Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that $\int_0^\infty e^{-a^2 s^2} \cos(2 b s) \,\mathrm ds=\frac{\sqrt{\pi}}{2a}e^{-b^2/a^2} $ Prove that
$$I(a,b)=\int_0^\infty e^{-a^2 s^2} \cos(2 b s) \,\mathrm ds=\frac{\sqrt{\pi}}{2a}e^{-b^2/a^2}\quad a>0 $$
I can prove it by differential-equations technique(taking the derivative with respect to $b$ to becom... | Note that we can complete the square and get the following integral:
$$\frac12 \Re{\left [\int_{-\infty}^{\infty} ds \, e^{-a^2 s^2} \, e^{i 2 b s} \right ]} = \frac12 e^{-b^2/a^2} \Re{\left [\int_{-\infty}^{\infty} ds \, e^{-a^2 (s-i b/a^2)^2} \right ]}$$
We can prove that the integral on the RHS is simply $\sqrt{\pi}... | {
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"timestamp": "2023-03-29T00:00:00",
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Does the series $ \sum_{n=1}^{\infty}\left( 1-\cos\big(\frac{1}{n} \big) \right)$ converge? I'm having trouble determining whether the series:
$$
\sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right]
$$
converges.
I have tried the root test:
$$\lim_{n\rightarrow\infty}\sqrt[n]{1-\cos\frac{1}{n}}=\lim_{n\rightarr... | We know:
$$\cos t \sim 1-\dfrac{t^2}{2}$$
Hence, for $n \to +\infty$
$$\sum_{n=1}^{+\infty}\left(1-\cos\left(\dfrac{1}{n}\right)\right)\sim \sum_{n=1}^{+\infty}\dfrac{1}{2n^2}=\frac{\pi^2}{12}$$
Thus it converges
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/625167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Do we know if there exist true mathematical statements that can not be proven? Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.
Phrased ... | Amongst the many excellent answers you have received, nobody appears to have directly answered your question.
Goldbach's conjecture can be true and provable, true but not provable using the "normal rules of arithmetic", or false.
There are strong statistical arguments which suggest it is almost certainly true.
Whethe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/625223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "166",
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"answer_id": 7
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Evaluating some Limits as Riemann sums. I really have difficulties with Riemann Sums, especially the ones as below:
$$\lim_{n\to\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n}\right)$$
When i try to write this as a sum, it becomes $$\frac { 1 }{ n } \sum _{ k=1 }^{ 2n } \frac { 1 }{ 1+\frac { k }{ n } } ... | With Eulero-Mascheroni : $$\sum_{k = 1}^{n}\frac{1}{k} - \log{n} \rightarrow \gamma$$ $$\sum_{k = 1}^{3n}\frac{1}{k} - \log{3n} \rightarrow \gamma$$ so $$\sum_{k = n+1}^{3n}\frac{1}{k} - \log{3n} +\log{n} \rightarrow 0$$ and then$$\sum_{k = n+1}^{3n}\frac{1}{k} \rightarrow \log{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/625306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The derivative of a holomorphic function on the boundary of the unit circle Let $f$ be holomorphic on $D(0,1)\cup \{1\}$, and $f(0)=0$, $f(1)=1$, $f(D(0,1))\subset D(0,1)$. Prove that $|f'(1)|\geq 1$.
I have no idea. Maybe $f'(\xi)=[f(1)-f(0)]/(1-0)$ for some $\xi \in (0,1)$, and the maximal modules principle applies? ... | It is easier to argue by contradiction. We are given that $f'(1)$ exists. Suppose $|f'(1)|<1$. Pick $c$ so that $|f'(1)|<c<1$. There is a neighborhood of $1$ in which $$|f(z)-f(1)|\le c|z-1|$$ By the reverse triangle $$|f(z)|\ge |f(1)|-c|z-1| = 1-c|z-1| \tag{1}$$
On the other hand, by the Schwarz lemma
$$|f(z)|\le |z... | {
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I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? Wikipedia defines the notion of an abelian category as follows (link).
A category is abelian iff
*
*it has a zero object,
*it has all binary products and binary coproducts, and
*it has all kernels ... | The claim is not trivial and requires a bit of ingenuity. For a guided solution, see Q6 here. The four steps are as follows:
*
*Show that the category has finite limits and finite colimits.
*Show that a morphism $f$ is monic (resp. epic) if and only if $\ker f = 0$ (resp. $\operatorname{coker} f = 0$).
*Show that ... | {
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"source": "stackexchange",
"question_score": "4",
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Manifolds and Topological Spaces from my understanding of manifolds they are structures defined on topological spaces. So if M is a manifold defined on a topological space $(X,\tau)$ and $X\subseteq\mathbb R^3$, does this mean $M$ is a $3$-manifold? If so does this generalize to higher dimensions such as if $X\subseteq... | It is usually said that the notion of manifolds was introduced by Riemann in 1854, but it wasn't until Whitney’s work in 1936 that people know what abstract manifolds are, other than being submanifolds of Euclidean space.
A plane in R^3 is a very trivial counterexample of your question.
| {
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How find this minimum of the $a$ such $x_{n}>0$ let real numbers $a>0$, and sequence $\{x_{n}\}$ such
$$x_{1}=1,ax_{n}=x_{1}+x_{2}+\cdots+x_{n+1}$$
Find the minimum of the $a$,such $x_{n}>0,\forall n\ge 1$
My try: since
$$ax_{n}=x_{1}+x_{2}+\cdots+x_{n+1}$$
$$ax_{n+1}=x_{1}+x_{2}+\cdots+x_{n+2}$$
$$\Longrightarrow a(x... | you've done a good work, do not feel bad about it and think your guess is also true( i havent' checked it yet).
If you continue , you will reach the answer eventually.
However, this way of approach forces us to do many caculations.
I would like to present to you the below problem and its proof to reduce your work:
Pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/625638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the improper integral $\int^\infty_0\frac{1}{x^3+1}\,dx$ $$\int^\infty_0\frac{1}{x^3+1}\,dx$$
The answer is $\frac{2\pi}{3\sqrt{3}}$.
How can I evaluate this integral?
| $$x^3+1 = (x+1)(x^2-x+1)$$
Logic: Do partial fraction decomposition.Find $A,B,C$.
$$\frac{1}{x^3+1} = \frac{A}{x+1}+\frac{Bx+C}{x^2-x+1}$$
By comparing corresponding co-efficients of different powers of $x$, you will end up with equations in A,B,C.After solving you get :
$$A=\frac{1}{3},B=\frac{-1}{3},C=\frac{2}{3}$$
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/625821",
"timestamp": "2023-03-29T00:00:00",
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Minimum distance between a disk in 3d space and a point above the disk How can I calculate the minimum distance between a point on the perimeter of a disk in 3d space and a point above the disk?
For example, there is a disk in 3d space with center [0,0,0]. It has radius 3 and lies flat on the x,y plane. If there is a p... | Your logic is wrong because you make the wrong thing a unit vector. The length of your vec2 is not necessarily equal to the radius of the disc, because it is equal to the radius multiplied by a vector which is some component of the unit vector unitvec1.
Instead of normalizing vec1, you should normalize vec2.
In this ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the equation of the normal to the curve $y = 8/(4 + x^2)$ , at $x = 1$. When you first differentiate the above, you get $-8/25$, right? Then you derive the gradient for a normal and proceed so on and so forth.
The textbook I'm using says when you differentiate, you get $-16/25$. I believe that's wrong...
| The textbook is correct.
You can use
$$\left\{\frac{f(x)}{g(x)}\right\}^\prime=\frac{f^\prime(x)g(x)-f(x)g^\prime(x)}{\{g(x)\}^2}.$$
Letting $$h(x)=\frac{8}{4+x^2},$$
we have
$$h^\prime(x)=\frac{0-8\cdot 2x}{(4+x^2)^2}=-\frac{16x}{(4+x^2)^2}.$$
Hence, we will have
$$h(1)=-\frac{16}{25}.$$
So...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/626034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Isolated singularities sort Can anyone help me out with finding the nature of the singularities of the following function:
$$g(z)=\frac{\cos z-1}{z^5}$$ without using Taylor expansions?
| You are trying to figure out the nature of the singularity $z_0=0$. Write $p(z)=\cos(z)-1$ and $q(z)=z^5$. Then $g(z)=\frac{p(z)}{q(z)}$. Notice that $z_0$ is a zero of order $2$ for $p$ and a zero of order $5$ for $q$. Then you must have seen a theorem that states that $z_0$ is then a pole of order $5-2=3$.
