Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Connectivity and Euler characteristic for surfaces I learn the concept of connectivity from Hilbert's Geometry and the Imagination as follows:
A polyhedron is said to have connectivity $h$ (or to be $h$-tuply connected) if $h-1$, but not $h$, chains of edges can be found on it in a certain order that do not cut the su... | It may be hard to find because I'm not sure that this term is used very much; at the very least, I have never really heard it used. In fact, using it we find the unusual descriptions that
*
*$S^{2}$ is 1-connected
*$T^2 = S^1 \times S^{1}$ is 3-connected
*More generally a Riemann surface of genus $g$ is ${2g + 1}$... | {
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Prove that a counterexample exists without knowing one I strive to find a statement $S(n)$ with $n \in N$ that can be proven to be not generally true despite the fact that noone knows a counterexample, i.e. it holds true for all $n$ ever tested so far. Any help?
| Define $k$ to be 42 if the Riemann Hypothesis is true, and 108 if it is false.
Now consider $S(n) \equiv n\ne k$.
Alternatively consider $S(n)$ to be "There is a two-symbol Turing machine with 100 states that runs for at least $n$ steps when started on an empty tape, but eventually terminates".
| {
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Rubik cube number of alternative solutions If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that only have the start and finish position in common?
| Of course there can be multiple possible solutions.
Before it was proved (via computer search) that every position required 20 moves or less, the Superflip was shown to require exactly 20 moves to solve. (See also here.) One sequence of moves which solves the Superfilp is
U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 ... | {
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Is this a poorly worded probability question? Unsolvable? The question says: "For a recent year, 0.99 of the incarcerated population is adults and 0.07 is female. If an incarcerated person is picked at random, find the probability that the person is female given they are an adult."
I've been thinking about this for mor... | On the assumption of independence the answer is obviously $0.07$.
That assumption is not necessarily reasonable. So indeed the question is poorly worded.
Imagine as an extreme case that no child females are put in jail. Then the probability a jailed person is female given the person is an adult is $\frac{0.07}{0.99}... | {
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Has this theorem (on existence of inverse) an analogous for unbounded operators? Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is invertible.
I would like to know if is there a ... | From the inequality $\lVert T-S\rVert \lVert T^{-1}\rVert < 1$, it follows that
$$R = (I - T^{-1}(T-S))$$
is an invertible bounded operator. Since $TR = T(I - T^{-1}(T-S)) = T - (T-S) = S$, we get the representation $S^{-1} = R^{-1}T^{-1}$ in the bounded case. Now we can verify that $R^{-1}T^{-1}$ is also the inverse o... | {
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Upper bound for $\Vert f \Vert^{2}$, where $f: [0,1] \to \mathbb{R}$ continuously differentiable. Let $f: [0,1] \to \mathbb{R}$ be continuously differentiable with $f(0)=0$. Prove that $$\Vert f \Vert^{2} \leq \int_{0}^{1} (f'(x))^{2}dx$$
Here $\Vert f \Vert$ is given by $\sup\{|f(t)|: t \in [0,1]\}$.
I'm just a bit un... | Let x = argsup f. Because f(0) = 0, f(x) is the integral up to x of f', which is less than or equal to the L2 norm of f' up to x by convexity of the squaring function, which is less than or equal to the L2 norm of f' on the entire interval. Now square both sides.
| {
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Is there a topological characterisation of non-Archimedean local fields? A local field is a locally compact field with a non-discrete topology. They classify as:
*
*Archimedean, Char=0 : The Real line, or the Complex plane
*Non-Archimedean, Char=0: Finite extensions of the p-adic rationals
*Non-Archimedean, Char=p... | Archimedian local fields are connected, non-archimedian local fields are totally disconnected.
| {
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Looking for the lowest number divisible by 1 to A. What would the math equation be for finding the lowest number divisible by 1 to A? I know factorial can make numbers divisible by 1 to A but that dosn't give me the lowest number.
Example of what I'm talking about:
the lowest number divisible by 1,2,3,4,5,6,7,8 = 840
E... | Look at the LCM. Note $lcm\{a,b\} = \frac{ab}{gcd(a, b)}$.
If you have a prime $p$ and a power of $p$, $p^{n}$, you will simply discard $p$ and keep $p^{n}$ instead. As $p^{n}|x \implies p|x$, but the converse is not true.
Note as well you can ignore $1$ in your $gcd$ calculations.
| {
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poisson distribution and the cdf $Y (t)$ is the number of events occurring in $[0,1]$ where for each $t> 0$, $Y (t)~\sim\operatorname{Poi} (\lambda t)$ and $X$ measures the time taken for the $r$th event to occur.
Am I right in saying that the event $(X \le t) = (Y(t) \ge t)$?
Also, how can I write the cdf of $X$ as th... | The $r$th event occurs before time $t$ if and only if the number of events before time $t$ is at least $r$.
So $[X<t] = [Y(t)\ge r]$.
| {
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Linear Algebra Self Study I'm currently a high school student with a love for math. I have taken Plane and Coordinate Geometry, both Algebra I and II, Trigonometry, and am halfway done with Calc A.
I want to major in quantum physics, and feel that a background with linear algebra would help. As there are no courses ava... | I'm in a similar situation, and I learn linear algebra from Axler's text, "Linear Algebra Done Right." The problem sets are very nice and I really like the book; it's very easy to understand and explanations are lucid.
The MIT OCW course uses Strang's text, I believe, which I'm not familiar with.
| {
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Epsilon-Delta More Confusion
Use Epsilon-Delta to prove:
$$
\lim_{x \to 1} (x^2 + 3) = 4
$$
So, we need to find a $\delta$ s.t.
$$
0 < x - 1 < \delta \; \implies \; 0 < |(x^2 + 3) - 4| < \epsilon
$$
We simplify
$$0< |(x^2 + 3) - 4| < \epsilon$$ to get $$0 < |x^2 - 1| < \epsilon$$
This is where I'm stuck.
How do I... | You can factor $x^2-1$. From there, the condition $|x-1|<\delta$ implies that $|x+1|<\delta +2$. Can you use this to find an expression for $\delta$?
| {
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Proof : An event is independent from every other event iff its probability is 0 or 1 As said in the title I need to prove that an event is independent from all other events iff its 0 or 1. One side is pretty simple, if I assume the event is 0 or 1 probability the answer is immediate.
I'm having trouble formulating the ... | I'll retract this as an answer but leave it here for those similarly confused
Not true it seems to me.
Consider the probablity of getting heads or tails (H, T) on the toss of a coin and also getting a number (1 to 6) on the throw of a dice. p(H) = 1/2 and p(2) = 1/6. But the events are surely indenpendent ?
From the su... | {
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Riemann Sphere/Surfaces Pre-Requisites I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on topology and complex and real analysis.
Can you recommend any good books that will ... | try
J. Jost Compact Riemann Surfaces, third edition Springer Universitext. (I think is a very good book)
http://www.zbmath.org/?q=an:05044797
and
S. Donaldson Riemann Surfaces. (this is beautiful but it is more "concentrated")
http://www.zbmath.org/?q=an:05900831
Also you may find interesting this fantastic book abou... | {
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Book on the first-order modal logic Is there a book on the metatheory for the first-order modal logic, or do I just need to take FOL as a base and use the standard translation?
| D. M. Gabbay & V. B. Shehtman & D. P. Skvortsov. Quantification in Nonclassical Logic (2009) (Series: Studies in Logic and the Foundations of Mathematics, Volume 153. Elsevier)
It covers a lot of material on first-order modal and first-order intuitionistic logic. E.g. Kripke semantics, algebraic semantics, completeness... | {
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Order of operations when using evaluation bar Suppose we have the function
\begin{align*}
f(x) = \sin(x)
\end{align*}
with first derivative
\begin{align*}
\frac{d}{dx}f(x) = \cos(x).
