Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Evaluate $\int_0 ^x \frac{\sin\left(\frac{1}{2}-n\right) t}{\sin\frac{t}{2}}\,dt$. Evaluate
$$\int_0 ^x \frac{\sin\left(\frac{1}{2}-n\right) t}{\sin\frac{t}{2}}\,dt$$
for $x\in(0,2\pi)$ and $n\in\mathbb Z$. From trigonometric formulas, we have:
$$\int_0 ^x\cos nt\, dt-\int_0 ^x \frac{\cos\frac{t}{2}}{\sin\frac{t}{2}}\,... | Only a partial answer for $n\in\mathbb{N}$, but maybe helpful nonetheless. Call $f_n(x)=\int_0^x dt \sin(nt)\cot(t/2)$ and consider the generating function
$$
F(z,x)=\sum_{n=0}^\infty f_n(x)z^n=\int_0^x dt\cot(t/2)\frac{\mathrm{i} \left(-1+e^{2 \mathrm{i} t}\right) z}{2 \left(-z+e^{\mathrm{i} t}\right) \left(-1+e^{\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Field with four elements If $F=\{0,1,a,b\}$ is a field (where the four elements are distinct), then:
1.What is the characteristic of $F$?
2.Write $b$ in terms of the other elements.
3.What are the multiplication and addition tableau of these operations?
| The addition: $\;a+b=0\implies a=-b=b\;$ , since $\;\text{char}\, F=2\;$ . It also can't be $\;a+b=a\;,\;\;a+b=b\;$ , else $\;b=0\;$ or $\;a=0\;$ . Thus it must be $$\;a+b=1\implies b=1-a=1+a\;$$
Generalize the above and get the addition table.
As for multiplication $\;ab\neq 0,a,b\;$ , else $\;a=0\;$ or $\;b=0\;$ , or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
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Is the space $R^2$ for a ring $R=\mathbb{Z}/p\mathbb{Z}$ the sum of two invariant lines? I didn't do a very good job of summarising the problem in the title, sorry, but here is the full question:
Let us define, for every ring $R$, the set
$S(R)=$ {$\rho : \mathbb{Z}/2\mathbb{Z} \to GL(2,R) | \rho (0+2\mathbb{Z}) \ne \r... | The answer is all if $p\neq 2$.
Consider the representation $\rho(0)=\pmatrix{1& 0\\0& 1}$ and $\rho(1)=\pmatrix{-1& -1\\ 0& 1}$
Then the lines $L_1=\langle\pmatrix{-1\\ 2}\rangle$ and $L_1=\langle\pmatrix{1\\ 0}\rangle$ (These are basically the eigenvectors of $\rho(1)$ with eigen values $1$ and $-1$ respectively) are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Number distinct powers of $cos(\alpha \pi) + i\sin(\alpha \pi)$ How many distinct powers of $\cos(\alpha \pi) + i\sin(\alpha \pi)$ are there if $\alpha$ is rational? Irrational?
Here's my thoughts. The answer is probably some finite number based on $\alpha$ if $\alpha$ is rational and infinitely many if $\alpha$ is i... | Remeber that you may write $\Bbb e ^{\Bbb i \alpha} = \cos \alpha + \Bbb i \sin \alpha$.
*
*If $\alpha = \frac p q \pi$ with $p,q \in \Bbb Z, \ q \ne 0$ and $p$ even, then $(\Bbb e ^{\Bbb i \alpha}) ^0, \dots, (\Bbb e ^{\Bbb i \alpha}) ^{q-1}$ are all distinct, the next power being $(\Bbb e ^{\Bbb i \alpha}) ^{q} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Rotation of the torus $T^2$ by irrational numbers linearly dependent over $\mathbb Z$ It is known that the rotation $x \to x + \alpha$ of $S^1 = \mathbb R / \mathbb Z$ with irrational $\alpha$ is ergodic and, in particular, $\alpha n$, $n = 1, 2,\dots$, are dense in $S^1$. In two dimensions, the corresponding result is... | No, that is not possible.
If $\alpha_1,\alpha_2$ are linearly dependent irrational numbers over $\mathbb{Z}$ then there exists another irrational number $\alpha'$, which is a $\mathbb{Z}$-linear combination of $\alpha_1$ and $\alpha_2$, such that your original set can be written as multiples of $(m_1\alpha',m_2\alpha'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to prove the following binomial identity How to prove that
$$\sum_{i=0}^n \binom{2i}{i} \left(\frac{1}{2}\right)^{2i} = (2n+1) \binom{2n}{n} \left(\frac{1}{2}\right)^{2n} $$
| Suppose we seek to verify that
$$\sum_{q=0}^n {2q\choose q} 4^{-q} =
(2n+1) {2n\choose n} 4^{-n}$$
using a method other than induction.
Introduce the Iverson bracket
$$[[0\le q\le n]]
= \frac{1}{2\pi i}
\int_{|w|=\epsilon}
\frac{w^q}{w^{n+1}} \frac{1}{1-w}
\; dw$$
This yields for the sum (we extend the sum to infin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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How is $\Bbb Z_0 = \{0, \pm m, \pm2m, \pm3m, \ldots\}$ denoted in set builder notation? $\Bbb Z_0$ has integers as its elements, and its elements are listed as follows:
$\Bbb Z_0 = \{0, \pm m, \pm2m, \pm3m, \ldots\}$
Then how is $\Bbb Z_0$ denoted in set builder notation?
Is there a correct representation for the set $... | A simple way would be to write it $m\Bbb Z=\{mn\in\Bbb Z:n\in\Bbb Z\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How to prove abelian group I am struggling answering this question for myself:
How can i prove that a Group $G$ is abelian, if $$g\circ g=e \ \forall g \in G $$
A group is abelian if this is true:
$$a\circ b = b\circ a\ \forall a,b \in G$$
But i dont understand how to prove this.
Hope someone can help me out with this!... | $a$ and $b$ commute iff $a \circ b = b \circ a$ iff $a \circ b \circ a^{-1} \circ b ^{-1} = e$.
However $a^{-1} = a$ and $b^{-1} = b$ so then $a \circ b \circ a^{-1} \circ b ^{-1} = a \circ b \circ a \circ b = (a \circ b) \circ (a \circ b) = e$ by hypothesis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1665945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Book(s) on Algebras (Quaternions)? Well, lately I've been looking for a book on quaternions but I've realized that quaternions are a particular case of the named Algebras(I think Geometric Algebra). Since here, I've found all kind of algebra you can imagine, Geometric Algebra, Lie Algebra, Clifford Algebra, conmutative... | The quaternions are a very specific object like the real numbers. There is no reason to plow into entire disciplines to 'understand' them. You can learn all their important properties from the quaternions wiki page.
IMO, despite previous recommendations above to the contrary, think it is a pretty terrible idea to try t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intuition behind cross-product and area of parallelogram The cross product in 2D is defined like that: $|(x_1, y_1) \times (x_2, y_2)| = x_1 y_2 - x_2 y_1.$
I perfectly understand the first part of the definition: $x_1 y_2$, which is simply the area of a rectangle:
I am struggling to understand the second part: $- x_2... | *
*The formula works fine for the standard unit vectors.
*Stretching one of the vectors by a constant $c$ should multiply the (oriented) area by $c$ and does indeed multiply the cross product by $c$
*Shearing along $(x_1,y_1)$, i.e., replacing $(x_2,y_2)$ with $(x_2+cx_1,y_2+cy_1)$ does not change the area and also ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Can there be an unbounded sequence of equicontinuous functions? I am trying to find a equicontinuous sequence of functions $f_n$ on $(a, b)$ that is bounded somewhere but not everywhere.
I am thinking along the lines of
$$f_n=\frac{1}{nx}$$ on $(0, 1)$, but this is obviously not equicontinuous.
Any hints?
| I understand/interpret your question as follows: You want to find
an equicontinuous sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$
of functions $f_{n}:\left(0,1\right)\to\mathbb{R}$ such that the
set
$$
U:=\left\{ x\in\left(0,1\right)\,\middle|\,\exists M_{x}\in\left(0,\infty\right)\,\forall n\in\mathbb{N}:\,\left|f_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show that if $p(x)$ is a polynomial, $|p(x)|$ attains its minimum.
Show that if $p(x)$ is a polynomial, $|p(x)|$ attains its minimum.
Attempt
Let $p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Then if $|p(x)| = |a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0|$. If $x > 0,$ then $|p(x)| = |a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a... | Does $|p(x)|$ always attain $p(x)$'s minimum?
Try using a $p(x)$ such that $a_0<0$ and $a_0$ happens to be its minimum [example: $p(x)=x^2-1$]. Then $|p(x)|$ will never attain $a_0$, but $|a_0|$.
Maybe I'm misunderstanding the question. Try to word it better.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Give a big-O estimate of $(x+1)\mathrm{log}(x^2+1) + 3x^2$ I wanted to know if the following solution demonstrates that the function $f(x) = (x+1)\mathrm{log}\, (x^2+1) + 3x^2 \in O(x^2)$, because my answer and the book's answer deviate slightly.
