Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Purpose Of Adding A Constant After Integrating A Function I would like to know the whole purpose of adding a constant termed constant of integration everytime we integrate an indefinite integral $\int f(x)dx$. I am aware that this constant "goes away" when evaluating definite integral $\int_{a}^{b}f(x)dx $. What has th... | There are many great answers here, but I just wanted to chime in with my favorite example of how things can go awry if one forgets about the constant of integration.
Consider
$$\int \sin(2x) dx.$$
We will find antiderivatives in two ways. First, a substitution $u=2x$ yields:
$$\int \frac{\sin(u)}{2}du = -\frac{\cos(u)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 9,
"answer_id": 7
} |
Simple trigonometry question (angles) I am starting again with trigonometry just for fun and remember the old days. I was not bad at maths, but however I remember nothing about trigonometry...
And I'm missing something in this simple question, and I hope you can tell me what.
One corner of a triangle has a 60º angle, ... | Since we are only interested in the angles, the actual lengths of the two sides do not matter, as long as we get their ratio right. So we can take the lengths of the adjacent sides to be $1$ and $3$, in whatever units you prefer. If you want the shorter of the two adjacent sides to be $20$ metres, then the other adjac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Need help deriving recurrence relation for even-valued Fibonacci numbers. That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$
Empirically one can check that:
$a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$.
If $f(n)$ is $\operatorname{Fibonacci}(n)$ (to make it short), then i... | The definition of $F_n$ is given:
*
*$F_0 = 0$
*$F_1 = 1$
*$F_{n+1} = F_{n-1} + F_{n}$ (for $n \ge 1$)
Now we define $G_n = F_{3n}$ and wish to find a recurrence relation for it.
Clearly
*
*$G_0 = F_0 = 0$
*$G_1 = F_3 = 2$
Now we can repeatedly use the definition of $F_{n+1}$ to try to find an expression f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 4
} |
Solving quadratic equation $$\frac{1}{x^2} - 1 = \frac{1}{x} -1$$
Rearranging it I get: $1-x^2=x-x^2$, and so $x=1$. But the question Im doing says to find 2 solutions. How would I find the 2nd solution?
Thanks.
| I think it should be emphasised what the salient point is here:
Given the equation
$$
\Phi =\Psi
$$
you may multiply both sides by the same non-zero number $a$ to obtain the equivalent equation
$$
a\Phi =a\Psi.
$$
Multiplying both sides of an equation by 0 may give an equation that's not equivalent to the original ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
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Conditional probability of a general Markov process given by its running process I have a question as follow:
"Let $X$ be a general Markov process, $M$ is a running maximum process of $X$ and $T$ be an exponential distribution, independent of $X$.
I learned that there is the following result:
Probability: $P_x(X_T\in ... | For real-valued diffusion processes, this is essentially a local form of David Williams' path decomposition, and can be deduced from
Theorem A in a paper "On the joint distribution of the maximum and its location for a linear diffusion" by Csaki, Foldes and Salminen
[Ann. Inst. H. Poincare Probab. Statist., vol. 23 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Proof that the set of odd positive integers greater then 3 is countable I found one problem which asks following:
Show that the set of odd positive integers greater then 3 is countable.
At the begining I was thinking that such numbers could be represented by $2k+1$,where $k>1$; but in the answers paper ... | If you know that a set $A$ is countable and you demonstrate a bijection $f:A\to B$ then you have also shown that the set $B$ is countable; when $A=\mathbb{Z}^+$ this is the very definition of countable. Both of the functions $2k+1$ and $2n+3$ can be used to show that the set of odds greater than $3$ are countable but t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Matrix/Vector Derivative I am trying to compute the derivative:$$\frac{d}{d\boldsymbol{\mu}}\left( (\mathbf{x} - \boldsymbol{\mu})^\top\boldsymbol{\Sigma} (\mathbf{x} - \boldsymbol{\mu})\right)$$where the size of all vectors ($\mathbf{x},\boldsymbol{\mu}$) is $n\times 1$ and the size of the matrix ($\boldsymbol{\Sigma}... | There is a very short and quick way to calculate it correctly. The object $(x-\mu)^T\Sigma(x-\mu)$ is called a quadratic form. It is well known that the derivative of such a form is (see e.g. here),
$$\frac{\partial x^TAx }{\partial x}=(A+A^T)x$$
This works even if $A$ is not symmetric. In your particular example, you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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How many smooth functions are non-analytic? We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather contrived, and most smooth functions that I've encountered in math and physics are... | In terms of cardinality, there are the same number of smooth and analytic functions, $2^{\aleph_0}$. The constant functions are enough to see that there are at least $2^{\aleph_0}$ analytic functions. The fact that a continuous function is determined by its values on a dense subspace, along with my presumption that y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
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Why is the pullback completely determined by $d f^\ast = f^\ast d$ in de Rham cohomology? Fix a smooth map $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$. Clearly this induces a pullback $f^\ast : C^\infty(\mathbb{R}^n) \rightarrow C^\infty(\mathbb{R}^m)$. Since $C^\infty(\mathbb{R}^n) = \Omega^0(\mathbb{R}^n)$ (the spa... | $\newcommand\RR{\mathbb{R}}$I don't have the book here, but it seems you are asking why there is a unique extension of $f^*:\Omega^0(\RR^n)\to\Omega^0(\RR^m)$ to an appropriate $\overline f^*:\Omega^\bullet(\RR^n)\to\Omega^\bullet(\RR^m)$ such that $f^*d=df^*$. Here appropriate should probably mean that the map $\overl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$p(x)$ divided by $x-c$ has remainder $p(c)$? [Polynomial Remainder Theorem] This is from Pinter, A Book of Abstract Algebra, p.265.
Given $p(x) \in F[x]$ where $F$ is a field, I would like to show that $p(x)$ divided by $x-c$ has remainder $p(c)$.
This is easy if $c$ is a root of $p$, but I don't see how to prove it ... | By the division algorithm, if $a(x)$ and $b(x)$ are any polynomials, and $a(x)\neq 0$, then there exist unique $q(x)$ and $r(x)$ such that
$$b(x) = q(x)a(x) + r(x),\qquad r(x)=0\text{ or }\deg(r)\lt \deg(a).$$
Let $b(x) = p(x)$, and $a(x)=x-c$. Then $r(x)$ must be constant (since it is either zero or of degree strictly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Why do we reverse inequality sign when dividing by negative number? We all learned in our early years that when dividing both sides by a negative number, we reverse the inequality sign.
Take
$-3x < 9$
To solve for $x$, we divide both sides by $-3$ and get
$$x > -3.$$
Why is the reversal of inequality? What is going in ... | Multiplying or dividing an inequality by $-1$ is exactly the same thing as moving each term to the other side. But then if you switch side for all terms, each term faces the opposite "side" of inequality sign...
For example:
$2x < -3$
Moving them on the other side yields:
$3 < -2x$ which is the same as $-2x > 3$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/94790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 9,
"answer_id": 0
} |
What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$? Find the limit:
$$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$
I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
| Since everybody was 'clever', I thought I'd add a method that doesn't really require much thinking if you're used to asymptotics.
The power series for $\sin x$
$$\sin x = x + O(x^3)$$
We can compute the inverse of this power series without trouble. In great detail:
$$\begin{align}\frac{1}{\sin x} &= \frac{1}{x + O(x^3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 8,
"answer_id": 0
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How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$? Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa.
One way of the proof.
We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ \lambda$ is an eigenv... | A shorter way of seeing this would be to observe that if
$$
(AB^{-1})x=\lambda x
$$
for some non-zero vector $x$, then by multiplying that equation by $B^{-1}$ (from the left) we get that
$$
(B^{-1}A)(B^{-1}x)=\lambda (B^{-1}x).
$$
In other words $(B^{-1}A)y=\lambda y$ for the non-zero vector $y=B^{-1}x$. This proc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Serving customers algorithm Well I have a problem with a Christmas assignment and my teacher is not responding(maybe he is skiing somewhere now) so I will need some help.
