Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that for all circle of radius $3^n$ we can have $7^n$ circles inside with a radius of 1 Prove that for all circle of radius $3^n$ we can have $7^n$ circles inside with radius of 1 and neither of them intersect.
For me, it sounds like using mathematic induction, but I have no clear idea or answer.
| Yes, induction should work. The picture below should give you a hint why:
(the radius of the green circles is three times those of the red ones; the radius of the blue circle is three times those of the green ones)
| {
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"url": "https://math.stackexchange.com/questions/2338124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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While solving for $nP5 = 42 \cdot nP3$, $n > 4$ ...if we cancel out $n!$ on both sides we get to a complex quadratic which gives a wrong result. But, if we cancel out the $(n-5)!$ and $(n-3)!$ on their respective sides of the equation and then solve the quadratic and use the constraint $n>4$ we arrive at an answer of $... | $\frac {(n-5)!}{(n-3)!} \ne (n-5)(n-4)$ as $(n-5)! \ne (n-3)!*[(n-4)(n-5)]$
Notice: $(n-5) < (n-3)$ and $(n-3)! = 1*2*3.....*(n-5)*(n-4)*(n-3) = (n-5)!*[(n-4)(n-3)]$ .
And therefore $\frac {(n-5)!}{(n-3)!} = \frac 1{(n-4)(n-3)}$
This will give you the right answer.
| {
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"url": "https://math.stackexchange.com/questions/2338202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Value of $f'(6)$ in given polynomial
A polynomial function $f(x)$ of degree $5$ with leading coefficients one $,$ increase in the interval
$(-\infty,1)$ and $(3,\infty)$ and decrease in the interval $(1,3).$ Given that $f'(2)=0\;, f(0)=4$
Then value of $f'(6) =$
Attempt: Given function increase in the interval $(-\i... | just a hint
$$f (x)=x^5+ax^4+bx^3+cx^2+dx+4$$
$$f'(2)=80+32a+12b+4c+d=0$$
$$f'(1)=5+4a+3b+2c+d=0$$
$$f'(3)=405+108a+27b+6c+d=0$$
$$f'(6)=5.6^4+4.6^3.a+3.6^2b+12c+d $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2338333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to express a matrix as sum of two square zero matrices I have a real square matrix $M$ that I'd like to express as $M=A+B$ such that
$A^2=0$,$B^2=0$. $M$ has an additional property that $M^2$ is a scalar matrix :
($M^2=s^2I$); and it's dimension is a power of 2 : $dim(M)=2^n,n>0$; Any suggestions?
| Take
$$
A=\begin{pmatrix} 1 & -r^{-1}\cr r & -1 \end{pmatrix},\quad
B=\begin{pmatrix} 1 & -s^{-1}\cr s & -1 \end{pmatrix}
$$
for non-zero $r,s$, and $M=A+B$.
Then $A^2=B^2=0$ and
$$
M^2=(A+B)^2=\frac{-r^2+2rs-s^2}{rs}I
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2338445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Binomial coefficients that are powers of 2 I would like a proof that
$$ {{n}\choose{k}} = \frac{n!}{k!(n-k)!} = 2^m $$
for $n,k,m\in \mathbb{N}$, only if $k=1$ or $k=n-1$.
It seems to me that this must be true since for other values of $k$ the numerator contains more factors that are not powers of 2 than the denominato... | Betrand's Postulate implies that for $n \ge 1$ there is always a prime $p$ with $n < p \le 2n$.
Sylvester strengthened this result to:
If $n \ge 2k$ then at least one of the numbers $n, n - 1, n - 2, \cdots, n - k + 1$ has a prime divisor $p > k$.
Hence if $n \ge 2k$, which we can always assume since $\displaystyle \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2338488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Singular value decomposition with zero eigenvalue. I want to calculate the SVD ($A = U\Sigma V^*$)of
$$A =
\begin{bmatrix}
0 & 2 \\
0 & 0 \\
0 & 0
\end{bmatrix}$$
but
$$A^TA =
\begin{bmatrix}
0 & 0 \\
0 & 4
\end{bmatrix}$$
which has a zero eigenvalue. The pro... | No, $Av_i=\sigma_iu_i$, which is perfectly well defined even when $\sigma_i=0$. The point is $U$ can be decomposed into vectors corresponding to $\sigma_1,\cdots,\sigma_k>0$ and, when $\sigma_i=0$, you pad $U$ with vectors spanning the cokernel (i.e. whatever the range of $A$ misses) of $A$. See the example calculatio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $B$ is bounded with $0< B\leqslant 1$ and $T$ is closed, is $BT$ closable? Let $H$ be a separable Hilbert space, let $B$ be a bounded operator on $H$ with $0< B \leqslant 1$ and let $T$ be a closed, densely defined operator in $H$. The notation $0<B\leqslant 1$ signifies that $0<\langle x,Bx\rangle \leqslant 1$ for ... | This answer is based on Nate's example. The operator $B$ is made injective by adding a compact multiplication operator. The coefficients had to be adjusted to make it self-adjoint, bounded, and positive definite. Moreover, the diagonal part is made to vanish faster than then rank-one part for $n\to\infty$.
Take an orth... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A simple inequality and root I need to use the following inequality
$$ \left(\sum_{i=1}^N X_i \right)^{1/2} \leq \left( \sum_{i=1}^N X_i^{1/2} \right),\hspace{0.5cm} 0 \leq X_i $$
But i can't remember its name.
Is the inequality correct?
If it's correct, then how can i prove it?
| I don't now about a name but this is a consequence of the triangle inequality.
The triangle inequality states that for any vectors $z_1,y \in \mathbb{R}^n$, we have:
$$\|z_1+y\| \leq \|z_1\|+\|y\|$$
Let $y=z_2+z_3$ then,
$$\|z_1+z_2+z_3 \| \leq \|z_1\|+\|z_2+z_3\| \leq \|z_1\|+\|z_2\|+\|z_3\|$$
Continuing in this fash... | {
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Proving that a harmonic function is bounded on a open connected set Let $u:G\to \mathbb{R}$ be harmonic and $K\subset G$ compact where $G$ is open and connected. If $u\leq c$ on $K^c$ then I want to prove that $u\leq c$ on $G$ where $c\in \mathbb{R}$. Here is my attempt.
Since $u$ is continuous on the compact set $K$, ... | Here's a start: let $v:G\to\mathbb{R}$ be the harmonic conjugate of $u$ so that the function $f:G\to \mathbb{R}$ defined by $f=u+iv$ is analytic. Now set $g(z)=e^{f(z)}$ and observe that $|g(z)|=|e^{u(z)}||e^{iv(z)}|=|e^{u(z)}|$. Since $u$ is bounded on $G\setminus K$, what does that tell you about $g$'s behaviour on t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does this proof use Axiom of Choice?
Here is a question from Munkres Topology:
and here is its solution:
I think that while choosing $b_1,b_2,\dots$ we are using Axiom of Choice. But again, since for each $b_n$ we are 'choosing' only one element from a nonempty set, I think we do not have to apply Axiom of Choi... | Since the integers are countable, we can enumerate them as use that as a choice. But for the negative integers it's even simpler. Just go down the order of the integers.
For a general linear order, though, this would require some choice. So you are right to be skeptical.
Do note, though, that just choosing one elemen... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Cross ratio on sphere Project from the north pole to have an identification of the sphere in real space $\mathbb{R}^3$ and the complex projective line. Given $4$ (say, different) points on the sphere, I can project and then compute their cross-ratio.
Where can I find a formula that directly computes this value from th... | Identify the sphere $S^2 \subset \mathbb{R}^3$ with the space of imaginary quaternions of norm $1$. Then an explicit expression for the cross-ratio can be found in the arXiv preprint On a Quaternionic Analogue of the Cross-Ratio, see in particular Section 3.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the probability of tossing two heads if the game started with a head? A player is tossing an unbiased coin until two heads or two tails occur in a row. What's the probability of heads winning the game if the game started with a head?
I looked at Two tails in a row - what's the probability that the game started w... | If $p$ is the probability you're looking for, then $p= 0.5 + 0.5(1-p)$. Can you see why?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2339274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Manipulation of calculus notation I'm trying to get my calculus back up to scratch after not using it for 20 odd years. During my research, I've just seen this on https://physics.info/kinematics-calculus/:
$$a = \frac{dv}{dt}$$
$$dv = a\ dt$$
$$\int_{v_0}^v dv = \int_0^{\Delta t} a\ dt$$
Is this a valid manipulation of... | It is fine, mostly. Infinitesmals are really defined under any number system, even when one considers proffs in calculaus/analysis, one deals with epsilon-delta proofs that skirt arounf the issue of infinitesmals.