Clearly,
$$3x^2 \in O(x^2) \tag{1}$$
$$x+1 \in O(x)\tag{2}$$
The followi... | If this is as
$x \to \infty$,
then
$\log(x^k+a)
=O(\log(x))
$
for any fixed $k$
and $a$.
Also,
$\log(x)
=O(x^c)
$
for
any $c > 0$.
This should be enough.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Evaluation of $\lim_{x\rightarrow \infty}\left\{\left[(x+1)(x+2)(x+3)(x+4)(x+5)\right]^{\frac{1}{5}}-x\right\}$
Evaluation of $\displaystyle \lim_{x\rightarrow \infty}\left\{\left[(x+1)(x+2)(x+3)(x+4)(x+5)\right]^{\frac{1}{5}}-x\right\}$
$\bf{My\; Try::}$ Here $(x+1)\;,(x+2)\;,(x+3)\;,(x+4)\;,(x+5)>0\;,$ when $x\righ... | $$\lim _{t\to 0}\left(\left[\left(\frac{1}{t}+1\right)\left(\frac{1}{t}+2\right)\left(\frac{1}{t}+3\right)\left(\frac{1}{t}+4\right)\left(\frac{1}{t}+5\right)\right]^{\frac{1}{5}}-\frac{1}{t}\right) = \lim _{t\to 0}\left(\frac{\sqrt[5]{1+15t+85t^2+225t^3+274t^4+120t^5}-1}{t}\right) $$
Now we use the Taylor's developmen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 2
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Limit of function of hyperbolic How can I - without using derivatives - find the limit of the function
$f(x)=\frac{1}{\cosh(x)}+\log \left(\frac{\cosh(x)}{1+\cosh(x)} \right)$
as $x \to \infty$ and as $x \to -\infty$?
We know that $\cosh(x) \to \infty$ as $x \to \pm \infty$ thus $\frac{1}{\cosh(x)} \to 0$ as $x \to \pm... | $$
\begin{aligned}
\lim _{x\to \infty }\left(\frac{1}{\cosh \left(x\right)}+\ln\left(\frac{\cosh \left(x\right)}{1+\cosh \left(x\right)}\right)\right)
& = \lim _{x\to \infty }\left(\frac{1+\ln \left(\frac{\left(\frac{e^x+e^{-x}}{2}\right)}{\left(\frac{e^x+e^{-x}}{2}\right)+1}\right)\left(\frac{e^x+e^{-x}}{2}\right)}{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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What do the (+) and (-) symbols after variables mean? This paper describing an Unscented Kalman Filter implementation uses notation that I am unfamiliar with nor can find on eg https://en.wikipedia.org/wiki/List_of_mathematical_symbols
Xu et al (2008)
An example line is:
$$\chi_{i,k-1}(+) = f(\chi_{i,k-1}), x_k(-)=\su... | They seem to refer to the a priori and a posteriori estimate. Compare the equations with these, Wikipedia uses $x_{k∣k−1}$ where your paper uses $x_k(−)$ and $x_{k∣k}$ instead of $x_k(+)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of $x^5\cos\left(\frac1x\right)$ as $x$ approaches $0$ Find the limit of $x^5\cos\left(\frac1x\right)$ as $x$ approaches $0$.
Can I just substitute $0$ to $x^5$? But what would be $\cos\left(\frac10\right)$ be?
I could solve for $-x^4\le x^4\cos(1/x)\leq x^4$
in which limit of $x^4$ is and $-x^4=0$ and by sandwic... | Notice, $-1\le \cos \left(\frac 1x\right)\le 1\ \ \forall \ \ x\in \mathbb{R}$, hence
$$\lim_{x\to 0}x^5\cos\left(\frac 1x\right)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1666981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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If $v_1,v_2,\dots,v_n$ are linearly independent then what about $v_1+v_2,v_2+v_3,\dots,v_{n-1}+v_n,v_n+v_1$? If $v_{1},v_{2},...v_{n}$ are linearly independent then are the following too?
$v_{1}+v_{2},v_{2}+v_{3},...v_{n-1}+v_{n},v_{n}+v_{1}$
I tried summing them to give 0 but had no success.
| $$a_1(v_1+v_2) + a_2(v_2+v_3) + \cdots + a_{n-1}(v_{n-1}+v_n) + a_n(v_n + v_1) \\ = a_1v_1 + a_1v_2 + a_2v_2 + a_2v_3 + a_3v_3 + \cdots + a_{n-1}v_n + a_nv_n + a_nv_1 \\ = (a_1+a_n)v_1 + (a_1+a_2)v_2 + \cdots (a_{n-2} +a_{n-1})v_{n-1}+(a_{n-1}+a_n)v_n \\ = b_1v_1 + b_2v_2 + \cdots + b_{n-1}v_{n-1}+b_nv_n$$
Now make con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Evaluating an integral of a form related to $\int_{-\infty}^{\infty} e^{-ax^2} \cdot e^{-2\pi i k x} dx$ Claim
\begin{equation}
\int_{\mathbb R} \exp\left(-2\pi \cdot \left(\frac{x}{\sqrt 2}\right)^2 \right) \cdot \exp\left(-2i \pi \frac{x}{\sqrt 2} \cdot f\right) \mathrm{d}\left( \frac{x}{\sqrt{2}} \right)
= \frac{1}... | Hint:
Let$$I=\int_{\mathbb R}e^{-x^2/2}\cos(fx)dx.$$
Then deriving on $f$ and integrating by parts,
$$I'_f=-\int_{\mathbb R}xe^{-x^2/2}\sin(fx)dx=\left.e^{-x^2/2}\sin(fx)\right|_{\mathbb R}-\int_{\mathbb R} fe^{-x^2/2}\cos(fx)dx=-fI.$$
The solution of this differential equation is
$$I=Ce^{-f^2/2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the orthogonal projection of a given vector on the given subspace W of the inner product space V. Let $V = R^3$ with the standard inner product
$u = (2,1,3)$ and $W = \{(x,y,z) : x + 3y - 2z = 0\}$
I came up with the basis $\{(-3,1,0), (2,0,1)\}$ but these are not orthogonal to each other. I'm not exactly sure... | There are many ways how to find an orthogonal projection.
You seem to want to use an orthogonal (or an orthonormal) basis of $W$ in some way.
If you already have a basis of $W$, you can get an orthogonal basis from it using Gram-Schmidt process.
Another way to do this. Let us choose $\vec b_1=(2,0,1)$ at the first vec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $2$ is a primitive root modulo $13$. Find all primitive roots modulo $13$.
We show $2$ is a primitive root first. Note that $\varphi(13)=12=2^2\cdot3$. So the order of $2$ modulo $13$ is $2,3,4,6$ or $12$.
\begin{align}
2^2\not\equiv1\mod{13}\\
2^3\not\equiv1\mod{13}\\
2^4\not\equiv1\mod{13}\\
2^6\not\equiv1... | 1 . This is Lagrange's theorem. If $G$ is the group $(\mathbb{Z}/13\mathbb{Z})^{\ast}$ (the group of units modulo $13$), then the order of an element $a$ (that is, the smallest number $t$ such that $a^t \equiv 1 \pmod{13}$) must divide the order of the group, which is $\varphi(13) = 12$. So we only check the divisor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Is the action of $\mathbb Z$ on $\mathbb R$ by translation the only such action? It is well known that $\mathbb Z$ acts on $\mathbb R$ by translation. That is by $n\cdot r=n+r$. The quotient space of this action is $S^1$.
Could someone give me an example where $\mathbb Z$ acts on $\mathbb R$ in some other (non trivial... | Recall that a continuous action of $\Bbb{Z}$ on $\Bbb{R}$ consists of a group morphism $\Bbb{Z} \to \operatorname{Aut}(\Bbb{R})$, where $\operatorname{Aut}(\Bbb{R})$ denotes the group of homeomorphisms of $\Bbb{R}$ into itself.
Now, $\Bbb{Z}$ is cyclic, hence any morphism $\mu:\Bbb{Z} \to \operatorname{Aut}(\Bbb{R})$ i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Problem finding Jacobian when computing density I'm having trouble finding the Jacobian when trying to compute a distribution. If $(X,Y)$ is a point on a unit disk with radius $1$, I'd like to find the density of the distance between the point and the centre of the disk.
So the joint density function of $X$ and $Y$ is... | Full solution, using Dirac delta:
$$
p_U(u)=\int_D dx dy \frac{1}{\pi}\delta\left(u-\sqrt{x^2+y^2}\right)\ .
$$
Making a change to polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$, whose Jacobian is $=r$,
$$
p_U(u)=\frac{1}{\pi}\int_0^{2\pi}d\theta\int_0^1 dr\ r\ \delta(r-u)=2u\qquad 0\leq u\leq 1\ ,
$$
which is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability - Can't understand hint professor gave You don't really need any expert knowledge in Machine Learning or linear regression for this question, just probability.
