Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is there an interval notation for complex numbers? Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$
Pictorially, the set of all $z \in \mathbb{C}$ lying in the green ar... | Maybe just define something as $[a,b]+[c,d]i$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/899077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 9,
"answer_id": 6
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Integration of $1/\sin^3 x$
I need a explanation of this problem:
$$ \int \frac{1}{\sin^3 x}\,dx $$
Change the variable $$ t = \tan (x/2) $$
With use of $\tan$, $\cos$, $\sin$ and $\cot$, only.
So how do I think of this and what are the steps?
Progress
Well I have $\sin x = 2t/(1+t^2)$ and $t = \tan x/2$ g... | More simple is to change the variable by $t=\cos x$ so $dt=-\sin x dx$ and then
$$\int\frac{dx}{\sin^3 x}=-\int\frac{dt}{(1-t^2)^2}$$
Can you take it frome here?
| {
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"url": "https://math.stackexchange.com/questions/899121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$f(t) = \cos t^{-1} + \int_t^\infty \frac{1}{\tau^2 + f(\tau)^2} d\tau$ implies the integral is $O(\frac{1}{t})$ The following is a quote from "asymptotic methods in analysis" by de Bruijn (p. 136).
If we know that the real function $f(t)$ satisfies the relation
$$f(t) = \cos t^{-1} + \int_t^\infty \frac{1}{\tau^2 +... | Since $f$ is real valued, we have, for $0 < t \leqslant \tau$,
$$0 < \tau^2 \leqslant \tau^2 + f(\tau)^2,$$
and therefore
$$0 < \int_t^\infty \frac{d\tau}{\tau^2 + f(\tau)^2} \leqslant \int_t^\infty \frac{d\tau}{\tau^2} = \frac{1}{t}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/899216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of subsets/open subsets/closed subsets of a metric space. Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces which have the same infinite set $X$, but the different metrics $d_1$ and $d_2$.
Denote the collection of subsets $X$ by $S$, and the collection of all open subsets of $(X,d)$ by $U_{d}$.
Then, is it possib... | Let $X$ be an infinite set.
Note that a topology $\tau$ on X is a collection of subsets of $X$, that is $\tau\subset P(X)$.
We consider the collection $T$ of all topologies on X.
Pick a topology $\tau$ in $T$ one at a time, and let $(X,\tau)$ be the corresponding topological space.
If $(X,\tau)$ is a metrizable space, ... | {
"language": "en",
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If a recursive sequence converges, must its inverse be divergent? Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = 2b_{n}$. While the sequence clearly diverges, could I know t... | Say the sequence $(a_n)$ defined by $a_{n+1} = f(a_n)$ for bijective $f$ converges to $a$.
If $f$ is continuous at $a$, then
$$0 = \lim\limits_{n \to \infty} a_
{n+1}-a_n = \lim\limits_{n \to \infty} f(a_n)-a_n = f(a)-a$$
So $f(a)=a$. But then it also follows that $f^{-1}(a)=a$.
So if the inverse sequence is started wi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you calculate the dimensions of the null space and column space of the following matrix?
I understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with:
1 -2 0 19 -6 0 -37
0 0 1 -6 2 0 6
0 0 0 0 0 1 3
0 0 0 0 0 0 0
Fr... | Because the matrix is already in row-echelon form:
*
*The number of leading $1$'s (three) is the rank; in fact, the columns containing leading $1$'s (i.e., the first, third, and sixth columns) form a basis of the column space.
*The number of columns not containing leading $1$'s (four) is the dimension of the null s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How many combinations from rolling 5 identical dice? Where, for example:
(1,3,1,4,6) is considered the same outcome as (1,1,3,6,4)
How many total outcomes are there?
Edit 1:
My hunch is that there are:
6 outcomes from choosing 1 dice to be missing.
6*5 = 30 outcomes from choosing 1 number to be the same and one missing... | Let $x_i$ be the number of dice with the $i$ face showing. Then, the $x_i$ are non-negative integers with $x_1+x_2+x_3+x_4+x_5+x_6 = 5$. This is now the standard Stars and Bars problem. You can simply use the formula given in that Wikipedia article to get the total number of solutions.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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general solution of elastic beam equation For the following equation;
$t^4 \dfrac{d^2u}{dt^2} + \lambda^2 u = 0, \quad \lambda >0, ~ t>0,$
where $u(t)$ is real valued function. by using the change of variables
$t=\dfrac{1}{\tau} , \quad u(t)=\dfrac{v(\tau)}{\tau} ,$
how can we find the general solution of the abov... | This shows a method to carry out the change of variable and the change of function, as requested in the wording. Then, you can solve the ODE which is obtained on a well-known form :
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Homology and topological propeties i have this theorem with it's proof but i don't understand the last part
They use this proposition:
My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where $\varphi^c=\lbrace x, \varphi(x)\leq c\rbrace$
Please thank you.
| The theorem is applied (see the middle in $6.6$ ) to $X=\varphi^c\cap C$ and $X_i=\varphi^c\cap U_i$ which is a closed subset in $X$.($X_i=X\cap(\cup_{j\neq i}U_j)^c$ where the c's power is the complementary ).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/900836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$ Let
$$
\begin{align}
I=\int_0^{2\pi}\frac{1}{4+\cos t}dt
\end{align}
$$
I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form like $\int_{\gamma}\frac{f(z)}{z-z_0}dz$ .
I tried using $\c... | Let $z=e^{it}$, then $dz=ie^{it}\ dt$ or $dt=\dfrac{dz}{iz}$, and $\cos t=\dfrac{e^{it}+e^{-it}}{2}=\dfrac{z+z^{-1}}{2}$.
\begin{align}
\int_0^{2\pi}\frac{1}{4+\cos t}dt&=\oint_C\frac{1}{4+\frac{z+z^{-1}}{2}}\cdot\frac{dz}{iz}\\
&=\frac2i\oint_C\frac{1}{z^2+8z+1}dz,
\end{align}
where $C$ is the circle of unit radius wi... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of images from $\mathbb{N}$ to {0, 1}.
Are the number of images from $\mathbb{N}$ to {0, 1} countably infinite or uncountably infinite?
I was thinking of counting in base 2 to make a bijection between $\mathbb{N}$ and {0, 1}. So, a table like this one:
Image | 0 1 2 3 ... <- Here is N
--------... | You can use that if $A\subseteq \mathbb{N}$ then $f(a)=1$ if $a\in A$ and $f(b)=0$ if $b\notin A$ then $\mid \{0,1\}^\mathbb{N}\mid=\mid P(\mathbb{N})\mid$ by Cantor's theorem, $\mid \mathbb{N}\mid<\mid P(\mathbb{N})\mid$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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What is the probability that the maximum number of shots fired successively from a type A gun is $2$? A gun salute always takes place at the funeral of a military leader who has died in a certain country.
(The $21$ gun salute where $21$ rounds are fired - is the most common for the most senior military leaders and the... | Your expression is mostly right. Take the case "3 A's in succession and 2 A's in succession". We have 2 subcases, either the AAA comes first or the AA comes first. These are equivalent.
Let X denote a B or a C (whether it's B or C, we decide at the end). We have 9 X's to place in 3 slots as shown:
AAA AA
^ ^ ^
Exc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Manifolds, charts and coordinates Let's consider the manifold $S^1$
It is well known that we need two charts to cover this manifold.
Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle from the center.
Now, is this a general feature? I mean, is it always possible to have ... | Perhaps you're "really" asking whether or not every $n$-manifold $M$ has $\mathbf{R}^{n}$ as a covering space, i.e., whether there's a single (smooth) map $\pi:\mathbf{R}^{n} \to M$ whose restrictions to suitably small subsets define coordinate charts on $M$.
If so, the answer is "no"; even among spheres, only $S^{1}$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Simulate repeated rolls of a 7-sided die with a 6-sided die What is the most efficient way to simulate a 7-sided die with a 6-sided die? I've put some thought into it but I'm not sure I get somewhere specifically.
