Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Classification of real algebra with zero divisors (subquotient of Clifford algebra?) Consider the even-dimensional real vector space $\Bbb R^{2N}$. We construct the algebra as follows:
*
*Pick a basis in this space.
*Partition the $2N$ basis elements into $N$ pairs of zero divisors, called $e_1, \hat e_1, e_2, \hat... | The algebra is a quotient of a polynomial algebra:
$$
A_N=\Bbb{R}[x_1,y_1,x_2,y_2,…,x_N,y_N]/\langle x^2_i=x_i,y^2_i=y_i,x_iy_i=0\rangle,
$$
and $\dim(A_N)=3^N$. In fact, the non zero monomials in that algebra can be reduced to monomials where every variable has at most power one, and if $x_i$ is in that monomial, th... | {
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Are $f(x)$ and $f(x+ \delta x)$ the same after Taylor series expansion? According to 15.2.1 from https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/lectfiniteDifference.pdf, the Taylor series of u(x) can be written as
However, according to wikipedia, the Taylor series is
The difference is in $\delta x$. ... | In the second formula:
$f(x)_{about\space x=a}= f(a) + \dfrac{f'(a)}{1!}(x-a) + \dfrac{f''(a)}{2!}(x-a)^2+\dfrac{f'''(a)}{3!}(x-a)^3$
Replace $x \rightarrow x+\Delta x$ & $a \rightarrow x$ to get:
$f(x+\Delta x)_{about\space x=x}= f(x) + \dfrac{f'(x)}{1!}(\Delta x) + \dfrac{f''(x)}{2!}(\Delta x)^2+\dfrac{f'''(x)}{3!}(\... | {
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How is the chain method used in finding this derivative?
Find the derivative of $\tan^3[\sin(2x^2-17)]$.
Sorry if my question is a little too specific but I am confused on this trig equation. After completing the derivative I was wondering why does the $3tan^2$ not distribute to $sec^2$? Is there a rule for this? Ho... | You're probably getting confused by trying to do too much at once. Rather than doing the whole calculation in a single step, apply them one at a time. For example, apply the power rule
$$ \mathrm{d}(u^n) = n u^{n-1} \mathrm{d}u $$
to get
$$ \mathrm{d}\left( \tan^3[\sin(2x^2-17)] \right)
= 3 \left( \tan[\sin(2x^2-17)] \... | {
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Line inside the hyperboloid going through a point of a circle I am having trouble with an absurdly simple problem. It has been a long time since I last dealt with this kind of problem.
Consider the one-sheet hyperboloid given by
$$
x^2+y^2-a^2z^2=c^2
$$
and let $(X,Y,0)$ be a point of the cir... | You're nearly there. Substituting your expressions for $x, y, z$ into the equation for the hyperboloid gives an polynomial in $A, B, C, t$.
$$(A t + X)^2 + (B t + Y)^2 - a^2 (C t)^2 = c^2.$$
Since all points on the line must be contained in the hyperboloid, this equation must hold for all times $t$, we can collect and ... | {
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A rank one matrix can be written in a special form Any rank one matrix can be written in the form $uv^{t}$, where $u,v$ are column vectors considered as matrices and $t$ denotes transposition.
Why? How?
| One way to see this is with SVD. This is overkill, though.
Another approach: suppose that $A$ is a rank $1$ matrix. Then $A$ has at least one non-zero row. Call this row $v^T$. Every row of $A$ must be a multiple of this row. That is, there exist coefficients $u_i$ such that the rows of $A$ are exactly $u_1v^T,u_2v^T,\... | {
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How combine two inequalities (complex numbers)? This is from a book. I don't understand how the inequalities are combined to one inequality. Are they added/subtracted?
$z_1$ and $z_2$ are complex numbers. We have the inequalities
$$\lvert z_2\rvert -\lvert z_1\rvert \leq \lvert z_2-z_1\rvert $$
$$\lvert z_1\rvert... | We have $|x|\leq y$ iff $-y\leq x\leq y$ and with
$$\lvert z_2\rvert -\lvert z_1\rvert \leq \lvert z_2-z_1\rvert $$
$$\lvert z_1\rvert -\lvert z_2\rvert \leq \lvert z_1-z_2\rvert $$
have
$$-\lvert z_1-z_2\rvert\leq \lvert z_1\rvert -\lvert z_2\rvert \leq \lvert z_1-z_2\rvert $$
iff
$$\Big|\lvert z_1\rvert -\lvert z_2\... | {
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Why is $ \emptyset$ considered a set? My question is short and concise. Here it goes -
In my book the definition of a set is given as a well defined collection of things and in mathematicse they are well defined collection of mathematical objects. Then why is $\emptyset$ which has nothing is even considered as a set. I... | The existence of the empty set is one of the Zermelo-Frankel axioms of set theory.
One can argue whether or not the concept of the "empty set" violates one's intuition. I can give you an intuitive argument to say that it is not a violation, along the following lines: "Think of a set as the contents of a bag. Just becau... | {
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A proof in real analysis If we let $f$ and $g$ be Riemann integrable functions on $[a,b]$ and $c\in \mathbb{R}$ be a constant.
I need to show that $$\displaystyle \int_a^b cf(x)\,dx=c\int_a^b f(x)\,dx.$$
My idea here was to consider both cases, when $c>0$ and when $c<0$.
Case 1:
Suppose $c>0$. Let $P$ be a partition.
... | Your proof is fine. You'd be having the same results, had you used Riemann sums.
| {
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evaluate $lim_{n \to \infty}\left (\frac{1+\sqrt{3}i}{2}\right)^n$
$$\lim_{n \to \infty} \left(\frac{1+\sqrt{3}i}{2}\right)^n$$
In general we look at $\lim(x+yi)$ as $\lim(x,y)$ which we obviously can not here.
So I looked at $$\lim_{n \to \infty} \frac{\left(1+\sqrt{3}i\right)^n}{2^n}$$
$1+\sqrt{3}i=2e^{\frac{\pi i}... | Hint: if $\,\omega=\cfrac{1 + i \sqrt{3}}{2}\,$ then $\omega^2=-\overline{\omega}\,$, $\omega^3 = -1\,$ and $\omega^6=1\,$, so the sequence $\omega^n$ is periodic and non-constant, therefore the limit doesn't exist.
| {
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Prove that $\lim_{x\to\infty} (\ln x) = \infty$ Can someone help me prove that the function $\ln(x)$ diverges to infinity as $x$ approaches infinity. I tried using the definition to show that $\lvert \ln(x) -∞ \rvert < \epsilon $ where $\epsilon > 0$ and there exists a number $N$ which is in the set of natural numbers ... | I guess you want an analysis using the definition of divergence at infinity.
Let $M > 0$; and note that $\log x > M$ if $x > e^{M}$.
This shows that for every $M > 0$ there is some $X > 0$ (take $X := e^{M}$, say) such that $x > X$ implies $\log x > M$.
| {
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A general topology textbook for a specific purpose and taste, from a specific set of choices I'm not much interested in algebraic/differential/geometric topology as I'm more geared towards analysis. A solid foundation for general topology (aka point-set topology) would do for now. I can't decide on which one to choose ... | You won't like Willard, it's for serious students of point set topology with little devoted to spaces analyists use.
| {
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Understanding the Existence and Uniqueness of the GCD Definitions
For $a,b \in \mathbb{Z}$, a positive integer $c$ is said to be a common divisor of $a$ and $b$ if $c\mid a$ and $c\mid b$.
$c$ is the greatest common divisor of $a$ and $b$ if it is a common divisor of $a,b$ and for any common divisor $d$ of $a$ and $b$,... | We are asked to show the existence and uniqueness of the GCD denoted as $c$ of two integers $a,b$. There are two parts of of this proof: Showing the existence and showing the uniqueness. To show the existence we must show there is a $c$ that divides $a,b$ and for any common divisor $d$ of $a,b$, $d|c$.
Part I: Existenc... | {
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Use Rouche`s theorem to prove # of zeros For a fixed $\lambda$ satisfying $\vert\lambda\vert$ < 1, show that $(z - 1)^n e^z + \lambda (z + 1)^n$ has
n zeros in the right half-plane, which are all simple if $\lambda \not=$ 0.
