Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
One to one correspondence between transcendental and uncomputable numbers I know that both sets are uncountable infinite but the transcendentals are not a subset of the uncomputables. I don’t know if there exist uncomputable numbers that are not transcendental. But my question is whether the two sets have the same card... | Here
https://en.wikipedia.org/wiki/Computable_number
under "properties" it is stated the the set of computable numbers is countable.
Hence, the set of uncountable numbers must be uncountable infinite and hence have the same cardinality as the set of transcendental numbers which is uncountable infinite as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A game with an $n$-sided die
Suppose you play a game with a fair $n$ sided die (that, if being rolled yields us a discrete random variable uniformly distributed on $\{k \in \mathbb{N}| k \leq n\}$).
You play the following game:
You start by rolling this die. Every time you roll it you may choose either to quit the gam... | Assuming for simplicity you are not allowed to play game infinitely (for example you are forced to quit if you roll $n$), then the game is "either take what you rolled or pay $1$ and play again". Lets say that we get maximum expected score using strategy $f$ "take roll if you rolled at least $k$, otherwise switch to st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Normal subgroup with index that divides n! Is this argument valid?
If $G$ is a finite group with $n$ Sylow $p$-subgroups (in particular $n = 1$ mod $p$ and $n$ divides $|G|$), then $G$ permutes them acting by conjugation. Therefore there is a homomorphism $\phi: G \to S_n$ (the symmetric group on $n$ elements). If we c... | Hint:
What does the first isomorphism theorem tell you?
Note that the size of the kernel is $|G|/|\phi(G)|$, not the index of the kernel.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve this probability/combinatorics question about balls in bins Below is the problem that I need to solve
Suppose you blindly place five balls labeled A, B, C, D and E inside five bins labeled A, B, C, D and E. What are the chances that, in your selection, no ball has a label that matches its box?
I tried t... | That sounds like a question of finding permutations without a fixed point.
The idea
Let's just assume we've got our balls sitting in a row, from A to E. Below, we've got our bins arranged from A to E. Once we've blindly done our permutation on the balls (more on that later), we'll put the leftmost ball in the row into ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
About an equality of fractional Laplacian on a bounded domain Let $0<s<1$. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain.
We know that $$\|(-\Delta)^{s/2}u\|_{L^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dxdy$$
See e.g. Hitchhiker's guide to the fractional Sobolev s... | Nice question, with negative answer (unless $s=0$ or $2$, of course).
(I am unsure about this formula. The non-shaded part of this post is fine). The correct answer is $$\|(-\Delta)^{s/2} f\|_{L^2(\Omega)}^2=\int_\Omega \int_{\mathbb R^n} \frac{ |f(x)-f(y)|^2}{|x-y|^{n+2s}}\, dxdy; $$ note that only one of the integr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3194900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$
$H$ is a subgroup of $G$ with $\phi(H) = H$ $\forall \phi \in Aut(G) \Rightarrow $ $H$ is normal in $G$
I'm not sure where to begin.
Perhaps we can use this corollary of the First Isomorphism Theorem
$N_G(H)/C_G... | Hint
$$h\mapsto ghg^{-1}$$
is an automorphism for all $g\in G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What method does a calculator use to calculate a linear regression line? Take three coordinates $(1,1)$, $(3,2)$ and $(4,3)$.
My calculator returns the linear regression line: $$y=0.6429x+0.2857$$ of the form $$y = ax +b$$ correct to four significant figures for constants $a$ and $b$.
How can I do this calculation by h... | (Taken from a previous writeup)
How to do
linear least squares fitting.
To fit
a linear sum of
$m$ functions
$f_k(x), k=1$ to $m$
to $n$ points
$(x_i, y_i), i=1$ to $n$,
we want to find the
$a_k, k=1$ to $m$
so that
$\sum_{k=1}^m a_kf_k(x)
$
best fits the data.
Let
$S
=\sum_{i=1}^n(y_i-\sum_{k=1}^m a_kf_k(x_i))^2$.
$\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Totally ramified extension of $\mathbb{Q}_{p}$ which is not of a form $\mathbb{Q}_{p}(\sqrt[n]{pu})$ It is known that a finite extension $K/\mathbb{Q}_{p}$ is totally ramified if and only if $K = \mathbb{Q}_{p}(\alpha)$ where $\alpha$ is a root of Eisenstein polynomial. Is there any totally ramified extension that is n... | There is a general theorem that every tamely totally ramified extension of $\mathbf Q_p$ with degree $n$ has the form $\mathbf Q_p(\sqrt[n]{\pi})$ for some prime $\pi$ in $\mathbf Z_p$, so $\pi = pu$ for a unit $u$ in $\mathbf Z_p$. (There is a similar theorem over other local fields.) So if you want a totally ramifie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Solve $f(x)=c \times f(\frac{x}{2})$ for $c$ Given:
*
*Function $f(x)$ is infinitely differentiable
*equation (1)
$f(x)=c \times f(\frac{x}{2})$
We have to find all $c$, for which the (1) has non-zero solutions
Any hints on theorems to apply here, I reckon it's somehow related to ODEs
| If
$ f(x)=c*f(x/2)
$
then
$\begin{array}\\
f(x)
&=cf(x/2)\\
&=c^2f(x/4)\\
&=c^3f(x/8)\\
&...\\
&=c^nf(x/2^n)\\
\end{array}
$
If $|c| < 1$ then
$f(x) \to 0$
so
$f(x) = 0$
for all $x$.
If $f(0) \ne 0$,
$\dfrac{f(x)}{c^n}
\to f(0)
$.
If $|c| > 1$,
$\dfrac{f(x)}{c^n}
\to 0
$
which contradicts
$f(0) \ne 0$.
If $f(0) = 0$,
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Probability of a false negative A population includes a fraction m of individuals carrying a disease exists and has the following characteristics:
P(positive test | individual with the disease) = p
P(positive test | individual without the disease) = r
What is the probability of a false negative : P(individual with the... | You have it almost right. You write $P(W|-)=P(-|W)P(W)/P(-)$, and you know everything except $P(-)$: $P(-|W)=1-p$ and $P(W)=m$.
On the other hand, you write the correct expression for $P(+)$, but then you know $P(-)=1-P(+)$. That is, $P(-)=1-pm-r(1-m)$.
Here is an alternate solution, where I compute directly $P(-)$ in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Mapping $\mathbb{Z}_k$ into $\{d,d+1,\ldots,d+k-1\}$ preserving value $\bmod k$ I want to give a (simple) map $f:\mathbb{Z}_k \to \{d,d+1,\ldots,d+k-1\}$ for $d,k \in \mathbb{N}$ and such that $\forall i \in \mathbb{Z}_k. f(i) \bmod k = i$.
Is there a particularly simple way of doing this?
Goal
The goal is to formally... | There's only one way to map $\Bbb{Z}_k$ onto a set of consecutive $k$ integer in such a way
$$
f(\bar x)\equiv x\bmod k.
$$
That's because in any given set $K$ of $k$ consecutive integers there's only one $y\in D$ satisfying the displayed congruence for any given $\bar x$.
Thus, let $f(\bar x)=y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3195804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Variance of x_i chosen from uniformly distributed hypersphere I'm looking for an expression of the variance of a single component of a point chosen from within a uniformly distributed n-ball with radius r for any n.
There are a few proofs showing that components of a point chosen this way become increasingly normally d... | I didn't totally follow @kimchi lover's answer (particularly his assumption of normality), so here is a similar approach.
Suppose X is hyperspherically uniformly distributed with r=1, so each $X_i$ has $E[X_i] = 0$, and we want to compute $VAR[X_i]$. Since $E[X_i]=0$, by the variance-squares identity, we have $VAR[X_i]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find $\operatorname{Cl }A$ in topological space $\mathbb{R^2}$ with dictionary order topology. Let $ A = \left\{ (x,y) \in \mathbb{R^2} \mid y= \sin ( \frac{1}{x}) , \ 0 < x \leq 1 \right\}$ . Find $\operatorname{Cl} A$ in topological space $\mathbb{R^2}$ with dictionary order topology.
