Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Derive a rotation from a 2D rotation matrix I have a rotation 2D rotation matrix. I know that this matrix will always ever only be a rotation matrix.
$$\left[
\begin{array}{@{}cc}
\cos a & -\sin a \\
\sin a & \cos a \\
\end{array}
\right]$$
How can I extract the rotation from this matrix?
The less steps, the better,... | Pick any non-zero vector $v$ and compute the angle between $v$ and $Av$, where $A$ is the matrix above.
A simple vector is $e_1 = \binom{1}{0}$, and $Ae_1 = \binom{\cos \alpha}{\sin \alpha} = \binom{A_{11}}{A_{21}}$, hence the angle $\alpha$ can be computed from $\text{atan2}(\sin \alpha, \cos \alpha) = \text{atan2}(A_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How many odd numbers with distinct digits between 1000 and 9999 How many numbers with distinct digits are there between 1000 and 9999. [1]
I came up with a solution like this.
Since we can't know what numbers have been used, in the tens, hundreds and thousands we start counting at the ones.
1s: { 1, 3, 5, 7, 9 }, s... | Your problem comes when you assume that there were $3$ taken from your last set. It's quite possible that the second or third choice was a $0$, so that you in fact have only eliminated $2$ options from the last set.
Instead, choose the last digit first ($5$ choices), then the first digit ($8$ choices, since we can't ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fractions in binary? How would you write a fraction in binary numbers; for example, $1/4$, which is $.25$? I know how to write binary of whole numbers such as 6, being $110$, but how would one write fractions?
| $1/4=0\cdot(1/2)^0+0\cdot(1/2)^1+1\cdot(1/2)^2=0.01$ in base $2$, you just go in reverse with powers $(1/2)^n, n=0,1,2,...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 8,
"answer_id": 1
} |
Combination / Permutation Question
There are 3 bags and 5 different marbles. In how many ways can the marbles be put into the bags?
(disclosure - the question is one of many in a teacher prep study guide. I am taking the qualification exam for teaching middle school math next month.)
The answer and explanation for t... | Hint: Think about the balls. In how many bags can one ball be put in? How many balls are there?
| {
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"url": "https://math.stackexchange.com/questions/301501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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showing compactness of the intersection of a compact set and disjoint open sets. Here's the problem I'm trying to solve.
"If $A\subseteq B_1\cup B_2$ where $B_1, B_2$ are disjoint open sets and $A$ is compact, show that $A\cap B_1$ is compact. Is the same true if $B_1$ and $B_2$ are not disjoint?"
Hope you can help... | Let $\{V_{\alpha}\}_{\alpha\in J}$ be a family of open sets such that:
$$A\cap B_1\subseteq\cup_{\alpha\in J}V_{\alpha}$$Since $A\subseteq B_1\cup B_2$, it follows that $A\subseteq\ B_2\cup(\cup_{\alpha\in J}V_{\alpha})$. Since $A$ is compact , therefore there is a finite subset of $\{{V_{\alpha}}|\alpha\in J\}\cup\{B_... | {
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Proof of $\displaystyle \lim_{z\to 1-i}[x+i(2x+y)]=1+i$ I am having some difficulty with the epsilon-delta proof of the limit above.
I know that $|x+i(2x+y)-(1+i)|<\epsilon$ when $|x+iy-(1-i)|<\delta$.
I tried splitting up the expressions above in this way:
$|x+i(2x+y)-(1+i)|\\
=|(x-1)+i(y+1)+i(2x+2)|\\
\le|(x-1)+i(y+1... | Hint: If $z\to 1-i$ and $z=x+iy$, then $x\to 1$ and $y\to -1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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What is mathematical research like? I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or biology one collects data and analyzes it... | For me as an independent mathematical researcher, it includes:
1) Trying to find new, more efficient algorithms.
2) Studying data sets as projected visually through different means to see if new patterns can be made visible, and how to describe them mathematically.
3) Developing new mathematical language and improving ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "71",
"answer_count": 9,
"answer_id": 7
} |
Is the Euler phi function bounded below? I am working on a question for my number theory class that asks:
Prove that for every integer $n \geq 1$, $\phi(n) \geq \frac{\sqrt{n}}{\sqrt{2}}$.
However, I was searching around Google, and on various websites I have found people explaining that the phi function has a define... | EEDDIITT: this gives a proof of my main claim in my first answer, that a certain function takes its minimum value at a certain primorial. I actually put that information, with a few examples, into the wikipedia article, but it was edited out within a minute as irrelevant. No accounting for taste.
ORIGINAL: We take as g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 4,
"answer_id": 3
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How to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$ I'm having troubles to solve this integration: $\int_0^1 \frac{x^{2012}}{1+e^x}dx$
I've tried a lot using so many techniques without success. I found $\int_{-1}^1 \frac{x^{2012}}{1+e^x}dx=1/2013$, but I couldn't solve from 0 to 1.
Thanks a lot.
| You have $$\int_{-1}^{1} \frac{x^{2012}}{1+e^{x}} \ dx =\underbrace{\int_{-1}^{0}\frac{x^{2012}}{1+e^{x}}}_{I_{1}} \ dx + \int_{0}^{1}\frac{x^{2012}}{1+e^{x}} \ dx \qquad \cdots (1)$$
In $I_{1}$ put $x=-t$, then you have $dx = -dt$, and so the limits range from $t=0$ to $t=1$. So you have $$I_{1}= -\int_{1}^{0} \frac{e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Definition of Tangents When we had not learnt Calculus, we met the concept of Tangent in Circle, which was defined as the line touching the circle at ONE point.
Then, after learning Calculus, we knew that a curve could intersect with its tangent at more than one point, and a line intersecting with a curve at only one p... | Limit means approaching, not coincidence. So, if you take two points on the circle, line that goes through them is not a tangent, of course. But if you make one point closer to another, that line goes closer to the tangent. If you take it even more close, then line will be also closer. And here comes the limit into the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Applications of prime-number theorem in algebraic number theory? Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory.
Since we might think of the density theorem as an analytic theorem, and as prime number... | I don't know if this can be considered algebraic number theory or if it is more algebraic geometry; but, here goes.
Deligne uses the methods of Hadamard-de la Vallée-Poussin to prove the Weil conjectures. Even if it is not an application of the ordinary PNT as such, the exact same methods are applied elsewhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/302048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is wrong with my proof: Pseudoinverse and SVD I was trying to prove the following:
Let $U\Sigma V$ be the $SVD$ decomposition of $A\in\mathbb R^{m\times n}$, where $\textrm{rank}(A)=k$. Show that the pseudoinverse of $A$ is given by,
$$
\displaystyle A^\dagger=\sum_{i=1}^k\sigma_i^{-1}v_iu_i^T.
$$
${\bf Proof:}$ L... | You shouldn't assume that $A^\dagger A$ is equal to $I$:
\begin{align*}
A &= U_1 \widetilde{\Sigma} V_1^T,\\
A^\dagger &= V_1 \widetilde{\Sigma}^{-1} U_1^T,\\
\Rightarrow A^\dagger A &= V_1V_1^T \ \text{ is symmetric}.
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/302096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$
I don't know how to do this ?
Note that $\gamma $ is the Euler-Mascheroni constant
| A standard trick is to use the reflection identity $$\Gamma(-n+x) \Gamma(1+n-x) = -\frac{\pi}{\sin(\pi n - \pi x)}$$
giving, under the assumption of $n\in \mathbb{Z}$
$$
\Gamma(-n+x) = (-1)^n \frac{\pi}{\sin(\pi x)} \frac{1}{\Gamma(n+1-x)} = (-1)^n \frac{\pi}{\sin(\pi x)} \frac{1}{\color\green{n!}} \frac{\color\gree... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Question about closure of the product of two sets Let $A$ be a subset of the topological space $X$ and let $B$ be a subset of the topological space $Y$. Show that in the space $X \times Y$, $\overline{(A \times B)} = \bar{A} \times \bar{B}$.
