Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Differential equation of $y = e^{rx}$ I am trying to find what values of r in $y = e^{rx}$ satsify $2y'' + y' - y = 0$
I thought I was being clever and knew how to do this so this is how I proceeded.
$$y' = re^{rx}$$
$$y'' = r^2 e^{rx}$$
$$2(r^2 e^{rx}) +re^{rx} -e^{rx} = 0 $$
I am not sure how to proceed from here, th... | Here's the best part: $e^{rx}$ is never zero. Thus, if we factor that out, it is simply a quadratic in $r$ that remains.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/158013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Infinite products - reference needed! I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to read Davenport's Multiplicative Number Theory, and the treatment of L-functions in ... | I will answer your question
"Most importantly I'd like to know why
$$
\prod (1+|a_n|) \to a < \infty \quad \Longrightarrow \quad \prod (1+ a_n) \to b \neq 0.
"$$
We will first prove that if $\sum \lvert a_n \rvert < \infty$, then the product $\prod_{n=1}^{\infty} (1+a_n)$ converges. Note that the condition you have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 1
} |
Matrix with no eigenvalues Here is another problem from Golan.
Problem: Let $F$ be a finite field. Show there exists a symmetric $2\times 2$ matrix over $F$ with no eigenvalues in $F$.
| The solution is necessarily split into two cases, because the theory of quadratic equations has a different appearance in characteristic two as opposed to odd characteristic.
Let $p=\mathrm{char}\, F$. Assume first that $p>2$. Consider the matrix
$$
M=\pmatrix{a&b\cr b&c\cr}.
$$
Its characteristic equation is
$$
\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Notation for an indecomposable module. If $V$ is a 21-dimensional indecomposable module for a group algebra $kG$ (21-dimensional when considered as a vector space over $k$), which has a single submodule of dimension 1, what is the most acceptable notation for the decomposition of $V$, as I have seen both $1\backslash 2... | My feeling is that this notation is not sufficiently standard for you to use either choice without explanation, hence whichever choice you make, you should signal it carefully in your paper. Given that, either choice looks fine to me.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/158192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Numerical Analysis over Finite Fields Notwithstanding that it isn't numerical analysis if it's over finite fields, but what topics that are traditionally considered part of numerical analysis still have some substance to them if the reals are replaced with finite fields or an algebraic closure thereof? Perhaps using H... | The people who factor large numbers using sieve algorithms (the quadratic sieve, the special and general number field sieves) wind up with enormous (millions by millions) systems of linear equations over the field of two elements, and they need to put a lot of thought into the most efficient ways to solve these systems... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true? Usually for modules $M_1,M_2,N$
$$M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$$
is wrong. I'm just curious, but are there any cases or additional conditions where it gets true?
James B.
| A standard result in this direction is the Krull-Schmidt Theorem:
Theorem (Krull-Schmidt for modules) Let $E$ be a nonzero module that has both ACC and DCC on submodules (that is, $E$ is both artinian and noetherian). Then $E$ is a direct sum of (finitely many) indecomposable modules, and the direct summands are unique... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers? The category $\mathbf{Set}$ contains as its objects all small sets and arrows all functions between them. A set is "small" if it belongs to a larger set $U$, the universe.
Let $\mathbf{Grp}$ be the category of small groups a... | This is very similar to Arturo Magidin's answer, but offers another point of view.
Consider the dihedral group $D_n=\mathbb Z_n\rtimes \mathbb Z_2$ with $2\nmid n$ (so the $Z(D_n)=1$). From the splitting lemma we get a short exact sequence
$$1\to\mathbb Z_n\rightarrow D_n\xrightarrow{\pi} \mathbb Z_2\to 1$$
and an arro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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"answer_id": 0
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Perturbation problem This is a mathematica exercise that I have to do, where $y(x) = x - \epsilon \sin(2y)$ and it wants me to express the solution $y$ of the equation as a power series in $ \epsilon$.
| We're looking for a perturbative expansion of the solution $y(x;\epsilon)$ to
$$ y(x) = x-\epsilon \sin(2y)$$
I don't know if you're asking for an expansion to all orders. If so, I have no closed form to offer. But, for illustration, the first four terms in the series may be found as follows by use of the addition theo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Orthonormal basis Consider $\mathbb{R}^3$ together with inner product $\langle (x_1, x_2, x_3), (y_1, y_2, y_3) \rangle = 2x_1 y_1+x_2 y_2+3 x_3 y_3$. Use the Gram-Schmidt procedure to find an orthonormal basis for $W=\text{span} \left\{(-1, 1, 0), (-1, 1, 2) \right\}$.
I don't get how the inner product $\langle (x_1, ... | The choice of inner product defines the notion of orthogonality.
The usual notion of being "perpendicular" depends on the notion of "angle" which turns out to depend on the notion of "dot product".
If you change the way we measure the "dot product" to give a more general inner product then we change what we mean by "an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp(n x-\frac{x^2}{2}) \sin(2 \pi x)\ dx$ I want to evaluate the following integral ($n \in \mathbb{N}\setminus \{0\}$):
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp\left(n x-\frac{x^2}{2}\right) \sin(2 \pi x)\ dx$$
Maple and WolframAlpha tell me that this ... | $$I = \frac{e^{n^2/2}}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} e^{-\frac{(x-n)^2}{2}} \sin (2\pi x) \, dx \stackrel{x = x-n}{=} \frac{e^{n^2/2}}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} e^{-\frac{x^2}{2}} \sin (2\pi x) \, dx$$
Now divide the integral into two parts:
$$\int_{-\infty}^{+\infty} e^{-\frac{x^2}{2}} \sin (2\pi x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158714",
"timestamp": "2023-03-29T00:00:00",
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What is the result of sum $\sum\limits_{i=0}^n 2^i$
Possible Duplicate:
the sum of powers of $2$ between $2^0$ and $2^n$
What is the result of
$$2^0 + 2^1 + 2^2 + \cdots + 2^{n-1} + 2^n\ ?$$
Is there a formula on this? and how to prove the formula?
(It is actually to compute the time complexity of a Fibonacci recur... | Let us take a particular example that is large enough to illustrate the general situation. Concrete experience should precede the abstract.
Let $n=8$. We want to show that $2^0+2^1+2^2+\cdots +2^8=2^9-1$. We could add up on a calculator, and verify that the result holds for $n=8$. However, we would not learn much d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Integral of $\int 2\,\sin^{2}{x}\cos{x}\,dx$ I am asked as a part of a question to integrate $$\int 2\,\sin^{2}{x}\cos{x}\,dx$$
I managed to integrate it using integration by inspection:
$$\begin{align}\text{let } y&=\sin^3 x\\
\frac{dy}{dx}&=3\,\sin^2{x}\cos{x}\\
\text{so }\int 2\,\sin^{2}{... | Another natural approach is the substitution $u=\sin x$.
The path your instructor chose is less simple. We can rewrite $\sin^2 x$ as $1-\cos^2x$, so we are integrating $2\cos x-2\cos^3 x$. Now use the identity $\cos 3x=4\cos^3 x-3\cos x$ to conclude that $2\cos^3 x=\frac{1}{2}\left(\cos 3x+3\cos x\right)$.
Remark: The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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A series with prime numbers and fractional parts Considering $p_{n}$ the nth prime number, then compute the limit:
$$\lim_{n\to\infty} \left\{ \dfrac{1}{p_{1}} + \frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \right\} - \{\log{\log n } \}$$
where $\{ x \}$ denotes the fractional part of $x$.
| This is by no means a complete answer but a sketch on how to possibly go about. To get the constant, you need some careful computations.
First get an asymptotic for $ \displaystyle \sum_{n \leq x} \dfrac{\Lambda(n)}{n}$ as $\log(x) - 2 + o(1)$.
