Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Summations of $\frac{1}{n^p}$ I wasn't able to find any information on this, so I'm just curious and am asking here.
We know the summation (n sums from 1 to infinity) of $\frac{1}{n^2}$, $\frac{1}{n^4}$, and I'm sure finding $\frac{1}{n^k}$ where k is a multiple of 2 shouldn't be an issue.
My question is, do we know ... | If you are concerned by the patrial sums$$S_n=\sum_{k=1}^n\frac{1}{k^s}=H_n^{(s)}$$ where appear the generalized harmonic numbers.
Usig their asymptotics, we have
$$S_n=n^{-s} \left(\frac{n}{1-s}+\frac{1}{2}-\frac{s}{12 n}+\frac{s (s+1) (s+2)}{720
n^3}+O\left(\frac{1}{n^5}\right)\right)+\zeta (s)$$
Just for the fun ... | {
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"timestamp": "2023-03-29T00:00:00",
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Struck in Diffeomorphism invariance of the line integral I have been trying to study The Cotangent bundle from An introduction to smooth manifolds by John M Lee. I have been struck at a specific point in the Diffeomorphism Invariance of the integral. The proposition is given below.
Let $\omega$ be a smooth covector fie... | The expression $f \circ \varphi$ is literally the same as the function $f(\varphi(s))$.
Now for the second part: the $d(t \circ \varphi)$. The function $t \circ \varphi$ means apply the function $\varphi$, and then take the $t$-coordinate of the result. Since everything is 1-dimensional, "taking the $t$-coordinate" is ... | {
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"timestamp": "2023-03-29T00:00:00",
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Using Laurent series to evaluate integral Use the Laurent series of $e^{1/z}$ on the punctured plane $\mathbb{C} - \{0\}$ to compute $1/\pi \int_0^\pi e^{\cos\theta}\cos(\sin\theta-n\theta) d\theta$ for $n=0,1,2,...$
I have been trying to attempt this question from my complex analysis book and I am not sure where to e... | Let $\gamma$ be the unit circle, and consider the integral
$$I_m=\int_\gamma z^m\exp(1/z)\,dz$$
where $m$ is an integer. Writing $z=e^{it}$ gives
$$I_m=i\int_0^{2\pi}e^{im t}e^{\cos t-i\sin t}e^{it}\,dt
=i\int_0^{2\pi}e^{\cos t}e^{i((m+1)t-\sin t)}\,dt.$$
The imaginary part of this integral is
$$\int_0^{2\pi}e^{\sin t}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3629534",
"timestamp": "2023-03-29T00:00:00",
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Is this proof correct convergence I'm trying to show that $(a_{n}) \to -\infty \Leftrightarrow (-a_{n}) \to \infty$
Attempt
$(\Rightarrow)$: We have $(a_{n}) \to -\infty$. By definition this means, for all $C < 0$ there exists $N$ such that $a_{n} < C$ for all $n > N$. Therefore we have that $ a_{n} < C < 0$. Therefore... | Your proofs are correct. Well done.
| {
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"timestamp": "2023-03-29T00:00:00",
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Cyclic subspace and one dimensional range
*$\therefore$.. Let $T$ be a linear operator on a finite dimensional vector space $V$ with $\operatorname{dim} V$; $\geq 2$, and let $R(T)$ be the range of $T$. If $\operatorname{dim} R(T)=1$, prove that there exists a scalar $k$ such that $T^{2018}=k^{2017} T$
What I am thin... | We can write $Tx=f(x)w$ where $w$ is some non-zero vector and $f$ is a linear map: $V \to \mathbb R$ (assuming that your vector space is over $\mathbb R$). Also $Tw=cw$ for some scalar $c$. This gives $T^{m}x=f(x)c^{m-1} w$ by induction . Hence $T^{m}=c^{m-1}T$ for all positive integers $m$. Put $m=2018$.
| {
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bug on the box problem A bug walks on the surface of a box ($L=B=1$, $H=2$), starting at a corner, $A$. You want to feed the bug but you also want the bug to walk the longest distance. The bug takes the shortest path possible. Where do you put the food to make it walk the longest?
The box floats in the air and the bu... | If a curve gives the shortest path between two points then it is a geodesic. But any running line between two arbitrary points is geodesic then the path length need not be the shortest. We can choose one among the many longer geodesics that are not unique on $\mathbb R^2$ including surfaces with discontinuous folds.
I... | {
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Let $G$ be a group and $H = \{x^{-1}\mid x\in G\}$. Show $G=H$.
Let $G$ be a group and $H = \{x^{-1}\mid x\in G\}$. Show $G=H$.
I have showed that $H \subseteq G$.
Can somebody give me a hint to show how an arbitrary $x \in G$ also belongs in $H$?
| Based on the Axiom of Inverse Element, if $a\in G$, then $b\in G$ if $a* b=b* a=e$, where $*$ represents the groupg operation. Hence the rest of the proof goes as follows:
$$a\in G\implies a^{-1}\in G\implies (a^{-1})^{-1}\in H\implies a\in H\implies G\subseteq H$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3630352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derivative parameric equation I would like to compute the derivative of the following parametric equations w.r.t $a$ and $b$:
$x=a~ \text{cos}(t)$ and $y= b~ \text{sin}(t)$ with $t \in [0, b]$.
Derivative w.r.t $a$ are easy to compute : $d_a x = \text{cos}(t)$, $d_a y = 0$ with $t \in [0, b]$.
However, the ones w.r.t ... | On the contrary, $t$ doesn't depend on $b.$ Or at least not for most of its values. It depends on $b$ only once in the interval $[0,1/b].$ Otherwise it doesn't. So the derivatives are as before.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3630494",
"timestamp": "2023-03-29T00:00:00",
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How do you show that $\lim_{x \rightarrow a} f(x)=0$? This is a simple proof of Real Analysis.
Show that $\lim_{x \rightarrow a} f(x)=0$, given $|f(x)|\leq g(x)$, $\forall x\in\mathbb R$ and $\lim_{x \rightarrow a} g(x)=0$.
How do you show that $\lim_{x \rightarrow a} f(x)=0$?
I tried by myself $\forall\epsilon>0\exis... | You know $\forall\epsilon>0\exists\delta>0\forall x\in\mathbb{R}:|a-x|<\delta\implies |g(x)|<\epsilon.$ Since $|f(x)|\leq g(x)\leq |g(x)|$ it follows
$$\forall\epsilon>0\exists\delta>0\forall x\in\mathbb{R}:|a-x|<\delta\implies |f(x)|\leq |g(x)|<\epsilon.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3630648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $f(H)=\{y\in G∶y=f(x)\text{ for some }x\in H\}\le G.$ Let $f∶ K \rightarrow G$ be an isomorphism of groups and Suppose $H$ is a subgroup of $K.$
So, I will do the subgroup test. Show its nonempty and that $ab^{-1} \in f(H).$ Maybe let $a \in f(H),$ then $a=f(x)$ for some $x\in H$ and $b \in f(H),$ then $b=f(... | You are almost there: $f(x)f(t)^{−1}=f(x)f(t^{−1})=f(xt^{−1})$. Now, as $x,t∈H$ and $H$ is a subgroup, then $xt^{−1}\in H$ and so $f(xt^{−1})\in f(H)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3630766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If the graph of an equation intersects the x-axis, is it possible for there to be a horizontal tangent I would add a picture of the equation that this question pertained to, but the file size is too large
The equation is $x^2 + 2x + y^4 + 4y = 5$.
The question was "Is it possible for this curve to have a horizontal t... | I guess it depends on your definition of "intersection", but my understanding is it means the two lines "touch" each other at least one point, regardless of whether or not these lines cross each other.
Note Intersection - math word definition states
An intersection is a single point where two lines meet or cross each ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A variable parabola touches the $x$-axis and $y$-axis at $A(1,0)$ and $B(0,1)$. Find the locus of its focus.
A variable parabola touches the $x$-axis and $y$-axis at $A(1,0)$ and $B(0,1)$ on the co-ordinate plane respectively. Now, we are required to find the locus of the focus of this variable parabola.
The process ... | Contrary to what you wrote, $x=y=\frac12$ satisfies your last equation: the circle isn’t imaginary but consists of a single point. This is as it should be: two points and the tangents at those points uniquely determine a parabola.
I’ve seen a version of this problem in which the parabola is tangent to the coordinate ... | {
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$\frac{da}{dt} \leq f(a(t))$ and $\frac{db}{dt}=f(b(t))$ imply $a(t) \leq b(t)$? Let $a(t)$ and $b(t)$ be nonnegative real-valued differentiable functions and $f$ be a strictly positive continuous function.
If $a(0)=b(0)$, $\frac{da}{dt} \leq f(a(t))$ and $\frac{db}{dt}=f(b(t))$ for $t \geq 0$. Then how do I show that ... | See
https://users.math.msu.edu/users/sen/Math_848/lec_6s.pdf
Inequality of differential equation implies inequality of functions.
| {
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"url": "https://math.stackexchange.com/questions/3631447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there any Richard Feynman parallel in mathematics world? This is a soft question. I was studying the Red Book by Richard Feynman which consists of his lectures which he delivered in 60's. We all know that what a brilliant teacher an explainer he was. Moreover his lectures consist of almost all the basic Physics whic... | As Shamim said, Walter Rudin is considered one of the great authors.
