Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$ Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$.
The result is g... | Let us rewrite the product rule as follows:
$$(fg)'=f'g+g'f=\frac{f'}{f}fg+\frac{g'}{g}fg=\left(\frac{f'}{f}+\frac{g'}{g}\right)fg$$
Yours is just the generalization to $n$ factors, but is handled in the exact same way.
| {
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"timestamp": "2023-03-29T00:00:00",
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Why is multivariable continuous differentiability defined in terms of partial derivatives? Both in my textbook and on Wikipedia, continuous differentiability of a function $f:\Bbb R^m \to \Bbb R^n$ is defined by the existence and continuity of all of the partial derivatives. Since there is a notion of a (total) deriva... | Continuous differentiability of the function $f: \mathbb{R}^m \to \mathbb{R}^n$ (in terms of partial derivatives) is equivalent to existence and continuity of the map
$$Df: \mathbb{R}^m \to L(\mathbb{R}^m, \mathbb{R}^n)$$
$$ x \to Df_x$$
which takes a point to the derivative at the point. Any book on analysis on $\mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that? If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have:
$$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} ... | Any function $g(x)$ such that $g(c+a)+g(c-a)=k$ for all $a$ on the interval $(0,b)$ with any function $f(x)$ such that $f(c-a)=f(c+a)$ for all $a$ on the interval $(0,b)$ will satisfy the equation
$$\int_{c-b}^{c+b}{f(x)g(x)dx}=k*\int_{c}^{c+b}{f(x)dx}$$
because, using a trapezoidal Riemann sum after splitting the inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1868929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Critical Number real life applications I've studying a lot Critical Number/Point and I have to give a presentation about it.
I am searching real life applications to explain the concept, but it's difficult to find.
Anyone here can give me some real life applications examples about critical number?
| There are many problems in physics that use this concept. For example, when two atoms come together to form a molecule. They come closer to each other because the energy of the system is smaller if they share electrons. But if they are too close, the electrons cannot screen the nuclei. The two nuclei will repel, so at ... | {
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"source": "stackexchange",
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Proof that two non-parallel planes must intersect? I managed to find, by enumeration, the intersection point of two planes $ax+by+cz+d=0$ and $ex+fy+gz+h=0$, in all possible cases (with the condition that the planes are not parallel). But this is a very ugly proof.
I wonder if there is a quicker and more elegant proof ... | You want to show that if $v,w$ are linearly independent vectors in $\mathbb R^3$, then the $2\times 3$ matrix $A$ formed by putting $v$ and $w$ in two rows defines a map $A:\mathbb R^3\to \mathbb R^2$ that is onto. It suffices you show that the kernel of $A$ has dimension $1$ when $v,w$ are linearly independent, from t... | {
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"source": "stackexchange",
"question_score": "4",
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Field with $125$ elements I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $\Bbb F_5[x]/(f)$ is a field with exactly $5^3=125$ elements.
How do I find an irreducible pol... | Belatedly, there is a way to "be lucky" here, and not so computational, in an informative way. Namely, of all the polynomials known to humans, the best understood are cyclotomic ones. If a cyclotomic polynomial or a relative can resolve an issue, that's good fortune.
After a moment's fooling around, we observe that the... | {
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Differentiate equation with parenthesis I have a problem. I'm studying calculus, but I don't have a good math background, so I have a problem: I don't know well how to differentiate an equation with parenthesis.
The equation is the following:
$f(x) = 25x^3(x-1)^2$
Is it correct to use the Differentiation Product Rule i... | A small (useful) trick when you face products, quotients, powers,.. : logarithmic differentiation.
Let us take your cas $$f(x) = 25x^3(x-1)^2\implies \log(f(x))=\log(25)+3\log(x)+2\log(x-1)$$ Now, differentiate $$\frac{f'(x)}{f(x)}=\frac 3 x+\frac{2}{x-1}=\frac{5x-3}{x(x-1)}$$ $$f'(x)=f(x)\frac{5x-3}{x(x-1)}=25x^3(x-1)... | {
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Find the Wrong Student There are 15 student in the class and each of them has a different number 1 to 15.
*
*Student #1: wrote the natural number on the board.
*Student #2 said : This number is divisible by my number(number 2)
*Student #3 said : This number is divisible by my number(number 3)
*Student #4 sa... | The answer is $17$, as students number $8$ and $9$ are wrong.
To see this, note that if student $i$ is wrong, then student $ki$ must be wrong for every $k \ge i$. As these will not be two consecutive numbers, this cannot be the case. This means students $2$ through $7$ must be right.
Given $pq$, with $p$ and $q$ copr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $\sec (x) + \tan (x) = 4$ $$\sec{x}+\tan{x}=4$$
Find $x$ for $0<x<2\pi$.
Eventually I get $$\cos x=\frac{8}{17}$$
$$x=61.9^{\circ}$$
The answer I obtained is the only answer, another respective value of $x$ in $4$-th quadrant does not solve the equation, how does this happen? I have been facing the same problem ... | Using $t$-formula
Let $\displaystyle t=\tan \frac{x}{2}$, then $\displaystyle \cos x=\frac{1-t^2}{1+t^2}$ and $\displaystyle \tan x=\frac{2t}{1-t^2}$.
Now
\begin{align*}
\frac{1+t^2}{1-t^2}+\frac{2t}{1-t^2} &=4 \\
\frac{(1+t)^{2}}{1-t^2} &= 4 \\
\frac{1+t}{1-t} &= 4 \quad \quad (t\neq -1) \\
t &= \frac{3}{5} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1869545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\mathbb{C}/\mathbb{Z}$ is isomorphic to multiplicative group $\mathbb{C}\setminus\{0\}$ I have to show that $\mathbb{C}/\mathbb{Z}$ is isomorphic to the multiplicative group $\mathbb{C} \setminus \{0\}$.
Proof.
Let $f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}/\mathbb{Z}$ be the map
$$ f(\alpha) = \alpha \math... | The proof, as written, is definitely wrong -- the inverse map you define does not vanish on $\mathbb{Z}$, so it cannot be a map out of $\mathbb{C}/\mathbb{Z}$.
I think the idea is to use the complex exponential function. Notice that $\alpha \mapsto \exp(2\pi i \alpha)$ defines a group homomorphism from $\mathbb{C}$ to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The Jeep Problem and Nash's Friends The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a desert $x$ miles wide in the most economic way, that is minimizing the... | Finally, I found the answer to my question.
First of all, we can easily give a recursive solution to the problem as follows.
Let us note that if $P$ is a feasible plan of trips which allows the jeep to arrive at $0$, then we can find another feasible plan $P'$ which arrives at $0$ such that $P'$ is made up a number of... | {
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If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2. I'm trying to prove this statement:
If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$.
I know that if $a\in G$ such that $a\neq e$, then as a consequence of Lagrange's theorem $|a|\in \{2,... | There are only 2 groups of order 10, namely the cyclic group and the dihedral group of symmetries of a regular pentagon. The reflections in the dihedral group give you the five desired elements of order $2$.
Now, to prove that there are only 2 groups of order 10, let $a,b$ be elements of orders $2,5$ respectively. Cons... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$f$ is continuous at $x_0 \Leftrightarrow$ for every monotonic sequence $x_n$ in $\text{dom}(f)$ converging to $x_0$, we have $\lim f(x_n) = f(x_0)$ $f$ is continuous at $x_0 \Leftrightarrow$ for every monotonic sequence $x_n$ in $\text{dom}(f)$ converging to $x_0$, we have $\lim f(x_n) = f(x_0)$
Note: There is one ans... | Hint
For every subsequence $(x_{n_k})_{k \in \mathbb N}$ of $(x_n)_{n \in \mathbb N}$ there exists a subsequence $(x_{n_{k_\ell}})_{\ell \in \mathbb N}$ of $(x_{n_k})_{k \in \mathbb N}$ with $f(x_{n_{k_\ell}}) \rightarrow f(x_0)$ (this can be proven similar to your idea, so that comes into play here)
It follows that $f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I find the terms of an expansion using combinatorial reasoning? From my textbook:
The expansion of $(x + y)^3$ can be found using combinatorial reasoning instead of multiplying the three terms out. When $(x + y)^3 = (x + y)(x + y)(x + y)$ is expanded, all products of a term in the first sum, a term in the seco... | Expanding $(x+y)(x+y)(x+y)$ amounts to adding up all the ways you can pick three factors to multiply together. For example, you could pick an $x$ from the first $(x+y)$, a $y$ from the second $(x+y)$, and another $x$ from the third $(x+y)$ to get $xyx=x^2 y$.
