Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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What is the area of the region $R$ if $R=\{(x,y):x^2+y^2\le100$, $\sin(x+y)>0\}$? $x^2+y^2\le100 \implies $ a circle with radius 10 and the region enclosed within.
Now given, $sin(x+y)>0 \implies y>-x$
Also we know $sin(x+y)\le1 \implies x+y\le \dfrac{\pi}{2} \implies y\le-x+\dfrac{\pi}{2}$
From this we get the area of... | By symmetry, assuming uniform distribution over the circle with radius $10$, $$Pr(\sin(x+y)>0)=Pr(\sin(x+y)<0)=\frac12$$
Hence the area is
$$\pi \frac{(10)^2}2=50\pi$$
Note that $\sin(x+y) > 0 $ does not imply $x+y >0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stuck at this definite integration: $\int_ {0} ^ {\infty} \frac {\log(x)} {x^2 + 2x + 4} \, \mathrm dx$
If $$\int_ {0} ^ {\infty} \frac {\log(x)} {x^2 + 2x + 4} \, \mathrm dx \ $$ is equal
to $\pi \ln p/ \sqrt {q} \ $, where $p$ and $q$ are coprimes, then
what is the value of $p + q$?
Ok, so I am stuck with this prob... | Jack has already covered the simple, real analysis way of evaluating this integral. So if anybody's curious, here's a way to solve your integral using complex analysis. Note that for these kinds of integral, contour integration is a bit overkill.
The function under consideration is$$f(z)=\frac 1{z^2+2z+4}$$And we are i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2750011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Must the dimensions of $V$ and $U$ be equal for $f: U \mapsto V$ to be a diffeomorphism? Suppose we have $U, V \subset \mathbb{R}^{n}$, and a map $f: U \mapsto V$. If $f$ is a diffeomorphism, must the dimensions of $U$ and $V$ be equal?
I'm thinking this is true since the tangent map $Df_{x}: T_{x}U \mapsto T_{f(x)}V$ ... | You are right, the explanation you gave is also the right one.
If $f$ is a diffeomorphism, then there exists $g\colon V\rightarrow U$ such that $g\circ f=\operatorname{id}_{U}$ and using the chain rule:
$$Dg_{f(x)}\circ Df_x=\operatorname{id}_{T_xU},$$
so that $Df_x$ is an invertible linear map.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2750195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Combinatorics - Removing Double Counted Cases
Here's my solutions:
$$1.)\,(2)(4!)=48$$
$$2.)\,(4)(3)(3!)=72$$
$$3.)\,48+48+3!=102$$
It's the last one I get wrong. The correct answer is 42. The solutions manual of the textbook says the following:
Someone mind explaining what exactly are the double counted cases. Also... | Let $A$ be the event that Anya is at the left end of the line; let $B$ be the event that Elena is at the right end of the line. Then the event that Anya is on the left or Elena is on the right or both is $A \cup B$.
We want to find $|A \cup B|$, the number of elements that are in the union of $A$ and $B$. Notice t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$A,B$ are orthogonal projections and $\|Ax\|^2+\|Bx\|^2=\|x\|^2$ show $A+B=I$ Here is the problem: $A,B:\mathbb{C}^n\to\mathbb{C}^n$ are two orthogonal projections satisfying for any $x\in\mathbb{C}^n$, $$\|Ax\|^2+\|Bx\|^2=\|x\|^2$$ Show that $A+B=I$.
I know that $\|Ax\|^2+\|Bx\|^2=\|x\|^2$ tells that $(Ax,Ax)+(Bx,Bx)... | Let me state two hints and one proposal:
*
*One has $\,\|Ax\|^2=(Ax,x)\,$ by hypothesis.
*What does $\,(Tx,x)=0\;\forall x\in\mathbb C^n\,$ imply for $T$?
(The conclusion would not hold when working in $\mathbb R^n$!)
*Please use the command "\|" to produce nice(-r) norm delimiters,
cf $\,\|\,$ versus $\,||$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2750473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability the driver has no accident in the next 365 days
The time until the next car accident for a particular driver is
exponentially distributed with a mean of 200 days. Calculate the
probability that the driver has no accidents in the next 365 days, but
then has at least one accident in the 365-day period that f... | You want the first accident to be between the first year and second year.
\begin{align}
P(365< T \leq 2 \cdot 365) &= F(2 \cdot 365) - F(365)
\end{align}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Construct a sequence of simple functions converging to $f\in C([a,b])$ I am aware of the result that a measurable function $f\in L^p(\mathbb{R}^d)$ can be approximated with simple functions of the form $f=\sum_{k=1}^{\infty}c_k\chi_k$
However, I am interested in the following:
Given some function $f\in L^p(\mathbb{R}^d... | Hint: If $f$ is defined in a compact its range is a compact and thus contained in some finite interval $I = [-A,A]$. Divide it in $n$ intervals $I_j$ of length $2|A|/n$. Their preimages are disjoint sets ready for building a simple function.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Weak convergence of a sequence of probability measures implies integrability of the limiting probability measure Let $(X_{n})_{n \in \mathbb{N}}$ be a sequence of uniformly integrable random vectors with values in some normed vector space $V$ with $\mathbb{E}[\|X_n\|] < \infty$. This means that
$$
\lim_{C \to \infty} \... | By the uniform integrability of $(X_n)_{n \in \mathbb{N}}$ we have $$M := \sup_{n \geq 1} \mathbb{E}(|X_n|) < \infty. \tag{1}$$ On the other hand, the weak convergence of $X_n \to X$ entails that
$$\mathbb{E}f(X_n) \to \mathbb{E}f(X) \tag{2}$$
for any bounded Lipschitz continuous function; in particular, we can choose ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Eigenvalue of $f \circ f = id_V$ An endomorphism $f$ in the vector space $V$ with the popperty $f \circ f = id_V$ is given. Is my assumption correct that the only Eigenvalues of $f$ are $\lambda_1=1$ and $\lambda_2 =-1$?
Reasoning
We have the defenition of the Eigenvalue:
$$f(v)=\lambda v \tag{1}$$
$$\Rightarrow_{\circ... | Although, @Doe's answers your question, the following method is a more general method for finding eigenvalues of a given map. Note that, by doing some extra computations, we can directly find the eigenvalues of $f$ for sure with this method.
First observe that the map $f$ is a zero of the the polynomial
$$p(\lambda) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2751009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Continuous limit of continuous functions with a distance condition Let $\{f_n\}_n$ be a sequence of continuous functions $f_n\colon[0,1]\to\Bbb R$ and let $U_n=\{\,x\in[0,1]\mid f_n(x)>1\,\}$.
We know that $\forall x\in[0,1]\colon \lim_{n\to\infty}f_n(x)=0$ does not imply that $f_n\to 0$ uniformly. However, the usual c... | We must have $\mu(U_n)\to 0$. It also doesn't matter that the $f_n$'s are continuous or that they converge pointwise everywhere, mere measurability and pointwise a.e. convergence is fine.
Proof. Let $\epsilon > 0$ be given. Since $f_n\to 0$ pointwise a.e. on the finite measure space $[0,1]$, Egorov's theorem tells us t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2751172",
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"source": "stackexchange",
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Result on the power of norm in Banach space? I want to inquire whether there exist any result of the type $$\|x+y\|^{\lambda} \leq c_1\|x\|^{\lambda}+ c_2\|y\|^{\lambda}$$ where $\lambda \in (0, 1]$, $c_1$ and $c_2$ are positive constants and $x, y$ are in banach space $X$? Any reference?
For $\lambda \geq 1$, i know t... | $||x+y||^{\lambda} \leq (2\max \{||x||,||y||\})^{\lambda}=2^{\lambda} \max \{||x||^{\lambda},||y||^{\lambda}\}) \leq 2^{\lambda} \{||x||^{\lambda}+||y||^{\lambda}\})$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A question on a proof that $ L^1 (E)$ is Complete I'm studying Capinski - Copp Measure - Integral Probability.
Specifically, at the proof of Thm 5.1 p.130 ($L^1(E)$ is Complete) they write:
Firstly, they consider a sequence ${f_n}$ in $L^1(E)$, after some work they produce a subsequence of ${f_n}$ that converges to som... | I do not know the full detail of the proof in your book so I will explain based on the proof I know. (The last page of following lecture note introduces the proof I know. It proves the completeness of $L^2$, but the argument works also for $L^1$.)
In fact, the sequence $\langle f_n\rangle$ need not be Cauchy. For exam... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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A finite set is open and closed in the same time. Let $(X,d)$ to be a arbitrary metric space. I know that: every finite subspace of a metric space is closed. Then $\{a\}$ ,with $a\in X$, is closed $\Rightarrow X \setminus \{a\}$ is open. (1)
But if $X$ is a finite metric space then $X\setminus \{a\}$ is also finite so ... | The distinction becomes less important in this finite case if, for $d$ the minimum distance between two points, one works with open or closed balls of radius $d/2.$
So every subset is both open and closed.
