let $f(t)=e^t+1$ Apply LMVT to this on the interval $(0,x)$ where $0<x<\ln 2$.
Then $$\frac{(e^x+1)-2}{x-0}=f'(c)=e^c,~~0 <c <\ln 2.~~~~(1)$$
Next $$ 0<c<\ln 2 \implies 1<e^c <2~~~~(2)$$
Using (2) in (1) we get
$$1<\frac{e^x-1}{x-0}<2 \implies x+1 <e^x < 2x+1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If the line $ax+by +c = 0$ touches the circle $x^2+y^2 -2x=\frac{3}{5}$ and is normal to $x^2+y^2+2x-4y+1=0$, what is (a,b)? I have a question that goes:
If the line $ax+by +c = 0$ touches the circle $x^2+y^2 -2x=\frac{3}{5}$ and is normal to $x^2+y^2+2x-4y+1=0$, what is (a,b)?
So what I tried was I know that since th... | Let $(x_0,y_0)$ be the point of contact to $(x-1)^2+y^2=\frac85$. The tangent line equation is:
$$y=y_0+y'(x_0)(x-x_0) \Rightarrow \\
y=y_0+\frac{1-x_0}{y_0}(x-x_0) \Rightarrow \\
\frac{1-x_0}{y_0}x-y+y_0-\frac{1-x_0}{y_0}x_0=0 \Rightarrow \\
a=\frac{1-x_0}{y_0};b=-1;c=y_0-\frac{1-x_0}{y_0}x_0$$
The tangent line passes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3608996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is the cartesian product of connected algebraic varieties a connected algebraic variety? I'm wondering about the following question:
Let $X \subset k^n$ and $Y \subset k^m$ be two algebraic varieties that are connected (in the Zariski topology). Is the cartesian product $X \times Y \subset k^{n+m}$ connected in the Zar... | Suppose $X\times Y=S\cup T$, with $S,T$ clopen disjoint subsets. Then every fiber $X\times \{y\}$ and $\{x\}\times Y$ must lie entirely inside one of either $S$ or $T$: we can write each fiber as the disjoint union of the clopen sets given by intersecting with $S$ and $T$, but each fiber is connected. So, WLOG, there e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3609146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show that the initial value problem has a unique solution in the given interval? Use Picard’s theorem to show that the initial value problem $(1+e^x)\frac{dy}{dx} = \sin(x + y^3)$, $y(1) = 3$,
has a unique solution on the interval $x ≥ 1$.
By Picard's Existence and Uniqueness Theorem; If $f$ is continuous on a d... | For existence theorem to be used f(x,y) needs to be continuous in the interval
\begin{equation}
R=\left\{(x, y):\left|x-x_{0}\right| \leq a,\left|y-y_{0}\right| \leq b\right\}, \quad(a, b>0)
\end{equation}
To find the interval actually you should randomly pick the area by yourself because you are analysing the equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3609557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why is $\zeta(s)=\lim_{x\to\infty}\left(\sum_{n\leqslant x} \frac{1}{n^s}-\frac{x^{1-s}}{1-s}\right)$ for $0I am currently reading the book Introduction to Algebraic Number Theory by Apostol. To introduce some important asymptotic formulas, Apostol gives a rough definition of the Riemann zeta function (for $s\in\mathbb... | For an elementary approach, you want to show that the limit is indeed analytic in $s$ (uniform limit of analytic functions) in an open subset of $D=\{s:\Re(s)>0\land s\ne1\}$. For $\Re(s)>1$ this is fairly trivial since $x^{1-s}\to0$. For $0<\Re(s)\le1$, a full asymptotic expansion makes this more obvious, but it suffi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3609704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Derivatives of $G(x)=\int^{e^x}_1(\log(t))^2dt$ and $H(x)=\int^{x^2}_{-x^2}e^{-t^5}dt$ I have $2$ tasks:
To evaluate $G(x)=\int^{e^x}_1(\log(t))^2dt$ for $x\gt 0$ and $H(x)=\int^{x^2}_{-x^2}e^{-t^5}dt$ for $x \in \Bbb R$
So by the fundamental theorem of calculus:
If $F(x)=\int^x_af$ is differentiable at $c$, then $F'(c... | You are almost correct. You also need to apply the chain rule. If $F(x) = \int_a^x f(t) dt$, then indeed $F'(x) = f(x)$, but suppose $\hat F(x) = \int_a^{x^2} f(t) dt$. Then $\hat F(x) = F(x^2)$ so the chain rule gives $$
{\hat F} {'(x)} = \left(\frac{d}{dx} x^2\right)F'(x^2) = 2x f(x^2) \ne f(x^2)
$$
Do you see how t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3609857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
On the Reduction Lemma (the existence of an $(n-1,M,d-1)$-code) We want to prove the following Lemma:
Lemma. Let $A$ be an alphabet of size $|A|:=q\in\Bbb Z_{\geq 2},n\in \Bbb Z^+$ be a positive integer and $d \geq 2$ be a positive integer. If a $q$-ary $(n,M,d)$-code exists, then a $q$-ary $(n-1,M,d-1)$-code also ex... | Your idea of removing one letter in the codewords is a good one. But it must not necessarily the last one. The trick is to look at words where the minimal distance is attained and then remove a position in all code words where the minimum is attained:
Let $C$ be a $[n,M,d]$-code. Fix code words $c, c'$ with $c \neq c'$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Newton's sum and complex numbers Does Newton's sums include imaginary numbers? For example, if $α,β$ are the real roots of $x^4-2x^2-1=0,$ and I want to calculate $\frac{(α^2+β^2)(α^8+β^8)}{α^4+β^4}$, can I use Newton's sums?
| Yes, it does.
However for this particular problem you can complete the square to obtain $$x^2=1\pm\sqrt2.$$ Then through away the complex roots :) to obtain the values of $\alpha^2$ and $\beta^2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Partial fractions decomposition. Why $cx+d$ instead of $cx$ for the numerator of $(x^2+2$). I understand that the aim of partial fractions decomp. is simply to reach (an) integrable functions, but then I have trouble wrapping my head around why you cannot make the numerator of something like $x^2+2$ equal to $Cx$ alone... | Of course you would try to look for $Cx$ in your case because in general to integrate $\frac 1{x^2+bx+c}$ where the denominator cannot be further factored in $\mathbb{R}$, you would aim for something like $\frac{C(2x+b)}{x^2+bx+c}$. In your case $b=0$ and hence you would look for $Cx$.
In your case your $Cx$ appears na... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Approximating $\frac{(n+1)H_n -n}{n^2}$ I have the following value
$$\frac{(n+1)H_n -n}{n^2}$$
But it is complicated to write, so I want to write a simple approximation to use it. I guess I can just write
$$\frac{(n+1)H_n -n}{n^2}\approx \frac{H_n-1}{n} \approx \frac{H_n}{n} \approx \frac{\ln n}{n}$$
Is this a good app... | Because $\gamma > 0.5$, the approximation $$\frac{\log n}{n}$$ is superior to $$\frac{\gamma + \log n}{n}$$ despite $$\gamma = \lim_{n \to \infty} H_n - \log n.$$ This is because you write $$\frac{H_n - 1}{n} \approx \frac{H_n}{n},$$ which means you are introducing an error on the order of $O(1/n)$ with that step. If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Joint distributions of the rvs $X_1,X_2,X_3$ which are exchangeable and only take values in $\{0,1\}$ Problem: Describe all possible joint distributions of the random variables $X_1,X_2,X_3$ which are exchangeable and only take values from $\{0,1\}.$
Thoughts: First let me note that I have previously posted a similar... | This time you have $3$ degrees of freedom. The probabilities are described by $4$ distinct variables, $p_n=P(n\text{ of the }X_i\text{ are true})$ for $n=0,1,2,3$, and that all probabilities sum to $1$ is encapsulated in the constraint $p_0+3p_1+3p_2+p_3=1$. So this easily leads to a description of all distributions.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Nonlinear first order PDEs intuition I hope you’re all successfully cracking THAT problem that doesn’t let you sleep during this lockdown. For me, it’s this problem right here. I’m afraid I’m having a hard time following the author of this book through the process he outlines.I have no problem replicating the steps he ... | This is the method of characteristics applied to the Cauchy problem (IVP) of scalar conservation laws -- i.e., quasi-linear transport equations. You'll find many examples similar to the Exercises 13-17 on this site, where graphical representations are provided. For Exercise 16., see e.g. this post.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Books for pure cryptography. I am trying to find a"good" book (or a series of "good" books) which covers the following parts in cryptography:
*
*Some background (such as Number Theory, Finite Fields, )
*Classical cryptography (I would say private key cryptography)
*Public key cryptography
*The security models (In... | I recommend you this book: A course of criptography that you can find here https://bookstore.ams.org/amstext-40. I think it's very good and covers (some in details some not as much) all your arguments.