To create a 7-sided die we can use a rejection technique. 3-bits give uniform 1-8 and we need uniform 1-7 ... | In the long run, just skip the binary conversion altogether and go with some form of arithmetic coding: use the $6$-dice rolls to generate a uniform base-$6$ real number in $[0,1]$ and then extract base-$7$ digits from that as they resolve. For instance:
int rand7()
{
static double a=0, width=7; // persistent state... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/901222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 12,
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best approximation polynomial $p_1(x)\in P_1$ for $x^3$ I want to find a best approximation polynomial $p_1(x)\in P_1$ for $f(x)=x^3$ in $[-1,1]$ w.r.t. $||\cdot||_{\infty}$. I want to use Chebyshev polynomial to do that, but I don't know how to hang on.
| Let $p(x)$ will be polynomial which you find. First note that $f(x)-p(x)$ is polynomial of degree $3$ with leading coefficient $1$. You know that among the polynomials of degree $3$ with leading coefficient $1$
$$w(x)=\frac{1}{4}T_{3}(x)$$
is the one of which the maximal absolute value on the interval $[−1, 1]$ is mini... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Interval of existence of a certain first-order ODE
Without solving the following initial value problem, determine the interval in which the solution is certain to exist: $$\dfrac{dy}{dx}+(\tan x)y=\sin x, \ \ \ y \left (\frac{\pi}{4} \right )=0.$$
Please help me also solve (if you can) under what conditions the solut... | Hint: Multiply the entire equation by $\sec{x}$ and note that
$$\frac{d}{dx}\left[\sec{x}\,y(x)\right]=\sec{x}\,y^\prime{(x)}+\sec{x}\tan{x}\,y(x).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/901459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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explicit solution for second order homogeneous linear differential equation If we consider the equation
$(1-x^2)\dfrac{d^2y}{dx^2} -2x \dfrac{dy}{dx}+2y=0, \quad -1<x<1$
how can we find the explicit solution, what should be the method for solution?
| There is one obvious particular solution which is $y=c_1 x$. The second one is much less obvious to me but, using a CAS, I found as general solution $$y=c_1 x+c_2 \left(x \tanh ^{-1}(x)-1\right)$$ I hope and wish this will give you some ideas. As said by achille hui, beside the solution in terms of Legendre polynomials... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to check for convexity of function that is not everywhere differentiable? I have a question. I have just been introduced to the subject of convex sets and convex functions.
I read this in wikipedia that a practical test for convexity is -
to check whether the 2nd derivative (Hessian matrix) of a continuous differ... | One option is to check directly that the definition of a convex function is satisfied.
It's useful to know that any norm on $\mathbb R^n$ is a convex function. Proof: If $x,y \in \mathbb R^n$ and $0 \leq \theta \leq 1$, then
\begin{align*}
\| \theta x + (1 - \theta) y \| & \leq \| \theta x \| + \| (1 - \theta) y \| \\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Estimate the discounts... Here is a problem on discount estimation,,
If marked price of an article is $\frac{8}{5}$ times cost price and its selling price is at least $\frac{4}{5}$th the cost price then discount percentage is...
multiple choices are possible...
options:
A) $25$
B) $40$
C) $50$
D) $60$
E) $80$
I am so... | HINT
Say the cost price is $x$,
the maximum possible discount price = $\frac{8}{5}(x) - \frac{4}{5}(x)$
$\implies $maximum possible discount percent = $\dfrac{\frac{8}{5}(x) - \frac{4}{5}(x)}{\frac{8}{5}(x)}\times 100$
Simplify and see which options are not greater than this percent
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show an open set has no minimum Suppose $A$ is an open subset of the real numbers. How to prove $A$ has no minimum?
I guess it should be done with Proof-by-contradiction, but I don't get anywhere at this point.
| Suppose $m \in A$ is the minimum of A. By definition of open set, you have that $(m-\delta , m + \delta) \subseteq A$ with $\delta > 0$ sufficiently small, then $m- \frac{\delta}{2} \in A$ but this is a contradiction (because we have found an element in A smaller than $m$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/901869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving integral that contain exponential function and lower incomplete gamma function I have the following integral;
$$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$
where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma function
Can anyone help me how to sol... | Hint:
$\int_0^\infty\dfrac{e^{-xf}}{m+x}\gamma(a,hx)~dx$
$=\int_0^\infty\dfrac{e^{-fx}}{x+m}\sum\limits_{n=0}^\infty\dfrac{(-1)^nh^{n+a}x^{n+a}}{n!(n+a)}dx$
$=\int_0^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^nh^{n+a}x^{n+a}e^{-fx}}{n!(n+a)(x+m)}dx$
$=\int_m^\infty\sum\limits_{n=0}^\infty\dfrac{(-1)^nh^{n+a}(x-m)^{n+a}e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $(n+1)^{n+1}>(n+2)^n$ for all positive integers Show that:
$(n+1)^{n+1}>(n+2)^n$ holds for all positive integers
I tried using induction:
for $n=1$ we have 4>3 then for $n+1$ we have to show that $(n+2)^{n+2}>(n+3)^{n+1}$ and here I stuck
| Since $(1+x)^n \geq 1 + nx, \forall x\geq -1$
$(\dfrac{n+1}{n+2})^n=(1-\dfrac{1}{n+2})^n \geq 1 -\dfrac{n}{n+2} = \dfrac{2}{n+2}$
$(n+1)\dfrac{2}{n+2} >1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$ Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$.
I appreciate your hints, Thanks
| 2 by 2 only. i really hope you will try the 3 by 3 case by hand, analogous to this:
Aright, eigenvalue $w,$
$$ J =
\left(
\begin{array}{rr}
w & 1 \\
0 & w
\end{array}
\right),
$$
trial
$$ M =
\left(
\begin{array}{rr}
a & b \\
c & d
\end{array}
\right),
$$
Next
$$ JM =
\left(
\begin{array}{rr}
wa + c & wb+d \\
wc & wd
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Circumference of separate circle So I have been out of Algebra for a while now. I am trying to help my wife prep for an entrance exam and we ran across this in the practice test:
A concrete walkway 2 feet wide surrounds a circular pool of radius 5 feet.
Find the area of the concrete walkway.
I know that the area of t... | In this kind of problem, there is an exterior larger shape enclosing an interior smaller shape. Taking the difference between the enclosed areas of the two will give you the area of the border.
So subtract the areas of the "small circle" (just the pool) from the "large circle" (which is basically circle formed by the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that the Binary Icosahedral Group (given by a presentation) has order $120$. Say we have a group generated by $a, b$, with the relations $(ab)^2=a^3=b^5$ (note that these are not necessarily equal to the identity). How do I show that the group has $120$ elements? Without having orders given to any of the genera... | Consider the group $G=\langle a, b, c; a^2=b^3=c^5=abc\rangle^{\ast}$. Clearly, $\langle abc\rangle$, which is the same subgroup as $\langle a^2\rangle$, as $\langle b^3\rangle$, and as $\langle c^5\rangle$, is a central subgroup of $G$. As we know from the earlier part of the problem, quotienting out this subgroup giv... | {
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"source": "stackexchange",
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Bactracking to find compound interest I'm trying to find what percentage 5000 dollars compounding monthly over 120 months will be if the final sum will be 7000 dollars.
So:
7000=5000(1+r/12)^120
When working backwards to find r I always get the percentage = 0.2% but I did trial and error and know the actual answer is ... | We have
$$1.4=\left(1+\frac{r}{12}\right)^{120}.$$
Now we can use a calculator to find $(1.4)^{1/120}$. I get about $1.0028079$. Subtract $1$, multiply by $12$. I get about $0.0336944$.
