Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
The Moore Plane's topology In the definition of the Moore plane $X=L{_1}\cup L{_2}$, where $L{_1}$ is the line $y=0$ and
$L{_2}=X\setminus L{_1}$ , I have a problem. In the Engelsking's book, for each $x\in L{_1}$
neghbourhood of $x$, is the form $U(x,1/i)\cup \{ x \}$ where $U(x,1/i)$ be the set of $X$ inside the circ... | You don’t need to cover $X$ with basic open nbhds of points of $L_1$: you also have the basic open nbhds of points of $L_2$, which are ordinary Euclidean balls small enough to stay within $L_2$. Specifically, the following collection is a base for $X$:
$$\left\{\{\langle x,0\rangle\}\cup B\left(\left\langle x,\frac1k\r... | {
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Problem related to continuous complex mapping. We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$.
1) I want to show that: $\partial(g(D))\subseteq g(\partial D).$
And further, I need two examples:
a) First, to... | It's better to assume the function $g$ to be bounded. If the image curve $f(\partial D)$ forms infinitely many loops everywhere, then $\partial f(D)\subsetneqq f(\partial D)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/571404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to solve this initial value problem on $(-\infty, \ +\infty)$? I've managed to solve the following initial-value problem on the interval $(0, +\infty)$: $$x y^\prime - 2y = 4x^3 y^{1/2} $$ with $y = 0$ when $x = 1$. The unique solution is $y = (x^3 - x)^2$.
How to solve this problem on the interval $(-\infty, \ +\... | With the scaling $x \to \alpha x$, $y \to \beta y$ we can see that the equation is invariant whenever $\alpha^{1/2} = \beta^{3}$. It means that $y^{1/2}/x^{3}$ is invariant over the above mentioned scaling. It suggests the variable change $u = y/x^{6}$ or/and $y = x^{6} u$. It leads to:
$$
{1 \over 4\sqrt{u} - 6u}\,{{\... | {
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"url": "https://math.stackexchange.com/questions/571474",
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Find the four digit number? Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
| The Number is of the form 1000A + 100A + 10B + B = 11( 100A + B ) = 11 ( 99A + A + B )
Since it is a perfect square number , (99A + A + B) should be divisible by 11 hence (A + B) is divisible by 11....(i)
Any perfect square has either the digits 1 , 4 , 9 , 6 , 5 , 0 at the units' place , ...(ii)
Only numbers which sat... | {
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Determinant of a n x n Matrix - Main Diagonal = 2, Sub- & Super-Diagonal = 1 I'm stuck with this one - Any tips?
The Problem:
Let $n \in \mathbb{N}.$ The following $n \times n$ matrix:
$$A = \left( \begin{array}{ccc}
2 & 1 & & & & ...\\
1 & 2 & 1 & & & ...\\
& 1 & 2 & 1 & & ...\\
& & 1 & 2 & 1 & ...\\
& & ... | $A_n=2A_{n-1}-A_{n-2}$, therefore $A_n=a\cdot n\cdot1^n+b$. The coefficients $a=1,b=1$ can be computed from $A_2=2$ and $A_3=3$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible to subtract a matrix from both side? I have this equation $AX + B = I$ and I want to find Matrix $X$.
$$(A^{-1})AX + B = (A^{-1})I$$
$$X + B = (A^{-1})I$$
My question is, is it legal to do $X + B - B = (A^{-1})I - B$?
| $AX+B=I$, $A^{-1}AX+A^{-1}B=A^{-1}I$. So:
$X=A^{-1}I-A^{-1}B$. It's legal.
| {
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"url": "https://math.stackexchange.com/questions/571789",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 2
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Limit: $\lim_{n\to \infty} \frac{n^5}{3^n}$ I need help on a homework assignment. How to show that $\lim_{n\to\infty} \left(\dfrac{n^5}{3^n}\right) = 0$? We've been trying some things but we can't seem to find the answer.
| Can you complete this?
$n>32 \to n^5<2^n \to \frac{n^5}{3^n}<\frac{2^n}{3^n}$
$n>\log_{\frac32}(\epsilon) \to (\frac32)^n>\epsilon \to (\frac{2}{3})^n<\epsilon$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof of a trigonometric expression Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$.
This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.
| Let $7\theta=\pi, 4\theta=\pi-3\theta\implies \sin4\theta=\sin(\pi-3\theta)=\sin3\theta$
$$\frac1{\sin3\theta}+\frac1{\sin2\theta}$$
$$=\frac1{\sin4\theta}+\frac1{\sin2\theta}$$
$$=\frac{\sin4\theta+\sin2\theta}{\sin4\theta\sin2\theta}$$
$$=\frac{2\sin3\theta\cos\theta}{\sin4\theta\sin2\theta}\text{ Using } \sin2C+\sin... | {
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"timestamp": "2023-03-29T00:00:00",
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Is every function from $\aleph_0 \to \aleph_2$ bounded? If $f$ is a function $f:\aleph_0 \to \aleph_2$, does it mean that the range of f is bounded in $\aleph_2$? Does this hold for all regular cardinals?
| One of the definitions of a cardinal $\kappa$ being regular is that, whenever $\alpha < \kappa$, every function $f : \alpha \to \kappa$ is bounded.
In any case, you can prove this directly, using the fact that a countable union of sets of cardinality $\aleph_1$ has cardinality $\aleph_1$: consider $$\bigcup_{n < \omega... | {
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Is my calculation correct about a probability problem? Suppose there are $300$ tickets in the pool, where $7$ of them belong to me. $20$ tickets are randomly taken out of the pool, and are declared as "winning tickets". What is the probability that exactly 4 of the winning tickets are mine?
When I tried to solve this I... | Your answer is correct but the way it is notated is not very elegant.
I should choose to write it as:
$$\frac{\binom{20}{4}\binom{280}{3}}{\binom{300}{7}}$$
Choosing $7$ from $300$ gives $\binom{300}{7}$ possibilities. $4$ of them belonging to your $20$ gives $\binom{20}{4}$ possibilities and $3$ of them belonging t... | {
"language": "en",
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Measurable functions are not polynomials The problem I have says:
If $f$ is measurable on $\mathbb R$, prove that there is at most a countable number of polynomials $P$ such that $P\overset{\text{a.e.}}{=}f$.
I think I need to show that if $f$ is not a polynomial then it is different almost everywhere from every poly... | Since the unique open set with $0$ measure is the empty set, the set where two continuous functions are different is either empty or of positive measure.
So given a measurable function $f$, there is at most one continuous function $g$ such that $f=g$ almost everywhere. (if $g_1$ and $g_2$ do the job, $g_1=g_2$ almost ... | {
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Proof that f is Riemann Integrable
Theorem 6.1.8a: If $f$ is continuous on $[a,b]$ then $f$ is Riemann Integrable on $[a,b]$.
Theorem 6.1.7: A bounded real-valued function $f$ is Riemann Integrable on $[a,b]$ if and only if for every $e > 0$, there exists a partition $P$ of $[a,b]$ such that $U(\mathcal{P},f) - L(\ma... | Since $f$ is continuous on $[a, c - \delta]$ and $[c + \delta, b]$ therefore it is integrable on both these intervals.
Let $P_{1}$ be partition of $[a, c - \delta]$ and $P_{2}$ be partition of $[c + \delta, b]$ such that $U(P_{1}, f) - L(P_{1}, f) < \epsilon / 3$ and $U(P_{2}, f) - L(P_{2}, f) < \epsilon / 3$. Now choo... | {
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Algorithm for finding a basis of a subgroup of a finitely generated free abelian group Let $G$ be a finitely generated free abelian group.
Let $\omega_1,\cdots, \omega_n$ be its basis.
Suppose we are given explicitly a finite sequence of elements $\alpha_1,\cdots, \alpha_m$ of $G$ in terms of this basis. Let $\alpha_i ... | Let $x_1,\cdots, x_m$ be a sequence of elements of $G$.