\end{align*}
If we evaluate $f'(x)$ at $x=0$, the result depends on whether you evaluate $f(0)$ or differentiate $f(x)$ first.
\begin{ali... | If the following were true:
$$\frac{d}{dx}f(x)\mid_{x = n} = \frac{d}{dx}\left(f(x)\mid_{x = n}\right)$$
Then it would always be 0. Why? Because once you evaluate a function of x at a particular value of x, it is no longer a function of x; it's a value (or, a constant function). The example (sin, cos) you gave is a lit... | {
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Prove $f(x) = x^2$ is continuous at $x=4$ I want to show that $f(x) = x^2$ is continuous at $x=4$
and here's how the proof goes:
$\forall\epsilon>0$, $\exists\delta>0$ s.t $\forall x$, $|x-4|<\delta$ $\implies |f(x)-16|<\epsilon$
So working backwards we get:
$$|f(x)-16|<\epsilon ⇔ |x^2 - 16| < \epsilon$$
... | If you assume $\delta < 1$, then you know that:
$$\begin{align*}
|x-4|<\delta \\
\Rightarrow |x-4|<1 \\
\Rightarrow -1 < x-4 < 1 \\
\Rightarrow 3 < x < 5
\end{align*} $$
But then, we can determine what this means about $|x+4|$:
$$\begin{align*}
&\Rightarrow 7 < x+4 < 9 \\
&\Rightarrow |x+4| < 9
\end{align*} $$
So this ... | {
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complex analysis: If $f$ is analytic and $\operatorname{Re}f(z) = \operatorname{Re}f(z+1)$ then $Im\;f(z) - Im\;f(z+1)$ is a constant I am having trouble deciphering the reason behind a line in a complex analysis textbook (Complex made Simple by Ullrich, page 360 5 lines down in Proof of Theorem B, for those who are in... | That is wrong. Consider $f(z) = \cos 2 \pi z$, there $f(z + 1) = f(z)$, but $\Im(f(z))$ isn't constant.
| {
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Prove $f(x) = \sqrt{x}$ is continuous at $x = 4$ Show that $f(x) = \sqrt{x}$ is continuous at $x = 4$
So my textbook has a proof for this and this is their scratch work:
$\forall\epsilon>0$ $\exists\delta>0$ s.t $\forall x$ $0<|x-4|<\delta \implies |\sqrt{x}-2|<\epsilon$
Working backwards:
$$|\sqrt{x}-2|<\epsilon \iff ... | If $\delta \gt 4$ then there are negative $x$ which satisfy $|x-4|\lt \delta$ but for which there is not real $\sqrt{x}$.
Using "let us assume $\delta \le 1$" would in fact have worked here, but would not have worked if the original question had been "Prove $f(x)=\sqrt{x}$ is continuous at $x=\frac12$." So it is reas... | {
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Normal Distribution in I am so confused with this problem:
The middle 95% of adults have an IQ between 60 and 140. Assume that IQ for adults is normally distributed.
a. What is the average IQ for adults? The standard deviation?
I got the average by subtracting the values given and then multiply it with 95%. But I dont... | Actually the average IQ is 100 and its standard deviation is 15.
Intelligence tests are scored in such a way the resulting IQ distribution conform to these properties.
http://en.wikipedia.org/wiki/Intelligence_quotient
| {
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In how many ways can a teacher divide a group of seven students into two teams each containing at least one student? Can someone please help me with this?
In how many ways can a teacher divide a group of seven students into two teams each containing at least one student? two students? What about when seven is replaced ... | well, the basic breakdown goes:
1-6, 2-5, 3-4
For the 1-6, there are 7 choices for the student that is by himself, and the other 6 are dictated by the one by himself, so there are 7 ways to break them up into 2 groups with one student in one and 6 in the other.
For 2-5, you have 7 options for the first person in the gr... | {
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A definite integral $$\int_0^1\sqrt{\left(3-3t^2\right)^2+\left(6t\right)^2}\,dt$$
I am trying to take this integral. I know the answer is 4.
But I am having trouble taking the integral itself.
I've tried foiling and the simplifying. I've tried u-sub. I just can't get the correct way to take the integral.
Any help woul... | Hint : By doing some manipulation and expansion, notice that
$$\begin{align}
(3-3t^2)^2 + (6t)^2 &= 3^2(1-2t^2 +t^4 +4t^2)\\
&= 9(1+ 2t^2 + t^4) \\
&=9(1+t^2)^2\end{align}$$
Now, just take the square root of this and integrate the result.
| {
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Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts? Sometimes I help my next door neighbor's daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and I found this h... | Let starting segment is $AB$.
As one can see from first link, starting condition is "segment-on-the-line".
Anyway one can add $1$ line at the start to get this starting condition.
Consider odd $n$.
Let coordinates of starting points are: $A(-1,0)$, $B(0,0)$.
If point $C$ has coordinates $C(n,0)$, then (see Figure 1) co... | {
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Minimizing the value of this integral $ I \equiv \int_{0}^{\pi/2}\left\vert\,\cos\left(x\right) - kx^{2}\,\right\vert \,{\rm d}x $
Given the integral
$$
I \equiv \int_{0}^{\pi/2}\left\vert\,\cos\left(x\right) - kx^{2}\,\right\vert
\,{\rm d}x
$$
Find the value of $k$ so that $I$ is minimum.
How do I start?
| We have to minimize the function
$$I(p):=\int_0^{\pi/2}\bigl|\cos x -p x^2\bigr|\ dx\qquad(p\in{\mathbb R})\ .$$
When $p<0$ then obviously $I(p)>I(0)=1$. When $p>0$ then the parabola $y=p\,x^2$ intersects the curve $y=\cos x$ at a point $x=t$ with $0<t<{\pi\over2}$ depending on $p$. We now let $t>0$ be our new paramete... | {
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Modular Arithmetic - Quadratic Solutions Problem I've just been given the following question in my crypto class, and I think I'm fairly sorted for it, but I was just wondering whether there might be any extra solutions to the ones I've worked out.
Compute all solutions of $x^2 + 4x - 21 \equiv 0\,\bmod\,33$
First, I ... | $(x+7)(x-3)\equiv0\mod 33$
Now, as you noticed $ x=-7,3$ are obvious solutions. Also, any number congrunet to $-7$ or $3$ modulo $33$ will be a solution, for instance, $26$, as it is congruent to $-7$ modulo $33$, or $36$, congruent to $3$ modulo $33$.
But there are other possible solutions, congruent to neither $-7$ ... | {
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Can we apply squeeze in that way? Claim:
if $a_n\leq b_n\leq c_n$ for all $n\in \mathbb{N}$ and $\displaystyle\sum\limits_{n=0}^{\infty} a_n,\displaystyle\sum\limits_{n=0}^{\infty} c_n$ are convergent then$\displaystyle\sum\limits_{n=0}^{\infty} b_n$ is convergent.
I think it is a wrong statement but I could not find ... | $$\sum_{i=k}^l a_i \le \sum_{i=k}^l b_i \le \sum_{i=k}^l c_i$$
This implies that
$$|\sum_{i=k}^l b_i| \le\max \{ |\sum_{i=k}^l a_i|, |\sum_{i=k}^l c_i|\}$$
Now $\sum a_i$, $\sum c_i$ exist, hence and right term becomes arbitrarily small (Cauchy-criterion), if $k,l$ are chosen large enough, hence the partial sums $... | {
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Evaluate a twice differentiable limit Evaluate $$\lim_{h \rightarrow 0} \frac{f(x+h) -2f(x) + f(x-h) } { h^2}$$
if $f$ is a twice differentiable function.
I'm not sure how to understand this problem. If I differentiate the numerator I get $f'(x+h) - 2f'(x) + f'(x-h)$ but that doesn't seem to take me anywhere?
| Hint
Rewrite
$$ [f(x+h) -2f(x) + f(x-h)] =[f(x+h)-f(x)] -[f(x)-f(x-h)]$$
Divide each portion by $h$. What does that become ? Do it again to arrive to ...