Remarks: $1.$ I did this on an ordinary calculator. But it could also have been done by Google. Just type in (1.4)^(1/120). It gives ... | {
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How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$? Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$
This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$
I say it is surjective because for all really $x$ there is all real $y$,
$x \in \mathbb R$ there $y \in \mathbb R$?
| To show surjectivity, you show that for each $y$, there is an $x$ such that $f(x) = y$. This means you need to find an $x$ : $x^5 = y$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Partial fraction decomposition of type $1/(x^2+k)$ I know that partial fraction of this can be written as:
$$\frac{3x}{(1+x)(2+x)}=\frac{-3}{1+x}+\frac{6}{2+x}$$
Which can be done in these ways:
$$\frac{3x}{(1+x)(2+x)}=\frac{A}{1+x}+\frac{B}{2+x}\implies3x=A(2+x)+B(1+x),\forall\;x$$
And now solving it to get A and B.Al... | These are two different beasts, at least on the reals. First, $$\frac1{(x-3)(x^2-1)}=\frac{a}{x-3}+\frac{b}{x-1}+\frac{c}{x+1},$$ for some suitable real numbers $(a,b,c)$. Second, on the reals, $$\frac1{(x-3)(x^2+1)}=\frac{A}{x-3}+\frac{Bx+C}{x^2+1},$$ for some suitable real numbers $(A,B,C)$. But, on the complex numb... | {
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} |
Complex Equations The Equation:
$$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$
has a root with a real part 1, solve the equation.
When it says a real part of 1, does this mean that we could use (z-1) and use polynomial division to extract the other rots? Hence:
$$ (z^{4} -2 z^{3} + 12z^{2} -14z + 35 ) / (z-1) $$
But i ... | This is not so different from Lucian's approach, and may (or may not) feel computationally more straightforward.
This equation has real coefficients so that roots come in complex conjugate pairs. We are give that there are roots $z=1\pm bi$. Now let $y=z-1$ so that $z=y+1$ whence $$(y+1)^4-2(y+1)^3+12(y+1)^2-14(y+1)+35... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/903016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Converting repeating decimal to fraction How do I convert $0.297$ to a fraction, if the $2$ and $9$ are repeating? The non-repeating number is in the middle, so I am not sure how to proceed from here.
| Note that parenthesized "phrases" should be considered to repeat ad infinitum.
$.297(29)=.(92)-.632$.
$.(92)={{92}\over{{10}^2}-1}={{92}\over{99}}$.
$.632={{632}\over{1000}}={{79}\over{125}}$.
So, the sought-after fraction $={{92}\over{99}}-{{79}\over{125}}=
{{92·125-79·99}\over{99·125}}={{11500-7821}\over{1237... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/903103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Simple use of log I am struggling to see how we can go from the first expression to the second:
$$\begin{align}
2\log_3 12 - 4\log_3 6 &= \log_3 \left ( \frac{4^2 \cdot 3^2}{2^4 \cdot 3^4} \right )\\
&= \log_3 (3^{-2}) = -2
\end{align}$$
| First note that
\[ \log_{b}(x^{n})=n\log_{b}(x) \]
\[ \log_{b}(x)-\log_{b}(y)=\log_{b}\left(\frac{x}{y}\right) \]
Here are the steps
\[
2\log_{3} 12 - 4\log_{3} 6 = 2\log_{3} (4\cdot 3) - 4\log_{3} (2\cdot 3)
\]
\[
=\log_{3} (4\cdot 3)^{2} - \log_{3} (2\cdot 3)^{4} = \log_{3} (4^{2}\cdot 3^{2}) - \log_{3} (2^{4}\cdot ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Can I prove this, or hopeless? Deviating too much from mean Can I prove this:
We have a sequence of vectors $\left(X_i(n)\right)$ for $i=1,\ldots,t$, where $n\rightarrow \infty$. $t$ does depend on $n$ and is Chosen such that $1 \ll t \ll n$, for instance, take $t=\log(n)$.
We know that $$X(n):=\sum_{i=1}^t X_i(n).$$
C... | Define
$$ \lambda:=\frac{\sum_{i\not\in P(i)} X_i(n)}{\sum_{i\in P(i)} X_i(n)} $$
and note that $\sum_{i\not\in P(i)} X_i(n)=\mathcal{O}(X(n)/t)$ and $\sum_{i\in P(i)} X_i(n)=\Omega(X(n)/t)$.
Your desired result requires that $\lambda=o(1)$. But this only necessarily follows if $\mathcal{O}(f)/\Omega(f)=o(1)$ for all f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What advanced methods in contour integration are there? It is well known how to evaluate a definite integral like
$$
\int_{0}^\infty dx\, R(x),
$$
where $R$ is a rational function, using contour integration around a semicircle or a keyhole.
Most complex analysis books only treat well-known and easy examples like this. ... | Most interesting examples come from physics, often quantum mechanics. There are interesting methods around asymptotic expansions (when you can't solve an ODE exactly, but still want to know the behavior of the solutions at large values of some parameter, for example), or representation of solutions of ODEs using contou... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Convergence of $\sum \frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$ Show that
$$ \sum\frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$$
Converges.
I've tried the telescopic property or even write it as
$$\sum \frac{1}{\ln n (\sqrt{n+1}+\sqrt{n})}$$
But didnt help.
Thanks in advance!
| It diverges. By comparison test,
\begin{align}
\frac{\sqrt{n+1} - \sqrt{n}}{\ln{n}} &= \frac{1}{\ln{n}\left(\sqrt{n+1} + \sqrt{n} \right)} \\
&> \frac{1}{n}
\end{align}
for $n > 100$ (say), and the harmonic series diverges.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Quotient set cardinal in $\mathbb{Z}_{12}$ In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$
Then what is the cardinal of the quotient set?
| Since there are only 12 elements, you can simply check this element by element. Clearly $\bar{0},\bar{1},\bar{2},\bar{3}$ form distinct equivalence classes since their squares are all different. Since $\bar{4}^2=\bar{2}^2, \bar{5}^2 = \bar{1}^2, \bar{6}^2=\bar{0}^2, \bar{7}^2=\bar{1}^2,\bar{8}^2=\bar{2}^2,\bar{9}^2=\ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/903518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Can the winding number be infinite? Let $z$ be a point in the complex plane, and $\gamma$ be a closed curve. Is it possible that
$$n(\gamma,z) = \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z}$$
becomes unbounded? In other words, is it possible to find a curve $\gamma$ such that it winds around a fixed point infinitely ma... | Without loss of generality, we show that the winding number around $0$ is finite (the general proof follows by translating the complex plane).
Remember the alternative definition of the winding number: if $\gamma$ is a closed path, we can choose a continuous choice of argument $\theta:[0,1] \to \mathbb R$ such that $$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Folland's proof of the Hahn Decomposition. Minor error? Theorem 3.3 of Folland's Real Analysis (ed 2) is the Hahn decomposition theorem. In the proof he assumes that the signed measure $\nu$ he is considering does not take the value $-\infty$. Then he argues:
Let $m$ be the supremum of $\nu(E)$ as $E$ ranges over all ... | Yes, you are right. Folland did give an errata for his book which shows that $-\infty$ is a typo in the proof and it should be $+\infty$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/903694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Compact opens in sober $T_1$ are closed? I am trying to establish some basic facts about spectral spaces. In relation to this I am looking for a proof of, or a counter example to, the statement that compact open subsets of a sober $T_1$ space are closed.
Recall that a topological space is sober if every closed irreduc... | You can find counterexamples in some fairly typical sober $T_1$ spaces that are not Hausdorff. For example, consider $\mathbb{N}\cup\{x,y\}$ where every point of $\mathbb{N}$ is isolated and the neighbourhoods of $x$ and $y$ are the cofinite subsets containing $x$ and $y$ respectively. Then $\mathbb{N}\cup\{x\}$ is c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/903854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find $\int {t^n \, e^{t}}\mathrm dt$? Consider:$$\int {t^n e^{t}}\ \mathrm dt$$
is there any closed formula for this?
