Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Let $A\in M_{1\times3}(\mathbb{R})$ be a arbitrary matrix. Find the eigenvalues and eigenvectors of matrix $A^TA$. Let $A\in M_{1\times3}(\mathbb{R})$ be a arbitrary matrix. Find the eigenvalues and eigenvectors of matrix $A^TA$.
My approach:
$$ A^TA = \begin{bmatrix}
a^2 & ab & ac\\
ab & b^2 & bc\\
ac & bc & c^2
\en... | $\DeclareMathOperator{\tr}{tr}$The columns of $A^TA$ are all scalar multiples of $A^T$, so for $A\ne0$, this matrix has rank 1: its column space is spanned by $A^T$ and two of its eigenvalues are $0$. The last eigenvalue you get “for free” since the trace is equal to the sum of the eigenvalues, so it is $\tr A^TA-0-0=A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386612",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Why is the word "complement" used in set theory? Maybe this should have been on the English Exchange, but why do we use the word "complement" in set theory? If I have:
$$(A \cup B)'$$
Why does "complement" mean everything but the union? Is it because it is "all the things" that the original operation is not, thus it ... | "X complements Y" in colloquial English means basically "X has what Y lacks". This is exactly what the complement is in set theory, except that the complement of $A$ also has none of what $A$ has.
That said, my suspicion is that this term actually originates in mathematical French and was borrowed directly from mathema... | {
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"timestamp": "2023-03-29T00:00:00",
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The number of the partition of the set $A$ into $k$ bounded blocks. Let $A=\{1,2,\cdots,n\}$ be a set. We want to partitions of this set into $k$ non-empty unlabelled subsets $B_1,B_2,\cdots ,B_k$ such that cardinality of each $B_i$ between positive integers $a$ and $b$, that means $a\leq |B_i|\leq b$.
Let $D_{a,b}(n... | Supposing that we are trying to generalize Stirling numbers here we
get the combinatorial class
$$\def\textsc#1{\dosc#1\csod}
\def\dosc#1#2\csod{{\rm #1{\small #2}}}
\textsc{SET}_{=k}(\textsc{SET}_{a\le\cdot\le b}(\mathcal{Z}))$$
which yields the generating function
$$G(z) = \frac{1}{k!} \left(\sum_{q=a}^b \frac{z^q... | {
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The integral of complex function is zero
Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function on $\mathbb{C}$ and holomorphic in $\mathbb{C}\setminus \mathbb{R}$. Prove that for every closed curve $\gamma$: $\int_\gamma f(z)\,dz=0$.
So if $\gamma$ does not intersect $\mathbb{R}$ at all then we know that ... | One thing you could do is define $F(z) = \int_{[0,z]}f(w)\,dw.$ If you can show $F'(z) = f(z)$ everywhere, then you'll know $F$ is entire, hence its derivative $f$ is entire. The conclusion then follows from Cauchy's theorem.
To get started, suppose $z$ is in the upper half plane $\mathbb H.$ Then $z+h\in \mathbb H$ if... | {
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Are there 4 consecutive numbers that are each the sum of 2 squares? Some numbers can be expressed as the sum of two squares (ex. $10=3^2+1^2$) such as:
$$0,1,2,4,5,8,9,10,13,16,17,18,20,25,...$$
Other numbers are not the sum of any two squares of integers:
$$3,6,7,11,12,14,15,19,21,22,23,24,27,...$$
There are a lot of ... | No, there aren't
Every odd integer that is the sum of two squares, is congruent to $1$ modulo $4$ because every prime factor of the form $4k+3$ must occur in a power with even exponent.
| {
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What will be the negation of this statement:
Every street in the city has at least one house in which we can I find a person who is rich and beautiful or highly educated and kind.
Negation: 'There exists a street in the city where in every house we can find no person who is rich and beautiful or highly educated and ... | The negation is NOT(Every street in the city has at least one house in which we can I find a person who is rich and beautiful or highly educated and kind.)
Which means --- NOT Every street in the city has at least one house in which we can I find a person who is rich and beautiful or highly educated and kind.
Which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2387120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the real and imaginary part of z let $z=$ $$ \left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta} \right)^n$$
Rationalizing the denominator:
$$\frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta}\cdot\left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta + i\cos\theta}\right) =... | HINT: Express the fraction as $r e^{i\theta}$ and compute $r^n e^{i n\theta}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Understanding the use of brackets in set theory notation I would appreciate help understanding the presence and absence of brackets in this particular example, which hopefully will clarify things for me.
It is a line from a short proof by contradiction based on the Axiom of Foundation that No Set is an Element of Itsel... | Given an arbitrary set $S$, the goal is to prove that $S$ is not an element of itself. The proposed proof proceeds by considering the set $T=\{S\}$, i.e., the set $T$ whose one and only member is the set $S$. Then $T$ is nonempty, because it has a member, namely $S$. So we can apply the axiom of regularity to conclude ... | {
"language": "en",
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solutions for $x^n = 1$ I'm supposed to solve this in terms of $n$, a natural number.
I'm really getting tripped up on this, and I don't really know why. The only way this can have a solution is if $n = 0$, specifically the algebra I wrote to show this is
$
\begin{gather*}
x^n = 1\\
\log_x(x^n) = \log_x(1)\\
n\log_x(x)... | Your question is a bit unclear. You say that you are solving this "in terms of $n$, a natural number. The following doesn't give you all the details, you will need to figure out what the question is actually asking before you write down your solution.
First question is: What is your definition of natural number? Or to... | {
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"timestamp": "2023-03-29T00:00:00",
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Does $\sum^{\infty}_{1}\frac{1}{k+1} = \sum^{\infty}_{2}\frac{1}{k}$ diverge because $\sum^{\infty}_{1}\frac{1}{k}$ diverges? If you manipulate the index of a series does it still converge/diverge?
For example:
Does $\sum^{\infty}_{1}\frac{1}{k+1} = \sum^{\infty}_{2}\frac{1}{k}$ diverge because $\sum^{\infty}_{1}\frac{... | $$\sum^{\infty}_{1}\frac{1}{k+1}=\frac 12 +\frac13 +\frac 14+\ldots$$
$$\sum^{\infty}_{1}\frac{1}{k}=1+\frac 12 +\frac13 +\frac 14+\ldots$$
Hence $$\sum^{\infty}_{1}\frac{1}{k+1}=\sum^{\infty}_{1}\frac{1}{k}-1$$
Since $$\sum^{\infty}_{1}\frac{1}{k}\to \infty \implies \sum^{\infty}_{1}\frac{1}{k}-\color{blue}1 \to \inft... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many squares does the diagonal of this rectangle go through?
I have a rectangle made of tiles measuring $9$ by $12$ tiles to get $108$ tiles. A diagonal line is cut through the top left corner down to the bottom right corner. How many tiles does the diagonal go through?
I had this question on a test today and I w... | The diagonal $d$ will go through two grid points which divide it into three equal parts. Each part $d'$ is a diagonal of a $4\times3$ grid rectangle $R$. It intersects three horizontal and two vertical interior grid lines of $R$. These $5$ intersection points partition $d'$ into $6$ parts. It follows that $d'$ traverse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2387729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving a linear matrix equation Could you please help me solve this problem?
Given a $n$ by $m$ matrix $Q$ whose columns are orthonormal ($rank(Q)=m$) and a $n$ by $n$ symmetric definite positive matrix $X$ (the unknown) and another $n$ by $n$ symmetric definite positive matrix $B$.
We want to solve the following prob... | If I understand the statement of the problem correctly, $A=Q Q^t$ is an orthonormal projection onto an $m$ dimensional subspace of ${\Bbb R}^n$: $A^2=A=A^t$.
You have a solution iff the image of $B$ is in the image of $A$. In this case $ABA=B$ as well (when $B$ is symmetric) so a solution is simply $X=B$. If $B\neq ABA... | {
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Is it possible to dissect a circle of radius 9 into 81 equal areas, using only circles? You can dissect a circle of radius $3$ into $9$ equal areas by placing within it $5$ unit circles in an orthogonal cross shape. You could then place $5$ of these radius $3$ circles into a circle of radius $9$ in a similar way. Is it... | My first thought was to make 80 concentric circles, all centered at the center of the big circle. Choose radii $r_1, r_2, \cdots, r_{80}$ so that each band of
the dart board has the area $1/81$ of the area of the big circle.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Difference between Ritz vectors and Eigenvectors This is probably a silly question, as it came from an error in the Eigenvectors I found using ARPACK (Fortran). In this case, the values of the Ritz vectors are identical in value to the theoretical Eigenvectors but different in sign. So what is the real difference betwe... | First of all, the eigenvectors corresponding to eigenvalues of multiplicity one are only unique up to a scalar. Hence, if $v$ is an eigenvector, so is $\mu v$ for $\mu \in \mathbb{C}$.