I would really appreciate any help.
| You took a semicircle in the right half plane. Good idea!
$$
(z - 1)^n e^z + \lambda (z + 1)^n=0\iff \left(\frac{z-1}{z+1}\right)^n+\lambda e^{-z}=0,$$
since $z+1\ne 0$ for $z$ with $\operatorname{Re}z>0.$
Let $f(z)=\left(\frac{z-1}{z+1}\right)^n,$ $g(z)=\lambda e^{-z}.$
Take a semicircle in the right half plane wi... | {
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prove metric space X isn't isometric to any subspace of $\Bbb E^n$ for any $n$ Let $X={A,B,C,D}$ with $d(A,D)=2$, but all other distances equal to 1. $d$ Is a metric. Prove that metric space $X$ is not isometric to any subset of $\Bbb E^n$, for any $n$.
I've only managed to prove that it's not an isometry when $n=1$. ... | Assume the contrary that there is $\Phi : X=\{A, B, C, D\} \to \mathbb E^n$ so that $d(x, y) = |\Phi(x)- \Phi(y)|$. Call $a = \Phi(A)$ (and similarly for $B, C, D$).
Note that $a, b, c$ and $b, c, d$ forms two equilateral triangles with side length one and sharing the same side $bc$. So the largest possible distance ... | {
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Attempt to prove the $∀d∀x∀y (d | x ∧ d | y ∧ x ≤ y → d | y- x)$ property of the “divides” relation for non-negative integers I am attempting to prove the $∀d∀x∀y (d | x ∧ d | y ∧ x ≤ y → d | y- x)$ property of the “divides” relation for non-negative integers, but am having a little difficulty and am hoping someone can... | The key formula you will have to use is:
$\forall x \forall y \forall z (z \not = 0 \rightarrow (x\cdot z \le y\cdot z \rightarrow x \le y))$
So, you will need to first consider the special case where $a=0$, but in that case you can easily show that it must be the case that $a1=0$ and $a2=0$, and it is also easy to sho... | {
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Prove that $2^{1/n}$ is irrational Proof by contradiction, Assume $2^{1/n}$ is rational so:
$$2^{1/n} = \frac ab $$
where a,b have no common factors.
$$2 = \frac{a^n}{b^n}$$
$2$ divides LHS, therefore $2$ divides RHS
so $2$ divides $a^n$ or $2$ divides $b^n$ which implies $2$ divides $a$ or $2$ divides $b$.
Stuck on w... | Factor $a$ and $b$ into products of primes.
We have the identity $2b^n = a^n$; compare the exponents of the primes on both sides of the equation (and look in particular at the exponent of 2).
| {
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How prove this inequality $\sum\limits_{cyc}\sqrt{\frac{yz}{x^2+2016}}\le\frac{3}{2}$ Given $x,y,z$ are positive real number satisfy $xy+yz+xz=2016$. Prove that $$\sqrt{\dfrac{yz}{x^2+2016}}+\sqrt{\dfrac{xy}{z^2+2016}}+\sqrt{\dfrac{xz}{y^2+2016}}\le\dfrac{3}{2}$$
I tried
$\sqrt{\frac{yz}{x^2+2016}}=\sqrt{\frac{yz}{x^... | I believe there is something called the Purkiss Principle which would imply that in this case the maximum of $f$ is achieved when $x=y=z=\sqrt{2016/3}$. Thus, $$f(\sqrt{2016/3},\sqrt{2016/3},\sqrt{2016/3}) = 3/2.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\int_{|z-2i|=1}\frac{(z-2i+\frac{1}{2})^2 \sin(2i\pi z)}{\overline{z}^{2} (z-2i)^4} \ dz$ Let $C$ be the circle $|z -2i|=1$
How Can I Compute this Integral :
$$\int_{C}\frac{(z-2i+\frac{1}{2})^2 \sin(2i\pi z)}{\overline{z}^{2} (z-2i)^4} \ dz$$
Thank you
| I have tried solving this using cauchy integral formula as follows :
$|z-2i|= 1\Rightarrow (z-2i)(\overline{z}+2i)=1 \Rightarrow (z-2i+\frac{i}
{2}-\frac{i}{2})(\overline{z}+2i)=1$
Now we have : $$(z-2i+\frac{i}{2}-\frac{i}{2})(\overline{z}+2i)=1 \Rightarrow(z-2i+\frac{i}{2})(\overline{z}+2i)=\frac{i \overline{z}}{2}$... | {
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Given two lines to find their intersection. I will fully disclose that this is a homework question. I would prefer not to be given an answer directly, and am looking for more of an indication as to whether I am on the right track. The problem with the courses I am working with is that they only show examples, and do no... | Here is another way
to find the distance
between two lines
in any number of dimensions.
If the lines intersect,
the distance will be zero.
Find shortest distance between lines in 3D
| {
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Universal property of images in category theory
Let $\mathcal{A}$ be an additive category with all kernels and cokernels and $f:A\to B$ a morphism. If $e:B\to \text{coker}(f)$ is the canonical epimorphism, define $\text{im}(f):=\ker(e)$, with a canonical monomorphism $i:\text{im}(f)\to B$. Prove that:
$1)$ There is a ... | This is not true in general. For instance, let $\mathcal{A}$ be the category of torsion-free abelian groups. This is an additive category with kernels and cokernels (to form a cokernel, first take the cokernel in $Ab$ and then mod out the torsion subgroup). Now consider the map $f:\mathbb{Z}\to\mathbb{Z}$ given by m... | {
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Different way's of approaching the order of an element When I was making one of the assignments from the book 'Groups and Symmetry' by M. A. Armstrong, I got a little confused about the order of an element of a group.
First there was exercise 4.2, where you had to find the order of each element of the group $\mathbb{Z}... | In the first group, the identity element is $0$ where as in the second group the identity element is $1$. The order of an element $g \in G$ is the smallest $n \in \mathbb{N}$ such that $g^n=e$ where $e$ is the identity element of $G$.
Note: In Example-1, $e=0$ and the operation is addition. So $g^n$ turned out to be $... | {
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Pairing $2n$ real numbers Let $\{l_1,l_2,\dots,l_{2n}\}$ be a set of real numbers.
I need to divide those numbers into -$n$- pairs such that the sum of their multiplications (of each pair) will be as maximum as possible.
I know that for $\{1,2,3,4,5,6,7,8\}$ the best pairing is: $(8,7),(6,5),(4,3),(2,1)$ because $8·7... | It's enough to observe any pairing that includes pairs $(a,b)$ and $(c,d)$ with $a > c > b > d$ is suboptimal: $(a,c)$ and $(b,d)$ would be better. This follows from the rearrangement inequality or, more directly, because $$(ac + bd) - (ab + cd) = a(c-b) + (b-c)d = (a-d)(c-b) > 0.$$
As a consequence, assuming $l_1 \le ... | {
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Infinite group theory recommendations What is a good book to start a journey in the field of infinite group theory ? I have already taken a first course in algebra where we studied the most important (finite) algebraic structures and I'm taking the second course so I'm used to the basic tools of abstract algebra, howev... | I can recommend the two volumes of Derek Robinson Finiteness Conditions and Generalized Soluble Groups (Part 1 and Part 2) which are probably discontinued, but possibly available in a math library near you. They are an excellent source to start with. Part 2/Contents I found on-line in .pdf format. Note that a lot of re... | {
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Basis for $\mathbb{Q}(\alpha, \beta)$ over $\mathbb{Q}$ One can prove that a basis for $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ is the set $\{1, \sqrt{2}, \sqrt{3}, \sqrt{6} \}$. This got me wondering if the following is true:
Let $\alpha, \beta$ be elements that are (1) not rational and (2) not scalar multiples of each othe... | (Compiled from the comments and posted as CW in order to mark the question as answered.)
The proposition does not hold true in general. Counterexamples:
*
*finite extension: $\;\mathbb{Q}(\sqrt{2}, \sqrt[3]{2})$
*infinite extension: $\;\mathbb{Q}(\pi, \sqrt{2})$
| {
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Prove $\sum_{d|n} \frac{\Phi(d)}{d} = \prod_{i=1}^r (1 + a_i - \frac{a_i}{p_i})$ I want to prove $\sum_{d|n} \frac{\Phi(d)}{d} = \prod_{i=1}^r (1 + a_i(1 - \frac{1}{p_i}))$, where $\Phi(n)$ is the Euler phi function and given the prime factorisation $n = \prod_{i=1}^r p_i^{a_i} $.
My instinct says to use the Möbius inv... | Observe that with your factorization we get for example
$$\sum_{d|n} d =
\prod_{q=1}^r (1+p_q+p_q^2+\cdots+p_q^{a_q})$$
Now we have with the product ranging over prime divisors that
$$\frac{\varphi(d)}{d} = \prod_{p|d} \left(1-\frac{1}{p}\right).$$
Using the same scheme again we thus obtain
$$\sum_{d|n} \frac{\varphi(d... | {
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Number of possibilities to arrange n objects between m objects. I have m objects and insert n indistinguishable objects between them.