I am sure that you can take from here.
| {
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Understanding the fundamental theorem of calculus
I'm having problems understanding why $$ \frac{d}{dx} \int_{a}^{x} f(t)\,dt = f(x)$$
I'm somewhat weirded out by the fact that there is a $dt$ at the end of $$F(x) = \int_a^x f(t)\,dt$$ too.
We are differentiating with respect to $x$...I understand that $ \frac{d}{dx} ... | http://en.wikipedia.org/wiki/Free_variables_and_bound_variables
The expression
$$
\sum_{i=1}^3 \cos(i^2 k^3)
$$
means
$$
\cos(1^2k^3) + \cos(2^2k^3)+\cos(3^2k^3),
$$
and that's the same as
$$
\sum_{j=1}^3 \cos(j^2 k^3),
$$
i.e. $i$ and $j$ are "bound variables", whereas $k$ is a "free variable".
The $t$ in
$$
\int_a^x ... | {
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How to prove whether a given matroid is a Gammoid? Statement : Given a matroid in some representation say $(E,I)$. How do we prove it is a gammoid?
For example to prove a matroid is transversal, we try to create a bipartite graph. If we are unable to(i.e. if we get some contradiction) then it is not transversal.
Simila... | According to Oxley, deciding whether a given matroid is a gammoid is still open. There is a way to check whether a given matroid is a strict gammoid, though, and you can find it in Oxley as well.
| {
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Is $\epsilon^2/\epsilon^2=1$ or $0/0$? Is it possible in the system of dual numbers ($a+\epsilon b$; $\epsilon^2=0$) to calculate $\epsilon/\epsilon =1$? How then does one deal with $\epsilon^2/\epsilon^2=1$ versus $\epsilon^2/\epsilon^2=0/0$?
The same question for infitesimal calculus using hyperreal numbers where: $\... | In the dual numbers, ${\mathbb R}[\epsilon]$ ($={\mathbb R}[X]/(X^2)$), $\epsilon$ is not invertible, so the expression $\epsilon / \epsilon$ ($= \epsilon \epsilon^{-1})$ is undefined.
In hyperreals, as Asaf Karagila mentions in the comments, $\epsilon^2 \neq 0$. There you do have $\epsilon / \epsilon = \epsilon^2 / \e... | {
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"timestamp": "2023-03-29T00:00:00",
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What values of a is the set of vectors linearly dependent? The question is is "determine conditions on the scalars so that the set of vectors is linearly dependent".
$$ v_1 = \begin{bmatrix} 1 \\ 2 \\ 1\\ \end{bmatrix}, v_2 = \begin{bmatrix} 1 \\ a \\ 3 \\ \end{bmatrix}, v_3 = \begin{bmatrix} 0 \\ 2 \\ b \\ \end{bma... | The Determinant Test is appropriate here, since you have three vectors from $\mathbb{R}^{3}$. The set of vectors is linearly dependent if and only if $det(M) = 0$, where $M = [v_{1} v_{2} v_{3}]$.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Minimum number of operations (divide by 2/3 or subtract 1) to reduce $n$ to $1$ This question is inspired by a Stack Overflow question which involves the task to find an algorithm to solve the following problem:
Given a natural number $n$, what is the least number of moves you need to reduce it to $1$? Valid moves are... | We will use a very straightforward method that produces a solution of order o(N), still though not polynomial in the size of the input.
We will use the following recursive function to calculate the result:
$$m(N) = 1 + min( Ν \ mod \ 2 + m( \frac{N}{2} ), N \ mod \ 3 + m( \frac{N}{3} ) )$$
$$m(1) = 0$$
The correctness ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find flux,using Cartesian and spherical coordinates Find the flux of the vector field $\overrightarrow{F}=-y \hat{i}+ x \hat{j}$ of the surface that consists of the first octant of the sphere $x^2+y^2+z^2=a^2(x,y,z \geq 0).$
Using the Cartesian coordinates,I did the following:
$$ \hat{n}=\frac{\nabla{G}}{|\nabla{G}|}=\... | Both of your methods are correct, and the flux through the sphere being $0$ is actually what we would expect, as your field is purely rotational and therefore the field vectors all point along the surface of the sphere. We can see this by observing:
$$\nabla \cdot \vec{F}=\frac{\partial(-y)}{\partial x}+\frac{\partial ... | {
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Convex function, sets and which of the following are true? (NBHM-$2014$) Let $f:]a,b[ \to\Bbb R$ be a given function. Which of the following statements are true?
a. If $f$ is convex in $]a,b[$, then the set $\tau=\{(x,y) \in\Bbb R^2| x\in ]a,b[, y\ge f(x)\}$ is a convex set.
b. If $f$ is convex in $]a,b[$, then the se... | a. True.
c. False. Example: take $f(x) = -1 + x^2$ on $[-1, 1]$. $f''(x) = 2 > 0$ on this interval so $f$ is
convex on this interval. But $g(x) = |f(x)| = |x^2 - 1|$ is not convex on this interval. Look
at the graph of $g$.
| {
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Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book).
I think it should be pretty much straight-... | For each $1\leq i \leq n$, the set $C_i = U_n(k)e_{i,i}$ is a left ideal, which is projective, since $\bigoplus C_i \cong U_n(k)$. Suppose $J$ is any ideal in $U_n(k)$ - let us show that we can write it as $J'\oplus C_i$ for some $i$, hence from the previous argument+induction it will be projective.
Let $i$ be the maxi... | {
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primitive root of residue modulo p I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication.
My Try:
So I first assumed that if there exists a generator, then from the set $\{a^1,a^2,...,a^{p-1}\}$ all those powers ... | Assuming $p$ is prime then $\mathbb{Z}/p\mathbb{Z}$ is the finite field $\mathbb{F}_p$ and the set you are interested in is the multiplicative group $\mathbb{F}_p^*$. In this context what you are looking for is a proof that the multiplicative group is cyclic. And with this formulation you can find a lot of answers, fo... | {
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What is infinity to the power zero I have this notation:
$$\lim_{k->\infty} k^ {1/k}$$
Is it correct to say that the output is 1, or is there some other result?
| $$\lim_{n\to\infty}n^{^\tfrac1n}=\lim_{n\to\infty}\Big(e^{\ln n}\Big)^{^{\tfrac1n}}=\lim_{n\to\infty}e^{^\tfrac{\ln n}n}=e^{^{\displaystyle{\lim_{n\to\infty}}\tfrac{\ln n}n}}=e^{^{\displaystyle{\lim_{t\to\infty}}\dfrac t{e^t}}}=e^0=1.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Number of $3$ letters words from $\bf{PROPOSAL}$ in which vowels are in the middle
The number of different words of three letters which can be formed from the word $\bf{"PROPOSAL"}$, if a vowel is always in the middle are?
My try: We have $3$ vowels $\bf{O,O,A}$ and $5$ consonants $\bf{P,P,R,S,L}$. Now we have to fo... | I am not very sure about the methodology. Please let me know if the logic is faulty anywhere.
We have $3$ slots.$$---$$
The middle one has to be a vowel. There are only two ways in which it can be filled: O, A.