We denote by $[x_1,\cdots,x_m]$ the subgroup of $G$ generated by $x_1,\cdots, x_m$.
We use induction on the rank $n$ of $G$.
Suppose $n = 1$.
Then $\alpha_1 = a_{11}\omega_1,\cdots, \alpha_m = a_{m1}\omega_1$.
We may suppose that not all of $\alpha_i$ are zero.
Le... | {
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Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$? I am trying to find the minimal value of the expression:
$log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$
I think experience gives that the variables should be equal, if so then the minimal value is 6, but this not true in general.
Any hints or help will be greatly appreciat... | Assuming $a > 1$ and $a <= b <= c$. Let $q = b/a$ and $r = c/b$, so b = $qa$ and $c = qra$, and $ q,r \ge 1$.
$$ f(a,b,c) = \lg_a a^2 q^2 r + \lg_{aq} a^2 q r + \lg_{aqr} a^2 q $$
$$ = \dfrac{2 \ln a + 2 \ln q + \ln r}{\ln a} + \dfrac{2 \ln a + \ln q + \ln r}{\ln a + \ln q} + \dfrac{2 \ln a + \ln q}{\ln a + \ln q + \l... | {
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Numerical integration of $\int_0^2 \frac{1}{x+4}dx.$ I have homework problem. Determine the number of intervals required to approximate
$$\int_0^2 \frac{1}{x+4}dx$$ to within $10^{-5}$ and computer the approximation using (a) Trapezoidal rule, (b) Simpson's rule, (c) Gaussian quadrature rule. I think the phrase "withi... | It might also be asking you to use the remainder term formula. I happen to remember that the remainder term for Simpson's rule using $n$ intervals (where here $n$ must be an even number) is $$ -\frac{(b-a)^5 f^{(4)}(\xi)}{180n^4}$$
where $a$ and $b$ are the limits of integration, $f$ is the integrand, and $\xi$ is bet... | {
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Project point onto line in Latitude/Longitude Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D.
Diagram:
| What to do
Express $A,B,C$ using Cartesian Coordinates in $\mathbb R^3$. Then compute
$$D=\bigl((A\times B)\times C\bigr)\times(A\times B)$$
Divide that vector by its length to project it onto the sphere (with the center of the sphere as center of projection). Check whether you have the correct signs; the computation m... | {
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Techniques for removing removable singularities (without resorting to series expansion)? Suppose $f: \mathbb{C} \supset U \to \mathbb{C}$ is a meromorphic function with a removable singularity at the point $z_0 \in U$. Then $f$ can be extended to a holomorphic function over all of $U$. However, the material I've enco... | The extension across the removable singularity simply coincides with the original function outside the singularitites (that's the very point!). At the removable singularity, the value of the extended function is just $\lim_{z\to a} f(z)$. In your particular case, we have that
$$
\lim_{z\to 0} \frac{e^z-1}{z} = 1,
$$
so... | {
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Poisson Distribution Lambda, Probability, and Looking for Exactly k Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with a rate of $ \lambda $ = 10 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.
What is the probability that ... | If $X$ has Poisson distribution with parameter $\lambda$, and $Y$ has binomial distribution with the number of trials equal to the random variable $X$, and $p$ any fixed probability $\ne 0$, then the number of "successes" has Poisson distribution with parameter $\lambda p$.
This has been proved repeatedly on MSE, at l... | {
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Describing where a Kleisli Triple fits into a Monad ontology I'm trying to map a Kleisli triple onto my existing understanding of Monads.
I can represent my understanding of Monads like this: (courtesy of Jim Duey's slides at 13)
Could you please point to the part on this diagram where Kleisli triples fit in - or eve... | Kleisli triples fit in the diagram exactly where you have "monads". Kleisli triples are equivalent to monads (you might say they are one presentation of monads).
I infer from the link that you are thinking about this in the context of programming languages? This question/answers might help: What is a monad in FP, in c... | {
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Why is the expected value (mean) of a variable written using square brackets? My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies for $Var[X]$.
| I don't.
You can write in both ways, it doesn't matter. Some don't even use brackets but might instead write $EX$ or $VX$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is there a means of analytically proving the following identity? Okay, so before I begin, my background is more in the world of applied, rather than pure, mathematics, so this question is motivated by a physics problem I'm looking at just now. Mathematically, it boils down to looking for a minimum of a real valued func... | A way to do this is to define a function $f: f(u)=\cosh^2(u)-u^2/8$ and to show that $\forall u, f(u)\geq 0$.
And how to do this? Take the derivative of $f(u)$, which is $2 \cosh(u) \cdot \sinh(u)- u/4$, find the minimum value of $f$ according to the value which makes the derivative $0$. As it's greater or equal than ... | {
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Proving all sufficiently large integers can be written in the form $a^2+pq$. This is one of those numerous questions I ask myself, and to which I seem unable to answer:
Can every integer greater then $657$ be written in the form $a^2+pq$, with $a\in\mathbb Z$ and $p,q$ prime?
A quick brute force check trough numbers ... | This is very hard.
(Notation: A semiprime is a product of exactly two distinct primes.)
Just consider this problem for perfect squares. We're looking at the equation $b^2=a^2+pq$, which easily translates to $$(b-a)(b+a)=pq.$$ For fixed $b$, this is soluble in $a$ (and $p$ and $q$), precisely if either: 1) $2b-1$ is a... | {
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Counting Problems. I'm having trouble with the following.
*
*A man has 10 distinct candies and puts them into two distinct bags such that each bag contains 5 candies. In how many ways can he do it?
a. For this problem I thought it would (10 choose 5) since we could place 5 candies in one box out of 10, and t... | I’m going to add a little more explanation of the difference between the first and second problems. In the first problem we’re told that the bags are distinct; given the wording of the last two problems, that almost certainly means that they are individually identifiable, not interchangeable. That’s as if in the second... | {
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Does there exist a semigroup such that every element factorizes in this way, which nonetheless lacks a left identity? If a semigroup $S$ has a left identity-element, then for any $y \in S$ we can write $y = xy$ for some $x \in S$. Just take $x$ to be any of the left identities, of which there is at least one, by hypoth... | To give a concrete example based on vadim123's answer:
Let $X$ denote an arbitrary set (for ease of imagining, assume non-empty). Then $2^X$ can be made into an idempotent monoid by defining composition as binary union. Now delete the empty set from $2^X$, obtaining a semigroup $S$. Since $S$ is idempotent, thus every ... | {
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Are these isomorphic $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ and $\mathbb{Z}_{9}^{*}$ Is $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ isomorphic to $\mathbb{Z}_{9}^{*}$
both have orders 6
both have elements with orders 1,2,3,6 (1 element of order 1, 2 elements of order 3, 1 element of order 2 and 2 elements of order 6)
Both are... | Yes, the two groups are isomorphic. And you are almost there in proving this.
Note that $\mathbb Z_2\times \mathbb Z_3 = \mathbb Z_6$, since $\gcd(2, 3) = 1$.
And since the order of $\mathbb Z^*_9 = 6$ and is cyclic, we know that $\mathbb Z^*_9 \cong \mathbb Z_6$.
There is no need to construct an explicit isomorphism ... | {
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Prove that the $x$-axis in $\Bbb R^2$ with the Euclidean metric is closed
I want to show that the $x$-axis is closed.
Below is my attempt - I would appreciate any tips on to improve my proof or corrections:
Let $(X,d)$ be a metric space with the usual metric.
Want to Show: $\{(x,y) | x ∈ \Bbb R, y = 0\}$ is closed
C... | Another fun way you might approach this problem is to let $f : \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x,y) = y^2$. If you know/can show that $f$ is continuous, then it will imply that $f^{-1}(\{0\})$ is closed.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Need help with permutations and combinations problems A woman has 6 friends each evening, for 5 days, she invites 3 of them so that the same group is never invited twice. How many ways are there to do this? ( Assume that the order in which groups are invited matters.)