Let me turn to the definition of Ritz vectors. Ritz vectors are usually approximations to the eigenvectors of a matrix $A$ that are obt... | {
"language": "en",
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Interpret this function notation? I have the following function notation
\begin{align}
f: &\, \mathbb{R} \rightarrow \mathbb R^2 \\ &x\mapsto y=f(x)
\end{align}
Does it actually mean
\begin{gather}
y=f(x) \\
(y_1,y_2)=(f_1(x),f_2(x)) \\
\begin{cases}
y_1=f_1(x) \\
y_2=f_2(x)\end{cases} \qquad ?
\end{gather}
Or ... | Yes. The notation $f:\mathbb{R}\to\mathbb{R}^2$ tells you that the function takes in a single real number and returns an ordered pair of real numbers. Hence the notation $x\mapsto y=f(x)$ is telling you that $y$ is an ordered pair of real numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2388272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I fit two equal-length arcs in the opposite corners of a right triangle? I am trying to make two arcs of the same length such that they will both fit in a right triangle like so:
I am given the first radius (r1) and a constant k that is the difference between the leg opposite angle one and the second radius (r2... | just hint
The same length means that
$$r_1\theta =r_2 (\frac {\pi}{2}-\theta) $$
or
$$\theta=\frac {\pi r_2}{2 (r_1+r_2)} $$
in the triangle,
$$\sin (\theta)=\frac {k+r_2}{r_1+r_2} $$
if we put $k=xr_2$ then
$$x=\frac {2\theta}{\pi}\sin (\theta)-1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Analytic Continuation for a Product I was trying to solve the functional equation
$$\phi(x)^2\phi(2x)=x^2+2x+1$$
and by assuming that $\phi(1)=1$, and setting up a recurrence relation, I found the solution
$$\phi(x)=\prod_{i=0}^{\log_2(x)-1} (2^i+1)^{x-i}$$
However, this only makes sense for values of $x$ that are perf... | $\phi$ is non-negative.
$f(x)=\log(\phi(x))$ satisfies
$$2f(x)+f(2x)=\log\left((x+1)^2\right)$$
Since you are talking about analytic continuation
For analytic solutions we can compute the derivatives at $x=0$
$$2f^{(n)}(x)+2^nf^{(n)}(2x)=\frac{d^n}{dx^n}\log\left((x+1)^2\right)$$
Therefore $$f^{(n)}(0)=\frac{1}{2+2^n... | {
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Show that $e^{-x}x^n$ is bounded on $[0,\infty)$ and hence prove that $\int_0^\infty e^{-x}x^n \, dx$ exists. Show that $e^{-x}x^n$ is bounded on $[0,\infty)$ for all positive integral values of $n$. Using this result show that $\int_0^\infty e^{-x}x^n \, dx$ exists.
My work:
I know $f$ is a continuous function and $\l... | For the boundedness, yoou can take the derivative and show that it is negative on some interval $[a,\infty)$ and thus show the function is bounded on $[a,\infty)$. It is bounded on $[0,a]$ since it's continuous.
The way you show the integral on the infinite interval exists is to remember the definition of an improper i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Decidability of irreducibility in $Z[X]$ I am interested in a naive algorithm for testing irreducibility in $\mathbb Z[X]$: given a polynomial $p=a_nX^n+\ldots+a_0$ in $\mathbb Z[X]$, is there a known explicit bound $M=M(n,\max_{i=0}^n|a_i|)$ such that if $p$ factorizes as $qr$ in $\mathbb Z[X]$, then the coefficients ... | An alternative idea that doesn't involve the complex roots of $p$:
Find $n+1$ distinct integers $x_0, x_1, \ldots, x_n$ such that $p(x_i) \neq 0$ for all i. Note that if $p$ factorizes as $p = qr$, then we have $p(x_i) = q(x_i) r(x_i)$ for all $i$. This gives us an upper bound on $\left|q(x_i)\right|$, and the values $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to evaluate $\int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy$
How to evaluate $$F(\Lambda)=\int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy$$ where $\Lambda$ is a positive real number?
I tried evaluating the innermost i... | By symmetry, for any $\Lambda>0$ we have
$$\begin{eqnarray*} F(\Lambda)=\iint_{(0,+\infty)}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2\,dx\,dy&=&2\int_{0}^{+\infty}\int_{0}^{x}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2\,dy\,dx \\ &\stackrel{y\mapsto xz}{=}&2\int_{0}^{+\infty}\int_{0}^{1}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2388892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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pick out the uniformly continous function Pick out the uniformly continuous function for $x \in (0,1)$
$$(1) \quad \quad\quad f(x)= \cos x \,\cos \frac {\pi}x$$
$$(2)\quad \quad \quad f(x) = \sin x \, \cos \frac {\pi}x$$
i was trying this question, i was think that $\sin x$ and $\cos x$ are periodic and continuous ,... | In case (2) you can define it as $0$ for $x=0$. This turns it into a continuous function on $[0,1]$.
In fact, $\sin(x)\cos(\pi/x)$ is continuous on $(0,1]$ and $\lim_{x\to0^+}\sin(x)\cos(\pi/x)=0$.
Therefore, by Cantor's theorem, it is uniformly continuous.
In case (1), for $\epsilon=1/2$ we can find $x_n=\frac{1}{2k... | {
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"timestamp": "2023-03-29T00:00:00",
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Subgroups of $\{\sigma^k,\tau \sigma^k\mid 0\le k \le 7\}$ I am reading in Dummit and Foote, page $579$. We have the following group:
$$G = \langle \tau,\sigma\mid \sigma^8,\tau^2,\sigma \tau = \tau \sigma^3\rangle = \{\sigma^k,\tau\sigma^k\mid 0\le k\le 7\}$$
which is just $Gal\left( \mathbb Q(i,\sqrt[8]{2})/\mathbb Q... | The Galois group of $K/\mathbb{Q}$ with $K:=\mathbb{Q}(\sqrt[8]{2},\zeta)$ has order $16$ and is an extension of $(\mathbb Z_8)^\times$ by $\mathbb Z_4$. For all groups of order $16$, in particular for this Galois group, the subgroups have been computed, see here, or here and similar references. One can use the classif... | {
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The Number Theoretic Statement is ..... Prove or Disprove :
There exists $A\subset\mathbb{N}$ with exactly FIVE elements, such that sum of any three elements of $A$ is a prime number.
I don't have any hint or insight about proving or disproving the above statement. I even don't know whether the statement is true or no... | Consider the residues of the elements $\pmod 3$. Note that if we have $a_1 \equiv 0 \pmod 3, a_2 \equiv 1 \pmod 3, a_3 \equiv 2 \pmod 3$, then we are done - their sum is divisible by 3. So only two of the residues $\pmod 3$ can be present.
But by the pigeonhole principle, that means that at least $\left\lceil\frac{5}{2... | {
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Why does the difference between squares of the $i^{th}$ even and odd integers increase by 4? I've been trying to find a formula for sum of the first n squares of odd numbers (a question from Spivak's calculus) and I was trying to subtract the the $i^{th}$ odd number from the $i^{th}$ even number, I've noticed that the ... | Choose an even number and write as $2k$. Then the first two differences between
even/odd squares are:
$4k^2 - (2k - 1)^2 = 4k^2 - ( 4k^2 - 4k +1) = 4k - 1$
$ (2k+2)^2 - (2k+1)^2 = 4k^2 +8k + 4 - (4k^2 +4k + 1) = 4k + 3 $
and clearly $4k+3 - (4k-1) = 4$
| {
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"timestamp": "2023-03-29T00:00:00",
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Is there a geometric method to show $\sin x \sim x- \frac{x^3}{6}$ I found a geometric method to show $$\text{when}\; x\to 0 \space , \space \cos x\sim 1-\frac{x^2}{2}$$ like below :
Suppose $R_{circle}=1 \to \overline{AB} =2$ in $\Delta AMB$ we have $$\overline{AM}^2=\overline{AB}\cdot\overline{AH} \tag{*}$$and
$$\o... | Not a complete answer, I am afraid. Approximating the arc $AB$ as a line segment leads to the correct approximation of $\cos(x) \sim 1 - \frac{x^2}{2}$ but leads to the inaccurate result for the corresponding sine as $ \sin(x) \sim x - \frac{x^3}{8}$. Nevertheless, I post my approach, since it has not been covered in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2389537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 6,
"answer_id": 4
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Finite symmetries for embeddings of genus $\geq 2$ surfaces in $\mathbb{R}^3$ Let $f : \Sigma \to \mathbb{R}^3$ be a genus $g \geq 2$ surface smoothly embedded in $\mathbb{R}^3$. Let
$$
G(f) = \{ \phi \in \text{Isom}(\mathbb{R}^3) : \phi(f(\Sigma)) = f(\Sigma)\}
$$
be the group of isometries of $\mathbb{R}^3$ that p... | Benson-Tilsen's paper "Isometry Groups of Compact Riemann Surfaces" seems to have part of the answer. Be careful, though; it only deals with orientation-preserving isometries ($G^+ \neq G$), so the numbers there are half of what they could be. And it doesn't deal with embedding.
