Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Map $\mathbb{C}\setminus [-1,1]$ onto the open unit disk Let $G = \mathbb{C}\setminus [-1,1]$. I wan't to find an analytic function $f:G\rightarrow \mathbb{D}$ where $\mathbb{D}$ denotes the unit disk such that $f$ is onto, and preferably if possible one-to-one.
Now I've seen that $g(z) = \frac{1}{2}(z+1/z)$ maps the o... | If $f:G\to \mathbb D$ is onto it cannot be one-to-one: else $G$ and $\mathbb D$ would be analytically isomorphic.
Indeed the inverse of a bijective analytic mapping between open subsets of $\mathbb C$ is automatically analytic.
Note that this is a non trivial result.
But this is absurd since these domains are not even ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3541832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Philosophy of simple field extensions In B. L. van der Waerden's Algebra stuck on the problem 6.9:
The polynomial $f(x) = x^4 + 1$ is irreducible in the field of rationals.
Adjoin a root $\theta$ and resolve the polynomial in the extended
field $\mathbb Q (\theta)$ into prime factors.
Seems like I haven't nailed ... | I think it's important to realize what's taken as obvious here and what not.
Obvious: there is a root in some extension, and there is a prime factorization in another extension
Non obvious: all roots are equivalent in the sense that:
Non obvious: the polynomial can be factored in the simple field extension $Q(\theta)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3541971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
} |
Prove $e^{z+w}=e^ze^w$ using the Taylor expansion for $ e^z.$ For all $z,w ∈ \Bbb{C}$, using the power series definition for $e^z$, prove $e^{z+w}=e^ze^w$.
| Starting with $e^{z+w}$, we have :
\begin{align}
e^{z+w} & = \sum_{n=0}^\infty \frac{(z+w)^n}{n!} \\ & = \sum_{n=0}^\infty \sum_{k=0}^n \binom{n}k \frac{z^kw^{n-k}}{n!}
\end{align}
Here's the point : we now need to collect terms of the form $z^K$ for fixed $K$. This will involve a change of variable.
To see how we do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3542281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Decreasing sequence term sign for zero convergent series Given a convergent series
$$\lim_{n \to \infty} \sum_{k=1}^n a_k = 0 $$ for an decreasing sequence $\{a_n\}$. Can we make any deductions on the sign of sequence terms $a_n$?
| If $(a_n)_{n \in \mathbb{Z}^+}$ is decreasing and $\sum_{k=1}^\infty a_k = 0$, then $a_n = 0$ $\forall n$.
To see this, we first show that $a_n \geq 0$. Suppose not, so $a_1 = q < 0$. Then since the sequence is decreasing, $a_n \leq q$ $\forall n$. Then:
$$
\sum_{k=1}^\infty a_k \leq \sum_{k=1}^\infty q = -\infty
$$
Wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3542456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Bounded function whose second derivative is non negative Is it true that a twice continuously differentiable bounded function from R to R with non negative second derivative for all x in R is necessarily a constant? If not give a counter example. The given function is convex throughout R since it’s second derivative is... | If $f''\ge 0$ then $f'$ is increasing. Because if $x<y$ and $f'(x)>f'(y)$ then by the MVT there exists $z\in (x,y)$ such that $0>\frac {f'(y)-f'(x)}{y-x}=(f')'(z)=f''(z)\ge 0,$ which is absurd.
If $f$ is differentiable and not constant then $f'$ is not everywhere $0.$ For if $f(x)\ne f(y)$ then by the MVT there exists ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3542619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit $\lim_{x\to 0^{-}}\frac{e^{40x}-1}{5x}$ $$\lim_{x\to 0^{-}}\frac{e^{40x}-1}{5x}$$
What I have does is:
$$\lim_{x\to 0^{-}}\frac{e^{40x}-1}{5x}=\lim_{x\to 0^{-}}\frac{e^{40x}-e^0}{5x}=\lim_{x\to 0^{-}}\frac{8}{8}\cdot\frac{e^{40x}-e^0}{5x}=\lim_{x\to 0^{-}}8\cdot\frac{e^{40x}-e^0}{40x-0}=\\=8\cdot(e^{40x})'_{x=0}=... | $$l=\lim_{x\to 0^{-}}\frac{e^{40x}-1}{5x}$$
Using the derivative definition:
$$l=\frac 1 5\lim_{x\to 0^{-}}\frac{e^{40x}-e^{40*0}}{x-0}$$
$$l=\frac 1 5(e^{40x})_{x=0}'$$
$$l=\frac 1 5(40e^{40x})_{x=0}$$
Therefore:
$$l=\frac {40} 5=8$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3542932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove that a maximal planar graph/triangulation of $V>3$ has minimum degree = $3$ I'm trying to prove that any maximal planar graph that has more than 3 vertices will have a minimum degree of 3, that is any vertex in such a graph has greater or equal to 3 edges connected to itself. So it doesn't have to have a degree 3... | Given a planar graph, embedded in the plane, you want to add a number of edges to get a maximal planar graph, and you want to be sure that resulting maximal planar graph won't have vertices with degree $=2$.
From this Wikipedia page:
A simple graph is called maximal planar if it is planar but adding any edge (on the g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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A double sum for the square of the natural logarithm of $2$. I am trying to show
\begin{eqnarray*}
\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{(n+m)^2 2^{n}} =(\ln(2))^2.
\end{eqnarray*}
Motivation : I want to use this to calculate $ \operatorname{Li}_2(1/2)$. So I want a solution to the above that does not use... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
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Examples of Dense\Not Dense sets Let the set of eventually zero sequences $c_{00} = \{x = (x_1, x_2, . . .) : x_n = 0 \ \text{for all but finitely many} \ n \}$ where $x_i$ are real numbers.
(a) Prove that $c_{00}$ is dense in $l^p, 1 \le p < \infty$.
(b) Prove that $c_{00}$ is not dense in $l^\infty$.
$\text{My attemp... | For (a), your argument does not prove that $c_{00}$ is dense in $\ell^p$, rather it proves that the set of elements of $c_{00}$ having rational coordinates is dense in $c_{00}$. To prove that $c_{00}$ is dense in $\ell^p$, you need to start with an arbitrary element of $\ell^p$ and show that the $\varepsilon$ ball cent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543364",
"timestamp": "2023-03-29T00:00:00",
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How to find the stationary point under constraints analitically? I am working with the optimization problem from the paper, eq.(5)
$$\max_{X=(x_1, x_2, \ldots, x_{n+1})} f(X)=(A-B\sum_{i=1}^n \frac{1}{x_i})\times x_{n+1}$$
subject
to $$x_{n+1}=1-2k\sum_{i=1}^n x_i,$$ $$x_i \geq 0, \quad i = 1,2,\ldots, n+1.$$
Here $A \... | Let’s follow your approach.
Unfortunately, the first $n$ equations of your system are wrong, we have $F'_{x_i}(X, \lambda)= x_{n+1}\frac{B}{x_i^2} +\color{red}{2\lambda k}=0$. Next, both values of $X^*$ from the paper and the first $n$ equations of the system (unless $\lambda=x_{n+1}=0$) says that for the stationary ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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probability priority of comma and bar In probability does , have priority over |?
I am asking because of below question.
Is below X and (Y given by Z)?
P(X,Y|Z)
or is it (X and Y) given by Z?
| To answer your question more directly, yes, comma (,) has higher priority than bar (|) when interpreting this notation.
So P(X,Y|Z) is the conditional probability of X and Y occurring together, given Z. It might help to think of this case as P((X,Y)|Z) (though such use of additional brackets is not common in the notati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Curry-Howard: Types with Logic vs Types as Logic In the paper Knowledge Representation in Bicategories of Relations there is a short remark on p47 that makes a distinction that seems quite far ranging regarding the Curry-Howard Correspondence. The paper is is interested in "types with logic" not "types as logic", where... | You might try these two posts by Mike Shulman, Propositions as Some Types and Algebraic Nonalgebraicity and Freedom From Logic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Estimate $n$ such that $\log(n^C)Given $C\in \Bbb N$ (which we can assume to be big), is there a simple way to estimate de value of $n$ such that the following formula is satisfied?$$\log(n^C)<n.$$
Equivalently, how can we estimate the index $n$ where the sequence $x_n=\frac{n}{\log(n)}$ exceed a given value $C\in\Bbb ... | The inequality $\log n^C < n$ is equivalent to $ \dfrac{\log n}{n} < \dfrac 1C$. Since $\dfrac{\log n}{n} \le \dfrac 1e$ for all $n \in \mathbf N$ we may as well assume that $C \ge e$.
This is a good place to apply the Lambert $W$-function, although not its usual branch.
The function $y = xe^x$ is decreasing on $(-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3543992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to find the intersection of two circles I am doing a question on coordinate geometry.
It asks to find the points of intersection of the two circles:
$\\(x+1)^2+(y-2)^2=10$
and
$\\(x-1)^2+(y-3)^2=5$
And then find the area of the triangle formed by the two points and the origin.
