Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is the *exact* definition of a bounded subset in a metric space? I see quite a lot of different definitions of a bounded space. For instance, from nLab:
Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that $d(x,y)<r$ for all $x,y∈B$.
From Wiki:
A subset $S$ of a metric... | Assuming the subset is nonempty, the two definitions are equivalent. Each implies the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/730533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
How to solve this Diff Eq (with multiple terms) $$\frac{dy}{dx}=7xy$$
I know this turns into
$$\frac{dy}{y}=7xdx$$
.....etc.
But, how do you solve the following:
$$\frac{dy}{dx}=7x+y$$
Not sure how to seperate the parts to respective sides.
| Here is another approach (guessing, otherwise known as undetermined coefficients).
We call this part the complementary solution, we have
$$y' - y = 0$$
Do you know a function that will provide a zero when differentiated and subtracted from itself?
Choose $y_c(x) = c e^x$.
We now have what is called the particular par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding the Limit in: $\lim\limits_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$ Need some help finding this limit:
$$\lim_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$$
Here is what I have so far:
$$\lim_{x\rightarrow1}\dfrac{\dfrac{1-\sqrt{x}}{\sqrt{x}}}{x-1}$$
$$\lim_{x\rightarrow1}\dfrac{1-\sqrt{x}}{\sqrt{x}... | Another approach is to make the change of variables $y=\sqrt{x}$; because this is a continuous function, we know that the limit as $x\rightarrow 1$ is the same as the limit as $y\rightarrow 1$, so that we have $$\lim_{x\rightarrow 1}{ \frac{1-\sqrt{x}}{x\sqrt{x}-\sqrt{x}}}= \lim_{y\rightarrow 1}{\frac{1-y}{y^3-y}}=\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving $\big(n!^{\frac1n}\big)_{n\in\mathbb N^*} \to \infty$ By definition, for any $a\in\mathbb R$, there exists $k\in\mathbb N^*$ such that, if $n\in\mathbb N+k$, then $n!^{\large\frac1n}>a$. Therefore, by induction, I must:
*
*find some $k$ that satisfies $k!>a^k$;
*show that, for each $n\ge k$, if $n!>a^n$, th... | Using Stirling's approximation, we have $n!> \sqrt{2\pi n}(n/e)^n$. Then $(n!)^{1/n}> \sqrt[n]{2\pi}n^{1/n} \frac{n}{e}$. Since $\sqrt[n]{2\pi}\rightarrow 1$ and $n^{1/n}\rightarrow 1$ but $\frac{n}{e}\rightarrow \infty$, we see that $(n!)^{1/n}\rightarrow \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/730823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
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Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
Prove the following integral
$$I=\int\limits_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\dfrac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$$
This in... | Let $f : [-1,1] \to \mathbb{R}$ be any continuous even function on $[-1,1]$.
Consider following integral
$$I_f \stackrel{def}{=}
\int_0^{\pi/2} f(\sin(\tan x)) dx
= \frac12 \int_{-\pi/2}^{\pi/2} f(\sin(\tan x)) dx
= \frac12 \int_{-\infty}^{\infty} f(\sin y)\frac{dy}{1+y^2}
$$
where $y = \tan x$. Since $f(\cdot)$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 3,
"answer_id": 2
} |
Describing asymptotic behaviour of a function
For question B!
x^2+x+1/x^2
= 1+ [x+1/x^2]
shouldnt the answer be
asymptote at x=0 and y=1 ??
i dont understand the textbook solution
| You are right : there is a vertical asymptote for $x=0$ and an horizontal asymptote at $y=1$.
But the answers given in the textbook correpond to the analysis of the behavior of $y$ when $x$ goes to $0^+$ or $0^-$ as well as when $x$ goes to $+ \infty$ or to $- \infty$. $$y=\frac{x^2+x+1}{x^2}=1+\frac{1}{x}+\frac{1}{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Distribution of different objects into different boxes We want to put n different objects into n different boxes. In how many ways can we do this if we want that exactly two boxes remain empty?
| *
*Select the $2$ of $n$ boxes you wish to remain empty. $\binom{n}{2}$ arrangements.
*Put the remaining $n$ objects, at least $1$ object into each of the $n-2$ remaining boxes and in either:
*
*1 box will have $3$ objects. Giving $\,^{n-2}C_1 \,^{n}P_{3}$ permutations.
*2 boxes will have 2 objects, giving $\,^{n-2}C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Closest distance between two quadratic curves I'm having trouble with the following problem:
Find the closest distance between $x^2+4y^2=4$ and $xy=4$.
I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted axes makes it hard, i think. I also thought about using a circle that has it... | My try : I selected two arbitrary points $(x_e,y_e)$ and $(x_h,y_h)$ and I computed the distance which I say to be minimum.
$y_e$ can be eliminated (expressed as a function of $x_e$ since the point is along the ellipse).
$y_h$ can be eliminated (expressed as a function of $x_h$ since the point is along the hyperbole)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Find the least value of x which when divided by 3 leaves remainder 1, ... A number when divided by 3 gives a remainder of 1; when divided by 4, gives a remainder of 2; when divided by 5, gives a remainder of 3; and when divided by 6, gives a remainder of 4. Find the smallest such number.
How to solve this question in 1... | Hint $\ $ Apply the ubiquitous constant case optimization of CRT
$$\ x\equiv m_i\!-\!2\!\!\!\pmod{m_i}\iff x\!+\!2\equiv 0\!\!\pmod {m_i}\iff m_i\mid x\!+\!2\iff {\rm lcm}\{m_i\}\!\mid x\!+\!2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ by residues We want to compute $\int_0^{\pi/2} \frac{1}{(\sinh t)^{2}+(\sin{\theta)^{2}}} d\theta$ with $t>0$ using residues.
The first thing I want to do is using $z=e^{i\theta}$ to transform the integral to an integral over the unit circle. ... | If you substitute $z = e^{i\theta}$ directly, you get a polynomial of degree $4$ in the denominator, and finding its roots may be a bit difficult. If you first transform the $(\sin\theta)^2$ using a trigonometric identity,
$$\sin^2\theta = \frac{1}{2}(1 - \cos (2\theta)),$$
the substitution leads to a quadratic polynom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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3 Questions concerning the fundamental group of $S^1\vee S^1\vee S^1$ I have the following questions:
1) What is the fundamental group of $S^1\vee S^1\vee S^1$ and why?
2) Is $S^1\vee S^1\vee S^1$ homeomorphic to the bouquet of 3 circles
3) Are $S^1\vee S^1\vee S^1$ and the bouquet of 3 circles homotopic to one another... | *
*The fundamental group of $S^1\lor S^1\lor S^1$ is a free group on three generators: each generator is the homotopy class of a loop about one of the three circles. More generally, if $X$ and $Y$ are path-connected spaces with fundamental group $\pi_1(X)$ and $\pi_1(Y)$ then the fundamental group $\pi_1(X\lor Y)$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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About $ \lim_{x\rightarrow 0}\frac {\sin x}{x} = 1$ I do not understand how $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ As if $$ x = 0, \frac{\sin (0)}{0} = \frac {0}{0} $$
So if someone could explain this I would appreciate it!
Thanks!
| Easy way to view: L'Hospital Rule:
$$ \lim_{x\rightarrow 0} \frac {\sin x}{x} = \lim_ {x \rightarrow 0} \frac {\cos x}{1} = \cos 0 = 1 $$
Another way: first take $ 0 <\delta <\frac {\pi}{2} $. Thus, $ \forall x \in (0, \delta) $ we have the following chain of valid inequalities:
$$ \sin x \leq x \leq \tan x $$
As $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
every subset of a measurable set is measurable Is it true that every subset of a measurable set is measurable? for any measure. So if A is a measurable set then, B as a subset of A must be measurable wrt the same measure.
| The answer to that question depends on a choice of the consistent system of axioms in which we prefer to work.
