Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Convergence to a distribution implies convergence of a logarithm? Let $X_n$ be a sequence of almost surely positive real-valued random variables s.t. $$\sqrt{n} \, \left( X_n -a \right) \to_D \mathcal{N} ( 0, 1)$$
where $\to_D$ denotes convergence in distribution and $a>0$. Now I'm interested in what happens to$$\sqrt{... | Indeed $\sqrt{n}\log(X_n/a)$ converges in distribution to a Gaussian $\mathcal{N}(0,1/a^2)$. One way to prove is to use the identity:
$$ \frac{x}{1+x} \leq \log(1+x) \leq x $$
which holds for all $x>-1$ (i.e., whenever $\log(1+x)$ is nicely defined).
So now define $G_n = \sqrt{n}(X_n-a)$. Then $\log(X_n/a) = \log(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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I am trying to find the limit of P(x) When I am looking for a $\lim\limits_{x \to -1} P(x)$
where P(x)$= \sum \limits_{n=1}^\infty \left( \arctan \frac{1}{\sqrt{n+1}} - \arctan \frac{1}{\sqrt{n+x}}\right) $ do I have to ignore a summation sign sigma and dealing with it like this $ \
\lim\limits_{x \to -1}(\arctan\sqrt... | We write $P(-1^+)$ as
$$\begin{align}
P(-1^+)&=\arctan(1/\sqrt{2})-\pi/2+\sum_{n=2}^{\infty}\left(\arctan\left(\frac{1}{\sqrt{n+1}}\right)-\arctan\left(\frac{1}{\sqrt{n-1}}\right)\right)\\\\
&=\sum_{n=1}^{\infty}\left(\arctan \sqrt{n-1}-\arctan \sqrt{n+1}\right)\\\\
&=\lim_{N\to \infty}\sum_{n=1}^{N}\left(\arctan \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is there a theory of transcendental functions? Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus books or special functions). On the other hand, there's many books and articl... | The class of all functions is just too wild to study in general, so usually we focus on studying large collections of functions that still have certain nice properties. For example: algebraic, continuous, differentiable, Borel, measurable, . . .
"Transcendental" just means "anything not algebraic," so that's too broad.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Uniform distribution on $\{1,\dots,7\}$ from rolling a die This was a job interview question someone asked about on Reddit. (Link)
How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die?
Presumably, you are to do this by combining repeated i.i.d draws from $\{1,\dots,6\}$ (and not some Rube-Gol... | The easiest way: roll the dice twice. Now, there are 36 possible pairs. Delegate five pairs to each of 1, 2, 3, 4, 5, 6, 7 - then, if you get the remaining pair, roll the dice again. This guarantees equal probability for each of the 7 values. Note that the probability you don't have to roll again is $35/36$, so the exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$A=\{A,\emptyset\}$ and axiom of regularity The axiom of regularity says:
(R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$.
From (R) it follows that there is no infinite membership chain (imc).
Consider this set: $A=\{A,\emptyset\}$.
I am confused because it seems to me that A violates and ... | Suppose there were a set $A$ with the property $A=\{A,\emptyset\}$. Clearly $A \not = \emptyset$ since $A$ would have at least one element and possibly two.
Now let $x=\{A\}$. Clearly $x \not = \emptyset$ since $x$ would have exactly one element. So the axiom of regularity would imply $\exists y\,(y\in x\land x\cap y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why does the derivative of sine only work for radians? I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm discovering that I've been wrong all along! I'm guessing that when you dif... | It works out precisely because
$$ \lim_{x \to 0} \frac{\sin x}{x} = 1,$$
which in turns happens precisely because we've chosen our angle to be the same as the arclength around a unit circle (and for small angles, the arc is essentially a straight line).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
"answer_count": 17,
"answer_id": 8
} |
Quotients of Solvable Groups are Solvable I just proved that subgroups of solvable groups are solvable. So given that $G$ is solvable there is $1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G$ where $G_{i+1}/G_i$ is abelian and for $N$ a normal subgroup we know it is solvable so there is $1=N_0 \unrhd N_1 \unrhd \cdots \un... | The "standard" proof:
Consider, for each $i$, the subgroups $G_iN/N$ of $G/N$. It is straight-forward to show that $G_iN$ is normal in $G_{i+1}N$ (and thus $G_iN/N$ is normal in $G_{i+1}N/N$).
Now $\dfrac{G_{i+1}N/N}{G_iN/N} \cong G_{i+1}N/G_iN$.
Taking any $x,y \in G_{i+1}$, and $n,n' \in N$, we see that:
$[xn(G_iN),y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
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Circular uses of L’Hôpital’s rule If you try to find the $\lim_{x\to \infty}\frac{\sqrt{x^2+1}}{x}$
using L’Hôpital’s rule, you’ll find that it flip-flops back and forth between $\frac{\sqrt{x^2+1}}{x}$ and $\frac{x}{\sqrt{x^2+1}}$.
Are there other expressions that do a similar thing when L’Hôpital’s rule is applied to... | You're basically asking if we can find functions $f, g$ such that:
$$\frac{f'}{g'} = \frac{g}{f}$$ i.e.
such that $$ff'=gg'$$
And in this particular case, you have $ff'=x$, of which the solutions are of the form $x \mapsto \sqrt{x^2 + c}$ (you can see that $x \mapsto x$, $x>0$ is a particular case when $c=0$).
Then you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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(Is it a set?) Set of all months having more than 28 days.
Set is a well defined collection of distinct objects.
Is the following is a set?
Set of all months having more than 28 days.
I'm confused here. Because on one hand I think that it is well defined because from person to person its meaning is not changed. On th... | Given that you say that the "Set of eleven best cricketers of the world [...] is not a set because the criteria for best cricketer changes from person to person," then wouldn't the "Set of all months having more than 28 days" also be not a set because the number of days of the month changes from year to year?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
find coordinate on line at given distance from given coordinate I got two coordinates of a straight line $(-2,-4)$ and $(3,4)$. How can i find a coordinate that lies on this line and is $5$ units away from the $(-2,-4)$ coordinate?
| The line passing through two points $(-2,-4),(3,4)$ can be expressed as
$$y-4=\frac{4-(-4)}{3-(-2)}(x-3),$$
i.e.
$$y=\frac{8}{5}x-\frac 45.$$
So, every point on this line can be expressed as $(t,\frac{8}{5}t-\frac 45)$ for some $t\in\mathbb R$.
Now, solve the following equation for $t$ :
$$5=\sqrt{(t-(-2))^2+\left(\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1339965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof: If $Ax=b$ has more than one solution, it has infinitely many solutions This is a follow up question to a question I asked yesterday: Link
I want to prove the following statement:
Let $A$ be an arbitrary $n \times n$ matrix, and let $Ax=0$ have more
than one solution then it follows that $Ax=b$ van be solved f... | The statement underlined in skin-tone is blatantly false. If $Ax=0$ has more than one solution, which implies: at least one solution $k\ne0$, then ${\rm dim\>ker}(A)\geq1$. Therefore $V:={\rm im}(A)$ has dimension $$\dim (V)=n-\dim\ker(A)<n\ .$$ It follows that ${\mathbb R}^n\setminus V$ contains infinitely many vecto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Galois group and traslations by rational numbers. Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$.
Now my question:
Given a polynomial $g(x) \in \mathbb{Q}[x]$ of degree $n$ is true
that... | No. Take $g(x)=x^4$, and consider the polynomial $x^4+q$, where $q$ is a rational number.
If $q$ is negative, then the roots of $x^4+q$ in $\mathbb{C}$ are $\sqrt[4]{-q}, i\sqrt[4]{-q}, -\sqrt[4]{-q}$ and $-i\sqrt[4]{-q}$, where $\sqrt[4]{-q}$ is the positive real fourth root of $-q$. Then the decomposition field of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Probability Modem is Defective
A store has 80 modems in its inventory, 30 coming from Source A and
the remainder from Source B. Of the modems from Source A, 20% are
defective. Of the modems from Source B, 8% are defective. Calculate
the probability that exactly two out of a random sample of five modems
from th... | I believe an approximation is involved.