For example, with m=4:
n=1: {0,x,1,2,3},{0,1,x,2,3},{0,1,2,x,3} (3 ways)
n=2:{0,x,x,1,2,3},{0,1,x,x,2,3},{0,1,2,x,x,3},{0,x,1,x,2,3},{0,x,1,2,x,3},{0,1,x,2,x,3} (6 ways)
By counting and... | This can be represented exactly as a modified stars and bars problem, where the distinguished numbers play the role of bars and the indistinguishable objects the role of stars.
The only restriction is that we cannot have any "stars" on the left of the most-left bar and on the right of the most-right bar.
Let the numbe... | {
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Find $r$ given that: $M=aY + B(r-x)^{-c}$ I have an equation like so:
$M=aY + B(r-x)^{-c}$
Assuming all the variables are positive, how do I find $r$? I've worked it out to about this point:
$M(r-x)^c=aY(r-x)^c+B$
$r-x=\sqrt[c]{\frac{aY(r-x)^c+B}{M}}$
$r=\sqrt[c]{\frac{aY(r-x)^c+B}{M}}+x$
But I don't think this is ri... | $$M=aY + B(r-x)^{-c}\\\frac {M-aY}B=(r-x)^{-c}\\\frac B{M-aY}=(r-x)^c\\
\sqrt[c]{\frac B{M-aY}}=r-x\\x+\sqrt[c]{\frac B{M-aY}}=r$$
| {
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"Which answer in this list is the correct answer to this question?" I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct solution?
Which answer in this list is the correct answer ... | You can use propositional logic to formalize the problem, then satisfying assignments help to find the solutions.
Let $a,b,c,d,e,f$ represent the six sentences, respectively.
*
*$a\leftrightarrow b\land c\land d\land e\land f$
*$b\leftrightarrow \neg c\land \neg d\land \neg e\land \neg f$
*$c\leftrightarrow a\land... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2217248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "259",
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Finding a single equation without use of words, only functions and math, that solves outputs patterns within patterns If you were given a sequence that went something like
$$1,1,3,2,5,3,7,4,9,5....$$
You would notice that there are two patterns alternating, one being 1,3,5,7,9... and the other being 1,2,3,4,5...
How wo... | $$f(n)=\frac{n}{2-n+2\lfloor\frac{n}2\rfloor}$$
Not very elegant, though.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2217313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of real solutions.
Question : Let $\{a_i\}$ be a sequence of real numbers such that $0<a_1<a_2\cdots <a_n$. Show that the equation :
$$\frac{a_1}{a_1−x}+\cdots+\frac{a_n}{a_n−x}=2015$$
has exactly $n$ real solutions.
My try:
I know that this is an nth degree polynomial. But I really have no idea how to sh... | Hint: the LHS is continuous in each $(a_i,a_{i+1})$ interval. Note that for a small enough $\varepsilon$, $\dfrac{a_i}{a_i-x} \ll 0$ for an $x\in (a_i,a_i+\varepsilon)$ and $\dfrac{a_{i+1}}{a_{i+1}-x} \gg 0$ for an $x\in (a_{i+1}-\varepsilon,a_{i+1})$.
Hint2: The fact that the RHS is nonzero and the fact that the LHS g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How many ways, can sum be equal to 12 of 3 dice? I've tried solving it in the below method,
$$X_1+X_2+X_3 =12$$
Using Stars and Bars method, we've to restrict one star for all the three entries,
$$X_1+X_2+X_3=9$$
Now we have $\binom{11}{9}$ possibilities where each dice value will be atleast or greater than $1$.
Now I... | There are two small errors I can see:
*
*Where you subtract the ways in which each $X_i$ can be greater than or equal to $7$ you subtract $7$ when you should subtract $6$ giving $X_1+X_2+X_3=3$, this is because you have already put $1$ star into each bin at the start.
*Also you have only subtracted $1$ such case wh... | {
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"source": "stackexchange",
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Integrating $3^{2x}$ ($\int 3^{2x} dx $) I am tring to integrate the following: $3^{2x}$
My working has been shown as the following
$$\int 3^{2x} dx $$
I used the substitution $u = 2x$, so
$$\frac{du}{2} = dx $$
hence
$$\int 3^{2x} dx = \int 3^{u} \frac{du}{2} = \frac{1}{2}\int 3^{u} du $$
However, I can't go any fur... | Hint: $\displaystyle\int{a^{mx}}\ dx=\dfrac{a^{mx}}{m\cdot\ln a}+c$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2217754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Let $G$ be a group and $K = \{x^2 \mid x\in G\}$
Let $G$ be a group such that $K = \{x^2 \mid x\in G \}$ is a subgroup of $G$.
(a) If $H$ is a subgroup of $G$ with index $2$ show that $K\subset H$.
(b) Show that the number of subgroups in $G$ with index $2$ is equal to
the number of subgroups in $G/K$ with index $2$... | Your proof of point a) is correct albeit that $x^2H \neq xH$ should be explained. Indeed $x^2 = xh \implies x = h \implies x \in H$, contradiction. For the proof of b) we first have to prove that $K$ is normal, the rest is simply a consequence of the "fourth" isomorphism theorem (see point nr. 3). We have $g^{-1}kg =... | {
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"source": "stackexchange",
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Simple examples of preorders that are not partial orders? What are some simple examples of preorders — that is, binary relations that are reflexive and transitive — that are not partial orders (and hence not total orders, either)?
I'm looking for a couple of examples that do not involve graph theory or other less basic... | In this answer I do not provide simple examples of preorders, but enable you to find them on base of simple examples of partial orders.
You can just start with a partial order $\langle B,\leq\rangle$ and a surjective function $\nu:A\to B$.
Then $\preceq$ defined by: $$x\preceq y\iff \nu(x)\leq \nu(y)$$ is a preorder o... | {
"language": "en",
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"source": "stackexchange",
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Show that the vector space of real functions is not finitely spanned Here's what I've come up with. Let $\mathcal B$ be a generating set of $\mathcal F(R)$. Let us assume that $\mathcal B$ is a finite set, thus: $\lt \mathcal B \gt = \lt \{b_1, ..., b_n \}\gt, n \in N$. Let $\mathcal f \in F(R).$ Thus: $\mathcal f = ... | I think I understand the spirit of your proof, but it doesn't seem rigorous.
What you are trying to prove is equivalent to saying that $\mathcal{F}(R)$ is an infinite-dimensional vector space (if $\mathcal{F}(R)$ were finite dimensional, any basis would be a finite spanning set). Therefore if we could display an infini... | {
"language": "en",
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Relationship between Spin(3), SU(2), unit quaternions, and SO(3) There may be some short-hand / informal statements that are tripping me up, but I am getting confused trying to understand the relationship between Spin(3), SU(2), SO(3), and the unit quaternions.
Trying to find information online, many discussions seem t... | $Spin(3), SU(2)$, and the unit quaternions $Sp(1)$ are all isomorphic; this Lie group is also sometimes referred to simply as its underlying manifold $S^3$. $SO(3)$ is diffeomorphic to $\mathbb{RP}^3$ and so is not diffeomorphic to $S^3$, although its double cover is $Spin(3)$ (and hence also $SU(2)$ and $Sp(1)$).
One ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "12",
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Given order of two points, determining the number of points on an elliptic curve My problem is the following:
$E$ is an elliptic curve $y^2 = x^3 + bx + c$ over integers modulo $221 = 13\cdot 17$.
There exist some points $P$ and $Q$ on $E$ such that $11P = \mathcal{O}$ and $7Q = \mathcal{O}$.
Can you determine $\sharp ... | I figured out the problem:
Recall Lagrange's Theorem, which states that the order of any element in a group divides the number of elements in the group. Thus, $7 \mid \sharp E$ and $11 \mid \sharp E$. Therefore, $7\cdot 11 = 77 \mid \sharp E$.
By Hasse's Theorem, $\sharp E$ lies in the range $(221+1-2\sqrt{221},221+1+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the price of the European call option For the Black-Scholes market model where the price of the riskless
asset (bond) satisfies
$$dB_t=rB_tdt, B_0 = 1$$
for some $r>0$ and the stock price evolves according to
$$dS_t = µS_tdt + σS_tdW_t, S_0 = 1,$$
where $µ, σ > 0$ constants and $W_t$ is a (standard) Brownian motion. Wi... | First of all you confuse the payoff of the option with its price. The payoff of this option is $C_T = \max\{S_T-K, 0\}$, you wish to find its price, that is, $C_0$.
So the price of the stock at maturity (expiry) $S_T$ is given.
