Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
A $k+1$-sphere containing a $k$-sphere and a point. Earlier I asked a question on whether it is possible to find a sphere passing through a circle and a point non-coplanar to it. I wanted to know whether this was possible to do in higher dimensions.
In $\mathbb{R}^{k+2}$, given a $k$-sphere and a point outside the $k$... | Use the algebric approach:
Equation of the hypersphere of dimension $k$:
$\{(x_1,...,x_{k+2}) | \sum_{i=1}^{k+1} x_i^2 = R^2,x_{k+2}=0 \}$
Point $M$ outside of the plan: $ 0 \not = a_{k+2}$
Can we find $(b_1,...,b_{n+2})$ the center and $R'$ the radius of a new sphere containing both?
By definition that sphere needs ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/919086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve $\cos{z}+\sin{z}=2$ I am trying to solve the question:
$\cos{z}+\sin{z}=2$
Where $z \in \mathbb{C}$
I think I know how to solve $\cos{z}+\sin{z}=-1$:
$1+2\cos^2{\frac{z}{2}}-1+2\sin \frac{z}{2}\cos{\frac{z}{2}}=0\\
2\cos{\frac{z}{2}}(\cos{\frac{z}{2}}+\sin{\frac{z}{2}})=0$
etc... (that is, if the double angl... | Note that
$$\cos(\arctan(z)) + \sin(\arctan(z)) = \frac{1}{\sqrt{1+z^2}} + \frac{z}{\sqrt{1+z^2}}.$$
So, if you can solve the equation
$$\frac{1}{\sqrt{1+z^2}} + \frac{z}{\sqrt{1+z^2}} = 2,$$
(which is quadratic) then you can solve the original.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/919230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
General formula of repeated roots. Prove that $$\underbrace{\sqrt{k\sqrt{k\sqrt{k\sqrt{\cdots\sqrt{k}}}}}}_{n\text { times}}=k^{1-1/2^n}$$
How do I derive this formula?
| Take the logarithm (as already suggested in comments):
$$ \frac{1}{2}\left(\ln k + \frac{1}{2}\left(\ln k + \frac{1}{2}\left( \ln k + \ldots \right) \right)\right)$$
$$ =\left(\frac{1}{2}+ \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^n}\right)\ln k$$
$$ =\left(1 - \frac{1}{2^n}\right) \ln k.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/919313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Proof by Induction - Algebra Problem (Steps included but not understood) I do not quite understand this proof, if anyone could explain the steps for me it would be greatly appreciated. It's probably something glaringly obvious I'm not seeing, thanks in advance.
Prove that for every integer $n \ge 0,$ the number $4^{2n+... | The glaringly obvious looks like the concept of proof by induction itself. To prove a proposition $P(n)$ is true for all $n\geq 0$, it suffices to prove that $P(k) \Rightarrow P(k+1)$ (which is done by assuming $P(k)$, then showing $P(k+1)$, after which you may discharge the assumption and write down $P(k) \Rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/919380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Matrix inversion using decomposition Why do I need $LU$, $QR$ and other matrix decomposition tech to find the inverse of a matrix?
Please if possible provide a ref for the answer.
Thanks in advance
| If you have the $A=LU$ decomposition, then you could get the the $A^{-1}$ inverse matrix of course with $A^{-1}=U^{-1}L^{-1}$. Computer algebra systems often use the method to calculate the inverse matrix. You can read about it at LU decomposotion wikipedia article. This paper also write about it. You can find details ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/919464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit using Stirling's formula Consider the limit
$$\lim_{n \to \infty} \left[n(1-2\log2) + \sum_{k=1}^{n} \log\left(1+\frac {k}{n}\right)\right] = \frac {1}{2} \log 2.$$
This can be shown by using Stirling's formula for $n!$. My question is if this is the only way or there is also an elementary solution.
| This is the case $f:x\mapsto\log(1+x)$ of the more general result that, for every $C^1$ function $f$ on $[0,1]$,
$$\lim_{n \to \infty} \sum_{k=1}^{n} f\left(\frac {k}{n}\right)-n\int_0^1f(x)\mathrm dx = \frac {1}{2}(f(1)-f(0)).$$
To show this, one can use Riemann sums and a first-order Taylor expansion on $f$.
More ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Upper bound on $\sum_{n=1}^N \frac{1}{n}$ are there any analytical upper bounds on
\begin{align}
\sum_{n=1}^N \frac{1}{n}
\end{align}
Clearly, one upper bound is $N$. But there has to be a better one. There is an approach with Harmonic number but it's not "analytic".
Thank you
| $\sum_{n=1}^N {1 \over n} \le 1+ \ln N$.
This follows from $\int_{n-1}^n {1 \over t} dt = \ln n - \ln (n-1) > {1 \over n}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Why does the definition of the functional limit involve a limit point? This may be an odd question, and I'm not sure if these type of questions are at all appreciated in the maths community. But given the definition of the functional limit:
Let $f: A \to \mathbb{R}$, and let $c$ be a limit point of the domain $A$. We ... | To see where things go wrong take $A=\mathbb{N}$ and some function
$f:A\to\mathbb{R}$ and say you want to prove that
$$
\lim_{x\to c}f(x)=L
$$
where $c\in A$ is a natural number.
Let $\epsilon>0$ and choose $\delta(\epsilon)=\delta=\frac{1}{2}$
Indeed - For every $x\in A$ s.t $0<|x-c|<\frac{1}{2}$ we have it
that $|f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/919807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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A $(3k+3)$-node graph with degrees $k+1,\dots,k+3$ has $k+3$ degree-$(k+1)$ nodes or $k+1$ degree-$(k+2)$ nodes or $k+2$ degree-$(k+3)$ nodes?
Graph $G$ has order $n=3k+3$ for some positive integer $k$ . Every vertex of $G$ has degree $k+1$, $k+2$, or $k+3$. Prove that $G$ has at least $k+3$ vertices of degree $k+1$ o... | Since $G$ has only vertices of degrees $k+1$, $k+2$ and $k+3$ you can split all of the vertices into 3 sets: $X$, $Y$ and $Z$ where $X$ is set of vertices with degree $k+1$, $Y$ with $k+2$ and $Z$ with $k+3$.
Now, since you already know the answers for cases:
$1)$ $|X|<k+3$ and $|Y|<k+1 \Rightarrow |Z|\geq k+2$
$2)$ $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/919894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluating Summation of $5^{-n}$ from $n=4$ to infinity The answer is $\frac1{500}$ but I don't understand why that is so.
I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that $x=\frac15$ and plugging in the values I have $\frac1{1-(\frac15)}... | The problem you have is that you do not know why the formula works to begin with. If you did the situation would be clear. Here's the thing:
$$\sum_{n=0}^{\infty}r^n=\frac{1}{1-r} \; \;\;\;\;\; |r|<1.$$
Let $r=\frac{1}{5}$, then you really have the following situation:
$$\left(\frac{1}{5}\right)^0+\left(\frac{1}{5}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Evaluating $\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}$ How to evaluate the following limit? $$\lim_{x \to 0}\frac{(1+x)^{1/x} - e}{x}.$$
| You may write
$$
\begin{align}
\frac{(1+x)^{1/x} - e}{x} &= \frac{\large e^{\large\frac{\log (1+x)}{x}} - e}{x}\\\\
&= \frac{e^{\large \frac{x-\frac{x^2}{2}+{\mathcal{O}}(x^3)}{x}} - e}{x}\\\\
&= \frac{ e^{1-\frac{x}{2}+{\mathcal{O}}(x^2)} - e}{x}\\\\
&= \frac{ e \:e^{-\frac{x}{2}+{\mathcal{O}}(x^2)} - e}{x}\\\\
&= \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Showing Surjectivitity of $f(x) = x^3$ I want to show that the function $f: \mathbb{R} \to \mathbb{R},\; f(x) = x^3$ is surjective.
First Question: If a function has an inverse, it is bijective yes?
Second Question: Is my process below, for showing surjectivity of $y = x^3$, incorrect because I am assuming that the fu... | (1) Yes, by definition, the inverse of a map $f: X \to Y$ is a map $f^{-1}: Y \to X$ such that $f^{-1} \circ f = id_X$ and $f \circ f^{-1} = id_Y$, so in particular $f$ must be bijective.