The weighted average % defective = $\frac{30}{80}\cdot{20} + \frac{50}{80}\cdot{8} = 12.5$
Now apply binomial(5,0.125) which yields P(2 defective) = $\frac{1715}{16384} \approx 0.105$,
Edit
There are differing thumb rules as to when the binomial approximation to the hypergeometr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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About positive operators being Hermitian... I have been asked to prove the following:
If $V$ is a finite-dimensional inner product space over $\mathbb C$, and if $A:V→V$ satisfies $⟨Av,v⟩≥0$ for all $v∈V$, then $A$ is Hermitian.
A proof is available at the address Show that a positive operator is also hermitian but I ... | Yes. The polarization identity shows that if $⟨Bv,v⟩ = 0$ for all $v$ then $⟨Bv,w⟩ = 0$ for all $v,w$, and thus $B=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Finding the angle of inclination of a cone. After my lecture on solving triple integrals with spherical coordinates, we defined $\phi$ as the angle of inclination from the positive z-axis such that $0\leq \phi \leq\pi$.
What I don't understand is given an equation of a cone:
$$z= c \, \sqrt{x^2+y^2}$$
for some constan... | $ z = c\, r $ is the generator of cone slanting side, apex at origin.
slope of cone generator = $$ \dfrac {dr}{dz} =\tan \phi =\dfrac{1}{c}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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What does this definition mean: $F_Y(y) =P(YI am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means.
For instance the following question:
Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x \geq 0$ (parameter $\lambda>0$)
Show that for $x>0$ $P(X>x)=e^{-\lambda x^2}$
I did ... | $F_Y(y) := P(Y\le y)$ is the probability that the random variable $Y$ is less than or equal to a given real value $y$.
This function is then known as the cumulative distribution function.
For a continuous random variable it is the integral of the probability density function up to $y$, while for a discrete random var... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Infimum taken over $\lambda$ in $\mathbb{C}$ I want to calculate the infimum of
$$ |\lambda-2|^2+|2\lambda-1|^2+|\lambda|^2 $$
over $\lambda\in\mathbb{C}.$
I choose $\lambda=2,1/2,0$ so that one term in the above expression becomes zeros and minimum comes out $5/2$ at $\lambda =1/2.$ Is this infimum also, please help.... | If we take $\lambda=x+i y$, we get (as commented above)
\begin{equation}
f(x,y)=6x^2+6y^2-8x+6.
\end{equation}
BUT, to be accurate, then we need to calculated the gradient, so we differentiate by $x$ and by $y$
\begin{alignat}{2}
f_x=12x-8 ~,\\
f_y=12y ~.
\end{alignat}
Hence we get a stationary point $(2/3,0)$. However... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to have caught any pattern I can ... | Outline: Write down the Maclaurin expansion of $\ln(1-x^2)$, of $\ln(1+x^2)$, and subtract term by term. For $\ln(1+x^2)$, write down the series for $\ln(1+t)$ and replace $t$ by $x^2$.
The radius of convergence can be found in one of the usual ways, for example by using the Ratio Test.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1340985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation of plane
Find the equation of the plane through the point $(1,−1,2)$ which is
perpendicular to the curve of intersection of the two surfaces
$x^2+y^2−z=0$ and $2x^2+3y^2+z^2−9=0$.
i've gotten as far as subbing one equation into the other but i'm stuck on the differentiation. it would be much appreciated ... | Outline: If we substitute in this way:
$$2\overbrace{(z-y^2)}^{x^2}+3y^2+z^2-9=0$$
and
$$2x^2+3\overbrace{(z-x^2)}^{y^2}+z^2-9=0$$
we can obtain equations for $x$ and $y$ in terms of $z$. So the curve through the point $(1,-1,2)$ can be parameterized by $$\big(\sqrt{\text{function of } t},-\sqrt{\text{function of } t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to translate set propositions involving power sets and cartesian products, into first-order logic statements? As seen from an earlier question of mine one can translate between set algebra and logic, as long as they speak about elements (a named set A is the same as {x ∣ x ∈ A}).
However I've stumbled upon proposit... | There will be differences of opinion about notation, but I like to use:
The Cartesian Product of sets $A$ and $B$ is given by $ A\times B\$ such that:
$\forall x,y:[(x,y)\in A\times B\iff x\in A\land y\in B]$
The power set of set A is given by $\mathcal P(A)$ such that:
$\forall x: [x\in \mathcal P(A)\iff \forall y:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Elementary number theory - prerequisites Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really caught my eye, but I have no prior training in it.
I've decided to conduct my s... | One of the best is An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 0
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What is the minimum degree of a polynomial for it to satisfy the following conditions? This is the first part of a problem in the high-school exit exam of this year, in Italy.
The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below:
$\Gamma$ exhibits horizontal tangents at $x=-1,1,2$. Th... | Here we have the unique nine-degree polynomial that fulfills the ten constraints $f(-2)=f(0)=f(2)=f'(-1)=f'(1)=f'(2)=0, (A,B,C,D)=(2,3,3,1)$
$$ p(x) = -\frac{13960909 x}{3829050}-\frac{224 x^2}{9525}+\frac{26462017 x^3}{38290500}+\frac{8 x^4}{1905}+\frac{17935383 x^5}{34036000}+\frac{14 x^6}{1905}-\frac{6421193 x^7}{38... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Does $\Bbb R-\Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$ Does $\Bbb R- \Bbb Q$ have a well ordered subset of type $\omega\cdot\omega$?
I thought of taking the subset to be A={$n\cdot \sqrt{m}:n\in\Bbb N,m\in P$} where P is the set of all prime numbers, with the well ordering -
$\sqrt{2}<\sqrt{3}<\sq... | There's an incredibly easy answer to this.
Theorem. If $(A,<)$ is a dense linearly ordered set, then countable linear order can be embedded into $A$.
This is really a theorem about $\Bbb Q$ itself, and then a consequence from the fact that every dense linear order has a subset isomorphic to $\Bbb Q$.
But to see this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. I need to prove, that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$.
Prove that $$x^m+x^{-m}=P_m (x+x^{-1} )=a_m (x+x^{-1} )^m+a_{m-1} (x+x^{-1} )^{m-1}+\cdots+a_1 (x+x^{-1} )+a_0$$ on induction in $m$.
*
*$m=1$;
*$m=k$;
*$⊐n=k+1$. ... | $$\cos m \theta = T_m(\cos \theta)$$
with $T_m$ the Chebyshev polynomial of first kind, so, taking $x = e^{i\theta}$
$$x^{m} + x^{-m} = P_m(x+ x^{-1})$$
where $P_m(t) = 2 T_m(\frac{t}{2})$. For instance
$$x^{12} + x^{-12}= P_{12}(x+x^{-1})$$
where $P_{12}(t) = t^{12}-12 t^{10}+54 t^8-112 t^6+105 t^4-36 t^2+2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Prove $e^x$ limit definition from limit definition of $e$. Is there an elementary way of proving
$$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n,$$
given
$$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n,$$
without using L"Hopital's rule, Binomial Theorem, derivatives, or power series?
