Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Spectral norm inequality Suppose I have two matrices A, B
$$
A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{m \times n}
$$
Then on what conditions on A and B will the followig ineqiality hold:
$$
||A+B||_2 \geq ||A||_2
$$
I for some reason feel that this would hold when the spaces spanned by the columns these matric... | You are wrong. In particular, if we take
$$
A = \pmatrix{1&100\\0&0}\\
B = \pmatrix{-1&-100\\1/100&1}
$$
Verify that the column space of the two matrices is different, but $\|A + B\| < \|A\|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
sums and product of density functions are density?
Given $f,g$ to be density functions. If the function $\psi = \lambda f +(1-\lambda)g$, for $\lambda \in [0,1]$, a density function? If the product $fg$ is also a density function?
I know that a random variable $X$ is called continuous if its distribution function can... | I think the example $f=g=1$ on $[0,1]$ is not a counterexample— it's a positive example of two density functions whose product is also a density function. A counterexample might be something like:
Let $f(x) = 1$ if $x \in [0, 1]$, and zero elsewhere. Let $g(x) = 1$ if $x \in [2,3]$ and zero elsewhere. The product $fg$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Tangent planes of a surface parallel to a fixed direction I'm learning differential geometry, specifically the theory of surfaces, and need help with the following problem:
Consider the surface $S$ given by $x_1 = x_2 - f(x_2 - x_3)$, where $f \in C^2$.
Show that every tangent plane of $S$ are parallel to a fixed dire... | Given
$$
x_{\,1} = x_{\,2} + f(x_{\,2} - x_{\,3} )\quad i.e.\quad F(x_{\,1} ,x_{\,2} ,x_{\,3} ) = 0
$$
we can rewrite it ,as
$$
x_{\,1} - x_{\,3} = x_{\,2} - x_{\,3} + f(x_{\,2} - x_{\,3} )\quad \Rightarrow \quad y_{\,1} = y_{\,2} + f(y_{\,2} )\quad i.e.\quad G(y_{\,1} ,y_{\,2} ) = 0
$$
or, otherwise, as
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The integral $\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy$ What is $$\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy?$$ I split it as $\frac{y^{2}}{(y^2+1)^{0.5}} + \sqrt{y^2+1}.$ Now I substituted $y^{2}=u $ thus $2y\,dy=du$ so we get $0.5 \sqrt{\frac{u}{u + 1}} + 0.5 \sqrt{\frac{1 + u}{u}}$ but now what to do? Another idea was doi... | $\displaystyle\int\frac{4y^2+2}{\sqrt{y^2+1}}dy=\int\frac{2y^2+2}{\sqrt{y^2+1}}dy+\int\frac{2y^2}{\sqrt{y^2+1}}dy=2\int\sqrt{y^2+1}dy+\int\frac{2y^2}{\sqrt{y^2+1}}dy.$
Now use $\displaystyle u=2y,\;dv=\frac{y}{\sqrt{y^2+1}}dy\;$ so $\;du=2dy,\;v=\sqrt{y^2+1}$ in the 2nd integral to obtain
$\displaystyle2\int\sqrt{y^2+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Does topology apply to the integers? What is the natural topology (or topologies) on the integers. Can we define a metric on the integers?
| We could add (since I don't see it or any equivalent when skimming the comments) the symmetric set topology on $\mathbb{Z}$: $U \subseteq \mathbb{Z}$ is open if $$n \in U \iff -n \in U$$ holds. This topology disconnects the integers and makes them non-compact, but does admit a countable basis (correct me if I'm messing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 4
} |
Using Green's theorem find line integral $\oint_C (-x^2+x) dy $ enclosed by $x=2y^2$ and $y=2x$
Using Green's theorem find line integral $\oint_C (-x^2+x)\, dy $ enclosed by $x=2y^2$ and $y=2x$
The intersection points between the line and the parabola are
$$P_1 = \left( 0,0 \right) \quad P_2 = \left( \frac{1}{8} , \f... | You changed $-2x-1$ to $-2x+1$ in your last integral.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can we specifically prove that if $n\in\mathbb{Z^+}$ is composite and square-free then the ring of integers of $\mathbb{Q}(\sqrt{-n})$ is not a UFD? Let $\mathcal{O}_K$ be the ring of integers of a field $K$. I have learnt about the Baker-Heegner-Stark theorem, which implies that if $K=\mathbb{Q}(\sqrt{-n})$ with $n\in... | This is not quite enough. It could be that the two factorizations can each be refined into the same factorization.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Revisit "How can I visualize the nuclear norm ball" Revisiting How can I visualize the nuclear norm ball?
Two eigenvalues are reproduced as following:
$$ s_{1,2}=\frac{1}{\sqrt{2}}\sqrt{x^2+2y^2+z^2\pm|x+z|\sqrt{(x-z)^2+4y^2}}. $$
According to the following (from a paper)
If a symmetric matrix: $$ A=\left( \begin{arr... | The answer to the linked question shows that the curved side of the "cylinder" satisfies the equation
$$(x-z)^2+4y^2=1$$
and its planar caps satisfy
$$(x+z)^2=1.$$
The red curves are the intersection of the two surfaces, so they satisfy both equations. Therefore, they must also satisfy their sum,
$$x^2+4y^2+z^2=2.$$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Intuition for $\lim_{x\to\infty}\sqrt{x^6 - 9x^3}-x^3$ Trying to get some intuition behind why: $$ \lim_{x\to\infty}\sqrt{x^6-9x^3}-x^3=-\frac{9}{2}. $$
First off, how would one calculate this? I tried maybe factoring out an $x^3$ from the inside of the square root, but the remainder is not factorable to make anything ... | The only thing I would add to T. Bongers' answer is that the key to these kinds of questions is generally to ask:
*
*How can I transform the limit to evaluate it using standard techniques?
*Especially with limits at infinity, what is 'dying off' quicker?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 4
} |
What is meant by the fact that $k\langle x_1, \ldots, x_n\rangle$ denotes a free associative algebra in indeterminate $x_1, \ldots, x_n$? As the question title suggests, what is meant by the fact that $k\langle x_1, \ldots, x_n\rangle$ denotes a free associative algebra in indeterminates $x_1, \ldots, x_n$? Could anybo... | Just like a polynomial algebra is spanned by monomials, the free associative algebra with indeterminates $x_1, \dots, x_n$ has a basis given by monomials of the type
$$x_{i_1} \dots x_{i_k}$$
where $k \ge 0$ and $i_1, \dots, i_k \in \{ 1, \dots, n \}$, the case $k=0$ corresponding to the empty monomial, i.e. the unit o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two total orders Let $\mathcal{R}$ and $\mathcal{S}$ be two total orders on $E$ such that $\forall(x,y) \in E\times{E}, x\mathcal{R}y \Rightarrow x\mathcal{S}y$. Show that $\forall(x,y) \in E\times{E}, x\mathcal{R}y\Leftrightarrow x\mathcal{S}y$.
I need some help with this problem. I know that a total order is a set w... | We are going to use the converse of
$$x\mathcal Ry \Rightarrow x\mathcal S y$$
which is
$$\neg (x\mathcal Ry) \Rightarrow \neg(x\mathcal S y).$$
The order is total, so if we have $\neg (x\mathcal Ry)$ we have $y\mathcal R x$.
By the first property it gives us
$$y\mathcal S x.$$
By anti-symetry we can conclude that $x=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Where to find the paper of G.H. Hardy called "Sur les zéros de la fonction Zeta de Riemann" please? Where to find the paper of G.H. Hardy called "Sur les zéros de la fonction Zeta de Riemann" please?
I've been looking for a while now, and I just don't find it, the basics of the hypothesis !
If anyone can help !
| The paper is listed in the Journal Comptes Rendus de l'Académie des Sciences and the full title of the article is Sur les zéros de la fonction $\zeta(s)$ de Riemann. The full reference is
Hardy G H. Sur les zéros de la fonction $\zeta(s)$ de Riemann. Comptes Rendus de l'Académie des Sciences,
1914, 158: 1012-1014
and y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $\int_{-\pi}^{\pi} f(t) \sin (nt) dt \rightarrow 0 , \int_{-\pi}^{\pi} f(t) \cos (nt) dt \rightarrow 0$ For any integrable function $f: S^1 \rightarrow \mathbb{C}$ , I need to show that $$ \int_{-\pi}^{\pi} f(t) \sin (nt) dt \rightarrow 0$$ $$ \int_{-\pi}^{\pi} f(t) \cos (nt) dt \rightarrow 0$$ as $n \righta... | Um, I think you can just break $f$ into real and imaginary parts, and apply Riemann Lebesgue Lemma to each, just as you did for real valued $f$ here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
no $k$-sparse vector in null space of $A$ where $y=Ax$ Here is a part of my notes on compressed sensing. Suppose we are given an underdetermined linear system of equations $Y=AX$. For a unique solution,
What we need:
subtraction of no two $k$-sparse vectors ($x_1$, $x_2$) must be in nullspace of $A$.