I guess $ \operatorname{Cl} A... | You're quite right. To prove it, I would take an arbitrary point not in $A,$ and find an open interval around it containing no points of $A.$ This shows that the complement of $A$ is open, so that $A$ is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability of matching pair of 10-item baskets out of 100M people shopping 100 times in a year This is from Exercise 1.2.2 of MMDS MMDS Book
Suppose we have information about the supermarket purchases of 100
million people. Each person goes to the supermarket 100 times in a
year and buys 10 of the 1000 items that... | There are ${1000 \choose 10} \approx 2.634 \cdot 10^{23}$ possible shopping baskets. This is a generalized birthday paradox problem. We must be intended to assume each customer takes a random independent sample of the possible baskets one each visit. There are $10^{10}$ samples taken. The number of samples needed t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding other generalized inverses besides the pseudoinverse? I have a $16\times 4$ matrix $A$ of rank $4$. Besides its Moore-Penrose pseudoinverse $A^+$, I'm also interested in other generalized inverses $A^g$ that satisfy $A^gA=I_4$.
Is there a way to get all of them (presumably in some analytical form with free vari... | All generalized inverses can be written as
$$
A^g = A^+ + U(I_{16}-AA^+),
$$
where $U\in {\mathbb C}^{4\times 16}$ is arbitrary.
Note: $I_{16}-AA^+$ is the orthogonal projection onto the null space of $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
degree extension over filed of $p$-adic numbers Let $K = \mathbb{Q}(\theta)$ be a numberfield and $[K:\mathbb{Q}]=n$. When $\mathbb{Q}_p$ is the field of $p$-adic numbers and $K_p=\mathbb{Q}_p(\theta)$, what about $[K_p : \mathbb{Q}_p]$?
| I’ll give the results, with no hint of a proof:
The general situation is that if $G(X)=\text{Irr}(\theta,\Bbb Q[X])$ splits as a product of $\Bbb Q_p$ irreducibles $G=g_ig_2\cdots g_m$, then there are essentially $m$ ways of embedding $\Bbb Q(\theta)$ into some finite extension of $\Bbb Q_p$. Each of these embeddings h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Converse linear quadratic optimal control It is well known that for a linear time invariant system
$$
\dot{x} = A x + B u \tag{1}
$$
with $(A, B)$ controllable, there exists a static state feedback $u = -K x$ such that the cost function
$$
J = \int_0^{\infty} x^T Q x + u^T R u \, dt \tag{2}
$$
is minimized, assuming $Q... | See the paper: Kalman, R. E. (1964). When is a linear control system optimal?. Journal of Basic Engineering, 86(1), 51-60.
The answer is positive at least for a class of systems. As far as I remember, the answer is also positive for a general LTI system, but I cannot find a reference at the moment.
UPDATE: Every linea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3196885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Distribution of prime numbers modulo $4$ Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
| In general, if $\gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $\dfrac{\pi(x)}{\varphi(a)}$ where $\pi(x)$ is the number of primes $\le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Intersection of connected components with discrete subgroup I am currently studying Harmonic Analysis and didn't quite understand part of the proofs for the structure theorems of locally-compact abelia groups (LCA).
Let $A$ be an LCA group such that $\hat{A}/\hat{A}_0$ is compact, where $A_0$ denotes the connected comp... | If by contradiction $A$ has an infinite compact subgroup $E$, then, as you said, $\hat{A}$ has the infinite discrete quotient $\hat{E}$. Since $\hat{A}_0$ has a trivial image in $\hat{E}$, this is an infinite discrete quotient of $\hat{A}/\hat{A}_0$. But the latter is compact, so its discrete image should be finite. Co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the interior $\{(x,y): 0 < x^2 + y^2 < 1\}$.
Find the interior $A = \{(x,y): 0 < x^2 + y^2 < 1\}$. I assume the metric is the standard Euclidean metric.
I know that the int$(A) = A$, but I don't know how to prove it. I can gather that the radius for any $x \in A$ should be $r = \min(\sqrt{x^2_1 + x_2^2},1 -\sqrt... | This works in any metric space. So you have a distance $d$ and the ball of radius $1$ around $x_0$.
Let $r=1-d(x_1,x_0)$. If $x$ is at distance less than $r$ from $x_1$, then by the triangle inequality
$$
d(x,x_0)\leq d(x,x_1)+d(x_1,x_0) < r + 1-r = 1.
$$
In your example $x_0=(0,0)$, $x_1= (x_1,x_2)$ and $d((a,b),(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Derivatives without limits Update: H/t David K for pointing out that my assumption that I can force $a^2+b^2=r^2$ is wrong. This led to analyzing a cubic equation which is now moot, but I think the bulk of the question remains sensible.
I think I'm making some mistakes in the below, but I'm not sure where. Even if ther... | This is very creative, but I think it's overly complicated. You could do something very similar by using lines instead of circles.
Consider the point $P=(a,a^2)$ on the parabola $y=x^2$. Which lines through $P$ only intersect the parabola once? The line is determined by its slope, so it has equation $y=m(x-a)+a^2$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that a number $n$ is divisible by 6 if and only if it can be written as a sum of three distinct divisors. If $6|n$ then $n=6k=3k+2k+k$. And $3k|n$, $2k|n$ and $k|n$.
Now let $p,q$ and $r$ be three distinct divisors of $n$ so that :
$$n=p+q+r$$
Because $p|n $, $ q|n$ and $r|n$ I figured that $p|q+r $, $ q|p+r$ and ... | Without, loss of generality, presume that $p > q > r$. Observe that $$p|(q+r) \implies (q+r) = k_1 p \implies 1 \le k_1 = \frac{q+r}{p} < 2 \implies k_1 =1.$$
As such, $p = q+r$, and $n = 2(q+r)$.
Since $q|n=2(q+r)$, we have $q|2r$. That is, $$2r = k_2 q \implies 1 \le k_2=\frac{2r}{q}<2 \implies k_2 = 1.$$
Therefore,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Cartan homotopy formula and curl In Topological Methods in Hydrodynamics, V. I. Arnol'd writes that the following expression
$$curl(\mathbf a \times \mathbf b)=[\mathbf a, \mathbf b]+ \mathbf a \ div \ \mathbf b - \mathbf b \ div \ \mathbf a$$
could be obtained "repeatedly applying" the Cartan homotopy formula
$$L_v = ... | Apply (plug in by means of the operator $i$) both sides to a volume 3-form $\tau$.
The equality of vector fields on both sides is equivalent to the equality of the obtained 2-forms.
On the left we have
$i_{curl(a\times b)}\tau= di_b i_a \tau$ (note the order: vector field $a$ is plugged in first).
On the right first u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$
If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$.
The graph of the domain of $f$ ... | Continuty at (3,0) : When $|(x,y)-(3,0)|=\sqrt{(x-3)^2+y^2}<\delta<1$ for some $\delta>0$ then $|f(x,y)|\leq 9-x^2-y^2=(3-x)(3+x)\leq 6 (3-x)\leq 6\sqrt{(x-3)^2+y^2}<6\delta.$ This implies that $f$ is continuous at $(3,0).$
Partial differential wrt $y$ : $f(3,y)=0$ for any $y$ and hence $f$ has a partial derivative at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3197878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Does $x^n$ converges uniformly on $ [0,1)?$ Doing the limit we can see that in the open interval it converges pointwise to the constant function $f(x) = 0$. In the closed interval it doesn't converge uniformly because in $x=1$ $f(x) =1$ and when $0<x<1$ then $f(x)=0$. It isn't a continuous function although $f_n(x)$ is... | No, $x^n$ does not converge to zero uniformly on $[0,1)$, because
$$\lim_{n\to\infty}\sup_{0\leq x<1}|x^n|=1\neq 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Intuition behind determinant of a matrix with $2$ equal rows In my linear algebra course, we have just proved that if a matrix $A$ contains $2$ equal rows, then $\det(A)=0$.