For example, if we consider the question asking whether every subset of the real axis $R$ is Lebesgue measurable, then we will get the following different pictures:
Theorem 1 (Vitali(1903)). (ZFC) There exi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
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Probability of choosing a subset of elements where each element has a different probability I am trying to write a C++ program to do this but nobody on Stackoverflow can seem to help me so I thought I'd try to do it myself with some help from you guys.
My post on Stackoverflow can be found here:
https://stackoverflow.... | Let $p_i$ be the probability of person $i$ saying yes, and let $q_k$ be the probability of exactly $k$ out of $n$ persons saying yes. Then,
$$
q_k = \left[\prod_{i=1}^n(1-p_i + p_ix)\right]_k,
$$
where $[f(x)]_k$ is the coefficient of $x^k$ in the series expansion of $f(x)$.
In your question, $p_i = \frac{1}{i}$, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
cyclic subgroup elements I'm having hard time finding elements of the cyclic subgroup $\langle a\rangle$ in $S_{10}$, where $a = (1\ 3\ 8\ 2\ 5\ 10)(4\ 7\ 6\ 9)$
This is my attempt:
\begin{align}
a^2 &= (1\ 8\ 5\ 10)(4\ 6\ 9) \\
a^3 &= (1\ 3\ 5\ 10)(4\ 7\ 9\ 6) \\
a^4 &= (1\ 5\ 10)(4\ 9\ 7) \\
a^5 &= (1\ 3\ 8\ 2\ 10)(7... | If you have $$a = (1\,3\,8\,2\,5\,10)(4\,7\,6\,9)$$
that means that $a$ is the permutation that takes 1 to 3, 3 to 8, 8 to 2, and so on.
The permutation $a^2$ is obtained by applying $a$ twice. Since $a$ takes 1 to 3, and then 3 to 8, $a^2$ takes 1 to 8. $$\begin{array}{ccc}
a^0 & a^1 & a^2 \\ \hline
1 & 3 & 8 \\
2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/731792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Are integers mod n a unique factorization domain? I am trying to learn abstract algebra from scratch, jolly stuff, but in the process of doing so this puzzles me:
Having a ring of integers mod $n$, where $n=pq$ is composite, as I understand we have that $\mathbb{Z}/n\mathbb{Z}$ is a Principal Ideal Domain (PID) (by th... | When $\,n\,$ is composite $\,\Bbb Z/n\,$ is not an integral domain. Factorization theory is much more complicated in non-domains, e.g. $\rm\:x = (3+2x)(2-3x)\in \Bbb Z_6[x].\:$ Basic notions such as associate and irreducible bifurcate into a few inequivalent notions, e.g. see
When are Associates Unit Multiples?
D.D. An... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
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Ergodic Rotation of the Torus Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel $\sigma$-algebra, the Lebesgue measure, and the rotation defines as
$$
R_{(\alpha, \beta)}(x,y) ... | Note that the system is ergodic if and only if the orbit of each point is equidistributed in
$\mathbb{R}^2 / \mathbb{Z}^2$.
The first condition $$\forall (k_1, k_2) \in \mathbb{Z}^2 \backslash \{(0,0)\}, \ k_1 \alpha + k_2 \beta \notin \mathbb{Z}$$ is a necessary and sufficient condition. You can use Weyl criterion fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Inequality of Class operators H S and P First few definitions:
$A \in I(K)$ iff $A$ is isomorphic to some member of $K$
$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$
$A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$
$A \in P(K)$ iff $A$ is a direct product of a nonempty family of algebras ... | The only homomorphic images of a field $F$ (when it is considered as a ring) are the field itself and the zero ring $\{0\}$.
If we consider the ring $(\mathbb{Q},+,\cdot,-,0,1)$ we see that $\mathsf{SH}(\mathbb{Q})$ will just contain $\{0\}$ and all the subrings of $\mathbb{Q}$. Note that every subring of $\mathbb{Q}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Generalization of $n$ mod $2 = \dfrac{1-(-1)^n}{2}$ I had this idea that
$n$ mod $2 = \dfrac{1-(-1)^n}{2}$ for $n \in \mathbb{N}$.
Are there any generalizations for this? For example for $n$ mod $3$ etc.?
I would prefer some answers containing basic arithmetic operations, although the use of analytic functions is also... | To give an alternate approach to Hurkyl's answer. If you let $\zeta$ be a primitive $n$-th root of unity, the function
$$f_n(x)=\sum_{i=0}^{n-1} i\frac{(\zeta^x-\zeta^0)...(\zeta^x-\zeta^{i-1})(\zeta^x-\zeta^{i+1})...(\zeta^x-\zeta^{n-1})}{(\zeta^i-\zeta^0)...(\zeta^i-\zeta^{i+1})(\zeta^i-\zeta^{i+1})...(\zeta^i-\zeta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Dividing 100% by 3 without any left In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left.
Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality of the 3 apples is 100%. Now, you can divide those 3 apples for 3 persons and you will... | Here, in a practical way – I understand all your concepts, sirs – I was using a simple calculator on my computer, and looks like mine's got a 10 decimal digit precision. Which make this possible:
9 decimal "3s"
33.333333333 * 3
= 99.999999999
10 decimal "3s"
33.3333333333 * 3
= 100
If I divide 100 by 3, it'll give me 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "78",
"answer_count": 13,
"answer_id": 10
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the continuous functions with norm I'm having trouble trying to understand what does means the first expression in particular
the last term in it
should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ($\|f\|_{\infty}$) here
\begin{align*}
{C}(\mathbb{R}^{n})&=\{f:\mathbb{R}^{n}\longmapsto \... | Well, to answer your question:
let $$f:\Bbb R^n\to \Bbb R, \quad \text{continuous},\quad \lim_{|x|\to\infty}f(x)=0.$$
Fix $\varepsilon >0$; then by definition of the limit there exists $R>0$ such that $|f(x)|<\varepsilon$ whenever $|x|>R$. On the compact set $\{|x|\le R\}$ the function $|f|$ is continuous, hence attain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Taylor series of $\sqrt{1+x}$ using sigma notation I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation
I got till
$1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on.
But I don't know what will come in sigma notation.
| The generalized binomial theorem says that
$$
(1+x)^{1/2}=\sum_{k=0}^\infty\binom{1/2}{k}x^k
$$
where $\binom{1/2}{0}=1$ and for $k\ge1$,
$$
\begin{align}
\binom{1/2}{k}
&=\frac{\frac12(\frac12-1)(\frac12-2)\cdots(\frac12-k+1)}{k!}\\
&=\frac{(-1)^{k-1}}{2^kk!}1\cdot3\cdot5\cdots(2k-3)\\
&=\frac{(-1)^{k-1}}{2^kk!}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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Show that $\,\,\, \lim_{n\to\infty} \sin\bigl(\pi\sqrt{n^2+1}\bigr)=0 $ Can anyone help me to solve this problem?
Show that
$$
\lim_{n\to\infty} \sin\bigl(\pi\sqrt{n^2+1}\bigr)=0
$$
| We have by the Taylor series:
$$\sqrt{n^2+1}=n\sqrt{1+\frac1{n^2}}=n\left(1+O\left(\frac1{n^2}\right)\right)=n+O\left(\frac1{n}\right)$$
hence
$$\sin\left(\pi\sqrt{n^2+1}\right)=\sin\left(n\pi+O\left(\frac1{n}\right)\right)=(-1)^n\sin\left(O\left(\frac1{n}\right)\right)\xrightarrow{n\to\infty}\ 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/732611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$ I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right?
If I can use the rational roots test here, then since it is monic I simply check fact... | Try showing that it has no roots in $\mathbb{Q}$, that is a good criteria for irreducibility.
Another trick, is that if it is irreducible, it is irreducible for all $x\in \mathbb{R}$ so let $x=\frac{1}{y}$ then try eisenstein on the new polynomial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/732713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Compute this factor group: $\mathbb Z_4\times\mathbb Z_6/\langle (0,2) \rangle$
So I'm going through example 15.10 in Fraleigh, which is computing $G/H$, where $G = \mathbb Z_4\times\mathbb Z_6$ and $H = \langle (0,2) \rangle$.
We have $H =\{(0,2), (0,4), (0,0)\}$, so the subgroup generated by $H$ has order $3$.
