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Can a Sylow $p$-subgroup be conjugate to anything other than another Sylow $p$-subgroup? |
In Tu's book, the tangent space at $p \in U \subseteq \mathbb{R}^n$ is defined as an arrow emanating from $p$. However, when passing to manifolds, such an approach cannot be used, since we should fix a chart $(U,\phi)$ about $p$ and, next, to decree that a tangent vector is an arrow emanating at $\phi(p)$. In such a ca... |
I am taking a course on Large deviation theory but I'm a bit stuck at the first place. The lecturer gave **an example to motivate the study of large deviation**.
**First**, if $X_1$ is standard gaussian, and $X_1,\cdots,X_n$ are i.i.d. r.v.s, then $S_n/\sqrt{n}$ is also standard gaussian, thus the event $S_n>an$ wh... |
What I mean, is, given a product measure space $(X \times Y, \mathcal{S} \otimes \mathcal{T})$, is any $E \in \mathcal{S} \otimes \mathcal{T}$ the limit of a sequence of sets ${E_n}$ composed of finite unions and intersections of rectangles? That is,
Can we find a sequence of sets ${E_i}$ s.t.
$$E = \bigcup_{i=1... |
In a product measure space $(X \times Y, \mathcal{S} \otimes \mathcal{T})$, is any measurable set $E$ the limit of a sequence of rectangles? |
The exponential generating function for the number of permutations of $n$ elements is (see [Wikipedia][1])
\begin{eqnarray*}
\frac1{1-z}
&=&
\exp\left(\log\left(\frac1{1-z}\right)\right)
\\
&=&
\exp\left(\sum_{k\ge1}\frac{z^k}k\right)\;,
\end{eqnarray*}
where the term $\frac{z^k}k=\frac{(k-1)!z^k}{k!}$ rep... |
This integral: $\int\left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$ seems to be complicated but it can be solved very easily if one know the formula $\tan(a+b+c)= \frac{\tan(a)+ \tan(b)+ \tan(c ) - (\tan(a) \tan(b)\tan(c))}{1-\tan(a)\tan(b)-\tan(a)\tan(c)-\tan(b)\tan(c)}$. so $\i... |
How one can notice a generalisation of famous formulas? |
Consider $n$ independent and identically distributed random variables $ \{X_i\}_{i=1,...n} $ with support on some interval $[a,b]$ and its $n$'th order statistic $\max_{i \in \{1,...n\}} X_i$ . The following "entropy-looking" measure of dispersion of the maximum is
$$ - \int_a^b F^n(x) \ln F^n(x) dx ,$$
where $F(x... |
This integral: $\int\left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$ seems to be complicated but it can be solved very easily if one know the formula $\tan(a+b+c)= \frac{\tan(a)+ \tan(b)+ \tan(c ) - (\tan(a) \tan(b)\tan(c))}{1-\tan(a)\tan(b)-\tan(a)\tan(c)-\tan(b)\tan(c)}$. so $\i... |
That means the process may be periodic by distribution. And in what condition could the process be present as a series of Fourier bases multiplied by random variables? |
How to construct a Wiener-like process on Tori? |
[![The example][1]][1]
I have the matrix in the image for a
reference.
I can't understand why using combinations work on finding all the minimal paths.
(In the image, I go from bottom-left to top-right).
For the first matrix we use Combinations of 2 elements taken once, but who those 2 elements are??
(Are ... |
this comes from a comment of [this][1]
where the comment use $A^{1/2}$ ,
I learned that a integer kth power of a matrix, if A can be diagonizled as $XDX^{-1}$
then $A^k=XD^kX^{-1}$, even when k<=0
so the OP means $A^{1/2}=XD^{1/2}X^{-1}$?
what's a general definition of matrix's exponent if the exponent is r... |
What's matrix's exponent if the exponent is real or complex? |
I am taking a course on Large deviation theory but I'm a bit stuck at the first place. The lecturer gave **an example to motivate the study of large deviation**.