I understand how the proof works, but could somebody offer a more intuitive explanation of why this is the case?
| If 2 rows are equal then they are linearly dependant so you'll have a zero eigen value and so the determinant (which is the product of the eigen values) will be zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 1
} |
For which $\alpha$ will take the cake ever be again with chocolate on the bottom and cream on the top Question: A bored kid left alone at home decides to take a chocolate cream cake (chocolate on the bottom, cream on top) and his protractor and spend the day as follows: He cuts a slice of angle $\alpha$ put it back up-... | EDIT: this answer assumes model bad model of piece inverting (that putting piece up-side-down is equal to reversing colors). In real pie we not only inverse colors but also mirror them. See @user2323232's link for correct answer.
You need $\alpha$ to be rational (I assume slice is half-open segment, otherwise we will n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$11x + 13 \equiv 4$ (mod 37) $11x + 13 \equiv 4$ (mod 37)
My "solution" to the problem
$11x + 13 \equiv 4$ (mod 37) $\rightarrow$
$11x + 13 = 4 + 37y $
$11x - 37y = - 9$
Euclid's algorithm.
$37 = 11*3 + 4$
$11 = 4*2 + 3$
$4 = 3*1 + 1 \rightarrow GCD(37,11) = 1$
$3 = 1*3 + 0$
Write as linear equation
$1 = 4 - 1*3$
$3 ... | Note: $90\mod37 =16$, so the answer you found was correct, just not fully reduced.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Polynomial Roots with no complex roots Let $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_n,b_n)$ be the ordered pairs $(a,b)$ of real numbers such that the polynomial
$$p(x) = (x^2 + ax + b)^2 +a(x^2 + ax + b) - b$$has exactly one real root and no nonreal complex roots. Find $a_1 + b_1 + a_2 + b_2 + \dots + a_n + b_n.$
I have... | We expand the OP's polynomial and write it as
$\tag 1 p(x) = (a b + b^2 - b) + (a^2 + 2 a b) x + (a^2 + a + 2 b) x^2 + 2 a x^3 + x^4 $
We also have (see Robert Israel's answer)
$\tag 2 (x-r)^4 = r^4 - 4 r^3 x + 6 r^2 x^2 - 4 r x^3 + x^4$
Fortunately, we can find an easy 'coefficient hook', and so $2a = -4r$, or
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Combinatorial proof for $\sum_{k=0}^p (-1)^k {n \choose k} = (-1)^p {n-1 \choose p}$ I am trying to give a combinatorial proof for: $$\sum_{k=0}^p (-1)^k {n \choose k} = (-1)^p {n-1 \choose p}$$
Where $p$ and $n$ are natural numbers.
We could easily see that if $p=n$ this reduces to the fact that a set has as many su... | The LHS counts subsets of $\{1,2,\dots,n\}$ whose size is at most $p$, with even subsets counted positively and odd subsets counted negatively.
We define a sign reversing involution of such subsets; if $1$ is in the subset, remove it, otherwise, add it in. This involution partitions almost all of the subsets of size a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proof of a Corollary to Hall's Marriage Theorem I'm trying to prove Corollary 3.3 from this paper (http://www.sfu.ca/~mdevos/notes/graph/345_matchings.pdf) and am lost at the step that says: "Similarly, every vertex in N(X) has degree k, so t is less than or equal to k|N(X)|. It follows that |X| is at most |N(X)|."
How... | Every vertex in $X$ has at least $k$-neighbors (that are all in $Y$ because the graph is bipartite
So if $|X| \le k$ we are done since $|N(X)| \ge k$
If $|X|=n>k$ then there is at least $n$ vertices in $N(X)$ since if there is less than $n$ vertices then you'll find a vertex in $N(X)$ of degree $> k$, a contradiction.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Vanishing of rotation of timelike unit vector in numerical relativity I'm a bit confused. I don't think it is that difficult, but still don't manage :-(
So the question is why the rotation of the timelike unit-vector in numerical relativity
$$n_\mu=-\alpha \nabla_\mu t$$
vanishes. $\alpha=\left(-g^{\mu\nu} \nabla_\mu \... | Hint:
$$n_{[i} \nabla_j n_{k]} = n_i \nabla_j n_k + n_j \nabla_k n_i + n_k \nabla_i n_j - n_i \nabla_k n_j - n_j \nabla_i n_k - n_k \nabla_j n_i $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3198864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$ Excercise
Let $X, Y$ be random variables such that their joint density function is defined by:
$f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$.
Find the distribution of Z defined as:
$Z=\frac{X+Y}{2}$.
Attempt of solution
I am using the fact that:
$F_Z(... | Note that marginal densities of $X$ and $Y$ are $f_X(x)=e^{-x}$ and $f_Y(y)=e^{-y}$.
Then, $Z=X+Y$ has density according to the convolution:
$$f_Z(z)=\int_0^\infty f_X(x)f_Y(z-x)dx = \int_0^z e^{-x}e^{-(z-x)}dx=ze^{-z}.$$
Density of $W=Z/2$ is
$$f_W(w)=f_Z(2w)|dz/dw| = 4we^{-2w}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3199998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Showing that for $x \ge 7$, $x\# \ge x^2+x$ Let $x\#$ be the primorial of $x$.
I am trying to show that if $x \ge 7$:
$$x\# \ge x^2+x$$
Is there a straight forward argument?
Here's what I came up with:
(1) From Bertrand's Postulate, for any $x$, there exists a prime $p$ such that $x < p < 2x$
(2) Base Case: $7\# = 2... | The idea is pretty good, but I feel the solution was difficult to read. I will just rephrase your ideas more clearly.
1) Claim: It is enough to prove that $p\#\geq (2p)^2+2p$ for $p$ prime.
Indeed, let $x$ any natural number, then we can find $p_{n}\leq x<p_{n+1}$. By Bertrand postulate, we have $ p_{n+1}<2p_n$. Henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3200120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Number of possible chess pairs where order and position matter
Given 11 chess players and 5 distinct tables, in how many ways can we pair them to play (color does matter)?
My problem is that I have found two approaches, both of which give different numbers, and I am not sure what is missing in one or double-counted i... | The first approach perfectly makes sense.
In the second approach, when we say first player $p$ has $10$ options, we explicitly choose a player and choose it's opponents from $10$ options. But then we skip the case where $p$ does not play at all. This shows there are some cases that are not counted but still does not co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3200256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Why we never use the product between vectors like between elements of direct groups product? We can say, that any field $\mathbb{K}$ -- $1$-dim vector space on itself: $\mathbb{K}_{\mathbb{K}}$. So any vector of one another finite-dimensional vector space $V_{\mathbb{K}}$, after choosing the some basis can be represent... | On the vector space, there is an addition that does not depend on the choice of basis you make, sometimes called a "natural" addition.
However, you usually can't define a natural multiplication in your way, it would always depend on a basis and different bases will give different multiplications. Therefore, it is mostl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3200431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Show that the order of an element g is well-defined Suppose $G$ is a group and let $g∈G$, explain why the order of $g$ is well-defined, while the definition of the order is the following:
The smallest positive r such that $g^r=e$, if no such r is found then we say g has infinite order.
My Question:
What strategy should... | Your approach is good, but not correct. You cannt say “Suppose the order of $g$ is finite”, since this already assumes that the order exists. You can consider two cases:
*
*$(\forall k\in\mathbb N):g^k\neq e$: then, by definition, $\operatorname{ord}f=\infty$.
*$g^k=e$ for some natural $k$. Then, by the Well-Orderi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3200560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Alpha-beta quadratic equation Question: The equation $$3x^2-6x-4=0$$ has roots α and β. Find the value of 1/α + 1/β.
I'd just like confirmation on my answer, as I've already found the answer but am not confident in it.
since αβ=c/a and
α+β= -b/a
1/α +1/β= (α+β)/αβ= (-b/a)/(c/a)=2/(-4/3)=-3/2
| Yes, it is correct.
You could do also like this. You have to find $a+b$ if $a,b$ are solution to $$3{1\over x^2}-6{1\over x}-4=0$$that is $$3-6x-4x^2=0$$
so $$a+b =-{-6\over -4}=-{3\over 2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3200865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that there is a differentiable function $f$ such that $[f(x)]^5 + f(x) + x = 0$. Spivak Ch. 12, Q. 14. Prove that there is a differentiable function $f$ such that $[f(x)]^5 + f(x) + x = 0$. (My textbook offers the following hint: Show that $f$ can be expressed as an inverse function.)