Sinc... | Note: direct product of quotients is isomorphic to quotient of direct products.
More clearly: $(G_1\times G_2)/(H_1\times H_2) \cong G_1/H_1\times G_2/H_2$ where $H_i\leq G_i$.
From that point; $$\mathbb Z_4\times \mathbb Z_6/\langle(0,2)\rangle\cong \mathbb Z_4/\langle(0)\rangle\times \mathbb Z_6/\langle(2)\rangle\con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/732792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Irreducible Variety of Irreducible Polynomial Prove that if $f \in k[x_1,...x_n]$ is an irreducible polynomial, then the variety $V(f) \subseteq A^n_k$ is an irreducible variety.
Basically, I think that I want to prove that the ideal which corresponds to the variety is prime (since there is a bijective correspondence ... | Since $k[x_1, \ldots, x_n]$ is a UFD, irreducible elements are prime. Thus any principal ideal generated by an irreducible polynomial is in fact a prime ideal, and so the corresponding variety will be irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/732879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Ratio and root test
Hi! I am working on some ratio and root test online homework problems for my calc2 class and I am not sure how to completely solve this problem. I guessed on the second part that it converges, but Im not sure how to solve of the value that it converges to. If someone could possibly help me with thi... | You aren't expected to figure out what the value of the series is (although in time you might figure out it has something to do with $e+e^{-1}$). Did you actually compute $\rho$? What is $$\lim_{n\to\infty} \frac{(2n)!}{(2n+2)!}?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/732996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Probability of rolling a 1 before a 6 on a dice what is the probability that I roll a "1" on a dice before rolling a "6". I do not know how to tackle this problem. I was thinking that this is a Geometric random variable but I do not know how to solve it.
Any help is appreciated.
| By symmetry, the probability is $\frac{1}{2}$.
Remark: We could also do it the long way. We win if we roll a $1$ before we roll a $6$. This can happen in several ways.
*
*We roll a $1$ immediately. The probability that happens is $\frac{1}{6}$.
*We roll something that is neither a $1$ nor a $6$, and then roll a $1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/733090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Problem with solving a complicated Integral I need to determine the $ \int \frac{\sin^3(x)}{8-\cos^3(x)} dx$. It's an indefinite integral.
| Hint: $\sin^3(x) = \sin(x)(\sin^2(x)) = \sin(x)(1-\cos^2(x))$
Take $u = \cos(x) \Rightarrow du = -\sin(x) \ dx$
So we have, $- \int \frac{1-u^2}{8-u^3} du = -\int \frac{1}{8-u^3} du - \int \frac{u^2}{8-u^3} du $
| {
"language": "en",
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"source": "stackexchange",
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Prove for each $a ∈ G, aHa^{-1}$ is a subgroup of G. Question:
Let $H$ be a subgroup of $G$. For any $a \in G$, let $aHa^{-1} = \{axa^{-1} : x \in H\}$; $aHa^{-1}$ is called a conjugate of $H$. Prove: For each $a \in G$, $aHa^{-1}$ is a subgroup of $G$.
I know in order to prove something is a subgroup it needs to be no... | One slick way to prove a subset $K$ of a group $G$ is a subgroup is to show:
1) The set $K$ is nonempty.
2) For every $x,y \in K, xy^{-1} \in K$.
(See if you can prove this criterion to be a subgroup!)
You've shown 1), since the identity $e \in K = aHa^{-1}$. Now let $x,y \in aHa^{-1}$. Then $x = aha^{-1}$ and $y = ah'... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How to find the order of the elements of $A_4$? Just wondering how to find the order of each element in this group:
$A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$
I tried writing each elements not in disjoint cycle but it didn't look right to me. I got 3 for all the cycles with... | Notice for any 3-cycle $(abc)$, $(abc)^2=(abc)(abc)=(acb)$, and $(abc)^3=(abc)^2(abc)=(acb)(abc)=e$, the identity. Thus the order of any 3-cycle is 3.
Noting that disjoint cycles commute, it is easy to see that $((ab)(cd))^2=(ab)(ab)(cd)(cd)=e*e=e$, so the order of any product of two disjoint transpositions is 2.
| {
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"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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proof using a recursive definition I am doing a 2-part question. Thus far, I have finished the first part, requiring me to make a recursive definition of a set "S" of all binary strings, starting with a 1. I have:
Base: 1
Recursion: S1 or S0
Restriction: nothing else in the set
Now, the next question is to show, using ... | Take your recursive strings definition
and append a "value" function for
each part of the definition:
Base: 1 - (val(1) = 1)
Recursion: S1 (val(S1) = 2val(S)+1) or S0 (val(S0) = 2 val(S))
This usually works for any recursive definition
that you want to assign additional properties or interpretions to.
| {
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I'm looking for the name of a transform that does the following (example images included) I'm in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I'm looking for:
Suppose I have the following "snake" of 10 quadrilaterals:
I ... | From the tag "linear algebra" I guess you are looking for a linear transformation.
Then it is impossible. Because, the inverse, which is again a linear transformation can only map a rectangle to a parallelogram and not to a 'snake'.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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How to show that this real function is not periodic?
How can one prove that $$\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$ is not periodic?
This question is motivated by the harmonic spectral representation of time series. Indeed, it is easy to show that a path of a time series given by $$ \cos(\lambda_1 t... | The function
$$f(t)=\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$
is even; so $f(t+T)=f(t) \Leftrightarrow f(t-T)=f(t),~~ \forall t$.
Adding the equations
$$f(t+T)=f(t), $$
$$f(t-T)=f(t), $$
we arrive at
$$\cos t \cos T + \cos\left(\frac{\pi}{2} t \right)\cos\left(\frac{\pi}{2} T \right)=
\cos t+ \cos\left... | {
"language": "en",
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"source": "stackexchange",
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Showing $2^{n_2} + 3^{n_3}+\cdots+9^{n_9}$ is dense in $\mathbb{R}^+$ I encountered this problem via a friend. He asked me to prove that
$$ \left\{u: u= \sum_{k=2}^9 k^{n_k} \quad n_k \in \mathbb{Z} \right\}$$
is dense in $\mathbb{R}^+$.
I was able to show that $0$ is approachable(can get arbitrarily close to) by nu... | I don't think this can be done. Think of it like this: Suppose you have $u \in \Bbb{R}_+$ and $u_n \to u$ with
$$ u_n = \sum_{k=2}^9 k^{m_k^n} $$
Now each $(u_n)$ is completely characterized by $(m_2^n,...,m_9^n)$. If $u$ is not zero then not all sequences $(m_k^n)_n$ go to $-\infty$, so some of them are bounded, which... | {
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"source": "stackexchange",
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Visually stunning math concepts which are easy to explain Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time.
Do you know of any other c... | Ulam Spiral:
Discovered by Stanislaw Ulam, the Ulam Spiral or the Prime Spiral depicts the certain quadratic polynomial's tendency to generate large number of primes.Ulam constructed the spiral by arranging the numbers in a rectangular grid . When he marked the prime numbers along this grid, he observed that the prim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1573",
"answer_count": 87,
"answer_id": 24
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Visually stunning math concepts which are easy to explain Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time.
Do you know of any other c... | I would like to add some explorations of the concept asked by the OP of my own:
*
*Visualization of the set of real roots of quadratic equations $ax^2+bx+c=0$, for the specific values of the intervals $a \in [-a_i,a_i]$, $b \in [-b_i,b_i]$, $c \in [-c_i,c_i]$, $a,b,c \in \Bbb N$.
By Cartesian coordinates $(x,y)=(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1573",
"answer_count": 87,
"answer_id": 56
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Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$). $d(x,y)=\|x-y\|_p$
$p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$.
This is not a metric because
if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$.
Hence $d(x,y)=p^{-0}=1 \ne0$.