Our model is this: We have our input matrix $X \in \mathbb R^{n \times d}$ and our output vector $y \in \mathbb R^n$ which is binary. $y_i$ is eithe... | It's just a way of writing two identities in one.
If $y_1 = 0$ you get what you had before, and similarly if $y_1= 1:$ there's always one of the two terms canceling. Since $0$ and $1$ are the only possible valuse you can get, the equality holds.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the delta function the continuous generalization of the kronecker delta and not the identity function? In a discrete $n$ dimensional vector space the Kronecker delta $\delta_{ij}$ is basically the $n \times n$ identity matrix. When generalizing from a discrete $n$ dimensional vector space to an infinite dimensio... | I) Let there be given an linear operator $A:V\to V$, where $V$ is a vector space.
*
*If $V$ is finite-dimensional, given a choice of basis, the operator $A$ can be represented
$$(Av)^i ~=~ \sum_j a^i{}_j v^j.\tag{1}$$
by a matrix $a^i{}_j$.
*If $V$ is infinite-dimensional, of the form of an appropriate function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1667753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Compute the matrix $A^n$, $n$ $\in$ $\mathbb{N}$. Compute $A^n$, $n$ $\in$ $\mathbb{N}$., where $
A=\left[\begin{array}{rr}
2&4\\3&13
\end{array}\right]$.
Hi guys, this is the question, I compute the diagonal matrix of $A$ and obtained this $
D=\left[\begin{array}{rr}
1&0\\0&14
\end{array}\right]$. I need use this resu... | If a matrix $A$ is diagonalizable, then what that means is there exists a decomposition of the vector space into eigenspaces. Then the form $A = MDM^{-1}$ is merely a change-of-basis from the basis in which $A$ was expressed into this new eigenbasis.
Therefore, the entries of $M$ will be the eigenvectors associated to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Negation of a statement without using (symbol ¬)
Write the negation of the following statement (without using the symbol $¬$ ):
$\mathrm P ~=~ (∃x ∈ \Bbb R)\Big(\big((∃y ∈ \Bbb R)(x = (1 − y)^2)\big) ∧ \big((∃z ∈ \Bbb R)(x = −z^2)\big)\Big)$
Which statement is true, $\rm P$ or $\rm ¬P$ ?
P.S.... i'm kinda confused as... | What you are supposed to do, is the following: Consider $\neg P$ and then iteratively transform this into an equivalent sentence $Q$ in which the symbols $\neg$ doesn't appear. I will show you how to begin:
(I'm assuming that your $R$ is supposed to be $\mathbb R$ - the set of all reals.)
$$
\begin{align}
&\neg \left( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1668049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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De Mere's Martingales In a casino, a player plays fair a game. If he bets $ k $ in one hand in that game, then he wins $ 2k$ with
probability $0.5$, but gets $0$ with probability $0.5$. He adopts the following strategy. He bets $1$ in the
first hand. If this first bet is lost he then bets $2$ at the second hand. If he ... | For each $n\geqslant 0$ we have
$$X_{n+1} = X_n + 2^{n+1}Y_n.$$ Clearly $\mathbb E[X_0]=0$, and hence $$\mathbb E[X_{n+1}] = \mathbb E[X_n] + 2^{n+1}\mathbb E[Y_n] = \mathbb E[X_n].$$ Therefore $\mathbb E[X_n]=0$ for all $n$. Further,
\begin{align}
\mathbb E[X_{n+1}\mid \mathcal F_n] &= \mathbb E[X_n + 2^{n+1}Y_n\mid\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1668138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Paths of even and odd lengths between cube vertices I have an ordinary cube with the standard 8 vertices and 12 edges. Say I define a path as a journey along the edges from one vertex to another such that no edge is used twice. Then I pick two vertices that are connected by a single edge, ie. a path of length 1. How wo... | Imagine to color the vertices of the cube in this way:
Now imagine you start from a red vertex. Doing one step, in any direction, you go in a blue vertex. It is easy to verify that this hold for each red vertex.
Now check the blue vertices. It is easy to see that from one blue vertex, with one step, you can only go to... | {
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Liouville numbers and continued fractions First, let me summarize continued fractions and Liouville numbers.
Continued fractions.
We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ]=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}$$
where for natural numbers $i$ we have... | Bounding the error.
The error between a continued fraction $[a_0;a_1,a_2,\ldots]$ and its truncation to the rational number $[a_0;a_1,a_2,\ldots,a_n]$ is given by
$$
|[a_0;a_1,a_2,a_3,\ldots] - [a_0;a_1,a_2,\ldots,a_n]|=\left|\left(a_0+\frac{1}{[a_1;a_2,a_3,\ldots]}\right) - \left(a_0 + \frac{1}{[a_1;a_2,a_3,\ldots,a_n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If the limit of a function when x goes to infinity is infinity, does that mean the integral from 1 to infinity is also infinity? If the limit of a function when x goes to infinity is infinity, does that mean the integral from 1 to infinity is also infinity?
Can we use the above to show that an unsolvable integral is in... | The integral
$$\int_{1}^\infty f(x)dx$$
is defined as
$$\lim_{b\to\infty} \int_1^b f(x)dx$$
so it should be simple for you to show that if there exist $x_0\geq 1$ and $M>0$ such that $f(x)>M$ for all $x>x_0$ (this is of course true of $\lim_{x\to\infty} f(x)=\infty$), then $$\int_{1}^\infty f(x)dx = \infty.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Extracting information from the graph of a polynomial Problem: Below is the graph of a polynomial with real coefficients
What can you say about the degree of the polynomial and about the sign of the first three and last three coefficients when written in the usual manner.
My attempt: let $P(x)=a_nx^n+a_{n-1}x^{n-1}+a_... | You're right about $a_0$, $a_1$, $a_2$.
It doesn't seem to be stated that all of the real zeroes are in the $x$-range shown on the graph, but if you do assume that, you can certainly also conclude that the degree is even, and that $a_n$ is negative.
You can say more about the degree than that, though. Clearly it must b... | {
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"source": "stackexchange",
"question_score": "3",
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Fokker Planck and SDE I have the following Fokker-Planck equation in spherical coordinates $(\theta,\phi)$:
$$ \partial f/ \partial t= D \cot\theta \quad \partial f/\partial \theta + \quad 1/\sin^2\theta \quad \partial^2 f/\partial \phi^2 - \quad A[\sin\theta \partial f/\partial\theta +2 \cos\theta f] \tag{1}$$
whe... | The relation between the Fokker-Planck equation and the associated SDE has been investigated by Figalli (2008) and is known as the (stochastic analogue of the) superposition principle.
Michael Röckner and colleagues extended the superposition principle to many different situations, e.g. for McKean-Vlasov SDEs or non-lo... | {
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What's the negation of “At least three of the sentence are false”? Following with the question I asked before
What's the negation of "One of the sentence is false"?
The negation of “At least three of the sentence are false” would be "any, one or two of the sentence is/are false"?
| With f-o logic, the original statement is:
$\exists x \ \exists y \ \exists z \ [(x \ne y \land x \ne z \land y \ne z) \ \land \ (False(x) \land False(y) \land False(z))]$.
Thus, negeatin it:
$\forall x \ \forall y \ \forall z \ \lnot [(x \ne y \land x \ne z \land y \ne z) \ \land \ (False(x) \land False(y) \land Fa... | {
"language": "en",
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Show that $lim_{n \rightarrow \infty}m(O_n)=m(E)$ when $E$ is compact. Below is an attempt at a proof of the following problem. Any feedback would be greatly appreciated. Thx!
Let $E$ be a set and $O_n = \{x: d(x, E) < \frac{1}{n}\}$.
Show
*
*If $E$ is compact then $m(E) = lim_{n \rightarrow \infty} m(O_n)$.
*This... | I refer to your three settings as (1), (2), and (3).
(1) What you write here is confusing. You only have to prove that $\bigcap_nO_n = E$. Clearly, $E\subset\bigcap_nO_n$. Now show that $\bigcap_n O_n\subset E$.
(2) This is a correct counterexample.
(3) The counterexample is false. You have $\bigcap_n O_n = [0,1]$ and ... | {
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Finding a number of twin primes less than a certain number I was doing some problems on number theory, and I came across the following question: "How many twin primes less than 100 exist?" I was wondering if anyone could tell me what method would be used to solve this problem and others too.
Thanks!
| HINT.Only I think to see Wilson's theorem and (because of primes less that $100$) looking for solutions of the two congruences
$$(p-1)!+1\equiv 0 \pmod p$$ $$(p+1)!+1\equiv 0 \pmod {p+2}$$
Paying some attention this could work, I guess. However for primes $p$ larger, verification in tables of primes or twin primes coul... | {
"language": "en",
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Find all possible values of $c^2$ in a system of equations. Numbers $x,y,z,c\in \Bbb R$ satisfy the following system of equations:
$$x(y+z)=20$$
$$y(z+x)=13$$
$$z(x+y)=c^2$$
Find all possible values of $c^2$.