I would also point you to Terrence Tao, who is both a prolific research mathematician as well as a writer of textbooks. His blog contains much that is "rigorous as well as motivating".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3631593",
"timestamp": "2023-03-29T00:00:00",
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Does there exist a sequence of sets such that the sequence of their cardinalities is strictly decreasing? Does there exist a sequence of sets such that the sequences of their cardinalities is strictly decreasing?
More explicitly, does there exist a sequence of sets $S_1,S_2...$ such that for each $i$ there exists an in... | No: cardinalities are well-ordered, and no well-order admits an infinite descending chain. Such a chain would be a non-empty set with no least element.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3631670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $\lim_{n\rightarrow\infty} I_n$ knowing that the sequence $(I_n)_{n\ge 2}$ is $I(n) = \int_1^{\sqrt{2}} \sqrt[n] {x^n+x^{n+2}} dx $
Calculate
$\lim_{n\rightarrow\infty} I_n$ knowing that the sequence $(I_n)_{n\ge 2}$ is $I_n = \int_1^{\sqrt{2}} \sqrt[n] {x^n+x^{n+2}} dx $
Now, my attempt was like this :... | $I_n=\int_{1}^{√2}(x^n+x^{n+2})^{1/n}dx;$
$\int_{1}^{√2}x(1+1^2)^{1/n} \lt I_n \lt \int_{1}^{√2}x(1+(√2)^2)^{1/n}dx;$
$2^{1/n}\int_{1}^{√2}xdx\lt I_n < (1+2)^{1/n}\int_{1}^{√2}xdx.$
Integrate and take the limit.
Used:
$0 \le f(x):=(1+1^2)^{1/n}x \le x(1+x^2)^{1/n} \le x(1+√2^2)=:g(x),$
then $\int_{1}^{√2}f(x)dx \le ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding primitive element of field extension in characteristic 2 corresponding under Galois correspondence to the group $G_f\cap A_n$ Let $F$ be a field and let $f(X)\in F[X]$ be a separable polynomial over $F$ of degree $n$. Let $F_f$ be the splitting field of $f$. Then the Galois group $Gal(F_f/F)=G_f$ acts as a grou... | Assume $\mbox{char}(F)=2$ and let $\alpha_1,\dots,\alpha_n$ be the distinct roots of $f(X)$. Define
$$
\delta_f = \sum_{i<j} \frac{\alpha_i}{\alpha_i + \alpha_j}
$$
and
$$
D(f) = \sum_{i<j} \frac{\alpha_i\alpha_j}{\alpha_i^2 + \alpha_j^2}.
$$
The element $D(f)$ is called the Berlekamp discriminant of $f(X)$. It is stra... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\lim_{t\rightarrow 0}\left[-t^{-4} + t^{-5}\left(1+\frac{t^2}{3}\right)\tan^{-1}t\right] = \frac{4}{45}$ The author shows the following limit being taken
$\lim_{t\rightarrow 0}\left[-t^{-4} + t^{-5}\left(1+\frac{t^2}{3}\right)\tan^{-1}t\right] = \frac{4}{45}$
I don't see how you could get anything but $\infty$..... | With little $o$ notation,$$\begin{align}\frac{-t+(1+\tfrac13t^2)\arctan t}{t^5}&=\frac{-1+(1+\tfrac13t^2)(1-\tfrac13t^2+\tfrac15t^4+o(t^4))}{t^4}\\&=\frac{-\tfrac19t^4+\tfrac15t^4+o(t^4)}{t^4}\\&=-\tfrac19+\tfrac15+o(1)\\&=\tfrac{4}{45}+o(1).\end{align}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Integration problem with shadowy hint. Solving some integration problems this quarantine I found this one:
Let $f, g:[0,\frac{\pi}{2}] \to \mathbb{R}$ be two class $C^1$ functions such that $$f(x)g(x) = \sin(x)$$ for each $x \in [0,\frac{\pi}{2}]$. Show that $$\int_{0}^{\frac{\pi}{2}} f'(x)g^2(x) \,\rm{d}x \neq -g(\f... | You have\begin{align}\int_0^{\pi/2}f'(x)g^2(x)\,\mathrm dx&=\left[f(x)g^2(x)\right]_{x=0}^{x=\pi/2}-2\int_0^{\pi/2}f(x)g(x)g'(x)\,\mathrm dx\\&=\left[\sin(x)g(x)\right]_{x=0}^{x=\pi/2}-2\int_0^{\pi/2}\sin(x)g'(x)\,\mathrm dx\\&=g\left(\frac\pi2\right)-2\int_0^{\pi/2}\sin(x)g'(x)\,\mathrm dx\\&=g\left(\frac\pi2\right)-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3632209",
"timestamp": "2023-03-29T00:00:00",
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prove that a measurable function is in $L^{\infty}([0,1])$ I am trying to solve this problem but I can not find a good idea:
Let $\phi$ be a measurable function on $[0,1]$ and assume that the linear transformation
A $\colon f \to f \cdot \phi$
maps $L^{2}([0,1])$ into itself .Prove that $\phi \in L^{\infty}([0,1])$.
... | The condition is that $$\int_{[0,1]}|\varphi|^2\,|f|^2<\infty$$ for all $f\in L^2$. So the operator $A_0:f\longmapsto |\varphi|\,f$ also maps $L^2$ into $L^2$.
We also have
$$
\int_{[0,1]}(1+|\varphi|)^2\,|f|^2\leq 2\int_{[0,1]}(1+|\varphi|^2)\,|f|^2\leq\int_{[0,1]}|f|^2+\int_{[0,1]}|\varphi|^2\,|f|^2<\infty,
$$
so t... | {
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How to find the remainder of big number divisions using congruences? I´m asked to find the remainder of dividing $\sum_{i=0}^{1080}i^5$ by $14$. How can I do this using only basic results from modular arithmetic? Only one thing comes to my mind, here's my idea:
we know that each number has a representative $r$ in the c... | We have
$\tag 1 \displaystyle \sum_{i=0}^{13}i^5 \equiv 0 + + 7^5 +\sum_{i=1}^{6}\bigr(i^5+(-i)^5\bigr) \equiv 7^5 \equiv 7 \pmod{14} $
Also, since
$\quad 1081=77\times14+3$ we can write
$ \tag 2 \displaystyle \sum_{i=0}^{1080}i^5 \equiv 77\cdot7 + 0^5 + 1^5 + 2^5 \equiv 7 + 0 + 1 + 4 \equiv 12\pmod{14} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3632437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Find the probability that a stick will lie entirely on the tile. A floor is paved with tiles, each tile being a parallelogram such that the distance between pairs of opposite sides are $a$ and $b$ respectively, the length of diagonal being $L$. A stick of length $C$ falls on the floor parallel to the diagonal. Show tha... |
My drawing skills are not so good but I think this should help you give a visual intuition for the problem.
If you have a stick of length $C$, parallel to the diagonal, you can see that the possible space where this stick lands is the trapezoid with one base the diagonal $L$ and the other base the stick of length $C$... | {
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"timestamp": "2023-03-29T00:00:00",
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How Many Ways Can Things be Grouped Under Conditions Let's say we have a sentence:
"A man walked his dog to the park."
How many groups can be made with only words that are next to each other (general formula)?
To explain this:
"A"
is considered a group.
"A man"
is considered a group because the words are next to each o... | I think this problem is the same as choosing 2 numbers from k numbers(k is the number of words in the sentence): which you choose the start and end for each group. So the answer is k(k-1)/2.
Edit: I found out I made I mistake which I didn't consider the single word groups, so the answer should be k(k-1)/2 +k, which is ... | {
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A sufficient condition for differentiability of $f:\mathbb{R}^n\rightarrow\mathbb{R}$ There is a theorem in my textbook which says:
Theorem: consider the function $f:\mathbb{R}^2\to\mathbb{R}$ such that both of its partial derivatives at a point exist and at least one of them is continuous, then $f$ is differentiable ... | Here is a counterexample.
$\displaystyle f(x,y,z)=\begin{cases} \frac{xy}{x^2+y^2}+c(z)\quad (x,y)\ne(0,0) \\ c(z) \quad (x,y)=(0,0) \end{cases}$
where $c(z)$ is any function with continuous derivative (so that $f_z$ exists).
Since $\frac{xy}{x^2+y^2}$ is zero along the $x-$ and $y-$ axes, the partial derivatives $f_x$... | {
"language": "en",
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"source": "stackexchange",
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How does $ a^{2^i}a^{2^i} $ become $ a^{2^{i+1}} $? I have this expression: $$ a^{2^i}a^{2^i} $$
My slides convert this into: $$ a^{2^{i+1}} $$
I however don't see how to convert $$a^{2^{i}+2^{i}}$$ into that resultant form.