... | You didn't specify what $m,n$ are in the definition of $\|.\|_p$. Supposedly, they integers that are not divisible by $p$. However, $0$ is divisible by $p$, hence you cannot compute $\|0\|_p$ this way. Indeed, one needs to define explicitly that $\|0\|_p=0$; with this addon, $d$ is a metric.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of a functional I'd like to find:
$$
\lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R})
$$
And I started with the definition:
$$
\left\langle \frac{\varepsilon}{\varepsilon^2+x^2},\varphi\right\rangle
$$
After doing the integration by parts I had some pro... | Define $$T_\varepsilon(\varphi):=\int_{-\infty}^{+\infty}\frac{\varepsilon}{\varepsilon^2+x^2}\varphi(x)\mathrm dx.$$
Then, after the substitution $x=\varepsilon t$, we obtain
$$T_\varepsilon(\varphi)=\int_{-\infty}^{+\infty}\frac{1}{1+t^2}\varphi(t\varepsilon)\mathrm dt.$$
Using dominated convergence, we obtain that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How find this function $f(x)$ such $f(a+f(b))=f(a+b)+f(b)$ let function $f:R_{+}\to R_{+}$,and such
$$f(a+f(b))=f(a+b)+f(b),\forall a,b\in R_{+}$$
Find $f(x)$.
my try: let $a=b=1$,then
$$f(1+f(1))=f(2)+f(1)$$
$a=1,b=2$,then
$$f(1+f(2))=f(3)+f(2)$$
then I can't find have any regular,so I can't.Thank you
Edit: This is I... | the hint is :
$f(f(2a))=f(a+f(a+f^{-1}(a)))=f(2a+f^{-1}(a))+a$
$f(2f(a))=f(f(a)+a)+f(a)=f(2a)+2a$
$\Longrightarrow$$f(2a)+a=f(2a+f^{-1}(a))$
$\Longrightarrow$$f(x)=2x$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Continuity of a partial derivative I have the function
$$f(x,y)=\begin{cases}
x^2ysin(\frac1x) & \text{if $x$ is not 0} \\
0 & \text{if $x=0$}\end{cases}$$
And I need to find the derivative and the partial derivatives, and see if they are continuos.
I´ve already proved that the function is cont... | Notice that $\sin$ and $\cos$ remain bounded, while the factors they have in front tend to zero as $(x,y)$ tends to $(0,0)$.
But now consider the limit to points $(0,y)$ with $y\neq0$.
| {
"language": "en",
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Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $? Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b> 0$ and $c\geq 0$ ?
Is there a general proof for that relation ?
| Good observation, these inequalities are quite useful. But you need a little bit more:
*
*if $a\ge b$, then $\dfrac ab\ge\dfrac{a+c}{b+c}$
*if $a\le b$, then $\dfrac ab\le\dfrac{a+c}{b+c}$
You can prove it by multiplying by the common denominator:
*
*$a(b+c)\ge b(a+c)\Longleftrightarrow ac\ge bc$
*$a(b+c)\le ... | {
"language": "en",
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"source": "stackexchange",
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Summation involving Fibonacci numbers Find:
$$
\sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n}
$$
where $F_n$ is $n$-th Fibonacci number.
| $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
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Induction question help. Let $x$ and $y$ belong to a commutative ring $R$ with prime characteristic $p$.
Show that, for all positive integers $n$
$$ (( x + y )^p)^n = (x^p)^n + (y^p)^n $$
I hope you can can understand notation.
We have to use induction on $n$.
For $n=1$ $ (x + y)^p $ = $ x^p $ + $ y^p $
Assu... | Note that by the binomial theorem
$$
(x+y)^{pn} = \sum_{k=0}^{pn}\binom{pn}{k}x^ky^{pn-k} = \sum_{k=0}^{pn}\frac{(pn)!}{k!(pn-k)!}x^ky^{pn-k} = \sum_{k=0}^{pn}\frac{pn\cdot\ldots\cdot(pn-k+1)}{k!}x^ky^{pn-k} = \sum_{k=0}^{pn}p\frac{n\cdot\ldots\cdot(pn-k+1)}{k!}x^ky^{pn-k} = x^{pn}+y^{pn},
$$
since only when $k=0$ or $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show $f$ is a complex polynomial of degree at most 2 Suppose $f:\mathbb{C}\rightarrow\mathbb{C}$ is an entire function and
$$\displaystyle\min\{\left|f'(z)\right|,\left|f(z)\right|\}\leq \left|z\right|+2$$ for all $z\in\mathbb{C}$.
How to see that $f$ is a polynomial in $z$ of degree at most 2?
I can only see it when ... | By adding a constant, you can assume that $f(0) = 0$ . For an arbitrary point $z \in \mathbb{C}$ consider the minimal part of the line segment $L_z$ from $z$ to $0$ which connects $z$ to a point $z_0$ with $|f(z_0)| \le |z_0| + 2$. (If this is already satisfied for $z_0 = z$, the line segment will just be the point $z ... | {
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Are contractive completely positive maps trace decreasing? Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is $\tau_M(1_M)=\tau_N(1_N)=1$). Do we have that $\tau_N(f(a))\leqslan... | For von Neumann algebras in general it is false, there is two much leeway in the choice of the traces. For instance let $M=N=\mathbb C\oplus\mathbb C$, with $f$ the identity map. Let the traces be
$$
\tau_M(x,y)=\frac x4+\frac{3y}4, \ \ \tau_N=\frac{3x}4+\frac y4.
$$
Then, for $a=(1,0)$,
$$
\tau_M(a)=\tau_M(1,0)=\fra... | {
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Multiplicative version of the principle of Archimedes
Any clear proof of the above theorem is greatly appreciated.
| Suppose first that $y\ge 1$. We prove that the set $S=\{x^n\}$, $n=0,1,2,\dots$ is unbounded. Suppose to the contrary that $S$ bounded. Let $b$ be the supremum. Then for some $n$, we have $x^n \gt \frac{b}{(x+1)/2}$. Then $x^{n+1}\gt b\frac{x}{(x+1)/2}\gt b$. This contradicts the fact that $b$ is an upper bound of $S$.... | {
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Building a partial injective relation Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$.
Example: $A =[a,b,c]$ and $B=[1,2,3,4]$.
Build a partial injection relation. Are there 3-4 elemen... | There are only injections from the smaller set to the larger, namely
$$
f: A \to B
$$
There are $4$ choices for the value $f(a)$, $3$ choices for $f(b)$, and $2$ choices for $f(c)$ for a total of $4 \cdot 3 \cdot 2 = 24$ possible injective functions. Each of these gives a distinct maximal injective relation, consistin... | {
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Uniform Distribution and Distribution function technique Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$.
(a)$y \le 0$
(b)$0<y<1$
(c)$1<y<2$
(d)$y\ge2$
I'm thinking $f(x_1)=f(x_2) = 1$ for $0 \le x_1\le 1$... | You have $P(X_1 \leq u) = P(X_2 \leq u) = u$ for $0 \leq u \leq 1$ and $X_1,X_2$ independent. Thus $$
P(X_1 + X_2 \leq u) = \int_{(x_1,x_2) \in A_u} \,d(P_{X_1} \times P_{X_2}) \text{ where } A_u = \{(x_1,x_2) \in [0,1]^2 \mid x_1 + x_2 \leq u\} \text{.}
$$
which yields the CDF (note that the density of $X_1$ and $X... | {
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Recurrence problem for $a_5$ Assume that the sequence $\{a_0,a_1,a_2,\ldots\}$ satisfies the recurrence $a_{n+1} = a_n + 2a_{n−1}$. We know that $a_0 = 4$ and $a_2 = 13$. What is $a_5$?
| Rearranging the recurrence relation gives $a_n = a_{n+1} - 2a_{n-1}$, so that $a_1 = 13 - 2 \cdot 4 = 5$. Now that we know $a_0,a_1$ and $a_2$ you are good to go by simply applying the recursion repeatedly until you get to the fifth term.
| {
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Possible difference between $\mathbb{Z}$-modules and vector spaces Suppose $G$ is a free abelian groups, i.e. a free $\mathbb{Z}$-module; we have a set $S \subset G $ such that $S$ spans $G$.