What is require... | I presume the formulation should be that every 2k columns of $A$ should be linearly independent, as this implies that there are no 2k sparse vector in the null space:
To see this, denote $(\vec{a}_1,...,\vec{a}_N)$ the column vectors of $A$ and let $X$ be a 2k-sparse vector, i.e. there are at most $2k$ nonvanishing com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Cauchy subsequence problem Consider the set $S_0$ of all continuous functions $f : [0,1] \to [0,1]$.
Define a metric on the set $S_0$ by setting
$$
\rho(f,g)=\sup\limits_{x\in{[0,1]}}|f(x)-g(x)|.
$$
(a) Give an example of a sequence in $S_0$ that does not contain a Cauchy subsequence with respect to the metric $\rho$.... | Hint for (a): Try a sequence of functions $f_n$ such that, e.g., $f_n(1/n) = 1$ but $f_n(1/m) = 0$ for all other $m$.
Hint for (b): if you have a sequence $f_n$ in $S_\epsilon$ such that $f_n(r)$ is Cauchy for each rational $r$, show that $f_n$ is Cauchy in the metric $\rho$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Are there integers $a,b,c$ such that $a$ divides $bc$, but $a$ does not divide $b$ and $a$ does not divide $c$?
Are there integers $a,b,c$ such that $a$ divides $bc$, but $a$ does not divide $b$ and $a$ does not divide $c$?
I am not quite sure what to do with the given information. I know I could easily find an examp... | The general result on this question is this generalisation of Euclid's lemma:
Gauß's lemma: If a number divides a product of two factors, a,d is coprime with one of them, it divides the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
Find $f^{(100)}(0) $ and $f^{(101)}(0) $ if $f(x)=xe^{\arctan{x}}$ $$f(x)=xe^{\arctan{x}}$$
Part of my solution
$$f^{(n)}(x)=\sum_{k=0}^{n}\binom{n}{k}x^{(k)}(e^{\arctan{x}})^{(n-k)}=x(e^{\arctan{x}})^{(n)}+(e^{\arctan{x}})^{(n-1)}$$
First term probably disappears because $x=0$ but i don't know what to do with second t... | Partial answer:
As you pointed out,
$$\begin{align}f^{[n]}(x)&=\sum_{i=0}^n{n\choose i}x^{[i]}\left(e^{\tan^{-1}(x)}\right)^{[n-i]}
\\
&=x\left(e^{\tan^{-1}(x)}\right)^{[n]}+n\left(e^{\tan^{-1}(x)}\right)^{[n-1]}
\end{align}$$
We set $g(x)=\left(e^{\tan^{-1}(x)}\right)$, $h(x)=\tan^{-1}(x)$ and consider $g^{[n]}(x)$.
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1915876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove each of the following conditions is sufficient to ensure that $f(x+y)≤f(x)+f(y)$ $f$ is increasing and $f$ satisfies $f(x)=0$ and $f(x)>0$ for all $x>0$.
Show that each of the following conditions is sufficient to ensure that $f(x+y)≤f(x)+f(y)$ for all $x,y≥0$.
*
*(a) $f$ has a second derivative satisfying $f'... | Hint: Assume (c). Then
$$x\frac{f(x+y)}{x+y}\le x\frac{f(x)}{x} $$
and
$$y\frac{f(x+y)}{x+y}\le y\frac{f(y)}{y}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is a reflection followed by another reflection is a rotation? I just started abstract algebra and we are working with dihedral groups. I've made Cayley tables for D3 and D4 but I can't explain why two reflections are the same as a rotation
| Consider the dihedral group $D_5$, and consider its action on the pentagon. In particular, every element of the group can be thought of as some combination of rotations and reflections of a pentagon whose corners are labeled $1,2,3,4,5$ going clockwise.
First, notice that no matter what we do, the numbers will be in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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What is the smallest and largest possible values for the variance? Suppose
$P( X \in \{1,2,3\}) = 1$ and $E(X) =2.5.$
What is the smallest and largest possible values for the variance?
My understand: So what I understand is variance finds the distance between each element and the mean. So the closer 2 is to E(X) the... | Although the two things are equal, I think it is easier to use $$\operatorname{Var}(X) = \mathbb{E}\Big[(X-\mathbb{E}X)^2\Big]$$
rather than $\operatorname{Var}(X)=\mathbb{E}X^2-(\mathbb{E}X)^2$.
Let $\mathbb{P}[X=i]=p_i$, then
\begin{align}
\operatorname{Var}(X) &= \mathbb{E}\Big[(X-\mathbb{E}X)^2\Big] \\
&= \sum_{i ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Question about condtion of MVT for integral. For $f \in C[a,b], g\ge0$ (or $g\le0$) on $[a,b]$, then there exists $c\in(a,b)$ $\int _a^b f(x)g(x)dx = f(c)\int_a^bg(x)dx$.
It is MVT for integral.
Why $c \in (a,b)$ ? How about $c\in[a,b]$? Is it wrong?
I am wondering this because, in proving progress,
by IVT, there exi... | In the book Mathematical Analysis I by Vladimir A. Zorich (p. 352) I found:
As far as I can tell from the proof in the book, the proof depends on the intermediate value theorem, which states
But $c \in (a,b)$, since if either $f(a) = 0$ or $f(b) = 0$ would imply $f(a)f(b) = 0$ and so the function would not fulfill t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Different seven-digit numbers could Sid have meant to type
Sid intended to type a seven-digit number, but the two 3's he meant to
type did not appear. What appeared instead was the five-digit number
$52115$. How many different seven-digit numbers could Sid have meant to type?
First time, I attempted this problem ... | Consider the following possibilities:
_ 3 5 _ _ 2 _ _ 1 _ _ 1_ _ 5 _ 3
_ 3 5 _ _ 2 _ _ 1 _ _ 1_ _ 5 3 _
3 _ 5 _ _ 2 _ _ 1 _ _ 1_ _ 5 _ 3
3 _ 5 _ _ 2 _ _ 1 _ _ 1_ _ 5 3 _
All of these would give the same result of 3521153, but are counted separately by your first method.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Number of ways of distributing $N$ balls into $M$ bins such that at least one bin has at least $n$ balls in it?
What is the number of ways of distributing $N$ indistinguishable balls into $M$ bins such that at least one bin has at least $n$ balls in it?
My attempt:
The number of ways of of placing $N$ balls into $M$... | Seems that it's not correct, since you going to have double counting for the case where cell 1 and cell 2 both have $n$ balls.
Try use inclusion-exclusion principle.
It's easy using the properties $p_i = \mathrm{The\space ith\space cell\space has\space at\space least\space n\space balls}$ to calculate $E(0)$ the case w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Does every theorem have a short proof? This question is somehow based on my belief that every theorem has a short and simple proof. By "proof" I mean:
*
*Proving an statement
*Disproving a statement
*Proving that a statement is undecidable
Once we have formalized what we understand for a "step" in a proof, could... | surely a theorem doesn't need to conform to any real quality requirements, so it could contain an huge number of steps that need to be proved, greater than any n you think of.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Why do parabolas' arms eventually become parallel? Here is what this site states
All parabolas are the same shape, no matter how big they are. Although they are infinite, meaning that the arms will never close up, the arms will eventually become parallel.
Now, I have an argument against it. Let $f(x) = ax^2 + bx + c$... | There is no contradiction. As you said, the slope keeps increasing, this means that as you go towards $x\rightarrow\pm\infty$, the slope also goes to $\pm\infty$, i.e the graph of the function gets more and more "vertical".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 13,
"answer_id": 8
} |
Translating $\forall$ and $\exists$ in a statement The following statement has two versions – one where $d$ is quantified by $\forall$ and the second where it's quantified by $\exists$. The task here is to find a counterexample where the statements below are false. The domain is all integers.