In order to find $C_0$ you have to use Black-Scholes formula
\begin{align}
C_t &= N(d_1)S_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2218417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Notation for Matrix Exponentials Just wondering if this is acceptable notation: For
$$
A=
\begin{bmatrix}
1 & 4\\
2 & 3
\end{bmatrix},
$$
$$
e^{At}=\sum_{n=0}^{\infty}\frac{(At)^n}{n!}=I+At+\frac{(At)^2}{2!}+\frac{(At)^3}{3!}+\cdots
$$
$$
I+
\begin{bmatrix}
1 & 4\\
2 & 3
\end{bmatrix}
t+
\frac{1}{3}
\sum_{n=2}^{\infty}... | Moo’s comment to your question provides some excellent links to materials on matrix exponentials. From them you can learn, among other things, that if $A$ is diagonalizable into $B\Lambda B^{-1}$, then $e^{tA}=Be^{t\Lambda}B^{-1}$, and that $e^{t\Lambda}=\operatorname{diag}(e^{\lambda_1t},\dots,e^{\lambda_nt})$ where t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Statistics finding p value from frozen dinners
For this question, for test statistic
$\frac{(sample\ mean - population\ mean)}{ (\frac{sample\ standard\ deviation} {\sqrt n})}$
$\frac{200.75 - 200}{(\frac{8.2586}{ \sqrt 12})}$
I have gotten $0.314590392$ for the test statistic
By using the free software from
https:... | It seems you are testing $H_0: \mu = 200$ against the two-sided alternative
$H_a: \mu \ne 200.$ The null distribution of the $T$-statistic is Student's t
distribution with 11 degrees of freedom, and you have computed the observed value of the
$T$-statistic to be $T = .3146.$ Consider the PDF of $\mathsf{T}(11)$ plotted... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$A, B$ are linear map and dim$null(A) = 3$, dim$null(B) = 5$ what about dim$null(AB)$ $A, B$ are linear map from $\mathbb{R}^{12} \to \mathbb{R}^{12}$ and dim$null(A) = 3$, dim$null(B) = 5$ what value could dim$null(AB)$ be?
I think it could be greater than or equal to 5 because
$$
Ker(B) \subset \{x\in \mathbb{R}^{12... | $\ker(AB)$ can have any dimension between $5$ and $8$. As you have already observed, $\ker(B) \subset \ker(AB)$, so that $\dim \ker(AB) \geq 5$. On the other hand: using the rank nullity theorem, we have
$$
\dim\operatorname{im}(AB) =
\dim\operatorname{im}(A|_{\operatorname{im}(B)}) =
\dim\operatorname{im}(B) - \di... | {
"language": "en",
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prove hyperbolic metric is independent of conformal map A conformal map $g(z)$ of a domain $D$ onto the open unit disk $\mathbb{D}$ induces the metric $\rho_D$ on $D$ defined by
$d \rho_D (z) =\frac {2\vert g'(z) \vert}{1- \vert g(z)\vert ^2}$$ \vert dz\vert $ for $z\in D$.
Show that $\rho_D$ is independent of the co... | Let $f(z)$ be another conformal map of $D$ onto the open unit disk $\mathbb{D}$. Then $(g\circ f^{-1})(z)$ is a bijective mapping of $\mathbb{D}$ to $\mathbb{D}$, so it can be expressed as $$
(g\circ f^{-1})(z)=e^{i\theta }\frac{z-a}{1-\bar{a}z},$$
where $a(|a|<1)$ is some point in $\mathbb{D}$ and $\theta \in\mathbb{R... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite models are atomic I want to show that any finite model is atomic. To prove this, it is enough to show that any type realized in a finite model is principal (isolated).
Let $T$ be a theory, $\mathcal{A}$ a finite model of $T$, and $p$ a type realized in $\mathcal{A}$. How can we show that $p$ is principal?
If $T$... | A (complete) $n$-type is just a complete theory over the extended language where you adjoin $n$ new constant symbols, and a model together with a realization of the type is just a model of the complete theory in the extended language. So it suffices to show that any complete theory (over a finite language) which has a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convergence of $1+\frac{1^2\cdot2^2}{1\cdot3\cdot5}+ \frac{1^2\cdot2^2\cdot3^2}{1\cdot3\cdot5\cdot7\cdot9}+...$ I am trying to use the ratio test, for that, I need the general formula for the series.
The general formula for the numerator is $(n!)^2$
The denominator is a sequence of odd numbers that grows by two terms e... | Lets try writing the general term
$$a_n=\frac{(n!)^2}{1\cdot 3\cdot 5\cdots (4n-5)(4n-3)}\\a_{n+1}=\frac{((n+1)!)^2}{1\cdot 3\cdot 5\cdots (4n-1)(4n+1)}\\\frac{a_{n+1}}{a_n}=\frac{((n+1)!)^2}{1\cdot 3\cdot 5\cdots(4n-1)(4n+1)}\cdot\frac{1\cdot 3\cdot 5\cdots(4n-5)(4n-3)}{(n!)^2}\\\frac{a_{n+1}}{a_n}=\frac{(n+1)^2}{(4n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2219077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prob. 19, Chap. 4 in Baby Rudin: Any real function on $\mathbb{R}$ whit the intermediate-value property for which ... is continuous Here is Prob. 19, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Suppose $f$ is a real function with domain $\mathbb{R}^1$ which has the intermedia... | Your proof looks fine, and I would agree it's the approach that Rudin was suggesting.
The one suggestion I would make is to explain why the sequence $\{t_n\}$ converges to $p$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What will be the Taylor series and the radius of the convergence of $\frac{1+x}{1-x}$ $\frac{1+x}{1-x}$, well it's pretty similar to the geometric series, which is $$1+x+x^2+x^3+...=\sum_{n=0}^{\infty} x^n=\frac{1}{1-x}$$ So if I multiple $$\sum_{n=0}^{\infty} x^n$$ by $x$ can I get the Taylor-series(which is in this c... | $\textbf{Hint}:$ $$\frac{1+x}{1-x} = \frac{2}{1-x} - 1 \ \ \ \text{(Why?)}$$
| {
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For $00$, show that $\left|\int_{a}^b \frac{\cos x}{x^n}\,dx\right|\leq \frac{2}{a^n}.$ For $0<a<b$ and $n>0$, show that $\left|\int_{a}^b \frac{\cos x}{x^n}\,dx\right|\leq \frac{2}{a^n}.$
I did some estimate, but it got much bigger bound:
$$
\left|\int_{a}^b \frac{\cos x}{x^n}\,dx\right| \leq \int_{a}^b\left|\cos x\ri... | By the Mean-Value theorem for integrals, there is $\xi\in(a,b)$ such that
\begin{eqnarray}
&&\left|\int_{a}^b \frac{\cos x}{x^n}dx\right|=\left|\frac{1}{\xi^n}\int_{a}^b \cos x dx\right|=\frac{1}{\xi^n}|\sin b-\sin a|\\
&\le& \frac{2}{\xi^n}\le\frac{2}{a^n}.
\end{eqnarray}
| {
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What is the position of the surviving mouse? I have this question that I think it may be very interesting to all maths' lovers.
A cat caught $n$ (integer) mice and put them in line, numbered them from 1 to $n$, from left to right.
He starts eating every other mouse, starting with the mouse at the 1st position, i.e. 1, ... | Following Arthur's hint: after the $k$th round, the only mice that are left are the multiples of $2^{k+1}$. This is because in the $k$th round, the cat eats all the mice that are not divisible by $2^k$.
Let $2^k$ be the largest power of $2$ that is $\le n$, i.e. $k=\lfloor \log_2 n\rfloor$.
Then the last mouse is the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2219700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Defective battery problem
A flashlight has $6$ batteries, $3$ of which are defective. If $3$ are selected at random without replacement, find the probability that all of them are defective.
I am finding the probability of getting all of them defective batteries which should be the probability of each when its drawn, ... | If three out of the six are defective and you select three without replacement, there is only one way to obtain all three defective batteries. But there are clearly more ways to select three batteries in which one or more is not defective.
To see this, it suffices to label the batteries as follows:
$$\{G_1, G_2, G_3, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2219812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Computing the expectation of $X^n e^{-\lambda X}$ Is there an elegant (probabilistic) way to compute $E(X^n e^{-\lambda X})$, where $n \in \mathbb{N}$, $\lambda > 0$ and $X$ is a random variable with normal distribution $N(\mu,\sigma^2)$?
