Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How one can show that $\int _0 ^1 ((h')^2-h^2)dx \ge0 $ for all $h\in C^1[0,1]$ and $h(0) = 0$? How one can show that $\int _0 ^1 ((h')^2-h^2)dx \ge0 $ for all $h\in C^1[0,1]$ and $h(0) = 0$?
| By the fundamental theorem of Calculus:
$$
\left|h(x)\right| = \left|\int_0^x h'(t) \,dt\right| \le \int_0^x \left|h'(t)\right| \,dt \le \int_0^1 \left|h'(x)\right| \,dx
$$
By Cauchy-Schwarz (or Jensen's) inequality:
$$
\left|h(x)\right|^2 \le \left(\int_0^1 \left|h'(x)\right| \,dx\right)^2 \le \int_0^1 \left|h'(x)\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Probability Question relating prison break I am stuck in a question regarding a prisoner trapped in a cell with 3 doors that actually has a probability associated with each door chosen(say $.5$ for door $A$, $.3$ for door $B$ and $.2$ for door $C$). The first door leads to his own cell after traveling $2$ days, whereas... | Let's go through it for part (a) first. Let $X$ denote the number of days until this prisoner gains freedom. I think you already have $E[X|D=1], E[X|D=2], E[X|D=3]$:
$E[X|D=1] = E[X + 2]$
$E[X|D=2] = E[X + 3]$
$E[X|D=3] = E[0]$
So we have
$E[X^2|D=1] = E[(X + 2)^2] = E[X^2] + 4E[X] + 4$
$E[X^2|D=2] = E[(X + 3)^2]= E[X^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Group theory - left/right $H$-cosets and quotient sets $G/H$ and $G \setminus H$. Let $G$ be a group and $H$ be a subgroup of $G$. The left $H$-cosets are the sets $gH, g \in G$. The set of left $H$-cosets is the quotient set $G/H$. The right $H$-cosets are the sets $Hg, g\in G$. The set of right $H$-cosets is the quo... | It means that $G=\cup_{g\in G} gH=\cup_{g\in G} Hg$, that the map $x\rightarrow gx$ is a bijection between $H$ and $gH$ and therefore $|gH|=|H|$. It can be deduced from this that $|G|=|G/H||H|$. A similar statement for right cosets.
| {
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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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Right translation - left coset - orbits We can remark that the left coset $gH$ of $g \in G$ relative to a subgroup $H$ of $G$ is the orbit of $g$ under the action of $H \subset G$ acting by right translation.
What is that right translation? and how can I prove that the orbit of $g$ under the action of $H \subset G$ a... | Right translation can equally be read as "right multiplication", except there is an implication of commutativity.
As to your second query, let the subgroup $H$ act on $G$ by right multiplication:
$$h \cdot g = gh \qquad \forall g \in G \quad \forall h \in H$$
For any $g \in G$, the orbit $H \cdot g$ is the set
$$\{ h ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Trouble with wording of this math question "Find the derivative as follows (you need not simplify)":
a) $y = 2^x f (x)$, where $f (x)$ is a differentiable function and so is $f '(x)$: find $\frac{d^2x}{dx^2}$.
That's the exact wording of the question, and no additional information is given outside of this question. Ca... | We are given:
$$y = 2^x f(x)$$
where $f(x)$ and $f'(x)$ are differentiable functions and asked to find the second derivative.
This is just an application of the product rule, so
$\displaystyle \frac{d}{dx} \left(2^x f(x)\right) = 2^x (f(x) \log(2) + f'(x))$
$\displaystyle \frac{d^2}{dx^2} \left(2^x f(x)\right) = 2^x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346919",
"timestamp": "2023-03-29T00:00:00",
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Determine and classify all singular points Determine and find residues for all singular points $z\in \mathbb{C}$ for
(i) $\frac{1}{z\sin(2z)}$
(ii) $\frac{1}{1-e^{-z}}$
Note: I have worked out (i), but (ii) seems still not easy.
| Thank Mhenni Benghorbal for the hint! I have worked out (i) so far:
The singular points for (i) are $\frac{k\pi}{2}, k \in \mathbb{Z}$,
The case $k=0$ was justified by Mhenni,
For $k \neq 0$ the singular points are simple poles, since $$\lim_{z \to \frac{k\pi}{2}}\frac{z-\frac{k\pi}{2}}{z\sin(2z)}=\lim_{z \to \frac{k\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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why $\log(n!)$ isn't zero? I have wondered that why the $\log (n!)$ isn't zero for $n \in N$.
Because I think that $\log (1)$ is zero so all following numbers after multiplying the result will become zero.
Thanks in advance.
| Might as well make an answer of it.
$$\begin{align*}
\lg(n!)&=\lg(1\cdot2\cdot3\cdot\ldots\cdot n)\\
&=\lg 1+\lg 2+\lg 3+\ldots+\lg n\\
&=\lg 2+\lg 3+\ldots+\lg n\;,
\end{align*}$$
so it won’t be $0$ unless $n=1$ (or $n=0$): you’re adding the logs, not multiplying them.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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How to prove $\sum\limits_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$? How do I prove the following identity directly?
$$\sum_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$$
I thought about using the binomial theorem for $(x+a)^n$, but got stuck, because I realized that my $x$ and $a$ in this case are dynamic va... | A nice proof uses the Lagrange-Bürman inversion formula. Start defining:
\begin{equation}
C(z) = z e^{C(z)}
\end{equation}
which gives the expansion:
\begin{equation}
e^{\alpha C(z)} = \alpha \sum_{n \ge 0} \frac{(\alpha + n)^{n - 1} z^n}{n!}
\end{equation}
Then you have:
\begin{equation}
e^{(\alpha + \beta) C(z)} = e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 6,
"answer_id": 2
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Conditional probabilities, urns I found this question interesting and apparently it has to do with conditional probabilities:
An urn contains six black balls and some white ones. Two balls are drawn simutaneously. They have the same color with probability 0.5. How many with balls are in
the urn?
As far as I am concerne... | Long hint/walkthrough: Let the number of white balls be denoted $w$. The probability of pulling two white balls will be $\frac{w}{6+w}\cdot\frac{w-1}{6+w-1}$ since the probability of choosing a white ball will be $P(w_1)=\frac{w}{w+6}$ and since there is one less white ball the probability of choosing another will be $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Strictly increasing function on positive integers giving value between $100$ and $200$ I'm looking for some sort of function $f$ that can take any integer $n>0$ and give a real number $100 \le m \lt 200$ such that if $a \lt b$ then $f(a) \lt f(b)$. How can I do that? I'm a programmer and I need this for an application ... | $f(n)=200-2^{-n}$ satisfies your criteria.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/347252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$ I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me decompose $V(F)$ when $F$ is a single poly... | $\newcommand{\rad}{\text{rad}\hspace{1mm}}
$
The ideal $(x^2 + y^2 - 1,x^2 - z^2 - 1)$ is equal to the ideal $(y^2 + z^2 ,x^2 - z^2 - 1)$. This is because
\begin{eqnarray*} (y^2 + z^2) + (x^2 - z^2 - 1) &=& y^2 + x^2 - 1\\
(x^2 + y^2 - 1) - (x^2 - z^2 - 1) &=& y^2 + z^2. \end{eqnarray*}
Thus we get
\begin{eqnarray*} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 1
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Funny integral inequality Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$
and how to find the smallest constant $C$ which satisfies
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq C\int_0^1[f'(x)]^2\text{... | solutin 2:
by Schwarz,we have
$$\int_{0}^{\frac{1}{2}}[f'(x)]^2dx\int_{0}^{\frac{1}{2}}x^2dx\ge\left(\int_{0}^{\frac{1}{2}}xf'(x)dx\right)^2=\left[\dfrac{1}{2}f(\dfrac{1}{2})-\int_{0}^{\frac{1}{2}}f(x)dx\right]^2$$
so
$$\int_{0}^{\frac{1}{2}}[f'(x)]^2dx\ge 24\left[\dfrac{1}{2}f(\dfrac{1}{2})-\int_{0}^{\frac{1}{2}}f(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
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Proving $\,f$ is constant. Let $\,f:[a,b] \rightarrow \Bbb R $ be continuous
and $\int_a^b f(x)g(x)\,dx=0$, whenever $g:[a,b] \rightarrow \Bbb R $ is continuous and $\int_a^b g(x)\,dx=0$.