**First**, if $X_1$ is standard gaussian, and $X_1,\cdots,X_n$ are i.i.d. r.v.s, then $S_n/\sqrt{n}$ is also standard gaussian, thus the event $S_n>an$ wh... |
I have learnt that QR decomposition and RREF can be used to find linearly independent vectors, but now I am confronted with a question that is one step further: How do I find linearly dependent GROUPS among vectors?
For example, I have a matrix composed of column vectors.
```
>>> import numpy as np
>>> import sym... |
How do I find linearly dependent groups among vectors? |
If we have some function, for example, $$y = ix^2$$, we can express that, (or any) function ($x,y$) with some kind of three-dimensional pixel ($t$ represents the pixel base whose length can be changed, and the values of the variables can also be changed, in my example the variable is $i$ ). I posted a link where you ... |
> Prove that every holomorphic function $f$ on the closed disk
> $\overline{D}(0,1)$ with $|f(z)|<1$ when $z\in
\overline{D}(0,1)$ has at least one fixed point in $D(0,1)$.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, $f$ is either constant or attains its maximum on boundary.\
If $f$ is constant, we f... |
Express $\left(sin\left(\frac{\pi }{3}\right)+icos\left(\frac{\pi }{3}\right)\right)^8$ in a+bi form. I used demoivre's, so i got sin(8pi/3)+i cos (8pi/3), does that yield my answer upon evaluation? |
> Prove that every holomorphic function $f$ on the closed disk
> $\overline{D}(0,1)$ with $|f(z)|<1$ when $z\in
\overline{D}(0,1)$ has at least one fixed point in $D(0,1)$.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, $f$ is either constant or attains its maximum on boundary.\
If $f$ is constant, we f... |
If we have some function, for example, $$y = ix^2$$, we can express that, (or any) function ($x,y$) with some kind of three-dimensional pixel ($t$ represents the pixel base whose length can be changed, and the values of $i$ can also be changed, in my example). I posted a link where you can see what it looks like. htt... |
Does a Riemannian submersion maps horizontal geodesics to geodesics, and a relevant question? |
Express $\left(\sin\left(\frac{\pi }{3}\right)+i\cos\left(\frac{\pi }{3}\right)\right)^8$ in $a+bi$ form. I used de Moivre's, so i got $\sin(8\pi/3)+i \cos (8\pi/3)$, does that yield my answer upon evaluation? |
Express $(\sin(\frac{\pi }{3})+i \cos(\frac{\pi }{3}))^8$ in $a+bi$ form. I used demoivre's, so I got $\sin(8 \pi/3)+i \cos (8 \pi/3)$, does that yield my answer upon evaluation? |
You need to be careful: DeMoivre's theorem says
$$(\color{red}{\cos} \theta + i \color{blue}{\sin} \theta)^n = \cos n \theta + i \sin n \theta.$$
Note that what you have instead is $$(\color{blue}{\sin} \theta + i \color{red}{\cos} \theta)^n.$$
So what you need to do first is find a way to switch the cosine a... |
I = $\int _{0}^{3}\left( 1+x^{2}\right) d[ x]$
a)12
b)17
c)15
d)19
where [x] is the greatest integer less than or equal to x,
shouldn't this integral be 0 since d[x] is 0 for all x in the domain of the fucntion.
what is the correct explanation ?? |
How to prove continuity of the inverse function $\alpha^{-1}$ in general? |
Let **V** be the set of all **nxn** orthogonal and symmetric matrices and define an equivalence relation on **V** by **A~B** iff there exists invertible matix **C** such that **C^-1 A C** = **B**. Since every symmetric matrices have real eigen values every **A** in **V** is similar to a diagonal matrices consisting of ... |
I'm trying to understand how the [LLL algorithm](https://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm) works and I've stumbled upon the following question:
Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process obtaining $(v_{1},\cdots... |
**Setup:**
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v),... |
I am not crazy, this is an actual geometry question that I found interesting as a non math person.
