Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Testing polynomial divisibility by evaluation at specific points This question is just something I got to wondering about. Assume
$$p(x), q(x) \in \Bbb Z[x], \deg (q) = m \lt \deg(p) =n.$$
Assume also $\forall a_k \in \{a_k~|~0 \leq k \leq n-m \} \subseteq \Bbb Z~ (\text{with } a_k \text{ distinct}), q(a_k)|p(a_k) \te... | No, this doesn't work. Consider $p(x) = x^4 + 1$ and $q(x) = x^2 + 1$. We have $p(a) = q(a)$ for $a=0,\pm 1$ but $p$ and $q$ have no roots in common over $\mathbb{C}$. In this example, it fails most obviously because the conditions on $k(x)$ determine it to have degree less than $2$.
But that's not the only thing that ... | {
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"url": "https://math.stackexchange.com/questions/3148405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Probability that lightbulb stops working in odd year I have a lightbulb that has an exponential lifetime distribution with mean $\mu$ months. So if I construct a pdf $f(x)$ and cdf $F(x)$ with parameter $\lambda$, and since $E[X] = \frac{1}{\lambda}$,
\begin{align}
f(x) &= \frac{1}{\mu}e^{-\frac{1}{\mu}x} \\
F(x) &= ... | There's only one parameter in this exponential model – $\lambda=\frac1\mu$. Thus
$$F_Y(y)=1-e^{-\lambda x}$$
$$P(12k<Y<12(k+1))=F_Y(12(k+1))-F_Y(12k)=1-e^{-12\lambda(k+1)}-1+e^{-12\lambda k}$$
$$=-e^{-12\lambda(k+1)}+e^{-12\lambda k}=e^{-12\lambda k}(1-e^{-12\lambda})$$
Then the probability the bulb fails in an odd yea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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checking the Solution of Bessel differential equation I want to check the first part of the solution and help in the second part.
Obtain the solution
$$y_{1}(x)=J_{0}(x)=1-\frac{x^{2}}{2^{2}}+\frac{x^{4}}{2^{2}\cdot 4^{2}}-\ldots+\frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}}+\ldots$$
of the differential equation $$x \... | Your series solution looks good.
As for finding the second solution... it would help a great deal if we actually used the definition of $u$.
\begin{align*}y_2(x) &= u(x)J_0(x)\\
y_2'(x) &= u(x)J_0'(x) + u'(x)J_0(x) = u(x)J_0'(x) + \frac{1}{xJ_0(x)}\\
y_2''(x) &= u(x)J_0''(x) + u'(x)J_0'(x) - \frac{J_0(x)+xJ_0'(x)}{x^2J... | {
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Finding a future co-linear midpoint of two moving objects I have three objects in a 2d space: S, E1, and E2.
E1 and E2 are at some location ((E1x, E1y) & (E2x, E2y)) and moving at constant velocities VE1 and VE2. They will be set initially and not change.
S starts at some location and needs to pick a direction. It has ... | Without loss of generality, assume $S$ starts at the origin. If $S$ has constant speed $s$ and chooses a constant heading $\vec h_S$ where $|\vec h_S|=1$ then its position at time $t$ is
$\vec S(t) = (st) \vec h_S$
Let's call the midpoint of $E_1$ and $E_2$ $F$, so at time $t$
$\vec F(t) = \frac 1 2 \left( \vec E_1(t) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3149063",
"timestamp": "2023-03-29T00:00:00",
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Using transition matrix and initial distribution to calculate probabilities. (Markov chains) Let a markov chain with state space $\{1,2,3,4\}$ and transition matrix
$$P=\begin{pmatrix}1/3 && 1/3 && 0 && 1/3 \\ 1/4 && 1/4 && 1/4 && 1/4 \\ 0 && 0 && 1/2 && 1/2 \\ 0 && 0 && 0 && 1\end{pmatrix}$$
and initial distribution $... | $P[X_0=2, X_2=2] = \sum_{i=1}^{4}P[X_2=2|(X_1=i,X_0=2) ].P[X_1=i,X_0=2] $
$= \left( \sum_{i=1}^{4}P[X_2=2|(X_1=i,X_0=2) ].P[X_1=i|X_0=2] \right) . P[X_0 =2] $
since $X_2$ only depends on the value of $X_1$
$= \left( \sum_{i=1}^{4}P[X_2=2|X_1=i].P[X_1=i|X_0=2]\right) . P[X_0 =2] $
so
$P[X_0=2, X_2=2, X_3 =1] = $
$P[X3=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3149167",
"timestamp": "2023-03-29T00:00:00",
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Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$
Let $H$ be a subgroup of a matrix group $G$.
Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$
Definition:
Let $\phi:G\rightarrow H$ be a smooth homomorphism of matrix groups.
If $\gamma '(0)$ is a tangent vector to $G... | It looks like your book is defining tangent vectors as derivaties of somoth curves. But if $\gamma$ is a curve landing in $H$ then a priori it lands in $G$, so the injectivity is automatic.
| {
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Probability minimum is "reached" Let $(X_i)_{i=1,\dots,n}$ be a finite sequence of random variables such that $X_i\sim\mathcal{E}(\lambda_i).$
We can prove that $Y:=\min_{1\le i\le n}X_i\sim\mathcal{E}(\lambda=\sum_{i=1}^n \lambda_i)$
Now I would like to compute $P(X_i=Y).$
We have
$$P(\min_{j\ne i}X_j>x)=P(\cap_{j\... | Manipulating the probabilities:
$$P(X_i = Y) = 1 - P(X_i \ne Y) = 1 - (P(X_i < Y) + P(X_i > Y)) = 1 - P(X_i > Y) =$$
$$1 - P(X_i > \text{min}_{j = 1, \ldots, n}X_j) = 1 - P(X_i > \text{min}_{j \ne i}X_j)$$
Now, $X_i$ and $Y_i = \text{min}_{j \ne i}X_j$ are two independent exponential random variables with parameters $\... | {
"language": "en",
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Is it generally true that $\langle a,b \rangle \cong \langle c,d \rangle\Rightarrow \text{ either }|a|=|c|,|b|=|d| \text{ or } |a|=|d|,|b|=|c|$? Background: We are given two groups $G,H$ generated by two elements, say $G=\langle a,b\rangle$ and $H=\langle c,d\rangle$. Further suppose that the orders of $a,b,c,d$ are fi... | No, take $G=H=\mathbb{Z}_4$ which has two sets of generators $\langle 1,2\rangle$ and $\langle 1, 3\rangle$. They satisfy the assumption but the order equalities don't follow.
For non-cyclic case note that if $\langle a, b\rangle$ generates a group then so does $\langle a, ab\rangle$. Now for any integers $n,m,r>1$ the... | {
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"url": "https://math.stackexchange.com/questions/3149590",
"timestamp": "2023-03-29T00:00:00",
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Show that there exists $i\in \lbrace 1, 2, 3 \rbrace $ s.t. there exists $a, b\in A_i $ s.t. $a+b\in B $. Let $A=\lbrace 1, 2, 3,..., 2019\rbrace= A_1\cup A_2\cup A_3$, where $A_1\cap A_2=A_2\cap A_3= A_1\cap A_3=\emptyset $ and $B=\lbrace 672, 1008, 1344, 1680, 2016\rbrace $.
Show that there exists $i\in \lbrace 1, 2... | Letting $b=168$, we have $B=\{4b,6b,8b,10b,12b\}$, while $2019=12b+3$. Thus,
$\{b,2b,3b,4b,5b,6b,7b,8b,9b,10b,11b,12b\}\subseteq A$, and restricting to these multiples of
$b$ only, it suffices to show that for any partition
$\{1,2,3,4,5,6,7,8,9,10,11,12\}=A_1\cup A_2\cup A_3$, there are $i\in\{1,2,3\}$ and $a,b\in A_i,... | {
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"timestamp": "2023-03-29T00:00:00",
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Trying to Solve A Differential Equation Using Variation of Parameters? A few of my classmates and I are trying to solve the following differential equation: $x'' - x'\sin(t) - x\cos(t) = 0$ using the variation of parameters. However, we're at a bit of a loss because we struggled to find a fundamental solution, and at t... | Note that
$$0=x''(t) - x'(t)\sin(t) - x(t)\cos(t)=D(x'(t)-x(t)\sin(t))$$
and it follows that
$$x'(t)-x(t)\sin(t)=c$$ which is a linear ODE of the first order.
Can you take it from here?
| {
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Conditional Expectation of Two Random Variable I'm new to conditional expectations and having trouble with the following problem:
*
*For $\Omega = \{ a, b, c, d \}$, with $P(a) = 4P(b) = 2P(c) = 3P(d)$.