To get this asymptotic, you need Stirling's formula and the fact that $\psi... | {
"language": "en",
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Is $A^{q+2}=A^2$ in $M_2(\mathbb{Z}/p\mathbb{Z})$? I'm wondering, why is it that for $q=(p^2-1)(p^2-p)$, that $A^{q+2}=A^2$ for any $A\in M_2(\mathbb{Z}/p\mathbb{Z})$?
It's not hard to see that $GL_2(\mathbb{Z}/p\mathbb{Z})$ has order $(p^2-1)(p^2-p)$, and so $A^q=1$ if $A\in GL_2(\mathbb{Z}/p\mathbb{Z})$, and so the e... | If $A$ is not invertible, then its characteristic polynomial is either $x^2$ or $x(x-a)$ for some $a\in\mathbb{Z}/p\mathbb{Z}$. In the former case, by the Cayley-Hamilton Theorem we have $A^2 = 0$, hence $A^{q+2}=A^2$. In the latter case, the matrix is similar to a diagonal matrix, with $0$ in one diagonal and $a$ in t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Parametric equation of a cone.
I have a cone with vertex (a, b, c) and base circumference with center $(x_0,y_0)$ and radius R. I can't understand what is the parametric representation of three dimensional space inside the cone. Any suggestions please?
| The parametric equation of the circle is:
$$
\gamma(u) = (x_0 + R\cos u, y_0 + R\sin u, 0)
$$
Each point on the cone lies on a line that passes through $p(a, b, c)$ and a point on the circle. Therefore, the direction vector of such a line is:
$$
\gamma(u) - p = (x_0 + R\cos u, y_0 + R\sin u, 0) - (a, b, c) = (x_0 - a +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Signature of a manifold as an invariant Could you help me to see why signature is a HOMOTOPY invariant? Definition is below (from Stasheff)
The \emph{signature (index)} $\sigma(M)$ of a compact and oriented $n$ manifold $M$ is defined as follows. If $n=4k$ for some $k$, we choose a basis $\{a_1,...,a_r\}$ for $H^{2k}(M... | You should be using a more invariant definition of the signature. First, cohomology and Poincaré duality are both homotopy invariant. It follows that the abstract vector space $H^{2k}$ equipped with the intersection pairing is a homotopy invariant. Now I further claim that the signature is an invariant of real vector s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159036",
"timestamp": "2023-03-29T00:00:00",
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Positive Semi-Definite matrices and subtraction I have been wondering about this for some time, and I haven't been able to answer the question myself. I also haven't been able to find anything about it on the internet. So I will ask the question here:
Question: Assume that $A$ and $B$ both are positive semi-definite. W... | There's a form of Sylvester's criterion for positive semi-definiteness, which unfortunately requires a lot more computations than the better known test for positive definiteness. Namely, all principal minors (not just the leading ones) must be nonnegative. Principal minors are obtained by deleting some of the rows and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Traces of all positive powers of a matrix are zero implies it is nilpotent Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$.
What I would like to show is the converse, that is,
if $\text{t... | If the eigenvalues of $A$ are $\lambda_1$, $\dots$, $\lambda_n$, then the eigenvalues of $A^k$ are $\lambda_1^k$, $\dots$, $\lambda_n^k$. It follows that if all powers of $A$ have zero trace, then $$\lambda_1^k+\dots+\lambda_n^k=0\qquad\text{for all $k\geq1$.}$$ Using Newton's identities to express the elementary symme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
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What is the correct way to solve $\sin(2x)=\sin(x)$
I've found two different ways to solve this trigonometric equation
$\begin{align*}
\sin(2x)=\sin(x) \Leftrightarrow \\\\ 2\sin(x)\cos(x)=\sin(x)\Leftrightarrow \\\\ 2\sin(x)\cos(x)-\sin(x)=0 \Leftrightarrow\\\\ \sin(x) \left[2\cos(x)-1 \right]=0 \Leftrightarrow \\\\... | These answers are equivalent and both are correct. Placing angle $x$ on a unit circle, your first decomposition gives all angles at the far west and east sides, then all the angles $60$ degrees north of east, then all the angles $60$ degrees south of east.
Your second decomposition takes all angles at the far east side... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Motivation for Koszul complex Koszul complex is important for homological theory of commutative rings.
However, it's hard to guess where it came from.
What was the motivation for Koszul complex?
| In this answer I would rather focus on why is the Koszul complex so widely used. In abstract terms, the Koszul complex arises as the easiest way to combine an algebra with a coalgebra in presence of quadratic data. You can find the modern generalization of the Koszul duality described in Aaron's comment by reading Loda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generating function of Lah numbers Let $L(n,k)\!\in\!\mathbb{N}_0$ be the Lah numbers. We know that they satisfy
$$L(n,k)=L(n\!-\!1,k\!-\!1)+(n\!+\!k\!-\!1)L(n\!-\!1,k)$$
for all $n,k\!\in\!\mathbb{Z}$. How can I prove
$$\sum_nL(n,k)\frac{x^n}{n!}=\frac{1}{k!}\Big(\frac{x}{1-x}\Big)^k$$
without using the explicit ... | We have
\begin{align}
f_k(x)&:=\sum_{n\in\Bbb Z}L(n,k)\frac{x^n}{n!}\\
&=\sum_{n\in \Bbb Z}L(n-1,k-1)\frac{x^n}{n!}+\sum_{n\in \Bbb Z}(n+k-1)L(n-1,k)\frac{x^n}{n!}\\
&=\sum_{j\in \Bbb Z}L(j,k-1)\frac{x^{j+1}}{(j+1)!}+\sum_{j\in \Bbb Z}(j+1+k-1)L(j,k)\frac{x^{j+1}}{(j+1)!}\\
&=\sum_{j\in \Bbb Z}L(j,k-1)\frac{x^{j+1}}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Prove the convergence/divergence of $\sum \limits_{k=1}^{\infty} \frac{\tan(k)}{k}$ Can be easily proved that the following series onverges/diverges?
$$\sum_{k=1}^{\infty} \frac{\tan(k)}{k}$$
I'd really appreciate your support on this problem. I'm looking for some easy proof here. Thanks.
| A proof that the sequence $\frac{\tan(n)}{n}$ does not have a limit for $n\to \infty$ is given in this article (Sequential tangents, Sam Coskey). This, of course, implies that the series does not converge.
The proof, based on this paper by Rosenholtz (*), uses the continued fraction of $\pi/2$, and, essentially, it sh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
"answer_count": 3,
"answer_id": 1
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Proving the Möbius formula for cyclotomic polynomials We want to prove that
$$ \Phi_n(x) = \prod_{d|n} \left( x^{\frac{n}{d}} - 1 \right)^{\mu(d)} $$
where $\Phi_n(x)$ in the n-th cyclotomic polynomial and $\mu(d)$ is the Möbius function defined on the natural numbers.
We were instructed to do it by the following stag... | I found a solution that is easy to understand for those who want to know how to solve without following the steps given in the problem.
Click to see the source.
First, we have a formula
$$
{x^{n}-1=\Pi_{d|n}\Phi_{d}(x)}
$$
Then, by taking the logarithm on the both sides,
$$
\log(x^{n}-1)=\log(\Pi_{d|n}\Phi_{d}(x))=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Partial Integration - Where did I go wrong? For a Homework, I need $\int \frac{x}{(x-1)^2} dx$ as an intermediate result. Using partial integration, I derive $x$ and integrate $\frac{1}{(x-1)^2}$, getting: $$ \frac{-x}{x-1} + \int \frac{1}{x-1} dx = \ln(x-1)+\frac{x}{x-1} $$
WolframAlpha tells me this is wrong (it give... | the result is : $\ln|x-1| - \frac x{x-1} + C$ where $C$ is some constant.
If $C=1$ you get : $\ln |x-1| + \frac 1{1-x}$
The finaly result can be expressed :
$$F(x) = \ln |x-1| + \frac 1{1-x} + \lambda$$ where $\lambda$ is some constant.