Show that $f$ is a constant function.
I tried a bunch of things including the mid-point integral theorem(?) but to no avail.
I... | Suppose $f$ is nonconstant.
Define $g(x) = f(x)-\bar{f}$, where $\bar{f}:= \frac{1}{b-a} \int_a^b f(x)dx$. Then $\int_a^b g = 0$.
Then $$\int_a^b f(x)g(x)dx = \int_a^b f(x) \big(f(x)-\bar{f}\big) dx = \int_a^b \big(f(x)-\bar{f}\big)^2 dx >0$$ The reason that this last term is larger than zero, is that $f$ is non-consta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Differential Forms, Exterior Derivative I have a question regarding differential forms.
Let $\omega = dx_1\wedge dx_2$. What would $d\omega$ equal? Would it be 0?
| The differential form $\omega = dx_1 \wedge dx_2$ is constant hence we have $$ d\omega = d(dx_1 \wedge dx_2) = d(1) \wedge dx_1 \wedge dx_2 \pm 1 \, ddx_1 \wedge dx_2 \pm 1 \, dx_1 \wedge ddx_2$$ and because $d^2 = 0$, we have $$ d \omega = 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/347494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Prove set is dense This is a pretty basic and general question.
I have to prove (if true) that the sum of two dense sets is dense as well.
Let $A, B$ be nonempty dense sets in $\mathbb R$. Then $A+B=\{a+b\mid a\in A, b\in B\}$ is also dense.
Can anyone give me a pointer as to how one may prove this (just the method)?... | Hint:
*
*If $A$ is dense in $\mathbb{R}$, then $A+2013 = \{a+2013 \mid a \in A\}$ is also dense in $\mathbb{R}$.
*Union of dense sets is dense, in particular, $A+B = \bigcup_{b \in B}A+b$.
Good luck!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Vector Fields Question 4 I am struggling with the following question:
Prove that any left invariant vector field on a Lie group is complete.
Any help would be great!
| Call your lie group $G$ and your vector field $V$.
It suffices to show that there exists $\epsilon > 0$ such that given $g \in G$ (notice the order of the quanitfiers!) there exists an integral curve $\gamma_g : (-\epsilon, \epsilon) \rightarrow G$ with $\gamma_g (0) = g$, I.e. a curve $\gamma$ starting at $p$ with $\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimize $\|A-XB\|_F$ subject to $Xv=0$ Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. Let $\|\cdot\|_F$ be the Frobenius norm of a matrix. How can we solve the following optimization problem in $X \in \mathbb R^{n \times n}$?
$$\begin{array}{ll} \text{minimize} & \... | As the others show, the answer to your question is affirmative. However, I don't see what's the point of doing so, when the problem can actually be converted into an unconstrained least square problem.
Let $Q$ be a real orthogonal matrix that has $\frac{v}{\|v\|}$ as its last column. For instance, you may take $Q$ as a... | {
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Vertical line test
A vertical line crossing the x-axis at a point $a$ will meet the set in exactly one point $(a, b)$ if $f(a)$ is defined, and $f(a) = b$.
If the vertical line meets the set of points in two points then $f(a)$ is undefined?
| The highlighted proposition is one way of describing the vertical line test, which determines whether $f$ is a function.
If there is one and only point of intersection between $x = a$ and $f(x)$, then $f$ is a function.
If there are two or more points of intersection between $x = a$ and $f(x)$, then $f$ maps a given ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Adjust percentage I'm stuck on what I would think is a simple problem.
A group of $3$ people are selling a product through a store.
Under the current arraignment, the store gets $30$% of the price the product is sold for. The group of $3$ get the remaining $70$%.
The group of $3$ split up the remaining $70$% as $25$%, ... | If you are confused with the percentages, it is always to better write down statements to make it easier.
If $70$% is equivalent to $100$% , then $25$% is equivalent to ?
$(25*100/70)$ = $(2500/70)$% = $35.715$%
Similarly if $70$% is equivalent to $100$% , then $25$% is equivalent to ?
$(25*100/70)$ = $(2500/70)$% = $3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In Search of a More Elegant Solution I was asked to determine the maximum and minimum value of $$f(x,y,z)=(3x+4y+5z^{2})e^{-x^{2}-y^{2}-z^{2}}$$ on $\mathbb{R}^{3}$.
Now, I employed the usually strategy; in other words calculating the partial derivatives, setting each to zero, and the solve for $x,y,z$ before comparin... | We have $$\frac\partial{\partial x}f(x,y,z)=(3-2x(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
$$\frac\partial{\partial y}f(x,y,z)=(4-2y(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
$$\frac\partial{\partial z}f(x,y,z)=(10z-2z(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
At a stationary point, either $z=0$ and then $3y=4x$, $x=\pm\frac3{10}\sqrt 2 $.
Or $3x+4y+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Find the value of : $\lim_{x\to\infty}x\left(\sqrt{x^2-1}-\sqrt{x^2+1}\right)=-1$ How can I show/explain the following limit?
$$\lim_{x\to\infty} \;x\left(\sqrt{x^2-1}-\sqrt{x^2+1}\right)=-1$$
Some trivial transformation seems to be eluding me.
| The expression can be multiplied with its conjugate and then:
$$\begin{align}
\lim_{x\to\infty} x\left(\sqrt{x^2-1}-\sqrt{x^2+1}\right)
&= \lim_{x\to\infty} x\left(\sqrt{x^2-1}-\sqrt{x^2+1}\right)\left(\frac{\sqrt{x^2-1}+\sqrt{x^2+1}}{\sqrt{x^2-1}+\sqrt{x^2+1}}\right) \cr
&=\lim_{x\to\infty} x\left(\frac{x^2-1-x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Does a such condition imply differentiability? Let function $f:\mathbb{R}\to \mathbb{R}$ be such that
$$
\lim_{\Large{(y,z)\rightarrow (x,x) \atop y\neq z}} \frac{f(y)-f(z)}{y-z}=0.
$$
Is it then $f'(x)=0$ ?
| What is $f'(x)=\displaystyle\lim_{y\to x}\dfrac{f(y)-f(x)}{y-x}$?
Let $\epsilon>0$. From the hypothesis it follows that exists a $\delta>0$ such that if $y\neq z$ and $0<\|(y,z)-(x,x)\|<\delta$ then $\left|\dfrac{f(y)-f(z)}{y-z}\right|<\epsilon$.
Now if $0<|y-x|<\delta$ then $0<\|(y,x)-(x,x)\|=|y-x|<\delta$ and $y\ne... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Best Fake Proofs? (A M.SE April Fools Day collection) In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen.
I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's very easy to draw t... | Let me prove that the number $1$ is a multiple of $3$.
To accomplish such a wonderful result we are going to use the symbol $\equiv$ to denote "congruent modulo $3$". Thus, what we need to prove is that $1 \equiv 0$. Next I give you the proof:
$1\equiv 4$
$\quad \Rightarrow \quad$
$2^1\equiv 2^4$
$\quad \Rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "242",
"answer_count": 27,
"answer_id": 21
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A question on estimates of surface measures If $\mathcal{H}^s $ is $s$ dimensional Hausdorff measure on $ \mathbb{R}^n$, is the following inequality true for all $ x \in \mathbb{R}^n,\ R,t > 0 $ ?
$$ \mathcal{H}^{n-1}(\partial B(x,t)\cap B(0,R)) \leq \mathcal{H}^{n-1}(\partial B(0,R)) $$ If the answer is not affirmativ... | The definition of $\mathcal H^{s}$ implies that it does not increase under $1$-Lipschitz maps: $\mathcal H^s(f(E))\le \mathcal H^s(E)$ if $f$ satisfies $|f(a)-f(b)|\le |a-b|$ for all $a,b\in E$.