The algorithm is about an office and the waiting time of the customers. We have one office that has to serve $n$ customers $a_1, a_2,\cdots ,a_n$. We... | Problems of this kind belong to the area of operations research known as scheduling problems (scheduling theory). Here is a short bibliography of books that deal with this topic: http://www.york.cuny.edu/~malk/biblio/scheduling2-biblio.html There is a lot of nice mathematics involved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/94976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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When is $[0,1]^K$ submetrizable or even metrizable? Let $I=[0,1]$ and $K$ is a compact space. Then could the function space $I^K$ be submetrizable, even metrizable? In other words, in general, if $I^A$ can be submetrizable (metrizable) for some space $A$, what's condition that $A$ should satisfying?
| If $A$ is compact, $I^A$ is metrizable with the metric being the uniform norm. That is, $d(f,g):=\sup_{a\in A} d(f(a),g(a))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/95092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof that $\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$ I try to prove the following
$$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$
with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas?
Any help is appreciated.
| Combinatorial proof of ${2n \choose n+1} \geq 2^n$ where $n \geq 2$:
Let's take set $\{x_1,y_1,\dots,x_{n-2},y_{n-2},a,b,c,d\}$ which has $2n$ elements; select three elements out of $\{a,b,c,d\}$ and for all $i$, a single element of $\{x_i,y_i\}$, you'll select $n+1$ in total. So
${2n \choose n+1} \geq {4 \choose 3} 2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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partial sum involving factorials Here is an interesting series I ran across.
It is a binomial-type identity.
$\displaystyle \sum_{k=0}^{n}\frac{(2n-k)!\cdot 2^{k}}{(n-k)!}=4^{n}\cdot n!$
I tried all sorts of playing around, but could not get it to work out.
This works out the same as $\displaystyle 2^{n}\prod_{k=1}... | This identity can be re-written as
$$\sum_{k=0}^n {2n-k \choose n-k} 2^k = 4^n.$$
Start from
$${2n-k \choose n-k} =
\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{2n-k}}{z^{n-k+1}} \; dz.$$
This yields for the sum
$$\frac{1}{2\pi i} \int_{|z|=\epsilon}
\sum_{k=0}^n \frac{(1+z)^{2n-k}}{z^{n-k+1}} 2^k \; dz
\\ = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
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Side-stepping contradiction in the proof of ; ab = 0 then a or b is 0. Suppose we need to show a field has no zero divisors - that is prove the title - then we head off exactly like the one common argument in the reals (unsurprisingly as they themselves are a field).
What I want to know is; how do we prove this not by ... | Let $m,n\in\mathbb{N}$ such that $m,n>0$ (I subscribe to $0\in\mathbb{N}$ but it really doesn't matter here). It can be shown by induction that $mn\neq 0$. That is, that $mn>0$.
Now, let $a,b\in\mathbb{Z}$. If $ab=0$, then $|ab|=|0|=0$. Therefore $|ab|>0$ implies $ab\neq 0$.
Suppose $a,b\neq 0$. Then $|a|=m>0$ and $|b|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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Clarifying the definition of "unstable" I would appreciate a definition clarification.
if a numerical method is "unstable", does it mean that if we introduce a small random error in one of the steps, the error would be magnified greatly after further steps? is this true for all unstable algorithms or are there some whe... | We say a method is stable when it is capable of controlling errors introduced in each computation. Stability allows the method to converge to a certain solution. Here's a simple example:
\begin{equation}
u_t = -u_x
\end{equation}
Suppose a set of equally distanced nodes on the x-axis in 1D. We assume $U_i$ denotes app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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An inequality about maximal function Consider the function on $\mathbb R$ defined by
$$f(x)=\begin{cases}\frac{1}{|x|\left(\log\frac{1}{|x|}\right)^2} & |x|\le \frac{1}{2}\\
0 & \text{otherwise}\end{cases}$$
Now suppose $f^*$ is the maximal function of $f$, then I want to show the inequality $f^*(x)\ge \frac{c}{|x|\l... | By definition
$$
f^*(x)=\sup_{B\in \text{Balls}(x)}\frac{1}{\mu(B)}\int\limits_B |f(y)|d\mu(y)\qquad(1)
$$
where $\text{Balls}(x)$ the set of all closed balls containing $x$. We can express $f^*$ in another form
$$
f^*(x)=\sup_{\alpha\leq x\leq\beta}\frac{1}{\beta-\alpha}\int\limits_{\alpha}^{\beta} |f(y)|d\mu(y)
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Meaning of $f:[1,5]\to\mathbb R$ I know $f:[1,5]\to\mathbb R$, means $f$ is a function from $[1,5]$ to $\mathbb R$. I am just abit unclear now on the exact interpretation of "to $\mathbb R$". Is $1\le x\le 5$ the domain? And is $\mathbb R$ the co-domain (or image?)?
Is my interpretation in words ---$f$ is a function wh... | The notation
$$
f: [1,5] \rightarrow \mathbb{R}
$$
means that $f$ is a function whose domain is to taken to be the interval $[1,5]$ and whose codomain is $\mathbb{R}$ (i.e. all the outputs of $f$ fall into $\mathbb{R}$). It makes no claims about surjectivity or injectivity; you must analyze the function itself to dec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Proof of $f \in C_C(X)$ where $X$ is a metric space implies $f$ is uniformly continuous Can you tell me if the following proof is correct?
Claim:
If $f$ is a continuous and compactly supported function from a metric space $X$ into $\mathbb{R}$ then $f$ is uniformly continuous.
Proof:
The proof is in two parts.
First we... | That looks good except for the correction that t.b. pointed out. In the spirit of Henning Makholm's comment, here is a "canned theorem" approach.
A continuous function on a compact metric space is uniformly continuous, so $f|_K$ is uniformly continuous. Let $\varepsilon>0$ be given. Then $K_\varepsilon:=\{x:|f(x)|\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A question on the sylow subgroups of a normal subgroup $H$ normal subgroup of a group $G$ with cardinality finite. $p$ prime number dividing $|H|$. $P$ a $p$-Sylow subgroup of $H$, how can I prove that then $G=HN_G(P)$ where $N_G(P)$ is the normalizer of $P$ in $G$?
| If $g\in G$, then $gHg^{-1}=H$, and so $gPg^{-1}\subseteq H$. Since $gPg^{-1}$ is a $p$-Sylow subgroup of $H$, by Sylow's Theorems we know that $gPg^{-1}$ is conjugate to $P$ in $H$. That is, there exists $h\in H$ such that $hPh^{-1} = gPg^{-1}$. Therefore, $g^{-1}hPh^{-1}g = P$, so $h^{-1}g\in N_G(P)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/95735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dedekind domain with a finite number of prime ideals is principal I am reading a proof of this result that uses the Chinese Remainder Theorem on (the finite number of) prime ideals $P_i$. In order to apply CRT we should assume that the prime ideals are coprime, i.e. the ring is equal to $P_h + P_k$ for $h \neq k$, but ... | Hint $\ $ Nonzero prime ideals are maximal, hence comaximal $\, P + Q\ =\ 1\, $ if $\, P\ne Q.$
Another (perhaps more natural) way to deduce that semi-local Dedekind domains are PIDs is to exploit the local characterization of invertibility of ideals. This yields a simpler yet more general result, see the theorem below... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
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software for algebraic simplifying expressions I have many huge algebraic expressions such as:
$$\frac{8Y}{1+x}-\frac{(1-Y)}{x}+\frac{Kx(1+5x)^{3/5}}{2}$$
where $\ Y=\dfrac{Kx(1+x)^{n+2}}{(n+4)(1+5x)^{2/5}}+\dfrac{7-10x-x^2}{7(1+x)^2}+\dfrac{Ax}{(1+5x)^{2/5}(1+x)^2}\ $ and $A,n$ are constants.