You are right, the only possible products we can get are $x^... | {
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"timestamp": "2023-03-29T00:00:00",
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Cylinder defined on 3d coordinate plane This is the first time, I have seen a problem like this:
I feel as though if i knew where to start i would be able to do this problem easily. In other words, question 1-4 make sence to me and i know what they are asking for, but i just can't visualize the cylinder. I'm not askin... | I don't have acesss to a plotting software or a scanner right now so I can't provide a precise plot, but you have the following ingriedents:
*
*The equation $x^2 + y^2 = r^2$ is the equation of an infinite cylinder of radius $r$ whose symmetry axis is the $z$-axis. The inequality $0 \leq x^2 + y^2 \leq r^2$ throws i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1870146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find $\lim_{x\to \infty}\left(\frac{n+2}{n-1}\right)^{2n+3}$
Find $$\lim_{n\to \infty}\left(\frac{n+2}{n-1}\right)^{2n+3}.$$
My attempt:
$$\lim_{n\to \infty}\left(\frac{n+2}{n-1}\right)^{2n+3}=\lim_{n\to \infty}\left(1+\frac{3}{n-1}\right)^{2n+3}=\lim_{n\to \infty}\left(1+\frac{1}{\frac{n-1}{3}}\right)^{2n+3}$$
Now ... | $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove this inequality $3^{n}\geq n^{2}$ for $n\geq 1$ with mathematical induction? Prove this inequality $3^{n}\geq n^{2}$ for $n\geq 1$ with mathematical induction.
Base step: When $n=1$
$3^{1}\geq1^{2}$, statement is true.
Inductive step: We need to prove that this statement $3^{n+1}\geq (n+1)^{2}$ is true. ... | Essentially, you want to show that $$3n^2 > (n+1)^2$$ which is not so hard since $$3n^2 - (n^2 + 2n + 1) > 0 \iff 2n^2 - 2n-1 > 0$$
But $2n^2 - 2n - 1 = 2(n^2 -n) - 1 = (n^2-2) + (n^2 - 2n+1) =(n^2 -2) + (n-1)^2$, so that we have for all $n \geq 2$ that $2n^2 - 2n - 1 \geq 0$ since $(n-1)^2$ is always $\geq 0$ and $n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1870539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this a valid proof that sine is continuous at the origin? $$
\text{Let } \left|\sin x - 0\right| < \epsilon. \\
-\epsilon < \sin x < \epsilon \\
\arcsin (-\epsilon) < x < \arcsin (\epsilon) \\
-\arcsin \epsilon < x < \arcsin \epsilon \\
\left|x\right| < \arcsin \epsilon \\
\left|x - 0\right| < \arcsin \epsilon \\
\t... | Provided you know properties of the arcsine your idea will be a proof. However, are you sure you do not need to know that $\sin$ is continuous to deduce properties of the arcsine?
Spivak's calculus book has a note about a faulty proof he had in there in one of the pre-publication drafts. It used the square-root fun... | {
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Integer solutions to $x^3+y^3+z^3 = x+y+z = 8$
Find all integers $x,y,z$ that satisfy $$x^3+y^3+z^3 = x+y+z = 8$$
Let $a = y+z, b = x+z, c = x+y$. Then $8 = x^3+y^3+z^3 = (x+y+z)^3-3abc$ and therefore $abc = 168$ and $a+b+c = 16$. Then do I just use the prime factorization of $168$?
| Hint:
Taking from where you left off: $ab \mid 168 \implies ab = \pm 1, \pm 2, \pm 4, \pm 6, \pm 7, \pm 8, \pm 12, \pm 14, \pm 21, \pm 24, \pm 28, \pm 42, \pm 56, \pm 84, \pm 168$. Even though it looks cumbersome, it is easy to solve. For example, $ab = 6 \implies c = \dfrac{168}{6} = 28 \implies a+b = 16-c = 16 - 28 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The mean of the of a sum is the sum of the means
Transcription:
The mean has good mathematical properties. The mean of a sum is the sum of the means. For example, if $y$ is total income, $u$ is "earned income" (wages and salaries), $v$ is "unearned income" (interest, dividends, rents), and $w$ is "other income" (soci... | In your example, you have $u_1, u_2, u_3$, $v_1, v_2$, and you have correctly showed that
$$
\text{mean}(u_1,u_2,u_3) + \text{mean}(v_1,v_2)
$$
is not necessarily equal to
$$
\text{mean}(u_1 + u_2 + u_3, v_1 + v_2),
$$
so in that sense you are exactly correct.
However, this is not what the statement was intended to exp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of selecting same factor. Willie Pikette randomly selects a factor of $144$. Betty Wheel selects a factor of $88$. What is the probability that they selected the same number?
This is my incorrect approach (and please feel free to bash at me):
$144$ has $15$ factors in total whereas $88$ has
$8$ factors.
Be... | You've got the factoring part right, but the combinatorical part wrong:
*
*The number of ways to pick a pair of factors is $15\cdot8$
*The number of ways to pick a pair of identical factors is $4$
*Hence the probability of picking a pair of identical factors is $\frac{4}{15\cdot8}$
You have answered correctly f... | {
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"url": "https://math.stackexchange.com/questions/1871002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$ I'm trying to prove the following statement:
Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let the coordinate functions of $f$ be called $f_n$, for... | Restrict attention to base elements. A base for the product topology is furnished by elements of the form $B=I_1\times\cdots \times I_N\times \prod_{k>N} {\mathbb R}$ with $N$ finite and each $I_k$ open. Each $f_k^{-1} (I_k)$, $1\leq k\leq N$ is open and their (finite) intersection which equals $f^{-1} (B)$ is thus ope... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Algebraic solution for the value of $x$. I solved this problem the fifteen years ago without numerically solving equations of degree 4, I was happy in a substitution that I avoid directly attacking equations of degree 4.
Today my nephew, who is an enthusiastic student of mathematics, proposes me the same problem. It ... | From $y = 1/x,$ then multiplying by $x^2,$ i got
$$ x^4 + 2 x^3 - x^2 - 2 x - 1. $$
This looks bad. However, set
$$ x = t - \frac{1}{2} $$
and you get rid of the cubic term, always worth a try. I was pleased to discover that the linear term also vanished, giving
$$ t^4 - \frac{5}{2} t^2 - \frac{7}{16}, $$
and you can s... | {
"language": "en",
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What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology? I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a lot of the re... | The paper
J.H.C. WHITEHEAD: A certain exact sequence. Ann. Math. 52
(1950), 51-110.
introduced a functor $\Gamma$ which is the ``universal
quadratic functor" from Abelian groups to Abelian groups. Let $A$ be
an Abelian group. Then $\Gamma(A)$ is the Abelian group with
generators $\gamma a, a \in A$, and the following... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show $\sum_{k=1}^{n}\frac{1}{k}\sim \ln(n)$ there is an example of how we apply Integral test for convergence
Theorem:
Consider an $n_{0}$ and a non-negative, continuous function $f$ defined on the unbounded $[n_0,+\infty[$, on which it is monotone decreasing. Then
$\forall (p,q)\in\mathbb{N}^{2}$ such that $n_o\... | The value of $p$ is different for each of the inequalities.
*
*The inequality $\displaystyle \int_1^{n+1} \frac{1}{x}\, dx \le \sum_{k=1}^n \frac{1}{k}$ comes from taking $p=0$ and $q=n$ in the theorem.
*The inequality $\displaystyle \sum_{k=2}^n \frac{1}{k} \le \int_1^n \frac{1}{x}, dx$ comes from taking $p=1$ and... | {
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"timestamp": "2023-03-29T00:00:00",
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What are examples of irreducible but not prime elements?
I am looking for a ring element which is irreducible but not prime.
So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$.