Note that one commonly one will say that a function $f$ defined by a rule (but without specifying the codomain) has an inverse, but ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Series $\sum_{n=0}^{\infty}\frac{1}{2^n-1+e^x}$ properties We need to prove $$f(x)=\sum_{n=0}^{\infty}\frac{1}{2^n-1+e^x}<\infty\ \ \forall_{x\in\mathbb{R}}$$
and then find all continuity points and all points in which $f$ is differentiable + calculate $f'(0)$. Especially the latter part is of difficult to me. It can b... | use ratio test,let $a_n=\frac{1}{2^n-1+e^x}$ and take,$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0.5<1 $$so$\sum_{n=1}^{\infty}a_n$ converges
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/920288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find vertices pointing to common vertex In a directed graph I'm interested in finding pairs of vertices pointing towards a common vertex. More in detail, from an adjacency matrix I want to derive a matrix where a positive entry denotes that the vertices represented by the entry's row and column both point to a common v... | Yes, you're correct (nice guess!)
I find it a bit easier to visualize the $(i,j)$-entry of $AA^T$ as the standard inner product of rows $i$ and $j$ of $A$. Since all entries of $A$ are non-negative, you will got a positive value if and only if there is a column $k$ so that both entries in coordinates $(i,k)$ and $(j,k)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Conditional expected value of the maximum of n normally distributed variables What is the conditional expected value of the maximum of $n$ normally distributed variables $x_i$, conditionally on exceeding some threshold $z$?
That is, for $x=\max(x_1,...,x_n)$, the probability that $x$ will exceed some threshold $z$ is... | $$E(x\mid x\gt z)=z+\frac1{1-F(z)^n}\int_z^\infty(1-F(t)^n)\mathrm dt$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/920458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How prove $g(x)$ is odd function :$g(x)=-g(-x)$ QUestion:
let $f(x),g(x)$ is continuous on $R$,and such
$$f(x-y)=f(x)f(y)-g(x)g(y)$$
and $f(0)=1$
show that: for any $x\in R$, have $g(x)=-g(-x)$
my try: let $x=y=0$,then
$$f(0)-[f(0)]^2=g(0)g(0)\Longrightarrow g(0)=0$$
and let $x=0$, note $f(0)=1,g(0)=0$,so
$$f(-y)... | First note that by using $f(x)=f(-x)$
$$
f(0) = f(x-x) = f(x)^2- g(x)^2 \\
f(0) = f(-x-(-x))=f(x)^2-g(-x)^2 \\
g(x)^2 = g(-x)^2
$$
Thus let one define into two sets (note an element may be in both)
$$
\Gamma^+=\{x \in \mathbb{R} : g(x) = g(-x) \} \\
\Gamma^-=\{x \in \mathbb{R} : g(x) = -g(-x) \} \\
$$
Clearly $\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
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Inequality: $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$ I need to determine the range of $p,q,r$ such that $2(p^2+q^2+r^2)+2(pq+qr+rp)\ge pqr$. I am not given any other information except that $p,q,r\in \mathbb{R}$. I haven't solved a problem like this before, so I'm really stuck. I'm thinking that it might be true for all rea... | This doesn't hold true in general.
For example, let $p=q=r$, you will get $$12p^2\geq p^3.$$ You can see it is not true when $p>12$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/920623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Limit of this recursive sequence and convergence $$a_{n+1}=\sqrt{4a_n+3}$$ $a_1=5$
I can solve simpler but I get stuck here because I cant find an upper bound or roots of the quadratic equation $a_{n+1} -a_n= \frac{4a_n+3 - a_n^2}{\sqrt{4a_n +3}+a_n}...$ to find monotony.
I tried this generic aproach but have difficu... | Prove it using induction. Whether the sequence is increasing or decreasing depends on the value of $a_1$. Observe that $$a_n \le a_{n+1} \implies 4a_n + 3 \le 4a_{n+1} + 3 \implies a_{n+1} \le a_{n+2}$$ and similarly $$a_n \ge a_{n+1} \implies 4a_n + 3 \ge 4a_{n+1} + 3 \implies a_{n+1} \ge a_{n+2}.$$ Thus if $a_1 \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Function that is absolutely continuous but not $C^1$
*
*I would like to know example of a function that is absolutely
continuous on a compact subset of real line but is not $C^1$.
*Does always $C^1$ imply absolute continuity on unbounded subsets of real line? I know that it is so on compact subsets.
|
Let $-\infty<a<b<\infty$ . A function $u:[a,b]\to\mathbb{R}$ will be absolutely continuous if and only if, $u'$ does exist almost everywherem $u'\in L^1(a,b)$ and $$u(x)=u(a)+\int_a^x u'(t)dt.$$
Therefore, take any function $v\in L^1(a,b)$ and let $$u(x)=\int_a^x v(t)dt.\tag{1}$$
You can verify that $u'=v$ a.e. and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that lcm$(a,b)= ab$ if and only if gcd$(a,b)=1$ Not sure how to begin. If gcd$(a,b)=1$ what can I deduce from that?
| Hint $ $ note that $n\mapsto ab/n\,$ bijects the common divisors of $\,a,b\,$ with the common multiples $\le ab.$ Being order-reversing it maps the greatest common divisor to the least common multiple, i.e. $\,\gcd(a,n)\mapsto ab/\gcd(a,b) = {\rm lcm}(a,b).\,$ Thus $\,\gcd(a,b)=1\iff {\rm lcm}(a,b) = ab$.
Remark $\ $ F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/920978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
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Second derivative of Hypergeometric function I'm looking for the following second derivative
$$
\kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = 0} ,
$$
where $\alpha$ is a real parameter in $[0,1]$. As you may have guess... | The definition of $\operatorname{Li}_2$ used in the last formula is
$$\operatorname{Li}_2(z) =
-\operatorname{Li}_2 \left( \frac 1 z \right) -
\frac {3 \ln^2(-z) + \pi^2} 6,
\quad |z| > 1.$$
Therefore
$$\kappa_2 =
4 \operatorname{Li}_2 \left( \frac \alpha {\omega - \alpha} \right) -
4 \operatorname{Li}_2 \left( \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Are these proofs logically equivalent? Here are two proofs, firstly:
x = 0.999...
10x = 9.999...
= 9 + 0.999...
= 9 + x
9x = 9
x = 1
And secondly:
x = 1 - 1 + 1 - 1 + 1 - 1 ...
= 1 - (1 - 1 + 1 - 1 + 1 ...
= 1 - x
2x = 1
x = 1/2
The fourth line in the first and the third line in the second use the same tr... | No. The first series is absolutely convergent, so its terms can be rearranged without changing the sum, while the second series is divergent.
The number $0.\overline{9}$ can be expressed in the form
\begin{align*}
0.\overline{9} & = \sum_{k = 1}^{\infty} \frac{9}{10^k}\\
& = \frac{9}{10} \sum_{k = 0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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golden ratio from new formula? perhaps from theory of modular units? Please consider the following infinite product series which I found by pure happenstance:
$$\frac{1+\sqrt{5}}{2}= e^{\pi/6} \prod_{k=1}^\infty \frac{1+e^{-5(2k-1)\pi}}{1+e^{-(2k-1)\pi}}$$
My question: Would this formula fit within, or be obtainable by... | It turns out that OP also posted this question on my blog page. And I provide the answer I posted there.
The formula can be expressed as $$\frac{1 + \sqrt{5}}{2} = e^{\pi/6}\prod_{k = 1}^{\infty}\frac{1 + e^{-5(2k - 1)\pi}}{1 + e^{-(2k - 1)\pi}} = \frac{2^{-1/4}e^{5\pi/24}}{2^{-1/4}e^{\pi/24}}\prod_{k = 1}^{\infty}\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Does the analytic continuation of $f$ always exist? Let $f(z)$ be a holomorphic funtion on region $\Omega$. Then, does the analytic continuation of $f$ always exist? Note that $f$ is always the analytic continuation of itself, so I exclude this case.
If the answer is yes, then what is the analytic continuation of $\su... | The function
$$f(z)=\sum_{n=0}^\infty z^{n!}$$
has no analytic continuation beyond $|z|<1$. The circle $|z|=1$ is a natural boundary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/921377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Bounding the size of all consecutive sums of i.i.d. random variables Let $X_1, X_2, \ldots$ by an i.i.d sequence of bounded (for simplicity), mean zero random variables. For any $a<b$, call $S_{a,b} = X_{a+1} + \cdots + X_b$.
I would like to show that for any $\epsilon >0$, there is a $\delta > 0$ such that for all $N... | Ok, I figured this out. The Bernstein inequality states that if $|X_i| \leq M$,
$$\mathbf{P} \left (\sum_{i=1}^n X_i \geq t \right ) \leq \exp \left ( -\frac{\frac{1}{2} t^2}{\sum \mathbf{E} \left[X_j^2 \right ]+\frac{1}{3} Mt} \right) $$.