*
*$\forall a \forall ... | Turning a comment into an answer:
For the first statement, you can remove the for all variables a, b, c, and d part, as the counterexample of $[a=1,b=2,c=3,d=4]$ already states this fact.
For the second statement, you need to prove that your counterexample of $[a=1,b=2,c=10]$ is true for ALL values of $d$.
I'm not sure... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Understanding why $\det(A) = \det(A^T)$ via the 3D Paralleliped I am trying to understand why, geometrically, we have that
$$
\det(A) = \det(A^T).
$$
To build intuition, I am thinking in 3 dimensions. So let $A$ be a $3 \times 3$ matrix of real numbers.
First, I know that if
$$
\det(A) = \left|
\begin{array}
xx_1 & y_... |
Here is a 2D example.
$$\begin{pmatrix}
2&3\\
1&4\\
\end{pmatrix} \mapsto_{transpose}
\begin{pmatrix}
2&1\\ 3&4\\
\end{pmatrix}$$
I'm still thinking of a satisfying (geometric) explanation as to why this always works, but pictures are always good.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1916929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How to compute this limit involving complementary error functions I am trying to take the following limit
$$\lim_{x\to \infty } \, \frac{2 x \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}{t \operatorname{erfc}\left[-\frac{x}{\sqrt{2}}\right]}$$
my first thoughts were to use LHospital's rule after making the top ... | By exploiting the continued fraction representation for the complementary error function
$$\frac{2x\int_{\frac{x}{t\sqrt{2}}}^{+\infty}e^{-z^2}\,dz}{t\int_{-\frac{x}{\sqrt{2}}}^{+\infty}e^{-z^2}\,dz}\approx \sqrt{\frac{2}{\pi}}e^{-\frac{x^2}{2t^2}}$$
hence the wanted limit is simply zero. You may also use concentration... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1917071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Proof Proving that aj=aj−1+aj−2 cannot converge to a finite limit writing question. I was trying this problem and was hoping for some feedback as I am not sure if I proved it or not... I am trying to figure out where I went wrong as I feel I didn't run a good argument.
For $\epsilon>0,\exists N:|a_j-a_{j+1}|<\epsilon$ ... | No, you do not arrive at a contradiction. For example, it is true that $|17|-|42|\le 0.001$ even though the left hand side is negative.
Instead, let $\phi$ be the larger solution of $\phi^2=\phi+1$. Show that $\phi>1$. Show by induction that $a_n>c\phi^n$ provided you pick $c>0$ such that $a_1>c\phi$ and $a_2>c\phi^2$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are there infinitely many $\alpha \times \beta$ Chomp boards where player 2 wins? Let $\alpha$ and $\beta$ be nonzero ordinals.
Infinite chomp (called ordinal chomp by Wikipedia) on an $\alpha \times \beta$ board is played as follows.
We consider the set $\alpha \times \beta$, partially ordered under the natural orderi... | Yes, for every nonzero ordinal $\alpha$ there is an ordinal $\beta$ such that $\alpha \times \beta$ is a second player win (I can't find the source sorry). So far the only pairs that are known are $1 \times 1$ (trivial), $2 \times \omega$, $3 \times \omega^\omega$, and $4 \times \omega^2$.
The situation with five or mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1917228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Sketching Functions
Sketch the region enclosed by $x+y^2=30$ and $x+y=0$. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region.
How do I start off sketching? Very first step?
I'm in Calc 2 and it's sad that I don't know what to do but I am trying to learn.
| Can you sketch $y=30-x^2$? This is fairly standard highschool stuff. Once you do that, flip it over the line $y=x$ to interchange the $x$ and $y$ coordinates, to get $x=30-y^2$. That’s your first graph. The other one is easier, it’s $y=-x$ (you can also think of it as $x=-y$), still standard highschool stuff. With your... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1917350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Naming scheme for the acceleration vector and acceleration when working with parametric space curves? The formula for the acceleration vector on a space curve is $$a=\kappa|v|^2N+\frac{d^2s}{dt^2}T$$
I understand the formula above and how to calculate the components. What I don't understand is the naming scheme betwee... | Perhaps try a name like "intrinsic acceleration." The motivation for this schema is that if you have an ant skating along a curve embedded in $\Bbb{R}^3$, its acceleration will come from two sources:
*
*the curvature of the embedding in ambient space (i.e., how hard the ant has to grip the curve in order to not fly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1917463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Extreme point of the unit ball in H(U) An extreme point of a convex subset $C$ of a vector space $X$ is a point which can not be expressed in the form $\lambda a+(1-\lambda)b$, with $a,b\in C$ and $0<\lambda<1$.
For an open subset $U$ of the complex plan, $\mathbb C$, we denote by $H(U)$ the set of all holomorhphic fu... | Partial answer: If $A={\Bbb C}\setminus U$ contains at most two points then $B_1(H(U))$ consists of constants only. So assume $|A|\geq 3$ and let $\phi:U\rightarrow {\Bbb D}$ be a conformal bijection with the unit disk. On the unit disk a set of extremal points are $e_{n,\theta}=\theta z^n$, $n\geq 0$, $|\theta|=1$. An... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\left(\sum_{i=1}^n x_i\right) \cdot \left(\sum_{i=1}^n \frac{1}{x_i}\right) \ge n^2$, for all integers $n\ge 1$. Let $x_1,\ldots, x_n$ be positive integers. Use mathematical induction to prove that
$$\left(\sum_{i=1}^n x_i\right) \cdot \left(\sum_{i=1}^n \frac{1}{x_i}\right) \ge n^2$$ for all integers $n \ge 1$.
(Give... | $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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Statement provable for all parameters, but unprovable when quantified I've been reading a book on Gödel's incompleteness theorems and it makes the following claim regarding provability of statements in Peano arithmetic (paraphrased):
There exists a formula $A(x)$ such that the statements $A(0), A(1), A(2), \dots$ are ... | If you simulate a Turing Machine using a Universal Turing Machine and you run the simulation for exactly N steps you will see either that the machine Halts in N steps or that the machine does not Halts in N steps.
However in general you cannot proove that a Turing Machine Halts by mere simulation, because if it does no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1917753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Variable chord of hyperbola If a variable chord of hyperbola $x^2$$/4$ - $y^2$$/8$ $=$ $1$ subtends a right angle at the centre of hyperbola . If the chord touches a fixed concentric circle with hyperbola then we have to find the radius of the circle .
I thought of doing it by homozenizing , but not able to do how ?
| Homogenise it and then
coeff.[x^2+y^2]=0 you will get constant term and since it is tangent to the circle x^2+y^2=r^2 then equate and u will get it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to factorise this expression $ x^2-y^2-x+y$ This part can be factorised as $x^2-y^2=(x+y)(x-y)$, How would the rest of the expression be factorised ?
:)
| Notice here,
$$x^2 - y^2 - x + y$$
$$(x+y)(x-y)-1(x-y)$$
$$(x-y) (x+y-1)$$
That's done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1918045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
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Injection - Bijection function Question Suppose $A, B$ are two non-empty sets and let $$f:A\to B$$ and $$g:B\to A$$ be two injective (one to one) functions.
QUESTIONS:
1. Is there exist a bijective function from $A$ to $B$?
2. If so, how can we prove it?
I think, there should be a such function. But I can not come up... | I have seen 2 proofs, one complicated, and this one: For $x\in A$ the possible sequence $$x,\; f^{-1}(x),\; g^{-1}f^{-1}(x),\; f^{-1}g^{-1}f^{-1}(x),...$$ may have just one term (if $f^{-1}(x)$ doesn't exist), or 2 terms (if $f^{-1}(x)$ exists but $g^{-1}f^{-1}(x)$ does not), et cetera.
Let $x\in E$ if the sequence st... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Supremum of the product of two sets Here is a GRE math subject problem:
63. For any nonempty sets $A,B\subseteq \Bbb R$, let $A\cdot B$ be the set defined by
$$A\cdot B=\{xy\,:\, x\in A\wedge y\in B\}$$
If $A$ and $B$ are nonempty bounded sets and if $\sup A>\sup B$, then $\sup(A\cdot B)=$
(A) $\quad\sup(A)\sup( B... | When $\inf B<0$, $|\inf B|\ge \sup B$, $\sup A>0$ and $\sup A\ge |\inf A|$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Basis of the topology of a Stone space I'm trying to understand the general idea of Stone Representation Theorem (or, at least, the existence of a functor from the category of boolean algebras to the category of compact totally disconnected spaces) with the example $B=\{0,1,2,3\}$.