This is irreducible because in any product $x+y=fg$ only one factor, say f, can have a $x$ in it (oth... | Let $\rm\ R = \mathbb Q + x\:\mathbb R[x],\ $ i.e. the ring of real polynomials having rational constant coefficient. Then $\,x\,$ is irreducible but not prime, since $\,x\mid (\sqrt 2 x)^2\,$ but $\,x\nmid \sqrt 2 x,\,$ by $\sqrt 2\not\in \Bbb Q$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + ... + 1/(x+a_n) = 1/x$ are all real Prove that the roots of the equation:
$$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$
are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
| We can prove a stronger statement: the equation above has $n - 1$ real positive roots and a negative real one, and there are no other roots.
Let
$$g(x) = \sum_{i = 1}^n \frac1{x-a_i} - \frac1x,\qquad a_i \in (0, +\infty).$$
Note that $g(x)$ is defined in $\mathbb R \setminus \{0, a_1, \ldots, a_n\}$ and it's also conti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1871711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
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How to know if a segment is completely included between two lines? I have three segments (not necessarily parralel):
*
*blue $((ax1, ay1), (ax2, ay2))$
*green$((bx1, by1), (bx2, by2))$
*red $((cx1, cy1), (cx2, cy2))$
and a $margin$ value which is the width of the sky blue band in the sketch bellow (with infinite... | Of course. Since the band is convex, to make sure the segment is completely in the band, you only need two endpoints in the band. (It is applicable for all convex domains)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1871814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Implied plus-minus sign in radical equation? Say we have: $$\sqrt{x+7}=5-x$$
Is it implicitly understood that the following also holds? $$-\sqrt{x+7}=5-x$$
I'm exploring the notion of "extraneous solutions." In this example, solving either equation leads to two results, namely x=2 and x=9.
Standard practice is to chec... | If what you say is true, then we should be able to add both sides,
$$\sqrt{x+7}-\sqrt{x+7}=5-x+5-x$$
$$0=2(5-x)$$
$$\implies x=5$$
Going back, this is not true in either equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1871940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Limit of function. How can it suddenly change it's domain after simple manipulations I'm trying to refresh my math at the moment and have quickly become very confused by the calculation of limits of functions.
For example, I solved the following
$\lim_{x \to 0} \frac{7x^2+4x^4}{3x^3-2x^2}$
by first manipulating it to ... | You are dividing with $x^2$ which lies in the denominator and hence $x\neq0$. So it can't be $f(0)$ anymore.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Determine if the following short exact sequence is split. Do the following short exact sequences split?
$$0\longrightarrow A\longrightarrow B\longrightarrow \mathbb{Z}^2 \longrightarrow 0$$
$$0\longrightarrow\mathbb{Z}\longrightarrow A\longrightarrow B\longrightarrow 0$$
This is a question on a Ph.D Topology exam. I k... | For the first short exact sequence, note that $\mathbb{Z}^2$ is a free $\mathbb{Z}$-module, it is thus projective and the short exact sequence thus splits. In fact, a characterizing property for projective modules is the following.
$M$ is a projective $R$-module if and only if any short exact sequence
$$0\longrightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Spanning 2-regular subgraphs in even regular graphs.
Theorem: Every regular graph of positive even degree has a spanning 2-regular subgraph.
This was taken from Corollary 5.10 of ETH Zurich's notes on graph theory. The proof constructs a Eulerian tour, splits the vertices into in and out vertices on the tour, then in... | A $2$-regular graph is a union of disjoint cycles.(So it doesn't have to be exactly one cycle)
You yourself have provided an example of a $4$-regular graph with a $2$-regular spanning subgraph but no hamiltonian cycle.
An example of a $2$-regular subgraph in your linked graph is the union of the following two cycles:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Derivative of a analytic function at its fixed point Let $D$ be a bounded domain, and let $f(z)$ be an analytic function from $D$ to $D$.Show that if $z_{0}$ is fixed point for $f(z)$,then $|f'(z_{0})|\leq 1$
All the conditions above make me think about Schwartz Lemma to solve this problem.But I don't know how to const... | Assume $D$ is simply connected. Let $\phi: D \to \Bbb U$ be a conformal map with $\phi(z_0)=0$, guaranteed by Riemann Mapping Theorem.
Define $g = \phi \circ f \circ \phi^{-1}: \Bbb U \to \Bbb U$. Then $g(0)=0$, so $|g'(0)| \leq 1$ by the Schwarz Lemma.
But $(\phi^{-1})'(z_0) = 1/\phi'(f(z_0))$ and therefore by the cha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine if certain operation is associative based on Cayley table I have the following table and I don't know how to determine if an operation is associative based on the table. Is there an easy way to do it? Or it's just brute force
\begin{array}{|c|c|c|c|c|c|}
\hline
*& a & b & c &d &e \\ \hline
a& a&b &c&b... | Light's associativity test is based on the following Lemma.
Let $*$ be a binary operation on the set $S$ (called product).
Definition:
A subset $G$ of $S$ generates $S$ if every element of $S$ can be generated as product of elements of $G$.
Lemma: If G generates S then * is associative on S if and only if
$$\foral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculation of $\frac{a_{20}}{a_{20}+b_{20}}$? The solution of $\frac{a_{20}}{a_{20}+b_{20}}$ is $-39$ (This is wrote by answer sheet) from the recursive system of equations :
\begin{cases}
a_{n+1}=-2a_n-4b_n \\
b_{n+1}=4a_n+6b_n\\
a_0=1,b_0=0
\end{cases}
This is taken from $2007$ GATE entrance exam in India.
anyone ... | $4b_n=-a_{n+1}-2a_n$, $4b_{n+1}=-a_{n+2}-2a_{n+1}$, $-a_{n+2}-2a_{n+1}=16a_n-6a_{n+1}-12a_n$, $a_{n+2}-4a_{n+1}+4a_n=0$. Do you know how to solve that kind of recurrence?
Here's an approach. $a_{n+2}-4a_{n+1}+4a_n=(a_{n+2}-2a_{n+1})-2(a_{n+1}-2a_n)=c_{n+1}-2c_n$, where we are defining $c_n$ by $c_n=a_{n+1}-2a_n$. Now w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Hadamard-like complex variable substitution \begin{align}
\frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\
\tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)}
\end{align}
So far so good. Now introduce a complex variable $z$ and its conjugate $z^*$ such that
$$
x = \frac{z+z^*}{2}, y=\frac{z-z^*}{2i}
$$
I su... | The substitution you made yields the Jacobian
$$\begin{vmatrix}\cfrac12&\;\;\cfrac12\\\cfrac1{2i}&-\cfrac1{2i}\end{vmatrix}=-\frac1{2i}\implies dxdy=-\frac1{2i}dzdz^*$$
The minus sign is not that relevant here as without a specific integration path, and thus some orientation, it rather is moot.
I can't tell anything ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Upper bound to a series with binomial coefficients Let $c>0$ and $m$ be a positive integer. The following sum is convergent, but how fast does it grow with $m$ as $m$ is large?
$$
f(m)= \sum\limits_{n=1}^{\infty} \binom{n + m}{n} e^{-c \, n}
$$
Is there a polinom in $m$, $g(m)$, such that
$$f(m) \leq g(m) ?$$
| $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counting the degrees of a face in planar graph I've been having trouble wrapping my head around this concept. How do I calculate the degree of a face in planar graphs. In our textbook, we are given this image:
where $f_1, f_2, f_3, f_4$ are the faces.
In the textbook, it gives the degrees of the four faces as
$$deg(f_... | Think of the edges as being two sided. As you move around the face of $f_1$ you see both sides of the leaf edge, so that edge is counted twice. Likewise, when you travel around the outer face you see the bridge edge twice (both sides of it) so it is counted twice. Hope this helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1872960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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How could a statement be true without proof? Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions;
1) How can we say that a statement is true without a proof?