W|A gave me this but I don't know what is Gamma function:
$$\int {t^n e^t\ \mathrm dt} = (-t)^{-n}\ t^n\ \Gamma(n+1, -t)+ \text{constant}$$
| It is easily proved by induction and integration by parts that
$$
\int {t^n e^{t}}\,dt = p_n(t) e^t
$$
where $p_n$ is a polynomial of degree $n$ and
$$
p_{n+1}(t) = t^{n+1}-(n+1)p_n(t)
$$
This will give you the formula mentioned by gammatester.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/903939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
First order ODE: $tx'(x'+2)=x$ $$tx'(x'+2)=x$$
First I multiplied it:
$$t(x')^2+2tx'=x$$
Then differentiated both sides:
$$(x')^2+2tx'x''+2tx''+x'=0$$
substituted $p=x'$ and rewrote it as a multiplication
$$(2p't+p)(p+1)=0$$
So either $(2p't+p)=0$ or $p+1=0$
The first one gives $p=\frac{C}{\sqrt{T}}$
The second one giv... | EDIT:
$x=t(x')^2+2tx'$
$p=x'$
$x=tp^2+2tp$
We differentiate in respect to $t$:
$p=p^2+t2pp'+2p+2tp' \Rightarrow p'(2tp+2t)=(p-p^2-2p) \Rightarrow p'(p+1)2t=-(p^2+p) \Rightarrow p'(p+1)2t=-p(p+1) \Rightarrow p'(p+1)2t+p(p+1)=0 \Rightarrow (p+1)(2tp'+p)=0 \\ \Rightarrow p+1=0 \text{ or } 2tp'+p=0 \\ \Rightarrow p=-1 \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/904041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Convergent or Divergent Integral Convergent or Divergent?
$$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$
I have problem with the fact that if we have integration from 0 to a say and a to infinity.
How does this change the way we do the comparison test ?
I have in my textbook: If we have from 0 to a and compare with the f... | Set $x=y^2$, then $dx=2ydy$. So the integral becomes:
$$I=\int_0^1 \frac {1}{(x+x^{5})^{1/2}}dx=\int_0^1 \frac {2y}{(y^2+y^{10})^{1/2}}dy=\int_0^1 \frac {2}{(1+y^5)^{1/2}}dy$$
Thus the integral is convergent.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Reduction modulo p I am going to begin the Tripos part III at Cambridge in October (after going to a different university for undergrad) and have been preparing by reading some part II lecture notes.
Here is an extract from some Galois theory notes:
'$f(X) = X^4 + 5X^2 − 2X − 3 = X^4 + X^2 + 1 = (X^2 + X + 1)^2$ (mod ... | First, note that if a polynomial $f$ is reducible, then you could witness these same factors modulo any prime. For instance, $x^2 - 1$ is reducible, and we see this mod 2: $(x+1)^2$ and mod 3: $(x+1)(x+2)$. Of course, it is possible that an irreducible factor of $f$ were further reduced mod $p$, for instance $x^2+1$ wi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$ $$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$$
Try 1:
Put $z= \ln(x+\sqrt{1+x^2})$, $\mathrm dz=1/\sqrt{1+x^2}\mathrm dx$
$$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb v}\mathrm dz=x\int zdz-\int (z^2/2)\mathrm dz\tag{... | Hint:
$$
\int x\frac{\ln({x+\sqrt{1+x^2})}}{\sqrt{1+x^2}}dx=\int\ln({x+\sqrt{1+x^2})}d\sqrt{1+x^2}
$$
and
$$
(\ln({x+\sqrt{1+x^2})})'=\frac{1}{\sqrt{1+x^2}}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/904296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
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When can an infinite sum and complex integral be interchanged? Are there some conditions under which the following two are equal?
$$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$
In the case of real valued functions, the condition $f_n(z) \geq 0$ suffices (see here). Is there a simple condition like that in ... | Since
$$
\int_C f(z)\,dz = \int_C \operatorname{Re} f(z)\,dz + i\int_C \operatorname{Im} f(z)\,dz,
$$
you can essentially apply the same criteria in the complex case as in the real case. Specifically, you just need the real and imaginary parts of the integrands to be single-signed.
Let $Q_j$, $j=1,2,3,4$, denote the f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$? Of course one can prove it by brute force.
But how to Understand it, in the simplest way?
Another issue is $\nabla \times (\nabla f) =0 $.
| It's just a way to remember/understand. Think that $\nabla $ is a vector. Then, by definition, $\nabla \times A$ is perpendicular to $\nabla $ and so the product scalar of $\nabla $ and $\nabla \times A$ is nul, therefore
$$\nabla \cdot (\nabla \times A)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/904423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Simplifying nested/complex fractions with variables I have the equation $$x = \frac{y+y}{\frac{y}{70} + \frac{y}{90}} $$ and I need to solve for x. My calculator has already shown me that it's not necessary to know y to solve this equation, but I can't seem to figure it out. This is how I try to solve it:
$$
x = \frac... | This is my solution:
$$ x=\frac{y+y}{\frac{y}{70}+\frac{y}{90}}={y}\cdot\frac{2}{{y}\cdot\left(\frac{1}{70}+\frac{1}{90}\right)}=\frac{2}{\frac{1}{70}+\frac{1}{90}}=\frac{2\cdot6300}{90+70}=12600/160=\boxed{78.75}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/904558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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How to prove that $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists I am trying to show that the integral $\int_{0}^{\infty}{\frac{e^{-nx}}{\sqrt{x}}}\mathrm dx$ exists ($n$ is a natural number). I tried to use the comparison theorem by bounding from above the integrand by another function which is integra... | The integrand is less than $1/\sqrt{x}$, so that $0<\int_\varepsilon^1 \frac{e^{-nx}dx}{\sqrt{x}}<\int_0^1 \frac{dx}{\sqrt{x}}=2$, and $\int_0^1 \frac{e^{-nx}dx}{\sqrt{x}}$ exists. As to $\int_1^\infty \frac{e^{-nx}dx}{\sqrt{x}}$, the integrand rapidly decreases to $0$ as $x\rightarrow\infty$, in particular, $0<\int_1^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Computing the standard part of $(3-\sqrt{c+2})/(c-7)$ where the standard part of $c$ is $7$ I'm working through Keisler's calculus book based on infinitesimals. The following problem has me a little bit stumped.
Compute the standard part of:
$$\frac{3-\sqrt{c+2}}{c-7}$$
Given that $c\ne7$ and $st(c) = 7$ where $st(x)$... | HINT: Try multiplying by $\frac{3+\sqrt{c+2}}{3+\sqrt{c+2}}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you solve for x in this equation? I tried expanding, but I still can't get rid of the exponents to isolate x.
$$\frac{(1+x)^4-1}{x}=4.374616$$
Thank you in advance for your help.
| I will present two general approaches, starting with the one that works best for this particular case:
Assuming small values of x, we have $(1+x)^4\approx1+4x+6x^2$. Subtracting $1$ and dividing by x, we are left with solving $4+6x=4+\alpha\iff x=\dfrac\alpha4=0.06,~$ where $\alpha=0.374616$.
Let $~f(x)=(1+x)^4.\quad... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? I am reading a chapter about mathematical proofs. As an example there is:
Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for $a,b>0$. There is stated that the thesis for the proof is short ... | Argument is fine as long as it is clear to you that your are using equivalence and not implication on your way down the bullets. That is for $a, b \in \Bbb R^{+}$
$$ \frac{a+b}{2} \geq \sqrt{ab} \iff a-2\sqrt{ab}+b \geq0 \iff (\sqrt a- \sqrt b)^2 \ge 0 $$
You must use "double arrows". Or "if and only if" connectives.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Kirchoff Matrix -Tree Theorem I'm reading a proof of the Kirchoff Matrix -Tree Theorem:
If $G$ is a simple connected graph, $D$ the diagonal matrix with the vertices' degrees and $A$ the adjacency matrix, then in $M = -A+D$ every cofactor is equal to the number of the covering trees of $G$.
I don't understand in the ... | Another definition of laplacian matrix is M( or L) = $QQ^t$, where $Q$ is the incidence matrix. By cauchy-binet theorem, one can calculate the determinant of $QQ^t$ by considering $Q$ and $Q^t$. So, if we take a cofactor, then it can contain twigs of a tree and since determinant of matrix containing (#nodes-1) and twig... | {
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The image of non-reflexive Banach space in its bidual is at large distance from some unit vector in the bidual I am having some difficulties with part of a problem, I am working on.
Let $X$ be a non-reflexive Banach space and let $i: X \to X^{**}$ be the canonical embedding.