Mathematically however, that is treating those quantities as manipulated 'variables' one could write $dy=f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2339398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving trigonometric identity $\frac{\sin(A)}{1+ \cos(A )}+\frac{1+ \cos(A )}{\sin(A)}=2 \csc(A)$
$$
\frac{\sin(A)}{1+\cos(A)}+\frac{1+\cos(A)}{\sin(A)}=2\csc(A)
$$
\begin{align}
\mathrm{L.H.S}&= \frac{\sin^2A+(1+\cos^2(A))}{\sin(A)(1+\cos(A))} \\[6px]
&= \frac{\sin^2A+2\sin(A)\cos(A)+\cos^2(A)+1}{\sin(A)(1+\cos(A))... | You make several mistakes, the main one being
$$
(a+b)^2=a^2+b^2
$$
The mistake is $(1+\cos(A))^2=1+\cos^2(A)$, whereas it should be
$$
(1+\cos(A))^2=1+2\cos(A)+\cos^2(A)
$$
Note that
$$
\frac{a}{b}+\frac{b}{a}=\frac{a^2+b^2}{ab}
$$
where $a=\sin(A)$ and $b=1+\cos(A)$.
In the second step you also arbitrarily insert a t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2339468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Finding the volume of the solid generated by revolving the given curve. The objective is to find the volume of the solid generated by revolving the curve $y=\dfrac{a^3}{a^2+x^2}$ about its asymptote.
Observing the given function yields that $y\ne0$, hence $y=0$ is the asymptote to the given curve. Thus, the volume of t... | Your integral is equal $\frac{\pi^2}{2}( \frac{1}{a^2})^{\frac{3}{2}}a^6$ according to wolfram alpha. Your algebra must be wrong somewhere. I recommend trying trigonometric substitutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2339578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Solve three equations with three unknowns Solve the system:
$$\begin{cases}a+b+c=6\\ab+ac+bc=11\\abc=6\end{cases}$$
The solution is:
$a=1,b=2,c=3$
How can I solve it?
| For class work it is likely that the roots are integers, so I would just try them. There are not many factorizations of $6$ and $1,2,3$ should jump out. Then just try it and you are done.
The routine approach is substitution. Write the first as $a=6-b-c$ and plug that into the other two. Solve the second for $b$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Expand $\frac{z}{z^4+9}$ To Taylor Series
expand $$\frac{z}{z^4+9}$$ to taylor series
$$\frac{z}{z^4+9}=\frac{z}{9}\frac{1}{1--\frac{z^4}{9}}$$
Can we write $$\frac{z}{9}\sum_{n=0}^{\infty}(-1)^n\left(\frac{z^4}{9}\right)^n=\sum_{n=0}^{\infty}(-1)^n\frac{z^{4n+1}}{9^{n+1}}$$?
| Yes, your solution is a good solution.
In fact, you are expanding around $0,$ but one can choose different points. Also, note that the radius of convergence is $\sqrt{3}.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can you use 2^n - 2 / n to check if a number is prime with 100% accuracy? According to the AKS primality test:
$$(x-1)^p - (x^p-1)$$
If all coefficients (which can be found in Pascal's triangle) are divisible by p then p is prime.
If we sum these coefficients we get:
$2$ for $p = 2$;
$6$ for $p = 3$;
$14$ for $p = 4$;
... | If a sum is divisble by $p,$ it does not mean the summands are.
The smallest counterexample to your claim is $p=341.$ We have $341=11\cdot 31,$ but $2^{341}=2\cdot(2^{10})^{34} = 2\cdot(1024)^{34} = 2\cdot(3\cdot 341+1)^{34} \equiv 2\cdot 1^{34} = 2 \pmod{341}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2340018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding roots of a trigonometric function I have a calculus problem that has some trigonometric difficulty to it.
It is $\ 20\sin{x}-10\sin{2x}-\frac{40}{\pi}=0$. I basically want to find two $x \in [0,\pi]$.
I got to $\ \pi\sin{x}(1-\cos{x})=2$
I don't know if there is some trig identity or trick I am missing out on h... | Similar to Robert Israel's answer.
Using the tangent half-angle substitution $$t=\tan(\frac x 2)\qquad \sin(x)=\frac{2t}{1+t^2}\qquad \cos(x)=\frac{1-t^2}{1+t^2}$$ the equation reduces to $$t^4-2 \pi t^3+2 t^2+1=0$$ Just as Robert Israel answered, solving analytically quartic equations is not the most pleasant thing ... | {
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"timestamp": "2023-03-29T00:00:00",
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Polynomial $x^5 + 5x^2 +1$ is irreducible or not
Which way I can determine whether the polynomial $x^5 + 5x^2 +1$ is irreducible over $\mathbb Q$ or not?
Mod $p$ Irreducibility Test and Eisenstein's criterion not applicable here.
Which way I should proceed?
| $7^5+5\cdot 7^2+1 = 17053$ which is a prime number. Thus the polynomial is irreducible by Cohn's irreducibility criterion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2340235",
"timestamp": "2023-03-29T00:00:00",
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Given $f(x) = e^{-x} \sin x, $ Find $\lim\limits_{x \rightarrow \infty} f(x)$ if it exists. Justify using limits definition.
Given $$f(x) = e^{-x} \sin x, $$
Find $\lim\limits_{x \rightarrow \infty} f(x)$ if it exists. Justify using directly the following definition:
$\lim\limits_{x \rightarrow \infty} f(x)=L$ if $f$ ... | Fix $\varepsilon>0$. Then
$$
\left|\frac{\sin x}{e^x}-0\right| \le \frac{1}{e^x} \le \varepsilon
$$
whenever $x \ge -\log \varepsilon$. (In particular, yes the reasoning is correct.)
| {
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"timestamp": "2023-03-29T00:00:00",
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Question on the subgroups and its cosets I am studying group theory on my own with the available resources online esp. wikipedia so please be kind.
I know that a subgroup of a group is isomorphic to any of its cosets.
The map
$$f:H\rightarrow xH$$
where $H$ is a subgroup of some group and $xH$ is a coset of $H$, is an ... | First of all that is not a isomorphism. That is just a set bijection. A coset is just a set, it has no group structure. So you cannot define a homomorphism. $f$ is just a bijection.
This bijection says that $|H|=|aH|$ for all $a \in G$.In case of finite groups, this result will later be used to prove the all- important... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the probability that the special ball is chosen? I know this question is already posted here, but I doubting my own solution which is I know is wrong. It is quite basic, but I want to learn probability from scratch, so I am posting my question.
Question:
An urn contains $n$ balls, one of which is special. If $... | You need to consider the number of ways of selecting $k$ balls such that the selected set of balls always contain the special ball. This number will be equal to the number of ways of selecting $k-1$ balls from $n-1$ balls.
| {
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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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Are the corresponding eigenspaces of $A$ and $A^n$ equal? Let $A$ be a square matrix and $n$ be a positive integer. If $\mu$ is an eigenvalue of $A^n$ such that there is a unique eigenvalue $\lambda$ of $A$ with $\lambda^n=\mu$, can we say that the eigenspaces of $A^n$ and $A$ corresponding to $\mu$ and $\lambda$ respe... | $$A=\left(\begin{matrix}0&1\\0&0\end{matrix}\right)$$
$$n=2$$
$$A^2=\left(\begin{matrix}0&0\\0&0\end{matrix}\right)$$
$$\lambda=\mu=0$$
$Ax=0$ for $x=(x_1,0)^T$, while $A^2x=0$ for all $x$.
If $\lambda\neq0$ then we can argue as follows:
It is enough to look at one Jordan block $A=\lambda I+N$ where $N$ has zeros ever... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculating $\text{PV}\int_{-\infty}^{+\infty}\frac{e^{\alpha x}}{e^{2x}-1}\mathrm d x$ I am trying to show that for $0 < \alpha < 2$:
$$
{\rm P.V.}\int_{-\infty}^{\infty}\frac{{\rm e}^{\alpha x}}
{{\rm e}^{2x} - 1}\,{\rm d}x
=
-\frac{\pi}{2}\,\cot\left(\frac{\alpha\pi}{2}\right )
\tag{$\star$}
$$
to gain some familiar... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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"question_score": "7",
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If $f(x) \in C^1$ then number of maxima is finite in any interval. I am wondering if the following statement is true.
If $f(x) \in C^1$ (not-constant) then a number of maxima are finite in any finite interval.