In other words, given the above rest... | If you accept that exponentiation is continuous, then certainly
$$\left(\lim_{n\to\infty}\left(1+\frac1n\right)^n\right)^x = \lim_{n\to\infty}\left(1+\frac1n\right)^{nx}$$
But if $u=nx$, then by substitution we have
$$
\lim_{n\to\infty}\left(1+\frac1n\right)^{nx}=\lim_{u\to\infty}\left(1+\frac{x}{u}\right)^u
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Summing divergent asymptotic series I found the sine integral si to be
$$Si (x)\sim \frac \pi 2+\sum _{n=1}^\infty (-1)^n \left(\frac{(2 n-1)! \sin (x)}{x^{2 n}}+\frac{(2 n-2)! \cos (x)}{x^{2 n-1}}\right)$$
Say I want to find $Si(\frac \pi 4)$ what options have I got to use this divergent series to find the actual valu... | In order to find a good approximation of $\text{Si}\left(\frac{\pi}{4}\right)$, I strongly suggest you to use a converging series and not a diverging one. For instance, the almost trivial:
$$\text{Si}\left(\frac{\pi}{4}\right)=\int_{0}^{\pi/4}\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}x^{2n}\,dx = \sum_{n\geq 0}\frac{(-1)^n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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action of a monoid on a mapping telescope In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15:
For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on $M_\infty$ on the left is by homology equivalences.
Notations:
(1). The space $M_\... | You are missing a hypothesis they are assuming, which is that $H_*(M)[\pi^{-1}]$ (which is just $H_*(M)[m_1^{-1}]$ in this case) can be constructed by right fractions. This implies that $H_*(M_\infty)=H_*(M)[m_1^{-1}]$, as the colimit that computes $H_*(M_\infty)$ is exactly the right fractions for $H_*(M)[m_1^{-1}]$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1341729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
What is Skorohod's Represntation Theorem Saying? From Wikipedia:
Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space
S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S
as $n \to \infty$. Suppose also that the support of μ is separable. Then there
exist random... | One example is a version of the continuous mapping theorem which states that if $X_n \rightsquigarrow X$ then $f(X_n) \rightsquigarrow f(X)$ for a continuous function $f$. Using the a.s. representation (Skorohod's Representation theorem) there is a sequence of random variables $Y_n$ and a random variable $Y$ defined on... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In how many ways I can write a number $n$ as sum of $4$ numbers? The precise problem is in how many ways I can write a number $n$ as sum of $4$ numbers say $a,b,c,d$ where $a \leq b \leq c \leq d$.
I know about Jacobi's $4$ square problem which is number of ways to write a number in form of sum of $4$ squares number. T... | Note that this problem is equivalent to the problem of partitioning $n$ into $4$ parts.
In number theory, a partition is one in which the order of the parts is unimportant. Under these conditions, we are under the illusion that the order is important, however we can only order any given combination of $4$ parts in $1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Definition of reducible matrix and relation with not strongly connected digraph I connot quite understand the definition of reducible matrix here.
We know $A_{n\times n}$ is reducible, when there exists a permutation matrix $\textbf{P}$ such that:
$$P^TAP=\begin{bmatrix}X & Y\\0 & Z\end{bmatrix},$$
where $X$ and $Z... | Consider your $3 \times 3$ matrix $A$, as a matrix with nodes $\{1,2,3\}$. More specifically, you can consider your matrix $A$ as an adjacency matrix of a graph. Assume that $a_{12}, a_{23}, a_{31}$ are strictly positive elements. Then it holds:
Since we can reach any node of the graph starting from any node, the ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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From the graph find the number of solutions. The figure below shows the function $f(x)$ .
How many solutions does the equation $f(f(x))=15$ have ?
$a.)\ 5 \\
b.)\ 6 \\
c.)\ 7 \\
d.)\ 8 \\
\color{green}{e.) \ \text{cannot be determined from the graph}}$
From figure $f(x)=15$ occurs at $x\approx \{4,12\}$ and
$f(x)=4$ ... | I don't think that it is obvious that there is a real number $\alpha$ such that $3\lt \alpha\lt 5$ and $f(\alpha)=15$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example for a norm on Hom(V,W) which is not determined by rank-one operators Assume $(V,\|\cdot \|_V),(W\|\cdot \|_W)$ are two finite dimensional normed spaces (over $\mathbb{R}$).
Any operator norm on $\text{Hom}(V,W)$ is determined by its value on rank-one operators. (This is a corollary from a reconstruction argume... | An operator norm on $\operatorname{Hom}(V,W)$ is determined by its value on rank-one operators once we know it's an operator norm.
Without this additional information, it is not so determined. First, I claim that the operators of rank $\le 1$ form a manifold $R_1(V,W)$ of dimension $$\dim R_1(V,W) = \dim V+\dim W-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the last digit of $2003^{2003}$?
What is the last digit of this number?
$$2003^{2003}$$
Thanks in advance.
I don't have any kind of idea how to solve this. Except that $3^3$ is $27$.
| Since we have
$$3^1=\color{red}{3},3^2=\color{red}{9},3^3=2\color{red}{7},3^4=8\color{red}{1},3^5=24\color{red}{3},\cdots.$$
and $$2003\equiv 3\pmod 4,$$
the right-most digit of $2003^{2003}$ is the same as the right-most digit of $3^3=27$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integration Of exponential Function I have tried almost everything, but can't solve this integral.
$$\int e^{-1/x^2} \, dx $$
| To some extent, you have in fact given the answer yourself, if you have (literally) tried everything then you have proven it is impossible to integrate in terms of elementary functions. This is indeed the case.
This statement is similar to the impossibility of solving a polynomial of degree $\geq$ 5 in terms of radical... | {
"language": "en",
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"source": "stackexchange",
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Proof of determinant formula I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my "solution" and then, hopefully, you can tell me where I went wrong. If my proof strategy fo... | The "holes-digging" method might be interesting to prove this.
On one hand, dig a hole at the lower-left corner of $A$,
$$A := \begin{bmatrix}I & X \\
-Y^T & 1 \end{bmatrix} = \begin{bmatrix} I & 0 \\
Y^T & 1\end{bmatrix}\begin{bmatrix}I & X \\
0 & 1 + Y^TX\end{bmatrix}$$
Take determinants on both sides to have $\det(A... | {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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Show the following set is connected For any $x \in \Bbb R^n$ how do I show that the set
$B_x := \{{kx\mid k \in \Bbb R}$} is connected.
It should also be concluded that $\Bbb R^n$ is connected.
I was thinking of starting by assuming that the set is not connected.Then there exist $U,V$ relatively open such that $\varnot... | You can see that $B_x$ is path-connected, hence connected. Also: $$\Bbb R^n = \bigcup_{x \in \Bbb R^n}B_x, \quad \bigcap_{x \in \Bbb R^n}B_x = \{0\} \neq \varnothing.$$Hence...
| {
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"source": "stackexchange",
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Probability that all colors are chosen A box contains $5$ white, $4$ red, and $8$ blue balls. You randomly select $6$ balls, without replacement, what is the probability that all three colours are present.
Most similar problems ask for the probability that at least one colour is missing. Which happens to be $1 - P(\t... | Look that you want that all the colors are present, so there are many posible cases. Just to name some:
*
*There are 4 white, 1 red, and 1 blue ball
*There are 3 white, 2 red, and 1 blue ball
*There are 3 white, 1 red, and 2 blue balls
And the list is much bigger, so what you can do is to use ${n\choose x}$, ho... | {
"language": "en",
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$K_S$ modulo $K_S^n$, where $K_S$ is the group of $S$-units Let $K$ be a field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones, and $K_S$ the group of $S$-units, i.e. those $x \in K^{\ast}$ which are units at all places outside $S$. It is a consequence of the unit th... | I don't think $K_S$ modulo the $n$th roots of unity is free of rank $s-1$ in general, but $[K_S : K_S^n] = n^s$ is still true. By the unit theorem, we can write $K_S$ as an internal direct sum $H \times T$, where $H$ is the group of units and $T$ is a free $\mathbb{Z}$-module of rank $s-1$. Here $T/T^n$ is isomorphic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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trouble in finding partial derivatives Following is my cost that I need to minimize wrt $\mathbf{y}$
\begin{equation}
J = (\mathbf{y^T\mathbf{z_1}})^2+(\mathbf{y^T\mathbf{z_2}})^2-\lambda(\mathbf{y}^T\mathbf{e}-1)
\end{equation}
$\lambda$ is a scalar varaible.