I am wondering how to do this - when I... | Hint:
By subtracting the equations you get the radical line of them: $$4x+2y = 10.$$ Now plug $y =5-2x$ in to one of them and solve a quadratic equation on $x$...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3544246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Effects of increasing a matrix's values on the eigenvalue decomposition Let $A \in \mathbb{R}^{n \times n}$ be a real, symmetric positive semi-definite matrix, and let $A = UVU^T$ be $A$'s eigendecomposition.
Suppose that the matrix $A'$ was obtained by $A$ by making some values of $A$ larger, in a way that $A'$ is sti... | You are asking if the eigenvalues of $(A+\Delta)$ are larger than the values of $A$. If the entries of the symmetric $\Delta$ are positive, this might not be true. We may have
$\det(A+\Delta)< \det A$, just increase some off-diagonal elements of a symmetric $2\times 2$ matrix. However, if $\Delta$ itself is positive s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3544413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of existence of $a$ in $\lim_{h\rightarrow0}\frac{a^h-1}{h}=1$ In deriving the derivative of function $f(x)=\exp(x)$, it is often pointed out that in the general case of $f_a(x)=a^x$ the following expression can be deduced from the definition of the derivative:
$$
\frac{d}{dx}a^x = a^x\cdot\left(\lim_{h\rightarro... | Assuming that basic theorems about limits and the logarithm function are available as well as $\lim_{x\to 0}\left(x+1\right)^{1/x}=e$, we can first prove that:
$$\lim_{y\to 0}\frac{\ln(y+1)}{y}=\lim_{y\to 0}\ln\left(y+1\right)^{1/y}=\ln\left(\lim_{y\to 0}\left(y+1\right)^{1/y}\right)=\ln(e)=1 $$
Switching the limit and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3544583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does $\int\frac{1}{2}\tanh\left(\frac{1}{x^{2}}\right)\,dx$ have a closed form? Context: I am looking for an activation function that is linear in the area surrounding $x=0$, while also staying within the range of -1 to 1. While I was messing around in Desmos, I stumbled across the function $f(x)=\frac{1}{2}\tanh\left(... | Simplest one I can think of is
$\dfrac{x}{1+|x|}
$.
There are also
$\tanh(x)
$
and
$\arctan(x)
$
scaled as
$(2/\pi)\arctan(\pi x/2)
$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3544710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Let V, W and Z be vector spaces, and let $T:V \rightarrow W$ and $U: W \rightarrow Z$ be linear.
Let V, W and Z be vector spaces, and let $T:V \rightarrow W$ and $U: W \rightarrow Z$ be linear.
a).Prove if UT is one-to-one, then T is one-to-one. Must U also be one-to-one?
b). Prove if UT is onto, then U is onto. Must ... | Both holds also for functions that are not linear, so I will prove this in the traditional way.
Recall that, if $f:A\to B$ is a function, $f$ is one-to-one if and only if for every $x$ and $y$ in $A$,
$$f(x) = f(y) \textrm{ implies that } x=y.$$
Also, recall that $f$ is onto if and only if for any $b\in B$ there exis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3544869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Ring homomorphism and characteristic Let $F$ be a finite field. If $f : F \to F$, given by $ f(x) = x^3$ is a ring homomorphism, then
*
*$ F = \mathbb{Z} / \mathbb{3Z}$
*$ F = \mathbb{Z}/ \mathbb{3Z}$ or $ F = \mathbb{Z}/ \mathbb{2Z}$
*$ F = \mathbb{Z}/ \mathbb{2Z}$ or characteristic of $F$ is 3.
*Characteris... | Consider $(\mathbb Z/3\mathbb Z)[x]/(x^2+1)$. It is a field $\not=\mathbb Z/3\mathbb Z$ of characteristic $3$, so option $(1)$ is not true.
Similarly option $(2)$ is not a necessary condition for $x\mapsto x^3$ to be a ring homomorphism.
For option $(3)$, we shall show that it is necessary for $x\mapsto x^3$ to be a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving recursive function with floor The recursive function is this:
$$ T(n) =
\begin{cases}
2 & \text{ for }n=1;\\
T \left( \lfloor \frac{n}{2} \rfloor \right) + 7 &,\text{ otherwise}
\end{cases}
$$
Based on the definition of the function, the right side becomes:
$T(n) = T( \lfloor \frac{n}{2^i} \rfloor) + 7 * i;$
... | We list first some terms:
$$\underbrace{2}_{1}, \underbrace{9, 9}_{2}, \underbrace{16, 16, 16, 16}_{4},
\underbrace{23, 23, 23, 23, 23, 23, 23, 23}_{8}, 30, 30, \cdots$$
Conjecture:
$$T(n) = 2 + 7\lfloor \log_2 n\rfloor, \quad n = 1, 2, 3, \cdots.$$
We need to prove it. To this end, let
$$S(n) = 2 + 7\lfloor \log_2 n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A Vandermonde Identity for Stirling Numbers? I'm facing the problem of trying to express a quantity in the simplest possible way (it is, using the least possible number of sum symbols).
$$ \sum_{j=0}^{n} \sum_{\ell=0}^m \frac{1}{j!}\binom{b+j}{j} {j+1 \brack {\ell+1}} {b+2 \brack {m-\ell+1}}$$
Of course, this can be ea... | As usual check carefully (although there is a code snippet to illustrate).
I seem to have a form of mathematical dyslexia.
Answer: $\left[x^{M}\right]\frac{\Gamma\left(x+a\right)\Gamma\left(x+b\right)}{\Gamma\left(x\right)\Gamma\left(x+a+b\right)}\left(x\right)_{\left(a+b\right)}$
Actually the expression gives the answ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
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What kind of surface is this? Is there a way to plot this? I am given the surface:
$$S=\{ \vec{x} \in \mathbb R^3: {\|\vec{x} \|}_2^2=4, x^2+y^2 \le 1, z >0 \}$$
and I want to calculate the Mass of $S$ given a density $\rho$. It sort of looks like the upper half of a sphere. The problem I have is that the first equatio... | In spherical coordinates (warning: notations may differ)
$$ \eqalign{x &= r \sin (\theta) \cos (\phi)\cr
y &= r \sin (\theta) \sin(\phi)\cr
z &= r \cos(\theta)}$$
you have $\|\vec{x}\| = r$ and $x^2 + y^2 = r^2 \sin^2(\theta)$, so in this case
you want $r = 2$ and $0 \le \theta \le \arcsin(1/2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Nilpotent Lie algebras are closed under extensions Problem: Let $L$ be a Lie algebra and $K$ an ideal such that $L/K$ is nilpotent and such that $ad(x)|_K$ is nilpotent for all $x \in L$. Prove that $L$ is nilpotent.
By Engel's Theorem, I know that $K$ is nilpotent, which implies that $ad(K)$ is also nilpotent. So, th... | The answer is yes provided that $L$ is finite-dimensional. Denote by $x^*$ a class in $L/K$, $x\in L$. Since $L/K$ is nilpotent, there exists $m$ such that $ad_{x^*}^r=0$. That is, for all $z\in L$ we have $ad_x^r(z)\in K$. But $(ad_x)|_K$ is nilpotent, hence there exists $s$ such that $0=ad_x^s(ad_x^r(z))=(ad^{r+s}_x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\frac{1}{x}-\frac{\ln 2}{2^x} > 0$ if $x>0$ Given that $x>0$, can we show that
$$\dfrac{1}{x}-\dfrac{\ln 2}{2^x} > 0$$
I've plotted it out and it appears to be monotonically approaching its $0$ limit, and when plugged into Wolfram it appears to have no real $0$s, but I do not know how to go about this anal... | We have: $\dfrac{1}{x} - \dfrac{\ln 2}{2^x}= \dfrac{2^x-\ln(2^x)}{x\cdot 2^x}> 0$ because $x > 0, 2^x > 0$ and $2^x > \ln(2^x)$. The last one holds because $y > \ln y$ is true due to $e^y > y$ which is clearly true for $y = 2^x > 0$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Intuition behind $\sin(\theta)$ when introducing this to high school students When first introducing trigonometry to students, the traditional setup is to start with a right-angled triangle with reference angle $\theta$ and we label the sides with "Hypotenuse, Opposite and Adjacent."
To keep students engaged with some ... | You can sell sine and cosine based on expressing how much of the right triangle in question aligns with the adjacent or opposite side.
Let us set notation,
*
*$A$ = adjacent side length
*$B$ = opposite side length
*$C$ = hypotenuse side length
Since the triangle is assumed to be a right triangle we know $A^2+B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3545998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How do I Transform a Quadratic expression into a Pell Equation? I have been told that a simple linear transformation (or a change of variables) can transform the quadratic $$x^2+45xy-216y^2$$ into the Pell equation $$p^2−321q^2=1$$ However I have been unable to achieve this. Can anyone help me find a simple linear tr... | Let $p=x+\dfrac{45}2y$ and $q=\dfrac32y$.