Can we conclude that the rank of $G$ as a $\mathbb{Z}$-module is $ \leq |S| $ as in the vector-space case ? Why ?
| There are also many differences between vector spaces and $\mathbb{Z}$-modules. Every vector space has a basis, but not every $\mathbb{Z}$-module. For example, any finite abelian group is not a free $\mathbb{Z}$-module, and the $\mathbb{Z}$-module $\mathbb{Q}$ is not free.
Furthermore a free $\mathbb{Z}$-module may hav... | {
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I need someone to explain this proof from James Munkres' Topology.
The author writes $(X-C)\cap Y = Y-A,$ and, also, $A=Y \cap (X-U)$. I was wondering how is that something anyone writing an original proof of the theorem saw and if there is an analytic proof that the equalities holds true. Thanks for your help.
| He defined $A = C \cap Y$; To see $(X-C) \cap Y = Y - A$, we can compute (as Frank did) or reason: a point is in the left hand side if it is not in $C$ but it is in $Y$. This means not in $A$ (to be in $A$, one has to be both in $C$ and $Y$), so it is in $Y - A$. On the other hand, if a point is in $Y - A$, it is in $Y... | {
"language": "en",
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Soft Question: Algorithms: Will We (One Day) No Longer Need to Study Algorithms? I'm just now getting into the study of algorithms and it seems like as computers get faster and faster the need to study algorithms may begin to diminish.
How likely is it that in 50 years there won't be much of a demand for the analysis o... | Euclidean geometry has existed for a long time and we certainly don't only study it for historical purposes.
Usually what happens with a field of mathematics is that it gets incorporated into something bigger and more complex when we attain total mastery of it. For example, from Euclidean geometry we've gotten a lot of... | {
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Nice book on geometry to gift an undergraduate in mathematics I would like some suggestions on a nice book on geometry to gift an undergraduate. I'm not searching for something that is common: I need something new and exciting. Suggestions?
| I love this one. It is one of the most stimulating books on geometry that I have ever read: it is strikingly innovative and really enjoyable (also, it does not require a particularly advanced background).
From the book description:
New Horizons in Geometry represents the fruits of 15 years of work in
geometry by a ... | {
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Is it possible to formalize areas such as image processing and computer vision? Is it possible to formalize areas such as image processing?
By formalize I mean setup axioms, then derive theorems, and
reason about image processing concepts and methods formally.
I would say now image processing is pretty informal and ill... | I would beg to disagree. Image processing is quite formal. Images are treated as discrete functions in 2D, and are manipulated using very formal methods from calculus, linear algebra, and statistics.
| {
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Question regarding notation involving vector spaces. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on $u+v=(u_1+v_1+1,u_2+v_2+1),$ $ku=(ku_1,ku_2)$.
Show that $(0,0)\neq0$.
I'm confused by their notation. Can anyone explain?
| By $\mathbf{0}$ they mean the zero vector. The zero vector has to be the additive identity in order for $V$ to be a vector space, in other words, $\mathbf{0}$ has to be a vector such that $\mathbf{v}+\mathbf{0}=\mathbf{v}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which is the topology generated by the neighborhood system $V(x)=\{\{x\}\}$ My question is the following:
Which is the topology generated by the neighborhood system $V(x)=\{\{x\}\}$ ?
I say that is the coarse topology but I don't know how is the mechanism to generate a topology from a basis
sorry if this is so trivial... | By definition a set $U$ is open if and only if for every $x\in U$, there is a neighborhood $B_x\in V(x)$ such that $B_x\subset U$. Particularly, $V(x)\ni\{x\}\subseteq\{x\}$.. You should recognize that this is a standard topology on a set after some thought.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How prove $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\cdots+\frac{1}{a_{n}}<2$ Let $$A=\{a_{1},a_{2},\ldots,a_{n}\}\subset N$$
Suppose that for any two distinct subsets $B, C\subseteq A$, we have
$$\sum_{x\in B}x\neq \sum_{x\in C}x$$
Then show that
$$\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\dfrac{1}{a_{3}}+\cdots+\dfra... | This is an old conjecture of Erdos, which was subsequently proved (Ryavek from my notes) - though I cannot find a handy online reference just now. The proof goes along the following lines, IIRC:
With $0 < x< 1$, we have by the distinct sum condition:
$$ \prod_{k=1}^n (1+x^{a_k}) < \sum_{k=0}^{\infty} x^k = \frac1{1-x}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Sum of probabilities or mean of probability My question is about being confused about two way of approaching a problem, which in this case lead me to the same solution. One method is very verbose, the other one is fast and clean.
Let's consider this problem where there are three only components of a computer with proba... | Both methods are correct!
To see that Method 1 is correct, define the independent Bernoulli random variables
$$X_i=\begin{cases} 1, &\text{component $i$ is defective} \\ 0, &\text{component $i$ is not defective} \end{cases}$$ for $i=1,2,3$. Then $$X_i \sim \mathrm{Bern}(p_i)$$ where the $p_i$'s are the given probabil... | {
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Solving a non linear ODE problem Please I will like to solve this non linear ODE problem
$$
y'(x) = e^{-b(x)} \times \left( p + \left( \frac{q}{n} y(x) \right)\right)\left( n - y(x) \right).
$$
Can anyone help? Thank you
I made some correction to the equation
| Using the same approach as izoec, we can arrive to$$\frac{n (n (p+q) \log (n p+q y)-q y)}{q^2}= \int e^{-b(x)} \, \mathrm{d} x = \xi (x)$$ and the solution of $y$ is $$y=- \frac {n} {q} \left(p+(p+q) W\left(-\frac{\left(e^{\frac{\xi(x) q^2}{n^2
p}-1}\right)^{\frac{p}{p+q}}}{n (p+q)}\right)\right)$$
I hope and wish ... | {
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Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected Question is :
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected.
What i have tried is :
Suppose ... | Continuing with your argument, We can prove that $B \cup C$ and $D$ form the partition of $X$.(Assuming your conclusion that $D \subset A$).
We have $X = (B \cup C) \cup D$.
$C$ and $D$ form separation of $Y \cup A$. So, using theorem (23.1), no limit point of $C$ is in $D$. Similarly, no limit point of $B$ is in $D$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Computing $\lim\limits_{n\to\infty}(1+1/n^2)^n$
Why is $\lim\limits_{n\to\infty}(1+\frac{1}{n^2})^n = 1$?
Could someone elaborate on this? I know that $\lim\limits_{n\to\infty}(1+\frac{1}{n})^n = e$.
| There are some good answers here, but let me share a method that is more computational (in that it doesn't require noticing certain inequalities and using the squeeze theorem), and may help you with other similar limits.
First, one convenient way of taking exponents out of a limit expression is to take a logarithm:
$$\... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving the Division Algorithm using induction Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing into $m$.
I'm having trouble proving this with induction. I believe the ... | Uniqueness doesn't need induction. Suppose $m=qn+r=q'n+r'$, where $0\le r\le n-1$. It's not restrictive to assume $r\le r'$, so we have $0\le r'-r\le n-1$; but $r'-r=n(q-q')$, so
$$
0\le n(q-q')<n
$$
As $n>0$, this implies
$$
0\le q-q'<1
$$
so $q=q'$ and therefore $r'=r$.
The proof of existence can be conveniently spl... | {
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Lagrange Polynomial Interpolation - Equation Help I understand the concept of Lagrange Interpolation but am having issues understanding how to interpret the following general equation (which I will be provided) for n points. For example, how would you get the equation for n = 4 points from the general equation below?
T... | Let $x_1,x_2,x_3,x_4$ be mutually distinct numbers and we need to fit the polynomial of degree 3 passing through $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. Then
$$p(x)=L_1(x)y_1+L_2(x)y_2+L_3(x)y_3+L_4(x)y_4,$$
where
$$L_1(x)=\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)},$$
$$L_2(x)=\frac{(x-x_1)(x-x_3)(... | {
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For $(x_n)$ increasing, $\sum_{n=1}^{\infty}\left(1-\frac{x_n}{x_{n+1}}\right)$ if $(x_n)$ is bounded and diverges if it is unbounded Let $\{x_n\}$ be monotone increasing sequence of positive real numbers. Show that if $\{x_n\}$ is bounded, then $\sum_{n=1}^{\infty}\left(1-\frac{x_n}{x_{n+1}}\right)$ converges. On the ... | I try to rewrite the proof cleaning some details.