Here the set $C^1$ means that on the domain of $f$ we have that
\begin{align}
\sup_{x \in dom(f(x))} |f(x)... | By interval do you mean a finite interval. If not, the statement is not true. Counter example would be $sinc(x)$ on $(0,\infty)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2340846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does Cantor's Theorem and the Continuum Hypothesis imply discrete levels of infinity? Cantor's Theorem shows that there are an infinite number of distinct infinite
set cardinalities, as there is at least one infinite set, and it provides a method for producing a set with a larger cardinality from another set that work... | See cardinal numbers, and probably also the ordinal numbers. This is a rich subject and your question is much too vague, but these are a good start.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate $\int ( 3x + \frac{6x^2 \sin^2(\frac{x}{2})}{x - \sin x} ) \frac{(x-\sin x)^{3/2}}{\sqrt{x}} \mathrm{d}x$ I have a really vague integration problem. It's some substitution and then integration by parts maybe.
I got this from a friend, but it seems it's unlikely to be solved.
Now my question is, how to appr... |
Easier way to solve this problem
| {
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"timestamp": "2023-03-29T00:00:00",
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Find all orthogonal $3\times 3$ matrices of the form... Find all orthogonal $3\times 3$ matrices of the form
\begin{bmatrix}a&b&0\\c&d&1\\e&f&0\end{bmatrix}
Using the fact that $A^TA$ = $I_n$, I set that all up and ended up with the following system of equations:
$$\left\{\begin{array}{l}a^2 + e^2 = 1\\
ab + ef = 0\\
... | You have
\begin{align}
a^2+e^2=b^2+f^2&=1\\
c=d=ab+ef&=0.
\end{align}
The first equation represents the lengths of the (unit) vectors $\pmatrix{a\\e}$ and $\pmatrix{b\\f}$ and the second equation represents the scalar product of these vectors, showing that they are perpendicular.
Wihtout loss of generality, let $\pma... | {
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"url": "https://math.stackexchange.com/questions/2341220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $ S^{1}\times S^{1}\times S^{1}$. What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $S^{1}\times S^{1}\times S^{1}$.
Where their homology groups are:
If $\;\;\;\;(S^{2}... | For $M,N$ with dimension $n>2$ the connected sum $M\# N$ has fundamental group $\pi_1(M\# N) \approx \pi_1(M) * \pi_1(N)$ as the boundary of the $n$-ball used for identification is simply connected. Thus $\pi_1(3-(S^2 \times S^1)) \approx \mathbb Z * \mathbb Z * \mathbb Z$ yet $\pi_1(T^3) \approx \mathbb Z \times \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2341346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find $a^2+b^2+2(a+b)$ minimum if $ab=2$ Let $a,b\in R$,and such
$$ab=2$$
Find the minimum of the $a^2+b^2+2(a+b)$.
I have used $a=\dfrac{2}{b}$, then
$$a^2+b^2+2(a+b)=\dfrac{4}{b^2}+b^2+\dfrac{4}{b}+2b=f'(b)$$
Let $$f'(b)=0,\,b=-\sqrt{2}$$
So $$a^2+b^2+2(a+b)\ge 4-4\sqrt{2}$$
I wanted to know if there is other way to ... | For $a=b=-\sqrt2$ we get a value $4-4\sqrt2$.
We'll prove that it's a minimal value.
Indeed, let $a+b=2k\sqrt{ab}$.
Hence, $|k|=\left|\frac{a+b}{2\sqrt{ab}}\right|\geq1$ and we need to prove that
$$a^2+b^2+2(a+b)\geq4-4\sqrt2$$ or
$$a^2+b^2+\sqrt{2ab}(a+b)\geq(2-2\sqrt2)ab$$ or
$$(a+b)^2+\sqrt{2ab}(a+b)\geq(4-2\sqrt2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2341443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration of $e^{it}$ I am pretty sure that
$$\bigg|\int_{A}e^{it}\,dt\bigg|\leq2$$
for every measurable set $A\subseteq[-\pi,\pi]$,
but I cannot prove this...
| We avoid arguments that use the modulus (i.e. triangle inequality) since for some measurable sets, for instance $A = [-\pi, \pi]$, we have an overestimate $$\Big|\int_A e^{it}dt\Big| \leq 2 < \int_A|e^{it}|dt = 2\pi$$
Simply by rotating the value of the integral (which is a complex number) back to the real numbers.
Le... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Question about baby rudin theorem 5.12 corollary The corollary says if $ f $ is differentiable on $ [a,b] $ then $ f' $ cannot have any simple discontinuities on $[a,b] $.
I just don't how to prove it.
I think it should be proved on both two cases of simple discontinuities(first type and second type of simple disconti... |
Let $g \colon (a,b) \to \mathbb{R}$ a function. If $g$ has a simple discontinuity at $c \in (a,b)$, then $g$ doesn't have the intermediate value property.
Let's look at the case of a jump discontinuity. Replacing $g$ with $-g$ if necessary, we can assume that
$$L := g(c^-) < R := g(c^+).$$
Let $\varepsilon = \frac{R-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2341778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A problem from Real analysis-Royden regarding the finite additivity of bounded disjoint sets for the Lebesgue outer measure Let $A$ and $B$ be bounded sets for which there exists an $\alpha > 0$ s.t $|a-b|\geq \alpha $ $\forall a \in A, b\in B$. Prove that
$$m^{*}(A\cup B)=m^{*}(A)+m^{*}(B)$$.
Where, $m^{*}$ is the L... | Hints: The point is that $A$ and $B$ are not only disjoint but they are well separated; so they can be covered by balls (or cubes or whatever you use to generate the Lebesgue outer measure in your favorite definition) independently. Draw a picture. That should help a lot.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2341861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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When is the probability of A given B equal to probability of B given A? Can they ever equal each other?
If not, then is it because the denominator ($P(A)$ vs $P(B)$) is not the same?
I'm asking because in Probability for the Enthusiastic Beginner (A wonderful book by the way), the author says they aren't equal ... In g... | They can be equal, but it would be a coincidence. To see they are not equal in general, just think about how conceptually $P(A|B)$ and $P(B|A)$ are different kinds of probabilities.
To use a standard example: the probability that I test positive given that I have a certain disease (a measure of how accurate the test... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate an indefinite integral Find the value of
$$\int{\frac{x^2e^x}{(x+2)^2}} dx$$
My Attempt: I tried to arrange the numerator as follows:
$$ e^xx^2 = e^x(x+2-2)^2 $$ but that didn't help.
Any guidance on this problem will be very helpful.
| Another method:
\begin{align}
\int \frac{x^2 \, e^{x}}{(x+2)^2} \, dx &= - \int x^2 \, e^{x} \, \frac{d}{dx} \left(\frac{1}{x+2}\right) \, dx \\
&= - \left[ \frac{x^2 \, e^{x}}{x + 2} \right] + \int x(x+2) \, e^{x} \cdot \frac{1}{x+2} \, dx \\
&= - \frac{x^2 \, e^{x}}{x + 2} + \int x \, e^{x} \, dx \\
&= - \frac{x^2 \,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Matrix norm of Kronecker product Is it true that $ \| A \otimes B \| = \|A\|\|B\| $ for any matrix norm $ \|\cdot \| $? If not, does this identity hold for matrix norms induced by $ \ell_p $ vector norms?
| On page 149 exercise 6 in book: Matrix analysis for scientists and engineers, this is true for operator norm. You can see chapter 13 of the book by the link: http://www.siam.org/books/textbooks/OT91sample.pdf
| {
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"source": "stackexchange",
"question_score": "9",
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How can we factor out the maximum value of f'(x) in an integral with an absolute value? I'm currently trying to understand a proof concerning the error term in the left- and right Riemann sums to approximate a definite integral. What I can't seem to understand is the last three lines of the proof where the author first... | By the Mean Value Theorem you have
$$
f(x)-f(x_k^*)=f'(\xi)(x-x_k^*)\leq(x-x_k^*)\max_{[a,b]}{f'}.
$$
Then, by integrating $x-x_k^*$ you get
$$
\frac{1}{2}(x-x_k^*)^2
$$
but you should be careful with the absolute value and separate the part in which $x\geq x_k^*$ and the other one where $x<x_k^*.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Why is absolute value function not a polynomial?
Why is absolute value function not a polynomial?
I need a clear answer to this question please,?