Let us put O in the middle.$$-\rm O-$$
Consider the remaining letters: O, A, $\bf P_1$, $\bf P_2$, R, S, L. S... | {
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Converting from radius of convergence to interval of convergence Using the root test I have determined that $$\sum n^{-n} x^n$$ has a radius of convergence of infinity and $$\sum n^{n} x^n$$ has a radius of convergence of 0. Does this mean that the respective intervals of convergence are $(-\infty,\infty)$ and $\emptys... | You're done for the first one; there are no endpoints to evaluate. The second one has interval of convergence either $\emptyset$ or $[0,0]$; you need to determine whether $x=0$ leads to a convergent series.
| {
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Prove that $\Gamma(p) \cdot \Gamma(1-p)=\frac{\pi}{\sin (p\pi)}$ for $p \in (0,\: 1)$ Prove that $$\Gamma(p)\cdot \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1),$$
where $$\Gamma (p)=\int_{0}^{\infty} x^{p-1} e^{-x}dx.$$
Here's what I tried:
We have
$$B(p, q)=\int_{0}^{1} x^{p-1} (1-x)^{q-1}dx=\frac{\Ga... | $\newcommand{\+}{^{\dagger}}
\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{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\newcom... | {
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Find the point on the y-axis which is equidistant from the points $(6, 2)$ and $ (2, 10)$. Find the point on the y-axis which is equidistant from the points $(6, 2) $ and $ (2, 10)$.
Please help, there are no examples of this kind of sum in my book! I don't know how to solve it.
| Find the locus of all the points that are equidistant from the points $(6, 2)$ and $(2, 10)$, that is, the line that passes through the middle point of $(6, 2)$ and $(2, 10)$. All the points of that line are equidistant from both points.
You are looking for the point that satisfies two condition: (1) It's equidistant f... | {
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Integrating $\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$ I am a little bit lost with integral: $$\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$$
I have already worked on in and done substitution $x = \sin(t)$:
This brings me to: $$\int{\frac{\cos(t)^2}{(\sin(t)+\cos(t))^2}dt}$$
Further treating denominator to ... | Instead of doing a trig substitution, expand the denominator and simplify the integrand.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Real roots plot of the modified bessel function Could anyone point me a program so i can calculate the roots of
$$ K_{ia}(2 \pi)=0 $$
here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D
My conjecture of exponential potential means that the solutions are $ s=2a $ with
$$ \z... | I can quantify somewhat Raymond's suggestion that the zeros of $K_{ia}(2\pi)$ are much more regular than the zeros of $\zeta(1/2+i2a)$. The calculations below are rough and I didn't verify the details, so this is perhaps more of a comment than an answer.
The Bessel function in question has the integral representation
... | {
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How to detect an asymptote So I'm trying to write a program to draw graphs that are entered by the user. The way I draw them is by finding y values at $x=a$ number of $x$ values across the graph and then connecting them by lines. However, some graphs (like $\tan(x)$) have asymptotes, so the bottom and top values are c... | There is a limit test at a point $x=a$. So if $\lim_{x\to 0}f(x)=\pm\infty$, then you say $x=a$ is a vertical asymptote.
| {
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With regards to vector spaces, what does it mean to be 'closed under addition?' My linear algebra book uses this term in the definition of a vector space with no prior explanation whatsoever. It proceeds to then use this term to explain the proofs.
Is there something painfully obvious I'm missing about this terminol... | Consider the collection of points that literally lie on the $x$-axis or on the $y$-axis. We could still use Cartesian vector addition to add two such things together, like $(2,0)+(0,3)=(2,3)$, but we end up with a result that is not part of the original set. So this set is not closed under this kind of addition.
If add... | {
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Modeling a chemical reaction with differential equations The problem says:
Two chemicals $A$ and $B$ are combined to form a chemical $C$. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of $A$ and $B$ not converted to chemical $C$. Initially, there are $40$ grams of $... | Solving our equation for x: $$
\frac{150-x}{60-x}=C_1e^{90kt}
$$
We obtain:
$$
X(t)=\frac{60 C_1e^{90kt}-150}{C_1e^{90kt}-1}
$$
And if we substitute $\ k=2.5184X10^{-4}$ and $\ C_1 = \frac{5}{2}$ this would be:
$$
X(t)=\frac{150 e^{0.0226656t}-150}{\frac{5}{2}e^{0.0226656t}-1}
$$
Now we're able to answer Part A:
$$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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tangent line vs secant line According to definition, secant lines intersect the curve on two different points say $P,Q$ while tangent lines intersect only at one point. Also according definition with $P$ fixed and $Q$ variable as $Q$ approaches $P$ along the curve direction of secant approaches that of tangent.
Now my... | The definition of tangent is not that it just intersects at one point. It has to do with precisely the way the line touches the curve at that point, and nothing to do with what happens anywhere else.
If you zoom in closer and closer to the point of tangency, and as you get closer, the curve and the line become indistin... | {
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Solving this recurrence relation Hi all I'm preparing for a midterm and the following appeared as a practice problem that I'm not quite sure how to solve. It asks to find a tight bound on the recurrence using induction
$$
{\rm T}\left(n\right)
={\rm T}\left(\left\lfloor{n \over 2}\right\rfloor\right)
+{\rm T}\left(\lef... | The solution is cleary increasing. Use the change of variables $n = 2^k$ and $T(2^k) = a_k$, so that:
$$
a_k = a_{k - 1} + a_{k - 2} + a_{k - 3} + 2^k
$$
This is the same as:
$$
a_{k + 3} = a_{k + 2} + a_{k + 1} + a_k + 8 \cdot 2^k
$$
Define $A(z) = \sum_{k \ge 0} a_k z^k$, multiply the recurrence by $z^k$ and sum over... | {
"language": "en",
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$N$ balls and $M$ boxes, probability of last $ i$ boxes are empty I encountered this problem. There are $M$ boxes and $N$ balls. Balls are thrown to the boxes randomly with probability of $\frac1M$. The boxes are numbered $1, 2, 3, ..., M$.
what is the probability of last $i$ slots are empty, $i = 1, 2, 3, ...,M-1$?
I ... | There are $M^N$ functions from the set of balls to the set of boxes, all equally likely. The number of functions that miss $i$ specific boxes is $(M-i)^N$.
Equivalently, the probability that the first ball misses $i$ specified boxes is $\frac{M-i}{M}$. By independence, the probability they all do is $\left(\frac{M-i}{M... | {
"language": "en",
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Completeness proof? First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is to say: if a metric space is complete, then every Cauchy sequence $(x_n)$ where $x_n\in M$ for all ... | In practice, you often construct your limit object. It can be by using the completeness of a well-known space, generally $\mathbb{R}$ itsel. You probably know the following examples but the principle is very general.
Consider $C_b$ the space of continuous and bounded functions from $\mathbb{R}$ to itself, with the nor... | {
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Is this node proper or improper We have the Jacobian matrix (note: $c > b$ and $a,b,c>0$): $$J\left(\frac{a}{b}, 0\right) = \begin{pmatrix} -a & -\frac { ca }{ b } \\ 0 & a-\frac { ca }{ b } \end{pmatrix}$$ which is lower triangular so has eigenvalues $\lambda_{1} = -a < 0, \lambda_{2} = a-\frac{ca}{b}$. Since $c > b... | We have the eigenvalues as:
$$\lambda_{1,2} = \left\{-a,\frac{a (b-c)}{b}\right\}$$
We have:
*
*For the first eigenvalue, $a \gt 0 \implies \lambda_1 \lt 0$.
*For the second eigenvalue, we have cases:
*Case 1: $c \le 0, c \gt b, ~\lambda_2 \gt 0$. This is a saddle.
*Case 2: $c \gt 0 \gt b, \lambda_2 \lt 0$. Eigen... | {
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Probability of multiple events: continued I have 5 independant events. How do I calculate the probability of event A occurring + 1 other? How do I calculate the probability of event A occurring + 2 others? What about the occurrence of A & B plus one other, but not the rest? I have multiple variations of this I'm tr... | The key here is to leverage independence. The first thing to note is that $A$ is independent of the event "at least one of $B,C,D,E$ occurs", so that
$$
P(A\cap\{\text{at least one of }B,C,D,E\})=P(A)\cdot P(\text{at least one of }B,C,D,E).
$$
So, how can you compute the probability that at least one of $B,C,D,E$ occur... | {
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recursive formulas I'm studying linear algebra. I don't know how to deal with recursive formulas, for example
*
*fibonacci
*when you should find a big determinant of size $n$ (I think Vandermonde uses a recursive formula is found there, or is it induction?)