2) What has the self reference to do with this? God... | *
*Assuming your formal system is consistent, Gödel shows there is a statement in that system whose interpretation is true but that is unprovable in the system. The statement is actually provable, but not in that system: you need the additional assumption that the system in consistent, and that is not provable in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
How does changing the log base affect the intepretation of information entropy Entropy of random variable is defined as:
$$H(X)= \sum_{i=1}^n p_i \log_2(p_i)$$
Which as far as I understand can be interpreted as how many yes/no questions one would have to ask on average, to find out the value of the random variable $X$.... | Obviously this interpretation breaks down (at least somewhat) for non-integer bases, but for any logarithm base $b$, not just base $b=2$, we can interpret the information with respect to that base, $$H_b(X) = \sum_{i=1}^n p_i \log_b(p_i)$$ to be the average number of $b-$ary questions one would have to ask on average i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $[\sin x]' = \cos x$ without using $\lim\limits_{x\to 0}\frac{\sin x}{x} = 1$ I came across this question: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
From the comments, Joren said:
L'Hopital Rule is easiest: $\displaystyle\lim_{x\to 0}\sin x = 0$ and $\displaystyle\lim_{x\to 0} = 0$, so $\displayst... | What is required here is not a proof of $\sin'=\cos$ without using $$\lim_{\phi\to0}{\sin\phi\over\phi}=1\tag{1}\ ,$$ but a proof of the basic limit $(1)$ using the "geometric definition" of sine provided by the OP. To this end we shall prove "geometrically" that
$$\sin\phi<\phi\leq\tan\phi\qquad\left(0<\phi<{\pi\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Matrix differential equation of the form $X'=CX$ Let $n \in \mathbb{N}^{\ast}$ and $\mathrm{Sym}(n)$ (respectively $\mathrm{Spd}(n)$) denote the linear space (respectively set) of real $n \times n$ symmetric (respectively positive definite) matrices. I am interested in the following matrix differential equation :
$$ \f... | In your generality, no.
Let $V = \begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$ and $S^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 1\end{pmatrix}$. Then
$$ VS^{-1} = \begin{pmatrix} 2& 1 \\ 2 & 1 \end{pmatrix} $$
Let $X(0) = \begin{pmatrix} 1 & 1 \\ 1 & -2 \end{pmatrix} $, the corresponding solution to the ODE is
$$ X(t) = \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Joint probability density function $(X^2,Y^2)$ Let $X$ and $Y$ be random variables having the following joint probability density function
$f(x,y)=\begin{cases}
\frac{3}{8}xy & x\geq0,\,y\geq0,\:x+y\leq2,\\
0 & \mbox{otherwise}.
\end{cases}$
Find the joint probability density function of $X^2$ and $Y^2$.
This is my... | I think the issue is the original pdf:
$$ \int_{\mathbb{R}^2}f(x,y)\;dydx=\frac{3}{8}\int_0^2\int_{0}^{2-x}xy\;dydx=\frac{3}{8}\int_0^2\frac{x(2-x)^2}{2}\;dx$$
$$ =\frac{3}{8}\int_0^2\frac{(2-x)x^2}{2}\;dx=\frac{3}{8}\cdot\frac{2}{3}=\frac{1}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Flipping coins- percentages of heads vs tails If I flip a coin multiple times and count the number of time it fell on heads and the number of times it fell on tails and keep a track of them. In how many flips on average will the delta between percentage of heads and percentage of tails will be less than 0.1%?
|
I want to check if a coin is fair(lands 50% of the times on each
side. I assume that delta of 0.1% between them is fair). How many
flips do i need in order to be 99% confident that the coin is fair?
For large n you can apply moivre laplace. So you approximate the binomial distribution by the normal distribution. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Help with simplification of a rational expression (with fractional powers) Can you please help me see what I don't see yet. Here's a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029):
$$ \frac{ (a^{1/m}-a^{1/n})^{2} \cdot 4a^{(m+n)/mn} }{ (a^{2/m}-a^{2/n}) (\sqrt[m]{a^{m+1}} + \sqrt[n]{a^{n+1}}) ... | After checking the older edition of the book, I'm quite sure that the original problem looked like this:
$$\frac{ (a^{1/m}-a^{1/n})^{2} \color{blue}{+} 4a^{(m+n)/mn} }{ (a^{2/m}-a^{2/n}) (\sqrt[m]{a^{m+1}} + \sqrt[n]{a^{n+1}}) }$$
Now we have:
$$(a^{1/m}-a^{1/n})^{2} \color{blue}{+} 4a^{(m+n)/mn} =(a^{1/m}+a^{1/n})^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute trigonometric limit without use of de L'Hospital's rule
$$ \lim_{x\to 0} \frac{(x+c)\sin(x^2)}{1-\cos(x)}, c \in \mathbb{R^+} $$
Using de L'Hospital's rule twice it is possible to show that this limit equals $2c$. However, without the use of de L'Hospital's rule I'm lost with the trigonometric identities.
I c... | Your first step is very good: by multiplying numerator and denominator by $1+\cos x$, the limit becomes
$$
\lim_{x\to0}(x+c)(1+\cos x)\frac{\sin(x^2)}{\sin^2 x}=
\lim_{x\to0}(x+c)(1+\cos x)\frac{\sin(x^2)}{x^2}\frac{x^2}{\sin^2 x}
$$
and it should be easy to finish.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Find a point R, such that angle increases 3 times Let $X=(4,0), Y=(4,3), O=(0,0) $ are points. I have to find point R with integer coordinates, such that $3|\angle$$XOY|$=$\angle$$|XOR|$. I think it's $R=(-3,8)$, but I am not sure. How can I prove it? Thanks for your help.
| If $v, w$ are vectors in $\mathbb{R}^2$, the cosine of the angle between them is equal to $$\frac{v \cdot w}{||v|| \cdot ||w||}$$
Also, the triple angle identity formula can be derived from the formula for $\cos x+ y$: $$\cos 3x = 4 \cos^3 x - 3 \cos x$$
Let $\phi$ be the angle between $X$ and $Y$, and $\theta$ be the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to factorize the polynomial $a^6+8a^3+27$?
I would like to factorize $a^6+8a^3+27$.
I got different answers but one of the answers is
$$(a^2-a+3)(a^4+a^3-2a^2+3a+9)$$
Can someone tell me how to get this answer? Thanks.
| By the rational root theorem, we guess the factors include at least one of the following:
$$x \pm 27$$
$$x \pm 9$$
$$x \pm 3$$
$$x \pm 1$$
Where did I get those?
*
*Take the absolute value of the last term (in descending order), $27$, and the absolute value of the last term, $1$.
*Take the factors of each:
$$27: \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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About the eigenvalues of a block Toeplitz (tridiagonal) matrix I have found the following $n\times n$ squared matrix in one stability analysis problem (i.e. I have to identify the sign of its eigenvalues)
$$
A(\theta) = \begin{bmatrix}
W(\theta)+W(\theta)^T & -W(\theta) & 0 & \dots & 0 & 0 \\
-W(\theta)^T & W(\theta)+W... | $A=B^TB$ for some $B$ if and only if $A$ is positive semidefinite. Let $c=\cos\theta$ and $w=e^{i\theta}$. Then $A$ is unitarily similar to $C\oplus \overline{C}$, where
$$
C=\pmatrix{
2c&-w\\
-\bar{w}&\ddots&\ddots\\
&\ddots&\ddots&-w\\
&&-\bar{w}&2c}.
$$
Now $C$ is a Hermitian tridiagonal Toeplitz matrix. Its eigenva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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$2^{n!}\bmod n$ if $n$ is odd Given an odd number $n$, find $2^{n!}\bmod n$ and what if $n$ is even?
I am not getting how to deal with that $n!$ in the power of $2$.
Any help will be truly appreciated.....
| For the first question, note that $\varphi(n)$ divides $n!$, and use Euler's Theorem.
The $n$ even problem is more interesting. Let $n=2^km$ where $m$ is odd. Then by the result for odd moduli, we have $2^{n!}\equiv 1\pmod{m}$. Also, $2^{n!}\equiv 0\pmod{2^k}$. Now use the Chinese Remainder Theorem.
Added: In more deta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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3-sphere complex co-ordinates I am currently trying to understand some mathematical physics papers that deal with torus knots.
I am trying to find the origin of a complex scalar field used. These fields are somehow related to the Hopf fibration.
I have spent the last week reading about and trying to understand the Hopf... | The following answer is not rigorous. To make it rigorous you may need to know more about smooth manifold.