Can someone explain the proof in detail? The book I have kind of skims throug... | $(\subseteq)$: The product of closed sets $\overline{A} \times \overline{B}$ is closed. For every closed $C$ that contains $\overline{A} \times \overline{B}$, $A \times B \subseteq C$ so $\overline{A \times B} \subseteq \overline{\overline{A} \times \overline{B}} = \overline{A} \times \overline{B}$.
$(\supseteq)$: Choo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/302213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
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Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0Evaluate by complex methods
$$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$
Sis.
| It can be done easily without complex, if we note that
$$ \frac{\sin x}{1-2a\cos x+a^2}=\sum_{n=0}^{+\infty}a^n\sin[(n+1)x]$$
Just saying.
EDIT: for proving this formula, we actually use complex method
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Vector Space with Trivial Dual How to construct a Vector Space $E$ (non trivial) such that, the only continuous linear functional in $E$ is the function $f=0$?
| In addition to what Asaf has written: There are non-trivial topological vector spaces with non-trivial topologies which have a trivial dual. I think in Rudin's "Functional Analysis" it is shown that the $L^p$-spaces with $0<p<1$ are an example of this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/302374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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Finding the smallest positive integer $N$ such that there are $25$ integers $x$ with $2 \leq \frac{N}{x} \leq 5$
Find the smallest positive integer $N$ such that there are exactly $25$ integers $x$ satisfying $2 \leq \frac{N}{x} \leq 5$.
| $x$ ranges from $N/5$ through $N/2$ (ignoring the breakage) so $N/2-N/5+1=25$ so $N=80$ and a check shows $x$ goes $16$ through $40$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to read mathematical formulas? I'm coming from a programmers background, trying to learn more about physics. Immediately, I was encountered by math, but unfortunately unable to read it.
Is there a good guide available for reading mathematical notation? I know symbols like exponents, square roots, factorials, but I... | I guess the most natural anwer to "How can I improve my ability to read and interpret mathematical formulas in notation?" is: through practice. If you're trying to read physics, you're probably familiar with Calculus. I would advise then that you do a Real Analysis course, only to get used to these notations, to mathem... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove $\lim_{j\rightarrow \infty}\sum_{k=1}^{\infty}\frac{a_k}{j+k}=0$ I am only looking for a hint to start this exercise, not a full answer to the problem, please take this into consideration.
Suppose that $a_k \geq 0$ for $k$ large and that $\sum_{k=1}^\infty\frac{a_k}k$ converges. Prove that
$$\lim_{j\rightarrow \i... | I gave the hint in my comment. For a full solution, read below:
Let $\epsilon>0$.
Choose $N>1$ so that $a_j\ge 0$ for $j\ge N$ and such that $\sum\limits_{k=N}^\infty {a_k\over k}<\epsilon/2$. Note that for $j>0$, we then have
$$\tag{1}0\le \sum\limits_{k=N}^\infty {a_k\over k+j} \le\sum\limits_{k=N}^\infty {a_k\o... | {
"language": "en",
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"source": "stackexchange",
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Automorphisms on Punctured Disc I have to find the automorphism group of the punctured unit disc $D = \{|z| <1\}\setminus \{0\}$.
I understand that if $f$ is an automorphism on $D$, then it will have either a (i) removable singularity or (ii) a pole of order 1 at $z=0$.
If it has a removable singularity at 0, then $f... | A bounded holomorphic function does not have a pole.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/303734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Combinatorics Catastrophe How will you solve
$$\sum_{i=1}^{n}{2i \choose i}\;?$$
I tried to use Coefficient Method but couldn't get it! Also I searched for Christmas Stocking Theorem but to no use ...
| Maple gives a "closed form" involving a hypergeometric function:
$$ -1-{2\,n+2\choose n+1}\; {\mbox{$_2$F$_1$}(1,n+\frac32;\,n+2;\,4)}-\frac{i
\sqrt {3}}{3}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/303816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Semigroup homomorphism and the relation $\mathcal{R}$ Let $S$ be a semigroup and for $a\in S$ let $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; b \Leftrightarrow aS^1 = bS^1 \;\;\;\;\forall \;\;a,b\in S\text{.... | Use that $a\mathcal R b \iff (a=b) \lor (a\in bS\land b\in aS)$.
(For this, in direction $\Leftarrow$, in the case of $a\ne b$ we conclude $aS^1\subseteq bS$ and $bS^1\subseteq aS$.)
We can assume $a\ne b$, then $a\mathcal Rb$ means $a=bs$ for some $s$ and $b=as'$ for some $s'\in S$, so $\phi(a)=\phi(b)\phi(s)\in\phi(b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the asymptotical bound of this recurrence relation? I have the recurrence relation, with two initial conditions
$$T(n) = T(n-1) + T(n-2) + O(1)$$
$$T(0) = 1, \qquad T(1) = 1$$
With the help of Wolfram Alpha, I managed to get the result of $O(\Phi^n)$, where $\Phi = \frac{1+\sqrt 5}{2} \approx 1.618$ is the gol... | You have essentially stated the Fibonacci sequence, or at least asymptotically. There are numberless references, here for instance. And your result is not correct; as you will see from the reference, the Fib sequence behaves as $\phi^n/\sqrt{5}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/304003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Superharmonic function and super martingale. The definition (from Durrett's "Probability: Theory and Examples"):
Superharmonic functions. The name (super martingale)
comes from the fact that if $f$ is superharmonic (i.e., f has continuous derivatives of
order $\le 2$ and $\partial^2 f /\partial^2 x_1^2 + · · · + \... | $f(S_{n+1})=f(S_n+\xi_{n+1})$
Using Did's Hint: If $g(x)=E(f(x,Y))$ then $g(X)=E(f(X,Y)|X)$
$\Rightarrow E(f(S_n+\xi_{n+1})|\mathcal F_n)=g(S_n)$
but $\displaystyle g(x)=\int f(x+y)\nu(dy)$
and in our case $\nu(dy)=\mathbf 1_{B(0,1)}\frac{1}{|B(0,1)|}dy$ (because $\xi_i$ are uniform on $B(0,1)$)
$\displaystyle\Rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are these exactly the abelian groups? I'm thinking about the following condition on a group $G$.
$$(\forall A\subseteq G)(\forall g\in G)(\exists h\in G)\ Ag=hA.$$
Obviously every abelian group $G$ satisfies this condition. Are there any other groups that do? Can we give a familiar characterization for them? Can we g... | Assume $ab\ne ba$. Let $A=\{1,a\}$, $g=b$. Then there is $h\in G$ such that $\{h,ha\}=\{b,ab\}$. This needs $h=b\lor h=ab$.
In the first case $ha=ba\ne ab$, so this fails.
Therefore $h=ab$ and $ha=aba=b$. Similarly, $bab=a$. This implies $aa=abab=bb$. We conclude
$$a=bab=bbabb=aaaaa, $$
hence $a^4=1$ and similarly $b^4... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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$\epsilon$-$\delta$ proof of discontinuity How can I prove that the function defined by
$$f(x) = \begin{cases} x^{2}, & \text{if $x \in \mathbb{Q}$;} \\ -x^{2}, & \text{if $x \notin \mathbb{Q}$;} \end{cases} $$ is discontinuous?
I see that it is true by using sequences but I cannot prove using only $\epsilon$'s and $... | Hint: Check if the sequence $f(1+\frac{\sqrt{2}}{n})$ becomes very close to $f(1)$ for large $n$
To do it using $\epsilon,\delta$ definition. Suppose there exists $\delta>0$ such that for $\forall y\in \mathbb{R}[|x-y|<\delta\implies|f(x)-f(y)|<0.1]$. Now choose $n$ sufficently large such that $|(1+\frac{\sqrt{2}}{n})-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are your favorite proofs using mathematical induction? I would like to get a list going of cool proofs using mathematical induction.