Why couldn't we consider absolute value function as a polynomial?
| Just quoting the definition of "polynomial" does not constitute a proof. Who knows, maybe there is a certain polynomial of degree $2017$ with particular coefficients that does the job. To be serious: We have to exhibit a property of ${\rm abs}$ that no polynomial can have. In this sense Reiner Martin's answer is fine.
... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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A geometry problem involving triangles In the figure, AE is the bisector of the exterior angle CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, then find the length of CE.
My Attempt: I tried to find out the existence of congruent triangles in the diagram, but couldn't find any.
Any help will be ... | Alternatively: Continue the line $D$ until $F$ and connect $F$ with $E$ so that $AC=AD$.
Note that $\Delta ACE$ is equal to $\Delta ADE$, because two sides and the angle between them are equal. It implies the line $AE$ is a bisector in $\Delta BDE$.
Using the property of bisector:
$$\frac{AB}{BE}=\frac{AD}{DE} \Rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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proving $h(1/z)= \overline{1/h(\overline{z})}$ Suppose $h$ is a holomorphic function on the disk $B_2(0)$ such that $|h(z)=1$ if $|z|=1$.
I want to prove that $h(1/z)= \overline{1/h(\overline{z})}$ when $1/2<|z|<2$.
I wanted to use schwarz Lemma but I don't know if the image of the disk is a disk or if $h(0)=0$. I tri... | A hint:
Use the reflection principle. Its basic version is the following: If $f$ is analytic in a disc $D_r$ around the origin, and if $f(x)\in{\mathbb R}$ for $-r<x<r$ then
$$f(\bar z)=\overline{f(z)}\quad\forall \ z\in D_r\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Moving along circles For each natural number
$k$. Let
$C_k$ denote the circle with radius
$k$ centimetres and centre at origin. On the circle
$C_k$
a particle moves
$k$ centimetres in the counter - clockwise direction. After completing its motion on
$C_k$ the particle moves to
$C_{k+1}$
in the radial direction.... | Hint. On each circle the point moves along an arc of $1$ radian. Now a complete revolution is $2\pi\approx 6.28$ radians.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Hyperbolas: Deriving $\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1$ from $\sqrt{(x + c)^2 + y^2} - \sqrt{(x - c)^2 + y^2} = \pm2a$ My textbook's section on Hyperbolas states the following:
If the foci are $F_1(-c, 0)$ and $F_2(c, 0)$ and the constant difference is $2a$, then a point $(x, y)$ lies on the hyperbola if an... | You have\begin{multline*}\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2}=\pm2a\Longleftrightarrow\\\Longleftrightarrow(x+c)^2+y^2+(x-c)^2+y^2-2\sqrt{(x+c)^2+y^2}\sqrt{(x-c)^2+y^2}=4a^2.\end{multline*}This is the same thing as saying that$$\sqrt{(x+c)^2+y^2}\sqrt{(x-c)^2+y^2}=-2a^2+c^2+x^2+y^2.$$Squaring both sides, one gets$$\bi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $f(x) = |x|$ not surjective? Can anyone explain to me why the function
$$
f(x)=|x|
$$
is not surjective (onto)?
I think it should be, but my teacher told me it's not.
| It depends on your definition of $f$. Consider $f : \mathbb R \to \mathbb R$ where $x \mapsto |x|$, this is certainly not surjective because every negative value $(-\infty, 0)$ is not mapped to by $f$.
Whereas one could define $f : \mathbb R \to [0, \infty)$, which would be surjective, but not injective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Is each integer $n > 2$ is divisible by $4$ or by an odd prime number While reading through the proof of Fermat's Last Theorem, I came across this statement. "Each integer $n > 2$ is divisible by $4$ or by an odd prime number"
But I don't know how to prove it.
| claim: Every integer $n$ at least $3$ is divisible either by $4$ or by an odd prime.
$3$ possible cases:
*
*The prime factorisation of $n$ contains exactly at least two ‘$2$’s.
So, $n$ is divisible by $4.$
*The prime factorisation of $n$ contains exactly one ‘$2$’, in which
case—since $n\geq3$—it must contain an odd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2342992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Without directly evaluting, show that the determinant of $A = 0$
Without directly evaluting, show that
$det \left[
\begin{array}{ccc}
b + c & c + a & b + a \\
a & b & c \\
1 & 1 & 1
\end{array}
\right]
=0$
I am stuck on this one. I can only do this by evuating.
... | The row vectors are linearly dependent.
Specifically, denoting the row vectors as $$\vec r_1=(b+c,a+c,a+b)\quad \vec r_2=(a,b,c)\quad \vec r_3=(1,1,1)$$ then we have $$\vec r_1+\vec r_2=(a+b+c)\,\vec r_3$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Questions about differential forms Let $\omega$ be a $k$-form on $M \subseteq \mathbb R^n$ a $d$-dimensional submanifold of $\mathbb R^n$ and $k,d \le n$.
1) Can I only integrate $\omega$ over $M$ if k = d? Why?
2) Let $\omega$ be a $1$-form and $\alpha: \mathbb R \to \mathbb R^n$ a curve. If I integrate $\omega$ alon... | The answer to your first question comes from measure theory . The way integration is defined is only possible for the forms which are of the same length as the dimension. Also , for me i think integration as a pairing of k forms and n-k forms .Thus then it makes sense only for the top forms.
For the third question ,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine whether the series converges or diverges. Determine whether the series converges or diverges.
$$
\sum _{n=1}^{\infty }\:\left(\frac{19}{n!}\right)
$$
I know that this question a lot easier if I use ratio test but I have not learned ratio test yet. The only option I have is divergence, comparison, limit com... | Then use the fact that $(\forall n\in\mathbb{N}\setminus\{2,3\}):\frac{19}{n!}\leqslant\frac{19}{n^2}$ and apply the integral test in order to prove that $\sum_{n=1}^\infty\frac{19}{n^2}$ converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2343287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
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is "commensurability" as simple as saying a and b are rational? I am trying to better understand commensurability. Wikipedia says:
two non-zero real numbers a and b are said to be commensurable if
$\frac{a}{b}$ is a rational number.
Richard Courant in Introduction to Calculus and Analysis says:
Two quantities who... | $x$ and $y$ are commensurable if there exists a real number, $r$ and positive integers $m$ and $n$ such that $x = mr$ and $y=nr$. If such an $r$ exists, it is called a common measure. If $x$ and $y$ are commensurable, we can aviod mention of a common measure by writing $x : y :: m : n$, or $\dfrac xy = \dfrac mn$. The ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve $Ax = b$ using Simulated Annealing? I have an idea how the Simulated-Annealing algorithm works w/ TSP, but I have no idea how to solve $Ax = b$, given an $n \times n$ matrix $A$ and a vector $b$. I know that it might sound stupid, but I really need some help.
| It works the same way as for the traveling salesman problem. Start with any candidate solution $x$ and
*
*Generate a new possible value for $x$
*Transition to the new value with a probability determined by the relative cost of the new value to the old and the temperature.
*Repeat 1 and 2 while gradually lowering t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2343497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the largest constant $k$ such that $\frac{kabc}{a+b+c}\leq(a+b)^2+(a+b+4c)^2$
Find the largest constant $k$ such that $$\frac{kabc}{a+b+c}\leq(a+b)^2+(a+b+4c)^2$$
My attempt,
By A.M-G.M, $$(a+b)^2+(a+b+4c)^2=(a+b)^2+(a+2c+b+2c)^2$$
$$\geq (2\sqrt{ab})^2+(2\sqrt{2ac}+2\sqrt{2bc})^2$$
$$=4ab+8ac+8bc+16c\sqrt{ab}$... | One more way...
Noting that replacing $a, b$ with $\frac{a+b}2, \frac{a+b}2$ leaves RHS unchanged but increases the LHS, we have to only check for the case $a=b$. Further as the inequality is homogeneous in $a, b, c$; WLOG we may set $a=1$. Hence we need only look for the minimum of the univariate
$$f(c) = (4+(2+4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2343592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Financial Mathematics: Annuity relating to loan You took a loan of 500,000 which required to pay 25 equal annual payments at 10% interest. The payments are due at the end of each year. The bank sold your loan to an investor immediately after receiving your 6th payment. With yield to the investor of 7% , the price the i... | Let $L=500,000$, $n=25$ and $i=10\%$ then
$$
P=\frac{L}{a_{\overline{n}|i}}=\frac{500,000}{a_{\overline{25}|10\%}}=55,084
$$
The bank sold the remaing loan after the 6th payment at interest $i'=7\%$, that is at price
$$
L'=P\,a_{\overline{n-6}|i'}=569,326
$$
So, for the bank we have the return $r$ is the solution of
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2343692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding the inverse of a matrix using gaussian elimination It's given the matrix A such that:
[ 0 1 1 ... 1 1]
|-1 0 1 ... 1 1|
|-1 -1 0 ... 1 1|
| . . |
| . . |
| . . |
|-1 -1 -1 ... 0 1|
[-1 -1 -1 ...-1 0]
Can someone help me find the inverse of this matrix using... | HINT
I am not certain that this will help you, but notice that for even matrix dimensions (i.e. your $n\times n$ matrix has $n=2k$, $k \in \mathbb{N}$) the determinant is equal to $0$ and thus the matrix is not invertible.