Alternatively, my question is simply how to compute the integral
$$
\frac{1}{\sqr... | $$\lambda x + \frac{(x-\mu)^2}{2\sigma^2} = \frac{x^2 - 2(\mu - \sigma^2 \lambda)x + \mu^2}{2\sigma^2} = \frac{(x-(\mu-\sigma^2 \lambda))^2}{2\sigma^2} + \frac{2 \lambda \mu -\sigma^2 \lambda^2}{2}$$
Ignoring constant factors, your integral becomes
$$\int x^n e^{-\frac{(x-(\mu-\sigma^2 \lambda))^2}{2\sigma^2}} \mathop{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2219919",
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multiplying a linear system by an invertible diagonal matrix let $Ax = b$ be a linear n by n system
if we multiply this system by a non-singular diagonal $D$ I can say that the new system still has the same solution as the previous one , right ?
I ran some tests on matlab and noticed that the condition number stays all... | we have $Ax=b$ we change this to
$$
DAx = b.
$$
This will be solvable if $A$ and $D$ are invertible since
$$
(DA)^{-1} = A^{-1}D^{-1}
$$
In terms of condition numbers, suppose $D=A^{-1}$
then the new system is
$$
x=b
$$
which is nicely conditioned. So, yes, the condition number can change.
| {
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degree of a map in terms of fundamental classes confusion Hatcher makes the following definition in exercise 7 on page 258.
For a map $f: M \rightarrow N $ between connected, closed, orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, the degree of $f$ is defined to be the integer $d$ such that $f_{*}[M... | For the definition of degree, you should consider homology with integer coefficients. Now, $$f_*[M]\in H_n(N,\mathbb{Z})=\mathbb{Z}\langle[N]\rangle,$$and so, there is a unique $d\in\mathbb{Z}$ such that $$f_*[M]=d[N].$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Straightforward application of Exactness of sequence I am trying to show that if the sequence $$0 \rightarrow M^* \xrightarrow{f^*} MCyl(\alpha)^* \rightarrow MC(id_L^*) \rightarrow 0$$
is exact ($\alpha^* : L^* \rightarrow M^*$, where $L^*, M^*$ are cochain complexes), then $f$ is a quasi-isomorphism.
$MCyl, MC$ are ... | There is a less messy approach. The mapping cone $MC(id_{L^*})$ of the identity is sometimes just called the "cone" of $L^*$, and denoted by $CL^*$. The cone has degree $n$ part $L^{n+1}\oplus L^n$, with differentials:
$$d^n:L^{n+1}\oplus L^n\to L^{n+2}\oplus L^{n+1},\quad d^n(x,y) = (-d_L^{n+1}(x),x+d_L^n(y)).$$
With ... | {
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Induced action from $SL$ to $\mathfrak sl$ Suppose $SL(V, \mathbb C)$ acts on $V$ as expected, and thus on $V \otimes (V\wedge V)$ as $g \cdot [ v \otimes (w \wedge z)] = g \cdot v \otimes (g \cdot w \wedge g \cdot z )$. Which is the induced action of $\mathfrak sl_n$ on this same spaces?
| The action of $\mathfrak{sl}(V)$ on $V^{\otimes n}$ is given by $$g\cdot (v_1\otimes \dots\otimes v_n)=gv_1\otimes v_2\otimes\dots \otimes v_n+\dots+ v_1 \otimes \dots \otimes v_{n-1}\otimes gv_n;$$
and hence the action on its quotients modules (such as $V^{\otimes (n-k)}\otimes\Lambda^k V$) has the same form.
It's ob... | {
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Proving that the number of compositions of n into positive odd summands is Fibonacci sequence I'm currently stuck with a problem of proving that the number of compositions of natural $n$ into positive odd summands generates a Fibonacci sequence. (i.e. $4=1+1+1+1=3+1=1+3$)
My guess is that solution should be similar to ... | Suppose the number of compositions of n into odd parts is c(n). For even n we have
$$c(n) = c(n-1) + c(n-3) + c(n-5) + ... + c(1)$$
because we can add a 1 to each of compositions of n-1, or add a 3 to each of the compositions of n-3 etc.
For odd n we have
$$c(n) = c(n-1) + c(n-3) + c(n-5) + ... + c(2) + 1$$
where the e... | {
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Negative flux when vector arrows are pointing outwards I'm doing some homework for my calculus class and I came across an exercise where I shall calculate the flux of a vectorfield out of a elliptic cylinder $\frac{x^2}{9}+\frac{y^2}{4} \leq 1$, limited by a parabolic function $z = x^2 + y^2$.
I drew a graph to help me... |
I simply use the total minus the cylinder, but then I get a negative flux. How come? When the arrows are clearly pointing outwards at any point inside the given volume?
Your calculations look alright, so I think you're wrong in your geometric interpretation or intuition. Notice that although the paraboloid is the 'to... | {
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How many SHA-256 hashes of emails are duplicates of each other?
There are $5$ billion unique email addresses in the World. If I created a database containing their SHA-256 hashes, how any unique hashes would we expect that database to contain?
By my crude methods, a SHA-256 hash is $256$ bits long so there are $k=2^{... | Your reasoning is basically right.
You can compute two related but different things.
First, the expected number of coincidences (or collisions). This is given by
$$C= \frac{n (n-1)}{2} \frac{1}{k} \approx \frac{n^2}{2k} $$
Then the probability that there is at least a collission. This is
$$P = 1- \frac{k!}{n!}\frac{1... | {
"language": "en",
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Is the vector $3 + x^2$ in the subspace spanned by $\sin^2 x$ and $\cos^2 x $? My idea was: if $3 + x^2$ was in the subspace spanned by the other two, then it would be some linear combination of those two. So what I did is I formed the Wronskian, found that it was not identically zero everywhere, and concluded that th... | Your approach seems to work, but there are simpler ways to do this : for example, notice that any linear combination of $\cos^2$ and $\sin^2$ is periodic (or bounded), while $3+x^2$ isn't...
| {
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Why is infinite Integration by Parts valid? Few days ago I wrote an answer to solving $\int x\exp(x^2)$ using integration by parts,the general formula for the integral is
$$\int x^{2n+1}e^{x^2}dx=\frac{x^{2n+2}}{2n+2}e^{x^2}-\frac{1}{n+1}\int x^{2n+3}e^{x^2}dx$$
If we label the integral $I_n$ we get
$$I_n=\frac{x^{2n+2... | To ask a precise question, consider the definite integral over an interval $[a,b]$.
Recall that a series is just a sequence of partial sums. And if $A_j$ is the general term of the series that you get in the end, you have something like:
$$
I_0 = \sum_{j=1}^n A_j - \frac{1}{(n+1)!}I_{n+1}
$$
And so the partial sums ... | {
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Exercise involving light cones and light rays in special relativity I'm having some trouble in understanding (and solving) a particular exercise given in Gregory L. Naber's book "The Geometry of Minkowski Spacetime". First, let me supply the required definitions. This exercise involves the quadratic form $\mathcal{Q}(v... | Condition $\cal{Q}(x-x_0)=0$ is written there just because $R_{x_0,x}$ is defined only when $x\in C_N(x_0)$. Your proof is correct.
| {
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Polynomial and Function I have been give a question if
$ f(x) = x^3-1 $
then show
$$ \frac {f(b) - f(a)} {b-a} =b^2+ab+a^2 $$
how to show that the above fraction is equal to that polynomial if $ f(X) = x^3-1 $ ?
| It results from the high-school identity:
$$x^3-y^3=(x-y)(x^2+xy+y^2).$$
More generally:
$$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+\dots+xy^{n-2}+y^{n-1}).$$
| {
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A problem on Minima. Proving sphere has minimum surface area for a given volume Has the below question been answered here before?
Prove: For a given enclosed volume, a sphere has minimum surface area.
Please provide link or ways to solve it. I know it is a problem of Minima and involves finding derivative and second de... | I recommend reading the first part of The Brunn-Minkowski inequality for nilpotent groups by Terence Tao. He first considers the simple one-dimensional Brunn-Minkowski inequality, then proves the Prékopa-Leindler inequality, which again implies the full version of Brunn-Minkowski:
Brunn-Minkowski inequality. Let $A,B... | {
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If matrix is diagonalizable, eigenvalue?
Let $A$ be an $n \times n$ matrix and suppose $A$ is diagonalizable and the only eigenvalue is $\lambda = k$, what can you say about matrix $D$ where $A = P^{-1} D P$, for invertible matrix $P$.
So if the only eigenvalue of $A$ is $\lambda = k$, what can I say about $D$?
I kn... | Yes, it is. One way to see this is that eigenvalues are invariant under conjugation. This is a fancy way to say that if
$$
A=PDP^{-1}
$$
then $A$ and $D$ have the same eigenvalues.