This gives exponential control of any window $|S_{a,b}|$ being greater than $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Quick way to find eigenvalues of anti-diagonal matrix If $A \in M_n(\mathbb{R})$ is an anti-diagonal $n \times n$ matrix, is there a quick way to find its eigenvalues in a way similar to finding the eigenvalues of a diagonal matrix? The standard way for finding the eigenvalues for any $n \times n$ is usually straightfo... | For ease of formatting and explanation, I'll be doing everything for the $5 \times 5$ example. However, the same trick works for any $n \times n$ antisymmetric matrix (though slightly differently for even $n$).
Suppose
$$
A =
\begin{pmatrix}0&0&0&0&a_{15}\\0&0&0&a_{24}&0\\0&0&a_{33}&0&0\\0&a_{42}&0&0&0\\a_{51}&0&0&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 3,
"answer_id": 2
} |
To find the determinant of a matrix Given $A_{n\times n}$=$(a_{ij}),$ n $\ge$ 3, where $a_{ij}$ = $b_{i}^{2}$-$b_{j}^2$ ,$i,j = 1,2,...,n$ for some distinct real numbers $b_{1},b_{2},...,b_{n}$. I have to find the determinant of A. I can see that A is a skew-symmetric matrix. So determinant of A is $0$ when n is odd. B... | Hint: The coefficients of the characteristic equation satisfied by $A$ are given by the equation: $c_{n} = - \frac{1}{n} \left(c_{n-1}\mathrm{Trace}(A) + c_{n-2}\mathrm{Trace}(A^{2})+ ... + c_{1}\mathrm{Trace}(A^{n-1}) + \mathrm{Trace}(A^{n}) \right)$ with $c_{0} = 1$. Now the conditions in the problem imply that $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality with prime numbers I found exercise in my book for number theory that I can't resolve. How do you show that
$$p_n < e^{1+n}$$
where $p_n$ is $n$-th prime number?
| Let's try induction. For the base case of $n=1$, it is clear that $p_{1}=2<e^{1+1}=e^2$. Now let's suppose for all $k$ where $n \geq k >1$ that our result holds. We know by Bertrand's Postulate that $p_{k+1}<2p_{k}$ for all $k \in \mathbb{N}$ and by induction we know $2p_{k}<2e^{k+1}$. It follows that $$p_{k+1}<2e^{k+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/921919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Any number $n>11$ is a sum of two composite numbers I have seen the following problem:
Any number $n>11$ can be formed by the sum of two composite numbers
and it says to prove it by contradiction. I have done the following:
Assume that the sum will be formed by non-composite numbers. Because a composite number is one t... | No, it is not fine. If $n$ is not the sum of two composite numbers, it does not follow that it must be the sum of two primes. It might be the sum of a prime and a composite number. All you can say is that it cannot be expressed as the sum of two composite numbers. Also 1 is not regarded as either prime or composite.
A... | {
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If $\space o(\ker\phi) = n$, then $\varphi$ is an $n$-to-$1$ mapping from $G$ onto $\varphi(G)$ If $G,H$ are groups and $\varphi\colon G\to H$ a group homomorphism, how can I prove that if the order of $\ker\phi$ is $n$ then $\varphi$ is an $n$-to-$1$ mapping from G onto $\varphi(G)$
| Hint: Given $g\in G$, which $x\in G$ satisfy $\phi(xg)=\phi(g)$?
| {
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Evaluating $\int \:\sqrt{1+e^x}dx$ , why I got different answers? I've got 2 steps to evaluating $\int \:\sqrt{1+e^x}dx$ which lead to different values
first step :
$\int \:\sqrt{1+e^x}dx$
let $u\:=\:\sqrt{1+e^x}$ , $du\:=\:\frac{e^x}{2\sqrt{1+e^x}}dx$ , but $e^x\:=\:u^2-1$ and substitute $u$ into $du$, i get $dx\:=\... | It is perfectly normal. It seems just that you forgot that we can rewrite $$\tanh ^{-1}(z)=\frac{1}{2}\log\frac{1+z}{1-z}$$ Have a look at http://en.wikipedia.org/wiki/Inverse_hyperbolic_function where you will find other very useful identities for inverse hyperbolic functions.
| {
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Trisecting a Triangle Given a (non-degenerate) triangle $PQR$ in the Euclidean plane, does there exists a point $A$ in the interior of the triangle such that, the triangles $APQ$, $AQR$, and $ARP$ have same area? If it exists, is it unique?
(I thought about this question while reading the book "Proof without Words- R. ... | An analytic solution, based on the fact that the area of the triangle with vertices at $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ is
$$
\frac{1}{2}\det
\begin{bmatrix}
1 & 1 & 1 \\
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3
\end{bmatrix}
$$
provided the path goes counterclockwise. Without loss of generality, we can assume the ve... | {
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Evaluate $\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$ Evaluate the following the limit:
$$\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$$
I tried expressing the limit in the form $f(x)g(x)\left[\frac{1}{f(x)} - \frac{1}{g(x)}\right]$ but it did not help... | You know that the product $(x - a_1)(a - a_2)\cdots(x - a_n)$ is equal to $x^n + \cdots + a_1a_2\cdots a_n$, where the $\cdots$ between $x^n$ and the product of the $a_i$'s is made of powers of $x$ with an exponent that is smaller than $n$. After you've seen this, it's easy to see that the only term that matters inside... | {
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Limit points of $(2,3)$. From this question: Constructing a set with exactly three limit points.
To answer this question
Construct a bounded set of real numbers with exactly three limit points.
But why $(1,2)\bigcup (2,3)$ can't be answer?
From my understand to limit point, $(2,3)$ only have two limit point $2$ and... | No.
If you understand why $(2,3)$ is an open set then,
Lemma: Let $X$ be an open set in $\mathbb{R}^k$ , if $p\in X$ then $p$ is a limit point of $X$. (Hint: every neighborhood of $p$ contains an element other than $p$)
Thus every point in $(2,3)$ is a limit point. Also we have two additional limit points, namely $2$ a... | {
"language": "en",
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A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency Suppose $a$ is an arc length-parametrized space curve with the property that
$\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that
$k(s_0) \geq 1/R$.
So, I was going to consider the... | Some intermediate hints:
*
*We are given that $\Vert a(s) \Vert \leq \Vert a(s_0)\Vert$ for all $s$ near $s_0$. Squaring both sides tells us that $f(s) \leq f(s_0)$ for all $s$ near $s_0$. In other words, $s = s_0$ is a local max for the function $f$. What does that imply about $f''(s_0)$?
*You should compute $f'... | {
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Dirac delta distribution & integration against locally integrable function I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally integrable function $f$ such that $T_f=\delta$, name... | Suppose there exists $f \in L^1_{\text{loc}}$ such that $T_f = \delta$. Define
$$\varphi_k(x) := \begin{cases} \exp \left(- \frac{1}{1-|kx|^2} \right) & |x| < \frac{1}{k} \\0 & \text{otherwise} \end{cases}.$$
for $k \in \mathbb{N}$. As $\varphi_k \in D$, we get
$$\int \varphi_k(x) f(x) \, dx = T_f(\varphi_k) = \delta(\... | {
"language": "en",
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What does this function notation mean? My text tells me that the general term of a sequence can be looked at like a function:
$
f:\mathbb{N}\rightarrow \mathbb{R}
$
What does that mean translated into common english?
| This specific case is called a sequence. The function $f$ takes in an input from the natural numbers (denoted $\mathbb N$), and gives an output in the real numbers (denoted $\mathbb R$).
In general, this notation contains three parts: the function name, the domain, and the codomain. The function name (in your case $f$... | {
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Help evaluating triple integral over tetrahedron I have a triple integral of $\iiint xyz\,dx\,dy\,dz$ over the volume of a tetrahedron with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
Normally I would just have limits 0 to 1 but that does not seem to work. How do I solve a problem like this?
| Using the facts that the projection of the solid in the xy-plane is the triangle with vertices (0,0), (0,1), and (1,0), and that the top of the solid is the plane $x+y+z=1$, we can set up the integral as
$\displaystyle\int_{T} f(x,y,z) \;dV =\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{1-x-y} f(x,y,z) \;dz dy dx$.
| {
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Circle diameter using only 3 points I've never been very good at math and confuse very easily so I need some help in doing a calculation regarding a circle.