Im not really interested in the standard proofs, like $1+3+5+...+(2n-1)=n^2$, that can be found in any discrete math text. I am looking for more interesting proofs.
Thanks a lot.
|
Let $a>0$ and $d\in\mathbb{N}$ and define the simplex $S_d(a)$ in $\mathbb{R}^d$ by
$$
S_d(a)=\{(x_1,\ldots,x_n)\in\mathbb{R}^d\mid x_1,\ldots,x_d\geq 0,\;\sum_{i=1}^d x_i\leq a\}.
$$
Then for every $a>0$ and $d\in\mathbb{N}$ we get the following
$$
\lambda_d(S_d(a))=\frac{a^d}{d!},\qquad (*)
$$
where $\lambda... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable? How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?
Edit: This answer came to mind. Is it correct?
This answer just came to mind. By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$. Define ... | Hint : use the diadic developpement of elements of $[0,1]$.
| {
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Decomposition of $C_0^{\infty}(\mathbb{R}^n)$ -function I got the following question as part of a fourier-analysis course..
Consider $\phi\in C_0^{\infty}(\mathbb{R}^n)$ with $\phi(0)=0$. Apparantly then we can write $$\phi =\sum_{j=1}^nx_j\psi_j $$ for functions $\psi_j$ in the same space, and I would like to prove th... | The result can be shown by induction on $n$.
For $n=1$, just write $\phi(x)=\int_0^x\phi'(t)dt=x\int_0^1\phi'(sx)ds$, and the map $x\mapsto \int_0^1\phi'(sx)ds$ is smooth, and has a compact support.
Assume the result is true for $n-1\geqslant 1$. We have
$$\phi(x)=x_n\int_0^1\partial_n\phi(x_1,\dots,x_{n-1},tx_n)dt+... | {
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Formal proof for $(-1) \times (-1) = 1$ Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
| In any ring, it holds, where $1$ denotes the unit element ($1x=x=x1$ for all $x$) and $-x$ denotes the additive inverse ($x+(-x)=0$ for all $x$).
$x=1\cdot x=(1+0)\cdot x=1\cdot x+0\cdot x=x+0\cdot x$. Then, using the additive group, it follows that $0\cdot x=0$ for all $x$.
Now use distributivity for
$$0=(1+(-1))(-1).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "93",
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Unions of disjoint open sets. Let $X$ be a compact metric space (hence separable) and $\mu$ a Borel
probability measure.
Given an open set $A$ and $r,\epsilon>0$ $\ $does there exist a finite set of
disjoint open balls $\left\{ B_{i}\right\} $ contained in $A$ and of radius smaller than $r$ , so that $\mu(\cup B_{i}... | Recall
Lemma (Finite Vitali covering lemma) Let $(X,d)$ be a metric space, $\{B(a_j,r_j),j\in [K]\}$ a finite collection of open balls. We can find a subset $J$ of $[K]$ such that the balls $B(a_j,r_j),j\in J$ are disjoint and
$$\bigcup_{i\in [K]}B(a_i,r_i)\subset \bigcup_{j\in J}B(a_j,3r_j).$$
A proof is given p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to evaluate a definite integral that involves $(dx)^2$? For example: $$\int_0^1(15-x)^2(\text{d}x)^2$$
| Just guessing, but maybe this came from $\frac {d^2y}{dx^2}=(15-x)^2$ The right way to see this is $\frac d{dx}\frac {dy}{dx}=(15-x)^2$. Then we can integrate both sides with respect to $x$, getting $\frac {dy}{dx}=\int (15-x)^2 dx=\int (225-30x+x^2)dx=C_1+225x-15x^2+\frac 13x^3$ and can integrate again to get $y=C_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on the proof of a property of the rank of a matrix The task is that I have to prove this statement:
Given $(m+1)\times(n+1)$ matrix $A$ like this:
$$
A=\left[\begin{array}{c |cc}
1 & \begin{array}{ccc}0 & \cdots & 0\end{array} \\
\hline
\begin{array}{c}0\\ \vdots\\0\end{array} & {\Large B} \\
\end{array... | In analogy, you're arguing: "$5 \leq 7, 4 \leq 7 \Rightarrow 5 \leq 4$", which, when put this way, is clearly not true.
The way to prove this exercise depends on how your class introduced these concepts. Absent that knowledge, I would argue as follows:
When reduced to row-echelon form, you can read off the rank of a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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About an extension of Riesz' Lemma for normed spaces The Riesz' Lemma is as follows:
Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for every real number $\theta$ in the open interval $(0,1)$, there is a... | I would apply the following trick in case $\dim Y<\infty$:
Let $z_0\in Z\setminus Y$ arbitrary and imagine the finte dimensional normed space $U$ spanned by $\langle Y,z_0\rangle$. Then there are many ways to continue, for example, the unit ball is compact in $U$, thus applying Riesz's lemma to $\theta_n:=1-\frac1n$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Open and Closed Set Problems using a Ball I am having trouble with these two questions. In particular, using a ball and choosing an $r$ to show that a set is open.
(a)
$$X = \left \{ \mathbf{x} \in \mathbb{R}^d | \: ||\mathbf{x}|| \leq 1 \right \} .$$
So, $X$ is closed if its compliment $X^c$ is open. So if I can... | Regarding you second question: Let $a\in X$, and you want to find $r>0$ such that $B_r(a)\subseteq X$. What could "go wrong"? Well, this ball might contain points of the form $(x,0)$, and these are not in $X$. So all we have to do is to eliminate this option. Try to figure out a general way of doing it, using an exampl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Construct a pentagon from the midpoints of its sides Let $p_{1},p_{2},p_{3},p_{4},p_{5}$ be five points in the euclidean plane such that no set of three of those points lie on the same line. It is easy to prove that there exists a unique pentagon such that $p_{1},p_{2},p_{3},p_{4},p_{5}$ are the midpoints of its sides ... | Let $A,B,C,D,E$ be the given midpoints opposite the unknown vertices $V,W,X,Y,Z$ respectively; both sets of points in rotational order. Then quadrilateral $WXYZ$ must have the midpoints of its sides on a parallelogram, and three of those midpoints are given by $E,A,B$. To find the midpoint $M$ of $\overline{ZW}$, const... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Existence of vector fields Does there exists two vector fields $X$ and $Y$ on $\mathbb R^2$ such that the following are satisfied?
*
*$X(0)= Y(0)= 0$, where $0\in \mathbb R^2$ and for others points
$q\in \mathbb R^2$, we have $X(q)\neq 0, Y(q)\neq 0$.
*For any curve $\gamma\in \mathbb R^2$, we have $\langle... | I think $ X(x,y)=(x^2+y^2)\frac{\partial}{\partial y} $ and $ Y(x,y)=(x^2+y^2)\frac{\partial}{\partial x} $ should work.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding second order linear homogenous ODE from the fundamental set of solutions
Find second order linear homogeneous ODE with constant coefficients if its fundamental set of solutions is {$e^{3t},te^{3t}$}.
Attempt: Had this question in my midterm. So, since the fundamental set of solutions is $$y=y_1+y_2=c_1e^{3t}+... | The eigenvalues and eigenvectors for the coefficient matrix $A$ in the linear homogeneous system:
$Y'= AY$ are $\lambda_{1} = 3$ with $v_1 =< a; b >$ and $\lambda_2 = 3$ with $v2 =< c; d >$
The fundamental form of the solution is:
$$ Y = c_1 e^{3t}v_1 + c_2t e^{3t}v_2$$
Take the second derivative, $Y''$ for the DEQ.
Y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Arcs of a circle Geometry and angles If a central angle of measure $30°$ is subtended by a circular arc of length $6\,\mathrm{m}$, how many meters in length is the radius of the circle?