Attempt:
I know if I did, 6 choose 3, I would get a... | The number to make groups of 3 people out of 6 is ...?
Then note that if a group is invited on the first evening, it can't be invited for the second, so there will be one less to choose and so on.
Can you solve it on your own now?
| {
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Enjoyable book to learn Topology. I believe Visual Group Theory - Nathan Carter is the best book for a non-mathematician (with high school math) to learn Group Theory.
Could someone please recommend me a similar book (if there is) to learn Topology?
Edit: I know many books in Topology, but someone who has read the abov... | Topology -James Munkres
I have been using James Munkres book for self study.
The proofs are well presented ,easy to follow and yet still rigorous.
The first few chapters give you Set Theory concepts to prepare you for the rest of the book
| {
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Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number.
Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number.
Actually, I know a way to solve this, but even if it is very large and cumbe... | It's possible to demonstrate that if N is a multiple of 100, N! ends with (N/4)-1 zeroes.
| {
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Probability of forming a 3-senator committee If the Senate has 47 Republicans and 53 Democrats,
in how many ways can you form a 3-senator committee
in which neither party holds all 3 seats?
The solution says that:
You can choose one Democrat, one Republican, and one more senator from either party. We can make these c... | Hint: find
[Number of ways of choosing arbitrary set of senators] -
[number of ways of choosing 3 Democrats] -
[number of ways of choosing 3 Republicans].
Finding each of these 3 quantities is a standard probability task (think combininations, permutations, etc.).
NOTE: I'm ignoring the given solution, since I thi... | {
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} |
Negative curvature compact manifolds I know there is a theorem about the existence of metrics with constant negative curvature in compact orientable surfaces with genus greater than 1. My intuition of the meaning of genus make me think that surfaces with genus greater that 1 cannot be simply-connected, but as my knowl... | Every two (classification of surfaces) and three (Poincare-Thurston-Perelman) closed simply-connected manifold is diffeomorphic to a sphere, hence does not admit any metric of negative curvature.
If a manifold has constant sectional curvature, its metric lifts to the universal cover, which is isometric to one of the sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/574053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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When do two functions differ by a constant throughout an interval (Fundamental Theorem of Calculus) I'm reading the proof of the Fundamental Theorem of Calculus here and I don't understand the following parts (at the bottom of page 2):
I don't know how to conclude that $G(x)-F(x)=C$ for a $x \in [a,b]$.
How do I prove... | This is a consequence of the following general fact:
If $f'(x) = 0$ for all $x$ in an interval $[a, b]$, then $f$ is constant on $[a, b]$.
One way to prove this is by the Mean Value Theorem: If there were to exist $x_1$ and $x_2$ in the interval for which $f(x_1) \ne f(x_2)$, there would exist a $c$ between $x_1$ and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of series $\frac{n}{(n+1)!}$ I'm encountering some difficulty on a question for finding the sum of the series $$\sum_{n=0}^\infty \dfrac{n}{(n+1)!}$$
The method I use to tackle this type of problem is generally to find a similar sum of a power series and algebraically manipulate it to match that of the original. I ... | hint:
$$ \sum_{n=0}^\infty \dfrac{n}{(n+1)!}= \sum_{n=1}^\infty \dfrac{n-1}{(n)!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/574211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Triangle Ratio/Proportions Problem I would like someone to verify that I am solving this problem correctly. I do not remember the theorem that allows me to make the two halves of the triangle proportional.
Because (h1/h2 = h1/h2) Triangles are proportional?
Here is the problem:
My work:
| $$\frac{Area(BML)}{Area(BCM)}=\frac{LM}{MC} \Rightarrow \frac{5}{10}=\frac{LM}{MC}$$
$$\frac{Area(MCK)}{Area(BCM)}=\frac{KM}{MB} \Rightarrow \frac{8}{10}=\frac{KM}{MB}$$
Let's say the area of $AMK$ be $2A$ then from $\frac{Area(ALM)}{Area(AMC)}=\frac{LM}{MC}$, the area of $ALM$ will be $4+A$. Now
$$\frac{Area(AMK)}{Ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/574268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do this algorithm terminates? Let $x \in \mathbb{R}^p$ denote a $p$ dimensional data point (a vector). I have two sets $A = \{x_1, .., x_n\}$ and $B = \{x_{n+1}, .., x_{n+m}\}$. So $|A| = n$, and $|B| = m$. Given $k \in \mathbb{N^*}$, let $d_x^{(A, k)}$ denote the mean Euclidean distance from $x$ to its $k$ nearest poi... | Take $p=1$ and $k=1$. Consider $A=\{0,3\}$ and $B=\{2,5\}$.
*
*$d_0^A=3$ and $d_0^B=2$
*$d_2^A=1$ and $d_2^B=3$
*$d_3^A=3$ and $d_3^B=1$
*$d_5^A=2$ and $d_5^B=3$
So $A'=A$ and $B'=B$, so $A$ becomes $B$ and $B$ becomes $A$, and the algorithm never stops.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/574344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability space proof
PROBLEM
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(E_n)$ be a sequence in the $\sigma$-algebra $\mathcal{F}$.
$a)$ If the sequence $(E_n)$ is increasing (in the sence that $E_n \subset E_{n+1}$) with limit $E = \cup_nE_n$, prove that $P(E_n) \rightarrow P(E)$ as $n \rightar... | a) Define $A_n:=E_{n}\setminus E_{n-1}$: and $A_0:=E_0$. Then
*
*if $i\neq j$, we have $A_i\cap A_j=\emptyset$;
*for each $N$, $\bigcup_{i=1}^NA_i=\bigcup_{i=1}^NE_i$;
*$\mathbb P\left(\bigcup_{i=1}^NA_i\right)=\sum_{i=1}^N\mathbb P(A_i)$.
b) Consider the sequence $(\Omega\setminus E_n)_n$. This forms a non-de... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Probability that exactly k of N people matched their hats [SRoss P63 Ex 2g]
The match problem stated in Example 5m of Chapter 2 (of A First Course in Pr, 8th Ed, Ross) showed that the probability of no matches when $N$ people randomly select from among their own $N$ hats $= P[N]= \sum_{0 \le i \le N}(-1)^i/i!$
What ... |
Since $E$ is the event that exactly these $k$ people, for some $k$, have a
match, how and why isn't the required probability?
Because
*
*You are asked for the probability that exactly $k$ people (no more and less) match their hats. The event that the (say) first $k$ match is not necessarily a "success", because... | {
"language": "en",
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How do you rotate a vector by $90^{\circ}$? Consider a vector $\vec{OA}$. How will I rotate this vector by $90^{\circ}$ and represent in algebraically?
| Calling the vector $\overrightarrow v$, with components $v_x,v_y$ the angle between the vector and the $x$ axis is: $\alpha=\arctan\frac{v_y}{v_x}$. So if you add $\frac{\pi}{2}$ to $\alpha$, you get:
$$v_x=\overrightarrow v\cos(\alpha+\frac{\pi}{2})$$
$$v_y=\overrightarrow v\sin(\alpha+\frac{\pi}{2})$$
If you wanto to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/574693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simplify Sum of Products: $\;A'B'C' + A'B'C + ABC'$ How would you simplify the following sum of products expression using algebraic manipulations in boolean algebra?
$$A'B'C' + A'B'C + ABC'$$
| Essentially, all that's involved here is using the distributive law (DL), once.
Distributive Law, multiplication over addition: $$PQ + PR = P(Q + R)\tag{DL}$$
In your expression, in the first two terms, put $P = A'B'$:
We also use the identity $$\;P + P' = 1\tag{+ID}$$
$$\begin{align} A'B'C' + A'B'C + ABC' & = A'B'(C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/574749",
"timestamp": "2023-03-29T00:00:00",
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Show that there is no natural number $n$ such that $3^7$ is the largest power of $3$ dividing $n!$
Show that there is no natural number $n$ such that $7$ is the largest power $a$ of $3$ for which $3^a$ divides $n!$
After doing some research, I could not understand how to start or what to do to demonstrate this.