Define the random variables:
$X(ω) = \begin{cases}~~~5 &:&\omega \in \{ a, b \}\\ −2&:& ω \in \{ c, d \}\end{c... | Firstly, you have $\mathsf P(a)=4\mathsf P(b)=2\mathsf P(c)=3\mathsf P(d)$, so you don't need to evaluate them since there will be cancelation.
Secondly, $\mathsf E(X\mid Y)$ is a random variable, measured over $Y$, so don't forget to leave in the indicator functions.. $$\begin{align}\mathsf E(X\mid Y)&=\begin{cases}\m... | {
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Verification of Alternate Proof for Identity Theorem in Conplex Analysis. Can't we prove Identity Theorem like this.
IDENTITY THEOREM.
Consider a function which is analytic on an open connected domain $D$, if the set of zeros of $f$ has a limit point in $D$ then $f=0$ on $D$.
My Proof Idea.
If it is not $0$ identic... | As far as I can tell, the error in your proof is in the last line, when you use the fact that a non-constant function has isolated zeros. This is definitely true if your domain is $\mathbb{C}, $ which is probably the version you are thinking of, but is false in general. For example, let $A$ and $B$ be disjoint and defi... | {
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Using Taylor series expansion to solve the equation $\frac{\tanh^{-1}(x)}{\beta} -2x =0$ I want to use the Taylor series epansion of $\tanh$ to get an approximate solution $\tilde{x}(\beta)$ for the equation
$$
\frac{\tanh^{-1}(x)}{\beta} -2x =0
$$
for $\beta >\frac{1}{2}$ such that I can calculate the limit
$$
\lim_{... | Write the problem first as $$\beta=\frac{\tanh ^{-1}(x)}{2 x}$$
Now, expand the rhs using the usual
$$\tanh ^{-1}(x)=\sum_{n=1}^\infty \frac {x^{2n+1}} {2n+1}$$ Using a few terms, we then have
$$\beta=\frac{1}{2}+\frac{x^2}{6}+\frac{x^4}{10}+O\left(x^{6}\right)$$ Now, using series reversion
$$\tilde{x} (\beta)=\sqrt{6}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$ f(\frac{x^2+1}{x^2})=x^2+\frac{1}{x^2}-2 \Rightarrow f(x)=?$
$ f(\frac{x^2+1}{x^2})=x^2+\frac{1}{x^2}-2 \Rightarrow f(x)=?$
I'm getting a wrong answer even though my solution looks valid to me:
Inverse of $(1+\frac{1}{x^2})$ is $(\frac{1}{{\sqrt{x-1}}})$, so I plug this expression in wherever I see $x$'s
$$f(x)=(\... | I think your answer is correct. I solved the problem like this:
$$f(\frac{x^2+1}{x^2})=f(1+\frac{1}{x^2})=x^2+\frac{1}{x^2}-2$$
Now $f(1+\frac{1}{x^2})$ can be written as a function of just $\frac{1}{x^2}$ so,
$$f(1+\frac{1}{x^2})=g(\frac{1}{x^2})=x^2+\frac{1}{x^2}-2$$
Put $t=\frac{1}{x^2}$, to get:
$$f(1+t)=g(t)$$
$$g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3150399",
"timestamp": "2023-03-29T00:00:00",
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What is the geometric meaning of this null-determinant? While reading about interpolation I came across the following equation in
Norlund.
It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton follow by using the Laplace expansion.
$$
P_n=-\det
\begin{bmatrix}
1 & x_0 & \dot... | In the hypotesis, you have that
$$\det
\begin{bmatrix}
1 & x_0 & \dots & x_0^n\\
1 & x_1 & \dots & x_1^n\\
\vdots & \vdots & \ddots & \vdots\\
1 & x_n & \dots & x_n^n\\
\end{bmatrix} \neq 0$$
It means that
$$\ \text{rank} \begin{bmatrix}
1 & x_0 & \dots & x_0^n\\
1 & x_1 & \dots & x_1^n\\
\vdots & \vdots & \ddots & ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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exponential null operator In $A$ banach algebra with unit, and $X\in A$. if i define
$e^X=\sum_{n=0}^{\infty} \frac{1}{n!}X^n$
why $e^0=Id$ , i am aassuming $O^0=Id $ with $0$ null operator
thanks
| Since$$e^X=\operatorname{Id}+X+\frac{X^2}{2!}+\frac{X^3}{3!}+\cdots,$$then$$e^0=\operatorname{Id}+0+\frac{0^2}{2!}+\frac{0^3}{3!}+\cdots=\operatorname{Id}.$$
| {
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Chordal Graph to Directed Acyclic Graph I have seen an exercise which says an undirected graph $G=(V,E)$ is chordal if and only if the edges of $G$ can be oriented with directions, such that the resulting graph $D=(V,A)$ has the following properties:
*
*$D$ is acyclic
*if $(x,y)$ and $(x,z)$ belong to $A$, then $(y... | If G is chordal, to construct D: take a perfect elimination ordering and label the vertices by their index in it. Now orient all the edges by increasing labels, that is, assuming $i < j$ : $\{v_i,v_j\} \Rightarrow (v_i,v_j)$.
The property is satisfied, and D is acyclic since any cycle would contain a decreasing edge, w... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding other eigenvector and matrix $A$ given eigenvalues I want to find a symmetric matrix $A$, whose eigenvalues are $4$ and $-1$. One of the eigenvectors corresponding to the eigenvalue $4$ is $(2,3)$. I want to find an eigenvector corresponding to the eigenvalue $-1$ and then find the matrix $A$.
| Recall that the eigenspaces of a symmetric matrix are mutually orthogonal. Thus, all eigenvectors with eigenvalue $-1$ are orthogonal to all eigenvectors with eigenvalue $4$. Can you come up with a nonzero vector that’s orthogonal to $(2,3)$? Once you’ve done that, you have a basis of $\mathbb R^2$ that consists of eig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to simplify $\sqrt{2+\sqrt{3}}$ $?$
Simplify $\dfrac{2\left(\sqrt2 + \sqrt6\right)}{3\sqrt{2+\sqrt3}}$
The answer to this question is $\frac{4}{3}$ in a workbook.
How would I simplify $\sqrt{2+\sqrt3}$ $?$ If it was something like $\sqrt{3 + 2\sqrt2}$ , I would have simplified it as follows:
$\sqrt{3 + 2\sqrt2... | Note that$$\left(\frac{2\left(\sqrt2+\sqrt6\right)}{3\sqrt{2+\sqrt3}}\right)^2=\frac{4\left(8+4\sqrt3\right)}{9\left(2+\sqrt3\right)}=\frac{16}9=\left(\frac43\right)^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Optimal Value of a Cost Function as a Function of the Constraining variable Consider the optimization problem :
$
\textrm{min } f(\mathbf{x})
$
$
\textrm{subject to } \sum_i b_ix_i \leq a
$
Using duality and numerical methods (with subgradient method) i.e.
$d = \textrm{max}_\lambda \{ \textrm{inf}_x ( f(\mathb... | It could perhaps be useful to know that you are essentially asking about properties of the value function in parametric programming, or multi-parametric programming to be completely general.
https://en.wikipedia.org/wiki/Parametric_programming
And regarding convexity, the answer is yes, see e.g.,
Convexity and Concavi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151310",
"timestamp": "2023-03-29T00:00:00",
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Differential Equations - Exact ODE's
Hi all, I've attempted the question part b) after rearranging the 1st expression to :
$$v(x,y)\, dx - u(x,y) \, dy = 0.$$
After this I tried using the method of differentiating each expression and trying to calculate if they were exact. It was very lengthy so I am unable to post on... | The equation can be rewritten as $$\frac{1}{(x^2+y^2)^2}\left(2xdx+2ydy\right)y-\frac{2y^2}{(x^2+y^2)^2}dy+\frac{y^2-x^2}{(x^2+y^2)^2}dy+dy=0$$Thus$$y\frac{1}{(x^2+y^2)^2}d(x^2+y^2)+\left(-\frac{1}{x^2+y^2}\right)d(y)+dy=0$$which can be put up as $$y\left\{d\left(-\frac{1}{x^2+y^2}\right)\right\}+\left(-\frac{1}{x^2+y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Union of two countable set is also countable This question is asked many time when I search it. But i didn't find what am I really looking for.
My approach:
Let $A_1,A_2$ be two countable sets. If they have some common elements then redefine it by $B_2=A_1\setminus A_2=\{x\in A_2:x\notin A_1 \}$. The point of this is ... | Sets are countable if you can enumerate the elements ($=$ assign an index to each), without omission.
