Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
On the behaviour of $\left(1+\frac{1}{n+1}\right)^{n+1}-\left(1+\frac{1}{n}\right)^n$ I have to find the limit : (let $k\in \mathbb{R}$)
$$\lim_{n\to \infty}n^k \left(\Big(1+\frac{1}{n+1}\Big)^{n+1}-\Big(1+\frac{1}{n}\Big)^n \right)=?$$
My Try :
$$\lim_{n\to \infty}\frac{n^k}{\Big(1+\frac{1}{n}\Big)^n} \left(\frac{\B... | $$\lim_{n\to \infty}n^k \left((1+\frac{1}{n+1})^{n+1}-(1+\frac{1}{n})^n \right)=
\lim_{n\to \infty}n^k \left(\frac{e}{2n^2}+O((\frac{1}{n^3})) \right)$$
for n<2 limit is 0, for n=2 limit is e/2, for n>2 limit is infinity
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Area of projected parallelogram onto a plane. Say you have a parallelogram which is defined by the to vectors: $\vec u$, $\vec v$. Prove that the area of its projection on a plane with a perpendicular vector $\vec n$ (where $|\vec n|=1$) is: $E=|(\vec u \times \vec v)\ \vec n|$. Now I know that the area of the origina... | The geometric intuition is that the projected area is equal to the original area multiplied by $\,\cos \theta\,$ where $\,\theta\,$ is the angle between the planes. But $\,\vec u \times \vec v\,$ is a vector along the normal to the plane spanned by$\,(\vec u, \vec v)\,$, so the angle between $\,\vec u \times \vec v\,$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Suppose $\lim _{x\to \infty} f(x) = \infty$. Calculate $\lim_{x\rightarrow \infty} \left(\frac{f(x)}{f(x)+1}\right)^{f(x)}$ Suppose $\lim _{x\to \infty} f(x) = \infty$. Calculate: $\lim_{x\rightarrow
\infty} \left(\dfrac{f(x)}{f(x)+1}\right)^{f(x)}$
I figured the limit is $\dfrac{1}{e}$, but I have to prove it using ... | Notice that $$\left(\frac{f(x)}{f(x)+1} \right)^{f(x)} = \frac{1}{\left(1+\frac{1}{f(x)} \right)^{f(x)}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Dedekind domain and converse of Krull's principal ideal theorem It is clear that Dedekind domain's all primes other than $(0)$ are maximal. Its dimension must be $\leq 1$. Suppose this domain is not a field. So $\dim=1$.
So all maximal must be minimal over a principal element.
Here is Krull's principal ideal theorem ... | The ideal $m$ is minimal over itself, but this does not contradict the theorem: it simply says that $m$ is minimal over a principal ideal. And indeed, since $(3) = (3, 1 + \sqrt{-5})(3, 2 + \sqrt{-5})$, $m$ is minimal over the principal ideal $(3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is the mapping $T : x \mapsto \int_{0}^{\bullet} \tau^{-1/2}x(\tau)\,d\tau$ uniformly continuous? Consider $X = C([0,1])$ with its natural metric induced by $\| \cdot \|_{\sup}$ and $Y = C([0,1])$ with the metric $d_1(x,y) = \int^1_0 |x(t)-y(t)| \, dt$. Let
$$T: X\to Y : x(t) \mapsto y(t) = \int_0^t \frac{1}{\sqrt \ta... | I believe this is uniformly continuous. We can estimate
$$d(T(x(t)), \ T(y(t))) = \int_0^1\bigg|\int_0^t \frac{1}{\sqrt \tau}\big[x(\tau) - y(\tau) \big] d\tau \bigg | dt \leq \int_0^1\int_0^t \bigg|\frac{1}{\sqrt \tau} \bigg| \ \delta \ d\tau \ dt$$
Where we assume that the distance between $x(t)$ and $y(t)$ is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Morphisms to categorical product completely determined by composition with canonical projections? Consider the diagram:
$$\begin{array}{} &&&& A_1 \\ &&& \overset{\pi_1}\nearrow \\ X & \overset f{\underset g \rightrightarrows} & A_1 \times A_2 \\ &&& \underset{\pi_2}\searrow \\ &&&& A_2 \end{array}$$
where $X$ is any a... | Yes, this is exactly the content of the universal property of the categorical product.
*
*For every pair of maps $f_1,f_2\colon X\to A_1,A_2$ there is a map $f\colon X\to A_1\times A_2$ satisfying $\pi_1\circ f=f_1$ and $\pi_2\circ f=f_2$
*If there are two maps $f,g\colon X\to A_1\times A_2$ satisfying $\pi_1\circ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Determining if points lie on a vertical plane This is slightly related to a recent post of mine. If I am dealing with three-dimensional Cartesian coordinates and have at least three points, how can I easily tell if the best-fit plane to the data points is vertical (or near-vertical) in the $z$ dimension?
My concern is ... | Consider three points with coordinates $\vec v_i=(x_i,y_i,z_i)$ with $i=1,2,3$. The obvious step for your problem would be to find the normal to this plane. You can do that by using the cross product $(\vec v_1 -\vec v_0)\times (\vec v_1 -\vec v_0)$. You can now look at the $z$ component of this vector product. It the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Almost surely convergence related with independent suppose that $ x_1 , x_2 $ are independent observation of a random variable x
prove that $x_1 +x_2$ has the same distribution as $x \iff x=0 $ a.s
please please help me for this problem
| If $X_{1}+X_{2}$ has the same distribution as $X_{1}$, then
\begin{align*}
Var[X_1]
&= Var[X_1+X_2] \\
&= Var[X_1]+Var[X_2]
& \text{Independence}
\end{align*}
Conclude that $Var[X_2]=0$. Since the variables are identically distributed,
$X_1=X_2=k$. Since $X_{1}+X_{2}$ has the same distribution as $X_{1}$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\big(\cot \frac{\pi}{18}-3\cot \frac{\pi}{6}\big)\cdot \big(\csc \frac{\pi}{9}+2\cot \frac{\pi}{9}\big)$
Finding value of $\displaystyle \bigg(\cot \frac{\pi}{18}-3\cot \frac{\pi}{6}\bigg)\cdot \bigg(\csc \frac{\pi}{9}+2\cot \frac{\pi}{9}\bigg)$
Try: $$\cot \frac{\pi}{18}\csc \frac{\pi}{9}-3\sqrt{3}\csc \... | $$\cot x-3\cot3x=\dfrac{\cos x\sin3x-3(\cos3x\sin x)}{\sin x\sin3x}$$
Again,
\begin{align}
2\cos x\sin3x-3(2\cos3x\sin x)
&=\sin4x+\sin2x-3(\sin4x-\sin2x)\\[4px]
&=4\sin2x-2\sin4x \\[4px]
&=4\sin2x(1-\cos2x)\\[4px]
&=4\sin2x(2\sin^2x)
\end{align}
Finally,
\begin{align}
\csc2x+2\cot2x
&=\dfrac{1+2\cos2x}{\sin2x}\\[4px]
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Problem about convergence of operator norm and compactness of an operator. Problem-$1$ Given the sequence of continuous linear operators $T_n : l^2 \to l^2$ defined by $$T_n(x) = (0, 0, \ldots, x_{n+1}, x_{n+2}, \ldots)$$ for every $x \in l^2$. Then for every $x \neq 0$ in $l^2$ i want to check whether $\|T_n\|$ and $\... | What you did for problem 1 is correct (maybe it would be better to say that for any positive $\varepsilon$, there exists an $N$ such that...). Your computation shows that $\left\lVert T_n\right\rVert\leqslant 1$. Considering $x$ as the vector whose $(n+1)$th coordinate is $1$ and all the others are zero, you can show t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Relation between integral, gamma function, elliptic integral, and AGM The integral $\displaystyle\int\limits_0^{\infty}\frac {\mathrm dx}{\sqrt{1+x^4}}$ is equal to $\displaystyle \frac{\Gamma \left(\frac{1}{4}\right)^2}{4 \sqrt{\pi }}$.
It is calculated or verified with a computer algebra system that $\displaystyle \f... | Let $t=\frac{1}{{1+x^4}}$ and then
$$ dx=-\frac14(1-t)^{-3/4}t^{5/4} $$
So
\begin{eqnarray}
&&\int\limits_0^{\infty}\frac {\mathrm dx}{\sqrt{1+x^4}}\\
&=&\frac14\int\limits_0^{1}(1-t)^{-3/4}t^{1/4}\mathrm dx\\
&=&\frac14B(\frac14,\frac54)\\
&=&\sqrt{\frac{\pi}{2}}\frac{\Gamma(\frac54)}{\Gamma(\frac34)}\\
&=&\frac{\Gamm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Pascal's rule : I don't understand why we can do this... So I understand that they are equal, but I can't get my head around the wording. Why can we choose "1 special element, and $k − 1$ from the remaining $n − 1$, or choose all $k$ from the $n − 1$"?