My Progress
$f(x) = -[f(x)]^... | I think the answer of Alex R. is on track, but I want to make things a bit clearer.
Let $g(y) = y^5 + y.$ Argue from one variable results that $g$ is a bijection of $\mathbb R$ onto $\mathbb R,$ with a differentiable inverse $g^{-1}.$ Define $f(x) = g^{-1}(-x).$ Then $f$ is differentiable on $\mathbb R,$ and we have $g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is the Lagrange multiplier $\lambda$ always positive? In the Lagrange function
$${\mathcal {L}}(x,y,\lambda )=f(x,y)-\lambda g(x,y),$$
is the Lagrange multiplier $\lambda$ term supposed to always be positive or can't it take negative values?
| Geometrically, the Lagrange multiplier method has the following interpretation:
*
*We want a point on the level curve $g(x,y)=0$. This comes from differentiating with respect to $\lambda$ and setting to zero.
*We want that, at the point $p$ under consideration, the level curve of $f(x,y)$ is tangent to the level cu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Can a nonabelian group $G$ have a normal abelian subgroup $H$ with $[G:H]=3$? If $H$ is a cyclic group, for instance, and $[G:H］=2$, then $G$ is a dihedral group with elements equal to the possible states of a regular polygon upon which rotations and reflections are applied. In this case, the quotient group $G/H$ of or... | Yes. The Klein subgroup $\{1, (1~2)(3~4), (1~3)(2~4), (1~4)(2~3)\} \triangleleft A_4$. This subgroup is abelian, as the title of the question requests, but it's not cyclic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Characterizing a module of Kahler differentials
Consider the $\mathbb C$-algebra $R=\mathbb C[x,y,z]/(z(y^2-x^3)-1)$. How to prove that the module of Kahler differentials $\Omega_{R/\mathbb C}$ of $R$ over $\mathbb C$ is a free $R$-module of rank 2?
This $R$-module is generated by $\{d(f):f\in R\}$ modulo the relatio... | There are two observations which make this calculation go much more smoothly.
*
*First, note that $$R\cong\Bbb{C}\left[x,y,\frac{1}{y^2 - x^3}\right]\cong S^{-1}\Bbb{C}[x,y],$$
where $S = \{1, y^2 - x^3, (y^2 - x^3)^2,\dots\}.$
*Secondly, K\"ahler differentials are compatible with localization: namely, if $B$ is an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I prove this combinatorial identity Show that
$${2n \choose n} + 3{2n-1 \choose n} + 3^2{2n-2 \choose n} + \cdots + 3^n{n \choose n} \\ = {2n+1 \choose n+1} + 2{2n+1 \choose n+2} + 2^2{2n+1 \choose n+3} + \cdots + 2^n{2n+1 \choose 2n+1}$$
One way that I did it was to use the idea of generating functions.
For ... | We seek to show that
$$\sum_{q=0}^n {2n-q\choose n} 3^q
= \sum_{q=0}^n {2n+1\choose n+1+q} 2^q.$$
We have for the LHS
$$\sum_{q=0}^n {2n-q\choose n-q} 3^q
= \sum_{q=0}^n 3^q [z^{n-q}] (1+z)^{2n-q}
\\ = [z^n] (1+z)^{2n} \sum_{q=0}^n 3^q z^q (1+z)^{-q}.$$
The coefficient extractor controls the range and we obtain
$$[z^n]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 0
} |
Matrix-valued function: understanding
$A(x)=(a_{i,j}(x))_{i,j}: \mathbb{R}^n\to \mathbb{R}^{m\times k}$ is a matrix-valued function.
From my understanding it is a function that maps a vector to a matrix. How does this happen? Is the 'function' a matrix of which the elements are functions? If so, are these just $\math... | As an example: Imagine $x=[x_1,x_2]^T\in \mathbb{R}^2$ Then the function
$$A(x)=
\begin{bmatrix}
x_1x_2 & x_1^2\sin x_2 & x_2 \\
x_1x_2^3 & x_2\exp(x_1) & 2x_2x_1\sin(x_1) \\
\end{bmatrix}$$
is a map from $\mathbb{R}^2\to \mathbb{R}^{2\times3}$. The entries of the matrix are multivariate scalar functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Asymptotic distribution of the MLE of $\theta$ such that $\log X_i$ is distributed as $N(\theta, \theta)$ Let $X_1, . . . , X_n$ be a random sample such that $\log X_i$ is distributed as $N(θ, θ),$ $θ > 0$ is unknown.
I've calculated the MLE and I got $$\hat\theta = \frac{-1 + \sqrt{1 + 4n^{-1} \sum_{i=1}^{n} Y_i^2}}{2... | Under certain regularity conditions (like the ones mentioned here on page 1), maximum likelihood estimators have an asymptotic normal distribution. In particular, distributions which are members of the regular exponential family satisfy these conditions.
For $Y_i=\log X_i$, joint density of $Y_1,\ldots,Y_n$ is
\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding minimum value of x such that GCD(A+x,B+x) = C where A , B ,C are given I need to add Minimum non-negative Integer such that I can get the desired GCD(a+x,b+x)
Let say A=12 & B=26
For GCD(12+x,26+x) = 1 , x should be 1
For GCD(12+x,26+x) = 2 , x should be 0
For GCD(12+x,26+x) = 7 , x should be 9
For GCD(12+x... | Your question title says you want to find the minimum value of a non-negative integer $x$ such that $\gcd(A + x, B + x) = C$, where $A, B, C$ are given. First, note there's no such integer in certain cases. In particular, since $C$ divides $A + x$ and $B + x$, it must also divide their difference, i.e., $C \mid (A + x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3201901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Logic: "the cubic root of a rational number is also a rational number" I was attempting an online logical and mathematical statements self-test from the University of Toronto and came across the following statement in question 1:
The cubic root of a rational number is also a rational number.
We are asked to select an... | Let's firstly put aside the completely irrelevant question herein of whether the statements are true; obviously, it's not true in general that a rational number's cube root is rational. Let's look into how we rewrite statements with algebra.
While "a feline is also a mammal" becomes "if $x$ is a feline, then $x$ is a m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Complex numbers $z=-3-4i$ polar form Let $z=-3-4i$ . In polar form this becomes $[5, 233° ]$. The question then asks for $z^2$, so the polar form becomes $[25,466]$ However in the solution they did $466° -360° $ and I am unsure why they did this. Is it a rule that needs to be applied.
| It's because$$\cos(x-360^\circ)=\cos(x)\text{ and }\sin(x-360^\circ)=\sin(x).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Convergence for power series $\sum_{n=0}^{\infty}z^n(5+e^{i\pi n})^n$? Consider the power series
$$\sum_{n=0}^{\infty}z^n(5+e^{i\pi n})^n \tag{1}.$$
Identifiying $e^{i\pi n}$ as $(-1)^n$ for all integers $n$, highlights the problem with using the ratio test or the root test for convergence of series, since $(-1)^n$ doe... | Let's set $a_n=(5+(-1)^n)z^n$.
We have $|4z|^n\le|a_n|\le |6z|^n$ so the series converges for sure whenever $|z|<\frac 16$ and diverges for sure whenever $|z|>\frac 14$
What about values in the ring $\frac 16\le |z|\le\frac 14$ ?
To show that $r=\frac 16$ is the actual radius of convergence, we just need to exhibit one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Using knowledge of derivatives, what is $f(x)$ if $f'(x) = 4x + e^x$ with initial condition $f(0)=2$? How to solve this with knowledge of derivatives? Is it asking to solve this without using anti-derivatives?
| Integrate both sides to get $f(x)$ and the equation becomes
$$f(x) = 2x^2 + e^x + c.$$
Put $x=0$ and $f(x=0)=2$ to get $c=1$. Substitute the value of $c$ and get
$$f(x)=2x^2+e^x+1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Any comprehensive books on global smooth optimization? Can you share any information about books that review most of the existing numerical methods for global minimization of a multivariable objective function? The objective and constraints are assumed to be smooth, I'm not into integer programming of any type at the ... | Try "Metaheuristics". Simulated Annealing, Genetic Algorithm, Tabu search, Ant colony algorithm, swarm intelligence and so on. There will be many books, but I do not know which book is best.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Stuck on candy bowl fraction I am really stuck on this problem because I'm not even sure where to start.