To try to solve this, I expanded the equations:
$$xy+xz=20$$
$$yz+xy=13$$
$$xz+yz=c^2$$
Then I subtracted the f... | Let $c^2 = s$. Eliminating $x$ and $y$, you get an equation in $s$ and $z$:
$$ s^2 + 2 z^2 s - 66 z^2 - 49 $$
Thus
$$ z^2 = \dfrac{s^2 - 49}{66 - 2 s}$$
Since $z^2 \ge 0$, we need either $s \le -7$ or $7 \le s < 33$. This corresponds to $\sqrt{7} \le c < \sqrt{33}$. We then have
$$ \eqalign{y &= \dfrac{z (33 - c^2... | {
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Limits: $\lim_{n\rightarrow\infty}\frac{nx}{1+n^2x^2}$ on $I=[0,1]$ I'm doing an assignment for my analysis course on the uniform convergence. And I have to assess the uniform convergence of the sequence $$f_n(x)=\frac{nx}{1+n^2x^2}$$ on the intervals $I=[1,2]$ and $I=[0,1]$. Before assessing whether there is uniform c... | For $x\in [1,2]$ we have
$$0 < f_n(x) = \frac{nx}{1+n^2x^2} < \frac{nx}{n^2x^2} = \frac{1}{nx} \le \frac{1}{n}.$$
This shows $f_n \to 0$ uniformly on $[1,2].$ (In fact $f_n \to 0$ on any $[a,\infty),$ where $a>0.$)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to solve Schrödinger equation numerically with time dependent potential How to solve the Schrödinger equation with time dependent potential in 1D or 3D (if it is easier):
$$i\hbar\dfrac{\partial \Psi}{\partial t}(x,t)=\left(-\dfrac{\hbar}{2m}\nabla^2-\frac{e^2}{x+\alpha}-exE(t)\right)\Psi(x,t)$$
where $E(t) = E_0 \... | this is not an easy topic to discus, since it would take some writing-up.. you can find in Google the detailed methods. $1D$ is simpler. You can use the split operator time-propagator, where the $x$ and $d/dx$-alike operators are treated differently. When applying the derivative operators, usually one makes use of the ... | {
"language": "en",
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Probability clarification $\chi^2$ distribution table I'm having some trouble understanding the solution of this probability question.
Ammeters produced by a manufacturer are marketed under the specification that the standard deviation of gauge readings is no larger than $.2$ amp. One of these ammeters was used to mak... | $\Bbb P\left(\frac{(n-1)S^2}{\sigma^2}\geq \frac{(n-1)s^2}{\sigma^2}\right) = \Bbb P({\raise{0.5ex}{\chi}}^2_{9}\geq 14.925) = ~$$0.0930...$ $~\approx 0.1{\small 0}$
The value comes from lookup tables or an online calculator.
| {
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Proving that the line joining $(at_1^2,2at_1),(at_2^2,2at_2)$ passes through a fixed point based on given conditions on $t_1,t_2$ Problem:If $t_1$ and $t_2$ are roots of the equation $t^2+kt+1=0$ , where $k$ is an arbitrary constant. Then prove that the line joining the points $(at_1^2,2at_1),(at_2^2,2at_2)$ always pas... | Tgts at ends Parb
With some reference to the above:
When the roots are solved, the tangent drawn at these particular slope pairs all cut on the directrix when the chord between $(t_1- t_2)$ points of tangency passes through the focus.
EDIT 1
There appears to be an incorrect tangent/slope supplying equation to start wit... | {
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Direct limits of simple C*-algebras are simple Let $S$ be a non-empty set of simple C$^*$-subbalgebras of a C$^*$-algebra $A$. Let us also suppose that $S$ is upwards-directed and that the union of all element of $S$ is dense in $A$. Then $A$ is simple.
I can't show this. Could you tell me how to show this?
| Write $S=\{A_j\}$. Then, for any $a\in A$, we have $a=\lim a_j$ with $a_j\in A_j$.
Now fix $I\subset A$, a nonzero ideal. As $A_j\cap I$ is an ideal of $A_j$, we either have $A_j\cap I=0$ or $A_j\cap I=A_j$. We also have that $\{A_j\cap I\}$ is an increasing net of ideals of $A$.
For $a\in I$ and $a=\lim a_j$ with $a... | {
"language": "en",
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Find $\sum r\binom{n-r}{2}$
Let $A=\{1,2,3,\cdots,n\}$. If $a_i$ is the minimum element of set $A_i$ where $A_i\subset A$ such that $n(A_i)=3$, find the sum of all $a_i$ for all possible $A_i$
Number of subsets with least element $1$ is $\binom{n-1}{2}$
Number of subsets with least element $r$ is $\binom{n-r}{2}$
Sum... | Two possibilities:
*
*write $r=\binom r1$ and apply a variation of the Vandermonde identity.
*If you add an element $0$ to your set, then $a_i$ counts the number of ways a fourth element can be chosen, less than $a_i$, and therefore less than all elements of $A_i$. One easily checks this gives a bijective correspon... | {
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Switching limits: $n \rightarrow \infty$ for $n\rightarrow 0$ I feel like this question may have already been asked, but despite my searches, I could not find it.
I am looking to prove that $\lim\limits_{\epsilon \rightarrow 0} \int_{[b, b+\epsilon]} g=0$ for an integrable function $g$. To do so, I would like to use ... | Your approach of setting $\epsilon = 1/k$ can be made to work. However, you need to make some adjustments to your argument:
*
*Note that it suffices to show that $\int_{[b,b+\epsilon]}|g|\to 0$
*Note that for $\epsilon \in [1/(k+1),1/k]$, we have
$$
\int_{[b,b+1/(k+1)]}|g| \leq
\int_{[b,b+\epsilon]}|g| \leq
\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1670122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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is $E(x),E(y) ↦ E(x+y)$ well defined?
Let $x,y\in \mathbb R$ and $x\sim y \iff x-y\in \mathbb Z$. $E(x)$ is
the equivalance class containing $x$.
a) Is $E((x),E(y)) ↦ E({x+y})$ well defined?
Where $\rightarrow$ means an operation
b) Is $(E(x),E(y)) ↦ E(xy)$?
| If the notation $(E(x),E(y))$ is mean an operation, then $(E(x),E(y))=(E(z),E(w))$.
If the notation $(E(x),E(y))$ is mean an pair, then $(E(x),E(y))\neq(E(z),E(w))$.
| {
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"timestamp": "2023-03-29T00:00:00",
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finding the area using integral
Find the area blocked between X-Y axis and $x=\pi$, $y=\sin x$ according to $x$ and $y$.
According to x: $\int_0^\pi \sin x \, dx=-\cos(\pi)+\cos(0)=2$
According to $y$: $\int_0^1 (\pi- \sin^{-1} y)\,dy=\left[\pi y-\sin^{-1}y + \sqrt{1-y^2}\right]_0^1 = \pi-\frac{2}{\pi}+1=\frac{2}{\pi... | $\sin^{-1}$ is the inverse of the restriction of the function $\sin$ to the interval $[-\pi/2,\pi/2]$. But you're working with areas outside of that interval.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1670307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Describing the motion of a particle (sphere) If I have the following position at time t : $\hat{r}(t) = 3\cos(t)\hat{i} + 4\cos(t)\hat{j} + 5\sin(t)\hat{k}$ , then how can I tell if the particle's path lies on a sphere or not? If e.g. the second term was simply $t$ and not a trigonometric function, I know that it would... | Let $t = \theta + \frac{\pi}{2}$, so that $\hat{r}(t)$ gives $x = 3\cos(\theta + \frac{\pi}{2}) = -3\sin\theta, y= -4\sin \theta, z = 5\cos\theta$.
Comparing to the spherical coordinates: $x = r\sin\theta \cos\phi, y = r\sin\theta \sin\phi, z = r\cos\theta$.
Then radius of sphere $r = 5$, but we need to find the fixed ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.
Theorem 2.1 : Suppose that $f$ is an integrable function on the circle with $\hat f(n)=0$ for all $n \in \Bbb Z$. Then $f(\theta_0)=0$ whenever $f$ is continuous at the point $\theta_0$.