I have a MacBook Pro 15 with dimensions $35 \times 24 \times 1.5$ and I want to buy an handbag and know if it will fit. The dimensions of the bag are $32 \times 20 \times 20$ (the typical maximum dimensions for the sec... |
In Tu's book, the tangent vector at $p \in U \subseteq \mathbb{R}^n$ is defined as an arrow emanating from $p$. However, when passing to manifolds, such an approach cannot be used, since we should fix a chart $(U,\phi)$ about $p$ and, next, to decree that a tangent vector is an arrow emanating at $\phi(p)$. In such a c... |
I'm reading Dedekind's [work](https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf#page=14) where he defines irrational number, using cuts. I'm stuck where he proves (p. 13) that
> there exist infinitely many cuts not produced by rational numbers
His proof include proof of "lemma" that there is no... |
I'm reading Dedekind's [work](https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf#page=14) where he defines irrational numbers, using cuts. I'm stuck where he proves (p. 13) that
> there exist infinitely many cuts not produced by rational numbers
His proof include proof of "lemma" that there is n... |
I'm reading Dedekind's [work](https://personal.math.ubc.ca/~cass/courses/m446-05b/dedekind-book.pdf#page=14) where he defines irrational numbers, using cuts. I'm stuck where he proves (p. 13) that
> there exist infinitely many cuts not produced by rational numbers
His proof includes proof of "lemma" that there is ... |
Introduction:
Find a such family of polynomial then we can always use Newton's method provided we have a procedure to start.
We introduce a rational function:
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots x}}}}\right)-C=0... |
suppose V is 4-dimensional vector space,
$v_1$ and $v_2$ are linear independent in V,
now for arbitrary plane S that goes through origin,
can I find two new linear independent vector
in plane S that
condition:
\begin{align}
v_1 \cdot v_3 \ge 0 \ \& \ v_2 \cdot v_3 \ge0,
& v_1 \cdot v_4 \ge0 \ \& \ v_2 \cdot... |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$.
Let $f: H^1(G_K,E[2])\to \bigoplus_v H^1(G_{K_v},E)[2]$ be a natural map given by
$f \mapsto (f\mid_{G_{K_v}})_v$
For $M$: profinite abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bb... |
[![The example][1]][1]
I have the matrix in the image for a
reference.
I can't understand why using combinations work on finding all the minimal paths.
(In the image, I go from bottom-left to top-right).
For the first matrix we use Combinations of 2 elements taken once, but who those 2 elements are??
(Are ... |
This equation is bugging me for a while now. It seems pretty simple but what is the general solution?
There are ways to find solutions like
$u(x,y)=Ae^{Bx+\frac{B}{1-B}y}+C(x-y)+D$
But I wonder if there is a way to represent all the solutions.
Any thoughts will be appreciated. |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$.
Let $f: H^1(G_K,E[2])\to \bigoplus_v H^1(G_{K_v},E)[2]$ be a natural map given by
$f \mapsto (f\mid_{G_{K_v}})_v$
For $M$: profinite abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bb... |
Are IVPs $x'=f(x)$, $x(0)=x_0$ uniquely solvable if $f$ is merely differentiable? |
On Banach spaces you mostly look at two different topologies: the strong topology induced by the norm and the weak topology induced by the continuous linear functionals.
This also results in the statement that a linear operator between Banach spaces is strong continuous if and only if it is weak continuous.
But in... |
Let $M$ be a non-compact contractible bounded submanifold of the Euclidean space. Is it possible to find a compact submanifold $S$ of the Euclidean space such that $M$ is a submanifold of $S$?
As a positive example, let $M = \{(x,y) \in \mathbb{R}^2: (x,y) \in S^1, x>0, y>0\}$ be the portion of the 1-sphere $S^1$ ly... |
If the improper integral exists show:
$\lim_{t\to\infty} \int_{-t}^{t} f(x)dx= \int_{-\infty}^{\infty} f(x)dx$
Isn't that just simple definition?
$\int_{-\infty}^{\infty} f(x)dx= \int_{-\infty}^{0} f(x)dx+\int_{0}^{\infty} f(x)dx$
because the integral exists i can write:
$\lim_{a\to\infty} \int_{-a}^{0... |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$.