To choose $k$ elements from $n$ we can either choose 1 special el... | To see the correspondence, note that the number $C_k^n = \binom{n}{k}$ is the number of a teams we can construct of size $k$ with $n$ players. Now assign one of the players as "special". This player can be either included or not included in the team, so if it's included there are $C_{k-1}^{n-1} = \binom{n-1}{k-1}$ ways... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 2
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Determine $\lim\limits_{x\to 0, x\neq 0}\frac{e^{\sin(x)}-1}{\sin(2x)}=\frac{1}{2}$ without using L'Hospital How to prove that
$$\lim\limits_{x\to 0, x\neq 0}\frac{e^{\sin(x)}-1}{\sin(2x)}=\frac{1}{2}$$
without using L'Hospital?
Using L'Hospital, it's quite easy. But without, I don't get this. I tried different approa... | Hint Let $f(x)=e^{sin(x)}-1$. Then
$$\lim_{x \to 0} \frac{f(x)-f(0)}{x-0}=f'(0)$$
Also,
$$\lim_{x \to 0} \frac{x}{\sin(2x)}$$ can be easily be deduced from the fundamental trigonometric limit.
Alternately canceling $\sin(x)$ you get
$$\frac{e^{\sin(x)}-1}{\sin(2x)}=\frac{\sin(x)+\sum\limits_{k=2}^\infty\frac{\sin(x)^k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 12,
"answer_id": 5
} |
Convergence/absolute convergence of $\sum_{n=1}^\infty \left(\sin \frac{1}{2n} - \sin \frac{1}{2n+1}\right)$
Does the following sum converge? Does it converge absolutely?
$$\sum_{n=1}^\infty \left(\sin \frac{1}{2n} - \sin
\frac{1}{2n+1}\right)$$
I promise this is the last one for today:
Using Simpson's rules:
$$\sum... | $$|\sin(\frac{1}{2n})-\sin(\frac{1}{2n+1})| = |cos(\xi)(\frac{1}{2n}-\frac{1}{2n+1})|$$
The above is followed by mean value theorem, then
$$|\sin(\frac{1}{2n})-\sin(\frac{1}{2n+1})| \leq \frac{1}{2n}-\frac{1}{2n+1}$$
So the series is absolutely convergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
$z ≤ x + y$ implies $z/(1 + z) ≤ x/(1 + x) + y/(1 + y)$
Suppose $x,y,z $ be nonnegative reals. Show that $z ≤ x + y\implies z/(1 + z) ≤ x/(1 + x) + y/(1 + y)$.
My Proof: If $z=0$, then we are done. So, suppose $z>0$. Since $z ≤ x + y$, and $x$ and $y$ nonnegative, $z ≤ x + y+2xy+xyz$, which leads to $z(1+x+y+xy)\le (... | Yes your proof is correct. I have provided another proof as follows. Notice that $f(t) = \dfrac{t}{1+t}$ is an increasing function on its domain. Therefore if $x$, $y$, and $z$ are non-negative real numbers and $z$ is less than or equal to $x+y$ then f(z) is less than or equal to f(x+y). That is $$\dfrac{z}{1+z} <= \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Isometries of $\mathbb R^2$ with maximum norm. I have been asked to prove that all isometries of $\mathbb R^2$ with suprenum norm
$$ \|(x,y)\|_{\infty}=\max \{ |x|,|y| \} $$
Are $T(x,y)=(ax+by, cx+dy)$ where
$b=c=0$ and $a, d \in \{1, -1\}$ or
$ a=d=0$ and $b, c \in \{1, -1\}$.
It is obvious that these are isometries.... | The concept of extreme points helps here, even though it does not involve the norm itself. The definition implies that if $p$ is an extreme point of a set $K$, and $T$ is an invertible linear map, then $T(p)$ is an extreme point of $T(K)$ .
Consider the closed unit ball of this space, which is a square $Q$. Its extreme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integrals of form $\int^{∞}_{-∞} x^{2k} \exp(-b^2 (x+x_0)^2)dx$, $k=1,2$. Please can anybody help me to solve these integrals:
$$\int^{∞}_{-∞} x^2 \exp(-b^2 (x+x_0)^2)dx\,,\,\,\int^{+∞}_{-∞} x^4 \exp(-b^2 (x+x_0)^2)dx\;\;?$$
| One may start with the gaussian integral
$$
\int^{∞}_{-∞} \exp(-b^2 u^2)du=\frac{\sqrt{\pi}}{b},\qquad b>0,
$$ getting, by differentiation with respect to the parameter $b$,
$$
\int^{∞}_{-∞} u^2\exp(-b^2 u^2)du=\frac{\sqrt{\pi}}{2b^3},
$$$$
\int^{∞}_{-∞} u^4\exp(-b^2 u^2)dx=\frac{\sqrt{\pi}}{4b^4}.
$$ Then, by the chan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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What does homomorphism mean in the GLOVE paper? It is mentioned in the GloVe: Global Vectors for Word Representation. It says:
where $w_i$, $w_j$ and $\tilde{w}_k$ are all word vectors and $F$ is just an unknown function. The author then assumes $F$ as $exp$. What is homomorphism exactly? Why equation (4) can make $F... | copying verbatim from Wikipedia
A homomorphism is a map between two algebraic structures of the same
type (that is of the same name), that preserves the operations of the
structures. This means a map ${\displaystyle f:A\to B}$ between two
sets $A$, $B$ equipped with the same structure such that, if $∗$ is an
o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$\lim_\limits{x \to 0}(x\sec x)=0$? $$\lim_{x \to 0}(x\sec x)$$
So putting in $x=0$ you get the answer $0$.
$$\lim_{x \to 0}(x\sec x)=0$$
My question is is this a correct way to solve?
edit : So from the answers below, I've understood that if a function is continuous, then $\lim_{x \to a}f(x)=f(a)$
But how do you figur... | Yes. For continuous functions, you can just plug in the value, because you can switch a limit and a continuous function.
I.e., if $g$ is continuous, then $$\lim_{x \to a} g(f(x)) = g \left(\lim_{x\to a}f(x)\right)$$
provided the limit on the right exists and gives a value on which $g$ is defined.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Need a help in finding the inverse of an operator . The question and part of its answer is given as follows:
13. Let $K$ be an operator of a finite rank on a Hilbert space $H$. For $\varphi \in H$,
$$ K\varphi = \sum_{j=1}^{n} \langle \varphi, \varphi_j\rangle\psi_j. $$
Suppose $\psi_j \in \operatorname{sp}\{ \var... | Your hypotheses on $\{\psi_1,\dotsc,\psi_n\}$ and $\{\varphi_1,\dotsc,\phi_n\}$ make an appeal to that theorem completely and utterly unnecessary. Since $\{\psi_1,\dotsc,\psi_n\} \subset \{\varphi_1,\dotsc,\phi_n\}^\perp$, it immediately follows that $K^2 = 0$. What, then, is $I^2 - \alpha^2 K^2$, and why should this s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Limit of recurrence sequence I have to find a limit (or prove it doesn't exist) for the following recurrence sequence.
$a_1 = 2; a_{n+1} = \frac{1}{2}(a_n + \frac{2}{a_n})$
Now I know, in order to find the limit, I first need to prove that the sequence is monotonic and bounded. I've made a partial table of values and c... | one prove easy that $$a_n>0$$ for all $n$ then we have by $AM-GM$
$$a_{n+1}=\frac{1}{2}\left(a_n+\frac{2}{a_n}\right)\geq \sqrt{a_n\cdot \frac{2}{a_n}}=\sqrt{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Equivalent formulations of Baire's Lemma / Theorem I use the following notation $int_Z(A)$ for the interior of $A \subset Z$ in a metric space $Z$.
Now I have the following problem understanding a proof:
Let $(M,d)$ be complete. Then we have that statement i) implies ii).
i) If $M= \bigcup_{j=0}^{\infty}A_j$ with $A_j... | Let $p\in int_X(X\cap A_{j_0})=S\cap X$ where $S$ is open in $M$. Let $b>0$ such that $B_b(p)\subset S.$
Since $p\in X=\overline {B_r(x)}$ we may take some $q\in B_b(p)\cap B_r(x).$
Let $c>0$ be small enough that $B_c(q)\subset B_b(p)$ and $B_c(q)\subset B_r(x).$
Then $B_c(q)\subset B_b(p)\subset S$ and $B_c(q)\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find: $\lim_{x\to\infty} \frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$
Find: $\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}.$
Question from a book on preparation for math contests. All the tricks I know to solve this limit are not working. Wolfram Alpha struggled to find $1$ as the solut... | A fun overkill: it is well known (at least among Ramanujan supporters) that for any $x>1$ we have
$$ \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} = \tfrac{1}{2}+\sqrt{x+\tfrac{1}{4}} $$
hence $\frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$ is bounded between $1$ and $\frac{\sqrt{x}}{\sqrt{x+\frac{1}{4}}+\frac{1}{2}}$, whos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 4
} |
Antisymmetry and Totality implies Reflexivity One can check that antisymmetry and totality imply re
exivity. Thus,
a totally ordered set is equivalent to a partially ordered set in which the binary
relation is total.