Which math rule am I missing here?
| $a^{2^i}\cdot a^{2^i}=a^{2^i+2^i}=a^{2\cdot 2^i}=a^{2^{i+1}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3633137",
"timestamp": "2023-03-29T00:00:00",
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Prove : $\forall x>0$, $\arctan(x)>\frac{x}{1+x^2}$ I think the question is itself wrong as, $\frac{x}{1+x^2}$ is the derivative of $\arctan x$. If not, I don't know how to proceed.
| Letting $x=\tan\theta$ for some $\theta\in\left(-\pi/2,\pi/2\right)$ we are left with
$$ 2\theta > \sin(2\theta) $$
which holds for any $\theta\in(0,\pi/2)$, by convexity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3633306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding irregular points of a homotopy between two parametrisations of the unit circle with varying speed The curve $d: [0, 2 \pi] \to \mathbb R$, $d(t) := \left(\cos\left(t \cdot e^{t - 2 \pi}\right), \sin\left(t \cdot e^{t - 2 \pi}\right)\right)$ is a parametrisation of the unit circle of non-constant velocity.
I sus... |
Thus we have to solve
$$
(c - 1) \sin(t) = c (t + 1) e^{t - 2 \pi} \sin\left(t \cdot e^{t - 2 \pi}\right)
\quad \text{and} \quad
(c - 1) \cos(t) = c (t + 1) e^{t - 2 \pi} \cos\left(t \cdot e^{t - 2 \pi}\right).
$$
It follows
$$(c - 1) \sin(t) c (t + 1) e^{t - 2 \pi} \cos\left(t \cdot e^{t - 2 \pi}\right)= (c - 1)... | {
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Show that $\int_{0}^{\infty}\frac{t^{\tau - 1}}{1+t}= \frac{\pi}{\sin(\pi \tau)}$, where $0I need to show that Mellin Transform of the function $\frac{1}{1+t}$ is $\frac{\pi}{\sin(\pi \tau)}$. So, by definition $(Mf)(\tau)=\int_{0}^{\infty}f(t)t^{\tau -1}dt=\int_{0}^{\infty}\frac{t^{\tau - 1}}{1+t}dt$. In my attemp, i ... | Hint: the simplest way to evaluate the integral is by contour integration. Use the cut along the positive real semi-axis and choose the contour running from $0$ to $+\infty $ along the upper bank of the cut, then along the full circle centered at $z=0$ and finally from $+\infty$ back to $0$ along the lower bank of the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there anything special about this summation I found that calculates out the square, cube, fourth, etc. power of any integer? let me start by saying that my formatting may be way off, but it's the best I can do, and has little to do with the question, and I will make sure I am as clear as humanly possible, including ... | You are implicitly using telescoping series with your sums giving the integer squares and cubes. For the first case, you have
$$\begin{equation}\begin{aligned}
& \sum_{n=0}^{x-1}(2n+1) \\
& = \sum_{n=0}^{x-1}((n^2 + 2n+1) - n^2) \\
& = \sum_{n=0}^{x-1}((n+1)^2 - n^2) \\
& = (1^2 - 0^2) + (2^2 - 1^2) + \ldots + ((x-1)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3634033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does axiom of choice not imply the set of real numbers is countable? The axiom of choice implies all sets can be well ordered. If that is true, you can well order the set of real numbers and the set of the integers. Now, why can one not just pair the set of real numbers off as follows: start at the smallest integer... | To understand the issue better consider $\mathbb N^2$ with the dictionary order.
That means that $(a,b) \leq (c,d)$ if and only if either $a <c$ or $a=c$ and $b \leq d$.
Then $(\mathbb N^2 , \leq)$ is a well ordered set.
Now, let us run through your argument, with this set instead of $\mathbb R$.
You pair the numbers t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rotation Matrix following Right-Hand Rule. In $\Bbb{R}^3$, let $L$ be the line spanned and oriented by $v=(−4,6,−7)$.
Let $R$ the rotation of $\Bbb{R}^3$ through the angle $\frac{\pi}{2}$ about the $v$ oriented line $L$ according to the Right-Hand Rule.
I know what a rotation matrix in $2$ dimensions look like in terms... | Note that $\alpha$ is counter clockwise for below matrices.
$R_x(\alpha)=\begin{pmatrix} 1&0&0 \\ 0&cos(\alpha)&-sin(\alpha) \\ 0&sin(\alpha)&cos(\alpha) \end{pmatrix}$.
Similarly,
$R_y(\alpha)=\begin{pmatrix} cos(\alpha)&0&sin(\alpha) \\ 0&1&0 \\ -sin(\alpha)&0&cos(\alpha) \end{pmatrix}$.
and
$R_z(\alpha)=\begin{pmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3634254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Subharmonic functions and monotonically increasing integrals Let $U \subset \mathbb{C}$ be open and $f : U \to \mathbb{R}\cup\{ - \infty\}$ be a subharmonic function. Define $\varphi(r) := \frac{1}{2\pi} \int_{0}^{2\pi} f(a + re^{i\theta}) \ d\theta$. Show that $\varphi$ is monotonically increaing by first showing that... | Since, $f$ is u.s.c it is infimum of a monotone decreasing sequence $\{g_k\}_{k \in \mathbb{N}}$ of continuous functions, that is $g_k \searrow f$ monotonically. Then we kave $f \le g_k$ on $\partial D_{r_2}(a)$ and consider the unique harmonic function $h$ in $D_{r_2}(a)$ s.t., $h\rvert_{\partial D_{r_2}(a)} = g_k$. T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3634422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is the minimum of this optimization problem essentially unique? Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s>0$ be a parameter, and define $
F(s)=\min_{xy=s,x,y>0} g(x,y)... | Yes, it's possible. Define $$h(x)=\begin{cases}
(x-1)^2 & x\in (0,2] \\
2-(\frac{4}{x}-1)^2 & x\in [2+\epsilon,3]
\end{cases}
$$
for some small $\epsilon$. We'll deal with the values on other domains in a bit, but let's first see what this gets us. For $s=4$, we have $g(2,2)=2$ and $g(x,y) = h(x)+h(4... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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find $\lim_{x \to 1} \frac{2x -\sqrt{x^2 +3}}{\sqrt{x+3} \ -\sqrt{2x+2}}$ without l'hospital rule EDITED VERSION
find $$\lim_{x \to 1} \frac{2x -\sqrt{x^2 +3}}{\sqrt{x+3}
\ -\sqrt{2x+2}}$$ without l'hospital rule.
using l'hospital rule, you'll have: $\lim_{x \to 1} \frac{2x -\sqrt{x^2 +3}}{\sqrt{x+3}
\ -\sqrt{... | I will assume that you meant$$\lim_{x\to1}\frac{2x-\sqrt{x^2+3}}{\sqrt{x+3}-\sqrt{2x+2}}.$$Multiplying the numerator and the denominator by $\sqrt{x+3}+\sqrt{2x+2}$, this becomes$$\lim_{x\to1}\frac{\left(2x-\sqrt{x^2+3}\right)\left(\sqrt{x+3}+\sqrt{2x+2}\right)}{1-x}.$$So, define $f(x)$ as $2x-\sqrt{x^2+3}$ and then th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\int_0^1|x-\mu|f(x)dx\le \frac{1}{2}, \text { where } \mu=\int_0^1xf(x)dx.$
Question: Let $f:[0,1]\to(0,\infty)$ be a function satisfying $$\int_0^1f(x)dx=1.$$ Show that the integral $$\int_0^1(x-a)^2f(x)dx\text{ is minimized when } a=\int_0^1xf(x)dx.$$ Hence or otherwise show that $$\int_0^1|x-\mu|f(x)dx\l... | By Cauchy-Schwarz inequality we have $$\left|\int_0^1|x-\mu|f(x)dx\right|=\int_0^1|x-\mu|f(x)dx=\int_0^1\left(|x-\mu|\sqrt{f(x)}\right)\left(\sqrt{f(x)}\right)dx\\\le \sqrt{\int_0^1|x-\mu|^2f(x)dx \int_0^1f(x)dx}\\= \sqrt{\int_0^1|x-\mu|^2f(x)dx}\\=\sqrt{\int_0^1(x-\mu)^2f(x)dx}\\=\sqrt{\int_0^1x^2f(x)dx-\mu^2}.$$
Now ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Alternative definition of Dirac function Can someone explain why the formula below is a definition of the Dirac function?
$$\int_I y(0) \delta(t)\, dt =\begin{cases}
y(0) \text{ if 0}\in \text{I} \\\\
0 \text{ otherwise}
\end{cases}$$
where $I$ is an interval.
I am used to the definition $\int \delta(t) \,dt = 1$ and $... | The definition you give ($\int \delta(t) dt = 1$ and $\delta(t) = \infty$ if $t = 0$, otherwise $0$) is more of a characterization its behavior, rather than a true definition.
Rewrite your expression as:
$$ \int_\mathbb{R} y(0) 1_I \delta(t)\, dt$$
where $1_I$ is the indicator function for I.