Proof : We suppose first that $f$ is r... | *
*Continuity tells us we can choose $\delta>0$ so that $|f(\theta) - f(0)|< \frac{f(0)}{2}$ if $|\theta - 0| < \delta$ (which in particular implies $f(\theta) > \frac{f(0)}{2}$). Once the existence of such a $\delta$ is established, we can assume it is as small as we need; in particular, we're free to take it to be l... | {
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Proving a sequence is increasing Prove the sequence defined by $a_0=1$ and $a_{n+1}=\sqrt{3a_n+4}$ is increasing for all $n\ge0$ and $0\le a_n\le4$
I know that a sequence is increasing if $a_n\le a_{n+1}$ but I don't know what information I can use to prove that since all I have is $a_{n+1}$ and a base case of n=0. Am ... | Observe that if $1 < x < 4$, then essentially, $x^2 - 3x + 4 < 0$. Prove by induction that $1 < a_n < 4$. Then, set $x = a_n$ to get that $\{a_n\}$ is increasing.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Strangely defined ball compact in $L^p(I)$ or not? Let $I = (0, 1)$ and $1 \le p \le \infty$. Set$$B_p = \{u \in W^{1, p}(I) : \|u\|_{L^p(I)} + \|u'\|_{L^p(I)} \le 1\}.$$When $1 < p \le \infty$, does it necessarily follow that $B_p$ is compact in $L^p(I)$?
| Note that
$B_p$ compact in $L^p$ $\Rightarrow$ $B_p$ closed in $L^p$ $\Rightarrow$ $B_p$ complete in $L^p$ $\Rightarrow$ $W^{1,p}(I)$ complete w.r.t. $\|\cdot\|_{L^p}$
So, $B_p$ isn't compact in $L^p$.
However, $B_p$ is relatively compact in $L^p$.
| {
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Equivalence Relations and Classes 3 I am studying for a discrete math exam that is tomorrow and the questions on equivalence classes are not making sense to me.
Practice Problem: Let $\sim$ be the relation defined on set of pairs $(x, y) \in R^2$ such that $(x, y) \sim (p, q)$ if and only if $x^2 + y^2 = p^2 + q^2$. Fi... | The equivalence relation is defined such that $(x,y) \sim (p,q)$ if $x^2 + y^2 = p^2 + q^2$.
If $(x,y) \in[(0,1)]$, then $(x,y) \sim (0,1)$, so it must satisfy
$$
x^2 + y^2 = 0^2 + 1^2 = 1
$$
Which is the equation of the unit circle, which all three of those points are a part of.
| {
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"timestamp": "2023-03-29T00:00:00",
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Analytic vs. Analytical I am trying write an article.
Little Summary:
I have developed some tools to analyze derivative of some function $f$. This characterization leads to better results than previous works that only studied the function itself.
I am trying to say that:
"Our analytic view of the problem provides a b... | If you have doubts, most probably some other people will also have!
It is always better to be clear, although it is sometimes a difficult decision where to stop (it depends on whether it is for a paper or for a book, whether it is for an abstract or introduction, etc, etc).
Summing up, much better something like: "By l... | {
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Combinatorics problem involving selection of digits How many 10-digit decimal sequences using $(0, 1, 2, . . . , 9)$ are there
in which digits 3, 4, 5, 6 all appear?
What I did to solve this question was this.
The number of ways to select $3,4,5,6$ from $10$ numbers is
$$\binom{10}{4}$$
and the ways to fill the rest ... | First, your description of $10 \choose 4$ is wrong because there is only one way to select specifically those digits. What you really should be saying is there are $10 \choose 4$ ways to select the positions for the $3,4,5,6$. Second you should multiply by $4!$ for the orders of $3,4,5,6$ Third, you are double count... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation for a smooth staircase function I am looking for a smooth staircase equation $f(h,w,x)$ that is a function of the step height $h$, step width $w$ in the range $x$.
I cannot use the unit step or other similar functions since they are just one step. I have been experimenting with various sigmoid curves and whil... | Here is an example based on Math536's answer: Wolfram link
$$f(h,w,a,x) = h \left[\frac{\tanh \left( \frac{ax}{w}-a\left\lfloor \frac{x}{w} \right\rfloor-\frac{a}{2}\right)}{2\tanh\left(\frac{a}{2}\right) } + \frac{1}{2} + \left\lfloor \frac{x}{w} \right\rfloor\right]$$
Where h is the step height, w is the period, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Can I change this summation to a sum of other summations? The form of the summation I have is
$$\sum _{ x=0 }^{ \infty }{ x{ a }^{ x } } $$
I need to somehow remove the $x$ from the original summation in order to achieve the geometric series in each other summation. For instance, $$\sum _{ x=? }^{ \infty }{ { a }^{... | This is one of my favourite tricks. Multiplying a series by a carefully chosen term. It is especially useful in dealing with arithmetic-geometric series.
$$\begin{align}
S &= a + 2a^2 + 3a^3 + \dots \\
aS &= a^2 + 2a^3 + 3 a^4 + \dots \\
S - aS &= a + a^2 + a^3 + \dots \\
S &= \frac {a + a^2 + a^3 + \dots}{1-a}\\
S ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Nonlinear optimization: Optimizing a matrix to make its square is close to a given matrix. I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.
Here is my problem:
Assume we have an unknown Q wi... | For the modified question, let me try to give an answer that can address situations with matrices having the structure that you have.
Basically, your matrix $G$ has the following structure $$G=uv^T$$ where I have taken $u$ as the all $1$'s vector and $v$ a vector of positive coordinates such that $v^Tu=1$, i.e. $G$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Opening and closing convex sets It seems true that, given $K \subseteq \mathbb{R}^n$ a convex set with $K^\circ \neq \emptyset$, then $\overline{K^{\circ}} = \overline{K}$ and $\left ( \overline{K} \right )^\circ = K^\circ$.
I am able to prove the first equality by making use of the "segment Lemma", which states that i... | Either use the suggestion by Andrea, or you can prove directly this way: $K \subseteq \bar K$, so $\mathring K \subseteq (\bar K)^\circ$. On the other hand, if $x \in (\bar K)^\circ$ then there is a neighborhood $x \in U_x \subseteq \bar K$. Now take a small simplex in $U_x$ that contains $x$ in its interior. Up to per... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What is the speed of the car given the time taken to receive an echo? I am trying to solve this question-
The driver of an engine produced a whistle sound from a distance $800m$ away a hill to which the engine was approaching.The driver heard the echo after $4.5s$.Find the speed of the car is speed of sound through ai... | See sound will travel $340*4.5=1530m$ but in this time train will also travel some distance. original distance between hill and train back and forth is $1600m$ but sound was heard at $1530m$ from hill ie $70m$ from original place of train so train travelled $70m$ in $4.5s$ thus speed is approximately $70/4.5=15.55m/s$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inverse Laplace Transform of $e^{\frac{1}{s}-s}$ doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert:
$$
F(s) = e^{\frac{1}{s}-s}
$$
I can't find it in any table and the strong singular growth for $s=0$ makes me think that perhaps the inverse doesn't exist? Does i... | The inverse most definitely exists. Write the integrand of the ILT in its Laurent expansion about $s=0$ as follows:
$$F(s) e^{s t} = e^{\frac1s} e^{(t-1) s} = \left (1+\frac1s +\frac1{2! s^2} + \frac1{3! s^3}+\cdots \right ) \left [1+(t-1) s+\frac1{2!} (t-1)^2 s^2 + \cdots \right ]$$
The ILT is simply the residue of t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Linear Algebra Coordinate Systems Isomorphism This is an excerpt from the book.
Let $B$ be the standard basis of the space $P_3$ of polynomials; that is let $B=\{1,t,t^2,t^3\}$. A typical element $p$ of $P_3$ has the form
$p(t) = a_0 + a_1t+ a_2t^2 + a_3t^3.$
Since $p$ is already displayed as a linear combination of ... | The polynomial is not turned into a vector. The vector $[P]B=[a_0,a_1,a_2,a_3]$ shows the coordinate of $f(x)=a_0+a_1x+a_2x^2+a_3x^3$ when you choose $B=[1,x,x^2,x^3]$ as your basis. As I understand, there is no intention to change the basis.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the volume of the solid generated by rotating $B$ about $l$
In a space, let $B$ be a sphere (including the inside) with radius of $1$. Line $l$ intersects with $B$, the length of the common part is the line segment with the length of $\sqrt{3}$. Find the volume of the solid generated by rotating $B$ about $l$.
T... | It can be shown that you are describing a circle with radius $1$ and centre $(0,0.5)$.
There are two parts to the curve:
Green: $y_1=\frac 12 + \sqrt{1-x^2}$
Red: $y_2=\frac 12 - \sqrt{1-x^2}$
Rotate the green part about the $x$-axis between $x=-1$ and $x=1$ to find a large volume.
Then rotate the red part between $x=... | {
"language": "en",
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Why is the Jacobian determinant continous in the proof of the Inverse function theorem.
Can anyone explain to me why the first sentence in the proof is valid.
| Alternatively to the answer of Oskar Linka, we can also see the continuity by looking at the Leibniz-formula for determinants.
First of all, note that $f \in C'$. (I guess $C'$ denotes the set of all maps whose partial derivatives exist in all variables and are continuous, i.e. all the maps $\frac{\partial f_i}{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1671981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Polynomial game problem: do we have winning strategy for this game? I'm thinking about some game theory problem. Here it is,
Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and "Girl" are playing game in following way:
*
*First,... | If the girl puts $0$ at position $B$, the boy can choose $1764$, resulting in roots $[-6, 7, 42]$ or $[-864, -1440, 540]$.