Let $f: H^1(G_K,E[2])\to \bigoplus_v H^1(G_{K_v},E[2])$ be a natural map.
For $M$: profinite abelian group $M$, we let $\widehat{M}=\text{Hom}(M, \Bbb{Q}/\Bbb{Z})$ be the Pontryagin dual of $... |
>Suppose $k$ is the signed curvature of a plane curve $C$ (parametrized by $\gamma$, say) expressed in terms of its arc length. Show that, if $C_a$ is the image of $C$ under the dilation mapping $\vec{v}\to a\vec{v}$, the signed curvature of $C_a$ in terms of *its* arc-length $s$ is $\frac{1}{a}k(\frac{s}{a})$.
This... |
I am not crazy, this is an actual geometry question that I found interesting as a non math person.
I have a MacBook Pro 15 with dimensions $35 \times 24 \times 1.5$ and I want to buy a handbag and know if it will fit. The dimensions of the bag are $32 \times 20 \times 20$ (the typical maximum dimensions for the seco... |
In FOL, a formula is interpreted in meta-language as $m,s\models\phi$, where m is a model/structure and s is an assignment function. If we continue to interpret this meta-language sentence, how can we do? How to determine a domain? I think this domain should at least include some models, some assignment functions, and ... |
How to interpret formulas in meta-language? |
>Suppose $k$ is the signed curvature of a plane curve $C$ (parametrized by $\gamma$, say) expressed in terms of its arc length. Show that, if $C_a$ is the image of $C$ under the dilation mapping $\vec{v}\to a\vec{v}$ where $a$ is a non-zero constant, the signed curvature of $C_a$ in terms of *its* arc-length $s$ is $\f... |
Let $R$ be a ring, $I$ a finitely generated ideal of $R$ and $J$ an ideal of $R$ with $J \subset I$. Show that $R$ has an ideal $M$ so that $J \subset M \subset I$ and for every ideal $N$ with $M \subset N \subset I$: $N=M$ or $N=I$.
If I choose $M:=I+J=\{a+b,a \in I, b \in J\}$ then the first part should be proved?... |
Let $f(x)$ be a cubic whose coefficient of the leading term is positive and $g(x) = e^{\sin(\pi x)} - 1$. The composite function $h(x) = g(f(x))$ is defined in the set of all numbers, and has a local maximum at $x=0$. In the open interval $(0,3)$ the function $h(x)$ intersects the line $y= 1$ seven times.
Given tha... |
Consider the following density function (unnormalized)
$$p(y;b)=\int_{0}^{\infty}\dfrac{\exp\left(-\frac{y^2}{2\left(b+\frac 5u\right)}\right)}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
Is it true that $p(y;b)$ is a log-concave function in $b$ when $y$ is considered fixed?
My attempt: First ... |
Is the density of scale mixtures of Normal-Gamma log-concave? |
I have an $m$-by-$m$ matrix $A = BCD$ where $B$ is an $m$-by-$n$ matrix, $C$ is an $n$-by-$n$ matrix, and $D$ is an $n$-by-$m$ matrix. My question is: what are the requirements on matrices $B$ and $C$ so that matrix $A$ is always invertible for every invertible matrix $C$? Note that $m < n$ and all matrices are real ma... |
In this https://web.abo.fi/fak/mnf/mate/kurser/fourieranalys/chap3.pdf lecture notes, it is said that the fourier transform of a convolution of distributions $\mathcal{F}(f*g)$ can be defined if the fourier transform of $g$ is in the space of tempered test functions, i.e. $\mathcal{F}(g)\in C^\infty_{pol}$. Where the t... |
We know that for every normalized Hecke eigenform $f$, its Fourier coefficients generate a number field. I wonder if we have an "inverse Galois" type conjecture regarding which number fields can be generated in this way. Moreover, do we have some sort of statistical results/conjectures regarding the frequency of a numb... |
Let $p$ be a prime and $U$ a subgroup of $\mathbb{Z}_p^d$ generated by $(a_1, \ldots, a_d) \in \mathbb{Z}_p^d$. Are there any theorems concerning the minimum of the sum of the natural representatives of an element of $U$, i.e. the term
$$(*) \min_{(b_1, \ldots, b_d) \in U} \phi(b_i),$$
where $\phi: \mathbb{Z}_p... |
A standard 6-sided fair die is rolled until the last 3 rolls are ascending. What is probability that the first such roll is a 1,2,3, or 4? |
Let $\mu(x)$ be a probability measure on $\mathbb{R}^d$ which we assume to be absolutely continous with respect to Lebesgue measure, $\mbox{d}\mu(x)=\omega(x)\mbox{d}x$.