Larissa has a bowl of candies. On the first day, she eats 1/2 of the candies plus one more. On the second day, she eats 1/3 of the remaining candies plus one more. On the third day, she eats 1/5 of the remaining ca... | Hints
Let $N$ be the number of candies.
Day 1: she eats $\frac{1}{2}N + 1$: remainder $N_{1}$
Day 2: she eats $\frac{1}{3}N_{1} + 1$: remainder $N_{2}$
Day 3: she eats $\frac{1}{5}N_{2} + 1$: remainder $N_{3}$
Day 4: she eats $3$: $N_{3} = 3$
Can you go from here? (answer $=20$)
Step 1: English $\to$ algebra
Create a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
limit of the sequence $(x_{n+1})=a^{x_n}$ with $x_1=a$ We are to find the limit of the sequence $(x_{n+1})=a^{x_n}$, where $x_1=a$ for $a>0$.
Attempt:
Case (1): $0<a<1$
We have $\frac{1}{a}=(1+ |r_1|)>1$ for some real $r$.
Hence, $\frac{1}{a^a}=(1+ |r_1|)^a=1+a|r_1|+\frac{a(a-1)}{2}|r_1|^2+...>1$
Evidently, $1+a|r... | If the limit $L$ does exist, it is given by the solution of
$$L=a^L\implies L=-\frac{W(-\log (a))}{\log (a)}$$ where appears Lambert function.
However, in the real domain, $W(t)$ requires $t \geq -\frac 1e$ that is to say $a \leq e^{\frac{1}{e}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does the order of the subgroup generated by two elements divide the product of the element orders? Let $(G, \cdot)$ be a group, $a,b\in G$ such that $\DeclareMathOperator{\ord}{ord}\ord(a),\ord(b)<\infty$.
Do we then have that $\left|\left<a,b\right>\right|$ divides $\ord(a)\ord(b)$? (Where $\left< a,b \right>$ denotes... | $S_3=\langle (12), (23)\rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3202891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derivation of a well-known property of the standard normal distribution I have found the following property of the standard normal distribution:
$ \int_r^\infty xf(x) dx = f(r) $
where $f(.)$ is the pmf of the standard normal distribution.
$f(r)$ and $F(r)$ are defined as follows:
$f(r) = \cfrac{1}{\sqrt {2π}} e^{-r^... | It is obtained by just noting that $-xe^{-x^{2}/2}$ is the derivative of $e^{-x^{2}/2}$, so $\int_r^{\infty} xf(x)dx =f(r)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $r+r’\leq d(a,a’)$ then $B(a,r)\cap B(a’,r’)= \emptyset$. I have to prove this assertion :
Suppose that $r+r’\leq d(a,a’)$
Show that $B(a,r)\cap B(a’,r’)= \emptyset$.
What I have to prove is that if I have $p \in B(a,r)\cap B(a’,r’) $ then $p=0$.
If $p \in B(a,r)\cap B(a’,r’)$ then :
$$d(a,p)<r $$
$$d(a’,p)<r’$... | I assume the balls are open, otherwise the statement is not true.
Let $p\in B(a,r)$, i.e. $d(a,p)\lt r$. Then the triangle inequality states that
$$d(a,p)+d(p,a')\ge d(a,a')\ge r+r'$$
Hence
$$d(p,a')\ge (r-d(a,p))+r'>r'$$
Hence $p\notin B(a',r')$.
Therefore, $B(a,r)\cap B(a',r')=\emptyset$.
Note that it's absolutely n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A consequence of Riemann-Lebesgue lemma Riemann Lebesgue lemma states that: For a function $f\in L_1([-\pi,\pi])$ we have
\begin{align}
\lim_{|n|\to \infty} \int_{[-\pi,\pi]} f(t) e^{-int} \text{ d}t =0.
\end{align}
The integral here is Lebesuge integral. As consequences of this lemma,
\begin{align}
\lim_{|n|\to \in... | Assuming $\lim_{|n|\rightarrow\infty}\int f e^{inx}dx = 0$ gives
$$
\lim_{n\rightarrow\infty}\int f e^{inx}dx = 0 \\
\lim_{n\rightarrow\infty}\int f e^{-inx}dx = 0
$$
Adding these results in
$$
\lim_{n\rightarrow\infty}\int f \cos(nx)dx = 0
$$
I'll let you tackle the other combinations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Geometric interpretation of an elliptic point on a Riemann surface / hyperbolic surface Let $\Gamma$ be a Fuchsian group of signature $[g;m_1,\dots,m_r;s]$. When we quotient $\mathbb{H}^2$ by $\Gamma$ we obtain a genus $g$ surface with $s$ cusps and $r$ elliptic points with orders $m_1,\dots,m_r$.
Now, the concepts ... | They are sometimes called cone points, because they look like taking a piece of paper with a corner of angle $2\pi/m_i$ and folding it over to identify opposite sides of the vertex. This produces something which looks like a cone at the singular point. They are also sometimes called pillowcase points as they look like ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding a pattern, I'm stuck I have a friend who knows how much I love math, (I imagine new problems just to do the math behind them and to see if I can expand my understanding) and so he brings me the stuff that stumps him. Usually I end up finding the answer for him and then explaining how to him, but he gave me one ... | If I could give some advice, these kinds of questions generally benefit from a look at the Online Encyclopedia of Integer Sequences.
Considering it has to do with ellipses, the sequence of denominators should be A056982.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
evaluating $\int_{-\infty}^{\infty}{\frac{x\sin(2x)}{x^2+4}}\mathrm dx$ I want to calculate $\int_{-\infty}^{\infty}{\frac{x\sin(2x)}{x^2+4}}\mathrm dx$. Now I think the answer is $\pi/e^4$. Now the poles are cleary $2i, -2i$. When I try to then use the residue theorem I get that the integral is $0$, which I think is w... | The trick to apply contours in this case is to write $\sin 2x = \frac1{2i} e^{2ix} + \frac1{2i} e^{-2ix} $ and do two separate integrals.
For the former, we have a contour going from $-\infty$ to $\infty$ along the $x$ axis, then returning n a large semicircle in the upper half plane. The semicircle contributes zer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Classification of surfaces The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $b$ the number of boundary components and $c$ the number of projective planes.
From there, it... | If I read correctly you are trying to determine $c$ and $g$, given a compact connected surface $M$ for which you know $b$ the number of boundary components, $\chi(M)$ the Euler characteristic the and whether or not $M$ is orientable.
This can't be done in a unique manner, as the connected sum of a tori and a projectiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3203841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Solving polynominals equations (relationship of roots)
The roots of $x^3-4x^2+x+6$ are $\alpha$, $\beta$, and $\omega$.
Find (evaluate):
$$\frac{\alpha+\beta}{\omega}+\frac{\alpha+\omega}{\beta}+\frac{\beta+\omega}{\alpha}$$
So far I have found:
$$\alpha+\beta+\omega=\frac{-b}{a} = 4 \\
\alpha\beta+\beta\omega+\... | Hint: We can write $$\frac{4-w}{w}+\frac{4-\beta}{\beta}+\frac{4-\alpha}{\alpha}$$ and this is $$4\left(\frac{\alpha\beta+\alpha w+w\beta}{\alpha \beta w}\right)-3$$ and this is $$-\frac{2}{3}\left(1-\beta w-\alpha w+\alpha w+\beta w\right)$$
This simplifies to $$-\frac{2}{3}-3=-\frac{11}{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Number of ways to pick a team of 4 with at least 1 girl and 1 boy from 4 girls and 4 boys. To find the number of ways to pick a team of $4$ with at least $1$ girl and $1$ boy from $4$ girls and $4$ boys, I thought to manually assign the team with $1$ girl and $1$ boy to begin with.
$$G, \quad B, \quad G \textrm{ or } B... | A team of four with at least one boy and at least one girl will have either three boys and one girl, two boys and two girls, or one boy and three girls. Hence, the number of admissible teams is
$$\binom{4}{3}\binom{4}{1} + \binom{4}{2}\binom{4}{2} + \binom{4}{1}\binom{4}{3} = 68$$
which agrees with the answer you obta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why isn't the definition of absolute value applied when squaring a radical containing a variable? I recently learned about the following definition of absolute value:
$|a| = \sqrt{a^2}$
Then I came across a solution to a problem that had the following step:
$5 \geq \sqrt{5 - x}$
In order to proceed, we had to squar... | From the fact that you can take $\sqrt {5-x}$ you know that $5-x \ge 0$ so you don't need the absolute value signs.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
What is the probability there is no prime between $n$ and $n+\ln(n)$? Consequences of the Prime Number Theorem tell us the probability of $n$ being prime is $1/\ln(n)$. This also means that the number of expected primes between $n$ and $n+\ln(n)$ is close to $1$, but not always. What is the probability there is no prim... | This is currently a question we cannot answer unconditionally; however, there is a heuristic answer, which we can prove if we assume another result.