EDIT: If the polynomial $(x-a)(x-b)(x-c)$ has $ab+bc+ac=0$, then $c = -ab/(a+b)$. Writing $t = a+b$, we need $t$ to divide $ab = at - a^2$. Thus $t$ is a divisor of $a^2$.
Here's a Maple program... | {
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Assumption and simple calculation I'm having an issue with what seems to be an simple question.
Here it is:
Two hockey teams, team A and team B played a game, Team A beat Team B by 2 goals. The crowd was pleased as there were 8 goals in total for the whole game.
What was the game's scored?
How many goals did team A sc... | If I may, how did you deduce that Team A scored 6 goals and team B scored 2. There were 8 goals TOTAL for the entire game. And we know the condition $$Ascore= Bscore+2$$ Knowing that, recall what the total score was and deduce your answer by system of equations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1672220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Restricting a map from $S^{2n-1}$ to $\mathbb{R}P^{2n-1}$ This might be really obvious, but I have constructed a continuous map $f:S^{2n-1} \rightarrow S^{2n-1}$ with no fixed points, and I want to use this to get a continuous map $g:\mathbb{R}P^{2n-1} \rightarrow \mathbb{R}P^{2n-1}$ with no fixed points (where $\mathb... | Hint: Let me put $m$ for $2n-1$ for brevity and generality. $\Bbb{R}P^m$ is a quotient space of $S^m$ obtained using the equivalence relation that identifies antipodal points. You need to show that your function $g : S^m \to S^m$ is compatible with the equivalence relation, so that it induces a function on the equivale... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1672314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Binomial distribution with random parameter I want to compute the probabbilty
$$
P(S = k) = \sum_{l = 0}^n P(S=k|N=l)P(N=l)
$$
where $N$ is a binomial random variable $B(n, p)$. And when $N = l$, $S$ is also a binomial $B(l, r)$. I tried to compute but it seems difficult, thank you for any anwser or suggestion.
| You can think of $S$ this way. There are $n$ potential coins. For each of them, we have a Bernoulli random variable $Y_i$ which equals 1 with probability $r$. Then we choose which of the coins we use, choosing each of them with probability $p$. We sum $Y_i$ over the remaining coins. We can set $X_i$ to be 1 if the $i$t... | {
"language": "en",
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Solve the recurrence relation $a_{k+2}-6a_{k+1}+9a_k=3(2^k)+7(3^k), k\geq 0, k_0=1,k_1=4.$
Solve the recurrence relation $a_{k+2}-6a_{k+1}+9a_k=3(2^k)+7(3^k), k\geq 0, k_0=1,k_1=4.$
I know that I need a general solution of the form $a_k=a^{(h)}_k+a^{(p)}_k$, where the first term is a general solution to the recurrenc... | The $3(2^k)$ part is easy to deal with. Look for a solution of the shape $c(2^k)$.
The $7(3^k)$ part is made more complicated by the fact that $3$ is a root, indeed a double root, of the characteristic polynomial. Look for a solution of the shape $dk^2(3^k)$. There will be very nice cancellation.
For a particular solut... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does the sequence $a^n/n!$ converge? The sequence when plotted converges to zero because a factorial grows faster than the numerator, but I can not prove that this sequence actually converges.
| METHODOLOGY $1$:
Here is another way forward that is very efficient.
We know from the $n$'th Term Test that if a series converges, then its terms must approach zero.
Inasmuch as the series
$$\sum_{n=0}^\infty \frac{a^n}{n!}=e^a$$
converges, then the $n$'th Term Test guarantees that $\lim_{n\to \infty}\frac{a^n}{n!}=... | {
"language": "en",
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Closed form 0f $I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \right) } \right) }^{ 3 } }{ 1-x } dx }$ While solving a problem, I got stuck at an integral. The integral is as follows:
Find the closed form of: $$I=\int _{ 0 }^{ 1 }{ \frac { \ln { x } { \left( \ln { \left( 1-{ x }^{ 2 } \rig... | We have $$ I=\int_{0}^{1}\frac{\log\left(x\right)\log^{3}\left(1-x^{2}\right)}{1-x}dx=\int_{0}^{1}\frac{\log\left(x\right)\log^{3}\left(1-x^{2}\right)}{1-x^{2}}dx+\int_{0}^{1}\frac{x\log\left(x\right)\log^{3}\left(1-x^{2}\right)}{1-x^{2}}dx
$$ and so if we put $x=\sqrt{y}$ we get $$I=\frac{1}{4}\int_{0}^{1}\frac{y^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1672759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing continuity of an operator from $L^p$ to $L^q$ Question: Let $1 \leq q \leq p < \infty$ and let $a(x)$ be a measurable function. Assume that $au \in L^q$ for all $u \in L^p$. Show that the map $u \to au$ is continuous.
My Approach: I have tried to use closed graph theorem to show continuity of $a$.
Let $\{u_n\} ... | This is not true. Let $b:L^p \to \mathbb{R}$ be a discontinuous linear functional (such functional always exists on infinite dimensional Banach spaces) and let $f\in L^q.$ Then the operator $a:L^p \to L^q ,$ defined by $a(f)(x) =b(x)\cdot f(x) $ satisfies assumptions of your exercise but $a$ is not continuous.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Limit of the function $\lim \limits_{(x,y)\to (0,0)} \sin (\frac{x^2}{x+y}) \ (x+y \neq 0)$. Since $\sin x$ is a continuous function at $(0,0)$ it suffices to check if the limit $\lim_{(x,y)\to (0,0)} \frac{x^2}{x+y}$ is finite.
I seem to be missing the idea in order to show that the limit $\lim_{(x,y)\to (0,0)} \frac... | We can see there's a problem with your method if
$\;\cos\theta+\sin\theta=0\iff\tan\theta=-1\iff \theta=-\frac\pi4\;$ , in the trigonometric circle.
We can try for example:
$$\begin{align*}&y=x:\implies \frac{x^2}{x+y}=\frac{x^2}{2x}=\frac x2\xrightarrow[(x,y)\to(0,0)]{}0\\{}\\
&y=x^2-x:\implies\frac{x^2}{x+y}=\frac{x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Two Subnormal subgroups with index of one and order of other relatively prime Let $H,K$ two subgroups of a finite group $G$. Suppose that $\gcd(|G:H|,|K|)=1$
Prove that if $K\triangleleft\triangleleft\; G$ then $K\subseteq H$.
My idea: Consider before the case $K\triangleleft G$. Then $HK\leq G$.
Remember that $|HK|=\d... | Let $\pi$ be the set of primes dividing $|K|$, and let $O_\pi(G)$ be the largest normal subgroup of $G$ whose order is divisible only by primes in $\pi$. Then $O_\pi(G)$ is characteristic and hence normal in $G$, so by the case you have solved, $O_\pi(G) \le H$.
So now we have to prove that $K \le O_\pi(G)$ You can do ... | {
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"timestamp": "2023-03-29T00:00:00",
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Coincidence Lemma and universal validity? I'm supposed to say whether the following statement is true or not:
$$ \exists x \exists y f(x) = y \equiv \forall x \exists y (( x = y \vee E(x,y)) \rightarrow \exists z (z=y \vee E(z,y)))$$
I have a couple of questions about the coincidence lemma here:
If you assume that your... | The lefthand side is true in any (nonempty!) model (= interpretation); the righthand side is true in any model. Function symbols are total functions, and the righthand side is true basically because $\forall x\exists y(whatever(x,y)\to\exists z(z=y \lor whateverElse(y,z)))$ is valid.
Re your question about the "coinci... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$T^3=\frac{1}{2}(T+T^*) \rightarrow$ T is self adjoint
Let $T$ be a normal transformation on a finite-dimensional Hilbert space; that is, $TT^*=T^*T$, where $T^*$ is the adjoint of $T$.
Prove that if $T^3=\frac{1}{2}(T+T^*)$, then $T$ is self adjoint.
I have tried to do some math on $(Tv,u)$ but I was not successful ... | $T^3=\frac{1}{2}(T+T^*)\Rightarrow T^4=\frac{1}{2}(T^2+TT^*)=\frac{1}{2}(T^2+T^*T)\text{ implies }TT^*=TT^*.$
Hence T is normal. Since $T$ is normal, there exists an orthonormal basis consists of its eigenvectors such that $T=UDU^*$, where $D$ is diagonal and $U$'s column vectors are $T$'s eigenvectors. $T$ is normal a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Integration involving $\arcsin$ How to integrate the following function:
$$\int_0^a \arcsin\sqrt{a \over a+x}dx$$
By using the substitution $x = a\tan^2\theta$, I managed to write the integral as:
$$2a\int_0^{\pi \over 4}\theta \frac{\sin\theta}{\cos^3\theta}d\theta$$
How would I proceed? Should I use by parts method?
| You are on the right track.