Otherwise... | {
"language": "en",
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Choose the branch for $(1-\zeta^2)^{1/2}$ that makes it holomorphic in the upper half-plane and positive when $-1<\zeta<1$ From Stein/Shakarchi's Complex Analysis page 232:
...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z
\frac{d\zeta}{(1-\zeta^2)^{1/2}},$$ where the integral is taken from
$0$ to $z$ along an... | Another (probably not as good) way to look at it is to think
$$
\sqrt{1-\zeta^2} = i \sqrt{\zeta -1} \sqrt{\zeta + 1}
$$
Choose the (possibly different) branches for each of the square roots on the right side to give you what you need. It might look at first like you will have problems on $z \in [-1,1]$, but the disco... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/626256",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Express $\int_0^1\frac{dt}{t^{1/3}(1-t)^{2/3}(1+t)}$ as a closed path integral enclosing the interval $(0,1)$ From an old complex analysis qualifier:
Define $$I=\int_0^1\frac{dt}{t^{1/3}(1-t)^{2/3}(1+t)}.$$
*
*Express $I$ as a closed path integral enclosing the interval
$(0,1)$.
*Evaluate $I$.
Ideas: A... | Consider $$f(z) = e^{-1/3\mathrm{LogA} z} e^{-2/3\mathrm{LogB} (1-z)} \frac{1}{1+z}$$
where $\mathrm{LogA}$ is the logarithm with branch cut on the negative real axis and argument from $-\pi$ to $\pi$ and $\mathrm{LogB}$ has the branch cut on the positive real axis and argument from $0$ to $2\pi.$
Then we have continui... | {
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"source": "stackexchange",
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For non-negative iid random variables, show that the max converges ip From Resnick's A Probability Path, exercise 6.7.11. (Studying for my comprehensive exam.)
Suppose $\{X_n, n \ge1\}$ are iid and non-negative and define $M_n=\bigvee^n_{i=1}X_i$.
a) Check that $P[M_n > x] \le nP[X_1 > x].$
b) Show that if $E(X^p_1) ... | Use this instead in the very last inequality.
$$ \frac{E(X_1^pI_{X_1 > n^{1/p}\epsilon})}{\epsilon^p} .$$
You should have a theorem somewhere that if $Y \ge 0$ and $EY < \infty$, then $E(YI_{Y>\alpha}) \to 0$ as $\alpha \to \infty$.
| {
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What's the proof that the sum and multiplication of natural numbers is a natural number? I'm explaining the construction of the natural numbers to someone and I'm asking him to show where to find $C$ and $F$ with $a,b,g,h\in \mathbb{N}$ in:
$$a+b=C$$
$$g*h=F$$
I know intuitively (and from some readings) that $C$ and $F... | What is the extent of the mathematical knowledge of your friend? That these statements are true are a consequence of the axiomatic construction of the natural numbers using the successor function. To say a little more, essentially, what makes $a$ and $b$ natural numbers? They are formed by adding $1$ to $1$ some number... | {
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Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? Why, conceptually, is it that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}?$$ I know how to prove that this is true, but I don't understand conceptually why it makes sense.
| Sure. You're dividing into 2 cases. On the one hand you're saying I'm stuck choosing this element over here. So now I have r-1 more choices to make out of n-1 things. In the other case you're refusing that element. Now you've eliminated a choice, but still must pick r elements. These two cases are exhaustive and ... | {
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Tangents at singularities Given an implicit polynomial function $f$ with singularity at the origin. How do I find the tangents to the curve at the point?
Wikipedia says that ignore all the terms except the ones with lowest degree. Why is this true?
Take for example the curve $x^3+x^4+y^3=0$. What are the tangents at th... | Wikipedia is right: the tangent cone to your curve $C$ is given by $x^3+y^3=0$.
If the base field is algebraically closed of characteristic $\neq 3$ and if $j\neq 1$ is a primitive cubic root of$1$, it consists of the three lines $x+y=0, x+jy=0, x+j^2y=0$, since $x^3+y^3=(x+y)(x+jy)(x+j^2y)$.
But where does the Wikip... | {
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"url": "https://math.stackexchange.com/questions/626745",
"timestamp": "2023-03-29T00:00:00",
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Use of L'hopital's rule Define $f:\mathbb{N} \to \mathbb{R}$ by $f(n)=\frac{sin (\frac{n\pi}{4})}{n}.$
May I know if we can use L'hopital's rule to evaluate $\lim_{n \to 0} f(n)$ ? If not, how can we evaluate the limit without the use of series?
Thank you.
| There is no such thing as $\lim_{n\to 0}f(n)$ if $f$ is only defined on $\mathbb N$.
| {
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What's the digit "a" in this number? a and b are digits in a four-digit natural number 7a5b. If 7a5b is divisible by 18, how many different possible values can "a" have?
| A number is divisible by $18$ iff it's divisible by $2$ and $9$. So, we must have $b \in \{0,2,4,6,8\}$ and $7+a+5+b$ divisible by $9$, since a number is divisible by $9$ iff the sum of its digits is divisible by $9$. I think you can solve it by now.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solution of $\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)$ Consider the PDE
$$\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)\tag{1} $$
with $t\ge0,\ x\in\mathbb R,\ f(0,x)=e^x$. I want to find $f(t,x)$.
I know that the heat equation
$$\frac{\partial p}{\part... | $\newcommand{\+}{^{\dagger}}%
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\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displaystyle... | {
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Why are compact sets called "compact" in topology? Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$.
Just curiosity:
I've done some search in Internet why compact sets are called compact, but it doesn't contain any good result. For ... | If you are curious about the history of compact sets (the definition of which dates back to Fréchet as mentioned in another response) and you can read French, then I suggest checking out the following historical article:
Pier, J. P. (1980). Historique de la notion de compacité. Historia mathematica, 7(4), 425-443. Retr... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving that there are $n-1$ roots in $a_1x^{b_1}+a_2x^{b_2}+...+a_nx^{b_n}=0 $ on $(0,\infty)$
We know that:
$a_1,...,a_n\in \mathbb R , \forall a_i\neq0 \\
b_1,...,b_n\in \mathbb R : b_j\neq b_k : \forall j\neq k$
Prove that there are $n-1$ roots in $(0,\infty)$:
$$a_1x^{b_1}+a_2x^{b_2}+...+a_nx^{b_n}=0 $$
Using... | Hint:
So now just work with $\;a_1 +a_2x^{b_2-b_1}+\ldots+a_nx^{b_n-b_1}\;$, differentiate and get less than $\;n\;$ summands. Now use what the other question mentions about the relation between the zeros of the derivative of a function and those of the function itself.
| {
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"timestamp": "2023-03-29T00:00:00",
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Conditional probability with 3 events i'm struggling to understand how this answer for a past paper is correct.