Edit: OP has asked for "more examples". Let $X$ be the finite metric space consisting of vertices of a square of s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2751605",
"timestamp": "2023-03-29T00:00:00",
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" Let $A$ be a symmetric $2 \times 2$ matrix with the property $A^{-1} = A$. Find all possible trace values of $\operatorname{tr}A$" I need some help solving this.
I have tried:
$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
=\frac{1}{\operatorname{det}A}\cdot \begin{bmatrix}
d & -b \\
-... | By the Cayley–Hamilton theorem or direct verification, we have $A^{2}-\operatorname {tr}(A)A+\det(A)I=0$.
From $A^2=I$, we get $\operatorname {tr}(A)A=(\det(A)+1)I$.
Taking traces on both sides, we get $\operatorname {tr}(A)^2=2(\det(A)+1)$.
From $A^2=I$, we also get $\det(A)^2=1$ and so $\det(A)=\pm1$.
If $\det(A)=1$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2751819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $f$ be a non-negative differentiable function such that $f'$ is continuous and $\int_{0}^{\infty}f(x)\,dx$ and $\int_{0}^{\infty}f'(x)\,dx$ exist.
Let $f$ be a non-negative differentiable function such that $f'$ is continuous and
$\displaystyle\int_{0}^{\infty}f(x)\,dx$ and $\displaystyle\int_{0}^{\infty}f'(x)\,... | Let $\varphi(x)=\exp\left(\dfrac{1}{3}-\dfrac{1}{4-x^{2}}\right)$ for $|x|\leq 2$, $\varphi(x)=0$ for $|x|>2$.
Let $f(x)=\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{2^{n}}\varphi\left(2^{n}(x-n)\right)$, one may check that $f\in C^{\infty}(0,\infty)$ and that $f,f'\in L^{1}(0,\infty)$.
For all $x$ with $1<2^{n}(x-n)\leq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Block diagonalization We know that not all matrices can be diagonalized, but all matrices can be block diagonalized (with just one block) How can we find a similarity transformation leading to block diagonalization with the greatest possible number of blocks?
| Every matrix with elements in $\mathbb C$ has a Jordan Normal Form. The transform in the canonical basis will have blocks of sizes equal to the sizes of the generalized eigenspaces of the matrix.
The Jordan blocks have a very particular structure:
$$\left[\begin{array}{ccc}\lambda&1&0&\cdots&0\\0&\lambda&1&0&0\\0&\ddot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2752024",
"timestamp": "2023-03-29T00:00:00",
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Find 2 idempotent matrix with some poperties. I want to find an example of two idempotent matrices $a,b$ with entries in $\mathbb{Z}_2$ with $a+b$ also idempotent, $ab\not=0$ and $a\not=b$. Can someone find one?
I have prove that if you work in a field with characteristic greater than 2 this is not possible. But I have... | Try:
$$
a = \left(\matrix{1&0&0\\0&1&0\\0&0&0}\right) \quad b = \left(\matrix{0&0&0\\0&1&0\\0&0&1}\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2752144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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show that orbits have the same order under the normal subgroup of a transitive group
Let $G$ be a group that works transitively on $X$, and let $N$ be a normal subgroup of $G$. Show that the orbits of $X$ under $N$ have the same order, that is $\operatorname{ord}(Nx)=\operatorname{ord}(Ny)$ for all $x,y\in X$.
I'm no... | By transitivity, there exists $g\in G$ such that $gx=y$. Define the map $\phi: Nx\to Ny$ by $\phi(n x)=gng^{-1}y$ for all $n\in N$. Note that $gng^{-1}\in N$ by normality of $N$.
To prove injectivity, assume $gn_1g^{-1} y=\phi(n_1x)=\phi(n_2x)=g n_2 g^{-1}y$. Cancelling the $g$ both sides, we get $n_1g^{-1}y=n_2g^{-1}y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can the Sum of Two Tensor Products Be Written as a Single Tensor Product? In general, if I have $|\Psi\rangle = (|\Psi_{1_1}\rangle \otimes |\Psi_{1_2}\rangle + |\Psi_{2_1}\rangle \otimes |\Psi_{2_2}\rangle)$, can I find $|\Psi_{3_1}\rangle$ and $|\Psi_{3_2}\rangle$, such that $|\Psi\rangle = |\Psi_{3_1}\rangle \otime... | In general the answer is no and you can clearly see why when you look at the dimensions: $\dim (V\otimes W)$ is much larger than $\dim (V\times W)$ whenever $V$ and $W$ are both of dimension greater than $1$. This tells you that in general a tensor is more than just a couple of vectors.
Tensors of the form $a\otimes b$... | {
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"timestamp": "2023-03-29T00:00:00",
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Euler -Lagrange equation: variational Problem min $K[u]$ for $K[u]=\int_D(|\nabla u|^2+\frac{1}{2}gu^4)dxdy$ Consider the variational Problem min $K[u]$ for $$K[u]=\int_D\left(|\nabla u|^2+\frac{1}{2}gu^4\right)\,{\rm d}x\,{\rm d}y$$ where $D \subset \mathbb{R}^2$ and $g(x,y)$ is a given positive function. Find the Eul... | It may be easier to see what is happening if we write $$K[u] = \int_D \underbrace{\left(u_x(x,y)^2 + u_y(x,y)^2 + \frac{1}{2}g(x,y)u(x,y)^4\right)}_{= L(x,y,u,u_x,u_y)}\,{\rm d}x\,{\rm d}y.$$For two variables, we have the Euler-Lagrange equation: $$\frac{\partial L}{\partial u} - \frac{\partial}{\partial x}\left(\frac{... | {
"language": "en",
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Finding the PDF of a random variable with the mean as the realisation of another random variable What method would I use to find the PDF of a random variable that has a parameter as a realisation of another random variable?
For example, I first have an exponential distribution $\Omega \sim exp(\lambda)$ which has a re... | Independence of the normal and the exponential is needed for computing the density function. In the present case the density function is $\frac 1 {\sqrt {2\pi}\sigma}\int e^{-(x-a)^{2}/2\sigma ^{2}} \lambda e^{-\lambda a}da$. I hope the general procedure is clear from this formula. Of course the PDF is obtained by inte... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $x^3-a$ is irreducible in $\mathbb{Z}_7$ unless $a=0$ or $\pm1$ Show that $x^3-a$ is irreducible in $\mathbb{Z}_7$ unless $a=0$ or $\pm1$
My Idea:
Suppose $x^3-a=(Ax+b)(Bx^2+cx+d).$
Then $A=B=1$ or $A=B=-1$ WLOG $A=B=1.$
Then $x^3-a=x^3+x^2(b+c)+x(bc+d)+bd\\
\Rightarrow c+b=0,d+bc=0,bd=-a.$
I can't go furthe... | Let $F$ be any field, and let
$p(x) = x^3 + ax^2 + bx + c \in F[x] \tag 1$
be any cubic polynomial over $F$. Then we have the following
Fact: $p(x)$ is reducible in $F[x]$ if and only if it has a zero in $F$.
Proof of Fact: Clearly if $p(x)$ has a zero $z \in F$, then $p(x) = (x - z)q(x)$ where $q(x) \in F[x]$ is of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Fraction in its lowest form I read that if $a = \frac mn$ is a positive rational number, it can be expressed in "lowest form" by cancelling common factors of $m$ and $n$, so that $a = \frac rs$ where r and s are relatively prime.
I'm wondering if we define the "lowest form" representation for a positive rational numbe... | Suppose, $$\frac{a}{b}=\frac{c}{d}$$ with coprime positive integers $a,b$ and coprime positive integers $c,d$. Then, we have $$ad=bc$$
Since $a$ and $b$ are coprime, we can conclude $a|c$ because of $a|bc$ and $b|d$ because of $b|ad$
Since $c$ and $d$ are coprime, we can conclude $c|a$ because of $c|ad$ and $d|b$ becau... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Dedekind Cuts and Rationals I have a question regarding how you might construct the reals from the rationals by taking Dedekind cuts.