$\mathbb S^3 = \{ (u, v) \in \mathbb C^2 : |u|^2 + |v|^2 =1\}$ is itself a $3$-manifold, which means that locally it looks like an open sets in $\mathbb R^3$. Mathematically speaking, it means that for each poin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to estimate the range of a normal distribution when the mean and standard deviation are given For example, how would you respond to this question?
The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4, find an approximate value... | Recall that about $99.7\%$ of data under a normal curve falls within three standard deviations of the mean. When you are given the mean and standard deviation, this seems like a pretty good way to approximate the range. So since the mean is $24$, we could estimate that most of the data falls in the interval $$[24-3(4),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$. We are asked to find the universal cover of the wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.
I am second guessing myself on this problem because I cam... | Edit (last answer was wrong)
The universal covering space of $S^2\lor S^2$ is itself. However, once you introduce the projective plane, the wedge point splits, so $S^2\lor \Bbb RP^2$ has a chain of three spheres as universal cover, where the middle sphere is a two-sheeted cover of $\Bbb RP^2$, and the two other spheres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to minimize $f(x)$ with the constraint that $x$ is an integer? I would like to find the integer x that minimizes a function. That is:
$$
x_{min} = \min_{x \in \mathbb{Z}}{(n - e^x)^2}
$$
The goal is to write a program that computes the integer $x$ such that $e^x$ is closest to $n$, preferably avoiding conditionals.... | The fact that the minimum is squared is irrelevant, if you take the absolute value instead. Therefore $$x_{min}=\min_{x\in Z}{|n-e^x|}$$
If x werent constrained to the integers, x would equal as you pointed out ln(n), but since it is constrained to integers, to find $x_{min}$, take $\min(e^{Floor(ln(n))},e^{Ceil(ln(n))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874539",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 1
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Corollary of Schur's Lemma - why abelian
Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional.
My question is now, why has the group to be abelian? As far as I know, we want the representation $\rho(g)$ to be a $Hom_G(V,V)$, where $V$ is the representat... | I had the same question, for maybe longer than a year, but because of a stupid mistake in understanding Schur's. Here it is, in case ( hoping ;) ) that someone else might do the same mistake:
Schur's lemma says something about >any< linear $\psi$ so that
(*) $ \;\;\;\; \psi \rho (g) = \rho (g) \psi$ $\;$ $\forall g$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Any smart ideas on finding the area of this shaded region?
Don't let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can't think of an easy/smart way of finding it.
Any ideas?
For reference, the Area is:
$$\bbox[10pt, border:2pt solid grey]{90−18.75\pi−25\cdot \arc... | The big black roundy-corner on the bottom right has area $(10^2 - \pi\cdot 5^2)/4$ and there are 3 complete copies of it and one copy trimmed by a small roundy-triangle. We will focus on this roundy-triangle which is the same as the one in the bottom left.
So the key is to compute the area of the small white roundy-tri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
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"answer_id": 9
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How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
How does one show that $$\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$$ for each nonnegative integer $n$?
I tried using the Snake oil technique but I guess I am applying it incorrectly. With the snake oil technique we have $$F(x)= \sum_{n=0}^{\infty}\le... | $$\begin{align}
\sum_{k=0}^n \binom {n+k}k\frac 1{2^k}
&=\frac 1{2^n}\sum_{k=0}^n \color{blue}{2^{n-k}}\binom {n+k}n\\
&=\frac 1{2^n}\sum_{k=0}^n\color{blue}{\sum_{r=0}^{n-k}\binom {n-k}r}\binom{n+k}r\\
&=\frac 1{2^n}\sum_{s=0}^n\sum_{r=0}^s\binom sr\binom {2n-s}n
&&\scriptsize (\text{Putting } s=n-k)\\
&=\frac 1{2^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 15,
"answer_id": 5
} |
prove inequality using methods of differential calculus Could you help me prove following inequality ?
$$(x+y)^{\alpha}\le x^{\alpha} + y^{\alpha} $$
$$x,y\ge 0, \alpha \le 1$$
I don't know from what start, I should use methods of differential calculus.
| We may assume $0<\alpha<1$, because otherwise the claim is trivial (or wrong). The function $f(x):=x^\alpha$ $(x\geq0)$ is concave, i.e., has a decreasing derivative $f'(x)=\alpha x^{\alpha-1}$. It follows that
$$f(x+y)-f(x)=\int_0^y f'(x+t)\>dt\leq \int_0^y f'(t)=f(y)\ ,$$
which is equivalent to the claim.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1874883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Lagrange theorem question I'm trying to teach myself group theory and this question is the final one in an exercise on Lagrange theorem and it has me currently stumped.
Finite group ${G}$ contains distinct elements ${a}$ and ${b}$ and identity ${e}$, such that: ${a*b=a^3*b^4}$ and ${a^4*b^3=e}$. Show that ${a^2=e}$ a... | Proof that $a^2=e$:
$a^4*b^3=e$. Multiply both sides by $b$ from right to get to $a^4*b^4=b$. This combined with the assumption that $a*b=a^3*b^4$, implies $a*a*b=b$ (more details: $a^4*b^4=a*a^3*b^4=a*a*b$). Multiply both sides by $b^{-1}$ to get to $a*a=e$ or $a^2=e$.
Proof that $|G|$ is a multiple of $6$:
First we ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is this lot drawing fair? Sorry for a stupid question, but it is bugging me a lot.
Let's say there are $30$ classmates in my class and one of us has to clean the classroom. No one wants to do that. So we decided to draw a lot - thirty pieces of paper in a hat, one of which is with "X" on it. The one who draws "X" has ... | Actually, it looks like its fair to me.
The probability that person 1 chooses an "X" is $\frac{1}{30}$, since only one of the 30 lots has an "X" on it.
The probability that person 2 chooses an "X" is $\frac{29}{30}*\frac{1}{29} =\frac{1}{30} $, since Person 1 must have not chosen a lot with an "X" on it.
The probabilit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1875045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 4
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$ \int_{-\infty}^{\infty} \frac{e^{2x}}{ae^{3x}+b} dx,$ where $a,b \gt 0$ Evaluate
$$ \int_{-\infty}^{\infty} \frac{e^{2x}}{ae^{3x}+b} dx,$$ where $a,b \gt 0$
I tried using $y=e^x$, but I still can't solve it.
I get $\displaystyle\int_0^\infty \frac y{ay^3+b} \, dy.$
Is there any different method to solve it?
| $$
\begin{align}
\int_{-\infty}^\infty\frac{e^{2x}}{ae^{3x}+b}\,\mathrm{d}x
&=\frac1{3b}\left(\frac ba\right)^{2/3}\int_0^\infty\frac{u^{-1/3}}{u+1}\,\mathrm{d}u\tag{1}\\
&=\frac13\left(\frac1{ba^2}\right)^{1/3}\pi\csc\left(\frac23\pi\right)\tag{2}\\
&=\frac{2\pi}{3\sqrt3}\left(\frac1{ba^2}\right)^{1/3}\tag{3}
\end{ali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1875122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Gradient of a real valued function defined on a sphere
Points on a sphere of radius $R$ is expressed in spherical coordinates as $\left(\varphi,\theta\right)$.
For a real valued, continuous and differentiable function $f:\mathbb R\times\left[0,\pi\right] \to \mathbb R$ evaluated on said sphere centered at $\left(0,0,0... | Rigorous formulation
This involves the definition of the surface gradient operator, which is defined as
$$
\nabla_{\Gamma} = \nabla -
{\mathbf e}_{r} \left({\mathbf e}_{r} \cdot \nabla\right).
$$
The projection of the gradient along the unit normal ${\mathbf e}_{r}$ is evaluated by
$$
{\mathbf e}_{r} \left({... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1875194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $(A^\top A \mathbf{x}, \mathbf{x}) = (A \mathbf{x}, A \mathbf{x})$? Let $(\mathbf{x}, \mathbf{y})$ denote the inner product between $\mathbf{x}$ and $\mathbf{y}$, and let $A$ be a real matrix. Why is $(A^\top A \mathbf{x}, \mathbf{x}) = (A \mathbf{x}, A \mathbf{x})$?