Question:
In a lake there are 10 fish, 3 of which are tagged. 3 fish are caught randomly from the lake without replacement. What is the probability that the first two fish that are caught are tagged but not t... | Define $A=\mbox{the first fish is tagged}$, $B = \mbox{the second fish is tagged}$ and $C = \mbox{the third fish isn't tagged}$. We are willing to calculate $P(A \cap B \cap C)$. Now, $P(A)=3/10$, because there are $3$ tagged fishes of $10$. Similarly, $P(B\mid A) = 2/9$, because there are now $2$ tagged fishes out of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How should I denote "undefined" in a functions definition While solving some limits, I thought this might be a nice strategy.
$$
\lim_{x \to -2} \frac{x^2-4}{x^2+3x+2} = \lim_{x \to -2} f(x) \\
f(x) = \frac{(x-2)(x+2)}{(x+2)(x+1)} =
\begin{cases}
\frac{x-2}{x+1},& x \neq -2 \\
\varnothing,& x = -2
\end{case... | You don't have to. This is all you need to write.
$$\lim_{x \to -2} \frac{x^2 - 4}{x^2 + 3x + 2} = \lim_{x \to -2} \frac{(x-2)(x+2)}{(x+1)(x+2)} = \lim_{x \to -2} \frac{x-2}{x+1} = \frac{-4}{-1}=4.$$
The point is, the functions $\frac{x^2 - 4}{x^2 + 3x + 2}$ and $\frac{x-2}{x+1}$ are not the same function, as the firs... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Two different problems with similar solutions Problem 1 :
Calculate the sum
$$S(n):=\sum_{s=0}^{\infty}s^nx^s$$
The solution is S(n)=$\frac{P(n)}{(1-x)^{n+1}}$, where the polynomials
satisfy the reccurence
P(0)=1 , P(1)=x , P(n+1) = x(1-x)P(n)'+x(n+1)P(n)
Problem 2 :
Calculate the probabilities that the sum of n random... | This continues to be true. The coefficients of your polynomials are the Eulerian Numbers (and the polynomials the Eulerian polynomials).
Compare the formula in terms of binomial coefficients for the Eulerian numbers given in the above link with the integral of the pdf of the Irwin-Hall Distribution between consecutive... | {
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Maximising with multiple constraints I have $$Z=f(x_1 ,x_2 ,x_3 ,... ,x_n)$$ function and $$\left[\begin{array}{r}c_1=g_1(x_1 ,x_2 ,x_3 ,... ,x_n) \\c_2=g_2(x_1 ,x_2 ,x_3 ,... ,x_n)\\c_3=g_3(x_1 ,x_2 ,x_3 ,... ,x_n) \\...\\c_m=g_m(x_1 ,x_2 ,x_3 ,... ,x_n)
\end{array}\right]$$
constraints.
How can I know critical points... | I'm going to use the notation
\begin{equation}
f(x) := f(x_1, x_2, \ldots, x_n)
\end{equation}
and
\begin{equation}
g(x) = \left[\begin{array}{c} g_1(x) \\ \vdots \\ g_m(x) \end{array}\right]
\end{equation}
along with
\begin{equation}
c = \left[\begin{array}{c} c_1 \\ \vdots \\ c_m\end{array}\right]
\end{equation}
to r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/627587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Every compact metric space is complete I need to prove that every compact metric space is complete. I think I need to use the following two facts:
*
*A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with finite intersection property has $\bigcap\{F:F\in\mathcal{F}\}\neq\emptyset$... | Let $X$ be a compact metric space and let $\{p_n\}$ be a Cauchy sequence in $X$. Then define $E_N$ as $\{p_N, p_{N+1}, p_{N+2}, \ldots\}$. Let $\overline{E_N}$ be the closure of $E_N$. Since it is a closed subset of compact metric space, it is compact as well.
By definition of Cauchy sequence, we have $\lim_{N\to\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/627667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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Interesting but short math papers? Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new result.
For example, I recently read and liked On the Uniqueness of the Cyclic Group of O... | On the Cohomology of Impossible Figures by Roger Penrose: Leonardo Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue (1992), pp. 245-247.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
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"answer_id": 6
} |
Proof - There're infinitely many primes of the form 3k + 2 — origin of $3q_1..q_n + 2$ Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6
To instigate a contradiction, postulate $q_1,q_2,\dots,q_n$ as all the primes $\neq 2 (=$ the only even prime) of the form $3k+2$. Consider $N=3q_1q_2\dots q_n+2.$ None ... | Let's see if the following observations help:
*
*Different statement $\Rightarrow$ different proof :-)
*$N$ was chosen to be of form $3k+2$ precisely to guarantee that it must be divisible by at least one prime of the form $3k+2$.
*Look at the form of $N$: It was chosen so that $(N-2)$ is divisible by all of $q_i$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The 6 generals problem 6 generals propose locking a safe containing secret stuff with a number of different locks. Each general will get a certain set of keys to these locks. How many locks are required and how many keys must each general have so that, unless 4 generals are present, the safe can't be opened. Generalize... | Sorry not to post this as a comment, but I don't have enough points.
Are they supposed to be able to open the safe if and only if at least four generals are present?
Edit: Here is the outline of a solution for the case of four out of six generals, but it generalizes easily.
Let the locks be numbered $1, \ldots, p$, and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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By finding solutions as power series in $x$ solve $4xy''+2(1-x)y'-y=0 .$ By finding solutions as power series in $x$ solve
$$4xy''+2(1-x)y'-y=0 .$$
What I did is the following. First I let the solution $y$ be equal to
$$y =\sum_{i=0}^{\infty} b_ix^i =b_0 +b_1x+b_2x^2+\ldots$$
for undetermined $b_i$. Then I found the ... | You have made your mistake in the power series. In particular, you need to end up with a recurrence relation and solve that.
$$y'=\sum_{i=0}^\infty{ib_ix^{i-1}}=0+b_1+2b_2x+3b_3x^2+...=\sum_{i=1}^\infty{ib_ix^{i-1}}$$
Now you need to get your lower bound so that it starts at $0$. Rewriting the sum using $i=0$, we ge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Numer of possibilities of placing people of different nationalities How many ways of sitting 3 people of nationality A, 3 of nationality B and 3 of nationality C there are if no two people of the same nationality can sit near each other (so such placings are prohibited: AABACBCBC, BCCAABCAB)
I came to such result:
$\fr... | This may not be a good way to solve it, but this will give the answer.
We have
$$\frac{6!}{3!3!}$$
patterns to arrange three $A$s and three $B$s.
1) In each of $BBBAAA,AAABBB,BAAABB$ case, since there are four places where two same letters are next to each other, we can't arrange $C$s in them.
2) In each of $ABBBAA, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is this geometric Probability In a circle of radius $R$ two points are chosen at random(the points can be anywhere, either within the circle or on the boundary). For a fixed number $c$, lying between $0$ and $R$, what is the probability that the distance between the two points will not exceed $c$?
| Unfortunately I don't have enough time to write down the whole thing. Basically you have to distinguish two cases:
A circle with radius c around the first point lies completely inside the circle that is span by R. Then the probability is $$\frac{c^2}{R^2}$$, but this only happens in $$\frac{(R-c)^2}{R^2}$$ of the cases... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving an irrational equation Solve for $x$ in:
$$\frac{\sqrt{3+x}+\sqrt{3-x}}{\sqrt{3+x}-\sqrt{3-x}}=\sqrt{5}$$
I used the property of proportions ($a=\sqrt{3+x}$, $b=\sqrt{3-x})$:
$$\frac{(a+b)+(a-b)}{(a+b)-(a-b)}=\frac{2a}{2b}=\frac{a}{b}$$
I'm not sure if that's correct.
Or maybe the notations $a^3=3+x$, $b^3=3-x$... | Here is another simple way, exploiting the innate symmetry.
Let $\ \bar c = \sqrt{3\!+\!x}+\sqrt{3\!-\!x},\,\ c = \sqrt{3\!+\!x}-\sqrt{3\!-\!x}.\,$ Then $\,\color{#0a0}{\bar c c} = 3\!+\!x-(3\!-\!x) = \color{#0a0}{2x},\ $ so
$\,\displaystyle\sqrt{5} = \frac{\bar c}c\, \Rightarrow \color{#c00}{\frac{6}{\sqrt{5}}} = {\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there anything "special" about elementary functions? I just found an article on Liouville's integrability criterion, which gave me a thought.
What makes functions like $\mathrm{Si}(x)$, $\mathrm{Ei}(x)$, $\mathrm{erfc}(x)$, etc. inherently different from $\sin{x}$, $\log{x}$, etc. ?
Related question: What is the ex... | Yes, beginning from constants and the identity function, the different elementary functions are obtained by applying closure with respect to different elementary operations. Close by sum and multiplication and we get polynomials. Close by solving linear differential equations or order one, with constant coefficients, a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove uniform contininuity (probably by Lipschitz continuity) Prove uniform continuity at $(0,\infty)$ for:
$$f(x) = x + \frac{\sin (x)}{x}$$
Derivative is:
$$f'(x) = \frac{x\cos (x) - \sin (x) + x^2}{x^2}$$
so, taking the limit at $\infty$ I got the value of $1$.