The nearest point projection $\pi:\mathbb R^n \to B(x,t)$ is a $1$-Lipschitz map. (Note that when $y\in B(x,t)$, the neares... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Le Cam's theorem and total variation distance Le Cam's theorem gives the total variation distance between the sum of independent Bernoilli variables and a Poisson random variable with the same mean. In particular it tells you that the sum is approximately Poisson in a specific sense.
Define
$$S_n = X_1+\dots+X_n \text... | Let $Y_n$ be any Poisson random variable with parameter $\lambda_n$. Then, for every $x$,
$$
\left|P(S_n < x)-P(Y_n < x)\right|=\left|\sum_{k\lt x} P(S_n=k)-P(Y_n=k)\right|\leqslant\sum_{k=0}^{\infty}\left| P(S_n=k)-P(Y_n=k)\right|.
$$
Hence,
$$
\left|P(S_n < x)-P(Y_n < x)\right| < 2\sum_{i=1}^n p_i^2.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Struggling with an integral with trig substitution I've got another problem with my CalcII homework. The problem deals with trig substitution in the integral for integrals following this pattern: $\sqrt{a^2 + x^2}$. So, here's the problem:
$$\int_{-2}^2 \frac{\mathrm{d}x}{4 + x^2}$$
I graphed the function and because... | Hint: you can cut your work considerably by using the trig substitution directly into the proper integral, and proceeding (no place for taking the square root of the denominator):
You have $$2\int_0^2 \frac{dx}{4+x^2}\quad\text{and NOT} \quad 2\int_0^2 \frac{dx}{\sqrt{4+x^2}}$$
But that's good, because this integral (o... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
} |
Cayley's Theorem question: examples of groups which aren't symmetric groups. Basically, Cayley's Theorem says that every finite group, say $G$, is isomorphic to a subgroup of the group $S_G$ of all permutations of $G$.
My question: why is there the word "subgroup of"? If we omit this word, is the statement wrong? brief... | The symmetric group $S_n$ has order $n!$ whereas there exists a group of any order (eg. $\mathbb{Z}_n$ has order $n$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/348510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Interior of closure of an open set The question is is the interior of closure of an open set equal the interior of the set?
That is, is this true:
$(\overline{E})^\circ=E^\circ$
($E$ open)
Thanks.
| Let $\varepsilon>0$, I claim there is an open set of measure (or total length, if you like) less than $\varepsilon$ whose closure is all of $\mathbb R$.
To see this, simply enumerate the rationals $\{r_n\}$ and then for each $n\in\mathbb N$ choose an open interval about $r_n$ of length $\varepsilon/2^n$. The union of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Learning Mathematics using only audio. Are there any mathematics audio books or other audio sources for learning mathematics, like for example math podcasts which really go into detail. I ask this because I make about 1 hour from my house to the school and staring at a screen on the car makes me dizzy. I know about pod... | I can't really point to a source, but I find the question quite relevant, as audiobooks of mathematic subject can be important also for blind people.
Learnoutloud has a repository of audiobooks and podcast about math and statistics, and related novels as well. Nevertheless it seems to offer no advanced math repository.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 0
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Double dot product vs double inner product Anything involving tensors has 47 different names and notations, and I am having trouble getting any consistency out of it.
This document (http://www.polymerprocessing.com/notes/root92a.pdf) clearly ascribes to the colon symbol (as "double dot product"):
$\mathbf{T}:\mathbf{U}... | I know this might not serve your question as it is very late, but I myself am struggling with this as part of a continuum mechanics graduate course. The way I want to think about this is to compare it to a 'single dot product.' For example:
\begin{align}
\textbf{A} \cdot \textbf{B} &= A_{ij}B_{kl} (e_i \otimes e_j) \cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 0
} |
Can we extend any metric space to any larger set? Let $(X,d)$ be metric space and $X\subset Y$. Can $d$ be extended to $Y^2$ so that $(Y,d)$ is a metric space?
Edit:
how about extending any $(\Bbb Z,d)$ to $(\Bbb R,d)$
| Let $Z=Y\setminus X$. Let $\kappa=|X|$. If $|Z|\ge\kappa$, we can index $Y=\{z(\xi,x):\langle\xi,x\rangle\in\kappa\times X\}$ in such a way that $z(0,x)=x$ for each $x\in X$. Now define
$$\overline d:Y\times Y\to\Bbb R:\langle z(\xi,x),z(\eta,y)\rangle\mapsto\begin{cases}
d(x,y),&\text{if }\xi=\eta\\
d(x,y)+1&\text{if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How do I show that these sums are the same? My textbook says that I should check that
$$ \sum_{i=0}^\infty \frac{\left( \lambda\mathtt{I} + \mathtt{J}_k \right)^i}{i!} $$
is in fact the same as the product of sums
$$ \left( \sum_{i=0}^\infty \frac{\left( \lambda\mathtt{I}\right)^i}{i!} \right) \cdot
\left( \sum_{j=0}^k... | Hints:
$$(1)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum_{k=0}^\infty\frac{X^k}{k!}=e^X$$
$$(2)\;\;\;\;\;\;\;\;J_k^{n}=0\;,\;\;\;\text{where$\,n\,$ is the number of rows of the matrix}\;J_k$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/348869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Does every section of $J^r L$ come from some section $s\in H^0(C,L)$, with $L$ line bundle on a compact Riemann surface? I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a bund... | This is rather old so maybe you figured out the answers already.
Answer to Q1 is No. Not every global section of $J^r L$ comes from the "prolongation" of a section of $L$, not even locally. Consider for example the section in $J^1(\mathcal{O}_\mathbb{C})$ given in coordinates by $(0,1)$ (constant sections $0$ and $1$).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Hyperbolic cosine I have an A level exam question I'm not too sure how to approach:
a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$
b) Deduce $ \cosh x > x$
c) Find the point P such that it lies on $y=\cosh x$ and its perpendicular distance from the line $y=x$ is a minimum.
I understand how to show the first st... | a) You'r right, you can do this with the discriminant and it is very natural. But you can also use the well-known inequality: $2ab\leq a^2+b^2$ which follows from the expansion of $(a-b)^2\geq 0$. So you get
$$
2x=2\cdot x\cdot 1\leq x^2+1^2=x^2+1<x^2+2\qquad \forall x\in\mathbb{R}.
$$
Then divide by $2$.
b) By definit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\mbox{Ker} \;S$ is T-invariant, when $TS=ST$ Let $T,S:V\to S$ linear transformations, s.t: $TS=ST$, then $\ker(S)$ is $T$-invariant.
My solution:
$$\{T(v)\in V:TS(v)=0 \}=\{T(v)\in V:ST(v)=0 \}\subseteq\ker(S)$$
If its right, then why $$\{T(v)\in V:ST(v)=0 \}=\ker(S)$$?
Thank you.
| What you wrote is not correct. You simply have to check that if $v$ belongs to $\mbox{Ker} S$, then $Tv$ also lies in $\mbox{Ker} S$. So assume $Sv=0$. Then
$$STv=TSv=T0=0$$
where the last equality holds because a linear transformation always sends $0$ to $0$.
Therefore $v \in \mbox{Ker} S$ implies $Tv\in \mbox{Ker} S... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true?
If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
Edit: Addionally, what happens in $M_n(\mathbb{R})$?
Nota Bene: As a corollary of the countere... | Here's an example over $\mathbb{R}$, modeled after Harald's answer: let
$$A=\pmatrix{0&-2\pi\\ 2\pi&0}.$$
Again, $e^A=I$. Now choose any $B$ that doesn't commute with $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/349180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
"answer_count": 5,
"answer_id": 2
} |
Is there an algorithm to find all subsets of a set? I'm trying to find a way to find all subsets of a set.