Now, what are the eigenvalues of a diagonal (or upper triangular) matrix?
Edit: Proof of fact mentioned in comments:
suppose
$$
\lambda... | {
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How to solve $uu_{x_1}+u_{x_2}=1$ with characteristic method? $$uu_{x_1}+u_{x_2}=1\text{ with initial condition }u(x_1,x_1)=\frac{1}{2}x_1$$
I have problem to use following characteristic method to solve it.
Let $$x(s) = (x_1(s), x_2(s))$$$$z(s) = u(x(s))$$$$p(s) = (u_{x_1}(x(s)), u_{x_2}(x(s))$$$$F(x,z,p) = zp_1 + p_2... | Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx_2}{dt}=1$ , letting $x_2(0)=0$ , we have $x_2=t$
$\dfrac{du}{dt}=1$ , letting $u(0)=u_0$ , we have $u=t+u_0=x_2+u_0$
$\dfrac{dx_1}{dt}=u=t+u_0$ , letting $x_1(0)=f(u_0)$ , we have $x_1=\dfrac{t^2}{2}+u_0t+f(u_0)=\dfrac{x_2^2... | {
"language": "en",
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"source": "stackexchange",
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Proprieties of Kernel of Subring of Field While exploring concepts related to field extensions, I came across the following statement:
"Let $K$ be an extension field of $F$ and $u\in K$ an algebraic element over $F$. Consider the homomorphism $F[x]\to K$ defined by evaluation of a polynomial at $u$. Since the image is ... | Hint: Every subring of a field is a domain.The image is isomorphic to $F[x]/kernel$.
| {
"language": "en",
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"source": "stackexchange",
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How do you find the area of the shaded area of this circle?
I need to find the area of the shaded area. The triangle is equilateral. So far, I have found the area of the triangle to be $\sqrt 3$, but I cannot figure out how to find the radius of the circle in order to find the area of the circle. Any advice would be a... | We're looking for the radius right? So let's draw them..
Please excuse the drawing.
Ok. Now we have an isosceles triangle $30-30-120$. If $r$ is the radius then law of sines tells us,
$$\frac{2}{\sin 120}=\frac{r}{\sin 30}$$
So $r=2 \frac{\sin 30}{\sin 120}=2\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}$.... | {
"language": "en",
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How did people figure out that parabolas, hyperbolas, circles, and ellipses were conic sections? Maybe it is not surprising if one thinks that parabolas, hyperbolas, circles, and ellipses are relatives because they all have kind of the same form of equations, i.e.,
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,
$$
where diffe... | It's the other way around: in ancient Greece parabolas, hyperbolas and ellipses were defined as sections of a cone. From that definition, one can easily derive the analogous of a modern cartesian equation: as far as I remember, that was done for the first time by Apollonius of Perga in 3rd Century b.C. See here for a d... | {
"language": "en",
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"source": "stackexchange",
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Help with boolean algebra simplification I have the following boolean expression:
$$(A \land B) \lor (\lnot A \land C) \lor (B \land C)$$
I know this can be simplified to
$$(A \land B) \lor (\lnot A \land C)$$
I can see that doing truth tables, drawing a circuit, a venn diagram. I understand it simplifies to that.
Wha... | By looking at the similar part, you just want to show $$ C\,\vee \,B \wedge \, C = C$$
This identity is described on wikipedia as the absorption property :https://en.wikipedia.org/wiki/Boolean_algebra#Laws.
Intuitively, you can say that $B$ is absorbed by $C$ (via truth table for example).
Indeed: If $C = 1$, then $C \... | {
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Tangent Points for Common Tangent to Two Ellipses This is somewhat similar to my other question here.
Consider the two ellipses given by the equations
\begin{equation}
\frac{x^2}{2^2} + \frac{(y-1)^2}{1^2} = 1
\end{equation}
and
\begin{equation}
\frac{x^2}{1^2} + \frac{(y-4)^2}{(1/2)^2} = 1.
\end{equation}
How do I... | Let's introduce the following new variables:
$$2u=x\ \text{ and }\ y-1=v.$$
With these new variables, we have
$$u^2+v^2=1\ \text{ and }\ u^2+(v-3)^2=\frac14$$
that is, we have two circles as shown in the figure below.
We have similar triangles and we can see that $OD=6$. Also, by the Pythagorean theorem $DC=\frac{\sq... | {
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Determine the type of isolated single points of a function
Determine the type of isolated single points of the function
$$f(z)=\frac{\sin(z)}{z^5+2z^3+z}.$$
I tried:
$$f(z)=\frac{\sin(z)}{z(z^2+1)^2}.$$ So, $f(z)$ has a isolated singularity at $z=0$.
$$\sin(z)=\sum^\infty_{n=0}\frac{(-1)^n}{(2n+1)!}z^{2n+1}$$ and
$... | You're only asked to determine they types of isolated singularities.
At $z=0$, the singularity is removable since $\lim_{z\to 0}\frac{\sin(z)}{z}=1$. In fact, one can see from the Taylor series of the sine function that $\frac{\sin(z)}{z}$ is an entire function.
We have isolated singularities at $z=i$ and $z=-i$. B... | {
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recurrence relation concrete way to solve it I have the following recurrence relation:
$$a_n = 2a_{n-1} + 2^n; a_0 = 0$$ I used the characteristic equation method and some method I found online by calculating the $n+1$ th term and subtracting accordingly the equation with $a_{n+1}$ minus the equation with $a_{n}$:
$$a_... | You can't use the characteristic equation methoded beacuse it is not a linear recurrence but you can use the generating function method, let $f(x)=\sum_{n=0}^\infty a_nx^n$ :
$$f(x)=\sum_{n=0}^\infty a_nx^n=0+\sum_{n=1}^\infty a_nx^n=\sum_{n=1}^\infty (2a_{n-1}+2^n)x^n\\=2x\sum_{n=0}^\infty a_nx^n+\sum_{n=1}^\infty (2x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the set of invertible functions $f:A\rightarrow A$ compact? I would like to know if the space of invertible, or 1-to-1, functions $f:A\rightarrow A$ is a compact function space, or if restrictions on $A$ are required?
Recommended resources would also be welcome.
| $\renewcommand{\Re}{\mathbb{R}}$As a simple counterexample, consider the space of linear functions $f:\Re\to \Re$ with the operator norm $\|g\|=\alpha$ whenever $g(x) = \alpha x$. This defines the space of functions $(X,\|\cdot\|)$. Let $Y$ be the subset of $X$ of all invertible functions; equipped with the same norm. ... | {
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Convergence of $\sum_{n=3}^\infty \frac {1}{n \ln n}$ I know I have seen something similar and there is a telescoping trick to the convergence but it is eluding me.
|
We circumvent using the integral test or its companion, the Cauchy condensation test. Rather, we use creative telescoping to show that the series $\sum_{n=3}^\infty \frac{1}{n\log(n)}$ diverges. To that end, we now proceed.
We will use the well-known inequalities for the logarithm (SEE THIS ANSWER)
$$\frac{x-1}{x... | {
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is a quotient of a free module free? Is a direct sum of free modules free? is that right that free module M over R is that M can be generated by a linear independent subset A, and every element of M is a finite sum of elements of A multiplied by coefficients in R(the expression should be unique)?
is a quotient of free ... | Yes, an $R$-module $M$ is free if it has a basis, i.e., a linearly independent generating set.
No, the quotient of free modules need not be free. Consider the $\mathbb Z$-module $\mathbb Z$. This is clearly free and $1$ is a free generator. Similarly, $2\mathbb Z$ is a free $\mathbb Z$-module, and $2$ is a free generat... | {
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Condition for monos out of an initial object to be isos. What does one need to assume in a category $\mathcal{C}$ with an initial object $0$, so that every morphism $f:0\to A$ out of $0$ (I am not assuming anything on the object $A$) satisfies the property that:
whenever $f$ is a monomorphism it is necessarily an isomo... | If every monomorphism out an initial object $0$ is an isomorphism, then every object is (canonically) isomorphic to $0$, i.e. the category is an indiscrete category, i.e. two objects have exactly one morphism between them.
To see this, first consider an initial object $0$ every monomorphism out of which is an isomorph... | {
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How to obtain the standard error of measurements that already have error bars? Suppose I make three measurements: 9, 10 and 11. Let's say there is no uncertainty associated with these values because they are counts. Numbers of children in a class say.