The corresponding Stone space contain... | In that case $S(B)$ is simply the discrete two-point space. The points are $\{1,2\}$ and $\{1,3\}$, and the open sets are $\varnothing,\big\{\{1,2\}\big\},\big\{\{1,3\}\big\}$, and $S(B)$ itself. The base is
$$\beta_{S(B)}=\big\{\{U\color{red}{\subseteq}S(B):b\in U\}:b\in B\big\}\;.$$
Here
$$\begin{align*}
\{U\subsete... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the mean value property for harmonic functions using Green's second identity Suppose that $\phi:\mathbb R^3\to\mathbb R$ is a harmonic function. I am asked to show that for any sphere centered at the origin, the average value of $\phi$ over the sphere is equal to $\phi(0)$. I am also directed to use Green's sec... | I appreciate the other answers, but I came up with my own answer which, in my humble opinion, is a bit simpler.
For posterity's sake, here is the answer: Let $g=1/|\mathbf r|$ and $f=\phi$. Then the following hold: $$\int_S g(\mathbf r)\nabla f(\mathbf r)\cdot d\mathbf S=-\frac{1}{r}\int_S \nabla f\cdot d\mathbf S=0,$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1918781",
"timestamp": "2023-03-29T00:00:00",
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Is a function a special kind of relation? Is a function a "special kind of relation", or, does it "describe a specific relation"?
My text on discrete mathematics explains:
A relation is a subset of a Cartesian product and a function is a
special kind of relation.
But it would make more sense to me if a function des... | Think of the relation "is less than" in $\{1,2,3\}^2$. That relation is $\mathrm{Less}=\{(1,2),(1,3),(2,3)\}$, however you normally will not write $(x,y)\in\mathrm{Less}$, but you will write $x<y$. You certainly have no problem with saying "$x<y$ is a relation". And of course that is true not only for $x<y$, but also f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral using a parameter $k\in \mathbb Z$. I am trying to solve an exercise that says:
Compute the following integral for $k\in \mathbb Z$: $$\int _{\vert z \vert =1} \! z^k \sin {\frac{1}{z}} \, \mathrm d z$$
The only thing that I can think is making $z=\mathrm e ^{\mathrm i t}$, so the integral would be
$$\mathrm... | $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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"timestamp": "2023-03-29T00:00:00",
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How can we parametrise this matricial hypersphere? What I call a matricial hypersphere for lack of a recognised name is the set in $\mathbb{R}^{p\times k}$ defined by
$$\mathfrak{H}=\left\{
a_1,\ldots,a_k\in \mathbb{R}^{p};\ \sum_{i=1}^k a_i a_i^\text{T} = \mathbf{A}
\right\}$$
where $\mathbf{A}$ is a $p\times p$ symme... | Point 1) Let $B$ be the matrix with columns $a_i$: your description is equivalent to
$$\tag{1}BB^T=A$$
Thus, being given a symmetrical semi-definite positive $n \times n$ matrix $A$ with rank $k$, $\frak{H}$ can be identified with the set of $n \times k$ matrices $B$ such that $A$ can be written under the form (1).
Re... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Generalization of Kronecker–Weber theorem: is an abelian extension always cyclotomic? It is well-known that any abelian extension of $\Bbb Q$ is contained within some cyclotomic field $\Bbb Q(\zeta_n)$. My question is about the more general statement:
If $L/K$ is an abelian extension, do we have $L \subset K(\zeta_n)... | No.
Consider the extension $\mathbb{Q}(t^{1/2})$ of the field $\mathbb{Q}(t)$. It is an extension of degree $2$, and is thus abelian. However, adding any root of unity $\zeta$ to $\mathbb{Q}(t)$ will give you the field $(\mathbb{Q}(\zeta))(t)$, in which $t$ has no square root.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does the linear transformation that a matrix encodes depend on a choice of basis? Context
Let $M$ be an $m \times n$ matrix of real numbers. Let $\mathbf{x}$ be column vector of length $n$ with elements $x_1, \ldots , x_n \in \mathbb{R}$. Let $\vec{x} = (x_1, \ldots , x_n) \in \mathbb{R}^n$ be its analogue in $\mathbb{... | Yes, the choice of basis makes a difference. For this reason, some books will write something like the following: If $T$ is a linear transformation, then the matrix representation of $T$ relative to a basis $\beta = \{\beta_1, \beta_2,...,\beta_n\}$ will be written
$$[T]_\beta$$
For example, let the matrix
$$A = \begi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Rank of $vv^\top + ww^\top$ Given two non-colinear real unit vectors $v,w$, I believe the rank of $M=vv^\top + ww^\top$ is 2 and I'd like to prove it. $vv^\top$ and $ww^\top$ are obviously of rank one, $v$ is not in the kernel of $M$ because $vv^\top v=v$ and $\|ww^\top v\|<1$ (because $v$ and $w$ are not colinear), sa... | Yet another proof: note that
$$
vv^T + ww^T = \pmatrix{v&w} \pmatrix{v & w}^T
$$
and for any matrix $M$, $M$ has the same rank as $MM^T$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Verify Alternative Formula for Expected Value I am studying for the first actuarial exam (Exam P) and came across a formula in my ACTEX prep manual that I had never seen before:
$$E[X] = a + \int_{a}^{b}{[1-F(x)]dx}$$
And the text said this was true as long as $x$ was continuously defined on the interval, and as long a... | For a continuous random variable
$$
E[X]=\int_a^b dx\ x\ f_X(x)\ ,
$$
where $f_X(x)$ is the probability density function. But
$$
f_X(x)=F'(x)\ ,
$$
where $F$ is the cumulative distribution function.
Substituting in the integral above, and integrating by parts we have
$$
E[X]=\int_a^b dx\ x\ F'(x)=x F(x)\Big|_a^b -\und... | {
"language": "en",
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Is a Rotation Matrix Still a Rotation Matrix under a Non-Standard Basis? Let us work in $\mathbb{R}^2$. Consider the following rotation matrix:
$$
R = \begin{bmatrix}
\cos \theta & - \sin \theta \\
\sin \theta & \cos \theta
\end{bmatrix}
$$
I agree that this represents a rotation of angle $\theta$ in $\mathbb{R}^2$ u... | Yes, it's possible. In particular, $R=\pm I_2$ is invariant under every change of basis.
If you need a nontrivial example, note that when $R\ne\pm I_2$, the product $P^{-1}RP$ is a $2\times2$ rotation matrix if and only if $P$ is a nonzero multiple of any $2\times2$ real orthogonal matrix. The "if" part can be easily v... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to write equation for half-life of caffeine consumed over time? If you ingest 100mg of caffeine instantly (say, as a pill), then given the six hour half-life of caffeine in the body, you'd calculate the milligrams of caffeine left in your system with $100(\frac{1}{2})^{\frac{t}{6}}$ (where $t$ is the time since ing... | A solution based on the Laplace transform: you have $dy/dt=-ky+100(1-u(t-1)),y(0)=0$ where $u$ is the Heaviside step function and $k=\ln(2)/6$. Taking Laplace transforms gives $sY=-kY+\frac{100-100e^{-s}}{s}$, so that $Y=\frac{100-100e^{-s}}{s(s+k)}$. The inverse Laplace transform is then $y(t)=\frac{100}{k} \left ( 1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1919839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine if $y = x^2$ is injective I realize that $y=x^2$ is not injective. It is not one-to-one ($1$ and $-1$ both map to 1, for example).
However, in class it was stated that a function is injective if $f(x) = f(y)$ implies $x = y$.
Or if $x$ doesn't equal $y$, then this implies that $f(x)$ doesn't equal $f(y)$.