Is there an algorithm to calculate this?
| An algorithm is type of finite procedure operating on finite data as input and generating a finite output. So you can only have an algorithm to find the subsets of $\Sigma$ if $\Sigma$ is finite. (You've been given some hints for that case, but it is important to stress that these hints only work for finite $\Sigma$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/349220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 10,
"answer_id": 8
} |
Is $\mathbb{R}$ a subset of $\mathbb{R}^2$? Is it correct to say that $\mathbb{R}$ is a subset of $\mathbb{R}^2$? Or, put more generally, given $n,m\in\mathbb{N}$, $n<m$, is $\mathbb{R}^n$ a subset of $\mathbb{R}^m$?
Also, strictly related to that: what is then the "relationship" between the set $\{(x,0)\in\mathbb{R}^2... | I wouldn't say so, even though every onedimensional subspace of $\mathbb{R}^n$ is isomorphic to $\mathbb{R}$, but there is no natural embedding.
But a more or less funny is, that even thought nearly everyone say that $\mathbb{R}\not\subset\mathbb{R}^2$ many mathematicans say that $\mathbb{R}\subset\mathbb{C}$
even tho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Lie Groups induce Lie Algebra homomorphisms I am having a difficult time showing that if $\phi: G \rightarrow H$ is a Lie group homomorphism, then $d\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ satisfies the property that for any $X, Y \in \mathfrak{g},$ we have that $d\phi([X, Y]_\mathfrak{g}) = [d\phi(X), d\phi(Y)]_\... | Let $x \in G$. Since $\phi$ is a Lie group homomorphism, we have that
$$\phi(xyx^{-1}) = \phi(x) \phi(y) \phi(x)^{-1} \tag{$\ast$}$$
for all $y \in G$. Differentiating $(\ast)$ with respect to $y$ at $y = 1$ in the direction of $Y \in \mathfrak{g}$ gives us
$$d\phi(\mathrm{Ad}(x) Y) = \mathrm{Ad}(\phi(x)) d\phi(Y). \ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
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Best estimate for random values Due to work related issues I can't discuss the exact question I want to ask, but I thought of a silly little example that conveys the same idea.
Lets say the number of candy that comes in a package is a random variable with mean $\mu$ and a standard deviation $s$, after about 2 months of... | It depends on your definition of "better."
You need to define your risk function. If your risk function is MSE, you can do better than simply using the sample means. The idea is to use shrinkage, which as the name suggests means to shrink all your $\mu_i$ estimates slightly towards 0. The amount of shrinkage shoul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Solve $\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+\cdots\right)$ If
$$\displaystyle \frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$
then what is $x$?
I was thinking of expanding the brackets and trying to notice a pattern but as it effectively goes to infinity. I d... | Expand. The first term is $\frac{1}{2x}$.
The sum of the first two terms is $\frac{1}{2x}+\frac{1}{4x}$.
The sum of the first three terms is $\frac{1}{2x}+\frac{1}{4x}+\frac{1}{8x}$.
And so on.
The sum of the first $n$ terms is
$$\frac{1}{2x}\left(1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n-1}}\right).$$
As $n\to\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)...\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$ I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take ... | You may be interested in the asymptotic formula,
$$
{}_1F_1(a,a+b,z) = \frac{\Gamma(a+b)}{\Gamma(b)} (-z)^{-a} + O(z^{-a-1})
$$
as $\operatorname{Re} z \to -\infty$ (see, e.g., [1]).
Note, in particular, that it is not true that ${}_1F_1(a,a+b,x) \leq e^x$ for $x \in \mathbb{R}$ large and negative.
[1] Bateman Manuscri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Larger Theory for root formula Consider the quadratic equation:
$$ax^2 + bx + c = 0$$
and the linear equation:
$$bx + c = 0$$.
We note the solution of the linear equation is
$$x = -\frac{c}{b}.$$
We note the solution of the quadratic equation is
$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.
Suppose we take the limit as $a$ ... | Hint $\ $ First rationalize the numerator, then take the limit as $\rm\:a\to 0.$
$$\rm \frac{-b + \sqrt{b^2 - 4ac}}{2a}\ =\ \frac{2c}{-b -\sqrt{b^2 - 4ac}}\ \to\ -\frac{c}b\ \ \ as\ \ \ a\to 0 $$
Remark $\ $ The quadratic equation for $\rm\,\ z = 1/x\,\ $ is $\rm\,\ c\ z^2+ b\ z + a = 0\,\ $ hence
$$\rm z\ =\ \dfrac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Restricted Permutations and Combinations Tino, Colin, Candice, Derek, Esther, Mary and Ronald are famous artist. Starting next week, they will take turns to display their work
and each artist's work will be on display at the London Show for
exactly one week so that the display of the artworks will last the next seven
w... | Where do you get $2*6!$ for (a)? I find $7!$
For the three week problem, let us start by assuming $3,2,2$. We can multiply by $3$ at the end to take care of cyclic permutations of weeks. The two pairs are differently named bins in this case. The two C's can be together in one of the twos in $2\text{(which week)}*{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving the symmetry of the Ricci tensor? Consider the Ricci tensor :
$R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho}
-\partial_{\nu}\Gamma_{\rho\mu}^{\rho}
+\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda}
-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\rho\mu}^{\lambda}$
In the most general case, is this tensor symmetric ?... | This misunderstands the previous post by assuming different conventions about the ordering of indices on the curvature tensor. The previous post assumes the convention that
$$2\nabla_{[j}\nabla_{k]}v^\ell = {R^\ell}_{ijk}v^i.$$
With this convention, the argument is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/349817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$ I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much
| Let $\ell= \text{lcm}(a,b), g=\text{gcd}(a,b)$ for some $a,b$ positive integers.
Division algorithm: exists $q,r$ integers with $0\leq r < \ell$ such $ab = q\ell + r$. Observing that both $a$ and $b$ divide both $ab$ and $q\ell$ we conclude they both divide $r$. As $r$ is a common multiple, we must have $\ell \leq r$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 11,
"answer_id": 9
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Concatenation of 2 finite Automata I have some problems understanding the algorithm of concatenation of two NFAs.
For example: How to concatenate A1 and A2?
A1:
# a b
- - -
-> s {s} {s,p}
p {r} {0}
*r {r} {r}
A2:
# a b
- - -
-> s {s} {p}
p ... | We connect the accepting states of A1 to the starting point of A2. Assuming that -> means start and * means accepting state.. (I labelled the states according to the original automata, and deleted * from r1 and -> from s2, but added s2 for each possible state change to r1 (once an A1-word would be accepted, we can jump... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Applications of group cohomology to algebra I started learning about group cohomology (of finite groups) from two books: Babakhanian and Hilton&Stammbach.
The theory is indeed natural and beautiful, but I could not find many examples to its uses in algebra.
I am looking for problems stated in more classical algebraic t... | Here's a simple example off the top of my head. A group is said to be finitely presentable if it has a presentation with finitely many generators and relations. This, in particular, implies that $H_2(G)$ is of finite rank. (You can take nontrivial coefficient systems here too.) So you get a nice necessary condition for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Integrate: $\oint_c (x^2 + iy^2)ds$ How do I integrate the following with $|z| = 2$ and $s$ is the arc length? The answer is $8\pi(1+i)$ but I can't seem to get it.
$$\oint_c (x^2 + iy^2)ds$$
| Parametrize $C$ as $\gamma(t) = 2e^{it} = 2(\cos t + i \sin t)$ for $t \in [0, 2\pi]$.
From the definition of the path integral, we have:
$$
\oint_C f(z) \,ds = \int_a^b f(\gamma(t)) \left|\gamma'(t)\right| \,dt
$$
Plug in the given values to get:
\begin{align}
\oint_C (x^2 + iy^2) \,ds &= \int_0^{2\pi} 4(\cos^2{t} + i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/350137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Square root is operator monotone This is a fact I've used a lot, but how would one actually prove this statement?
