Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
About convex function The exercise is about convex functions:
How to prove that $f(t)=\int_0^t g(s)ds$ is convex in $(a,b)$ whenever $0\in (a,b)$ and $g$ is increasing in $[a,b]$?
I proved that
$$f(x)\leq \frac{x-a'}{b'-x}f(b')+\left(1-\frac{x-a'}{b'-x}\right)f(a')$$
when we have
$$x=\left(1-\frac{x-a'}{b'-a'}\right)... | A slightly different approach:
We need to show $f(x+\lambda(y-x)) \leq f(x) + \lambda (f(y)-f(x))$, with $\lambda \in (0,1)$. Suppose $x<y$. Then
$$f(x+\lambda(y-x)) - f(x) = \int_{x}^{x+\lambda(y-x)} g(s) \; ds$$
Using the change of variables $t=\frac{s-x}{\lambda}+x$, we get
$$\int_{x}^{x+\lambda(y-x)} g(s) \; ds = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Hypergeometric series If found that : "Assume further that this equation has e series solution $\sum a_ix^i$ whose coefficients are connected by two term recurrence formula. Then, such a series can be expressed in terms of hypergeometric series." [Bragg, 1969]
how can we do this conversion?
thanks in advance
| The (freely downloadable) book A = B, by Petkovsek, Wilf, and Zeilberger, is, generally speaking, a must-read. The authors explain, in particular how to deduce a hypergeometric series from a recurrence relation and the other way round.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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inequality in a differential equation Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that satisfies $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean norm. According to Temam's book paragraph 2.2 on page 32 number (2.10), inequality (1) implies $$|u(t)|^2\... | The basic argument would go like this. Go ahead and let $f(t) = |u(t)|^2$, so that equation (1) says $f'(t) + f(t) \leq 1$. We can rewrite this as $$\frac{f'(t)}{1-f(t)}\leq 1.$$ Let $g(t) = \log(1 - f(t))$. Then this inequality is exactly that $$-g'(t)\leq 1.$$ It follows that $$g(t) = g(0) + \int_0^t g'(s)\,ds\geq g(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143914",
"timestamp": "2023-03-29T00:00:00",
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A simple Riemann mapping question Let $\Delta$ denote the open unit disc.
Let $G$ be a simply connected region and $G\neq\mathbb{C}$. Suppose $f:G\rightarrow\Delta$ is a one-to-one holomorphic map with $f(a)=0$ and $f'(a)>0$ for some $a$ in $G$. Let $g$ be any other holomorphic, one-to-one map of $G$ onto $\Delta$. Ex... | Your solution works for any fixed $f$ whether or not $f'(a)>0$. The condition $f'(a)>0$ is just a convenient way to fix the phase of the derivative; this affects the exact value of $\theta$ in your argument, but not its existence or uniqueness.
| {
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$T :\mathbb {R^7}\rightarrow \mathbb {R^7} $ is defined by $T(x_1,x_2,\ldots x_6,x_7) = (x_7,x_6,\ldots x_2,x_1)$ pick out the true statements. Consider the linear transformations $T :\mathbb {R^7}\rightarrow \mathbb {R^7} $ defined by $T(x_1,x_2,\ldots x_6,x_7) = (x_7,x_6,\ldots x_2,x_1)$. Which of the following stat... | We can start guessing the eigenvectors: with eigenvalue $1$, we have eigenvectors $e_1 + e_7$, $e_2 + e_6$, $e_3 + e_5$, and $e_4$; with eigenvalue $-1$, we have eigenvectors $e_1 - e_7$, $e_2 - e_6$, $e_3 - e_5$. These seven eigenvectors form a basis of $\mathbb{R}^7$, so with respect to this basis $T$ will be diagona... | {
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Finding the rank of a matrix Let $A$ be a $5\times 4$ matrix with real entries such that the space of all solutions of the linear system $AX^t = (1,2,3,4,5)^t$ is given by$\{(1+2s, 2+3s, 3+4s, 4+5s)^t :s\in \mathbb{R}\}$ where $t$ denotes the transpose of a matrix. Then what would be the rank of $A$?
Here is my attemp... | Rank theorem says that if $A$ be the coefficient matrix of a consistent system of linear equations with $n$ variables then
number of free variables (parameters) = $n$ - $rank(A)$
By using this we have $1 = 4 - rank(A)$
Thus $rank(A) = 3$
thanks to Dr Arturo sir for clearing my doubt.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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A probabilistic method I am trying to study for a exam and i found a assignmet, that i cant solve.
Consider a board of $n$ x $n$ cells, where $n = 2k, k≥2$. Each of the numbers from $S = \{1,...,\frac{n^2}{2}\}$ is written to two cells so that each cell contains exactly one number.
How can i show that $n$ cells $c_{i, ... | The standard probabilistic approach would be the following:
For each $i$, calculate the probability that both $i$s are in the permutation provided that they are not already in the same row/column (then the probability is zero, of course). This gives $\dfrac 1 {n(n-1)}$, since there are $n(n-1)$ possible selections from... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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A strange characterization of continuity The problem I'm going to post, may appear a bit routine at first sight but it is not so! Suppose that a,b are two real numbers and $f:(a,b)\rightarrow \mathbb{R}$ satisfies: $f((c,d))$ is a bounded open interval for EVERY subinterval $(c,d)$ of $(a,b)$. Can we conclude that $f$... | This answer is based on the answer Brian M. Scott gave at the link Siminore mentioned. Take the interval $(0,1)$. Let $\equiv$ be the equivalence relation on $\mathbb{R}$ given by $x\equiv y$ if and only if $x-y\in\mathbb{Q}$. Each equivalence class is countable and $|(0,1)|=|\mathbb{R}|=|\mathbb{R}^\mathbb{N}|$, so th... | {
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Are there infinitely many primes of the form $n^2 - d$, for any $d$ not a square? Clearly, for $d$ a square number, there is at most one prime of the form $n^2 - d$, since $n^2-d=(n+\sqrt d)(n-\sqrt d)$.
What about when $d$ is not a square number?
| There's a host of conjectures that assert that there an infinite number of primes of the form $n^2-d$ for fixed non-square $d$. For example Hardy and Littlewood's Conjecture F, the Bunyakovsky Conjecture, Schinzel's Hypothesis H and the Bateman-Horn Conjecture.
As given by Shanks 1960, a special case of Hardy and Litt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are there any situations where you can only memorize rather than understand? I realize that you should understand theorems, equations etc. rather than just memorizing them, but are there any circumstances where memorizing in necessary? (I have always considered math a logical subject, where every fact can be deducted u... | If there are any, I'm certain that humans haven't discovered them. If there truly is a situation in mathematics where you can only memorize and there is no logical reasoning you can use to get there, then there comes an interesting question: how do we know that this formula is true? Everything we know about in mathemat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Original author of an exponential generating function for the Bernoulli numbers? The Bernoulli numbers were being used long before Bernoulli wrote about them, but according to Wikipedia, "The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1,... | I highly recommend the book Sources in the Development of Mathematics: Infinite Series and Products from the Fifteenth to the Twenty-first Century, by Ranjan Roy (Cambridge University Press, 2011). Got it from the library a couple of weeks ago. It has almost $1000$ pages of treasures.
On page $23$, Roy writes: "In the... | {
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"timestamp": "2023-03-29T00:00:00",
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Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$? I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old.
After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answ... | The usual meaning of $a/b=c$ is that $a=b\cdot c$. Since for $b=0$ we have $0\cdot x=0$ for any $x$, there simply isn't any $c$ such that $1=0\cdot c$, unless we throw the properties of arithmetic to the garbage (i.e. adding new elements which do not respect laws like $a(x+y)=ax+ay$).