I am reading Khovanov's notes for Representation theory of finite groups, and confused by this statemen... | Let $X$ be a set and $R \subseteq X \times X$ be a binary relation on $X$. We say that:
*
*$R$ is reflexive if $x \, R \, x$ for all $x \in X$;
*$R$ is antisymmetric if, for all $x, y \in X$, if $x \, R \, y$ and $y \, R \, x$ then $x = y$;
*$R$ is total if, for all $x, y, \in X$, either $x \, R \, y$ or $y \, R \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Constant of integration change So, sometimes the constant of integration changes, and it confuses me a bit when and why it does. So for example, we have a simple antiderivative such as $$\int \frac{1}{x} dx $$ and we know that the result is $$\log|x| + C$$ and the domain is $$x\in\mathbb R \backslash \{0\} $$ If we wan... | Here is a 1-line compactification of Βασίλης Μάρκος' answer $\ddot\smile$
It can be shown that on every sub-interval of the domain of the original function, any two anti-derivatives differ by an additive constant, but constants for different sub-intervals can very well be different.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
} |
The identification $(\pi^{-1}\mathcal{G})_p \to \mathcal{G}_{\pi(p)} $ induces a continuous map between the "space of sections" Let $\pi:X\to Y$ be a continuous map and $\mathcal{G}$ be a sheaf on $Y$, then there is a natural isomorphism $f_p:(\pi^{-1}\mathcal{G})_p \to \mathcal{G}_{\pi(p)} $ (by using the adjunction o... | Yes it is. Moreover, if we let $\mathcal E=\coprod_{q\in Y}\mathcal{G}_q$ be the etale space of $\mathcal G$ and $\mathcal{E}'
=\coprod_{p\in X}\mathcal{G}_{\pi (p)}$ be the etale space of $\pi^{-1}\mathcal G$
then we have a commutative square
$\require{AMScd}$
\begin{CD}
\mathcal{E'} @>f>> \mathcal{E}\\
@V V V @VV V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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A question about AP.
How is the encircled step justifiable? According to my knowledge I can substitute m=any variable but how can I substitute m=2m-1, isn't it the same as assuming m=1?
| The question itself is confusing. What it means is that the condition is true for any $m,n$ you care to choose, rather than a specific $m,n$.
Given that it is true for any $m,n$ it is somewhat confusing to use the same symbols in the suggested answer, as has been done.
Maybe clearer to say that if it is true for any $m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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The value of $f'(\sqrt{\pi})+g'(\sqrt{\pi}).$ Let $f(x)=(\int_{0}^{x}e^{-t^{2}}dt)^{2}$ and $g(x)=\int_{0}^{1}\frac{e^{-x^{2(1+t^{2})}}}{1+t^{2}}dt.$ Then what is the value of $f'(\sqrt{\pi})+g'(\sqrt{\pi})?$ According to me $g'(\sqrt{\pi})$ is equal to zero. But i dont't know how to find $f'(\sqrt{\pi})$. I am thinkin... |
I dont't know how to find $f'(\sqrt{\pi}).$
By applying the chain rule to
$$
f(x)=\left(\int_{0}^{x}e^{-t^{2}}dt\right)^{2}
$$ one gets
$$
f'(x)=2e^{-x^{2}}\int_{0}^{x}e^{-t^{2}}dt
$$ putting $x=\sqrt{\pi}$ gives a result in terms of the error function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Circular permutation on seating There are 21 people ,15 boys and 6 girls.how many ways are there to seat at least 2 boys between any two adjacent girls in a round table?.
I get my answer 708480.i m wrong ,i think.please help me.
| Distinguishing only by gender, with the usual assumption of unnumbered seating since it is not explicitly specified otherwise, the problem simplifies to placing $15$ green marbles and $6$ red marbles on an unnumbered circle with the given stipulations.
The $6$ red marbles in a circle will have $6$ gaps or "compartments... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Circles intersecting at two points orthogonally. I am finding the following much harder than it probably is!
If a circle $A$ intersects the circle $B$ at two points orthogonally, then why can't $A$ pass through the centre of B?
| Let $X $ and $Y $ be the centres of $A $ and $ B $, respectively, and let $Z $ be one of the two points where those circles intersect each other. The triangle $\triangle XZY$ is a right-angled triangle with hypotenuse $XY $, thus $XY $ is the longest side of the triangle: $XY\gt XZ $ and, since $XZ $ is the radius of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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find the limit of this function? Evaluate $$\lim_{n\to \infty}\frac{1}{(n!)^{1/n}}={?}$$
My try:
$$\lim_{n\to \infty}\frac{1}{(n!)^{1/n}}=\exp{\lim_{n\to \infty}\frac{-\ln(n!)}{n}}$$
$$=\exp{\lim_{n\to \infty}\frac{-\ln(1\times2\times3\ldots(n-1)\times n)}{n}}$$
$$=\exp{\lim_{n\to \infty}\frac{-(1+2+3+\ldots+(n-1)+n)}... | You were almost there! Remember that: $$1 + 2 + \cdots + n = \frac{n(n+1)}{2}.$$ If you use this, then you will see that the expression inside $\operatorname{exp}$ actually goes to $- \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove the geodesic on 2-sphere is the great circle I want to use the Killing vector fields to prove the geodesic on the sphere is the great circle. First of all, the given metric is $$ds^2=d\theta^{2}+\sin^2\theta d\phi^{2},$$ where I set the radius to be $1$. By Killing equation, $$\nabla_\mu K_\nu+\nabla_\nu K_\mu =0... | This should probably be a comment, but I do not yet have the reputation to make a comment. However, I believe your final form of the solution is incorrect. A great circle does not pass through the origin of a sphere: the center of the circle does, but the center of the circle does not belong to the curve. If $X$ and $Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Plotting $f(x) = x\lfloor 1/x \rfloor$ I want to plot $f(x) = x\lfloor 1/x \rfloor$ near the point zero for finding its limit but I can't choose proper intervals and plot it .
| I switched to $1/x$ for large positive $x,$ picture gives better idea this way. I would have preferred using a different letter, say $t = 1/x,$ but I do not yet know how to do that. So, think of the graphs as indicating that the limit as $t \rightarrow +\infty \; \; \; \lfloor t \rfloor /t = 1,$ and
$t \rightarrow -\i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple. If $G$ is a group of order $250,000 = 2^4 5^6$, show that $G$ is not simple.
By the Sylow theorem, we have that the number of $2$-sylow subgroups of $G$ $n_2$ satisfy that
$$
n_2 \equiv1\mod2\mbox{ and } n_2\mid5^6
$$
Similarly for $n_5$ we h... | As you state, $n_5$ is either $1$ or $16$. If it's the former, we are done. If it is $16$, then there is a homomorphism from $G\to S_{16}$ given by the fact that $G$ acts on the $5$-sylow subgroups by conjugation. But notice that the prime factorization of $|S_{16}|=16!$ contains exactly three copies of $5$ (coming fro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Will assuming the existence of a solution ever lead to a contradiction? I'm reading Manfredo Do Carmo's differential geometry book. In section 1-7, he discusses the "Isoperimetric Inequality" which is related to the question of what 2-dimensional shape maximizes the enclosed area for a closed curve of constant length. ... | A famous example is the existence of a solution for $$p^2 = 2q^2, p,q \in \mathbb Z$$
and $p,q$ share no common prime factors.
It is a rephrasing of a classical proof that $\sqrt{2}$ is irrational by assuming it can be written $\sqrt{2} = p/q$ for two such integers.
One comes to the conclusion that a square integer mus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "71",
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"answer_id": 10
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Intuition behind logarithm change of base I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplication and thereby also the division properties very nicely:
Suppose you have a table of powers of 2, which looks like this: (after revision)
... | Intuition is always tricky to get across, but I can try.
$\log_bx$, as you noted, tells you how many $b$s you need to multiply together to get $x$. Now if you need $\log_ba$ number of $b$s to multiply to get $a$, and you need $\log_ax$ number of $a$s to multiply to get $x$, we can "expand" each of those $a$s into a num... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Change of variable in Elliptic Curve using Maple I used Maple to get the change of variable for the quartic
v^2 = p^4 - 2p^3 + 5p^2 + 8p + 4
This is the output :
In other words, from the output I obtained from Maple:
x^3-(121/3)x-1690/27+y^2
x=-(1/3)*(5*p^2+24*p-12*v+24)/p^2
y=-(4*(p^3-5*p^2+2*p*v-12*p+4*v-8))/p^... | I don't understand why you say that the the substitutions produced by the Weierstrassform command do not satisfy the elliptic curve.
The entry k[1] is,
x^3 - (121/3)*x - 1690/27 + y^2
not,
x^3 - (121/3)*x - 1690/27 - y^2
Let's do some substitutions, using those results given by the Weierstrassform command.
restart;
f... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving that the function $f(x) = \int_0^\infty \cos (w^3/3 - x w ) d w$ satisfies the equation $f'' + x f = 0$ It is assumed that $x$ is real.
Formally, we have
$$ f'' = \int_0^\infty -\cos (w^3/3 - x w ) w^2 d w , $$
and hence
$$f'' + x f = \int_0^\infty \cos (w^3/3 - x w ) (-w^2 + x ) d w \\ = -\int_0^\infty... | Your derivation is fine and leads to an ambiguous form, as you state. To disambiguate this, consider the following.