Show that for given $\epsilon > 0$ there ... | Simply use the Riesz lemma as $i(X)$ is norm-closed in $X^{**}$.
| {
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Indefinite integral of $\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}$ How do I find $$\int\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}\mathrm dx$$
I used partial fractions by breaking up $x^2 + 2x - 2$ into $(x+1)^2 - 3$ and split it into $(a+b)(a-b)$ but as u can see it's extreme tedious. I was won... | HINT:
Write $$\frac{2x^3 + 5x^2 +2x +2)}{(x^2 +2x + 2)(x^2 + 2x - 2)}=\frac{Ax+B}{x^2+2x+2}+\frac{Cx+D}{x^2+2x-2}$$
For the ease of calculation, we can write
$$\frac{Ax+B}{x^2+2x+2}=\frac A2\cdot\frac{\left(\dfrac{d(x^2+2x+2)}{dx}\right)}{x^2+2x+2}+\frac C{(x+1)^2+1^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/905084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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The number of divisors of any positive number $n$ is $\le 2\sqrt{n}$ How to prove that the number of divisors of any positive number $n$ is $\le 2\sqrt{n}$?
I have started something like below:
$$ n^{\tau(n)/2} = \prod_{d|n} d$$
But not getting ideas on how to take this further and conclude. Any help is appreciated.... | There are at most $\sqrt n$ divisors $d\mid n$ with $d\le \sqrt n$, and there are at most $\sqrt n$ divisors with $d>\sqrt n$ as each of them corresponds uniquely to a divisor $\frac nd<\sqrt n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/905170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$X$ is contained in an annulus containing a circle in the annulus.Show that $\pi_1(X)$ contains a subgroup isomorphic to $\mathbb{Z}$. In $\mathbb{R^2}$ let $C=\{|x|=2\}$ and $A=\{1<|x|<3\}$, let $X$ be a path connected set such that $C \subset X \subset A$. Show that $\pi_1(X)$ contains a subgroup isomorphic to $\math... | The inclusion $X \hookrightarrow A$ induces a homomorphism on fundamental groups $\pi_1(X) \rightarrow \pi_1(A)$. As you said, pick the element $\gamma$ of $\pi_1(X)$ corresponding to $\{|x| = 2\}$ (i.e., a path traversing this circle once counterclockwise). The image of $\gamma$ in $\pi_1(A)$ represents a generator of... | {
"language": "en",
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Show that $ax^2+2hxy+by^2$ is positive definite when $h^2The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$."
Now I've seen questions similar to this before where it's a two variab... | You know that $P(x,y)$ and $a$ have the same sign, so $b$ has the same sign as $a$ (consider $P(0,1)$, so you know that $ab \geq 0$. Next:
case 1: $xy\leq 0$
$$P(x,y)=ax^2+2hxy+by^2=ax^2+by^2+2\sqrt{ab}xy-2\sqrt{ab}xy+2hxy= \\ =(\sqrt{|a|}x+\sqrt{|b|}y)^2-2\sqrt{ab}xy+2hxy=(\sqrt{|a|}x+\sqrt{|b|}y)^2+2xy(h-\sqrt{ab})$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, n$.
Remark. See details in this Hadamard product... | Proof: (assuming $n=m$)
($\Rightarrow$) by assumption, $A$ and $B$ are diagonal. Then $AB = diag(a_{11}b_{11}, a_{22}b_{22}, \dots, a_{nn}b_{nn})$ and $[A\circ B]_{i,j} = [A]_{i,j} [B]_{i,j}$. Since $A$ and $B$ are diagonal, then $[A\circ B]_{i,j} = 0$ for all $i\neq j$, and $[A\circ B]_{i,i} = a_{ii}b_{ii}$ $\Righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Proving combinatorial identity with the product of Stirling numbers of the first and second kinds $$ \sum_{k} \left[\begin{array}{c} n\\k \end{array}\right] \left\{\begin{array}{c} k\\m \end{array}\right\} = {n \choose m} \frac{\left( n-1\right)!}{\left(m-1 \right)!}, \quad \text{for } n,m > 0 $$
$ \left[\begin{array}{... | Recall the species of permutations marked by cycle count
which is
$$\mathfrak{P}(\mathcal{U}\mathfrak{C}_{\ge 1}(\mathcal{Z}))$$
so that it has the generating function
$$\exp\left(u \log\frac{1}{1-z}\right)$$
and in particular
$$\sum_{n\ge q} \left[n\atop q\right] \frac{w^n}{n!}
= \frac{1}{q!} \left(\log\frac{1}{1-w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof of Incircle A circle is drawn that intersects all three sides of $\triangle PQR$ as shown below. Prove that if AB = CD = EF, then the center of the circle is the incenter of $\triangle PQR$.
Designate the center of the circle $G$.
Thinking about it a bit, we realize if we could prove the incircle of the triangl... | I would develop your point just using circle symmetry.
Take a circle and an intersecting line. Make the radius of the circle shrink, then by the symmetry of the circle, the chord will shrink symmetrically to the bisecting line (its axis). Always by the circle symmetry, any other chord of same (original) length will shr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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What is the expected value of A? The Happy Animals Kennel has 18 cages in a row. They allocate these cages at random to 6 dogs, 6 cats, and 6 pot-bellied pigs (with one animal per cage). All arrangements are equally likely.
Let A be the number of times in the row of cages that two animals of the same species are adjac... | The probability that Fido and Rover (two dogs) are in adjacent cages is
$$\frac{17}{\binom{18}2}=\frac19,$$
because there are $\binom{18}2$ choices for the (unordered) pair of cages to put them in, and $17$ pairs of adjacent cages.
To get the expected value, multiply this probability by the number of same-species pairs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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For what $p$ does the series $\sum \frac1{n^p \ln(n)}$ converge "Find the values of $p$ s.t. the following series converges: $\sum_{n=2}^{\infty} \frac{1}{n^p \ln(n)}$"
I am trying to do this problem through using the Integral Test to find the values of $p$. I know that for $p = 0$, the series diverges so I will only b... | Comparison test is best here. But here's how to use that substitution idea. Notice how it comes down to comparison in the end.
Using $x=e^u$ we have $dx=e^u du$. Thus $$\int_2^\infty \frac{dx}{x^p \ln x} = \int_{\ln2}^\infty \frac{e^u}{u e^{p u}} du = \int_{\ln2}^\infty e^{(1-p) u} \frac{1}{u} du.$$
It is easy to see ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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If $(X,\tau)^n$ is Hausdorff, is $(X,\tau)$ also Hausdorff?
If $(X,\tau)^n$ is Hausdorff, is $(X,\tau)$ also Hausdorff?
I know that product of Hausdorff space is Hausdorff, but I want to know if this weaker converse of it is true. Thanks.
| Note that $(X,\tau)$ is homeomorphic to a subspace of $(X,\tau)^n$ by mapping $x \mapsto (x,x_0,x_0,\dots,x_0)$ for some fixed $x_0 \in X$ (ignoring the case $X = \emptyset$ which is even easier). Now use the fact that subspaces of Hausdorff spaces are Hausdorff spaces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/905786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Is there any proof for this formula $\lim_{n \to \infty} \prod_{k=1}^n \left (1+\frac {kx}{n^2} \right) =e^{x/2}$? Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would be app... | Notice that $x-x^2/2\leq \ln(1+x) \leq x$ for $x\geq 0$ then : $$\frac{xk }{n^2} \geq \ln\left(1+\frac{ xk}{n^2}\right) \geq \frac{x k}{n^2} -\frac{x^2 k^2}{n^4} $$ Sum form $k=1$ to $n$ : $$\frac{(n+1)x}{2n} \geq \sum_{k=1}^n \ln\left(1+\frac{ xk}{n^2}\right) \geq \frac{(n+1)x}{2n} - \frac{(n+1)(2n+1)x}{6n^3} $$
Then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Continuity almost everywhere Let
$$f(x) = \left\{\begin{array}{ll}
1 & \text{if }x\in\mathbb{Q},\\
0 & \text{if }x\notin\mathbb{Q}.