My work so far
\begin{align}
J &= (\mathbf{y^T\mathbf{z_1}... | Notice that $A_1$ and $A_2$ are symmetric, since $(zz^T)^T=(zz^T)$.
So going back to the derivative wrt $y$, you can write
$$ \eqalign{
\lambda e &= 2\,(A_1+A_2)y \cr
}$$
Next, find the gradient with respect to $\lambda$ and set it to zero
$$ \eqalign{
\frac{\partial J}{\partial\lambda} &= y^Te-1 &= 0 \cr
1 &= y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1342815",
"timestamp": "2023-03-29T00:00:00",
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Prime ideals in $R[x]$, $R$ a PID
Let $R$ be a PID. Show that if $\mathfrak p \subset R[x]$ is a prime ideal, $(r) = \left\{h(0) \colon h(x) \in \mathfrak p \right\}$, and $$\mathfrak p = (r, f(x), g(x)),$$ where $f(x), g(x) \in R[x]$ are nonconstant irreducible polynomials, then $f(x)$ and $g(x)$ are associates. So $... | As stated, the claim being made in the post is false, as user26857 makes clear in the comments. To reiterate, a counterexample is $$\mathfrak{p} = (2, x, x+2) \subset \mathbf{Z}[x].$$
| {
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"timestamp": "2023-03-29T00:00:00",
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why only square root approach to check number is prime Why do we use only square root approach to find a number is prime or not?
why not cube root & 4rth root?
| Consider the number $143$. This number is composite, not prime, because
$$11 \times 13 = 143.$$
But
$$\sqrt[3]{143} \approx 5.229 < 11 < 13,$$
so if you test only possible factors up to $\sqrt[3]{143}$
then you will not check whether $11$ or $13$ divide $143$,
and you will not discover that $143$ is not prime.
On the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1343171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$? The constraints are
*
*$0 \le a_1,a_2....a_n,b_1,b_2....b_n \le 1$.
*$|a_i-b_i|< \delta$ for all $1 \le i \le n $
How do I go about proving the following
$$|\prod_{i=1}^n a_i-\prod_{i=1}^n b_i| < n\delta$$
I tried reduc... | You can prove that:
$$\left|\prod\limits_{i=1}^{n}a_i-\prod\limits_{i=1}^{n}b_i\right|\le \sum\limits_{i=1}^{n} |a_i-b_i|.$$
The proof goes by induction. For $n=1$ it's obvious. Let's assume that it's true for some $n.$ We will prove this for $n+1.$ Let $A_n=\prod\limits_{i=1}^{n}a_i$, $B_n=\prod\limits_{i=1}^{n}b_i$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it allowed to define a number system where a number has more than 1 representation? I was just curious;
Is it allowed for a number system to allow more than one representation for a number?
For example, if I define a number system as follows:
The 1st digit (from right) is worth 1.
The 2nd digit is worth 2.
The 3rd... | Sure, you can define anything you care to. In fact many commonly used systems, like decimal numbers, have elements with multiple representations like $0.999\ldots=1.$ Another example would be fractions, where $1/3=2/6.$
See A066352 in the OEIS for an example of an early use (by S. S. Pillai) of this particular system, ... | {
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How can I prove irreducibility of polynomial over a finite field?
I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$.
As far as I know Eisenstein criteria won't help here.
I have heard about a criteria of irreducibility... | I think that the criterion that you allude to is the following:
Assume that $X^p - a$ has no roots in $\Bbb F _{p^n}$. Then $X^{p^m} - a$ is irreducible in $\Bbb F _{p^n}[X]$ $\forall m \ge 1$.
In any case, you can't use it here.
1) Let us see how to prove that the second polynomial (call it $P$) is reducible. First,... | {
"language": "en",
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Rotate a unit sphere such as to align it two orthogonal unit vectors I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors).
I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed to $d$, where $c$ and $d$ are two other orthogonal ... | Of course, if you can do it with two or more rotations, you can do it with one: the composition of rotations is a rotation.
Conceptually the strategy should be clear: $a$ and $c$ lie in a plane, and you can find a rotation $R_1$ that turns that plane carrying $a$ onto $c$. After that, $R_1(b)$ and $d$ lie in the plane ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the equation of base of Isosceles Traingle Given the two Legs $AB$ and $AC$ of an Isosceles Traingle as $7x-y=3$ and $x-y+3=0$ Respectively. if area of $\Delta ABC$ is $5$ Square units, Find the Equation of the base $BC$
My Try:
The coordinates of $A$ is $(1,4)$. Let the Slope of $BC$ is $m$. Since angle between $... | You have the vertex $A$ and the equations for the congruent sides.
So you can compute an apex angle $\theta$ with the dot product of two vectors from $A$.
Then, using that angle (or its complement) you can calculate the length of the congruent side $d$ from
$$\frac{d^2 \sin \theta}{2} = 5.$$
This gives you the length t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Contracted version of "isomorphic" Had a look around and I can't find a word which acts as a contraction for "isomorphic" in the same way that "monic/epic" is a contraction of "monomorphic/epimorphic". For some reason this strikes me as strange, considering that, at least in Awodey's text, iso is used as often as mono ... | Although there's a distinction between the noun and adjective abbreviations of monomorphism and epimorphism (mono vs. monic and epi vs. epic), it's fairly common to use iso for both the noun and adjective abbreviation of isomorphism.
For example, both of these seem normal to me:
*
*$\mathcal{C}$ is a balanced catego... | {
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Find the value of this series what is the value of this series $$\sum_{n=1}^\infty \frac{n^2}{2^n} = \frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots$$
I really tried, but I couldn't, help guys?
| Hint:
$$\sum_{n=1}^{\infty} \frac{n^2}{2^n} = \sum_{i=1}^{\infty} (2i - 1) \sum_{j=i}^{\infty} \frac{1}{2^j}.$$
Start by using the geometric series formula on $\displaystyle \sum_{j=1}^{\infty} \frac{1}{2^j}$ to simplify the double series into a singular series. Then you will have a series that looks like $\displaystyl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1343721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Binomial coefficients in Geometric summation Guys please help me find the sum given below.
$$\sum_{k=j}^i\binom{i}{k}\binom{k}{j}\cdot 2^{k-j}$$
(NOTE):The two coefficients are multiplied by 2 power (k-j)
I am using the formula: $\binom{r}{m}\binom{m}{q}=\binom{r}{q}\binom{r-q}{m-q}$
But not able to reach something fru... | Well you were totally in right direction
$\binom{i}{j}\cdot \sum_{k=j}^{k=i}$$\binom{i-j}{k-j}\cdot 2^{k-j}$ can be written as
$\binom{i}{j} \cdot \sum_{e=0}^{e=i-j}$$\binom{i-j}{e} \cdot 2^e$ .Apply binomial theorem to get the answer $\binom ij \cdot (1+2)^{i-j}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove by Induction ( a Limit) I think I did a lot wrong in my attempt to solve this exercise. I think I did solve it, in that case I'd like to know others way to solve the problem.
(Introduction to calculus and analysis vol 1, Courant page 113, exersice 16 )
Prove the relation
$$ \lim_{n\to \infty}\frac{1}{n^{k+1}} \s... | The general technique is to attempt to find a sufficiently good approximation to the anti-difference (summation). In this case a first-order approximation is enough.
$(i+1)^{k+1} - i^{k+1} = (k+1) i^k + \sum_{j=0}^{k-1} \binom{k+1}{j} i^j$. [The most significant term is the one we want.]