Call $r_n=\frac{x_n}{x_{n+1}}\in(0,1]$. If there's an infinite number of terms such that $1-r_n=1-\frac{x_n}{x_{n+1}}\geq \frac12$ then the series clearly diverges, while by rewriting the condition as $x_{n+1}\geq 2 x_n$ also the sequence diverges.
If instead eventually... | {
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Ordinary Generating functions for $b_n$ Problem
Let $f(x)$ be a ordinary generating function for the sequence $ \{\ a_0, a_1, a_2... \}\ $ Find the ordinary generating function for $b_0 = b_1 = 0, b_2 = 1$ $b_n = a_n$ for $n \geq 3$.
Also find the generating function for $b_n = 0$ for even $n$, $b_n = a_n$ for odd $n$.... | So for the first one there are a couple of issues, when you go from summing from $n = 3$ to $n=0$ you add 3 on to the power, but not the index of the coefficient.
Also when you factor out the $x^3$ it turns into $x^{-3}$. Fix these things and see how far you can get.
For the second I'd recommend looking at $f(-x)$ and ... | {
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Solving the recurrence relation $T(n) = T(n-\sqrt n) + 1$ I have an algorithm that at each step can discard $\lceil\sqrt(n)\rceil$ possibilities at a cost $1$. The solution to the recurrence relation below is related to the question of complexity of such algorithm:
$$T(n) = T\left(n-\lceil\sqrt n\rceil\right) + 1$$
I k... | Here is a geometric intuition.
We started with a width of $\sqrt{n}$. At each step, the width reduced by (1/2). Hence within roughly $2\sqrt{n}$ steps we reach the base case.
| {
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If two cycles are disjoints, then they commute I'm trying to prove that two $\tau$-cycles commute provided that these cycles are disjoints. Hungerford in his book says the following remark about this fact:
Intuitively clearly this is true, but how can we prove this formally?
Thanks in advance
| You can probably do this more formally by considering the permutations as elements which act on a set, namely the set of letters $\{1, \ldots, n\}$. In such a case, a cycle cyclically permutes a sub-collection of letters $\{\ell_1, \ldots, \ell_r\}$, and the fact that two cycles commute tell us that these sets which ar... | {
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Linear functional in Banach space Let $X$ be a Banach space, $(f_n)\in X^{*}$ a sequence with $f_n\neq 0 $ $ \forall n\in \Bbb N$. Show that there is a $x\in X$ such that $f_n(x)\neq 0 $ $\forall n$.
Need some help. Thank you!
|
It's something with the kernel of the functionals,right?
Right. What you want is an $x$ with $f_n(x) \neq 0$ for all $n$, so
$$x \in \bigcap_{n\in\mathbb{N}} \left(X\setminus \ker f_n\right) = X \setminus \bigcup_{n\in\mathbb{N}} \ker f_n.$$
The $f_n$ are continuous, so $\ker f_n$ is closed. $f_n \neq 0$, hence $\ker... | {
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Does there exist an $x$ such that $3^x = x^2$? I tried solving for $x$ by using $x \log(3) = \log(x^2)
$$\log(3) = \frac{\log(x^2)}{x}$$
I'm stuck on this part. how do I isolate $x$ by itself?
Any help would be appreciated.
| The unique real solution is $$-2 \dfrac{W(\ln(3)/2)}{\ln(3)}$$
where $W$ is the Lambert W function. There are also complex solutions, corresponding to the different branches of $-2 \dfrac{W(\pm\ln(3)/2)}{\ln(3)}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How prove this inequality $\frac{1}{(a+1)^2+\sqrt{2(b^4+1)}}+\frac{1}{(b+1)^2+\sqrt{2(c^4+1)}}+\frac{1}{(c+1)^2+\sqrt{2(a^4+1)}}\le\frac{1}{2}$ let $a,b,c>0$,and such $abc=1$, show that
$$\dfrac{1}{(a+1)^2+\sqrt{2(b^4+1)}}+\dfrac{1}{(b+1)^2+\sqrt{2(c^4+1)}}+\dfrac{1}{(c+1)^2+\sqrt{2(a^4+1)}}\le\dfrac{1}{2}$$
My idea :... | since Use Cauchy-Schwarz inequality and AM-GM inequality,we have
$$\dfrac{1}{(a+1)^2+\sqrt{2(b^4+1)}}\le\dfrac{1}{(a+1)^2+b^2+1}\le\dfrac{1}{2ab+2a+2}=\dfrac{1}{2}\cdot\dfrac{1}{ab+a+1}$$
Use follow well know reslut,if $abc=1$,then we have
$$\sum_{cyc}\dfrac{1}{ab+a+1}=1$$
poof: since
\begin{align*}\dfrac{1}{ab+a+1}+\... | {
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Fibonacci trick and proving it. I am trying to learn Fibonacci tricks and I have one that I can not prove. I know it works because Ive tried it multiple times but I have not a clue how to prove. Here it is:
f(0)^2 + f(1)^2 + f(2)^2 + f(3)^2 = f(3)f(3+1)
0 + 1 + 1 + 4 = 2 * 3
= 6 ... | You've already proven that it works for one example $k=3$. Now write up the general equation
$$
\sum_{k=0}^n F_k^2 = F_n\cdot F_{n+1}
$$
and add $F_{n+1}^2$ on both sides. You get
$$
F_{n+1}^2+ \sum_{k=0}^n F_k^2 =\sum_{k=0}^{n+1} F_k^2 = F_{n+1}^2+ F_n\cdot F_{n+1}=\left(F_{n+1}+ F_n\right)\cdot F_{n+1}
$$
and use $F_... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving equality of sigma-algebras Let $C_1$ and $C_2$ are two collections of subsets of the set $\Omega$. We want to show that if $C_2$ $\subset$ $\sigma$[$C_1$] and $C_1$ $\subset$ $\sigma$[$C_2$], then $\sigma$[$C_1$]=$\sigma$[$C_2$] where $\sigma$[$C$] is defined to be the sigma-algebra generated by a collection $C... | Use twice the fact that, if $A$ and $B$ are two subsets of $2^\Omega$ such that $A\subseteq\sigma(B)$, then $\sigma(A)\subseteq\sigma(B)$, once for $(A,B)=(C_1,C_2)$, and once for $(B,A)=(C_1,C_2)$.
The fact itself is direct using the identity
$$\sigma(B)=\bigcap_{F\in\mathfrak F} F,
$$
where $\mathfrak F$ is the colle... | {
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History of Morse theory. How can I get good references which give many information about history of Morse theory? Now I am interested in how and who found that Hessian have a lot of data. Thank you for your helping!!
| You may want to try these references:
*
*R. Bott, "Marston Morse and his mathematical works", Bull. Amer. Math. Soc. 3 (1980) 907–950.
*Dieudonné's A history of algebraic and differential topology.