Looking at the graph I see infinite numbers of maximum ... | $f'(x) = \frac{x\cos (x) - \sin (x) + x^2}{x^2}$ can be extended to a continuous function on $\mathbb{R}$ (check that $\lim _{x \rightarrow 0}f'(x)=1$). Thus it is bounded on the interval $[-1,1]$.
To show that $f'$ is bounded on $\mathbb{R}$ it remains to show that it is bounded on $[1,\infty]$ and $[-\infty,-1]$.
Let... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then $x$ is not integral over $S$
Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then I want to show that $x$ is not integral over $S$
I'm not se... | Let $\bar{R} = R / rR$. Then $R[x] / rR[x] \cong \bar{R}[X]$ and the image $\bar{S}$ of $S$ in $\bar{R}[\bar{x}]$ is contained in $\bar{R}$ by assumption.
Suppose $x$ is integral over $S$. Then $X$ is integral over $\bar{S}$. Hence $X$ is integral over $\bar{R}$. The indeterminate $X$ is integral over $\bar{R}$ inside ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$ I want to show that for all $a \in \mathbb{R }$
$$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$
So far I've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i go on to rearrange this, i come after a few steps b... | This is essentially the inverse of
$$
e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n
$$
The sequence of functions $f_n(x)=\left(1+\frac xn\right)^n$ converge equicontinuously; simply note that $f_n'(x)=\left(1+\frac xn\right)^{n-1}\sim e^x$. Thus, we get
$$
\lim_{n\to\infty}n\left(e^{x/n}-1\right)=x
$$
which is the sam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve $7x^3+2=y^3$ over integers I need to solve the following
solve $7 x^3 + 2 = y^3$ over integers.
How can I do that?
| To solve this kind of equations, we have several 'tools' such as
using mod, using inequalities, using factorization...
In your question, using mod will help you.
Since we have
$$y^3-2=7x^3,$$
the following has to be satisfied :
$$y^3\equiv 2\ \ \ (\text{mod $7$}).$$
However, in mod $7$,
$$0^3\equiv 0,$$
$$1^3\equiv 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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every element of $V_{\omega}$ is definable My attempt by $\in$-induction. I am trying find formula that will work:
$N=(V_{\omega},\in)\models rank(\varnothing) =0<\omega$
Assume,given $x\in V_\omega$ that $\forall y\in x$ are definable too $N\models rank(y)<\omega$. Then since $x\in V_\omega$, $|x|<\omega\Rightarrow x$... | Let $N$ be an arbitrary structure of an arbitrary language. Recall that $n\in N$ is called definable (without parameters) if there exists a formula $\varphi(x)$ such that $N\models\varphi(u)\iff u=n$.
We want to show that in $V_\omega$ in the language including only $\in$, every element is definable. We do this by $\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$
Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$
Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$.
Where $\alpha$ is the solution for $\cos x=x$.
Hint: check that $(a_n)$ is a cauchy sequence and use Lagrange's theorem.
Well I tried to sh... | Note that
$$
\cos(x):\left[\frac1{\sqrt2},\frac\pi4\right]\mapsto\left[\frac1{\sqrt2},\frac\pi4\right]
$$
and on $\left[\frac1{\sqrt2},\frac\pi4\right]$,
$$
\left|\frac{\mathrm{d}}{\mathrm{d}x}\cos(x)\right|=|\sin(x)|\le\frac1{\sqrt2}\lt1
$$
Thus, the Mean Value Theorem ensures that $\cos(x)$ is a contraction mapping o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/628885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find all homomorphisms from a quotient polynomial ring $\mathbb{Z}[X] /(15X^2+10X-2)$ to $\mathbb{Z}_7$ I'm completely lost, what my problem is I don't get the gist of a quotient polynomial ring nor ANY homomorphisms between it and some $\mathbb{Z}_n$, much less ALL of them.
I know there is something to be done with a... | Hint: a ring homomorphism $\Bbb Z[X]/(\cdots)\to\Bbb Z/7\Bbb Z$ will be determined by where $X$ is sent. It can't be sent just anywhere; it still has to satisfy $15X^2+10X-2=0$ (does this have roots in $\Bbb Z/7\Bbb Z$?).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving that expression is equivalent to the definition of derivative Let $f$ be differentiable at $x=a$.
Prove that if $x_n \to a^+$ and $y_n \to a^-$ then:
$$\lim_{n\to \infty} \frac{f(x_n)-f(y_n)}{x_n-y_n}=f'(a).$$
Every option that I think about seems to my very trivial, so I believe that I am doing something wron... | First remark that the result is easy if you assume that $f$ is derivable with continuous derivative in a neighborhood of $a$. Indeed, by the mean value theorem, you can write $\frac{f(x_n)-f(y_n)}{x_n - y_n}$ as $f'(c_n)$ for some $c_n$ between $y_n$ and $x_n$ and the result follows by continuity of the derivative.
For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/629122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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1-st countability of a uncountable topological product Let $X = \displaystyle\prod_{\lambda \in \Lambda} X_{\lambda}$, where each $X_\lambda$ is $T_2$ and has at least two points. Prove that if $\Lambda$ is uncountable then $X$ is not 1st-countable.
I dont even know how to start proving. Maybe in the opposite direction... | Hint: Membership in a countable family of open sets depends only on countably many coordinates, as is being a superset of a member of such a family. Being a neighbourhood of a point in the product depends on $\lvert \Lambda\rvert$ many coordinates (due to each $X_\lambda$ being nontrivial).
In fact, if you choose your ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$ Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are self-duals, it is an abelian category itself and thus,... | I think that if Vopěnka's principle is true, then $\mathsf{Mod}(R)^{op}$ can't fully embed in $\mathsf{Mod}(S)$ for any non-zero $R$ and $S$ (all the references that follow are to "Locally Presentable and Accessible Categories" by Adámek and Rosický): If it did, then $\mathsf{Mod}(R)^{op}$ would be bounded (Theorem 6.6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/629357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Proving a determinant inequality Let $A$ be a square matrix in $M_n(\mathbb R)$. Prove that:
$$det(A^2+I_n) \ge 0$$
I wrote $A^2+I_n=A^2 I_n+I_n=I_n(A^2+1)$:
$$det(I_n)\cdot det(A^2+1)=det(A^2+1)$$
How can I prove that is $\ge 0$ ? Thank you.
| The problem in the OP approach is that we don't know what $A^2+1$ is.
It's better to use complex numbers and the relatioships:
*
*$A^2+I_n=(A-iI_n)(A+iI_n)$, and
*$\det(\overline B)=\overline{\det B}$ for any complex square matrix $B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does $\lim_{x\to 0^+} x^{\sin x}=1$? I tried writing $y=x^{\sin x}$, so $\ln y=\sin x\ln x$. I tried to rewrite the function as $\ln(x)/\csc x$ and apply l'Hopital to the last function, but it's a mess.
Is there a by hand way to do it?
| Write $x^{\sin{x}} = \exp{\frac{\log{x}}{\frac{1}{\sin{x}}}}$ and apply rule of l'hopital:
$$\lim_{x\rightarrow 0}x^{\sin{x}} = \exp{\left(\lim_{x\rightarrow 0}\frac{\log{x}}{\frac{1}{\sin{x}}}\right)}$$
Now (by 2$\times$ l'hopital)$$\lim_{x\rightarrow 0}\frac{\log{x}}{\frac{1}{\sin{x}}} = \lim_{x\rightarrow 0}\frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/629459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\displaystyle \int_0^1x^a \ln(x)^m \mathrm{d}x$ Find $$\int_0^1x^a \ln(x)^m\ \mathrm{d}x$$ where $a>-1$ and $m$ is a nonnegative integer. I did a subsitiution and changed this into a multiple of the gamma function. I get $(-1)^m m! e^a$ as the solution but Mathematica does not agree with me. Can someone confirm m... | $\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{\ds}[1]{\displaystyle... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simple proof Euler–Mascheroni $\gamma$ constant I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges.
At first I want to know if my answer is OK.