$\sum_{i=1}^n (k+1)i^k = \sum_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the number of mappings? It is given that there are two sets of real numbers $A = \{a_1, a_2, ..., a_{100}\}$ and $B= \{b_1, b_2, ..., b_{50}\}.$ If there is a mapping $f$ from $A$ to $B$ such that every element in $B$ has an inverse image and
$f(a_1)\leq f(a_2) \leq ...\leq f(a_{100})$
Then, what is the number... | EDIT:My idea is to do an order-preserving partition of the set {$1,2,..,100$} into 50 parts, e.g., (1,2,3), (4,5,6,7,..10),..., (98, 99,100). Then every number in the i-th part would map to $a_i$. I think this takes care of all maps.
This is a matter of balls-in-boxes , i.e., of finding the number of solutions to $x_1... | {
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$x^2+y^2<1, x+y<3$ is open or closed? I'm trying to figure out if
$$\{x^2+y^2<1, x+y<3|(x,y)\in \mathbb R^2\}$$
is open or closed.
I tried to imagine this set. It looks, for me, as a 'pizza', or a circular sector, which have two 'straigth' sides (closed) and a circular side (open). So I'm really confused...
I also ne... | Since this problem is a bit trivial (one condition defining the set implies the other), let us consider a more general scenario where the second condition is $ax+by < c$ for some $a,b,c \in \mathbb{R}.$
It is easy to see that the set $X := \{ x^2 + y^2 < 1 \}$ is open using the open ball definition, since it's already ... | {
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Question about using arbitrary $\epsilon$ in real analysis proofs I've notice that in a lot of the proofs that are assigned in an undergraduate analysis course, we are often trying to show that some quantity is bounded by an arbitrary epsilon.
For example, if I want to show that
$$\lim_{n\to\infty}\frac{2}{\sqrt{n}}+... | Generally from the analysis point of view when we say that $a=b$, we generally mean $|a-b|<\epsilon \quad \forall \epsilon >0$. The way of looking at this is, since the distance between $a$ and $b$ is fixed it cannot be varied arbitrarily. If you are able to do it indefinitely by a small positive quantity which is this... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the relationship between $L_{P}(0,1)$ and $L_{P}[0,1]$? $L_{P}[0,1]$ be the set of measurable functions $f : [0,1]\rightarrow R$ such that
$\int |f(x)|^{p} dx<\infty$. What is the relationship between $L_{P}(0,1)$ and $L_{P}[0,1]$?
| Assuming you mean the normal(Lebesgue) measure, then noting that the endpoints are sets of measure zero, we can thow them out of the integral and not change its value. Note that in $L^p[0,1]$, if we fix a function $f$ on $(0,1)$ and then assign arbitrary values to their endpoints, then they are the same element in $L^... | {
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Why does the following integral starts from $0$? Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$
It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem:
$$\lim_{x\to 1^-} f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1}$$
Evaluating the derivative
$$f'(x) = \sum_{n=0}^\i... | The fundamental theorem tells you that
$$f(x)=f(a)+\int_a^x f'(t) dt.$$
It's convenient to choose $a$ such that $f(a)=0$, because then
$$f(x)=f(a)+\int_a^x f'(t) dt = \int_a^x f'(t) dt.$$
Since your function is a power series with no constant term, it's not hard to see that you can use $a=0$ for this purpose.
| {
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Formula to represent 'equality' I am trying to find the appropriate formula in order to calculate and represent the 'equality' between a set of values. Let me explain with an example:
Imagine 3 people speaking on an a TV Show that lasts 45 minutes:
Person 1 -> 30 minutes spoken
Person 2 -> 5 minutes spoken
Person 3 -> ... | Finally, thanks to @Rahul, found Gini Coefficient which works great for my case.
The Gini coefficient (also known as the Gini index or Gini ratio)
(/dʒini/ jee-nee) is a measure of statistical dispersion intended to
represent the income distribution of a nation's residents, and is the
most commonly used measure... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there any notation for general $n$-th root $r$ such that $r^n=x$? As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal root is always positive but there is also another n... | In complex analysis, $\sqrt[n]{x}$ is regarded as a multivalued function. Or you can write it as $$\sqrt[n]{x}=\exp{\frac{\operatorname{Log}(x)}{n}},\space x\ne0.$$
$\operatorname{Log}(x)$ is the inverse function of $\exp(x)$, see here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the integral of rational function of sines $$ \int \frac{\sin x}{1+\sin x} \, \mathrm{d}x$$ How do I integrate this? I tried multiplying and dividing by $ (1- \sin x) $.
| First, let's simplify things just a bit and write
$$\frac{\sin x}{1+\sin x}=1-\frac{1}{1+\sin x}$$
Then, applying the Wierstrauss substitution $u=\tan(x/2)$ with $\sin x=\frac{2u}{1+u^2}$ and $dx=\frac{2du}{1+u^2}$ reveals that
$$\begin{align}
\int\frac{\sin x}{1+\sin x}dx&=\int \left(1-\frac{1}{1+\sin x}\right)dx\\\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$? This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the following:
Let $R$ be a ring. Are there any simp... | This is well-studied under the heading of "cancellability," and Lam's crash course on the topic is very nice.
Are there any simple conditions on $R$-modules $M,A$ and $B$...
The readiest one is that if $R$ has stable range 1 and $M$ is finitely generated and projective, then it cancels from $M\oplus A\cong M\oplus B$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that $\{(x,y)\mid xy>0\}$ is open I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the set. However, I don't know how to find a radius for this open ball. Can somebody hel... | If you can use continuous functions, then the set in question is the inverse image of the open interval $(0,\infty)$ under the continuous function $\mathbb R^2 \to \mathbb R$ with $(x,y)\mapsto xy$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1344891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is the Householder matrix orthogonal? A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank.
*
*Why is $H$ orthogonal? I am looking for an intuitive proof rather than a rigorous one.
*How come it is full rank when $\mbox{rank} \left( ... | If $H$ is orthogonal, then $H^TH = I$, so let's compute that:
$$ H^TH = I - 2cuu^T + c^2 uu^Tuu^T $$
Since $u$ is a unit vector, then $u^Tu = 1$, so
$$ H^TH = I - 2cuu^T + c^2 uu^T $$
This shows that $c$ cannot be arbitrary, it must satisfy $c^2-2c = 0$ or $c = 2$ ($c=0$ is a trivial solution).
Intuitively, it represen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Approximating $\tan61^\circ$ using a Taylor polynomial centered at $\frac \pi 3$ : how to proceed? Here's what I have so far...
I wrote a general approximation of $f(x)=\tan(x)$ , which then simplified a bit to this:
$$\tan \left(\frac{61π}{180}\right) + \sec^2\left(\frac{61π}{180}\right)\left(\frac{π}{180}\right) + \t... | There are many ways to approximate (even very accurately) functions close to a point.
The simplest is Taylor expansion; in the case of the tangent, assuming $b<<a$, the expansion is $$\tan(a+b)=\tan (a)+ \left(\tan ^2(a)+1\right)b+ \left(\tan ^3(a)+\tan (a)\right)b^2+$$ $$
\left(\tan ^4(a)+\frac{4 \tan ^2(a)}{3}+\fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why a transcendental equation can not be analytically evaluated I'm reading this book in Classical Mechanics and they derive an equation for the time a projectile takes to reach the ground once is fired (accounting for air resistance):
$$T=\frac{kV+g}{gk}(1-e^{-kT})$$
I do not have any questions on how they derive the ... | Typically, equations which involve mixtures of polynomial, trigonometric, exponential terms do not show explicit solutions in terms of elementary functions (the solution of $x=\cos(x)$ is the simplest example I have in mind).