Then $p^2-321q^2=x^2+45xy+\dfrac{2025}4y^2-321\dfrac94y^2=x^2+45xy-216y^2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $G$ is isomorphic to the direct product of its subgroups $H$ and $K$, are $H$ and $K$ normal in $G$? Is this proposition true? Or give a counter example? It comes from this question: if $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$, then $H$ is normal in $G$. If the proposition is true then we are ... | Take any group $H$ with a non-normal subgroup $K\subseteq H$ and consider $G:=H\times K$.
Now let $H':=H\times\{e\}$ and let $K':=K\times\{e\}$. Both are subgroups of $G$ but $K'$ is not normal. However $K'$ is isomorphic to $K$ and $H'$ is isomorphic to $H$ and thus $G\simeq H'\times K'$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Showing $(a_k)_{k=1}^{\infty}$ is in $l^1$ My lecturer claimed this following without a proof. If anybody could give me some idea why this is true, it would be great.
Let $a_k\in\mathbb{R}$, be a real sequence, such that the series $\sum_{k=1}^{\infty}a_kx_k$ converges for all sequences $(x_k)_{k=1}^{\infty}$ with $\li... | It suffices to show that $(a_n)_{n=1}^\infty\in (c_0)^*$. To show this, observe that the operators $x\mapsto \sum_{n=1}^N a(n)x(n)$ are continuous on $c_0$. And since for each fixed $x\in c_0$, the quantity $$\sup_N |\sum_{n=1}^N a(n)x(n)|$$ is bounded. So by the Uniform Boundedness Theorem, $a$ is continuous on $c_0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $\lim_{t \to \infty} \int_1^t \sin(x)\sin(x^2)\,dx$ converges Question_
Prove that $$\lim_{t \to \infty} \int_1^t \sin(x) \sin(x^2) \, dx$$
converges.
I think the indefinite integration of $\sin(x)\sin(x^2)$ is impossible. Besides, I've wondered whether the definite integration of it is possible or not. ... | This is an old Putnam problem [2000, A4]: show that $\displaystyle{\lim_{B\to \infty}\int _0^B \sin(x)\sin(x^2)\,dx}$ exists.
Since we are interested in the limit, we can assume $B>1$. To simplify matters, we introduce a factor of 4.
$$
\lim_{B\to \infty}2\int _0^B \sin(x)\cdot 2\sin(x^2)\,dx
$$$\sin(x^2)$ doesn't hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
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Finding all integer solutions to $A\times B\times C = A! + B!+ C!$ with $0\leq A, B, C\leq 9$ without trying every combination?
Let $A$, $B$, $C$ be integers such that $0 \le A, B, C \le 9$. Find all the solutions for $$A\times B\times C = A! + B!+ C!$$
I have tried some values for $A$, $B$, and $C$ and found a solu... | We may suppose that $A\le B\le C$. Then $ABC\le C^3$. But if $C\ge 6$, then $C!>C^3$, so $$ABC<A!+B!+C!$$ Hence $C\le 5$. And if $C=5$, then we have
$$5AB=A!+B!+120$$
If $A\le 4$, then the LHS is $\le 100$; and if $A=5$, then the RHS is $>125$. Both cases lead to a contradiction; therefore $C$ can't be $5$.
Hence $C\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Use integer/quadratic programming to maximize consecutive zeros in a binary array A binary array $x = [x_1, x_2, x_3, x_4, x_5]$ with each element a binary integer variable taking values 0 or 1. One constraint:
$$x_1 + x_2 + x_3 + x_4 + x_5 == 1$$
Basically one of the variables must be 1. I am trying to maximize the nu... | Suppose you have the binary vector $x = (x_1,\cdots,x_n) \in \{0,1\}^{n}$, where $x_i = 1$ if the $i^{\text{th}}$ slot is filled and zero otherwise. I can think of the following naive and nasty formulation for the objective (maximum number of consecutive zeros):
$$ \underset{J \subset \{1,\cdots,n\}}{\max}\left\lbrace ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3546943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Jump discontinuity for Burgers equation $u_{y} +uu_{x} =0$
Let $u$ be a $C^{1}$-solution of $u_{y} +uu_{x} =0$ in each of two regions separated by a curve $x =\xi(y)$. Let $u$ be continuous, but $u_{x}$ have a discontinuity on the curve. Prove that $\frac{d\xi}{dy} = u$ and hence the curve is a characteristic.
I expr... | The Rankine-Hugoniot condition reads
$$
\frac{\text d \xi}{\text d y} = \frac12(u^++u^-)
$$
where $u^\pm$ are the values of $u$ on each side of the curve $x=\xi(y)$. Since $u$ is continuous across the curve, we have $u^\pm = u|_{x=\xi}$, which ends the proof (recall that characteristics satisfy the Lagrange-Charpit equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3547075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given a (Deterministic Finite Automata) DFA that recognizes a language $L$, show how to construct another DFA that recognizes the language $\max(L)$. If $L$ is any language, then
$$\max(L)= \{w \mid \text{$w$ is in $L$ and there is no non-empty string $x$ such that $wx$ is in $L$} \}.$$
I am really confused about what... | This new language is composed of the "maximal elements" of $L$. For example, if $L = \{ab, ba, aba\}$, then max$(L) = \{ba, aba\}$. It doesn't include $ab$ since that can be extended to $aba$ and still be $L$.
So how can we construct a DFA recognizing this? Since $L$ is regular, it has a DFA $D = (Q, \Sigma, \delta, q_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3547212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a bijection between uncountable sets? I know that there is a bijection between naturals and rationals. I also know that there is no bijection between naturals and reals (diagonal argument).
But, I have never heard of the existence of a bijection between uncountable sets (ex aleph-one). Is there a way to creat... | For a bijection between $A$ and $B$, consider the application that sends every natural $n$ to $e^n$, and $e^n+m$ to $e^n+m+1$ for non-negative integer $m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3547399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Gaussian width of sparse balls The Gaussian width of a set $T\subset \mathbb{R}^n$ is defined as,
$$
G(T) = E\left[\sup_{\theta \in T} \sum_{i=1}^n \theta_i W_i\right],
$$
where, $\mathbf{W}=(W_1,\ldots,W_n)$ is a sequence of i.i.d. $N(0,1)$ random variables. I am interested in finding the value of $G(T)$ for
$$
T(s) ... | Notation: $C$ below denotes (possibly different) absolute constants.
Recall that for $N$ sub-gaussian variable $X_i$ (independence not required) with $\max_i \| X_i\|_{\psi_2}\le K$, $E \max_{i\le N} X_i \le CK \sqrt{\log{N}}.$
For our problem, max in $E \max_{|S| \le s} |W_S|$ enumerates over $N:=\sum_{k=1}^s \binom{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3547487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f=f(x(s),y(t))$, then $\frac{\partial f}{\partial t}=0$ is wrong? enter link description here
You may see through the link that someone pointed out that if$$f=f(x(t),y(t))$$then$$\cfrac{\partial f}{\partial t}=0$$So now i want to challenge this by giving an example:
Let $f=f(x,y)$, where $x=x(t)$,$y=y(s)$. Because ... | $t$ affects $f$. in other words $f$ and $t$ is dependent since $f$=$f$($x$($s$),$y$($t$)) hope this helps! :)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3547619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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A formula for any finite sequence of number In Advanced Problems in Mathematics by Stephen Siklos, pg24, he writes
"Given any finite sequence of numbers, a formula can always be found
which will fit all given numbers and which makes the next number
(e.g.) 42."
Is there a source or proof for this statement?
| Given a sequence of $a_0 ... a_{n-1}$, all you have to do is find $n$ linearly independent functions $f_0$ through $f_{n-1}$. Then define a sequence $c_i$ by $\sum_{i=0}^{n-1} c_i f_i(k)= a_k$ for all $k$. In matrix form, that's $M$c = a where $M$ is a matrix whose entries are given by $f_i(k)$ (the two parameters $i$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given $f(x) = x^n e^{-x}$, show that $\int_0^1 f(x)\, dx$ is equal to a given expression. Consider the function:
$$f : \mathbb{R} \rightarrow \mathbb{R} \hspace{2cm} f(x) = x^n e^{-x}$$
I have to show the following:
$$\int_0^1 f(x) dx = n! \bigg [ 1 - \dfrac{1}{e} \bigg ( 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3... | My intuition first goes to repeated substitution:
$$ \begin{split}
I_n &= - \frac 1 e + n\left(-\frac 1e + (n-1) \left(-\frac 1e+\dots\right) \right) \\ &= -\frac 1e - \frac n e - \frac{n(n-1)}e - \dots - \frac{n(n-1)\cdots (2)}e + n! I_0.