Precesely :
$$F(x) = \ln (x-1) + \frac 1{1-x} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate all the four solutions to $(p+5)(p-1) \equiv 0 \pmod {16}$? This is a kind of a plain question, but I just can't get something.
For the congruence and a prime number $p$: $(p+5)(p-1) \equiv 0\pmod {16}$.
How come that the in addition to the solutions
$$\begin{align*}
p &\equiv 11\pmod{16}\\
p &\equiv ... | First note that $p$ has to be odd. Else, $(p+5)$ and $(p-1)$ are both odd.
Let $p = 2k+1$. Then we need $16 \vert (2k+6)(2k)$ i.e. $4 \vert k(k+3)$.
Since $k$ and $k+3$ are of opposite parity, we need $4|k$ or $4|(k+3)$.
Hence, $k = 4m$ or $k = 4m+1$. This gives us $ p = 2(4m) + 1$ or $p = 2(4m+1)+1$.
Hence, we get tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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constructive proof of the infinititude of primes There are infinitely many prime numbers. Euclides gave a constructive proof as follows.
For any set of prime numbers $\{p_1,\ldots,p_n\}$, the prime factors of $p_1\cdot \ldots \cdot p_n +1$ do not belong to the set $\{p_1,\ldots,p_n\}$.
I'm wondering if the following ca... | As noted in the comments, we can take $\delta_n=p_{n-1}$. In fact, there are improvements on that in the literature. But if you want something really easy to prove, you can take $\delta_n$ to be the factorial of $p_{n-1}$, since that gives you an interval which includes Euclid's $p_1\times p_2\times\cdots\times p_{n-1}... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there any geometric way to characterize $e$? Let me explain it better: after this question, I've been looking for a way to put famous constants in the real line in a geometrical way -- just for fun. Putting $\sqrt2$ is really easy: constructing a $45^\circ$-$90^\circ$-$45^\circ$ triangle with unitary sides will make... | Another approach might be finding a polar curve such that it's tangent line forms a constant angle with the segment from $(0,0)$ to $(\theta,\rho(\theta))$. The solution is the logarithmic spiral, defined by
$$\rho =c_0 e^{a\theta}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/159707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
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Diameter of wheel
If a wheel travels 1 mile in 1 minute at a rate of 600 revolutions per minute. What is the diameter of the wheel in feet ? The answer to this question is 2.8 feet.
Could someone please explain how to solve this problem ?
| The distance travelled by a wheel in one revolution is nothing but the circumference. If the circumference of the circle is $d$, then the distance travelled by the wheel on one revolution is $\pi d$.
It does $600$ revolutions per minute i.e. it travels a distance of $600 \times \pi d$ in one minute. We are also given t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
$\gcd(n!+1,(n+1)!)$ The recent post didn't really provide sufficient help. It was too vague, most of it went over my head.
Anyway, I'm trying to find the $\gcd(n!+1,(n+1)!)$.
First I let $d=ab\mid(n!+1)$ and $d=ab\mid(n+1)n!$ where $d=ab$ is the GCD.
From $ab\mid(n+1)n!$ I get $a\mid(n+1)$ and $b|n!$.
Because $b\mid n... | By Euclid $\rm\,(k,k\!+\!1)=1\:\Rightarrow\:(p\!\ k,k\!+\!1) = (p,k\!+\!1)\ [= p\ $ if $\rm\:p\:$ prime, $\rm\:k=(p\!-\!1)!\:$ by Wilson. See here for $\rm\:p = n\!+\!1\:$ composite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/159804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Automorphisms of the field of complex numbers Using AC one may prove that there are $2^{\mathfrak{c}}$ field automorphisms of the field $\mathbb{C}$. Certainly, only the identity map is $\mathbb{C}$-linear ($\mathbb{C}$-homogenous) among them but are all these automorphisms $\mathbb{R}$-linear?
| An automorphism of $\mathbb C$ must take $i$ into $i$ or $-i$. Thus an automorphism that is $\mathbb R$-linear must be the identity or conjugation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/159863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do mathematicians care so much about zeta functions? Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions?
What is its purpose? Is it just to develop small areas of pure mathematics?
| For one thing, the Riemann Zeta function has many interesting properties. No one knew of a closed form of $\zeta (2)$ until Euler famously found it, along with all the even positive integers:
$$\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$$
However, to this day, no nice closed form is known for values in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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norm for estimating the error of the numerical method In most of the books on numerical methods and finite difference methods the error is measured in discrete $L^2$ norm. I was wondering if people do the in Sobolev norm. I have never see that done and I want to know why no one uses that.
To be more specific look at t... | For one thing, it's a question of what norm measures how "accurate" the solution is. Which of the two error terms would you rather have: $0.1\sin(x)$ or $0.0001\sin(10000x)$? The first is smaller in the Sobolev norm, the second is smaller in the $L^2$ norm.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are There Any Symbols for Contradictions? Perhaps, this question has been answered already but I am not aware of any existing answer. Is there any international icon or symbol for showing Contradiction or reaching a contradiction in Mathematical contexts? The same story can be seen for showing that someone reached to t... | The symbols are:
$\top$ for truth (example: $100 \in \mathbb{R} \to \top$)
and $\bot$ for false (example: $\sqrt{2} \in \mathbb{Q} \to \bot$)
In Latex, \top is $\top$ and \bot is $\bot$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/160039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
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"answer_id": 11
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Continued fraction question I have been given an continued fraction for a number x:
$$x = 1+\frac{1}{1+}\frac{1}{1+}\frac{1}{1+}\cdots$$
How can I show that $x = 1 + \frac{1}{x}$? I played around some with the first few convergents of this continued fraction, but I don't get close.
| Just look at it. OK, if you want something more proofy-looking: if $x_n$ is the $n$'th convergent, then $x_{n+1} = 1 + 1/x_n$. Take limits.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
| I think you can do it this way.
\begin{align*}
\int \frac{1}{x^4 +1} \ dx & = \frac{1}{2} \cdot \int\frac{2}{1+x^{4}} \ dx \\\
&= \frac{1}{2} \cdot \int\frac{(1-x^{2}) + (1+x^{2})}{1+x^{4}} \ dx \\\ &=\frac{1}{2} \cdot \int \frac{1-x^2}{1+x^{4}} \ dx + \frac{1}{2} \int \frac{1+x^{2}}{1+x^{4}} \ dx \\\ &= \frac{1}{2} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 20,
"answer_id": 12
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Apply Cauchy-Riemann equations on $f(z)=z+|z|$? I am trying to check if the function $f(z)=z+|z|$ is analytic by using the Cauchy-Riemann equation.
I made
$z = x +jy$
and therefore
$$f(z)= (x + jy) + \sqrt{x^2 + y^2}$$
put into $f(z) = u+ jv$ form:
$$f(z)= x + \sqrt{x^2 + y^2} + jy$$
where
$u = x + \sqrt{x^2 + y^2... | In order for your function to be analytic, it must satisfy the Cauchy-Riemann equations (right? it's good to think about why this is true). So, what are the equations?
Well, du/dx = dv/dy.
Does this hold? Or you could consider du/dy = -dv/dx.
If either of these equations do not hold, then the function is not analyti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ I'm supposed to calculate:
$$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice result. I wonder in which ways we may appro... | Edited. I justified the application of the dominated convergence theorem.