Using the scalar product it's easy to see th... | We have $$(A^TAx,x)=[A^TAx]^Tx=[A^T(Ax)]^Tx=[(Ax)^TA]x=(Ax)^T(Ax)=(Ax,Ax)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1875312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Analytic continuation for $\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$ Define a function $F(s)$ by:
$$F(s)=\sum_{n=0}^{\infty}(\sqrt n+1/3)^{-s}$$
Is there a closed form expression for the analytic continuation of $F(s)$ to $F(-s)$?
| Not sure whether this matches the query for closed form, but Euler-Maclaurin summation gives an exact expression for the analytic continuation to any $s\in\mathbb{C}\setminus\{1,2\}$:
For abbreviation, let's write $f(s,x):=(\sqrt{x}+a)^{-s}$ for $x\in\mathbb{R}$, $0<a\in\mathbb{R}$. For $a=\frac{1}{3}$ we have the OP.
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $X$ is compact metric then $B(X)$ is non-separabale I've seen the following argument:
Let $X$ be a compact metric space, then denote by $B(X)$ the set of bounded, measurable functions on $X$. $B(X)$ with the sup-norm is a $C^*$-algebra. Then, $B(X)$ is not separable.
Now, if I take $X=[0,1]$, and denote $A=span_{\ma... | This is false as stated. For example, $X$ could be finite, in which case $B(X)$ may as well be some $\mathbb R ^n.$
Assume $X$ is infinite. Then it contains a countably infinite subset $\{x_1,x_2,\dots\}.$ Note that singletons are closed, hence Borel, hence all countable subsets of $X$ are Borel sets. Define a map $F$ ... | {
"language": "en",
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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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If $\text{Area} (A) = \text{Area} (B)$ and $\text{Perimeter}(A) = \text{Perimeter}(B) \implies A \cong B$? If I have an $n$-gon $A$ and a convex $n$-gon $B$ with the same perimeter and the same area, does $A\cong B$?
Edit : What becomes the answer if I replace convex by regular?
| If they're both regular then just one of the given equalities is enough to show they're congruent.
For example, let's denote $a_1, a_2$ the edges of the two equal perimiter polygons, then we get $na_1=na_2$. Clearly $a_1=a_2$. Same goes for area.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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The number of possible ways to select the four vertices of eleven vertices A cycle graph is a graph that consists of a single cycle or in the other words some number of vertices connected in a close chain and denoted by $c_n$.
I need the number of possibility ways for to select the four vertices of eleven vertices such... | An admissible selection of $k$ nonadjacent vertices from an $n$-cycle can be realized as follows: Choose an arbitrary first vertex, determining a train of $n-1$ consecutive vertices in between. In the end there will be $n-k$ unchosen vertices, and $k-1$ more chosen vertices in $k-1$ different slots between the unchos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.
Problem. Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational.
I ha... | It's actually possible to demonstrate that this is NOT a graph-theoretic problem.
The graph-theoretic condition equivalent to having a finite collection of irrational numbers as vertices, and recording (with edges) which pairs have rational sums is
Graph that is a disjoint union of complete bipartite graphs $K_{m,n}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "92",
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Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$. So I have seen the similar question and answers on here for $x^4 +1$, but I am having trouble extending anything there to this polynomial... I understand it is fairly trivial with Galois theory, but my class has just barely covered... | Another simple explanation comes from the fact that the zeros of $m(x)=x^4-10x^2+1$ are
$$
x=\pm\sqrt2\pm\sqrt3
$$
with all four sign combinations. So if $p$ is a prime, then you get the splitting
field of $m(x)$ over $K=\Bbb{F}_p$ by adjoining $\sqrt2$ and $\sqrt3$. Because up to isomorphism the field $K$ has only a s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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If $(w + 1)(w - 1) = w$, find $ { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } } $. Recently I was asked a question by my student that completely stumped me.
$$\text{If }(w + 1)(w - 1) = w\text{, find } { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } }. $$
One "cheat" method that I used was to solve for the exact value of $w$ from th... | We have $w-\frac{1}{w}=1$, and then $w^2+\frac{1}{w^2}=3$. Put $u_n=w^{2n}+\frac{1}{w^{2n}}$; we have $u_0=2$, $u_1=3$, and
$$u_{n+1}(w^2+\frac{1}{w^2})=u_{n+2}+u_n$$
Hence $u_{n+2}=3u_{n+1}-u_n$, it is easy to compute $u_2,u_3,u_4$, and finally $u_5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1875995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to show the existence of a subgroup of an abelian group?? If an abelian group has subgroups of orders m and n, respectively, then show that it has a subgroup whose order is the least common multiple of m and n.
I tried solving this on assumption that if an abelian group has a subgroup of order, say, N then it has a... | Let $H_0$ and $H_1$ be subgroups of size $m,n$ and let $H_2=H_0\cup H_1=\{h_1,h_2\dots h_s\}$ and suppose the elements have orders $o_1,o_2\dots o_s$ respectively .Then the set $H_3=\{h_1^{e_1}h_2^{e_2}\dots h_2^{e_s}| e_1,e_2\dots e_s\in \mathbb Z\}$ is a subgroup, because $G$ is abelian. It is also finite as it has a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1876063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the proof, that $\sum_{n=0}^{\infty} \frac{1}{2^n}\frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}$ is a metric? What is the proof, that $d(x,y) = \sum_{n=0}^{\infty} \frac{1}{2^n}\frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}$ is a metric? Where $x=(x_1,x_2,\cdots)$ , $y=(y_1,y_2,\cdots)$ and $d_n$ is a metric for $X_n$.
How does one... | Since $d_i$ is metric, inequality $$d_i(x_i,z_i)\leq d_i(x_i,y_i)+d_i(y_i,z_i)$$ holds in i-th metric space from the product space.
Also, metrics $d(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are equivalent.
(If $d$ is a metric, then $d/(1+d)$ is also a metric).
Therefore, we have $$\frac{d_i(x_i,z_i)}{1+d_i(x_i,z_i)}\leq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ways of showing $\sum_\limits{n=1}^{\infty}\ln(1+1/n)$ to be divergent
Show that the following sum is divergent
$$\sum_{n=1}^{\infty}\ln\left(1+\frac1n\right)$$
I thought to do this using Taylor series using the fact that
$$
\ln\left(1+\frac1n\right)=\frac1n+O\left(\frac1{n^2}\right)
$$
Which then makes it clear th... | "Sophisticated" does not mean "complicated". In my opinion, despite using more sophisticated ideas (asymptotic analysis), your proof is simpler than all of the other current answers — even the one expressing it as a telescoping series.
Incidentally, you possibly made an oversight: to complete the proof,
$$ \sum_{n=1}^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Formal proof that weak partial order difference equivalence relation is a strict partial order. I'm having difficulty with the following problem:
Prove: if $R$ is a weak partial (linear) order on $X$, then $R^− = R \; \backslash\ Id_X$ is a strict partial (linear) order.
I know that as a weak partial order, $R$ is ref... | Suppose that $\langle x,y\rangle\in R^-$ and $\langle y,z\rangle\in R^-$; then $\langle x,y\rangle,\langle y,z\rangle\in R$, and $R$ is transitive, so $\langle x,z\rangle\in R$. This means that $\langle x,z\rangle$ will be in $R^-$, as desired, unless $x=z$, so we want to rule out that possibility. But if $x=z$, then $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Characterize an analytic function with restriction of its growth Characterize all analytic functions $f(x)$ in $|z|<1$ such that $|f(z)|\leq|\sin(1/z)|$ for all points in punctured disk.
I think we should change the form of $\sin(1/z)$ to find a connection with polynomial which $f(x)$ can be expanded into. But I don't ... | Hint:
Take the sequence $\{z_n\}=\left\{\frac{1}{n\pi}\right\}$ and use Identity theorem.
What is $f(z_n)$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1876520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Parity of a generalized characteristic polynomial Let
$$Q(\lambda)=\det\begin{pmatrix}-\lambda C_{11}&C_{12}\\C_{21}&-\lambda C_{22}\end{pmatrix}$$
where $C_{ij}$ are (not necessary square) matrices.