Let's assume we're drawing the letter L.
The top point $A$ of the long leg is at $0$ the bottom point of the long leg $B$ is at $3$ feet and the point to the r... | In your special case, you have to use the Pythagorean theorem, as the letter $L$ has a right angle.
You have then $AC^2=AB^2+BC^2$
And $AC$ is the diameter of the circumcircle.
For a more generic solution (working with any three points), you will need to find out the coordinates of the center $O$ of the circle, and th... | {
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How to get started with solving basic Differential Equations? I've just started learning Differential Equations and am having general difficulties with a bit of concepts and on how to actually get started. The problem I have is that the books and sources I find always launch into slope fields, and while I get what a fi... | (1). In a slope field,
the direction of the arrow
at each point
represents the direction of the derivitive
(or slope, hence its name),
and the length sometimes
represents the magnitude of the derivative
at the point.
This enables you
to draw an approximate solution
to the differential equation
starting at any point,
ju... | {
"language": "en",
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Find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$ I have to find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$.
I know that $ \delta =\frac{1}{|x+2|} $ has this behaviour, but it is not guaranteed for it to be less than two. I don't know what else to do. ... | Observe that : $|x^2 - 4| = |x-2||x+2| < \delta|x+2| < \delta\left(|x - 2| + 4\right) < \delta(\delta + 4) = \delta^2 + 4\delta$. Thus all we need is to solve for $\delta$:
$\delta^2 + 4\delta < 1 \iff (\delta + 2)^2 < 5 \iff \delta + 2 < \sqrt{5} \iff \delta < \sqrt{5} - 2$. For example we can take: $\delta = \dfrac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/923244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dot product and its representation as sum.
If I define the dot product $u\cdot v $ where $u,v\in \mathbb{R}^n$ as $u\cdot v= |u||v|\cos \theta$ where $\theta$ is the angle between $u$ and $v$. How can I get that
$$u\cdot v= u_1v_1+u_2v_2+u_3v_3 $$
with $u=u_1e_1+u_2e_2+\cdots+u_ne_n$ and $v=v_1e_1+v_2e_2+\cdots+v... | Letting $\vec u = (u_1, u_2, \dots, u_n)$ and
$\vec v = (v_1, v_2, \dots, v_n)$,
then $\displaystyle |u|^2 = \sum_{i=1}^n u_i^2$,
$\displaystyle |v|^2 = \sum_{i=1}^n v_i^2$, and
$\displaystyle |u-v|^2 = \sum_{i=1}^n (u_i - v_i)^2$.
By the law of cosines,
$\cos \theta = \dfrac{|u|^2 + |v|^2 - |u-v|^2... | {
"language": "en",
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How was this approximation of transcendent equation solution found? I have an equation for $\xi$:
$$\xi\gamma=\cos\xi,$$
where $\gamma\gg1$. I've tried solving it assuming that $\xi\approx0$ and approximating $\cos$ by Taylor's second order formula:
$$\xi\gamma\approx1-\frac{\xi^2}2,\tag1$$
then I get
$$\xi\approx2\lef... | $f(x)=\cos x-\gamma x$ is a concave decreasing function over $(0,\pi/2)$, hence Newton's method gives that the first positive root of $f(x)$ is less than:
$$ 0-\frac{f(0)}{f'(0)} = \frac{1}{\gamma} $$
as well as it is less than:
$$\frac{1}{\gamma}-\frac{f(1/\gamma)}{f'(1/\gamma)}=\frac{1}{\gamma}-\frac{1}{2\gamma^3}+O\... | {
"language": "en",
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Relation of ellipse semi-axes with rotation angle and projection length In the following setup, assume $w$ (length of the projection of the ellipse) and $\theta$ (the rotation angle) are known. I want to know what equation(s) do I have here that helps me to derive the semi-axes ($a$ and $b$) of the ellipse based on $w$... | $$\large 4(a^2\sin^2\theta+b^2\cos^2\theta)=w^2$$
Equation of an ellipse is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Diffrentiate:
$$\frac{dy}{dx}=-\frac{b^2}{a^2}.\frac{x}{y}$$
Polar form of ellipse:
$$P(\phi)\equiv(a\cos\phi,b\sin\phi)$$
Slope of tangent in polar form:
$$m=-\frac ba\cot\phi$$
Equation of tangent:
$$\fr... | {
"language": "en",
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retraction induced homomorphism is surjective I am having a hard time proving this although it looks trivial...
Let $r:X\to A$ be a retraction between a topological space $X$ and $A\subset X$ such that $r(a_0)=a_0$ for $a_0\in A$
then the induced homomorphism $r_*:\pi_1(X,a_0)\to \pi_1(A,a_0)$ is surjective.
I tried to... | Consider $i$ canonical injection $i\colon A \to X$.
The fact that $r$ is a retract means
$$r \circ i = \mathrm{id}_A$$
From this we get
$$r_* \circ i_* = \mathrm{id}_{\pi_1(A)}$$
and this implies $r_*$ surjective since it has a right inverse.
| {
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How do I prove that numbers not divisible by 3 can be represented as 3x+1 or 3x-1? I saw that some proofs used the fact that numbers not divisible by $3$ can be represented as $3x+1$ or $3x-1$. But how do I prove that it is true?
| If $k=3x+1$ then $$k\equiv 1\mod 3$$ and if $k=3x-1$ then $$k\equiv 2\mod 3.$$ Moreover $$\mathbb Z/3\mathbb Z=\{[0],[1],[2]\}$$
where $$[1]=\{3x+1\mid x\in\mathbb Z\}$$ and $$[2]=\{3x+2\mid x\in\mathbb Z\}=\{3x-1\mid x\in\mathbb Z\}$$
what conclude the proof.
| {
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Relationship between Continuity and Countability This is a consequence of one of the problems in elementary real analysis that I am attempting to solve. I have this doubt.
Suppose $f$ is a continuous map from the reals to the reals. If the set $S=${$f(x)|x\in A$} is countable, is the set $S'=${$f(x+1)|x\in A$} countabl... | Let $f$ be zero on the nonpositive numbers and then the identity on the nonnegative numbers.
If you take $A=[-1,0]$, $|f(A)|=|\{0\}|=1$, but $|f(A+1)|=|[0,1]|=|\Bbb R|$.
(Here I abuse notation a little by writing $A+1=\{x+1\mid x\in A\}$)
| {
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Some issues concerning joint random variables Let the joint random variable $P[x;y]$ be
$P[x;y] = c[2x^2 + y^2], x=-1;0;1, y=1;2;3;4$
$=0$ $elsewhere$
So I had to find the value of $c$ that makes $P[x;y]$ a joint discrete random variable.
I think I did that right. I just add up all the probabilities where $x = -1;0;1... | I have worked out the problem for you and I hope it helps to check with your answer. I have done it in EXCEL what other responders have alluded to. To verify your answer.
| {
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root of the function $f(x)=\sqrt{2}-x$ by using fixed point iteration How can I find the approximate value of $\sqrt{2}$ by using the fixed point iteration? I have tried
$x-\sqrt{2}=0$,
$x^2=2$,
$x^2-2+x=x$,
$g(x)=x^2-2+x$,
$g\prime(x)=2x+1$
And i choose $x_0=-\frac{1}{2}$.
But i cant find the approximate value correc... | In general, you can produce a function $g(x)$ with $\sqrt2$ as a fixed point by letting
$$g(x)=x+(x^2-2)h(x)$$
with pretty much any function $h(x)$. However, as André Nicolas pointed out, if you want $\sqrt2$ to be an attracting fixed point for $g$, which is what you need if you want to approximate $\sqrt2$ by iterati... | {
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Proper way to make transition from $k$ to $k+1$ in proofs by induction First off my apologies for asking really simple questions, but I'm having some trouble wrapping my head around proof of induction when trying to prove a statement involving a natural number n holds for all values of n.
In the induction step lets as... | It does matter. However for this problem both methods amount to the same thing.