However, you will be pleased (I hope !) to know that any equation which can write or rewrite $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A Problem That Involves Differential Equations, Implicit Differentiation, and Tangent Lines of Circles Here is the Statement of the Problem:
Consider the family $\mathbb F$ of circles given by
$$ \mathbb F:x^2+(y-c)^2=c^2, c \in \mathbb R. $$
(a) Write down an ODE $y'=F(x,y)$ which defines the direction field of the tr... | For part (a) you have $2x+2(y-c)y'=0,$ so
$$y'=-x/(y-c) \tag{1}$$
as you say. However this is not the differential equation for the whole family since it still mentions the specific constant $c.$
From the initial relation $x^2+(y-c)^2=c^2$ you can solve for $c$ after multiplying it through to $x^2+y^2-2cy=0$ since t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Matrices derivative I have a linear product of matrices, I did solve most of it, however, I stop at this component $(X^T W^T D W X)^{-1}$.
Given that $X$ is $n \times p$ matrix and $D$ is $n\times n$ matrix. $W$ is a diagonal matrix $n\times n$
what is the derivative of this component with respect of $W$.
$\frac{\p... | For convenience, define $G=X^TWDWX$.
Since {$W,D$} are diagonal, they are symmetric and therefore $G$ is symmetric, too.
Then your matrix function and its differential are
$$ \eqalign{
F &= G^{-1} \cr
dF &= -F\,dG\,F \cr
&= -FX^T\,d(WDW)\,XF \cr
&= -FX^T\,(dW)\,DWXF - FX^TWD\,(dW)\,XF \cr
}$$
Apply the vec ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$A$ is a doubly stochastic matrix, how about $A^TA$ I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix.
However, the paper says $A^TA$ is symmetric and stochastic.
*
*Since $A^TA$ is symmetric, if $A^TA$ is stochastic, it must be doubly stochastic.
*The $(j,j)$ entry of... | I'm not sure I follow what you're trying to prove from which assumptions; I'll assume that the aim is to show that $A^TA$ is doubly stochastic if $A$ is doubly stochastic. You've already shown that since $A^TA$ is symmetric it suffices to show that it is right stochastic. Being right (left) stochastic means having the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there a way to figure out the number of possible combinations in a given total using specific units I'm not professional mathematician but I do love a math problem - this one, however has me stumped.
I'm a UX Designer trying to figure out some guidelines for using tables in a page layout. The thing I want to know is... | Making Change for a Dollar (and other number partitioning problems) is a related question that provides a lot of background on how to solve this sort of problem. In your case, you want the coefficient of $x^{12}$ in the generating function
$$\frac1{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^6)(1-x^{12})}\;,$$
for which Wolfram|Alp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a matrix metric Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ unitary matrices of the same size.
We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i V_i$.
I'd like to show that $E(U,V) \leq \sum_i E(U_i,V_i) $.
At first I thought proving this by ... | Let $||.||$ be the matrix norm induced by the $l_2$ norm over $\mathbb{C}^n$. Then (if I correctly understand the question) $E(U,V)=||U-V||$. Note that, if $U$ is unitary, then $||U||=1$ and that the set of unitary matrices is a group.
Case $n=2$. $||U_1U_2-V_1V_2||=||(U_1-V_1)U_2+V_1(U_2-V_2)||\leq ||U_1-V_1||||U_2||+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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function bounded by an exponential has a bounded derivative? here's the question. I want to be sure of that. Let $v:[0,\infty) \rightarrow \mathbb{R}_+$ a positive function satisfying
$$\forall t \ge 0,\qquad v(t)\le kv(0) e^{-c t}$$
for some positive constants $c$ and $k$. Can I conclude that
$$\dot{v}(t) \le -c v(t)$... | No. I will show a non-negative function which does the job (it's easy enough to turn it into a positive one). Take $v(x)=e^{-x}\cos^2(e^x)$. Clearly we have $v(x)\leq v(0)e^{-x}$. The derivative, which is $-e^x\sin(2e^x)$, is not bounded.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that certain points lie on an ellipse I have the equation
$$r(\phi) = \frac{es}{1-e \cos{\phi}}$$
with $e,s>0$, $e<1$ and want to show that the points
$$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = \begin{pmatrix}r(\phi)\cos{\phi}\\r(\phi)\sin{\phi}\end{pmatrix}$$
lie on an ellipse.
I have a basic and a spec... | Your work seems correct. I suppose that from $x^2+y^2=e^2(s+x)^2$ you find:
$x^2(1-e^2)+y^2-2se^2x-e^2s^2=0$, and this is an equation of the form:
$$
Ax^2+Bxy+Cy^2+Dx+Ey+F=0
$$
that represents a conic section, and it is an ellipse if $B^2-4AC<0$, as it is easely verified since $0<e<1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1345923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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solve $|x-6|>|x^2-5x+9|$
solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$
I have done $4$ cases.
$1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\
2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\
3.)\ -(x-6)>x^2-5x+9\ ,\implies 1<x<3\\
4.)\ (x-6)>-(x^2-5x+9),\ \implies x>3\cup x<1 $
I am confused on how I proceed.
Or... | It usually helps to draw a graph of the functions involved.
First of all, note that the discriminant of the parabola $P: x^2 - 5x + 9 = 0$ is $5^2 - 4\cdot9 < 0$, thus $x^2 - 5x + 9$ is always positive. This means that you are looking for the points $x \in \Bbb{R}$ for which the graph of $|x-6|$ lies above $P$.
Then ob... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Infinitesimal Generator for Stochastic Processes Suppose one has the an Ito process of the form:
$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$
The infinitesimal generator $LV(x)$ is defined by:
$$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) \right]-V(x)}{t}$$
One can show that $$LV(x) = \sum_{i}b_i(x)\frac{\partial V}{\partial ... | You can look at your process $X_{t}$ as a two dimensional stochastic process
$$Y_{t}=\left[\begin{array}{cc}X_{t}\\ \eta_{t}\end{array}\right]$$
Then
$$dY_{t}=\left[\begin{array}{cc}dX_{t}\\ d\eta_{t}\end{array}\right]=\left[\begin{array}{cc}b(X_t)+\lambda\eta_{t}\sigma(X_{t})\\ \lambda\eta_{t}\end{array}\right]dt+\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1346149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Unknown both as a exponent and as a term in an equation Let's say I have an equation $e^{x-1}(x+1)=2$. According to Solving an equation when the unknown is both a term and exponent it's impossible to solve this using elemetary functions. If so, then how do you solve it? Could you give me some keywords for further resea... | Seems to be a case for the Lambert W-function:
\begin{align}
e^{x-1}(x+1) &=2 \iff \\
e^{x+1}(x+1) &=2e^2 \iff \\
f(x+1) &= 2e^2 \Rightarrow \\
x + 1 &= f^{-1}(2e^2) \Rightarrow \\
x &= f^{-1}(2e^2) - 1
\end{align}
where $f(x) = x e^x$ and $f^{-1} = W$, which is that Lambert W-function.
We need the branch with positive... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist? How can I show that $\displaystyle\lim_{x\to 0}\frac{\sin(x)}{\sqrt{x\sin(4x)}} $does not exist ?
| Since $\sqrt{x\sin(4x)}>0$ in a (punctured) neighborhood of $0$, while $\sin x<0$ for $-\pi<x<0$ and $\sin x>0$ for $0<x<\pi$, the limit exists if and only if it is $0$ or, equivalently,
$$
\lim_{x\to0}\left|\frac{\sin x}{\sqrt{x\sin 4x}}\right|=0
$$
However
$$
\lim_{x\to0}\left|\frac{\sin x}{\sqrt{x\sin 4x}}\right|=
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1346342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 4
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Number Theory- Mathematical Proof I am trying to prove a theorem in my textbook using another theorem. What I need to show that
if a,b, and c are positive integers, show that the least positive integer linear combination of a and b equals the least positive integer linear combination of a+cb and b.