By parts,
$$\int\theta \frac{\sin\theta}{\cos^3\theta}d\theta=\frac\theta{2\cos^2(\theta)}-\int\frac{d\theta}{2\cos^2(\theta)}=\frac\theta{2\cos^2(\theta)}-\frac{\tan(\theta)}2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$ Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} \phi_{\sqrt{t}}(p)$$for $t \in (-\epsilon , \epsilon)$ where $\epsilon$ i... | Let me please put $\alpha(t):=\beta(t^2)$. We are asked to prove
$$\left.\frac{d}{dt}\right|_{t=0}\alpha(\sqrt{t})=[X.Y]_p.$$
Fact: $$\left.\frac{d}{dt}\right|_{t=0}\alpha(t)=
\left.\frac{d}{dt}\right|_{t=0}\psi_{-t}\phi_{-t}\psi_t\phi_t(p)
=-\psi'_0\phi_0\psi_0\phi_0(p)-\psi_0\phi'_0\psi_0\phi_0(p)
+\psi_0\phi_0\psi'_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a number n as pa+qb How can we express a number $n$ as $pa+qb$ where $p \geq0$ and $q \geq 0$ and $p$ and $q$ can't be fraction. In contest I got a puzzle as if we can express $c$ as sum of $a$ and $b$ in form $pa+qb$.
Suppose $a$ is $3$ and $b$ is $4$ and $c$ is $7$ so we can express $7$ as $3+4$.
Suppose $a$ is $4$ a... | If $a,b$ are fixed natural numbers, then the set of all integers $n$ which can be written in the form $n=pa+qb$ for integers $p$ and $q$ is precisely the set of multiples of the greatest common divisor of $a$ and $b$. So $n$ can be written in the desired form if and only if $n$ is divisible by $\gcd(a,b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$ p \in Q[x] $ has as a root a fifth primitive root of unity, then every fifth primitive root of unity is a root of $p$. I'm extremely stuck. Can't figure it.
The conjugate is easy: let $w$ be a primitive root of unity, then $w^{-1}$ will also be a root, that's easy. But I'm missing $w^2$ and $w^3$. Why would they be ... | Since $\;p(x)\in\Bbb Q[x]\;$ and it vanishes on $\;\zeta:=e^{2\pi i/5}\;$ , the minimal polynomial of $\;\zeta\;$ over the rationals (also known as a cyclotomic polynomial) also divides $\;p(x)\;$ , and thus
$$\Phi_5(x)\,|\,p(x)\implies p(x)=\overbrace{(x^4+x^3+x^2+x+1)}^{=\Phi_5(x)}\cdot h(x)\;,\;\;h(x)\in\Bbb Q[x]$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Symplectic manifold, flow $\phi_t$ generated by unique vector fiel $X_f$ preserves symplectic form $\omega$? Let $M$ be a symplectic manifold with symplectic form $\omega$, let $f$ be a smooth function on $M$, and let $X_f$ be the unique vector field on $M$ so that $df(Y) = \omega(X_f, Y)$ for all vector fields $Y$. Do... | I'm just going to call your vector field $X$. It suffices to show that the Lie derivative $\mathcal L_X \omega = 0$ (do you see why?) To do this, use Cartan's formula for the Lie derivative of a form: $\mathcal L_X \omega = d\iota_X \omega + \iota_X d\omega = d\iota_X \omega$, because $\omega$ is closed; now by your as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$ isomorphic to? This question came up after I'd solved the following exercise:
Determine the order of $\mathbb Z \oplus \mathbb Z / \langle (2,2) \rangle$. Is the group cyclic?
I had no trouble solving the exercise: The answer is the group is infinite and no... | Let $G = \dfrac{\mathbb Z \oplus \mathbb Z}{ \langle (2,2) \rangle}$.
*
*$v_1=(1,1)$ is mapped to an element of order $2$ in $G$.
*$v_2=(2,1)$ is mapped to an element of infinite order in $G$.
This means that $G$ is infinite and non-cyclic.
Note that $\mathbb Z \oplus \mathbb Z = v_1 \mathbb Z \oplus v_2\mathbb Z$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1673889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Arithmetic growth versus exponential decay I have a kilogram of an element that has a long half-life - say, 1 year - and I put it in a container. Now every day after that I add another kilogram of the element to the container.
Does the exponential decay eventually "dominate" or does the amount of the substance in the c... | On the first day, you have 1 kilogram of decayium. On the second day, that 1 kilogram decays into $1/\sqrt[365]{2}$ kilograms of decayium, on the third day you have $1/(\sqrt[365]{2})^2$ kilograms from the original... And so on, until after a year has passed, you finally have $1/(\sqrt[365]{2})^{365} = 1/2$ kilograms f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Continuous Poisson distribution: $\int_0^\infty \frac{t^n}{n!}dn$ I was thinking poission distribution, actually i like it. Then i thought there is no reason for some events to be integers. We can define occurences as half finished homeworks for example, or 3.7 apples etc.
So when i give wolfram an example, it actually... | Since $n!=\Gamma(n+1)$, you are then asking about a closed form of
$$
\int_0^\infty \frac{t^x}{\Gamma(x+1)}dx, \quad t>0. \tag1
$$ There is no known closed form of $(1)$, but this integral has been studied by Ramanujan who proved that
$$
\int_0^{\infty} \frac{t^x}{\Gamma(1+x)} \, dx = e^t - \int_{-\infty}^{\infty} \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What are the elements of a filtration generated by a Wiener process? I understand the concept of filtration intuitively, and I can wrtite down the elements of a filtration for example in the case of a coin toss game, but what are the sets in the filtration of a Wiener process at a given time? How do they look like?
| For example, a stock price $S_t$ is usually modelled by the following
$$dS_t=S_tdt+S_tdW_t$$
where $W_t$ is a Wiener process. In this context, roughly speaking, events that generate the filtration (natural filtration of $W_t$) are any random events that can influence the stock price $S_t$. For instance, at time $t$ the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Metric embedding of negatively curved surfaces Suppose a metric surface, simply connected and locally isometric to the hyperbolic plane; Do you can embed this surface on the hyperbolic plane?
The fact is known to be false in the spherical case (that is, there exist a surface, simply connected and locally isometric to t... | For a counterexample, take any point $p \in \mathbb{H}^2$, and take the universal cover of $\mathbb{H}^2 - p$.
A spherical counterexample is similar, except removing a single point does not work because the result is already simply connected and so nothing changes when you take its universal cover. So you simply remove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Summing the terms of a series I have a really confusing question from an investigation. It states-
Find the value of:
$$\sqrt{1^3+2^3+3^3+\ldots+100^3}$$
How would I go about answering this??
| And if you don't know the formula
and don't need it exactly,
$\sum_{k=1}^{100} k^3
\approx \int_0^{100} x^3 dx
=\frac{100^4}{4}
$
so the result is
$\sqrt{\frac{100^4}{4}}
=\frac{100^2}{2}
=5000
$.
If you add in the
usual correction of
$\frac12 f(n)$,
the result is
$\sqrt{\frac{100^4}{4}+\frac12 100^3}
=\frac{100^2}{2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Error bound in the sum of chords approximation to arc length We are currently covering arc length in the calculus class I'm teaching, and since most of the integrals involved are impossible to solve analytically, I'd like to have my students do some approximations instead. Of course we could use the usual rectangle-ba... | \begin{align*}
e_{h}
&= \int_{a}^{a+h} \sqrt{1+f'(x)^{2}} \, dx-\sqrt{h^{2}+[f(a+h)-f(a)]^{2}} \\
&= \frac{h^{3}}{24}
\frac{f''(a)^{2}}{\left[ 1+f'(a)^{2} \right]^{\frac{3}{2}}}+O(h^{4}) \\
E &=\frac{(b-a)^{3}}{24n^{2}}
\frac{f''(\xi)^{2}}{\left[ 1+f'(\xi)^{2} \right]^{\frac{3}{2}}}
\end{align*}
Alte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is it true that if $(n\Delta s_n)$ is bounded then $(s_n)$ is bounded? Let $(s_n)$ be a sequence of real numbers and $\Delta s_n=s_n-s_{n-1}.$ Is the following always true?
If $(n\Delta s_n)$ is bounded then $(s_n)$ is bounded.
| Let $s_n=\sum_{k=1}^n\frac1k$. Then $$Δs_n=\frac1n$$ hence $nΔs_n=1$ for all $n\in \Bbb N$ but $s_n\to +\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\liminf_{n→∞} s_n \le \liminf_{n→∞} σ_n$
I am trying to prove that $$\liminf_{n→∞} s_n \le \liminf_{n→∞} σ_n$$
given that $σ_n=\frac1n(s_1+s_2+\dots+s_n)$.