So "undefined" or "not a number" i... | {
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Constructing Young subgroups of $S_4$ Given the symmetric group $S_4$ and a subgroup $H\subset S_4$ I want to construct a Young subgroup $Y\subset S_4$ such that $Y$ is minimal, meaning that there is no other young subgroup $Y'$ such that $H\subset Y'\subset Y$. My understanding of the problem is in two ways:
1) Consid... | It might be easier to think in terms of the orbits of the actions of $H$ and $Y$ on the set $\{1,2,3,4\}$, as the $Y$-orbits must contain the $H$-orbits (since $H$ is a subgroup of $Y$).
The $Y$-orbits are simply the partition of $\{1,2,3,4\}$ which defines $Y$. This obviously agrees with your answer to the first exam... | {
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One sided limit of an increasing function defined on an open interval Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$? If so, is it true that $\ell=\lim_{x\to a}f(x)\le f(x) \ \ \forall x\in (a,b)$? Please provide p... | As for the second statement, I think there is a more precise result. Below is my trial, please check if it is correct.
Lemma Let $f\colon D\subset\mathbb{R}\to\mathbb{R}.$ Suppose that $a, b\in D$ with $a<b, $ and the point $a$ is a right-sided limit point of $D.$ Let $f$ is strictly increasing on $(a,b)\cap D.$... | {
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Multiple choice question from general topology Let $X =\mathbb{N}\times \mathbb{Q}$ with the subspace topology of $\mathbb{R}^2$ and $P = \{(n, \frac{1}{n}): n\in \mathbb{N}\}$ .
Then in the space $X$
Pick out the true statements
1 $P$ is closed but not open
2 $P$ is open but not closed
3 $P$ is both open and closed
... | For example, 1 is true. You can see that $P$ is not open by looking at an $\varepsilon$-ball around any point $p = (n, \frac1n )$ in $P$. Then there will be a rational $q$ such that $(n,q)$ is inside the ball hence $P$ is not open. (because $\mathbb Q$ is dense in $\mathbb R$)
Also, it's closed: think about why its com... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Could someone please explain what this question is asking? I have some trouble understanding the following question:
Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ are geodesics. Show that ${\partial \over \partial y}\left({F\... | Remember that $(u,v)$ is a local system of coordinates of a neighborhood of your surface. If you have a first fundamental form given, implicitly the system of local coordinates is given wich is a diffeomorphism. $x=u$ and $y=v$ meaning that you are looking the images of coordinate axis, this images must be geodesics fo... | {
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Subgroups of finite solvable groups. Solvable? I am attempting to prove that, given a non-trivial normal subgroup $N$ of a finite group $G$, we have that $G$ is solvable iff both $N$, $G/N$ are solvable. I was able to show that if $N,G/N$ are solvable, then $G$ is; also, that if $G$ is solvable, then $G/N$ is. I am stu... | With your definition, to show that if $G$ is solvable then $N$ is solvable, let
$$ 1 =G_0 \triangleleft G_1\triangleleft\cdots\triangleleft G_{m-1}\triangleleft G_m=G$$
be such that $G_{i+1}/G_{i}$ is abelian for each $i$.
Note: We do not need to assume that $N$ is normal; the argument below works just as well for any ... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$ Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$
My Trig is really rusty and weak so I don't understand the given answer:
$(2+i)^n + (2-i)^n $
$= \left ( \sqrt{5} ... | Hint $\ $ Scaling the equation by $\sqrt{5}^{\:-n}$ and using Euler's $\: e^{{\it i}\:\!x} = \cos(x) + {\it i}\: \sin(x),\ $ it becomes
$$\smash[b]{\left(\frac{2+i}{\sqrt{5}}\right)^n + \left(\frac{2-i}{\sqrt{5}}\right)^n} =\: (e^{{\it i}\:\!\theta})^n + (e^{- {\it i}\:\!\theta})^n $$
But
$$\smash[t]{ \left|\frac{2+i}... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve this inequation Given two real numbers $0<a<1$ and $0<\delta<1$, I want to find a positive integer $i$ (it is better to a smaller $i$) such that
$$\frac{a^i}{i!} \le \delta.$$
| Here is a not-very-good answer. Let $i$ be the result of rounding $\log\delta/\log a$ up to the nearest integer. Then $i\ge\log\delta/\log a$, so $i\log a\le\log\delta$ (remember, $\log a\lt0$), so $a^i\le\delta$, so $a^i/i!\le\delta$.
| {
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Path lifting theorem http://www.maths.manchester.ac.uk/~jelena/teaching/AlgebraicTopology/PathLifting.pdf
I'm trying to generalize this theorem. But, was wondering in the proof given here and similarly in Hatchers. Can you replace the $S^{1}$ with a general space X. As it seems to not be that important in the proof. So... | We have the following generalization (this can e.g. be found in Munkres "Topology", 3rd edition):
All spaces are assumed to be connected, locally path connected.
Lemma 79.1 (The general lifting lemma):
Let $p: E\to X$ be a covering map; let $p(e_0) = x_0$. Let $f: Y\to X$ be a continuous map with $f(y_0) = x_0$. Th... | {
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How are the "real" spherical harmonics derived? How were the real spherical harmonics derived?
The complex spherical harmonics:
$$
Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi}
$$
But the "real" spherical harmonics are given on this wiki page as
$$
Y_{lm} =
\begin{cases}
\frac{1}{\sqrt{2}} ( Y_l^m + ... |
Why are the real spherical harmonics defined this way and not simply as $\Re{(Y_l^m)}$?
Well yes it is! The real spherical harmonics can be rewritten as followed:
$$Y_{lm} =
\begin{cases}
\sqrt{2}\Re{(Y_l^m)}=\sqrt{2}N_l^m\cos{(m\phi)}P_l^m(\cos \theta) & \text{if } m > 0 \\
Y_l^0=N_l^0P_l^0(\cos \theta) & \te... | {
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Complex eigenvalues of real matrices
Given a matrix
$$A = \begin{pmatrix}
40 & -29 & -11\\ \ -18 & 30\ & -12 \\\ \ 26 &24 & -50 \end{pmatrix}$$
has a certain complex number $l\neq0$ as an eigenvalue. Which of the following must also be an eigenvalue of $A$:
$$l+20, l-20, 20-l, -20-l?$$
It seems that comp... | Hint: The trace of the matrix is $40+30+(-50)$. As you observed, $0$ is an eigenvalue.
| {
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Examples of mathematical induction What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ where the number of terms depends on $n$ and you're doing induction on $n$?
Postscript t... | I like the ones that involve division.
For instance, prove that $7 \mid 11^n-4^n$ for $n=1, 2, 3, \cdots$
Another example would be perhaps proving that
$$(3+\sqrt{5})^n+(3-\sqrt{5})^n$$
is an even integer for all natural numbers $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/145189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 13
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Area of a Spherical Triangle from Side Lengths I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing to find the answer to the following embarrassingly easy (I think) questio... | As an equilateral spherical triangle gets arbitrarily small, its angles all approach π/3. So one might say there is a degenerate spherical triangle whose angles are in fact all π/3 and whose area is 0.
| {
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Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian. I am revising for my Rings and Modules exam and am stuck on the following two questions:
$1.$ Let $M$ be a noetherian module and $ \ f : M \rightarrow M \ $ a surjective homomorphism. Sho... | $1.$ Let $\,f:M\to M\,$ be an epimorphism or $\,R\,$-modules, with $\,M\,$ Noetherian.
i) Show that $\,M\,$ can be made into $\,R[t]\,$-module, defining $\,tm:=f(m)\,,\,\forall m\in M$
ii) Putting $\,I:=\langle t\rangle=tR[t]\,$ , show that $\,MI=M\,$
iii) Apply Nakayama's Lemma to deduce that there exists $\,1+g(t)t\i... | {
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The topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.
Let $\tau$ be to topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.
How would you find the closure of $(0,1)$ in $\tau$?
I'm trying to find the smallest closed set containing $(0,1)$ in that ... | Hints:
(i) Show that $(-\infty,b)$ is in $\tau$ for every $b$.