My try:
$\lim\limits_{n\to\infty} \left(\sum\limits_{k=1}^n \frac{1}{k} ... | Upper Bound
Note that
$$
\begin{align}
\frac1n-\log\left(\frac{n+1}n\right)
&=\int_0^{1/n}\frac{t\,\mathrm{d}t}{1+t}\\
&\le\int_0^{1/n}t\,\mathrm{d}t\\[3pt]
&=\frac1{2n^2}
\end{align}
$$
Therefore,
$$
\begin{align}
\gamma
&=\sum_{n=1}^\infty\left(\frac1n-\log\left(\frac{n+1}n\right)\right)\\
&\le\sum_{n=1}^\infty\frac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/629630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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"answer_id": 0
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How would you explain to a 9th grader the negative exponent rule? Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = \frac{1}{a^x}$ from the pattern method: $2^2 = 4, 2^1 = 2, 2^... | To a 9th grader, I would say "whenever you see a minus sign in the exponent, you always flip the number."
$$
2^{-3} = \frac{1}{2^{3}} = \frac{1}{8}
$$
I would simply do 10-20 examples on the board, and hammer the point until they start to get it.
You may have to review fractions with them here too.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
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Algebra 2-Factoring sum of cubes by grouping Factor the sum of cubes: $81x^3+192$
After finding the prime factorization of both numbers I found that $81$ is $3^4$
and $192$ is $2^6 \cdot 3$.
The problem is I tried grouping and found $3$ is the LCM so it would outside in parenthesis. The formula for the sum of cubes i... | $$81x^3+192 = 3 (27 x^3 + 64) = 3 ((3x)^3+4^3) \\= 3 (3x + 4) ((3x)^2 - 3x\cdot 4 + 4^2) = 3 (3x+4)(9x^2-12x+16)$$
Since $12^2-4\cdot9\cdot16$ does not have a nice square root further factorization is not possible.
| {
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"source": "stackexchange",
"question_score": "1",
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What is a supremum? I'm reading here about sequence of functions in Calculus II book,
and there's a theorem that says:
A sequence of functions $\{f_n(x)\}_0^\infty$ converges uniformly to $f(x)$ in domain $D$ $\iff$ $\lim_{n \to \infty} \sup_{x \in D} |f_n(x) - f(x)| = 0.$
I really serached a lot , in Google, Wikip... | supremum means the least upper bound. Let $S$ be a subset of $\mathbb{R}$
$$
x = \sup(S) \iff ~ x \geq y~\forall y \in S \mbox{ and } \forall \varepsilon > 0, x - \varepsilon \mbox{ is not an upper bound of } S
$$
You may also define $\sup(S) = +\infty$ when $S$ is not bounded above.
The reason why we have supremum ins... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rank of homology basis in Ahlfors' Complex Analysis In Ahlfors' Complex Analysis book Section 4.4.7, he decomposes the complement in the extended plane of a region $\Omega$ into connected components. He then constructs a collection of cycles $\gamma_i$ in $\Omega$, one for each component, such that every cycle $\gamma$... | Concerning your edit, yes, you can embed the integer homology group into the homology with rational coefficients. You can define chains with rational coefficients for the paths, a cycle is a chain with empty boundary (the boundary of a path is $\text{endpoint} - \text{startpoint}$, the boundary of a chain is the corres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/630138",
"timestamp": "2023-03-29T00:00:00",
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Is $\sin{(\log{x})}$ uniformly continuous on $(0,\infty)$? Is $\sin{(\log{x})}$ uniformly continuous on $(0,\infty)$?
Let $x,y \in (0,\infty)$.
$$
|f(x)-f(y)| = |\sin{\log{x}} - \sin{\log{y}}| = \left|2 \cos{\frac{\log{xy}}{2}}\sin{\log{\frac{x}{y}}{2}} \right| \leq 2 \left|\sin{\frac{\log{\frac{x}{y}}}{2}} \right| \l... | No it is not uniformly continuous because $\log x$ is not uniformly continuous on $(0,1)$.
If we take the sequence of numbers $(e^{-n})$ then $|e^{-n} -e^{-n-1}|$ tends to zero but $|\log(e^{-n})-\log(e^{-n-1})|=1$
| {
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Applying Open Mapping Theorem Let $X$ and $Y$ Banach spaces and $F: X \to Y$ a linear, continuous and surjective mapping. Show that if $K$ is a compact subset of $Y$ then there exists an $L$, a compact subset of $X$ such that $F(L)= K$.
I know by the Open Mapping Theorem that $F$ is open. What else can I do? Thank yoU!... | I suspect there must be a simpler proof for Banach spaces, but as I don't see one, here is what I came up with, using a faint recollection of the proof of Schwartz' lemma (simplified, since we're dealing with Banach spaces, not Fréchet spaces):
By the open mapping theorem, we know that there is a $C > 1$ such that
$$\b... | {
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calculation of Stefan's constant In the calculation of Stefan's constant one has the integral
$$J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx$$
which according to Wikipedia is equal to $\frac{\pi^4}{15}$.
In this page of Wikipedia there is a (long) method of calculation using the Taylor expansion of $f(k) ... | Writing the integral as
$$
J = \int_0^\infty \frac{x^3}{\mathrm{e}^{x} -1} \mathrm{d}x = \int_0^\infty \frac{x^3 \mathrm{e}^{-x} }{ 1 - \mathrm{e}^{-x}} \mathrm{d}x = \lim_{\epsilon \downarrow 0} \int_\epsilon^\infty \frac{x^3 \mathrm{e}^{-x} }{ 1 - \mathrm{e}^{-x}} \mathrm{d}x
$$
Now for $x >\epsilon$, $\left(1-\m... | {
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Help with a contour integration I've been trying to derive the following formula
$$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$
for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the way to go (and would prefer a solution by this method), but I keep getting stuck either procee... | Note that
$$|1+(x+i y)t|^2=1+2 x t +(x^2+y^2) t^2$$
The integral is therefore a straightforward application of the residue theorem, if you want. That is, evaluate
$$\int_{-\infty}^{\infty} \frac{dt}{1+2 x t +(x^2+y^2) t^2}$$
The poles are at $t_{\pm}=(-x \pm i y)/(x^2+y^2)$. If we close in the upper half plane with ... | {
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What's the explanation for why n^2+1 is never divisible by 3? What's the explanation for why $n^2+1$ is never divisible by $3$?
There are proofs on this site, but they are either wrong or overcomplicated.
It can be proved very easily by imagining 3 consecutive numbers, $n-1$, $n$, and $n+1$. We know that exactly one of... | one of $n-1,n$ or $n+1$ is divisible by $3$.
If it is $n$ then so is $n^2$.
If it is not $n$, then one of $n-1$ or $n+1$ is divisible by $3$, and hence so is their product $n^2-1$.
Thus, either $n^2$ or $n^2-1$ is a multiple of $3$. If$n^2+1$ would be a multiple of three, then one of $2=(n^2+1)-(n^2-1)$ or $1=(n^2+1)-... | {
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How to rewrite $7-\sqrt 5$ in root form without a minus sign? How to rewrite $7-\sqrt 5$ in root form without a minus sign ?
For clarity "root form " means an expression that only contains a finite amount of positive integers , additions , substractions , multiplications and root extractions (sqrt, cuberoot etc).
For ... | Let $n$ be some positive integer.
$(7-\sqrt 5)^n = r$ where $r$ is in the desired form but there is no root symbol over the entire expression on the RHS.
Now every root symbol in $r$ denotes a principal root.
The number of ways that we can write the same identity $r$ but replace one or more of its roots by a nonprincip... | {
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I flip M coins, my opponent flips N coins. Who has more heads wins. Is there a closed form for probability? In this game, I flip M fair coins and my opponent flips N coins. If I get more heads from my coins than my opponent, I win, otherwise I lose. I wish to know the probability that I win the game.
I came to this:
$$... | If you toss $M$ coins, there are ${M \choose m}$ ways to get exactly $m$ heads.
Then, the number of ways you can win when your opponent tosses $N$ coins, given that you've tossed $m$ heads, is
$${M \choose m}\sum_{i=0}^{m-1}{N \choose i}.$$
Now, just sum over all possible values of $m$, and divide by the total number o... | {
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Necessary condition for have same rank Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix.