I am interested in tractable sufficient conditions for $\mu$ to support a Poincare inequality, which means that there exists a finite constant $C_P$ ... |
Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$.
I was thinking that I could use Maximum Modulus Principle for function $g = \dfrac{1}{f(z)-z}$, but I got stuck at here (because I can't... |
What about the following number consisting of 85 decimal digits found on http://factordb.com/listtype.php?t=3&mindig=85&perpage=100&start=100 as the third number on the list:
$\dfrac{15019567^{17}-1}{33547074128942890979279691620109514914} = 300276513709773843874226028841063399787134368375660480809424143984059723991... |
Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$.
I was thinking that I could use Maximum Modulus Principle for function $g = \dfrac{1}{f(z)-z}$, but I got stuck at here (because I can't... |
Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$.
I was thinking that I could use Maximum Modulus Principle for function $g = \dfrac{1}{f(z)-z}$, but I got stuck at here (because I can't... |
I am currently studying a system characterized by a particularly structured matrix, as shown below, $$ \dot{ \boldsymbol{x}} = \begin{pmatrix}
0 & -2c & \\
c & 0& 3c & \\
&-2c & 0& -4c & \\
& & 3c & 0 & 5c \\
& & & \ddots & \ddots &\ddots
\end{pmatrix} \boldsymbol{x}.$$
Let's denote this matrix as $cF$. I am pa... |
Is it true that the space $H^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2,(1+|x|^2)dx)$ is compactly embedded in $L^1(\mathbb{R}^2)?$ I don't think that the dimension $2$ is important. Here $H^1$ is the usual Sobolev space and $L^2(\mathbb{R}^2,(1+|x|^2)dx)$ is the weighted $L^2$ space, i.e. $\int_{\mathbb{R}^2}|f|^2(1+|x|^2)d... |
> We have $100$ indistinguishable tickets and $40$ children. In how many ways can we distribute tickets if each child is to receive at least one ticket and not all tickets have to be distributed?
Below is a solution which I am not sure is correct and would like someone to check it:
1) We give each child $1$ ticke... |
In how many ways can we distribute tickets? |
Let $f(x)$ be a cubic whose coefficient of the leading term is positive and $g(x) = e^{\sin(\pi x)} - 1$. The composite function $h(x) = g(f(x))$ is defined in the set of all numbers, and has a local maximum at $x=0$. In the open interval $(0,3)$ the function $h(x)$ intersects the line $y= 1$ seven times.
Given tha... |
Let $f(x)$ be a cubic whose coefficient of the leading term is positive and $g(x) = e^{\sin(\pi x)} - 1$. The composite function $h(x) = g(f(x))$ is defined in the set of all numbers, and has a local maximum at $x=0$. In the open interval $(0,3)$ the function $h(x)$ intersects the line $y= 1$ seven times.
Given tha... |
I want to find a 'nice' drawing of the lipids and genes in my database. Lipids belong to one one of several classes, while genes belong to one of several regions. Each gene/lipid pair has an associated strength value, which indicates how closely related they are.