By a simple calculation,
$$ \begin{align*}
e^{-n}\sum_{k=0}^{n} \frac{n^k}{k!}
&= \frac{e^{-n}}{n!} \sum_{k=0}^{n}\binom{n}{k} n^k (n-k)! \\
(1) \cdots \quad &= \frac{e^{-n}}{n!} \sum_{k=0}^{n}\binom{n}{k} n^k \int_{0}^{\infty} t^{n-k}e^{-t} \, d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "257",
"answer_count": 9,
"answer_id": 7
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$k$th power of ideal of of germs Well,We denote set of germs at $m$ by $\bar{F_m}$ A germ $f$ has a well defined value $f(m)$ at m namely the value at $m$ of any representative of the germ. Let $F_m\subseteq \bar{F_m}$ be the set of germs which vanish at $m$. Then $F_m$ is an ideal of $\bar{F_m}$ and let $F_m^k$ denote... | If $x_1,\ldots,x_n$ are local coordinates at the point $m$, then any smooth germ at $m$ has an associated Taylor series in the coordiantes $x_i$. The power $F_m^k$ is precisely the set of germs whose degree $k$ Taylor polynomial vanishes, i.e. whose Taylor series has no non-zero terms of degree $\leq k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/160320",
"timestamp": "2023-03-29T00:00:00",
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Well definition of multiplicity of a root in a polynomial of ring. Let $R$ be a conmutative ring with identity, let $\displaystyle f=\sum_{k=0}^{n}a_k x^k \in R[x]$ and $r\in R$. If $f=(x-r)^m g$, $m\in\mathbb{N}$ and $g\in R[x]$ with $g(r)\neq 0$, then the root $r$ is said to have $multiplicity\,\,\, m$. If the multip... | More generally, for $\rm\!\ c\in R\!\ $ any ring, every $\rm\!\ r\ne 0\,$ may be written uniquely in the form $\rm\!\ r = c^n\,\! b,\,$ where $\rm\,c\nmid b,\,$ assuming $\rm\,c\,$ is cancellable, and only $0\,$ is divisible by arbitrarily high powers of $\rm\,c.\,$ Indeed, by hypothesis there exists a largest natura... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Cutting cake into 5 equal pieces
If a cake is cut into $5$ equal pieces, each piece would be $80$ grams
heavier than when the cake is cut into $7$ equal pieces. How heavy is
the cake?
How would I solve this problem? Do I have to try to find an algebraic expression for this? $5x = 7y + 400$?
| The first step is to turn the word problem into an equation; one-fifth of the cake is $80$ grams heavier than one-seventh of the cake, so one-fifth of the cake equals one-seventh of the cake plus 80. "The cake" (specifically its mass) is $x$, and we can work from there:
$$\dfrac{x}{5} = \dfrac{x}{7}+80$$
$$\dfrac{x}{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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number prime to $bc$ with system of congruences Can you please help me to understand why all numbers $x$ prime to $bc$ are all the solutions of this system?
$$\begin{align*}
x&\equiv k\pmod{b}\\
x&\equiv t\pmod{c}
\end{align*}$$
Here $k$ is prime to $b$, and $t$ is prime to $c$.
| Hint $\rm\quad \begin{eqnarray} x\equiv k\,\ (mod\ b)&\Rightarrow&\rm\:(x,b) = (k,b) = 1 \\
\rm x\equiv t\,\,\ (mod\ c)\,&\Rightarrow&\rm\:(x,c) =\, (t,c) = 1\end{eqnarray} \Bigg\}\ \Rightarrow\ (x,bc) = 1\ \ by\ Euclid's\ Lemma$
| {
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Probably simple factoring problem I came across this in a friend's 12th grade math homework and couldn't solve it. I want to factor the following trinomial:
$$3x^2 -8x + 1.$$
How to solve this is far from immediately clear to me, but it is surely very easy. How is it done?
| A standard way of factorizing, when it is hard to guess the factors, is by completing the square.
\begin{align}
3x^2 - 8x + 1 & = 3 \left(x^2 - \dfrac83x + \dfrac13 \right)\\
& (\text{Pull out the coefficient of $x^2$})\\
& = 3 \left(x^2 - 2 \cdot \dfrac43 \cdot x + \dfrac13 \right)\\
& (\text{Multiply and divide by $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about harmonic form of trigonometric functions. The question is:
i) Find the maximum and minimum values.
ii) the smallest non-negative value of x for which this occurs.
12cos(a)-9sin(a)
I think it should be changed into the form of Rcos(a+x) and it should be 15cos(a+36.87), and I get the answer i)+15 / -15 i... | We review the (correct) procedure that you went through. We have $12^2+9^2=15^2$, so we rewrite our expression as
$$15\left(\frac{12}{15}\cos a -\frac{9}{15}\sin a\right).$$
Now if $b$ is any angle whose cosine is $\frac{12}{15}$ and whose sine is $\frac{9}{15}$, we can rewrite our expression as
$$15\left(\cos a \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute: $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$ I try to solve the following sum:
$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$
I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion is v... | Here's another approach.
It depends primarily on the properties of telescoping series, partial fraction expansion, and the following identity for the $m$th harmonic number
$$\begin{eqnarray*}
\sum_{k=1}^\infty \frac{1}{k(k+m)}
&=& \frac{1}{m}\sum_{k=1}^\infty \left(\frac{1}{k} - \frac{1}{k+m}\right) \\
&=& \frac{1}{m}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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what is the use of derivatives Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in real life most of the things we do are not linear functions and derivatives helps to make a real life functions int... | There can be also economic interpretations of derivatives. For example, let's assume that there is a function which measures the utility from consumption.
$U(C)$ where $C$ is the consumption. It is straightforward to say that your utility increases with consumption. This means that when you increase your consumption on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 4
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Tensor product of sets The cartesian product of two sets $A$ and $B$ can be seen as a tensor product.
Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ?
The context is the following: assume one has a set-valued presheaf $F$ on a monoidal category, knowing $F(A)$ a... | Simply being in a monoidal category is a rather liberal condition on the tensor product; it tells you very little about what the tensor product actually looks like.
Here is a (perhaps slightly contrived?) example:
Let $C$ be the category of vector spaces over a finite field $\mathbb F_p$ with linear transformations. Th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$. Let $R$ be a UFD and $P$ a prime ideal. Here we are defining a UFD with primes and not irreducibles.
Is the following true and what is the justification?
If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.
| I think thats wrong. If $ a \in P $ holds, then $ a + P = 0 + P \in R/P$ and therefore $a+P$ not prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/160937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intersection of compositum of fields with another field Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1 \cap K = F_2 \cap K = M$, the extensions $F_i / (F_i \cap K)$ are Galois, and $[F_1 \cap F_2 : M ]$ is finite. Then is $[F_1 F_2 \cap K : M]$ finite?
| No. First, the extension $\mathbb{C}(z)/\mathbb{C}$ is Galois for any $z$ that is transcendental over $\mathbb{C}$: if $p(z)$ is a nonconstant polynomial, and $\alpha$ is a root and $\beta$ is a nonroot, then the automorphism $\sigma\colon z\mapsto z+\alpha-\beta$ maps $z-\alpha$ to $z-\beta$, so that $p(z)$ has a fact... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tangent line of parametric curve I have not seen a problem like this so I have no idea what to do.
Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.
$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$
I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{... | One way to do this is by considering the parametric form of the curve:
$(x,y)(t) = (1 + \log t, t^2 + 2)$, so $(x,y)'(t) = (\frac{1}{t}, 2t)$
We need to find the value of $t$ when $(x,y)(t) = (1 + \log t, t^2 + 2) = (1,3)$, from where we deduce $t=1$. The tangent line at $(1,3)$ has direction vector $(x,y)'(1) = (1,2)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$ Сould any one help me how to show $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$ can not be injective?
| If we remove three points from the domain it will be connected. In $\mathbb{R}$ the connected sets are intervals, so if we remove three point from an interval it will be disconnected. So there can not exist a continuous injective function from $\mathbb{R}^2$ to $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/161144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
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Limit of exponentials Why is $n^n (n+m)^{-{\left(n+m\over 2\right)}}(n-m)^{-{\left(n-m\over 2\right)}}$ asymptotically equal to $\exp\left(-{m^2\over 2n}\right)$ as $n,m\to \infty$?
| Let $m = n x$.