Prove that $P$ and $Q$ have the same rank.
Some help with this please , happy year and thanks.
| Let $V$ be the vector space on which all these matrices act. First, note that $V = P(V) \oplus (I-P)V$ (and in fact, $P(V) = \ker (I-P),$ $(I-P)V = \ker P.$ Similarly for $Q$ instead of $P.$
Now, notice that the third condition states that for no vector is it true that $(I-P)v = Q v.$ This means that $\dim Q (V) \leq \... | {
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Direct proof of empty set being subset of every set Recently I finished my first pure mathematics course but with some intrigue about some proofs of definitions by contradiction and contrapositive but not direct proofs (the existence of infinite primes for example), I think most of them because the direct proof extends... | Yes, there's a direct proof:
The way that we show that a set $A$ is a subset of a set $B$, i.e. $A \subseteq B$, is that we show that all of the elements of $A$ are also in $B$, i.e. $\forall a \in A, a\in B$.
So we want to show that $\emptyset \subseteq A$. So consider all the elements of the empty set. There are non... | {
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Evaluate the limit $\lim_{x\to 0} \frac{(\tan(x+\pi/4))^{1/3}-1}{\sin(2x)}$ Evaluate the limit
$$\lim_{x\to 0} \frac{(\tan(x+\pi/4))^{1/3}-1}{\sin(2x)}$$
I know the limit is $1\over3$ by looking at the graph of the function, but how can I algebraically show that that is the limit. using this limit: $$\lim_{x \rightarro... | Use the definition of derivative: $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.$$ Second hint: multiply numerator and denominator by $x$.
| {
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Proving that two sets are diffeomorphic I have the following two sets
$\mathcal{S}= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$
and $\mathcal{S}' = \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = r \right\rbrace$
for some non-zero real number $r$.
I need to show t... | As you wrote yourself, your argument works only for $r\gt 0$.
If $r\lt 0$ the trick is to notice that the change of variables $x=Z,y=W,z=X, w=Y$ shows that $S$ is diffeomorphic to $Z^2+W^2-X^2-Y^2=1$ and thus to $x^2+y^2-z^2-w^2=-1$.
You can then apply your argument to show that you can replace the $-1$ on the right... | {
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Intersection of conics By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given smooth conics $Q_1$ and $Q_2$ (given by $q_1,q_2\in S^2V^*$ respectively). So, we are to find the se... |
First, is this number necessarily finite?
No: a degenerate conic factors into two lines. Two such degenerate conics might have a line in common. This line would be a continuum of intersections. Barring such a common component, the number of intersections is limited to four.
And second, how to describe such intersect... | {
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Book request: mathematical logic with a semantical emphasis. Suppose I am interested in the semantical aspect of logic; especially the satisfaction $\models$ relation between models and sentences, and the induced semantic consequence relation $\implies,$ defined by asserting that $\Gamma \implies \varphi$ iff whenever ... | The book "Éléments de Logique Mathématique" by Kreisel and Krivine (which I believe has an English translation, probably with the obvious title "Elements of Mathematical Logic") takes a fiercely semantical approach to the basic parts of mathematical logic. The material you don't want, about the axiomatic method and fo... | {
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Prove that $\lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $. My attempt:
We prove that $$\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $$
It is sufficient to show that for an arbitrary real number $\epsilon\gt0$, there is a $K$
... |
Is this proof correct? What are some other ways of proving this?
Thanks!
Your proof is correct with the caveat that you are a bit more precise about what $\epsilon$ and $K$ mean. Another way to prove this is using l'Hôpital's rule. Let $f(n)=23n+2$, $g(n)=4n+1$, then we can see that
$$
\lim_{n\rightarrow\infty} f(n... | {
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Square of Bernoulli Random Variable I was wondering about the distribution of the square of a Bernoulli RV. My background in statistics is not too good, so maybe this doesn't even make sense, or it is a trivial problem.
Let, $Z\sim X^2$, where $X\sim \text{Ber}(p)$.
$F_Z(z)=\Pr(X^2\leq z)$
$=\Pr(-z^{1/2}\leq X\leq z^{1... | If $X$ is Bernoulli, then $X^2=X$.${}{}{}{}{}{}$
| {
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Does this orbifold embed into $\mathbb{R}^3$? Let $X$ be the space obtained by gluing together two congruent equilateral triangles along corresponding edges.
Note that $X$ has the structure of a Riemannian manifold except at the three cone points. In particular, $X$ is a Riemannian orbifold.
Is there an isometric embe... | If you want the image to be convex, then no. Otherwise, my guess is that Kuiper's theorem gives you an embedding...
| {
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What is the purpose of universal quantifier? The universal generalization rule of predicate logic says that whenever formula M(x) is valid for its free variable x, we can prefix it with the universal quantifier,
M(x) ⊢ ∀x M(x).
But it seems then makes no sense. Why do you introduce a notion that does not mean anythin... | $M(x)$ doesn't strictly have a truth value; $\forall x.M(x)$ does. So, different beasts. Not to mention "$T\vdash M(x)$" is not a sentence in the language, while again, "$\forall x.M(x)$" is.
The universal generalization rule connects a piece of metatheory ("for any variable '$x$', '$M(x)$' is provable", or some such),... | {
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Fixed point of tree automorphism Given the tree $T$ and its automorphism $\phi$ prove that there exists a vertex $v$ such that $\phi(v)=v$ or an edge $\{{u,v\}}$ such that $\phi(\{{u,v}\}) = \{{u,v}\}$
| See this wikipedia article. The center is preserved by any automorphism. There is also the baricenter of a tree (which is a vertex for which the sum of the distances to the other points is minimized). Again, there is either one such or two adjacent ones. Are the center and the baricenter the same? For you to find out.
| {
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Can someone explain how this linear equation was solved [proof provided] Can someone please explain to me the steps taken in the proof provided to solve the linear equation? [1]: http://i.imgur.com/2N52occ.jpg "Proof"
What I don't understand is how he removed the denominator of both fractions (3,5 respectively). I know... | Yes. The proof just multiplied everything by $15$ which is the least common multiple of $3,5$.
Note that in this case $3\times 5=15$ is the least common multiple of $3,5$, but if you have two denominators $3,6$, then the least common multiple is $6$. ($3\times 6=18$ is not the least common multiple.)
In general, if you... | {
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Invertible function $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ How can I prove that $f(x) = \frac{x^3}{3} + \frac{5x}{3} + 2 $ is invertible.
First I choose variable $x$ for $y$ and tried to switch and simplified the function but I am stuck. Need some help please.
| Both $g(x)=\frac{x^3}3$ and $h(x)=\frac{5x}3+2$ are increasing functions. (This should be clear if you know graphs of some basic functions.)
Sum of two (strictly) increasing functions is again an increasing function, therefore $f(x)=g(x)+h(x)$ is strictly increasing.
If a function is strictly increasing, then it is inj... | {
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Ansatz of particular solution, 2nd order ODE Find the particular solution of $y'' -4y' +4y = e^{x}$
Helping a student with single variable calculus but perhaps I need some brushing up myself. I suggested y should have the form $Ce^{x}$. This produced the correct answer, but the solution sheet said the correct ansatz wo... | Exp[x] alone is a particular equation of the ODE. On the othe hand, the canonical equation has two identical roots corresponding to r=2. Then the general equation will contain a term Exp[2x] and, because of the degeneracy a term x Exp[2x].
So, the general solution of the ODE is y[x] = Exp[x] + (C1 + C2 x) Exp[2 x]
| {
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Number of zeros equals number of poles The following is an old qualifying exam problem I cannot solve:
Let $f$ be a meromorphic function (quotient of two holomorphic functions) on an open neighborhood of the closed unit disk. Suppose that the imaginary part of $f$ does not have any zeros on the unit circle, then the nu... | This is not quite Rouche though everything in the end boils down to the Cauchy integral formula. To count zeros and poles you pass to the logarithmic derivative i.e. $\frac{f'}{f}$ and then integrate over the unit circle (which you are allowed to do since $f$ does not vanish there by hypothesis). Why you can take a co... | {
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Prove or disprove the implication: Prove or disprove the implication:
$a^2\cdot \tan(B-C)+ b^2\cdot \tan(C-A)+ c^2\cdot \tan(A-B)=0 \implies$
$ ABC$ is an isosceles triangle.