Let $C:=A\cup B$. From the enumerations
$$a_1,a_2,a_3,\cdots$$ and $$b_1,b_2,b_3,\cdots,$$
we form the enumeration
$$a_1,b_1,a_2,b_2,a_3,b_3,\cdots$$
also written
$$c_1,c_2,c_3,c_4,c_5,c_6,\cdots$$
In other terms, the ... | {
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"url": "https://math.stackexchange.com/questions/3151699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}=K(a-b)(b-c)(c-a)$, solve for $K$ Qestion: $\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}=K(a-b)(b-c)(c-a)$, solve for $K$
Answer: $K=(ab+bc+ca)$
My attempt: $$\begin{align}\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}&=\begin{vmatr... | Consider the matrix
\begin{bmatrix} 1& 1 &1 &1 \\ X & a&b&c \\X^2 & a^2 & b^2 & c^2 \\ X^3 & a^3 & b^3 & c^3 \end{bmatrix}
Its determinant is a Vandermonde determinant, equal to
$$(a-X)(b-X)(c-X)(b-a)(c-a)(c-b)$$
But developping with respect to the first column, you see that the determinant you are looking for is just... | {
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For any natural $m \neq n$ show that $|\sqrt[n]{m} - \sqrt[m]{n}| > \frac{1}{mn}$. Here's my try.
The inequality above is equivalent to $$|m^{\frac{1}{n}} - n^{\frac{1}{m}}|> \frac{1}{mn}$$ First, I want to get rid of the absolute value.
Assume without loss of generality that $m>n$. Then $m^m > n^n$. Raising this inequ... | Suppose that $m>n$, as a result of which $m^m>n^n$. If $m=2$, then $n=1$ and the proof is easy to complete; suppose thus that $m\ge 3$.
Applying Lagrange's mean value theorem to the function $f(x)=x^{1/(mn)}$ on the interval $[n^n,m^m]$, we get
$$ \sqrt[n]m-\sqrt[m]n = f(m^m)-f(n^n) = \frac1{mn}\,c^{-1+1/(mn)}(m^m-n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3151907",
"timestamp": "2023-03-29T00:00:00",
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Computing last two digits of $27^{2018}$ For abstract algebra I have to find the last two digits of $27^{2018}$, without the use of a calculator, and as a hint it says you should work in $\mathbb{Z}/100\mathbb{Z}$.
I thought breaking up the problem into $\mod(100)$ arguments. Thus:
$27^{2}=729\equiv 29 \mod (100)$, an... | Because $\varphi(100)=40$ certainly the sequence of powers of $27$ will repeat after $40$ terms. A quick calculation shows that in fact it already repeats after $20$ terms. You could also use the Chinese remainder theorem to reduce the problem to computing $27^{2018}$ mod $25$ and mod $4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3152033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Reducible and Irreducible polynomials over $\Bbb Q$ Consider polynomials of the form: $$x^r-(1-x)^k,$$
for $r,k\ge2.$
$x\in(0,1).$
When $r=k$ the polynomial seems to be reducible, except at $r=k=2.$
Do the irreducible and reducible polynomials form a pattern when plotted?
To try to visualise what was going on, I mad... |
Do the irreducible and reducible polynomials form a pattern when plotted?
Depends what you consider a pattern, but here are few quick observations. Let $f_{r,k}(x)=x^r-(1-x)^k$, then:
*
*The graph is vertically symmetrical. This follows simply from fact that if $f(x)$ is irreducible, then $f(-x)$ is irreducible, a... | {
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Automata|The mid 1/3 of regular language is still regular Define $$L_{\frac{1}{3}}=\{w \in \Sigma^*\ |\ \exists x,y\in \Sigma^*,\ xwy\in L,\ |x|=|w|=|y|\}$$L is a regular language, is $L_{\frac{1}{3}}$ a regular language?
I think it might be similar to the question of half of L. Automata | Prove that if $L$ is regular ... | A powerful method to prove this result (and many similar ones) is to use the fact that a language is regular if and only if it is recognized by a finite monoid. A language $L$ of $A^*$ is recognized by a finite monoid $M$ if there is a surjective monoid morphism
$f:A^* \to M$ and a subset $P$ of $M$ such that $f^{-1}(P... | {
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Weakly convergence If $g \in L^p(\mathbb{R})$ be a given non-trivial function, show that following sequences converge weakly in $L^p$ but not strongly in $L^p$.
(a) $g_k(x)=k^{1/p}g(kx)$.
(b) $h_k(x)=g(x+k)$.
I need to show that for every $f \in L^q$ where q is the conjugate exponent to $p$, we have
$$\int k^{1/p}g(kx... | You need $1<p<\infty$. The arguments for a) and b) are similar. Here are some hints for a): Claim: the weak limit is $0$. Start with $\int k^{1/p} g(kx)f(x)dx=\int k^{1/p-1} g(y)f(\frac y k)dy$. Split the integral into integral over $|y| \leq M$ and $|y| >M$. Use Holder's inequality for the second part. Observe that th... | {
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"source": "stackexchange",
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is this convex inequality possibly true? Let $a_1,\dots,a_k$ be non-negative and sum to $1$ and let $x_1,y_1,\dots,x_k,y_k$ be positive. Then is it true that $$\prod_{i=1}^kx_i^{\alpha_i}+\prod_{i=1}^ky_i^{\alpha_i}\leq\sum_{i=1}^k(x_i+y_i)^{\alpha_i}?$$
This is example $1.2.3$ in Convex Analysis and Minimization Algor... | Consider what happens when all the $x_i$'s and $y_i$'s equal some value $x$ and the $\alpha_i$'s equal $1/k$. Then the claim is that
$$
2x=2\prod_{i=1}^kx^{1/k}\leqslant \sum_{i=1}^k(2x)^{1/k}=k(2x)^{1/k}.
$$
But if $x>(2^{1/k - 1} k)^{1/(1 - 1/k)}$ this is false.
| {
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Definitionally prove that $\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}$ $$\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}\quad (f'(0) = 0)$$
It seems quite a rudimentary problem, but I can't find an appropriate solution without using L'hospital's rule and Maclaurin series.
Is it possible that a probl... | Proof using MVT: let $g(x)=f(x)-\frac 1 2 x^{2}f''(0)$. Then $g''(0)=0$. If we prove the result for $g$ then result for $f$ follows immediately. Now $\frac {g(x)-g(0)} {x^{2}}=\frac {g'(\xi_x)} {x} $for some $\xi_x$ between $0$ and $x$. But $\frac {g'(\xi_x)} {x} =\frac {g'(\xi_x)} {\xi_x} \frac {\xi_x} x \to 0$ becaus... | {
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The Proximity Operator of a Function with Multiple Affine Mapping Let $f(\mathbf{x}) = g(\mathbf{A}\mathbf{x})$, where $\mathbf{A} \in \mathbb{R}^{M \times N}$ is a linear transformation satisfying $\mathbf{A}\mathbf{A}^T = \mathbf{I}$. Then for any $\mathbf{x} \in \mathbb{R}^{N}$,
\begin{equation}
\text{prox}_f (\math... | Even in the simpler case where $P=2$ and $A_1=A_2= \textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
| {
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What is the tangent at a sharp point on a curve? How to know which line represents tangent to a curve $y=f(x)$ (in RED) ?From the diagram , I cannot decide which line to take as tangent , all seem to touch at a single point.
| Since we're doing geometry, let's think kinematically, too. Imagine a point moving along your curve, in the direction of increasing $x.$ Furthermore, let a ray project from the point, tangent to the curve there (you might think of your curve as a lane, and the point a car in the night; imagine the headlights beaming st... | {
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Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, ..., 0, \operatorname{Tr(}A))$ respectively. Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar t... | Hint Prove the claim by induction on the size of the matrix, with inductive hypothesis that the claim holds for $n \times n$ matrices.
For an $(n + 1) \times (n + 1)$ matrix $A$, decompose
$$A = \pmatrix{\lambda&\ast\\\ast&B} ,$$ where $B$ has dimension $n \times n$. By the inductive hypothesis there is a matrix $Q$ su... | {
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How to form Matrix pairs of $m$ friends and to find common friend from all possible pairs. The below is a problem given in entrance exam.
Problem:
A golf club has $m$ members with serial numbers $1, 2 . . . , m$. If members with serial numbers $i$ and $j$ are friends, then $A(i, j) = A(j, i) = 1$, otherwise $A(i, j) = ... | Not a complete solution, but some thoughts.
The well-known fact (you can easily prove it by induction) is that $k$th power of adjacency matrix $A$ is a matrix of pathes that have length $k$. Thus, $A^{k}(i, j)$ equals to number of pathes from $i$ to $j$ that have length $k$. In this terms, $A^9(i, j) > 0$ means ther... | {
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Number of ways to distribute 20 identical pencils to 6 non-identical children with restrictions
How many ways are there to pass out 20 pencils (assume all the pencils are identical the same) to six children?
Based on the following condition:
a) No restriction. ( i.e. each kid may receive zero to 20 pencils.)
b) Every... | I realized that there is no way to generalize the answer for question c). it needs a special solution.