The important result, for the maximum po... | {
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Implicit differentiation for $x^2+y^2-x^2y^2=1$ while the slope of the curve must be ($0$) $x^2+y^2+cx^2y^2=1$
1- What happens to the curve when $c=-1$? Describe what appears on the screen. Can you prove it algebraically?
2- Find $y'$ by implicit differentiation. For the case $c=-1$, is your expression for $y'$ consis... | When $c=-1$ then the curve reduces to $(y^2-1)(x^2-1)=0.$ That means either $y= 1$ or $y=-1$ or $x=1$ or $x=-1.$ So the graph consists of two vertical lines and two horizontal lines.
When $y=1$ or $y=-1$ then the expression you've got for $y'$ becomes $0,$ as you'd expect given that those are two horizontal lines. When... | {
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"url": "https://math.stackexchange.com/questions/2389848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $x^2\geq 0$ an axiom? For real numbers, $x^2\geq 0$ is always true, but why actually?
Is it an axiom, definition or is there a proof?
| Any of the above, depending on exposition.
There are two typical axiomatic approaches to ordered fields, which boil down to how to relate multiplication to the ordering.
One version axiomatizes the ordering, and including the requirement that if $0 \leq a$ and $0 \leq b$, then $0 \leq a \cdot b$.
One can then use this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given two stochastic processes on a probability space, will their compound process be a valid stochastic process on the same probability space? Let the stochastic process $M=(M_t, t\ge 0)$ and the stochastic pathwise continuous increasing process $Y=(Y_t,t\ge 0)$ be defined on the probability space $(\Omega, \mathcal F... | Your "compound" process $M_Y$ at a fixed $t\ge 0$ is the composition of two mappings $\psi_t(\omega):=(\omega,Y_t(\omega))$ and $\varphi(\omega,u):=M_u(\omega)$. The former is an $\mathcal F / \mathcal F\otimes\mathcal B$ measurable mapping of $\Omega$ to $\Omega\times[0,\infty)$. (Where $\mathcal B$ denotes the Borel ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why a definite integral is not infinite If i consider a definite integral of $f(x)=1$ from $[-1,1]$ in $\mathbb{R}$, why the integral is equal to $2$ even if from $-1$ to $1$ there are infinite points ?
Thanks
| Because the definite integral is calculating the area between the curve and the horizontal axis. It isn't just counting points.
In the case of $f(x)=1$ on $[-1,1]$, the region under the curve is a rectangle with length $2$ and height $1$. This gives an area of $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to tell the number of solutions to a simultaeneous equation?
How many solutions are there to the following simultaeneous equations?:
$$
\begin{align}
x - 2y + 3z = 1\\
2x + 2y - z = 4\\
4x - y + 5z = 6
\end{align}
$$
How can I know the number of solutions that there are?
EDIT:
I have found z = 0, y = 2/7, x = ... | Find the augmented matrix corresponding to the system of equations, then, using Gaussian elimination, manipulate it into row echelon form. This will give you information about the number of solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Curvature of a curve is independent of the parametrization My professor defined the curvature $\kappa$ of a regular curve $\gamma:[a,b]\to\mathbb{R}^n$ at a point $t\in(a,b)$ to be $$k(t) = \frac{\left|\gamma''(t)\right|}{\left|\gamma'(t)\right|^2}$$ but there was not enough time to prove that this definition is indepe... | This expression is correct for any constant-speed parametrization, which are all related by affine diffeomorphisms (i.e. $\phi'' = 0$); so your calculation shows invariance within this restricted class.
For formulae in a general coordinate see e.g. wikipedia.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Matrix with spectral radius greater or equal 1 such that fixed point iteration converges Let $M \in \mathbb{R}^{n \times n}$. In a lecture on numerical linear algebra we had a theorem which states that the iteration $\phi(x) = Mx+b$ converges for all $b \in \mathbb{C}^n$ and initial values $x_0 \in \mathbb{C}^n$ if and... | If $v$ is an eigenvector for $\lambda$, then $\text{Re}(M^n v) = M^n \text{Re}(v)$ and $\text{Im}(M^n v) = M^n \text{Im}(v)$. If $\|M^n v\| > N$, then at least one of these has norm $> N/2$. Thus for $b = 0$ and at least one of $x_0 = \text{Re}(v)$ and $x_0 = \text{Im}(v)$, the iteration doesn't converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the derivative of $y=x^{\sin x}$ Could someone please explain step 3 for the following: Why do they multiply $1/y$ with $y'$? I understand that the derivative of $\ln y$ is $1/y$, but I don't understand why it is multiplied with $y'$ in step 3.
Find the derivative for $y=x^{\sin x}$
Step 1: $\ln y=\ln x^{\sin ... | The derivative of $\ln y$ with respect to $y$ is certainly $1/y$, but in this case the derivative needs to be taken with respect to x, so by the chain rule, the derivative of $\ln y$ with respect to $x$ is $y'/y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Scaling a set of reals to be nearly integers I have a set $P$ of $n$ positive real numbers, for example:
$$
P = \{ \pi, e, \sqrt{2} \} \approx \{3.14159, 2.71828, 1.41421\} \;.
$$
Given some $\epsilon > 0$, I would like to find the smallest scale
factor $s \ge 1$ so that, for each $x \in P$, $s x$ is within $\epsilon$ ... | Such $s$ leads to good rational approximations for the quotients of the components (here, $\frac \pi e\approx\frac{22}{19}$, $\frac\pi{\sqrt 2}\approx\frac{22}{10}$, and $\frac e{\sqrt 2}\approx\frac{19}{10}$).
Generalizing, we look for rational approximatons $\alpha\approx \frac nm$ where more precisely $\frac{n-\epsi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sum of unitary operators converges to projection operator So let $\mathcal{H}$ be a hilbert space and $U$ is a unitary operator on $\mathcal{H}$. Let $I=\{v\in\mathcal{H}:U(v)=v\}$. Show that $\frac{1}{N}\sum_{n=1}^NU^n(v)\rightarrow Pv$ where $P:\mathcal{H}\rightarrow\mathcal{H}$ is the projection operator onto $I$. I... | if $v \in I$ the result is trivial. Let $v \in I^{\perp}$ since the restriction of $Id - U$ on $I^{\perp}$ is bijective (Indeed it is injective : if $x \in I^{\perp}$, $(Id-U)(x) = 0 \Rightarrow x \in I \Rightarrow x = 0$ it is also surjective because if $z \in Im(Id-U)$ then there is $y \in H$ such that $z = (Id -U)(y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving $c A + I = (\det A) S$ I have a simple matrix equation $$c A + I = (\det A) S$$ which seems linear (or perhaps quadratic) and involves the determinant of the matrix being solved for. Here, $c$ is a constant, and $S$ is a matrix. How could I solve for $A$, a symmetric matrix in $\mathbb{R}^{k\times k}$? An extre... | I'm not sure if this will help or not.
So, let us do case $k=2$.
Let $A$ be a solution and let $Q$ such that $QAQ^{-1}$ is $A$'s Jordan form. Multiplying the equation we get something of form
$$\begin{pmatrix}
c\lambda_1+1 & 0\\
0 & c\lambda_2+1
\end{pmatrix} = \lambda_1 \lambda_2 QSQ^{-1} = \lambda_1 \lambda_2 \begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2390812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Finding a solution to a system Let $i$ be of the form $i=2^a3^b5^c$, where $a,b,c\ge 0$ are integers.
Consider numbers $x_{i,4},x_{i,6}$ where $x_{i,6}$ is defined when $2$ or $3$ divides $i$, $x_{i,4}$ is defined only when $2$ divides $i$.
The constraints are
*
*If $2$ divides $i$, then $$x_{i, 4} + x_{i,6} \le \... | Actually this system have no solution, first $x_2,y_2,x_3,y_3,x_5,y_5 \in \mathbb{R} \geq 0$.
And $x_2+y_2=1 = \sum \limits_{k=1}^{\infty} \frac{1}{2^k}$
And $x_3+y_3 = \frac{1}{2} = \sum \limits_{k=1}^{\infty} \frac{1}{3^k}$
And $x_5 +y_5 = \frac{1}{4} = \sum \limits_{k=1}^{\infty} \frac{1}{5^k}$.
So $\sum x_{i,4} = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Matrix chain rule question: what is $\frac{d}{dX} f(S)$ where $S = (A+X)^{-1}$ I'm trying to find the following derivative: $$\frac{d}{dX} f(S)$$ where $f$ is a function that takes a matrix and returns a scalar, and $S=(A+X)^{-1}$. Assume that we know what $\frac{d}{dS} f(S)$ is (for example, if $f(S)=\exp(u^\intercal ... | We know how to calculate the gradient with respect to $S$
$$G=\frac{\partial f}{\partial S}$$
We also know that
$$\eqalign{
X &= S^{-1} - A\cr
dX &= -S^{-1}\,dS\,S^{-1} &\implies dS = -S\,dX\,S \cr
}$$
Let's use this to write the differential of the function, and then perform a change of variables to find a result ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When is a graph which is a combination of two graphs connected Let $G$ and $H$ be simple graphs, we build the graph $G \times H$ such that every vertex $(u,v)$ in $G \times H$ is an ordered pair of one vertex from $G$ (the first) and another from $H$ (the latter).