Th... | I think that the syntax of the definition from your class is the point of confusion. The class definition depends on functions being defined as
$$
f(x)=(\text{expression of }x)
$$ rather than as
$$
y=(\text{expression of }x).
$$ Consequently, the "$y$" in "$f(y)$" is just some dummy variable that gets input into $f$ ... | {
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"timestamp": "2023-03-29T00:00:00",
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Integral with modulus in the denominator. I want to solve this integral:
Let $f$ be a holomorphic function in $\Omega$ such that $\bar D (0,1) \subset \Omega$ and let $a\in D(0,1)$. Compute $$\int_{\vert z\vert =1}\! \frac{f(z)}{\vert z-a \vert ^2}\, \mathrm d z$$
If the modulus wasn't in the denominator, I know this... | In the unit circle, $\bar{z} = z^{-1}$, and thus $\lvert z - a\rvert^2 = (z - a)(\bar{z}-\bar{a}) = (z - a)(z^{-1} - \bar{a})$ in the unit circle. Hence
$$\int_{\lvert z\rvert = 1} \frac{f(z)}{\lvert z - a\rvert^2}\, dz = \int_{\lvert z\rvert = 1} \frac{f(z)}{(z - a)(z^{-1}-\bar{a})}\, dz = \int_{\lvert z\rvert = 1} \f... | {
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Is the function ring $C^{\infty}(M)$ noetherian? Let $M$ be a smooth manifold and $C^{\infty}(M)$ be its function ring. Is this a noetherian ring?
| I think that this is not true in general.
Consider the ring $C^{\infty}(\mathbb{R})$ and the ideal $I$ consisting of smooth functions that vanish on a neighborhood of $0$. I claim that $I$ is not finitely generated.
Assume by contradiction that $I=(f_{1},\ldots,f_{n})$, where each $f_{i}$ vanishes on a neighborhood $... | {
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"timestamp": "2023-03-29T00:00:00",
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Write the plane $y + 2z = 1$ in a parametric form I'm asked to write the plane
$$y + 2z = 1$$
in a parametric form.
I know how to do this with variables like
$$ax + by + cz = d$$
but the lack of an $x$-variable stumps me. How should I go about doing it?
| Since $x$ does not appear in the equation $y+2z=1$, $x$ is simply independent of the values of $y$ and the corresponding $z$. We can thus use any other variables to parametrize $x$. For instance, $t:=x$.
Then we first parametrize $y$ or $z$. Here, we let $u=y$. Since $y$ and $z$ are dependent of each other, $z$ can be ... | {
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Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$. Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$.
I can't see any way a subring of $\Bbb Z \oplus \Bbb Z$ can NOT be an ideal. Subrings of $\Bbb Z \oplus \Bbb Z$ are of the form $n\Bbb Z \opl... | Any subring would I presume contain $(1,1)$, the ring's unity (this is not the only possible interpretation).
Then any proper subring will do.
The proper subrings are the $R_n=\{(a,b)\in\Bbb Z^2\mid a\equiv b\pmod n\},$ for $n\gt1.$ See @Qiaochu's proof here.
$R_\infty$ is @Arthur's example.
| {
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Does 3-partite graph with at least n+1 edges per vertex have a triangle? I need help for one problem.
In every of 3 schools there are n students (in total = 3n vertex, n per school).
Every student knows at least n+1 students from the other two schools.
Prove that there are 3 students, one from each school, who kno... | Let $G$ be a counterexample, i.e. a 3-partite graph with $n$ vertices in each partite set
and minimum degree $n+1$ that has no triangle.
Let $v$ be a vertex that maximizes the number of neighbors it has in one partite set, say $k$.
We may label the partite sets $A,B,C$, such that $v\in A$ and $v$ has $k$ neighbors in $... | {
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"question_score": "4",
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"answer_id": 0
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Repeating or clarifying proofs in cited papers Is it bad form to repeat proofs or arguments that exist already in cited papers (while making it clear this is not my own argument but from the cited paper)? Instead of just citing the argument... I find it more 'readable' if I rewrite the argument within my paper in my ow... | It depends on how long the proof is (longer is worse, for your question) and how much of an improvement you are making (more is better).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1920601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Vandermonde's identity check in probability The book on probability I'm reading states Vandermonde's identity as: $\binom{m+n}{k} = \sum_{i=0}^k \binom{m}{i} \binom{n}{k-i}$, but further in the book I'm seeing it being used as: $\binom{m+n}{k} = \sum_{i=0}^n \binom{m}{k-i} \binom{n}{i}$ and I can't seem to show that th... | The identity is correct both ways. All that matters is that the index run over all non-zero products of the two binomials. In the first version we know that $\binom{n}{k-i}$ is $0$ if $i<0$ or $i>k$, so having $i$ run from $0$ to $k$, inclusive, ensures that we include all of the non-zero terms. We may also include som... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1920677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Why Two's Complement works About to read computer science, I have just stumbled accross the concept of "Two's complement". I understand how to apply the "algorithm" to calculate these on paper, but I have not yet obtained an understanding of why it works. I think this site: https://www.cs.cornell.edu/~tomf/notes/cps104... | With the help of the other answers on this post (especially Ben Grossmann's), I managed to figure out Two's Complement and why it works for myself, but I wanted to add another complete barebones answer for anyone who still can't understand. This is my first post, so thank you for reading in advance. Also, much of my ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1920772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 4,
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Find the centre of volume, defined by $\frac 1V\int_V {\bf x}$ $\mathrm d\mkern1muV$ of the tetrahedron. Anyone who has seen any of the questions I have posted recently will be getting sick of this introduction, but; I'm currently teaching myself some Vector Calculus in preparation for university and I'm finding it sig... | The integral of function $f(x,y,z)$ on that tetrahedron is given by
$$\int_V f dV=\int_{x=0}^1\int_{y=0}^{1-x}\int_{z=0}^{1-x-y} f(x,y,z) dz dy dx.$$
Note that the volume is given by $\int_V 1 dV=1/6$ and, by symmetry
$$\int_V x dV=\int_V y dV=\int_V z dV.$$
By using the above parametrization, the easiest one is
$$\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1920881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\mathbb{Q}$ and $\mathbb{Q} \cup \{\pi, e\}$ have the same cardinality
Prove that $\mathbb{Q}$ and $\mathbb{Q} \cup \{\pi, e\}$ have the same cardinality.
I know I must show that there exists a bijection between these two sets but I'm having a difficult time trying to come up with a function that relates... | Here's an outline of a proof:
*
*Show that the sets $\{0, 1, 2, \dots\}$ and $\{-2, -1, 0, 1, 2, \dots\}$ have the same cardinality.
*Show that one of these sets has the same cardinality as $\Bbb{Q}$, and the other has the same cardinality as $\Bbb{Q} \cup \{\pi, e\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Prove: $\cos^3{A} + \cos^3{(120°+A)} + \cos^3{(240°+A)}=\frac {3}{4} \cos{3A}$ Prove that:
$$\cos^3{A} + \cos^3{(120°+A)} + \cos^3{(240°+A)}=\frac {3}{4} \cos{3A}$$
My Approach:
$$\mathrm{R.H.S.}=\frac {3}{4} \cos{3A}$$
$$=\frac {3}{4} (4 \cos^3{A}-3\cos{A})$$
$$=\frac {12\cos^3{A} - 9\cos{A}}{4}$$
Now, please help m... | use that $$\cos(120^{\circ}+x)=-1/2\,\cos \left( x \right) -1/2\,\sqrt {3}\sin \left( x \right) $$
and $$\cos(240^{\circ}+x)=-1/2\,\cos \left( x \right) +1/2\,\sqrt {3}\sin \left( x \right) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 4
} |
How can I improve the proof of this statement with convexity? $A\in\mathbb{R}^{m\times n}$. How do I prove that if $S\subseteq \mathbb{R}^m$ is
convex then so is $A^{-1}(S) =\{ x \in \mathbb{R}^n : Ax
\in S\}$ ?
My proof:
$\forall t \in [0,1], ta+(1-t)b \in S$
Since $Ax\in S$, define $Aa'\in S, Ab'\in S$,
$\forall ... | I think you're overcomplicating a touch,
let $x,y\in A^{-1}(S)$ then if
$$tx + (1-t)y \in A^{-1}(S)$$ for all $t\in [0,1]$, the set is convex.
Now, let $z = tx+(1-t)y$, then $z\in A^{-1}(S)$ iff $Az \in S$. You can now write:
$$Az = ta + (1-t)b$$
letting $a=Ax, b=Ay$, both $a,b\in S$ since $x,y\in A^{-1}(S)$, by conv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Laplace transform of $\sin (t) / t$ I need to find the Laplace transform of $$\frac{\sin(t)}{t}$$ without using the following rule $\mathcal{L}\left(\frac{f(t)}{t}\right)=\int_s^{\infty}F(u)du$.