First of all, out of 6 children, 1 child has to get 0 pencils.
After that 5 children are remaining. So, there are just 7 ways to distribute 20 pencils among them, which are:
(1,2,3,4,10)
(1,2,3,5,9)
(1,2,3,6,8)
(1,2,... | {
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Find the maximum integer value $n$ such that $2^n\mid 3^{1024} -1$ Since $2^{10} = 1024$ then we can $3^{1024}-1 = ( 3 - 1 )\Pi_{i=0}^{9} {3^{2^i}+1}$
And then we can start eliminating the $3-1=2$ and $(3^2-1)= 2\times 5$
But then? I guess I could calculate, but this is not different as starting calculating in first pl... | Hint
For $i \geq 1$ you have:
$$3^{2^i}+1 \equiv (-1)^{2^i}+1 \equiv 1+1 \equiv 2 \pmod{4}$$
| {
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Prove that there exist $135$ consecutive positive integers so that the $n$th least is divisible by a perfect $n$th power greater than $1$ Prove that there exist 135 consecutive positive integers so that the second least is divisible by a perfect square $> 1$, the third least is divisible by a perfect cube... | Use the Chinese Remainder Theorem. Pick $134$ distinct primes. The perfect square is the square of the first, the cube is the cube of the second, and so on. All your moduli are distinct, so CRT guarantees a solution. If you use the smallest primes in order and $N$ is the least of your $135$ numbers, you have $N+1 \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find the angle in a protein which is inside of a triangle which appears inscribed in a circle? I'm confused at which property or identity can be used to find the angle in a triangle when it looks inscribed in a circle but one of its sides doesn't appear to pass through the center. I'm also confused about the oth... | Well, as you know, $\omega=36^\circ$. Now, the bottom two $\omega$’s and $\phi$ cut out the same arc of the circle, namely the arc $AED$, so that $\phi=2\omega=72^\circ$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3153777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Using spherical coordinates for triple integral $ \int_{0}^3 \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} \frac{\sqrt{x^2+y^2+z^2}}{1+x^2+y^2+z^2} \ dz \ dy \ dx$
Using spherical co-ord's this becomes :
$ \int_{0}^{2\pi} \int_{0}^\pi \int_{0}^3 \frac{r^3\sin(\theta)}{1+r^2} \ dr \ d\theta \ d\phi $
Is this cor... | It looks fine to me, except for the limits of integration. The next step is to write it as$$\frac\pi2\left(\int_0^{\frac\pi2}\sin(\theta)\,\mathrm d\theta\right)\left(\int_0^3\frac{r^3}{1+r^2}\,\mathrm dr\right).$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sheafification of a given presheaf Let $\mathcal{F}$ be a presheaf on $\mathbb{R}$ such that $\mathcal{F}(U)$ is the abelian group of continuous functions with bounded support on $U$. Then what is the sheafification of $\mathcal{F}$?
I guess the sheafification should be the abelian group of continuous functions, but ho... | Let $F$ be our presheaf of continuous functions with bounded support, $G$ be the sheaf of continuous functions, $F^+$ the sheafification of $F$.
Every continuous function with bounded support is a continuous function, so we have an injective morphism $F\to G$, which induces a map $F^+\to G$ by definition of the sheafif... | {
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Definition linear ODE An ordinary differential equation is said to be linear if
$$F(t,y(t),...,y^{(n)}(t))=0$$
is linear in every derivative. I run into a little problem when using this definition for the equation
$$y'y=0$$
because we have that $$F(t,\alpha y_1(t)+\beta y_2(t),y'(t))=(\alpha y_1(t)+\beta y_2(t))y'(t)=... | We have $F(t,y(t),y’(t))=yy’$. Note that $$F(\alpha t, \alpha y(t), \alpha y’(t)) =\alpha^2yy’ \neq \alpha yy’=\alpha F(t,y(t),y’(t))$$
and thus $F$ is not linear in its arguments, implying that it is not a linear differential equation using your first definition.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A sum of a product of binomial coefficients I am trying to simplify the following summation of products of binomial coefficients:
$$\sum_{k=0}^n \binom{k+b}{a} \binom{y+n-k}{y}$$
where $b > a$ (specifically, $b = 2a+1$).
I have searched through some of the usual resources (Gradshteyn and Ryzhik, HW Gould, the DLMF, Abr... | You can show that
$$
\sum_{k=a-b}^n \binom{k+b}{a} \binom{y+n-k}{y}
%=\sum_{h=0}^{n+b-a}\binom{h+a}a\binom{y+n+b-a-h}{y}
=\binom{b+n+y+1}{a+y+1}.\tag{*}
$$
That is, your summation is a nice one minus some missing terms. Therefore, your summation is
$$
\boxed{\binom{b+n+y+1}{a+y+1}-\sum_{k=a-b}^{-1}\binom{k+b}a\binom{y+... | {
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Solve tan(x)+cos(x)=1/2 Is it possible (not numerically) to find the $x$ such as:
$$
tan(x)+cos(x)=1/2
$$
?
All my tries finishes in a 4 degree polynomial. By example, calling c = cos(x):
$$
\frac{\sqrt{1-c^2}}{c}+c=\frac{1}{2}
$$
$$
\sqrt{1-c^2}+c^2=\frac{1}{2}c
$$
$$
1-c^2=c^2(\frac{1}{2}-c)^2=c^2(\frac{1}{4}-c+c^2)
... | If we set $X=\cos x$ and $Y=\sin x$, the equation becomes
$$
Y=\frac{1}{2}X-X^2
$$
so the problem becomes intersecting the parabola with the circle $X^2+Y^2=1$.
This is generally a degree four problem. The image suggests there is no really elementary way to find the intersections.
The equation becomes
$$
X^4-X^3+\frac... | {
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Show if $n \sqrt n \in O(n^4)$ I'm quite confused as to how to solve this step by step. What I have done was graph this using Desmos graphing software to visually see it, and I can see that the $n \sqrt n $ is below $n^4$.
I thought I could do something like this:
$n \sqrt n \lt n^4$
$ \sqrt n \lt n^3$
$n^{1/2} \lt n^4... | ${{n\sqrt n}\over n^4}={1\over n^{5/2}}$ and $\lim_\limits{n\rightarrow +\infty}{1\over n^{5/2}}=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection between them.
| Hint: first, show that $\mathbb{R} \simeq (0,1)$. Then, observe that if $(0,1) \simeq (0,1)^2$, then $\mathbb{R} \simeq \mathbb{R}^2$. For the former, one can consider a map such as $0.a_1 a_2 a_3 a_4 \dots \mapsto (0.a_1 a_3 \dots,0.a_2a_4\dots) $.
| {
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Finding 01 or 00 A and B play a bit game. Unbiased bit generator is generating 0 or 1 repeatedly until one of the following happens.
*
*The bit patterns to '00' (i.e., a 0 is immediately followed by a 0) for the first time. In this case A wins.
*The bit patterns to '01' (i.e., a 0 is immediately followed by a 1) f... | The first $0$ is followed by a $0$ or a $1$ equiprobably and the game is over.
| {
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Prove without induction that $2×7^n+3×5^n-5$ is divisible by $24$. I proved this by induction. But I want to show it using modular arithmetic. I tried for sometime as follows
$$2×7^n-2+3×5^n-3\\
2(7^n-1)+3(5^n-1)\\
2×6a+3×4b\\
12(a+b)$$
In this way I just proved that it is divisible by 12 but it is not enough. Am I mis... | Note that you have
$$
7^n - 1 = 6a\\
5^n - 1 = 4b
$$
Now we're interested in whether $a$ and $b$ are even or odd. Which is to say we want to know when $7^n - 1$ is divisible by $4$ (so that when you divide it by $6$ you get an even number), and when $5^n-1$ is divisible by $8$ (so that when you divide it by $4$, you ge... | {
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When is $YE(Y\mid\mathcal{G}) = E(Y^2\mid\mathcal{G})$? I know this is true when $Y \in \mathcal{G}$, but is there a better (less restrictive) condition that can allow this?
In particular, I want to know when I can say that $E(YE(Y\mid\mathcal{G})) = E(E(Y^2\mid\mathcal{G}))$, which is, of course, equal to $EY^2$.
| No.
If I understand correctly your notation, then $Y\in\mathcal{G}$ means that $Y$ is measurable with respect to $\mathcal{G}$. In particular you have $E(Y|\mathcal{G}),E(Y^2|\mathcal{G})\in\mathcal{G}$.