Any help would be... | Let the least positive linear combination of $a,b$ be $f$. You can write $f=da+eb$ Now show you can write $f$ as a linear combination of $a+cb, b$. You should be able to find explicit coefficients for the combination. To show this is minimum, assume you can write $g \lt f$ as a linear combination of $a+cb, b$. Show... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin? Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair?
Furthermor... | For "almost all" priors, the answer is 100%: you have to specify an interval of "fairness" in order to get a nonzero value. Why? Because the bias $p$ can be any real number between $0$ and $1$, so the probability that it is a very specific real number is $1/\infty = 0$; hence the probability that it is $1/2$ is 0.
Howe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the coefficient of $x^i$ in $(1+x+\dots+x^i)^j$ is $\binom{i+j-1}{j-1}$ Show that $$\text{ The coefficient of } x^i \text{ in } (1+x+\dots+x^i)^j \text{ is } \binom{i+j-1}{j-1}$$
I know that we have:
$\underbrace{(1+x+\dots+x^i) \cdots (1+x+\dots+x^i)}_{j\text{ times}}$
My first problem how to explain that a... | As far as the coefficient of $x^i$ is concerned, we might just as well look at
$$(1+x+x^2+x^3+\cdots)^j,$$
that is, at $\left(\frac{1}{1-x}\right)^j$. The Maclaurin series expansion of $(1-x)^{-j}$ has coefficient of $x^i$ equal to $\frac{j(j+1)(j+2)\cdots (j+i-1)}{i!}$. This is $\frac{(j+i-1)!}{(j-1)!i!}$, which is $\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question in proof from James Milne's Algebraic Number Theory I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ is the unique prime ideal dividing $(p)$, and it's been ass... | By the assumption (to be contradicted) there is some prime $\frak{q}$ that divides $x+ \zeta^i y$ and $x+\zeta^j y$. In particular $\frak{q}$ divides $(x+ \zeta^i y)$, which is to say $x+\zeta^i y \in \frak{q}$. The first part of the proof shows $\frak{q}$ has to be $\frak{p}$.
| {
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Non-standard model of $Th(\mathbb{R})$ with the same cardinality of $\mathbb{R}$ Let $\mathfrak{R}= ⟨\mathbb{R},<,+,-,\cdot,0,1⟩$ be the standard model of $Th(\mathbb{R})$ in the language of ordered fields. I need to show that there exists a (non standard) model of $Th(\mathbb{R})$ with the same cardinality of $\mathbb... | Instead of compactness, you can try an ultrapower argument.
Let $\cal U$ be a nonprincipal ultrafilter on $\Bbb N$. Then $\Bbb{R^N}/\cal U$ is elementarily equivalent to the reals. And it is not hard to show it has the wanted cardinality, and that it is non-standard.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1346811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Zeros of Dirichlet L-functions on the line $\Re(s)=1$ in proof of Dirichlet's theorem In the proof of Dirichlet's theorem, we show that $L(s,\chi_0)$ has a simple pole at $s=1$ where $\chi_0$ is the principal character and that $L(1,\chi)\neq 0$ otherwise. Therefore the logarithmic pole at $s=1$ of $\log L(s,\chi_0)$ i... | To prove only the statement that there are infinitely primes in arithmetic progressions, you only look at the limit
$$ \lim_{s \to 1} \sum_{p \equiv a \pmod n} \frac{1}{p^s} \tag{1}$$
and show that it's infinite. So the only point of analytic interest is $s = 1$. To understand $(1)$, one ends up looking at a sum of the... | {
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Evaluating $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$
Evaluate $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$ where $\gamma$ is the positively oriented boundary of $\{x+iy \in \Bbb{C} : y^2 < (4\pi^2 -1)(1-x^2)\}$.
I just learned the residue theorem, so I assume I have to apply that here.
My trouble is that the integrad $h(z... | We can rewrite the domain in more familiar form,
$$\{ x+iy \in \mathbb{C} : (4\pi^2-1)\cdot x^2 + y^2 < 4\pi^2-1\} = \biggl\{ x + iy \in \mathbb{C} : x^2 + \biggl(\frac{y}{\sqrt{4\pi^2-1}}\biggr)^2 < 1\biggr\}.$$
In that form, it is not too difficult to see what the domain is, and hence what $\gamma$ is. Then you just ... | {
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Properties preserved under equivalence of categories I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?
| Basically any property that can be considered categorical in nature. Any textbook would list a warning if a property isn't preserved. Wikipedia lists some simple examples.
Here are some things that aren't necessarily preserved:
*
*Number of objects
*Number of morphisms (total)
*Underlying graph
*Other evil propert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347048",
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"answer_id": 0
} |
Given a Line Parametrization, Finding another Equation So I am given a line $l$ with the parameterization, $x=t, y=2t, z=3t$. Now let some point, $p$ be a plane that contains the line $l$ and the point $(2,2,2)$. So given this, how do I find an equation for $p$ in the form $ax+by+cz=d$?
My thoughts:
I think that first ... | in this particular case the point $L_0=(1,1,1)$ lies on the line ($t = 0$). since this point and the point $P=(2,2,2)$ lie in the plane, so does the origin $(0,0,0)$. so its equation is:
$$
ax+by+cz=0 \tag{1}
$$
now any point in the plane is a linear combination $\lambda P + (1-\lambda) L_t$, where $L_t$ is the point o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to choose a left-add$(X)$-approximation with a certain property Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact.
Set $C:=Coker(\alpha)$ (as module) and $c:=coker(\alpha)$ (as a map).
Why is it then possible to c... | First, note that there always exists a left-$add(X)$-approximation $\varphi:C\to X^{\widehat{m}}$ for any module $C$ in mod-$A$. This comes from the fact that $Hom_A(C,X)$ is a finitely generated module over the (implicit) base ring $k$; if $f_1, \ldots, f_r$ are generators, then the map given in matrix form by $(f_1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A question on convex hull Let $a_1, a_2,\ldots, a_n$ be $n$ points in the $d$-dimensional Euclidean space. Suppose that $x$ is a point which does not belong to the convex hull of $a_1, a_2,\ldots, a_n$.
My question is, does there exist a vector $v$ such that $\langle v, x-a_i\rangle < 0$ for all $i$? My geometric intui... | I figured I might as well expand on my comment. Let $C$ be the convex hull of the given points (or indeed, any closed convex set). Choose $p$ to be a point of minimal distance from $x$ in $C$ (in fact, it will be unique).
Suppose, for the sake of contradiction, there exists some $c \in C$ such that $\langle x - p, c - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Groups of the from $gMg$ in a monoid where $g$ is an idempotent Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no idempotent other than $g$, it must be a group, since it is finite... | This is true for any semigroup (even if it is not a monoid) and is easy to prove if you know about Green's relations.
If $S$ be a semigroup and $e$ is an idempotent of $S$, then $eSe$ is a monoid
(with identity $e = eee$). Suppose that $e$ and $f$ are idempotents such that $eSe$ and $fSf$ are groups. Then in particular... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Finite double sum: Improve index transformation In order to prove a rather complicated binomial identity a small part of it implies a transformation of a double sum.
The double sum and its transformation have the following shape:
\begin{align*}
\sum_{k=0}^{l}\sum_{j=\max\{1,k\}}^{\min\{l,k+c\}}1=\sum_{j=1}^{l}\sum_{... | Note: It would be nice to provide a more elegant solution than my answer below.
The following identity is valid for $l\geq 1, c\geq 1$
\begin{align*}
\sum_{k=0}^{l}\sum_{j=\max\{1,k\}}^{\min\{l,k+c\}}1=\sum_{j=1}^{l}\sum_{k=0}^{\min\{j,c\}}1
\end{align*}
In Order to prove this identity we consider two cases: $c<l$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Determinant proof question.