Paraphrased: given two positive operators $X, Y \geq 0$, how can you show that $X^2 \leq Y^2 \Rightarrow X \leq Y$ (or that $X \leq Y \Rightarrow \sqrt X \leq \sqrt Y$, but I feel like the first version wou... | Here is a proof which works more generally for $x,y\ge 0$ in a $C^*$-algebra, where the spectral radius satisfies $r(z)\le \|z\|=\sqrt{\|z^*z\|}$ for every element $z$, and $r(t)=\|t\|$ for every normal element $t$. The main difference with @user1551's argument is that we will use the invertibility of $y$. Other than t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/350188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Order of an element in a group G Suppose that $a$ is an element of order $n$ in a group $G$. Prove:
i) $a^i = a^j$ if and only if $i \equiv j \pmod n$;
ii) if $d = (m,n)$, then the order of $a^m$ is $n/d$;
I was trying to self teach myself this and came to this question. How would you solve this? Can someone please sho... | If $i \equiv j \pmod n$, then $i = j + kn$ for some $k \in \Bbb Z$. It follows that:
$$
a^i = a^{j + kn} = a^ja^{kn} = a^j (a^n)^k = a^j
$$
If you haven't proved the power properties $a^{p+q} = a^pa^q$ and $a^{pq} = (a^p)^q$ for $p, q \in \Bbb Z^+$, this is a good exercise to do now. Try using induction.
Now, if $a^i =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What series test would you use on these and why? I just started learning series , I am trying to put everything together...I have few some few random problems just to see what kind of strategy you would use here...
*
*$\displaystyle\sum_{n=1}^\infty\frac{n^n}{(2^n)^2}$
*$\displaystyle\sum_{n=1}^\infty\frac2{(2n - 1... | General and mixed hints:
$$\frac{2}{(2n-1)(2n+1)}=\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)$$
$$\frac{n}{4}\xrightarrow[n\to\infty]{}\infty$$
$$\frac{2^n}{2^n(1+n^2\log^22)}\le\frac{1}{\log^22}\cdot\frac{1}{n^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/350401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Can the matrix $A=\begin{bmatrix} 0 & 1\\ 3 & 3 \end{bmatrix}$ be diagonalized over $\mathbb{Z}_5$? Im stuck on finding eigenvalues that are in the field please help.
Given matrix:
$$
A= \left[\begin{matrix}
0 & 1\\
3 & 3
\end{matrix}\right]
$$
whose entries are from $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$, find, if possi... | yes over $\Bbb Z_5$ because:
$\lambda^2 -3\lambda-3=o$ at Z_5 we will have $\Delta=9+12=4+2=6$ (9~4 and 12~2 at Z_5)
so $\Delta=1$
and so $\lambda_1=\frac{3+1}{2}=2$ and $\lambda_2=\frac{3-1}{2}=1$
about:
$\lambda_1$ we have :$ ( \left[\begin{matrix}
0 & 1\\
3 &3
\end{matrix}\right]-\left[\begin{matrix}
2 & \\
0 &2
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/350470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Separating $\frac{1}{1-x^2}$ into multiple terms I'm working through an example that contains the following steps:
$$\int\frac{1}{1-x^2}dx$$
$$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$
$$\ldots$$
$$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$
I don't understand why the separation works. If I attempt to re-combine the ter... | The thing is $$\frac{1}{1-x}\color{red}{+}\frac 1 {1+x}=\frac{2}{1-x^2}$$
What you might have seen is
$$\frac{1}{x-1}\color{red}{-}\frac 1 {x+1}=\frac{2}{1-x^2}$$
Note the denominator is reversed in the sense $1-x=-(x-1)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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PRA: Rare event approximation with $P(A\cup B \cup \neg C)$? The rare event approximation for event $A$ and $B$ means the upper-bound approximation $P(A\cup B)=P(A)+P(B)-P(A\cap B)\leq P(A)+P(B)$. Now by inclusion-exclusion-principle $$P(A\cup B\cup \neg C)= P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap \neg C) -P(B\cap\neg C) +P(... | Removing all the negatives certainly gives an upper bound. But if one looks at the logic of the inclusion-exclusion argument, whenever we have just added, we have added too much (except possibly, at the very end). So at any stage just before we start subtracting again, our truncated expression gives an upper bound.
Th... | {
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We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece We break a unit length rod into two pieces at a uniformly chosen point. Find the
expected length of the smaller piece
| With probability ${1\over2}$ each the break takes place in the left half, resp. in the right half of the rod. In both cases the average length of the smaller piece is ${1\over4}$. Therefore the overall expected length of the smaller piece is ${1\over4}$.
| {
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Section of unions of open subschemes I'm stuck at a line in Hartshorne's text (p.g. 82). Could someone help me please?
Fact. Suppose that $X$ is a scheme having $U$ and $V$ as two non-empty disjoint open subsets of $X$. Then $\mathcal{O}_X(U \cup V) = \mathcal{O}(U) \times \mathcal{O}_X(V)$.
I know how to prove this wh... | There is a canonical homomorphism $(\rho^{U\cup V}_U, \rho^{U\cup V}_V) : \Gamma(U\cup V) \to \Gamma(U) \times \Gamma(V)$ induced by the restriction homomorphisms. This being an isomorphism follows directly from the fact that the structure sheaf is a sheaf: injectivity is precisely the fact that a section of $U \cup V... | {
"language": "en",
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Reasoning about the gamma function using the digamma function I am working on evaluating the following equation:
$\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$
If I'm understanding correctly, the above is an increasing function which can be demonstrated by the following argument using the digamma function $\frac... | This answer is provided with help from J.M.
$\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$ is an increasing function. This can be shown using this series for $\psi$:
The function is increasing if we can show: $\frac{d}{dx}(\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)) > 0$
We can show this using the diga... | {
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Name of $a*b=c$ and $b*a=-c$ $A_+=(A,+,0,-)$ is a noncommutative group where inverse elements are $-a$
$A_*=(A,*)$ is not associative and is not commutative
$\mathbf A=(A,+,*)$ is a structure where
1) if $a*b=c$ then $b*a=-c$ holds and
2) $(a*b)+a=b+(a*b)$
A-How is called the structure $\mathbf A$?
B-What is the name ... | From 2) we have $(a*0)+a = 0+(a*0) = a*0 = (a*0)+0$ which contradicts the fact that $(A,+)$ is a group (which implies "left-multiplication" by $(a*0)$, i.e. $x \mapsto (a*0)+x$, is injective). Thus the structure $(A,+,*)$ cannot exist.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rank of a matrix. Let a non-zero column matrix $A_{m\times 1}$ be multiplied with a non-zero row matrix $B_{1\times n}$ to get a matrix $X_{m\times n}$ . Then how to find rank of $X$?
| Let me discuss a shortcut for finding the rank of a matrix .
Rank of a matrix is always equal to the number of independent equations .
The number of equations are equal to the number of rows and the variables in one equation are equal to number of columns .
Suppose there is a 3X3 matrix with elements as :
row 1 : 1 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to compute the area of the shadow?
If we can not use the integral, then how to compute the area of the shadow?
It seems easy, but actually not?
Thanks!
| Let $a$ be a side of the square. Consider the following diagram
The area we need to calculate is as follows.
$$\begin{eqnarray} \color{Black}{\text{Black}}=(\color{blue}{\text{Blue}}+\color{black}{\text{Black}})-\color{blue}{\text{Blue}}. \end{eqnarray}$$
Note that the blue area can be calculated as
$$\begin{eqnarray}... | {
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Rudin Theorem 1.35 - Cauchy Schwarz Inequality Any motivation for the sum that Rudin considers in his proof of the Cauchy-Schwarz Inequality?