Take the logarithm:
$$
n \log(n) - n \frac{1+x}{2} \left(\log n + \log\left(1+x\right) \right) - n \frac{1-x}{2} \left( \log n + \log\left(1-x\right) \right)
$$
Notice that all the terms with $\log(n)$ cancel out, so we are left with
$$
-\frac{n}{2} \left( (1+x) \log(1+x) - (1-x) \log(1-x) \right) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Looking for some simple topology spaces such that $nw(X)\le\omega$ and $|X|>2^\omega$ I believe there are some topology spaces which satisfying the network weight is less than $\omega$, and its cardinality is more than $2^\omega$ (not equal to $2^\omega$), even much larger.
*
*Network: a family $N$ of subsets of a t... | If $X$ is $T_0$ and $nw(X)=\omega$ (network weight) then $|X|\leq 2^\omega$. Let $\mathcal N$ be a countable network. For each $x\in X$ consider $N_x=\{N\in\mathcal N: x\in N\}$. Since $X$ is $T_0$ it follows that $N_x\ne N_y$ for $x\ne y$. Thus, $|X|\leq |P(\mathcal N)|=2^\omega$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/161462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant? Alternately, let $M$ be an $n \times n$ matrix with entries in a commutative ring $R$. If $M$ has trivial kernel, is it true that $\det(M) \neq 0$?
This math.SE question deals with the case that $R$ is a polynomial ring ove... | Lam's Exercises in modules and rings includes the following:
which tells us that your determinant is not a zero-divisor.
The paper where McCoy does that is [Remarks on divisors of zero, MAA Monthly 49 (1942), 286--295] If you have JStor access, this is at http://www.jstor.org/stable/2303094
There is a pretty corollary... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the total number of combinations of 5 items together when there are no duplicates? I have 5 categories - A, B, C, D & E.
I want to basically create groups that reflect every single combination of these categories without there being duplicates.
So groups would look like this:
*
*A
*B
*C
*D
*E
*A, B
*A,... | There are $\binom{5}{1}$ combinations with 1 item, $\binom{5}{2}$ combinations with $2$ items,...
So, you want :
$$\binom{5}{1}+\cdots+\binom{5}{5}=\left(\binom{5}{0}+\cdots+\binom{5}{5}\right)-1=2^5-1=31$$
I used that $$\sum_{k=0}^n\binom{n}{k}=(1+1)^n=2^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/161565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Course for self-study I have basically completed a good deal of Single Variable Calculus from Spivak's Calculus and since I leave school in May next year,I intend to put in some effort to pick up college mathematics.I am a bit confused as to what to study next.I did buy Herstein's Topics in Algebra.
Question: So can a... | Hoffman and Kunze is a great book. If you understand the abstractions in Spivak's book, you should be able to handle it. The problems are great. I have worked many of them.
Go there next.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I convert between powers for different number bases? I am writing a program to convert between megabits per second and mebibits per second; A user would enter 1 Mebibits p/s and get 1.05 Megabits p/s as the output.
These are two units of computer data transfer rate. A megabit (SI unit of measurement in deny) is... | If you have $x \text{ Mebibits p/s}$, since a Mebibit is $\displaystyle \frac{2^{20}}{10^6} = 1.048576$ Megabits, you have to multiply by $1.048576$, getting $1.048576x \text{ Megabits p/s}$. Likewise, if you have $y\text{ Megabits p/s}$, since a Megabit is $\displaystyle \frac{10^6}{2^{20}} = 0.95367431640625$ Mebibit... | {
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What is the distribution of a random variable that is the product of the two normal random variables ? What is the distribution of a random variable that is the product of the two normal random variables ?
Let $X\sim N(\mu_1,\sigma_1), Y\sim N(\mu_2,\sigma_2)$
and $Z=XY$
That is, what is its probability density funct... | For the special case that both Gaussian random variables $X$ and $Y$ have zero mean and unit variance, and are independent, the answer is that $Z=XY$ has the probability density $p_Z(z)={\rm K}_0(|z|)/\pi$. The brute force way to do this is via the transformation theorem:
\begin{align}
p_Z(z)&=\frac{1}{2\pi}\int_{-\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 5,
"answer_id": 0
} |
What's an intuitive explanation of the max-flow min-cut theorem? I'm about to read the proof of the max-flow min-cut theorem that helps solve the maximum network flow problem. Could someone please suggest an intuitive way to understand the theorem?
| Imagine a complex pipeline with a common source and common sink. You start to pump the water up, but you can't exceed some maximum flow. Why is that? Because there is some kind of bottleneck, i.e. a subset of pipes that transfer the fluid at their maximum capacity--you can't push more through. This bottleneck will be p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Irreducible polynomial over an algebraically closed field of characteristic distinct from 2 Let $k$ be an algebraically closed field such that $\textrm{char(k)} \neq 2$ and let $n$ be a fixed positive integer greater than $3$
Suppose that $m$ is a positive integer such that $3 \leq m \leq n$.
Is it always true that $f... | Let $A$ be a UFD.
Let $a$ be a non-zero square-free non-unit element of $A$.
Then $X^n - a \in A[X]$ is irreducible by Eisenstein.
$Y^2 + Z^2 = (Y + iZ)(Y - iZ)$ is square-free in $k[Y, Z]$.
Hence $X^2 + Y^2 + Z^2$ is irreducible in $k[X, Y, Z]$ by the above result.
Let $m \gt 2$.
By the induction hypothesis, $X_{1}^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Algorithm for computing Smith normal form in an algebraic number field of class number 1 Let $K$ be an algebraic number field of class number 1.
Let $\frak{O}$ be the ring of algebraic integers in $K$.
Let $A$ be a nonzero $m\times n$ matrix over $\frak{O}$.
Since $\frak{O}$ is a PID, $A$ has Smith normal form $S$.
I'm... | This is routine and is already implemented in several computer algebra systems, including Sage, Pari and (I think) Magma. (I wrote the Sage version some while back). As you point out, the standard existence proof for Smith form is completely algorithmic once you know how to find a GCD of two elements, which any of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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When does the vanishing wedge product of two forms require one form to be zero? Let $\alpha$ and $\beta$ be two complex $(1,1)$ forms defined as:
$\alpha = \alpha_{ij} dx^i \wedge d\bar x^j$
$\beta= \beta_{ij} dx^i \wedge d\bar x^j$
Let's say, I know the following:
1) $\alpha \wedge \beta = 0$
2) $\beta \neq 0$
I want ... | \begin{eqnarray}
\alpha\wedge\beta
&=&\sum_{i,j,k,l}\alpha_{ij}\beta_{kl}dx^i\wedge d\bar{x}^j\wedge dx^k\wedge d\bar{x}^l\cr
&=&(\sum_{i<k,j<l}+\sum_{i<k,l<j}+\sum_{k<i,j<l}+\sum_{k<i,l<j})\alpha_{ij}\beta_{kl}dx^i\wedge d\bar{x}^j\wedge dx^k\wedge d\bar{x}^l\cr
&=&\sum_{i<k,j<l}(-\alpha_{ij}\beta_{kl}+\alpha_{kj}\bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If a function has a finite limit at infinity, does that imply its derivative goes to zero? I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L < \infty$. Then must it be the case that $\lim\limits_{x\to +\infty}f'(x) = ... | Let a function oscillate between $y=1/x$ and $y=-1/x$ in such a way that it's slope oscillates between $1$ and $-1$. Draw the picture. It's easy to see that such functions exist. Then the function approaches $0$ but the slope doesn't approach anything.
One could ask: If the derivative also has a limit, must it be $0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
"answer_count": 6,
"answer_id": 5
} |
Techniques of Integration w/ absolute value I cannot solve this integral.
$$
\int_{-3}^3 \frac{x}{1+|x|} ~dx
$$
I have tried rewriting it as:
$$
\int_{-3}^3 1 - \frac{1}{x+1}~dx,
$$
From which I obtain:
$$
x - \ln|x+1|\;\;\bigg\vert_{-3}^{\;3}
$$
My book (Stewart's Calculus 7e) has the answer as 0, and I can intuitivel... | You have an odd function integrated over an interval symmetric about the origin. The answer is $0$. This is, as you point out, intuitively reasonable from the geometry. A general analytic argument is given in a remark at the end.