We ha... | Hint: What is the smallest value $n_1$ such that $3^7\mid (n_1)!$? What is the largest value $n_0$ such that $3^7\nmid (n_0)!$? What is the largest exponent $k$ such that $3^k\mid (n_1)!$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/574898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Asymptotics of logarithms of functions If I know that $\lim\limits_{x\to \infty} \dfrac{f(x)}{g(x)}=1$, does it follow that $\lim\limits_{x\to\infty} \dfrac{\log f(x)}{\log g(x)}=1$ as well? I see that this definitely doesn't hold for $\dfrac{e^{f(x)}}{e^{g(x)}}$ (take $f(x)=x+1$ and $g(x)=x$), but I'm not sure how to ... | It does not follow. Take the example of $f(x)=e^{-x}+1$ and $g(x)=1$. Then
$$
\lim_{x \to \infty}\frac{f(x)}{g(x)}= \lim_{x \to \infty} \frac{e^{-x}+1}{1}=1
$$
However,
$$
\lim_{x \to \infty} \frac{\log f(x)}{\log g(x)}=\lim_{x \to \infty} \frac{\log(e^{-x}+1)}{\log 1}
$$
Does not exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/575008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
} |
Find a generating function for $a_r = n^3$ What is the generating function for $a_r = n^3$? I computed an answer, just wanted to double check my answer.
| Here is how you advance. Assume
$$ F(x) = \sum_{r=0}^{\infty} a_r x^r \implies F(x)=\sum_{r=0}^{\infty} r^3 x^r $$
$$ \implies F(x)= (xD)(xD)(xD)\sum_{r=0}^{\infty} x^r = (xD)^3 \frac{1}{1-x}, $$
where $D=\frac{d}{dx}$.
Can you finished now?
Added Here is the final answer
$$ F(x)={\frac {x \left( 1+4\,x+{x}^{2} \righ... | {
"language": "en",
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Continuity of a function defined by an integral Ok, Here's my question:
Let $f(x,y)$ be defined and continuous on a $\le x \le b, c \le y\le d$, and $F(x)$ be defined >by the integral $$\int_c^d f(x,y)dy.$$ Prove that $F(x)$ is continuous on $[a,b]$.
I think I want to show that since $f(x,y)$ is continuous on $[a,b]$... | This is quite relevant for Rudin 10.1 from Real Analysis on page 246. Function $ f(x,y) $ is a continuous function on a compact set $[a,b]$. Therefore it is uniformly continuous, so the integrated expression is some real value, which can be made arbitrarily small.
In other words, uniform continuity of $f(x,y)$ indeed i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Proofs with Induction Imply Proofs Without Induction? Assume we can prove $\forall x P(x)$ in first order Peano Arithmetic (PA) using induction and modus ponens. Does this mean we can prove $\forall x P(x)$ from the other axioms of PA without using induction?
Given the induction axiom $(P(0) \land \forall x(P(x) \right... | The part:
"This can be converted to $P(0) \land \forall x( \neg P(x) \lor P(Sx))$."
is correct.
The part:
"We better not be able to prove $\exists x \neg P(x)$ from the other axioms so this reduces to $P(0) \land \forall x P(Sx)$."
is not. In general $\forall x(A(x)\lor B(x))$ is not equivalent to $\forall x A(x) \lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Factorial lower bound: $n! \ge {\left(\frac n2\right)}^{\frac n2}$ A professor in class gave the following lower bound for the factorial
$$
n! \ge {\left(\frac n2\right)}^{\frac n2}
$$
but I don't know how he came up with this formula. The upper bound of $n^n$ was quite easy to understand. It makes sense. Can anyone ex... | Suppose first that $n$ is even, say $n=2m$. Then
$$n!=\underbrace{(2m)(2m-1)\ldots(m+1)}_{m\text{ factors}}m!\ge(2m)(2m-1)\ldots(m+1)>m^m=\left(\frac{n}2\right)^{n/2}\;.$$
Now suppose that $n=2m+1$. Then
$$n!=\underbrace{(2m+1)(2m)\ldots(m+1)}_{m+1\text{ factors}}m!\ge(m+1)^{m+1}>\left(\frac{n}2\right)^{n/2}\;.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/575389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
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Getting angles for rotating $3$D vector to point in direction of another $3$D vector I've been trying to solve this in Mathematica for $2$ hours, but got the wrong result.
I have a vector, in my case $\{0, 0, -1\}$. I want a function that, given a different vector, gives me angles DX and DY, so if I rotate the original... | So you need a 3×3 rotation matrix $E$ such that
$$ E\,\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix} = -\hat{k} $$
This rotation matrix consists of two elementary rotations
$$ \begin{aligned}
E & = {\rm Rot}(\hat{i},\varphi_x){\rm Rot}(\hat{j},\varphi_y) \\
& = \begin{pma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $x$ and $y$ be two vectors in $\mathbb{R}^d$ with $\mid x \mid = \mid y \mid$. Find a unit vector $u$ such that $P_u x = y$ Let $u \in \mathbb{R}^d = V$ be a unit vector and set $W = \text{span}(u) ^{\bot}$ (with respect to the dot product). The reflector across W is $P_u = I_d - 2uu^T$.
Let $x$ and $y$ be two vect... | If $x=y$ choose any non-zero vector perpendicular to $x$. Otherwise $u:=(x-y)/\|x-y\|$. In this case
$$x\mapsto x-2\frac{\langle x,x-y\rangle}{\|x-y\|^2}(x-y)=\frac{\|x\|^2x-2\langle x,y\rangle x+\|y\|^2x-(2\|x\|^2x-2\|x\|^2y-2\langle x,y\rangle x+2\langle x,y\rangle y)}{\|x\|^2-2\langle x,y\rangle+\|y\|^2}.$$
As $\|x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a $3 \times 3$ Matrix that maps points in $\mathbb{R}^3$ onto the a given line Give a $3 \times 3$ matrix that maps all points in $\mathbb{R}^3$ onto the line $[x,y,z] = t[a,b,c]$ and does not move the points that are on that line. Prove your matrix has these properties.
Can someone verify if I am doing this co... | Since it maps all the vectors into directions of single vector, hence it must be rank 1; in particular following solution will work
$\left[\begin{array}{ccc}
\gamma a & \alpha a & \beta a\\
\gamma b & \alpha b & \beta b\\
\gamma c & \alpha c & \beta c
\end{array}\right]\left[\begin{array}{c}
x\\
y\\
z
\end{array}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575647",
"timestamp": "2023-03-29T00:00:00",
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How many permutations of $\{1,2,3,4,5\}$ leave at least two elements fixed?
How many permutations $f: \{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ have the property that $f(i)=i$ for at least two values of $i$?
I'm just struggling with this inclusion/exclusion question. I figured the best way would be to subtract (the ca... | There is $1$ permuatation that fixes $5$ elements.
There are no permutations that fix $4$ elements.
There are ${5 \choose 3} = 10$ permuations that fix $3$ elements (the other two are switched around).
There are $2 {5 \choose 2} = 20$ permutations that fix $2$ elements, because there are $5 \choose 2$ ways to pick the ... | {
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"timestamp": "2023-03-29T00:00:00",
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Are there any non-constructive proofs for which an example was never constructed? By non-constructive I mean the following:
A mathematical object is proven to exist yet it is not constructed in the proof.
Are there any examples of proofs like this where the mathematical object was never constructed? (by which i mean ev... | On the same line of thought but, imo, more striking, is the use of Zermelo's Theorem to prove there must exist a well-ordering of the reals (and thus, that golden dream of having a grip on that elusive first positive real number seems to be closer...).