We aren't allowed to use this rule unless we can prove it, and I'm assuming our lecturer does not want us to use it.
However,... | I think you should write
$$ \int_0^\infty e^{-ut} \frac{\sin{t}}{t} = \int_0^\infty \int_0^\infty e^{-(u+s)t} \frac1{2i}( e^{it}-e^{-it}) \;ds \; dt$$
and change the order of integration. Details depends on the expected level of math rigor.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How to express sum of areas of triangular elements meeting a criteria I have a metric for rating the results of a finite element simulation.
I am summing the area of the triangular elements which have a single value associated with them.
If the value associated with each triangle meets my criteria, I add the area of ... | You can use the Indicator function. Let $A_i$ be the area of the $i$th triangle with associated value $x_i$.
Then your sum is
$$\sum_i A_i \mathbf{1}_C(x_i),$$
where $C$ is set of values for $x_i$ that meets your criteria.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Can every number be represented as a sum of different reciprocal numbers? Can every rational number be represented as a finite sum of reciprocal numbers? You are only allowed to use each reciprocal number one time per expression (So for example 3/2 cannot be 1/2+1/2+1/2).
You could then express these numbers in a kind... | Yes, and there is a simple algorithm to achieve this. First, we subtract successive reciprocal numbers $(1/1,\ 1/2,\ 1/3,\ \ldots)$ until the remainder is less than the next reciprocal number. This is always possible because the harmonic series is divergent.
If the remainder is zero then we are done, so let us assume t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Powers of Möbius transformations equal to identity? I'm looking at "Mobius transformations" where $a,b,c,d\in\mathbb R$. I want to know for which $n$ there exists $a,b,c,d$ such that for $f(x) = \dfrac{ax+b}{cx+d}$,
$$f^n(x) = f(f(...(f(x)))) = x$$
and what relationships between $a,b,c,d$ are required. Or if it is for ... | As said by @Did, you need to find coefficients a,b,c,d such that
$$M(f)^n=\begin{pmatrix}a&b\\c&d\end{pmatrix}^n=k\begin{pmatrix}1&0\\0&1\end{pmatrix}$$
Thinking to rotation matrices, there is an evident solution :
$$M=\begin{pmatrix}\cos(\frac{\pi}{n})&-\sin(\frac{\pi}{n})\\ \sin(\frac{\pi}{n})&\cos(\frac{\pi}{n})\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating an indefinite integral with exponents and logarithms I was taking a GRE practice exam and came across
$$ \int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1 + e^{bx})} dx $$
I noted that this can be expressed as
$$ \int_0^{\infty} \frac{1}{(1 + e^{bx})} - \frac{1}{(1 + e^{ax})} dx $$
And
$$ \int \frac{1... | Hint:
$$
\ln \left(\frac{e^{cx}}{1+e^{cx}}\right)\Big|_{x=0}^{x=\infty} = \ln 1 - \ln \frac{1}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Trouble Identifying Counting Problem Formula? Would this be the correct formula for the counting problem
Partition with Identical Items
$$
\begin{pmatrix}
n \\
r \\
\end{pmatrix}= \begin{pmatrix}
n-1 \\
r \\
\end{pmatrix} + \begin{pmatrix}
n-1 \\
... | I am a member of a group of $n$ people in a room, of whom $r \le n$ will be awarded
one of $r$ identical prizes. Number of ways: ${n \choose r}.$
But I am interested in whether I will get a prize. There are two
cases:
I do not win a one of the prizes: Number of ways: ${n-1 \choose r}.$
Send me out of the room. Award ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many functions are there from $\mathbb Z$ to $\mathbb Z$? Repeating the question,
How many functions are there from $\mathbb Z$ to $\mathbb Z$?
A function $f \colon A \to B$ is a subset of $A \times B$ satisfying
$$(a,b) = (a,c) \qquad \Rightarrow \qquad b = c,$$
so it's enough (maybe) to look at subsets of $\mat... | You’ve made a start. As you say, each function from $\Bbb Z$ to $\Bbb Z$ is a subset of $\Bbb Z\times\Bbb Z$, so there are at most $2^{|\Bbb Z|}$ of them. To finish the argument you could prove that there are also at least $2^{|\Bbb Z|}$ such functions by actually finding that many that you can clearly identify as dist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Axioms for category theory I'm confused of background logic on category theory.
In ZFC set theory, we can construct new sets from existing sets by axioms, such as power set axiom, axiom of pairing etx.
I read first few pages of MacLane's category theory text and now I'm reading Tom Leinster's category theory text. Neit... | The standard axioms vary: they're either ZFC with an axiom of choice for proper classes, some set theory such as NBG that axiomatizes classes more thoroughly, or ZFC with Grothendieck universes, so that "large" categories are interpreted as still being small, but relative to a larger "universe" of sets. There have been... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Difference between f(x) and f(x,y)? I recently started doing calculus and came across terms such as f(x) and f(x,y). What is the difference between them? Are they the same thing?
| No, they are not the same thing. $f(x,y)$ is a function of two variables $x$ and $y$, e.g., $f(x,y) = 3x + \sin(y)$. But $f(x)$ is a function of only one variable, e.g., $f(x) = x^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Proof of quadratic inequality using AM-GM Proof of quadratic inequality using AM-GM
| we have to show that $$x^2y^2+x^2+y^2+4-6xy\geq 0$$ dividing by $x^2+1$ gives
$$y^2-\frac{6xy}{x^2+1}+\frac{4+x^2}{x^2+1}\geq 0$$ this is equivalent to
$$\left(y-\frac{3x}{x^2+1}\right)^2+\frac{(4+x^2)(x^2+1)-9x^2}{(x^2+1)^2}\geq 0$$
and this is equivalent to $$\left(y-\frac{3x}{x^2+1}\right)^{ 2 }+\frac{(x^2-2)^2}{(x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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why is probability not considered part of pure mathematics? wikipedia considers probability a part of applied mathematics, and it doesn't seem to fall under one of the four areas of mathematics (algebra, number theory, topology/geometry,analysis).
Nevertheless it seems to me to be a very fundamental mathematical concep... | The way I interpret the applied and pure dichotomy in math is:
Applied math is taking established mathematical results and utilizing them in describing, understanding and solving real-world problems.
Pure math is a collection of mathematical results that serve the purpose describing, understanding and solving abstract ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prime field of a characteristic zero field is isomorphic to $\mathbb{Q}$ I was attempting to prove that the prime field $P$ of a characteristic 0 field $K$ is isomorphic to $\mathbb{Q}$. Here, I will use the notation
$nx=\left\{\begin{array}{cc}\underbrace{x+\cdots+x}_{n \text{ times}} & \text{if }n>0\\ 0 & \text{if }... | If you define a map $\phi : S\to T$, you must always check that $\phi$ is well defined; if it isn't well defined, it's not a function at all! More accurately, whenever you define $\phi$ based on a presentation of an element $x\in S$, you must check that if $x'$ is a different presentation of $x$, then $\phi(x) = \phi(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ways in which 38 can be divided into 3 positive parts such that the first is divisible by 8, the second by 7 and the third part by 3? I am stuck with the question, i have tried a couple of random approaches but none of them is correct. The answer is 2. Please help if you know how to solve this question.
| Assuming that each part must be an integer, this is equivalent to the number of non negative integral solutions of
$$8x+7y+3z=38$$
We have $x,y,z\ge1$. So, let $a=x-1,b=y-1,c=z-1$ such that $a,b,c\ge0$.
Then,
$$8a-8+7b-7+3c-3=38$$
$$8a+7b+3c=56$$
Let $X=8a,Y=7b,Z=3c$.
$$X+Y+Z=56$$
Allowed values of $X=0,8,16,24,32,40,4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a linear transformation $T$ from $ R^3$ into $R^2$ such that $T(1,-1,1) = (1,0)$ and $T(1,1,1)= (0,1)$ . Is there a linear transformation $T$ from $ R^3$ into $R^2$ such that
$T(1,-1,1) = (1,0)$ and $T(1,1,1)= (0,1)$ .
| A linear transformation is uniquely specified by its action on a basis. We can extend the set of linearly independent vectors $\{(1, -1, 1), (1,1,1) \}$ to a basis for $\mathbb{R}^3$ by adjoining some vector $v\in \mathbb{R}^3$ to the set. The required linear transformation can then be specified by setting $T(v)$ to be... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Induction proof $\sum_{k=1}^{n} k^{2} = \frac{1}{6} n(n+1)(2n+1)$ I'm asked to prove $$\sum_{k=1}^{n} k^{2} = \frac{1}{6} n(n+1)(2n+1)$$
using proof by induction.