Your integral is
$$
-\int_0^\infty \cos (w^3/3 - x w ) d(w^3/3 - x w )
$$
If you make the transformation $t = w^3/3 - x w$ you have, for any given (finite) $x$,
$$
-\int_0^\infty \cos (t) d t
$$
whi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that $f(x) = x^7 + x^5 + x^3 + x$ is bijective I want to show that the real polynomial function $f: \mathbb R \to \mathbb R, f(x) = x^7 + x^5 + x^3 + x$ is bijective. I want to show this without using the inverse or the derivative.
I'm struggling to prove injectivity, because I see no easy way to arrive at $x = y$... | Hint: This is an odd function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
find $a_1+a_3+a_5+\cdots+a_{37}+a_{39}$
Let $(1+x-2x^2)^{20} = \sum_{r=0}^{40} a_r x^r.\;\;$ Find
$$a_1+a_3+a_5+\cdots+a_{37}+a_{39}$$
I mean, at first, I don't have any idea how this can be solved. I tried factorizing the LHS: $((2x-1)(1-x))^{20}$ and then using binomial theorem, I get something like:
$$\Biggl(\b... | thanks guys, i have solved it too.
$$(1+x-2x^2)^{20} = \sum_{r=0}^{40} a_r x^r$$
putting $x = 1$
$$\sum_{r=0}^{40} a_r=0\;\;\;\;\;\;\;\;\;\;\;\;\;-(1)$$
putting $x = -1$
$$-a_1+a_2-a_3+a_4-a_5+\cdots+a_{40}=2^{20}\;\;\;\;\;\;\;\;\;\;\;\;\;-(2)$$
calculating $(1)-(2)$
$$a_1+a_3+a_5+a_7+\cdots+a_{39}=-2^{19}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
linearly independent elements of W I want to ask about the number of linearly independant elements of
$W=(u_{1},u_{2},...,u_{m},u_{1}-u_{2},u_{2}-u_{3},...,u_{m-1}-u_{m},u_{m}-u_{1})$.
I say that it is $m$ and i need someone to confirm or not that.
Thanks
| The answer depends on the number of linearly independent vectors among $u_1, \ldots, u_m$.
Notice that $$\operatorname{span}\{u_1, u_2, \ldots, u_m, u_1 - u_2, u_2 -
u_3, \ldots, u_{m-1} - u_m, u_m - u_1\} = \operatorname{span}\{u_1, u_2, \ldots, u_m\}$$
because all $u_i - u_{i+1}$ are linear combinations of $u_1, u_2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Spectral radius is not matrix norm. I have seen an example of matrix
$$A = \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}$$
whose spectral radius is zero therefore the spectral radius is not matrix norm. Why the spectral radius is not matrix norm in this case Is it possible that $\|A\|=\epsilon$?
| From this Wikipedia page, the spectral norm of a matrix $A\in\mathbb{C}^{n\times n}$ is defined as
$$\rho\left(A\right)=\max_{1\leq i\leq n}\left\{\left|\lambda_{i}\right|\right\}$$
where the $\lambda_{i}$'s are the eigenvalues of the matrix. In your case
$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}$$
is triangular, so the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given $x_1 := \sqrt{2}$ and $x_{n+1} :=\sqrt{2x_n} $, prove $\sqrt{2} ≤ x_n ≤ 2$
Let $x_1 := \sqrt{2}$ and $x_{n+1} :=\sqrt{2x_n} $ for all $n \in \mathbb{N}$.
By using proof by induction:
(i) Prove that $\sqrt{2} ≤ x_n ≤ 2$ for all $n \in \mathbb{N}$.
(ii) Prove that $x_n ≤ x_{n+1}$ for all $n \in \mathbb{N}$.
For... | From
$$\sqrt2<x_n<2$$ you can deduce
$$\sqrt{2\sqrt2}<\sqrt{2x_n}<2$$
and obviously
$$\sqrt{2}<\sqrt{2x_n}<2,$$ which is nothing but
$$\sqrt{2}<x_{n+1}<2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Is there formulas for $\sum_{g\in G}|\operatorname{Fix}(g)|^n$? I saw a problem in Show $r(x)|G|=\sum_{g\in G}|Fix(g)|^2$ where $(G,X)$ is a transitive group and $r(x)$=#{different orbits of X under $Stab(x)$}
Let $(G,X)$ be a transitive group action, that is, for any $x,y\in X$,there exists an element $g\in G$ s.t. $g... | $|\text{Fix}(g)|^n$ is the number of fixed points of $G$ acting diagonally on $X^n$. By Burnside's lemma, it follows that
$$\frac{1}{|G|} \sum_{g \in G} |\text{Fix}(g)|^n$$
is the number of orbits of $G$ acting on $X^n$.
If $G$ acts transitively on $X$ then we can count the number of orbits of $X^2$ as follows. First, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Why is this combination of nearest-integer functions --- surprisingly --- continuous? Alright, I didn't know the best way to formulate my question. Basically, whilst doing some physics research, I naturally came upon the function
$$ f(x) = 2x[x] - [x]^2 $$
where I use $[x]$ as notation for the `nearest-integer function... | Let $g(x,y)$ be any continuous function such that $g(x,-\frac12)=g(x,\frac12)$. Then $f(x)=g(x,x-[x])$ is continuous.
In particular, your function is given by $g(x,y)=x^2-y^2$. Consequently, we can write $f(x)=x^2 - (x-[x])^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 6,
"answer_id": 1
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How to estimate a limit value of ordinary differential equations $$\frac{dx}{dt}=(x-y)(1-x^2-y^2)\\
\frac{dy}{dt}=(x+y)(1-x^2-y^2)$$
Initial condition $x(0),y(0)$ are nonzero real numbers.
How to estimate the solution $x(t)$ as $t \to \infty$?
| Note that if $(x(0),y(0))= (0,0)$ or $x^2(0)+y^2(0) = 1$, then
$(x(t),y(t))=(x(0),y(0))$ for all $t$.
Let $s(t) = x^2(t)+y^2(t)$, then
$\dot{s} = f(s)=2(1-s)s$.
Note that $f(0)=f(1) = 0$, if $s \in [0,1]$ then $f(s) \ge0$ and
if $s \ge 1$ then $f(s) \le 0$.
In particular, if $s(0) >0$ then $\lim_{t \to \infty} s(t) = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proper subset of the set of irrationals such that it is countable and dense in $\Bbb R$ We know that $\Bbb R$ is separable i.e. it contains a dense subset which is countable. We have $\Bbb Q$ and ${\Bbb R} - {\Bbb Q}$ to be dense subsets respectively countable and uncountable. I was looking for a countable dense subse... | You may also pick a countable dense set whose elements are all trascendental (and in such a way that this set is not the set of all rational multiples of some fixed transcendental, which would be the right boring answer). As algebraic numbers are countable, the set $T$ of all transcendentals is dense in $\mathbb{R}$. "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2584004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Finding the limit of the sequence $a_n\cdot a_{n+1}=n,\,n=1,2,3,\cdots.$ Let $(a_n)_{n>=1}$ be a sequence of real numbers defined by the below recurrence relation:
$$a_n\cdot a_{n+1}=n,\quad n=1,2,3,\cdots.$$
Prove that $\lim_{n\to \infty}a_n=+\infty.$
Edit: $a_1>0$
| Since $a_n\cdot a_{n+1}=n$ and $a_{n+1}\cdot a_{n+2}=n+1$,
$$
\frac{a_{n+2}}{a_n}=\frac{n+1}n\ge\sqrt{\frac{n+2}n}
$$
Therefore, for even $n$
$$
\begin{align}
a_n
&\ge a_2\sqrt{\frac n2}\\
&=\frac1{a_1}\sqrt{\frac n2}
\end{align}
$$
and for odd $n$
$$
a_n\ge a_1\sqrt{n}
$$
Thus,
$$
\bbox[5px,border:2px solid #C0A000]{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2584120",
"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
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Why Do Mersenne Primes Only Occur at Terms of Prime Index? A Mersenne prime is a prime of the form $2^n-1$.
Only when $n$ is a prime itself is there a chance that $2^n-1$ is a Mersenne primes. The largest primes discovered are almost always Mersenne primes. Some of the more known Mersenne primes are $3, 7, 31, 127$, e.... | Because if $m=kl$, with $k,l>1$, then\begin{align}2^n-1&=2^{kl}-1\\&=(2^k)^l-1^l\\&=(2^k-1)\bigl((2^k)^{l-1}+(2^k)^{l-2}+\cdots+1\bigr).\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2584200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Discuss the monotonicity of $\sqrt[n]{n!}$. It seems that $\sqrt[n]{n!}$ is increasing because it turns to $+\infty$ as $n\to+\infty$. How to prove it?
| It is enough to show that $\frac{1}{n}\log(n!)=\frac{1}{n}\sum_{k=1}^{n}\log(k)$ is increasing.