I have tried to define my dataset formally as follows... |
I = $\int _{0}^{3}\left( 1+x^{2}\right) d[ x]$
$a)12$
$b)17$
$c)15$
$d)19$
where $[x]$ is the greatest integer less than or equal to $x$,
shouldn't this integral be $0$ since $d[x]$ is 0 for all $x$ in the domain of the function.
what is the correct explanation ?? |
While self-studying a probability book, I came up with a modified version of a problem that I don't know how to solve. In the original statement we have a salesman that visites houses selling books, and his probability of selling a book in a visit is 0.3. The problem asks what his sales average per day is, if he does a... |
Let $p$ be a prime and $U$ a subgroup of $\mathbb{Z}_p^d$ generated by $(a_1, \ldots, a_d) \in \mathbb{Z}_p^d$. Are there any theorems concerning the minimum of the sum of the natural representatives of an element of $U$, i.e. the term
$$(*) \min_{(b_1, \ldots, b_d) \in U} \sum_{i=1, \ldots, d}\phi(b_i),$$
wher... |
Is it true that the space $H^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2,(1+|x|^2)^2dx)$ is compactly embedded in $L^1(\mathbb{R}^2)?$ I don't think that the dimension $2$ is important. Here $H^1$ is the usual Sobolev space and $L^2(\mathbb{R}^2,(1+|x|^2)^2dx)$ is the weighted $L^2$ space, i.e. $\int_{\mathbb{R}^2}|f|^2(1+|x|... |
We know that for every normalized Hecke eigenform $f$, its Fourier coefficients generate a number field. I wonder if we have an "inverse Galois" type conjecture regarding which number fields can be generated in this way. Moreover, do we have some sort of statistical results/conjectures regarding the frequency of a numb... |
>If $a,b,c ≥ 0$ and $ab+bc+ca+2abc=1$, prove that $$a+b+c ≥ 3/2$$ and $$a^2 + b^2 + c^2 ≥ 1$$
I tried using the expansion of $(a+b+c)^2$ and $a^3+b^3+c^3$, but nothing helped.
Any hints$?$ |
Let $X \sim f_X(x)$ and $Y = g(X)$. If $g(x)$ is a differentiable, monotonic function with inverse such that $X = g^{-1}(Y)$ then the PDF of $Y$ can be described:
$$
f_Y(y) = f_X(g^{-1}(y)) \bigg| \dfrac{d}{dy}g^{-1}(y) \bigg|
$$
In the multidimensional case, I have seen a very similar expression where the abso... |
When stuying a 1D differential equation, I tried to solve the problem by finding the Green's function and then solving with the inverse Fourier transform.
To make the problem well-behaved, I used a regularisation paremter $\epsilon$. **My problem is that when $\epsilon\to 0$, the amplitude becomes constant whereas ... |
I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to tensor products of elements from a vector space $V$ and its extension. The operation $\otimes$ is understood as the tenso... |
I can see that the map $f(z) = z^k$ is an homeomorphism for neighbourhoods around non-zero points not including zero. So I need to check only around neighbourhoods of zero.
Can anyone help me out?
**Edit: As mentioned in the comments, I forgot to mention $k$ here is a positive integer.** |
I calculated $ L= lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers. $ \\ 1. ~ L = lim_{n \to \infty}[1 + (\frac {n-1}{n})^n + (\frac {n-2}{n})^n + \cdots + (\frac {n-n}{n})^n] \\ = 1+ \frac{1}{e} + \frac {1}{e^2} + \cdots = \frac {e}{e-1}. $
$\\ 2. ... |
Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$
My attempt:- We know that $D^\alpha u$ exists in the weak sense and $D^\alpha u \in L^p(U), \forall |\alpha|\leq k.$ ($\because$ By definition)
In order $D^\alpha \in W^{k-|\alpha|,p}, $ $D^{\beta}D^{\alpha}... |
I calculated $ L= \lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers.
$ 1. ~ L = \lim_{n \to \infty}[1 + (\frac {n-1}{n})^n + (\frac {n-2}{n})^n + \cdots + (\frac {n-n}{n})^n] \\ = 1+ \frac{1}{e} + \frac {1}{e^2} + \cdots = \frac {e}{e-1}. $
$... |
I calculated $ L= \lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers.