(ii) Show that $[a,+\infty)$ is in $\tau$ for every $a$.
(iii) Deduce that $[a,b)$ is closed in $\tau$ for every $a\lt b$.
(iv) Show that $a$ is in the closure of $(a,b)$ with respect to $\tau$ for every $a\lt b$.
(v) Conclude that the closure of $(a,b)$ is... | {
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Solve equations using the $\max$ function How do you solve equations that involve the $\max$ function? For example:
$$\max(8-x, 0) + \max(272-x, 0) + \max(-100-x, 0) = 180$$
In this case, I can work out in my head that $x = 92.$ But what is the general procedure to use when the number of $\max$ terms are arbitrary? Tha... | Check each of the possible cases. In your equations the "critical" points (i. e. the points where one of the max's switches) are $8$, $272$ and $-100$. For $x \le -100$ your equation reads
\[ 8-x + 272 - x + (-100-x) = 180 \iff 180 - 3x = 180 \]
which doesn't have a solution in $(-\infty, -100]$.
For $-100 \le x \le ... | {
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Metrization of the cofinite topology Can you help me please with this question?
Let $X$ be a non-empty set with the cofinite topology.
Is $\left ( X,\tau_{\operatorname{cofinite}} \right ) $ a metrizable space?
Thanks a lot!
| *
*If $X$ is finite, the cofinite topology is the discrete one, which is metrizable, for example using the distance $d$ defined by $d(x,y):=\begin{cases}0&\mbox{ if }x=y,\\
1&\mbox{ otherwise}
.\end{cases}$
*If $X$ is infinite, it's not a Hausdorff space. Indeed, let $x,y\in X$, and assume that $U$ and $V$ are two d... | {
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"answer_id": 0
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What is the $x(t)$ function of $\dot{v} = a v² + bv + c$ to obtain $x(t)$ How to solve $$\frac{dv}{dt} = av^2 + bv + c$$ to obtain $x(t)$, where $a$, $b$ and $c$ are constants, $v$ is velocity, $t$ is time and $x$ is position. Boundaries for the first integral are $v_0$, $v_t$ and $0$, $t$ and boundaries for the second... | "Separating variables" means writing
$${dv\over av^2+bv +c}=dt\ .\qquad(1)$$
The next step depends on the values of $a$, $b$, $c$. Assuming $a>0$ one has
$$a v^2+bv +c={1\over a}\Bigl(\bigl(av +{b\over 2}\bigr)^2+{4ac -b^2\over 4}\Bigr)\ ,$$
so that after a linear substitution of the dependent variable $v$ the equation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Choosing squares from a grid so that no two chosen squares are in the same row or column
How many ways can 3 squares be chosen from a 5x5 grid so that no two
chosen squares are in the same row or column?
Why is this not simply $\binom{5}{3}\cdot\binom{5}{3}$?
I figured that there were $\binom{5}{3}$ ways to choose ... | Here's the hard way to do the problem: inclusion-exclusion.
There are $25\choose3$ ways to choose 3 squares from the 25.
Now you have to subtract the ways that have two squares in the same row or column. There are 10 ways to choose the row/column, $5\choose2$ ways to choose the two squares in the row/column, and 23 c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\log(n) = O(\sqrt{n})$ How to prove $\log(n) = O(\sqrt{n})$?
How do I find the $c$ and the $n_0$?
I understand to start, I need to find something that $\log(n)$ is smaller to, but I m having a hard time coming up with the example.
| $\log(x) < \sqrt{x}$ for all $x>0$ because $\log(x) /\sqrt{x}$ has a single maximum value $2/e<1$ (at $x=e^2$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/145739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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"answer_id": 2
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Cantor set: Lebesgue measure and uncountability I have to prove two things.
First is that the Cantor set has a lebesgue measure of 0. If we regard the supersets $C_n$, where $C_0 = [0,1]$, $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and so on. Each containig interals of length $3^{-n}$ and by construction there are $2... | I decided I would answer the latter part of your question in a really pretty way.
You can easily create a surjection from the Cantor set to $[0,1]$ by using binary numbers. Binary numbers are simply numbers represented in base $2$, so that the only digits that can be used are $0$ and $1$. If you take any number in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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A finite-dimensional vector space cannot be covered by finitely many proper subspaces? Let $V$ be a finite-dimensional vector space, $V_i$ is a proper subspace of $V$ for every $1\leq i\leq m$ for some integer $m$. In my linear algebra text, I've seen a result that $V$ can never be covered by $\{V_i\}$, but I don't kno... | This question was asked on MathOverflow several years ago and received many answers: please see here.
One of these answers was mine. I referred to this expository note, which has since appeared in the January 2012 issue of the American Mathematical Monthly.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Compact complex surfaces with $h^{1,0} < h^{0,1}$ I am looking for an example of a compact complex surface with $h^{1,0} < h^{0,1}$. The bound that $h^{1,0} \leq h^{0,1}$ is known. In the Kähler case, $h^{p,q}=h^{q,p}$, so the example cannot be (for example) a projective variety or a complex torus. Does anyone know of ... | For a compact Kähler manifold, $h^{p,q} = h^{q, p}$, so the odd Betti numbers are even. For a compact complex surface, the only potentially non-zero odd Betti numbers are $b_1$ and $b_3$; note that by Poincaré duality, they are equal.
So if $X$ is a compact complex surface, and $X$ is Kähler, then $b_1$ is even. Surpr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove or disprove: $(\mathbb{Q}, +)$ is isomorphic to $(\mathbb{Z} \times \mathbb{Z}, +)$? Prove or disprove: $\mathbb{Q}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$. I mean the groups $(\mathbb Q, +)$ and $(\mathbb Z \times \mathbb Z,+).$ Is there an isomorphism?
| Yet another way to see the two cannot be isomorphic as additive groups: if $a,b\in\mathbb{Q}$, and neither $a$ nor $b$ are equal to $0$, then $\langle a\rangle\cap\langle b\rangle\neq\{0\}$; that is, any two nontrivial subgroups intersect nontrivially. To see this, write $a=\frac{r}{s}$, $b=\frac{u}{v}$, with $r,s,u,v\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Terminology question What do you call a set of points with the following property?
For any point and any number $\epsilon$, you can find another point in the set that is less than $\epsilon$ away from the first point.
An example would be the rationals, because for any $\epsilon$ there is some positive rational number... | Turning my comment into an answer:
Such a set is said to be dense-in-itself. The term perfect is also sometimes used, but I prefer to avoid it, since it has other meanings in general topology. One can also describe such a set by saying that it has no isolated points. All of this terminology applies to topological space... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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sequence of decreasing compact sets In Royden 3rd P192,
Assertion 1: Let $K_n$ be a decreasing sequence compact sets, that is, $K_{n+1} \subset K_n$. Let $O$ be an open set with $\bigcap_1^\infty K_n \subset O$. Then $K_n \subset O$ for some $n$.