\end{split}$$
Direct computation of $I_0$ and collecting $n!$ results in the for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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UMVUE of $\frac{p}{1-p}$ when $X\sim bin(n,p)$ Uniform Minimum Variance Unbiased Estimate of $\frac{p}{1-p}$ when $X\sim bin(n,p)$
Note: $Bin(n,p)$ is an one parameter exponential family member with min complete sufficient statistic $X$. Then If can find $E[T(X)]=\frac{p}{1-p}$ Then by Scheffes theorem then UMVUE, but... | I think the UMVUE does not exist for $\frac{p}{1-p}$.
$$E[T(X)]=\sum_{t=0}^{n}T(t){n\choose t}p^t(1-p)^{n-t}=\frac{p}{1-p}$$
$$\sum_{t=0}^{n}T(t){n\choose t}(\frac{p}{1-p})^t(1-p)^{n}=\frac{p}{1-p}$$
$$\sum_{t=0}^{n}T(t){n\choose t}(\frac{p}{1-p})^t=\frac{p}{1-p}*\frac{1}{(1-p)^{n}}$$
by choosing $\lambda=\frac{p}{1-p}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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rank deficient least squares with minimum $\ell_1$ norm When a least square solution of a rank deficient least square problem is sought, there are multitude of solutions that give the same minimal residual vector $r=\mathbf{A}x-b$. I am familiar with Moore-Penrose pseudo-inverse that picks minimum $\ell_2$-norm least s... | First solve the unconstrained least squares problem, for example by using the Moore-Penrose pseudo-inverse (which picks the minimum $\ell_2$-norm least square solution), or any other method. Evaluating the two-norm of the residual at that optimal value of $x$ provides the $\epsilon$. Then solve the $\ell_1$-norm minim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Can anyone explain a compact set intuitive way? I know that there are many ways to define the compactness of the set in a topological space, e.g., $X$ is compact if and only if every open cover of $X$ has a finite subcover. I also know that it is the extension of the concept of the bounded closed set in an Euclidean sp... | Compact sets have many nice properties of finite sets but they can be infinite. For example, in finite sets:
*
*All functions have a maximum
*All functions are bounded
*All sequences have a constant subsequence
In compact sets:
*
*All continuous functions have a maximum
*All continuous functions are bounded
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If the ratio of areas of two polygons is the square of the ratio of their perimeters, are the polygons similar? It is known that for two similar polygons $A$ and $B$, the ratio of the areas is the square of the ratio of the perimeters. That is, for example, if the ratio of perimeter A to perimeter B is 5:7, then the ra... | HINT.-Let $p_1$ be the perimeter and $A_1$ the area of the polygone $P_1$ and similarly $p_2$ and $A_2$ for the polygone $P_2$ with $$\frac{p_1^2}{p_2^2}=\frac{A_1}{A_2}$$ You have $$p_2=s_1+s_2+\cdots+s_n$$ How many changes of sides $s_i$ preserving the value $A_2$ are there? The answer to your problem is NO.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Eigenpairs of Normal Matrices
Let's say $A$ is a normal matrix, that is, $AA^*=A^*A$ - then what can be said about the eigen-pairs of $A$ and $A^*$?
I'm trying to show that if $Ax = kx$, then $A^*x = \overline{k}x$. How do I proceed?
I tried the following:
*
*$Ax = kx$ implies
*$x^*A^* = \overline{k}x^*$
*Multip... | Hint: We have $\|Ax-kx\|^2 =\|A^*x-\bar kx\|^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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complex integral - residuum theorem or something else? I have to compute integral of complex function:
$$ \int_{|z|=3} \frac{z^9}{z^{10} - 1} $$
I thought that I should after I found points where $z^{10} = 1$ use the residue theorem but I do not know how to execute that.
So I thought that maybe I should integrate by su... | Residues sum to zero hence the sum of the residues from the poles on
the unit circle is minus the residue at infinity:
$$-\mathrm{Res}_{z=\infty} f(z) =
\mathrm{Res}_{z=0} \frac{1}{z^2} f\left(\frac{1}{z}\right).$$
In the present case we find with $f(z) = z^9/(z^{10}-1)$
$$\mathrm{Res}_{z=0} \frac{1}{z^2} \frac{1/z^9}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3548822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Complex number inequality $|z-1| \ge \frac{2}{n-1}$
If $n \ge 3$ is an odd number and $z\in\mathbb{C}, z\neq -1$ such that $z^n=-1$, prove that
$$|z-1|\ge \frac{2}{n-1}$$
I was thinking that $-2=z^n-1=(z-1)(z^{n-1}+z^{n-2}+...+z^2+z+1)$ and because $z\neq -1$, the second factor can not be $0$:
$$|z-1|=\frac{2}{|z^{n... | Let $n=2m+1$. Observe that $$z^n+1=(z+1)(z^{2m}-z^{2m-1}+\cdots+z^2-z+1)$$ has simple roots (consisting of roots of unity). Let $f(z)$ be the second factor of the right hand side. Then $$f(z)=(z-1)(z^{2m-1}+z^{2m-3}+\cdots+z)+1=0$$ if $z$ is a root of $z^n+1$ and $z\neq -1$.
Applying similar argument as yours, one has ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the infinite sum of a Fourier Series at a given $x$ My question is this:
Let $f(x)=3x-1$, with period $1$ and $x\in(0,1)$.
Is the Fourier series of $f(x)$ convergent at $x=2/3$? If yes, what is the corresponding value of the sum of the Fourier series?
I thought that every Fourier series is convergent for all ... | The Fourier series converges to $f(x)$ at points of continuity, including $x=\frac{2}{3}.$
At points of discontinuity, $x =k\in \mathbb{Z}$ the series will converge to the average of the limits from the left and right:
$$ \textrm{Fourier series } \rightarrow \frac{1}{2} (\lim_{x\uparrow k} f(x) + \lim_{x \downarrow k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549086",
"timestamp": "2023-03-29T00:00:00",
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Given by a graph is the function $f(x)$. How many solutions does $f(f(f(x)))=0$ have? This problem, like the other ones I've posted so far, is from a 2017 olympiad. It goes like this:
Given by the following graph is the function $f(x)$:
How many solutions does $f(f(f(x)))=0$ have?
In other words, how many time does th... | You don't really need to obtain a different representation of $f$ besides the graph, to solve this problem.
If $f^3(x)=0$, then $f^2(x)$ must be either $0$ or $2$.
So, the problem becomes computing how many solutions are there of $f^2(x)=0$ union the solutions of $f^2(x)=2$.
For $f^2(x)=0$ we have again $f(x)=0$ or ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find the largest semi-circle that fits inside a polygon? I have seen (and implemented) algorithms that find the 'Pole of Inaccessibility' for a polygon - that allows you to draw the largest circle within it. However, if I wanted to find the largest semi-circle that fits inside a polygon, is there a similar metho... | My Approach Thus Far
Using the largest rect routine above, I find the largest 2:1 rectangle that will fit.
And (if necessary) move it parallel to the short edge until it is on the boundary.
I then draw a semicircle on each of the long edges, and see if I can make it bigger.
I then move the rectangle parallel to the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $\frac{\cos(n\theta)-\cos((n+1)\theta)}{2-2\cos(\theta)}=\frac{\sin((n+\frac{1}{2})\theta)}{2\sin(\theta/2)}$ In working to prove that
$$1+\cos\theta+\cos(2\theta)+\dots+\cos(n\theta)=\frac{1}{2}+\frac{\sin((n+\frac{1}{2})\theta)}{2\sin(\theta/2)} \tag{1}$$
I have shown
$$\begin{align}
1+\cos(\theta)+\cos(2\theta)... | $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+\dots+\cos(n\theta)=$
$\dfrac{2+e^{i\theta}+e^{-i\theta}+e^{2i\theta}+e^{-2i\theta}+e^{3i\theta}+e^{-3i\theta}\dots+e^{ni\theta}+e^{-ni\theta}}2=$
$\dfrac{1+e^{-ni\theta}+\dots+e^{-3i\theta}+e^{-2i\theta}+e^{-i\theta}+1+e^{i\theta}+e^{2i\theta}+e^{3i\theta}+\cdots+e^{ni\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Number of odd and even permutation. I needed to find number of odd and even permutations in a symmetric group $S_n$ (having $n$ elements).
What we do is select a arbitrary fixed odd permutation $h \epsilon S_n $. We know that $hS_n=${$hg:g\epsilon S_n$} = $ S_n$.
Let's say there are $x$ odd permutations and $y$ even p... | By way of enrichment I would like to show how to solve this using
analytic combinatorics. We have that the sign of a permutation $\pi$ is
given by
$$\sigma(\pi) = \prod_{c\in \pi} (-1)^{|c|-1}$$
where $c$ iterates over the cycles of the permutation and $|c|$ is the
length of the cycle. Therefore we use the following c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Lipschitz continuity of conformal maps I have a basic question on Lipschitz continuity of maps.
Let $\mathbb{D}$ be the unit disk and $T$ be an equilateral triangle. We have a conformal map $\phi :\mathbb{D} \to T$, which is extended to a homeomorphism from $\overline{\mathbb{D}}$ to $\overline{T}$. In fact, $\phi$ ca... | What you wrote is almost correct but your logic is muddled: You did not prove that $\phi^{-1}$ is $C^1$ on $\bar{T}$.
What one observes is that there is a constant $L$ such that $\phi^{-1}$ is $L$-Lipschitz on $T$. (Use the inverse function theorem to prove that $\phi^{-1}$ has uniformly bounded derivative.)