1. $$L = \lim_{n \to \infty}\left[1 + (\frac {n-1}{n})^n + (\frac {n-2}{n})^n + \cdots + (\frac {n-n}{n})^n\right] \\ = 1+ \frac{1}{e} + \frac {1}{e^2} + \cdots = \frac {e}{e-1}. $$... |
Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\lim_{n\to\infty} \mu(f^n(A))=1.$$
How to prove, as simply as possible, that $$\lim_{n\to \infty} \mu(A\cap f^{-n}(B))=\mu... |
>If $a,b,c ≥ 0$ and $ab+bc+ca+2abc=1$, prove that $$a+b+c ≥ 3/2$$ and $$a^2 + b + c^2 ≥ 1$$
I tried using the expansion of $(a+b+c)^2$ and $a^3+b^3+c^3$, but nothing helped.
Any hints$?$ |
I calculated $ L= \lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers.
1. $$L = \lim_{n \to \infty}\left[1 + (\frac {n-1}{n})^n + (\frac {n-2}{n})^n + \cdots + (\frac {n-n}{n})^n\right] \\ = 1+ \frac{1}{e} + \frac {1}{e^2} + \cdots = \frac {e}{e-1}. $$
... |
If for a ring $R$ a positive integer $n$ exists such that $n\cdot r = 0$ for all $r\in R$, then the least such positive integer is the characteristic of the ring $R$, denoted by ${\rm char}(R)$. If no such positive integer
exists, then $R$ is of characteristic $0$.
I have seen repeatedly in papers that the authors ... |
Why do we sometimes assume ${\rm char}(R)\neq 2$ for a ring $R$? |
Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\lim_{n\to\infty} \mu(f^n(A))=1.$$
How to prove, as simply as possible, that $$\lim_{n\to \infty} \mu(A\cap f^{-n}(B))=\mu... |
I want to find a 'nice' drawing of the lipids and genes in my database. Lipids belong to one one of several classes, while genes belong to one of several regions. Each gene/lipid pair has an associated strength value, which indicates how closely related they are.
I have tried to define my dataset formally as follows... |
Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$.
I was thinking that I could use Maximum Modulus Principle for function $g = \dfrac{1}{f(z)-z}$, but I got stuck at here (because I can't... |
I am on a closed and bounded surface $S$ without boundary.
Then it holds that
$$
\int_S \nabla f \cdot \nabla f \mathrm dA = -\int_S f\Delta f \mathrm dA.
$$
This is the $H^1(S)$ seminorm, meaning that I can express the $H^1(S)$ norm as
$$
\|f\|_1 = \int_S f(1-\Delta) f \mathrm dA.
$$
Imagine now I am... |
I calculated $ L= \lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers.
1. $$L
= \lim_{n \to \infty}\left[1 + (\frac {n-1}{n})^n + (\frac {n-2}{n})^n + \cdots + (\frac {n-n}{n})^n\right]
= 1+ \frac{1}{e} + \frac {1}{e^2} + \cdots = \frac {e}{e-1}.
$$... |
The first answer is incomplete because you need to justify the exchange of limit and sum
$$
\begin{align}
L &= \lim_{n\to\infty}\sum_{r=1}^n\frac{r^n}{n^n}\\
&\stackrel{?}{=} \sum_{r=1}^n\lim_{n\to\infty}\frac{r^n}{n^n}\\
&= \sum_{r=1}^n\lim_{n\to\infty}(1 - (n-r)/n)^n\\
&= \sum_{j=0}^{n-1}\lim_{n\to\infty}(1 -... |
The question is: \
Let $C$ be a plane curve and let $T$ be the tangent line at a point $p \in C$. Draw a line $L$ parallel to the normal line at $p$ and at distance $d$ of $p$ (Fig. 1-36). Let $h$ be the length of the segment determined on $L$ by $C$ and $T$ (thus, $h$ is the "height" of $C$ relative to $T$ ). Prove t... |
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