*how do you go from a recursive formula to an explicit ... | (From a bit of a computer science perspective)
A recursive formula has two parts: a terminal condition, and a recursive call.
For instance, let's say Fibonacci numbers. Let's say that $f(n)$ will calculate the $n$-th Fibonacci number. Our domain is the natural numbers.
The function is therefore piecewise:
$$
f(n) =
\be... | {
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Isomorphism between this subgroup of complex numbers and all finitely generated abelian groups ? Every cyclic (abelian) group of infinite order is isomorphic to $G=(\mathbb{Z},+)$. Is there a corresponding set of groups $S_G=\{G\}$ such that every finitely generated abelian group of infinite order is isomorphic to at l... | Take
$$
S_G=\left\{\mathbb Z^r\times \prod_{i=1}^n \mathbb Z/p_i^{a_i}\mathbb Z:n,r,a_i\in\mathbb Z^+,p_i\text{ prime}\right\}.
$$
This is using the classification theorem for finitely generated abelian groups.
| {
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Proof that $f(x)=x^{1/n}$ is continuous. Here's what I've done:
According to the definition, a function is continuous at $c$ if, for any $\epsilon>0$, there exists a $\delta>0$ so that, if $|x-c| < \delta$, then $|f(x)-f(c)| < \epsilon$.
$$\begin{split}
|f(x)-f(c)| < \epsilon
& \Leftrightarrow \left|x^{1/n}-c^{1/n}\r... | I think you want to take $\delta = \min\{(c^{1/n}+\epsilon)^n-c, c-(c^{1/n}-\epsilon)^n\}$ for $c\ne0$. Otherwise just take $\delta=\epsilon^n$.
EDIT: Looks like you made the correction as I was posting.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Limits involving logarithm and argument in the complex plane
*
*$\operatorname{Log}((2/n) + 2i)$ as $n \to \infty$
*$\operatorname{Log}(2 + (2i/n))$ as $n \to \infty$
*$\operatorname{Arg}((1+i)/n)$ as $n \to \infty$
*$(\operatorname{Arg}(1+i))/(n)$ as $n \to \infty$
For the Log questions, I am getting $(i\pi)/2 ... | The meaning of capitalized names such as $\operatorname{Log}$ varies by source. I assume that $ \operatorname{Log}$ has been defined so that it's continuous at $2i$ and at $2$; this is the case for the common definitions I'm familiar with. Check your definition. Then
*
*$\operatorname{Log}((2/n) + 2i) \to \operat... | {
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Show that $A \cong \mathbb{C}^n$ with A a commutative algebra
Let A be a commutative algebra of finite dimension, and if $A$ has no nilpotent elements other than $0$, is true that $A \cong \mathbb{C}^n$ ?
The question emerge to my mind, I thought that the finite dimension tell us that the scheme is Artinian (geomet... | An artinian ring $A$ has only finitely many prime ideals, which are all maximal. Thus, by the Chinese remainder theorem,
$$A / \mathfrak{n} \cong A / \mathfrak{m}_1 \times \cdots \times A / \mathfrak{m}_r$$
where $\mathfrak{m}_1, \ldots, \mathfrak{m}_r$ are the distinct prime/maximal ideals of $A$ and $\mathfrak{n}$ i... | {
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Combining results with Chinese Remainder Theorem? $9x^2 + 27x + 27 \equiv 0 \pmod{21}$
What is the "correct" way to solve this using the Chinese Remainder Theorem? How do I correctly solve this modulo $3$ and modulo $7$ without brute force?
| First, modulo $3$, your congruence reduces to $0\equiv 0$, because all coefficients are multiples of $3$. Therefore there are three solutions: $x\equiv 0,1,2 (\mod 3)$.
Working modulo $7$ the congruence becomes $2x^2+6x+6\equiv 0$, or $x^2+3x+3\equiv 0$, since we can multiply both sides by the inverse of $2$. To solve ... | {
"language": "en",
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Question about $T_n = n + (n-1)(n-2)(n-3)(n-4)$
The formula $T_n = n + (n-1)(n-2)(n-3)(n-4)$ will produce an arithmetic sequence for $n < 5$ but not for $n \ge 5$. Explain why.
I think it is because if n is less than five the term with multiplication will be equal to zero and the common difference will be one. If n ... | Your reasoning is correct. You can generalize it to say that
$$T_n = (an + b) + (n-1)(n-2)\cdots(n-k)$$
is an arithmetic progression for $1 \le n \le k$, for some arbitrary integer $k \ge 1$. This is because similarly, the product evaluates to $0$ for these values of $n$, to leave
$$T_n = an + b$$
This remainder essen... | {
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"timestamp": "2023-03-29T00:00:00",
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A Flat Tire Excuse I have this multi-part question on an assignment that I don't understand. Hopefully someone can help.
There's a story that 4 students missed their final and asked their professor for a make-up exam claiming a flat tire as their excuse. The professor agreed and put them in separate rooms. The first qu... | Hm ... how about this:
H0: flat tire
HA: students lied
If H0 holds then the probablity of different answers is exactly 0 (assuming students are in a good memory).
1) Hence, reject H0 if there are any different answers and the p-value is exactly 0.
2) Keep H0 if answers agree and the power of the test is 1-1/256.
| {
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Why is $\pi_1(\Bbb{R}^n,x_0)$ the trivial group in $\Bbb{R}^n$? My Algebraic Topology book says
Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone).
I wonder why that is. I can imagine infinite continuous "loops" in $\Bbb{R}^3$ th... | The problem with your last sentence is that $\pi_1(X,x_0)$ is not the set of loops based on $x_0$, but of homotopy classes of loops based on $x_0$.
Can you see why every loop in $\mathbb R^n$ based on $x_0\in \mathbb R^n$ is homotopic to the constant map based in $x_0$?
| {
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"source": "stackexchange",
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Proving Vector Subspaces Question 1:
The set $\mathbb R^3$ of all column vectors of length three, with real entries, is a vector space. Is the subset $$B=\{xyz \in \mathbb R^3 \mid xy+yz= 0\}$$ a subspace of $\mathbb R^3$?
Justify your answer.
Attempted answer:
$(0)$: let $0$ vector be in set $B$, then $0\cdot0 + 0\cd... | So for question 2:
A1: $\sin(x)(f+g)'' + x^2 (f+g)(x)= 0 \implies
\sin(x) [f''(x) + g''(x)] + x^2\cdot f(x)+x^2\cdot g(x) = 0$ $ \implies \sin(x)\cdot f''(x) + \sin(x)\cdot g''(x) + x^2\cdot f(x) +x^2\cdot g(x)$ so clearly $f +g \in F$.
S1: $\alpha \cdot (\sin(x)\cdot f''(x) + x^2\cdot f(x)) \implies \alpha \sin(x)\... | {
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Minimum number of different clues in a Sudoku I wonder if there are proper $9\times9$ Sudokus having $7$ or less different clues. I know that $17$ is the minimum number of clues. In most Sudokus there are $1$ to $4$ clues of every number. Sometimes I found a Sudoku with only $8$ different clues.
In this example the n... | Does the mere interchangeability of 2 or more missing values really result in more than one solution?
In a standard sudoku these 'values' are only symbols and bear no arithmetical value or meaning. Follow the reasoning above and not any standard sudoku with even 8 different clues can have a unique solution, because you... | {
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Proof by induction: $n! > n^2$ for $n\ge 4$ Proof by induction: $n! > n^2$ for $n\ge 4$
Basis step:
if $n=4$
$4\cdot3\cdot2\cdot1 > 4^2$
$24 > 16$
I don't know how to do the inductive step.
| Inductive step:
$$(n+1)! = (n+1)n! > (n+1)n^2 $$
But clearly, $n^2 > n + 1$ for integer $n \ge 2$. You could prove this rigorously by showing that the curve $n^2 - n - 1$ lies strictly above the $x$ axis for $n \ge 2$.