From t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability of Finding Oil
An oil prospector will drill a succession of holes in a given area until he finds r productive wells. The probability that he is successful on a given trial is 0.3. The prospector can only afford to drill 6 wells. What is the probability the inspector fails to find r productive wells if:
(... | The probability of success for a given $r$ is the probability to have at least $r$ successes when attempting $6$ times an expriment with a probability of success equal to $p=0.3$.
That is, if $X$ is random variable following a binomial ditribution $B(6,0.3)$, the probability $p_r$ of success, for a given $r$, is
$$p_r=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3549998",
"timestamp": "2023-03-29T00:00:00",
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Solution for a nonlinear ODE During my PhD thesis im facing the following ODE:
$$\frac{C}{\gamma}\frac{f'(y)}{\sqrt{1+(f'(y))^2}}=y-K_1$$
Where $y$ is a positive variable$(y\geq0)$, $C$ and $\gamma$ are parameters, $K_1$ is a constant (due to previous integration and is necessarily $\leq0$) and $f'(y)=df/dy$ is the fu... | Let us re-write the ODE as
$$\frac{A y'(x)}{\sqrt{1+y'^2(x)}}=x-B \implies A^2 y'^2(x)=(x-B)^2(1+y'^2(x)).$$
$$\implies y'^2(x)(A^2-(x-B)^2)=(x-B)^2.$$
$$\implies y'(x)=\pm \frac{x-B}{\sqrt{A^2-(x-B)^2}}.$$
$$\int dy= \pm \int \frac{x-B}{\sqrt{A^2-(x-B)^2}}dx.$$
$$\implies y(x)=\pm \sqrt{A^2-(x-B)^2}+D$$
$$\implies (y-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3550182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Derivation of D'Alembert's Solution Could anyone provide references wherein D'Alembert's solution to the one-dimensional wave equation is derived from basic properties? I keep running into the approach where D'Alembert's solution is mysteriously presented out thin air and then shown to satisfy the wave equation. I am... | For the derivation of D'Alembert's solution to the one-dimensional wave equation you may follow the following references:
$(1)~~$"Linear Partial Differential Equations for Scientists and Engineers" by Tyn Myint-U & Lokenath Debnath (Chapter $5$, section $5.3$)
$(2)~~$d’Alembert’s solution of the wave equation / energy
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How Many 3 letter words from characters in "ABRACADABRA"? The words cannot have a repetition of letters and each character of the type is distinct i.e. the first 'B' is distinct from the second 'B', and both the 'B's can not be in a word simultaneously.
I need an algorithm/formula that answers for any word(provided as ... | As an algorithm:
loop over (0-9) as A
loop over (0-9 excluding A) as B
loop over(0-9 excluding B) as C
output ABC
end
end
end
You could try asking this question on stackoverflow as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3551029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$T\in \mbox{End}(V)$. If $p|m_T,$ then there is a vector $v$ such that the minimal polynomial of $v$ is exactly $p$. Let $T:V\rightarrow V$ be linear a linear map and let $p\in \mathcal P(\mathbb C)$ be a non-constant polynomial such that $p | m_T$, where $m_T$ is the minimal polynomial of the endomorphism $T$. Prove t... | Use the fact that $V^T$ has cyclic submodule $C \cong \mathbb C[x]/(m_T)$ and then
$h$ has order $p$ in it.
Here are more details:
Cyclic Modules, Characteristic Polynomial and Minimal Polynomial
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3551180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})$ Can someone help me out with $$\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})?$$ I tried Taylor's expansion to no avail.
| $$\lim_{x\rightarrow 0}\ln\left(1+\frac{1}{x}\right)^{x}=\lim_{x\rightarrow 0}\frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}=\frac{\frac{-\frac{1}{x^{2}}}{1+\frac{1}{x}}}{-\frac{1}{x^{2}}}=\frac{x+1}{x}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3551494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Question on Primitive Ring and $\operatorname{End}(V)$ A ring $R$ is primitive if it has a faithful irreducible module. Let's say that $R$ is primitive and it's faithful irreducible module is $V_R$. Since this module is faithful, we have that $R$ embeds naturally into $\operatorname{End}_R(V)$. Since $V_R$ is irreducib... | Hm, a couple things need to be straightened out.
First of all, given a faithful left $R$ module $M$, one can always talk about the natural embedding of $R$ into $\mathrm{End}(M_\mathbb Z)$. If it is a faithful right $R$ module, then you get an embedding of $R^{op}\to \mathrm{End}(M_\mathbb Z)$.
When $M$ is a faithful ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\mathscr A$ has all products and equalizers, then it has all limits This question is about part (a) of this proposition:
Here is a plan of the proof.
Here's what the picture looks like, from what I understand:
But I don't understand what the condition $s\circ p=t\circ p$ means. I guess this means componentwise e... | $s \circ p = t \circ p$ are both arrows into a product, the equality here is equivalent to them being equal after each projection, i.e., $\pi_u \circ s \circ p = \pi_u \circ t \circ p$ for all $u : J \to K$ in $\mathbf I$.
Now, $\pi_u \circ s$ is the $u$-component of $s$, hence is equal to $D(u) \circ \text{pr}_J$. Sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3551869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine the value of the sum to infinity of $(u_{r+1} + \frac{1}{2^r})$ Determine the value of $$\sum_{r=1}^\infty\Bigl(u_{r+1} + \frac{1}{2^r}\Bigr)$$.
In earlier parts of the question, it is given $\displaystyle u_r= \frac{2}{(r+1)(r+3)}$, which when expressed in partial fractions is $\displaystyle\frac{1}{r+1}-\fr... | Write $$U_{r+1}=\frac{2}{(r+2)(r+4)}= 2 \left[\left(\frac{1}{r+2}-\frac{1}{r+3}\right)+\left(\frac{1}{r+3}-\frac{1}{r+4}\right) \right]= [(F_r-F_{r+1})+(F_{r+1}-F_{r+2})], F_r=\frac{1}{r+2}.$$
Now telescoping summation can be done easily.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3552067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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find matrix element from matrix equation How to find "x" from this equation
$$
\begin{bmatrix}
a_1 & a_1^2 & \cdots & a_1^n \\
a_2 & a_2^2 & \cdots & a_2^n \\
\vdots & \vdots& \ddots & \vdots \\
a_m & a_m^2 & \cdots & a_m^n \\
\end{bmatrix}
\begin{bmatrix}
x \\
b_2\\
\vdots\\
b_m \\
\end{bm... | If $n \ne m$, then the given equation makes no sense.
So, let $n=m$ and $j \in\{1,2,...,m\}.$ Then we have
$$a_jx+a_j^2b_2+...+a_j^mb_m=c_j.$$
If $a_j \ne 0$ we get
$$x= \frac{c_j}{a_j}-(a_jb_2+...+a_{j}^{m-1}b_m).$$
If $a_1=a_2=...=a_m=0$, then you can't find $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3552274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An identity for double integrals Let $f$ be a real valued continuous function on $\mathbb{R}_+^2$ and $F(x,y)=\int_0^x\int_0^y f(s,t)\,ds\,dt$.
How to show that
$$\frac{1}{uv}\int_0^u\int_0^vF(x,y)\,dx\,dy=\int_0^u\int_0^v\left(1-\frac{x}{u}\right)\left(1-\frac{y}{v}\right)f(x,y)\,dx\,dy$$
is satisfied.
| Let's look at the one variable situation first: If
$$G(x):=\int_0^x g(s)\>ds$$
then
$$\int_0^u G(x)\>dx=\int_0^u\int_0^x g(s)\>ds\>dx=\int_0^u\int_s^u g(s)\>dx\>ds=\int_0^u(u-s)g(s)\>ds$$
(draw a sketch of the $(x,s)$-plane!), and therefore
$${1\over u}\int_0^uG(x)\>dx=\int_0^u\left(1-{s\over u}\right)g(s)\>ds\ .$$
Now... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finite equational basis for trigonometric identities Consider the structure $(\mathbb{R}, +,-,*,\sin,\cos,0,1)$, where $+$ is addition, $-$ is additive inverse, $*$ is multiplication, $\sin$ is the sine function, and $\cos$ is the cosine function.
Is there a finite basis for the equational identities of that structure?... | I don't know the answer to this question, but it reminds me of a lemma I read in
Equations on real intervals
Walter Taylor
Algebra universalis 55 (2006) 409-456.
Lemma 5.2. (Expanded, so that it makes sense here.)