A. $\frac{π}{36}$
В. $\frac{1}{5}$
С. $π$
D. $\frac{36}{π}$
E. $180$
How do I find out what the radius length of the angle is? The a... | HINT: The circumference of a circle of radius $r$ is $\pi r$. The $30^\circ$ angle is $\frac{30}{360}=\frac1{12}$ of the total angle at the centre of the circle, so $6$ metres is $\frac1{12}$ of the circumference of the circle. The whole circumference is therefore $6\cdot12=72$ metres, which, as already noted, is $2\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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trying to verify pdf for distance between normally distributed points Math people:
I am trying to find the probability density function for the distance between two points in $\mathbb{R}^3$ selected independently according to the Gaussian pdf $F(\mathbf{z}) = \left(\frac{1}{\sqrt{2\pi}}\right)^3 \exp(-\frac{1}{2}|\math... | The square of the distance is $r^2=(X_1-X_2)^2+(Y_1-Y_2)^2+(Z_1-Z_2)^2$, where the $X_i$s, $Y_i$s, and $Z_i$s are independent standard normal r.v.s. So, $r^2/2$ is $\chi^2$ distributed with $3$ degrees of freedom, and as you say, the distance $r$ has density $r^2\exp(-\frac{r^2}{4})/(2\sqrt{\pi}).$ p. 10 of the pape... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\lim_{n \rightarrow \infty} \frac{(2n)!}{n^{2n}}$ I am trying to show $$\lim_{n \rightarrow \infty} \frac{(2n)!}{n^{2n}}$$
I tried breaking it down, and got stuck when trying to $\left( \frac{2^{n}n!}{n^{n}} \right)$ goes to 0.
| It is not true that $\left( \frac{(2n)!}{n^{2n}} \right)=\left( \frac{2^{n}n!}{n^{n}} \right)$ The left side has a factor $2n-1$ in the numerator while the right side does not. But you can use Stirling's approximation to say $$\frac {(2n)!}{n^{2n}}\approx \frac {(2n)^{2n}}{(ne)^{2n}}\sqrt{4 \pi n}$$ and the powers of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$ Is there a way to show, that the only solution of
$$\sin(x)=y$$
is $x=y=0$ with $x,y\in \mathbb{Q}$.
I am seaching a way to prove it with the things you learn in linear algebra and analysis 1+2 (with the knowledge of a second semester student).
| Sorry for the previous spam. I shall prove for $\cos$, cosine of any rational numbers except for 0 cannot get rational numbers.
By using polynomial argument, we shall only have to prove for integers.
Suppose that $m\in\mathbb{N}$, $\cos(m)\in\mathbb{Q}$.
For any fixed prime number $p>m$, consider polynomial $x\in(0,m)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Property kept under base change and composition is preserved by products The following is true? Why?
Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$.
Then the unique morphism $X\times_S X' \to Y\times_S Y'$ has pro... | Yes this is true.
The canonical morphism $X\times_S X'\to Y\times_S Y'$ is the composition of $X\times_S X'\to Y\times_S X'$ and $Y\times_S X'\to Y\times_S Y'$. The latter verify property P because they are obtained by base change (the first one is $X\to Y$ base changed to $Y\times_S X'$, the second one is similar). A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Schoen Estimates (part 3) I'm referring to the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of Richard Schoen
In the first paragraph of the proof of theorem 2 the author seems to assert that the universal covering of $ M $ is conformally equivalent to the unit disk (with standard metri... | Indeed, $M$ could be simply the plane, which is its own universal cover. So yes, their statement
we may assume that $M$ is represented by a conformal immersion $f:D\to N$ [where $D=D_1$ is presumably the unit disk from page 116]
needs to be modified. But I think it suffices to replace $M$ with $B_R(P_0)$, which is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find (or 'generate') combinatorial meaning for the given expression $\left(\dfrac{6(k-n)(k-1)}{(n-2)(n-1)}+1\right)\dfrac{30}{n(n+1)(n+2)}$
(for $n\geq 3$ and $1\leq k \leq n$)
The expression comes from question https://math.stackexchange.com/questions/304876/please-help-to-find-function-for-given-inputs-and-out... | This post does not show a meaning of the expression, but explains how one might arrive at it.
I want a quadratic function with zero mean defined on $\{1,\dots,n\}$. Naturally, it should be symmetric about the midpoint of the interval. The obvious symmetric function is $k(k-n-1)$, but it does not have mean zero. The mea... | {
"language": "en",
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"source": "stackexchange",
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A improper integral with Glaisher-Kinkelin constant Show that :
$$\int_0^\infty \frac{\text{e}^{-x}}{x^2} \left( \frac{1}{1-\text{e}^{-x}} - \frac{1}{x} - \frac{1}{2} \right)^2 \, \text{d}x = \frac{7}{36}-\ln A+\frac{\zeta \left( 3 \right)}{2\pi ^2}$$
Where $\displaystyle A$ is Glaisher-Kinkelin constant
I see Chris's ... | Here is an identity for log(A) that may assist.
$\displaystyle \ln(A)-\frac{1}{4}=\int_{0}^{\infty}\frac{e^{-t}}{t^{2}}\left(\frac{1}{e^{t}-1}-\frac{1}{t}+\frac{1}{2}-\frac{t}{12}\right)dt$.
I think Coffey has done work in this area. Try searching for his papers on the Stieltjes constant, log integrals, Barnes G, log G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Compound interest derivation of $e$ I'm reviewing stats and probability, including Poisson processes, and I came across:
$$e=\displaystyle \lim_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$
I'd like to understand this more fully, but so far I'm struggling. I guess what I'm trying to understand is how you prove t... | It's not too hard to prove, but it does rely on a few things. (In particular the validity of the taylor expansion of $\ln$ around 1 and that $\exp$ is continuous.)
Consider in general the sequence $n\ln(1+x/n)$ which is defined for all $x$, positive or negative provided $n$ is large enough. (In fact the proof that foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Wave Equation, Energy methods. I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem:
Theorem 5 (Uniqueness for wave equation). There exists at most one function $u \in C^{2}(\overline{U}_{T})$ solving
$u_{tt} -\Delta u=f... | This quantity, the definition of $e(t)$, can be easily recognized as the Hamiltonian (basically another name of energy) of the system by a physics student. Let's explore more details.
The PDE $w_{tt}-\Delta w=0$ is equivalent to a variation problem (under some boundary conditions, maybe), whose Lagrangian (or Lagrangia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to compute the SVD of $2\times 2$ matrices? What's an efficient algorithm to get the SVD of $2\times 2$ matrices?
I've found papers about doing SVD on $2\times 2$ triangular matrices, and I've seen the analytic formula to get the singular values of a $2\times 2$ matrix. But how to use either of these to get the SVD... | The SVD of a $2\times 2$ matrix has a closed-form formula, which can be worked out by writing the rotation matrices in terms of a single unknown angle each, and then solving for those angles as well as the singular values.
It is worked out here, for instance.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Functions of algebra that deal with real number
If the function $f$ satisfies the equation $f(x+y)=f(x)+f(y)$ for every pair of real numbers $x$ and $y$, what are the possible values of $f(0)$?
A. Any real number
B. Any positive real number
C. $0$ and $1$ only
D. $1$ only
E. $0$ only
The answer for this problem ... | Just to add to the collection above, if the vector space is $V$:
$$f(v)=f(v+0)=f(v)+f(0)\Longrightarrow f(0)=0$$
for any $v\in V$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/305788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Exercise of complex variable, polynomials. Calculate the number of zeros in the right half-plane of the following polynomial:
$$z^4+2z^3-2z+10$$
Please, it's the last exercise that I have to do. Help TT. PD: I don't know how do it.
| Proceed like the previous problem for first quadrant.
You will find one root.
And note that roots will be conjugates.
So 2 roots in the right half-plane..
For zero in the first quadrant, consider the argument principle: if $Z$ is the number of zeroes of $f$ inside the plane region delimited by the contour $\gamma$, the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Ask for a question about independence This is the question I met while reading Shannon's channel coding theorem. Assume a random variable $X$ is transmitted through a noisy channel with transition probability $p(y|x)$. At the receiver a random variable $Y$ is obtained. Assume we have an additional random variable $X'$ ... | Two given conditions are:
(1)$X'$ is independent of $X$,
(2)$Y$ is generated only from $X$, i.e., $p(y|x)=p(y|x,x')$. This acutually means $X'\to X\to Y$ forms a Markov chain.