Hence,
$$(n+1)! > (n+1)^2$$
| {
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Determine if R is an equivalence relation I've got this question: Consider the relationship $R$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d)$ (denoted by $(a, b) R (c, d)$) if and only if $ad = bc$.
Discuss whether $R$ is an equivalence relation.
I'm pretty new to set theory and am... | *
*Check reflexivity: Is it the case that for all $(a, b)\in \mathbb
N\times \mathbb N$, it is true that $(a, b) R (a, b)$? That is, is it
true that for all such $(a, b)$, $ab = ba$?
*Check symmetry: Is it the case that for all $(a, b), (c, d) \in \mathbb N\times \mathbb N,$ that If $(a, b) R (c, d)$, then $(c, d... | {
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Application of Composition of Functions: Real world examples? Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug and chug where you take f(g(4) and run it through both fun... | First example of Algorythms: You have a list, compose by a head (an element) and a tail (a list). A composition of functions could return the second element of the list, let's say, L:
$ Head(Tail(L)) $
This is a simple examen in my field of study, I don't know if that's what you're looking for.
| {
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The Affine Tangent Cone I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)?
Could someone explain this, or at least redirect me to a source I could read about? I tried to look online... | Initial remark. Suppose someone gives you a continuous function of two variables, and asks you to calculate the Taylor series in $(0,0)$. If we happen to notice that the first derivatives vanish at the origin, we will call it a critical point.
The idea of the tangent cone is that we are doing exactly this: taking the T... | {
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How is $\{\emptyset,\{\emptyset,\emptyset\}\} = \{\{\emptyset\},\emptyset,\{\emptyset\}\}$? I'm slightly confused as to how
$$\{\emptyset,\{\emptyset,\emptyset\}\} = \{\{\emptyset\},\emptyset,\{\emptyset\}\}$$
are equivalent. I thought two sets were equivalent if and only if "$A$" and "$B$" have exactly the same eleme... | may be it is better to use that $\{ \emptyset, \emptyset\} = \{\emptyset\} \cup \{\emptyset\}$
| {
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Proof that certain number is an integer Let $k$ be an integer and let
$$
n=\sqrt[3]{k+\sqrt{k^2-1}}+\sqrt[3]{k-\sqrt{k^2-1}}+1
$$
Prove that $n^3-3n^2$ is an integer.
(I have started posting any problem I get stuck on and then subsequently find a good solution to here on math.se, primarily to get new solutions which m... | Since $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
$$a+b+c=0\implies a^3+b^3+c^3=3abc$$
Let $a=\sqrt[3]{k+\sqrt{k^2-1}}$, $b=\sqrt[3]{k-\sqrt{k^2-1}}$, $c=1-n$
Clearly we have $a+b+c=0$
$$a^3+b^3+c^3=3abc\tag{1}$$
$$ab=\sqrt[3]{k+\sqrt{k^2-1}}\cdot\sqrt[3]{k-\sqrt{k^2-1}}$$
$$ab=\sqrt[3]{k^2-(\sqrt{k^2-1})^2}$$
$... | {
"language": "en",
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$\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/\mathbb{2Z}$ When I first looked at this problem, I thought that $\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/5\mathbb{Z}$, but apparently the correct answer is $\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}/2\mathbb{Z}$.
Here's where I'm confused: saying that $\mathbb{Z}[i]/(1+i) \cong \mathbb{Z}... | There is a natural ring image of $\,\Bbb Z\,$ in $\,R = \Bbb Z[i]/(1\!+\!i)\,$ by mapping integer $\,n\,$ to $\ n \pmod{1\!+\!i}$ by composing the natural maps $\,\Bbb Z\to \Bbb Z[i]\to \Bbb Z[i]/(1+i).\,$ This map $\, h\color{#0a0}{ \ {\rm is\ surjective\ (onto)}}$ since $\,{\rm mod}\ 1\!+\!i\!:\ \,1\!+\!i\equiv 0\,... | {
"language": "en",
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Why is variance squared?
The mean absolute deviation is:
$$\dfrac{\sum_{i=1}^{n}|x_i-\bar x|}{n}$$
The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$
*
*So the mean deviation and the variance are measuring the same thing, yet variance requires squaring the difference. Why? Squaring always gives a no... | The long and the short is that the squared deviation has a unique, easily obtainable minimizer (the arithmetic mean), and an inherent connection to the normal distribution. The absolute deviation, on the other hand, can admit multiple non-unique, potentially laborious to obtain minimizers (medians). For a simple illust... | {
"language": "en",
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"source": "stackexchange",
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Is a brute force method considered a proof? Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$".
If we let a computer check every single member of $X$ and conclude that the condition $Y$ holds for all of them, can we call this a proof? Or is it possib... | Using a computer to brute-force can be the first step to a proof. The next step is to prove that the program is correct!
A few ways you might do this are:
*
*Have the program output a proof for each member of the set. We can then check these proofs without having to trust the program at all. We could even send them ... | {
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easy inequality to prove Prove that $\log_2(x)+\frac{1-x}{x} > 0$
I think the answer is easy but I've no clue how to go about it.
| Take the exponents of both sides, it is equivalent to showing $xe^{\frac{1}{x}-1} > 1$. But $e^x = 1+x+\cdots > 1+x$ whenever $x$ is positive, and thus $xe^{\frac{1}{x}-1} > x(1+\frac{1}{x}-1) = 1$.
| {
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How do I reduce (2i+2)/(1-i) with step-by-step please? I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the co... | Try multiplying the numerator and denominator by $1+i$. This will give you $\frac{(2i+2)(1+i)}{1^2+1^2}$. Then, FOIL the numerator and note $i^2=-1$.
| {
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modern analysis: integrals and continuity Let $$f(x) = \sum_1 ^\infty n*e^{-nx}$$ Where is $f$ continuous? Compute $$\int_1^2f(x) dx$$
I am having trouble proving where $f$ is continuous. For the second part, so far I have been able to compute the derivative.. although I basically had to move the summation from inside... | When the real part of $x$ is greater than $0$, $f(x)=\frac{e^x}{(e^x-1)^2}$. Clearly this is continuous so simply show the relation.
When the real part is less than $0$, use the limit test.
| {
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Gluing Lemma when A and B aren't both closed or open. Gluing Lemma:
$Let X = A \cup B \text{ and } f: A \rightarrow Y$ be continuous and $g: B \rightarrow Y$ be continuous with $A,B$ closed. Also $\left.f\right|_{A \cap B} = \left.g\right|_{A \cap B}$. Then $h: x \rightarrow y$ such that $\left.h\right|_A = f$ and $\le... | The relevant part here is that $A\cap B$ can be empty and yet have have the functions need to agree at the boundary of one because it's a limit point of the other. Let $A=[0,1]$ and $B=(1,2)$. Then $A\cup B=[0,2)$. Let $f(x)=x$ and $g(x)=-x$. Then $h(x)$ is discontinuous at $1$. But the criterion that they agree on the... | {
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Minimizing a convex cost function I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function
$$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 + \sin\left(\frac{\pi k}{n}\right) x_2\right)^2.$$
I'd like to show some basic propert... | Looking at your derivatives is quite interesting since
$$\sum _{k=1}^n \cos\left(\frac{\pi k}{n}\right) \cos \left(\frac{\pi
k}{n}\right) =\frac{n} {2}$$
$$\sum _{k=1}^n \cos\left(\frac{\pi k}{n}\right) \sin \left(\frac{\pi
k}{n}\right) =0$$
$$\sum _{k=1}^n \sin\left(\frac{\pi k}{n}\right) \sin \left(\frac{\... | {
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Lagrange polynomials sum to one I've been stuck on this problem for a few weeks now. Any help?