Also, if $n=2k+1$, $k \in \mathbb{N}$, then the determinant is equal to $1$.
So you need to focus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2343851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Logistic regression and cross-entropy Cross-entropy is a good perspective to understand logistic regression, but I have the following question:
the objective function of LR:
$$\max L(\theta) = \max \sum_{i=1}^N y_i \log \hat y_i + (1-y_i) \log (1- \hat y_i)$$
where $y_i$ is the probability of true label，$\hat y_i$ is t... | The notion of cross entropy is related to KL-divergence and entropy:
$$
H_q(p)=\sum p\log\frac 1q=\sum p\log\frac pq+\sum p\log \frac 1p=-D(p||q)+H(p).
$$
Maximizing the cross entropy over $q$ is equivalent to minimizing KL-divergence. Since the KL-divergence is non-negative, the maximum cross entropy is $H(p)$ and it ... | {
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"url": "https://math.stackexchange.com/questions/2343991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Can the closure of a countable set be characterized sequentially? Suppose that I have a countable subset $S \subset X$, where $(X, \tau)$ is a topological space that is NOT first countable (so that convergence is characterized by nets and not sequences). I'm interested in the closure $cl(S)$, which is defined as the co... | No. A classical counterexample is the space described in this question. (Arens space). The closure of all points $\mathbb{N} \times \mathbb{N}\setminus \{(0,0\}$ is all of the space, but no sequence from it converges to $(0,0)$.
See also this blog post
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Hoffman and Kunze, Linear algebra sec 3.5 exercise 9
Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers and let $$B=\begin{pmatrix}2&-2\\-1&1\end{pmatrix}.$$
Let $W$ be the subspace of $V$ consisting of all $A$ such that $AB=0.$ Let $f$ be a linear functional on $V$ which is in th... | There is nothing wrong with your solution.
Perhaps the author had the following approach in mind for which $C$ could be useful. Also it helps to determine the functional completely. Let us consider the following standard basis of $V$:
$$\mathcal{B}=\{E_1,E_2,E_3,E_4\}=\left\{\begin{bmatrix}1&0\\0&0\end{bmatrix}, \begi... | {
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I've searched and cannot find this pattern anywhere concerning integers and their factors Over the last few months, I've been studying a pattern that I stumbled on concerning integers and their factors. First, I noticed that the number of factors a number has, follows an extremely regular pattern based on prime numbers... | Your 'less interesting' pattern has a very simple explanation.
The number of divisors $d(n)$ for a number $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is equal to $(e_1+1)(e_2+2)\cdots(e_k+1)$. This includes the $1$ and $n$ divisors, which is standard mathematical convention.
That means that any multiple $pk$ of a prime nu... | {
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Does the "field" over which a vector space is defined have to be a Field? I was reviewing the definition of a vector space recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space.
Take for example the set of all ord... | These things are studied: they are called modules over the ring instead of vector spaces.
The main difference is that the elements of general modules do not allow a lot of the geometric intuition we have for vector spaces, so we still retain the traditional term "vector space" because it is still a useful term.
So, mod... | {
"language": "en",
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"source": "stackexchange",
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If $\frac {\sin A + \tan A}{\cos A}=9$, find the value of $\sin A$. If $\dfrac {\sin A + \tan A}{\cos A}=9$, find the value of $\sin A$.
My Attempt:
$$\dfrac {\sin A+\tan A}{\cos A}=9$$
$$\dfrac {\sin A+ \dfrac {\sin A}{\cos A}}{\cos A}=9$$
$$\dfrac {\sin A.\cos A+\sin A}{\cos^2 A}=9$$
$$\dfrac {\sin A(1+\cos A)}{\cos^... | Hint: using the tangent half-angle formulas, let $\,t=\tan(A/2)\,$, then the equation becomes:
$$
\frac{2t}{1+t^2} + \frac{2t}{1-t^2}=9 \,\frac{1-t^2}{1+t^2} \;\;\iff\;\; 9 t^4 - 18 t^2 - 4 t + 9 = 0
$$
The quartic has $2$ real roots which can be solved in radicals, but the calculations are not pretty.
[ EDIT ] Onc... | {
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norms simplification. I encountered this in a book. $||.||$ is the norm of a real normed space.
$$\lim_{\lambda\rightarrow\pm 0}\frac{||x+\lambda y||^2-||x||^2}{2\lambda}=||x||.\lim_{\lambda\rightarrow\pm 0}\frac{||x+\lambda y||-||x||}{\lambda}.$$ I could not understand the derivation. Any help would be appreciated.
Af... | For brevity, let the LHS and RHS expressions be $L$ and $R.$
(1). If $x\ne 0$ then $L=MR$ where $M=(\|x\|+\|x+\lambda y\|)/2\|x\|.$
We have $\lim_{\lambda\to 0}M=1$ because $\|x\|-|\lambda|\cdot \|y\|\leq \|x+\lambda y\|\leq \|x\|+|\lambda|\cdot \|y\|.$
So either $\lim_{\lambda\to 0}L=\lim_{\lambda \to 0}R$ or neither... | {
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Where is my mistake using the Banach theorem for $x^2 - 2 = 0$? Consider example $x^2 - 2 = 0$. I can rewrite so I get $x^2 + x - 2 = x$. If I define $\phi(x) = x^2 + x - 2$, I need to solve $\phi(x) = x$. $\phi$ is Lipschitz-continuous, since it's differentiable. On $[-\frac34,-\frac14]$ we have, using the mean value ... | For the Banach fixed point theorem to apply, you need that your to satisfy $\phi : X \rightarrow X$. In the example you provide, you chose the interval in such a way that this is not the case and the range of $\phi$ over $X$ is not contained in $X$.
| {
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Conditions that a complex matrix has real and identical eigenvalues Given a matrix $H \in \mathbb{C}^{N \times N}$, is there some condition on its elements such that all the $N$ eigenvalues are real and have the same value? Obviously, the trivial case of the identity matrix is not acceptable.
Thanks in advance.
| Elaborating on Hagen von Eitzen's comment:
If an $N\times N$ matrix has $N$ eigenvalues equal to $\lambda$, we usually speak of a single eigenvalue $\lambda$ with geometric multiplicity $N$. The latter means that there exist $N$ linearly independent eigenvectors with eigenvalue $\lambda$. Since the space if $N$-dimensi... | {
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A Equivalence of Uniform Continuity Using Distance (Rudin's exercise 4.20)
If $E$ is a nonempty subset of a metric space $X$, define distance from $x\in X$ to $E$ by $$\rho_E(x) := \inf_{z\in E}d(x,z)$$
Prove that $\rho_E$ is uniformly continuous function on $X$ by showing that $$|\rho_E(x) - \rho_E(y)| \le d(x,y)$$... | The argument proves that $\rho_E$ is Lipschitz which implies in particular that it is uniformly continuous.
| {
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Proof for an integral inequality For a function $f:\mathbb{R}\to\mathbb{R}$, $|f|\leq M$, $f(s)\equiv 0$ for $s<0$, define
\begin{equation}
I^n(t)= n \int_{t-1/n}^{t} f(s)ds.
\end{equation}
In my book they conclude that
\begin{equation}\label{eq}
|I^n(t')-I^n(t)|\leq 2n|t'-t|M.
\end{equation}
How can I proof this inequ... | hint
Put $u=t-s $.
then
$$I^n (t)=\int_0^\frac 1nf (t-u)du $$
and
$$I^n (t)-I^n (t')=\int_0^\frac 1n (f (t-u)-f (t'-u))du $$
| {
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In a convex hexagon, Diagonal intersect at interior point of an hexagon.
In a convex hexagon two diagonal are drawn at random. The probability that the
diagonal intersect at an interior point of an hexagon is
$\bf{Attempt}$ I have a doubt,
Diagonal of Convex hexagon always intersect at interior of an hexagon. So prob... | Hint
I am considering that a vertex is not an interior point.
Consider just those black points as an interior point.