I want to know the average - 10 - and I want to know the error of th... | This is a very important topic in physics, usually named as data reconciliation. So, you want to minimize $$S=\frac12\sum_{i=1}^n\left(\frac {y_i-Y}{\sigma_i}\right)^2$$ Differentiate with respect to $Y$ to get $$S'=-\sum_{i=1}^n \frac{y_i}{\sigma_i}+Y\sum_{i=1}^n \frac{1}{\sigma_i^2}$$ Since you want the minimum, set... | {
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It is the same $f(x):=x^2\sum\limits_{k=0}^\infty(\cos x)^k$ than $g(x):=\sum\limits_{k=0}^\infty x^2(\cos x)^k$? It is the same $f(x):=x^2\sum\limits_{k=0}^\infty(\cos x)^k$ than $g(x):=\sum\limits_{k=0}^\infty x^2(\cos x)^k$?
It seems that $f(0)$ is not defined because $\sum_{k=0}^\infty 1^k=\infty$, however $g(0)=0... | They are same. The key is to remember that:
\begin{align*}
\sum_{k=1}^{\infty}a_k = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k
\end{align*}
If you consider the sequence:
\begin{align*}
(a_n) = \{0^2\cdot1, 0^2\cdot(1+1), 0^2\cdot(1+1+1), \cdots\} = \{0, 0, 0, \cdots\}
\end{align*}
You will see that for each $n\in \math... | {
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Limit of a 2-variable function Given problem:
$$
\lim_{x\to0,y\to0}(1+x^2y^2)^{-1/(x^2+y^2)}
.$$
I tried to do it with assigning y to $y = kx$, but that didn't help me at all. Also one point, I can't use L'Hospital's rule.
| The expression equals
$$\tag 1\left ((1+x^2y^2)^{1/(x^2y^2)}\right )^{-x^2y^2/(x^2+y^2)}.$$
Because $(1+u)^{1/u} \to e$ as $u\to 0,$ the expression inside the parentheses $\to e.$ Since
$$0\le x^2y^2/(x^2+y^2) \le x^2[y^2/(x^2+y^2)] \le x^2,$$
the outside exponent in $(1)$ goes to $0.$ The desired limit is therefore $e... | {
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Number of Solutions to Apollonius's LLC Problem I have been practicing my straightedge and compass constructions over the last few days and I'am trying to reproduce the solutions to the ten Apollonius problems (constructing circles which are tangent to three given objects which can be some combination of points, lines,... | You'll have eight solutions only for some positions of given lines and circle, see picture below for an example.
| {
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how to find parametrization for an intersection of a plane and one sheet hyperboloid I need to find a parametrization for intersection of a plane and one sheet hyperboloid.
one sheet hyperboloid equation: $x^2+y^2-z^2=1$
plane equation: $x-1=0$
I don't know how to parametrize the intersection, but I do know that it is ... | First parametrize the hyperboloid putting $$x = \cosh u \cos v, \quad y = \cosh u \sin v, \quad z = \sinh u $$Now force $x=1$, meaning $\cosh u \cos v = 1$. Write, say, $v = \arccos({\rm sech}\, u)$. Then one parametrization can by $$x = 1, \quad y = \cosh u \sin(\arccos({\rm sech}\, u)), \quad z = \sinh u.$$You can si... | {
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Finding the recurrence relation for a binary string that contain an even number of $0$'s A computer system considers a string of decimal digits a valid codeword if it contains an even number of $0$ digits. Let $a_n$ be the number of valid n-digit codewords. Find the recurrence relation for $a_n$.
The total possibilit... | If we know the first digit is $1$ then we need the rest of the $n-1$ digits to be valid on there own. There are $a_{n-1}$ of those.
If we know the first digit is $0$, then we need the rest not to be valid on their own (that way they contain an odd amount of zeros). Then the question becomes how many $n-1$ bit strings a... | {
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Calculating $\sum_{n=1}^N \frac{1}{(N+1+n)(N+n)}$ by hand In a recent proof I used induction to prove an identity concerning the harmonic progressions:
$$ \sum_{n=1}^{2N}\frac{(-1)^{n-1}}{n}=\sum_{n=1}^{N}\frac{1}{N+n} $$
I needed to know what the following sum equaled so I used Wolfram Alpha to find, $N \in \mathbb{N... | $$ \frac{1}{(N+n+1)(N+n)} = \frac{1}{N+n} - \frac{1}{N+n+1}, $$
and summing, the middle terms cancel and one is left with
$$ \sum_{n=1}^N \frac{1}{(N+n+1)(N+n)} = \frac{1}{N+1} - \frac{1}{2N+1} = \frac{N}{(2N+1)(N+1)}. $$
| {
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How to prove $\gcd(a,b) \cdot \gcd(c,d)=\gcd(ac,ad,bc,bd)$? I want to prove the identity $\gcd(a,b) \cdot \gcd(c,d)=\gcd(ac,ad,bc,bd)$. I tried this: if $x=\gcd(a,b)$ and $y=\gcd(c,d)$ then I must show $xy=\gcd(ac,ad,bc,bd)$ so I think I have to use the property $\gcd(ar,br)=r\cdot \gcd(a,b)$ but I don't know how to ap... | Another way to think about this problem is the following fact: two integers $a$ and $b$ are equal when $r \mid a \implies r \mid b$ and $r \mid b \implies r \mid a$.
(This is similar to how equality works for sets, where $S=T$ when $x \in S \implies x \in T$ and $x \in T \implies x \in S$.)
Here's how you can prove one... | {
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Why can't this matrix have a right inverse?
Let $A$ be an $m \times n$ matrix with m > n. Why can't $A$ have a right inverse.
We want $AB = I_m$, why is this impossible if $m > n$?
| Think about $A$ as the linear transformation that it represents: $T:\mathbf{R}^n\to \mathbf{R}^m$. We know that the number of columns corresponds to the dimension of the domain, while the number of rows corresponds to the dimension of the codomain. We know, then, that $m>n$. So, we know that $T$ can not be surjective. ... | {
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Proof: if $a$ and $b$ are integers, then $a^2-4b-3\neq 0$. I was wondering if someone could take the time to look over this proof and make sure it is correct. I greatly appreciate the help.
Proposition: If $a$ and $b$ are integers, then $a^2-4b-3\neq 0$.
Proof: Assume $a,b\in\mathbb{Z}$ and, for contradiction's sake, $... | Yes, your proof is correct! Also, when you discorver that a has a remainder of 3 when divided by 4, you can go straight to the fact that a is odd.
| {
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Homomorphism always exists between modules over an integral domain
Let $R$ be an integral domain. Let $F$ be a free module over $R$, and let $M$ be an arbitrary nonzero $R$-module. Is it true that there always exists a nonzero module homomorphism from $M$ to $F$?
I know that there always exists one from $F$ to $M$ b... | Not necessarily. Consider $R=\mathbb{Z}$, $F=\mathbb{Z}$, and $M = \mathbb{Z}/2\mathbb{Z}$. If $\varphi: M \to F$ is a module homomorphism, then $2\varphi(1)=\varphi(2\cdot 1) = \varphi(0)=0$ implies that $\varphi(1)=0$, and hence there are no non-zero module homomorphisms from $M$ into $F$.
| {
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If $a$ is a primitive root modulo $p$, then $(p-1) | ord(a)$ in $\mathbb{Z}/p^e\mathbb{Z}$ I'm a bit confused on a fact that my book has been using rather liberally without proof and I'm sure I'm missing something incredibly simple. Plainly stated, my question is if $a$ is a primitive root modulo $p$ so that
$$a^{p-1}\... | Max basically answered this question so just to reiterate what he posted in the comments, if $a$ is a primitive root modulo $p$ then let $m$ denote the order of $a$ in $\mathbb{Z}/p^e\mathbb{Z}$, then we have
$$a^m \equiv 1 \pmod {p^e} \implies p^e | a^m - 1 \implies p | a^m - 1 \implies a^m \equiv 1 \pmod p$$
But then... | {
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Partition of number vs dividing into blocks Let $P(n)$ be the number of partitions of integer $n$ and let $A(n,k)$ be the number of ways to put $n$ indistinguishable toys into $k$ distinguishable boxes but with the following restriction: the number of toys in $i$-th box must be divisible by $i$.
For which $(n,k)$ we ha... | Your reasoning looks fine. Nice bijection.
| {
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"timestamp": "2023-03-29T00:00:00",
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Surjective map on compact metric space Is there a surjective map $f:X\to X$ on compact metric spcace $(X, d)$ with the following condition?