This integrand is a repre... | {
"language": "en",
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General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$. QUESTION: General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$. Then the quotient $G/H_G$ is isomorphic to a subgroup of $S_n$.
ANSWER: In the answer I found, there is this isomorphism $\pi... | Since $H \leq G$ is a subgroup of index $[G:H]=n$, the set $G/H$ has size $n$ and you can identify $\text{Sym}(G/H)$ with $S_n$.
Now $G$ acts on the set $G/H$ by left multiplication. This gives a homomorphism $\phi: G \to \text{Sym}(G/H)$ where $\phi(g): yH \mapsto gyH$. The kernel of this map is $\ker \phi = \{g \in G... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Estimate the probability that N random 4-digit pin numbers are all distinct for N=10, 1000, 100 As per title, the question is "estimate the probability that N random 4-digit pin numbers are all distinct for N=10, 1000, 100".
My current working is as follows:
There are $10^{4}$ total 0-10 4 digit combinations. For N pin... | Yes, that's correct so far. So now the difficulty is estimating the numerator, namely the three products $10000 \cdots 9991$, $10000 \cdots 9901$, $10000 \cdots 9001$. In other words, you're looking for $10000!/9990!, 10000!/9900!, 10000!/9000!$.
Factorials can be approximated using some variant of Stirling's approxima... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Hall's Marriage Theorem $2n$ Rooks on $2n \times 2n$ board I saw this question on https://brilliant.org/wiki/applications-of-hall-marriage-theorem/. I can't work it out, any help would be appreciated!
In a $2n \times 2n$ chess board, there are $n$ rooks in each row and each column of the board. Show that there exist $... | Consider the bipartite graph in which the first set of vertices is the set of rows, the second set of vertices is the set of columns, and there is an edge between a row and a column if and only if the square corresponding to that row and column has a rook in it.
We must verify that this graph satisphies the Hall condit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the following matrix that represents the relation on a set transitive in which would make it a partial order? I understand that this is both reflexive and anti-symmetric. However, I don't think it's transitive. I understand that the classic rule is aRb, bRc, aRc. I can't seem to piece that together with this matr... | Hint:
If we let your matrix be $R$, find $R^2$. If the nonzero entries in $R^2$ are in the same positions as those in $R$, then $R$ represents a transitive relation.
| {
"language": "en",
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Condition for triviality of group morphism $\tau : \mathbb{Z}_p \to\mathrm{Aut}(\mathbb{Z}_q)$ : $\tau_\overline{k}(\overline{n}) = \bar{r}^k \bar{n}$ I'm trying to show the following:
Let $p,q$ be prime numbers, $\tau : \mathbb{Z}_p \to \operatorname{Aut}(\mathbb{Z}_q)$ be a group morphism such that $$\overline{k} \... | If $\tau_{\bar k}\bar n=\bar n,\forall\bar n\in\Bbb Z_q$, then $r^k\cong1\pmod q,\forall k\implies r\cong1\pmod q$.
Conversely, if $r=\bar1\in\Bbb Z_q$, then $\tau_{\bar k}\bar n=\bar r^k\bar n=\bar1^k\cdot\bar n=\bar n$ is trivial.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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how prove $\varepsilon_i \in \left \{ -1,1 \right \}$ such $\left | \sum_{i=1}^{n} \varepsilon_i z_i \right | \leqslant 1.$ Let $n\geqslant 3$ be an odd number, and $z_1,z_2,...,z_n$ be complex numbers such that $|z_i|=1$ for all $i$.
Prove that there exist $\varepsilon_i \in \left \{ -1,1 \right \}$ satisfying $$\left... | Note that the convex polygon generated by $\mathrm{C} = \text{conv}\{\pm z_j: j = 1, 2, \cdots, 2k+1\}$ lies in the closed unit disk $\overline{\mathbb{D}}$. We are trying to find a vector $\overline{\varepsilon} \in \{1,-1\}^{2k+1}$ s.t., $\displaystyle J(\overline{\varepsilon}) = \left|\sum\limits_{j=1}^{2k+1} \varep... | {
"language": "en",
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Determine whether $(0,1)$ and $[0,1]$ are homeomorphic or not I'm studying Gemignani's Elementary Topology. I need to determine that the open interval $(0,1)$ and the closed interval $[0,1]$ are homeomorphic or not when considered as subspaces of the real numbers with the absolute value topology.
My claim is that they'... | A property that $[0,1]$ has that $(0,1)$ has not not, is compactness, indeed. It's not one you use in your argument, though.
$(0,1)$ has the property "every point of $X$ is a cutpoint" (where a cutpoint (of a connected space) is a point $x$ such that $X\setminus \{x\}$ is not connected).
$[0,1]$ does not have this prop... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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why $\int_0^1f= \lim_{n\to+\infty}\sum_{k=2}^n\int_{\frac{1}{k}}^{\frac{1}{k-1}}f$ I'm looking for an explanation for why this equality is true assuming f is integrable. obviously it is true for a finite amount of integrals covering exactly [a,b], but how do we know it is also true for the infinite case?
My intuition i... | Let us assume that the antiderivative of $f$ is $F$.
By the Fundamental Theorem of Calculus,
$$\therefore \int_{\frac{1}{k}}^{\frac{1}{k-1}}f(x)dx =F(\frac{1}{k-1})-F(\frac{1}{k})$$
$$\therefore \sum_{k=2}^{n}\int_{\frac{1}{k}}^{\frac{1}{k-1}}f(x)dx=\sum_{k=2}^{n}F(\frac{1}{k-1})-F(\frac{1}{k})=F(1)-F(\frac{1}{n-1})$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3636558",
"timestamp": "2023-03-29T00:00:00",
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Prove that A - I is nonsingular If matrix $A^3 = O$, then $A - I$ is nonsingular? True or false?
I tried to solve it like this:
Given a linear transformation $T: V \rightarrow V$, such that $$m(T) = A, m(O) = O, m(I) = I, m(T^3) = A^3$$
where $m(.)$ is an operator that converts a linear transformation into the matrix.... | Hint:
If $A^3=0$, what is $(A-I)(A^2+A+I)$?
| {
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Existence of a maximal subgroup I'm looking for a "fake" proof of the existence of a maximal subgroup using Zorn's lemma. Do either of you know where I might find this?
| Here’s a trivial and obvious gappy proof:
Let $G$ be a nontrivial group. Let $P$ be the collection of all proper subgroups of $G$, partially ordered by inclusion. It is nonempty, since $\{e\}\in P$. If $\mathcal{C}$ is a nonempty chain in $P$, then let $K=\cup_{H\in\mathcal{C}}H$. Then $K$ is a subgroup: there exists $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the continuity of Fourier Transform between Schwartz spaces via sequences Prove that if $f_k \rightarrow f$ in the Schwartz space $\mathcal{S}(\mathbb{R}^n)$, then $\hat{f_k} \rightarrow \hat{f}$ in $\mathcal{S}(\mathbb{R}^n)$.
This is the Exercise 2.2.2 in Loukas Grafakos's book named Classical Fourier Analys... | $\hat{g} = x^\alpha \partial^\beta(\hat{f})(x)$ $$ \sup |\hat{g}(x)|\le \|g\|_{L^1}\le \pi\|(1+x^2)g\|_{\infty} \le \pi\|g\|_{\infty}+\pi\|x^2g\|_{\infty}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Given FOL=Turing machines, why is SOL different than FOL? [1] Every SOL (second order logic axiom system) has a corresponding Turing machine that verifies SOL statements, given a proof and axioms. (If this weren't the case, how could we be sure that our SOL proofs are correct?)
[2] FOL = Turing machines
[3] From [1] an... | Short answer: Second order logic with full semantics is not recursively axiomatizable. In other words, a given recursively axiomatized second order proof system (one whose proofs a Turing machine could check) always falls short of capturing all the valid second order logical consequences. That's why SOL is essentially ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove constructive dilemma without using additional assumptions
Prove that if $p,q,r$ are propositions, then the following rule of inference holds: $$\begin{array}{l}p\to q\\r\to s\\p\lor r\\\hline q\lor s\end{array}$$ Note 1. Prove it not using additional assumptions, such as $p\quad\text{Assumption}$.
Note 2. You mu... | Rewrite the $p \lor r$ as $\neg \neg p \lor r$, which can then be rewritten as $\neg p \to r$
Also, $p \to q$ can be rewritten as $\neg q \to \neg p$
And now it is just a bunch of Hypothetical Syllogisms and you're pretty much there.
Formally:
$$
\begin{array}{lll}
1)&p\to q&\text{Premise}\\
2)&r\to s&\text{Premise}\\
... | {
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Let $p$ be prime. What is the biggest number $m$ such that $\{1,\zeta_p,\zeta_p^2,...,\zeta_p^m\}$ is linearly independent over $\mathbb{Q}$? Let $p$ be prime. What is the biggest number $m$ such that $\{1,\zeta_p,\zeta_p^2,...,\zeta_p^m\}$ is linearly independent over $\mathbb{Q}$? Here, $\zeta_p$ is a primitive $p$th... | The set $\{1,\zeta_p,\zeta_p^2,\dots,\zeta^{p-2}\}$ is a basis for $\Bbb Q(\zeta_p)$ over $\Bbb Q$.