If you really want to calculate, break up the region into two parts, where (i) $x$ is po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Dixon's random squares algorithm: a step in the proof of its subexp. running time I am currently working to understand Dixon's running time proof of his subexp integer factorization algorithm (random squares).
But unfortunately I can not follow him at a certain point in his work. His work is available for free here: ht... | The $4hv+2$ bound is indeed deterministic, but only applies when we're not in a "bad" case. So the question we need to ask is how "bad" cases are defined.
I think the idea of the author is that the previous paragraph can be read by relaxing the condition $N=v^2+1$ to any $N\ge 4hv$, and in particular $N=4hv$. But then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded
Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded.
I am not getting what theorem or result to apply. Could anyone help me?
| Let $u(x)=x$, for all $x \in \mathbb{R}$. Then $u''(x)=0$ for all $x$. But $u^{-1}(\{c\}) = \{c\}$. I think somebody is cheating you :-)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/162292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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how to change polar coordinate into cartesian coordinate using transformation matrix I would like to change $(3,4,12)$ in $xyz$ coordinate to spherical coordinate using the following relation
It is from the this link. I do not understand the significance of this matrix (if not for coordinate transformation) or how it ... | The transformation from Cartesian to polar coordinates is not a linear function, so it cannot be achieved by means of a matrix multiplication.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/162366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
How can I calculate this limit: $\lim\limits_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$? I was given this limit to solve, without using L'Hospital rule. It's killing me !! Can I have the solution please ?
$$\lim_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$$
| The solution below may come perilously close to the forbidden L'Hospital's Rule, though the Marquis is not mentioned.
To make things look more familiar, change signs, and then divide top and bottom by $x-2$. The expression we want the limit of becomes
$$\frac{\sqrt{2+x}-2}{x-2} \cdot \frac{x-2}{(4-x)^{1/3}-2^{1/3}}.$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A limit involving polynomials Let be the polynomial:
$$P_n (x)=x^{n+1} - (x^{n-1}+x^{n-2}+\cdots+x+1)$$
I want to prove that it has a single positive real root we'll denote by $x_n$, and then to compute:
$$\lim_{n\to\infty} x_{n}$$
| Since it's not much more work, let's study the roots in $\mathbb{C}$.
Note that $x=1$ is not a solution unless $n=1$, since $P_n(1) = 1-n$.
Since we are interested in the limit $n\to\infty$, we can assume $x\ne 1$.
Sum the geometric series,
$$\begin{eqnarray*}
P_n (x) &=& x^{n+1} - (x^{n-1}+x^{n-2}+\cdots+x+1) \\
&=& x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Which theories and concepts exist where one calculates with sets? Recently I thought about concepts for calculating with sets instead of numbers. There you might have axioms like:
*
*For every $a\in\mathbb{R}$ (or $a\in\mathbb{C}$) we identify the term $a$ with $\{a\}$.
*For any operator $\circ$ we define $A\circ B... | I like Minkowski addition, aka vector addition. It is a basic operation in the geometry of convex sets. See: zonotopes & zonoids, Brunn-Minkowski inequality, polar sets... and here's a neat inequality for an arbitrary convex set $A\subset\mathbb R^n$:
$$
\mathrm{volume}\,(A-A)\le \binom{2n}{n}\mathrm{volume}\,(A)
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Describing multivariable functions So I am presented with the following question:
Describe and sketch the largest region in the $xy$-plane that corresponds to the domain of the function:
$$g(x,y) = \sqrt{4 - x^2 - y^2} \ln(x-y).$$
Now to be I can find different restrictions like $4 - x^2 - y^2 \geq 0$... but I'm honest... | So, you need two things:
$$ 4 - x^2 - y^2 \geq 0 $$
to get make the square root work, and also
$$ x-y > 0 $$
to make the logarithm work.
You will be graphing two regions in the $xy$-plane, and your answer will be the area which is in both regions.
A good technique for graphing a region given by an inequality is to firs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to determine if 2 points are on opposite sides of a line How can I determine whether the 2 points $(a_x, a_y)$ and $(b_x, b_y)$ are on opposite sides of the line $(x_1,y_1)\to(x_2,y_2)$?
| Writing $A$ and $B$ for the points in question, and $P_1$ and $P_2$ for the points determining the line ...
Compute the "signed" areas of the $\triangle P_1 P_2 A$ and $\triangle P_1 P_2 B$ via the formula (equation 16 here)
$$\frac{1}{2}\left|\begin{array}{ccc}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{arr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
smooth quotient Let $X$ be a smooth projective curve over the field of complex numbers. Assume it comes with an action of $\mu_3$. Could someone explain to me why is the quotient $X/\mu_3$ a smooth curve of genus zero?
| No, the result is not true for genus $1$.
Consider an elliptic curve $E$ (which has genus one) and a non-zero point $a\in E$ of order three .
Translation by $a$ is an automorphism $\tau_a:E\to E: x\mapsto x+a$ of order $3$ of the Riemann surface $E$.
It generates a group $G=\langle \tau_a\rangle\cong \mu_3$ of or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Direct limit of localizations of a ring at elements not in a prime ideal For a prime ideal $P$ of a commutative ring $A$, consider the direct limit of the family of localizations $A_f$ indexed by the set $A \setminus P$ with partial order $\le$ such that $f \le g$ iff $V(f) \subseteq V(g)$. (We have for such $f \le g$... | If $a/f^n \in A_f$ is mapped to $0$ in $A_p,$ then there is a $g \not \in p,$ s.t. $ga=0,$ therefore, $a/f^n=0 \in A_{gf}.$ Hence the injectivity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/162861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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principal value as distribution, written as integral over singularity Let $C_0^\infty(\mathbb{R})$ be the set of smooth functions with compact support on the real line $\mathbb{R}.$ Then, the map
$$\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: C_0^\infty(\mathbb{R}) \to \mathbb{C}$$
defined via the Cauchy principa... | We can write
$$I(\varepsilon):=\int_{\Bbb R\setminus [-\varepsilon,\varepsilon]}\frac{u(x)}xdx=\int_{-\infty}^{-\varepsilon}\frac{u(x)}xdx+\int_{\varepsilon}^{\infty}\frac{u(x)}xdx.$$
In the first integral of the RHs, we do the substitution $t=-x$, then
$$I(\varepsilon)=-\int_{\varepsilon}^{+\infty}\frac{u(t)}tdt+\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Closest point of line segment without endpoints I know of a formula to determine shortest line segment between two given line segments, but that works only when endpoints are included. I'd like to know if there is a solution when endpoints are not included or if I'm mixing disciplines incorrectly.
Example : Line segme... | There would not be a shortest line segment. Look at line segments from $(0,1)$ to points very close to $(1,1)$ on the segment that joins $(1,1)$ and $(1,4)$. These segments get shorter and shorter, approaching length $1$ but never reaching it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/162996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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First order ordinary differential equations involving powers of the slope
Are there any general approaches to differential equations like
$$x-x\ y(x)+y'(x)\ (y'(x)+x\ y(x))=0,$$
or that equation specifically?
The problem seems to be the term $y'(x)^2$. Solving the equation for $y'(x)$ like a qudratic equation gives s... | Hint:
$x-xy+y'(y'+xy)=0$
$(y')^2+xyy'+x-xy=0$
$(y')^2=x(y-1-yy')$
$x=\dfrac{(y')^2}{y-1-yy'}$
Follow the method in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=234:
Let $t=y'$ ,
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Is it true that $\max\limits_D |f(x)|=\max\limits\{|\max\limits_D f(x)|, |\min\limits_D f(x)|\}$? I came across an equality, which states that
If $D\subset\mathbb{R}^n, n\geq 2$ is compact, for each $ f\in C(D)$, we have the following equality
$$\max\limits_D |f(x)|=\max\limits\{|\max\limits_D f(x)|, |\min\limits_D... | I assume that you are assuming that $\max$ and $\min$ exists or that you are assuming that $f(x)$ is continuous which in-turn guarantees that $\max$ and $\min$ exists, since $D$ is compact.