The naive probabilistic model is that each integer $k$ between $n$ and $n+\log n$ independently has a probability $1/\log k \sim 1/\log n$ of being prime. According to thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Big $O$ small $o$ notation True or False I'm stuck on a few big-$O$ and small $o$ notation True or False questions, any insight would be appreciated.
$4n^3+6n+17=O(n^4)$ True. The degree of complexity is directly related to n, but it is not represented there. The function grows much faster (left side is like n^2, since... | [When using this notation, it is important to specify which limit you are thinking about. I assume from your attempted solutions that you mean $n\to\infty$.]
You have got the correct answers. For the second one, you're right that it is false. How can you prove it? Suppose it's true. $$4n^3+6n+17=O(n^2)$$ Then $\exists ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Reference Request: Convergence of Series over Integer Lattice I am writing a school paper on Modular forms, and I want to use the following result without including a proof, since it is not directly related to my main topic.
For $s>0$, $d \in \mathbb{N}$, the series $\sum\limits_{v\in\mathbb{Z}^d\setminus\{0\}} \frac... | It is enough to prove the convergence of the sum over tuples of nonzero integers.
$$
\sum_{v\in (\mathbb{Z}-\{0\})^d} \frac1{\|v\|^s} \ \ \ \tag{1}
$$
Apply the AM-GM inequality, we have for $v=(v_1,v_2,\ldots, v_d)$,
$$
\frac1{\|v\|^2} \leq \frac d{(v_1^2v_2^2\cdots v_d^2)^{1/d} }
$$
The $s/2$-th powers of these sat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convergence of series of quotients of gamma functions I am interested in determining if the following series converges:
$$\sum_{k=0}^{\infty} \frac{\Gamma(k+2\delta)}{\Gamma(k+1)} \;,$$
where $\delta > 0$ is a real number and the gamma function is defined by $$\Gamma(\alpha) = \int_0^{\infty} x^{\alpha-1} e^{-x} \, dx ... | For $\delta>0$ and $k\geq1$ we have $\Gamma(k+2\delta)>\Gamma(k)>0$ and $\Gamma(k+1)>0$, so
$$\frac{\Gamma(k+2\delta)}{\Gamma(k+1)}>\frac{\Gamma(k)}{\Gamma(k+1)}=\frac1k.$$
It follows that the series diverges because the harmonic series diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3204970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$(b_n) \rightarrow b$ then $(1/b_n) \rightarrow 1/b$ proof. Let $(b_n)$ be a sequence that converges to $b$. Show that $(1/b_n)$ converges to $1/b$ when $b_n \neq 0$ for all $n$ and $b \neq 0$.
First see that
$$
|\frac{1}{b_n} - \frac{1}{b}| = |\frac{b - b_n}{b_nb}| = \frac{|b_n - b|}{|b_n||b|}
$$
Now from convergence ... | I would recommend not assuming anything is positive. It's fine to do that, but is very important to be able to work with absolute values carefully. And absolute value inequalities specifically.
You need to be sure to know when you want a lower vs upper bound. Often the game is bounding something by a series of upper bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the largest square that can fit in a dodecagon and not rotate? I am an engineer in the oil field and I am trying to find a socket that can fit around a square drive pin. I am trying to prove this for fun before I just draw it on CAD and measure it. I believe sockets have six or twelve sides. I have had some suc... | Wouldn't it be easier with a regular dodecagon, since 4 divides evenly into 12?
The largest square which will not rotate will fit snugly into four corners of the dodecagon.
Note that in $\triangle AOM$, side $OM$ must be the largest side. Since $\angle MOA<30^\circ$ then $\angle MAO>75^\circ$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Minimum value of $4$-digit number divided by sum of its digits? If x was a positive 4 digit number and you divide it by the sum of its digits to get the smallest value possible, what is the value of x? For example (1234 = 1234/10)
I got 1099 as my answer however I don't know if this is right or how to prove it.
| Let $x$ be the 4-digit number ending in digit not equal to 9 with the sum of digits equal to $s$. If you increase the last digit by one, the number becomes $x+1$ and the sum of digits becomes $s+1$.
You can easily prove that:
$$\frac xs>\frac{x+1}{s+1}$$
This is true because $x>s$. So to reduce the ratio as much as pos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Can a function be of class C1 even if its partial derivatives are not continuous? I know that if all partial derivatives of a function f exist and are continuous then the function is said to be of class C1 (continuously differentiable).
However, I was not able to find whether this is a necessary or a sufficient condit... | No. A function is of class $C^{1}$ if and only if it has continuous partial derivatives. You can refer to Rudin's book for a proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that the limit $\lim_{z \to i}{\frac{1-|z|}{i-z}}$ does not exist. Show that the limit $\lim_{z \to i}{\frac{1-|z|}{i-z}}$ does not exist.
I tried by putting $z =ix $, where $x \to 1$ and got that the limit is $\frac{1}{i}$, but can't think of another example.
| Hint:
$$\lim_{z \to i}{\frac{1-|z|}{i-z}}= \lim_{s \to 0}{\frac{|s+i|-1}{s}}.$$
Choose $s$ real number then the limit will be 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$ Calculate
$$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$$
My try: $x-1<[x]<x$
$-x<-[x]<1-x$
$0<x-[x]<1$ then I applied an integral but I get the result pi which is the wrong answer.
| Break the integral into four parts; from $0$ to $1$, from $1$ to $2$, from $2$ to $3$ and from $3$ to $\pi$. Then $\{x\}=x,\ \{x\}=x-1,\ \{x\}=x-2,\ \{x\}=x-3$ in those intervals respectively. Then, you can easily calculate the three integrals to find that
$$n\int_{0}^{\pi}\{x\}^ndx=\frac{3n}{n+1}+\frac{n(\pi-1)^{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Calculating tangent line at $t=2$
Calculate the tangent line at $t=2$ of the curve
$$x(t)=t^2+2t+4,\ y(t)=1+te^t$$
How would you go about determining this? So far I have
$$x'(t)=2t + 2\\
y'(t) = te^t+e^t$$
Where do I go from here to get the tangent line?
| The following expression is how you can find the slope of a parametric curve:
$$
\frac{dy}{dx}=\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)},\ \frac{dx}{dt}\ne0.
$$
When you have found the slope of the curve at $t=2$, use the equation of a straight line to form the actual line:
$$
y-y_0=m(x-x_0).
$$
Use... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3205963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How can $\pi$ be defined as a surreal number. I want to express $\pi$ in the Surreal Number notation $\{L|R\}$. What is the most natural or intuitive way of doing so, seeing as there are many (possibly infinite) ways of expressing the same surreal number.
| If you want to have a more explicit expression, start with the largest integer below $\pi$, that is $a_0=3$, and then iterate the following algorithm:
*
*If $a_n<\pi$, it goes into the left set, and $a_{n+1} = a_n + 1/2^{n+1}$.
*If $a_n>\pi$, it goes to the right set, and $a_{n+1} = a_n - 1/2^{n+1}$.
This gives
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that if $\alpha \vDash \beta$ and $\beta \vDash \alpha$, then $\vDash (\alpha \leftrightarrow \beta)$ I would like to show that if $\alpha \vDash \beta$ and $\beta \vDash \alpha$, then $\vDash (\alpha \leftrightarrow \beta)$, and I'm thinking that the argument below fails.