In induction, you prove a base case and then prove that a next case will be true. Depending on your problem, this can take a range of approaches, as long as the approach proves the next case.
| {
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Calculate domain of $f(x) = \sqrt{\frac{1}{x-4} + 1 }^2 -9$ I ran this through the MathWorld domain/range widget, and got $x \leq 3$ & x > 4. I understand that x=4 is undefined, therefore the domain of x includes x > 4, but I cannot for the life of me figure out how to determine the $x \leq 3$. When I set up the inequa... | Note that $$\frac{1}{x-4} + 1 \geq 0 \\ \Rightarrow \frac{1+(x-4)}{x-4} \geq 0 \\ \Rightarrow \frac{x-3}{x-4} \geq 0$$
Can you figure out the rest? (Sign chart)
| {
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Evaluating $ \lim_{x\to0} \frac{\tan(2x)}{\sin x}$ $$\lim_{x\to0} \frac{\tan(2x)}{\sin x}$$
How would I evaluate that? I was thinking changing the tan to sin/cos, but when I tried that, it did not work.
| Since $\displaystyle\tan(2x)=\frac{2\tan x}{1-\tan^2x}$
Set $t=\frac x2$
Using tangent half-angle formula we get;
$\tan x=\frac{2t}{1-t^2}$, $\quad$$\sin x=\frac{2t}{1+t^2}$
Hence $\displaystyle\lim\limits_{x\to 0}\frac{\tan(2x)}{\sin(x)}=\displaystyle\lim\limits_{x\to 0}\frac{2\tan x}{1-\tan^2x}\cdot\frac{1}{\sin x}$
... | {
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Prove that the limit exists and then find its limit We are given a function $h(x)$ which strictly positive. The function $h$ is defined on $\mathbb{R}$ and that satisfies the following:\
$$\lim_{x\to 0}(h(x)+\frac{1}{h(x)})=2$$
The question is to prove that the limit of $h$ exists at$ 0$ and then find its limit as $x\t... | Let $g(x) = x + x^{-1}$. Then your limit is
$$ \lim_{x \to 0} g(h(x)) = 2 $$
$g$ isn't quite an invertible function: for every point $a \in (2, \infty)$ there are two solutions to $g(x) = a$. But in some sense it's a continuous two-valued 'function': if I let $f^+$ and $f_-$ be the larger and smaller of the two solutio... | {
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Solve the equation:$ \bar{z}=z^{n-1}$ Solve the following equation:
$\bar{z}=z^{n-1}$
Where $\bar{z}$ is the complex conjugate of z, and n is a natural number such that $n\neq 2$.
I have tried to write z in rectangular form and polar form. I have tried to play with De Moivre's formula.
But I still do not see where to ... | Use polar coordinates, $z = re^{i\theta}$:
$$re^{-i\theta} = r^{n-1}e^{i(n-1)\theta},$$
$$r = r^{n-1}\implies\cdots,$$
$$\cdots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/924466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
open sets in the order topological space I have a question. I am really confused about determining if a set is open. First, the idea of a set being closed has nothing to do with homomorphic ideas of closure: if $x,y \in F$ then $x+ y$ in $F$ . (This is not the idea of closure.)
My question is from an example fro... | Your question isn’t entirely clear, but if you’re starting with $\Bbb R\times\Bbb R$ with the dictionary order and then giving $I\times I$ the subspace topology that it inherits from $\Bbb R\times\Bbb R$, then it’s true that $Y$ is open in $I\times I$. To see this, let $p=\left\langle\frac12,\frac12\right\rangle$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/924579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Applying angle addition formulas for subtraction The angle addition formula says that:
$\sin(\phi + \theta) = \sin(\phi) \cdot \cos(\theta) + \cos(\phi) \cdot \sin(\theta)$
Why are the following steps valid?:
$\sin(\phi − \theta) = \sin(\phi) \cdot \cos(−\theta) + \cos(\phi) \cdot \sin(−\theta)= \sin(\phi) \cdot \cos(\... | Hint:
$\cos(\theta)$ is an even function while $\sin(\theta)$ is an odd function.
Hence:
$$\cos(-\theta) = \cos(\theta),$$
while
$$\sin(-\theta) = -\sin(\theta).$$
(See this Wikipedia link on Even and odd functions for more information.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/924774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Solvable groups in group theory If $N \unlhd G$, and $M,K \leq G$ such that $M \unlhd K$, then does it imply that $MN \unlhd KN$? If yes, how?
| First of all, $N$ normalizes $MN$. In fact, if $n_{1}, n_{2} \in N$, and $m \in M$, then
$$
(m n_{1})^{n_{2}} = n_{2}^{-1} m n_{1} n_{2} = m (m^{-1} n_{2}^{-1} m) n_{1} n_{2} = m (n_{2}^{-1})^{m} n_{1} n_{2}\in MN,
$$
as $N$ is normal in $G$.
And then $K$ also normalizes $MN$. If $k \in K$, then
$$
(m n_{1})^{k} = m^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/924898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Containment of $c_0$ or $\ell_p$ Suppose that $(x_n)$ is a sequence of unit vectors in a Banach space $X$ such that $$\mbox{dist}(x_m, S_{X_n})=1$$
for all $m > n$. Here $S_{X_n}$ stands for the unit sphere $\mbox{span}\{x_1, \ldots, x_n\}$. Can we conclude that $X$ contains an isomorphic copy of $c_0$ or $\ell_p$?
Thi... | No.
You can find such a sequence in any infinite dimensional normed space. See the proof of Lemma 1.4.22 in Megginson's An Introduction to Banach Space Theory. But, there are infinite dimensional Banach spaces that contain neither $c_0$ nor any $\ell_p$, such as Tsirelson's space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/925004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Partial differential equation, mixed derivatives What can be concluded from following equation:
$$\frac{\partial f(x,y)}{\partial x}-\frac{\partial g(x,y)}{\partial y} = 0$$
where $f(x,y)$ and $g(x,y)$ are functions of two independent variables $x,y$. Does it generally imply that
$$\frac{\partial f(x,y)}{\partial x}=\f... | No.
It implies that there exists a function $F(x,y)$ such that $\partial_x F(x,y)=g(x,y),\partial_y F(x,y)=f(x,y)$. Since
$$\partial_x\partial_y F(x,y)=\partial_y g(x,y)=\partial_x f(x,y)=\partial_y\partial_x F(x,y)$$
In general you can not separate $F(x,y)$ into $a(x)b(y)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/925083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How "big" are the mathematical disciplines? This question may be difficult to answer, but is it possible to estimate the "size" of the individual mathematical disciplines; measures of size could be number of professionals dealing with it, the number of magazines and new theorems etc.? For instance, I usually consider a... | For a rough graphical representation see for example here and here.
(Related questions with interesting links can be found here and here and here.)
| {
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"source": "stackexchange",
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Making a Piecewise Function a Single Function Is there a way to turn a piecewise function into one function. For example:
$$\ f(x)=\begin{cases} g(x) & \text{if $a≤x<b $} \\ h(x) & \text{if $b≤x≤d$} \end{cases}$$
(Can you use the Heaviside Step Function? $\theta(x))$
| You can rewrite $f(x)$ as $$f(x)=\chi_{[a,b)}(x)g(x)+\chi_{[c,d)}(x)h(x),$$ where $\chi_{[a,b)}(x)$ is $1$ if $x\in[a,b)$ and $0$ otherwise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/925407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Are there any mathematics "problem websites" similar to Project Euler? Are there any mathematics websites similar to Projet Euler, a website which hosts math-heavy programming questions, many of which can be solved with a pen and paper?
I've become almost addicted to Project Euler's progress tracking system, and I also... | The AwesomeMath website--www.awesomemath.com--will lead you to the journal Mathematical Reflections, each issue of which has one or two short articles and lots of problems divided into Junior, Senior, Undergraduate, and Olympiad problems. Solutions appear in later issues.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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Write $\vec{a}=(3,2,-6)$ as the sum of two vectors, one parallel, and one perpendicular, to vector $\vec{d}=(2,-4,1)$. The solution is $$\vec{a}=(-8/21)\vec{d}+(79/21,10/21,-118/21)$$
| Consider the vector
$$
v=a-\frac{a\cdot d}{d\cdot d}d
$$
Note that
$$
\begin{align}
v\cdot d
&=a\cdot d-\frac{a\cdot d}{d\cdot d}d\cdot d\\
&=0
\end{align}
$$
Therefore,
$$
a=v+\frac{a\cdot d}{d\cdot d}d
$$
where $v$ is perpendicular to $d$ and $\frac{a\cdot d}{d\cdot d}d$ is parallel to $d$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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} |
I'm not sure what this is exactly asking Without using words of negation, write the meaning of : "f is not an increasing function"
I did:
$$"f\ is\ not\ an\ increasing\ function" \ \equiv\ "f\ is\ a\ decreasing\ function"$$
Is this what it's asking or am I completely missing it?
| No this is not true, for example any function from $\mathbb{R}$ to itself which increases on one interval and decreases on another is neither increasing nor decreasing.