Yet no such ordering on $\;\Bbb R\;$, as far as I am aware, is know... | {
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"url": "https://math.stackexchange.com/questions/575835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Laplace transform of the following function find the laplace transform of the function :
$$f(t) =\begin{cases}
t^2, & 0<t<1 \\
2\cos t+2, & t>1 \\
\end{cases}$$
My attempt:
$$L\{f(t)\}=\int_{0}^{1}e^{-st} \ t^2 \ \text{d}t+\int_{1}^{\infty}e^{-st} \ (2\cos t+2) \ \text{d}t$$
Now,
$$\int_{0}^{1}e^{-st} \ t^2 \ \text{d... | $$\begin{align} \int_1^{\infty} dt \, e^{-s t} \cos{t} &= \Re{\left [\int_1^{\infty} dt \, e^{-(s-i) t} \right ]}\\ &= \Re{\left [\frac{e^{-(s-i)}}{s-i} \right ]}\\ &= e^{-s} \Re{\left [(\cos{1}+i \sin{1}) \frac{s+i}{s^2+1}\right ]} \\ &= \frac{s \cos{1}-\sin{1}}{s^2+1} e^{-s} \end{align}$$
$$ \int_1^{\infty} dt \, e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/575905",
"timestamp": "2023-03-29T00:00:00",
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Rational translates of the unit circle cover the plane Is it true that the translations of the unit circle by vectors
with both coordinates rational cover the plane?
This comes to solving
$$ x=a+\cos \theta, \ y=b+\sin \theta$$
with unknowns $a,b$ rational and $\theta$
between $0,2\pi$.
I couldn't find a positive answe... | No, the above statement is false.This is because it would imply that every real number is algebraic over $\mathbb{Q}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/575986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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$X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq\varnothing$ Let $X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq \varnothing$ then show that $E$ spans $X$.
I am trying it in a following way....
Let be the norm $\|\cdot\|:X \rightarrow... | "$E$ has a non-empty interior" means that there is $x_0\in E$ and $r\gt 0$ such that $B(x_0,r)\subset E$. We thus have $B(0,r)\subset\operatorname{span}(E)$. Since $E$ is a subspace, it's invariant by multiplication by scalars, so...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/576074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there any nonconstant function that grows (at infinity) slower than all iterations of the (natural) logarithm? Is there any nonconstant function that grows at infinity slower than all iterations of the (natural) logarithm?
| In fact there are functions that go to $\infty$ more slowly than any function you can write down a formula for. For positive integers $n$ let $f(BB(n)) = n$ where $BB$ is the Busy Beaver function. Extend to $[1,\infty)$ by interpolation.
EDIT: Stated more technically, a "function you can write down a formula for" is ... | {
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"url": "https://math.stackexchange.com/questions/576130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 4
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Beautiful Mathematical Images My Maths department is re-branding itself, and we've been asked to find suitable images for the departmental sign. Do you have a favourite mathematical image that could be used for the background of an A1-sized sign?
| Penrose tiling, an example:
http://en.wikipedia.org/wiki/Penrose_tiling
Or any basic first year theorems (or more advanced) theorems like MVT, Taylor, BW, cardinalities, ordinals etc, in nice fonts or even as sculpture.
Complex but symmetric 3D objects like the Wolframram Alpha star for example, see also George Hart.... | {
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"url": "https://math.stackexchange.com/questions/576306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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"answer_id": 0
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find all integers x such that $4x^2 - 1$ is prime I tried factoring it and got $(2x+1)(2x-1)$, however I do not know how to prove for all integers from here.
| Well, you're trying to show that $4x^2-1$ is prime for some integer $x$, but you just factored it! If $4x^2-1$ is going to be prime, your factorization has to a trivial factorization. We can always factor primes as $p=p\cdot 1$, so maybe this will lead you to the correct answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/576356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit $\lim_{n\to \infty} n(\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\cdots+\frac{1}{(2n)^2})$ I need some help finding the limit of the following sequence:
$$\lim_{n\to \infty} a_n=n\left(\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\cdots+\frac{1}{(2n)^2}\right)$$
I can tell it is bounded by $\frac{1}{4}$ from below and decreasin... | $$ \lim_{n\to\infty} n\sum_{k=n+1}^{2n} \frac{1}{k^2}=\lim_{n\to\infty} \frac{1}{n}\sum_{k=n+1}^{2n}\frac{1}{\left(\frac{k}{n}\right)^2}=\int_1^2 \frac{1}{x^2} \mathrm dx=\left[-\frac{1}{x}\right]_1^2=\frac{1}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/576458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solution to the limit of a series I'm strugling with the following problem:
$$\lim_{n\to \infty}(n(\sqrt{n^2+3}-\sqrt{n^2-1})), n \in \mathbb{N}$$
Wolfram Alpha says the answer is 2, but I don't know to calculate the answer.
Any help is appreciated.
| For the limit: We take advantage of obtaining a difference of squares.
We have a factor of the form $a - b$, so we multiply it by $\dfrac{a+b}{a+b}$ to get $\dfrac{a^2 - b^2}{a+b}.$
Here, we multiply by $$\dfrac{\sqrt{n^2+3}+ \sqrt{n^2 - 1}}{\sqrt{n^2+3}+ \sqrt{n^2 - 1}}$$
$$n(\sqrt{n^2+3}-\sqrt{n^2-1})\cdot\dfrac{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/576550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reed Solomon Code Working in GF(32). Polynomial is $x^5+x^2+1$. $\alpha$ is primitive element. $t = 3$. RS code. $n = 31$, $k = 25$. I have obtained generator polynomial $x^6+\alpha^{10}x^5+\alpha^{30}x^4+\dots$
How do I obtain generator matrix? I believe I write down coefficients in increasing order of $x$ over $31$ c... | The answer given by Sudarsan is one possible generator matrix. If
$${\bf d} = (d_0, d_1, \ldots , d_{k-1}) \longleftrightarrow
d(x) = d_0 + d_1x + \cdots + d_{k-1}x^{k-1}$$
is the data polynomial, then the codeword polynomial corresponding
to the codeword ${\bf c} = {\bf d}G$ (where $G$ is the generator
matrix in Suda... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/576612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the eigenvectors and eigenvalues of A geometrically $$A=\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}=\begin{bmatrix} 1/2 & 1/2 \\ 1/2 & 1/2\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix} \begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}.$$
Scale by 2 in the $x$- direction, then scale by 2 in the $y$- direction, th... | The geometry is what makes things easier (for me). Without the geometry, it would be a mechanical computation which I would not like doing, and might get wrong.
Note that the vector $(1,1)$ gets scaled by our two scalings to $(2,2)$, and projection on $y=x$ leaves it at $(2,2)$. So the vector $(1,1)$ is an eigenvector ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing a numerical sequence converges How could I show that the following sequence converges?
$$\sum_{n = 1}^{\infty} \frac{\sqrt{n} \log n}{n^2 + 3n + 1}$$
I tried the ratio and nth-root tests and both were inconclusive. I was thinking there might be a way to use the limit comparison test, but I'm not sure. Any hints... | Hint:
$$\sum_{n = 1}^{\infty} \frac{\sqrt{n} \log n}{n^2 + 3n + 1} < \sum_{n = 1}^{\infty} \frac{ \log n}{n^{3/2}}$$
Then by integral test, since $\int_{1}^{\infty}\frac{ \log n}{n^{3/2}}=4$ (converges), so the given series converges.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof of Heron's Formula for the area of a triangle
Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula:
$$A = \sqrt{p(p-a)(p-b)(p-c)},$$
where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$.
Could you please provide the proof of this formula?