To simplify these express... | Note that if you set $\rm\ z = (5x+1)^{1/5}\ $ then your computations reduce to rational function arithmetic combined with the rewrite rule $\rm\: z^5\ \to\ 5x+1\ $ with the following expressions
$$\frac{8Y}{1+x}-\frac{(1-Y)}{x}+\frac{Kxz^3}{2}$$
where $\ Y\ =\ \dfrac{Kx(1+x)^{n+2}}{(n+4)z^2}+\dfrac{7-10x-x^2}{7(1+x)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Isomorphism of quotient modules implies isomorphism of submodules? Let $A$ be a commutative ring, $M$ an $A$-module and $N_1, N_2$ two submodules of $M$.
If we have $M/N_1 \cong M/N_2$, does this imply $N_1 \cong N_2$?
This seems so trivial, but I just don't see a proof... Thanks!
| The implication is false for all commutative non-zero rings $A$.
Indeed, just take $M=\oplus_{i=0}^{i=\infty} A$ , $N_1=A\oplus 0\oplus0...$ and $N_2=A\oplus A\oplus 0\oplus 0...$.
Since $N_1$ is isomorphic to $A$ and $N_2$ is isomorphic to $A^2$, they are not isomorphic.
However $M/N_1$ and $M/N_2$ are isomorph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Odds of guessing suit from a deck of cards, with perfect memory While teaching my daughter why drawing to an inside straight is almost always a bad idea, we stumbled upon what I think is a far more difficult problem:
You have a standard 52-card deck with 4 suits and I ask you to guess the suit of the top card. The odds... |
overall odds for any given draw over the course of 52 picks
If I rephrased your question as "how much should you be willing to pay to play the game where I will show you $n$ cards out of 52 and if you guess the next remaining card then I give you a dollar" would an answer to this question be suitable? Just to be cle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
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Probability of an odd number in 10/20 lotto Say you have a lotto game 10/20, which means that 10 balls are drawn from 20.
How can I calculate what are the odds that the lowest drawn number is odd (and also how can I calculate the odds if it's even)?
So a detailed explanation:
we have numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, ... | The total number of outcomes is ${20 \choose 10}$. Now count the total number of favorable outcomes:
*
*outcomes with lowest element 1 : ${19 \choose 9}$ ;
*outcomes with lowest element 3 : ${17 \choose 9}$ ;
*outcomes with lowest element 5 : ${15 \choose 9}$ ;
*outcomes with lowest element 7 : ${13 \choose 9}$ ;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is every Mersenne prime of the form : $x^2+3 \cdot y^2$? How to prove or disprove following statement :
Conjecture :
Every Mersenne prime number can be uniquely written in the form : $x^2+3 \cdot y^2$ ,
where $\gcd(x,y)=1$ and $x,y \geq 0$
Since $M_p$ is an odd number it follows that : $M_p \equiv 1 \pmod 2$
Accordin... | (Outline of proof that, for prime $p\equiv 1\pmod 6$, there is one positive solution to $x^2+3y^2=p$.)
It helps to recall the Gaussian integer proof that, for a prime $p\equiv 1\pmod 4$, $x^2+y^2=p$ has an integer solution. It starts with the fact that there is an $a$ such that $a^2+1$ is divisible by $p$, then uses u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Do "imaginary" and "complex" angles exist? During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting:
$ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $
$ \sin ( 1 + i ) \approx 1.30 + 0.63 i $
Does this mean that there is such a thing as ... | A fundamental equation of trigonometry is $x^2+y^2 = 1$, where $x$ is the "adjacent side" and $y$ the "opposite side".
If you experiment plot $f(x)$ out of the real domain - for example to $x=1.5$ you obtain $y$ imaginary - you will get an imaginary shape situated in a plane perpendicular to the plane $x,y$ and contain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 7,
"answer_id": 5
} |
question on fourier transform. I ask myself what
$$
{\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)'(\xi) )(s)
$$
is. If it was just about
$$
{\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)(\xi) )(s)
$$
it would be clear (a shift by $t$), the same is with
$$
{\mathscr F}^{-1}( ({\mathscr F} \phi)'(\x... | Why don't you just compute it?
$$
{\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)'(\xi) )(s)={\mathscr F}^{-1}( ({\mathscr F} \phi)'(\xi) )(s+t)=-i(s+t)\phi(s+t)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/96217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Question about implication and probablity Let $A, B$ be two event. My question is as follows:
Will the following relation holds:
$$A \to B \Rightarrow \Pr(A) \le\Pr(B) $$
And why?
| In terms of intuition, the fact that some event $A$ implies some event $B$ means that whenever $A$ happens, $B$ happens. But if the event $B$ happens, we might not have event $A$. So in other words, we have that $\mathbb P(A) \le \mathbb P(B)$ because the probability that $A$ happens is also "the probability that $B$ h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Motivation for a particular integration substitution In an old Italian calculus problem book, there is an example presented:
$$\int\frac{dx}{x\sqrt{2x-1}}$$
The solution given uses the strange substitution $$x=\frac{1}{1-u}$$
Some preliminary work in trying to determine the motivation as to why one would come up with s... | This won't answer the question, but it takes the geometry a bit beyond where the question left it. Consider the circle of unit radius in the Cartesian plane centered at $O=(0,1)$. Let $A=(1,0)$ and $B=(x,0)$. Let $C=(1,\sqrt{2x-1})$. Your right triangle is $ABC$, with angle $\alpha$ at vertex $B$. Another right t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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quadratic reciprocity happy new year
I have this statement:
"By quadratic reciprocity there are the integers $a$ and $b$ such that $(a,b)=1$, $(a-1,b)=2$, and all prime $p$ with $p\equiv a$ (mod $b$) splits in $K$ (where $K$ is a real quadratic field)".
I have tried with many properties of quadratic reciprocity but cou... | Edited to address some bizarrely horrible errors in the first version.
Here's a simple case from which it should not be too hard to generalize. Suppose that $K$ is a quadratic field of prime discriminant $q$. Since $q\equiv 1\pmod{4}$, note that a prime $p$ splits in $K$ if and only if $\left(\frac{p}{q}\right)=1$.
L... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Equivalence class on real numbers Call two real numbers equivalent if their binary decimal expansion differ in a finite amount of places, if S is a set which contains an element of every equivalence class, must S contain an interval?
How to show that every interval contains an (uncountable number of?) element of every ... | Added part: We produce a bounded set $S$ that contains a member of every equivalence class but does not contain an interval. Every equivalence class meets $[0,1]$, since for any $x$, we can, by making a finite number of changes to the bits of $x$, produce an $x'\in [0,1]$.
Use the Axiom of Choice to select $S\subset [0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In a Boolean algebra B and $Y\subseteq B$, $p$ is an upper bound for $Y$ but not the supremum. Is $qI don't think that this is the case. I am reading over one of my professor's proof, and he seems to use this fact. Here is the proof:
Let $B$ be a Boolean algebra, and suppose that $X$ is a dense subset of $B$ in the sen... | The set of upper bounds is closed under intersection, so $p \cap q$ is an upper bound less than $p$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Jordan decomposition/Levi decomposition in GL(n) in positive characteristic Let $k$ be a non archimedean field of positive characteristic. Lets consider a parabolic subgroup $P \subset GL(n, k)$.