Now let's prove $X'$ is independent of bivariate random variable $(X,Y)$, i.e., $p(x,y|x')=p(x,y)$:
$p(x,y|x')=p(x|x')p(y|x,x')=p(x)p(y|x,x')$(f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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what is the logic behind the 'UPC -A' check digit? In UPC-A barcode symbology, the 12th digit is known as check digit and it is used by the scanner to determine if it scanned the number correctly or not.
the check digit is calculated as follows:
1.Add the digits in the odd-numbered positions (first, third, fifth, e... | Just a remark on your question
Can't two different combination of digits produce the same check digit?
Of course this happens, after all there are $10^{11}$ possibilities for the first $11$ digits, and only $10$ for the $12$-th one. So plenty of valid codes will share the same $12$-th digit, and a mistake that takes yo... | {
"language": "en",
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Vantage point of character theory I am not sure whether I can frame my question properly, or whether at this point my understandings permit me to comprehend the perspectives of the answers to come, but somehow I find it pretty amazing that while doing representation of finite groups over characteristic 0 fields, the ch... | I know I'm digging up an old question, but one seems to have brought up this point, although darij grinberg did briefly allude to it in the comments:
For any fixed $g \in G$, knowing the trace of $\rho(g)$ doesn't tell you much. However, the character contains the additional information of the trace of $\rho(g^k)$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306094",
"timestamp": "2023-03-29T00:00:00",
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Solve $\frac{(b-c)(1+a^2)}{x+a^2}+\frac{(c-a)(1+b^2)}{x+b^2}+\frac{(a-b)(1+c^2)}{x+c^2}=0$ for $x$ Is there any smart way to solve the equation: $$\frac{(b-c)(1+a^2)}{x+a^2}+\frac{(c-a)(1+b^2)}{x+b^2}+\frac{(a-b)(1+c^2)}{x+c^2}=0$$
Use Maple I can find $x \in \{1;ab+bc+ca\}$
| I have a partial solution, as follows:
Note that $\frac{(b-c)(1+a^2)}{x+a^2}=\frac{(b-c)\left((x+a^2)+(1-x)\right)}{x+a^2}=(b-c)+\frac{{(b-c)}(1-x)}{x+a^2}$. Likewise, $\frac{(c-a)(1+b^2)}{x+b^2}=(c-a)+\frac{(c-a)(1-x)}{x+b^2}$ and $\frac{(a-b)(1+c^2)}{x+c^2}=(c-a)+\frac{(a-b)(1-x)}{x+c^2}$.
Now, $\frac{(b-c)(1+a^2)}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Proving that $\|A\|$ is finite. Let $|v|$ be the Euclidean norm on $\mathbb{R^n} $. For $A\in \mathrm{Mat}_{n\times n}(\mathbb{R})$ we define $\displaystyle \|A\|:= \sup_{\large v\in \mathbb{R^n},\,v \neq 0}\frac{|Av|}{|v|}$. How to show that $\|A\|$ is finite for every $A$?
It would be very helpful if someone could g... | Hint:
Let $S=\{v\in\mathbb{R}^n\;|\;|v| = 1\}, N = \{\frac{|Av|}{|v|}\;|\;v\in\mathbb{R}^n.\;v\ne 0\}, N' = \{|Av|\;|\;v\in\mathbb{R}^n.\;|v|=1\}$
Step 1: $||A|| = \sup N = \sup N'$
Step 2: Show that $x\to|Ax|$ is a continuous map. $S$ is closed and bounded in $\mathbb{R}^n$ therefore compact, |Ax| attains max on $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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} |
Subgroups of order $p$ and $p^{n-1}$ in a group of order $p^n$. I have a group $G$ of order $p^n$ for $n \ge 1$ and $p$ a prime. I am looking for two specific subgroups within $G$: one of order $p$ and one of order $p^{n-1}$. I don't think I would use the Sylow theorems here because those seem to apply to groups with ... |
Let $P$ act on itself by conjugation. $1$ appears in an orbit of size $1$, and everything else appears in an orbit of size $p^k$ for some $k$. Since the sum of the orbit sizes is equal to $|P|$, which is congruent to $0\mod{p}$, that means there has to be at least one more orbit of size $1$. Orbits of size $1$ unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Show that $(m^2 - n^2, 2mn, m^2 + n^2)$ is a primitive Pythagorean triplet Show that $(m^2 - n^2, 2mn, m^2 + n^2)$ is a primitive Pythagorean triplet
First, I showed that $(m^2 - n^2, 2mn, m^2 + n^2)$ is in fact a Pythagorean triplet.
$$\begin{align*} (m^2 - n^2)^2 + (2mn)^2 &= (m^2 + n^2)^2 \\
&= m^4 -2m^2n^2 + n^4 + ... | To show $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ is equivalent to showing $(m^2 - n^2)^2 + (2mn)^2 - (m^2 + n^2)^2 = 0$ so \begin{align*} && (m^2 - n^2)^2 + (2mn)^2 - (m^2 + n^2)^2 \\ &=& m^4 - 2m^2n^2 + n^4 + 4m^2n^2 - m^4 - 2m^2n^2 - n^4 \\ &=& m^4 + n^4 - m^4 - n^4 \\ &=& 0\end{align*}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sine not a Rational Function Spivak This is Chapter 15 Question 31 in Spivak:
a) Show sin is not a rational function.
By definition of a rational function, a rational function cannot be $0$ at infinite points unless it is $0$ everywhere. Obviously, sin have infinite points that are 0 and infinite points that are not ze... | Hint: Use continuity of $\sin$ and rational functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/306464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Stadium Seating A circular stadium consists fo 11 sections with aisles in between. With 13 tiers of concrete steps for the final section, section K. Seats are places along every concrete step, with each step 0.45 m wide. The arc AB at the fron of the first row is 14.4 m long while the arc CD at the back of the back row... | If the inner radius of row $i$ is $r_i$, then the arc length of the inner arc of that row is given by
$$L_i = \frac{2\pi r_i}{11}.$$
If the width $w$ of each step is constant, then
$$r_i = r_1 + (i-1)w$$
and the arc length of the outer arc of step $i$ is
$$M_i = \frac{2\pi (r_i+w)}{11}.$$
You now have two unknowns -- ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If the matrix of a linear map is independent from the basis, then the map is a multiple of the identity map. Let $V$ be a finite dimensional vector space over $F$, and let
$$T:V\to V$$
be a linear map.
Suppose that given any two bases $B$ and $C$ for $V$, we have that the matrix of $T$ with respect $B$ is equal to that... | Hint: This is equivalent to say that any nonzero vector is an eigenvector of $T$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Legendre symbol- what is the proof that it is a homomorphism? I know that one property of the Legendre symbol is that it is a homomorphism. However, I have not been able to find a proof that this is the case. If someone could give me or show me to a thorough proof of this, that would be great.
I am going with the defin... | I am wondering if it is permissible to use the primitive roots modulo a prime $p$.
Suppose so, and then we look at the definition of a Legendre symbol, and then give a proof that it is a homomorphism. So fix a prime $p$ first.
Let $x$ be a number not divisible by $p$. Suppose that we already have at disposal a primitiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Tricky T/F... convergence T or F?
1) If $x_n \rightarrow 0$ and $x_n \neq 0$ for all $n$, then the sequence {$1/n$} is unbounded.
Also similarly...
2) If {$x_n$} is unbounded and $x_n \neq 0$ for all $n$, then $1/x_n \rightarrow 0$.
For the first one, I would say that is true because the limit of {$1/n$} would approa... | The first one is true; but note the sequence $(1/x_n)$ need not converge to $\infty$. The sequence $(1/|x_n|)$, however, would. Consider here, for example, the sequence $(1/2,-1/3,1/4,-1/5,\ldots)$.