First note that if we have $a \leq y \leq b$, then $\vert y \vert \leq \max\{\vert a \vert, \vert b \vert\}$, where $a,b,y \in \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Sum of Natural Number Ranges? Given a positive integer $n$, some positive integers $x$ can be represented as follows:
$$1 \le i \le j \le n$$
$$x = \sum_{k=i}^{j}k$$
Given $n$ and $x$ determine if it can be represented as the above sum (if $\exists{i,j}$), and if so determine the $i$ and $j$ such that the sum has the s... | A start: Note that
$$2x=j^2+j-i^2+i=(j+i)(j-i+1).$$
The two numbers $j+i$ and $j-i$ have the same parity (both are even or both are odd). So we must express $2x$ as a product of two numbers of different parities.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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for what value of $a$ has equation rational roots? Suppose that we have following quadratic equation containing some constant $a$
$$ax^2-(1-2a)x+a-2=0.$$
We have to find all integers $a$,for which this equation has rational roots.
First I have tried to determine for which $a$ this equation has a real solution, so I ... | Edited in response to Simon's comment.
Take any odd number; you can write it as $2m+1$, for some integer $m$; its square is $4m^2+4m+1$; so if $a=m^2+m$ then, no matter what $m$ is, you get rational roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why does the Gauss-Bonnet theorem apply only to even number of dimensons? One can use the Gauss-Bonnet theorem in 2D or 4D to deduce topological characteristics of a manifold by doing integrals over the curvature at each point.
First, why isn't there an equivalent theorem in 3D? Why can not the theorem be proved for od... | First, for a discussion involving Chern's original proof, check here, page 18.
I think the reason is that the original Chern-Gauss-Bonnet theorem can be treated topologically as
$$
\int_{M} e(TM)=\chi_{M}
$$
and for odd dimensional manifolds, the Euler class is "almost zero" as $e(TM)+e(TM)=0$. So it vanishes in de rh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Is the range of this operator closed? I think I am stuck with showing closedness of the range of a given operator. Given a sequence $(X_n)$ of closed subspaces of a Banach space $X$. Define $Y=(\oplus_n X_n)_{\ell_2}$ and set $T\colon Y\to X$ by $T(x_n)_{n=1}^\infty = \sum_{n=1}^\infty \frac{x_n}{n}$. Is the range of $... | The range is not necessarily closed. For example, if $X=(\oplus_{n \in \mathbb{N}} X_n)_{\ell_2}$=Y:
if $T(Y)$ is closed, $T(Y)$ is a Banach space. $T$ is a continuous bijective map from $X$ onto $T(Y)$, so is an homeomorphism (open map theorem) . But $T^{-1}$ is not continuous, because $T^{-1}(x_n)=nx_n$, for $x_n \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Projection onto Singular Vector Subspace for Singular Value Decomposition I am not very sophisticated in my linear algebra, so please excuse any messiness in my terminology, etc.
I am attempting to reduce the dimensionality of a dataset using Singular Value Decomposition, and I am having a little trouble figuring out h... | If you want to do the $100\times80$ to $100\times5$ conversion, you can just multiply $U$ with the reduced $S$ (after eliminating low singular values). What you will be left with is a $100\times80$ matrix, but the last $75$ columns are $0$ (provided your singular value threshold left you with only $5$ values). You can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$p=4n+3$ never has a Decomposition into $2$ Squares, right? Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma.
Is it correct to say that, primes of the form $p=4n+3$, never have a decomposition into $2$ squares, because sum of the quadratic resi... | Yes, as it has been pointed out, if $a^2+b^2$ is odd, one of $a$ and $b$ must be even and the other odd. Then
$$
(2m)^2+(2n+1)^2=(4m^2)+(4n^2+4n+1)=4(m^2+n^2+n)+1\equiv1\pmod{4}
$$
Thus, it is impossible to have $a^2+b^2\equiv3\pmod{4}$.
In fact, suppose that a prime $p\equiv3\pmod{4}$ divides $a^2+b^2$. Since $p$ cann... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How many subsets are there? I'm having trouble simplifying the expression for how many sets I can possibly have.
It's a very specific problem for which the specifics don't actually matter, but for $q$, some power of $2$ greater than $4$, I have a set of $q - 3$ elements. I am finding all subsets which contain at least ... | You're most of the way there; now just shave a couple of terms from the upper limit of your sum. Setting $r=q-3$ for clarity, and noting that $\lfloor r/2\rfloor = q/2-2$ (and that $r$ is odd since $q$ is a power of $2$),
$$\sum_{i=0}^{q/2}\binom{r}{i} = \binom{r}{q/2}+\binom{r}{q/2-1} + \sum_{i=0}^{\lfloor r/2\rfloo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Examples of logs with other bases than 10 From a teaching perspective, sometimes it can be difficult to explain how logarithms work in Mathematics. I came to the point where I tried to explain binary and hexadecimal to someone who did not have a strong background in Mathematics. Are there some common examples that can ... | The most common scales of non-decimal logrithms are the music scale.
For example, the octave is a doubling of frequency over 12 semitones. The harmonics are based on integer ratios, where the logrithms of 2, 3, and 5 approximate to 12, 19 and 28 semitones. One can do things like look at the ratios represented by the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integrating multiple times. I am having problem in integrating the equation below. If I integrate it w.r.t x then w.r.t y and then w.r.t z, the answers comes out to be 0 but the actual answer is 52. Please help out. Thanks
| $$\begin{align}
\int_1^3 (6yz^3+6x^2y)\,dx &= \left[6xyz^3+2x^3y\right]_{x=1}^3
=12yz^3+52y
\\
\int_0^1 (12yz^3+52y)\,dy &=\left[6y^2z^3+26y^2\right]_{y=0}^1
=6z^3+26
\\
\int_{-1}^1 (6z^3+26)\,dz &= \left[\frac{3}{2}z^4+26z\right]_{z=-1}^1 = 52 .
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Saturated Boolean algebras in terms of model theory and in terms of partitions Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ and least upper bound is $1$) of $\mathbb... | As far as I know, there is no connection; it's just an unfortunate clash of terminology. It's especially unfortunate because the model-theoretic notion of saturation comes up in the theory of Boolean algebras. For example, the Boolean algebra of subsets of the natural numbers modulo finite sets is $\aleph_1$-saturate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Common factors of the ideals $(x - \zeta_p^k)$, $x \in \mathbb Z$, in $\mathbb Z[\zeta_p]$ I'm trying to understand a proof of the following Lemma (regarding Catalan's conjecture):
Lemma:
Let $x\in \mathbb{Z}$, $2<q\not=p>2$ prime, $G:=\text{Gal}(\mathbb{Q}(\zeta_p):\mathbb{Q})$, $x\equiv 1\pmod{p}$ and $\lvert x\lv... | In the world of ideals gcd is the sum. Pick any two ideals $(x-\sigma(\zeta_p))$ and $(x-\sigma'(\zeta_p))$. Then $\sigma(\zeta_p)-\sigma'(\zeta_p)$ will be in the sum ideal. This is a difference of two distinct powers of $\zeta_p$, so generates the same ideal as one of the $1-\zeta_p^k$, $0<k<p$. All of these generate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Silly question about Fourier Transform What is the Fourier Transform of :
$$\sum_{n=1}^N A_ne^{\large-a_nt} u(t)~?$$
This is a time domain function, how can I find its Fourier Transform (continuous not discrete) ?
| Tips:
*
*The Fourier transform is linear; $$\mathcal{F}\left\{\sum_l a_lf_l(t)\right\}=\sum_l a_l\mathcal{F}\{f_l(t)\}.$$
*Plug $e^{-ct}u(t)$ into $\mathcal{F}$ and then discard part of the region of integration ($u(t)=0$ when $t<0$):
$$\int_{-\infty}^\infty e^{-ct}u(t)e^{-2\pi i st}dt=\int_0^\infty e^{(c-2\pi is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does this mean: $\mathbb{Z_{q}^{n}}$? I can't understand the notation $\mathbb{Z}_{q}^{n} \times \mathbb{T}$ as defined below. As far as I know $\mathbb{Z_{q}}$ comprises all integers modulo $q$. But with $n$ as a power symbol I can't understand it. Also: $\mathbb{R/Z}$, what does it denote?