On the other hand, $1+\zeta_p+\dots\zeta_p^{p-1}=0$, so $\{1,\zeta_p,\dots,\zeta_p^{p-1}\}$ is dependent.
| {
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To find the range of $f(x)=\frac {x^2-1}{x^2+3x+2}$, the "discriminant method" doesn't work—why and how can we fix it? Define $f: \mathbb R \setminus \{-1,-2\} \rightarrow \mathbb R $ by $$f(x)=\frac {x^2-1}{x^2+3x+2}$$
To find the range of $f$, we use the "discriminant method" (used in e.g. 1, 2, 3):
*
*Write $y=\f... | You multiplied with $$x^2+3x+2$$ which is $\ 0\ $ for $\ x=-2\ $ and $\ x=-1\ $.
| {
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Why are natural numbers defined with union in Zermelo–Fraenkel (ZF) set theory? In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined as $0 = \emptyset$ and $n + 1 = n \cup \{n\}$
$$\begin{alignat}{2}
0 & {} = \{\} && {} = \emptyset,\\
1 & {} = \{0\} && {} = \{\emptyset\},\\
2 & {} = \{0,1\} && ... | Zermelo's original definition of the natural numbers was actually your "intuitive" definition – it's not a new idea. The standard definition (the first one) is due to von Neumann, and is used because it makes defining order on the natural numbers much easier using the subset relation.
For example, $3\le5$ because the v... | {
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Linear program solutions with three decision variables Suppose we have a linear program which has exactly three non-negative decision variables
x1, x2, x3 and exactly three functional constraints, each containing a single variable: xi ≤ 1,
i ∈ {1, 2, 3}.
How do we find the number of basic and feasible solutions exactly... | At most one of $x_i\ge0$ and $x_i\le1$ can be tight for each $i$. To get a basic solution, with three tight constraints, we have to choose for each variable which of the two constraints is tight – exactly one for each variable. This gives us $2^3=8$ basic solutions. They may all be easily checked to be feasible and non... | {
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Borell-Cantelli lemma If $\sum_{n} P(|X_{n}|>n)<\infty$, then prove that the $\limsup_{n}$ $|X_{n}|/n \leq 1$ a.s.
My approach
Let $E_{n}=|X_{n}>n|.$
$\sum_{n}P(E_{n})<\infty$ implies $P(E_{n} \text{ i.o})=0$
which further implies $P(\limsup_{n} E_{n})=0$.
Can anyone throw some light on how to approach further!
| Your choice of $E_n$ should be $E_n=(|X_n|>n)$. Since $P(E_{n}\, \text{i.o})=0$, it follows that with probability one $|X_n|/n\leq 1$ eventually whence $\limsup |X_n|/n\leq 1$.
| {
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Counting problem (combination). How many ways to create 2 indistinguishable piles from 21 identical elements. You have 21 identical elements, let's say books. How many ways to place the books into 2 indistinguishable piles? (This means that putting 16 books in the first pile and 5 books in the second is the same as 5 b... | Lets say there are $F$ books in the first pile and $S$ books in the second pile
$$F+S=21$$
$$F=21-S$$
Since $F,S \geq 1$,
for each $S,1\leq S\leq 20 $ there is a corresponding $F$ pile.
This gives $20$ solutions.
But each case was overcounted twice.
Hence there are a total of $10$ ways
| {
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"timestamp": "2023-03-29T00:00:00",
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Why does TTH occur before THT more often in a sequence of coin flips? Premise: Flip a coin $n$ times, and record the flips in a string (e.g. $THTTHTTTTHHHTHTTTH$). Then, check if $TTH$ occurs before $THT$. In the aforementioned case, it does not.
I performed this with increasing $n$, from $4\to100$, each $10000$ times,... | This is easiest to see via states. There are only $4$ relevant states (other than the end states). We'll label these by how much of either $TTH$ or $THT$ you have in the current string. Thus the states are $\emptyset, T, TT, TH$. If $S$ is a state we'll denote by $P_S$ the probability that $TTH$ will come first giv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating the product $\prod_{p}\, \left(1- \dfrac{2}{p^s}\right)^{-1}\,\left(1 - \dfrac{1}{p^s} \right)^2\,$ I need to evaluate $$ f(s) = \prod_{p}\, \left(1- \dfrac{2}{p^s}\right)^{-1}\,\left(1 - \dfrac{1}{p^s} \right)^2\,.$$ This arises from $$\sum_{n=1}^{\infty} \dfrac{2^{\Omega(n)}}{n^s} = \zeta^2 (s) \, \prod_p\... | What do you mean with "evaluate" ? $$\log f(s) = \sum_{p^k} \frac{2^k - 2 }k p^{-sk}$$ converges and is analytic for $\Re(s) > 1/2$.
$f(s)$ has a simple pole at $s=1/2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find 'r' in the NCR equation with n and answer known The value of $n = 105$. $\binom nr = 187460$. How do i find the value of r?
| $\binom{n}{k}$ is close to $\frac{n^k}{k!}$ when $n$ is considerably larger than $k$.
So it goes like $100,100^2/2,100^3/6\dots100^4/24$
The number you want is close to the third one and far from the other.
Also note that $\binom{n}{k}$ increases as $k$ gets close to $n/2$
| {
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Degree of determinants Consider the below determinant $\triangle$:
$$ \triangle = \begin{vmatrix}
a^3&a^2&a&1\\
b^3&b^2&b&1\\
c^3&c^2&c&1\\
d^3&d^2&d&1\\
\end{vmatrix} $$
"$\triangle$ is of the sixth degree in a,b,c and d" - This is the statement given about the above determinant.
What does "degree" of a determinant m... | It is its degree as a polynomial in the a,b,c and d variables. In order to count this degree easier, you can look at
$$ \triangle (x)= \begin{vmatrix}
(ax)^3&(ax)^2&ax&1\\
(bx)^3&(bx)^2&bx&1\\
(cx)^3&(cx)^2&cx&1\\
(dx)^3&(dx)^2&dx&1\\
\end{vmatrix} $$
Now you can think about polynomials only in the variable x rather t... | {
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If a set of sentences $\Delta$ is satisfiable and $\Delta \subseteq \Gamma$, then $\Gamma$ must also be satisfiable. Show that this does not hold. Let $\Delta$ and $\Gamma$ be sets of sentences. Definition of satisfiable: If a set of sentences is satisfiable, then it has model.
If a set of sentences $\Delta$ is satisfi... | Assume $\Delta$ is satisfiable. Choose any sentence $\psi \in \Delta$, and let $\Gamma = \Delta \cup \{ \lnot \psi \}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3639642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Two overlapping circles with tangents drawn at their intersection points intersecting at each others' centres.
So I'm stumped by what should be a rather simple problem. There are two circles whose tangents intersect at each others' centres. The tangents are at right angle. If I know the distance between the centres, t... | Let the centers of the circles be $O_1$ and $O_2$. Draw a circle with the diameter $O_1O_2$. Take an arbitrary point $C$ on the circle. Draw through $C$ two circles centered at $O_1$ and $O_2$. The circles will be orthogonal (i.e. correspond to the requirement that the tangent of one circle drawn at intersection point ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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limit $\lim_{x\rightarrow 0^{+}}\frac{1}{x^2}\cdot \int^{x}_{\sin(x)}\frac{1}{\sqrt{1+\sin(u)}}du$
Evaluation of $$\lim_{x\rightarrow 0^{+}}\frac{1}{x^2}\cdot \int^{x}_{\sin(x)}\frac{1}{\sqrt{1+\sin(u)}}du$$
What I try
Applying D L'Hospital's Rule,
$$\lim_{x\rightarrow 0^+}\frac{1}{x}\bigg(\frac{1}{\sqrt{1+\sin x}}-\... | If you know Taylor's formulas / $O$ notation, this is as usual faster and less painful than L'Hospital. Informally, since you are looking at $x$ small, the integrand is essentially 1, so the limit is the same as
$$
\frac{x - \sin x}{x^2},
$$
and as $x - \sin x = O(x^3)$, then this tends to $0$. To be slightly more prec... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's the Taylor series for $f(x+h,y+k)$? My course notes give the Taylor series up to degree 2 as
$$f(x+h,y+k)\approx f(x,y)+\frac{\partial f}{\partial x}h+\frac{\partial f}{\partial y}k+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}h^2+\frac{\partial^2 f}{\partial x \partial y}hk+\frac{1}{2}\frac{\partial^2 f}{\partia... | If we denote $x^1 := x, x^2 := y$ and $h^1 := h, h^2 := k$, then the terms of total degree $i$ are those of the sum
$$\frac{1}{i!} \sum_{k_i = 1}^2 \cdots \sum_{k_1 = 1}^2 \frac{\partial^i f}{\partial x^{k_1} \cdots \partial x^{k_i}}(x^1_0, x^2_0) h^{k_1} \cdots h^{k_i} .$$
Expanding this in the case $i = 2$ indeed giv... | {
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Lower bound for The chromatic index of the complete graph of order n, where n is odd. How can I prove that the chromatic index i of the complete graph of order n, where n is odd, is i > n-1.