But then, from the equation $YE(Y|\mathcal{G}) = E(Y^2|\mathcal{G})$ you necessarily have that $Y\in\mathcal{G}$.
| {
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Why don't we have to use polynomial long division on quadratic irreducible partial fractions in this case? For $(x^2+3x+1)
/(x^2+1)^2 $
Why don't we have to use polynomial long division in this case since the degree of the denominator has the same magnitude of degree as the numerator before doing partial fraction decom... | We don't have to, we can go backwards, make ansatz, multiply by least common multiple for denominator and solve linear equation system that arises.
$$\frac{a}{x^2+1} + \frac{bx+c}{(x^2+1)^2}$$
Now multiply both numerator and denominator of first term by $x^2+1$
$$\frac{a(x^2+1) + bx+c}{(x^2+1)^2}$$
Now set up equations... | {
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"url": "https://math.stackexchange.com/questions/3155560",
"timestamp": "2023-03-29T00:00:00",
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About the fact that $\frac{a^2}{b} + \frac{b^2}{a} + 7(a + b) \ge 8\sqrt{2(a^2 + b^2)}$ There's a math competition I participated yesterday (19/3/2019).
In these kinds of competitions, there will always be at least one problem about inequalities.
Now this year's problem about inequality is very easy. I am more interest... | Let $a^2+b^2=2k^2ab,$ where $k>0$.
Thus, by AM-GM $k\geq1$ and we need to prove that:
$$\frac{\sqrt{(a+b)^2}(a^2-ab+b^2)}{ab}+7\sqrt{(a+b)^2}\geq8\sqrt{2(a^2+b^2)}$$ or
$$\sqrt{2k^2+2}(2k^2-1)+7\sqrt{2k^2+2}\geq16k$$ or
$$\sqrt{2k^2+2}(k^2+3)\geq8k,$$ which is true by AM-GM:
$$\sqrt{2k^2+2}(k^2+3)\geq\sqrt{4k}\cdot4\sq... | {
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Dimensions of paper needed to roll a cone (Updated with clarifications) I'm looking for a way to calculate the dimensions of a piece of paper needed to roll up into a cone shape. Please consider the following diagram I created (nothing is drawn to scale):
In this example, I'm trying to create a cone shape, that is 100... | If we slightly simplify the situation, we can say that at the top (and at the bottom) the paper is arranged in concentric circles. If $t=0.1$mm is the thickness of the paper, we get a radius $r_n = n \cdot t$ for the $n$th circle. The circumference of the $n$th circle is $2 \pi r_n$. The total length of the paper (for ... | {
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People arranged in a queue, probability of $A$ always being in front of $B$. There are $n\geq 2$ people including $A$ and $B$. They form a line uniformly at random and we want to find the probability that $A$ is in front of $B$.
So far I've determined my sample space to be $|\Omega|=n!$.
I also let $A$'s position in th... |
Intuitively I know that the number of ways that A is in front of B is the same as that if B is in front of A and the probability I should be getting is 1/2 but I'm not sure how to get there from the information I have so far.
Your intuition is the way to go. The other people are a distraction. All you care about ... | {
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"answer_id": 0
} |
Confused about how to show independent random variable $Y$ has the Poisson distribution with parameter $t\lambda$ Assume there are $N$ independent exponential random variable $(X_1, X_2,\ldots, X_N)$ with parameter $\lambda$. Fix a real number $t > 0$. Let $Y$ be the largest $N$ so that $X_1 + X_2 + \ldots + X_N \leqsl... | Let $S_k:=\sum_{i=1}^k X_i$ (note that $S_k\sim \Gamma(k,\lambda^{-1})$). Then
\begin{align}
\mathsf{P}(Y=k)&=\mathsf{P}(S_k\le t, S_k+X_{k+1}>t)=\mathsf{E}\!\left[1\{S_k\le t\} \mathsf{P}(X_{k+1}>t-S_k\mid S_k)\right] \\
&=\mathsf{E}\!\left[1\{S_k\le t\}e^{-\lambda(t-S_k)}\right]=\int_0^t\frac{\lambda^k}{(k-1)!}x^{k-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Given: $\int_{a}^{b}f=\int_{a}^{b}g$, Can we always find some point $\theta$ in $[a,b]$ such that $f(\theta)=g(\theta)$ I noted something today, I don't know whether the result holds in general setting or not.
Suppose $f$ and $g$ are continuous function on $[a,b]$ such that
$\int_{a}^{b}f=\int_{a}^{b}g$.
Can we alw... | An equivalent formulation (via $h=f-g$) is
Suppose $h$ is a continuous function on $[a,b]$ such that
$\int_{a}^{b}h(x) \, dx=0$.
Can we always find some point $\theta$ in $[a,b]$ such that
$h(\theta)=0$?
and that is true. If $h$ has no zeros then the intermediate value theorem implies that $h$ is strictly posi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Cartier divisor example in Harthsorne This question is about example 6.11.4 in Chapter II of Hartshorne. The example is about computing the Cartier divisor class group of the cuspidal cubic curve $y^2z = x^3$ in $\mathbb{P}_{k}^{2}$. He begins by saying "note that any Cartier divisor is linearly equivalent to one whose... | I agree he did not phrase it very well. Let $U$ be an open chart containing the cusp. By removing the cuspidal point from the other charts, we may assume it is the only one containing the cuspidal point. Let $f$ be the rational function on $U$. What he means, is that we may assume that $f$ is invertible in $\mathcal{O}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Easier way to solve a LES (only pen/paper, no calculator) I want to solve the following Linear Equation System with only pen and paper;
$$
470 = x_A - \frac{3}{10}x_B \tag{1}
$$
$$
940 = x_B - \frac{2}{10}x_A \tag{2}
$$
I attempt to solve for $x_A$ by inserting equation (2) into (1) and rewriting it, etc. I got somethi... | First notice $94=47\cdot 2$. You can simplify before adding/multiplying:
$$x_A\frac{94}{100}=470+\frac{3}{10}940\color{red}{=470+\frac{2820}{10}} \Rightarrow \\
x_A\frac{47\cdot 2}{100}=47\cdot 10+\frac3{10}\cdot 47\cdot 20 \Rightarrow \\
x_A\frac2{100}=10+\frac{60}{10} \Rightarrow \\
2x_A=1000+600 \Rightarrow \\
x_A=8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluate limit without using L'H rule I was revising teacher's notes about L'H rule and I came across this limit.
$$\lim_{x \to \frac{\pi}{2}} \frac{x-\frac{\pi}{2}}{\sqrt{1-\sin x}}$$
The teacher tried to evaluate without L'H to demonstrate L'H is necessary here but I can't explain the first passage he did:
$$\lim_{... | Hint
$\dfrac\pi2-x=2y$
$$\lim_{y\to0}\dfrac{-2y}{\sqrt{1-\cos2y}}=-\sqrt2\lim_{...}\dfrac y{|\sin y|}$$
So, the limit doesn't exist
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find general term of $1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$
Find general term of $1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$
However it has been ask to check convergence but how can i do that before knowing the general term. I can't see any pattern,comment quickly!
| Hint
The numerator is easy.
For the denominator, see the succesive differences. If you still can't figure out see Arithmetico–geometric sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3156962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How can vector space $V\subsetneq W$ but $V\cong W$, where $V$ and $W$ don't have finite dimensions? Let $V$ and $W$ two vector spaces. If they have finite dimension, then $V\subset W$ and $V$ and $W$ have same dimension will imply that $V=W$. But I heard that in infinite dimensions, this doesn't hold anymore in the se... | A standard example is with the space of real polynomials $\mathbb{R}[x]$. If you define a map $T:\mathbb{R}[x]\to \mathbb{R}[x]$ by $p(x)\to p(x^2)$ then it is an isomorphism between $\mathbb{R}[x]$ and the image of $T$. However $T(\mathbb{R}[x])$ is a proper subspace.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
the following claim true or false: $\bigtriangledown f(x)$ always points in the same direction as $x-\hat{x}$ if $f(\hat{x})=0$ Take $f: \mathbb{R}^{n}\rightarrow \mathbb{R}, f\geq0 \ \forall x \in \mathbb{R}^{n}$.
A statement in an exercise claims the following:
given an $x' \in \mathbb{R}^{n}$, let $\hat{x}$ be the p... | I suspect that the claim might be a typo, and that it's talking about $\nabla (x')$ instead of $\nabla f (\hat{x})$. And "point in the same direction" is clearly false if taken literally, but suppose instead it meant "lie in the same half-space" (i.e., have positive dot product).
Even so, you'd need more conditions on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Computing a Basis of Holomorphic Differential Forms for a Given Curve I find myself repeatedly having to ask this question, because no one seems to have answered it. I put this up on Math Overflow with a bounty, got a guy who said he would answer it, but—though he helped with other things—never actually got around to a... | If your plane curve is smooth and irreducible, then Theorem 1 on p630 of Brieskorn and Knorrer's "Plane algebraic curves" will answer your question (see also the Corollary on p634).