Additionally, two vertices $(u_1,v_1),(u_2,v_2)$ are co... | Note that $(u_1,v_1)$ and $(u_1,v_2)$ are not connected by an edge in $G\times H$.
The requirement that the graphs $G$ and $H$ are connected is fairly straightforward. If say G can be decomposed into two graphs $G_1$ and $G_2$ which are not connected, then clearly there will also be no connections between $G_1\times H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Normal approximations: What is the probability that player A will have at least 10 points more than player B? Two players $A$ and $B$ play the following game. Player $A$ uses a fair 8-sided die with numbers of $1, ...,8$ and rolls it to earn points. For each roll player $A$ collects a numbers of points corresponding to... | To start I give you some hints:
a) Firstly the inequlity $Y_A > Y_B + 10$ can be transformed to $Y_A-Y_B>10$. Then using the converse probability:
$P(Y_A-Y_B>10)=1-P(Y_a-Y_B\leq 9)$
The expected values of the random variables are $E(Y_A)=E(Y_B)=50\cdot 4.5$.
The variances of the random variables are $Var(Y_A)=50\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391358",
"timestamp": "2023-03-29T00:00:00",
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Finding the remainder of $N= 10^{10}+10^{100}+10^{1000}+\cdots+10^{10000000}$ divided by $7$ $$N= 10^{10}+10^{100}+10^{1000}+\cdots+10^{10000000}$$. What is the remainder when N is divided by 7?
$$N=(10^{10}+10^{100}+10^{1000}+\cdots+10^{10000000})/7$$
$$Rem[3^{10}+3^{100}+\cdots+3^{10000000}]/7$$
Now I did not underst... | 3^9 is a multiple of 3^3 which can be written as 27 and what is 27=28-1 now if you divide 27 by 7 what is the remainder?. It is -1 so now you multiply it with 3 in every step
so= $-3 (7times )/7$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to change the appearence of the correct answer of $\cos55^\circ\cdot\cos65^\circ\cdot\cos175^\circ$ I represented the problem in the following view and solved it: $$\begin{align}-\sin35^\circ\cdot\sin25^\circ\cdot\sin85^\circ\cdot\sin45^\circ&=A\cdot\sin45^\circ\\ -\frac{1}{2}(\cos20^\circ-\cos70^\circ)\cdot\frac{1... | As $\cos175^\circ=\cos(180^\circ-5^\circ)=-\cos5^\circ,$
Like prove that : cosx.cos(x-60).cos(x+60)= (1/4)cos3x
$$4\cos(60^\circ-5^\circ)\cos5^\circ\cos(60^\circ+5^\circ)=\cos(3\cdot5^\circ)$$
Now use $15=60-45$ or $=45-30$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Showing that $\lim_{x \rightarrow \infty}\frac{e^x}{x^n} = \infty$ I have been told that I can show this by showing two things, first that
$$f(x) = \frac{e^x}{x^n}>\frac{e^n}{n^n}, \quad (x > n)$$
then
$$f'(x) = \frac{e^x(x-n)}{x^{n+1}}>\frac{e^{n+1}}{n^{n+1}}, \quad (x > n+1)$$
I have managed to show both of these thi... | $e^x =$
$1 + x + \frac{x^2}{2!} + ...\frac{x^{n+1}}{(n+1)!} .....$
Hence:
$e^x \gt \frac{x^{n+1}}{(n+1)!}$ , for $x \gt 0$, $ n \in \mathbb{N}$, $n \ge 1$.
$\frac{e^x}{x^n} \gt \frac{x}{(n+1)!}$.
Finally:
$\lim_{x \rightarrow \infty} \frac{e^x}{x^n} \ge \lim_{x \rightarrow \infty} \frac{x}{(n+1)!} = \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Can a set of surreal numbers be defined with arbitrary cardinality? It is my understanding that the surreal numbers form a class rather than a set, because their collection is larger than any set. Thus it would seem to follow that for any cardinality, such as $\aleph_n$ or $\beth_n$ for a fixed $n$, a set of surreal nu... | Yes: Since every ordinal is a surreal number, the sets you're looking for can be taken to be the initial ordinals that represent $\aleph_n$, $\beth_n$, and so forth.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Conditional probability on 3 events. Say you have 3 events $A, B$, and $C$. Then you have to calculate the probability of $B$ given $A$.
The formula that the answer key states:
$$P(B|A)=P(B|A,C)P(C) + P(B|A,C^\complement)P(C^\complement)$$
I understand that for just two events $B$ and $A$ it is:
$$P(B)=P(B|A)P(A) + P(B... | Sometimes it's easier to work with intersections rather than conditionals. The key formula here is that $$ P(A) = P(A\cap B)+P(A\cap B^c)$$ which follows from the fact that $A = (A\cap B)\cup(A\cap B^c)$ and $(A\cap B)\cap(A\cap B^c) = \emptyset$ and that disjoint unions are additive. The second formula that you write... | {
"language": "en",
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Maximum Likelihood Estimator of Uniform($-2 \theta, 5 \theta$)
Let $X = (X_1, \dots, X_n)$ be a random sample from the Uniform($-2 \theta, 5 \theta$) distribution with $\theta > 0$ unknown. Find the maximum likelihood estimator (MLE) for $\theta.$ Furthermore, determine whether the MLE $\hat{\theta}$ is a function of ... | The support of $L(\theta; x)$ is given by $L\ge -2 \theta$ , $M\le 5\theta$; or, equivalently $\theta \ge M/5$ and $\theta \ge -L/2$. Or $$\theta \ge T \triangleq
\max(M/5,-L/2)$$
Because over its support $L(\theta; x)$ (for $n>1$) is decreasing, then $ \theta_{ML}=T$
Regarding $(M,L)$ being or not minimal sufficie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If $0\le x_1\le\dots\le x_n\le1$, then $\sum\limits_{i=1}^n\left(x_i-\frac{i}{n+1}\right)^2\le\sum\limits_{i=1}^n\left(\frac{i}{n+1}\right)^2$
Given positive integer $n$, $0\leq x_1 \leq \dots \leq x_n \leq 1$. Prove that $$
\sum_{i=1}^n \left(x_i- \frac{i}{n+1} \right)^2 \leq \sum_{i=1}^n \left(\frac{i}{n+1}\right) ^... | Hint:
you can use the Chebyshev sum inequality to prove that
$$
\frac{1}{n}\sum_{i=1}^n x_i \cdot \frac{i}{n+1}
\geq \left(\frac{1}{n}\sum_{i=1}^n x_i\right) \cdot \left(\frac{1}{n}\sum_{i=1}^n
\frac{i}{n+1}\right) = \frac{1}{2n} \sum_{i=1}^n x_i.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2391997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find all functions of positive integers for $f(f(n))=n+2$ This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with induction, which yields the shortest, simplest proofs for proving the finite amount of functions, but other than that, the textbook gave... | Note that $f(f(1))=3$ and $f(1)\ge 1$
We can't have $f(1)=1$ because that would make $3=f(f(1))=f(1)=1$
If we had $f(1)=3$, we'd have $3=f(f(1))=f(3)$, but $f$ is strictly increasing, so this is a contradiction. As would be if $f(1)=n\gt 3$ when we'd have $3=f(f(1))=f(n)\gt f(1)\gt 3$.
So we have $f(1)=2$, and then $f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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If $\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n} -\frac {1}{n^2}$ then $\sum\limits_na_n$ diverges
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that, for every $n\ge1$,
$$\frac {a_{n+1}}{a_n} \ge 1 -\frac {1}{n} -\frac {1}{n^2} \tag 2$$ Prove that $x_n=a_1 + a_2 + .. + a_n$ diverges.
It is clear t... | It's easy to show that, for every $n\ge3$,
$$ 1 -\frac {1}{n} -\frac {1}{n^2}
\ge \frac {n-2}{n-1}$$ It follows that, for every $n\ge3$,
$$\frac{a_{n+1}}{a_n}\ge \frac {n-2}{n-1}$$
Thus,
$$\frac {a_4}{a_3} \ge \frac 1 2\qquad
\frac {a_5}{a_4} \ge \frac 2 3\qquad
\ldots\qquad
\frac {a_{n-1}}{a_{n-2}} \ge \frac {n-4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
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How to determine whether a curve lies on a plane?
Given that a sphere $x^2+y^2+z^2=1$ and a cylinder $x^2+y^2=x$ intersect at point $(1/2,1/2,1/\sqrt 2)$ determine whether the curve from the intersection lies on a plane.