Setting $\alpha = \liminf_{n→∞}s_n$, hence $$\forall \epsilon>0 \ \exists N: \forall n \geq N, \ \alpha - \epsilon <\inf s_n \le s_n$$ Now, I kn... | Note: this is a response to the first version whose details were a little hard to read.
Fix any $N$. For any $n>N$ we have
$$\sigma_n =\frac 1n (s_1+\dots+s_n) \ge \frac 1 n ( s_1+\dots+ s_N + (n-N) \min\{s_{N+1},\dots,s_n\}) \ge \frac{s_1+\dots+s_N}{n} + \frac{n-N}{n} \inf \{s_{N+1},s_{N+2},\dots\}.$$
Therefore $$\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Optimal rounding a sequence of reals to integers I'm given positive real numbers $c_1,\dots,c_m \in \mathbb{R}$ and an integer $d \in \mathbb{N}$. My goal is to find non-negative integers $x_1,\dots,x_m \in \mathbb{N}$ that minimize $\sum_i (x_i - c_i)^2$, subject to the requirement $\sum_i x_i = d$.
I'm inclined to s... | This is an MIQP (Mixed Integer Quadratic Programming) problem. This version is the easy one: convex. That means there are quite a few good solvers available to handle this. Still, finding proven global optimal solutions is often difficult. On the other hand, solvers find typically good solutions very quickly. For $n=10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1674971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Is there a way to obtain exactly 2 quarts in the 8-quart or 5-quart pitcher? Suppose we are given pitchers of waters, of sizes $12$ quarts, $8$ quarts, and $5$ quarts. Initially the $12$ quart pitcher is full and the other two empty. We can pour water from one pitcher to another, pouring until the receiving pitcher is ... | I would go to (7, 0, 5), then (7, 5, 0), (2, 5, 5) and (2, 8, 2). This is four pours.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Could anyone explain why this is a general case of Weierstrass Approximation? Suppose $X_1, X_2 ...$ are independent Bernoulli random variables. with probability $p$ and $1-p$.
Let $\bar{X}_n = \frac{1}{n} \sum\limits_{i=1}^nX_i$.
If $U \in C^0([0,1],\mathbb{R})$, then $E(U(\bar{X_n}))$ converges uniformly to $ U(E(... | The polynomial here is $E(U(\bar X_n))$. Writing it out, this is
$$
E(U(\bar X_n))=\sum_{k=0}^n U(k/n)P(\bar X_n=k/n)\tag1$$
since $\bar X_n$ takes values in $0/n, 1/n,\ldots k/n$. If we write
$$P(\bar X_n=k/n)=P(n\bar X_n=k)$$
we see that $n\bar X_n=\sum_i X_i$ is a binomial($n,p$) random variable, so the sum (1) beco... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
show quadratic polynomial cannot solve differential equation I have a differential equation$ y\prime + 2xy = 1 $and I need to show that there is no quadratic polynomial that solves this equation. I set $y=Ax^2+Bx+C $ and solved for $y\prime$ plugged in $y$ and $y\prime$ into my differential equation. Is this the corre... | You do not need to
equate coefficients,
and you can easily show that
no polynomial can be
a solution to
$y' + 2xy = 1
$.
Suppose $y$
is a polynomial
of degree $d$.
Then $2xy$
is a polynomial of
degree $d+1$
and
$y'$ is a polynomial
of degree $d-1$.
Therefore
$y' + 2xy
$
is a polynomial
of degree $d+1$,
since the term o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
What are the advantages of outer measure? I am learning about measure theory. I have studied on outer measure then i am learning about Lebesgue measure. But i have a question why we learn outer measure since we have Lebesgue measure? That is What are the advantages of outer measure ?
| Another reason besides the excellent answer given is that every set has an outer measure, whereas not every set has a measure. If a set can't be proven to be measurable, it's common to investigate it with the outer measure. If the set turns out to be measurable, the outer measure results still apply because they agree
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
proof - Bézout Coefficients are always relatively prime I had been researching over the Extended Euclidean Algorithm when I happened to observe that the Bézout Coefficients were always relatively prime.
Let $a$ and $b$ be two integers and $d$ their GCD. Now, $d = ax + by$ where x and y are two integers.
$$d = ax + by ... | You are partially right.Not necessarily. Bezout's identity also mentioned, "more generally, the integers of the form $$ n=ax + by$$ are exactly the multiples of $d$."
This implies if $\gcd(x,y)=d'$, then $n$ is also a multiple of $d'$.
Therefore,
$$n=ax+by=\gcd(a,b)\gcd(x,y)n'$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Diagonalizable Linear Operator - Is $Ker(T)=Ker(T^2)$? Been spending a lot of time on this one.
Given a linear operator T, in a vector space V, having a finite dimension and is diagonalizable - Is $Ker(T)=Ker(T^2)$?
One way is trivial, apply $T$ on $T(v)$ to get $0$, but I cannot find the other way around.
As always, ... | Express $T$ as diagonal matrix. Then the matrix of $T^2$ has the diagonal entries of $T$ squared. In particular, the number of zeroes on the diagonal is the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
The j-function of a point on the boundary of the fundamental domain is real valued. I am trying to prove the following statement:
Let $z \in \mathbb{D} $ (the standard fundamental domain for $SL_{2}(\mathbb{Z})$). Prove that if $z$ lies on the boundary of $\mathbb{D}$, or if $Re(z)=0$, then $j(z) \in \mathbb{R}$.
So fa... | Hint: Show that if $f(\tau)=g(\exp(\alpha\mathrm{i}\tau))$ for some Laurent series $g$ with real coefficients and real $\alpha>0$, then $f(-\bar{\tau}) = \bar{f}(\tau)$. Then use the symmetries of $j$ to show that on the boundary of the fundamental domain, as well as for $\operatorname{Re}\tau=0$, you get $j(\tau) = j(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Ordered integral domain
If $a>0$ and $b>0$, both $a$ and $b$ are integers, and $a|b$. Use ordered integral domain to prove $a<b$.
I wrote: We can write that $b=an$, where $b$ is some positive integer and
we get $b\left(\frac1n\right)=a$;
$\frac1n < 1$;
that proves $b>a$.
Is this correct?
| The problem as stated cannot be proved, e.g. for $a=3$ and $b=3$ we have $a\mid b$ but $a \not < b$.
Assuming the problem was to prove $a\le b$, you should be careful how you conclude that $\frac 1 n \le 1$. When you write $b=an$ for some integer $n$, you should explain why $n$ cannot be negative or $0$.
With rings an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Bijection from $A \rightarrow \varnothing$ My thoughts. We need to prove that:
1 $\forall x,y \in A, \text{ if } f(x) = f(y) \rightarrow x = y$
2 $\forall y \in \varnothing, \exists x \in A, f(x) = y$.
In (1), $f(x) = f(y)$ is false, since neither $f(x)$ nor $f(y)$ have a value, so (1) is vacuously true.
Also, $\f... | Functions $f:A\rightarrow B$ can be thought of as particular subsets of $A\times B$ (ones that satisfy the well-defined property). Since $A\times\emptyset=\emptyset$, there is only one subset of $A\times\emptyset$.
Additionally, for the domain of $f:A\rightarrow B$ to be $A$, for all $a\in A$, there must exist $b\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Normed vector space with a closed subspace Suppose that $X$ is a normed vector space and that $M$ is a closed subspace of $X$ with $M\neq X$. Show that there is an $x\in X$ with $x\neq 0$ and $$\inf_{y\in M}\lVert x - y\rVert \geq \frac{1}{2}\lVert x \rVert$$
I am not exactly sure how to prove this. I believe since $M\... | The quotient space $X/M$ is a non-trivial Banach space with elements that are cosets of the form $x+M$. And $\|x+M\|=\inf_{m\in M}\|x+m\|$. Choose any non-zero coset $x'+M$. Then there exists $m'\in M$ such that
$$
\|x'+m'\| \le 2\|x'+M\|_{X/M}
$$
Then
$$
\|(x'+m')\| \le 2\inf_{m\in M}\|x'+m'+m\| =2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1675944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving $6 \sec\phi \tan\phi = \frac{3}{1-\sin\phi} - \frac{3}{1+\sin \phi}$
$$6 \sec\phi \tan\phi = \frac{3}{1-\sin\phi} - \frac{3}{1+\sin \phi}$$
I can't seem to figure out how to prove this.
Whenever I try to prove the left side, I end up with $\frac{6\sin\theta}{\cos\theta}$, which I think might be right.
As fo... | You can do this either from LHS to RHS or from RHS to LHS.
Solution 1: LHS $\rightarrow$ RHS
$$\require{cancel}\begin{aligned}6\sec\phi\tan\phi&=6\frac{1}{\cos\phi}\frac{\sin\phi}{\cos\phi}\\&=\frac{6\sin\phi}{\cos^2\phi}\\&=\frac{3\sin\phi+3\sin\phi}{\left(1-\sin\phi\right)\left(1+\sin\phi\right)}\\&=\frac{3\left(1+\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
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