*Morse theory in Tu's The life and works of Raoul Bott [pdf]. Also in Notice of AMS.
| {
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if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$ Question:
Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall n\in\ma... | First let's prove that $(a_n)$ is increasing for $n \geq 2$. We have $a_1=1,a_2=0.5, a_3=1/2+1/3$ so $a_3\geq a_2$. We have
$$ a_{n+1}-a_n = \sum_{k=2}^n \frac{a_{[(n+1)/k]}-a_{[n/k]}}{k}+1/(n+1).$$
Suppose that for every $2\leq i,j \leq n$ with $i<j$ we have $a_i\leq a_j$. Therefore in the above sum the only way that ... | {
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Proving whether a squared function has double pole at $z_0$ Let D be a domain in $\Bbb C$ and let $z_0 \in$ D. If a function $f:$ D \ {$z_0$} $\rightarrow \Bbb C$ has a simple pole at $z_0$, is it true that $g$ has a double pole at $z_0$, where $g(z) = [f(z)]^2 $ $\forall z \in$ D\ {$z_0$}
Definition of a pole: If ther... | $\;z_0\;$ is a simple pole of $\;f(z)\;$ iff there's a holomorphic $\;h\;$ in some domain $\;D_0-\{z_0\}\;$ s.t.
$$f(z)=\frac{h(z)}{(z-z_0)}\;\;,\;\;\forall\,z\in D_0\;\;\;\text{and}\;\;\;h(z_0)\neq 0$$
From here, we have that
$$\forall\,z\in D_0-\{z_0\}\;\;,\;\;\;g(z):=f(z)^2=\frac{h^2(z)}{(z-z_0)^2}$$
Prove now $\;h(... | {
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Demonstrate that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} \simeq 1.7 \sqrt{n}$ As in the title, I know that
$\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} = \frac{(2n - 2)(2n - 4)\cdots 4 \cdot 2}{(2n - 3)(2n - 5) \cdots 3 \cdot 1} \simeq 1.7 \sqrt{n}$
Could you give some hint to prove it?
(should I look the series e... | Hint Take the square root of the Wallis product...
| {
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if $gcd(a,b)=gcd(a,b,c)$ then I need to prove that $ax+by=c$ has solution in $\mathbb Z$ if $gcd(a,b)=gcd(a,b,c)$ then I need to prove that $ax+by=c$ has solution in $\mathbb Z$ that is: $gcd(a,b)|c$
but how can I prove it with the given hypothesis?
| First divide $a,b,c$ by $\gcd(a,b)$ so that we can assume that $a,b$ (and therefore $a,b,c$) are relatively prime.
Apply now Euclid's algorithm to $a,b$. This will give you equations:
$a=bq_1+r_1$, $b=q_2r_1+r_2$, ..., $r_n=q_{n+1}r_{n-1}+r_{n+1}$, where $r_{n+1}=1$ is the $\gcd(a,b)$.
Substituting backwards these equ... | {
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Counterexample for the normalizer being a normal subgroup Let $G$ be a group, and $H$ a subgroup.
$H$'s normalizer is defined: $N(H):=\{g\in G| gHg^{-1}=H \}$.
Prove $N(H)$ is a normal subgroup of G, or give counterexample.
Intuitively it seems to me that this claim is wrong, however, I'm having trouble with finding a... | Let $G=\langle f,r : f^2 = r^8 = 1, rf =fr^7 \rangle$ be the dihedral group of order 16, and let $H=\langle f \rangle$. Then $N_G(H) = \langle f,r^4 \rangle$ and $N_G( N_G(H) ) = \langle f, r^2 \rangle$ and $N_G( N_G( N_G(H) ) ) = \langle f,r \rangle =G$.
In other words, $H$ is not normal, neither is its normalizer, bu... | {
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Closed and Connected Subset of a Metric Space My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any.
I've been reading Conway's "Functions of One Complex Variable", and encountered following exercise in the book (p. 17). $X$ here is a metric space and... | Your approach seems correct. Your set contains the point $a$, so it's nonempty. For the openness pick any point and show that an open ball of radius $\varepsilon$ is contained in the set too. For the closedness pick a point in the closure and show there's a point from the set that is $\varepsilon$ close to it.
So your ... | {
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Are the entries in matrix/vector product independent This question has been confusing me. Choose a random square matrix $M$ with $M_{i,j} \in \{1,-1\}$ so that $M_{i,j} = 1$ with probability $1/2$ and $M_{i,j} = -1$ with probability $1/2$. Say all the $M_{i,j}$ are i.i.d. Now select a random vector $v$ so that $v_i = ... | Since the matrix rows of $M$ are i.i.d. and the column vector $v$ is i.i.d and $y_i$ is the $i$th row of $M$ times $v$, you won't get any dependence among $y_i$, since there is no dependence among the rows of $M$.
So the answer is yes!
| {
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Manipulating identities I'm having some trouble deriving certain identities. If
$$S(z) = \prod_{i=1}^n (z-z_i)$$
then how can I write
$$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq i}^n\frac{2}{z_i-z_j}$$
and
$$ \frac{1}{S(z)}\frac{dS}{dz}= \sum_{i=1}^n\frac{1}{z-z_i} $$
Truth be told, I'm ... | Hint for the second:
$${\frac {{
\frac {d}{dz}}S \left( z \right) }{S \left( z \right) }}={\frac {d}{dz}}\ln \left( S \left( z \right) \right) $$
Hint A for the first:
$${\frac {{\frac {d^{2}}{d{z}^{2}}}S \left( z \right) }{S \left( z
\right) }}={\frac {d^{2}}{d{z}^{2}}}\ln \left( S \left( z \right)
\right) + \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/738612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Probability exercise Bernoulli. Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video clips of your favorite artists. You normally download video clips from the Web site http... | You have started essentially correctly. We have $X-Y=X-(n-X)=2X-n$. So
$$\Pr(Z=k)=\Pr(2X-n=k)=\Pr\left(X=\frac{n+k}{2}\right)=\binom{n}{(n+k)/2}p^{(n+k)/2}(1-p)^{(n-k)/2}.$$
This makes sense only when $-n\le k\le n$ and $n$ and $k$ are of the same parity (both even or both odd).
The rest of the problems are more mech... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/738737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What type of triangle satisfies the equation $\cos(A)-\cos(B)+\sin(C)=0$? A triangle with angle $A,B,C$ satisfies the equation $\cos(A)-\cos(B)+\sin(C)=0$.
What type of triangle is this? Regular, acute, right, obtuse etc.
I tried using sine and cosine rule, but no result.
| Using Prosthaphaeresis Formulas,
$$\sin C=\cos B-\cos A=2\sin\dfrac{A+B}2\sin\dfrac{A-B}2$$
Now, $\displaystyle \sin\dfrac{A+B}2=\sin\dfrac{\pi-C}2=\cos\dfrac C2$
Using double angle formula the given relation becomes, $$2\sin\dfrac C2\cos\dfrac C2=2\cos\dfrac C2\sin\dfrac{A-B}2$$
which implies
$(1)$ either $\display... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/738925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Can you take off a sweater while wearing headphones? This seems like a graph theory problem, but I'm not sure how to approach it.
To clarify potential ambiguities, let's set up the situation.
You are wearing a sweater (with one arm through each sleeve). You are also wearing a pair of earphones, which are connected to y... | Theoretically, yes, if your electronic device is in your pocket. While it seemingly forms a closed loop, it doesn't actually.
Thus, if you can pull your sweater from over your head and follow the wire into your pocket and fit the entire sweater into your picket to leave the electronic device stationary and exit the 'l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Ideal of a polynomial ring, and an isomorphism between $R[x]/I$ and $R$ Let $R$ be a ring. $I\subset R[x]$ is the ideal of all elements with a zero constant term. Show that $I$ is an ideal, and show that $R[x]/I\cong R$.
Attempt: $R[x]=\{a_0+a_1x+...+a_nx^n:a_i\in R\}$. Then $I=\{a_1x+...+a_nx^n : a_i \in R\}$. Conside... | For the first part you also need to show that $I$ is closed under addition.
For the second part, the intuition is that taking the quotient by $I$ removes all $x$ terms from $R[x]$ and results in $R$. To rigorously show that $R[x]/I\cong R$ I recommend constructing a surjective map $R[x]\to R$ whose kernel is $I$, which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Continuity and lower bound I was initially presented with the following problem:
Show that $x < \tan{x}$ for all $x \in (0, \pi/2)$.
My solution involved setting $f(x) = \tan{x}-x$ and then showing that $f(x)$ is strictly increasing on $(0, \pi/2)$. However, I need the following lemma proved to complete (I know that th... | First let me give an insight to the proof. I am trying to prove that if there is an $x_0$ such that $f(a)>f(x_0)$ then under those circumstances $f$ cannot be continuous at $a$.