Thank you in advance.
| It is actually quite simple. Especially if you allow using trigonometry, which, judging by the tags, you do. If $\alpha$ is the angle between sides $a$ and $b$, then it is known that
$$
\begin{align}
A &= \frac{ab\sin \alpha}{2},\\
A^2 &= \frac{a^2b^2\sin^2 \alpha}{4}.
\end{align}
$$
Now, $\sin^2 \alpha = 1 - \cos^2 \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 2
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Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and $\lambda$?
This seems like a simple calculus problem but my usual... | In discrete case, it may be useful to look at ratio of successful terms. Here, let $a_k = \binom{n}k \lambda^k$. Then:
$$\frac{a_{k+1}}{a_k} = \lambda\frac{n-k}{k+1}$$
As $k$ increases from $1$ to $n$, it is easily seen that the numerator decreases and the denominator increases, so the fraction decreases steadily fro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the group $\Gamma$ such that $\mathbb{H}/\Gamma$ is a genus-n torus We know that the universal cover of genus-n torus is a unit disk ($n\ge2$), which is conformal to upper half plane $\mathbb{H}$, with automorphism group $SL(2,\mathbb{R})$. Thus the genus-n torus can be identified with $\mathbb{H}/\Gamma$. with... | First, the group of orientation-preserving isometries of the upperhalf plane is not $SL(2,R)$ but rather $PSL(2,R)$. Second, this surface is not called a genus-$n$ torus but rather a genus-$n$ surface; the term torus is generally reserved for the genus-$1$ case.
So what you are asking for is an explicit Fuchsian group ... | {
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"timestamp": "2023-03-29T00:00:00",
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Virtues of Presentation of FO Logic in Kleene's Mathematical Logic I refer to Stephen Cole Kleene, Mathematical Logic (1967 - Dover reprint : 2002).
What are the "pedagogical benefits" (if any) of the presentation chosen by Kleene, mixing Natural Deduction and Hilbert-style ?
Propositional Calculus - at pag.33 he refer... | @Peter Smith wrote: So it is worth noting that e.g. John Corcoran can write "Three Logical Theories" as late as 1969 (Philosophy of Science, Vol. 36, No. 2 (Jun., 1969), pp. 153-177), finding it still novel and necessary to stress the distinctions between different types of logical theory. Here it is https://www.academ... | {
"language": "en",
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"source": "stackexchange",
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On idempotent elements that are contained in center of a ring Let $e$ and $f$ be idempotent elements of a ring $R$. Assume that $e,f$ are contained in center of $R$. Show that $Re=Rf$ if and only if $e=f$
Help me a hint to prove it.
Thank in advanced.
| HINT : show that the sum and product of $R$ induce a sum and product on $Re$. What can you say about $e\in Re$ with respect to multiplication?
| {
"language": "en",
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Fixed point in plane transformation. Some one give me a idea to solve this one.
It's a problem from Vladimir Zorich mathematical analysis I.
Problem 9.c from 1.3.5:
A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if $f(p)=p$. Verify that any composition of a shift, a rotation, and a similarity transformati... | Shift and rotation are just special cases of similarity transformations. A generic similarity can be written e.g. in the following form:
$$
\begin{pmatrix}x\\y\end{pmatrix}\mapsto
\begin{pmatrix}a&-b\\b&a\end{pmatrix}\cdot
\begin{pmatrix}x\\y\end{pmatrix}+
\begin{pmatrix}c\\d\end{pmatrix}
$$
With this you can solve the... | {
"language": "en",
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"source": "stackexchange",
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Almost sure convergence proof Cud someone please explain the proof of $ P(X_n \to X)=1 $ iff $$ \lim_{n \to \infty}P(\sup_{m \ge n} |X_m -X|>\epsilon) \to 0 $$. Im not able to understand the meaning of the various sets they take during the course of the proof.
| Intuitively, the result means that to converge almost everywhere is equivalent to "bound the probability of the $\omega$'s for which $|X_n-X|$ is infinitely often larger than a positive number". Here is a more formal argument.
Assume that $X_n\to X$ almost surely and fix $\varepsilon\gt 0$. Define $A_m:=\{|X_m-X|\gt \... | {
"language": "en",
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Can You Construct a Syndetic Set with an Undefined Density? Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i =1,2,..$ (that is, "the gaps of $A$ are uniformly bounded"). Th... | Take the union of the even integers and a subset of odd integers whose density fluctuates (say between 1/4 and 1/8 of odd numbers, to meet the other conditions).
| {
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"source": "stackexchange",
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Proofs from the Book - need quick explanation I've been recently reading this amazing book, namely the chapter on Bertrand's postulate - that for every $n\geq1$ there is a prime $p$ such that $n<p\leq2n$.
As an intermediate result, they prove that $\prod_{p\leq x}p \le 4^{x-1}$ for any real $x\geq2$, where the product ... | Not sure if you're analysing too much for the last part?
If we look at the inequality
$$
\prod_{m+1<p<=2m+1}p\leq\binom{2m+1}{m},
$$
we see that for any prime $p \in (m+1,2m+1]$, we have $p|(2m+1)!$ but $p\nmid m!$ and $p \nmid (m+1)!$. ["The last part" that you mention is simply because $p > m+1$.]
So $p$ is indeed a... | {
"language": "en",
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"source": "stackexchange",
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Equal balls in metric space Let $x$ and $y$ be points in a metric space and let $B(x,r)$ and $B(y,s)$ be usual open balls. Suppose $B(x,r)=B(y,s)$. Must $x=y$? Must $s=r$?
What I got so far is that: $$r \neq s \implies x \neq y$$ but that's it.
| Think minimally: Let $X=\{x,y\}$ be a set with two points. Define $d(x,y)=d(y,x)=1$, and $d(x,x)=0=d(y,y)$.
Then, $B(x,2)=B(y,3)$, yet $x\ne y$ and $2\ne 3$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ for all $x$? I'm pretty convinced that the Taylor Series (or better: Maclaurin Series):
$$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$
Is exactly equal the sine function at $x=0$
I'm also pretty sure that this function converges for all $x$
What ... | First, let's take the Taylor's polynomial $\displaystyle T_n(x) = \sum\limits_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$ at a given point $a$. We can now say:
$\displaystyle R_n(x) = f(x) - T_n(x)$, where $R_n$ can be called the remainder function.
If we can prove that $\lim\limits_{n\rightarrow \infty}R_n(x) = 0$, then $f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Proofs without words of some well-known historical values of $\pi$? Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. I've read that the Egyptian figure at least could be ju... | Straight edge and compass construction of a quadrature of a circle is possible only with Babylonian value of pi. Try this link : https://www.academia.edu/8084209/Ancient_Values_of_Pi. Though, Egyptian value of Pi (22/7 or 256/81) is rational, compass and straight edge construction of a quadrature of a circle is not pos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine whether series is convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{n^2+4}$ I still haven't gotten the hang of how to solve these problems, but when I first saw this one I thought partial fraction or limit. So I went with taking the limit but the solution manual shows them using the integral test.
Was I... | We only have the following statement to be true:
$$\text{If $\sum_{n=1}^{\infty} a_n$ converges, then $a_n \to 0$.}$$ The converse of the above statement is not true, i.e.,
$$\text{if $a_n \to 0$, then $\displaystyle \sum_{n=1}^{\infty} a_n$ converges is an incorrect statement.}$$
For instance, $\displaystyle \sum_{n=1... | {
"language": "en",
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$2\times2$ matrices are not big enough Olga Tausky-Todd had once said that
"If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this."
There are, however, assertions about matrices that are true for $2\times2$ matrices but not for the larger ones. I came across one nice little example ... | Interesting although quite elementary is property for matrices made from consecutive integers numbers (or more generally from values of arithmetic progression) where rows make an arithmetic progression.
Only $2 \times 2$ matrices made from consecutive integers numbers are non-singular, matrices of higher dimension are ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$ I'm reading Real Analysis by Royden 4th Edition.