We can find the curve using the following parametrization:
$$
c(t)=\bigg( \frac{1}{2}+\frac{1}{2}... | First, of all, draw some pictures and you'll see that a sphere-cylinder intersection is only planar (it's a circle) when the centerline of the cylinder passes through the center of the sphere. That's not the case, here.
If you want "proof" as opposed to pictures and intuition, just use your parametric equations to comp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is this an issue with the law of the excluded middle, or an issue with the proof? Part of a proof requiring you to prove that if $x^2$ is odd then $x$ is odd (given that $x \in \mathbb{N}$). It is my understanding that the contrapositive is used for this as follows.
$x=2n, n \in \mathbb{N}$
$\Rightarrow x^2 = 4n^2$
$\R... | Ignore the broader proof - do you agree with the assertion "If $x$ is divisible by $4$, then $x$ is even?" This is all that's going on. We're always allowed to "forget" information in a proof, and this has nothing to do with the excluded middle. When you conclude "$x^2$ is even," this in no way implies that you've conc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist
Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that a gcd of $x^5$ and $x^6$ does not exist in $R$.
I am reasoning ... | Suppose $\,d\,$ is a gcd of $\,x^5,x^6$ in $R.\,$ By the general (universal) definition of the gcd
$\ \ \ c\mid x^5,x^6\! \iff c\mid d.\, $ Taking $c = d\,$ $\rm\color{#0a0}{shows}$ $\,d\mid \color{#c00}{x^5},x^6,\,$ i.e. the gcd is a common divisor.
$x^3\mid x^5,x^6\ \ \Rightarrow\ \ x^3\mid d,\ $ hence $\ d = x^3,\,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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relationship between the $2$-norm of a symmetric matrix and its maximum eigenvalue. If I have an $N\times N$ symmetric matrix $Q$, what is the relationship between $\|Q\|$ and its maximum eigenvalue, where $\|\|$ is the $2$nd norm?
| The general answer to this is this $\|Q\| = \max \{|\lambda_{\max}(Q)|, |\lambda_{\min}(Q)|\}$. To see why this is true, consider $Q = I_{N}$ for the first case and $Q = -I_{N}$ for the second case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Proving that the limit of an abstract function with certain properties is zero. Let $$F(x) = \sum_{a \leq x} f(a)$$ where $f: A \subseteq \mathbb{R} \rightarrow \mathbb{R} $ such that:
*
*$f(a) \geq 0$ for any $a \in A$ and
*$\displaystyle\sum_{a \in A} f(a) = 1$
How do I prove that $$\lim_{x\to -\infty} F(x) = 0\... | I use the following fact below, so I thought it would be useful to prove it first.
If $f:A\to\mathbb{R}$ is such that $f(a)\geq 0$ for every $a\in A$ and $\sum_{a\in A}f(a)<\infty$, then $[f>0]$ (notation for $\{a\in A \mid f(a)>0\}$) is countable.
Proof:
For each $x>0$, consider the set $[f>x]$. Note that
$$
\operat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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The length of an arc within two intersecting circles I found this mathematical expression for the length of a arc $l(r)$ i.e the shorter arc ACB. In other words, why is $l(r)$ equal to that expression with respect to R i.e $l (r) = 2r \arccos (r/2R)$? I have tried hard to prove it but I couldn't. I hope someone could g... | We have
$$
l(r) = 2r \angle ADC.
$$
So we just need to compute $\angle ADC$. Let $Q$ be the center of the circle on the left. Note that $AD=r$ and $DQ=AQ=R$. It follows that $\angle ADQ=\angle ADC=\arccos(r/2R)$. To see why, note that $\triangle ADQ$ is isosceles and drop a perpendicular from $Q$ to the midpoint of $AD... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Determine the highest order of an element of a Rubik's Cube group The period of a sequence of moves on a Rubik's Cube is the number of times it must be performed on a solved cube before the cube returns to its solved state. For example, a $90$° clockwise turn on the right face has a period of four; a $180$° clockwise t... | Notations are from https://ruwix.com/the-rubiks-cube/notation/
$RY$ is an element of order 1260.
It's easy to check that $(RY)^{36}$ keeps all the corners intact but permutes the edges in 2 disjoint groups of order 5 and 7, respectively. So, the total order is $36*5*7=1260$.
$RY$ applied 36 times - https://ruwix.com/sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 1,
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Evaluating $\int x\sin^{-1}x dx$ I was integrating $$\int x\sin^{-1}x dx.$$
After applying integration by parts and some rearrangement I got stuck at $$\int \sqrt{1-x^2}dx.$$
Now I have two questions:
*
*Please, suggest any further approach from where I have stuck;
*Please, provide an alternative way to solve th... | Alternative way: Let $x=\sin t$ so
$$\int x\sin^{-1}x dx=\int t\sin t\cos t dt$$
by parts $t=u$ and $\sin t\cos t dt=dv$ and finish it!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2392990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Goodwin Staton integral $G(x) = \int_0^\infty \frac{e^{-t^2}}{t+x}dt$ and its symmetry The Goodwin Staton integral
$$G(x) = \int_0^\infty \frac{e^{-t^2}}{t+x}dt$$
is said on Wikipedia to have the symmetry
$$G(x) = -G(-x)$$
I'm not convinced by this symmetry... indeed if we consider $G(-x)$ and we choose $k = -t$ thi... | I think Wikipedia is wrong. The integral does not converge for $x<0, $ try e.g. int(e^(-t^2)/(t-2),t=0..infinity) in Wolfram Alpha. It can be interpreted as a Cauchy principle value (see Nico Temme's answer http://mathforum.org/kb/message.jspa?messageID=7389647). I use
$$G(-x) = - \frac{1}{2} e^{-x^2}\left(\pi\; \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $X$ is a CW complex, then the path components of $X$ are the components of $X$. I'm self-learning Algebraic Topology from Rotman's Introduction to Algebraic Topology and I've come across this problem:
If $X$ is a CW complex, then the path components of $X$ are the components of $X$.
The proof states: If $A$ is a ... | A subset $A$ of a topolgical set $X$ which is open and closed is a union of connected components. To see this, consider $x\in A$ and $C$ is connected component, $C\cap A$ is closed and $C\cap (X-A)$ is also closed, you deduce that $C\cap (X-A)$ is empty since $C$ is connected, henceforth $C\subset A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Dualizing simple module which is given by primitive idempotent Let $e$ be a primitive idempotent in an associative finite-dimensional $k$-algebra $A$. Then the two modules $Ae/\text{rad}(Ae)$ and $D(eA/\text{rad}(eA))$ are both simple, where $D: \text{mod}(A^{\text{op}}) \to \text{mod}(A)$ is the standard dualization.
... | Denote by $k$ the ground field of the algebra $A$. Since both modules are simple, every nonzero homomorphism of $A$-modules $\varphi: Ae/\text{rad}(Ae) \to D(eA/\text{rad}(eA))$ will be an isomorphism. In order to find this, we first construct some $\tilde{\varphi}: Ae \to D(eA/\text{rad}(eA))$:
Let $g \in D(eA/\text{r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to approximate $\pi$ using the Maclaurin series for $\sin(x)$ We have that
$$\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
Now plugging in $x=\pi$,
$$0=\pi - \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + \cdots$$
Is there a way we can use this beautiful result to calculate bette... | It is easy to find better approximations to $\pi$ by iteration. Let $a_{n+1}=a_n+\sin(a_n)$. If you start with $a_1$ close enough to $\pi$ the sequence converges to $\pi$. For example, $a_1=3$ will work. You can replace $\sin(x)$ with a truncated Taylor series $f(x)$ and the iteration will converge to the root of $f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Find the angle NMC In triangle $ABC$, $\measuredangle B = 70^{\circ}$, $\measuredangle C = 50^{\circ} $. On $AB$ and $AC$ take points $M$ and $N$ such that $\measuredangle MCB = 40^{\circ}$, $\measuredangle NBC= 50^{\circ}$. Find $\measuredangle NMC$.