\end{array}\right.$$
$$g(x) = \left\{\begin{array}{ll}
\frac{1}{n} & \text{if }x = \frac{m}{n}\in\mathbb{Q},\\
0 & \text{if }x\notin\mathbb{Q}.
\end{array}\right.$$
I have some difficulti... | A function $f:X\to Y$ is continuous almost everywhere if it is only discontinuous on a set of measure $0$. Informally, this means that if we choose a random point on the function, the probability that it is continuous is exactly $1$.
For example, any countable set, such as $\mathbb Q$, will be of measure $0$ in an unc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/905965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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$Ker(T) \subseteq V$ Is A Subspace Let $V,W$ be a vector space over a field $\mathbb F$, and $T$ a linear transformation $T:V \rightarrow W$
$Ker(T) \subseteq V $ to prove that $Ker(T)$ is a subspace can we say that:
by definition $0\in Ker(T)$ and because V is a vector space, and in any vector space there are two tr... | Your reasoning isn't correct. Try the following hints:
To start with, one can show that for any linear map $T$, $T(0) = 0$ and so $0 \in \text{ker }T$ (you should prove this). Thus, since $\text{ker }T$ is non-empty, it suffices to show that it's closed under linear combinations. So suppose $u,v \in \text{ker }T$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A number is a perfect square if and only if it has odd number of positive divisors I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it.
For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ has odd # of positive divisors.
$\Righ... | If two integers multiply to equal N, you will add two divisors to your total for that number. This will be true for all values except when the two integers are the same. Divisors will always be added in pairs except in this case. So the only way to get to an odd total, is when the two divisors are equal, or N is a p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 2
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How do you transpose tensors? We transpose a matrix $A$ by replacing $A_{ij}$ with $A_{ji}$, for all $i$ and $j$.
However, in case $A$ has more than two dimensions (that is, it is a tensor), I don't know how to apply the transpose operation.
*
*If A has dimensions $3\times 3 \times 8$, then what will replace $A_{ijk... | The operation of taking a transpose is closely related to the concept of symmetry. One paper that addresses this is http://www.iaeng.org/publication/WCE2010/WCE2010_pp1838-1841.pdf. I have been researching $2^m$ dimensional matrices where the indices are zeros and ones. The transpose is found by changing all zeros to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Convergence of a sequence of continuous function Len $g_n:\mathbb{R}\rightarrow\mathbb{R}$ continuous with $g_n(x)=0$ for $|x|\geq 1/n$, $g_n(x)\geq 0$, $\int_{-1}^1 g_n(x)\,dx=1$. Consider $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous and:
$$
f_n(x)=\int_{-\infty}^\infty g_n(x-y)f(y)\,dy
$$
I want to show that $f_n$ ... | As you have noted $$\int\limits_{ - \infty }^{ + \infty } {{g_n}(x - y)f(y)dy} = \int\limits_{ - \infty }^{ + \infty } {{g_n}(z)f(x - z)dz} = \int\limits_{ - \frac{1}{n}}^{ + \frac{1}{n}} {{g_n}(z)f(x - z)dz} $$Assuming continuity of $f$ and the existence of it's derivative, for large enough $n$ (small enough $z$ in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$? Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$.
$$
\begin{array}{c|ccr}
& 0 & 1 & 2 \\ \hline
\chi_1 & 0 & 1 & 1 \\
\chi_2 & 0 & 1 & -1
\end{array}
$$
I read the L-functions for these series have special va... | We have:
$$L(2,\chi_1)=\sum_{j=0}^{+\infty}\left(\frac{1}{(3j+1)^2}+\frac{1}{(3j+2)^2}\right)=-\int_{0}^{1}\frac{(1+x)\log x}{1-x^3}\,dx$$
and integration by parts gives:
$$\begin{eqnarray*}\color{red}{L(2,\chi_1)}&=&-\int_{0}^{1}\frac{\log(1-x)}{x}\,dx+\frac{1}{3}\int_{0}^{1}\frac{\log(1-x^3)}{x}\,dx\\&=&-\frac{8}{9}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Solution of an equation involving even integers If $x$ is any positive even integer $> 1$. I compute:
$$ c = x + x! $$
Now assume instead $c$ (even integer) is given, and I want to get back the value $x$.
Is it possible to find a simple expression for $x$ ?
(or, if not, a convergent procedure to find it)
| The $x$ is very small compared to $x!$, so basically you want to invert the Gamma function. See e.g. this MathOverflow question
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/906520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluation of a dilogarithmic integral
Problem. Prove that the following dilogarithmic integral has the indicated value:
$$\int_{0}^{1}\mathrm{d}x \frac{\ln^2{(x)}\operatorname{Li}_2{(x)}}{1-x}\stackrel{?}{=}-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$
My attempt:
I began by using the polylogarithmic expansion in terms ... | This integral belongs to a wider class of integrals that can always be reduced to single zeta values and to bi-variate zeta values which in turn -- provided the weight of the zeta function in question is not too big-- can also be reduced to single zeta values. My standard way of doing this integrals is as follows. We n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
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Linearizing systems about critical points. $$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$
Find all the critical points of the following systems and derive the linearised system about each
critical point. Use this to determine the nature of the critical points and whether t... | The system is $y'=Ay+u$, where $A= \begin{bmatrix} 2 & 1 \\ -5 & 0 \end{bmatrix}$, and $u = \begin{bmatrix} 0 \\ 5 \end{bmatrix}$.
The critical points are where $y'=0$, this gives $y^* = -A^{-1} u = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$.
The linearised system can be written as $(\upsilon+y^*)' = \upsilon' = A(\upsilon... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do you paragraph a proof? When writing out a proof of moderate length, i.e. a proof taking less than or equal to 5 A4 papers and with normal spacing (please avoid asking the criterion for "normal"), do you tend to paragraph it or not? Why and Why not? Is there a style guide on such question?
| Proofs should follow the same rules as any other kind of writing. If the text naturally forms paragraphs then that's how it should be written. Now, usually if you have five pages of text forming a single paragraph that would be poor style, but I suppose it's possible that you could have a five-page-long paragraph if,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is a complex number plus infinity equal to infinity? Why is $$2 + 3 i + \infty = \infty$$
according to Mathematica and Wolfram Alpha?
Shouldn't it be:
$$2 + 3 i + \infty = \infty + 3 i$$
?
After all:
$$2 + 3 i + 10 = 12 + 3 i$$
and not:
$$2 + 3 i + 10 = 12$$
| In the context of the complex plane there is a very useful notion of a "point at infinity" but it is in infinity "in every direction". This gives rise to a notion of the Riemann Sphere, where two copies of the complex plane give coordinate charts with transition $z\mapsto 1/z$. Complex functions with poles can then be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/906830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Plotting a polar curve The question is, to generate a polar graph using a graphing utility, and to choose parameter interval so that the complete graph is generated.
$$r=\cos\frac{\theta}{5}$$
To find such an interval, we are looking for smallest number of complete revolutions until value of $r$ begins to repeat. Algeb... | Since the question has been already answered, I post here a visual answer that I hope may help you to understand the problem better:
Cheers!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f,g$ are entire functions such that $f(g(z))=0, \forall z, $ then $g$ is constant or $f(z) =0, \forall z \ ?$ Let $f,g$ be entire functions such that $f(g(z))=0, \forall z.$ Could anyone advise me on how to prove/disprove: either $g(z)$ is constant or $f(z) =0, \forall z \ ?$
Hints will suffice, thank you.
| Another way to make it: Assume that $g$ is not constant in particular $g'\neq 0$:
As $f\circ g=0$, then by derivation we get $g'\times (f'\circ g)=0$, but the ring of entires functions ( $\Bbb C$ connected) is an integral domain, it follow that $f'\circ g=0$, another derivation we get $g'\times (f''\circ g)=0$, hence ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $p : E \to B$ is a covering map and $U \subset B$ is connected and evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique.
Now, if I assume that $B$ i... | Let $p: E \rightarrow B$ be a covering map. For a fixed finite number $k$, define $A_k = \{ b \in B: |p^{-1}(b)|=k \}$. I claim this set is open and closed in $B$.