My basic understanding is that a Dedekind cut is a bipartition of the rationals such that the two partitions $X, Y$ satisfy certain properties:
$\forall x \in X,\; \exists y \in X \tex... | Note that it is not part of your axioms that $X\cup Y = \Bbb Q$. For instance, the rational number $0$ is given by
$$
X = \{q\in \Bbb Q\mid q<0\}\\
Y = \{q\in \Bbb Q\mid q>0\}
$$
and the number $0$ isn't contained in either of them. On the other hand, the axiom $$\forall x,y, x<y \Rightarrow \text{ either } x \in X \te... | {
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Prove $n$ is prime. (Fermat's little theorem probably)
Let $x$ and $n$ be positive integers such that $1+x+x^2\dots x^{n-1}$ is prime. Prove $n$ is prime
My attempt:
Say the above summation equal to $p$ $$1+x+x^2\dots x^{n-1}\equiv 0\text{(mod p)}\\
{x^n-1\over x-1}\equiv0\\
\implies x^n\equiv1\text{ (as $p$ can't di... | Hint:
$$x^{ab}-1=(x^a-1)(x^{a(b-1)}+x^{a(b-2)}+\dots+x^a+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$I(\alpha)=\int_\limits{0}^{\infty}\frac{\sin(\alpha x)}{x}dx$
Prove $$I(\alpha)=\int_\limits{0}^{\infty}\frac{\sin(\alpha x)}{x}dx$$ converges uniformly for $0<a\leqslant \alpha\leqslant b$ and it does not converge uniformly for $0\leqslant \alpha\leqslant b$
I know that $\int_\limits{0}^{\infty}\frac{\sin( x)}{x}dx... | Firstly, we have to be careful. The observation $$\int_0^{+\infty} \frac{\sin \alpha x}{x} = \int_0^{+\infty} \frac{\sin x}{x} = \frac{\pi}{2}$$ only holds if $\alpha \ne 0$ since we cannot divide by $0$. Otherwise, $$\int_0^{+\infty} \frac{\sin \alpha x}{x} = 0$$ as we are integrating the zero function. In fact, this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integer solutions to $y=x^2+\frac{(1-x)^2}{(1+x)^2}$ As part of another problem I've been trying to find the greatest integer solutions to $$y=x^2+\frac{(1-x)^2}{(1+x)^2}$$ but am getting very stuck... Would the fact that it asymptotes to $y=x^2$ help at all? Does this mean it won't pass through any integer coordinates... | $$y=x^2+\frac{(1-x)^2}{(1+x)^2}=x^2+\left(\frac{2}{1+x}-1\right)^2$$
If $y\in \Bbb Z$ then $1+x|2$ so $x\in\{0,1,-2,-3\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Solve the initial value problem $y''-6y'+13y=0,\;y(0)=y'(0)=1$ using the Laplace transform. Solve the initial value problem
$$
\begin{cases}
y''-6y'+13y=0 \\
y(0)=y'(0)=1
\end{cases}
$$
using the Laplace transform.
I cannot figure out how to factor or get around factoring $L(Y)$.
| Taking the LT gives you
$$13Y+s^2Y-6(sY-y(0))-sy(0)-y'(0)=0.$$
Plugging in the IC's yields
$$13Y+s^2Y-6(sY-1)-s-1=0.$$
Solving for $Y$ gives you
$$
Y(13+s^2-6s)=s+1-6=s-5,
$$
making
$$Y=\frac{s-5}{s^2-6s+13}=\frac{s-5}{s^2-6s+9+4}=\frac{s-5}{(s-3)^2+4}.$$
Computing the inverse LT yields
$$y(t)=[\cos(2t)-\sin(2t)][\cosh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integral inequality with derivative bound. Suppose $f \in C^1([0,1])$ and $f(0) = f(1) = 1/2$. Also $|f'(x)| \leq 1$ for all $x \in [0,1]$.
Is it possible that $$\frac{-1}{4} \leq \int_0^1 f(x) \, dx \leq \frac{1}{4} \:?$$
My attempt: Using $-1 \leq f'(x) \leq 1$ and $\displaystyle f(x) - f(0) = \int_0^x f'(t)\,dt$ I... | No, it is not possible. Using the fact that $f(0) = 1/2 = f(1)$ and the derivative bound, we have that
$$f(x) \ge \max\{1 - x, x - 1\}$$
(draw the picture of what this means!). Hence $\int_0^1 f(x) \, dx \ge \frac 1 4$. On the other hand, one can argue that (again because of the derivative bound, together with the fac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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The indefinite integral $\int\frac{\operatorname{Li}_2(x)}{1+\sqrt{x}}\,dx$: what is the strategy to get such indefinite integral Here there is an integral that I've found playing with Wolfram Alpha online calculator (thus to me is a curiosity that it has indefinite integral) $$\int\frac{\operatorname{Li}_2(x)}{1+\sqrt... | A natural temptation is to remove the square root from the denominator of the integrand function by enforcing the substitution $x=u^2$, then expanding $\text{Li}_2(u^2)$ as a Maclaurin series and convert the whole thing into a combination of Euler sums, hopefully with a low weight. Indeed
$$ \int_{0}^{1}\frac{\text{Li}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Ordinal $\alpha$ such that $\alpha = \omega_{\alpha}$? I am asked whether or not there exists an ordinal $\alpha$ such that $\omega_{\alpha}$ where we define:
1). $\omega_{0} = \omega$
2). $\omega_{\alpha+1} = \gamma(\omega_{\alpha})$
3). $\omega_{\lambda} = sup\{\omega_{\alpha} \mid \alpha < \lambda\}$ for a non-zero ... | The answer is yes. The mapping function $\alpha\mapsto\omega_\alpha$ is both continuous and increasing, which means it must have a fixed point. We can find this fixed point by taking the supremum - $\sup\{0,\omega,\omega_\omega,\omega_{\omega_\omega},...\}$ and in fact if we start with any two ordinals $\alpha_{1,2}$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Euler Differential Equation General Solution Given the initial value problem of Euler Differential Equation:
$$x^2y''+\beta xy'+\alpha y=0$$
$$y(-1)=2 , y'(-1)=3$$
According to my book, the general solution for x<0 is the same as that of x > 0 so all the possible general solutions should be expressed with absolute valu... | To explain, we must go back to why the trial solution $y=x^r$ works here. Substitute $x = e^t$ to get
$$ \frac{d^2y}{dt^2} + (\beta-1)\frac{dy}{dt} + \alpha y = 0 $$
This is a linear equation with constant coefficients, therefore the general solution has the form $y(t) = e^{rt}$, which leads to $y(x) = x^r$. A double ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2753987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Every function f: A $\rightarrow$ P(A) is not surjective In this case the P(A) is the power set of A.
I want to prove this by contradiction, even though it's easier to say that the power set of A is a bigger infinity, I am not allowed to assume that.
So I want to proceed by contradiction.
This is a proof by contradict... | You're almost done!
Let $A$ be a set and $f\colon A\to{\cal P}(A)$ a function.
Consider $$S = \{x\in A : x\notin f(x)\} \in {\cal P}(A).$$
If $f$ were surjective, there would exist some $z\in A$ such that $f(z)=S$.
One might ask: does $z$ belong to $S$?
We have:
$$z \in S \iff z\notin f(z) \iff z\notin S$$
a contradict... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Group Homomorphism Involving with The circle group in C^x If U = {${z\in C^x | |z|=1}$}, how do I show that $C^x$/U $\cong$ $\Bbb R^+$.
I know that the Fundamental Theorem of Group Homomorphisms has to come into play, given a function $$f:C^x-> \Bbb R^+$$
where f(z) = |z|.
| I take it that our OP means what is more usually written "$\Bbb C^\times$" by "$\Bbb C^x$" and that his
$\Bbb R^+ = \{ r \in \Bbb R \mid r > 0 \}, \tag 1$
so that $\Bbb R^+$ is the multiplicative subgroup of positive reals. Then the map he calls
$f(z) = \vert z \vert, \; f: \Bbb C^\times \to \Bbb R^+ \tag 2$
obeys
$f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimization of functional using Euler-Lagrange We've recently started doing Calculus of Variations in my analysis class and we're applying it to minimizing/maximizing functions. So the way we generally were taught to tackle the problem is to first find the Euler-Lagrange equation, solve the differential equation, then... | Here
$$
L(y,\dot y,t) = y^2+2t y\dot y +4t^2 \dot y^2
$$
$$
\frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial \dot y} = 8t^2\ddot y+16t \dot y = 0
$$
or
$$
t\ddot y + 2\dot y = 0
$$
now making $z = \dot y \Rightarrow t\dot z + 2 z = 0 \Rightarrow z = C_0 t^{-2}\Rightarrow y = -C_0 t^{-1}+C_1$
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Range of $k$ for which equation has positive roots
Range of $k$ for which both the roots of the equation $(k-2)x^2+(2k-8)x+3k-17=0$ are positive.
Try: if $\alpha,\beta>0$ be the roots of the equation. Then $$\alpha+\beta=\frac{8-2k}{k-2}>0\Rightarrow k\in(2,4)$$
And $$\frac{3k-17}{k-2}>0\Rightarrow k\in(-\infty,2)\cu... | You do not have to compute the roots and solve irrational inequations. Just a little thinking to use theorems on quadratic polynomials.
*
*First, it has to be a quadratic equation, which means $k\ne 2$.
*This condition being satisfied, it must have real roots, i.e. its reduced discriminant has to be non-negative:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Finding probability for general cases For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is $p$. If he fails in one of the exams then the probability of passing in the next exam is $p/2$ otherwise it remains the same. Find the probability that he will ... | All three passed: $p^3$.
First and second passed, third failed: $p^2(1-p)$.