Theorem 1.35 If $a_1,...,a_n$ and $b_1, ..., b_n$ are complex numbers, then
$$\Biggl\vert\sum_{j=1}^n a_j\overline{b_j}\Biggr\vert^2 \leq \sum_{j=1}^n|a_j|^2\sum_{j=1}^n|b_j|^2.... | He does it because it works. Essentially, as you see, $$\sum_{j=1}^n |Ba_j-Cb_j|^{2}$$ is always greater or equal to zero. He then shows that $$\tag 1 \sum_{j=1}^n |Ba_j-Cb_j|^{2}=B(AB-|C|^2)$$
and having assumed $B>0$; this means $AB-|C|^2\geq 0$, which is the Cauchy Schwarz inequality.
ADD Let's compare two differen... | {
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Find a simple formula for $\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$
$$\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$$
All I could think of so far is to turn this expression into a sum. But that does not necessarily simplify the expression. Pl... | Hint: it's the coefficient of $T$ in the binomial expansion of $(1+T)^n(1+T^{-1})^n$, which is equivalent to saying that it's the coefficient of $T^{n+1}$ in the expansion of $(1+T)^n(1+T^{-1})^nT^n=(1+T)^{2n}$.
| {
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Solve recursive equation $ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1$ Solve recursive equation:
$$ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1$$
$f_0 = 0, f_1 = 1$
What I have done so far:
$$ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1- [n=0]$$
I multiplied it by $n$ and I have obtained:
$$... | Let's take a shot at this:
$$
f_n - f_{n - 1} = \frac{n - 1}{n} (f_{n - 1} - f_{n - 2}) + 1
$$
This immediately suggests the substitution $g_n = f_n - f_{n - 1}$, so $g_1 = f_1 - f_0 = 1$:
$$
g_n - \frac{n - 1}{n} g_{n - 1} = 1
$$
First order linear non-homogeneous recurrence, the summing factor $n$ is simple to see he... | {
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Integral solutions of hyperboloid $x^2+y^2-z^2=1$ Are there integral solutions to the equation $x^2+y^2-z^2=1$?
| We can take the equation to $x^2 + y^2 = 1 + z^2$ so if we pick a $z$ then we just need to find all possible ways of expressing $z^2 + 1$ as a sum of two squares (as noted in the comments we always have one way: $z^2 + 1$). This is a relatively well known problem and there will be multiple possible solutions for $x$ an... | {
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Bounded partial derivatives imply continuity As stated in my notes:
Remark: Suppose $f: E \to \mathbb{R}$, $E \subseteq \mathbb{R}^n$, and $p \in E$. Also, suppose that $D_if$ exists in some neighborhood of $p$, say, $N(p, h)$ where $h>0$. If all partial derivatives of $f$ are bounded, then $f$ is continuous on $E$.... | The proof is a combination of two facts:
*
*A function of one real variable with a bounded derivative is Lipschitz.
*Let $Q\subset \mathbb R^n$ be a cube aligned to coordinate axes. If a function $f:Q\to\mathbb R$ is Lipschitz in each variable separately, then it is Lipschitz.
The proof of 2 involves a telescopi... | {
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How to prove boundary of a subset is closed in $X$? Suppose $A\subseteq X$. Prove that the boundary $\partial A$ of $A$ is closed in $X$.
My knowledge:
*
*$A^{\circ}$ is the interior
*$A^{\circ}\subseteq A \subseteq \overline{A}\subseteq X$
My proof was as follows:
To show $\partial A = \overline{A} \setminus A^... | From your definition, directly,
$$
\partial A=\overline{A}\setminus \mathring{A}=\overline{A}\cap (X\setminus \mathring{A})
$$
is the intersection of two closed sets. Hence it is closed.
No need to prove that the complement is open, it just makes it longer and more complicated.
Also, keep in mind that a set $S$ is ope... | {
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Matrix Manifolds Question I am not sure at all how to do the following question. Any help is appreciated. Thank you.
Consider $SL_n \mathbb{R}$ as a group and as a topological space with
the topology induced from $R^{n^2}$. Show that if $H \subset SL_n \mathbb{R}$ is an abelian subgroup, then the closure $H$ of $SL_... | Hint: The map $\overline{H}\times \overline{H}\to \overline{H}$ defined by $(a,b)\mapsto aba^{-1}b^{-1}$ is continuous. Since $\overline{H}$ is Hausdorff, and the map is constant on a dense subset of its domain, it must be constant everywhere.
| {
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"question_score": "2",
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Why does the series $\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$ converge? Why does this series
$$\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$$
converge? Can't you use a limit comparison with $1/n$?
| Note that $$\cos(n\pi/3) = 1/2, \ -1/2, \ -1, \ -1/2, \ 1/2, \ 1, \ 1/2, \ -1/2, \ -1, \ \cdots $$ so your series is just 3 alternating (and convergent) series inter-weaved. Exercise: Prove that if $\sum a_n, \sum b_n$ are both convergent, then the sequence $$a_1, a_1+b_1, a_1+b_1+a_2, a_1+b_1+a_2+b_2, \cdots ... | {
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Probability that a stick randomly broken in five places can form a tetrahedron Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here.
Randomly break a stick in five places.
Question: What is the probability that the res... | if stick pieces are s1 (longest) to s6 shortest.
Picture the tetrahedron with longest side s1 out of view. Then s2 is the spine and any combination of pairs from {s3,s4,s5,s6} can make the two side triangles Hence s3+s6 needs to be longer than s2 (P=0.25) And s4+s5 needs to be longer than s2. (P=0.25)
so P(can form)=0.... | {
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Transitive closure proof (Pierce, ex. 2.2.7) Simple exercise taken from the book Types and Programming Languages by Benjamin C. Pierce.
This is a definition of the transitive closure of a relation R.
First, we define the sequence of sets of pairs:
$$R_0 = R$$
$$R_{i+1} = R_i \cup \{ (s, u) | \exists t, (s, t) \in R... | We need to show that $R^+$ contains $R$, is transitive, and is minmal among all such relations.
$R\subseteq R^+$ is clear from $R=R_0\subseteq \bigcup R_i=R^+$.
Transitivity:
By induction on $j$, show that $R_i\subseteq R_j$ if $i\le j$.
Assume $(a,b), (b,c)\in R^+$. Then $(a,b)\in R_i$ for some $i$ and $(b,c)\in R_j$ ... | {
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Orthogonality of Legendre Functions The Legendre Polynomials satisfy the following orthogonality condition:
The definite integral of $P(n,x) \cdot P(m,x)$ from $-1$ to $1$ equals $0$, if $m$ is not equal to $n$: $$\int_{-1}^1 P(n,x) \cdot P(m,x) dx = 0. \qquad (m \neq n)$$
Based on this, I am trying to evaluate the i... | Integration by parts is $\int f'g+\int fg'=fg\ (+C)$, so for the definite integral, it is
$$\int_a^b f'g+\int_a^b fg'=[fg]_a^b=f(b)g(b)-f(a)g(a)\,.$$
Now we have $f=x$ and $g'=P_{n-1}(x)\cdot P_n(x)$. That is, $g$ is the antiderivative of $P_{n-1}\cdot P_n$. By the definite integral of this, it only allows us to conclu... | {
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Absolute convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$ This sum
$$\sum_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$$
apparently converges absolutely, but I'm having trouble understanding how so.
First of all, doesn't it already fa... | Since $$1-\cos{x}\underset{x\to{0}}{\sim}{\dfrac{x^2}{2}}\;\; \Rightarrow \;\; \cos{\dfrac{1}{n}}={1-\dfrac{1}{2n^2}} +o\left(\dfrac{1}{n^2} \right),\;\; n\to\infty$$
and
$$\ln(1+x)\underset{x\to{0}}{\sim}{x},$$
thus $$\ln\left(\cos { \dfrac{1}{n} }\right)\underset{n\to{\infty}}{\sim}{-\dfrac{1}{2n^2}}.$$
| {
"language": "en",
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How will studying "stochastic process" help me as mathematician?? I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University.