Now, I know how to do induction proofs and I end up at this step, needing to prove that:
$$\frac{1}{6} n(n+1)(2n+1) + (n+1)^2 = \frac{1}{6} (n+1)(n+2)(2n+3... | Dividing by $(n+1)$ was good, but when you got rid of the $\frac16$ factor, you forgot to multiply the $(n+1)^2$ term by $6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Can 2 + 2 ≈ 5 be true? I was wondering a little, about how to proof that 2+2 =5
And here I am:
2.4 + 2.4 = 4.8
If we approximated numbers in each side individually then :
2 + 2 ≈ 5
I know this may not be right, but I don't know why it's wrong.
| When you are talking about approximations you are usually doing so in a real world context. And so whether or not $4 \approx 5$ will depend on what you are measuring.
For example, the distance from the earth to the moon is $239,000$ miles. This is approximately equal to $240,000$ miles. So there $1,000$ is basically eq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing isomorphism of field extensions I need to show that if $\alpha$ has minimal polynomial $t^2-2$ over $\mathbb{Q}$ and $\beta$ has minimal polynomial $t^2-4t+2$ over $\mathbb{Q}$, then the extensions $\mathbb{Q}(\alpha):\mathbb{Q}$ and $\mathbb{Q}(\beta):\mathbb{Q}$ are isomorphic.
I want to say that somehow $t^... | Write $\Bbb Q(\alpha)\cong\Bbb Q[t]/(t^2 - 2)$ and $\Bbb Q(\beta)\cong\Bbb Q[x]/(x^2 -4x + 2)$. The roots of $t^2 - 2$ are $\pm\sqrt{2}$ and the roots of $t^2 - 4t + 2$ are $2\pm\sqrt{2}$. Since $t$ in the first quotient represents $\sqrt{2}$ or $-\sqrt{2}$, and $x$ in the second represents either $2 + \sqrt{2}$ or $2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Question about proof that $A\subset B \implies \overline{A}\subset \overline{B}$ Look at this simple proof that $A\subset B \implies \overline{A}\subset \overline{B}$:
$\overline{B}$ is a set that contains $B$, therefore it contains $A$. Since it's closed and contains $A$, it must contain $\overline{A}$.
This made sen... | $\bar C$ is the intersection of the set of all closed sets that have $C$ as a subset. Let $A^*$ be the set of all closed sets that have $A$ as a subset. Let $B^*$ be the set of all closed sets that have $B$ as a subset. If $A\subset B$ then $A^*\supset B^*,$ which implies $\bar A=\cap A^*\subset \cap B^*=\bar B.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to determine level curves/contours of a 3D function? I have the function $f(x,y) = e^{y/x^2}$ and I need to draw a contour map for levels $e^{-2}$, $e^{-1}$, $1$, $e$, and $e^2$.
I set $e^{-2}$ equal to the function, and solved for $y$ so that $y = -2x^2$.
Isn't this curve impossible? What am I doing wrong?
Thank... | You are lucky in the sense that $f(t)=e^{t}$ is an invertible function. That means that you can just set the argument equal to each other just like you have done. But you would not have been allowed to do so if it wasn't invertible!
For example if you set $(x)^2=(-1)^2$ you can't just assume $x=-1$, because $f(t)=t^2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A function in which addition and multiplication behave the same way Exponents have a well-known property:
$$x^ax^b = x^{a+b}$$
but
$$x^{a} + x^{b} \neq x^{a+b}$$
Similarly,
$$\log(a) + \log(b) = \log(ab) $$
But
$$\log(a)\log(b) \neq \log(ab)$$
So my question is this:
Is there a function $f$ on $\mathbb{R}$ or some infi... | Your title expresses interest in "a function in which addition and multiplication behave the same way". That's condition (3) alone. Conditions (1) and (2) are unnecessarily-strong requirements that artificially restrict the possible solutions. Be that as it may ...
Let's invoke condition (3) with three arbitrary values... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 7,
"answer_id": 5
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Show that this random variable is uniformly distributed in $\left(0,1\right)$ Let $X$ be a continuous random variable with distribution function $F_{X}\left(x\right)$ and let $Y=F_{X}\left(x\right)$, show that $Y\sim U\left(0,1\right)$, where $U$ is the uniform distribution.
Since the density function $f_{X}\left(x\rig... | As Did wrote, let $Y=F_X(X)$ and note that $F_X$ need not be strictly increasing for the following arguments to holds. If we define the generalized inverse of $F_X$ by $F_X^{\leftarrow}(y) =\inf \{ x \in \mathbb{R}: F_X(x)\geq y \}$, then try to prove the following steps
*
*If $F_X$ is continuous then (iff actually ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Each integer appears once in difference sequence Does there exist an increasing sequence $a_1,a_2,\dots$ of positive integers such that both of the following are fulfilled:
*
*Each positive integer appears exactly once among $a_2-a_1,a_3-a_2,\dots$
*For some $n$, each positive integer at least $n$ appears exactly o... | To fulfil the first condition, the differences $a_i-a_{i-1}$ must be distinct positive integers. Thus, to avoid repeating a difference, $$a_n\ge a_0+\frac{n(n+1)}2.$$ The second condition requires that all but finitely many positive integers appear among the $a_i-a_{i-2}$. But the above growth is too fast to enable tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve $\int \sqrt{a+x^2} dx $, $a>0$? By setting $a+x^2 = t^2$ I can get $2xdx=2dt$ so $dx=dt/\sqrt{t^2-a}$. But such substitutions only lead to iterate integrals of the form
$$
\int \frac{t^2}{\sqrt{t^2-a}}dt,
$$
substituting again $t^2-a=u^2$ we get
$$
\int udu +\int\frac{2}{\sqrt{u^2+a}}du.
$$
How can I sol... | Hint: substitute $$x=\sqrt{a}\sinh(t)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Avoiding circularity in explaining the meaning of real exponents In an answer to 'What does $2^x$ really mean when $x$ is not an integer?' Álvaro Lozano-Robledo explains that we can understand real number exponents in terms of the definition of $\log(x)$:
$$\log(x) := \int_1^x \frac{1}{t} dt$$
I understand that one can... | There are many approaches to define $a^{x}$ when $x$ is irrational. Note that if we are restricted to real numbers then we must have $a > 0$ for $a^{x}$ to make sense. The simplest (but non-intuitive) approach is to first develop a theory of logarithmic and exponential functions namely $\log x$ and $\exp(x)$. Since the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Correct method of integration involving two exponential terms I have an Integrand involving two exponential terms:
$$
\int_{0}^{\infty} \frac{\exp(x^2)}{(1+\exp(x^2))^2} dx
$$
I would like to know what is the best way to integrate such a function without blowing it up?
What if $x^2$ is replaced by two variables $(x^2 ... | $$
\begin{align}
\int_0^\infty\frac{e^{x^2}}{\left(1+e^{x^2}\right)^2}\,\mathrm{d}x
&=\int_0^\infty\frac{e^{-x^2}}{\left(1+e^{-x^2}\right)^2}\,\mathrm{d}x\\
&=\frac12\int_0^\infty\frac{e^{-x}}{(1+e^{-x})^2}\frac{\mathrm{d}x}{\sqrt{x}}\tag{1}
\end{align}
$$
Consider
$$
\begin{align}
&\int_0^\infty\frac{e^{-x}}{(1+e^{-x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1923997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Let A be an $n*n$ matrix. Prove that if $rank(A) = 1$, then $det(A + E) = 1 + trace(A)$ I feel like I've got the answer, but I've never been good at putting what I think into words.
$\begin{vmatrix}
n_{11} & n_{12} \\
n_{21} & n_{22}
\end{vmatrix} = 0 = n_{11}n_{22} - n_{12}n_{21}$
$\begin{vmatrix}
n_{11} + 1 & n_{12} ... | The value of a determinant as well of a trace is independent of the choice of basis. So suppose that the image of $A$ is generated by a vector $v_1$. Complement this vector with $v_2,...,v_n$ to form a base. In this base the matrix of $A$ takes the form:
$$ \underline{A} =
\left( \begin{matrix} a_{11} & a_{12} & ..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
show sequence $n^2+1$ diverges
Question: Show $S_n = n^2+1$ diverges.