Assertion 2: From this, we can easily see that $\bigcap_1^\infty K_n$ is ... | Here's a T_1 space for which Assertion 2 fails. Take the set of integers. Say that a set is open iff it is either a subset of the negative integers or else is cofinite. Then let K_n be the complement of {0, 1, ..., n}. Then each K_n is compact, but the intersection of K_n from n=1 to infinity is the set of negative... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Relating Gamma and factorial function for non-integer values. We have
$$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$
for integers, so if $\Delta$ is some real value with
$$0<\Delta<1,$$
then
$$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$
because $\Gamma$ is monotone there and so there is another number $f$ with
$$0<f<1,$$
... | Asymptotically, as $n \to \infty$ with fixed $\Delta$,
$$ f(n,\Delta) = \dfrac{\Gamma(n+1+\Delta)-\Gamma(n+1)}{\Gamma(n+2)-\Gamma(n+1)} =
n^\Delta \left( \dfrac{1}{n} + \dfrac{\Delta(1+\Delta)}{2n^2} + \dfrac{\Delta(-1+\Delta)(3\Delta+2)(1+\Delta)}{24n^3} + \ldots \right) $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$? Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
| The notation $\rm\:R[\alpha]\:$ denotes a ring-adjunction, and, analogously, $\rm\:F(\alpha)\:$ denotes a field adjunction. Generally if $\alpha$ is a root of a monic $\rm\:f(x)\:$ over a domain $\rm\:D\:$ then $\rm\:D[\alpha]\:$ is a field iff $\rm\:D\:$ is a field. The same is true for arbitrary integral extensions o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Determine the conditional probability mass function of the size of a randomly chosen family containing 2 girls. Suppose that 15 percent of the families in a certain community have no children, 20 percent have 1, 35 percent have 2, and 30 percent have 3 children; suppose further that each child is equally likely (and in... | It’s correct as far as it goes, but it’s incomplete. You’ve shown that $23.75$% of the families have at least two girls, but that doesn’t answer the question. What you’re to find is probability mass function of the family size given that the family has two girls. In other words, you want to calculate $$\Bbb P(B+G=x\mid... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"Weierstrass preparation" of $\mathbb{C}[[X,Y]]$ In Lang's book "Algebra", theorem 9.2, it said that suppose $f\in \mathbb{C}[[X,Y]]$, then by some conditions imposed to $f$, $f$ can be written as a product of a polynomial $g\in \mathbb{C}[[X]][Y]$ and a unit $u$ in $\mathbb{C}[[X,Y]]$.
It suggests the following claim ... | It is known that there are transcendental power series $h(X)\in \mathbb C[[X]]$ over $\mathbb C[X]$. Note that $Xh(X)$ is also transcendental. Let
$$f(X,Y)=Y-Xh(X)\in\mathbb C[[X,Y]].$$
Suppose $f=gu$ with $g$ polynomial and $u$ invertible. Consider the ring homomorphism $\phi: \mathbb C[[X,Y]]\to \mathbb C[[X]]$ whic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove $\mathcal{l}(D+P) \leq \mathcal l{(D)} + 1$ Let $X$ be an irreducible curve, and define $\mathcal{L}(D)$ as usual for $D \in \mathrm{Div}(X)$. Define $l(D) = \mathrm{dim} \ \mathcal{L}(D)$. I'd like to show that for any divisor $D$ and point $P$, $\mathcal{l}(D+P) \leq \mathcal l{(D)} + 1$.
Say $D = \sum ... | Here is an elementary formulation, without sheaves.
Let $t\in Rat(X)$ be a uniformizing parameter at $P$ (that is, $t$ vanishes with order $1$ at $P$) and let $n_P\in \mathbb Z$ be the coefficient of $D=\sum n_QQ$ at $P$.
You then have en evaluation map $$\lambda: \mathcal L(D+P)\to k:f\mapsto (t^{n_P +1}\cdot f)(P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Scheduling 12 teams competing at 6 different events I have a seemingly simple question. There are 12 teams competing in 6 different events. Each event is seeing two teams compete. Is there a way to arrange the schedule so that no two teams meet twice and no teams repeat an event.
Thanks.
Edit: Round 1: All 6 events ... | A solution to the specific problem is here:
Event 1 Event 2 Event 3 Event 4 Event 5 Event 6
1 - 2 11 - 1 1 - 3 6 - 1 10 - 1 1 - 9
3 - 4 2 - 3 4 - 2 2 - 11 2 - 9 10 - 2
5 - 6 4 - 5 5 - 7 7 - 4 3 - 11 4 - 8
7 - 8... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given an integral symplectic matrix and a primitive vector, is their product also primitive? Given a matrix $A \in Sp(k,\mathbb{Z})$, and a column k-vector $g$ that is primitive ( $g \neq kr$ for any integer k and any column k-vector $r$), why does it follow that $Ag$ is also primitive? Can we take A from a larger spac... | Suppose $\,Ag\,$ is non-primitive, then $\,Ag=mr\,\,,\,\,m\in\mathbb{Z}\,\Longrightarrow g=mA^{-1}r\, $ , which means $\,g\,$ is not primitive
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is solving non-linear recurrence relations "hopeless"? I came across a non-linear recurrence relation I want to solve, and most of the places I look for help will say things like "it's hopeless to solve non-linear recurrence relations in general." Is there a rigorous reason or an illustrative example as to why this... | Although it is possible to solve selected non-linear recurrence relations if you happen to be lucky, in general all sorts of peculiar and difficult-to-characterize things can happen.
One example is found in chaotic systems. These are hypersensitive to initial conditions, meaning that the behavior after many iterations... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that a space is disconnected
Show that a subspace $T$ of a topological space $S$ is disconnected iff there are nonempty sets $A,B \subset T$ such that $T= A\cup B$ and $\overline{A} \cap B = A \cap \overline{B} = \emptyset$. Where the closure is taken in $S$.
I've used this relatively simple proof for many of... | It looks fine to me, in particular because as $\,A\subset \overline{A}\Longrightarrow A\cap B\subset \overline{A}\cap B\,$ , so if the rightmost intersection is empty then also the leftmost one is, which is the usual definition
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Birational map between product of projective varieties What is an example of a birational morphism between $\mathbb{P}^{n} \times \mathbb{P}^{m} \rightarrow \mathbb{P}^{n+m}$?
| The subset $\mathbb A^n\times \mathbb A^m$ is open dense in $\mathbb P^n\times \mathbb P^m$ and the subset $\mathbb A^{n+m}$ is open dense in $\mathbb P^n\times \mathbb P^m$.
Hence the isomorphism $\mathbb A^n\times \mathbb A^m\stackrel {\cong}{\to} \mathbb A^{n+m}$ is the required birational isomorphism.
The aston... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Need to understand question about not-a-knot spline I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since not-a-knot is a boundary condition, what does it mean to derived it for $... | If $S$ is a N-a-K spline with knots $x_1, \dotsc, x_4$ then it satisfies the spline conditions: twelve equations in twelve unknowns. (Twelve coefficients, six equations to prescribe values at the knots and six more to force continuity of derivatives up to third order at $x_2$ and $x_3$.) Since $p_1, p_2, p_3$ fit up... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Weierstrass Factorization Theorem Are there any generalisations of the Weierstrass Factorization Theorem, and if so where can I find information on them? I'm trying to investigate infinite products of the form
$$\prod_{k=1}^\infty f(z)^{k^a}e^{g(z)},$$
where $g\in\mathbb{Z}[z]$ and $a\in\mathbb{N}$.
| The Weierstrass factorization theorem provides a way of constructing an entire function with any prescribed set of zeros, provided the set of zeros does not have a limit point in $\mathbb{C}$. I know that this generalizes to being able to construct a function holomorphic on a region $G$ with any prescribed set of zeros... | {
"language": "en",
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Symmetric Matrix as the Difference of Two Positive Definite Symmetric Matrices
Prove that any real symmetric matrix can be expressed as the difference of two
positive definite symmetric matrices.
I was trying to use the fact that real symmetric matrices are diagonalisable , but the confusion I am having is that 'if... | Let
$
A^{*}
$
be the adjoint of
$
A
$
and $S$ the positive square root of the positive self-adjoint operator
$
S^{2}=A^{*}A
$
(e.g. Rudin, ``Functional Analysis'', Mc Graw-Hill, New York 1973, p. 313-314, Th. 12.32 and 12.33) and write
$
P=S+A
$,
$
N=S-A
$.