First passed, second failed, third passed: $p(1-p)p/2$.
First failed, next two passed: $(1-p)(p/2)^2$.
Then add all four probabilities. The answer would be $7p^2/4-3p^3/4$.
UPDATE. I understood it so that if once failed, the probability of succ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Question on speed.
Speed of a bus is 45 km/h. If it stops for a few minutes in an hour
then its average speed becomes 42 km/h. Find out the time duration it
stops for in an hour.
My attempt:
Let Distance be D.
Let the time duration for which it halts be x.
$45=\frac{D}{1}$
$42=\frac{D}{1+x}$
Therefore,
$45=42+42x... | While there have been several answers posted already, I think that this approach is a bit more intuitive:
Every hour, the bus travels 42 miles. But its full speed is 45 m/hr. So it's traveling at 42/45 of its full speed. Which means that it's driving only 42/45 of the time, and is stopped the rest of the time. So the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 2
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Finding density of a sum of two variables
Let $X$ be exponential r.v with parameter $\lambda$ and $Y$ also
exponential with parameter $2 \lambda$ and independent of $Y$. Find
probability density of $X+Y$.
We know
$$ f_{X+Y}(a) = \int\limits_{-\infty}^{\infty} f_X(x)f_Y(a-x) dx $$
Now, my problem here is to put t... | Note that the exponential RV can only take positive values. Hence, your limits should ensure that $$X,Y>0$$
As such,
$$a-x>0\implies x<a$$
and from above, $$x>0$$
Thus the limits should be from $0$ to $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Resolvent of semicircle law I am trying to approximate the Stieltjes transform of the semicircle law. In particular, it is known that the Stieltjes transform m(z), for z in the upper half plane, is exactly
$$
m(z) = \frac{-z + \sqrt{z^2 - 4}}{2}
$$
I would like to show that
$$
Im(m(z)) \sim \sqrt{K + y}
$$
for $z = x+... | It is known that $\sqrt{a + ib} = p + iq$ with $p = \frac{1}{\sqrt{2}} \sqrt{\sqrt{a^2 + b^2} + a}$ and $q = \frac{sign(b)}{\sqrt{2}}\sqrt{ \sqrt{a^2 + b^2} - a}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2754919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Meaning of the Fourier transform of $1$ According to Wolfram, Fourier transform of $1$ is $\sqrt{2\pi} \, \delta(\omega)$. Can someone explain what this means?
The only frequency the function $1$ should just be $0$ since $\cos(0) = 1$.
So, shouldn't the result have just been $0$ instead of that expression?
| Short intuitive answer: The Fourier transform breaks functions down into their constituent frequencies, and frequency is the inverse of wavelength. A constant function $1$ is so spread out that it effectively has infinite wavelength, or zero frequency. Hence its Fourier transform is concentrated at $0$, giving a spike ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Topological Manofold is Hausdorf, Second Countable, Locally Homeomorphic to $R^n$ Hier is a result from Topology and Differential Geometry:
A topological manifold is a topological space such that the three conditions are met:
*
*Hausdorff
*second countable, and
*covered by charts homeomorphic to open subsets in $R... | Hausdorf, second countable, not locally homeomorphic to $\mathbb R$:
*
*$\mathbb Q$.
Hausdorf, not second countable, locally homeomorphic to $\mathbb R$:
*
*The disjoint union of uncountably many copies of $\mathbb R$.
Not Hausdorf, second countable, locally homeomorphic to $\mathbb R$:
*
*The line with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2755198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Divisibility of $n^2+9$ by $n+3$ How would one find all integers $n$ such that $n+3 \vert n^2 +9$? I assume it is important that $(n+3)^2 - 6n = n+3$, but I am struggling to see how you can find all $n$, and confirm an upper bound such that there are no more such $n$.
| Hint
$$n^2+9=(n+3)^2-6(n+3)+18$$
so
$$\frac{n^2+9}{n+3}=(n+3)-6+\frac{18}{n+3}$$
and then $n+3|18$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2755326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Inequalities involving means and probabilities of paths of cadlag stochastic processes Let's say we can prove that for a cadlag process, $X, $ there exists $\alpha > 1, \beta > 0, $ and a non-decreasing continuous function $H:[0, 1] \mapsto \mathbb{R} $ such that
$$ E\biggl[\bigl|X_s^{(n)} - X_r^{(n)}\bigl|^\beta\cdot... | Edited on April 27.
You are absolutely right Zhoraster. It is as simple as recognizing that for any
two random variables, $X $ and $Y, $ and $a> 0, $:
$$ a^2\cdot 1_{|X|\ge a, |Y|\ge a} \le |XY| $$
and then going from here it is straightforward.
For one reason or another, I managed to mess up that basic inequality. Sor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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British Maths Olympiad (BMO) 2006 Round 1 Question 5, alternate solution possible? The question states
For positive real numbers $a,b,c$ prove that
$(a^2 + b^2)^2 ≥ (a + b + c)(a + b − c)(b + c − a)(c + a − b)$
After some algebraic wrangling we can get to the point where:
$(a^2 + b^2)^2 + (a + b)^2(a − b)^2 + c^4 ≥ 2... | Suppose that $a,b,c$ can form the sides of a triangle. Let $s=\frac{a+b+c}{2}$ be the semiperimeter. The inequality becomes $$
(a^2+b^2)^2\ge2s\cdot 2(s-a)\cdot 2(s-b)\cdot 2(s-c)
$$
or by Heron's formula, $$a^2+b^2\ge 4A$$
where $A$ is the area of the triangle. If $\theta$ is the angle between sides $a$ and $b$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2755581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Least number of rounds to find all hidden pairs In my country there's a TV reality show in which $n+1$ men and $n$ women live in a house and, over the course of the show, they have to, as a group, find out who their 'match' is. $($In the actual TV show, $n=10)$.
Each man is assigned to a single woman such that every wo... | A very slight improvement on the circle idea of @saulspatz makes $n$ rounds sufficient for a win, as follows.
Terminology: There is one woman matched with 2 men. We'll call her the "special" woman, and her 2 men, the "special" men.
Algorithm: randomly set one man aside. This leaves $n$ men and $n$ women. Sit them in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Meet-irreducible element of lattice I have a question concerning the exercise 5.7 of the Davey & Priestley's book. Here are the questions:
Let $L$ be a finite distributive lattice. Prove by the steps below that $\mathcal{J}(L)\cong\mathcal{M}(L)$.
(i) Let $x\in\mathcal{J}(L)$. Show that there exists $\hat{x}\in L$ such... | Relating to your question
"Can we show first that...": no, that's not true.
As an example, take $L$ to be the power-set of $\{a,b,c\}$, which is distributive.
Singletons are join-irreducible. For example, take $x=\{a\}$; $y=\{b\} \in L\setminus\uparrow x$.
Then $y=\{a,b\}\cap\{b,c\}$, so it is not meet-irreducible.
No... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Geometric solution? Given coordinates of $A$, $B$, $C$, find $M$ on $y=x$ minimizing $AM+BM+CM$ I have the problem:
Let be given three points $A(1,2)$, $B(3,4)$, $C(5,6)$. Find point $M$ on the line $y=x$ so that sum of distances $P=AM+BM+CM$ is smallest.
I tried. We have
$$P=\sqrt{(x-1)^2 + (x-2)^2} + \sqrt{(x-... | Let's $a$ be a line $y=x$.
So, we claim that for the basis $M$ of the perpendicular dropped from $B$ to $a$ sum $AM + BM + CM$ is the smallest. It is easy to demonstrate this using additional point $A'$ which is symmetrical to the point $A$ with respect to the line $a$. One may see that $A'MC$ is the line segment (beca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2755885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Some Integral Estimate I am currently trying to figure out the following estimate:
Let $\gamma: [0, 2 \pi] \to \mathbb C, \gamma(t) = e^{\mathrm i t}$, $\gamma^* := \gamma[0, 2 \pi]$ and $f:
\gamma^* \to \mathbb R$ be continuous. Then it holds
$$\left\vert \int_{\gamma} f(z) \, dz \right\vert \leq 4 \max_{z \in \gamma... | We want to estimate $|\int_0^{2\pi}f(t)e^{it}\,dt|.$ This equals
$$e^{is}\int_0^{2\pi}f(t)e^{it}\,dt=\int_0^{2\pi}e^{i(s+t)}f(t)\,dt$$
for some real $s.$ Now the above is nonnegative, so it equals
$$\int_0^{2\pi}\text { Re}\left (e^{i(s+t)}f(t)\right)\,dt. $$
Since $f$ is real valued (!), the last integral equals
$$\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2755973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sufficient condition for graph isomorphism assuming same degree sequence We assume graph to be simple undirected.
In general, having the same degree sequence is not sufficient for two graphs to be isomorphic. A trivial example is a hexagon which is connected and two separated triangles, which is obviously not connected... | As you pointed out in your question, the graphs $G=C_6$ and $H=C_3+C_3$ are a trivial example of two nonisomorphic graphs with the same degree sequence. They are the only $2$-regular graphs on $6$ vertices.