Since $\alpha \vDash \beta$ and $\beta \v... | In your attempt, the problem is with the line
Since $\alpha \vDash \beta$ and $\beta \vDash \alpha$, we have for every truth assignment $v$ that $\overline{v}(\alpha) = \overline{v}(\beta) = T$.
We do not have $v(\alpha) = v(\beta) = T$ for every $v$. We only know If $v(\alpha) = T$ then $v(\beta) = T$ and vice v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that if $f$ is not surjective, then $f$ is homotopic to a constant via a homotopy that fixes a point I'm trying to prove that if $f:S^1 \rightarrow S^1$ is not surjective, then $f$ is homotopic to a constant function via a homotopy that fixes a point $\theta \in S^1$. Showing that it is homotopic to a constant fu... | More generally consider $f:S^n\to S^n$ and let $P\in S^n$ be such that $P\not\in f(S^n)$. For any $\theta\in S^n$ we have a homotopy
$$H:I\times S^n\to S^n$$
$$H(t, s)=\pi^{-1}\big(t\cdot \pi(f(s))+(1-t)\cdot \pi(f(\theta))\big)$$
where $\pi:S^n\backslash\{P\}\to\mathbb{R}^n$ is the stereographic projection which is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Use derivative to find maxima, minima of the function : $n^\frac 1n: n \in \mathbb{N}$ This post is in ref. to my earlier post's row #10.
*$\{n^\frac 1n: n \in \mathbb{N} \}$: Domain of values is in the set of naturals.
I am unable to find minimum, maximum (in range), and list a few values below:
$$\begin{array}{... | Hint: Check the derivatives at n=2 and 3
(specifically at $2.718281828459045235360\cdots$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Proving $\int_{S^{n-1}}x_1^2dS =\int_{S^{n-1}}x_k^2dS$ Denote $x = (x_1,...,x_n)$. I'm trying to prove the following:
$$\int_{S^{n-1}}x_1^2dS =\int_{S^{n-1}}x_k^2dS \; , \; 2\leq k\leq n $$
Intuitively this equality is due to the symmetry of the sphere, but I'm looking for a formal explanation. I thought about using ... | A formal explanation uses the invariance of the measure $dS$ under orthogonal transformations. That is, given a continuous function $f:\mathbb{S}^{n-1}\to\mathbb{R}$ and an orthogonal matrix $U$ (that is, $U^tU=I$), one has
$$\int_{\mathbb{S}^{n-1}}f(Ux)\,dS(x)=\int_{\mathbb{S}^{n-1}}f(x)\,dS(x)\quad (*)$$
In your case... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Express a trig equation in terms of $k$ The acute angle $x$ radians is such that $\tan x$ = $k$, where $k$ is a positive integer. Express in terms of $k$.
i) $\tan (\pi - x)$
ii) $\tan (\frac{1}{2}\pi - x )$
iii) $\sin x$
I don't understand what the question is stating or asking. I got the answer $\tan \pi - k$ for the... | Hint: Use $\tan (\alpha +\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to find the following summation How to find the following sum $$\sum\limits_{n=1}^{\infty}\dfrac{\left(\dfrac{3-\sqrt{5}}{2}\right)^n}{n^3}$$
I have tried by rationalizing. But after that I got stuck.
| By definition, for $|r|<1$,
$$ \sum_{n=1}^\infty r^n/n^3 = \text{polylog}(3,r)$$
This is not an elementary function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3206947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $T \mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^{2n}$ $T \mathbb{R}^n$ is the tangent bundle to $\mathbb{R}^n$. How do I show that is is diffeomorphic to $\mathbb{R}^{2n}$? I know that there is an isomorphism (and therefore a diffeomorphism) between $\mathbb{R}^n$ and $T_p \mathbb{R}^n$, can I conclude from ... | Going by Lee's definition, the tangent bundle is, by definition, disjoint union of all $T_pM $s.
Let $(U,\phi)$ be in atlas of M. Define map $f: \pi^{-1}(U) \rightarrow R^{2n} $ as $$f\Bigg(v^i|\frac{\partial}{\partial x^i}\Bigg|_p \Bigg)=(p,v) $$
Now write down the inverse of $f$ and observe that it is a bijection. N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$e(3,6,17)\geq 40$, minimum number of edges possible in an $(3, 6)$-graph on $17$ points the article is Some Graph Theoretic Results
Associated with Ramsey's Theorem for JACK E. GRAVER AND JAMES YACKEL pp: 144-145: https://core.ac.uk/download/pdf/82034211.pdf
I'm studying graph theory and the author makes a statement t... | I don't know if my argument is the same as the authors' "easy" argument, but here is a proof.
(Note that in modern terminology, "points" are called "vertices", we say "degree" rather than "valence", and probably other terminology in the paper is similarly outdated. I will stick to the paper's terminology.)
We assume fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If your category has pullbacks and equalizers, do you get products? I've proven that products + pullback gives equalizers, and products + equalizer gives pullback. So, can we get products out of pullback + equalizer?
| Short answer: no. There is actually a very trivial counterexample. Take the category $\mathbf 2$ with just two objects (say $A$ and $B$) and only the identity arrows. This category has all pullbacks and equalizers, but there is no product $A \times B$. This also indicates the general problem: a product is a limit of a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Locally connected and connected Let X be locally connected, $A \subset X$ arbitrary. Let $S \subset A$ be connected and open in A. Show $S=U\cap A$ where $U$ is connected and open.
I think that I have solved already but I never used that X was locally connected so if you could help me a little bit....
Like $S$ is open ... | As far as what you've done goes, I'd remark that
... since $J\cap K\cap A$ happens to be empty because I said so, there is no absurd here.
Rather, I think you should observe that connected components of open sets in a locally connected topological space are open, and then you should consider the connected component o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
scheme without global sections Let $R$ be an arbitrary non zero ring and $X$ a $R$-scheme. Is it possible that the ring of global sections $H^0(X, \mathcal{O}_X)$ might be a zero ring? The cruical point is that despite of the $R$-scheme structure for $X$ providing the existence of ring map $R \to H^0(X, \mathcal{O}_X)$... | Yes. The empty scheme has zero global sections, because for any scheme $X$, $\mathcal{O}_X(\emptyset)$ is the zero ring.
Conversely, any scheme with zero global sections is empty.
Proof: Suppose $X$ has zero global sections. Let $U\subseteq X$ be an affine open set, isomorphic to $\text{Spec}(A)$. Then there is a re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the general solution of differential equation $y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$ What is the general solution of differential equation $y\frac{d^{2}y}{dx^2} - (\frac{dy}{dx})^2 = y^2 log(y)$.
The answer to this DE is $log(y) = c_1 e^x + c_2 e^{-x}$
I don't know the method to solve different... | Hint. The equation can be written as
$$\frac{d}{dx}\Bigl(\frac{\frac{dy}{dx}}{y}\Bigr)=\log x\ .$$
There is in general no specific procedure for solving this kind of thing and you have to rely on "spotting" something like the above.
BTW I don't think the answer you have given is correct. Probably the $\log x$ should ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integral of $\frac{1}{100-x}$ I don't understand how the integral below:
\begin{align}
\int{\frac{1}{100-x}dx} = \ -\log|x-100| +c
\end{align}
When I integrate I get the answer:
\begin{align}
\ -\log|100-x| +c
\end{align}
I understand that it probably has something to do with the absolute value function, would someone ... | The absolute value function disregards sign, so |4| = |-4| for example.
That also means that $|x-100| = |100-x|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3207934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
estimating $\prod_{i=1}^k a_i\leq \max_i a_i^{p_i}$, where $\sum_i 1/p_i=1$ Let $1<p_1,\dots,p_k<\infty$ such that $\sum_{i=1}^k\frac{1}{p_i}=1$.
Moreover, let $a_1,\dots,a_k\geq 0$.
I have to show that
$$\prod_{i=1}^k a_i\leq\max_i a_i^{p_i}$$
I want to use induction over $k$, but I am struggling with it.