Let $n$ be the finite dimension of $A$ and $\lambda_{i}, i=1\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Change of Basis Calculation I've just been looking through my Linear Algebra notes recently, and while revising the topic of change of basis matrices I've been trying something:
"Suppose that our coordinates are $x$ in the standard basis and $y$ in a different basis, so that $x = Fy$, where $F$ is our change of basis m... | Without saying much, here is how I usually remember the statement and also the proof in one big picture:
\begin{array}{ccc}
x_{1},\dots,x_{n} & \underrightarrow{\;\;\; A\;\;\;} & Ax_{1},\dots,Ax_{n}\\
\\
\uparrow F & & \downarrow F^{-1}\\
\\
y_{1},\dots,y_{n} & \underrightarrow{\;\;\; B\;\;\;} & By_{1},\dots,By_{n}
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove Continuous functions are borel functions Take $f: (a,b) \to \mathbb{R}$ , continuous for all $x_{0}\in (a,b)$ and take $(Ω = (a,b) , F = ( (a,b) ⋂ B(\mathbb{R}))$ where $B(\mathbb{R})$ is the Borel $\sigma$-algebra.
Prove $f$ is a borel function by showing that $\{x \in(a,b): f(x) < c \}$ is in $F$.
I know that ... | To expand on Thomas E.'s comment: if $f$ is continuous, $f^{-1}(O)$ for $O$ open is again open.
$\{x \in (a,b) : f(x) < c \} = f^{-1}((- \infty , c)) \cap (a,b)$. Now all you need to show to finish this proof is that $f^{-1}((- \infty , c))$ is in the Borel sigma algebra of $\mathbb R$.
Edit (in response to comment)
R... | {
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Prove that the order of an element in the group N is the lcm(order of the element in N's factors p and q) How would you prove that
$$\operatorname{ord}_N(\alpha) = \operatorname{lcm}(\operatorname{ord}_p(\alpha),\operatorname{ord}_q(\alpha))$$
where $N=pq$ ($p$ and $q$ are distinct primes) and $\alpha \in \mathbb{Z}^... | Hint. There are natural maps $\mathbb{Z}^*_N\to\mathbb{Z}^*_p$ and $\mathbb{Z}^*_N\to\mathbb{Z}^*_q$ given by reduction modulo $p$ and reduction modulo $q$. This gives you a homomorphism $\mathbb{Z}^*_N\to \mathbb{Z}^*_p\times\mathbb{Z}^*_q$.
What is the kernel of the map into the product? What is the order of an eleme... | {
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Differential equation problem I am looking at the differential equation:
$$\frac{dR}{d\theta} + R = e^{-\theta} \sec^2 \theta.$$
I understand how to use $e^{\int 1 d\theta}$ to multiply both sides which gives me:
(looking at left hand side of equation only)
$$e^\theta \frac{dR}{d\theta} + e^\theta R.$$
However I am not... | We have
$$\frac{d R(\theta)}{d \theta} + R(\theta) = \exp(-\theta) \sec^2(\theta)$$
Multiply throughout by $\exp(\theta)$, we get $$\exp(\theta) \frac{dR(\theta)}{d \theta} + \exp(\theta) R(\theta) = \sec^{2}(\theta)$$
Note that $$\frac{d (R(\theta) \exp(\theta))}{d \theta} = R(\theta) \exp(\theta) + \exp(\theta) \frac... | {
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Find the intersection of these two planes.
Find the intersection of $8x + 8y +z = 35$ and
$x = \left(\begin{array}{cc} 6\\
-2\\ 3\\ \end{array}\right)
+$ $ \lambda_1 \left(\begin{array}{cc}
-2\\ 1\\ 3\\ \end{array}\right)
+$ $ \lambda_2 \left(\begin{array}{cc} 1\\ 1\\
-1\\ \end{array}\right) $
So, I have been ... | It's just a simple sign mistake. The equation should be
$$-4x+y-3z=-35$$
instead of
$$-4x+y-3z=35.$$
Your solution will work fine then.
| {
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Product Measures Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = S_0$ $\ u^{\Sigma X_i(\omega)}d^{n-\Sigma X_i(\omega)} \omega \in \Omega $, $\ n=1,\ldots,6 $.
Choose the... | Yes, that is correct. You have to show that $\sigma(X_i)$ and $\sigma(X_j)$ are independent, when $j\neq i$ (note that I have omitted the $\omega$ in $\sigma(X_i(\omega))$, because that is not what you want). Now, recall that
$$
\sigma(X_i)=\sigma(\{X_i^{-1}(A)\mid A\in \mathcal{B}(\mathbb{R})\}),
$$
and hence it is en... | {
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} |
Computing conditional probability out of joint probability If I have given a complete table for the joint probability $$P(A,B,C,D,E)$$ how can I compute an arbitrary conditional probability out of it, for instance: $$P(A|B)$$
| $$\mathbb{P}(A=a \vert B=b) = \frac{\mathbb{P}(A=a, B=b)}{\mathbb{P}(B=b)} = \frac{\displaystyle \sum_{c,d,e} \mathbb{P}(A=a, B=b, C=c, D=d, E=e)}{\displaystyle \sum_{a,c,d,e} \mathbb{P}(A=a, B=b, C=c, D=d, E=e)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/147831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why are zeros/roots (real) solutions to an equation of an n-degree polynomial? I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that $ y = 0$ mean for the solutions, the inte... | One reason is that it makes solving an equation simple, especially if $f(x)$ is written only as the product of a few terms. This is because $a\cdot b = 0$ implies either $a = 0$ or $b = 0$. For example, take $f(x) = (x-5)(x+2)(x-2)$. To find the values of $x$ where $f(x) = 0$ we see that $x$ must be $5$, $-2$, or $2$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Graph decomposition What is the smallest $n \in \mathbb{N}$ with $ n \geq5$ such that the edge set of the complete graph $K_n$ can be partitioned (decomposed) to edge disjoint copies of $K_4$?
I got a necesary condition for the decomposition is that $12 |n(n-1)$ and $3|n-1$, thus it implies $n \geq 13$. But can $K_{13}... | The degree of $K_9$ is 8, whereas the degree of $K_4$ is 3. Since $3$ does not divide $8$, there is no $K_4$ decomposition of $K_9$.
$K_n$ has a decomposition into edge-disjoint copies of $K_4$ whenever $n \equiv 1 \text{ or 4 } (\text{mod} 12)$, so the next smallest example after $K_4$ is $K_{13}$.
| {
"language": "en",
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Solving polynomial differential equation I have $a(v)$ where $a$ is acceleration and $v$ is velocity. $a$ can be described as a polynomial of degree 3:
$$a(v) = \sum\limits_{i=0}^3 p_i v^i = \sum\limits_{i=0}^3 p_i \left(\frac{dd(t)}{dt}\right)^i,$$
where $d(t)$ is distance with respect to time.
I want to solve (or ap... | Since the acceleration is the derivative of velocity, you can write
$$ \frac{\mathrm{d} v}{\mathrm{d} t} = p_0 + p_1 v + p_2 v^2 + p_3 v^3 $$
separating the variables we get the integral form
$$ \int \frac{\mathrm{d}v}{p_0 + p_1 v + p_2 v^2 + p_3 v^3} = \int \mathrm{d}t = t + c$$
Which we can integrate using partial fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Curve arc length parametrization definition I did some assignments related to curve arc length parametrization.
But what I can't seem to find online is a formal definition of it.
I've found procedures and ways to find a curve's equation by arc length parametrization, but I'm still missing a formal definition which I ha... | Suppose $\gamma:[a,b]\rightarrow {\Bbb R}$ is a smooth curve with $\gamma'(t) \not = 0$ for $t\in[a,b]$.
Define $$s(t) = \int_a^t ||\gamma'(\xi)||\,d\xi$$ for $t\in[a,b]$. This function $s$ has a positive derivative, so it possesses a differentiable inverse. You can use it to get a unit-speed reparametrization of ... | {
"language": "en",
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Given n raffles, what is the chance of winning k in a row? I was reading this interesting article about the probability of of tossing heads k times in a row out of n tosses. The final result was
$$P = 1-\frac{\operatorname{fib}_k(n+2)}{2^n}\;,$$
where $\operatorname{fib}_k(n)$ is the $n$-th $k$-step Fibonacci number.