Their complements $\overline G$ and $\overline H$ are another example of two nonisomorphic graphs with the same d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How did the author get to these inequalities?
Context: Here $\tau = \langle X, T \rangle $ and $v = \langle X, N \rangle$. I understood everything in the proof except for the blue part. Why is it that $\displaystyle{\lim_{x \to \infty} y'(x)} > 0$? Where do the final inequalities come from?
This is the thesis where I... | It is written that $y$ is convex, so $y'$ is increasing,more over $y'(0)=0$. So $y'$ has a limit and this limit is $\geq 0$.
More over if this limit is $0$ then $y'(x)=0 \forall x$, so $y''=0$ which leads to $y(x)-xy'(x)=y(x)=0$. But $\alpha=y(0)>0$ so there is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Weak$^*$ convergence on dense subspace of Hilbert space Let $H$ be a separable infinite dimensional Hilbert space over $\mathbb{C}$
Let $V$ be a dense subspace of $H$
Let $\{T_n\}_{n \in \mathbb{N}} \subset H^*$ be a sequence of continuos linear functionals such that
$$
\forall v \in V: \lim_{n \to \infty} T_n(v) = 0
... | No. Let $H=\ell^2(\mathbb N)$, and $V$ the subspace of sequences with finitely many nonzero elements. Denote by $\{e_n\}$ the canonical basis, and let
$$
T_n(x)=n\,x_n.
$$
Then, for any $x\in V$, eventually $x_n=0$, so $T_n(x)\to0$. On the other hand, if $x=(1/n)_n$, then $T_n(x)=1$ for all $n$.
The assertion become... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Question about Galois Theory. Extension of a field of odd characteristic.
Let $F$ be a field of characteristic $\ne 2$, and let $K$ be an extension of $F$ with $[K:F]=2$. Show that $K = F(\sqrt{a})$ for $a \in F$; that is, $K = F(\alpha)$ with $\alpha^{2}=a$. Moreover, show that $K$ is Galois over $F$.
My doubt is in... | $$F\subsetneq F[\sqrt{a}]\subseteq K$$
What can you say about the degrees of extensions of $F[\sqrt{a}]/F$, $K/F[\sqrt{a}]$ and $K/F$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Two approaches to the method of integration by substitution I came across two approaches to the method of integration by substitution (in two different books).
Approach I
Let $I=\int f(\phi(x))\phi'(x) dx$
Let $z=\phi(x)$
$\therefore \phi'(x)dx=dz$
$\therefore I=\int f(z)dz$
Approach II
Let $I=\int f(x) dx$
Let $x=\phi... | A concrete example of approach 1 may be something like $\int\frac{1}{1+\sqrt x}\,\mathrm{d}x$ and you make the substitution $x=z^2$ in order to get rid of the square root. In this case our $\phi(z)=z^2$ and $\phi’(z)=2z\,\mathrm{d}z$, this makes our integral solvable by some trivial algebra and is already completely in... | {
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"answer_count": 3,
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Solution to second order linear differential equation with only second order differential has 2 trial solutions? $$ m\frac {d^{2}y}{d^{2}x} = 1 $$
homogenous linear equation for this ode is
$$ m\frac {d^{2}y}{d^{2}x} = 0 $$
trial solution is $Ae^{kx}$ but clearly in this case $Bx+C$ is a trial solution that works. Wha... | The solution $Bx+C$ is a limit for the combination of the solutions $Ae^{\pm \omega\,x}$, when $\omega \,\to \,0$.
In fact, when you consider the general linear 2nd order ODE
$$
m{{d^2 y} \over {d^2 t}} + r{{dy} \over {dt}} + ky = 0
$$
and write the general solution to it, in case of an under-damped system, as
$$
\eqal... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Taylor expansion and Distribution Let $u(x)$ be the step function and $p_u(x)$ be the distribution defined by
$$\forall \varphi \in D,
\langle p_u , \varphi \rangle
= \lim_{\epsilon \to 0} \left (\varphi(0) \ln(\epsilon) + \int^{+ \infty } _ {\epsilon} \frac{\varphi(x)}{x} dx \right )$$
Using a Taylor expansion ... | Since you already have the "part (a)", this is much simpler than what you are trying to do. In short, we want to "replace" in the limit the $\varphi(\epsilon)$ by $\varphi(0)$. To do this, we only need to prove that $\lim_{\epsilon\to 0}(\varphi(0)\ln(\epsilon) - \varphi(\epsilon)\ln(\epsilon)) = 0$.
In more details, w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Group size as a function of presentation length? Given a presentation of a group, together with the promise that the group is finite, is there a computable upper bound on the size of the group? Edited to add: by "presentation" we mean a set of generators and relations. See e.g. Wikipedia.
Define the length of a present... | I found a paper here: https://www.math.auckland.ac.nz/~obrien/research/an-sn-present.pdf in which the authors prove that the symmetric group $S_n$ (of order $n!$) has a presentation of length $O(\log n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2756802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $F$ is a $1$-to-$1$ imersion and is proper then $F$ is an imbedding
Let $F:N\rightarrow M$ be a one-to-one immersion which is proper, i.e. the inverse image of any compact set is compact. Show that $F$ is an imbedding and that its image is a closed regular submanifold of $M$ and conversely.
This is an exercise fro... | Assume that it is not an embedding. For $p\in f(N)$, consider $$ f(N)\ \bigcap\ B\bigg(p,\frac{1}{n}\bigg)
\supseteq U\cup \{p_n\}$$ where $U$ is homeomorphic to
${\rm dim}\ N$-dimensional open ball.
Hence $p_n\rightarrow p$ so that $\{p_1,\cdots \}\cup B(p,\varepsilon) $ is compact. Its preimage contains
$U'$ and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to see wether numbers are distributed "evenly" ([1,2,18,35,36]) or "cluttered to one side" ([1,2,3,30,31], [7,9,17,16,36])? I have a set of 5 integer numbers {1,23, 17, 33, 35}. Elements can take values only from [1..36], and happen only once within the set.
What math can I use to understand, wether the numbers are... | An easy fast way to check:
Order $x_1 < x_2 < x_3 < x_4 < x_5$, then consider
\begin{equation}
\sum_{i=1}^5\left(|x_{i}| - |37-x_i|\right)
\end{equation}
If they are 'evenly distributed', this sum is close to 0.
Worst case is $\pm 155$. You can set a treshold somewehere in between.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2757084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Inequality between two functions I have a two functions defined for $x > 1$, and $c \in (0,1)$:
$$
f(x) = 1-\exp\left(-\frac{c}{x^2} \right),
$$
and
$$
g(x) = \exp\left(-\frac{x}{c} \right).
$$
From graphical tool (
https://www.desmos.com/calculator/hr8n8kkpym ), I know $f(x) > g(x)$. How can I prove this inequality an... | Calling
$$
\left\{ \begin{array}{rcl}
u & = & 1-e^{-\frac{c}{x^{2}}}\\
v & = & e^{-\frac{x}{c}}
\end{array}\right. (1)
$$
we have
$$
\left\{ \begin{array}{rcl}
\log(u) & = & \log(1-e^{-\frac{c}{x^{2}}})\\
\log(v) & = & -\frac{x}{c}
\end{array}\right. (2)
$$
and also
$$
\left\{ \begin{array}{rcl}
\frac{d}{dx}\log(u) & =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2757177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Complex numbers polar form change I have a pretty straight forward question.
Change $z = (-1+i\sqrt3)^{2017}$ to $a+bi$ $\;$ form & polar form. Where $i = \sqrt{-1}$.
So i want to change it to $z = re^{iv}$.
$r$ is easy to calculate. $r = \sqrt4 = 2$.
However the angle is where im struggeling.
I know all the "stan... | Hint: The point $-1+\sqrt3i$ makes an equilateral triangle together with $0$ and $-2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Strong law of large numbers for the conditional expectation of functions of random vectors Consider a collection of 0-1 random variables $Y_{n,N}$, for all $n$ and $N$.
The random variable $Y_{n,N}$ is a deterministic function of the collection of random variables in $\mathcal{F}_n = \{(U_k)_{k=1,\ldots,n}, (V_k)_{k=1,... | With the notations of the opening post,let $\mathcal G_n:=\mathcal F_{n+1}\setminus V_{n+1}$ and $D_{n,N}:=Y_{n,N}-\mathbb{E}\left[Y_{n,N} | \mathcal{F}_n\setminus V_n\right]$. Then $D_{n,N}$ is $\mathcal G_n$-measurable, $\mathbb{E}\left[D_{n,N} | \mathcal{G}_{n-1} \right]=0$ and $\left\lvert D_{n,N}\right\vert\leqsla... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does $x \in A$ and $x \notin B$ imply that $x \notin (A \cap B)$? For any two sets $A$ and $B$. Is it true that if $x \in A$ and $x \notin B$, then $x \notin (A \cap B)$?
| It's true that $x \in A \land x \not\in B \implies x \not\in A \cap B$, but its converse is false: if $x \not\in A \cap B$, it's possible that $x \notin A \land x \notin B$, and this renders $x \in A \land x \not\in B$ false.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is any extension ring $S \supset R$ an $R$-algebra? Is any extension ring $S \supset R$ an $R$-algebra?