Suppose that $\mathbb R$ is the real line considered as a topological space. Suppose also that
$$
\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3552599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it necessary to write limits for a substituted integral? To solve the following integral, one can use u-substitution:
$$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$
With $u = \sqrt[4]{x-2}$, our bounds become 0 and 1 respectively. Thus, we end up with the following:
$$36\int_0^1{u^2} \,du$$
In the first case, the lower ... | Suppose you must. Then we have $$9\lim_{a\to2^+}\int_a^3\frac1{\sqrt[4]{x-2}}\,dx.$$
Let $u=x-2$. Then we have $$9\lim_{a\to2^+}\int_{a-2}^1 u^{-1/4}\, du.$$
By the power rule, we have $$9\lim_{a\to2^+} \left.\frac43u^{3/4}\right]_{a-2}^1=9\left[\frac43(1-\lim_{a\to2^+}(a-2)^{3/4})\right]=9\left[\frac43(1-0)\right]=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3552753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,...$. Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,...$. Define $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)...\left(1+\frac{1}{a_n}\right).$$ Find $\lim\limits_{n\to\infty} P_n$.
My approach: $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right... | $$\begin{align}
a_{n}&=na_{n-1}+n\\
&=n[(n-1)a_{n-2}+(n-1)]+n\\
&=n(n-1)[(n-2)a_{n-3} + (n-2)]+n(n-1)+n\\
& \phantom{a bit of space here would be nice}\vdots\\
&=\left[n(n-1)(n-2)\dots 3 \cdot 2\cdot 1\right]\left( 1+\frac{1}{1!}+\frac{1}{2!}+\cdots +\frac{1}{(n-1)!} \right)\\
&=n!\left(1+\frac{1}{1!}+\frac{1}{2!}+\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3552895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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In how many ways can $6$ prizes be distributed among $4$ persons such that each one gets at least one prize? My understanding:
First select $4$ prizes and distribute it among the $4$ people in $^6C_4\times4!$ ways and then distribute the remaining $2$ prizes in two cases: when $2$ people have $2$ prizes: $\frac{4!}{2!}... | Someone could get $3$ prizes & everyone else gets $1$: There are $4$ ways to choose the person who gets $3$ prizes and then $ (6 \times 5\times 4)/3! \times 3 \times 2 \times 1$ ways to distribute the prizes. So $480$ ways in this case.
OR
Two people get $2$ prizes each & everyone else gets $1$: There are $6$ ways ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Lagrange multiplier pedagogy From what I can tell, the traditional way to teach Lagrange multipliers is to start with a function $f(x,y,z)$ and to look for extrema of $f$ subject to $g(x,y,z)=k$.
That is, we restrict $(x,y,z)$ to be on the level curve $g(x,y,z)=k$.
We then look at the level curves of $f$ and find the o... | Preliminary note. In the first tutorial the objective is a quadratic function and the constraint is a sphere. While in the second tutorial the objective is a sphere and the constraint a cylinder (or a quadratic function).
The main difference is that in the first tutorial only the variables $(x,y)$ are considered, and b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Composition Proof I am trying to study functions in math and learning some basic proofs. In numerous places I have seen this: $$(f \circ\ g) ^{-1}(u) = g^{-1}(f^{-1}(u))$$
I know this is true as well, having used it in numerous places in middle and high school. Is there any way of proving this definition though using l... | Assuming that $g : A \to B$ and $f : B \to C$, then, by definition, $(f\circ g)^{-1} : C\to A$ is the unique function such that
*
*$(f\circ g)^{-1}\circ(f\circ g) = \textrm{id}_A$, and
*$(f\circ g)\circ(f\circ g)^{-1} = \textrm{id}_C$.
But observe that the function $g^{-1}\circ f^{-1} : C\to A$ has the same propert... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Consider the following sequence: $a_j = 4a_{j-1} - 4a_{j-2}$
$a_0=0;\ a_1 = 1$
For all $j\geqslant2$, come up with a general formula for the term $a_j$.
Use mathematical induction to prove your claim.
I have calculated the first few terms of this sequence, and determined from them that $a_j = j · 2^{j-1}$.
However... | In your case, since your linear recurrence relation uses $2$ smaller subscripts in the definition for $a_j$, you want to use strong induction to prove that
$$a_j = j\left(2^{j-1}\right), \; j \ge 0 \tag{1}\label{eq1A}$$
In this situation, you need to verify $2$ base cases. Although the question asks you prove \eqref{eq... | {
"language": "en",
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How can derivatives represent tangents? I am taking a Introduction to Calculus course and am struggling to understand how derivatives can represent tangent lines.
I learned that derivatives are the rate of change of a function but they can also represent the slope of the tangent to a point. I also learned that a deriv... | *
*The formula defining the derivative function is not itself the equation of the tangent; this formula gives you , for each tangent ( one tangent for each point $(x, f(x))$ of the graph of $f$ ), the slope of this line. And a slope is a number.
The main point here is that the derivative function is a function that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is $O_n(\mathbb{R})\leq GL_{n}(\mathbb{R})$ A Sub Group? Is $O_n(\mathbb{R})\leq GL_{n}(\mathbb{R})$ where $O_n(\mathbb{R})$ are orthonormal matrices.
*
*$I_n^t=I^{-1}_n=I_n$ and therefore: $I_n\in O_n(\mathbb{R})$
*Let $A,B\in O_n(\mathbb{R})$ then we have to prove that $AB^{-1}\in O_n(\mathbb{R})$
We have that if... | Note that the condition of $A$ being invertible and satisfying $A^t=A^{-1}$ is equivalent to $AA^t = I$. With this it is easy to check the properties of being a subgroup.
*
*$II^t = II = I$, hence $I\in O_n$
*Given $A,B\in O_n$ we have $AA^t=BB^t=I$ and hence
$$\begin{align}
(AB^{-1})(AB^{-1})^t&= AB^{-1}(B^{-1})^t... | {
"language": "en",
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Minimize the number of non-zero rows I am trying to formulate a shipping problem as a mathematical optimization one, and I'm having some trouble determining my optimization objective (cost function). Without getting too far into the details of the particular problem, the variable I am trying to optimize over is an $K\t... | For each row $i$, introduce a binary variable $y_i$ to indicate whether the row is nonempty. The objective is to minimize $\sum_i w_i y_i$, and the constraints are:
$$X_{i,j} \le A_{i,j} y_i$$
for each row $i$.
This formulation enforces the logical implication $X_{i,j} > 0 \implies y_i = 1$, which is sufficient if $w_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3554192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expected number of dice that are all 6 Suppose that a group of n fair dice is rolled $4*6^{n-1}$ times.
a) Find the expected number of times that “all sixes” is achieved (i.e., how often among the $4*6^{n-1}$ rolls it happens that all n dice land 6 simultaneously).
The answer is $\frac{4}{6}$. (Binomial distribution)
... | For b)
The first roll can be all sixes if all the dice roll sixes. The second roll can be all sixes if the first roll is all sixes and the second roll does not change or the second roll is a reroll and you roll all sixes (whether or not the first roll is all sixes).
So, for the second roll, the probability that you get... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In the epsilon-delta definition, what is wrong if I said: "given delta, there exists an epsilon"? WHY are we always given $\epsilon > 0$ first, then solving for a $\delta>0$? This is in the limit definition.
I want to ask:
Can we say "given $\delta>0$, there exists $\epsilon>0$"? Since we can always solve for one give... | Because $\epsilon$ can be unbounded. Let's say we choose $\delta=3$., then is $\epsilon = 6$. But, does it have to be 6? No! It can be 7, 8, 9, 1000. In fact, any number bigger than 6 will do. So, choosing $\delta$ first leaves $\epsilon$ unbounded.
Choosing $\epsilon$ first is like reverse engineering. To me, changin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3554346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 10,
"answer_id": 8
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Backgammon variant — two player double or nothing on random walk?
Two players, A and B, play the following backgammon variant.
A series of steps is labelled $-2$, $-1$, $0$, $1$ and $2$, and a
chess piece is placed on step $0$. A fair coin is flipped, and it
lands on heads, the chess piece is moved forward by one ... | The analysis in quarague’s answer isn’t correct because it only takes the possibility of doubling once into account, whereas future doubling opportunities in fact increase the expected payoff.
Denote by $x_k$ the expected value of the game for $A$ when $A$ has score $k$ and has the right to challenge and $B$ doesn’t. W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3554502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I'm struggling with the general equation of this set of vectors algebraically. I'm struggling with Part B in the problem below. I thought the answer would be x(-1, 0, 1) + y(1, -1, 0), but that doesn't seem to be correct. Any assistance would be greatly appreciated!
| $u$ and $v$ are linearly independent, but $w=-u-v$, so the span of these $3$ vectors is a ($2$-dimensional) plane through the origin; that is, the coordinates $(x,y,z)$ of the space spanned by these vectors satisfy $Ax+By+Cz=0$ for some $A, B, $ and $C$.
Since $u$ and $v$ are in the plane, we know that $A\times-1+C\tim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why only either of $A^{-1}A$ and $AA^{-1}$ is equal to identity matrix? Consider $6\times 4$ matrix $A$.
Why either of $A^{-1}A$ and $AA^{-1}$ is equal to identity matrix, but the other one is not equal to identity matrix?
Note: I try to do it numerically as follows. But I don’t understand why I get such a result. I ... | Inverse is defined for square matrices and it is unique. For rectanglar matrices
either right or left inverse exists and it is also not unique.