H... | We can proceed as follows. Let $p$ be the probability that we flip a head, and $q=1-p$ the probability that we flip tails. Let us search for the probability that we do NOT have at least $k$ heads in a row at some point after $n$ flips, which we will denote $P(n,k)$.
Given a sequence of coin tosses (of length at least... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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cohomology of a finite cyclic group I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title.
Let $G=\langle\sigma\rangle$ where $\sigma^m=1$. Let $N=1+\sigma+\sigma^2+\cdots+\sigma^{m-1}$. Then it is claimed in Dummit... | As $(\sigma-1)(c_0+c_1\sigma+\dots c_{n-1}\sigma^{n-1})=(c_n-c_0)+(c_0-c_1)\sigma+\dots (c_{n-2}-c_{n-1})\sigma^{n-1}$, the element $a=c_0+c_1\sigma+\dots c_{n-1}\sigma^{n-1}$ is in the kernel of $\sigma-1$ iff all $c_i$'s are equal, i.e. iff $a=Nc$ for some $c\in\mathbb{Z}$. Similarly, $Na=(\sum c_i)N$, so here the ke... | {
"language": "en",
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Why is unique ergodicity important or interesting? I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be decomposed into smaller subsystems (the only invariant pieces are real... | Unique ergodicity is defined for topological dynamical systems and it tells you that the time average of any function converges pointwise to a constant (see Walters: Introduction to Ergodic Theory, th 6.19). This property is often useful.
Any ergodic measure preserving system is isomorphic to a uniquely ergodic (minim... | {
"language": "en",
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Density of the set $S=\{m/2^n| n\in\mathbb{N}, m\in\mathbb{Z}\}$ on $\mathbb{R}$? Let $S=\{\frac{m}{2^n}| n\in\mathbb{N}, m\in\mathbb{Z}\}$, is $S$ a dense set on $\mathbb{R}$?
| Yes, is it, given open interval $(a,b)$ (suppose $a$ and $b$ positives) you can find $n\in\mathbb{N}$ such that $1/2^n<|b-a|$. Then consider the set:
$$X=\{k\in \mathbb{N}; k/2^n > b\}$$
This is a subset of $\mathbb{N}$, for well ordering principe $X$ has a least element $k_0$ then is enought taking $(k_0-1)/2^n\in(a,b... | {
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What exactly is nonstandard about Nonstandard Analysis? I have only a vague understanding of nonstandard analysis from reading Reuben Hersh & Philip Davis, The Mathematical Experience. As a physics major I do have some education in standard analysis, but wonder what the properties are that the nonstandardness (is that ... | To complement the fine answers given earlier, I would like to address directly the question of the title: "What exactly is nonstandard about Nonstandard Analysis?"
The answer is: "Nothing" (the name "nonstandard analysis" is merely a descriptive title of a field of research, chosen by Robinson). This is why some scho... | {
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Homotopic to a Constant I'm having a little trouble understanding several topics from algebraic topology. This question covers a range of topics I have been looking at.
Can anyone help? Thanks!
Suppose $X$ and $Y$ are connected manifolds, $X$ is simply connected, and the universal cover of $Y$ is contractible. Why ... | Let $\tilde{Y} \xrightarrow{\pi} Y$ be the universal cover of $Y$.
Since $X$ is simply connected, any continuous map $X \xrightarrow{f} Y$ can be factorized as a continuous map $X \xrightarrow{\tilde{f}} \tilde{Y} \xrightarrow{\pi} Y$.
Since $\tilde{Y}$ is contractible, there is a point $y \in \tilde{Y}$ and an homotop... | {
"language": "en",
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Cumulative probability and predicted take in a raffle? Not sure if this is the right term!
If I have a raffle with 100 tickets in at $5 each, and people pull a ticket sequentially, how do I calculate the likely return before the winning ticket is drawn?
I'm half way there. I get that you work out the cumulative probab... | The calculations seem to involve a strange kind of raffle, in which the first ticket is sold (for $5$ dollars). We check whether this is the winning ticket. If it is not, we sell another ticket, and check whether it is the winner. And so on.
You seem to be asking for the expected return. This is $5$ times $E(X)$, wher... | {
"language": "en",
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Exponential objects in a cartesian closed category: $a^1 \cong a$ Hi I'm having problems with coming up with a proof for this simple property of cartesian closed categories (CCC) and exponential objects, namely that for any object $a$ in a CCC $C$ with an initial object $0$, $a$ is isomorphic to $a^1$ where $1$ is the... | You can also reason as follows, without the Yoneda lemma. But proving uniqueness of right adjoints is cumbersome without using Yoneda, and easy with. Anyway, here it goes:
The functor $(-)\times 1$ is isomorphic to the identity functor. The identity functor is a right adjoint of itself, so the identity functor is also ... | {
"language": "en",
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Sum of three primes Can all natural numbers ($n\ge 6$) be represented as the sum of three primes?
With computer I checked up to $10000$, but couldn't prove it.
| It was proved by Vinogradov that every large enough odd integer is the sum of at most $3$ primes, and it seems essentially certain that apart from a few uninteresting small cases, every odd integer is the sum of $3$ primes.
Even integers are a different matter. To prove that every even integer $n$ is the sum of three p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Spectra of restrictions of bounded operators Suppose $T$ is a bounded operator on a Banach Space $X$ and $Y$ is a non-trivial closed invariant subspace for $T$. It is fairly easy to show that for the point spectrum one has $\sigma_p(T_{|Y})\subseteq\sigma_p(T)$ and this is also true for the approximate point spectrum... | For example, consider the right shift operator $R$ on $X = \ell^2({\mathbb Z})$, $Y = \{y \in X: y_j = 0 \ \text{for}\ j < 0\}$. Then $Y$ is invariant under $R$, and $\sigma(R)$ is the unit circle while $\sigma(R|_Y)$ is the closed unit disk.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Analyze the convergence or divergence of the sequence $\left\{\frac1n+\sin\frac{n\pi}{2}\right\}$
Analyze the convergence or divergence of the following sequence
a) $\left\{\frac{1}{n}+\sin\frac{n\pi}{2}\right\}$
The first one is divergent because of the in $\sin\frac{n\pi}{2}$ term, which takes the values, for $n = ... | You’re on the right track, but you’ve left out an important step: you haven’t said anything to take the $1/n$ term into account. It’s obvious what’s happening, but you still have to say something.
Let $a_n=\frac1n+\sin\frac{n\pi}2$. If $\langle a_n:n\in\Bbb Z^+\rangle$ converged, say to $L$, then the sequence $\left\la... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$ Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
| The result is more general.
Fact: For any function $f$ regular enough on $[0,1]$, introduce
$$
A_n=\sum_{k=1}^nf\left(\frac{k}n\right)\qquad B=\int_0^1f(x)\mathrm dx\qquad C=f(1)-f(0)
$$
Then,
$$
\lim\limits_{n\to\infty}A_n-nB=\frac12C
$$
For any real number $p\gt0$, if $f(x)=x^p$, one sees that $B=\frac1{p+1}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding the second-degree polynomial that is the best approximation for cos(x) So, I need to find the second-degree polynomial that is the best approximation for $f(x) = cos(x)$ in $L^2_w[a, b]$, where $w(x) = e^{-x}$, $a=0$, $b=\infty$.
"Best approximation" for f is a function $\hat{\varphi} \in \Phi$ such that:
$||f ... | In your $L^2$ space the Laguerre polynomials form an orthonormal family, so if you use the polynomial
$$
P(x)=\sum_{i=0}^n a_i L_i(x),
$$
you will get the approximation error
$$
||P(x)-\cos x||^2=\sum_{i=0}^n(a_i-b_i)^2+\sum_{i>n}b_i^2,
$$
(Possibly you need to add a constant to account for the squared norm of the comp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Limit points of sets Find all limit points of given sets:
$A = \left\{ (x,y)\in\mathbb{R}^2 : x\in \mathbb{Z}\right\}$
$B = \left\{ (x,y)\in\mathbb{R}^2 : x^2+y^2 >1 \right\}$
I don't know how to do that. Are there any standard ways to do this?
| 1) Is set A closed or not? If it is we're then done, otherwise there's some point not in it that is a limit point of A
2) As before but perhaps even easier.
| {
"language": "en",
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Notation for infinite product in reverse order This question is related to notation of infinite product.