For the second one, consider the sequence $(1,1,2,1,3,1,4,1,5,\ldots)$. Note this sequence is unbounded, but the sequenc... | {
"language": "en",
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Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$ Find
$$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
| Here is another sketch of proof.
Let
$$J_n = \{(j, k) : 0 \leq j, k < n \text{ and } (j, k) \neq (0, 0) \}.$$
Then for each $(j, k) \in J_n$ and $(x_0, y_0) = (j/n, k/n)$, we have
$$ \frac{x_0 + y_0}{(x_0+\frac{1}{n})^{3} + (y_0 + \frac{1}{n})^3} \leq \frac{x+y}{x^3 + y^3} \leq \frac{x_0 + y_0 + \frac{2}{n}}{x_0^3 + y_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Powers and the logarithm By example:
*
*$4^{\log_2(n)}$ evaluates to $n^2$
*$2^{\log_2(n)}$ evaluates to $n$
What is the rule behind this?
| Here's a method that relies more on applying an appropriate strategy than on formulas.
If you want to rewrite $4^{\log_2(n)}$ as a power of $n$, then you simply want to solve for $u$ in the following equation:
$$4^{\log_2(n)} = n^u$$
The standard way of solving for something that appears in the exponent of an exponenti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$E^2[X]$ vs.$E[X^2]$, what's the difference? This has come up in a homework problem, but I've never seen exponents defined in terms of random variables and expected values. I've tried googling this, but I must not be using the right words. If anyone could define both terms for me, or briefly explain the difference, i... | The difference is exactly the same as with $$ a^2 + b^2 \neq (a+b)^2$$
Another thing to see is that $\rm E[X^2]$ is the $\rm L^2$-norm is $\rm X$ is centered ; $(\rm E[X])^2$ is the square of the $\rm L^1$-norm.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/306991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$ What ways would you propose for the limit below?
$$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$
Thanks in advance for your suggestions, hints!
Sis.
| OK, it turns out that
$$\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right) = \sum_{k=1}^{n-1} \frac{1}{(k+n)^2}$$
This may be shown by observing that
$$\sum_{k=1}^n \frac{1}{(2k-1)^2} = \sum_{k=1}^{2 n-1} \frac{1}{k^2} - \frac{1}{2^2} \sum_{k=1}^n \frac{1}{k^2}$$
The desired limit may then be rewritten as
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Differentiation in 3d of sin root and fractions in one! -> to find the normal to a function I have to find the normal of this function at a defined point $x$ and $z$, I have done A Level maths but that was some time ago but I don't think it was covered to this level, I am now doing a CS degree. I thought the best way w... | The normal to a curve at a point is perpendicular to the gradient at that point. In your case:
$$f(x,z) = \frac{\sin{\sqrt{x^2+z^2}}}{\sqrt{x^2+z^2}}$$
$$\begin{align}\nabla f &= \left ( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial z}\right ) \\ &= \left ( \frac{x \cos
\left(\sqrt{x^2+z^2}\right)}{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307171",
"timestamp": "2023-03-29T00:00:00",
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Why isn't this a valid argument to the "proof" of the Axiom of Countable Choice? I am having a little trouble identifying the problem with this argument:
Let $\{A_1, A_2, \ldots, A_n, \ldots\}$ be a sequence of sets.
Let $X:= \{n \in \mathbb{N} : $ there is an element of the set $A_n$ associated to $n \}$
(1) $A_1$ is ... | To avoid Choice, you need to have a definitive way of choosing an element from each set.
For example, if each $A_n$ is a pair of shoes, you may always choose the left.
A typical useful case is when each $A_n$ has a distinguished member such as a unique minimum that you can choose. For example, Baire Category Theorem f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Combination of three items with no adjacent items the same I am looking for a closed expression for calculating the number of combinations of $n = n_1 + n_2 + n_3$ objects arranged in an ordered list, where there are $n_1$ $a$, $n_2$ $b$ and $n_3$ $c$, under the constraint that $a$ may not appear adjacent to $a$, $b$ n... | In addition to the formula linked above there is a nice generating function. The number of solutions with $i$ $a$'s, $j$ $b$'s and $k$ $c$'s is the coefficient of $a^ib^jc^k$ in $$\frac{1}{1 - \frac{a}{1+a} - \frac{b}{1+b} - \frac{c}{1+c}}.$$ This is a consequence of the beautiful "Carlitz-Scoville-Vaughan" theorem; ... | {
"language": "en",
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Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$. Prove that $x^2=e$ and $y^8=e$ for all $y\in G$. To put this in context, this is my first week abstract algebra.
Let $G$ a group. Let $x\in G$. Assume that for every $y\in G, xyx=y^3$.
Prove that $x^2=e$ and $y^8=e$ for all $y\in G$.
A hint would b... | $\textbf{Full answer:}$
Let $x\in G$. Suppose $(\forall y\in G)(xyx=y^3)$.
(i) Set $y=e$ to get $xyx=xex=x^2=e=e^3=y^3$. (The OP already knew this part).
(ii) The hypothesis is $(\forall y\in G)(xyx=y^3)$. Let $y\in G$ be taken arbitrarily. By the hypothesis we have $xyx=y^3$. Cubing both sides we get $(xyx)(xyx)(xyx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Appearance of Formal Derivative in Algebra When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are coprime. My question is, should we be surprised to see the formal derivative here?... | Formal derivatives appear naturally when trying to rewrite a polynomial in $ x $ ( over a characteristic $ 0 $ field ) as a polynomial in $ (x-c) $ :
Let $ f(x) = a_n x^n + \ldots + a_1 x + a_0 \in K[x] $ ( with $ a_n \neq 0 $ ). Given any $ c \in K $, we can expand $ f(x) $ as a polynomial in $ (x-c) $ to get
$ \displ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 8,
"answer_id": 4
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A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$ The following question comes from Some integral with sine post
$$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$
but now I'd be curious to know how to deal with it by methods of complex analysis.
Some suggestions, hints? ... | A complete asymptotic expansion for large $n$ may be derived as follows. We write
$$
\int_0^{ + \infty } {\left( {\frac{{\sin t}}{t}} \right)^n {\rm d}t} = \int_0^\pi {\left( {\frac{{\sin t}}{t}} \right)^n {\rm d}t} + \int_\pi ^{ + \infty } {\left( {\frac{{\sin t}}{t}} \right)^n {\rm d}t} .
$$
Notice that
$$
\int_\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$? How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$?
I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got:
$$H = \displaystyle\pmatrix{\frac{2}{y} & -\frac{2x}{y^2} \\-\frac{2x}{y^2} & \frac{2x... | You're basically done:
A symmetric matrix $(a_{ij})_{i,j\le n}$ is positive semidefinite iff the subdeterminants $\det(a_{ij})_{i,j\in H}$ are $\ge 0$ for all $H\subseteq\{1,2,..,n\}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Convergence of a particular double series For the double series
$$ \sum_{m,n=1}^{\infty} \frac{1}{(m+n)^p} , $$
I was wondering when it converges. I want to use double integrals to estimate it, but I don't know how to write the process accurately... Could you show me a detailed computation? Thanks!
| While we wait for someone to do it using double integrals, here's another way:
Given a positive integer $k$, the number of pairs $m,n$ with $m\ge1$, $n\ge1$, and $m+n=k$ is $k-1$. So your sum is $\sum_{k=2}^{\infty}(k-1)/k^p$, and the usual single-series methods apply.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/307646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Really Stuck on Partial derivatives question Ok so im really stuck on a question. It goes:
Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$.
calculate $\frac{\partial u} {\partial x} (x,y)$ and $\frac{\partial u} {\partial y} (x,y)$ for all $ (x,y) \in \Bbb R^2. $ show ... | We are given:
$$u(x, y)=\begin{cases}
xy \frac {x^2-y^2}{x^2+y^2}, ~(x, y) \ne (0,0)\\\\
~~~0, ~~~~~~~~~~~~~~~(x, y) = (0,0)\;.