I am a little bit confused by the following statement in "Laumon - Cohomology of Drinfeld Modular ... ":
I have an issue wi... | Both those assertions describe a Levi decomposition. In general, groups need not have Levi decompositions, but parabolic subgroups of reductive groups do. This is proven for connected reductive groups (e.g. $GL(n, k)$) in Borel "Linear Algebraic Groups": see 20.5 for the decomposition over k.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Subgroup(s) of a group of order 25 I am working on a problem (self-study) from Artin - 2.8.8 which goes:
"Let G be a group of order 25. Prove that G has at least one subgroup of order 5, and that if it contains only one subgroup of order 5, then it is a cyclic group."
I can see that there is an element of order 5, and ... | It is an easy exercise to show that if $c_d(G)$ denotes the number of cyclic subgroups of $G$ of order $d$ then $\displaystyle \sum_{d\mid |G|}c_d(G)\varphi(d)=|G|$ (just partition your group according to the elements orders). Now, if $G$ had only one subgroup of order $5$, then via the fact that all groups of order $5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Chance for picking a series of numbers (with repetition, order doesn't matter) I want to calculate the chance for the following situation:
You throw a die 5 times. How big is the chance to get the numbers "1,2,3,3,5" if the order does not matter (i.e. 12335 = 21335 =31235 etc.)?
I have 4 different solutions here, so I ... | There are $5$ options for placing the $1$, then $4$ for placing the $2$, and then $3$ for placing the $5$, for a total of $5\cdot4\cdot3=60$. Alternatively, there are $5!=120$ permutations in all, and pairs of these are identical because you can exchange the $3$s, which also gives $120/2=60$. The total number of combin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lie Algebra of $SL_n(\mathbb H)$ The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions?
| The obvious canditate for $\mathfrak{sl}_2(H)$ is the space of $2\times 2$ matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with quaternion entries such that $a+d=0$, with bracket the commutator of matrices, but... that is not a Lie algebra.
For example, the trace of the commutator of $\begin{pmatrix}i&0\\0&-i\end{pmatr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$ In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is incomplete, he defines a sequence of elements in $X$ by
$$... | It's relatively easy to see that for $m<n$ we have $x_n(t)\le x_m(t)$ for each $t$. Hence
$$\|x_m-x_n\|=\int_0^1 x_m(t) \mathrm{d}t-\int_0^1 x_n(t) \mathrm{d}t.$$
We can disregard intervals $\langle 0,1/2-1/m\rangle$, since both functions are zero there. We can also disregard $\langle 1/2,1\rangle$, since $x_m(t)=x_n(t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this true about integrating composite functions? Let's say that I'm integrating a composite function, say $f(g(x))$, that is in a form to which I can apply the substitution rule. Is it true to say that both $f$ and $g$ must be differentiable?
I understand that the substitution rule requires $g$ to be differentiable... | if you talk about the Riemann-integral: if $f : [a,b] \to [c,d]$ R-integrable and $g : [c,d] \to \mathbb{R}$ continuous, than $g \circ f : [a,b] \to \mathbb R$ is R-integrable.
Differentiable implies continuous, so if f,g are differentiable they are R-integrable
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving $\int\frac{\ln(1+e^x)}{e^x} \space dx$ I'm trying to solve this integral.
$$\int\frac{\ln(1+e^x)}{e^x} \space dx$$
I try to solve it using partial integration twice, but then I get to this point (where $t = e^x$ and $dx = \frac{1}{t} dt$):
$$\int\frac{\ln(1+t)}{t^2} \space dt = \frac{1}{t} \cdot \ln(1+t) - \fra... | Edit:
Your problem is having integrated by parts twice. Doing it one time looks like this:
$$\int \frac{\ln(1+t)}{t^2} \ \mathrm{d}t = -\frac1{t}\ln(1+t) - \int -\frac1{t}\cdot\frac1{1+t} \ \mathrm{d}t = -\frac1{t}\ln(1+t) + \int \frac1{t}\cdot\frac1{1+t} \ \mathrm{d}t$$
That new integral should be evaluated with par... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partitioning of geometric net into equivalence classes Fellow Puny Humans,
A geometric net is a system of points and lines that obeys three axioms:
*
*Each line is a set of points.
*Distinct line has at most one point in common.
*If $p$ is a point and $L$ is a line with $p \notin L$, then there is exactly one lin... | It is exactly the number of lines you have in a class, simply because equal or parallel is an equivalence relation. Let me clarify:
Say $C_1, C_2,\dots, C_m$ are your equivalence classes. Say $L\in C_i$ is a line in $C_i$, for some $i\in\{1,\dots,m\}$. Say $1\leq j \leq m$, $j\neq i$, and $M\in C_j$. If $L\cap M = \emp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Secret Number Problem Ten students are seated around a (circular) table. Each student selects his or her own secret number and tells the person on his or her right side the number and the person his or her left side the number (without disclosing it to anyone else). Each student, upon hearing two numbers, then calculat... | Let us denote secret numbers as $x_i$ , where $i$ is announced number ,then we have following system of equations :
$\begin{cases}
x_1+x_3=4 \\
x_2+x_4=6 \\
x_3+x_5=8 \\
x_4+x_6=10 \\
x_5+x_7=12 \\
x_6+x_8=14 \\
x_7+x_9=16 \\
x_8+x_{10}=18 \\
x_9+x_1=20 \\
x_{10}+x_2=2
\end{cases}$
According to Map... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Number of subgroups of prime order I've been doing some exercises from my introductory algebra text and came across a problem which I reduced to proving that:
The number of distinct subgroups of prime order $p$ of a finite group $G$ is either $0$ or congruent to $1\pmod{p} $.
With my little experience I was unable t... | Here's another approach. Consider the solutions to the equation $x_1x_2\cdots x_p=1$ in the group $G$ of order divisible by $p$. Since there is a unique solution for any $x_1,\ldots,x_{p-1}$, the total number of solutions is $|G|^{p-1}$, which is divisible by $p$. If $x_1,x_2,\ldots,x_p$ is a solution, then so is $x_2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Probability that ace of spades is at bottom of deck IF ace of hearts is NOT at top
What is the probability that the ace of spades is at the bottom of a standard deck of 52 cards given that the ace of hearts is not at the top?
I asked my older brother, and he said it should be $\frac{50}{51} \cdot \frac{1}{51}$ becaus... | The ace of hearts has 51 positions available (since it's not at the top).
Having placed it somewhere, there are 51 positions available for ace of spades, so
Pr = P(ace of spades not at bottom)*P(ace of diamonds at bottom)
= 50/51 *1/51 = 50/51²
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/97527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solution of a polynomial of degree n with soluble galois group. Background: Given the fundamental theorem of algebra every polynomial of degree n has n roots. From Galois Theory we know that we can only find exact solutions of polynomials if their corresponding Galois group is soluble. I am studying Galois Theory ( Ian... | By exact roots you probably mean radical expressions. Even for equations whose Galois group is unsolvable there might be exact trigonometric expressions for the roots.
If you know German, the diploma thesis "Ein Algorithmus zum Lösen einer
Polynomgleichung durch Radikale" (An algorithm for the solution of a polynomial ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$? $$\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$$
Is there any formula that tells this or why is it like that?
| \begin{equation}
\text{You can use De Moivre's identity:}
\end{equation}
\begin{equation}
\text{Let's Call:}\\\\
\end{equation}
\begin{equation}
\mathrm{z=\cos x+i \sin x}\\
\mathrm{\frac{1}{z}=\cos x-i \sin x}\\
\end{equation}
\begin{equation}
\text{Now subtracting both equations together, we get:}\\
\end{equation}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 4
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Showing $f(x)/x \to 0$ when $\lvert f(x) - f(\lambda x)\rvert/x \to 0$ I would like to solve this problem, but I do not know how ...