If your curve isn't smooth then you can't avoid blow ups and resolutions of singularities. Brieskorn and Knorrer's book also has a section... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Does $d(x,y) = \log(1 + | x - y |)$ define a metric on $\mathbb{R}$? Help would be much appreciated here! I got as far as evaluating the following:
*
*$d(x,y) \geq 0$. Argument is that $(1 + |x - y|)$ doesn't produce negative values, since the $|x-y| \geq 0$ itself and, at most, it would be added to $1$. So this con... | The $d\left(x,y\right)=0$ proof is correct. For the triangle inequality, I would start out with the right hand side. Notice that
\begin{align*}
\log\left(1+\left|x-z\right|\right)+\log\left(1+\left|z-y\right|\right)&=\log\left(\left(1+\left|x-z\right|\right)\left(1+\left|z-y\right|\right)\right)\\
&=\log\left(1+\left|x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Find weak solution to Riemann problem for conservation law
Find the weak solution of the following conservation law
$$ u_t + (u^2)_x = 0 $$
with the initial condition
$$ u(x,0) = \left\lbrace
\begin{aligned}
&u_l & &\text{if } x < 0 ,\\
&u_r & &\text{if } x > 0.
\end{aligned}\right. $$
Consider both cases $u_l>u_r$ an... | This is very similar to the Riemann problem of the inviscid Burgers' equation (see e.g. (1), (2), (3), (4) and related posts). For this type of problem, weak solutions are not unique. Thus, I guess that the problem statement asks for the entropy solution. I will provide a detailed general answer for the case of conserv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Using Archimedian property to prove that infimum of set is $0$ Given set A = [ $\frac{1}{n} | n \in \mathbb{N}$]
It seems to me i have to prove two things :
*
*$\frac{1}{n} \geq 0 , \forall n \in \mathbb{N}$
2, Assuming $b$ be another lower bound , and so $b \leq 0$
Work :
1 Now $\frac{1}{n} \geq 0 , \forall n \in N... | $0$ is a lower bound, $ 1/n >0$, $n \in \mathbb{Z^{+}}$.
Assume $b >0$, real, is a lower bound.
Archimedean principle:
There is a $n_0 \in \mathbb{Z^+}$ s.t.
$n_0 >1/b$, i.e.
$b > 1/n_0$, a contradiction(Why?).
Hence $\inf (A)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Three dice: conditional probability After throwing $3$ dice, we know that on every die there is a different number. What is the probability that there is a $6$ on exactly one dice?
I figured that P(A) - we get 6 on some dice, P(B) - different number on every dice. Then
$$P(B)=\dfrac{6∗5∗4}{6^3}\text{ and }
P(A\cap B)... | The required probability is $$\frac {\binom 3 1 \times 5 \times 4} {6 \times 5 \times 4} = \frac 3 6 = \frac 1 2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3157989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Soft Question: Self-made graphs vs. Desmos I once heard that in real analysis you should expect to draw lots of pictures. Boy were they right. My question is, is it better to draw those pictures yourself or to rely on Desmos? On the one hand, drawing it yourself (or even better, mentally picturing it yourself) is good ... | There can be benefits to both. Sketching the graph yourself may allow you to notice some characteristics of a function that you wouldn’t otherwise just looking at the graph, because drawing it yourself forces you to pay attention to the details. On the other hand, Desmos will give a more accurate depiction of the graph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
probability with martingales 12.2 sum of zero-mean independent variables in L^2 I am struggling with the following theorem from David Williams, Probability with Martingales:
THEOREM
Suppose that $(X_{k}:k\in\mathbb{N})$ is a sequence of independent random variables such that, for every $k$, $E(X_{k})=0, \sigma_{k}^2:=V... | Using the notation from Williams, $M_n = X_1 + \dots + X_n$, we know that $M_n$ is a bounded sequence almost surely since it converges almost surely.
I will write $T_n = \inf\{r : |M_r| > n\}$. Notice that we have
$$\{\sup_{r \geq 1} |M_r| < \infty\} = \bigcup_{n \geq 1} \{T_n = \infty\}.$$
So if $\mathbb{P}(T_n = \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to compute a primitive element for the splitting field of $x^3-2 \in \Bbb{Q}[x]$? Let $\alpha:=\sqrt[3]{2}\in\mathbb{R}$ and $\omega:=e^{2\pi i/3}\in\mathbb{C}$. Then the splitting field for the polynomial $x^3-2\in\mathbb{Q}[x]$ is $$\mathbb{Q}(\alpha,\omega\alpha,\omega^2\alpha)=\mathbb{Q}(\alpha,\omega).$$
Since... |
The primitive element theorem asserts that for $u \in K$ and $\alpha,\omega$ separables, $\alpha+\omega u$ is a primitive element of $K(\alpha,\omega)/K$ iff $\forall \sigma \in Gal(\overline{K}/K)$, $$\sigma(\alpha)+\sigma(\omega)u = \alpha +\omega u \implies \sigma(\alpha)=\alpha,\sigma(\omega) = \omega$$
only finit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Probability, balls from boxes Consider two boxes: there are 15 white and 12 black balls in the first box, and 14 white
and 18 black balls in the second box. Anna puts
her hand in the first box, takes at once two balls and places them in the second box. Then,
she takes one ball wihout looking from the second box.
Knowi... | HINT
I would say
$P(X) = \frac{20}{39}$ (see colleagues 'calculus' comment)
$P(Y|X)=\frac{19}{34}$
Knowing that she took a black ball from the second box: $\Rightarrow P(Y)=1$
$ \Rightarrow P(X|Y)=\frac{P(Y|X)P(X)}{P(Y)}=\frac{P(Y|X)P(X)}{1}=\cdots$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
On the Lagrange Interpolation Error for the Trapezoidal Quadrature Rule
This is part of a document on numerical analysis. I have included its link below. In the derivation for the error in the linear interpolation approximation on page 84, the author "brings" the ξ(x) term of the integral outside using the mean value ... | This is the weighted or extended version of the mean value theorem, $\int_a^bw(x)g(x)dx=g(c)\int_a^bw(x)dx$ if $w$ has a uniform sign. But you are right, one needs that $f$ is continuous for that.
Here we can show that the function $g$ in
$$
f(x)-P_1(x)=(x-a)(x-b)g(x)
$$
is continuous on $[a,b]$ if $f$ is continuously... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to integrate $\frac{1}{\sqrt{x^2+x+1}}$
How to integrate $$\frac{1}{\sqrt{x^2+x+1}}$$
I tried to solve this integral as follows
$\displaystyle
\int \frac{1}{\sqrt{x^2+x+1}} \ dx=
\int \frac{1}{\sqrt{(x+\frac{1}{2})^2+\frac{3}{4}}} \ dx=
\int \frac{1}{\sqrt{(\frac{2x+1}{2})^2+\frac{3}{4}}} \ dx=
\int \frac... | As you did, by completing the square and with $y:=\dfrac2{\sqrt3}\left(x+\dfrac12\right)$,
$$I:=\int\frac{dx}{\sqrt{x^2+x+1}}=\int\frac{dy}{\sqrt{y^2+1}}.$$
Then by some magic $y:=\dfrac12\left(t-\dfrac1t\right)$ yields $dy=\dfrac12\left(1+\dfrac1{t^2}\right)$ and
$$\int\frac{dy}{\sqrt{y^2+1}}=\int\frac{\dfrac12\left(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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How might I quickly determine the equation of the parabola, given the coordinates of its focus and vertex? I have this MCQ.
Which of the following is the equation of the parabola with focus at
$(1,2)$ and vertex at
$(3,2)$ :
A. $ y^2 - 4y + 8x - 20 = 0 $
B. $ y^2 + y + 8x -20 = 0 $
C. $ y^2 + 4y + 8x - 20 = 0... | You don't need any particular formula here: just plug the coordinates of the vertex into the equation and you'll see that only in case A the point belongs to the parabola.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is it ever feasible to actually pump water out of the top of a tank? In math/calculus classes, a problem frequently posed asks how much work (J) is required to pump water out of the top of differently shaped water tanks.
If you are unsure of what I am referring to, here are some example problems:
*
*Emptying water... | Most soap dispensers are examples of tanks where you pump a liquid from the surface. The tube extending to the bottom is irrelevant - the work required is the same as if the tube magically expanded and contracted to just reach the surface, because the pressure will cause the liquid inside and outside the tube to be at ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3158930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$? If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$?