Remember that we have $9$ diagonals. Looking to the answer, the statement is
also considering that any pair of diagonals have a intersection, even if it happens at an exterior.
Can you finish?
| {
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Upper Bound on Aliquot Sequence Let $s_1(n)=\big{(}\sum_{d|n}d\big{)}-n=\sigma_1(n)-n$ be the restricted divisor sum, and define $s_k(n)=s_1(s_{k-1}(n))$ as the $k^{th}$ term of the aliquot sequence starting at $n$. What is the best proven upper bound on $s_k(n)$? In other words, if the sequence starting at $n$ seems t... | Erdos, On asymptotic properties of aliquot sequences, Math Comp 30 (1976), no. 135, 641-645, MR0404115 (53 #7919), proved that for every fixed $k$ and every $\delta>0$ and for all $n$ except a sequence of density zero one has $$(1-\delta)n(s(n)/n)^i<s_i(n)<(1+\delta)n(s(n)/n)^i$$ for $1\le i\le k$.
| {
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What is the probability that four players who each receive ten cards together receive less than four aces? A deck of cards contain 52 cards, including 4 aces. Suppose each player gets 10 cards and the other 12 cards are kept aside, what is the probability that the four players together have less than four aces?
My answ... | The players collectively receive $40$ cards. If they collectively receive all four aces, then they receive $36$ of the other $48$ cards in the deck. Hence, the probability that the players collectively receive all four aces is
$$\frac{\dbinom{4}{4}\dbinom{48}{36}}{\dbinom{52}{40}}$$
Hence, the probability that the p... | {
"language": "en",
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Find the Roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$ Once I came across the following problem: find the roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$.
Here it is how I proceeded:
\begin{align*}
(x+1)(x+3)(x+5)(x+7) + 15 & = [(x+1)(x+7)][(x+3)(x+5)] + 15\\
& = (x^2 + 8x + 7)(x^2 + 8x + 15) + 15\\
& = (x^2 + 8x + 7)[(x^2 + 8x +... | HINT.-Looking about integer solutions for $f(x)=(x+1)(x+3)(x+5)(x+7) + 15 = 0$ possible values should be even and negative so the only candidates are $-2,-4$ and $-6$. We verified that $-2$ and $-6$ are roots. The other two roots are solutions of $$\frac{x^4+16x^3+86x^2+176x+120}{x^2+8x+12}=x^2+8x+10$$
| {
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On how to do a division in a different base. I was wondering if there was a quick way to compute multiplication and division in a base different from base $10$.
Say for example we are in base $12$ then $3*16=40$ the way I do this is by noticing that in base $10$ we have that $3*16=48$ and that $48= 4*12$ so in base twe... | Faster approach is digit by digit. Just you would in base $10$.
$3*16$. You multiply the the $3*6$. As six is half of 12 (just like 5 is half of ten) $3*6 = 12 + 6 = 16$. We write the $6$ down and carry the one. $3*1 = 3$ and we add the one we carried. So we get $3*16 = 46$.
Notice: $3_{10}*16_{10}=48_{10} = 40_{... | {
"language": "en",
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Closed form of function composition Given that $f(x)=\dfrac{x+6}{x+2}$, find $f^{n}(x)$ where $f^{n}(x)$ indicates the $n$th iteration of the function.
I first tried to find a pattern but there didn't seem to be an obvious one:
$$f(x) = \dfrac{x+6}{x+2}$$
$$f^2(x) = \dfrac{7x + 18}{3x + 10}$$
$$f^3(x) = \dfrac{25x + 7... | \begin{align}
a_{n+1} &= a_n + b_n \\
b_{n+1} &= 6a_n + 2b_n\\
c_{n+1} &= c_n + d_n \\
d_{n+1} &= 6c_n + 2d_n
\end{align}
Let's write it in matrix form:
$$\begin{bmatrix} a_{n+1} & c_{n+1}\\ b_{n+1} & d_{n+1} \end{bmatrix}= \begin{bmatrix} 1 & 1 \\6 & 2 \\ \end{bmatrix}\begin{bmatrix} a_{n} & c_{n}\\ b_{n} & d_{n} \... | {
"language": "en",
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Maximum of $\vert f(z)\vert$ on $|z|\leqslant1$, where $f(z)= 1+z^2$
Let $f(z)=z^2+1$. Determine the maximum of $\vert f(z)\vert$ on $\overline{D_1(0)}$
First I want to use the maximum theorem for a complex holomorphic function. But I thought this is an easier way:
$$\vert f(z)\vert = \vert 1+z^2\vert \leq 1+ \vert z... | After you show that $$|f(z)| \leq 2$$
Show a point $z$ that makes $|f(z)|$ attain that value, in particular we can let $z=1$.
Hence the maximum value is indeed $2$.
| {
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Solving equations with complex numbers I have to solve:
$Z^3+\bar{\omega^7} = 0$ and $Z^5\omega^{11}=1$
From the second equation, I got $Z^5=\omega$ and from the first I got $Z^3=-\omega^2$. I plugged in omega from the first result into $Z^3=-\omega^2$, giving me $Z=0$ or $Z^7=-1$, finally giving me 8 solutions:$0,-1,-... | Let's assume that $\omega$ is a root of unity, but not necessarily a cube root of unity, and see what happens.
It's clear that $Z\not=0$. The equation $Z^3+\overline\omega^7=0$ can be rewritten as $Z^3\omega^7=-1$, which, by squaring both sides, implies $Z^6\omega^{14}=1$. Combining with the other equation, $Z^5\omeg... | {
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How existence of an irreducible polynomial in $F_p(x)$ of degree $n$ guarantees existence of finite field of order $p^n$ I read somewhere that if $\pi$ is an irreducible polynomial of degree $m$ then $F_p(x)\ \backslash \left< \pi \right>$ is a finite field of order $p$. What is $F_p(x)\ \backslash \left< \pi \right>$?... | If $p$ is irreducible in $F[X]$ ($F$ being any field), the quotient ring $F[X]/(p)$ is a field because an irreducible polynomial generates a maximal ideal of $F[X]$, and an $F$-vector space of dimension $n=\deg p$.
If $F=\mathbf F_p$ is the prime field with characteristic $p$, a vector space of dimension $n$ has exactl... | {
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$A\in M_n(\mathbb{R})$ is symmetric s.t. $A^{10}=I.$ Prove $A^2=I$
Let $A\in M_n(\mathbb{R})$ be symmetric, such that $A^{10}=I.$ Prove $A^2=I$
My thoughts:
Since $A$ is symmetric, $A^2$ is symmetric, so there exists an orthogonal $P\in M_n(\mathbb{R})$ such that $D=P^{-1}A^2P$ is a diagonal matrix.
I tried to work w... | The minimal polynomial of $A$ divides $x^{10}-1$.
$x^{10}-1=(x^2-1)q(x)$, where $q(x)$ has no real roots.
The eigenvalues of a real symmetric matrix are all real
and so its minimal polynomial is a product of linear real factors.
Therefore, the minimal polynomial of $A$ divides $x^2-1$ and so $A^2=I$.