It should be noted that "nonincreasing" is actually a perfectly valid adjective for describing a function, but that is probably not what it being sough... | {
"language": "en",
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Why are degree maps for cellular boundary formula from $S^{n-1}\to S^{n-1}$? For a CW-complex, there's the cellular boundary formula that
$$
d_n(e^n_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_\beta
$$
where the coefficients $d_{\alpha\beta}$ are the degrees of the map
$$
S^{n-1}\to X_{n-1}\to S^{n-1}
$$
where the first... | You're left with $S^1$. You've already mapped the boundary of the $2$-cell to itself, and the other $1$-cell is collapsed to the $0$-cell.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/925739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that something is irrational I'm trying to evaluate the following claim:
$$ \sqrt{2} + \sqrt{n} $$ is irrational.
This is what I tried:
Proof by contrapositive: Suppose $$ r = \sqrt{2} + \sqrt{n} $$
and r is rational. Then $$ \frac{m}{l} = \sqrt{2} + \sqrt{n} $$
$$\frac{m^2}{l^2} = 2 + 2\sqrt{2n} + n $$
I'm no... | Case 1: $n$ is of form $2u^2$ , $u$ being an integer
Then our number is equal to $\sqrt 2 (u+1)$, clearly this is irrational
Otherwise $2n$ is not a square, hence you get contradiction in the last identity you wrote.(since $\sqrt {2n}$ is irrational)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recurrence Problem involving multiple dependencies. I have 3 equations :-
*
*$r_n=r_{n-1}+5m_{n-1}$
*$m_n = r_{n-1} + 3m_{n-1}$
*$p_n = 5m_{n-1}$
The initial values of the sequences are
$$r_0=3, m_0=1, p_0=0$$
How can I get the formula to get the nth term of the series without the recurrence?
A ready-made will ... | It is given that,
$$r_n=r_{n-1}+5m_{n-1}$$
$$m_n = r_{n-1} + 3m_{n-1}.$$
By subtracting we can obtain that,
$$r_n=m_n+2m_{n-1}$$
$$r_{n-1}=m_{n-1}+2m_{n-2}.$$
Hence
$$m_n=4m_{n-1}+2m_{n-2}.$$
This second order liner recurrence with $m_0=1$ and $m_1=6.$
I think you can solve this one and try to find $r_n$ and $p_n$ vi... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the order of $2$ in $\mod 2^{n} -1 $ Find the order of $2$ in $\mod 2^n-1$
I know that the order of $2$ in $\mod 2^n-1$ is the smallest positive integer $k$ such that
$$2^k \equiv 1 \pmod {2^n-1}$$
How to proceed from here ? Any help/hints ?
| Since you we use $\mod 2^n-1$ it is clear that $k$ must be greater than $n-1$ because $2^{n-1}\leq 2^n-1$ If you try $k=n$ you see that $2^k = 2^n \equiv 1 (\mod 2^n-1)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Spectrum of integral operator
Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) =
\int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the
spectrum of T.
My attempt:
First I can show that $T$ is a compact operator. Given a sequence $u_n$ bounded in $C([0,1])$$\|u_n \|_\infty \leq M$, and ... | In order to show that your operator is compact, show that it maps a bounded sequence $\{ f_{n} \}_{n=1}^{\infty}\subset C[0,1]$ to an equicontinuous sequence of functions. So, let $\{ f_{n} \}_{n=1}^{\infty}$ satisfy $\|f_{n}\|_{C[0,1]}\le M$ for all $n$ and some fixed $M$; then, for every $\epsilon > 0$, show that the... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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STEPS: Proving $\displaystyle \lim_{x\to a} f(x)g(x) = -\infty$ Given $\displaystyle \lim_{x \to a} f(x) = \infty$ and $\displaystyle\lim_{x \to a} g(x) = c$ where $c<0$.
Prove that $\displaystyle \lim_{x \to a} f(x)g(x) = -\infty$ only using the precise definitions of limit and infinite limit.
I get the intuitive idea... | These exercises are almost always solved in a standard way.
*
*Write down the assumptions, using their definition. For instance, $\lim_{x \to a} f(x)=\infty$ means: for every $M>0$ there exists $\delta >0$ with the property that $0<|x-a|<\delta$ implies $f(x)>M$. Now please write down the meaning of your second assu... | {
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Finding marginal density from a joint density when range of random variables are dependent on one another. I have two joint density problems, where I would like to find the marginal density.
The first one: $f(x,y) = 24xy, 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq x+y \leq 1$
So, I "integrate out y" and get $f_x(x)=\int_... | $$f(x,y)=24xy$$
First integrate with respect to $y$, because $x+y \leq 1$ then $y$ goes from $0$ to $1-x$
$$\int_0^{1-x} 24xy dy=12x(1-x)^2$$
Second to check it's every okay we integrate with respect to $x$:
$$\int_0^1 12x(1-x)^2dx=1$$
As you thought when you are integrating you can take the variable out just when you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Probability of "either/or" and "neither" for two independent events This is a problem from GRE quantitative section practice book.
The probability of rain in Greg's town on Tuesday is $0.3$. The probability that Greg's teacher will give him a pop quiz on Tuesday is $0.2$. The events occur independently of each other.
... | The interpretation given by the practice exam is incorrect as adding the probability of both events happening overall double counts that event as you've probably already knew.
This interpretation is clearly bogus if you increase both event's probability to 50%: Going by the practice book's explanation, then the probab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/926430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Summation of series Find $\sum_1^n$ $\frac {2r+1}{r^2(r+1)^2}$
Also, find the sum to infinity of the series.
I tried decomposing it into partial fractions of the form $\frac Ar$ + $\frac{B}{r^2}$ + $\frac{C}{(r+1)}$ + $\frac{D}{(r+1)^2}$ but it was getting too complicated and tedious. Is there some trick here that i'm... | $$
\frac1{r^2} - \frac1{(r+1)^2} = \frac{(r+1)^2-r^2}{r^2(r+1)^2} = \frac{(r^2+2r+1)-r^2}{r^2(r+1)^2} = \frac{2r+1}{r^2(r+1)^2}
$$
The thing that suggested this to me is that $\displaystyle 2r+1 = 2\left(r+\frac12\right)$, and $r+\dfrac12$ is half-way between $r+0$ and $r+1$, the two expressions in the denominator.
Nex... | {
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"source": "stackexchange",
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If $c\mid ab$ and $\gcd(c,a)=d$, then $c\mid db$ I came across this problem in my number theory text and am having a bit of trouble with it:
Prove if $c\mid ab$ and $\gcd(c,a)=d$, then $c\mid db$.
Here's what I have so far:
If $c\mid ab$, then there exists an integer $x$ such that $cx=ab$.
Because $\gcd(c,a)=d$, $d\mid... | There are no mistakes in what you've done so far, but I agree that it's not immediately apparent why $dx/a$ is an integer.
I would start by noting that the answer to this problem is well known when $d = 1$, so maybe I should try to turn it into a problem about relatively prime integers. So write $a = da'$ and $c = dc'$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Closed form sum for the series given below? Does the following series have a closed form sum?
$$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
| Pascal's Identity states that $\dbinom{r+i}{r} + \dbinom{r+i}{r+1} = \dbinom{r+i+1}{r+1}$.
Hence, $\displaystyle\sum_{i = 1}^{n}\dbinom{r+i}{r} = \sum_{i = 1}^{n}\left[\dbinom{r+i+1}{r+1} - \dbinom{r+i}{r+1}\right]$.
This sum telescopes to $\dbinom{r+n+1}{r+1} - \dbinom{r+1}{r+1} = \dbinom{r+n+1}{r+1} - 1$.
Now, add ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/926698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve system of equations with $\sin$ and $\cos$ Solve system of equations
$\begin{cases}
3x^2 + \sin 2y - \cos y - 3 = 0 \\
x^3 - 3x - \sin y - \cos 2y + 3 = 0
\end{cases}$
I tried to use substitution $x = \cos t$ or sth, but I get literally nothing
| The equations are $x^2=A$ and $x^3-3x=B$ where
$$A=\frac{3+\cos y - \sin 2y}{3},\quad B=\sin y +\cos 2y - 3.$$
Now note that $x^3-3x=x(x^2-3),$ so one can substitute the value from $x^2=A$ here, obtaining $x(A-3)=B,$ that is, $x=B/(A-3).$ Then putting this $x$ back into $x^2=A$ gives the relation, involving only $A,B,$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it necessarily true that the following intersection has a positive measure? suppose that a set $S$ in $\mathbb{R}$ is a measurable set with measure greater than zero, and let $\mathcal{O}$ be an open cover of $S$, consisting of disjoint open intervals whose existence we know. It is then necessarily true that there e... | If I understand correctly, you have $S \subset \mathcal{O} := \bigsqcup_n I_n$ (disjoint union). Since $S$ is measurable, $\mu(S)=\sum_n \mu(S \cap I_n)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The limit of $f(x)= \sqrt{x^2+4x+3} +x$ as $x\to\infty$ The problem is to find $\lim_{x\to\infty} \sqrt{x^2+4x+3} +x$.