| Another solutions:
Observe that by angle chasing $BN = CN$ and $BC = MC$. Let point $D$ be chosen on the line $AB$ so that the points $B$ and $M$ lie on the segment $AD$ and $MA = BD$. Consequently, triangles $ACM$ and $DCB$ are congruent by construction and therefore the triangle $ACD$ is equilateral. Draw the line p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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By using the definition of limit only, prove that $\lim_{x\rightarrow 0} \frac1{3x+1} = 1$
By using the definition of a limit only, prove that
$\lim_{x\rightarrow 0} \dfrac{1}{3x+1} = 1$
We need to find $$0<\left|x\right|<\delta\quad\implies\quad\left|\dfrac{1}{3x+1}-1\right|<\epsilon.$$
I have simplified $\left|\dfr... | Note that
$$\left|\frac{1}{1+3x}-1\right|=\left|\frac{3x}{1+3x}\right| \tag 1$$
Now, we restrict $x$ such that $x\in [-1/4,1/4]$. And with this restriction, it is easy to see that $1/4/ \le 1+3x$. Using this in $(1)$ reveals that
$$\left|\frac{1}{1+3x}-1\right|\le 12|x|\tag 2$$
Finally, given any $\epsilon>0$,
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that for $a_i>0$ $\frac{a_1+\cdots+a_n}{n}$ converges to $0$ if and only if $\frac{a_1^2+\cdots+a_n^2}{n}$ converges to $0$. Let $\{a_n\}$ be a bounded and positive sequence. Show that
$$\lim_{n\to \infty}\frac{a_1+\cdots+a_n}{n}=0$$
if and only if
$$\lim_{n\to \infty}\frac{a_1^2+\cdots+a_n^2}{n}=0.$$
My attempt:... | By Cauchy—Schwarz inequality,
$$
\sum_{k=1}^n \frac{1}{n}\cdot a_k \leq \sqrt{\sum_{k=1}^n a_k^2}\cdot \sqrt{\sum_{k=1}^n \frac{1}{n^2}}
= \sqrt{\sum_{k=1}^n a_k^2}\cdot\sqrt{\frac{1}{n}}
= \sqrt{\frac{1}{n}\sum_{k=1}^n a_k^2}
$$
and you can conclude by the squeeze theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Why is $\omega+1+\omega+1 = \omega+\omega+1$? Why isn't it $\omega+\omega+2$?
On the other hand, is it true that $\omega+\omega = 2\omega$?
Also why is it that $2 \omega=\omega$?
Here $\omega$ is taken to be the limit ordinal which is just $\mathbb{N}$.
I am really confused as to why ordinals can't add/multiply just li... |
I am really confused as to why ordinals can't add/multiply just like natural numbers do, since they are essentially the same thing?
The ordinals generalize the natural numbers in a certain sense, but that does not mean that every property of the natural numbers carries over to the ordinals. Weird though it may be, ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2393922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Basis of a topology Does every topology have a unique basis? What I mean is if there is more than one topology that can be obtained from a given basis, which topology should I consider when they say the topology generated by the given basis?
| No, there can be many basis for the same topology (but a basis generates a unique topology).
For example in $\mathbb{R}^2$, for $r>0$, let
$$B((x_0,y_0),r):=\{(x,y)\in \mathbb{R}^2:(x-x_0)^2+(y-y_0)^2<r^2\}$$
and
$$S((x_0,y_0),r):=\{(x,y)\in \mathbb{R}^2:|x-x_0|+|y-y_0|<r\}.$$
Then $\{B((x_0,y_0),r): (x_0,y_0)\in \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On the necessity of being a *dense* subset in completion of a metric space Quoted from the book Introductory functional Analysis by Erwin Kreyszig :
1.6-2 Theorem (Completion). For a metric space $X = (X, d)$ there
exists a complete metric space $\bar{X}=(\bar{X}, \bar{d})$ which has a subspace $W$ that is
isom... | I am not sure what is your question.
Why $W$ must be dense?
This is by the construction in the proof of the theorem.
Why is it important?
Because it basically tells you that in every metric space, only adding a 'few' limit points will make it complete. This is remarkable! Assume for a second you ignore the density ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Question about Gödel's Incompleteness Theorem Gödel used prime factorization to encode each statement with a unique number (which is Gödel numbering).
But I wonder if this statement can be encoded:
"If this statement can be encoded, then this statement is false."
If it can be encoded, then it will be false and the tr... | Godel proved his result by first setting up a system of mathematical logic that can do the basics of arithmetic. It includes the symbols $\Rightarrow$ "if then", $\wedge$ "and", etc. It also includes the natural numbers and some of their operations. In order for your statement to work, you would need to encode "If t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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What seemingly innocuous results in mathematics require advanced proofs? I'm interested in finding a collection of basic results in mathematics that require rather advanced methods of proof. In this list we're not interested in basic results that have tedious simple proofs which can be shorted through more advanced met... | Perhaps the parallel postulate?
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point
It was only through trying to prove this "obvious" theorem that we discovered non-Euclidean geometries.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "97",
"answer_count": 18,
"answer_id": 8
} |
How to calculate line integral over the intersection of paraboloid $z=x^2+y^2$ and plane $z=2x$
Calculate $\oint_C xyz\,dx+x^2\,dy+xz\,dz$ over the curve from the intersection of paraboloid $z=x^2+y^2$ and plane $z=2x$. The direction of the curve may be chosen as you see fit.
It looks like if we chose to parametrize... | I'll start with some critique. First of all, your normal vector isn't quite correct: from the equation of the plane $-2x+z=0$, we get the normal vector $\mathbf{n}=\langle-2,0,1\rangle$ (or it could be its opposite, but this one gives the upward orientation, consistent with the counterclockwise orientation of the curve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Entire number continuum is equivalent to any finite segment: Courant and Robbins Book I am reading What is Mathematics? by Richard Courant and Herbert Robbins.
They discuss about the fact the $\Bbb{R}$ is not countable and after they say
«it is easy to show that the entire number continuum is equivalent to any finite... | Imagine that the dot at the center is a light bulb. Then every point on the cup-shaped part of the figure has a shadow point on the line. (That's what "projection" means in this context.)
The points on the cup shaped figure correspond to the points on the unit interval using the "bending" Courant and Robbins describe.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Show that $\left( \frac {11} {10}\right) ^{n}$ is divergent.
Show that $\left( \dfrac {11} {10}\right) ^{n}$ is divergent.
My proof. Let $B\in\mathbb{R}$. By the Archimedean property there is a $N$ in $\mathbb{N}$ such that $N>B$.
Let $\varepsilon >0$ By the Bernoulli inequality, we have $\left( 1+\varepsilon \rig... | Easy to think solution:
Note that $\ln$ is increasing function.
Note that $\ln\Big(\dfrac{11}{10}\Big)=\ln11-\ln10=c>0$
Now $\ln\Big(\dfrac{11}{10}\Big)^n=n(\ln11-\ln10)=nc$
Now since $c>0$, for every $N\in \mathbb{N}$ and $N>\Big\lfloor\dfrac{1}{c}\Big\rfloor+1$, you can find a $n\in\mathbb{N}$ such that $nc>N$. Hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finite Recursion Theorem Does there exists a version of the Recursion Theorem for finite totally ordered sets (instead of natural numbers)?
There are many cases where we have a finite totally ordered set and we have to define a thing recursively over that set, but how can it be formalized?. For example, if we have an o... | Any finite strictly totally ordered set has a unique strict order-isomorphism to a unique initial segment of $\mathbb N$. More precisely: If $(I; \prec)$ is a strict finite total order there is a unique $n \in \mathbb N$ (namely $n = \operatorname{card}(I)$) with a unique strict order isomorphism
$$
\pi \colon (I; \pre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Calculation of the $s$-energy of the Middle Third Cantor Set As the title suggests, I am trying to calculate the $s$-energy of the middle third Cantor set. I am reading Falconer's Fractal Geometry book, available here:
http://www.dm.uba.ar/materias/optativas/geometria_fractal/2006/1/Fractales/1.pdf
and this is an exerc... | First, a general observation: if a finite measure $\mu$ on $\mathbb{R}$ has no atoms, then the diagonal $\{(x,x)\in\mathbb{R}^2\}$ has zero measure with respect to the product measure $\mu\times \mu$. To see why, partition $\mathbb R$ into $n$ intervals of measure $1/n$, and observe that the diagonal is covered by $n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to apply Cauchy's MVT to evaluate the following?
Use Cauchy's Mean Value theorem to evaluate
$$\lim_{x\rightarrow 1} \left[\frac{\cos(\frac{1}{2}\pi
x)}{\ln(1/x)}\right]$$
I can't understand how to apply Cauchy's MVT over here. Any hints?
| $$\lim_{x\to 1}\frac{\cos\left(\frac{\pi}{2}x\right)}{-\log x}\stackrel{x\mapsto 1-z}{=}\lim_{z\to 0}\frac{\sin\left(\frac{\pi}{2}z\right)}{-\log(1-z)}$$
and since $\lim_{z\to 0}\frac{\sin z}{z}=1=\lim_{z\to 0}\frac{z}{-\log(1-z)}$ the wanted limit is $\frac{\pi}{2}$, you do not need anything fancy. If you like, you ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2394970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $x\in \Bbb Z$ such that $x=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
Find $x\in \Bbb Z$ such that $x=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$
Tried (without sucess) two different approaches: (a) finding $x^3$ by raising the right expression to power 3, but was not able to find something useful in the result t... | $$x=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\\x^3=2+\sqrt5+2-\sqrt5+3\sqrt[3]{2+\sqrt{5}}\cdot\sqrt[3]{2-\sqrt{5}}(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}})\\x^3=4+3\cdot(-1)\cdot(x)$$so $$x^3+3x-4=0 \\(x-1)(x^2+x+4)\to\\ x=1,x^2+x+4=0 ,\Delta <0\\x=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 0
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$\sqrt[3]{8\div{\sqrt[3]{8\div{\sqrt[3]{8\div{\sqrt[3]{8\div ...} }} }} }} $=? Suppose $$a=\sqrt[3]{8\div{\sqrt[3]{8\div{\sqrt[3]{8\div{\sqrt[3]{8\div ...} }} }} }} $$
$\bf{Question}:$Is it possible to find the value of $a$
Thanks in advance for any hint,idea or solution.