In our lecture note an $R$-algebra $A$ is defined as follows $:$
An $R$-algebra $A$ is a ring $A$, which is also an $R$-module satisfying the condition
$$a(xy)=(ax)y=x(ay),\ a \in R,\ x,y \in A.$$
It is clear that $... | No: take for example $\mathbb C \subseteq \mathbb H$, the complex numbers in the quaternions. (This is an example if what Pedro was getting at in the comments.)
The axioms of an $R$ algebra $A$ require that $R$ is contained in the center of $A$, which is not the case for this example.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\frac{1}{n+1}\le \log\left(1+\frac{1}{n}\right)\le \frac{1}{n} ,\forall n\ge1$ if $\log x= \int_{1}^{x}\frac{dt}{t},x>0$ I know it is very simple but something is going amiss.
We can see here that $\log(1+\dfrac{1}{n})=\displaystyle \int_{n}^{n+1}\dfrac{dt}{t}$
if $t \in[n,n+1]$, $\dfrac{1}{n+1}\le\dfrac{1}{t}\l... | Yes, because $\frac{1}{n+1} \leq \frac{1}{t} \leq \frac{1}{n}$ for $t \in [n, n+1]$, so
$$\int_{n}^{n+1} \frac{1}{n+1} \, dt \leq \int_{n}^{n+1} \frac{1}{t} \, dt \leq \int_{n}^{n+1} \frac{1}{n} \, dt$$
then your inequality follows.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving for angle of hyperbolic triangle in Poincare disk I am working out an example problem trying to find the angles of a hyperbolic triangle in the Poincare disk model. I am getting inconsistent answers.
For the sake of simplicity, I am using these coordinates for $\triangle OPQ$:
$O(0,0), P(\frac{1}{2},0),$ and $Q... | For points $P = (a,b)$ and $Q=(c,d)$ in the Poincaré disk model, suppose that the hyperbolic distance, $\delta$, between them is given by a formula of the form
$$\delta = \ln \frac{u+v}{u-v} \tag{0}$$
for some expressions $u$ and $v$. Then
$$\cosh \delta = \frac12\left(e^\delta + e^{-\delta}\right) = \frac12\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2757958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine $p$ such that $x^2 \equiv a \pmod{p}$ using Legendre symbols(for specific values of $a$) Determine $p$ such that $x^2 \equiv a \pmod{p}$ has a solution.(where $p$ is
a prime)
How would you approach this for "bigger" numbers, if you would want to solve this using Legendre symbols and their properties.
E.g. ... | I'll follow up on GNU Supporter's comment, with your given explicit example. If we have $(\frac{3}{p})=1$, then the law of quadratic reciprocity says
*
*if $p\equiv 1 \pmod{4}$, then $(\frac{p}{3})=1$, so $p \equiv 1\pmod{3}$ (a priori $p =3$ case is easy to deal with separately). Combining we have $p \equiv 1 \pmo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does it mean when I say that addition/multiplication for an equivalence relation is well defined? I have trouble understanding this concept. Why is it necessary to prove that addition or multiplication is well defined in equivalence classes? My understanding of equivalence classes is that it must be reflexive, sym... | Just because a relation is an equivalence, this doesn't mean it has to be "nice" with respect to any operation you'd like to put on its classes. To see this, look at a non-example of something being well-defined.
Let $V=\mathbb{R}^2$ be the plane with its usual vector space structure. Put a relation on $V$ by defin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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If $ u $ satisfy $ u_{t} = ku_{xx} $ then so does $ u_{\alpha, \beta, \gamma} $, provided $ \beta = \alpha^{2} $ If $ u $ satisfy the heat equation $ u_{t} = ku_{xx} $ then so does $ u_{\alpha, \beta, \gamma} $ ( where $ u_{\alpha, \beta, \gamma}(x,t) = \gamma u(\alpha x, \beta t) $), provided $ \beta = \alpha^{2} $.
M... | Note that
$$
\partial_{t}u_{\alpha,\beta,\gamma}=\partial_t \gamma u(\alpha x,\beta t)\\
=\gamma \beta u_t(\alpha x,\beta t)
$$
and
$$
\partial_{xx}u_{\alpha,\beta,\gamma}=\partial_{xx} \gamma u(\alpha x,\beta t)\\
=\gamma \alpha^2 u_{xx}(\alpha x,\beta t)
$$
so
$$
\partial_{t}u_{\alpha,\beta,\gamma}-k\partial_{xx}u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758351",
"timestamp": "2023-03-29T00:00:00",
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How does $S \otimes_R A$ become an $S$-algebra? Let $f : R \longrightarrow S$ is a ring homomorphism and $A$ is an $R$-algebra then in our lecture note it has been stated that its scalar extension $S \otimes_R A$ is an $S$-algebra.
How is that possible? I know that the scalar extension is a $S$-module given by the wel... | You have to be careful about actions being left or right if the rings are not commutative. In general, if you have $AB$-bimodule $M$ (i.e. left $A$-action and right $B$-action) and $BC$-bimodule $N$ (left $B$-action and right $C$-action), then the tensor product $M\otimes_B N$ is $AC$-module:
$$a(m\otimes n) = (am)\oti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Math. Logic in calculus Assume f : D -> S function, where D, S are subsets of R, different from R.
We know that statement for any A from D, f(A) belongs to S is correct.
I wonder whether this statement is correct as well: for any A from R\D, f(A) does not belong to S
If it is correct, whether we can say, that for any A... | As астон вілла олоф мэллбэрг stated, the best way to think of this conceptually is that $f(a)$ is just nonsense when $a$ is not in the domain of $f$. As such your second statement has no truth-value one way or the other.
Set theory, usually considered the "foundation" of math, does, however, give such statements meanin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a complex binomial identity I would like to prove an identity:
$$\binom{\alpha}{n} = \sum_{k=0}^n(-1)^k(k+1)\binom{\alpha + 2}{n-k}$$
Where $\alpha$ is complex.
I have already found that if you have two sequences related by the identity $$b_n = \sum_{k=0}^n(-1)^{k}(1+k)a_{n-k}$$ you can write the generating fun... | We apply the Cauchy product formula. It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series. This way we can write for instance
\begin{align*}
[z^n](1+z)^\alpha=\binom{\alpha}{n}
\end{align*}
We obtain for $\alpha\in\mathbb{C}$ and $n\in\mathbb{N}$:
\begin{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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On a closed form for $\int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$ Consider the following function of a real variable $p$ , defined for $p>\frac{1}{2}$:
$$I(p) = \int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$$
Playing around in Wolfram Alpha, I have conjectured that we have the following closed form:
$$... | By substituting into the definition of the $\Gamma$-function,
$$ \frac{1}{(1+x^2)^p} = \frac{1}{\Gamma(p)} \int_0^{\infty} \lambda^{p-1} e^{-\lambda(1+x^2)} \, d\lambda. $$
Interchanging the order of integration,
$$ I(p) = \frac{1}{\Gamma(p)} \int_0^{\infty} \lambda^{p-1} e^{-\lambda} \left( \int_{-\infty}^{\infty}e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to factor in the 'immediately prior to the administration of the next dose' statement in this question? For part b I have found that the stationary state is $a_{n}=d/k$. For part $d)$ however I am not sure on how to take into account the statement 'immediately prior to the administration of the next dose'. Without ... | The idea is that the amount in the bloodstream is slowly decreasing between doses, then jumps up when a dose is administered. $a_n$ is defined as the amount in the bloodstream just after a dose, so is at the peak of the step. The amount just before dose $n$ is $a_n-d$. If $a$ is the stationary value, you need to mak... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2758898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Probability that the two segments intersect P and Q are uniformly distributed in a square of side AB. What is the probability that segments AP and BQ intersect?
|
Suppose we start with $Q = (x, y)$. The admissible positions for $P$ will be in the triangle $BQR$ if $y < x$ or in the quadrilateral $BQRC$ if $y > x$. The areas are calculated from the sides and the heights, and
$$p = \int_0^1 \int_0^x \frac {y (1 - x)} {2 x} dy dx +
\int_0^1 \int_x^1 \left( 1 - \frac x {2 y} - \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2759011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Distribution of minimum of random variable and its square
Suppose $X \sim U[0,2]$ is a uniformly distributed random variable. What are the distributions of
$\min\{X,X^2\}$ and $\max\{1,X\}?$
The density function of $X$ has to be $f(x)=\frac{1}{2}$ and
$$F_{X^2}(x)=P_{X^2}(X^2\leq x)=P_{X^2}(-\sqrt{x}\leq X \leq\sq... | $$F_{\max\{1,X\}}(x)=P(\max\{1,X\}\le x) \tag{1}$$
$$=P(1 \le x \cap X \le x) \tag{2}$$
$$=P(1 \le x) \times P(X \le x) \tag{3}$$
$$=1_{1 \le x}(x) \times P(X \le x) \tag{4}$$
$$=1_{1 \le x}(x) \times F_X(x) \tag{5}$$
Remarks and explanations:
$(1)$ Say $P_{X^2}(x)$ or $P(X_2 \le x)$ but not $P_{X^2}(X^2 \le x)$
$(2)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Interchangability of arbitrary sums and linear operator Let $\mathcal H$ be a Hilbert space, $\{x_i : i \in I \}$ be a orthonormal base in $\mathcal H$ and $T \in L(\mathcal H)$.