My understanding of this question is proof by contradiction.
We first assume that $n^2+1$ converges to $s$. This implies that fix $\epsilon >0$, there exists $N$ s.t. $|n^2+1-s|< \epsilon$. I'm stuck here. anyhelp?
| Hint: show that for any natural number $N$ there is some $k$ such that $S_{n} > N$ for all $n > k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Did I solve $z^{n-1}=\overline{z}$ correctly?
Please could someone tell me if my solution of $z^{n-1}=\overline{z}$
is correct?
My solution:
We have $z^{n-1} =\overline{z}$ if and only if $z^{n-1}z =\overline{z}z$, so
$z^n = |z|^2$
In polar coordinates,
$$ r^n e^{i n \varphi} = r^2$$
or, equivalently,
$$ r^{n-2}... | I assume $n$ stands for a positive integer.
If $n=1$, the equation is $1=\bar{z}$.
If $n=2$, the equation is $z=\bar{z}$, so any real number is a solution.
Suppose $n>2$. First of all, $z=0$ is a solution. Assume now $z\ne0$ and write it as $z=re^{i\varphi}$. The equation becomes
$$
r^{n-1}e^{i(n-1)\varphi}=re^{-i\varp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to factorize $5$ in $\mathbb{Z}[\root 3 \of 2]$? Since $5$ has a norm of $125$ in this domain, and $N(1 + (\root 3 \of 2)^2) = 5$, it seems like a sensible proposition that $5 = (1 + (\root 3 \of 2)^2) \pi_2 \pi_3$, where $\pi_2, \pi_3$ are two other numbers in this domain having norms of $5$ or $-5$. This is suppo... | We have
$$ \mathbf Z[\sqrt[3]{2}]/(5) \cong \mathbf Z[x]/(x^3 - 2, 5) \cong \mathbf Z_5[x]/(x^3 - 2) \cong \mathbf Z_5[x]/(x+2) \times \mathbf Z_5[x]/(x^2 + 3x + 4) $$
so that the ideal $ (5) $ factors as $ (5) = \mathfrak p_1 \mathfrak p_2 $. To find the ideals $ \mathfrak p_1 $ and $ \mathfrak p_2 $, note that they c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Let $p,q$ be irrational numbers, such that, $p^2$ and $q^2$ are relatively prime. Show that $\sqrt{pq}$ is also irrational. Progress:
Since, $p,q$ are irrationals and $p^2$ and $q^2$ are relatively prime, thus, $p^2\cdot{q^2}$ cannot be a proper square, so, $pq$ is also irrational. Suppose, $pq=k$, then: $$\sqrt{k}\cd... | Except that you haven't proved $p^2 * q^2 $ cannot be perfect square, rest of the proof is correct.
Since $p^2 $ and $q^2$ are relatively prime, they do not have common prime factors. And they are not perfect squares. Else p,q would be rational numbers. This is necessary since 25, 4 are relatively prime but their produ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Derivative of scalar function $\rm v^\top A^n v$ w.r.t. matrix $\rm A$ I would like to derivative of a scalar function with respect to matrix.
In pariticular,
For given vector $v \in \mathbb{R}^n$, let $f(A)=v^\top A^n v$ for any integer $n\in\mathbb{N}$. I want to find $\nabla_A (v^\top A^n v)$
When $n=1$, $$\nabla_A ... | Let
$$f (\mathrm X) := \mathrm a^{\top} \mathrm X^n \, \mathrm a = \mbox{tr} (\mathrm a^{\top} \mathrm X^n \, \mathrm a) = \mbox{tr} (\mathrm a \mathrm a^{\top} \mathrm X^n)$$
Since
$$(\mathrm X + h \mathrm M)^n = \mathrm X^n + h \left( \sum_{k=0}^{n-1} \mathrm X^k \mathrm M \mathrm X^{n-1-k} \right) + O (h^2)$$
then t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Confusion on Summation Notation in Lagrange's Identity I'm working on a proof (the proof of Lagrange's Identity), but it includes a sum notation I'm not familiar with:
$$\sum_{1\le k\lt j\le n} (a_kb_j-a_jb_k)^2$$
I would appreciate any explanations of what this is saying, specifically in regards to the inequalities be... | ${1\leq j<k\leq n}$ is the domain of the operator. We sum the terms for all integer values of the bound variables, $(j,k)$, where this domain holds true.
This is sometimes more convenient than the double sum notation with which you might be more familiar.
$$\sum_{1\leq j<k\leq n} (a_k b_j −a_j b_k )^2
\\ ~=~ \sum_{j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How can we express the statement "$f$ is a bijection from $A$ to $B$" in predicate logic? How can we express the statement
$f$ is a bijection from $A$ to $B$
in predicate logic? Can it be expressed in first-order logic?
A problem seems to be that the sets $A$ and $B$ are different sets, while predicate logic applies ... | Establish Domain and Codomain:
$$\forall y~ \forall x~ F(x) = y \implies (x \in A \land y \in B)$$
Surjection:
$$\forall y \in B ~ \exists x ~F(x)=y$$
Injection:
$$\forall y ~ \forall x_1 ~ \forall x_2 ~ (F(x_1) = y \land F(x_2) = y) \implies (x_1 = x_2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof that columns of an invertible matrix are linearly independent
Explain why the columns of an $n \times n$ matrix $A$ are linearly
independent when $A$ is invertible.
The proof that I thought of was:
If $A$ is invertible, then $A \sim I$ ($A$ is row equivalent to the identity matrix). Therefore, $A$ has $n$... | I’ll use rows instead of columns, but the argument for columns holds similarly. Suppose a square matrix A is invertible that means it is row equivalent to the identity matrix. Now, observing that the rows of the identity are linearly independent, you can reapply the reverse operations on the rows of the identity to get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 6,
"answer_id": 5
} |
Selling price of coffee mixture
A trader buys $15$ Kg of Arabian coffee powder at $x$ dollars per kg and $25$ kg of Brazilian coffee at $y$ dollars per kg. He mixes the two types of coffee powder in a ratio $3 : 5$ and packs the mixture into packets each of which contains $100$ grams of the mixture. He sells the packs... | "I done up a ratio of the price of the coffee powder = 3x:5y.
Why is the cost price of the coffee powder not 3x+5y? "
It is. What makes you think it isn't? Of course, that is not what the question asked so would not be the "answer" to the question. The question asked for the selling price.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Extremality of $*$-automorphisms of $C^*$-algebras In a paper, I came across the notion of the extremality of a $*$-automorphism of a $C^*$-algebra. What is the definition of this property?
Edit: Basically, I have a unital completely positive map $\varphi: A \rightarrow A$, where $A$ is a $C^*$-algebra, such that $\psi... | The author is saying that automorphisms are extreme among the ucp maps.
Write $\psi=(\phi_1+\phi_2)/2$, with $\phi_1,\phi_2$ ucp. Let $a\in A$ selfadjoint. Then, as $\phi_1(a)$ and $\phi_2(a)$ are selfadjoint, $(\phi_1(a)-\phi_2(a))^2\geq0$. So
\begin{align}
\psi(a)^2&=\frac{(\phi_1(a)+\phi_2(a))^2}4
=\frac{\phi_1(a)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Cayley graph interpretation D3 I am trying to understand the Cayley graph for the group $D_3$, which from Mathematica, I got:
I tried to get the multiplication table in Mathematica:
$\left(
\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 1 & 4 & 3 & 6 & 5 \\
3 & 5 & 1 & 6 & 2 & 4 \\
4 & 6 & 2 & 5 & 1 & 3 \\
5 ... | The table and graph are related as follows. Let $X=\{1,2,3,4,5,6\}$ and let $S=\{2,4\}$. The table is missing its "headings" so that the actual table looks like this
$$
\begin{array}{c|cccccc}
& 1 & 2& 3& 4& 5&6\\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 \\
2 & 2 & 1 & 4 & 3 & 6 & 5 \\
3 & 3 & 5 & 1 & 6 & 2 & 4 \\
4 & 4 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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