As $f$ is increasing on $\left(a,b\right)$, hence note that for every $x_1,x_0\in (a,b)$ and $x_1 < x_0 \implies f\left(x_1\right)\le f\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Use convergence theorem to prove the limit Suppose $f:\mathbb{R}\to\mathbb{R}$ is $L^1$ and it is continuous at the origin. Let $g_n(x)=\frac{1}{1+x^2}f(\frac{x}{\sqrt{n}})$. The problem is to prove $$\lim_n\int_0^\infty g_ndx=\frac{\pi}{2}f(0)$$ The method I tried so far is using the dominated convergence theorem but ... | EDIT: the fact that the sequence is unbounded for $y = 0$ is not a problem since we don't care about sets of measure $0$.
Clearly $$\lim_{n \to \infty} \frac{1}{1 + x^2}f\Big(\frac{x}{\sqrt{n}}\Big) = \frac{1}{1 + x^2}f(0).$$
Moreover we can find $N$ such that if $n \ge N$ then $\Big|f\Big(\frac{x}{\sqrt{n}}\Big) - f(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]
$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $?
User Semsem below kindly identified the problem: The normal to the disk is on the direction $-j$ so we have to reve... | I will show you how the double integral works out no matter what you select for $z$.
Consider $$\begin{align}I &= \int_0^{2\pi}\!\int_0^1 \!\! r^3 + 2r^3f^2(\theta)\, \mathrm{dr} \, \mathrm{d}\theta \\ &= \int_0^{2\pi}\!\dfrac{1}{4} + \dfrac{1}{2}f^2(\theta)\;\mathrm{d}\theta\end{align}$$
$f(\theta)$ is either $\cos(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
A combinatorics question on a $n \times n$ grid In the book 'Foundations of Data Science' by Hopcroft and Kannan, they have the following exercise (Ex. 5.46):
Let G be a $n \times n$ lattice and let $S$ be a subset of $G$ with cardinality at most $\frac{n^2}2$. Define $$N = \{(i,j) \in S \,\, | \text{all elements in r... | Here's an entirely different approach (although somewhat related to Alex' solution) that can give a much stronger bound
$$
\sqrt{|N|}+\sqrt{n^2-|S|}\le n
$$
from which the result follows.
If $S$ contains $p$ full rows and $q$ full columns, then $|N=pq|$. However, with $p$ rows and $q$ columns full, if $k$ of the remain... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
cartesian product $A^2 = A$, possible? Do there exist non-empty sets $A$ such that $A\times A = A$?
$A\times A = A$ looks a little strange to me, since $A\times A$ seems somehow more complicated than $A$, hence it is unlike that they are equal, but then, I cannot think of any reason why there should not be such a (non-... | Under the usual axioms of set theory, the answer is negative. The reason is that one of these axioms, the axiom of foundation, allows us to assign to each set a rank $\alpha$ (an ordinal number) that indicates at what stage of the transfinite construction of the universe the set appeared. Ordinals have the property tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Demonstrate the following .... The diagonals of a trapezoid (ABCD where AD parallel to BC ) divide it into four triangles all having one vertice in O. Knowing that the area of triangle $\triangle AOD$ is $A_1$ and of $\triangle BOC$ is $A_2$, demonstrate that the area of the trapezoid is
$$(\sqrt{A_1} + \sqrt{A_2})^2$... | Note that $\triangle ABD$ and $\triangle ADC$ have the same area (same base, equal heights). They have $\triangle AOD$ in common, so the areas of the two parts where they differ are equal, That is, $\triangle AOB$ has the same area as $\triangle DOC$.
Call the area of each by the name $x$.
Look at $\triangle ABC$. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determining if a number is an nth root I am working on a proof that depends on if an adversary can determine if a number is an $nth$ power for some large prime $p$. My intuition tells me that for a sufficiently large value of $n$ this is impossible. However I am unaware of any theorems that state this. Could you point ... | Here is a simple way to test whether $r$ is an $n$th power modulo $p$.
Given $n,p,r$, compute the least common multiple $s=nt$ of $n$ and $p-1$, then compute $r^t$. $r$ is an $n$th power modulo $p$ if and only if $r^t\equiv1\pmod p$.
Proof. Let $g$ be a primitive root mod $p$. Then $r=g^u$ for some $u$, and $r^t=g^{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculating time to arbitrary points of distance with initial velocity and non-uniform acceleration I've been trying to tackle this question for a while now, but I'm afraid it's not going anywhere without some outside help.
So let's say we have some body falling towards a planet without any atmosphere, and let's assum... | Let me try to help you, but I cannot assure you that my approach is correct. Let $m$ be the mass of falling body and $M$ be the mass of planet. First, we start from the relation of the Newton's second law and Newton's law of universal gravitation.
$$
\begin{align}
ma&=-\frac{GMm}{r^2}\\
\frac{dv}{dt}&=-\frac{GM}{r^2}\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/739914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Doubly infinite sequence limit I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as
$$F^n_m=\prod_{k=1}^na_{mk}$$
For each finite $n$ and $k$, as $m \to \infty, a_{mk}$ converges to a finite nonzero l... | Looks like an iterated limit, right?
There has not been given an accepted (i.e. satisfactory) answer to the following question:
Commutativity of iterated limits
Please read the question, the answer and the comments in there and make up your mind.
Anyway, it is not a standard result in common mathematics - don't ask me ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Evaluating: $ \lim\limits_{x\to 1^{-}}\left[\ln\left(1-x \over 1 + x\right) - \ln\left(1 - x^{2}\right)\right] $ I would like to evaluate the following limit:
$$
\lim_{x\to 1^{-}}\left[%
\ln\left(1-x \over 1 + x\right) - \ln\left(1 - x^{2}\right)\right]
$$
I put this into a limit calculator and the calculator gave as a... | Your expression $-\infty + \infty$ is an indeterminate form, much like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. So, on its own, it has no meaning.
In your specific case, using the property of logs that $\ln(AB) = \ln(A) + \ln(B)$ will likely be helpful here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/740156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Functions of the form $\int_a^x f(t) dt$ that are commonly used. I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus. I'm using James Stewart's Calculus textbook, and I was hoping to give students several "re... | In cartography in the year 1599 the following problem arose. Mercator wanted a map of the world on which compass bearings on the earth correspond to those on the map. E.g. $13^\circ$ east of north on the surface of the earth corresponds to $13^\circ$ counterclockwise from straight up on the map, at every point. Goin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 1
} |
Connections between prime numbers and geometry This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?
PS: feel free to interpret t... | A simple example: number p>2 is prime iff any equiangular p-gon with rational side lengths is regular, see, e.g., http://www.cut-the-knot.org/Outline/Geometry/EquiangularP-gon.shtml
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/740323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 9,
"answer_id": 3
} |
Prove that if $f(x) = Ax^2 + Bx + C$ is an integer whenever $x$ is an integer, then $2A$, $A+B$ and $C$ are also integers.
Prove that if $f(x) = Ax^2 + Bx + C$ is an integer whenever $x$ is an integer, then $2A$, $A+B$ and $C$ are also integers.
I've tried a lot to do it, but can't get it exactly right.
| We have
$f(x) = Ax^2 + Bx + C; \tag{1}$
set $x = 0$, an integer. Then we have
$f(0) = C, \tag{2}$
so $C$ is an integer. Set $x = 1$; then
$f(1) - C = A + B, \tag{3}$
showing $A + B$ is an integer. Set $x = -1$; then
$f(-1) - C = A - B \tag{4}$
is also an integer. Since
$2A = (A + B) + (A - B), \tag{4}$
we see that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
a spider has 1 sock and 1 shoe for each leg. then find out the the total possibilities. a spider has one sock and one shoe for each of its 8 legs.in how many different orders can the spider put on its shocks and shoes; assuming that on each leg ;the shock must be put on before the shoe? i have tried the problem in foll... | Let the act of putting on a sock be $1,2,3,4,5,6,7,8$, each corresponding to a leg. Let the act of putting on a shoe be $A,B,C,D,E,F,G,H$, each corresponding to a leg as well.
This problem is now bijective to counting the number of strings such that $A,B,C,D,E,F,G,H$ comes after $1,2,3,4,5,6,7, 8$ respectively.
We sta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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