I tried to break down the left hand side in factors, but all efforts were in vain.
Does anyone have a suggestion?
Thank you very much!
| I believe I have an example in which the identity is true but the triangle is not isosceles. All the algebra in the following was done by Maple.
Let $a=1$ and $b=2$, and take $c^2$ to be a root of the cubic $x^3-5x^2-25x+45$. The cubic has two roots between $1$ and $9$, either of which will give a valid set of sides ... | {
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$\mathbb P $- convergence implies $L^2$-convergence for gaussian sequences Consider $(X_n)_{n \in \mathbb N}$ a sequence of gaussian random variables whose limit in probability exists and is given by $X$.
I was interested in showing that in this particular case we have always that this sequence converge also in $\mathb... | We have to show that for any subsequence $(X_{n_k})$ we can extract a further subsequence $(X_{n'_k})$ such $\mathbb E|X_{n'_k}-X|^p\to 0$. We can assume that $X_{n_k}\to X$ almost surely by passing to a further subsequence. Then $X$ is Gaussian.
Write $X_n=a_nN+b_n$, $X=aN+b$, where $N\sim N(0,1)$.
If $a\gt 0$, then... | {
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The number $n^4 + 4$ is never prime for $n>1$ I am taking a basic algebra course, and one of the proposed problems asks to prove that $n^4 + 4$ is never a prime number for $n>1$.
I am able to prove it in some particular cases, but I am not able to do it when $n$ is an odd multiple of $5$.
| $n^4 + 4 = (2 - 2 n + n^2) (2 + 2 n + n^2)$
| {
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Why are mathematical proofs that rely on computers controversial? There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some mathematicians. Why is it so?
I my opinion, shifting from manual... | I think a computer-assisted or computer-generated proof can be less convincing if it consists of a lot of dense information that all has to be correct. A proof should have one important property: a person reading it must be convinced that the proof really proves the proposition it is supposed to prove.
Now with most c... | {
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Complex numbers - proof Let z and w be complex numbers such that $|z| = |w| = 1$ and $zw \neq -1$. Prove that $\frac{z + w}{zw + 1}$ is a real number.
I let z = a + bi and w = c+ di so we have that $\sqrt{a^2+b^2} = \sqrt{c^2+d^2} = 1$ and $a^2+b^2 = c^2 + d^2 = 1$. I plugged it into the equation but I didn't really ge... | Since $|z|=|w|=1$,
$$z\bar z=w\bar w=1.$$
Letting $u$ be the given number,
$$u=\frac{z+w}{zw+1}=\frac{(1/\bar z)+(1/\bar w)}{zw+1}=\frac{\bar w+\bar z}{\bar z\bar w(zw+1)}=\frac{\bar z+\bar w}{\bar z\bar w+1}=\bar u.$$
| {
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Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also
known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that
$|f'(x)| \le A/2$ for every $x \in [0,1]$.
I'll explain what I did so far. First using Rolle's theorem, there i... | If $f''$ exists on $[0,1]$ and $|f''(x)| \le A$, then $f'$ is Lipschitz continuous, which implies that $f'$ is absolutely continuous. So $f'(a)-f'(b)=\int_{a}^{b}f''(t)\,dt$ for any $0 \le a, b \le 1$. And, of course, the same is true of $f$ because $f$ is continuously differentiable.
Because $f(0)=0$, one has the foll... | {
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What if $\operatorname{div}f=0$? Say, we have a function $f\in C^1(\mathbb R^2, \mathbb R^2)$ such that $\operatorname{div}f=0$. According to the divergence theorem the flux through the boundary surface of any solid region equals zero.
So for $f(x,y)=(y^2,x^2)$ the flux through the boundary surface on the picture (sor... | The divergence theorem is a statement about 3-dimensional vector fields, the 2-dimensional version sometimes being called the normal version of Green's theorem. In your second example, the vector field
$$g(x,y) = \left( -\frac{x}{x^2+y^2}, -\frac{y}{x^2+y^2} \right)$$
is not even defined at the origin, which is why Gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/632912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, is it true that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$. If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, p,rove or disprove that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$.
I think it is true. It is easy ... | Take $X= [0,\infty]$ then define take $f_n=n^2 \chi_{[0, \frac{1}{n}]}$
then you $\lim _{n\rightarrow \infty} \int _{X} f_n =\infty$ but $f = 0$ almost everywhere so
$\int _{X} f =0$ to fix that define $\tilde{f}_n =n^2 \chi_{[0, \frac{1}{n}]} + \chi _{[1,\infty]} $. Now $\tilde {f} = \chi _{[1, \infty]}$ almost everyw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/632957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
prove that $a^{nb}-1=(a^n-1)((a^n)^{b-1}+...+1)$ Prove that $a^{nb}-1=(a^n-1)\cdot ((a^n)^{b-1}+(a^n)^{b-2}+...+1)$
We can simplify it, like this:
$$(a^{n})^b-1=(a^n-1)\cdot \sum_{i=1}^{b}(a^{n})^{b-i}$$
How can we prove this?
| $$
\begin{align}
&(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+\dots+x+1)\\
=&\quad\,x^n+x^{n-1}+x^{n-2}+x^{n-3}+\dots+x\\
&\quad\quad\;\:\,-x^{n-1}-x^{n-2}-x^{n-3}-\dots-x-1\\
=&\quad\,x^n\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;-1
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/633035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Simple pendulum as Hamiltonian system I am unable to understand how to put the equation of the simple pendulum in the generalized coordinates and generalized momenta in order to check if it is or not a Hamiltonian System.
Having
$$E_T = E_k + E_u = \frac{1}{2}ml^2\dot\theta^2 + mgl(1-cos\theta)$$
How can I found what a... | The Lagrangian is
$${\cal L}=\frac{1}{2}ml^2\dot{\theta}^2-mgl(1-\cos\theta).$$
The conjugate momentum is
$$p_\theta=\frac{\partial{\cal L}}{\partial\dot{\theta}}=ml^2\dot{\theta}$$
and so the Hamiltonian is
$${\cal H}=\sum_q \dot{q}p_q-{\cal L}=\frac{1}{2}ml^2\dot{\theta}^2+mgl(1-\cos\theta)=\frac{p_\theta^2}{2ml^2}+m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Where is the world of Imaginary numbers? Complex numbers have two parts, Real and Imaginary parts.
Real world is base of Real numbers.
but
where is (or what is) the world of Imaginary numbers?
| Complex numbers can be thought of as 'rotation numbers'. Real matrices with complex eigenvalues always involve some sort of rotation. Multiplying two complex numbers composes their rotations and multiplies their lengths. They find frequent use with alternating current which fluctuates periodically. Also, $e^{ix}=\cos x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to evaluate the following summation I am trying to find the definite integral of $a^x$ between $b$ and $c$ as the limit of a Riemann sum (where $a > 0$):
$I = \displaystyle\int_b^c \! a^{x} \, \mathrm{d}x.$
However, I'm currently stuck in the following part, in order to find S:
$S = \displaystyle\sum\limits_{i=1}^n... | Note that:
$$\begin{align}
S &= \sum\limits_{i=1}^n \displaystyle{a^{\displaystyle\frac{i(c-b)}{n}}} \\
&= \sum\limits_{i=1}^n \left(a^{\left(\dfrac{(c-b)}{n}\right)}\right)^i
\end{align}$$
Now, we can use the finite form of the geometric series formula:
$$S = \frac{\left(a^{\left(\dfrac{(c-b)}{n}\right)}\right)-\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
What does this modular relation mean, if anything?
Let $x$ and $y$ be real numbers.
Suppose $\dfrac{x}{y \bmod x}$ is a natural number.
What does that say about the relationship between $x$ and $y$?