I found here a construction with n colors, but am having a hard time to prove lower bound.
| Hint Assume by contradiction that you can color $K_n$ with $n-1$ colors. Then, at each vertex each color appears exactly once.
Now, pick one color, and draw the subgraph of all edges of this fixed color. This is a subgraph on $n$ vertices, where each vertex has degree 1. But this is not possible by handshaking lemma, s... | {
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Is this a property of the sine function? Because of the equidistribution property of $\{n\mod{2\pi}\}_{n\in\mathbb{Z}}$ in $[0,2\pi]$, my intuition tells me that the following statement about the sine function must be true.
We can find an absolute constant $C>0$ such that for all $\epsilon>0$ and all $n\in\mathbb{Z}$... | Your question is equivalent to $\{n \mod 2\pi\}$ equidistributing quickly.
Writing $n' = n+k$, we want some $|k| \le C\epsilon^{-1}$ with $\epsilon > \sin(n') = \sin(n+k) = \sin(2\pi(\frac{n+k}{2\pi}))$ $= \sin\left(2\pi(\lfloor \frac{n}{2\pi}\rfloor+\lfloor\frac{k}{2\pi}\rfloor+\{\frac{n}{2\pi}\}+\{\frac{k}{2\pi}\})\r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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An empty set is always a subset of another set, but can it also be an element of a set too? I know an empty set ∅ is defined as a set that has no elements and is also a subset of every set, but can you say that it is an element of another set? I think not, but I'm not comepletly sure.
For example, consider the statemen... | Actually yes, any set can be put as element of other sets.
This is ensured for example by the axiom of pairing in the axioms of the ZF set theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms.
You can for example have $Z=\{\emptyset\}$, and then $\emptyset\in Z$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find an ellipse tangent to triangle given centre position of the ellipse I have a fully defined triangle, $PT_1T_2$, of side lengths $a$, $b$, & $c$ with corresponding opposite angles $A$, $B$, & $C$. I want to find an ellipse that is tangent to the lines $PT_1$ and $PT_2$ at points $T_1$ and $T_2$ respectively.
As the... | If tangents at $T_1$ and $T_2$ meet at $P$, and $M$ is the midpoint of $T_1T_2$, then line $PM$ passes through the centre $O$ of the ellipse.
Hence in our case centre $O$ must lie on the given line $PM$.
The ellipse is determined when five of its points are known, but once $O$ is fixed symmetric points $T_1'$ and $T_2'... | {
"language": "en",
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Showing a subspace is not closed I have been asked the following problem which I have so far failed to solve rigourosly.
Let $M=\{ (a_n)_{n=1}^\infty\in l^2\vert \sum_{n=1}^\infty \frac{a_n}{\sqrt{n}}=0 \}$. Determine whether $M$ is a closed subspace of $l^2$.
My attempt:
I think that the answer is no because this seem... | For an explicit example, consider the sequence $a^{(n)}$ of $\ell^2$ defined as follows.
*
*For $k \in \{2, \dots, n-1 \}$, $a^{(n)}_k = \frac{1}{\sqrt{k}\log k}$.
*For $k \in \{n, \dots, r_n\}$, $a^{(n)}_k = - \frac{x_n}{\sqrt{k}\log k}$.
*Otherwise, $a^{(n)}_k = 0$.
We denote here $r_n = \lfloor e^{(\log n)^2}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Bounded probability implies convergence in probability Let $(X_n)$ be a sequence of random variables and $(a_n),(b_n)$ be two sequences of non-negative real numbers such that $a_n\downarrow 0$ and $b_n\downarrow 0$ when $n\to\infty$.
If for any $t>0$,
$$
P(|X_n|\geq a_n+t)\leq b_n,
$$
can we conclude that $X_n\overset... | Fix $s > 0$. We want to show that $\mathbb{P}(|X_n| \geq s) \to 0$ so fix $\varepsilon > 0$ and we aim to show that for large enough $n$, $\mathbb{P}(|X_n| \geq s) < \varepsilon$.
By the assumption applied with $t = \frac{s}{2}$, $$\mathbb{P}(|X_n| \geq a_n + \frac{s}{2}) \to 0$$
Combining this fact with the assumptio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does the topology of a space describe the closeness of the open subsets of a given set $X$? I've been trying to learn about topology recently and there is a thing that I couldn't understand. I know that given the topological space $(X,\tau)$, the $\tau$ contains open subsets of $X$. To my understanding, the $\tau$ ... | *
*A set is closed if its complement with respect to X is open. I.e. {},{} and {,} are all open but not closed wrt $\tau_1$.
*I dont get what you mean by difference between $\tau_1$ and $\tau_2$ those are just families of subsets of X.
*As noted in 1. any set whose complement wrt X is open is called closed. You can ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find improper integral Find improper integral $\int_{0}^{1}-\frac{\log(x)}{x^2}\cdot (xe)^\frac{1}{x}dx$.
I used the following substitution $t=\frac{1}{x}$,
$x=\frac{1}{t}\mspace{10mu},dx=-x^2dt,\mspace{10mu} \log(x)=-\log(t)$.
$\int_{0}^{1}-\frac{\log(x)}{x^2}\cdot (xe)^\frac{1}{x}dx=\int_{1}^{\infty}\log(t)(\frac{e... | Put $x=e^z$ and $dx=e^zdz$. Then
$$
-\int _{0}^{1} \frac{\log(x)}{x^2} (xe)^{x^{-1}}\,dx =
$$
$$
=- \int _{-\infty}^{0} e^{e^{-z} (z+1)-z} z\,dz
$$
$$
= \left.e^{e^{-z} (z+1)}\right|_{-\infty}^{0} = e-0=e
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Properties lost when going from real number system to quaternions and octonions What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?
| From reals to complex you lose order. From complex to quaternions you lose commutativity. From quaternions to octonions you lose associativity. From octonions to ..?
I had written this as a comment, then I followed Noah Schweber's link, which essentially says this plus more. And pregunton's answer is meaty.
Wha... | {
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Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite. Problem:For fixed positive integers $s, t$, define $a_n$ as the following.
$a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$.
Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is ... | Let $n > 20080$ (large enough).
Let $ b_n = a_n - n$.
We want to show that for some $N$, for any $ n > N$, $ b_n = 0 $.
Claim: $b_{n+2} \leq \frac{ b_{n+1} + \frac{2b_{n+1} + b_n + 2010 } { 20080 }} {2} $
Proof: $ (a_{n+2})^2 \leq n + b_n + (n+2)(n+ 1+b_{n+1}) + 2008 $
$= n^2 + n (4 + b_{n+1} ) + 2b_{n+1} + ... | {
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"timestamp": "2023-03-29T00:00:00",
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Cardinal of a set of functions between two finite sets of integers
Let $ A= \{a_1, a_2, a_3,\cdots, a_{10}\}$, $B=\{ 1,2 \}$. Find the number of functions $ f: A \to B $ such that $ f(a_1) +f(a_2)+\cdots+f(a_{10})$ is an even number.
I have tried to find the number of functions $ 2^{10} $ But not getting any clue to ... | Hint
Intuitively, the number of such maps is $2^{10}/2$: there is the same number of maps such that the sum is even, vs. the maps for which the sum is odd.
To prove it, prove that there is a one to one correspondence between the maps such that $f(a_1)=1$, vs the ones for which $f(a_1)=2$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Surjective homomorphism from indefinite orthogonal group $O(p,q)$ to ${\{-1,1}\}^2$ For each $A\in O(p,q)$, we may write
$$A=\begin{bmatrix}A_{11} &A_{12}\\A_{21}&A_{22}\end{bmatrix}.$$
Then it is said that $\det(A_{11})\neq 0$ and $\det(A_{22})\neq 0$.
Furthermore, the map
$$O(p,q)\rightarrow{\{-1,1}\}^2,\quad A\m... | A matrix $M$ lies in $O(p,q)$ if, by definition, $M^t D M=D$, where $D=\pmatrix{ I_p & 0 \cr 0 & -I_q}.$
Assume further that $M$ is block diagonal, so $A_{12}=A_{21}=0$.Then $M$ lies in $O(p,q)$ if and only if $A_{11}^tA_{11}=I_p$ and $A_{22}^tA_{22}=I_q$, that is $A_{11}\in O(p)$ and $A_{22}\in O(q)$.
But it is know... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving the equivalence of the Criss - Cross theorem statement(ordinary one) to another statement. Here is the Criss - Cross theorem statement:
If $\alpha, \beta : I \rightarrow I \times I$ are paths such that $\alpha(0) = (0,0),\alpha(1) = (1,1),$ and $\beta(0) = (0,1),\beta(1) = (1,0).$ Then there are $a,b \in I $ su... | $\implies$: since $A$ is path connected and $(0,0),\, (1,1)\in A$, there's a path $\alpha:I\to A\subseteq I\times I$ connecting them. Similarly, there's a $\beta:I\to B\subseteq I\times I$ connecting $(0,1)$ and $(1,0)$.