Does the following hold:
$T( \sum_{i \in I} \lambda_i x_i) = \sum_{i \in I} \lambda_i T(x_i)$ ?
For example, consider $\{ e_i : i \in I\}$,... | We have for finite sums that:
$$
T\left(\sum_{ k = 1}^{N} \lambda x_k \right) = \sum_{k = 1}^{N} \lambda_kT(x_k)
$$
Because $T$ is continuous, we have that for some sequence $x_n \in \mathcal{H}$, $x_n \rightarrow x$ that:
$$
T(x_n) \rightarrow T(x)
$$
Hence, becuase:
$$
\sum_{ k = 1}^{\infty} \lambda_k x_k= \lim_{N \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2759217",
"timestamp": "2023-03-29T00:00:00",
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Bound for the Brownian motion exit time Suppose $T = \inf\{t: B_t\not\in (a,b)$ where $a<0<b$ and $a\neq -b$. I would like to show $$\mathbb{E}T^2 \leq C \mathbb{E}B_T^4.$$
The problem also says to apply Cauchy-Schwarz inequality to $\mathbb{E}(TB_T^2)$.
Now I know $\{B_t^4 - 6tB_t^2 + 3t^2\}_t$ is a martingale, and i... | It is just a scaling argument now. Write $E[T^2]=C(a,b) E[B_T^4]$ and then your inequality becomes
$$(3C(a,b)+1)E[B_T^4] \leq 6 C(a,b)^{1/2} E[B_T^4].$$
Thus $3C(a,b)+1 \leq 6C(a,b)^{1/2}$. There is a maximal solution to this inequality.
The point is that $E[T^2]$ contributes more to the left side of the inequality th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Subset of matrix rows with half of column sums Consider the following problem. We are given a matrix $A = (a_{ij})_{i,j = 1}^{m,n}$ with $m$ rows and $n$ columns, all $a_{ij}$ are nonnegative. Prove that there exists a subset $S$ of rows, $|S| \leq m/2 + n/2$, such that, for every column $j$, sum of all elements from $... | Ok, I think the following works so I will sketch out the idea. Maybe you can check the details?
We consider the following system: Let $C_1, \cdots, C_n$ be the $n$ column sums. Let $x_1, \cdots, x_m$ be $m$ variables, one for each row an consider the following constriants:
\begin{align*}
x_1 + \cdots + x_m &\le \frac{m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why is HCF of $x^2-1$ and $x-1$ is $x-1$, "why can't it be $1-x$?" I've been to different site and tried to find it using Mathematica also, but everywhere they put the answer $x-1$ before me.
I even tried to find for $1-x^2$ and $1-x$, then also I got the answer $x-1$.
Can anyone explain please!
| A $\gcd$ (or $\operatorname{hcf}$) is only defined up to associates in the respective ring. For real polynomials, the usual convention is to define the $\gcd$ to be the monic polynomial among those, since the units are the (non-zero) constant polynomials.
| {
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Fourier transform of a function with exponential and powers How can I calculate this Fourier transform $F(y)$ ?
$$F(y)= \int_0^{\infty}(1+x)^{\frac{1}{2}} x^{-\frac{1}{2}-a} e^{-a x} \cos(2 \pi xy) dx$$
with $a$ complex ($0<Re(a)<\frac{1}{2}$)
This is in fact a Cosine transform.
| $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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How to prove this series for convergence?
Let $(a_n)$ be a sequence of real numbers satisfying $$a_1 \geq 1 \;\;\;\text{and}\;\;\;a_{n+1}\geq a_n+1$$ for all $n \geq 1$. Then which one of the following is necessarily true?
a) The series $\sum \frac{1}{(a_n)^2}$ diverges.
b) The sequence $a_n$ is bounded.
c) The series... | HINT: $a_n\geq n$, hence
$$
\sum_{n=1}^{\infty}\frac{1}{a_n^2}\leq\sum_{n=1}^{\infty}\frac{1}{n^2}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Degree of the eighth vertex given the other degrees Consider a graph with $8$ vertices. If the degrees of seven of the vertices are $1,2,3,4,5,6,7$, find the degree of the eighth vertex. I also have to check the graph's planarity and its chromatic number.
I know that the sum of degrees of vertices is twice the number o... | Drawing the graph works, but here is a more formal argument.
The degree 7 vertex must be connected to each of the other vertices. So the degree 1 vertex is connected to the degree 7 vertex only.
Therefore the degree 6 vertex must be connected to every vertex apart from the degree 1 vertex. So the degree 2 vertex is con... | {
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"url": "https://math.stackexchange.com/questions/2760008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove that if $f^3 = f$, $f$ is diagonalizable? Prove that
Let $V$ be a finite dim. vector space over a field of characteristic zero, and $f: V \to V$ be a linear
map.Then if $$f^3 = f,$$ then $f$ is diagonalizable.
Since $f$ is zero of the polynomial $$p(x) = x^3 - x$$
the minimal polynomial $m_f$ should di... | Since the matrix $f$ is a zero of the polynomial $$p(x) = x^3 - x = x (x-1)(x+1),$$ the minimal polynomial of $f$ has to divide the polynomial $p$, but this means that $m$ is some combinations of the factors $x$, $(x-1)$, and $(x+1)$ with each having the multiplicity 1, but this implies that $f$ is diagonalisable.
QED.... | {
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Prove that $f^2$ is Lebesgue integrable if and only if $\sum_{k=1}^\infty k \cdot m\{x\in A: |f(x)|>k\}<\infty.$
Question: Let $f$ be a Lebesgue measurable function on $A$ with $m(A)<\infty.$
Prove that $f^2$ is Lebesgue integrable if and only if
$$\sum_{k=0}^\infty k\cdot m\{x\in A: |f(x)|>k\}<\infty.$$
My atte... | You only have
$$k^2 \lambda(A_k \setminus A_{k+1}) \leq \int_{A_k \setminus A_{k+1}} f^2 \mathrm{d} \mu \leq \lambda \leq (k+1)^2 \lambda(A_k \setminus A_{k+1}).$$
However, noting that
\begin{align}
\sum_{k=1}^\infty k \lambda(x \in A \colon |f(x)| >k) &= \sum_{k=1}^\infty k \sum_{i=k}^\infty \lambda(A_i \setminus A_{i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Group action defined on generators. In a question for an assignment (specific group but I've abstracted the question for academic integrity). I was given a group $G=\langle\ a,b \mid R\ \rangle$ where $R$ is some set of relations. Then, the question defines a group action on $X$ for each generating element. The first q... | What you need to check is that every relator will act (by concatenating the action for the generators as given in the relator) as the identity.
An example to show that this is needed would be a cyclic group $\langle g \rangle$ of order $3$, which acts on $4$ points via $g\mapsto (1,2,3,4)$.
Then $g^3$, the identity, wo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2760397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Weak control of FWER Suppose I have $p$ null hypotheses $H_1,\ldots,H_p$ and the global hypothesis:
$$
H_0:H_1 \cap H_2 \cap \cdots \cap H_p.
$$
I'm interested in level controlling when testing for $H_0$ (using multiple testing): bounding
$$
\Pr(\text{reject }H_0|H_0\text{ is true})
$$
regardless of the underlying depe... | According to Remark 3.2 from [1], FDR coincides with weak FWER:
Given that all null hypotheses are null, any discovery is a false discovery, and therefore, $$FDP = V/R = 1.$$ ($V$ = # of false discoveries, $R$ = # of discoveries)
As you can see, under the condition that all nulls are true, the definitions of Type I er... | {
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Uniform Convergence of $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^2}{(1+x^2)^n}$ in $\mathbb{R}$
Uniform convergence of $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^2}{(1+x^2)^n}$ in $\mathbb{R}$
Using Dirichlet Test, it can be shown that the it uniformly converges for $\mathbb{R}$ \ $\{0\}$.
In $x=0$ there is obviously a point... | In short: here, you shouldn't have to worry about $0$ in the first place.