In this answer, it is claimed that $Q(\lambda)$ has a parity,i.e. $Q(\lambda)=\pm Q(-\lambda)$, but no explanation is gi... | Suppose $C_{11}$ is $m \times k$, so that $C_{22}$ is $(n-m) \times (n-k)$ where the overall matrix is $n \times n$.
Consider the Leibniz formula. Each term has $n$ factors that are matrix elements, one factor in each row and one factor in each column. If a term has $a$ factors in the top left block, it must have $m-... | {
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Are there names for $A_n$, $Z_n$ and $B_n$ in a chain complex? I mean similar name like $n$-th homology group. If there would be, say "$n$-th component" for $A_n$, "The cycles of $A_n$" for $Z_n$, and "The boundaries of $A_n$" for $B_n$ than I could say that "Any cycle and boundary preserving homomorphism between the c... | If $C=(C_p)$ is a (co)cochain complex then elements in $C_p$ are usually called $p$-(co)chains.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1876653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral over $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$ What is
$$\int_{S}(x+y+z)dS,$$ where $S$ is the region $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$?
We can change the region to $0\leq x,y,z\leq 1$ and $x+y+z\geq 2$, because the total of the two integrals is just
$$\int_0^1\int_0^1\int_0^1(x+y+z)dxdydz=3\int_0^1xdxdydz=\fr... | You could write
\begin{equation}
\int_S x \, dx dy dz = \int_0^1 \left( \int_{y+z \leq 2-x; \, 0\leq y,z \leq 1} dy dz \right) x dx
\end{equation}
Now, you can interpret $y+z \leq 2-x$ with $y,z \geq 0$ as a triangle in the plane, whose area is $\frac{(2-x)^2}{2}$.
From this triangle, you subtract two smaller triangle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Interesting probability distribution of a mixed type random variable $Y$ Let $X$ and $U$ be independent random variables with:
$$P(X=k)=\frac1{N+1} \text{ for } k=0,1,2,\ldots,N$$
and $U$ having uniform $(0,1)$ distribution.
Let $Y=X+U$.
Find distribution function of $Y$.
I have tried to solve the problem by conditioni... | Suppose $k\in\{0,1,2,\ldots, N\}$ and $0\le a<b\le 1$. Then
$$
\begin{align}
& \Pr(a<X+Y<b) = \Pr(X=k\ \&\ a<Y<b) = \frac 1 {N+1}\cdot(b-a) \\[10pt]
= {} & \frac{\text{length of the interval }(a,b)}{\text{length of the interval} (0,N+1)} \\[10pt]
= {} & \frac{\text{length of the interval }(k+a,k+b)}{\text{length of th... | {
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Spivak's Calculus - Chapter 1 Question 1.5 - Proof by Induction In Spivak's Calculus Fourth Edition, Chapter 1 Question 1.5 is as follows:
Prove $x^n - y^n = (x - y)(x^{n-1}+x^{n-2}y+ \cdots + xy^{n-2}+y^{n-1})$ using only the following properties:
It's easy enough for me to use P9 to expand the right-hand side:
$$... | The basic steps of proof by induction are:
*
*Identify the Induction Hypothesis
In this case, it is $P(n) : x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1})$
Note that it is often a good idea to write the Induction Hypothesis using iterative operators, like $\sum$ and $\prod$; doing so usually makes the a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $x^{-0}$ defined? In the form of mathematics that most of humanity is taught, the following operation is undefined:
$\Large{\frac{x}{0}}$
But, how about the following operation?
$\Large{x^{-0}}$
Is the following statement true?
$\Large{x^{-0}=\frac{1}{x^0}=\frac{1}{1}=1}$
| $x^y=e^{ylog(x)}$ so $x^0=x^{-0}$ are defined for $x>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Binomial identity: Clever idea for a short proof? When answering this question I had to cope with this binomial identity:
The following holds true
\begin{align*}
\sum_{i=0}^k\binom{k}{i}\sum_{j=0}^i\binom{i}{j}(-1)^{k-i}j^{i-j}
=\sum_{i=0}^k\binom{k}{i}\sum_{j=0}^i\binom{i}{j}(-1)^{i-j}j^{k-i}\qquad\qquad k\geq 0
\e... | Let $K$ be a set with $k$ elements. Both sides can be interpreted as counting, with signs, choices of
*
*a subset $I$ of $K$,
*a subset $J$ of $I$, and
*either a function $K \setminus I \to J$ (on the LHS) or a function $I \setminus J \to J$ (on the RHS).
Picking $I$ and $J$ is equivalent to picking a partitio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
triple vector product: vector vs gradient I think there's a simple explanation for this, but could not find one from a few online searches. The triple vector product and the curl of $\mathbf{A}\times \mathbf{B}$ have very similar forms, however there are additional terms in the differentiation case:
$
\mathbf{A} \times... | As stated above, the first expression given is simply product of vectors, which can be expressed in terms of the dot product. The second involves differentiation, acting on a product. The product rule for vector differentiation will inevitably lead to the extra terms.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is $\mathbb{R}$ with the particular point topology and open half line topology compact? Consider $(\mathbb{R}, \tau_{p})$ and $(\mathbb{R}, \tau_l)$ (or sometimes called "ray topology")
Where $\tau_{p} = \{U \subseteq \mathbb{R}, p \in U\}\cup\{\varnothing\}$, and $\tau_l = \{(a, \infty)| a \in \mathbb{R}\}\cup\{\varno... | *
*Careful! not every open cover needs to contain $\mathbb{R}$. You will have to use a different approach.
*This is fine but actually I don't think you need uncountablility of $\mathbb{R}$ here. For instance, I think your argument would work for $\mathbb{Z}$ as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluating the definite integral $\int_{-1}^1 \lfloor \arccos x \rfloor \,dx$ involving the greatest integer function
Evaluating the definite integral $$\int_{-1}^1 \lfloor \arccos x \rfloor \,dx .$$
I tried it but unable to do as it is discontinuous. Somebody told me that you should do this by drawing graph---how?
C... | Hint:
Use that arccosine is a decreasing function, and that $\arccos(cos x)=x$ if $0\le x\le \pi$. Hence $f(x)=\lfloor\arccos x\rfloor$ is a step function, defined by:
$$f(x)=\begin{cases}3 &\text{if }\quad\!{-1}\le x\le\cos 3,\\2 &\text{if } \cos 3< x \le \cos 2, \\ 1 &\text{if } \cos 2< x\le \cos 1,\\ 0 &\text{if }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Truth tables for extremely long expressions One of the questions from my text book says to
Write the truth table for the expression:
$$
p \vee ( \neg (((\neg p \vee q) \rightarrow q) \land p ))
$$
and state whether it is a tautology, contradiction or neither.
Please do not think I am asking you to do my work for me, b... | Since there are only two variables in your expression, you will only need to evaluate it four times. Using the method you described, you can evaluate the expression outwards one step at a time using the following columns:
$p\ |\ q\ |\ \lnot p\lor q\ |\ \left(\lnot p\lor q\right)\rightarrow q\ |\ \left(\left(\lnot p\lor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Number of eigenvalues and their eigenspaces So Let matrix $A$ have eigenvalues as follows :
$$
e_1=0\\
e_2=0\\
e_3=2\\
e_4=2\\
$$
From here can we deduce that dimension of the eigenspace when the eigenvalues is $2$ is 2? can we deduce this?