Using determinants, prove that if $A_1,A_2,...,A_m$ are invertible $nxn$ matrices, where $m$ is a positive integer, then $A_1A_2...A_m$ is an invertible matrix.
Need help starting the proof. Do I show that $A_1,A_2,..,A_m$'s determinant is not equal to zero and use the determinant property... | A matrix is invertible if and only if its determinant is nonzero. Since $\det(A_i)\neq 0$, and $\det(A_1...A_n)=\det(A_1)...\det(A_n)$, we know that $\det(A_1...A_n)\neq 0$, so $A_1...A_n$ is invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
let $\phi (x) =\lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is not differentiable at $1$. For $x \geq 0$, let $\phi (x) = \lim_{n \to \infty} \frac{x^n +2}{x^n +1}$; and $f(x) = \int_0^x \phi(t)dt$. Then $f$ is continuous at $1$ but not differentiable at $1$.
First we calculate ... | First, we note that
$$\begin{align}
\phi(x) &= \lim_{n\to \infty} \frac{x^n+2}{x^n+1}\\\\
&=\begin{cases}
1,\,\,\text{for}\,\,x>1\\\\
\frac32 \,\,\text{for}\,\,x=1\\\\
2,\,\,\text{for}\,\,0\le x<1\\\\
\end{cases}
\end{align}$$
Next, let's form the difference quotients for $f$ at $x=1$. Thus, for $h>0$, we have
$$\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to prove the non-existence of a polynomial series that uniformly converges to this function? I was asked to prove the following.
There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to $p(x)$ on $(a,b)$.
I considered $p(... | Your example certainly suffices, but I don't think you'll have any luck working with the expression $|p_n(\frac{1}n)-e^n|$ since, for appropriate choice of $p_n$, we can make that $0$ for all $n$ (even if it approximates every other point terribly).
The important thing to note is that $p(x)$ is unbounded, whereas all p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to show that this function gets every real value exactly once? $$f(x) = {e^{\frac{1}{x}}} - \ln x$$
I thought maybe to use the Intermediate value theorem.
Thought to create 2 functions from this one. and subtracting.
And finally I will get something in the form of
$$f(c) - c = 0$$
$$f(c) = c$$
But how I proof that... | you can use the derivative to show that the function is monotonic
the derivative is
$ \frac{-1}{x^2} e ^{\frac{1}{x}} - \frac{1}{x} < 0$
so the function is monotonic
when x tends to 0 the function tends to ${\infty}$
and when x tends to ${\infty}$ the function tends to $-{\infty}$
so the function will get every real ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1347929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Definition of the vector cross product As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane spanned by $\mathbf{a}$ and $\mathbf{b}$. What I was wondering though is, if this is... | Yes, it is.
Another motivation is the following: in three dimensions (and only in three dimensions) a plane has only one normal direction. The angle between the normal direction and any vector $v$ has a geometric interpretation as "exposure" of the plane to $v$.
For example, a photo film is maximally exposed to the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Bounds for double exponential integrals I understand that the double-exponential integral
$$
F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)}
$$
can in general not be solved in closed-form.
I wonder wether there are 'simple expressions' in terms of $a,b,C$ as upper and low... | Replace $x$ with $\frac{y}{b}$, then $y$ with $\log z$. Then you are left with an exponential integral, for which there are many well-known approximations, for instance the one given by the Gauss continued fraction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is integration of $x\operatorname{cosec}(x)$ defined? Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
| Using $u=\tan(x/2)$ so that $\sin(x)=\frac{2u}{1+u^2}$, $\mathrm{d}x=\frac{2\,\mathrm{d}u}{1+u^2}$
$$
\begin{align}
&\int\frac{x}{\sin(x)}\,\mathrm{d}x\\
&=2\int\frac{\arctan(u)}u\,\mathrm{d}u\\
&=2\sum_{k=0}^\infty(-1)^k\int\frac{u^{2k}}{2k+1}\,\mathrm{d}u\\
&=2\sum_{k=0}^\infty(-1)^k\frac{u^{2k+1}}{(2k+1)^2}+C\\
&=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Joining two graphs Suppose I have $f_1(x)=x$
And i restrict its domain as $\color{blue}{(-\infty,0]}$ using $g_1(x)=\dfrac{x}{\frac{1}{2\left(x-0\right)}\left(x-0-\left|x-0\right|\right)}$
Resulting in :
Now, suppose i have $f_2(x)=\sin{(x)}$
And i restrict its domain to $\color{blue}{(0,\infty)}$ using $g_2(x)=\dfrac... | I found shorter ways of generalizing this. I am as old as you are so I am glad to know someone has a similar kind of interest as I do. I notice that when $f(0)\neq{0}$ and $g(0)\neq{0}$ my method could be quicker.
Basically you can take two functions and "fuse" them by the floor function with exponents (as long as the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proof that intervals of the form $[x, x+1)$ or $(x, x+1]$ must contain a integer. Show that any real interval of the form $[x, x+1)$ or $(x, x+1]$ must contain a integer.
Here is my proof (by contradiction)
We start by saying, assume the interval of the form $[x, x+1)$ or $(x, x+1]$ does not contain a integer. Now supp... | You have proved the statement only for $x\in\Bbb Z$.
If $x\notin\Bbb Z$, consider the set $A=\{t\in\Bbb Z: t>x\}$. This set is bounded below and not empty, and it is well-ordered. Let $n=\min A$. Try to prove that $n$ is in the interval.
Perhaps you should also visit this question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
One-One Correspondences Adam the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves $1$ unit up; if he flips tails, he moves $1$ unit right.
Betty the beetle starts at $(2,4)$. Each minute, she flips a fair coin. If she flips heads, she moves $1$ unit down; if she flips tails, she mov... | I assume (correctly?) that the question means "what is the probability that, on finishing a round of coin tosses, the two find themselves on the same grid point." If that's correct, it is not hard to see that they can only meet at three points: (1,2), (2,1), or (0,3). Let's analyze (1,2), as an example. For Adam to g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The ordinals as a free monoid-like entity Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set whose every $\kappa$-small subset has a join. Now recall that $\mathbb{N}$ can... | Let $P$ be a $\kappa$-monarchy and $p \in P$. We want to show that there is a unique morphism $f : \kappa \to P$ of $\kappa$-monarchies such that $f(1)=p$. We define $f$ via transfinite recursion as follows. We let $f(0)=0$, $f(x+1)=f(x)+p$, and for limit ordinals $x$ we let $f(x) = \sup_{y<x} f(y)$. It is clear that w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proof of why $\sqrt[x]{x}$ is greatest when $x=e$ Stated above question. If the mathjax I used was wrong, it should be: Why does the xth root of x reach the greatest y at x=e
| So, you're asking to maximize $x^{1/x}$. We'll restrict out attention to positive $x$'s because otherwise we need to deal with complex numbers.
Taking the derivative of this expression is sometimes tricky. There are more fun ways to do this, but you could write this as
$$
x^{1/x}=e^{(\ln x)/x}.
$$
The derivative of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Change of order of integration of a triple integral Consider $$ I = \int_0^{\omega}\int_0^{\alpha}\int_0^{\alpha}F(\beta){\tilde{F}(\gamma)}e^{i\beta t}e^{-i\gamma t}R(\alpha)d\beta d\gamma d\alpha$$
In this triple integral,I want to bring about, a change of order of integration, where in I take integration with respec... | What you have is wrong.
Hint. The original integral is over the domain consisting of all triples $(\alpha,\beta,\gamma)$ for which
$$0 \leq \beta, \gamma \leq \alpha \leq \omega.$$
To change the order, note that this is equivalent to picking $\beta$ and $\gamma$ anywhere in $[0,\omega]$, and then picking $\alpha$ in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1348910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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