We know that,
$$
\prod_{i=1}^{\infty}x_{i}=x_{1}x_{2}x_{3}\cdots
$$
How do I denote
$$
\cdots x_{3}x_{2}x_{1} ?
$$
One approach could be
$$
\prod_{i=\infty}^{1}x_{i}=\cdots x_{3}x_{2}x_{1}
$$
I need to use... | (With tongue in cheek:) what about this?
$$\left(x_n\prod_{i=1}^\infty \right)\;$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/149398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Calculate $\int_\gamma \frac{1}{(z-z_0)^2}dz$ This is the definition of the fundamental theorem of contour integration that I have:
If $f:D\subseteq\mathbb{C}\rightarrow \mathbb{C}$ is a continuous function on a domain $D \subseteq \mathbb{C}$ and $F:D\subseteq \mathbb{C} \rightarrow \mathbb{C}$ satisfies $F'=f$ on $D... | $\gamma(2\pi)=Re^{2\pi i}=R=Re^0=\gamma(0)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\lim_{x\rightarrow a}\|f(x)\|$ and $\lim_{x\rightarrow a}\frac{\|f(x)\|}{\|x-a\|}$ Given any function $f: \mathbb{R^n} \to \mathbb{R^m}$ , if $$\lim_{x\rightarrow a}\|f(x)\| = 0$$ then does $$\lim_{x\rightarrow a}\frac{\|f(x)\|}{\|x-a\|} = 0 $$
as well? Is the converse true?
| For the first part, consider e.g. the case $m = n$ with $f$ defined by $f(x) = x - a$ for all $x$ to see that the answer is no.
For the second part, the answer is yes. If $\lim_{x \to a} \|f(x)\|/\|x-a\| = L$ exists (we do not need to assume that it is $0$), then since $\lim_{x \to a} \|x - a\| = 0$ clearly exists, we... | {
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Is there a uniform way to define angle bisectors using vectors?
Look at the left figure. $x_1$ and $x_2$ are two vectors with the same length (norm). Then $x_1+x_2$ is along the bisector of the angle subtended by $x_1$ and $x_2$. But look at the upper right figure. When $x_1$ and $x_2$ are collinear and in reverse dir... | I also would like to give a solution, which I am currently using in my work.
The key idea is to use a rotation matrix.
Suppose the angle between $x_1$ and $x_2$ is $\theta$. Let $R(\theta/2)$ be a rotation matrix, which can rotate a vector $\theta/2$. Then
$$y=R(\theta/2)x_1$$
is a unified way to express the bisector.
... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is the expected value $E(X^2) \neq E(X)^2$? I wish to use the Computational formula of the variance to calculate the variance of a normal-distributed function. For this, I need the expected value of $X$ as well as the one of $X^2$. Intuitively, I would have assumed that $E(X^2)$ is always equal to $E(X)^2$. In fact... | May as well chime in :)
Expectations are linear pretty much by definition, so $E(aX + b) = aE(X) + b$. Also linear is the function $f(x) = ax$. If we take a look at $f(x^2)$, we get
$f(x^2) = a(x^2) \not= (ax)^2 = f(x)^2$.
If $E(X^2) = E(X)^2$, then $E(X)$ could not be linear, which is a contradiction of its definition... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
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Positive Operator Value Measurement Question I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators $\{E_m\}$ for states $|\psi_1\rangle = |0\rangle, |\psi_2\rangle = (|0\rangle + |1\rangle)/... | Yes, those are the results but you have the subindexes swaped.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/149794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is critical Haudorff measure a Frostman measure? Let $K$ be a compact set in $\mathbb{R}^d$ of Hausdorff dimension $\alpha<d$, $H_\alpha(\cdot)$ the $\alpha$-dimensional Hausdorff measure. If $0<H_\alpha(K)<\infty$, is it necessarily true that $H_\alpha(K\cap B)\lesssim r(B)^\alpha$ for any open ball $B$? Here $r(B)$ d... | Consider e.g. $\alpha=1$, $d=2$. Given $p > 1$, let $K$ be the union of a sequence of line segments of lengths $1/n^2$, $n = 1,2,3,\ldots$, all with one endpoint at $0$. Then for $0 < r < 1$, if $B$ is the ball of radius $r$ centred at $0$,
$H_1(K \cap B) = \sum_{n \le r^{-1/2}} r + \sum_{n > r^{-1/2}} n^{-2} \appr... | {
"language": "en",
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Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$
I have managed to prove $N... | Hint $\rm\ \ unit\ \alpha\iff \alpha\:|\: 1\iff \alpha\alpha'\:|\:1 \iff unit\ \alpha\alpha',\ $ since $\rm\:\alpha\:|\:1\iff\alpha'\:|\:1' = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/149886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Peano postulates I'm looking for a set containing an element 0 and a successor function s that satisfies the first two Peano postulates (s is injective and 0 is not in its image), but not the third (the one about induction). This is of course exercise 1.4.9 in MacLane's Algebra book, so it's more or less homework, so i... | Since your set has 0 and a successor function, it must contain $\Bbb N$. The induction axiom is what ensures that every element is reachable from 0. So throw in some extra non-$\Bbb N$ elements that are not reachable from 0 and give them successors. There are several ways to do this.
Geometrically, $\Bbb N$ is a ray... | {
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"timestamp": "2023-03-29T00:00:00",
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Removing redundant sets from an intersection Let $I$ be a non-empty set and $(A_i)_{i\in I}$ a family of sets.
Is it true that there exists a subset $J\subset I$ such that $\bigcap_{j\in J}A_j=\bigcap_{i\in I}A_i$ and, for any $j_0\in J$, $\bigcap_{j\in J-\{j_0\}}A_j\neq\bigcap_{j\in J}A_j$?
If $I=\mathbb{N}$, the ans... | The answer is no, even in the case $I=\mathbb N$. to see this, consider the collection $A_i=[i,\infty)\subset \mathbb R$. Then $\bigcap\limits_{i\in I}A_i=\emptyset$ and this remains true if we intersect over any infinite subset $J\subseteq I$, yet is false if we intersect over a finite subset. Thus there is no minimal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Alternative proof of the limitof the quotient of two sums. I found the following problem by Apostol: Let $a \in \Bbb R$ and $s_n(a)=\sum\limits_{k=1}^n k^a$. Find
$$\lim_{n\to +\infty} \frac{s_n(a+1)}{ns_n(a)}$$
After some struggling and helpless ideas I considered the following solution.
If $a > -1$, then
$$\int_0^1... | The argument below works for any real $a > -1$.
We are given that
$$s_n(a) = \sum_{k=1}^{n} k^a$$ Let $a_n = 1$ and $A(t) = \displaystyle \sum_{k \leq t} a_n = \left \lfloor t \right \rfloor$. Hence, $$s_n(a) = \int_{1^-}^{n^+} t^a dA(t)$$ The integral is to be interpreted as the Riemann Stieltjes integral. Now integr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Showing that if $R$ is a commutative ring and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$. Let $R$ be a local ring, and let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be an $R$-module. I understand that $M \otimes_R (R / \mathfrak m)$ is isomorphic to $M / \mathfrak m M$, but I... | Morover, let $I$ be a right ideal of a ring $R$ (noncommutative ring) and $M$ a left $R$-module, then $M/IM\cong R/I\otimes_R M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/150114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Area of ellipse given foci? Is it possible to get the area of an ellipse from the foci alone? Or do I need at least one point on the ellipse too?
| If the foci are points $p,q\in\mathbb{R}^{2}$ on a horizontal line
and a point on the ellipse is $c\in\mathbb{R}^{2}$, then the string
length $\ell=\left|p-c\right|+\left|q-c\right|$ (the distance from
the first focus to the point on the ellipse to the second focus) determines
the semi-axis lengths. Using the Pythagore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove an L$^p$ type inequality Let $a,b\in[0,\infty)$ and let $p\in[1,\infty)$. How can I prove$$a^p+b^p\le(a^2+b^2)^{p/2}.$$
| Some hints:
*
*By homogeneity, we can assume that $b=1$.