The entire problem statement is:
Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ and $f$ is integrable over $E$. Show th... | Fatou's Lemma is your friend. By Fatou,
\begin{align*}
\int_{E} 2|f|
&= \int_{E} \liminf_{n\to\infty} (|f| + |f_n| - |f-f_n|) \\
&\leq \liminf_{n\to\infty} \int_{E} (|f| + |f_n| - |f-f_n|) \\
&= 2\int_{E} |f| - \limsup_{n\to\infty} \int_{E} |f-f_n|.
\end{align*}
So it follows that $\limsup_{n\to\infty} \int_{E} |f-f_n|... | {
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Can we make rectangle from this parts? I have next problem:
Can we using all parts from picture (every part exactly one time) to make rectangle?
I was thinking like: we have $20$ small square, so we have three possibility: $1 \times 20$, $2 \times 10$ and $4 \times 5$. I can see clearly that $1 \times 20$ and $2 \time... | For the 4 by 5, suppose 4 rows and 5 columns, and consider rows 1,3 as blue and columns 1,3,5 as red. Then there are 10 blues and 12 reds. Now except for the T and L shapes, the other three contribute even numbers to either blue or red rows/columns. The L contributes an odd number to either blue or red, and the T contr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are all subspaces of equal dimension (of a vector space) the same? I haven't quite gotten my head around dimension, bases, and subspaces. It seems intuitively true, but are all subspaces of equal dimension of the same vector space the same?
If so, does it follow from the definitions of dimension, subspace, and vector s... | Here are some more intuitive "definitions".
Dimension - Number of degrees of freedom of movement. One-dimensional implies only one direction of movement: up and down a line. Two-dimensional means two distinct directions of movement, spanning a plane, etc.
Basis - The distinct directions of movement. One-dimensional mov... | {
"language": "en",
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Functions and convergence in law Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ converges in law to a random variable $Y$. Must it be the case that the pairs $(X, \phi_n(X))... | A more elementary counterexample: Let $X$ have uniform distribution over $[0,1]$ and define $\phi_n(x):=x$ when $n$ is odd, and $\phi_n(x):=1-x$ when $n$ is even. It's clear that every $\phi_n(X)$ has the same distribution (namely, that of $X$), hence we have convergence in law. However, the pair $(X,\phi_n(X))$ doesn'... | {
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Show that the following set has the same cardinality as $\mathbb R$ using CSB We have to show that the following set has the same cardinality as $\mathbb R$ using CSB (Cantor–Bernstein–Schroeder theorem).
$\{(x,y)\in \Bbb{R^2}\mid x^2+y^2=1 \}$
I think that these are the two functions:
$f:(x,y)\to \Bbb{R} \\f(x)=x,\\f(... | HINT: There is no continuous bijection between the two sets. Find a bijection from the unit circle to $[0,2\pi)$, and an injection from $\Bbb R$ into $[0,2\pi)$.
Also, when you define a function $g\colon\Bbb R\to\Bbb R^2$ you don't write $g(x)=\cos x$ and $g(y)=\sin y$. You should write $g(x)=(\cos x,\sin x)$ instead. ... | {
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Is the ring of polynomial invariants of a finite perfect group an UFD? Let $G$ be a finite group. $G$ acts on $\mathbb K[x_1,...,x_n]$ by automorphisms fixing $K$. $\mathbb K[x_1,...,x_n]^G=\{ T\in \mathbb K[x_1,...,x_n],\forall \sigma \in G, T^{\sigma}=T\}$ is the ring of invariants.
Is it true that $\mathbb K[x_1,.... | Let $P$ be an irreducible polynomial. Any element $g$ of $G$ maps $P$ to some irreducible polynomial, because action is invertible. This polynomial $gP$ may be either proportional to $P$ or coprime to $P$. Let $Q = P \cdot (g_1 P) \cdot \ldots \cdot (g_k P)$ be a product of polynomials in the "essential orbit" of $P$: ... | {
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Need help with finding domains, intercepts, max/min of function Here is a picture for clarification of the question:
So far, I have gotten this: the root is all non-negative numbers; x is greater than or equal to 0.
In order to find the asymptotes I set the denominator to zero and so the asymptotic is zero?? I don't ... | You can see that the domain is $\{x\in \mathbb{R} : x\geq 0 \}$. To find the max/min (which eventually gets you the range), we find out the first derivative.
$f'(x)=\frac{4-x}{2\sqrt(x)(x+4)^2}$. Here the critical point is $x=4$.
Also $f''(x)=-\frac{1}{4x^{3/2}}-\frac{4-x}{x^{1/2}(x+4)^2}$, then $f''(4)=-\frac{1}{32}$... | {
"language": "en",
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What is this curve formed by latticing lines from the $x$ and $y$ axes? Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts.
Question 1: What is the curve $f$ when $n\rightarrow \infty$?
Update: According to the answers this curve is not a par... | The OP's curve is (a portion of) the parabola with $(1/2,1/2)$ for its focus and $x+y=0$ for its directrix. With a slight modification (see below), the lines in the OP's drawing are tangent lines to the parabola, which can be thought of as (origami) crease lines created when the focus is "folded" to lie atop various p... | {
"language": "en",
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Differential Equations and Newtons method How can I approach this question?
For problem one this is what I did:
Given the DE,
$$p'(x) = p''(x) + \left(2\pi*\frac{f}{c}\right)^2p(x) = 0,$$
and its solution, $p(x) = \sin(kx)$, I substituted the things on the right hand side of the DE to get
$$p'(x) = -\sin(kx)\,k^2 + \co... | You are being asked to find a relationship between $k$ and $f$ and $c$. $\sin{(kx)}$ was given to show you the form of the equation but now you are asked to determine exactly what $k$ should be in this case, in terms of the other quantities in the problem.
To do this, generate the needed derivates of $\sin{(kx)}$ an... | {
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"source": "stackexchange",
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If $\int_{-\infty}^{\infty}f(x)\ \mathrm dx=100$ then $\int_{-\infty}^{\infty}f(100x+9)\ \mathrm dx =?$
Given $\displaystyle\int_{-\infty}^{\infty}f(x)\ dx=100$, evaluate $\displaystyle\int_{-\infty}^{\infty}f(100x+9)\ dx.$
Question is as above. I'm not sure how to even start. Is the answer $100$? Seems like if the f... | Hint: Consider a particular example. What if $f(x) = 1$ for $0 < x < 100$ and $f(x) = 0$ elsewhere? What is $f(100x+9)$ in this case? What is the value of the second integral?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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sequence with infinitely many limit points I am looking for a sequence with infinitely many limit points. I don't want to use $\log,\sin,\cos$ etc.!
It's easy to find a sequence like above, e.g. $1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots$
But how can you prove the limit points? The problem I am having is the recursion or d... | I realize this question was asked a long time ago, but for posterity, you may formally describe the sequence $<1, 1, 2, 1, 2, 3, 1, 2, 3, 4, ...>$ as $<x_i>$ where $x_i = n$ precisely when $i = \frac {(n^2 + (2k-1)n + (k^2 - 3k + 2))}{2}$ for some $k \in \mathbb{N}$.
To justify this formula, we start by showing that $... | {
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"timestamp": "2023-03-29T00:00:00",
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Can someone example and give an example? Given an example of a function $f $ such that $\lim_{x\rightarrow \infty } f(x)$ exists, but $\lim_{x\rightarrow \infty } f'(x)$ does not exist.
| Consider
$$
f(x) = \frac{\sin(x^2)}{x}.
$$
Using the Squeeze Theorem, you can show that
$$
\lim_{x \to \infty} f(x) = 0.
$$
However, its derivative
$$
f'(x) = 2\cos(x^2) - \frac{\sin(x^2)}{x^2}
$$
never settles down as $x \to \infty$.
| {
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Normal domain is equivalent to integrally closed domain. Is it true?
Normal domain is equivalent to integrally closed domain. Is it true?