| {
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"source": "stackexchange",
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Symmetric Matrix. $\mathcal{S}_{n}(\mathbb{R}) $ is the set of symmetric matrices of $ \mathcal{M}_{n}(\mathbb{R})$
*
*Show that if $ A \in \mathcal{S}_{n}(\mathbb{R})$, then: $ A = \sum_{i=1}^{n} a_{ii}E_{ii} + \sum_{1 \leq i < j \leq n}^{n}(2a_{ij})\big(\frac{1}{2}(E_{ji}+E_{ij}) \big)$
$E_{ii}$ is the elementar... | We can always decompose a matrix as a sum of elementary matrices, that is,
$$A= \sum_{1\leqslant i,j\leqslant n}a_{i,j}E_{i,j}=\sum_{i=1}^na_{ii}E_{ii}+\sum_{1\leqslant i\neq j\leqslant n}a_{i,j}E_{i,j}.$$
The second sum can be written as
$$\sum_{1\leqslant i\neq j\leqslant n}a_{i,j}E_{i,j}=\sum_{1\leqslant i\lt j\le... | {
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Determinant is alternating over a commutative ring with $1$ In Section 11.4 of Dummit and Foote, they introduce a determinant function $\det$ on the ring of $n\times n$ matrices over a commutative ring $R$ with $1$ as
*
*Any $n$-multilinear alternating form, where the $n$-tuples are the $n$ columns of the matrices ... | For a fully elaborated proof, I shall be lazy and just refer to Exercise 6.7 (e) in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. Or the proof of property (iii) in §5.3.4 of Hartmut Laue, Determinants. The main idea is to split the sum $\sum\limits_{\sigma \in S_n} \ldots$ into a su... | {
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Factoring within a proof In the proof text I am using, I am trying to understand a proof of the fact that the geometric mean is less than or equal to the arithmetic mean by showing that:
rst $\le$ (r$^3$ + s$^3$ + t$^3$)/3
The answer in the back says to note that:
r$^3$ + s$^3$ + t$^3$ - 3rst = $\frac 12$(r + s + t)[(r... | $$r^3+s^3+t^3-3trs=r^3+3r^2s+3rs^2+s^3+t^3-3r^2s-3rs^2-3rts=$$
$$=(r+s)^3+t^3-3rs(r+s+t)=(r+s+t)((r+s)^2-(r+s)t+t^2)-3rs(r+s+t)=$$
$$(r+s+t)(r^2+2rs+s^2-rt-st-3ts)=$$
$$=(r+s+t)(r^2+s^2+t^2-rs-rt-st)=\frac{1}{2}(r+s+t)((r-s)^2+(r-t)^2+(s-t)^2).$$
| {
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Expectation value of exponential of a function - first two moments of function known I have a function $f(t)$ and know that $\langle f(t)\rangle=0$ and $\langle f(t)f(t')\rangle=C(t-t')$;
Now i want to calculate:
$$\left\langle\exp\left(\int\limits_{0}^t f(t') \, \mathrm{d}t'\right)\right\rangle$$
I tried to look at th... | I think what you mean is that $f$ is a stochastic process (rather than a function) of mean $0$ and covariance $C(t - t')$. If it's a Gaussian process, that's all you need to determine it, but if it's non-Gaussian you really don't know enough to say anything about expectations of exponentials. They might not even exis... | {
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Where is the use of continuity in this Munkres Topology question?
*Let $A \subset X$; let $f : A \rightarrow Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g : \bar A \rightarrow Y$, then $g$ is uniquely determined by $f$.
I believe I've solved this, here's a... | You're taking in $U$ the intersection of two inverse images of open sets under $g$ and $h$. The latter are open by continuity of $g$ resp. $h$ (the intersection is open by the topology axioms). The openness is needed for it to intersect $A$.
The continuity of $f$ is needed because if $f$ has a continuous extension $g$ ... | {
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Have incompletable Sudokus ever been studied? Weird question, I know, but this is in relation to an extension of Sudoku into a set of sequential, partisan games which always results in incompletable Sudoku. (i.e. the requirements of strategic play lead to choices that create "dead cells" in which no integer may be plac... | There has been a bit of work on this problem, I think this 2012 result is the most well known. It basically says that if there are less than 17 filled entries, then a Sudoku cannot be uniquely completed (there will be more than one way to complete it).
http://www.ucd.ie/news/2012/01JAN12/100111-There-is-no-16-clue-or-l... | {
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How can estimate $\pi$ with differential equation? I know some method for estimation of $\pi$ .We can use Monte Carlo method to estimate $\pi $.We can use series to estimate $\pi $.
And my question is : Is there exist a (1 st order) differential equation or stochastic differential equation that can estimate $\pi $ ? or... | For a first-order solution, you could use:
Proposition 0. If $$f'(x) = \sqrt{1-x^2}, \qquad f(-1)=0$$
then $2f(1) = \pi$.
However, I think a second-order solution is a bit more conceptually satisfying. You could use:
Proposition 1. $$\pi = \min_{\theta > 0} \,\left(\sin\theta \leq 0\right).$$
So just write a progra... | {
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Smallest Possible Power When working on improving my skills with indices, I came across the following question:
Find the smallest positive integers $m$ and $n$ for which: $12<2^{m/n}<13$
On my first attempt, I split this into two parts and then using logarithms found the two values $m/n$ had to be between. However I ... | The inequality is equivalent to $\,12^n \lt 2^m \lt 13^n\,$. By brute force, looking for powers of $2$ between $12^n$ and $13^n$ starting from the lowest possible $n=1$ up:
*
*$\;n=1\,$: no solutions, since $\,2^3 = 8 \lt 12^1 \lt 13^1 \lt 16=2^4\,$
*$\;n=2\,$: no solutions, since $\,2^7 = 128 \lt 144 = 12^2 \lt 13... | {
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What is the number of ways to select ten distinct letters from the alphabet $\{a, b, c, \ldots, z\}$, if no two consecutive letters can be selected?
What is the number of ways to select ten distinct letters from the alphabet $\{a, b, c, \ldots, z\}$, if no two consecutive letters can be selected?
There are a couple o... | Here is another ways of visualizing the problem. First, we arrange $16$ blue balls and $10$ green balls so that no two of the green balls are consecutive. Then we label the balls.
Line up $16$ blue balls, leaving spaces between successive balls and at the ends of the row. There are $17$ such spaces, $15$ between suc... | {
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"answer_id": 0
} |
How to evaluate this integral $I=\int^{b}_{0}\sqrt{a-x^6}dx$ How to evaluate this integral
$$I=\int^{b}_{0}\sqrt{a-x^6}dx$$
Initially I tried substituting $x^3=\sin(t)$, but the integral becomes messy when finding $dx$. so is there any trick to evaluate this integral?
| As said in comments, the antiderivative is quite messy (involving elliptic integral of the first kind) plus many nasty terms.
For the definite integral, as Raffaele commented,
$$I=\int^{b}_{0}\sqrt{a-x^6}\,dx=b\,\sqrt{a} \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};\frac{b^6}{a}\right)$$ provided that $b\geq 0\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What two input functions satisify an output between 0 and 1? What are common functions that take two input variables and make the output between 0 and 1?
Question is as simple as that, two inputs and one output, output needs to stay between 0 and 1!
| This question is way way way way way way too broad. There are infinitelly many such functions.
Examples:
*
*$f(x,y)=|\sin(x)|$
*$f(x,y)=\sin^2(x)\sin^2(y)$
*$f(x,y)=1$
*$f(x,y) = e^{-x^2-y^2}$
There are many more. Without further details, it's hard to give a more accurate answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Group Theory: Proof of the cycle decomposition of the conjugate The following theorem is proven on this site:
Let $\pi$ and $\rho$ be permutations of $\{ 1,...n\}$. The cycle decomposition of $\rho \pi \rho^{-1}$ is obtained by replacing each integer $j$ in the cycle decomposition of $\pi$ with the integer $\rho (i)$.... | You only have to check it for cycles. Write $\pi = (\rho^{-1}(a_1),\rho^{-1}(a_2),\dots, \rho^{-1}(a_n))$. This is possible, because $\rho$ is a permutation. Now let $\rho\pi\rho^{-1}$ act on $a_i$ and you will see that indeed it is mapped to $a_{i+1}$. And that's exactly what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Nonlinear ODE of second order with Boundary Conditions defined. The problem is:
$y''(x)-a\cdot(y(x))^2=0, a>0$
S.t. $ y(0)=b,
\lim_{x\to\infty } y'(x)=0$
That problem results from a catalyst which has a chemical reaction of second order occuring within it - the book Transport Phenomena of Bird at. al. contains that que... | $$y''-a\:y^2=0\quad\to\quad 2y''y'-2a\:y^2y'=0$$
$$y'^2-\frac{2a}{3}y^3=c_1$$
$$y'=\pm \sqrt{c_1+\frac{2a}{3}y^3}$$
$$dx=\pm\frac{dy}{\sqrt{c_1+\frac{2a}{3}y^3}}\quad\to\quad x=\pm\int \frac{dy}{\sqrt{c_1+\frac{2a}{3}y^3}}$$
This integral involves the elliptic functions, which would be rather arduous.
By luck, we will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How do I calculate the integral of $\lfloor1/x\rfloor$ from $x=\frac{1}{n+2}$ to $x=\frac{1}{n}$? How do I calculate integral : $$\int_{\frac{1}{n+2}}^{\frac{1}{n}}\lfloor1/x\rfloor dx$$ where $\lfloor t\rfloor$ means the integer part (I believe that's how it should be translated) or floor function of $t$.
| I think this should be possible,to solve by decomposing the integrale from $1/(n+2)$ to $1/(n+1)$ and so on. On each of these Integrals, your function should be constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Wave re-construction - What are my mathematical options While my question applies to both DSP and Math, I feel it has more depth in mathematics. Here is a sample photo of some samples I have captured over time.