For $A_{6 \times 4}$ if B is the inverse of $A$, then $$A_{6 \times 4} B_{4 \times 6}=I_{6 \times 6}~~~~~(1)$$ Here $B$ has 24 unknowns whereas there are 36 equations. This sys... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3554844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every nonzero element in a Banach space has a norming extreme point For any Banach space $F,$ let $B_F$ be the closed unit ball of $F,$ that is, $B_F = \{x\in F: \|x\| \leq 1\}.$ Also, let $ext B_F$ be the set of extreme points of $B_F$ (Recall that $x$ is an extreme point of $B_F$ if $x = \frac{1}{2}(x_1+x_2)$ for som... |
I would like to show that $S$ is weak-star closed in $B_{F^*}$ so that it is weak-star compact
For a fixed $x\in F$, the map
$$
\phi_x : F^*\to\mathbb K,\;
x^*\mapsto x^*(x)
$$
is continuous from the weak-star topology of $F^*$ to the ordinary topology of $\mathbb K$, where $\mathbb K\in\{\mathbb R,\mathbb C\}$.
T... | {
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"url": "https://math.stackexchange.com/questions/3554988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a limit with sqrt $\lim_{x \to \infty}x^2\left(x^2 - x \cdot \sqrt{x^2 + 6} + 3\right)$ $$\lim_{x \to \infty}x^2\left(x^2 - x \cdot \sqrt{x^2 + 6} + 3\right)$$
I don't know how to rewrite or rationalize in order to find the limit.
| Just to give yet another approach, let $u=x^2+3$. Then, for $x\ge0$,
$$x^2(x^2-x\sqrt{x^2+6}+3)=(u-3)\left(u-\sqrt{u^2-9}\right)={9u\over u+\sqrt{u^2-9}}-{27\over u+\sqrt{u^2-9}}\to{9\over1+1}-0={9\over2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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Term of rotation matrix entries equals 1 - proof concept?! I derived some stuff and it is happening that i come to the following expression:
$\frac{r_{13}^2 + r_{23}^2}{(r_{11}r_{23} - r_{13}r_{21})^2 + (r_{12}r_{23} - r_{13}r_{22})^2}$
that must equal 1 for all first two rows of a rotation matrix (orthogonal matrix).... | The denominator shows the sum of squares of two components of a cross product, which are precisely the components of the vector at the denominator.
| {
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"url": "https://math.stackexchange.com/questions/3555227",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 1
} |
Find $\lim_{x \to \infty} x^3 \left ( \sin\frac{1}{x + 2} - 2 \sin\frac{1}{x + 1} + \sin\frac{1}{x} \right )$ I have the following limit to find:
$$\lim\limits_{x \to \infty} x^3 \bigg ( \sin\dfrac{1}{x + 2} - 2 \sin\dfrac{1}{x + 1} + \sin\dfrac{1}{x} \bigg )$$
What approah should I use? Since it's an $\infty \cdot 0$ ... | Let $y=x+1$ then
$$\begin{align}\sum\sin&=\sin\left(\frac1{x+2}\right)-2\sin\left(\frac1{x+1}\right)+\sin\left(\frac1x\right)\\
&=\sin\left(\frac1y-\frac1{y^2}+\frac1{y^2(y+1)}\right)-2\sin\left(\frac1y\right)+\sin\left(\frac1y+\frac1{y^2}+\frac1{y^2(y-1)}\right)\\
&=\sin\left(\frac1{y^2(y+1)}\right)\cos\left(\frac1y-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Check if $\lim_{n\rightarrow\infty}\sum_{k=1}^n\ln\Big(1-\frac{1}{k}+\frac{1}{k}\cos\Big(\frac{\theta}{\sqrt{\ln n}}\Big)\Big)$ converges. How can I check if the following expression converges as $n \rightarrow\infty$? I am confused because $n$ appears twice...
$$\lim_{n\rightarrow\infty} \sum_{k=1}^n \ln \Big(1 - \fr... | Since $\cos$ is even, you may assume WLOG that $\theta \geq 0$.
Note that
$\displaystyle \sum_{k=1}^n\frac{-\theta^2}{2k\ln n} = \frac{-\theta^2}{2}\frac{1}{\ln n}\sum_{k=1}^n \frac 1k \xrightarrow[n\to \infty]{}\frac{-\theta^2}{2}$ and
$$\left|\sum_{k=1}^n \ln \Big(1 - \frac{1}{k} + \frac{1}{k}\cos\Big(\frac{\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Principal divisor $P+Q+R-3\infty$ on elliptic curve comes from a straight line Let $E=V(y^2-(x-a)(x-b)(x-c))$ be an elliptic curve. Let $D=P+Q+R-3\infty$ be a divisor. Then $D$ is principal if and only if $P$, $Q$ and $R$ lie on a straight line.
One direction is straightforward - if the three points $P$, $Q$, $R$ are c... |
There is an argument specific to elliptic curves in Weierstrass form, it doesn't generalize well to other curves.
Let $E:y^2=x^3-ax-b, 4a^3-27b^2\ne 0$ be a smooth affine cubic curve defined over an algebraized closed field $k$. Its field of rational functions is $k(x)[y]/(y^2-x^3+ax+b)$. The projective closure is $C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How will matrix $A^n$ affect the original eigenvector and eigenvalue? For example, a matrix $A$ has three distinct eigenvalues and has 3 eigenvectors $v_1,v_2,v_3$ corresponding to the three distinct eigenvalues.
So, am I right to say $A^4$ will result in eigenvalue^4 and the eigenvectors remain unchanged?
| The claim that the eigenvectors remain unchanged can be interpreted in two different ways, one of which is true, while the other is false.
The first interpretation is
If $v$ is an eigenvector of $A$, then $v$ is an eigenvector of $A^4$.
This claim is true, and easily verified:
If $v$ is an eigenvector of $A$, then th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For $x\in G$, define $H_x=\{g^{-1}xg \mid g\in G\}$. Under what conditions on $x$ will $H_x \leq G$. Further, if $H_x \leq G$, will $H_x \lhd G$? For $x\in G$, define $H_x=\{g^{-1}xg \mid g\in G\}$. Under what conditions on $x$ will $H_x \leq G$. Further, if $H_x \leq G$, will $H_x \lhd G$?
My attempt is as below:
$G$... | Your reasoning is correct. To answer your final question, think that something to hold for every $g_1, g_2$ must hold in particular for $g_1=g_2$. Explicitly:
\begin{alignat}{1}
H_x \le G &\iff \forall g_1,g_2 \in G, \exists g\in G \mid g_1^{-1}xg_1(g_2^{-1}xg_2)^{-1}=g^{-1}xg \\
&\Longrightarrow \forall g_1 \in G, \ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3555895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Value of $ \lim_{n \to \infty} \int \limits_{0}^{1}nx^n e^{ x^2} ?$ How to find the value of $$ \lim_{n \to \infty}\int \limits_{0}^{1} nx^n e^{ x^2} ?$$
From wolfram the limit approaches to $e$ for larger values of $n$. I substituted $x^2 $ with $u$ and obtained
$$ \frac{ n} {2} \int \limits_{0}^{1} u^{\frac{n-1}{2}}... | You have the right idea about changing the variable, just a different change: $u=x^{n+1}$
$$
\begin{align}
\lim_{n\to\infty}\int_0^1nx^ne^{x^2}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^1\frac{n}{n+1}e^{u^{\frac2{n+1}}}\,\mathrm{d}u\\
&=\int_0^11\cdot e^1\,\mathrm{d}u\\[6pt]
&=e
\end{align}
$$
Note that $\frac{n}{n+1}e^{u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3556059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary? Question:
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63} $. Why is this assumption necessary?
Proof:
Since $\gcd(a,3)=1$ $\Leftrightarrow a\equiv 1\pmod 3$ $\Leftrightarrow a^7\equiv 1\pmod3\equiv a\pmod3$
Then using Fer... | Fermat's Little Theorem states an integer $a$ and prime $p$ satisfy $p|a^p-a$, and if further $p\nmid a$ we can cancel this to $p|a^{p-1}-1$. So we always have $7|a^7-a$, but if $3\nmid a$ we can reason$$3|a^2-1\implies 3^2|(a^3+2a)(a^2-1)^2+3(a^3-a)=a^7-a.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3556188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Calculus of variations Euler-Lagrange equation and variational problem
Find all the extrema (local minima and maxima) of the function $$J[y] = \int\limits_1^2(xy' + y)^2\,\mathrm dx;\qquad y(1) = 1, y(2) = \dfrac12.$$
Hint. Once you've found the solution of the Euler-Lagrange equation with the boundary conditions, rem... | Rather than going through your work line by line, let's see if I get the same answer: $$L=x^2y^{\prime2}+2xyy^\prime+y^2\implies 0=\frac{(\partial_{y^\prime}L)^\prime-\partial_yL}{2x^2}=y^{\prime\prime}+\frac2xy^\prime\implies y=A+\frac{B}{x}.$$The boundary conditions give $y=\frac1x$, as you said. With $y=\frac1x+\eta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3556367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving that $\frac{\sum_{n=0}^\infty a_n n z^n}{\sum_{n=0}^\infty a_n z^n}$ is strictly increasing. Im considering the quotient from the title:
$$g(z)=\frac{\sum_{n=0}^\infty n\cdot a_n z^n}{\sum_{n=0}^\infty a_n z^n}$$
with $a_n>0$ $\forall n\in \mathbb{N}$ and for all $z\in(0,1)$. And assuming that both sums suffice... | Even when the inequality changed to $g''(z) \ge 0$, it need not be true.