\end{cases}$$
I am going to multiply out the numerator for ease in calculations, so we have:
$$\tag 1 u(x, y)=\begin{cases}
\frac {x^3y - xy^3}{x^2+y^2}, ~(x, y) \ne (0,0)\\\\
~~~0, ~~~~~~~~~~... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of a sum of rational functions I have the complex functions $f_n(z) = 1/(1+z^n)$ and I'm supposed determine where $\sum_{n=1}^\infty f_n(z)$ converges for $z \in\mathbb{C}$
Extra Info:
I was only able to determine convergence for $|z| \gt 1$. The argument applies the ratio test:
$|\frac{f_{n+1}}{f_n}| = ... | For any point $z \in \bar{\mathbb D}$ you can apply the divergence test to your summation to conclude that it does not converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/307830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about normal subgroup $H<G,aHa^{-1}<G$. Then, $H$ is isomorphic to $aHa^{-1}$.
I want to show $aha^{-1}\in H,\forall h\in H,a\in G$, but I cannot figure it out. Any hint is appreciated.
| To show that $H\cong gHg^{-1}$, you need to show that for any $a\in G$, the conjugation $$\kappa_a : G\to G,\quad g\mapsto aga^{-1}$$ is an automorphism of $G$. The proof is easy: For $g,h\in G$ we have $$\kappa_a(gh) = agha^{-1} = ag(a^{-1}a)ha^{-1} = (aga^{-1})(aha^{-1}) = \kappa_a(g)\kappa_a(h)\text{,}$$
so $\kappa_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Multidimensional Hensel lifting I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ...
Let $p$ be an odd prime: we work over the $p$-adic numbers $Q_p$ with ring of integers $Z_p$ and res... | The same proof method works in higher dimension.
Let $F: \mathbb{Z}_p^m \to \mathbb{Z}_p^n$ be our system of polynomial functions.
If $x \in \mathbb{Z}_p^m$ satisfies
$$ F(x) \equiv 0 \pmod {p^e} $$
then
$$ F(x) \equiv p^e y \pmod {p^{e+1}} $$
for some vector $y$. Furthermore, the Taylor expansion about $x$ tells us
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308031",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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Roots of the equation $I_1(b x) - x I_0(b x) = 0$ I'm interested in the roots of the equation:
$I_1(bx) - x I_0(bx) = 0$
Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant.
More specifically, I'm interested in the behaviour of the largest non-negative root for $x \geq 0$.... | Notice that the equation also can be rewritten as
$$
\frac{I_1(b x)}{I_0(b x)} = x
$$
Using the asymptotic series expansion for large $b$ and some fixed $x>0$ we get:
$$
1 - \frac{1}{2 x b} - \frac{3}{8 b^2 x^2} - \frac{1}{8 b^3 x^3} + \mathcal{o}\left(b^{-1}\right) = x
$$
resulting in
$$
x(b) = 1 - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308119",
"timestamp": "2023-03-29T00:00:00",
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Proving {$b_n$}$_{n=1}^\infty$ converges given {$a_n$}$_{n=1}^\infty$ and {$a_n b_n$}$_{n=1}^\infty$ Suppose {$a_n$}$_{n=1}^\infty$ and {$b_n$}$_{n=1}^\infty$ are sequences such that {$a_n$}$_{n=1}^\infty$ coverges to A$\neq$0 and {$a_n b_n$}$_{n=1}^\infty$ converges. Prove that {$b_n$}$_{n=1}^\infty$ converges.
What... | Hint: Let $A=\lim\limits_{n\to\infty}a_n$ and $B=\frac{\lim\limits_{n\to\infty}a_nb_n}{A}$.
Since $|A|>0$, for $n$ large enough, $|a_n-A|\le\frac{|A|}{2}$. Show that then, $|a_n|\ge\frac{|A|}{2}$.
Then note that
$$
a_n(b_n-B)=(a_nb_n-AB)-B(a_n-A)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/308158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Linear Transformations and Identity Matrix Suppose that $V$ is a finite dimensional vector space and $T$ is a linear transformation from $V$ to $V$. Prove that $T$ is a scalar multiple of the identity matrix iff $ST=TS$ for every linear transformation $S$ from $V$ to $V$.
| One direction is trivial. For the other, take $\{v_1,...,v_n\}$ to be any basis for $V$, and consider the transformations $E_{i,j}:V\to V$ induced by $$v_k\mapsto\begin{cases} v_j & \text{if }k=i,\\0 & \text{otherwise.}\end{cases}$$ What can we conclude about $T$ from the fact that $E_{i,j}T=TE_{i,j}$ for all $i,j\in\{... | {
"language": "en",
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How to rewrite $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta,\cos4\theta$? I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$.
My attempts so far has been unsuccessful and I constantly get developments that are way to cumbersome and not elegant... | Write
$$\sin^4{\theta} = \left ( \frac{e^{i \theta} - e^{-i \theta}}{2 i} \right )^4$$
and use the binomial theorem.
$$\begin{align}\left ( \frac{e^{i \theta} - e^{-i \theta}}{2 i} \right )^4 &= \frac{1}{16} (e^{i 4 \theta} - 4 e^{i 2 \theta} + 6 - 4 e^{-i 2 \theta} + e^{-i 4 \theta}) \\ &= \frac{1}{8} (\cos{4 \theta} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
If $P(A \cup B \cup C) = 1$, $P(B) = 2P(A) $, $P(C) = 3P(A) $, $P(A \cap B) = P(A \cap C) = P(B \cap C) $, then $P(A) \le \frac14$ We have ($P$ is probability):
$P(A \cup B \cup C) = 1$ ; $P(B) = 2P(A) $ ; $P(C) = 3P(A) $ and $P(A \cap B) = P(A \cap C) = P(B \cap C) $. Prove that $P(A) \le \frac{1}{4} $.
Well, I tried ... | This solution is quite possibly messier than necessary, but it’s what first occurred to me. For convenience let $x=P\big((A\cap B)\setminus C\big)$ and $y=P(A\cap B\cap C)$. Let $a=P(A)-2x-y$, $b=P(B)-2x-y$, and $c=P(C)-2x-y$; these are the probabilities of $A\setminus(B\cup C)$, $B\setminus(A\cup C)$, and $C\setminus(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd. Find and example of two elements $a,b$ in a finite group $G$ such that $|a| = |b| = 2, a \ne b$ and $|ab|$ is odd. Any ideas as to how I would go about finding it?
Thanks
| There is a very general example you should know about, that of dihedral groups. A dihedral group has order $2n$, for any $n \ge 2$, and it is generated by two elements of order $2$, whose product has order $n$.
Probably the simplest way to see these groups is as a group of bijective maps on $\mathbf{Z}_{n}$,
$$
a : x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Is this the way to estimate the amount of lucky twins? To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does this imply that a good way to estimate the amount of lucky twins b... | I think that to answer this question one would need to start from the paper by Bui and Keating on the random sieve (of which the lucky sequence is a particular realization) and then carefully examine the proofs that they cite of Mertens' theorems for that sieve. The issue is whether there is a correction factor like M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Isomorphism with Lie algebra $\mathfrak{sl}(2)$ Let $L$ be a Lie algebra on $\mathbb{R}$. We consider $L_{\mathbb{C}}:= L \otimes_{\mathbb{R}} \mathbb{C}$ with bracket operation
$$ [x \otimes z, y \otimes w] = [x,y] \otimes zw $$
far all $x,y \in L$ and $z,w \in \mathbb{C}$. We have that $L_{\mathbb{C}}$ is a Lie algeb... | For (1):
Let
$$i:=(1,0,0),\quad j:=(0,1,0)\text{ and } k:=(0,0,1)\in\mathbb R^3$$
and
$$
A:=\left(\begin{array}{ccc}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0\end{array}\right),\quad
B:=\left(\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{array}\right)
\text{ and }
C:=\left(\begin{array}{ccc}0 & 0 & 1 \\ 0 & 0 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
quotient of finitely presented module $\DeclareMathOperator\Coker{Coker}$Assume the exact sequence
$$A \xrightarrow{f} B \xrightarrow{} \Coker f \xrightarrow{} \{0\} $$ where $A$ is a finitely generated module and $B$ is a finitely presented module.