If $x$ and $y$ are naturals themselves, then I think it means that $x$ is some multiple of $y$ plus some divisor of $y$... | I think it means that $y$ is $x$ divided by an integer plus some integral multiple of $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/633320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to compute $\lim_{n\to \infty}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{n^2+n-k^2}}$ Find this follow limit
$$I=\lim_{n\to \infty}\sum_{k=1}^{n}\dfrac{1}{\sqrt{n^2+n-k^2}}$$
since
$$I=\lim_{n\to\infty}\dfrac{1}{n}\sum_{k=1}^{n}\dfrac{1}{\sqrt{1+\dfrac{1}{n}-\left(\dfrac{k}{n}\right)^2}}$$
I guess we have
$$I=\lim_{n\to... | The difference term-wise is
\begin{align}
\frac{1}{ \sqrt {1+({\frac{k}{n}})^2}} - \frac{1}{ \sqrt {1 + \frac{1}{n} - {(\frac{k}{n}})^2}}&= \frac{\frac{1}{n}}{ \sqrt {1+({\frac{k}{n}})^2}\cdot \sqrt {1 + \frac{1}{n} - ({\frac{k}{n}})^2} ( \sqrt {1+({\frac{k}{n}})^2}+ \sqrt {1 + \frac{1}{n} - ({\frac{k}{n}})^2}) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Compute $\lim_{n \rightarrow \infty} \left(\left(\frac{9}{4} \right)^n+\left(1+\frac{1}{n} \right)^{n^2} \right)^{1/n}$ may someone show how to compute $\lim_{n \rightarrow \infty} \left(\left(\frac{9}{4} \right)^n+\left(1+\frac{1}{n} \right)^{n^2} \right)^{\frac{1}{n}}$?
According to W|A it's e, but I don't know even ... | Clearly,
$$
\left(1+\frac{1}{n}\right)^{\!n^2}<
\left(\frac{9}{4}\right)^n+\left(1+\frac{1}{n}\right)^{\!n^2}<2\,\mathrm{e}^n,
$$
and therefore
$$
\left(1+\frac{1}{n}\right)^{\!n}<
\left(\left(\frac{9}{4}\right)^n+\left(1+\frac{1}{n}\right)^{n^2}\right)^{1/n}\le 2^{1/n}\mathrm{e}
$$
which implies that
$$
\mathrm{e}=\li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What is the name of this function $f(x) = \frac{1}{1+x^n}$? $f(x\in\mathbb{R}) = \frac{1}{1+x^n}$
| In the particular case where $n$ is even, this looks like the pdf of the Cauchy distribution, so you might want to say that $f(x) = \frac{1}{1+x^{2n}}$ is some kind of a generalized Cauchy...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/633573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$ Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$?
Additionally, what are any solutions for $f\left(\dfrac{1}{x}\right)-f(x+1)=x$?
| A few hints that might help...
*
*$1/x = x+1$ when $x = \frac{\pm\sqrt{5}-1}2$
*Differentiating gives: $-\frac{f'(1/x)}{x^2}+f'(1+x)=1$
*Differentiating again gives: $f''(1+x)+\frac{f''(1/x)}{x^4}+\frac{2f'(1/x)}{x^3}=0$ - this can then be continued.
*An "analytic function" has a Taylor series at any point that i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Prove $e^x, e^{2x}..., e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$ Prove $e^x, e^{2x}..., e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$
isn't it suffice to say that $e^y$ for any $y \in \mathbb{R}$ is in $\mathbb{R}^+$
Therefore, there aren't $\gam... | The exercise is
$$
f\alpha = \left\{
\begin{array}{ll}
\mathbb{R}\rightarrow\mathbb{R} \\
\ t\mapsto e^{\alpha t}
\end{array}
\right.
$$ Prove $(f_\alpha)_{\alpha \in\mathbb{R}} $is linear independent.
Let $(f_{\alpha_k})_{1\leq k \leq n} $ a finite number of vectors as $\alpha_1<\alpha_2<...<\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 0
} |
How to calculate complicated geometrical series? I have a geometrical series (I don't know if its geometrical series or not):
$$
\sum_{n=1}^{\infty }n\rho ^{n}(1-\rho)
$$
how can I simplify it ? ( assume that $ 0 \le \rho \le 1$ )
The last answer in my calculatio should be $\frac{\rho}{1-\rho}$. But I really don't know... | $$\sum_{n=1}^{\infty }n\rho ^{n}(1-\rho)=(1-\rho)\rho\frac{d(\sum_{n=1}^{\infty }\rho^n)}{d \rho}$$
Now using Infinite Geometric Series $$\sum_{n=1}^{\infty }\rho^n=\frac \rho{1-\rho}=-1-\frac1{\rho-1}$$ as $|\rho|<1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/633784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Sum of weighted squared distances is minimized by the weighted average? Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights.
In some paper I came across the following equation:
$$\arg\min_{c \in \mathbb{R}^... | Let $f(x) = \sum_i w_i (c - x_i) \cdot (c - x_i)$. Then the partial derivative of $f$ wrt $c_j$ is
$$
2\sum_i w_i (c - x_i)\cdot e_j
$$
where $e_j$ is the $j$th standard basis vector. Setting this to zero gives
$$
\sum_i w_i (c_j - x_{i,j}) = 0 \\
c_j\sum_i w_i = \sum_i w_i x_{i,j} \\
c_j= \frac{\sum_i w_i x_{i,j}}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Lower bound on the size of a maximal matching in a simple cycle Let $C_n$ denote an undirected simple cycle of $n$ nodes. I want to determine a lower bound on the size of a maximal matching $M$ of $C_n$.
Please note: A subset $M$ of the edges in $C_n$ is called a matching $\Leftrightarrow$ every node $v\in C_n$ is inci... | A maximal matching can have less than $n/2$ edges. Imagine a $C_6$ with the vertices named in order $a,b,c,d,e,f$. Then, the $ab$ and $ed$ edges form a maximal matching, and here it's $n/3$ edges.
So what's the most stupid way of choosing your matching ?
Let $X$ denote the set of vertices of $C_n$ that are incident t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/633947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that there does not exist an integer $n\in\mathbb{N}$ s.t $\phi(n)=\frac{n}{6}$ Show that there does not exist an integer $n\in\mathbb{N}$ s.t $$\phi(n)=\frac{n}{6}$$.
My solution:
Using the Euler's product formula:
$$\phi(n)=n\prod_{p|n}\Bigl(\frac{p-1}{p}\Bigr)$$
We have:
$$\frac{\phi(n)}{n}=\prod_{p|n}\Bigl(\fr... | Let $n=p_{1}^{a_1}\cdots p_{r}^{a_r}$, so $\;\;\phi(n)=p_{1}^{a_{1}-1}(p_{1}-1)\cdots p_{r}^{a_{r}-1}(p_{r}-1)$.
If $n=6\phi(n)$, then $\;\;p_{1}^{a_1}\cdots p_{r}^{a_r}=6\big[p_{1}^{a_{1}-1}(p_{1}-1)\cdots p_{r}^{a_{r}-1}(p_{r}-1)\big]$, so
$\;\;p_{1}\cdots p_{r}=6(p_1-1)\cdots(p_{r}-1)=2\cdot3(p_1-1)\cdots(p_{r}-1)$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/634012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
calculator issue: radians or degrees for inverse trig It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
| It helps to understand that there are several different functions called cosine. I find it useful to refer to "cos" (the thing for which $\cos^{-1}(0) = \pi/2$) and "cosd" for which cosd(90) = 0.
Your calculator (if you're lucky) will mean "cos" when you press the "cos" button; if you've got the option of "degrees/ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/634076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
PDF of a sum of exponential random variables Let $X_i$ for $i=1,2,...$ be a sequence of i.i.d exponential random variables with common parameter $\lambda$. Let $N$ be a geometric random variable with parameter $p$ that is independent of the sequence $X_i$. What is the pdf of the random variable $Y=\Sigma_{i=1}^N X_i$.
| We can also answer this with the following consideration:
The expected value of $Y$ is
$$E(\sum_{i=1}^N T_i) = E_{geom}\left(E_{exp}\left(\sum_{i=1}^N T_i | N\right)\right) = \frac{1}{p\lambda}.$$
So if $Y$ is exponentially distributed, it is so with parameter $p\lambda$. That is, we are left with the need to prove it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/634158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
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