Since $\log(x)$ is concave on $\mathbb{R}^+$, this is a simple consequence of Karamata's inequality:
$$ \tfrac{1}{n}\cdot\log(1)+\tfrac{1}{n}\cdot\log(2)+\ldots+\tfrac{1}{n}\cdot\log(n)+0\cdot \log(n+1)\\ \leq \tfrac{1}{n+1}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simplify this equation if $x$ is negative I am trying to simplify $7 \cdot\sqrt{5x}\cdot\sqrt{180x^5}$, given that $x$ is negative. The answer is $-210x^3$, but I am getting $210x^3$. Below is my reasoning:
For a $ k > 0 $, let $ k = - x $. Then, $7\cdot\sqrt{5x}\cdot\sqrt{180x^5}$ = $7\cdot5\cdot6\cdot i\sqrt{k} \cdot... | $$\text{$7 \cdot\sqrt{5x}\cdot\sqrt{180x^5} \ $ where $ \ x<0$}$$
Lets let $x=-1$ to see what we should expect.
\begin{align}
\left. \left( 7 \cdot\sqrt{5x}\cdot\sqrt{180x^5}\right)\right|_{x=-1}
&= 7 \cdot\sqrt{-5}\cdot\sqrt{-180} \\
&= 7 \cdot i \cdot\sqrt{5}\cdot i \cdot \sqrt{180} \\
&= -7 \cdot \sqrt{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sum of exactly n perfect square divisors I'm doing a Number Theory question, and if someone could offer a hint, that would be greatly appreciated;
The question is: Find the sum of the perfect square divisors of the smallest integer with exactly 6 perfect square divisors.
My reasonings:
*
*My method so far has been ... | Conversation in the comments is correct, but I would suggest the following approach. First we find the lowest number with exactly $6$ divisors, and then we square that number. This squared number will then have exactly $6$ square divisors.
To find the number, we use the number of divisors formula
$$d\left(\prod p_i^{a_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the density of a brownian bridge $X_t=B_t-tB_1$? The original question is:
For $t\in [0,1]$, we define $X_t=B_t-tB_1$, where $\{B_t:t\geq 0\}$ is a standard Brownian motion. Find the density of $X_t$ .
After reading several resources, I think $X_t$ is a normal distribution. However, since most of the books conc... | Yes, it is normal. Brownian motion is a Guassian process, which means that its finite dimensional distributions are all multivariate normal, and in particular any linear combination of $B_{t_1},\ldots,B_{t_n}$ is normal for any indices $t_1,\ldots,t_n$.
To find the density of $X_t$, we need only calculate its mean and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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$f : \mathbb{R} \to \mathbb{R}$ continuous and positive. Show $\lim\limits_{t\to 0^+} \frac{1}{t^3} \int_t^{t+t^2} xf(x)dx =f(0)$
$f : \mathbb{R} \to \mathbb{R}$ continuous and positive. Show $$\lim_{t\to 0^+} \frac{1}{t^3} \int_t^{t+t^2} xf(x)dx =f(0)$$
$f$ is continuous so $\forall \epsilon >0\,, \exists \delta > 0... | Hint: $$\frac1{t^3}\int_t^{t+t^2}x\,dx=\frac1{2t^3}((t+t^2)^2-t^2)
=1+\frac t2,$$so $$f(0)-\frac1{t^3}\int_t^{t+t^2}xf(x)\,dx
=-t\frac{f(0)}{2}+\frac1{t^3}\int_{t}^{t+t^2}x(f(0)-f(x))\,dx.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is the sideways tension on a hanging chain under gravity? I'm sure you are familiar with the question of deducing the shape of a free hanging chain but if not, here's my solution so far summarised:
By taking $\tan(\theta)$ to be $f'(x)$ and considering the forces on a
general particle, one can show that the sha... | The answer on Physics SE, linked by cgiovanardi in a comment, considers a non-symmetric configuration, where the two ends are at different heights. If you are only interested in the symmetric case, it is somewhat simpler than that, though you can still apply the same approach, and in the end the equation is transcenden... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Proving particular solution in differential equation
Given the second-order ordinary differential equation:
$$
{y}''+y=f(x)
$$
prove that:
$$
y_p(x)=\int_{0}^{x}f(u)\sin(x-u)du
$$
is the particular solution of the equation.
I know this is homework but I've been trying to solve it for the past few days and I ... | Hint. We assume our $f$ is nice enough to be allowed to use the Leibniz rule,
$$
\frac{d}{dx} \left (\int_{0}^{b(x)}f(x,u)\,du \right) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) + \int_{0}^{b(x)}\frac{\partial}{\partial x} f(x,u) \,du
$$ giving here
$$
\begin{align}
y'_P(x)&=\frac{d}{dx} \left (\int_{0}^{x}f(u)\sin(x-u... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many tries on average before I see the same value $N$ times in a row? If I repeatedly roll a fair, $X$-sided die, on average how many rolls should I expect to make before I happen to roll the same value $N$ times in a row?
I've found questions on here with answers when $X=2$, or where $N=2$, or where you're looking... | $N_1$ - No of tosses for 1st Heads
$N_2$ - No of tosses for 2 consecutive Heads
$N_3$ - No of tosses for 3 consecutive Heads
$\mathbb E[N_1] = 2$
$\mathbb E[N_2] = (1+\mathbb E[N_1])2 = 6$
$\mathbb E[N_3] = (1+\mathbb E[N_2])2 = 14$
A general formula can be created
$\mathbb E(1) = X$
$\mathbb E(2) = (X+1)X$
$\mathbb E(... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $x_n\to y $ and $d(x_n, x)<\epsilon$, $\forall x_n$. Is it true $d(y, x)<\epsilon$? Let the sequence $\{x_n\}_{n=0}^\infty$ be given and for all $x_n$, we have $d(x_n, x)<\epsilon$ and $x_n\to y$.
Is it true that $d(y, x)<\epsilon$? ( It is important for me to know that $d(y, x)\neq \epsilon$)
| Try $x_n=1/n\in\Bbb R$, $y=0$ and $x=\epsilon$. (Start the sequence with $n$ sufficiently large.) ... So, no, you may well have $d(y,x) = \epsilon$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determining: $\lim_{n\rightarrow \infty}e^{-n}\sum_{k=n}^\infty n^k/k!$ What is the value of $\lim_{n\rightarrow \infty}e^{-n}\sum_{k=n}^\infty n^k/k!$ ?
I have tried initially but could not proceed any further. What I have tried is:
$$\lim_{n\rightarrow \infty}e^{-n}\sum_{k=n}^\infty{n^k \over k!}\\
=\lim_{n\rightarro... |
we have $$e^{-n}\sum_{k=n}^\infty \frac{n^k}{k!}
=e^{-n}\left[e^n-\sum_{k=0}^{n-1} \frac{n^k}{k!}\right]= \left[1+ \frac{e^{-n}n^n}{n!}-e^{-n}\sum_{k=0}^{n} \frac{n^k}{k!}\right] $$
By Stirling formula $$ \frac{e^{-n}n^n}{n!}\sim \frac{1}{\sqrt{2n\pi}}\to0$$
from and from here:Evaluating $\lim\limits_{n\to\infty} e^{... | {
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Prove that $\int_0^\infty\frac1{x^x}\, dx<2$
Prove that $$\int_0^\infty\frac1{x^x}\, dx<2.$$
Note: This inequality is rather tight. The integral approximates to $1.9955$.
Integration by parts is out of the question. If we let $f(x)=\dfrac1{x^x}$ and $g'(x)=1$ then $f'(x)=-x^{-x}(\ln x + 1)$ by implicit differentiat... | $$\int_{0}^{+\infty}e^{-x\log x}\,dx = \underbrace{\int_{0}^{1}e^{-x\log x}\,dx}_{I_1}+\underbrace{\int_{1}^{+\infty}e^{-x\log x}\,dx}_{I_2} $$
$$ I_1=\sum_{n\geq 0}\frac{(-1)^n}{n!}\int_{0}^{1}x^n\left(\log x\right)^n\,dx = \sum_{n\geq 0}\frac{1}{(n+1)^{n+1}}=\sum_{n\geq 1}\frac{1}{n^n}\tag{A}$$
$$ I_2 = \int_{0}^{+\i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate the determinant of $A-5I$ Question
Let $ A =
\begin{bmatrix}
1 & 2 & 3 & 4 & 5 \\
6 & 7 & 8 & 9 & 10 \\
11 & 12 & 13 & 14 & 15 \\
16 & 17 & 18 & 19 & 20 \\
21& 22 & 23 & 24 & 25
\end{bmatrix}
$.
Calculate the determinant of $A-5I$.
My approach
the nullity of $A$ is $3$, so the algebraic multiplicity of $\lam... | Let the standard basis of $\mathbb R^5$ be denoted $\{e_1,e_2,\dotsc,e_5\}$. Let $\vec l$ denote the leftmost column of $A$, and let $\vec 1$ be the column vector of all $1$s. We have that for all $i \in \{1,\dotsc,5\}$: $$\begin{align}&Ae_i = \vec l+(i-1)\vec1 \\&\vec l = \sum_{i=1}^5 ie_i \\&\vec 1 = \sum_{i=1}^5 e_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2585742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Taylor approximation of inverse square root Given the function $f(x)=\sqrt{1+mx+\mathcal{O}(x^2)}$ I am reading that $g(x) = \frac{1}{f(x)}$, the inverse square root, can be computed with first order Taylor approximation and take $g(x) = 1 - \frac{m}{2}x + \mathcal{O}(x^2)$. So given, $f'(x) = \frac{m+\mathcal{O}(x^2)}... | It is easier to write $g(x)=\frac{1}{f(x)}=(1 + m x + O(x^2))^{-1/2}$. Doing the derivative on g(x) (not f(x)) gives you $g'(x)= -m/2 (1 + m x +O(x^2))^{-3/2} +O(x^2)$, so the Taylor expansion (you have a sign error and should read $h(x)=h(a)+h'(a)(x-a)+O(x^2))$ at $a=0$ is:
$g(x)=g(0) + g'(x) (x-0) + O(x^2)= 1 - \frac... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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What is undefined times zero? Einstein's energy equation (after substituting the equation of relativistic momentum) takes this form:
$$E = \frac{1}{{\sqrt {1 - {v^2}/{c^2}} }}{m_0}{c^2}
% $$
Now if you apply this form to a photon (I know this is controversial, in fact I would not do it, but I just want to understand ... | Undefined is not a number. There is no such number as undefined, for which you could define the multiplication operation.