Is it true that $\Coker f$ is finitely presented module ?
In general, ... | This is true.
I'll denote the ring by $R$, while remaining silent about whether these are left or right modules.
Write $B = R^n/I$ for some finitely generated submodule $I \subseteq R^n$. Then the image of $f \colon A \to B$ is of the form $J/I$ for some submodule $J \subseteq R^n$, and $J/I$ is a finitely generated... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
limit of the error in approximating definite integral with midpoint rule I want to calculate $\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$ where $I_n$ is the integral approximation by midpoint rule:
$I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$ and $c_k$ is the point in the middle of $k^{th}$ interval.
My attempt:
... | Assume $f''$ is continuous. The error in a single Midpoint Rule interval of length $h$ is
$$ \int_a^{a+h} f(x)\ dx - h f(a+h/2) = -\frac{h^3}{24} f''(\xi)$$
for some $\xi \in [a,a+h]$. The error for $n$ equal subintervals of $[0,1]$ is the sum of the errors in each: if $h = 1/n$ and $x_j = j h$ we get
$$\int_0^1 f(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Applications of the Isomorphism theorems In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I can't recall having ever used the other two. Can anyone give some examples where... | E.g., in studying solvable groups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/308942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 7,
"answer_id": 6
} |
Cantor set + Cantor set =$[0,2]$ I am trying to prove that
$C+C =[0,2]$ ,where $C$ is the Cantor set.
My attempt:
If $x\in C,$ then $x= \sum_{n=1}^{\infty}\frac{a_n}{3^n}$ where $a_n=0,2$
so any element of $C+C $ is of the form $$\sum_{n=1}^{\infty}\frac{a_n}{3^n} +\sum_{n=1}^{\infty}\frac{b_n}{3^n}= \sum_{n=1}^{... | Short answer for this question is "your argument is correct ". To justify the answer consider some particular $n_{0}\in \mathbb{N}$. Since $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{3^{n}},\sum_{n=1}^{\infty}\dfrac{b_{n}}{3^{n}}\in C$$ we have that $ x_{n_{0}}=\dfrac{a_{n_{0}}+b_{n_{0}}}{2}\in \{0,1,2\} $. ($ x_{n_{0}}=0 $ if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 6,
"answer_id": 3
} |
Riesz representation theorem on dual space The Riesz representation theorem on Hilbert spaces is well known, It asserts we can represent a bounded linear function on a Hilbert space $H$ with an inner product on $H$ and vice-versa.
My question: Given an inner product in $H^*$, say $(a,b)_{H^*}$, can I write it as $$(a... | Well, for an arbitrary inner product on $X:=H^*$ it is not going to work, since then $X^*$ need not be isomorphic to $H$.
On the other hand, the Riesz representation gives a linear isomorphism $H\to H^*$, and if the inner product is defined via this isomorphism, i.e. if
$$(\langle x,-\rangle,\langle y,-\rangle)_{H^*}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Expectation of $X$ given a Cumulative Function I've (hard) to find the expectation of $X$:
CDF (Cumulative Function) = $1 - x^{-a}$, $1 \leqslant x < \infty$
$E[X]$ = ??
| First you have to establish the proper domain for the random variable $X$. That can be done by using $F(x_{\min}) = 0$. You random variable will be supported on $[x_{\min}, \infty)$.
*
*Find the probability density function, $f_X(x)$ by differentiating the cumulative function, $f_X(x) = F_X^\prime(x)$.
*Apply the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$T:P_3 → P_3$ be the linear transformation such that... Let $T:P_3 \to P_3$ be the linear transformation such that $T(2 x^2)= -2 x^2 - 4 x$, $T(-0.5 x - 5)= 2 x^2 + 4 x + 3$, and $T(2 x^2 - 1) = 4 x - 4.$
Find $T(1)$, $T(x)$, $T(x^2)$, and $T(a x^2 + b x + c)$, where $a$, $b$, and $c$ are arbitrary real numbers.
| hint:T is linear transformation $T(a+bx+cx^2)=aT(1)+bT(x) +cT(x^2) $ $$T(2 x^2)= -2 x^2 - 4 x \to T( x^2)= - x^2 - 2 x$$$$T(-0.5 x - 5)= 2 x^2 + 4 x + 3\to-0.5T( x) - 5T(1)= 2 x^2 + 4 x + 3\to 16x^2+72x-46$$$$T(2 x^2 - 1) = 4 x - 4.\to T( x^2) - \frac12T(1) = 2 x - 2\to T(1)=-2 x^2 - 8x +4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/309279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Truth set of $-|x| \lt 2$? An exercise in my Algebra I book (Pearson and Allen, 1970, p. 261) asks for the graph of the truth set for $-\left|x\right| \lt 2; x \in \mathbb{R}$.
I've re-stated the inequality in the equivalent form of $\left|x\right| \gt -2$. I know that the truth set of $\left|x\right| = -2$ is $\emptys... | $$-|x|<2\stackrel{\text{multiplication by}\,\,(-1)}\Longleftrightarrow |x|>-2$$
So: for what values of (real) $\,x\,$ it is true that $\,|x|>-2\,$ ? Hint: this is a rather huge subset of the reals...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/309341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Can any mathematical relation be called an 'operator'? Mathematics authors agree that $+,-,/,\times$ are basic operators. There are also logical operators like $\text{or, and, xor}$ and the unary negation operator $\neg$.
Where there seems to be a disagreement, however, is whether certain symbols used in the compositio... | A serious disadvantage of treating logical connectives as operators returning a value (that is, as functions) is that in non-classical logics their "range" may fail to correspond to any two-element set $\{\top,\bot\}$, and according to the Wikipedia page Truth value, may even fail to correspond to any set whatsoever: "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1...w_m$? DFA
Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1...w_m$?
For part 2, wouldn't it require M states if the word length is M?
| A language over the alphabet $\Sigma$ will be a regular language given that it follows the following clauses:
*
*$\epsilon$, {$a$} for $a\in\Sigma$.
*If $L_1$ and $L_2$ are regular languages, then $L_1\cup L_2$, $L_1L_2$, and $L^*$ are also regular.
*$L$ is not a regular language unless taken from those two claus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How is the set of all programs countable? I'm having a hard time seeing how the number of programs is not uncountable, since for every real number, you can create a program that's prints out that number. Doesn't that immediately establish uncountably many programs?
| All the answers to fit the question number of programs are countable use the discrete finite definition of program , using either finite memory, finite ( countable ) instruction etc.
How ever in the old analog days where voltage was considered as output it was a trivial task to construct a circuit that prdouced all the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 12,
"answer_id": 7
} |
Examples of categories where morphisms are not functions Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
| Let $G$ be a directed graph. Then we can think of $G$ as a category whose objects are the vertices in $G$. Given vertices $a, b \in G$ the morphisms from $a$ to $b$ are the set of paths in the graph $G$ from $a$ to $b$ with composition being concatenation of paths. Note that we allow "trivial" paths that start at a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 9,
"answer_id": 8
} |
Finding the standard deviation of a set of angles. My question is given a set of angles in the form of degrees minutes and seconds once finding the mean how do you find the standard deviation.
I know how to find the average or mean of a set (see below) but i'm not sure how to find the standard deviation.
For example sa... | Your first number is $39 + \dfrac{15}{60} + \dfrac{1}{60^2} = 39.250277777\ldots$. Deal similarly with the others.
And remember to do as little rounding as possible at least until the last step. Rounding should always wait until the last step except when you know how much effect it will have on the bottom line. One ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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