Prove:
$\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$
The sum of lagrange polynomials should be one, otherwise affine combinations of with these make no sense.
EDIT:
Can anybody prove this by actually working... | HINT: Throw in the $f(x_i)$ and what happens if they are all 1?
| {
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Flatness of a manifold (or a connection) Suppose we have an $n$-dimensional manifold $S$ (with a global coordinate system) with a metric $g$ and a connection $\nabla$ with connection coefficients (Christoffel symbols) $\Gamma_{i,j}^k$ given. Suppose that the $\nabla$-geodesic connecting any two points of the manifold c... | I think you are misunderstanding what a flat (affine) connection is: It is a connection on a manifold $M$ such that at each point of $M$ there exists a coordinate system with zero Christoffel symbols (vanishing depends heavily on which coordinates you use). Equivalently, a connection is flat if it has zero curvature. E... | {
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Show that, if $g(x)=\frac{1}{1+x^2}$ and $x_{n+1}:=g(x_n)$ then $x_n$ converges
Why does the sequence $\{x_n\}$ converge ?
If $x_{n+1}:=g(x_n)$, where $g(x)=\frac{1}{1+x^2}$
(We have a startpoint in $[0.5,1]$)
The sequence is bounded by $1$ independant of the startpoint (Is it necessary that $x_0\in[0.5,1]$ ?)
We hav... | See related approachs and techniques (I), (II).
| {
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The effects of requiring a recursive vs. a recursively enumerable axiomatization in the incompleteness theorem I believe that the (paraphrased) original statement of Gödels first incompleteness theorem (including Rosser's trick) is
If T is a sufficiently strong recursive axiomatization of the natural numbers (e.g. the... | *
*Just for the record, the original version of Gödel's incompleteness theorem was about theories where the class of axioms and the rules of inference are rekursiv, which for Gödel at the time meant, as we would now put it, primitive recursive. [Recall, Gödel was initially writing in 1930/1931, a few years before the ... | {
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Why is $f'(x)$ the annihilator of $dx$? Let $B=A[x]$ be an integral extension of a Dedekind ring $A$ where $x$ has minimal (monic) polynomial $f(x)$. Then the module of Kahler differentials $\Omega_A^1 (B)$ is generated by $dx$. Why is its annihilator $f'(x)$?
$\Omega_A^1 (B)=I/I^2$, where $I:=\{b\otimes_A 1 -1\otimes_... | Let us write $B = A[t]/(f(t))$ where $f(t)$ is some monic polynomial with coefficients in $A$. Then (you should prove!) that the Kahler differentials are precisely $\Omega_{B/A} = B[dt]/ (f'(t)dt)$. It is now clear the annihilator of this $B$-module is precisely $(f'(t))$.
The fact I am using is this: Let $B$ be an ... | {
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Is there a name for this type of integer? An integer $n$ such that $\exists$ at least one prime $p$ such that, $p|n$ but $p^2$ does not divide $n$.
i.e. : an integer with at least one prime that has a single power in the prime factorization.
Do these numbers have a special name, and have they been studied?
| These are sometimes called weak numbers, presumably because not weak naturals are precisely the powerful naturals. However the weak terminology does not appear to be anywhere near as widely used as is the powerful terminology (e.g. Granville and Ribenboim use "not powerful"). I also recall seeing other names besides we... | {
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Ramanujan's 'well known' integral, $\int\limits_{-\pi/2}^{\pi/2} (\cos x)^m e^{in x}dx$. $$\int_{-\pi/2}^{\pi/2} (\cos x)^m e^{in x}dx=\frac{\pi}{2^m} \frac{\Gamma(1+m)}{\Gamma \left(1+ \frac{m+n}{2}\right)\Gamma \left(1+ \frac{m-n}{2}\right)}$$
Appearing at the start of Ramanujan's paper 'A Class of Definite Integrals... | Assume that $n >m>-1$.
Then $$ \begin{align} \int_{-\pi /2}^{\pi /2} (\cos x)^{m} e^{inx} \, dx &= \int_{- \pi/2}^{\pi /2} \left( \frac{e^{ix}+e^{-ix}}{2} \right)^{m} e^{inx} \ dx \\ &= \frac{1}{i 2^{m}} \int_{C} (z+z^{-1})^{m} z^{n-1} \, dz \\ &= \frac{1}{i2^{m}} \int_{C} \left(z^{2}+1 \right)^{m} z^{n-m-1} \, dz \\ ... | {
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Square Idempotent matrix: efficient algorithms for finding eigenvectors Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ non-zero eigenvalues?
Information:
*
*Matrix is ide... | Since the matrix is idempotent $A^2=A$ the eigenvectors corresponding to the eigenvalue $1$ are exactly the elements of the image of the linear transformation described by $A$. Hence you could choose a basis of the column space of $A$ (or row space if you are thinking of rows...) and be done. Alas I do not know if your... | {
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Probability of winning a head on a coin The problem I am asking is generated from this problem:
Carla and Dave each toss a coin twice. The one who tosses the greater number of heads wins a prize. Suppose that Dave has a fair coin $[P_d(H)=.5]$, while Carla has a coin for which the probability of heasd on a single toss ... | The easy way is to recognize that you can ignore the event of a tie. If Dave's chance of winning on one turn is $d$, Carla's chance of winning on one turn is $c$, then Dave's chance of winning overall is $\frac d{c+d}$ and Carla's is $\frac c{c+d}$ You can show this by summing the geometric series as you suggest. Th... | {
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What is the sum of $1^4 + 2^4 + 3^4+ \dots + n^4$ and the derivation for that expression What is the sum of $1^4 + 2^4 + 3^4+ \dots + n^4$ and the derivation for that expression using sums $\sum$ and double sums $\sum$$\sum$?
| Here is my favorite method which works for any polynomial summand and you only need to remember two basic facts, one from calculus one and one about polynomials. First, since summations are analogous to integration, we have that
$$\int x^k \approx x^{k+1} \Rightarrow \sum x^k \approx x^{k+1}.$$
For your problem, let us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/718939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
} |
Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\frac{a_1}{2}+\ldots+\frac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0$ Show that if $f(x)= \sum\limits_{i=0}^n a_i x^i$ and $a_0+\dfrac{a_1}{2}+\ldots+\dfrac{a_n}{n+1}=0$, then there is an $x \in (0,1)$ with $f(x)=0.$
| you can use the zero point theorem!
if $a_0>0$ and $\dfrac{a_1}{2}+\ldots+\dfrac{a_n}{n+1}<0$
then the condition you write will turn out to be a critical condition:
$a_0+\dfrac{a_1}{2}+\ldots+\dfrac{a_n}{n+1}=0$
since $f(0)=a_0>0$ and $f(1)=$$-na_0-(n-1)\dfrac{a_1}{2}-\ldots-\dfrac{a_n}{n+1}<0$$\Longrightarrow$$\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving that $\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$ I want to prove for any positive integer $n$, the following equation holds:
$$\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$$
I tried to expand $2^k$ as $\sum_{i=0}^k{k\choose i}$ and int... | Note that successive terms in the sum on the left turn out have a simple expression for their sum: when $k \ne 0$, we have
$$\frac{1}{\binom{n}{k-1}} + \frac1{\binom{n}{k}} = \frac{\binom{n}{k}+\binom{n}{k-1}}{\binom{n}{k-1}\binom{n}{k}} = \frac{\binom{n+1}{k}}{\binom{n}{k-1}\binom{n}{k}} = \frac{\frac{n+1}{k}\binom{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Is a matrix that is symmetric and has all positive eigenvalues always positive definite? I know a symmetric matrix is positive definite if and only if every eigenvalue is positive. However, is a matrix that is symmetric and has all positive eigenvalues always positive definite? More specifically, I have the following m... | Yes in the real case.