Let $f:(0;1) \rightarrow \mathbb{R}$ be a function such that:
$$\lim_{x \to0^+}f(x)=0$$
and such that there exists $0<\lambda<1$ such that:
$$\lim_{x \to0^+} \frac{ \left [ f(x)-f(\lamb... | Since
$$ \frac{f(x) - f(\lambda x)}{x} \to 0,$$
for any $\epsilon > 0$, we can restrict $x$ near enough $0$ so that we have $\lvert f(x) - f(\lambda x)\rvert \leq \epsilon \lvert x \rvert$. Since $0 < \lambda < 1$, this means that we also have $\lvert f(\lambda^n x) - f(\lambda^{n+1} x) \rvert \leq \epsilon \lvert x \r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit of the sequence of regular n-gons. Let $A_n$ be the regular $n$-gon inscribed in the unit circle.
It appears intuitively obvious that as $n$ grows, the resulting polygon approximates a circle ever closer.
Can it be shown that the limit as $n \rightarrow \infty $ of $A_n$ is a circle?
| Given a sequence of sets $(A_n)_{n\geq3}$ there is a natural $\lim\inf_{n\to\infty} A_n=:\underline{A}$ and a natural $\lim\sup_{n\to\infty}A_n=:\overline{A}$ of this sequence.
In the problem at hand the $A_n$ are closed regular $n$-gons inscribed in the unit circle, all sharing the point $P:=(1,0)$.
The set $\underli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Changing the argument for a higher order derivative I start with the following:
$$\frac{d^n}{dx^n} \left[(1-x^2)^{n+\alpha-1/2}\right]$$
Which is part of the Rodrigues definition of a Gegenbauer polynomial. Gegenbauer polynomials are also useful in terms of trigonometric functions so I want to use the substitution $x ... | Instead of Faa di bruno's formula, you can try generalizing the formula for $n^{th}$ derivative of inverse function.
Let $f,g$ be functions of $x$ and inverses of each other. We know that $\displaystyle f'=\frac{1}{g'}$ i.e. $f'g'=1$. Using Leibniz' rule, we get
$\displaystyle (f'g')^{(n)}(\theta)=\sum_{k=0}^n \binom{... | {
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"timestamp": "2023-03-29T00:00:00",
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Derivative of a function is odd prove the function is even. $f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f'(x)$ exists $\forall x.$
And $f'(-x)=-f'(x)$
I would like to show $f(-x)=f(x)$
In other words a function with odd derivative is even.
If I could apply the fundamental theorem of calculus
$\int_{-x}^{x}f'(t)... | *
*Define functions $f_0(x)=(f(x)+f(-x))/2$ and $f_1(x)=(f(x)-f(-x))/2$. Then $f_0$ and $f_1$ are also differentiable, and $f_0$ is even and $f_1$ is odd.
*Show that the derivative of an odd function is even, and that of an even function is odd.
*From the equality $f'=f_0'+f_1'$ conclude that $f_1$ is constant and, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98003",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "11",
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Inequality for modulus Let $a$ and $b$ be complex numbers with modulus $< 1$.
How can I prove that
$\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ?
Thank you
| Here are some hints: Calculate $|a-b|^2$ and $|1-\overline{a}b|^2$ using the formula $|z|^2=z\overline{z}$. To show that $\displaystyle\left | \frac{a-b}{1-\bar{a}b} \right |<1$, it's equivalent to show that
$$\tag{1}|1-\overline{a}b|^2-|a-b|^2>0.$$
To show $(1)$, you need to use the fact that $|a|<1$ and $|b|<1$.
If... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the purpose of Stirling's approximation to a factorial? Stirling approximation to a factorial is
$$
n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.
$$
I wonder what benefit can be got from it?
From computational perspective (I admit I don't know too much about how each arithmetic operation is implemente... | A Stirling inequality
$$(n!)^{\frac{1}{n}} \le \frac{e}{n+1}$$
can be used to derive Carleman's inequality from the AM-GM inequality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/98171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
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Is there a simple formula for this simple question about a circle? What is the average distance of the points within a circle of radius $r$ from a point a distance $d$ from the centre of the circle (with $d>r$, though a general solution without this constraint would be nice)?
The question arose as an operational resear... | I guess it involves calculus. Let $(x,y)$ be a point within the circle of radius $R$ and $(d,0)$ the coordinates of the point a distance $d$ away from the origin (because of the symmetry we can choose it to lie on the $x$-axis). Then the distance between the two points is given by $$\ell = \sqrt{(x-d)^2 + y^2}.$$
Avera... | {
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"answer_id": 0
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Easy way to determine the primes for which $3$ is a cube in $\mathbb{Q}_p$? This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it:
For which $p$ is $3$ a cube root in $\mathbb{Q}_p$?
The case $p=3$ for which $X^3-3$ is not separable modulo $p$ can easily be ruled out by ch... | For odd primes $q \equiv 2 \pmod 3,$ the cubing map is a bijection, 3 is always a cube $\pmod q.$
For odd primes $p \equiv 1 \pmod 3,$ by cubic reciprocity, 3 is a cube $\pmod p$ if and only if there is an integer representation
$$ p = x^2 + x y + 61 y^2, $$ or $4p=u^2 + 243 v^2.$ In this form this is Exercise 4.15... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $D_{12}$ is isomorphic to $D_6\times C_2$ Show that $D_{12}$ is isomorphic to $D_6 \times C_2$, where $D_{2n}$ is the dihedral group of order $2n$ and $C_2$ is the cyclic group of order $2$.
I'm somewhat in over my head with my first year groups course. This is a question from an example sheet which I think ... | Assuming $D_{n}$ is the dihedral group of order $n$, I would proceed as follows. Note that $D_{6} \cong S_{3}$, and $S_{3}$ is generated by $(12)$ and $(123)$. Therefore $D_{6} \times C_{2} = \langle ((12),[0]),((123),[1]) \rangle$. Next note that $D_{12} = \langle r,s | \, r^{6}=s^{2}=e , s^{-1}rs=r^{-1} \rangle$, map... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Exercise about semidirect product This is exercise 7.12 from Algebra, Isaacs.
$ G= N \rtimes H \ $ is a semidirect product; no nonidentity element of H fixes any nonidentity element of N; identify N and H with the corresponding subgroups of G. Show that:
a) $ H \bigcap H^{g } = 1 $ for all $ g \in G - H $
b) If G i... | (a) Let $x\in H\cap H^g$, where $g=hn$, $h\in H$, $n\in N$, $n\neq 1$. Then there exists $y\in H$ such that $x=g^{-1}yg = n^{-1}(h^{-1}yh)n$. Since $h^{-1}yh\in H$, it suffices to consider the case of $g\in N$. So we set $g=n$.
(The intuition is that we want to go to some expression like $x^{-1}nx = n$, because this wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Estimate probabilities from its moments I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its distribution type).
| As I had pointed out in my comments, it's hard to answer this question in generality. So, I'll just point you to a resource online.
But, that said, the magic words are generating functions-Probability generating functions and Moment Generating Functions.
The probability generating functions $\Phi_X$ exists only for non... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Continuous extension of a real function defined on an open interval Let $I\subset\mathbb{R}$ be a compact interval and let $J$ denote its interior.
Consider $f:J\to\mathbb{R}$ being continuous.
*
*Under which conditions does the following statement hold?
$$
\text{There exists a continuous extension $g:I\to\mathbb{R}... | Call $I = [a, b]$ with $-\infty < a < b < \infty$. Such an extension exists if and only if both $\lim_{x\to a^+} f(x)$ and $\lim_{x\to b^-} f(x)$ exist, and in fact these values become the values of the extension. (The proof is left as a simple exercise.) With this in mind, boundedness is not sufficient due to previous... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$? How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$ in Kunen's book, I 10.19?
I will appreciate any help on this question. Thanks ahead.
| I took the trouble to read through Kunen in order to understand the problem, as well the definitions which you can use for this.