When you apply the Dirichlet test, take
$$
a_n(x) = (-1)^{n-1}, \qquad b_n(x) = \frac{x^2}{(1+x^2)^n}
$$
for $x\in\mathbb{R}$ and $n\geq 1$. Then
*
*For $M\stackrel{\rm def}{=} 1$, we have
$$
\left\lvert \sum_{n=1}^N a_n(x) \right\rvert \leq M... | {
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When ODE tells more than the explicit solution. "ODE is not just about 'solving' an equation and spitting out a (probably nasty) formula" -- this is what I want my (undergraduate) students to learn from my course this summer.
One example I am looking for is a scenario where one extracts info about a function from the O... | I would use mathematical models with biological application. Their ODE solutions are often very complex and many of them do not even have closed form solution; however, the formulation of the ODE itself is very intuitive. For instance, the logistic equation:
$$\frac{dx}{dt} = r x \left(1 - \frac{x}{K} \right)$$
Or a sp... | {
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"url": "https://math.stackexchange.com/questions/2760888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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$u^{-1}$ is integral over $R\subset S$ if and only if $u^{-1}\in R[u]$ I need to prove the following:
Let $R\subset S$ be a commutative ring and $u$ be any invertible element in $S$. Then $u^{-1}$ is integral over $R$ is and only if $u^{-1}\in R[u].$
Proof: Let $u^{-1}$ is integral over $R \implies \exists\ $ a monic... | Suppose $u^{-1} \in R[u]$. Then there exists some polynomial $f(x) \in R[x]$ such that $u^{-1}=f(u)$, or equivalently
$$uf(u)-1=0$$
Write $f(x)= \sum_{i=0}^n a_ix^i$. Then
$$0=\sum_{i=0}^n a_iu^{i+1}-1 = u^{n+1} \cdot (u^{-(n+1)}+a_0u^{-n} + a_1 u^{-(n-1)}+ \dots + a_n)$$
Since $u^{n+1}$ is a unit, you have
$$u^{-(n+1)... | {
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"timestamp": "2023-03-29T00:00:00",
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Famous convex maximization problems Which are the most famous problems having an objective of maximizing a nonlinear convex function (or minimizing a concave function)? As far as I know such an objective with respect to linear constraints is np-hard.
| THE most famous problem having an objective of maximizing a convex function (or minimizing a concave function), and having linear constraints, is Linear Programming, which is NOT np-hard.
Linear Programming is both a convex optimization problem (minimizing a convex function subject to convex constraints) and a concave ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$J_n(x)=0$ has no repeated root except zero. How to prove it? Suppose $J_n(x)=0$ has repeated roots.
Then it must have at least two equal roots, say $x=x_0(\neq 0)$
i.e., $x_0$ is a double root of $J_n(x)=0$.
Therefore $J_n(x_0)=0$ & $J_n’(x_0)=0$
Then what should I do?
| $J_n$ solves the Sturm–Liouville equation
$$ -(xy')' +\frac{n^2}{x} y = x^2y, $$
or more commonly
$$ x^2 y'' + xy' +(x^2-n^2)y = 0. $$
If $J_n(a)$ has a multiple zero at $a \neq 0$, $J_n'(a) = 0$. But then the differential equation implies that $J_n''(a)=0$, and repeated differentiation and substitution implies that $J... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Show that $u_{n+1}=6u_n-4u_{n-1}$ if $u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$ for $n=1,2,...$ I have already shown that for each $n,u_n$ is an integer.
Now we can show that $u_n=\dfrac{1}{2^n}(\sqrt{5}+1)^{2n}+\dfrac{2^{3n}}{(\sqrt{5}+1)^{2n}}$
But the problem is while showing $u_{n+1}=6u_n-4u_{n-1}$ , I can see that the wh... | Write $a=3+\sqrt5$, $b=3-\sqrt5$. Then $u_n=a^n+b^n$. Also
$$u_{n+1}-6u_n+4u_{n-1}=a^{n+1}-6a^n+4a^{n-1}+b^{n+1}-6b^n+4b^{n-1}
=(a^2-6a+4)a^{n-1}+(b^2-6b+4)b^{n-1}$$
etc. You need this to equal zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How do I prove that the series of $((1+\frac 1n)^n)a_n $ converges iff $\sum_n a_n$ converges? With sequences it simply arises from limit arithmetic, but with the series I cannot make it work. We have not learned integrals yet, so I can't use that. Any advice?
| If the $a_n$ are positive, the result follows from the limit comparison test.
If the assumption is dropped, summation by parts can be used. Let us prove that if $\sum_n a_n$ converges, then $\sum_n \left(1+\frac 1n \right)^na_n$ converges. Let $A_n=\sum_{k=0}^n a_k$, so that $$\sum_{n=0}^N \left(1+\frac 1n \right)^na_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Maximum Likelihood Estimation for Zero-inflated Poisson distribution I am trying to do exactly what the title says. What I have is the log-likelihood function as follows:
Likelihood function, where $I_i = 1$ when $X_i = 0$, and $I_i = 0$ otherwise. Then I took the partial derivatives of that like this. I tried simplify... | I think the inconsistency is caused by an error in the derivative of the profile likelihood in Ben's answer: The $\lambda e^{-\lambda}$ in the numerator should only be $ e^{-\lambda}$. In this case the final equation is
$$ \bar{x}(1-e^{-\hat{\lambda}})=\hat{\lambda}(1-r_0)(1-e^{\hat{\lambda }}+e^{\hat{\lambda }})=\hat{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\ i+i^2+i^3+i^4+\cdots+i^{100}$ $$i+i^2+i^3+i^4+\cdots+i^{100}$$
I figured out that every four terms add up to zero where $i^2=-1$, $i^3=-i$, $i^4=1$, so
$$i+i^2+i^3+i^4 = i-1-i+1 = 0$$
Thus, the whole series eventually adds up to zero. But how do I approach this problem in a more mathematical way?
| $$
\sum_{k=1}^n i^k = \frac{i^{n+1}-1}{i-1}-1 = i\frac{i^n-1}{i-1}
$$
but $i^{100} = 1$ hence $\sum_{k=1}^{100} i^k = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
solving differential equations : $ay' + by^2 + cy = u$ solving differential equations: $ay' + by^2 + cy = u$ with $a,b,c$ are positive constants and $u$ is arbitrary constant.
I can solve this equation:
$ay' + by^2 + cy = 0$
$\Rightarrow -a\frac{dy}{y(by+c)}=dx $
$\Rightarrow \frac{a}{c}(\frac{bdy}{by+c}-\frac{dy}{y}... | With a constant in the RHS, the equation is still separable and you solve it the same way !
$$\frac{a\,dy}{by^2+cy-u}=-dx.$$
After integration,
$$\frac{2a}{\sqrt{c^2+4 b u}}\arctan\frac{c + 2 b y}{\sqrt{c^2 + 4 b u}}=c-x$$ from which you can draw $y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Completing A proof of a question related to Radon-Nikodym
Let $\nu$ be a absolutely continous with respect to the measure $\mu$, where both of them are $\sigma$ finite and are on $( \omega , X)$. Prove that then. $\forall$ $\epsilon>0$ $\exists$ $\delta>0$ such that $\mu(A)<\delta$ $\implies$ $\nu(A)<\epsilon$, $\fora... | You certainly don't have to use Radon-Nikodym to prove this, as the other answer pointed out. But if you want to use it, then the result follows immediately from the fact that $f$ is $\mu$-integrable. See here (and note that the proof is essentially the same as the one given by harmonicuser).
Indeed, integrability impl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving function is strictly positive in a interval defined by coefficients
Let $f(x)=a_nx^n+a_{n-1}x^{n-1}\dots+a_1x+1$. Prove that $f(x)$ is strictly positive if $$0<|x|<{1\over1+\sum_{i=1}^{n}|a_i|}.$$
Any hints on how to start?
| Let $h(x)=f(x)-1=x\sum_{i=1}^{n}a_ix^{i-1}$ then for $|x| < \dfrac{1}{1 + \sum_{i=1}^n |a_i|}$we have that $|x|<1$ and
$$|h(x)|\leq |x|\sum_{i=1}^{n}|a_i|<1.$$
Therefore
$$f(x)=1+h(x)\geq 1-|h(x)|>0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2761988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that if $2x + 4y = 1$ where x and y are real numbers. Show that if $2x + 4y = 1$ where $$x, y \in \mathbb R $$, then $$x^2+y^2\ge \frac{1}{20}$$
I did this exercise using the Cauchy inequality, I do not know if I did it correctly, so I decided to publish it to see my mistakes. Thank you!
If $x=2$ and $y=4$, then $... | Your work is correct up to $$(2^2+4^2)(x^2+y^2)\ge (2x+4y)^2$$
From which you get $$ 20(x^2+y^2)\ge (1)^2$$
Therefore, $$x^2+y^2\ge \frac{1}{20}$$
At this point your are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2762114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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