Can anyone tell me?
| Here's a solution to this problem for the sake of completeness.
Let $A$ be an integral domain with $K=Quot(A)$. We'll show it's normal (this is, for every prime ideal $p\subset A$ the localization $A_p$ is integrally closed) if and only if it's integrally closed.
*
*Let $A$ be normal and consider any $r\in K$ satisf... | {
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Why is a certain subset of a regular uncountable cardinal stationary? this is an excerpt from Jech's Set Theory (page 94).
For a regular uncountable cardinal $\kappa$ and a regular $\lambda<\kappa$ let $$E^\kappa_\lambda= \{\alpha<\kappa:\mbox{cf}\ \alpha=\lambda \}$$
It is easy to see that each $E^\kappa_\lambda$ i... | Note that if $C$ is a club, then $C$ is unbounded, and therefore has order type $\kappa$. Since $\lambda<\kappa$ we have some initial segment of $C$ of order type $\lambda$.
Show that this initial segment cannot have a last element. Its limit is in $C$ and by the regularity of $\lambda$ must have cofinality $\lambda$.... | {
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Action on Pairs, On Sets and on points in GAP I am trying to understand GAP in group action. I am confused in few things what is the difference between action on pairs, on sets, with the domain sometimes on list, and on blocks. Please help me to clarify these things. Thanks.
| The "Group Actions" chapter of the GAP reference manual documents standard actions, and if you scroll until OnTuplesTuples, there will be a common example covering all of them. I will just take from there examples of the three actions in question, and try to shed more light.
First, create $A_4$ as g:
gap> g:=Group((1,... | {
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Brownian Motion conditional distribution Let $\{X(u),u\geq0\}$ be a standard Brownian motion. What is the conditional distribution of $X(t)$ given $\{X(t_{1}),\dots,X(t_{n})\}$, where $0<t_{1}<\cdots<t_{n}<t_{n+1}=t$?
--So far, I have derived the joint pdf of $X(t_{n+1})$ and $X(t_{1}),\dots,X(t_{n})$ using the fact th... | One knows that the marginal distributions of Brownian motion are normal and that $X(t)-X(t_n)$ is independent of $\sigma(X(s);s\leqslant t_n)$. Hence, the conditional distribution of $X(t)$ conditionally on $\sigma(X(t_k);1\leqslant k\leqslant t_n)$, for every $t_k\leqslant t_n$ (or, conditionally on $X(t_n)$ only) is ... | {
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Rationalizing a denominator. The question instructs to rationalize the denominator in the following fraction:
My solution is as follows:
The book's solution is
which is exactly the numerator in my solution.
Can someone confirm my solution or point what what I'm doing wrong?
Thank you.
| Going from the first to the second line of your working, you seem to have said
$$(\sqrt{6} -2)(\sqrt{6}+2) = 6+4$$
on the denominator of the fraction.
This isn't true: in general $(a-b)(a+b)=a^2-b^2$, not $a^2+b^2$.
| {
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Does closed imply bounded?
Definitions:1. A set $S$ in $\mathbb{R}^m$ is bounded if there exists a number $B$ such that $\mathbf{||x||}\leq B$ for all $\mathbf{x}\in S$, that is , if $S$ is contained in some ball in $\mathbb{R}^m$.2. A set in $\mathbb{R}^m$ is closed if, whenever $\{\mathbf{x}_n\}_{n=1}^{\infty}$ is c... | $\mathbb{R}^m$ itself is a closed set. is it bounded?
But in case of Compact sets, they are closed as well as bounded in $\mathbb{R}^n$.
| {
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How do i find the lapalace transorm of this intergral using the convolution theorem? $$\int_0^{t} e^{-x}\cos x \, dx$$
In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but $g(t)$ is not clear to me.
Please help. ... | So the transform becomes $\frac{s}{s^2 + 1} \frac{1}{s-1}$ with the shift $s+1$.
| {
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Prove that [0,1] is equivalent to (0,1) and give an explicit description of a 1-1 function from [0,1] onto (0,1) The problem is stated as follows:
Show that there is a one-to-one correspondence between the points of the closed interval $[0,1]$ and the points of the open interval $(0,1)$. Give an explicit description ... | Steps 2 and 3 are not necessary. The function $g:(0,1] \to [0,1]$ defined by $g(1) = 0$ and $g(x) = f(x)$ if $x \neq 1$ is a bijection. This shows that $(0,1]$ is equivalent to $[0,1]$ and, by transitivity, that $(0,1)$ is equivalent to $[0,1]$. Furthermore, the function $g \circ f$ is a one-to-one correspondence betwe... | {
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finding inverse of sin with a variable in the argument $$h(t) = 18\sin\left(\pi\frac{t}{7}\right) + 20$$
Where $h(t)$ is height in meters and $t$ is the elapsed time in seconds.
If $h$ is restricted to the domain $[3.5,10.5]$ find and interpret the meaning of $h^{-1}(20)$.
In the Facit the answer is $7$. This means tha... | Let $h(t) = h$ and $t = h^{-1}$. In this case:
$h = 20 + 18sin(\frac{\pi h^{-1}}{7})$
$h - 20 = 18sin(\frac{\pi h^{-1}}{7})$
$\frac{h - 20}{18} = sin(\frac{\pi h^{-1}}{7})$
$arcsin(\frac{h - 20}{18}) = \frac{\pi h^{-1}}{7}$
$\frac{7 arcsin(\frac{h - 20}{18})}{\pi} = h^{-1}$
Now substitute 20 to $h$:
$h^{-1}(20) = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/579743",
"timestamp": "2023-03-29T00:00:00",
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How many transitive relations on a set of $n$ elements? If a set has $n$ elements, how many transitive relations are there on it?
For example if set $A$ has $2$ elements then how many transitive relations. I know the total number of relations is $16$ but how to find only the transitive relations? Is there a formula for... | As noticed by @universalset, there are 13 transitive relations among a total of 16 relations on a set with cardinal 2. And here are they :)
| {
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"url": "https://math.stackexchange.com/questions/579817",
"timestamp": "2023-03-29T00:00:00",
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example of a connected matrix lie group with a nondiscrete normal subgroup H? What is an example of a connected matrix lie group with a non discrete normal subgroup H such that its tangent space at the identity is the zero matrix?
| Take the group $G$ to be $SO(2)\simeq S^1$, a $1$-dimensional circle. Since this group is abelian, any subgroup will be normal. Then take dense subgroup $H$ isomorphic to $\mathbb{Z}$, which is generated by the image of $\sqrt{2}$ (or any irrational number) in $S^1$ via the parametrization $\mathbb{R}\to S^1$, $x\mapst... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal? I know this can be easily proved with simple matrix tricks,
But I don't know the insight for this, and just feels it amazing that if I pick up 3 orthogonormal vectors in 3d space, their corresponding x,y,z portions automatic... | A square real matrix $M$ has columns ON if and only if $M^{-1}=^tM$ if and only if $(^tM)^{-1}=M=^t(^tM)$ if and only if $^tM$ columns are ON if and only if $M$ lines are ON.
ON = orthonormal.
| {
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Family of Straight line : Consider a family of straight lines $(x+y) +\lambda (2x-y +1) =0$. Find the equation of the straight .... Problem : Consider a family of straight lines $(x+y) +\lambda ( 2x-y +1) =0$.
Find the equation of the straight line belonging to this family that is farthest from $(1,-3)$.
Solution:
Let... | HINT:
We can rewrite the equation as $$x(1+2\lambda)+y(1-\lambda)+\lambda=0$$
If $d$ is the perpendicular distance from $(1,-3)$
$$d^2=\frac{\{1(1+2\lambda)+(-3)(1-\lambda)+\lambda\}^2}{(1+2\lambda)^2+(1-\lambda)^2}$$
We need to maximize this which can be done using the pattern described here or here
| {
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