For $k=2$ I... | Assuming $a_i > 0$ for $i=1,\ldots , k$ (otherwise the inequality is trivial) you may set
*
*$a_i = x_i^{\frac{1}{p_i}}$
So, it is enough to show for $x_i > 0$
$$ \prod_{i=1}^k x_i^{\frac{1}{p_i}}\leq\max_i x_i$$
But this follows immediately by the concavity and monotonicity of $\log x$:
$$\sum_{i=1}^k\frac{1}{p_i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Can't use Euler's Theorem? I want to use Euler's theorem to calculate the following:
$14^{(2019^{2019})} \mod 60$
$a^{\phi(n)} \equiv 1 \mod n \iff \gcd(a,n) = 1$
Now I start with the outer one first:
$14^{(2019)}\mod 60 \iff \gcd(14,60) = 2$
Now I don't know how can I solve it ...
Can anyone suggest me any hints/furth... | Like How to find last two digits of $2^{2016}$,
Get the last two digits of $16^{100}$ and $17^{100}$
what are the last two digits of $2016^{2017}$?
last two digits of $14^{5532}$?,
As $(14^n,60)=2^2$ for $n\ge2$
and as $14\equiv-1\pmod{15}\implies14^m\equiv(-1)^m$
$\implies F_{m+2}=14^{m+2}=14^2\cdot14^m\equiv14^2(-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Universality of uniform: plugging a random variable into its CDF? Consider the universality theorem of the uniform distribution. One way to formulate it is the following: Let $F:\mathbb{R}\rightarrow [0,1]$ be a right continuous, increasing function. Then, if $X\sim F$ (ie. $X$ is a random variable that has CDF $F$) th... | $X$ is a function from $S$ to $\mathbb R$, $F$ is a function from $\mathbb R$ to $\mathbb R$, so $F(X)$ is just a composition of functions $F(X(s))$.
For example, let $X$ be an exponential random variable with unit mean. Then $F(t)=1-e^{-t}$ for $t\geq 0$. And $F(X)=1-e^{-X}$.
More concrete, for each $s\in S$, $X(s)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculating Launch Angle Given Distance with Added Height (which itself depends on launch angle) In my physics class we have learned to calculate a desired launch angle to allow a projectile to hit a target given the target’s distance away and the initial velocity.
Now in this case the initial velocity of the projecti... | In the simple case
$x(\theta, t) = v (\cos \theta) t\\
y(\theta, t) = v (\sin \theta)t - \frac 12 g t^2$
Where $v$ is your launch velocity. $\theta$ is your launch angle, $t$ is time, and $g$ is your gravitational constant.
Based on the picture, you have the cannon at some initial altitude, $y_0$.
$x(\theta, t) = v (\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Another combinatorial identity involving product of binomial coefficients While answering a question I came in a side calculation across the following identity valid by numerical evidence for integer $m$ and $n$:
$$
\sum_{k=m}^{\left\lfloor\frac n2\right\rfloor} \binom km\binom n{2k}=\frac{n(n-m-1)!}{m!(n-2m)!}2^{n-2m-... | Here is an algebraic proof. With $n\ge 2m$ we claim that
$$\sum_{k=m}^{\lfloor n/2 \rfloor}
{n\choose 2k} {k\choose m} =
\frac{n}{m} {n-m-1\choose m-1} 2^{n-2m-1}.$$
The LHS is
$$\sum_{k=m}^{\lfloor n/2 \rfloor}
{k\choose m} {n\choose n-2k}
= [z^n] (1+z)^n \sum_{k=m}^{\lfloor n/2 \rfloor}
{k\choose m} z^{2k}.$$
The coe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How is the Tonnetz grid isomorphic to a torus? In Western music, there are $12$ notes up to transposition. In other words, music notes can be represented as integers modulo $12$ : $\big\{[0],[1],\dots,[11]\big\}$.
In Tonnetz grid, moving one step to the right gives the note "perfect fifth" above. That is, given a note ... | There are actually only two independent directions. If you can go $\nearrow$ and $\searrow$, then you create a step in the rightward direction. So there are two independent circles; the circle of three steps in the $\nearrow$ direction, and the circle of four steps in the $\searrow$ direction. This forms a torus:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Lagrange's theorem in Elementary Number Theory Can someone please explain it to me like you would explain to an idiot?
tried to read about it in Burton's, watch videos and read answer's from here on questions about the subject and I don't get it yet.
Here's a document presenting the proof I found on the internet:
http... | A few things:
*
*not all functions are polynomials, it applies to polynomials.
*by division by $x-a$ we have $g(x)=\frac{f(x)-f(a)}{x-a}$
*$f(x)=a_nx^
n + a_{n−1}x
^{n−1} +\cdots + a_0$ which evaluated at a is $f(a)=a_na^
n + a_{n−1}a
^{n−1} +\cdots + a_0$
*This makes their difference, the coefficients times a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why irreducible manifolds are prime? In wikipedia there is a proof for 3-manifolds that I don't understand. It says that if $M$ is an irreducible manifold and we express $M=N_1\sharp N_2$, then $M$ is obtained by removing a ball each from $N_1$ and $N_2$ and then gluing the resulting spheres together. These united sphe... | This is not about "gluing a 2-sphere to one of the two factors". Instead its about reconstructing one of the two factors $N_1$ or $N_2$ by gluing two 3-balls together. Here's a bit more detail.
Suppose that $B \subset M$ is the 3-ball bounded by the "united sphere".
Also, for $k \in \{1,2\}$ let $B_k \subset N_k$ be t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3208971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Using Euler's Formula to prove $e^{i\theta}e^{i\alpha}=e^{i(\theta+ \alpha)}$ I have a homework question thats been puzzling me. It says:
Using Euler's Formula, prove the product property of the complex exponential:
$$e^{i\theta}e^{i\alpha}=e^{i(\theta+ \alpha)}$$
Besides knowing Euler's formula, I have no idea wh... | Fix $y$ and let $f(x) = e^{ix} e^{iy}- e^{i(x+y)}$.
Note that $f(0)=0$ and
$f'(x) = (\sin x - i \cos x) (\cos y + i \sin y) - \sin (x+y) +i \cos (x+y) = 0$.
Hence $f(x) = 0$ for all $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3209084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Prove that $\overline{ \lim} ~u_n\overline{\lim}~v_n\geq \overline{\lim}~u_nv_n$ Problem: Let $\{u_n\}_n$ and $\{v_n\}_n$ be two bounded sequences and $u_n>0, v_n>0$ for all $n\in \mathbb{N}$. Prove that $$\overline{\lim }~u_n\cdot \overline{\lim}~v_n\geq \overline{\lim}~u_n v_n.$$
Progress:
Since $\{u_n\}_n$ and $\{v_... | For $n \geq m$ we have $u_ny_n \leq (\sup \{u_k: k \geq m\}) (\sup \{v_k: k \geq m\})$. Taking sup over $n$ we get $(\sup \{u_kv_k: k \geq m\}\leq (\sup \{u_k: k \geq m\}) (\sup \{v_k: k \geq m\})$. Now take limit as $m \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3209182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Minimizing $\left ( \sin^2(x) + \frac{1}{\sin^2(x)} \right )^2 + \left ( \cos^2(x) + \frac{1}{\cos^2(x)} \right )^2$ While solving a problem I came across this task, minimizing
\begin{align}
\left ( \sin^2(x) + \frac{1}{\sin^2(x)} \right )^2 + \left ( \cos^2(x) + \frac{1}{\cos^2(x)} \right )^2.
\end{align}
One can eas... | Another method:
$$\left ( \sin^2(x) + \frac{1}{\sin^2(x)} \right )^2 + \left ( \cos^2(x) + \frac{1}{\cos^2(x)} \right )^2=\\
\sin ^4x+\cos ^4x+4+\frac1{\sin^4 x}+\frac1{\cos^4 x}=\\
(\sin ^2x+\cos ^2 x)^2-2\sin^2x\cos^2x+4+\frac{(\sin ^2x+\cos ^2 x)^2-2\sin^2x\cos^2x}{\sin^4x\cos^4x}=\\
1-\frac12\sin^2 2x+4+\frac{1-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3209521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.