There is $0<L<1$ such that $d(f(x), f(y))<Ld(x, y)$ for all $x,y\in X$
| Let $x, y \in X$. Since $f$ is surjective, there are $x_1, y_1 \in X$ such that $f(x_1)=x, f(y_1)=y$, similarly $f(x_2)=x_1, f(y_2)=y_1$, ..., $f(x_{n+1})=x_n, f(y_{n+1})=y_n$. From the assumption,
$$d(f(x), f(y))<Ld(x, y)=Ld(f(x_1), f(y_1))<L^2d(x_1, y_1)<\cdots <L^{n+1}d(x_{n}, y_{n})\leqslant L^{n+1} M,$$
for all $n... | {
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"timestamp": "2023-03-29T00:00:00",
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A integral involving Riemann zeta function and Gamma function: $\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\,dx=\zeta(s)\Gamma(s) $ I need to prove this, today my Instructor solved an integral using this formula but didn't gave a proof $$\displaystyle \int_{0}^{\infty}\dfrac{x^{s-1}}{e^{x}-1}\,\mathrm dx=\zeta(s)\cdot\Gam... | We have $\int_{0}^{+\infty}z^{s-1}e^{-z}\,dz = \Gamma(s)$ for any $s>0$ by the very definition of the $\Gamma$ function.
Moreover
$$ \frac{1}{e^x-1} = e^{-x}+e^{-2x}+e^{-3x}+\ldots $$
with uniform convergence over any compact subset of $\mathbb{R}^+$. By the dominated convergence theorem it follows that
$$\begin{eqnarr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Lines from the centers of squares on two sides of a triangle to the third side? I have been working on the following problem from Visual Complex Analysis. My question is not necesarily if the solution is right, but more of a meta question about the solution and complex numbers. I apologize in advance if the question is... | I think the difficulty you are having is the misconception that a complex number, say z, is a point in the complex plane. Rather, think of it as a vector (hence all those arrows). And they add and subtract just like vectors and they have scalar and vector products as well. For example, given two complex numbers, say $z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many points with integer coordinates lie on at least one of these paths? A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many poin... | All coordinates on or within the outer border of points shown below, could be reached. I make it $195$ coordinates.
| {
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"timestamp": "2023-03-29T00:00:00",
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a question on hereditary $C^*$- subalgebras Let $A$ be a $C^*$-algebra and $a\in A$ be positive. It is known that $\overline{aAa}$ is the hereditary subalgebra generated by $a$. Now, let $f$ be a continuous function on $[0,\|a\|]$ such that $f(0)=0$ and $f(x)>0$ whenever $x>0$.
My question is whether $\overline{f(a)Af(... | What follows is an incomplete solution, but perhaps has some merit. One can apply Proposition 2.5 of this paper here, but there appears to be a problem I describe below.
Lemma: Let $X$ be a compact Hausdorff space, $f,g\in C(X)_+$ be two positive functions such that $\text{supp}(g) \subset \text{supp}(f)$. Then, for a... | {
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Find the volume between the cone $y = \sqrt {x^2 + z^2} $ and the sphere $x^2 + y^2 + z^2 = 49$.
Find the volume between the cone $y = \sqrt {x^2 + z^2} $ and the sphere $x^2 + y^2 + z^2 = 49$.
I know that the volume we're interested in is the volume of the intersection between the sphere of radius $7$ and a an upsid... | For given input eliminate $x,z$ etc. and you are left with a circle on the sphere:
$$ x= 7/\sqrt2 ,y= 7/\sqrt2 \,\cos t, z= 7/\sqrt2 \, \sin t $$
You can use established result Gauss Bonnet thm to advantage, since $k_g , K $ are constant as a differential geometry approach.
$$ k_g= \frac{1}{7},\, s= 2 \pi \frac{7}{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability of picking more balls of one color Suppose you flip a 3-sided coin $n$ times. The sides are denoted: $A$, $B$, and $C$. The probability of a coin flip turning one of the sides is given by $p_A$, $p_B$, and $p_C$, respectively.
What is the probability that you end up flipping side $A$ more times than side $... | The probability of flipping side $A$ $j$ times and side $B$ $k$ times is
$$\frac{n!}{j!k!(n-j-k)!}p_A^jp_B^kp_C^{n-j-k}$$
and so the probability that you flip $A$ more than you flip $B$ is
$$\sum_{k=0}^{n-1}\sum_{j=k+1}^n\frac{n!}{j!k!(n-j-k)!}p_A^jp_B^kp_C^{n-j-k}.$$
This does not strike me as a very useful expression... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\prod_1^\infty (1+p_n)$ converges Let $p_{2n-1} = \frac{-1}{\sqrt{n}}$, and $p_{2n} = \frac{1}{n}+\frac{1}{\sqrt{n}}$.
Prove $\prod_1^\infty (1+p_n)$ converges.
By numerical simulations, it appears to converge (to something around $0.759$). However, I'm not sure how to prove this. I know we can skip the first t... | Note that
$$\prod\limits_{k=2}^{2n}(1+p_k) = \prod\limits_{k=2}^{n}(1+p_{2k-1})(1+p_{2k}) = \prod\limits_{k=2}^{n}\left(1-\dfrac{1}{\sqrt{k}}\right)\left(1+\dfrac{1}{k} +\dfrac{1}{\sqrt{k}}\right) = \prod\limits_{k=2}^{n} \left(1- \dfrac{1}{k\sqrt{k}}\right)$$
It is also known that for any sequence $\{a_k\}$ such that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2227021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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For $v ∈ \mathbb{R}^m$, prove $\operatorname{rank}(vv^T) = 1$, where $v \ne 0$. I have this information from my notes:$\def\rk{\operatorname{rank}}$
Let $A ∈ \mathbb{R}^{m\times n}$. Then
*
*$\rk(A) = n$
*$\rk(A^TA) = n$
*$A^TA$ is invertible.
In my case, $n = 1$, so I would need to show $\rk(vv^T) = \rk(v^Tv) ... | Note that $v v^T v = \|v\|^2 v \neq 0$, hence $\operatorname{rk} (v v^T) \ge 1$.
Note that $v v^T x = (v^T x) v \in \operatorname{sp} \{ v \}$ for all $x$. Hence ${R (v v^T)} = \operatorname{sp} \{ v \}$ and hence
$\operatorname{rk} (v v^T) = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2227135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Symplifying a sum with multiple indices IM trying to understand the following simplification
$$ \sum_{k,n,m} [k^3 \leq n < (k+1)^3 ][n=km][1 \leq n \leq 1000] = 1 + \sum_{k,m} [k^3 \leq km < (k+1)^3][1 \leq k <10] $$
where [P(x)] = 1 if $P(x)$ is true statement a $0$ otherwise. Why is the above true? Im having hard ti... | The necessary and sufficient condition for $n$ to exist is
$$(k+1)^3\gt 1\quad\text{and}\quad 1000\ge k^3\iff 1\le k\le 10$$
So, we have
$$\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000]$$
$$=\sum_{k,n,m}[k^3\le n\lt (k+1)^3][n=km][1\le n\le 1000][1\le k\le 10]\tag1$$
Separating this sum into two cases, the ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2227205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\int_0^1(\ln\Gamma)(x)\mathrm dx=\ln(\sqrt{2\pi})$
Show that $\int_0^1(\ln\Gamma)(x)\mathrm dx=\ln(\sqrt{2\pi})$
Im totally stuck with this exercise. It is supposed that I must solve it using the reflection formula for the Gamma function. My work so far:
Using the reflection formula we have
$$\int_0^1(\ln ... | $$\frac{1}{2}\int_{0}^{1}\log \sin (\pi z)dz = \int_{0}^{1/2}\log \sin (\pi z)dz$$
$$\int_{0}^{1/2}\log \sin (\pi z)dz = \int_{0}^{1/2}\log \cos (\pi z)dz$$
Therefore,
$$2\int_{0}^{1/2}\log \sin (\pi z)dz = \int_{0}^{1/2}\log \frac{\sin (2\pi z)}{2}dz = \int_{0}^{1/2}\log \sin (2\pi z)dz - \frac{1}{2}\ln 2$$
Substitute... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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$0I want to show that given that $x,y>0$, we can deduce that $0<x^2<y^2 \Rightarrow x<y$. I am having problems with squarerooting the inequalities here.
$\sqrt{x^2}=|x|$ and $|x|=x$ here since x is positive so can we just squareroot the double inequality and deduce that the implication holds here.
In other words $$0<x^... | If $x \geq y > 0$, then $x^{2} \geq yx \geq y^{2}$; so $x^{2} < y^{2}$ implies $x<y$.
For a why you did not do it right, check out the first comment below.
| {
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"url": "https://math.stackexchange.com/questions/2227513",
"timestamp": "2023-03-29T00:00:00",
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