$A^{\perp} = \{x \in X|\langle x,a\rangle=0, \fora... | If $X$ is the space of all sequences $(x_n)_{n\in\mathbb N}$ of real numbers such that $n\gg0\implies x_n=0$, if$$\left\langle(x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N}\right\rangle=\sum_{n=1}^\infty x_ny_n,$$you can take$$A=\left\{(x_n)_{n\in\mathbb N}\in X\,\middle|\,\sum_{n=0}^\infty \frac{x_n}n=0\right\}.$$Then $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Give a counterexample or prove this statement We all know that in Real Analysis, if $E$ is Lebesgue measurable, then there is a $F_{\sigma}$ set $F$ such that:
$$F\subset E,\ m(E\backslash F)=0$$
But I want to know whether the next statement hold:
Suppose $E$ is a Lebesgue measurable set, is there a $F_{\sigma}$ set $F... | The general answer to your question is NO. For a measurable set $E$ what we can to is this:
$$
A \subseteq E \subseteq B,\qquad m(E\setminus A)=0,\qquad m(B\setminus E)=0
$$
where $A$ is an $F_\sigma$ set and $B$ is a $G_\delta$ set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Expanding $x^x$ to series with $o(x)$ polynomials I have some doubts in finding $x^x$ series. I know that from Taylor theorem I have
$$ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k + o(x^k) $$
I want to have $o(x)$ polynomials so:
Let $$ f(x) = x^x $$
$$ f'(x) = (e^{x\cdo... | As you have written
$$x^x=e^{x\ln{x}}$$
and as the series expansion of $e^x$ converges for all values of $x\in\mathbb{C}$ one can write
$$e^x=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+...$$
$$e^{x\ln{x}}=\frac{(x\ln{x})^0}{0!}+\frac{(x\ln{x})^1}{1!}+\frac{(x\ln{x})^2}{2!}+...$$
$$\therefore x^x=1+x\ln{x}+\frac{x^2\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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A certain functor in Hakim's "Topos annellés et schemas relatifs" is a sheaf for the canonical topology In M. Hakim's Topos annellés et schemas relatifs, page 43, (3.4.7), the Author wants to define a sheaf $f_{0A}^*(X)$ over a topos $T$, with respect to the canonical topology. A scheme $X$ is given, together with a co... | The argument provided in your edit is correct: The functor $Hom_{\text{Rings}}(R, -)$ preserves limits (this is an entirely general fact, valid for homming out of any object in any category), hence in particular equalizer diagrams.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of getting "tails-tails" for the first time.
A fair coin is tossed until the coin lands "tails-tails" (i.e. a tails followed by a tails) for the first time. Let $X$ counts the number of tosses required. Find the probability of the event $X=n.$ What will be the expectation of $X$?
I think this problem h... | Hints:
Let $g_n$ be the number of sequences of length $n$ made up of $H$ and $T$ (heads and tails) that contain no two consecutive tails. Then the sequence you want is such a sequence (but of length $n - 2$) followed by $TT$. Assume you know the value of $g_n$ in general. In terms of $g_{n - 3}$, what is the desired pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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If $\chi(g)$ generates a dense subgroup of $\chi(G)$ for all $\chi$ then $g$ generates a dense subgroup of $G$. This question arised from something in Ergodic theory, however this is not necessary to state or answer the question.
Suppose that $G$ is a compact abelian group and $g\in G$.
Are the following two equivalent... | The answer is Yes!.
Suppose by contradiction that $1$ holds but $2$ doesn't. Let $H=\overline{\left<g\right>}$, then by assumption $H\leq G$ is a proper subgroup.
Look at $G/H$. This is a non-trivial group and so admits a non-trivial character $\chi:G/H\rightarrow S^1$. We may compose with $G\rightarrow G/H$ to obtain ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a Givens rotation matrix such that $y=Gx$ Assume that $x,y \in \mathbb{R}^2$ with $||x||_2=||y||_2=1$.
Find a Givens rotation matrix $G=\begin{bmatrix}c & s \\ -s & c \end{bmatrix}$ (i.e., find $c$ and $d$ with $c^2+d^2=1$) such that $y=Gx$.
Answer:
Let $x=(x_1,x_2) \ $ and $ \ y=(y_1,y_2)$. Then,
$\begin{bmat... | HINT
show that $<x,y>=c$ then $cos \theta=c=<x,y>$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove $\text{det}(I+xy^{\top}+wz^{\top})=(1+y^{\top}x)(1+z^{\top}w)-(x^{\top}z)(y^{\top}w)$? Suppose $x,y,z,w$ are vectors in $\mathbb{R}^n$ and $I$ is the identity matrix.
Show that $\text{det}(I+xy^{\top}+wz^{\top})=(1+y^{\top}x)(1+z^{\top}w)-(x^{\top}z)(y^{\top}w)$.
| Consider the matrix
$$\tag1
\begin{bmatrix} 1+y^Tx & y^Tw \\ z^Tx & 1+z^Tw \end{bmatrix}
=I + \begin{bmatrix} y^T\\ z^T \end{bmatrix} \begin{bmatrix} x & w\end{bmatrix}
$$
For any $A,B$, we have the equality $\det(I+AB)=\det(I+BA)$. So the determinant in $(1)$ is equal to the determinant of
$$
I + \begin{bmatrix} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3159899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that each $y\in Y_1$ has a unique $x_y\in X.$ Setting: Let $X,Y$ be compact Hausdorff spaces and $E,F$ be any Banach spaces.
Let $C(X,E)$ be the collection of all $E$-valued continuous functions on $X.$
$C(Y,F)$ is defined similarly.
Endow sup-norm to both $C(X,E)$ and $C(Y,F).$
Let $T:C(X,E)\to C(Y,F)$ be a con... | As $y\notin Y_3$, we know that there exists $g\in C(X,E)$ such that $(Tg)(y)\neq 0$ and thus $g(x_y')\neq 0$ and $g(x_y')\neq 0$ (as both are in $Y_1$). Set $a:= g(x_y')$ and recall that in compact spaces the Urysohn lemma holds. Hence, there exists a continuous function $h: X \rightarrow \mathbb{R}$ such that $$h(x_y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3160032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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By means of an example, show that $P(A) + P(B) = 1$ does not mean that $B$ is the complement of $A$ I'm in grade 10, and I've just started to learn about complementary events. I am rather perplexed with this question. Isn't this question kinda contradictory, since $P(A) + P(A') = 1$?
This is what I got to:
$P(A) + P(B)... | From the two statements you obtained (correctly), you can further obtain $P(A') = P(B)$. But that does not imply $A'=B$. Just as $x^2 = y^2$ for reals $x,y$ does not imply $x = y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3160157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 3
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Does there exist a characteristically simple group, which is not a direct product of simple groups? Does there exist a characteristically simple group, which is not a direct product of simple groups? A characteristically simple group is a group without non-trivial proper characteristic subgroups.
The only thing I know... | Actually, as it was pointed out in the comments $(\mathbb{Q}, +)$ is an example of such group.
It is non-simple $(\mathbb{Q}, +)$ as $(\mathbb{Z}, +) \triangleleft (\mathbb{Q}, +)$.
It is characteristically simple, as for any field $\mathbb{F}$, $Aut(\mathbb{F}, +)$ acts transitively on $\mathbb{F}\setminus\{0\}$.
It i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3160323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Evaluating $\int\frac{dx}{\sqrt{4-9x^2}}$ with different trig substitutions ($\sin$ vs $\cos$) gives different results I am trying to solve the following integral with trig substitutions. However, I get a different answer for two substitutions that should yield the same result.
$$\int\frac{dx}{\sqrt{4-9x^2}}$$
*
*F... | Since $(-\arcsin)'(x)=\arccos'(x)=-\dfrac1{\sqrt{1-x^2}}$, you got twice the same thing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3160694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Set theory question regarding $A\overline{\sim} B = \{x+y : \in A\times B\}$ Given $A,B \in P(N)$ We mark $\overline{\sim}$ as
$$A\overline{\sim} B = \{x+y : \langle x,y\rangle\in A\times B\}$$
Now, order R will be as following
$ARB$ iff $\exists M\in P(N)$ so $A\overline{\sim} M=B$.
The question is
R is reflexive? sy... | You got off to a great start with your transitivity proof!
Note that $M=T\overline{\sim} S.$ Having chosen $q\in A\overline{\sim}M,$ we know that $q=a+m$ for some $a\in A$ and some $m\in M.$ Since $M=T\overline{\sim} S,$ then $m=t+s$ for some $t\in T$ and some $s\in S,$ whence $$q=a+(t+s)=(a+t)+s.$$ Since $a\in A$ and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why we can't differentiate both sides of a polynomial equation? Suppose we had the equation below and we are going to differentiate it both sides:
\begin{align}
&2x^2-x=1\\
&4x-1=0\\
&4=0
\end{align}
This problem doesn't seems to happens with other equation like $\ln x =1$ or $\sin x = 0$, we can keep differentiating t... | The kicker is that our domain of truth isn't "big enough" to allow it.
From your example, the functions on both sides only agree on $\left\{-\frac12,1\right\}$ However, we can't differentiate functions at isolated points of their domains!
On the other hand, consider the equation $$\sin x=\cos x\tan x.$$ The functions h... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proof of Lemma: Every integer can be written as a product of primes I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is.