Then just apply the criss-cross theorem.
$\impliedby$: for given paths $\alpha$ and $\beta$, consid... | {
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How does a function conveex on an open interval has both left and right hand derivatives at every point of that interval? Let $(a, b)$ be an open interval on the real line, let $f \colon (a, b) \rightarrow \mathbb{R}$ be a convex function, and let $c \in (a, b)$. Then how to show rigorously that the right- and left-han... | Making appropriate coordinate changes if necessary,we may assume,without loss of generality that we are checking left and right differentiability at $0$,with $f(0)=0$.
I shall demonstrate a geometric argument here.
See, a convex function is one for which given any two points $x$ and $y$ in the domain, the graph of the ... | {
"language": "en",
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How to Solve Quadratic Matrix Programming with Matrix Inequality Constraint? Let $\mathbf{B} \in \mathbb{R}^{M \times M}$, $\mathbf{X} \in \mathbb{R}^{N \times M}$, and $\mathbf{A} \in \mathbb{R}^{M \times N}$. We intend to solve for $\mathbf{X}$ by solving the following optimization problem
\begin{align}
\arg \min_{\m... | In case $ B $ is a Positive Definite Matrix then there is $ {C}^{T} C = B $ by the Cholesky Decomposition.
So the problem can be rewritten as:
$$\begin{aligned}
\arg \min_{X} \quad & \frac{1}{2} {\left\| A X C \right\|}_{F}^{2} - \operatorname{Tr} \left( D X \right) \\
\text{subject to} \quad & L \leq X \leq U \quad \t... | {
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Find CDF of $U^2$ where $U\sim{}\text{Unif}(-1,1)$. How am I misapplying Universality of the Uniform? Note: This is the same question posted here, but I am seeking clarification on how my attempt is incorrect (i.e., where I went wrong).
If we let $S\sim{}\text{Unif}(0,1)$, then $U = 2S-1$ and $U^2 = (2S-1)^2$ by locat... | You are misusing this property of the uniform distribution, which w.r.t. the present problem reads
*
*if $F$ is the CDF of some continuous random variable $X=U^2$, then $F(X) = S$ satisfies $S\sim\text{Unif}(0,1)$.
The key point is that the function $F$ must be the CDF of $X=U^2$. In particular, it must satisfy th... | {
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If $f : \mathbb R \rightarrow \mathbb R $ such that $f(x^2+x)+2f(x^2-3x+2) = 9x^2-15x$. Find $f(2016)$.
Determine all $f : \mathbb R \rightarrow \mathbb R $ such that $$f(x^2+x)+2f(x^2-3x+2) = 9x^2-15x$$ for all $x$. Find $f(2016)$.
Similar problem appeared on this site before: $f(x^2 + x)+2f(x^2 - 3x + 2)=9x^2 - 15x... | First, we solve $x^2 + x = 2016$ and (separately) $x^2 - 3x + 2 = 2016$ and write down the solutions. Then observe that, luckily,
When $x = \dfrac{-1 - \sqrt{8065}}{2}$:
*
*$x^2 + x = 2016$
*$x^2 - 3x + 2 = 2020 + 2\sqrt{8065} = a$ (say)
*$9x^2 - 15x = 18156 + 12\sqrt{8065}$
$$f(2016) + 2f(a) = 18156 + 12\sqrt{8... | {
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"timestamp": "2023-03-29T00:00:00",
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What is a relation in plain simple English? Can someone explain in simple English what a relation is? I found this definition: A relation between two sets is a collection of ordered pairs containing one object from each set. If the object x
x
is from the first set and the object y
y
is from the second set, then the o... | Just to get an informal intuition, a relation between two sets is a way to extract pairs from them. For example, $\textit{being married}$ is a relation between the set of men and the set of women.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3644468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\mathbb{Z}[i]/(3)$ and $\mathbb{Z}[i]/(7)$ are finite fields and find their cardinality In my rings subject's test I had to prove that $\mathbb{Z}[i]/(21)$ was decomposed as a product of two finite fields, and that was easy to prove for me because $21 = 3\cdot 7$, and $\mathbb{Z}[i]$ is a PID and $3$ and $7... | There is a short way to see it:
$$\mathbb{Z}[i]/(3) \cong \mathbb{Z}[X]/(X^2+1,3) \cong (\mathbb{Z}[X]/(3)) /(X^2+1) \cong \mathbb{F}_3 / (X^2+1)$$
Where $\mathbb{F}_3$ is the field with three elements.
As one can see easily, this is just $\mathbb{F}_3 [i]$, a field with 9 elements. The other case with 7 works out simi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3644650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
$\int_{0}^{\infty}(-1)^{[x^2]}$ converge? I need to tell if $\int_{0}^{\infty}(-1)^{[x^2]}dx$ converge, diverge or absolutely converges.
I managed to say it does not absolutely converges.
for it's converges, diverges i tried substituting $t=x^2$ so i get $\int_{0}^{\infty}\frac{(-1)^{[t]}}{2\sqrt{t}}dt$ but I still can... | Hint
$$\int_{\sqrt{0.5}}^{\sqrt{0.5+M}}(-1)^{[x^2]}dx=\sum_{n=1}^M\int_{\sqrt{n-0.5}}^{\sqrt{n+0.5}}(-1)^{[x^2]}dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3644770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Probability - Describing mathematical events in words Consider a target that consists of $5$ concentric circles with radii $1, 2, 3, 4, 5$ respectively. An event $A_j$, $j \in \{1,..,5\}$ describes a shot landing in the circle with the radius $j$. Describe the following events in words:
*
*$\bigcap\limits_{k=1}^{... | Your answer to 1 is correct. If you added in the other two circles it wouldn't change anything because $A_1$ is a subset of all the others. Any intersection that includes $A_1$ will just be $A_1$ in this case.
Your answer to $2$ is correct but may be marked down. As in 1, you are expected to realize that $A_4$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3644881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove that for every $n \in \mathbb{N}$ $\sum\limits_{k=2}^{n}{\frac{1}{k^2}}<1$ $$\sum\limits_{k=2}^{n}{\frac{1}{k^2}}<1$$
*
*First step would be proving that the statement is true for n=2
On the LHS for $n=2$ we would have $\frac{1}{4}$ therefore the statement is true for $n=2$
*Now we must assume the stateme... | $f(x)= \frac{1}{x^2}$, $x >0$, stricly decreasing.
$ \displaystyle{\sum_{k=2}^{n}}\frac{1}{k^2} <\displaystyle{ \int_{1}^{n}}\frac{1}{x^2}dx =-\frac{1}{x}\big ]_1^n=$
$1-1/n. $
As suggested by Calvin you can use the above inequality for the induction proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Proof of explicit formula for recursive sequence by induction I am new to proof-writing and have just started working on Math for Computer Science on MIT OCW. I encountered the following problem on one of the assignments and would like feedback on my proof.
Let the sequence $G_{0},G_{1},G_{2},...$ be defined recursive... | Your base case is correct. There is a fundamental error at the beginning of your induction step, however: when you assume that $G_n=3^n-2^n$ for all $n\in\Bbb N$, you are assuming precisely the result that you’re supposed to be proving, which makes your argument circular. What you should be assuming as your induction h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed form of the recursive function $F(1):=1,\;F(n):=\sum_{k=1}^{n-1}-F(k)\sin\left(\pi/2^{n-k+1}\right)$ Suppose that $F$ is defined via the recurrence relation
$$F(1)=1, \qquad F(n)=\sum_{k=1}^{n-1}-F(k)\sin\biggl(\frac{\pi}{2^{n-k+1}}\biggr)$$ What is $F(N)$? I don't have any idea how to solve this problem. Only o... | The $F(n+1)$ is the coefficient of $x^n$ in the power series expansion of
$$
\dfrac{1}{{\displaystyle\sum\limits_{n = 0}^\infty {\sin \left( {\dfrac{\pi }{{2^{n + 1} }}} \right)x^n } }}.
$$
Consequently,
$$
F(n + 1) = \sum\limits_{\substack{k_1 + 2k_2 + \cdots + nk_n = n \\
k_1 ,k_2 , \ldots ,k_n \in \mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
help with proving convergence or divergence of series I need to prove the convergence or divergence of:
$$\sum_{n=1}^\infty \frac{(-1)^{n-1}\cos(\frac{\pi n}{3})}{n!}$$
I've tried:
$$\sum_{n=1}^\infty \frac{(-1)^{n-1}\cos(\frac{\pi n}{3})}{n!} = \lim_{n \to \infty} \biggl|\frac{(-1)^n\cos(\frac{\pi (n+1)}{3})}{(n+1)!} ... | Actually in your argument $\frac{(-1)^{n-1}\cos(\frac{\pi n}{3})}{n!}<\frac{(-1)^{n-1}}{n!}$ is not true, since these are alternating terms. However, it is true once you take the absolute values. Then you can show $a_n$ converges absolutely.