Let $ A $ be a matrix that is symmetric and has all positive eigenvalues. Then, $\forall x\in V, x^TAx > 0 $. This is true for all eigenvectors $ x_i $ of $ A $. Hence $0<x^T_iAx_i=x_i\alpha_i x_i=||x_i||^2\alpha_i $, so $\alpha_i> 0 $.
Suppose $\alpha_i> 0\forall i $. Then, on eigenvectors, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A urn containing $n$ balls, numbered $1,2,...,n$, and $k$ balls are chosen at random without replacement. I have a homework question with specific $n$ and $k$ given for the below question, but I would rather understand how this works for any given $n$ or $k$ to build my intuition for these questions.
A urn containing $... | The largest, as you pointed out, is one of $k,k+1,\dots,n$.
There are $\binom{n}{k}$ equally likely ways to choose $k$ balls.
Let us count the number of "favourables," that is, the number of ways to choose $k$ balls with $m$ the largest.
So we must choose $k-1$ balls from the balls $1,2,\dots,m-1$ to keep $m$ company... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that if R is an integral domain then Torsion of M is a submodule of R-module M I can't quite get this one. I can show that two non-zero elements m,n of M have a non-zero product but they belong to the R-module, not the integral domain so I don't know if it's necessarily true that their product be nonzero.
| Your prove should go like this: If $m_1,m_2$ are torsion elements, then there exist two elements $a_1,a_2$ of $R$, $a_1\neq 0, a_2 \neq 0$ such that
$$
a_1m_1=0 \qquad a_2 m_2=0
$$
have to show: for any $r_1,r_2 \in R$ the Element $m:=r_1m_1+r_2m_2 \in M$ is a torsion element.
first note that $a_1a_2 \neq 0$ because $R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is $f$ differentiable at $(x,y)$? I am working on a practice question, and I am not sure if what I have done would be considered, 'complete justification'. I would greatly appreciate some feedback or helpful advice on how it could be better etc. The question is here:
Let $f: \mathbb {R}^2 \to \mathbb{R} $ be a function... | You've already found
$$\lim \limits_{(x,y)\to (0,0)}\left(\dfrac{f(x,y) - [f(0,0) + f_{x}(0,0)(x-0) + f_{y}(0,0)(y-0)]}{\sqrt{x^2 + y^2}}\right)=\lim \limits_{(x,y)\to (0,0)}\left( \dfrac{\frac{\sin(x^2 + y^2)}{x^2 + y^2} - 1}{\sqrt{x^2 + y^2}}\right).$$
To evaluate $\lim \limits_{(x,y)\to (0,0)}\left( \dfrac{\frac{\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integrate the following function:
Evaluate:
$$\int \frac{1}{ \cos^4x+ \sin^4x}dx$$
Tried making numerator $\sin^2x+\cos^2x$
making numerator $(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x$
Dividing throughout by $cos^4x$
Thank you in advance
| Another way:
$$I=\int\frac{dx}{\cos^4x+\sin^4x}=\int\frac{(1+\tan^2x)\sec^2x dx}{1+\tan^4x}$$
Setting $\displaystyle\tan x=u,$
$$I=\frac{(1+u^2)du}{1+u^4}=\int\frac{1+\dfrac1{u^2}}{u^2+\dfrac1{u^2}}du$$
$$=\int\frac{1+\dfrac1{u^2}}{\left(u-\dfrac1u\right)^2+2}du$$
Set $\displaystyle u-\dfrac1u=v$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/719585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Visual explanation of the following statement: Can somebody fill me in on a visual explanation for the following:
If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$
I know why it is true (ex. take $w = x-y, z = x+y$), but I would think there is a v... | It will take me forever to post the diagram so here is a description.
Draw the circle with centre $(0,0)$ and radius $\sqrt c\,$. Locate the point $(x,y)$ on this circle: by assumption, $x$ and $y$ are integers. Draw the tangent to this circle starting at $(x,y)$ and having length $\sqrt c\,$. This will give a point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 2
} |
Calculate percentage given value, minim and maximum It's my first time on Math.stack; be gentle.
I have slider with a range between -1 and 1.
If my slider is at 0 I'd expect it to be at 0%
If it were at either -1 or 1 I'd expect it to be 100%
However it must take into account those won't always be the max & min
When I... | Alright, so let's take $u$ to be the upper bound. Lets make $l$ the lower bound. When you go to the right, the percentage of the area swept from x=0 to some $x$ the right is:
$$\frac{x}{u}\times 100\%$$
Similarly, on the left you'll just use your lower bound. You don't even need absolute value because the negatives wil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to integrate $\displaystyle 1-e^{-1/x^2}$?
How to integrate $\displaystyle 1-e^{-1/x^2}$ ?
as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$
If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything:
$\,\displaystyle\int\limits_{-\infty}^{\infty}\left(1-e^{-1/x^2}\right)dx=\int\limits_{-\i... | $\int\left(1-e^{-\frac{1}{x^2}}\right)dx$
$=\int\left(1-\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{-2n}}{n!}\right)dx$
$=\int-\sum\limits_{n=1}^\infty\dfrac{(-1)^nx^{-2n}}{n!}dx$
$=-\sum\limits_{n=1}^\infty\dfrac{(-1)^nx^{1-2n}}{n!(1-2n)}+c$
$=\sum\limits_{n=1}^\infty\dfrac{(-1)^n}{n!(2n-1)x^{2n-1}}+c$
$\because\int_{-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/719963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
What is the relation between this binary number with no two 1 side by side and fibonacci sequence? I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side.
Now as an example
1 = 1
10 = 1
100,101 = 2
1000,1001,1010 = 3
10000,10001, 10010, 10100, 10101 = 5
Str... | Suppose we make an $n$-digit string with no consecutive $1$s. Then it either ends with a $0$ or a $1$.
If it ends with a $0$, we can add (from the front) any $(n-1)$ digit string with no consecutive $1$s. There are $a_{n-1}$ of these.
If it ends with a $1$, then the previous digit must be a $0$ because there are no c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Bifurcation values for logistic map To find the bifurcation values for $$x_{i+1}=f(x_i) = rx_i(1-x_i)$$first I set $rx(1-x) = 0$ and found the x values and then used the x values to find $r = 0$ and $r = 1$.
Do you think what I did here is correct? If not, can you help me find the mistakes here?
| To study bifurcations of maps, begin by looking for fixed points. The logistic equation is
$$x_{i+1} = f(x_i) = rx_i(1-x_i)$$
so a fixed point satisfies
$$x = rx(1-x) \Rightarrow x(rx - r+1) = 0$$
which implies that there are fixed points at $x=0$ and $x=1-1/r$. To analyze bifurcations as $r$ varies, consider the linea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces? I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there are a... | Bezier curves are just polynomials. From a mathematical point of view, the $m+1$ Bernstein polynomials of degree $m$ are just a basis for the vector space $\mathbb{P}_m$ of all polynomials of degree $m$. So, of course, you can use other bases of $\mathbb{P}_m$, instead. This won't give you new types of curves and surfa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Interesting question about functions I saw the following question and I would like to share. I don't know the answer.
Suppose that the function $f:\Bbb{N}\to\Bbb{N}$ has the property $f(f(n))<f(n+1)$ for any $n\in\Bbb{N}$. Prove that $f(n)=n$.
| Consider the set $A=\{ f ( f (1)), f (2), f ( f (2)), f (3), f ( f (3)),\ldots, f (n), f ( f (n)), . . .\}$.
That is the set of all numbers appearing in the inequality $f ( f (n)) < f (n + 1)$.
This set has a smallest element, which cannot be of the form $f (n + 1)$
because then it would be larger than $f ( f (n))$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/720313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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