I encountered the classic lemma about every nonzero integer being the product of primes in Ireland and Rosen's textbook A Classical Introduction to Modern Number T... | Although the proof by contradiction is correct, your feeling of unease is fine, because the direct proof by induction is so much clearer:
Take an integer $N$. If $N$ is prime, there is nothing to prove. Otherwise, we must have $N = mn$, where $1 < m, n < N$. By induction, since $m, n$ are smaller than $N$, they must e... | {
"language": "en",
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"source": "stackexchange",
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Why does the integral domain "being trapped between a finite field extension" implies that it is a field? The following is an exercise from Qing Liu's Algebraic Geometry and Arithmetic Curves.
Exercise 1.2.
Let $\varphi : A \to B$ be a homomorphism of finitely generated algebras over a field. Show that the image of a ... | $A$ and $B$ be finitely generated algebras over $k$. Let $\mathfrak m $ be maximal ideal of $B$. We have an injective map $A/\phi ^{-1}(\mathfrak m) \rightarrow B/\mathfrak m $. Identify $A/\phi ^{-1}(\mathfrak m)$ to its image via this map. Let $T\in A/\phi ^{-1}(\mathfrak m) $, then $1/T \in B/ \mathfrak m $- which i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3161381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
} |
Proving matrices equation when all the matrices in it may not be invertible I'm reviewing linear algebra for my exams this year, and I just encountered this problem.
For an arbitrary matrix, $\boldsymbol{A} \in \mathcal{R}^{m \times n}$, prove there must be a unique matrix $\boldsymbol{P} \in \mathcal{R}^{n \times m}$ ... | The matrix $A$ induces a decomposition $\ker A\oplus(\ker A)^\perp$ in the domain and $\mathrm{im}A\oplus(\mathrm{im}A)^\perp$ in the codomain. The core of the matrix is the square matrix $U:(\ker A)^\perp\to\mathrm{im}A, x\mapsto Ax$.
Since $AP$ is to act as an 'left-identity' on $A$ we have to define $Px=U^{-1}x$ for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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If $\mathbb E[|X_n-X|^2]\to 0$ can we say that $X_{n_k}(\omega )\to X(\omega )$ for a.e. $\omega $? Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and let $X_n\to X$ in $L^2(\Omega )$ where $(X_n)$ is a sequence of random variable and $X$ is a random variable as well.
By a theorem of Lebesgue measure theory,... | Yes it's true. And it's indeed quite strong, but it's weaker than $$\mathbb P\left\{\lim_{n\to \infty }X_n=X\right\}=1,$$
But it's at least better (stronger) than convergence in distribution or convergence in probability, since we indeed have more information of the sequence $(X_n)$.
By the way, when I say that $L^2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3161623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quick but not simple question. $2^\sqrt2$ or e, which is greater?
$2^\sqrt2$ vs $e$, which is greater?
$(2^\sqrt2)^\sqrt2 = 4\quad $ & $\quad e^\sqrt2$ = ?
$\log(2^\sqrt2) = \sqrt2\log(2)\quad$ & $\quad \log(e) = 1$
I tried but can't induce comparable form.
Is anybody know how to prove it?
| If you know that $\ln(2)\approx0.69$ and $1/\sqrt2=\sqrt2/2\approx1.414/2=0.707$, then you have $\ln(2)\lt1/\sqrt2$, in which case $\ln(2^\sqrt2)=\sqrt2\ln2\lt1=\ln(e)$, hence $2^\sqrt2\lt e$.
It's not hard to show that $\sqrt2\gt1.4$, since $1.4^2=1.96\lt2$. It's a little trickier to show that $\ln(2)\lt0.7$, but thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3161742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Investigate whether $ f $ meets Lipschitz continuity and whether it is uniform continuity
Let $f: (-\infty,0]\rightarrow\mathbb R$ which is a function: $f(t)=\arcsin(e^{t})$ for $t\le0$.Investigate whether $ f $ meets Lipschitz continuity and whether it is uniform continuity
My try:
For $t\neq0$: $$f'(t)=\frac{e^{t}}... | For the first part, note that if $f$ were Lipschitz continuous on $(-\infty,0]$ with constant $L$, then the derivative $f'$, where it exists, would satisfy $|f'(x)|\leq L$. But this does not hold since $|f'(t)|\rightarrow +\infty$ for $t\rightarrow 0^-$.
Another argument to prove that $f$ is not Lipschitz continuous on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3161878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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An application of Fredholm Alternative I have just started reading the Fredholm Alternative for finite dimensional spaces and I came towards an excersise that it reads as follows:
If $H$ is a Hilbert space and $T:H\to H$ is bounded, linear map, with
$\langle Tx,x\rangle >0$ for $x\neq0$ then prove that $T$ is surjectiv... | 1) Yes. It will be helpful to know the other terms for adjoint: "dual" and "transpose" (even for operators that are not matrices). See Transpose of a linear map. For a full, excellent exposition (which addresses infinite-dimensional linear algebra despite the title), see FDVS. I don't recommend learning linear alg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3162153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible to "mod" the action of a symmetric group on a symmetric operad? I am relatively new to category theory, so only have a rough understanding of the technicalities behind operads. My understanding is that symmetric operads are defined so that they are "nicely" acted on by the symmetric group.
My question i... | Yes. (This paper formulates extra structure on a sequence $G_0,G_1,\dots$ of groups to get a theory analogous to the theory of operads.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3162416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Summation of 'for loop' with conditional? I am trying to convert instances of nested 'for' loops into a summation expression. The current code fragment I have is:
for i = 1 to n:
for j = 1 to n:
if (i*j >= n):
for k = 1 to n:
sum++
endif
Basically, the 'if' conditional i... | Hint:
The "for" cycle in $k$ is very easy to turn into something simpler...
As for the other parts, see if you can split the problem into easier steps. For example, what happens for $i=1$? And what happens for $i=2$? And $i=3$? And...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3162547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove that if there are integers $m$ and $n$ such that $am +bn =1$ then $a$ and $b$ are coprime. Suppose $a,b \in \mathbb{N}$. Prove that if there are integers $m$ and $n$ such that $am +bn =1$ then $a$ and $b$ are coprime.
I came up with the following proof, but I am sure a shorter argument is possible.
To prove: $\... | Suppose $am+bn=1$. If $k$ divides both $a$ and $b$ then there exist $p$ and $q$ such that $a=kp$ and $b=kq$.
Substituting that into our first equation gives $kpm+kqn=1\implies k$ divides $1$
Therefore, $k=1$ and $a$ and $b$ are coprime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3162667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Let $T$ be an exponential random variable with parameter $\theta$. For $t \gt0$, compute $\Bbb{E}(T|T\le t)$
Let $T$ be an exponential random variable with parameter $\theta$. For $t \gt0$, compute $\Bbb{E}(T|T\le t)$.
My work: First $$P(T\le s|T\le t)=\frac{\int_0^s\theta e^{-\theta x}dx}{\int_0^t\theta e^{-\theta x... | Your computation looks correct.
Alternative way: using the absence of memory property, we have
$$
\mathrm E[T; T> t] = \mathrm E[T\mid T> t] \mathrm P(T>t) = (t+ \mathrm E[T])e^{-\theta t} = (t+ \theta^{-1})e^{-\theta t},
$$
whence
$$
\mathrm E[T\mid T\le t] = \frac{\mathrm E[T; T\le t]}{\mathrm P(T\le t)} = \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3162919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the value of $\lim\limits_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(x)\mathrm dx$
Finding the value of $\displaystyle \lim\limits_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(x)\mathrm dx$
What I tried
Let $\displaystyle I_{k} =\int^{\frac{\pi}{4}}_{0}\cos^{k}(x)\mat... | Put
\begin{equation*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}\cos^{2n-2}(x)\,\mathrm{d}x.
\end{equation*}
Via the substitutions $ y=\sin x $ and $ y=\frac{z}{\sqrt{n-1}} $ we get
\begin{gather*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}(1-\sin^2(x))^{n-1}\,\mathrm{d}x = \sqrt{n}\int_{0}^{1/\sqrt{2}}(1-y^2)^{n-1}\cdot\dfrac{1}{\sqrt{1-y^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3163039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Meaning of linear independence with row vectors So far I have understood that a set of vectors $S = {v_1, v_2, . . . , v_k }$ in a vector space V is linearly independent
when the vector equation
$c_1v_1 + c_2v_2 + . . . + c_kv_k = 0$
has only the trivial solution$c_1 = 0, c_2 = 0, . . . , c_k = 0.$
An example in matrix... | Linear independence is linear independence is linear independence. It's defined entirely independently of matrices. It is, instead, defined in terms of a vector equation, which in finite dimensions, can be turned into a system of linear equations. As such, matrices are an excellent tool to determine linear independence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3163109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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