$\bf{remark}:$ I changed the first question , B... | Consider the sequence,
$$x_{n+1}=\sqrt[3]{\frac{8}{x_n}}=2(x_n)^{-\frac{1}{3}}$$
With $x_1=1$. Our value of interest is $\lim_{n \to \infty} x_n$.
Such a sequence follows,
$$\ln x_{n+1}=\ln 2-\frac{1}{3} \ln x_n$$
Hence letting $\ln x_n=a_n$ we have the linear recurrence,
$$a_{n+1}+\frac{1}{3}a_{n}=\ln 2$$
$$(a_{n+1}-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Consider $ \ A \cap (B-C)$ and $ \ (A\cap B) - (A \cap C)$. Question:
Consider $ \ A \cap (B-C)$ and $ \ (A\cap B) - (A \cap C)$.
Are the two sets equal or is one a subset of the other?
My attempt:
We know that if $ \ x\in (A \cap B) - (A \cap C) \implies x \in A \cap B$ and $ \ x \notin A \cap C \implies x \in A$ and... | Through logic it should be true using definition $\forall x: x\in (X-A) \iff (x \in X) \land (x \notin A)$ that
$$\forall x: (x \in A) \land ((x \in B) \land (x\notin C)) =\forall x: ((x \in A) \land (x \in B)) \land ((x \notin C) \lor (x\notin A))$$
because in order to satisfy the formula, $x$ must be in $A$, theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Easy exercise with boundary could I have a confirm or a suggestion about this little exercise?
$\partial A=\emptyset$ if and only if $A$ is open and closed.
Sol.:
If $A$ is "clopen", then $Int(A)=A$ and $Cl(A)=A$, so $\partial A=A \setminus A=\emptyset $.
If $\partial A=\emptyset$, then $Cl(A) \setminus Int(A)=\empty... | Your first part is O.K. Your second part is incomplete.
If $\partial A=\emptyset$ then $Cl(A) \setminus Int(A)=\emptyset$. It follows that
$Cl(A) \subseteq Int(A)$.
Since $Int(A) \subseteq A \subseteq Cl(A)$, we get
$$Int(A) = A = Cl(A).$$
Hence $A$ is clopen.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $\sinh z = 2i$ This is my attempt at the question (I stopped early because it did not work out...)
$$\sinh z = 2i \\ e^{iz} - e^{-iz} = 4i \\ e^{2iz} - 4ie^{iz} - 1 = 0 $$
solving the quadratic gives
$$e^{iz} = i(2\pm \sqrt{3})$$
I stop here to check:
$$\sinh z = \frac{e^{iz} - e^{-iz}}{2}= \frac{i(2\pm\sqrt{3}... | Actually, correct version of the last expression is
$$ \sinh(z)=\frac{e^{iz}-e^{-iz}}{2}=\frac{i(2\pm\sqrt{3})+i(2\mp\sqrt{3})}{2}=2i $$
Use $z=(2+\sqrt{3})i$ or $z=(2-\sqrt{3})i $ uniformly in the all part of expression
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find all positive integers n for which the number obtained by erasing the last digit of n is a divisor of n? I know, through this, , that all numbers ending on 0 and 11, 12..19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99 are solutions. But how to prove that all 3- and more-digit numbers which do not end on ... | First case:
$ 100 \leq n $.
In this case we $\color{Green}{\text{claim}}$ that
$\color{Green}{\text{the last digit is equal to zero}}$ ,
conversly every integer with the last digit equal to zero
has the above property.
Let
$$ 100 \leq
n=\overline{ a_m a_{m-1} ... a_1 a_0 }=
a_m10^m + a_{m-1}10^{m-1} + ...... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Problem of diagonalizability and vector space I've been reading the solution and want you to help me understand it.
problem: Let V be a real vector space with 100-by-100 real matrices. Let $A \in V$, $W_A={B \in V | AB=BA}$, and $d_A$ be the dimension of $W_A$.
Assume that $A^4-5A^2+4I=0$. Find the minimum of $d_A$.
so... | Since $\psi_a(x) = (x-1)(x+1)(x-2)(x+2)$ we see that all Jordan blocks of $A$ are of size one and so $A=V \Lambda V^{-1}$ where $\Lambda$ is diagonal and $\{ [\Lambda]_{kk} \}_k = \{ \pm 1, \pm 2 \}$.
It is straightforward to see that
$W_A = V W_{\Lambda} V^{-1}$, in particular the dimensions of $W_A, W_\Lambda$ are th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\lim_{x \rightarrow a} \frac{x^2 + ax - 2a^2}{\sqrt{2x^2 - ax} -a}$ I need to calculate:
$$\lim_{x \rightarrow a} \frac{x^2 + ax - 2a^2}{\sqrt{2x^2 - ax} -a}$$
I get $0/0$ and can then use l'hopital's rule to find the limit, I can do this but someone asked me how I can do this without using l'hopital's rule.... | Hint. Note that for $x\not=a$,
$$\frac{x^2 + ax - 2a^2}{\sqrt{2x^2 - ax} -a}=\frac{(x+2a)(x-a)(\sqrt{2x^2 - ax} +a)}{(2x+a)(x-a)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Understanding the definition of a natural isomorphism On Wikipedia I can read this:
If, for every object $X$ in $C$, the morphism $η_X$ is an isomorphism in $D$, then $η$ is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors).
But I don't understand this. I thought that, if ... | A natural transformation $\eta$ maps objects of a category $\mathcal C$ to arrows of a category $\mathcal D$.
Specifically, if $F,G:\mathcal C\to\mathcal D,\ \ x\in Ob\mathcal C\ $ and $\eta:F\to G$, we have $\eta_x$ is an arrow $F(x)\to G(x)$ in $\mathcal D$.
A natural transformation $\eta$ is a natural isomorphism i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simple statistics exercise... Too simple? I have to solve the following exercise:
There are three different parts needed to construct a machine: part $1$ ($1000$ pieces), part $2$ ($400$ pieces), part $3$ ($600$ pieces).
The probability for a part $1$ or part $2$ piece to be defective is $2 \%$. The probability for a p... | Nice intuition. To make your argument more rigourous, consider using Bayes rule.
\begin{align}P(\text{part 2}|\text{defective})=\frac{P(\text{part 2})}{P(\text{defective})}P(\text{defective}|\text{part 2}) \end{align}
$$P(\text{part 2})=\frac{400}{2000}$$
$$P(\text{defective})=\frac{2}{100}\frac{1000}{2000}+\frac{2}{10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2395963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
which one is bigger $100^n+99^n$ or $101^n$ Suppose $n \in \mathbb{N} , n>1000$ now how can we prove :which one is bigger $$100^n+99^n \text{ or } 101^n \text{ ? }$$
I tried to use $\log$ but get nothing . Then I tried for binomial expansion...but I get stuck on this .
can someone help me ? thanks in advance.
| Of course
$$1.01^n>1+\frac{n}{100}>2$$
for $n>100$, and obviously $1+0.99^n<2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
Polar integrals with trigonometric Problem The prompt is to evaluate the double integrals using polar system. The given constraints are: $$ \iint_D (x^2 + y^2) \, dx \, dy $$
$$ D = \{(x,y): x \ge 0, y \ge x, x^2 + y^2 \le 2y \}$$
Upon graphing we have a circle placed at the origin of 1 on the y-axis and line cutting ... | The double integral set up should be: $$ \int_{\pi/4}^{\pi/2} \int_0^{2\sin \theta} r^3 \, dr \, d\theta= \int_{\pi/4}^{\pi/2} 4\sin^4\theta d\theta= \frac 1 4 \int_{\pi/4}^{\pi/2} (1-\cos(2\theta))^2 \, d\theta=\cdots.$$ Also use $\cos^2(2\theta) = \dfrac{1+\cos(4\theta)}{2}$ in expanding the integrand above.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How one knows it is $(n-1)$ versus $(n+1)$ I have had a problem with this concept all my life so I thought I would reach out to the experts for help!
Here is the problem statement:
Quote: "Consider how much work is required to multiply two -digit numbers using the usual grade-school method. There are two phases to wo... | When you add two $n$ digit numbers, you have to do $n$ additions, one for each digit, but there might be a carry at each digit, so that's another $n$ operations, and there's your $2n$.
Now you have to add $n$ of these $n$-digit numbers, which means you have to do $n-1$ of these additions of pairs of numbers – right? T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Recursive derangement proof clarification I saw a proof of a recursive definition of $D_n$ = the number of derangements of a set of size $n$.