To see it is open, let $b \in A_k$ and let $U$ be an evenly covered neighbourhood of $b$. So $p^{-1}[U]= \cup_{i \in I} V_i$, where the $V_i$ are pairwise ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is the history behind the development of the term "coefficient"? Why are coefficients called "coefficients"?
For example I learned that squaring a number is called "squaring" because it actually refers to "making a square". That's how it was developed.
|<----+-----+---->| 3
squared
+-----+-----+-----+... | The Oxford English Dictionary says:
According to Hutton, Vieta, who died in 1603, and wrote in Latin, introduced coefficiens in this sense.
and general meaning of the word around that time seems to have been
Cooperating to produce a result.
So perhaps the meaning is that the coefficient on $5 x$, namely the numeral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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How to solve $\tan x=x^2$ in radians? How to solve $\tan x=x^2$ with $x \in [0, 2\pi]$? I try with trigonometry and many ways but the numerical solutions seems to be difficult.
| Solving $\tan(x)=x^2$ is the same as finding the intersection of the two curves $y=\tan(x)$ and $y=x^2$.
If you plot these two curves for $x \in [0, 2\pi]$, you will notice that, beside the trivial root $x=0$, they intersect just before $x=\frac{3\pi}{2}$. The function being very stiff, it is then better to search for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I show why this equation has no complex roots? I've been asked to show why an equation has no complex roots but i'm at a complete loss.
The equation is
$F_{n+2}=F_n$
Where $F_n=(x-1)(x-2)...(x-n)$ and n is a positive integer.
I'd really appreciate if someone could explain how I could go about showing this becau... | $F_{n}=(x-1)(x-2) \cdots (x-n)$
$F_{n+2}=(x-1)(x-2) \cdots (x-n)(x-(n+1))(x-(n+2))$
$F_{n} = F_{n+2}$
$(x-1)(x-2) \cdots (x-n)=(x-1)(x-2) \cdots (x-n)(x-(n+1))(x-(n+2))$
$(x-1)(x-2) \cdots (x-n)[(x-(n+1))(x-(n+2)) - 1] = 0$
$(x-1)(x-2) \cdots (x-n)[(x^2-[(n+2)+(n+1)]x+(n+1)(n+2) - 1] = 0$
$(x-1)(x-2) \cdots (x-n)[x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Sum the series $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$ This sum is from Ramanujan's letters to G. H. Hardy and Ramanujan gives the summation formula as
\begin{align} &\frac{1}{1^{3}}\left(\coth \pi x + x^{2}\coth\frac{\pi}{x}\right) + \frac{1}{2^{3}}\left(\coth 2\pi x + x^{2}\coth\frac{2\pi... | Recall the well known Mittag-Leffler expansion of hyperbolic cotangent function (denote $\mathbb{W}=\mathbb{Z}/\{0\}$) :
$$\sum_{m\in\mathbb{W}}\frac{1}{m^2+z^2}=\frac{\pi\coth\pi z}{z}-\frac{1}{z^2}\tag{ML}$$
Hence, your sum is by its symmetry :
$$\begin{align}
S&=\frac{1}{2}\sum_{n \in \mathbb{W}}\{\coth (n\pi x) +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Discontinuous seminorm on Banach space We have known that if $X$ is a Banach space and $\sum_{n=0}^{\infty}x_n$ is an absolutely convergent series in $X$ then $\sum_{n=0}^{\infty}x_n$ is a convergent series. Moreover, we have
$$
(*)\quad \left\|\sum_{n=0}^{\infty}x_n\right\|\leq\sum_{n=0}^{\infty}\|x_n\|
$$
by the cont... | There exists a discontinuous functional $f:X\to \mathbb{R}$ take $p(x) =|f(x)|$ . Since $f$ is discontinuous then $\mbox{Ker}(f)$ is dense in $X.$ Take any $v\in X\setminus \mbox{Ker}(f)$ and construct absolutely convergent (with respect to norm) series such that $v=\sum_{k=1}^{\infty} v_k$ then you have $$p(v)>\sum_{k... | {
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"answer_id": 0
} |
How to express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k=\log_2 (\sqrt{9} + \sqrt{5})$? If $$k=\log_2 (\sqrt{9} + \sqrt{5})$$
express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k$.
| Adding the logarithms $\log_2{(\sqrt{9}-\sqrt{5})}$ and $\log_2{(\sqrt{9}+\sqrt{5})}$ we get the following:
$$\log_2{(\sqrt{9}-\sqrt{5})}+\log_2{(\sqrt{9}+\sqrt{5})}=\log_2{(\sqrt{9}-\sqrt{5})(\sqrt{9}+\sqrt{5})}=\log_2{(9-5)}=\log_2{4}=\log_2{2^2}=2 \cdot \log_2{2}=2$$
Knowing that $k=\log_2{(\sqrt{9}+\sqrt{5})}$ we h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Image of centralizer under an isomorphism Suppose we have a group isomorphism $\phi: G\rightarrow K$ between two finite groups and let $H$ a subgroup of $G$. Are there any known facts about the image of the centralizer $C_G(H)$ of $H$ in $G$ under $\phi$?
Also, same question for the special case of $K=G$ i.e. when $\p... | Morally, you can view isomorphism as just relabeling the elements of a group without changing any structure. So the best we can hope to say is that $$\phi(C_G(H)) = C_K(\phi(H))$$
Edit: I realize now that "best we can hope to say" is a bit strange. Basically any structure you define in group theoretic terms will carry ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is the result of $-2^2 = -4$ but $(-2)^2 =4$? I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $?
| It is true that $-2^2$ is ambiguous (unless you know the convention), because $-(2^2) \ne (-2)^2$. (And, indeed, some calculators or programing languages may do it using their own convention, different from the mathematicians' convention.) The mathematicians' convention is $-2^2 = -(2^2)$. Why? Presumably because we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Question about queues A queue is implemented with a sequence $Q[1\ldots n]$ and two indices $\def\head{\operatorname{head}}\head[Q]$ and $\def\end{\operatorname{end}}\end[Q]$ such that the elements of the queue are the following:
$$Q[\head[Q]], Q[\head[Q]+1], \ldots , Q[\end[Q]-1]$$
Initial $\head[Q]=\end[Q]=1$
When $\... | When $head=end$ we consider the queue empty.
If there is 1 element in the queue, the queue looks like:
$$-\ \ - \ - \underset{\stackrel\uparrow{head}}o \underset{\stackrel\uparrow{end}}- \ \ -$$
That means that:
$$head+1 \equiv end\pmod{n}$$
At some point in time the queue might look like:
$$o \underset{\stackrel\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/907921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Help Evaluating $\lim_{x\to0}(\frac{\sin{x}}{{x}})^{\frac{1}{x}}$ Does anyone know how to evaluate the following limit?
$\lim_{x\to0}(\frac{\sin{x}}{{x}})^{\frac{1}{x}}$
The answer is 1 , but I want to see a step by step solution if possible.
| Let $\displaystyle y=(\frac{\sin x}{x})^{\frac{1}{x}}$. Then $\displaystyle\lim_{x\to 0} \ln y=\lim_{x\to 0}\ln \left(\frac{\sin x}{x}\right)^{\frac{1}{x}}=\lim_{x\to 0}\frac{1}{x}\ln\left(\frac{\sin x}{x}\right)=\lim_{x\to 0}\frac{\ln\sin x-\ln x}{x}=^{(LH)}\lim_{x\to 0}\frac{\frac{\cos x}{\sin x}-\frac{1}{x}}{1}$
$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
A linear map that is multiplication by a matrix The problem statement, all given variables and data
Let $T$ be multiplication by matrix $A$:
$$A=
\begin{bmatrix}
1 & -1 & 3 \\
5 & 6 & -4 \\
7 & 4 & 2 \\
\end{bmatrix}
$$
Find the range, kernel, rank and nullity of T.
Attempt at a ... | Any matrix $A$ represents a linear function $T$, or more precisely $T_A$, defined by multiplication, that is, by $T(x)=Ax$. Now it makes sense to talk about the range and kernel of $T$. Rank and nullity are simply the dimension of range and kernel of $T$, respectively.