I recommend you also this site (https://bookauthority.org/books/new-cryptography-books) which gives you some good ref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3610922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Asymptotic estimation of a divergent integral I am looking for an equivalent of the following integral : $$ \int_{0}^{A}{e^{t^\alpha}}dt $$ with $0<\alpha$ when $A$ tends to infinity.
I call an equivalent a simpler function $f$ of $A$ such that $$ \int_{0}^{A}{e^{t^\alpha}}dt = f(A)(1 + o(1)) $$ when $A$ tends to infi... | By the L'Hospital rule
$$
\mathop {\lim }\limits_{A \to + \infty } \frac{{\int_0^A {e^{t^\alpha } dt} }}{{\frac{1}{\alpha }A^{1 - \alpha } e^{A^\alpha } }} = \mathop {\lim }\limits_{A \to + \infty } \frac{1}{{1 + \frac{{1 - \alpha }}{{\alpha A^\alpha }}}} = 1,
$$
hence
$$
\int_0^A {e^{t^\alpha } dt} = \frac{1}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3611125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding Basis of Kernel $\Bbb R^4 \to \Bbb R^2$
The linear transformation $T:\Bbb R^4→\Bbb R^2$ defined by
$$T(x,y,z,s)=(2x−2y+z+2s,4y−4x−5s)$$
Give a basis for the kernel of T.
Basis of $\operatorname{Ker}(T)$ is { ... }
Enter your answers as comma separated lists of vectors,
for example $(1,2,3),(4,5,6)$
So, I h... | First, we need to find the elements $(x,y,z,s)$ of the kernel. That is find the elements $(x,y,z,s)$ such that:
\begin{cases} 2x-2y+z+2s = 0 \\ 4y-4x-5s=0\end{cases}
Solving that, we get:
$$x = t-\frac{5}{4}w\quad ; \quad z=\frac{1}{2}w$$
Where $t,w \in \mathbb{R}$
So the elements in the kernel are:
\begin{align}
(x,y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3611284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
At what values of $\alpha$ and $\beta$ does $\int_0^1x^\alpha(1-x)^\beta \ln xdx$ converge? $$\int_0^1x^\alpha(1-x)^\beta \ln x dx$$
When $x\to 0$:
$(1-x)^\beta=1-\beta x+o(x)$
$$\int_0^1x^\alpha(1-\beta x)\ln xdx=\int_0^1x^\alpha\ln xdx-\int_0^1\beta x^{\alpha+1}\ln xdx$$
If we integrate by parts:
$$\int x^\alpha\ln x... | Remember that if $a>0$, then $x^a\log(x)\to0$ as $x\to0$ (you can check that easily). So if $a>0$ and $\beta\geq0$, the integral converges.
If $a\leq0$, then $\int_0^1x^a(1-x)^b\log(x)dx\geq\int_0^1(1-x)^b\log(x)dx\geq\int_0^{1/2}(1-x)^b\log(x)dx\geq c_b\int_0^{1/2}\log(x)dx=\infty$ for all $b\in\mathbb{R}$, where $c_b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3611465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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