For an counterexample, take $a_n = \begin{cases}\frac1{n!},& n \ne 1,\\ \frac12 & n = 1\end{cases}$. The denominator of this $g(z)$ is
$$D(z) \stackrel{def}{=} \sum_{n=0}^\infty a_n z^n = e^z - \frac{z}{2}$$
From this, we can deduce
$$g(z) = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3556624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is the value of $x$, given that $f\left(x\right)+f\left(\frac{1}{1-x}\right)=x$ and $f^{-1}\left(x\right)=2$? I tried to approach the problem by using the equation $f\left(2\right)=x$ but I always get stuck in the middle of the process.
| We have
$$x=2\Rightarrow f(2)+f(-1)=2$$
$$x=1/2\Rightarrow f(1/2)+f(2)=1/2$$
$$x=-1\Rightarrow f(-1)+f(1/2)=-1$$
This is a system of three equations with three unknowns. If it helps, we can rewrite it as
$$s+y=2$$
$$s+z=1/2$$
$$y+z=-1$$
where $f(2)=s$, $f(-1)=y$, and $f(1/2)=z$. We can easily solve this by noting that
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3556891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Find general term of recursive sequences $ x_{n+1}=\frac{1}{2-x_n}, x_1=1/2,$ Please help to solve:
*
*$ x_{n+1}=\frac{1}{2-x_n}, x_1=1/2,$
*$x_{n+1}= \frac{2}{3-x_n}, x_1=1/2$
I know answers, but can't figure out the solution.
The first one is obvious if you calculate first 3-5 terms by hand. But how can I get th... | If we recursively apply the recursive relation we get
$$x_{n+1} = \frac{1}{2-x_n} = \frac{2-x_{n-2}}{3-2x_{n-2}} = \frac{3-2x_{n-3}}{4-3x_{n-3}}$$
and in general
$$x_{n+1} = \frac{k-(k-1)x_{n-k}}{k+1-kx_{n-k}}$$
Setting $k=n-1$ we get
$$x_{n+1} = \frac{n-1-(n-2)\frac{1}{2}}{n-(n-1)\frac{1}{2}} = \frac{n}{n+1}$$
I haven... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Abelian group of finite order This problem was submitted to me as part of Abstract Algebra Homework.
Let $(G,\cdot)$ be an abelian group of finite order.
Show that if $o(G)=n$ then $a^n=e$ for all $a\in G $.
Hint : Show that if $G=\{e,a_1,a_2,\cdots,a_{n-1} \}$ then $G=\{a,a\cdot a_1, a \cdot a_2 , \cdots , a \cdot a_... | Use the hint: let $b=ea_1a_2\dotsm a_{n-1}$. Then
$$
b=ae\cdot aa_1\cdot \dotsm aa_{n-1}=a^nb
$$
(because the group is abelian). Hence $a^n=e$ by cancelling $b$.
Why can we write $G=\{a,aa_1,\dots,aa_{n-1}\}$? Consider the map $\mu_a\colon G\to G$, $\mu_a(x)=ax$. This map is injective, hence also surjective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Does the series diverge or converge: $\sum\limits_{n=1}^{\infty} \frac{1+(-1)^n}{\sqrt{n}}$? I'm asked to find out if this series converges or diverges:
$$\sum_{n=1}^{\infty} \frac{1+(-1)^n}{\sqrt{n}}=0+\frac{2}{\sqrt{2}}+0+\frac{2}{\sqrt{4}}...$$
So I thought I could use a direct comparison test, so
$$\sum_{n=1}^{\inf... | Hint:
Since $1+(-1)^n=0$ if $n$ is odd and $2$ if $n$ is even, it's
$$\sum\limits_{k=1}^\infty\dfrac{2}{\sqrt{2k}}=
\sqrt2\sum\limits_{k=1}^\infty \dfrac1{\sqrt k}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
$D$ is a point inside $\triangle ABC$, $\angle CAD=\angle DAB=10$, $\angle CBD=40$, $\angle DBA=20$, what is $\angle CDB$? $D$ is a point inside $\triangle ABC$, $\angle CAD=\angle DAB=10$, $\angle CBD=40$, $\angle DBA=20$, what is $\angle CDB$?
I'm sure I'm supposed to use trigonometry to obtain the value of $\angle C... | Take E, reflection of B in AD; it belongs to AC and $\triangle BED$ is equilateral; easily BE=BC, BEC being a Langley triangle and required $\widehat{BDC}=70^\circ$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$\operatorname{Hom}_A(K,A)\neq0$ iff $A=K$ I can prove this result in one way that is if $A=K$. Then there exists a natural morphism from $K$ to $K$. But I have no idea how to prove the converse part. I am new to this course please help by providing a good explanation.
Here A is integral domain and K is quotient field.... | Suppose that we have a non-zero $A$-module morphism $\phi : K \to A$. Notice that $\phi (1) \neq 0$ (because otherwise $\phi = 0$). We have $\phi (1) \phi (\frac{1}{\phi (1)}) = \phi (\phi (1) \cdot \frac{1}{\phi (1)}) = \phi (1)$ and therefore $\phi (\frac{1}{\phi (1)}) = 1$.
Let $0 \neq a \in A$. We want to show that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The equation of the tangent line at the point (1,-1) is $y=\frac{4x}{3}-\frac{7}{3}$. Given the equation, $x^2y+ay^2=b$, find the values of a and b. So I got this question:
"The equation of the tangent line at the point $(1,-1)$ is $y= \dfrac{4x}{3} - \dfrac{7}{3}$. Given the equation, $x^2y + ay^2 = b$, find the value... | First, you have to differentiate both sides implicitly:
$$\frac{\mathrm d}{\mathrm dx} \left(x^2y+ay^2\right) = \frac{\mathrm d}{\mathrm dx} (b)$$
$x^2y$ is a product, so $\dfrac{\mathrm d}{\mathrm dx} \left(x^2y\right) = y\dfrac{\mathrm d}{\mathrm dx} \left(x^2\right)+x^2\dfrac{\mathrm dy}{\mathrm dx} = 2xy+x^2\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3557872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can you switch $g'$ and $f'$ in the integration by parts formula? I'm practicing calculus for the future and have a question about the integration by parts formula.
I was taught: $\int f(x)g'(x) =f(x)g(x)-\int f'(x)g(x)dx $
But would switching $f'$ and $g'$ in the formula to this still work?
$\int f'(x)g(x) = f(x)g(x)... | Both will yield the same answer, but some integrals makes the math work out nicer.
Changing what you choose as the "parts" is a common strategy to try when solving these.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3558011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Differentiable function $f(x)=4x^7 −14x^4 +30x−17$ I am trying to prove that the function $f:\Bbb R→\Bbb R$, $f(x)=4x^7 −14x^4 +30x−17$,is injective. To do this I need to prove it is differentiable from first principles. I can then prove its derivative is strictly increasing to show it is injective. Any help on the pro... | Since we can evaluate a derivative, our function is differentiable.
Also, $$f'(x)=28x^6-56x^3+30=28\left(x^6-2x^3+\frac{15}{14}\right)=28\left((x^3-1)^2+\frac{1}{14}\right)>0.$$
$$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=$$
$$=\lim_{h\rightarrow0}\frac{4((x+h)^7-x^7)-14((x+h)^4-x^4)+30(x+h-x)}{h}=$$
$$=4(7x^6)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3558256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to separate variables in DiffEq of $y'=x-y$ $$\frac{dy}{dx} = x-y$$
How do I separate so I can integrate both sides?
Thanks for getting me started.
I know the solution is $$ y = x-1+2e^{-x}$$
| Hint
Your equation is $$y'(x)+y(x)=x.$$
Multiply both side by $e^x$ gives $$\big(y(x)e^x\big)'=xe^x.$$
I let you conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3558401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 1
} |
Are there finite lattices which are not order isomorphic to a sublattice of $\mathbb Z^n$? Consider the lattice $\mathbb Z^n$ for some finite $n$ ordered by $x\leq y \leftrightarrow \forall i: x_i \leq y_i$. I'm having difficulty thinking of all lattice which is not a sublattice of this.
For example, the diamond lattic... | While not every finite lattice is a sublattice of $\mathbb Z^n$, as amrsa pointed out, it is true that every finite poset is a subposet of $\mathbb Z^n$.
Given a finite poset $P$ consider the map $L\colon P\to\mathbb Z^P$ given by
$$
L(p)_q =
\begin{cases}
1 & \text{if $q\le p$}, \\
0 & \text{otherwise}.
\end{cases}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3558505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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