Do I just divide everything by $x^2$ and get limit $= \sqrt{1}+0=1$?
| According to what was given, with the assumption that the function $f: x \mapsto \sqrt{x^{2} + 4x + 3} + x$ is defined on $\mathbb{R}$, it is concluded that for every $\varepsilon > 0$ there is a real $X$ such that if $x \geq X$ then
$$|\sqrt{x^{2} + 4x + 3} + x - l| \geq \varepsilon,$$
viz, the function in question g... | {
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Concluding that a function is not analytic at a point? I have the function $f$ defined on $\mathbb{R}$ by:
$f(x)= \begin{cases} 0 & \text{if $x \le 0$} \\ e^{-1/x^2} & \text{if $x > 0$} \end{cases}$
I've inductively proved that $f$ is indefinitely differentiable on $\mathbb{R}$ and proved that $f^{(n)}(0)=0$ for all $n... | Well, if you have that $F(x)=\displaystyle \sum_{n=0}^{\infty}\frac{1}{n!}x^n f^n(0)$, then since $f^n(0)=0\;\;\forall n \in \mathbb{N}$, you would have that $f$ is identical zero, and this is not true since your function is not the zero function. So you can not have a expansion on Taylor series centred at zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/927109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Attempt to prove that every real number is a limit of a sequence of rational numbers Prove that given a real number $x$, there exists a rational sequence $r_n$ such that $r_n \to x$ as $n$ grows.
Proof: Suppose $x$ is a real number. Then we know by definition, there exists a rational number such that $x < q < x + \fra... | Well, you can simply show an example of such sequence for any $r\in\mathbb{R}$:
$r_n=\dfrac{\lfloor{r\cdot10^n}\rfloor}{10^n}$, i.e., the rational number that is given by the first $n$ digits of $r$ on base $10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/927198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Show that $\frac{(b+c)^2} {3bc}+\frac{(c+a)^2}{3ac}+\frac{(a+b)^2}{3ab}=1$ If $a^3+b^3+c^3=3abc$ and $a+b+c=0$ show that $\frac{(b+c)^2} {3bc}+\frac{(c+a)^2}{3ac}+\frac{(a+b)^2}{3ab}=1$
| $a+b+c=0\iff b+c=-a\implies (b+c)^2=a^2$
$\implies\dfrac{(b+c)^2}{3bc}=\dfrac{a^3}{3abc}$
Actually, $b+c=-a\implies(b+c)^3=(-a)^3$
$\implies -a^3=b^3+c^3+3bc(b+c)=b^3+c^3+3bc(-a)\iff\sum a^3=3abc$
| {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
} |
Recursion relation and initial conditions Write a recursion relation and initial conditions for the number of words of length n using the letters A,B,C such that the number of the letter 'A' is even.
(A occur an even number of times)
| $o_{n}=$ number of words having $n$ letters and with odd number of
letter $A$
$e_{n}=$ number of words having $n$ letters and with even number
of letter $A$
Then $o_{0}=0$ and $e_{0}=1$ (empty word counts).
$o_{n+1}=2o_{n}+e_{n}$
$e_{n+1}=o_{n}+2e_{n}$
Then $o_{n}+e_{n}=3^{n}$ leads to $e_{n+1}=3^{n}+e_{n}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/927409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Prove that the binary representation of a number n will use floor(lg(n)) + 1 bits. I'm taking Computer Algorithms class and one of my problems is from Skiena's Algorithm Design Manual, 2-41:
Prove that the binary representation of $n \ge 1$ has $\lfloor \lg n \rfloor +1$ bits ($\lg$ is base 2)
Some base cases:
$n = 1... | Hint
Note that the binary representation of $2^n$ has $n+1$ bits.
Find then that the binary representation of a sum of $2^{k_i}$ with distinct $k_i$ has $\max_i k_i + 1$ bits.
Finally conclude that any integer in the interval $[2^n, 2^{n+1})$ has a binary representation of exactly $n+1$ bits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/927468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
linear solution of curve fitting on multiple linear functions differing by a multiplier I recently posted this question here but I thought this could be of interest also in mathematics, given I found a partially related question here
I am facing the following problem. I know nonlinear least squares can provide a soluti... | I think it is not possible (but I may be wrong) to solve this directly. But I would do following:
*
*Calculate $b,c$ via least squares with the blue dataset only.
Then iterate over those two steps:
*
*Take $b,c$ from previous step and calculate $a',a''$ via least squares.
*Take $a',b'$ from previous step and c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/927588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Doob's decomposition and submartingale with bounded increments Let $(X_n)_{n \geq 0}$ be a submartingale defined on some filtered probability space $(\Omega, \mathcal{F}, ({\mathcal{F}}_n)_{n \geq 0}, \mathbb{P})$. It is a standard fact that $X_n = X_0 + M_n + A_n$, where $(M_n)_{n \geq 0}$ is a martingale null at $0$ ... | Define a sequence of stopping times $(\tau_k)_k$ by
$$\tau_k := \inf\{n \geq 0; X_n \geq k\}.$$
From
$$M_{n \wedge \tau_k} = X_{n \wedge \tau_k}-X_0 - \underbrace{A_{n \wedge \tau_k}}_{\geq 0} \leq 2k$$
it follows that $(M_{n \wedge \tau_k})_{n \in \mathbb{N}}$ is a martingale which is bounded above. Consequently, by a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/927688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to show that $a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$ How do you show that
$$a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$$
I could write $\sqrt{a^2+b^2-ab}=\sqrt{(a+b)^2-3ab}$, but this seems to lead nowhere.
| $$a+b> \sqrt{a^2+b^2-ab} \iff (a+b)^2>\left (\sqrt{a^2+b^2-ab}\right )^2$$ $$ \iff\quad a^2+2ab+b^2>a^2+b^2-ab$$ $$\iff\quad 3ab>0 \quad \iff\quad ab>0$$ which is correct because of the hypothesis $a,b >0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/927765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How many of these are surjective? Let $A=\{a,b,c,d\}$ and $B=\{e,f,g\}$.
*
*How many maps are there from A to B?
*How many of these maps are surjective?
$\textbf{Part 1:}$ There are 4 elements in A and 3 elements in B. Thus there are $3^4=81$ maps from A to B.
$\textbf{Part 2:}$ I can listed in order pair form th... | If $h:A\rightarrow B$ is a function then it induces a partition $\mathcal P_h$ on $A$. Elements of $\mathcal P_h$ are the sets of the form $\{a\in A\mid h(a)=b\}$ where $b$ belongs to the image of $h$. Notations for this sets are $h^{-1}(\{b\})$ or shortly $h^{-1}(b)$. $\mathcal P_h$ and the image of $h$ have equal car... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/927871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
2 similar question about how to find the $a_n$s and $b_n$ of a Fourier series Find the terms $b_n,\ n\geq 1$ so that $$x-\frac{\pi}{2}=\sum_{n=1}^{\infty}b_n \sin nx$$ for all $x\in (0,\pi)$.
A similar one:
Find the term $a_n, \ n \geq 0$ so that $$x-\frac{\pi}{2}=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n \cos nx$$ for all ... | How much do you know already?
A $\sin$ series of a function defined on the interval $(0,\pi)$ is the Fourier series of an odd function defined on $(-\pi,\pi)$. The $\cos$ series being the Fourier series of an even function.
If we suppose that we have the full Fourier series of a function $$f(x) = \frac{a_0}{2} + \sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/927951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve the integral $\int \frac {(x^2 +1)}{x^4- x^2 +1} dx$ I have started this problem but I'm not completely sure I'm going down the right path with it.
So far I have completed the square in the denominator.