A combinatorial proof of the recurrence for $D_n$:
Here's a combinatorial proof of $D_n = (n-1)(D_{n-1} + D_{n-2})$ for $n \geq 2$ due to Euler.
For any derangement $(j_1, j_2, ... | We do not have to pick the last element. We can pick other element as well and the proof will still work.
If $j_n = n$, then it is not a derangement anymore. Hence $j_n = k$ where $k \neq n$.
We can change the proof.
You can pick a particular index $p \in \{ 1, \ldots, n\}$
For any derangement $(j_1, j_2, \ldots, j_n)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can't tell the difference between permutation and combination In an exam, a student has to answer 6 out of 8 questions. How many ways can she answer the 6 questions if
a) there are no restrictions?
b) The first 3 questions are compulsory?
c) she must answer at least 3 of the first 4 questions?
I am confused. Why is t... | I'm not giving actual answers but trying to resolve your confusions.
I wouldn't say the questions are 'specific'. Each one is either answered or not answered, but the statistically important bit is that they are added together in aggregation for consideration (answer $6$ questions in total). Thus, for example, answer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Inverse elliptic integral, Weierstrass function, in other fields Take a separable cubic polynomial $4x^3-ax-b = 4 (x-e_1)(x-e_2)(x-e_3)$, let $h'(x) = (4x^3-ax-b)^{-1/2}$ and define its elliptic integral $h(x)= \int h'(x)dx$. Let $P(z) = h^{-1}(z)$ its inverse function. Then $\displaystyle P'(z) = \frac{1}{h'(P(z))}$ a... | One way to do this is to prove there is a lattice $\Lambda$ in $\Bbb C$
whose $\wp$-function satisfies
$$\wp'(z)^2=4\wp(z)^3-a\wp(z)-b.$$
Cox gives a proof in Primes of the form $x^2+ny^2$. This involves
the $j$ modular function. It is a fact that $j$ is surjective, so there
is $\tau$ with
$$j(\tau)=1728\frac{a^3}{a^3-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Taylor expansion of $\cos^2(\frac{iz}{2})$
Expand $\cos^2(\frac{iz}{2})$ around $a=0$
We know that $$\cos t=\sum_{n=0}^{\infty}(-1)^n\frac{t^{2n}}{{2n!}}$$
So $$\cos^2t=[\sum_{n=0}^{\infty}(-1)^n\frac{t^{2n}}{{2n!}}]^2=\sum_{n=0}^{\infty}(-1)^{2n}\frac{t^{4n}}{{4n^2!}}$$
We have $t=\frac{iz}{2}$
$$\sum_{n=0}^{\infty}... | $$ \cos^2(\frac{iz}{2}) = \frac{1}{2}(1+\cos(2*\frac{iz}{2})) = \frac{1}{2}(1+\cos(iz)) = \frac{1}{2}(1+\sum_{n=0}^{+\infty} (-1)^n\frac{(iz)^{2n}}{(2n)!}) = \frac{1}{2}(1+\sum_{n=0}^{+\infty} (-1)^n (i)^{2n}\frac{(z)^{2n}}{(2n)!}) = \frac{1}{2}(1+\sum_{n=0}^{+\infty} (-1)^{2n}\frac{(z)^{2n}}{(2n)!}) = \frac{1}{2}(1+\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
If $A$ is diagonalizable, find $\alpha$ and $\beta$
Let $A$ be a $5 \times 5$ matrix whose characteristic polynomial is given by
$$p_A(\lambda)=(λ + 2)^2 (λ − 2)^3$$
If $A$ is diagonalizable, find $\alpha$ and $\beta$ such that
$$A^{-1} = \alpha A + \beta I$$
I am unable to find the inverse of $5\times 5$ m... | The fact that A is diagonalizable means that there exist an invertible matrix, P, such that $PAP^{1}= \begin{bmatrix}2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & -2 \end{bmatrix}$. So $(PAP^{1})^{-1}= PA^{-1}P^{-1}= \begin{bmatrix}\frac{1}{2} & 0 & 0 & 0 & 0 \\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\int_{-\infty}^{\infty}\frac{1}{(x^2+4)^5}dx$ $$\int_{-\infty}^{\infty}\frac{1}{(x^2+4)^5}dx$$
I am trying to use residue.
We first need to find the singularities, $x^2+4=0\iff x=\pm 2i$
Just $2i$ is in the positive part of $i$ so we take the limit
$lim_{z\to 2i}\frac{1}{(z^2+4)^5}$ but the limit is $0$
| For any $a>0$ we have
$$ \int_{-\infty}^{+\infty}\frac{dx}{x^2+a}=\frac{\pi}{\sqrt{a}} $$
and by applying $\frac{d^4}{da^4}$ to both sides we get:
$$ 24\int_{-\infty}^{+\infty}\frac{dx}{(x^2+a)^5}=\frac{105 \pi}{16 a^4\sqrt{a}} $$
so by rearranging and evaluating at $a=4$ we get:
$$ \int_{-\infty}^{+\infty}\frac{dx}{(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2396943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Has this variant on multiplication by a natural number been studied before? Let $X$ denote an additively-denote commutative monoid. Then we get an action $\star$ of $\mathbb{N}$ on the powerset $\mathcal{P}(X)$ as follows: given a natural number $n$ and a set $A \subseteq X$, define $$n \star A = \left\{x \in X : \exis... | This looks like some sort of convolution on sets. For any two sets $A, B \in P(X)$, define the product
$$ A \cdot B = \{a + b \mid a \in A, b \in B\}$$
For example, we have that
$$ \{x, y\} \cdot \{x, y\} = \{2x, x + y, 2y\} $$
and in general, the star operator $n \star A$ is the $n$-fold product of $A$.
Note that if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2397029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Polynomial $x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1$ is reducible over $\mathbb{Q}$? Clearly, there are no roots, but how can I find factors of higher degree?
| This is
$$(x^6+x^3+1)(x^2+x+1).$$
In fact a polynomial
$$\sum_{j=0}^m x^j=x^m+x^{m-1}+\cdots+x+1$$
for $m\ge1$
is irreducible over $\Bbb Q$ iff $m+1$ is prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2397116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Limit of $\int_0^1\frac{f(hx)}{x^2+1}dx$ when $h\to0$
Let $f\in \mathcal{C}^0\big([0,1],\mathbb{R}\big)$ and, for every $h\in(0,1]$, $$I(h)=\int_0^1\dfrac{f(hx)}{x^2+1}dx$$
For $\varepsilon >0$, show there exists $\eta>0$ such that for every $h\in(0,\eta)$,
$$\left|I(h)-f(0)\frac{\pi}{4}\right|\leq \varepsilon$$
Sinc... | Note that we can write
$$\begin{align}
\left|\int_0^1 \frac{f(hx)}{x^2+1}\,dx-f(0)\frac\pi4\right|&=\left|\int_0^1 \frac{f(hx)-f(0)}{x^2+1}\,dx\right|\\\\
&\le \int_0^1 \frac{|f(hx)-f(0)|}{x^2+1}\,dx\\\\
&
\le\frac\pi4 \sup_{x\in [0,1]}|f(hx)-f(0)|
\end{align}$$
For any $\epsilon>0$, there exists a number $\eta>0$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2397228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Construct a bijection $\mathrm{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C\setminus \{0\}$ The question is :
Construct a bijection $\mathrm{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C\setminus \{0\}$.
Here $\text{Hom}_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C)$ is the set o... | A homomorphism from $R=\Bbb C[x,y]/(xy-1)$ to $\Bbb C$ is a homomorphism
$\Phi$ from $\Bbb C[x,y]$ to $\Bbb C$ with $\Phi(xy-1)=0$. Each homomorphism $\Phi:\Bbb C[x,y]\to \Bbb C$ has has the form
$\Phi_{a,b}:f(x,y)\to f(a,b)$ where $a$, $b\in\Bbb C$. Then
$\Phi_{a,b}(xy-1)=ab-1$. So $\Phi_{a,b}$ defines a homomorphism
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2397337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Does this sum to $2^n-1\ $? $$N=\sum_{i=1}^{n} C^i_n = \sum_{i=1}^n\frac{n(n-1)\cdots(n-i+1)}{i!}$$
Does $N = 2^n-1$ hold ?
I mean, $C^i_n = \binom{n}{i}$. According to the binomial formula, if this summation sums from $i=0$ instead of $i=1$, then it's equal to $2^n$.
Because of this, does this sum to $2^n-1$?
| We know that
$$(a+b)^n=\sum_{i=0}^n \binom {n}{i}a^{n-i}b^i$$
with $a=b=1$, it becomes
$$2^n=\sum_{i=0}^n\binom {n}{i} $$
$$=\sum_{i=\color {red} {1}}^n\binom {n}{i}+\frac {n!}{0!(n-0)!}$$
$$=N+1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2397466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
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