"... $ \mathbb{T} = ... | You write "afaik $\mathbb Z_q$ comprises..." You have to be careful here what is meant by this notation. There are two common options:
1) $\mathbb Z_q$ is the ring of integers module $q$. Many people think this should be better written as $\mathbb Z/q$, to avoid confusion wth 2). However it is not uncommon to write $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to solve $3x+\sin x = e^x$ How doese one solve this equation?
$$
3x+\sin x = e^x
$$
I tried graphing it and could only find approximate solutions, not the exact solutions.
My friends said to use Newton-Raphson, Lagrange interpolation, etc., but I don't know what these are as they are much beyond the high school syl... | sorry
f(b)=3(1)+sin(1)−e1= is Not Equal to ---> 1.12...
f(b)=3(1)+sin(1)−e1= is Equal to ----> 0.299170578
And
f(0.5)=3(0.5)+sin(0.5)−e0.5= is Not Equal to ---> 0.33...>0
f(0.5)=3(0.5)+sin(0.5)−e0.5= is Equal to ----> -0.1399947352
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/164162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Asymptotic behavior of the expression: $(1-\frac{\ln n}{n})^n$ when $n\rightarrow\infty$ The well known results states that:
$\lim_{n\rightarrow \infty}(1-\frac{c}{n})^n=(1/e)^c$ for any constant $c$.
I need the following limit: $\lim_{n\rightarrow \infty}(1-\frac{\ln n}{n})^n$.
Can I prove it in the following way? Let... | According to the comments, your real aim is to prove that $x_n=n\left(1-\frac{\log n}n\right)^n$ has a non degenerate limit.
Note that $\log x_n=\log n+n\log\left(1-\frac{\log n}n\right)$ and that $\log(1-u)=-u+O(u^2)$ when $u\to0$ hence $n\log\left(1-\frac{\log n}n\right)=-\log n+O\left(\frac{(\log n)^2}n\right)$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
} |
How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$ How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$
| Apply $\,[p]+1\,$ times L'Hospital's rule to$\,\displaystyle{f(x):=\frac{\log^px}{x}}$:
$$\lim_{x\to\infty}\frac{\log^px}{x}\stackrel{L'H}=\lim_{x\to\infty}p\frac{\log^{p-1}(x)}{x}\stackrel{L'H}=\lim_{x\to\infty}p(p-1)\frac{\log^{p-2}(x)}{x}\stackrel{L'H}=...\stackrel{L'H}=$$
$$\stackrel{L'H}=\lim_{x\to\infty}p(p-1)(p-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 4
} |
Calculate $\lim\limits_{ x\rightarrow 100 }{ \frac { 10-\sqrt { x } }{ x+5 } }$ $$\lim_{ x\rightarrow 100 }{ \frac { 10-\sqrt { x } }{ x+5 } } $$
Could you explain how to do this without using a calculator and using basic rules of finding limits?
Thanks
| I suppose that you asked this question not because it's a difficult question, but because you don't know very well the rules to take care of over the limits.
First of all you need to know what a limit
is, what the indefinite case are, and
why they are indefinite, what's the meaning behind this word (i.e. $ \frac{\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Markov and independent random variables This is a part of an exercise in Durrett's probability book.
Consider the Markov chain on $\{1,2,\cdots,N\}$ with $p_{ij}=1/(i-1)$ when $j<i, p_{11}=1$ and $p_{ij}=0$ otherwise. Suppose that we start at point $k$. We let $I_j=1$ if $X_n$ visits $j$. Then $I_1,I_2,\cdots,I_{k-1}$... | For any $j$, observe that $X_{3}|X_{2}=j-1,X_{1}=j$ has the same distribution as $X_{2}|X_{2} \neq j-1, X_{1}=j$. Since $X_{2}=j-1$ iif $I_{j-1}=1$, by Markovianity conclude that $I_{j-1}$ is independent of $(I_{j-2},\ldots,I_{1})$ given that $X_{1}=j$.
Let's prove by induction that $I_{j-1}$ independent of $(I_{j-2},\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Difference between power law distribution and exponential decay This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like the one in the Power law article, which is the same as ... | $$
\begin{array}{rl}
\text{power law:} & y = x^{(\text{constant})}\\
\text{exponential:} & y = (\text{constant})^x
\end{array}
$$
That's the difference.
As for "looking the same", they're pretty different: Both are positive and go asymptotically to $0$, but with, for example $y=(1/2)^x$, the value of $y$ actually cuts ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 5,
"answer_id": 3
} |
Test for convergence $\sum_{n=1}^{\infty}\{(n^3+1)^{1/3} - n\}$ I want to expand and test this $\{(n^3+1)^{1/3} - n\}$ for convergence/divergence.
The edited version is: Test for convergence $\sum_{n=1}^{\infty}\{(n^3+1)^{1/3} - n\}$
| Let $y=1/n$, then $\lim_{n\to \infty}{(n^3+1)}^{1/3}-n=\lim_{y\to0^+}\frac{(1+y^3)^{1/3}-1}{y}$. Using L'Hopital's Rule, this limit evaluates to $0$.Hence, the expression converges.
You can also see that as $n$ increases, the importance of $1$ in the expression ${(n^3+1)}^{1/3}-n$ decreases and $(n^3+1)^{1/3}$ approach... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Finding a point having the radius, chord length and another point Me and a friend have been trying to find a way to get the position of a second point (B on the picture) having the first point (A), the length of the chord (d) and the radius (r).
It must be possible right? We know the solution will be two possible point... | There are two variables $(a,b)$ the coordinates of B. Since B lies on the circle,it satisfies the equation of the circle. Also,the distance of $B$ from $A$ is $d$.You can apply distance formula to get an equation from this condition.Now you have two variables and two equations from two conditions,you can solve it now y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Implicit Differentiation $y''$ I'm trying to find $y''$ by implicit differentiation of this problem: $4x^2 + y^2 = 3$
So far, I was able to get $y'$ which is $\frac{-4x}{y}$
How do I go about getting $y''$? I am kind of lost on that part.
| You have $$y'=-\frac{4x}y\;.$$ Differentiate both sides with respect to $x$:
$$y''=-\frac{4y-4xy'}{y^2}=\frac{4xy'-4y}{y^2}\;.$$
Finally, substitute the known value of $y'$:
$$y''=\frac4{y^2}\left(x\left(-\frac{4x}y\right)-y\right)=-\frac4{y^2}\cdot\frac{4x^2+y^2}y=-\frac{4(4x^2+y^2)}{y^3}\;.$$
But from the original eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Derivative of $f(x)= (\sin x)^{\ln x}$ I am just wondering if i went ahead to solve this correctly?
I am trying to find the derivative of $f(x)= (\sin x)^{\ln x}$
Here is what i got below.
$$f(x)= (\sin x)^{\ln x}$$
$$f'(x)=\ln x(\sin x) \Rightarrow f'(x)=\frac{1}{x}\cdot\sin x + \ln x \cdot \cos x$$
Would that be th... | It's instructive to look at this particular logarithmic-differentiation situation generally:
$$\begin{align}
y&=u^{v}\\[0.5em]
\implies \qquad \ln y &= v \ln u & \text{take logarithm of both sides}\\[0.5em]
\implies \qquad \frac{y^{\prime}}{y} &= v \cdot \frac{u^{\prime}}{u}+v^{\prime}\ln u & \text{differentiate}\\
\im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 0
} |
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