*
*Cardinal is defined to be an ordinal $\kappa$ that there is no $\beta<\kappa$ and a bijection between $\kappa$ and $\beta$.
*The successor cardinal $\kappa^+$ is the least cardinal whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Computing taylor series for trigonometric exponential function How do I compute the taylor series for $\cos(x)^{\sin(x)}$ ? I tried using the $e^x$ rule but I still am not getting to the result:
$$\cos(x)^{\sin(x)}=1-\frac{x^3}{2}+\frac{x^6}{8}+o(x^6).$$
| Your formula ($\cos(x)^{\sin(x)}=1-\frac{x^3}{2}+\frac{x^6}{8}+o(x^6).$) has been achieved from the definition of The Taylor Series:
$$f(x) = \sum_{i=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$
Where $f^{(n)}(x)$ is $n$th derivative of $f(x)$ with respect to $x$.
(Notice that $f\in c^{\infty}$)
put $x_0=0$ and calcul... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Examples of Galois connections? On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end :
But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If you've got a little gadget k sitting in a big gadget K, you get a "Galois group" Gal(K/k) c... | This wikipage gives a good list of examples of both monotone and antitone Galois connections.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Differentiability of Moreau-Yosida approximation. I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is differentiable of class $\mathcal{C}^1$.
(where $f_{\lambda}:X\to\mathbb{R}\c... | Observe that the subdifferential of the function $y\to \frac{\|x-y\|^2}{2\lambda}+f(y)$ is the operator
$$y\to F(y-x)+\partial f(y),$$
where $F:X\to X^*$ is a duality mapping ($Fx=\{f^*\in X^*\,|\,\langle f,x\rangle=\|x\|^2=\|f\|^2\}$). Now, recall that a point $y$ is a minimazer of a convex function $g$ iff $0\in \par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Probability of choosing the correct stick out of a hundred. Challenge from reality show. So I was watching the amazing race last night and they had a mission in which the
contestants had to eat from a bin with 100 popsicles where only one of those popsicles had a writing on its stick containing the clue.
Immediately I... | The probability of getting the first wrong is $\dfrac{99}{100}$. The probability of getting the second right given the first is wrong is wrong $\dfrac{1}{99}$; the probability of getting the second wrong given that the first is wrong $\dfrac{98}{99}$. And this pattern continues.
Let's work out the probability of gett... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does the following set have no subspaces but {0} and itself? Here's the statement:
The following set, $V$, only has subspaces $\{0\}$ and $V$.
$$V=\{f(t) \colon \mathbb R \to \mathbb R \mid f'(t) = k\cdot f(t) \text{ where } k \text{ is a constant}\}$$
I'm having trouble understanding why there are no oth... | HINT $\rm\displaystyle\ \begin{align} f{\:'} &=\ \rm k\ f \\ \rm \:\ g' &=\ \rm k\ g \end{align}\ \Rightarrow\ \dfrac{f{\:'}}f\: =\: \dfrac{g'}g\: \iff\: \bigg(\!\!\dfrac{g}f\bigg)' =\ 0\ \iff \ g\: =\: c\ f,\ \ \ c'\: =\ 0,\ $ i.e. $\rm\ c\:$ "constant".
This is a special case of the the Wronskian test for linear dep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields Some days ago, I was thinking on a problem, which states that $$AB-BA=I$$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes the set of all $n\times n$ matrices with entri... | HINT $\ $ Extending a 1936 result of Shoda for characteristic $0,$ Benjamin Muckenhoupt, a 2nd year graduate student of A. Adrian Albert, proved in the mid fifties that in the matrix algebra $\rm\ \mathbb M_n(F)\ $ over a field $\rm\:F\:$, a matrix $\rm\:M\:$ is a commutator $\rm\ M\: = \: A\:B - B\:A\ $ iff $\rm\:M\:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "55",
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Integrate $\log(x)$ with Riemann sum In a homework problem I am asked to calculate $\int_1^a \log(x) \mathrm dx$ using a Riemann sum. It also says to use $x_k := a^{k/n}$ as steps for the stair functions.
So far I have this:
My step size is $x_k - x_{k-1}$ which can be reduced to $a^{\frac{k-1}{n}} (a^{\frac{1}{n}} -1)... | First, notice that $1-a+a \ln (a)$ can't be the (final) answer. It is an antiderivative of $\ln (a)$, but it is not the antiderivative you are looking for : it does not vanish at $0$. The subtelty is that the Riemann sums approximate the integral of the logarithm between $1$ and $a$, and not between $0$ and $a$.
1) The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Spaces with equal homotopy groups but different homology groups? Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of spaces going the other way around? Are there any obvious wa... | Standard example is $\mathbb RP^2\times S^3$ and $\mathbb RP^3\times S^2$ (they have same homotopy groups since they both have $\pi_1=\mathbb Z/2$ and the universal cover is in both cases $S^2\times S^3$).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Absolute value of Brownian motion I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion.
I am trying to find a bound for $\mathbb{E}[\int_{t=1}^{\infty}R^2_t]$.
Asymptotically $B_t^i$ is between $\sqrt{t}$ and $t$. I also know ... | Since $B_t$ and $\sqrt{t}B_1$ are identically distributed, $\mathrm E(R_t^2)=t^{-1}\mathrm E(R_1^2)$, hence $\mathrm E(R_t^2)\leqslant\mathrm E(R_1^2)$ for every $t\geqslant1$ and it remains to show that $\mathrm E(R_1^2)$ is finite. Now, the density of the distribution of $B_1$ is proportional to $\mathrm e^{-\|x\|^2/... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Mathematics understood through poems? Along with Diophantus mathematics has been represented in form of poems often times. Bhaskara II composes in Lilavati:
Whilst making love a necklace broke.
A row of pearls mislaid.
One sixth fell to the floor.
One fifth upon the bed.
The young woman saved one third of them.
One ten... | Prof. Geoffrey K. Pullum's "Scooping the Loop Snooper: A proof that the Halting Problem is undecidable", in the style of Dr. Seuss.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
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truth table equivalency I am stuck on this question and attempting to answer it makes me feel that its equivalent to searching for a needle in a large pond...
I need help with this, can someone explain how I even attempt to find the solution to this?
Question: Find a logical statement equivalent to $(A \to B) \& \sim ... | I will assume that "|" is NAND operator defined as :
$A | B \Leftrightarrow \lnot(A \land B)$
If it is so then we can write :
$(A \rightarrow B) \land \lnot C \Leftrightarrow (\lnot A \lor B) \land \lnot C \Leftrightarrow (\lnot A \land \lnot C) \lor (B \land \lnot C) \Leftrightarrow$
$\Leftrightarrow \lnot(\lnot A \mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of the next series I'm trying to determine the convergence of this series:
$$\sum \limits_{n=1}^\infty\left(\frac12·\frac34·\frac56·...\frac{2n-3}{2n-2}·\frac{2n-1}{2n}\right)^a$$
I've tried using D'Alambert criteria for solving it.
$$\lim_{n->\infty}\frac{(\frac12·\frac34·\frac56·...\frac{2n-3}{2n-2}·\frac... | Rewrite the summand in terms of factorials like this: $$ \left( \frac{(2n)!}{ 2^{2n} (n!)^2} \right)^a .$$ Applying Stirling's approximation gives $$ \frac{(2n)!}{ 2^{2n} (n!)^2} \sim \frac{1}{\sqrt{\pi n} } $$ so to finish off, apply what you know about the convergence of $ \displaystyle \sum \frac{1}{n^p} $ for vari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating $\prod (\omega^j - \omega^k)$ where $\omega^n=1$. Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate
$$ \prod_{j \ne k} (\omega^j - \omega^k)$$
where the product runs over all $j \ne k$ with $0... | First, note that
$$\prod_{k=1}^{n-1} (1-w^k) = n$$
The proof is that $\prod_{k=1}^{n-1}(x-w^k) = 1+x+x^2+...+x^{n-1}$, then substitute $x=1$.