If we could deduce that we could also deduce that dimension of the nullspace i... | No, think of
$$
\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 2 & 1 \\
0 & 0 & 0 & 2 \\
\end{bmatrix}.
$$
The eigenspace relative to the eigenvalue $0$ has dimension $1$, generated by
$$
\begin{bmatrix}
1\\
0 \\
0 \\
0\\
\end{bmatrix}.
$$
The eigenspace relative to the eigenvalue $2$ has dimension $1$, gen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Landau notation on $f(n) = \frac{1}{4n \tan(\frac{\pi}{n})}$ Does someone would have time to show me how to use the Landau notation "Big O"? A useful example could be on $f(n) = \frac{1}{4n \tan(\frac{\pi}{n})}$.
| One has, by the Taylor series expansion, as $x \to 0$,
$$
\frac{1}{1+x}=1+O(x),\qquad \tan x=x+O(x^3),\tag1
$$ then
$$
\frac{x}{\tan x}=\frac{x}{x+O(x^3)}=\frac1{1+O(x^2)}=1+O(x^2)\tag2
$$ Hence, as $n \to \infty$, we have $\dfrac{\pi}n \to 0$, and
$$
\begin{align}
f(n) := \frac{1}{4n \tan(\frac{\pi}{n})}
=\frac1{4\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1877994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derivative of $\operatorname{trace}(XWW^{T}X^{T})$ with respect to $W$ Compute
$$\frac{d}{dW}\operatorname{trace}(XWW^T X^T)$$
where $X$, $W$ are $n\times n$ real matrices.
| A different solution from the one I proposed using the Matrix Cookbook equation $(116)$ (if you are not too familiar with matrix calculus) involves taking these products, then writing them out using index notation:
$$V=WW^T$$
$$A=XV$$
$$B=AX^T$$
Hence:
$$v_{ij}=\sum_kw_{ik}w^T_{kj}$$
$$a_{mj}=\sum_ix_{mi}v_{ij}$$
$$b_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Question about whether $(x^2)^{0.5} = x$. The Wikipedia page on exponentiation suggests that the following identity holds provided the base $b$ is non-zero:
$$(b^m)^n = b^{mn}$$
Consider the following function:
$$y = (x^2)^{0.5}$$
According to the identity above the following should hold:
$$y = (x^2)^{0.5} = x^1$$
Howe... | The page you refer to actually says the following. (The emphasis below is mine.)
The following identities hold for all integer exponents, provided that the base is non-zero
$0.5$ is not an integer, so the property does not apply.
But as you've seen, $(x^2)^{0.5} = x$ is true for $x \ge 0$. The reason it fails for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Spivak Calculus - Chapter 1 Question 4.6 In Spivak's Calculus, Chapter 1 Question 4.6:
Find all the numbers $x$ for which $x^2+x+1>2$
The chapter focuses on using the following properties of numbers to prove solutions are correct:
Based on those properties, I am able to perform the following algebra:
$
\begin{align... | Complete the Square
$
\begin{align}
x^2+x+1&>2 & \text{Given}\\
x^2+x+1+0&>2+0 & \text{By Addition}\\
x^2+x+1+0&>2 & \text{By P2}\\
x^2+x+0+1&>2 & \text{By P4}\\
x^2+x+\left( \frac{1}{2} \right)^2+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P3}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+(-1)\left( \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find the least squares solution for rank deficient system
Find the least squares solution to the system
$$x - y = 4$$
$$x - y = 6$$
Normally if I knew what the matrix $A$ was and what $b$ was I could just do $(A^TA)^{-1} A^Tb$, but in this case I'm not sure how to set up my matrices. How can I find the least square s... | Your matrix is just the coefficients of your system of equations. In this case
$$ x-y = 4 $$
$$ x-y = 6 $$
leads to
$$
\begin{bmatrix}
1 & -1 \\ 1 & -1
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix}
=
\begin{bmatrix}
4 \\ 6
\end{bmatrix}
$$
but you should see that there is no solution to this since you can't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Example of $2\times2$ matrix with singular values $0$ and $5$ but only has one eigenvalue $0$. My understanding of singular values is that they are the square roots of eigenvalues, but I am definitely missing something here in the definition.
The problem I am trying to work on is:
Find an example of $T \in L(\mathbb{... | The singular values of $A$ are the square roots of the eigenvalues of $A^*A$ (or of $AA^*$), not those of $A$ itself. For one thing, $A$ could have negative eigenvalues.
For your specific problem, try $A=\begin{bmatrix}0&5\\0&0\end{bmatrix}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve for transform parameters given original and transformed vectors I have some transformation in 3D homogeneous coordinates that includes three-axis rotation, translation and strain (linear deformation):
$$\\
T P = P' \\
T = D R_z R_y R_x S
$$
$$
D =
\begin{bmatrix}
1 & 0 & 0 & d_x \\
0 & 1 & 0 & d_y \\
0 & 0 & 1 & ... | @dovalojd's answer is pretty good. There are better methods, though, of determining the nautical angles. You can use a test vector to eliminate the roll, $\theta_x$:
$
\hat n = Q\hat x \\
= R_z R_y R_x \hat x \\
= R_z R_y \hat x \\
= R_z \begin{bmatrix}
\cos \theta_y \\
0 \\
-\sin \theta_y
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability Book For Beginner. I am a graduate school freshman.
I did not take a probability lecture.
So I don't have anything about Probability.
Could you suggest Probability book No matter What book level?
| I think the book Probability Theory by Heinz Bauer is a very good text on probability theory. It contains an extensive discussion of all the basic parts of the theory and is very readable. The book requires, however, a modest background in measure theory.
The original version of the book from 1973, Probability Theory ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Why we can add an element on both sides like this? Suppose $A$ is a set equipped with a binary operation $+$.
How I can prove if $a=b$ then $a+c=b+c$? Why we can add an element on both sides like this?
| A simple explanation is that saying $a=b$ literally says that $a$ and $b$ are the exact same mathematical object. Hence 'adding' $c$ to $a$ on the right is one the same as 'adding' $c$ to $b$ on the right (adding in quotes because what I am really referring to is the given binary operation).
Note that the converse, $a+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
How to compute the limit of $\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\cdots+\frac{n}{n^2+n^2}$ without using Riemann sums?
How to compute the limit of $\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\cdots+\frac{n}{n^2+n^2}$ without using Riemann sums?
My Try: I have Solved It using Limit as a Su... | One may use the digamma function, from the standard $
\psi(z+1)-\psi(z)=\dfrac1z$ one obtains easily
$$
\psi(z+n+1)-\psi(z+1)=\sum_{k=1}^n\frac1{z+k},
$$
inserting $z:=in$ and considering imaginary parts gives
$$
\sum_{k=1}^n\frac{n}{n^2+k^2}=-\text{Im}\left[\psi(in+n+1)-\psi(in+1)\right]
$$ then one may recall that (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What does being "Linear" mean for a transformation and a function intuitively/graphically? I was wondering what is the geometric meaning or intuition behind a transformation and function(separately)being linear.
An example(or graph) illustrating the characteristics of a linear function/map would be much appreciated.
Th... | If you have a linear transformation on a space $X$ then the image is a subspace of the space $X$. Geometrically that means that the image of the transformation is a flat that contains the origin in the space.
Examples:
The easiest example is the 0-transformation.
Let $X = \mathbb R ^n$ and $f: X \to X, v \mapsto 0$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Finding complex eigenvalues and its corresponding eigenvectors
$$A = \begin{bmatrix}3&6\\-15&-15\end{bmatrix}$$ has complex eigenvalues $\lambda_{1,2} = a \pm bi$ where $a =$____ and $b = $ ____. The corresponding eigenvectors are $v_{1,2} = c \pm di$ where $c =$ (____ , _____ ) and $d =$ (____ , ___)
So I got the ch... | I'll cover the how to find the eigenvectors part.
$$A = \begin{bmatrix}3&6\\-15&-15\end{bmatrix}$$
has eigenvalues $\lambda_{1,2} = -6\pm 3i$.
Now to find the associated eigenvectors, we find the nullspace of $$A-\lambda_{1,2}I = \begin{bmatrix}3-(-6\pm 3i)&6\\-15&-15-(-6\pm 3i)\end{bmatrix} = \begin{bmatrix}9\mp 3i & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to find the Cartesian equation of a plane curve from a parametric equation? More specifically, how to express
$$\begin{aligned}x(t) &=\frac{2t}{1+t^2}\\
y(t) &=\frac{1-t^2}{1+t^2}\end{aligned}$$
in terms of $x$ and $y$? I attempted adding the two, getting a square from the numerator and a few other methods, but ru... | Those are the parametric equations of the unit circle $x^2+y^2=1$.
In fact
$$x^2+y^2=\frac{(2t)^2+(1-t^2)^2}{(1+t^2)^2}=\frac{4t^2+1-2t^2+t^4}{(1+t^2)^2}=\frac{1+2t^2+t^4}{(1+t^2)^2}=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1879452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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