More precicesly, set $a_n = \frac{(-1)^{n-1}cos(\frac{\pi n}{3})}{n!}$, so $|a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that $f(x) = e^x $ is a function. Can I simply say the following? I feel like I'm either over or under-thinking this, but it makes sense to me since a function has one y for every x and a one-to-one function has the same criteria.
Let $e^a = e^b$
then ln $(e^a) = $ ln$(e^b)$ $\Rightarrow a = b$
Therefore, $e^x$ is... | You proved the relationship is one-to-one, but did not prove it is a function. The usual definition is that a relation between two sets (domain and codomain, here both $\mathbb{R}$) defines a function if any element of the domain is associated with exactly one element of the codomain. Here, the definition $f(x) = e^x$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Model theory substitution notation Let $\mathcal M$ be a structure in language $\mathcal L$ and let expanded language $\mathcal L_\mathcal M$ be the language with added constants $c_a$ for $a \in \mathcal M$.
I thought I understood notation but I keep finding myself questioning if my understanding is actually accurate.... |
Does this mean before the substitution $\phi$ has free variables $x_1, x_2, \ldots, x_n$ and after the substitution $\phi(a_1, \ldots, a_n)$ is a sentence?
Yes, that's precisely what it means.
On a related note, what does $\phi(c_{a_1}, \dots, c_{a_n})$ in the expanded language mean? Is it just indicating that $\phi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3645847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Understanding inequalities and how not to blindly apply famous inequalities
If $x \in (0, \pi/2)$, prove that
$$ 2 (\sin x )^{(1-\sin x)} \cdot (1-\sin x)^{\sin x} \leq 1 $$
Attempt:
We use Youngs inequality: if $\alpha+\beta = 1$ and $x,y $ are positive, then $x^{\alpha} + y^{\beta} \leq \alpha x + \beta y $.
In our... | Your first solution is correct. It becomes a bit simpler to read with the substitution $u = \sin(x) \in (0, 1)$:
$$
2u^{1-u} (1-u)^u \le 4 u(1-u) \le 1 \, .
$$
The first estimate is Young's inequality (as you noticed), this is also a special case of the Weighted AM–GM inequality.
For the second estimate you determined... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3646029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to show that $f(x)=\sin(x)+\cos(x)$ is sinusoidal(alternative) There was this question in our trig homework; it was for plotting a graph but I found it far more interesting than that. When drawing the graph of $\sin(x)+\cos(x)$ (by hand, which I find rather pointless), I found that it looked like some sort of sine ... | Geometric interpretation:
Rotate the right triangle. The projected length of the $\sqrt 2$ hypothenuses will vary following a sinusoid.
At the same time, this projected length is $\cos\theta+\sin\theta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3646388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove that $(a_1-a_2)+(a_2-a_3)+....$ converges iff ${a_n}$ converges Let $b_n=a_n-a_{n+1}$.
We first assume that ${a_n}$ converges so ,$\lim(b_n)=\lim(a_n)-\lim(a_{n+1})$ , hence $\lim(b_n)=0$.
Now let $s_m$ and $s_n$ be the consecutive partial sums of ${b_n}$.
So $|s_n-s_m| = |a_{n+1}-a_{m}|=|a_{n+1}-L+L-a_m|< \epsil... | It's a telescoping series.
$\sum_{k=1}^n (a_k - a_{k+1}) = a_1 - a_{n+1}$.
so $\lim\limits_{n\to \infty} \sum_{k=1}^n (a_k - a_{k-1}) = \lim\limits_{n\to \infty} (a_1 - a_{n+1}) = a_1 - \lim\limits_{n\to \infty} a_n$
Which converges and exists if and only if the $a_n$ converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3646706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Help with $\int_{0}^{\infty}\frac{1}{\sqrt{x^{5}+1}}dx$ $$\int_{0}^{\infty}\frac{1}{\sqrt{x^{5}+1}}dx$$
So far,
$$\int\frac{1}{\sqrt{x^{5}+1}}dx=\int\frac{1}{\sqrt{x^{4}\left(x+\frac{4}{x^{4}}\right)}}dx=\int\frac{1}{x^{2}\sqrt{x+\frac{4}{x^{4}}}}dx$$
but from there I do not know how to proceed, I have tried several
ch... | The convergence is straightforward since $\frac{1}{\sqrt{x^5+1}}$ is positive and bounded by $\min\left(1,\frac{1}{x^{5/2}}\right)$, so the integral is positive and less than $\frac{5}{3}$. Of course we can state something fairly more accurate. By Euler's Beta function our integral equals
$$ \int_{0}^{+\infty}\frac{z^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3646854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
An attempt at proving "continuous function on a closed interval (I) is uniformly continuous" I am familiar with some of the standard proofs of the statement. However, I was trying to construct a proof that fits best with my natural intuition. To this end, given $\varepsilon >0$, I define a function $$\delta^*(c) = \s... | The function $\delta^{\ast}$ need not be continuous. Fix $0 < \varepsilon < 1$, and consider $f \colon [-10,10] \to \mathbb{R}$ defined by
$$f(x) = \begin{cases} -\frac{3}{4} \varepsilon &\text{if } x \leqslant - \frac{3}{4}\varepsilon, \\ \quad x &\text{if } \lvert x\rvert \leqslant \frac{3}{4}\varepsilon, \\ +\frac{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solution to Laguerre differential equation I am slightly confused by the solution to the Laguerre differential equation
$$xy''+(\alpha +1-x)y' + ny=0 .$$
The solution is
$$y = c_1U(-n,1+\alpha, x) +c_2 L_n^\alpha(x),$$
where $U$ is a hypergeometric function and $L$ a Laguerre polynomial (see https://mathworld.wolfram.... | As commented by Somos, these two solutions are correct in cases where $n$ is not an integer. Because we frequently use associated Laguerre polynomials (which are of integer order $n$), the second solution you list $U(-n, \alpha + 1, z)$ becomes proportional to $L_n^\alpha(z)$. In these cases, a second logarithmic solut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditions that may make a function continuous at the origin
ATTEMPT:
a) My answer is afirmative. Since for any $x$ we have the condition, then if we put $x = \dfrac{\epsilon + f(0) }{\epsilon} $, then
$$ |f(x)| < \epsilon + f(0) \implies |f(x)-f(0)| < \epsilon $$
and since $\epsilon$ is arbitrary then $f(x)$ is conti... | As @QC_QAOA said, the statement of the problem in (a) should be: Suppose that for every $\epsilon>0$ there exists $\delta > 0$ such that $|x|<\delta$ implies $|f(x)|\le \epsilon|x|.$ Is $f$ continuous at $0?$
The answer is yes. Here's a preliminary result: Suppose $a>0$ and $|f(x)|\le |x|$ for $|x|<a.$ Then $f$ is cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Determine that this multivariable function is differentiable
$$
f(x,y) = \begin{cases}
0, & x = 0 = y \\
\frac{x^3}{x^2 + y^2}, & \text{otherwise}
\end{cases}
$$
So, I did determine that this function is continuous(correct me if I'm wrong here). But, the issue is that the first order partial derivative with respect t... | $∆f=f(0+h,0+k)-f(0,0)$
$= \frac{h^3}{h^2+k^2}$
And $f_x = 1$, $f_y=0$.
Therefore $df = hf_x+kf_y=h$.
If the function is differentiable then $\lim_{(h,k)\to (0,0)} \frac{∆f-df}{\sqrt{h^2+k^2}}$ must be $0$.
Now, $\frac{∆f-df}{\sqrt{h
^2+k^2}}= -\frac{hk^2}{\sqrt{h^2+k^2}(h^2+k^2)}....(i)$
Putting $h=r cos\theta, k=rsin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Questions about null sets I'm working in an arbitrary measure space, denoted by $<X,M,\mu>$, where X is a non-empty set, $M$ is a sigma-algebra and $\mu$ is the measure, $\mu:M\rightarrow[0,\infty]$. Suppose $N$ is a null set, ie, $\mu(N)=0$, my questions are:
*
*Why does $N$ belongs to $M$? Is it because $\mu(N)$ i... | *
*Yes, if $\mu(A)$ is defined that means that $A\in M$, since $M$ is the domain of the function $\mu$.
*This isn't necessarily true. For example, let $M$ be the sigma-algebra of Borel sets on $\Bbb R$ and $\mu$ Lebesgue measure. Then $|M|=2^{\aleph_0}$. The Cantor set $C$
has measure zero and $|C|=2^{\aleph_0}$, so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
One dimensional representation and Commutator subgroup Let G* be the set of all one dimensional representations i.e all group homomorphism from G to C* and G' denotes the commutator subgroup of a finite group G. I need to prove that G* is isomorphic to the quotient group G/G'.
I can say that for $\phi\in$G* then G' l... | This question as stated is a bit trickier than it ought to be, as it combines two facts without telling you:
*
*The group of linear characters $\text{Pic}(G)$ under $\otimes$ is an abelian group, with a canonical isomorphism to the group $\text{Hom}_{Ab}(G^{\text{ab}},\mathbb C^*)$.
*For any finite abelian group $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3647767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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