Now, you can rewrite:
$$\prod_{j\neq k} (w^j-w^k) = \prod_{j=0}^{n-1} \prod_{i=1}^{n-1} (w^j-w^{i+j})$$
$$= \prod_{j=0}^{n-1} w^{j(n-1)} n = n^n w^{\frac{n(n-1)^2}2}$$
If $n$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Essays on the real line? Are there any essays on real numbers (in general?).
Specifically I want to learn more about:
*
*The history of (the system of) numbers;
*their philosophical significance through history;
*any good essays on their use in physics and the problems of modeling a 'physical' line.
Cheers.
I left... | You might try consulting The World of Mathematics, edited by James R. Newman. This is a four-volume compendium of articles on various topics in mathematics. It was published in 1956 so is not exactly cutting-edge, but then again, neither is our understanding of the construction of the real numbers. It contains an essay... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The square of an integer is congruent to 0 or 1 mod 4 This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post.
This problem is from assignment 6. The notes from this lecture can be found here.
a) P... | $$\begin{align}
x^2 \mod 4 &\equiv (x \mod 4)(x \mod 4) \pmod 4 \\
&\equiv \begin{cases}0^2 \mod 4 \\
1^2 \mod 4 \\
2^2 \mod 4 \\
3^2 \mod 4 \end{cases} \\
&\equiv \begin{cases}0 \mod 4 \\
1 \mod 4 \\
4 \mod 4 \\
9 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Morita equivalence of acyclic categories (Crossposted to MathOverflow.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. Suppose that they are Morita equivalent in the sense that the abelian categor... | On MO, Benjamin Steinberg links to a paper of Leroux with a counterexample.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/99784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How can I solve the differential equation $y'+y^{2}=f(x)$? $$y'+y^{2}=f(x)$$
I know how to find endless series solution via endless integral or endless derivatives and power series solution if we know $f(x)$. I also know how to find general solution if we know one particular solution ($y_0$).
I am looking for an exac... | Interesting. In Maple I tried $y'+y^2 = \sin(x)$, and the solution involves Mathieu functions $S, C, S', C'$.
I tried $y'+y^2=x$, and the solution involves Airy functions Ai, Bi.
I tried $y'+y^2=1/x$, and the solution involves Bessel functions $I_0, I_1, K_0, K_1$.
This is a Riccati equation. For more info, in ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If $xy$ is a unit, are $x$ and $y$ units? I know if $x$ and $y$ are units, in say a commutative ring, then $xy$ is a unit with $(xy)^{-1}=y^{-1}x^{-1}$.
But if $xy$ is a unit, does it necessarily follow that $x$ and $y$ are units?
| Yes. Let $z=xy$. If $z$ is a unit with inverse $z^{-1}$, then $x$ is a unit with inverse $yz^{-1}$, and $y$ is a unit with inverse $xz^{-1}$, because
$$x(yz^{-1})=(xy)z^{-1}=zz^{-1}=1$$
$$y(xz^{-1})=(yx)z^{-1}=(xy)z^{-1}=zz^{-1}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/99949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 1
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If $B \ (\supseteq A)$ is a finitely-generated $A$-module, then $B$ is integral over $A$. I'm going through a proof of the statement:
Let $A$ and $B$ be commutative rings.
If $A \subseteq B$ and $B$ is a finitely generated $A$-module, then all $b \in B$ are integral over $A$.
Proof:
Let $\{c_1, ... , c_n\} \subseteq... | Another way to phrase it, slightly different to Georges's answer and comments,
is as follows:
In the first paragraph of the proof, $B$ could be replaced by any f.g. $A$-module $M$, and $b$ could any endomorphism of that $A$-module. What we conclude is that every $\varphi \in End_A(M)$ is integral over $A$.
In particula... | {
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"url": "https://math.stackexchange.com/questions/100124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Restricted Integer Partitions Two Integer Partition Problems
Let $P(n,k,m)$ be the number of partitions of $n$ into $k$ parts with all parts $\leq m$.
So $P(10,3,4) = 2$, i.e., (4,4,2); (4,3,3).
I need help proving the following:
$P(2n,3,n-1) = P(2n-3,3, n-2)$
$P(4n+3, 3, 2n+1) = P(4n,3,2n-1) + n + 1$.
| For the first one: For any partition $2n=a+b+c$ where $a,b,c \leq n-1$, we have a partition $2n-3=(a-1)+(b-1)+(c-1)$ where $a-1,b-1,c-1 \leq n-2$ and vice versa.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/100186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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problem with continuous functions $f,g:\mathbb{R}\longrightarrow\mathbb{R}$
f,g are continuous functions. $\forall q\in\mathbb{Q}$ $f\left(q\right)\leq g\left(q\right)$
I need to prove that $\forall x\in\mathbb{R}$ $f\left(x\right)\leq g\left(x\right)$
| Hint: Note that if $(x_n)_{n\in\mathbb{N}}$ is a convergent sequence of real numbers and $f,g$ are continuous functions, $\lim\limits_{n\to\infty} f(x_n)=f(\lim\limits_{n\to\infty}x_n)$ and $\lim\limits_{n\to\infty} g(x_n)=g(\lim\limits_{n\to\infty}x_n)$, and that for any real number $x$ we have some sequence $(x_n)_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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The meaning of Implication in Logic How do I remember Implication Logic $(P \to Q)$ in simple English?
I read some sentence like
*
*If $P$ then $Q$.
*$P$ only if $Q$.
*$Q$ if $P$.
But I am unable to correlate these sentences with the following logic.
Even though the truth table is very simple, I don't want to re... | I would like to share my own understanding of this.
I like to think of Implication as a Promise rather than Causality which is the natural tendency when you come across it the first time.
Example:
You have a nice kid and you make him the following promise to him:
If you get an A in your exam, then I will buy you a car.... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "11",
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Is there a formula for solving integrals of this form? I was wondering if there was a substitution formula to solve integrals of this form:
$\int f(g(x))g''(x)dx$
| No, not a nice one, anyway. It is worthwhile, I think, to point out that integration rules, such as the usual substitution rule, do not always "solve" ("evaluate" is the proper term) the given integral. The usual substitution rule, for instance, only transforms the integral into another integral which may or may not be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $n\ge 3$, $4^n \nmid 9^n-1$ Could anyone give me a hint to prove the following?
If $n\ge 3$, $4^n \nmid 9^n-1$
| Hint :
Try to prove using induction :
$1.$ $9^3 \not \equiv 1 \pmod {4^3}$
$2.$ suppose : $9^k \not \equiv 1 \pmod {4^k}$
$3.$ $9^k \not \equiv 1 \pmod {4^k} \Rightarrow 9^{k+1} \not \equiv 9 \pmod {4^k}$
So you have to prove :
$ 9^{k+1} \not \equiv 9 \pmod {4^k} \Rightarrow 9^{k+1} \not \equiv 1 \pmod {4^{k+1}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/100393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Every non zero commutative ring with identity has a minimal prime.
Let $A$ be a non zero commutative ring with identity. Show that the set of prime ideals of $A$ has minimal elements with respect to inclusion.
I don´t know how to prove that, I can suppose that the ring is an integral domain, otherwise the ideal $(0)$... | Below is a hint, with further remarks on the structure of the set of prime ideals, from Kaplansky's excellent textbook Commutative Rings. For a recent survey on the poset structure of prime ideals in commutative rings see R & S Wiegand, Prime ideals in Noetherian rings: a survey, in T. Albu, Ring and Module Theory, 20... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $ I would like to show that
$$ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dx=\ln2 $$
What annoys me is that $ x-1 $ is the numerator so the geometric power series is useless.