Regarding your question, note that, by definition ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Pólya's urn scheme, proof using conditional probability and induction Problem
An urn contains $B$ blue balls and $R$ red balls. Suppose that one extracts successively $n$ balls at random such that when a ball is chosen, it is returned to the urn again along with $c$ extra balls of the same color. For each $n \in \math... | I would actually still advocate the approach suggested here with a small change in the way it is presented:
$P(R_1)=\frac{R}{R+B}$, now we need to prove that $P(R_n)=P(R_{n+1})$.
$P(R_{n+1})=P(R_{n+1}|R_n)P(R_n)+P(R_{n+1}|B_n)(1-P(R_n))$
$X_n$, the number of red balls in the urn at step $n$, is $P(R_n)T_n$, where $T_n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Are all endpoints discontinuous? I learned that something is a limit if the left limit and right limit exist and are equal. But then doesn't this mean that if I have a function on $[a,b]$, that the endpoints $a$ and $b$ are discontinuous because $a$ doesn't have a left limit and $b$ doesn't have a right limit?
| Given $f : A \subset \Bbb R \to \Bbb R$, we say that $f$ is continuous at $x_0 \in A $ if for every $\epsilon > 0 $, exists $\delta > 0 $ such that: for all $x \in A$ that $|x - x_0| < \delta$ implies $|f(x) - f(x_0)| < \epsilon$.
The points $x$ considered in the last implication must first of all be in the function's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Differentiation under the integral sign (one complex variable) Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define
$$g(s,z) = f((1-s)u(z) + su'(z))$$
and
$$h(z) = \int_0^1 g(s,z) ... | Since $g$ is smooth, we see that ${\partial g(s,z) \over \partial z}$ is bounded on $[0,1] \times U$, where $U$ is a bounded neighbourhood of $\Omega$.
The mean value theorem gives $|g(s,z+h)-g(s,z)| \le \left( \sup_{(s,z) \in [0,1] \times U}|{\partial g(s,z) \over \partial z}| \right) | h| $.
Hence ${1 \over h} |g(s,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding solutions of $y'''-4y''+5y'-2y=-x^2+5x+2$ Find all solutions of $$y'''-4y''+5y'-2y=-x^2+5x+2.$$
I know how to find the solutions of the corresponding homogenous differential equation $y'''-4y''+5y'-2y=0$. I've done that in the following way:
The characteristic equation is $$P(x) = x^3 - 4x^2 +5x-2=(x-1)^2(x-2).... | Hint
It may not be a very academic solution but since you found the solution for the homogenous part, let us set, because the rhs is a polynomial, that $$y=c_1 e^x+c_2 x e^{x}+c_3 e^{2x}+P(x)$$ Differentiate three times and substitute. You end with $$P'''(x)-4 P''(x)+5 P'(x)-2 P(x)=-x^2+5x+2$$
Try $P(x)=a+bx+x^2+dx^3$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to deduce a closed formula given an equivalent recursive one? I know how to prove that a closed formula is equivalent to a recursive one with induction, but what about ways of deducing the closed form initially?
For example:
$$ f(n) = 2 f(n-1) + 1 $$
I know how to use induction to prove that $\forall n \ge1$:
$$ f(... | $$f(n) = 2f(n-1) + k \\ f(n-1) = 2f(n-2) + k \\ f(n-2) = 2f(n-3) + k \\ \dots \\ f(1) = 2f(0) + k$$
$$$$
$$\Rightarrow f(n) = 2f(n-1) + k= \\ 2(2f(n-2) + k)+k= \\ 2^2f(n-2)+(2+1)k=2^2(2f(n-3) + k)+(2+1)k= \\ 2^3f(n-3)+(2^2+2+1)k= \\ \dots \overset{*}{=} \\ 2^nf(0)+(2^{n-1}+2^{n-2}+\dots +2+1)k$$
Therefore, $$f(n)=2^nf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
} |
Factoring $x^{4} +1$, using real factoring to the second degree Factoring to the second degree using real numbers
$$x^{4} +1$$
I know that $ x^{4} +1=(x^{2} + i)(x^{2}-i).\;$ But these are complex but I thought using these in some kind of way? Got no where!
And then I tried to guess, two solutions are $\pm (-1)^{1/4},... | HINT :$$x^4+1+\color{red}{2x^2}-\color{red}{2x^2}=(x^4+2x^2+1)-2x^2=(x^2+1)^2-(\sqrt 2x)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/908637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
If $\sin( 2 \theta) = \cos( 3)$ and $\theta \leq 90°$, find $\theta$ Find $\theta\leq90°$ if
$$\sin( 2 \theta) = \cos( 3)$$
I know that $\sin 2\theta = 2\sin\theta\cos \theta$, or alternatively, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$.
Can somebody help me?
| Assuming you mean $3^{\circ}$, here are the steps
$$ \sin 2\theta=\cos 3^{\circ} $$
$$ \arcsin(\sin 2\theta)=\arcsin(\cos 3^{\circ}) $$
$$ 2\theta=\arcsin(\cos 3^{\circ}) $$
$$ \theta=\frac{1}{2}\arcsin(\cos 3^{\circ})=\frac{87^{\circ}}{2}=43.5^{\circ} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/908703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
$\ln (2 x-5)>\ln (7-2 x)$ Solve
$$\ln (2 x-5)>\ln (7-2 x)$$
The answer is given as $$3<x<7/2$$
This is what I have done $$\ln (2 x-5)-\ln (7-2 x)>0$$
$$\ln \left(\frac{2 x-5}{7-2 x}\right)>0$$
However I am not able to understand how to get to the answer provided.
| Hint
If $$\ln \left(\frac{2 x-5}{7-2 x}\right)>0$$ it implies that $$\frac{2 x-5}{7-2 x}>1$$ But take care : the logarithm is such that its argument must be positive.
I am sure that you can take from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/908819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to maximize pay with repeated toss of coin repeated toss a coin and you can stop anytime and payoff is just #times you got head divided by total number of throws, how do you maximize your pay.
Does anyone have a clever strategy for this? This was changed from another problem I solved involving flipping over cards... | So I think that you should stop whenever the ratio of heads to flips is greater than 1/2. Like obviously since there is no maximal number of flips you can "eventually" get this as the ratio so stoping when your ratio is lower is silly but you can't guarantee you will "eventually" get a higher ratio so if you ever do ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Does validity of contrapostive proofs require the Law of Excluded Middle? I remember that during my first proofs class the hardest thing I had accepting were the logic we had to learn, and it seems I still have questions about.
So I was thinking about why when the contrapositive is proved true then it implies that the ... | In classical logic every statement has a truth value, even if it is nonsense like "if the Moon is made of cheese then the sky is made of rubber" (this one is true because any implication with a false premise is defined to be true). The law of excluded middle is a law of classical logic, so it is always true as an axio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/908984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$ For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$
My try :
Left hand side :
$\begin{align} \sum_{d|p^k}\sigma (d) &= \sigma(p^0) + \sigma(p^1) + \sigma(p^2) + \cdots + \sigma(p^k) \\ &= \dfrac{p^... | looking at this intuitively, firstly we note that:
$$
\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d) \\
= \sum_{d|n} d\tau(n/d)
$$
so now we are summing the divisors $d$ of $n$, each divisor being counted with multiplicity $\tau(n/d)$. so you just have to persuade yourself that this multiplicity is appropriate.
but this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/909131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Does $G$ always have a subgroup isomorphic to $G/N$? Let $G$ be a group and $N$ a normal subgroup of $G$. Must $G$ contain a subgroup isomorphic to $G/N$? My first guess is no, but by the fundamental theorem of abelian groups it is true for finite abelian groups, so finding a counterexample has been a little tough. The... | Let $G$ be the reals $\mathbb{R}$under addition. Let $N$ be the integers $\mathbb{Z}$ under addition. $G/N$ is the group of reals modulo $1$. This has a non-zero element whose square is $1$, the equivalence class of $0.5$. So $G/N$ is not isomorphic to any subgroup of $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/909224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
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