*Let $f(t):=(t^2+1)^{p/2}-t^p-1$ for $t\geq 0$. We have $f'(t)=p((t^2+1)^{p/2-1}-t^{p-1})$. We have $t^2+1\geq t^2$, so the derivative is non-negative/non-positive if $p\geq 2$ or $p<2$.
*Deduce the wanted inequality (when it is reversed or not).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/150316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finitely Generated Group Let be $G$ finitely generated; My question is:
Does always exist $H\leq G,H\not=G$ with finite index?
Of course if G is finite it is true. But $G$ is infinite?
| No.
I suspect there are easier and more elegant ways to answer this question, but the following argument is one way to see it:
*
*There are finitely generated infinite simple groups:
*
*In 1951, Higman constructed the first example in A Finitely Generated Infinite Simple Group, J. London Math. Soc. (1951) s1-26 (1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simplifying quotient or localisation of a polynomial ring Let $R$ be a commutative unital ring and $g\in R[X]$ a polynomial with the property that $g(0)$ is a unit in $R$ and $g(1)=1$. Is there any possible way to understand either
$$R[X]/g$$ or $$g^{-1}R[X]$$ better? Here $g^{-1}R[X]$ is the localised ring for the mu... | The only thing which comes into my mind is the following: $g(1)=1$ (or $g(0)$ is a unit) ensures that $R[X] \to R$, $x \mapsto 1$ (or $x \mapsto 0$), extends to a homomorphism $g^{-1} R[X] \to R$. For more specific answers, a more specific question is necessary ;).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/150435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What are some books that I should read on 3D mathematics? I'm a first-grade highschool student who has been making games in 2D most of the time, but I started working on a 3D project for a change. I'm using a high-level engine that abstracts most of the math away from me, but I'd like to know what I'm dealing with!
Wha... | "Computer Graphics: Principles and Practice, Third Edition, remains the most authoritative introduction to the field. The first edition, the original “Foley and van Dam,” helped to define computer graphics and how it could be taught. The second edition became an even more comprehensive resource for practitioners and st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
UFDs are integrally closed Let $A$ be a UFD, $K$ its field of fractions, and $f$ an element of $A[T]$ a monic polynomial.
I'm trying to prove that if $f$ has a root $\alpha \in K$, then in fact $\alpha \in A$.
I'm trying to exploit the fact of something about irreducibility, will it help? I havent done anything with sp... | Overkill: It suffices to show that $A$ is integrally closed, which we prove using Serre's criterion. For this, we recall some of the definitions, including the definitions of the properties $R_n$ and $S_n$ that appear in Serre's criterion. We will assume $A$ is locally Noetherian (can one assume less for this approach ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 3,
"answer_id": 1
} |
Classification of automorphisms of projective space Let $k$ be a field, n a positive integer.
Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo scalars.
His hint is to show that $f^\star \mathcal{O}(1) \cong \math... | Well, $f^*(\mathcal{O}(1))$ must be a line bundle on $\mathbb{P}^n$. In fact, $f^*$ gives a group automorphism of $\text{Pic}(\mathbb{P}^n) \cong \mathbb{Z}$, with inverse $(f^{-1})^*$. Thus, $f^*(\mathcal{O}(1))$ must be a generator of $\text{Pic}(\mathbb{P}^n)$, either $\mathcal{O}(1)$ or $\mathcal{O}(-1)$. But $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Some interesting questions on completeness (interesting to me anyway) Suppose $(Y,\Vert\cdot\Vert)$ is a complete normed linear space. If the vector space $X\supset Y$ with the same norm $\Vert\cdot\Vert$ is a normed linear space, then is $(X,\Vert\cdot\Vert)$ necessarily complete?
My guess is no. However, I am not awa... | You have to look at infinite dimensional Banach spaces. For example, $X=\ell^2$, vector space of square-summable sequence of real numbers. Let $Y:=\{(x_n)_n, \exists k\in\Bbb N, x_n=0\mbox{ if }n\geq k\}$. It's a vector subspace of $X$, but not complete since it's not closed (it's in fact a strict and dense subset).
H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Ex${{{}}}$tremely Tricky Probability Question? Here's the question. It's quite difficult:
David is given a deck of 40 cards. There are 3 gold cards in the deck, 3 silver cards in the deck, 3 bronze cards in the deck and 3 black cards in the deck. If David draws the gold card on his first turn, he will win $50. (The ob... | We can ignore the silver cards-each should be replaced whenever we see it with another. Similarly, if you have a bronze, you should draw immediately (but subsequent bronzes don't let you replace them). So the deck is really $36$ cards, $3$ gold, $3$ black, and $30$ other (including the 2 bronzes after the first). Yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Relation between min max of a bounded with compact and continuity While reading through Kantorovitz's book on functional analysis, I had a query that need clarification. If $X$ is compact, $C_{B}(X)$ - bounded continuous function, with the sup-norm coincides with $C(X)$ - continuous real valued function, with the sup-n... | That is exactly what you said, but changing it a little:
Every continuous real function over a compact space is bounded.
We know that the image of a compact set by a continuous function is compact, and that implies boundedness of the image. A function is bounded exactly when its image is bounded, so it's proved!
Then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditions for integrability Michael Spivak, in his "Calculus" writes
Although it is possible to say precisely which functions are integrable,the criterion for integrability is too difficult to be stated here
I request someone to please state that condition.Thank you very much!
| This is commonly called the Riemann-Lebesgue Theorem, or the Lebesgue Criterion for Riemann Integration (the wiki article).
The statement is that a function on $[a,b]$ is Riemann integrable iff
*
*It is bounded
*It is continuous almost everywhere, or equivalently that the set of discontinuities is of zero lebesgue ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
An alternating series ... Find the limit of the following series:
$$ 1 - \frac{1}{4} + \frac{1}{6} - \frac{1}{9} + \frac{1}{11} - \frac{1}{14} + \cdot \cdot \cdot $$
If i go the integration way all is fine for a while but then things become pretty ugly. I'm trying to find out if there is some easier way to follow.
| Let $S = 1 - x^{3} + x^{5} -x^{8} + x^{10} - x^{13} + \cdots$. Then what you want is $\int_{0}^{1} S \ dx$. But we have
\begin{align*}
S &= 1 - x^{3} + x^{5} -x^{8} + x^{10} - x^{13} + \cdots \\\ &= -(x^{3}+x^{8} + x^{13} + \cdots) + (1+x^{5} + x^{10} + \cdots) \\\ &= -\frac{x^{3}}{1-x^{5}} + \frac{1}{1-x^{5}}
\end{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Simple recurrence relation in three dimensions I have the following recurrence relation:
$$f[i,j,k] = f[i-1,j,k] + f[i,j-1,k] + f[i,j,k-1],\quad \mbox{for } i \geq j+k,$$
starting with $f[0,0,0]=1$, for $i$, $j$, and $k$ non-negative.
Is there any way to find a closed form expression for $f[i,j,k]$?
Note that this basi... | With the constraint $i \geq j+k$ I got following formula (inspired by the Fuss-Catalan tetrahedra formula page 10 and with my thanks to Brian M. Scott for pointing out my errancies...) :
$$f[i,j,k]=\binom{i+1+j}{j} \binom{i+j+k}{k} \frac{i+1-j-k}{i+1+j}\ \ \text{for}\ i \geq j+k\ \ \text{and}\ \ 0\ \ \text{else}$$
pla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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