You could extend a set of a numbers (the set of the real numbers or the set of the complex numbers) with a new element, what you call "undefined", and then define a multiplication on this set as usu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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validity of a formula with one existential quantifier and one variable Let $\sigma$ a dictionary without equlity symbol that contains at least one constant symbol.
Let $\varphi$ a formula over $\sigma$ without quantifiers such that $FV(\varphi)=\{x\}$.
Prove that: $\exists x\varphi$ is valid $\iff$
There exist $s_1,... | I assume that in your notation, the statement "$\psi_1,\dots,\psi_k$ is valid" is equivalent to "$\bigvee_{i=1}^k \psi_i$ is valid". Is that right?
For the converse, suppose $\exists x\,\varphi(x)$ is valid. Consider the set of sentences $T = \{\lnot \varphi(t)\mid t\text{ is a ground term}\}$. Suppose for contradictio... | {
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"source": "stackexchange",
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Why does Hausdorff measure go to zero as diameter power increases? For example, why does the Hausdorff measure of a flat disc go to zero when the power that the diameter is raised to (in the definition of the Hausdorff measure) reaches 3?
| A flat disc $D$ of radius $1$ has area $\pi.$ For $n\in \Bbb N$ let $S_n$ be a set of open discs, each of diameter $1/n$ or less, with $\cup S_n \supset D$ and $$\sum_{t\in S_n}A(t)<2\pi,$$ where $A(t)$ is the area of $t.$
For $t\in S_n$ let $d(t)$ be the diameter of $t.$ Then $A(t)=\pi d(t)^2/4.$
Let $r>0.$ Then $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question on surds rule Through the textbook, I've been taught the rule $\frac{\sqrt a}{\sqrt b} = \sqrt\frac{a}{b}$, however I realized that if all numbers are assumed to be real, and $a<0 ,b<0$, then the rule is not true as $\frac{\sqrt{-a}}{\sqrt{-b}} = \sqrt\frac{a}{b}$, whereas in $\frac{\sqrt{-a}}{\sqrt{-b}}$, the... | For all nonnegative real numbers $a$ and $b$ (with $b > 0$), we have
$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}. $$
After that, all bets are off. The basic problem is that the function $x \mapsto x^2$ is not injective (one-to-one) over the real numbers. Thus to define the principal square root, which is what t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Evaluation of limiting value of given series For a given sequence, $a_1=1$ and $a_n=n(1+a_{n-1})$ $\forall n\geq 2$, then value of given limit:
$$\lim_{n\to \infty} \bigg(1+\frac{1}{a_1}\bigg)\bigg(1+\frac{1}{a_2}\bigg)\cdots\bigg(1+\frac{1}{a_n}\bigg)$$
Usually such type of questions are solved by squeeze theorem or ... | Hint: $\displaystyle \frac{a_n}{n!} = \frac{a_{n - 1}}{(n - 1)!} + \frac{1}{(n - 1)!} \ (n \geqslant 2)$ and$$
\prod_{k = 1}^n \left(1 + \frac{1}{a_k}\right) = \left. \prod_{k = 1}^n (1 + a_k) \middle/ \prod_{k = 1}^n a_k\right. = \left.\prod_{k = 1}^n \frac{a_{k + 1}}{k + 1} \middle/ \prod_{k = 1}^n a_k\right. = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Prime divisors of sequences of integers I have the following problem and I need your support for it
Problem. Let P(x) be a polynomial with integer coefficients, such that $\deg P>0$ and $\lim_{x\to+\infty}P(x)=+\infty$. Prove that there exist infinitely many prime numbers $p$ such that for some natural number $n$
$$... | As I pointed out in my comments, what matters here is the growth. First we prove :
Proposition : Given $f : \mathbb{N} \mapsto \mathbb{N}$ strictly increasing, and $k \ge 0$ such that $f(n) = O_{n \to +\infty} \big( n^k \big)$, the set $\{ p \mbox{ prime } | \ \exists n:\ p|f(n) \}$ is infinite.
Proof : ad absurdum, ... | {
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Given a set $A \subseteq \{10,11,12,13...98,99\}$ such that $|A|=10$. Prove using Pigeonhole Principle there are 2 disjoint subsets with the same Sum. Given a set $A \subseteq \{10,11,12,...98,99\}$ such that $|A|=10$. Prove using Pigeonhole Principle there are 2 disjoint non-empty subsets of $A$ with the same Sum.
Dir... | The range of sums of a subset of $|A| \le 10$ is $\le 21 (10+11)$ and $945\ge (90+91+...+99)$, therefore $925$ different sums you can get from a subset of $\{10,11,...,98,99\}$ if $|A|\le10.$
In a set $A, |A| = 10$, there are $2^{|A|} - 1$ non-empty subsets = $1023$.
$1023 \gt 945$, therefore considering $945$ "pigeon... | {
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"timestamp": "2023-03-29T00:00:00",
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How many $3$-digit numbers can be formed using the digits $ 2,3,4,5,6,8 $ such that the number contains the digits $5$ and repetitions are allowed? The solution I have is by counting the complement which gives an answer of $91$. But I think that it should be solved as follows-
Let the $3$-digit number be denoted by $3$... | Take all the possible numbers with $5$
then substract all the numbers without $ 5$
Result: $6^3-5^3=91$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of integer triangles.
Number of integer isosceles or equilateral triangle none of whose
sides exceed $2c$ is?
I substituted $c =3$ checked, and got $27= 3 \times (3^2)$ i.e. $3c^2$ triplets. It gives the right answer($3c^2$) but how do I write a proper formal proof of this?
I tried it this way: We have three... | Let $b$ be the length of the base and $a$ be the length of the legs of such a triangle. Given $c\geq1$ we want
$$1\leq b\leq 2c, \quad {b\over2}<a\leq 2c\ .\tag{1}$$
If $b=2k-1$ with $1\leq k\leq c$ is odd the condition $(1)$ enforces $k\leq a\leq 2c$ and allows of $2c-(k-1)$ different integer values for $a$.
If $b=2k$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find possible values of angle C In an acute triangle $ABC$, $O$ is the circumcenter, $H$ is the orthocenter and $G$ is the
centroid. Let $OD$ be perpendicular to $BC$ and $HE$ be perpendicular to $CA$, with $D$ on $BC$ and $E$
on $CA$. Let $F$ be the midpoint of $AB$. Suppose the areas of triangles $ODC, HEA$ and $GFB$... | From $[ODC]=[HEA]$ one gets
$$
\tag{1}
\sin C\cos C={1\over4}{\sin A\over \cos A}.
$$
From $[ODC]=[GFB]$, taking into account that $\sin B=\sin(A+C)=\sin A\cos C+\cos A\sin C$, one gets
$$
\sin A \sin C\cos C+\cos A \sin^2C={3\over2}\cos A,
$$
that is, using $(1)$:
$$
{1\over4}{\sin^2 A\over \cos A}+\cos A \sin^2C={3\o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving $\cos x + \cos 2x - \cos 3x = 1$ with the substitution $z = \cos x + i \sin x$ I need to solve
$$\cos x+\cos 2x-\cos 3x=1$$
using the substitution$$z= \cos x + i \sin x $$
I fiddled around with the first equation using the double angle formula and addition formula to get
$$\cos^2 x+4 \sin^2x\cos x-\sin^2 x=1$... | Using the hint,
$$\Re(z+z^2-z^3)=1,$$ or
$$z+z^2-z^3=1+iw.$$
This can be factored as
$$-(z+1)(z-1)^2=iw$$ but I see no easy way to exploit it.
Direct resolution of the cubic equation looks terrible.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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System of homogeneous linear equations with coefficients in a field In the book I started to read, at almost the very beginning it is stated (without proof):
If $m<n$ and we have a system $$a_{11}x_1+...+a_{1n}x_n=0$$
$$\vdots$$
$$a_{m1}x_1+...+a_{mn}x_n=0$$
of homogeneous linear equations, with coefficients in a fiel... | To start, your last sentence is not correct in that if a matrix has an inverse it will only have the trivial solution to the homogenous system. Also, an invertable matrix must be square.