About the data:
I have a sensor that monitors pressure. When the pressure increases, the y axis value increa... | The way I would approach that would be through a Fourier series, you are going to have to look into a bit of calculus etc. to do that though.
Wikipedia - Fourier Series
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find $x-\sqrt{7x}$ given that $x - \sqrt{\frac7x}=8$
If $ x - \sqrt{\frac{7}{x}}=8$ then $x-\sqrt{7x}=\text{?}$
I used some ways, but couldn't get the right form :) by the way, the answer is $1$.
Thanks in advance.
| Let $x=7y^2,$ where $y>0$.
Thus, $$7y^2-\frac{1}{y}=8$$ or
$$7y^3-8y-1=0$$ or
$$7y^3+7y^2-7y^2-7y-y-1=0$$ or
$$(y+1)(7y^2-7y-1)=0.$$
Thus,
$$x-\sqrt{7x}=7y^2-7y=1.$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Derivative as difference quotient with tricky limits due to square roots I have a function that I want to find the derivative of using the difference quotient definition of a derivative. The function is:
$$f(x)=\frac{\sqrt{x}}{x+1}$$
therefore, using the difference quotient definition, we have:
$$f'(x)=\lim_{h\to0}\fra... | Using your idea, just ignore the denominator until later. If you go through the numerator, you will see that
$$\begin{align*}
&\left((x+1)\sqrt{x+h}-(x+h+1)\sqrt{x}\right)\cdot\left((x+1)\sqrt{x+h}+(x+h+1)\sqrt{x}\right)\\
=&(x+h)(x+1)^2-x(x+h+1)^2\\
=&hx^2+2hx+h+x^3+2x^2+x-h^2x-2hx^2-2hx-x^3-2x^2-x\\
=&-hx^2-h^2x+h\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Total Derivatives and Total Differential I am confused between total derivatives and total differential. What is the difference between total derivatives and
total differential?
| Let $f: U \subset \Bbb R^n \to \Bbb R^m$ be differentiable.
The total derivative of $f$ at $a$ is the linear map $df_a$ such that $f(a+t) - f(a) = df_a(t) + o(t)$.
For $m=1$, the total differential of $f$ is
$$df = \sum_{i=1}^m \frac{\partial f}{\partial x_i} dx_i$$
Hope this helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2347902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Riccati D.E., vertical asymptotes
For the D.E.
$$y'=x^2+y^2$$
show that the solution with $y(0) = 0$ has a vertical asymptote at some point $x_0$. Try to find upper and lower bounds for $x_0$:
$$y'=x^2+y^2$$
$$x\in \left [ a,b \right ]$$
$$b> a> 0$$
$$a^2+y^2\leq x^2+y^2\leq b^2+y^2$$
$$a^2+y^2\leq y'\leq b^2+y^... | 1. $x_0$ exists
First note that $y'''(x)$ is increasing$^{[1]}$. It is also easy to see that $y'(0)=y''(0)=0$ but $y'''(0)=2$$^{[2]}$, so by Taylor's theorem$^{[3]}$,
$$
y(x)=\frac{x^3}{6}y'''(c)\ge \frac{x^3}{3},\qquad (*)
$$
for all $x>0$ such that $y$ is defined. Choose one such $x=\epsilon>0$. Then if $x>\epsilo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2348022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find critical points of $\langle Av,v\rangle$
Let $A$ be an $n \times n$ real symmmetric matrix. Find the critical points of the function $\langle Av,v\rangle$ restricted to the unit sphere in $\mathbb{R}^n$.
I would think you just use Lagrange multipliers, and $\nabla\langle Av,v\rangle=2Ax$, since $A$ is symmetric.... | Actually the restriction of a function to the unit sphere does not have directional derivates perpendicular to the sphere surface. This means that the relevant gradient is the gradient of the extended function projected onto the tangent plane of the surface that is $\nabla A - \langle \nabla A, \hat n\rangle$:
$$2Ax - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2348172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Concentration of two independent sub-Gaussian random variables Suppose $X$ and $Y$ are independent sub-Gaussian random variables with 0 mean and $\sigma^2$ sub-Gaussian parameter. More specifically, $\mathbb E[\exp(a^T X)]\leq \exp\{\|a\|_2^2\sigma^2/2\}$ for all $a$, and the same holds for $Y$ as well.
I wish to upper... | Moment generating functions of subgaussian vector can often be bounded from above by the same moment generating function, with the subgaussian vector replaced by a standard normal. This is equivalent to zhoraster's answer.
Take $\sigma=1$ without loss of generality (otherwise, consider $X/\sigma$ and $Y/\sigma$ and sca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2348343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\sum_{x=0}^{n}(-1)^x\binom{n}{n-x} (n+1-x)^n=n!$ I figure out these thing when "playing" with numbers:
$$3^2-2.2^2+1^2=2=2!$$
$$4^3-3.3^3+3.2^3-1^3=6=3!$$
$$5^4-4.4^4+6.3^4-4.2^4+1^4=24=4!$$
So I go to the conjecture that:
$$\binom{n}{n}(n+1)^n-\binom{n}{n-1}n^n+\binom{n}{n-2}(n-1)^n-...=n!$$
or
$$\sum_{x=0... | First of all note that $$\sum_{k=0}^{n}\dbinom{n}{n-k}\left(-1\right)^{k}\left(n-k+1\right)^{n}=\sum_{k=0}^{n}\dbinom{n}{k}\left(-1\right)^{k}\left(n-k+1\right)^{n}$$ then from the special case of the Melzak's identity: $$\sum_{k=0}^{n}\left(-1\right)^{k}\dbinom{n}{k}\frac{f\left(y-k\right)}{x+k},=\frac{f\left(x+y\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2348444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 4
} |
What is the exterior algebra of $\textbf{R}^2$ Let $V$ be an $n$-dimensional vector space.
The exterior algebra $\Lambda V$ of $V$ is the direct sum of the exterior powers $\Lambda^kV$. It comes with a product (called the exterior product) which is bilinear, alternating and anticommutative. The dimension of $\Lambda V$... | The exterior algebra $\Lambda \mathbb{R}^2$ is a real vector space of dimension 4 with basis $1, e_1, e_2, e_1 \wedge e_2$. So its every element is a unique linear combination of these basis elements, say $a_1 \cdot 1 + a_2 e_1 + a_3 e_2 + a_4 e_1 \wedge e_2$, for real numbers $a_1, a_2, a_3, a_4$, which can be chosen ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2348787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why are critical points called critical? For a function $y = f(x)$, a number $x_0$ is called $\textit{critical}$ if either $f'(x_0) = 0$ or $f'(x)$ does not exist. Sometimes the term $\textit{stationary}$ is used, but it is by far less popular. My question is
Why is the word "critical" used in this case as terminology... | Critical as in "important" or "key" (for analyzing the behavior of the function).
For a continuous function from $\mathbb{R}$ to $\mathbb{R}$, the critical points may or may not correspond to actual turning points, but they are the only places where a turning point is possible. Thus, one could say that analyzing the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2349001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 1
} |
Other ways to evaluate the integral $\int_{-\infty}^{\infty} \frac{1}{1+x^{2}} \, dx$?
$$\int_{-\infty}^{\infty}\frac{1}{x^2+1}\,dx=\pi $$
I can do it with the substitution $x= \tan u$ or complex analysis. Are there any other ways to evaluate this?
| You can use partial fractions:
$$
\begin{align}
\int_{-\infty}^\infty \frac{dx}{1+x^2}
& = \int_{-\infty}^\infty \frac{1}{2} \left( \frac{1}{1+ix} + \frac{1}{1-ix} \right) dx \\
& = \frac{1}{2i} \bigg[\log(1+ix) - \log(1-ix)\bigg]_{-\infty}^\infty \\
& = \frac{1}{2i} \left[ \lim_{x\to\infty} \log\left( \frac{1+ix}{1-ix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2349224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 1
} |
Question about 'Archimedes parabola' In the Wikipedia article on lever mechanics from the Archimedes codex where
"The first proposition states:
The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB."
The wikipedia proof ends prematurely in my view:
"In other wo... | An analytic proof. Let $y=ax^2$ be the equation of a generic parabola, and
$A=(x_1,ax_1^2)$, $B=(x_2,ax_2^2)$ any two points on it. The equation of tangent $BC$ is then $y-ax_2^2=2ax_2(x-x_2)$ and $C=(x_1,2ax_1x_2-ax_2^2)$. Point $D$ is the midpoint of $AC$, thus: $D=(x_1,ax_1x_2+a(x_1^2-x_2^2)/2)$.
Let now $x$ be the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2349294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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