I can make an un-educated guess that stochastic processes are important in mathematics. But I am also curious to ... | such a similar question
stochastic process is very usefull in Acturial Sience, Mathematical finance.
| {
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Counting strictly increasing and non-decreasing functions $f$ is non-decreasing if $x \lt y$ implies $f(x) \leq f(y)$ and increasing if $x < y$ implies $f(x) < f(y)$.
*
*How many $f: [a]\to [b]$ are nondecreasing?
*How many $f: [a] \to [b]$ are strictly increasing?
Where $[a]=\{1,2\ldots a\}$ and $[b]=\{1,2... | Strictly increasing is easy: we need to choose the $n$ items in $[k]$ that will be the range of our function.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Parallel transport for a conformally equivalent metric Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a smooth positive function. Let $P_c'$ denote parallel transport alo... | Both the parallel transport and the connection map are determined by the connection, in your case this is the Levi-Civita connection of metric $g$ whose transformation is known (see e.g. this answer).
For the connection map you already have a formula in the definition, just use the facts and get the expression.
With re... | {
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Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$ How can I prove that for every $ x \notin \mathbb Z$ the series
$$ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$$
converges uniformly in a neighborhood of $ x $?
| Apart from the first few summands, we have $n^2-y^2>\frac12n^2$ for all $y\approx x$, hence the tail is (uniformly near $x$) bounded by $2\sum_{n>N}\frac1{n^2}$.
| {
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Law of Quadratic Reciprocity Equivalent Statement
Let $p,q$ be two distinct odd primes. Then $(\frac{q}p)=1 \iff p=\pm\beta^2 \pmod{4q}$ for some odd $\beta$. Show that this statement is eqivalent to the Law of Quadratic Reciprocity.
I'm trying to grapple with what the question is actually asking me to show.
Do I sp... | We do one of the four cases. Because $p$ and $q$ both of the shape $4k+1$ is "too easy" and does not fully illustrate the problems we can bump into, we deal with the case $p$ of the form $4k+3$ and $q$ of the form $4k+1$.
Suppose that $(q/p)=1$, with $p$ of the form $4k+3$ and $q$ of the form $4k+1$. We want to show ... | {
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Notation for "absolute value" in multiplicative group. In an additive number group (e.g. $(\mathbb{Z},+)$) there is a well known notation for absolute value, namely $|a|$, which coincides with $\max(a,-a)$, for $a \in \mathbb{Z}$.
When the context is a multiplicative number group instead, is there a similar notation, w... | If you're working with a multiplicative group $G\subseteq\Bbb{R}$, you can definitely say
$$
\operatorname{abs}(g) := \max\{g,g^{-1}\}\quad\textrm{for }g\in G.
$$
The question is whether or not it is useful to the study of the group $G$ in any way.
Also, when it comes to the question of notation, $\left|g\right|$ is no... | {
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Prove that $2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$ I am utterly new to modular arithmetic and I am having trouble with this proof.
$$2222^{5555}+5555^{2222}=3333^{5555}+4444^{2222} \pmod 7$$
It's because $2+5=3+4=7$, but it's not so clear for me with the presence of powers.
Maybe some explanation woul... | First recall that as $7$ is prime, then $x^6 = 1 \pmod{7}$. Now, we have
$$ 2222 = \begin{cases} 2 \pmod{6} \\ 3 \pmod{7} \end{cases}, \quad 3333 = 1 \pmod{7}$$
$$4444 = -1 \pmod{7}, \quad 5555 = \begin{cases} 5 \pmod{6} \\ 4 \pmod{7} \end{cases}$$
Then we can reduce each side of the equation to
$$ 3^5 + 4^2 = 1^5 + (... | {
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Darts on a ruler probability If two points are selected at random on an interval from 0 to 1.5 inches, what is
the probability that the distance between them is less than or equal to 1/4"?
| Draw the square with corners $(0,0)$, $(1.5.0)$, $(1.5,1.5)$, and $(0,1.5)$.
Imagine the points are chosen one at a time. Let random variable $X$ be the first chosen point, and $Y$ the second chosen point. We are invited to assume that $X$ and $Y$ are uniformly distributed in the interval $[0,1.5]$ and independent. (Un... | {
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Exact number of events to get the expected outcoms Suppose in a competition 11 matches are to be played, each having one of 3
distinct outcomes as possibilities. How many number of ways one can predict the
outcomes of all 11 matches such that exactly 6 of the predictions turn out to
be correct?
| The $6$ matches on which our prediction is correct can be chosen in $\binom{11}{6}$ ways. For each of these choices, we can make wrong predictions on the remaining $5$ matches in $2^5$ ways. Thus the total number is
$$\binom{11}{6}2^5.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/352771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$
I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)... | Hint:
$$ 0 \leq \lim_{n\to \infty}\frac{n!}{(2n)!} \leq \lim_{n\to \infty} \frac{n!}{(n!)^2} = \lim_{k \to \infty, k = n!}\frac{k}{k^2} = \lim_{k \to \infty}\frac{1}{k} = 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/352849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$? Is the vector $(3,-1,0,-1)$ in the subspace of $\Bbb R^4$ spanned by the vectors $(2,-1,3,2)$, $(-1,1,1,-3)$, $(1,1,9,-5)$?
| To find wether $(3,-1,0,-1)$ is in the span of the other vectors, solve the system:
$$(3,-1,0,-1)=\lambda_1(2,-1,3,2)+\lambda _2(-1,1,1,-3)+\lambda _3(1,1,9,-5)$$
If you get a solution, then the vector is the span. If you don't get a solution, then it isn't.
It's worth noting that the span of $(2,-1,3,2), (-1,1,1,-3), ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Rate of change of a cubes area in respect to the space diagonal The space diagonal of a cube shrinks with $0.02\rm m/s$. How fast is the area shrinking when the space diagonal is $0.8\rm m$ long?
I try:
Space Diagonal = $s_d = \sqrt{a^2+b^2+c^2}=\sqrt{3a^2}$ Where $a$ is the length of one side.
Area = $a^2$
Rate of ... | The simplest way is to express the area $a$ of the cube as a function of the length $d$ of the space diagonal. Given $d$ the side length $s$ of the cube is
$$s={1\over\sqrt{3}} \ d\ ,$$
and the total surface area $a$ then becomes
$$a=6s^2=2 d^2\ .$$
Now all quantities appearing here are in fact functions of $t$; theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Fourier series of a function Consider $$ f(t)= \begin{cases} 1 \mbox{ ; } 0<t<1\\ 2-t \mbox{ ; } 1<t<2 \end{cases}$$
Let $f_1(t)$ be the Fourier sine series and $f_2(t)$ be the Fourier cosine series of $f$, $f_1(t)=f_2(t), 0<t<2$. Write the form of the series (without computing the coefficients) and graph $f_1$ and $f... | Ok at first we gonna plot our function
We know that on jump discontinuities it will converge to the arithmetic mean of them, so the first approximation is just taking $\frac{1}{2}$.
This gonna look like
The Cos terms gonna look like
The Sin terms are looking like
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/353044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generating functions. Number of solutions of equation.
Let's consider two equations
$x_1+x_2+\cdots+x_{19}=9$, where
$x_i \le 1$
and
$x_1+x_2+\cdots+x_{10}=10, $ where $ x_i \le 5$
The point is to find whose equation has greater number of solutions
What I have found is:
number of solutions for first equation: $\binom... | Generating functions is a generic way.
To continue on your attempt, you can apply Binomial theorem, which applies to negative exponents too!
We have that
$$ (1-x)^{-r} = \sum_{n=0}^{\infty} \binom{-r}{n} (-x)^n$$
where
$$\binom{-r}{n} = \dfrac{-r \times (-r -1) \times \dots \times (-r - n +1)}{n!} = (-1)^n\dfrac{r(r+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
True or False: Every finite dimensional vector space can made into an inner product space with the same dimension. Every finite dimensional vector space can made into an inner product space with the
same dimension.
| I think it depends on what field you are using for your vector space. If it is $\mathbf{R}$ or $\mathbb{C}$, the answer is definitely "yes" (see the comments, which are correct). I am pretty sure it is "yes" if your field is a subfield of $\mathbb{C}$ that is closed (the word "stable" also seems to be standard) under... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Solution of a Sylvester equation? I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer product matrix (i.e, as $vv^T$ for some real vector $v$) while $D$ is 'not' symmetric. $... | More generally, Sylvester's equation of the form
$$AX+XB=C$$ can be put into the form
$$M\cdot \textrm{vec}X=L$$ for larger matrices $M$ and $L$.