Any idea?
| This is a classic example of differentiating inside the integral sign.
In particular, let $$J(\alpha)=\int_0^1\frac{x^\alpha-1}{\log(x)}\;dx$$. Then one has that $$\frac{\partial}{\partial\alpha}J(\alpha)=\int_0^1\frac{\partial}{\partial\alpha}\frac{x^\alpha-1}{\log(x)}\;dx=\int_0^1x^\alpha\;dx=\frac{1}{\alpha+1}$$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 3,
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One vs multiple servers - problem Consider the following problem:
We have a simple queueing system with $\lambda%$ - probabilistic intensity of queries per some predefined time interval.
Now, we can arrange the system as a single high-end server ($M/M/1$, which can handle the queries with the intensity of $2\mu$) or as... | You need to specify what you mean by "overall performance", but for most measures the two server system will have better performance. Intuitively, a "complicated" customer, one that has a long service time will shut down the M/M/1 queue but only criple the M/M/2 queue.
If we let the utiliztion be $$\rho=\frac{\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Linear operator's linearity $$f:R^n \to R^3 \ \ \ \ \ \ \ \ \ f(x,y,z)=(x-z,y,az^2)$$
I have to find $n$ and $a$ such that $f$ is a linear operator.
$$x-z=0$$
$$y=0$$
$$az^2=0$$
I found $n$ to be 3.
For $az^2$ to be equal to $0$, either $z$ is $0$ or $a$ is $0$, right? The $z^2$ is confusing me, I don't know from w... | The fact that $n=3$ comes from inspection.
In order for $f:\mathbb{R}^3\to\mathbb{R}^3:(x,y,z)\mapsto(x-z,y,az^2)$ to be a linear operator you need
$$f(\vec{x}+\vec{u})=f(\vec{x})+f(\vec{u}), \quad\text{or}$$
$$\forall \vec{x},\vec{u}\in\mathbb{R}^3:\quad\begin{cases}(x+u)-(z+w)=(x-z)+(u-w) \\ (y+v)=(y)+(v) \\ a(z+w)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Multiplying Infinite Cardinals (by Zero Specifically) On the Wikipedia page on Cardinal Numbers, Cardinal Arithmetic including multiplication is defined. For finite cardinals there is multiplication by zero, but for infinite cardinals only defines multiplication for nonzero cardinals. Is multiplication of an infinite c... | For any cardinal $\kappa$ whatsoever, $0\cdot\kappa=\kappa\cdot 0=0$. This is an immediate consequence of the definition and the fact that for any set $X$, $\varnothing\times X=\varnothing$.
Yes, if one assumes the axiom of choice, the product of two infinite cardinals is simply the larger of them; so is their sum. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Local diffeomorphism from $\mathbb R^2$ onto $S^2$ Is there any local diffeomorphism from $\mathbb R^2$ onto $S^2$?
| First note that there actually are no covering maps $\mathbb{R}^2 \to S^2$. This is because $S^2$ is simply connected and hence is its own universal cover; if there were a covering map $\mathbb{R}^2 \to S^2$, then by the universal property of the universal cover there would be a covering map $S^2 \to \mathbb{R}^2$. B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Hindu calendar (lunar) to Gregorian calendar We have to convert the Hindu calendar (the lunar one) to the Gregorian calendar. According to the (Dutch) Wikipedia (sorry for that, but it has more information than other websites), it is based on he angle between the sun and moon. Now I have many questions about that, such... | I don't have the entire answer but I hope this will at least help a bit you if not more.
Have you looked at http://www.webexhibits.org/calendars/calendar-indian.html
Quoting:
"All astronomical calculations are performed with respect to a Central Station at longitude 82°30’ East, latitude 23°11’ North."
Why do you wish ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
On the set of integer solutions of $x^2+y^2-z^2=-1$. Let
$$
\mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}.
$$
The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication.
It's known that there is only one $\Gamma$-orbits in $\mathcal R$, i.e. $\Gamma \cdot e_3=\math... | Do you know about Frink's paper?
http://www.maa.org/sites/default/files/Orrin_Frink01279.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/101001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Compute: $\int_{0}^{1}\frac{x^4+1}{x^6+1} dx$ I'm trying to compute: $$\int_{0}^{1}\frac{x^4+1}{x^6+1}dx.$$
I tried to change $x^4$ into $t^2$ or $t$, but it didn't work for me.
Any suggestions?
Thanks!
| First substitute $x=\tan\theta$. Simplify the integrand, noticing that $\sec^2\theta$ is a factor of the original denominator. Use the identity connecting $\tan^2\theta$ and $\cos2\theta$ to write the integrand in terms of $\cos^22\theta$. Now the substitution $t=\tan2\theta$ reduces the integral to a standard form, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 2
} |
Showing $C=\bigg\{ \sum_{n=1}^\infty a_n 3^{-n}: a_n=0,2 \bigg\}$ is uncountable Let us define the set
$$C=\bigg\{ \sum_{n=1}^\infty a_n 3^{-n}: a_n=0,2 \bigg\}$$
This is the Cantor set, could anyone help me prove it is uncountable? I've been trying a couple of approaches, for instance assume it is countable, list the ... | Note the "obvious" bijection between $C$ and $P(\mathbb N)$ defined as:
$$f(A)= 2\sum_{n=1}^\infty\frac{\chi_A(n)}{3^n}$$
Where $\chi_A$ is the characteristics function of $A$ ($1$ for $n\in A$, $0$ otherwise).
Suppose that $A\neq B$ and $x=\min (A\Delta B)$, wlog $x\in A$ then $f(A)-f(B)\ge\dfrac2{3^x}>0$.
In the othe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that $u+v$ bisects $u$ and $v$ only if $|u|=|v|$ I want to show that if I have two Euclidian vectors in $\mathbb{R}^n$ than the sum of these two vectors bisects the angle between the two vectors. Said more mathematically.
Let $(u,v) \in \mathbb{R}^n$
Then $\angle(u,v+u) = \angle(u+v,v)$ if and only if $|u|=|v|$
... | The first prove:
Two vectors form a parallelogram ABCD, we know that in a parallelogram diagonals bisect by their point of intersection. It means that segment from A to point of intersections of diagonals is a middle line of triangle ABD. There is a criteria that middle line is bisecting line if and only if the triangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is a linear equation with three variables a plane? In linear algebra, why is the graph of a three variable equation of the form $ax+by+cz+d=0$ a plane? With two variables, it is easy to convince oneself that the graph is a line (using similar triangles, for example). However with three variables, this same techniqu... | Look at the equation as a dot product or inner product:
$$
\left[ \begin{array}{ccc}
a & b & c \end{array} \right]
\left[ \begin{array}{c}
x \\
y \\
z \end{array} \right] = -d.
$$
Then it is clear to see that the point $(x, y, z)$ that satisfies the equation is any point in the plane that is perpendicular to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
terminology: euler form and trigonometric form Am I right, that the following is the so-called trigonometric form of the complex number $c \in \mathbb{C}$?
$|c| \cdot (\cos \alpha + \mathbf{i} \sin \alpha)$
And the following is the Euler form of the very same number, right?
$|c|\cdot \mathbf{e}^{\mathbf{i}\alpha}$
I th... | They are the same, and can also be called "polar coordinates" for the complex number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/101311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding how many terms of the harmonic series must be summed to exceed x? The harmonic series is the sum
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... + 1/n + ...
It is known that this sum diverges, meaning (informally) that the sum is infinite and (more formally) that for any real number x, there there is some number n suc... | The DiGamma function (the derivative of the logarithm of the Gamma function) is directly related to the harmonic numbers: ψ(n) = Hn-1 - γ, where γ is Euler's constant (0.577...).
You can use one of the approximations in the Wikipedia article to compute an approximate value for Hn, and then use that in a standard root f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 0
} |
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