The result itself follows from the fact that the system is underdetermined and will have multiple solutions (proof via Gaussian algor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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surface area of implicit surface Ok, so I read ( on Thomas' book) that the surface area of an implicit surface is the double integral of the magnitude of the function's gradient divided by the magnitude of the dot product between such gradient and the unit normal vector p over the projection of this surface onto a plan... | If you project a piece $S$ of a planar region in the plane with unit normal vector $\vec n$ onto a region $R$ in the $xy$-plane (with normal vector $\vec k$), then it's a little bit of geometry to see that
$$\text{area}(S) = \frac1{|\cos\gamma|}\text{area}(R) = \frac1{|\vec n\cdot\vec k|}\text{area}(R).$$
(Here $\gamma... | {
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"timestamp": "2023-03-29T00:00:00",
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Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$ If $a,b,c$ are sides of triangle Find Minimum value of
$$S=\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$$
My Try:
Let $$P=\sqrt{a}+\sqrt{b}+\sqrt{c}$$
we have $$S=\sum \frac{1}{\frac{\sqrt{b}}{\sqrt{a}}+\frac{\sqrt{c}}{\sqrt{a}}-1}$$
$$S=\sum ... | When $a = b = c$, $S = 3$. Next it will be proved that $S \geqslant 3$ for all possible $a, b, c$.
Denote $\displaystyle u = \frac{\sqrt{a}}{\sqrt{a} + \sqrt{b} + \sqrt{c}}$, $\displaystyle v = \frac{\sqrt{b}}{\sqrt{a} + \sqrt{b} + \sqrt{c}}$, $\displaystyle w = \frac{\sqrt{c}}{\sqrt{a} + \sqrt{b} + \sqrt{c}}$, then $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Perpendicular from incenter of a triangle to any side is equal to the radius of the incircle Given a triangle $ABC$ with incenter $I$, it is said that the perpendicular line segment from $I$ to any of the sides $AB$, $AC$, or $BC$ is equal to the radius of the incircle. (See the second picture on this page: http://math... | Well the definition of an incenter is the center of the largest circle that fits into the triangle. So the circle is externally tangent to each side of the triangle. A well-known circle theorem is that the radius at the point where a tangent touches the circle is perpendicular to the tangent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is there an algebraic proof for $\sum_{m=k}^{n-k} \binom{m}{k}\binom{n-m}{k} = \binom{n+1}{2k+1}, n\ge2k\ge0$ $\sum_{m=k}^{n-k} \binom{m}{k}\binom{n-m}{k} = \binom{n+1}{2k+1}, n\ge2k\ge0$
An combinatorial proof of the identity above states as follow:
(1)Number of ways of picking (2k+1) numbers from 1 to (n+1) should be... | Here is an algebraic proof based upon generating functions. It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series. This way we can write for instance
\begin{align*}
[z^k](1+z)^n=\binom{n}{k}
\end{align*}
We obtain
\begin{align*}
\color{blue}{\sum_{m=k}^{n-k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Group of order $3$ acting on the tetrahedron It is well-known that the group of (orientation-preserving) symmetries of the tetrahedron is isomorphic to $A_4$. Since $\mathbb{Z}/3$ is a quotient of $A_4$, $\mathbb{Z}/3$ also acts on the tetrahedron.
Is there a way to see this actions geometrically? i.e., are there $3$ ... | The tetrahedron has three (unordered) pairs of opposite edges. The stabiliser of each is the fours-group whose quotient is cyclic of order $3$. I think this is what you are interested in.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Existence of a special continuous function I have stuck at a problem. " does there exist any continuous function that takes every real value exactly twice?"
Intuitively, I think such function cannot exist, as $x^2$, $|x|$, although they take every possitive value exactly twice, but they take $0$ only once. I tried to a... | You're right that no such function exists. If $f$ were one then there would be two values $a < b$ such that $f(a)=f(b)=1$. Then on the closed interval $[a,b]$ the function $f$ would have a maximum value $M$. We can assume $M > 1$ (if not, then use the minimum value).
If the maximum value $M$ appeared twice, at $c < d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show $\int_{0}^{\infty} x^tf(x)dx$ is uniformly convergent for $t\in[a,b]$. Assume $\int_{0}^{\infty} x^tf(x)dx$ converges at $t=a$ and $t=b$. Show
$$\int_{0}^{\infty} x^tf(x)dx$$
is uniformly convergent for $t\in[a,b]$.
It seems that the common tests cannot work. May I consider Cauchy's convergence test？
| Absolute convegence case:
Here we consider the case where $\int^\infty_0 x^t\,|f(x)|\,dx<\infty$ for $t\in\{a,b\}$.
Since $\left|\int_A x^t\,f(x)\,dx\right|\leq \int_A x^t|f(x)|\,dx$ for all $A\subset [0,\infty)$, it suffices to assume $f\geq0$.
Given $\varepsilon>0$, there exists $C>0$ such that
$$
\begin{align}
\int^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the radius of curvature at the origin
Find the radius of curvature at the origin for the curve $3x^2+4y^2=2x$.
Below is my attempt:
We know that the radius of curvature $\rho=\dfrac{{(1+y_1^2)}^{\frac{3}{2}}}{y_2}$ .
Computing $y_1$ at $(0,0)$ : Differentiating we get $3x+4y\frac{dy}{dx}=1\implies \dfrac{dy}{dx}... | If we consider the differential $(6x-2)dx+8ydy$, then at $(0,0)$, we see $(6x-2) \neq 0$. By the implicit function theorem, we may write $x$ as a function of $y$.
Complete the square:
$$3x^2 -2x +4y^2=0$$
$$3(x-\frac{1}{3})^2+4y^2=\frac{1}{3}$$
From here we get:
$$x(y)=\frac{1}{3}+\sqrt{\frac{1}{9}-\frac{4}{3}y^2}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Examples of when a "new way of thinking" led to a solution I was reading William Thurston's On Proof and Progress in Mathematics in which he discusses the value of the different ways people think about the same mathematical structure. He claims that many mathematical solutions are the result of different ways of thinki... | The probabilistic method has been successfully used for proving the existence of mathematical objects non constructively, by proving that the probability of choosing an object in that class is not zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Product of a exponential and discrete Distribution Let $U \sim e(1)$ and $V$ a discrete random variable independent of $U$ such that
$p_V(v)=1/2$ if $v \in \{-1,1\}$ and $p_V(v)=0$ otherwise.
Problem:
Let $W=UV$. Find the distribution function of $W$ $\forall w\in \mathbb{R}$.
My try:
\begin{align}
P(W\leq w)&=P(UV\leq... | You wrote:
\begin{align}
& P(W\leq w) = P(UV\leq w) \\[10pt]
= {} &P(U\leq w \mid V=1)P(V=1)+P(-U\leq w \mid V=-1)P(V=-1).
\end{align}
What is $P(U\le w\mid V=1)$?
It is $\begin{cases} 1-e^{-w} & \text{if } w\ge0, \\ 0 & \text{if } w<0. \end{cases}$
What is $P(-U\le w\mid V=-1)$?
It is $\begin{cases} e^w & \text{if } w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Compute $A^n$ where $A^2+bA+cI=0$ Let $A$ be a complex matrix such that $$A^2+bA+cI=0,$$
where $I$ is the identity matrix and $b,c\in \mathbb{C}$. I am interested in finding a formula for $A^n$ in terms of $A$ and $I$. The binomial formula is not giving an answer I think. Maybe using $A^2=-bA-cI$, then $A^3=-bA^2-cA$,... | You have $$ A^{n+2} = b_nA + c_nI $$
where $b_0 = -b, c_0 = -c$
Multiplying both sides by $A$ gives
$$ A^{n+3} = b_nA^2 + c_n A = b_n(-bA-cI)+c_nA = (c_n-bb_n)A - cb_nI $$
therefore
$$ b_{n+1} = -bb_n + c_n, \ c_{n+1} = -cb_n $$
You may find an implicit relation by solving the linear system
$$ \left(\matrix{b_n\\c_n}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
When is the symmetric product of a matrix and a diagonal matrix negative definite on $\sum_i x_i=0$? Let $A$ be a matrix and $b$ be a strictly positive vector. Denote by $B=\text{diag}(b)$, a matrix with $b$ on its diagonal and zeros elsewhere. I am interested in the symmetric matrix
$$
C = AB+(AB)'.
$$
Now, consider t... | With your definition of negative definiteness, $C$ is never negative definite when $n>2$. Note that your example is not correct: you do have $x'Cx=-2x'Bx<0$ for all nonzero $x\in S$, but you cannot find $d>0$ such that $-2x'Bx\le-d\|x\|^2$ on $S$ for every positive diagonal matrix $B$. In fact, when $x'=(1,-1,0)$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can I solve $a_{n+2} + 4a_n = n \cdot 2^n$ for non homogeneous solution?
Solve $a_{n+2} + 4a_n = n \cdot 2^n\;$ for non homogeneous $a_0 =1$ and $a_1 =0$
I was trying but I think it's wrong. I solved a question with $2^n$ but that had a form with $A \cdot 2^n$ but I think this requires a different solution than t... | We have here a linear recurrence relation.
Solve first the homogeneous equation : $$h_{n+2}+4h_n=0$$
It has characteristic equation $x^2+4=0$ thus $\pm 2i$ are roots.