Here $\textrm{vec}X$ is a stack of all columns of matrix $X$.
How to find the matrix $M$ and $L$, is shown in chapter 4 of this book: http://www.amazon.com/Topics-Matrix-A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int_0^\pi\frac{1}{1+(\tan x)^\sqrt2}\ dx$ How can we evaluate $$\int_0^\pi\frac{1}{1+(\tan x)^\sqrt2}\ dx$$ Can you keep this at Calculus 1 level please? Please include a full solution if possible. I tried this every way I knew and I couldn't get it.
| This is a Putnam problem from years ago. There is no Calc I solution of which I'm aware. You need to put a parameter (new variable) in place of $\sqrt 2$ and then differentiate the resulting function of the parameter (this is usually called "differentiating under the integral sign"). Most students don't even learn this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Homeomorphism vs diffeomorphism in the definition of k-chain In "Analysis and Algebra on Differentiable Manifolds", 1st Ed., by Gadea and Masqué, in Problem 3.2.4, the student is asked to prove that circles can not be boundaries of any 2-chain in $\mathbb{R}^2-\{0\}$. I understand the solution which makes use of the di... | As discussed in the comments, the unit circle defines a singular 1-simplex, $\sigma$ (i.e. a continuous map from the closed interval) which is not the boundary of any singular 2-chain (i.e. any formal sum of continuous maps from the standard 2-simplex) in the punctured plane. One way to see this is by noting that the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Write in array form, product of disjoint cycles, product of 2-cycles... In symmetric group $S_7$, let $A= (2 3 5)(2 7 5 4)$ and $B= (3 7)(3 6)(1 2 5)(1 5)$.
Write $A^{-1}$, $AB$, and $BA$ in the following ways:
(i) Array Form
(ii) Product of Disjoint Cycles
(iii) Product of $2$-Cycles
Also are any of $A^{-1}, AB$, or... | I won’t do the problem, but I will answer the same questions for the permutation $A$; see if you can use that as a model. I assume throughout that cycles are applied from left to right.
$A=(235)(2754)=(2345)(2754)(1)(6)$; that means that $A$ sends $1$ to $1$, $2$ to $3$, $3$ to $5$ to $4$, $4$ to $2$, $5$ to $2$ to $7$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show measurability of a function implies existence of bounding simple functions If $(X,\mathscr{M},\mu)$ is a measure space with $\mu(X) < \infty$, and $(X,\overline{\mathscr{M}},\overline{\mu})$ is its completion and $f\colon X \to \mathbb{R}$ is bounded. Then $f$ is $\overline{\mathscr{M}}$-measurable (and hen... | A set $N$ is $(\mathcal M,\mu)$-negligible if we can find $N'\in\mathcal M$ such that $\mu(N')=0$ and $N\subset N'$.
Recall that
$$\overline{\mathcal M}^{\mu}=\{B\cup N,B\in\mathcal M,N\mbox{ is }(\mathcal M,\mu)-\mbox{negligible}\}.$$
It can indeed be shown that the latter collection is a $\sigma$-algebra, the small... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How use Maple12 to solve a differential equation by using Euler's method? Consider the differential equation $y^{\prime}=y-2$ with initial condition $y\left(0\right)=1$.
a) Use Euler's method with 4 steps of size 0.2 to estimate $y\left(0.8\right)$
I know how to do this by hand; however, I have maple 12 installed and... | maybe this would help ,just change initially condition and step size
http://homepages.math.uic.edu/~hanson/MAPLE/euler.html
i am not sure that is is for maple12,but i think commands would be same,just try it and if there is errors,post here
use also
https://stackoverflow.com/questions
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/353704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Notation for $X - \mathbb{E}(X)$? Let $X$ be a random variable with expectation value $\mathbb{E}(X)=\mu$.
Is there a (reasonably standard) notation to denote the "centered" random variable $X - \mu$?
And, while I'm at it, if $X_i$ is a random variable, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, and if $\overl... | I've never seen any specific notation for these. They are such simple expressions that there wouldn't be much to gain by abbreviating them further. If you feel you must, you could invent your own, or just say "let $Y = X - \mu$".
One way people often avoid writing out $X - \mu$ is by a statement like "without loss of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Clarification of sequence space definition Let $(x_n)$ denote a sequence whose $n$th term is $x_n$, and $\{x_n\,:\,n\in\mathbb{N}\}$ denote the set of all elements of the sequence. I have a text that states
Note that $\{x_n\,:\,n\in\mathbb{N}\}$ can be a finite set even though $(x_n)$ is an infinite sequence.
To me ... | A sequence of real numbers is a function, not a set. Thus, for instance, the sequence $(x_n)$ is actually a function $f:\mathbb N \to \mathbb R$, where we have the equality $x_n =f(n)$. Now, the image of the function is the set $\{x_n\mid n\in \mathbb N\}$, which is a very different thing. An example where this associa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Euclidean cirle question Let $c_1$ be a circle with center $O$. Let angle $ABC$ be an inscribed angle of the circle $c_1$.
i) If $O$ and $B$ are on the same side of the line $AC$, what is the relationship between $\angle ABC$ and $ \angle AOC$?
ii) If $O$ and $B$ are on opposite side of the line $AC$, what is the rela... |
Here $O$ and $B$ are on the same side of of the line $AC$, you can figure out the other part.
$\angle AOB=180-2y$ and $\angle COB=180-2x$
$\angle AOB+ \angle COB=360-2(x+y) \implies \angle AOC=2(x+y) \implies 2 \angle ABC$.
This is known widely as Inscribed Angle theorem as RobJohn said in his comment.:)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/353876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Mystery about irrational numbers I'm new here as you can see.
There is a mystery about $\pi$ that I heard before and want to check if its true. They told me that if I convert the digits of $\pi$ in letters eventually I could read the Bible, any book written and even the history of my life! This happens because $\pi$ is... |
$\pi$ is just another number like $5.243424974950134566032 \dots$, you can use your argument here. Continue your number with number of particles of universe, number of stars, number of pages in bible, number of letters in the bible, and so on. And 'DO NOT STOP DOING SO, if you do the number becomes rational'.
There a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Calculus world problem expansion of air How would I solve this problem?
The adiabatic law for expansion of air is $P(V)^{1.4}=C$ when P is pressure V is volume and C is a certain constant.At a given instant the volume is 30 cubic feet and the pressure is 60 psi. At what rate is the pressure changing if the volume is de... | Note:
$PV^{\gamma}=C \implies \dfrac{dP}{dt}\cdot V^{\gamma}+(\gamma)V^{(\gamma-1)} \cdot \dfrac{dV}{dt}\cdot P=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/353989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove two inequalities about limit inferior and limit superior I wish to prove the following two inequalities:
Suppose $X$ is a subset in $\Bbb R$, and functions $f$ and $g$: $X\to \Bbb R$, and $x_{0}\in X$ is a limit point. Then: $$\lim\sup_{x\to x_0}(f(x)+g(x))\le \lim\sup_{x\to x_0}(f(x))+\lim\sup_{x\to x_0}(g(x))... | Contradiction is not recommended, as there is a natural direct approach.
1- limsup: recall that
$$
\limsup_{x\rightarrow x_0}h(x)=\inf_{\epsilon>0}\sup_{0<|x-x_0|<\epsilon}h(x)=\lim_{\epsilon>0}\sup_{0<|x-x_0|<\epsilon}h(x)
$$
where the rhs is the limit of a nonincreasing function of $\epsilon$. Note that the conditio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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