$x^4-x^2+1= (x^2-1/2)^2+\frac{3}{4}$
Then, let $u=x^2-\frac{1}{2}$ so $x=\sqrt(u+\frac{1}{2})$
$\int\f... | $$\frac{x^2+1}{x^4-x^2+1}=\frac{1+\dfrac1{x^2}}{x^2-1+\dfrac1{x^2}}$$
Now $\displaystyle\int\left(1+\dfrac1{x^2}\right)dx=x-\dfrac1x$
and $\displaystyle x^2-1+\dfrac1{x^2}=\left(x-\dfrac1x\right)^2+2-1$
Hope you can take it from here
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/928040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Which of the following series will converge and which one will diverge? Can anyone help me out that which of the following series will converge and which one will diverge, with some explanation?
A) $\sum_{n=1}^\infty \sin\left(\frac{\pi}n\right)$
B) $\sum_{n=1}^\infty (-1)^n \cos\left(\frac{\pi}n\right)$
Thanks a lot... | Euler said that if $x$ is an infinitely small positive number than $\sin x=x$. Today we way $\lim\limits_{x\to0}\dfrac{\sin x}x=1$. At any rate we know that if $x$ is a sufficiently small positive number then
$$
\frac x 2 <\sin x < x,
$$
so
$$
\frac\pi2\sum_n \frac 1 n =\sum_n \frac\pi{2n}\le\sum_n \sin\frac \pi n \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Alternative definition for order-embedding I realize that the traditional definition for an order-embedding f is that
$a_1 \sqsubseteq a_2 \iff f(a_1) \sqsubseteq f(a_2)$
However, is it also fair to say that if $(A, \sqsubseteq)$ is a lattice, a definition for meet and this partial-order uniquely define one another:
$a... | Embeddings are functions from one structure to another which preserve the structure (and do no add additional structure to the image either).
For partially ordered sets this means, as you suggest that $a\sqsubseteq b\iff f(a)\sqsubseteq f(b)$. Lattices have more structure, so we can expect embeddings of lattices to pre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If a subgroup acts transitively on a set, then the index of the subgroup equals the index of the stabilizer? I am trying to prove the following: If a subgroup $H < G$ acts transitively on a set $X$, then $[G:H] = [G_x:H_x]$ for any $x \in X$ ($H_x$ denotes the point stabilizer of $x$ in $H$.) Any hints would be appreci... | For the case where $G, H$ can be infinite:
Consider $H< H<G \Rightarrow [H:H_x][G:H] =[G:H_x]$ and $H_x<G_x<G \Rightarrow [G_x:H_x][G: G_x]= [G:H_x]$ .
$\Rightarrow [G_x:H_x][G: G_x] = [H:H_x][G:H]$. Since $[G: G_x]=[H: H_x] = |X|$, we're done.
Note: Can I assume $[G: G_x]=[H: H_x]$ even if $X$ is not finite?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/928304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Computing a messy convolution Consider the functions
$$
x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2})
$$
and
$$
h(t) = tu(t)
$$
where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$.
I'm trying to compute
$$
(x*h)(t) = \int\limits_{-\infty}^{\infty} x(t)h(\tau - t) \, d\tau.
$$
However, the problem gets nasty very ... | An easy (and pretty standard) way of computing convolutions is by taking the Laplace transform of the functions, multiplying them (convolution transforms to multiplication in Laplace transformed space), and taking the inverse Laplace transform. As for this question $$\eqalign{
& X(s) = {{{e^{ - {s \over 2}}} - {e^{ -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge? The integral is $\int\left(\,1 + n^{2}\,\right)^{-1/4}\,{\rm d}n$ is not quite possible, so I should make a comparison test.
What is your suggestion?
EDIT:
And what about the series
$$
\sum\left(\, 1 + n^{2}\,\right)^{-1/4} \cos\left(\, n\pi \over 6\,\right)
$$... | For the second part, notice that there is a pattern to the factor of $\cos\left(\frac{\pi n}{6}\right)$ which repeats every 12 terms. Use this pattern to think of the sequence as alternating.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/928663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian Let $G$ be a group. If $(ab)^n=a^nb^n$ $\forall a,b \in G$ and $(|G|, n(n-1))=1$ then prove that $G$ is abelian.
What I have proven is that:
If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in G$, then $G$ is abeli... | We can assume that $n>2$. Since $(ab)^n = a^nb^n$, for all $a,b\in G$, we can write $(ab)^{n+1}$ in two different ways:
$$(ab)^{n+1} = a(ba)^nb = ab^na^nb,$$
and
$$(ab)^{n+1} = ab(ab)^n = aba^nb^n.$$
Hence,
$$ab^na^nb = aba^nb^n.$$
Cancel $ab$ on the left and $b$ on the right to obtain
$$b^{n-1}a^n = a^nb^{n-1}.$$
Not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Can I pick any N in the epsilon N - definition? If I were asked to pick a $N$ that would satisfy the definition such that for any $n > N$, the distance between $f(n)$ and the limit would be smaller than a specific epsilon....
...can I just pick $N$'s at random that would satisfy this, and then say that for any $n > N$ ... | [In the general case]
Suppose you somehow found some $N$ such that for any $n > N$, we have $|f(n)-L| < \epsilon$.
Then any $N' \ge N$ will also "work," that is, for any $n > N'$, we have $|f(n)-L| < \epsilon$.
However, as Jean-Claude pointed out it is not true that $|f(n)-L|<\epsilon$ implies that for all $n' \ge n$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that the the variance estimator $\widehat{\sigma}^2=MSE/(n-2)$ is biased is the simple linear regression model This is in scope of the simple linear model. Im trying to prove that $\mathbb{E}\left(\widehat{\sigma}^2\right) = \sigma^2$ for
$$\widehat{\sigma}^2 = \frac{1}{n-2}\sum^n_{i=1} \left(y_i-\widehat{y}_i\ri... | $\newcommand{\b}{\begin{bmatrix}}\newcommand{\eb}{\end{bmatrix}}$
The vector of fitted values
$$
\hat Y = \b \hat y_1 \\ \vdots \\ \hat y_n\eb
$$
is the orthogonal projection of
$$
Y = \b y_1 \\ \vdots \\ y_n \eb
$$
onto the column space of the design matrix
$$
X = \b 1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \eb.
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/928946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$\nabla \cdot f + w \cdot f = 0$ Let $w(x,y,z)$ be a fixed vector field on $\mathbb{R}^3$. What are the solutions of the equation
$$
\nabla \cdot f + w \cdot f = 0 \, ?
$$
Note that if $w = \nabla \phi $, then the above equation is equivalent to
$$
\nabla \cdot (e^\phi f) = 0,
$$
for which the solutions are of the for... | You can try to proceed along the following lines:
\begin{equation*}
(\partial _{\mathbf{x}}+\mathbf{w})\cdot \mathbf{f}=0
\end{equation*}
Special case $\mathbf{w}$ is constant. Then
\begin{equation*}
\exp [-\mathbf{w\cdot x}]\partial _{\mathbf{x}}\exp [+\mathbf{w\cdot x}
]=\partial _{\mathbf{x}}+\mathbf{w}
\end{equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/929032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
General solution for intersection of line and circle If the equation for a circle is $|c-x|^2 = r^2$ and the equation for the line is $n \cdot x=d $, and assuming that the circle and line intersect in two points, how can I find these points?
Also as kind of a side note, is there also a general formula for obtaining t... | Here I present a method, I don't think it was not used yet, but don't usually find it in mathematics textbooks :
$$\begin{cases}
ax + by + c = 0 \\
(x-\alpha)² + (y-\beta)² = R_{0}^2
\end{cases}
$$
As it shows, the first equation describes the line $\Delta$ with a normal vector $\left( \begin{matrix}
a \\
b
\end{matrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/929193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Automorphisms of the group of integers $\mathbb Z$ Can anyone help me showing $\operatorname{Aut}(\mathbb Z)\simeq \mathbb Z_2$?
I guess I should define an homomorfism $\phi:\mathbb Z\longrightarrow S(\mathbb Z)$ with kernel $2\mathbb Z$ and image $\operatorname{Aut}(\mathbb Z)$. Here $S(\mathbb Z)$ are the bijections... | Let $\varphi \in \operatorname{Aut}(\mathbb Z)$. We have
$$
\varphi(n) = \varphi(\underbrace{1 + \cdots + 1}_{n \text{ times}}) = \underbrace{\varphi(1) + \cdots + \varphi(1)}_{n \text{ times}} = n \ \varphi(1).
$$
Thus, $\varphi$ is completely determined by its value at $1$. Only two values make $\varphi$ surjective, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/929258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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