The solutions of homogeneous equation are then given by $h_n=2^n(A\cos(\frac{n\pi}2)+B\sin(\frac{n\pi}2))$
Now we have to find a particular solution of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Conflicting definitions of isogeneous and the relevance of separability The following questions seem to be related.
Firstly, let $E_1$ and $E_2$ denote elliptic curves. Silverman defines that $E_1$ and $E_2$ are isogeneous if and only if there exists a basepoint preserving regular map $E_1 \rightarrow E_2$ that is non-... | *
*I don't think they're equivalent, since you can use google to see that authors use the phrase "separably isogenous." I don't know how to construct a counterexample but Frobenius twists might work.
*I think Galbraith is taking the kernel on $\overline{\mathbb{F}_p}$-points, not the kernel as a group scheme. The ke... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculating probability using combinations We have 50 chairs. 2 out of them are broken. What is probability that out of 5 taken chairs 4 or 5 are not broken?
We can calculate it by using $P(4 not broken) + P( 5 not broken)$
$P( 5 not broken ) = \frac{48}{50} * \frac{47}{49} * \frac{46}{48} * \frac{45}{47}* \frac{44}{46... | Note that the broken chair can be chosen either:
*
*first with probability: $$P_1 = \frac {2}{50}\times \frac {48}{49}\times \frac {47}{48}\times \frac {46}{47}\times \frac {45}{46} $$
*second with probability: $$P_2 = \frac {48}{50}\times \frac {2}{49}\times \frac {47}{48}\times \frac {46}{47}\times \frac {45}{46}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability of a letter coming from a city I got this question:
A letter is known to have come either from $\text{TATANAGAR}$ or from $\text{CALCUTTA}$. On the envelope just two consecutive Letters $\text{TA}$ are visible. What is the probability that the letters came from $\text{TATANAGAR}$?
My attempt:
Total number... | $$T\equiv \text{comes from Tatanagar}\quad \text{and}\quad C\equiv \text{Comes from Calcutta}\rightarrow \left\{ \begin{array}{lcc}
p(T)=\dfrac{1}{2} \\
\\ p(C)=\dfrac{1}{2}
\end{array}
\right.$$
$$\text{Possible choices of two consecutive letters:}\left\{ \begin{array}{lcc}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Evaluate $\int (2x+3) \sqrt {3x+1} dx$ Evaluate $\int (2x+3) \sqrt {3x+1} dx$
My Attempt:
Let $u=\sqrt {3x+1}$
$$\dfrac {du}{dx}= \dfrac {d(3x+1)^\dfrac {1}{2}}{dx}$$
$$\dfrac {du}{dx}=\dfrac {3}{2\sqrt {3x+1}}$$
$$du=\dfrac {3}{2\sqrt {3x+1}} dx$$
| Let us just split $2x+3 = \frac23 (3x+1)+\frac73$ and simplify to give us: $$I = \int (2x+3)\sqrt{3x+1}\, dx = \frac23 \int (3x+1)^{\frac32}\, dx+ \frac73 \int \sqrt {3x+1}\, dx$$ which can be easily solved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Directional derivative doubt I have two functions.
$e(t,z)=\cos(\omega t)\cos(\beta z)$, $h(t,z)=\frac{1}{\eta}\sin(\omega t)\sin(\beta z)$
Let's set the direction $\hat{u} = \hat{z} - c\hat{t}$, with $\dfrac{\epsilon_0}{c}=\dfrac{1}{\eta}$ and $\dfrac{\omega}{c}=\beta$. I can calculate $\dfrac{\partial( h-\frac{1}{\et... | There seems to be some confusion regarding directional derivatives vs. partial derivates.
*
*Your function $f(t,z)=h-\frac{1}{\eta} e= -\frac{1}{\eta} \cos(\beta(z+ct))$ is constant in the direction $\hat{u}=(-\frac{1}{c},1)$ in the $(t,z)$-plane, since:
$$ -\frac{1}{c} \partial_t f + \partial_z f = 0$$
*If you m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that $RU$ intersects $AB$ at the midpoint (of $AB$). $PA, PB$ are tangents. $PU$ is a reflection of $PS$ over $PO$ where $O$ is the center of the circle. Show that $RU$ intersects $AB$ at the midpoint (of $AB$).
It looks so obvious. And I can see a lot of similar triangles. But I can't really solve it.
Please avo... | Note that $TS$ and $RU$ meets on $AB$ because $P$ is the pole of the line $AB$.
On the other hand $TS$ and $RU$ meets on the simmetry axis $PO$ because they corresponds under the reflection.
Thus $TS\cap RU=AB\cap PO$ and $AB\cap PO$ is the mid point of $AB$ because $OP$ is axis of $AB$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Entropy solution - Burgers' equation Why doesn't the following problem have a solution for $t\ge1$?
$u_{t}+uu_{x}=0\\
u(0,x)=-x$.
The characteristics don't intersect and they cover the whole space above t=1.
| All characteristics intersect at $(t,x) = (1, 0)$.
Indeed, the characteristic starting from $(0, x_0)$ is $x(t) = x_0 - x_0 \, t = (1-t) x_0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding solutions to a non homogeneous differential equation knowing the solutions of its homogenous counterpart
Let $f(x)$ and $xf(x)$ be the particular solutions of a differential
equation $$y''+R(x)y'+S(x)y=0$$ Find the solution of the differential
equation $$y''+R(x)y'+S(x)y=f(x)$$ in terms of $f(x)$.
I was t... | Once you found two linearly independent solution of the homogeneous equation say $y_1$ and $y_2$, the particular solution is $$ y_p= u_1y_1+u_2 y_2$$Substitute in your inhomogeneous equation and solve for $u_1$ and $u_2.$ You may study the method of varion of constant for specific examples.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Any example of a connected space that is not locally connected? A topological space $X$ is said to be locally connected at a point $x \in X$ if, for every neighborhood $U$ of $x$ (i.e. open set $U$ such that $x \in U$), there exists a connected neighborhood $V$ of $x$ such that $V \subset U$. If $X$ is locally connecte... | Another standard example that is even path connected but not locally connected is the comb space
$$ [0,1]\times\{0\} ~\cup~ \{0\}\times[0,1] ~\cup~ \bigcup_{k\ge 1} \{\tfrac1k\}\times[0,1] $$
Every neighborhood of the point $(0,1)$ contains the top end of infinitely many of the teeth, but if the neighborhood is so smal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Continuous $f$ such that $f(x)=f(x^2)$ is constant? Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x)=f(x^2)$ for all $x\in\mathbb{R}$. I've proven that also $f\left(y^{(2^{-n})}\right)=f(y)$ for all $n\in\mathbb{N}, y\geq0$. How can I deduce that $f$ is constant?
| If $f$ is continous, for all sequence $(x_n)_{n \in \mathbb{N}}$ such as $x_n \underset{n \rightarrow +\infty}{\rightarrow}x$ then
$$
f\left(x_n\right) \underset{n \rightarrow +\infty}{\rightarrow}f\left(x\right)
$$
It should help you conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Counterexample to $M$ a $R$-module or rank $n$ then if $N\leq M$, $N$ is a $R$-module of rank $\leq n$ I've seen the following theorem
Let $R$ be a PID (principal ideal domain), If now $M$ is a free $R$-module of rank $n$ and $N\leq M$ a submodule then $N$ is a free $R$-module of rank $\leq n$.
I would like to see th... | Take any non-PID $R$ and a projective $R$-module that isn't free. One definition of projective module is that $P$ is projective if there exists $N$ such that $P \oplus N$ is a free $R$-module. In this case $P$ is a submodule of the free module $P \oplus N$ but it's not free.
If $R$ is local or a PID then a projective $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$P(X \geq 2Y) = ?$ where $P(n) = 2^{-n} , n =1,2,3,...$. Let $X,Y$ are independent identically distributed random variables.Then $P(X \geq 2Y) = ?$ where $P(n) = 2^{-n} , n =1,2,3,...$.
So it is the discrete case, So in order to calculate this -
$P(X \geq 2Y) = \sum_{y=1}^{\infty} \sum_{x = 2y}^{\infty} 2^{-x-y} = \su... | Above Answer is not Correct. It should be
$P(X \geq 2Y) = \sum_{y=1}^{\infty} \sum_{x = 2y}^{\infty} 2^{-x-y}
= \sum_{y=1}^{\infty}2^{-y} (\frac{2^{-2y}}{1 - 2^{-1}})
= \sum_{y=1}^{\infty} \frac{2}{1}. 2^{-3y} =\frac{2}{1}. \frac{2^{-3}}{1 - 2^{-3}}
= \frac{2}{7} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $a(u,v)=\int_\Omega A\nabla u\cdot \nabla v$ is continuous if $A$ uniformly elliptic. Let $\Omega \subset \mathbb R^d$ smooth, bounded and connected domain. Let $A\in \mathbb R^{d\times d}$ symetric and uniformly elliptic, i.e. there is $C>0$ such that $$C^{-1}\|x\|^2\leq Ax\cdot x\leq C\|x\|^2.$$
How can I ... | If $A$ is a symmetric matrix, then $Ax \cdot x \le C \|x\|^2$ implies that all eigenvalues of $A$ are bounded from above by $C$. Hence, $\|A x\| \le C \|x\|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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