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If we define $f(x)= \sum_{m=0}^{\infty} x^{2^m} = \lim_{n\to\infty}f_n(x)= \lim_{n\to\infty}\sum_{m=0}^{n} x^{2^m}$.
It's easy to see that $|{f(x)}|$ converge in the interval $(-1,1)$. (We can easily born it with the geometric series)
Supose I want to look at the limit :
$$\lim_{x\to1^-}|f(x)|$$
**How could a ... |
In mathematics we do not consider questions like "are there objects that are XYZ?", which is a rather philosophical question. In mathematics we consider questions like "are there objects that are XYZ under ABC axioms?". Everything in maths is relative to some axiomatic theory.
In particular the answer to this questi... |
The question is this:
> Prove that the Galois group of the algebraic closure of a finite field is Abelian and the order of every element in the Galois group is infinite.
I have a trouble with the first part. How can I prove the Galois group is Abelian?
(Sorry, I'm very beginner in Galois theory.) |
These are the information about this question.
[![this is the first function][1]][1]
[![This is what phi is][2]][2]
[1]: https://i.stack.imgur.com/gaiEf.jpg
[2]: https://i.stack.imgur.com/QDjgr.jpg
In this case, f is a twice differentiable function and compact support in $\mathbb{R}^n$.
My question... |
I am teaching myself ai / machine learning without libraries
I understand most of the derivative of the softmax activation function
if I have 3 nodes in a layer
then the softmax activation equation becomes
e^x_i / sigma e^x_i
and its derivative returns a vector of partial derivatives that can be descr... |
How do you get from a vector of partial derivatives to a scalar? |
These are the information about this question.
[![this is the first function][1]][1]
[![This is what phi is][2]][2]
[1]: https://i.stack.imgur.com/gaiEf.jpg
[2]: https://i.stack.imgur.com/QDjgr.jpg
In this case, f is a twice differentiable function and compact support in $\mathbb{R}^n$.
My question... |
I am reading [this paper](https://www.sciencedirect.com/science/article/pii/S0167668799000268?ref=cra_js_challenge&fr=RR-1) of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me? |
You could use ordinary generating functions, and find the coefficient of $x^5$ in the expansion of
$\left[\binom81x+\binom82x^2+\binom83x^3\right]\left[\binom71x + \binom72x^2+\binom73x^3\right]+\left[\binom61x+\binom62x^2 + \binom63x^3\right]$
which [evaluates][2] to the expected $14140$
Note that we have re... |
In mathematics we do not consider questions like "are there objects that are XYZ?", which is a rather philosophical question. In mathematics we consider questions like "are there objects that are XYZ under ABC axioms?". Everything in maths is relative to some axiomatic theory.
In particular the answer to this questi... |
>Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $\mathrm{Aut}(F/E)$, and let $f$ be an element of $F$. Suppose that
(i) $[F:E]=3$,
(ii) $F=\mathbb{Q}(f)$,
(iii) $u(f)=f^2$.
Prove that $f$ is a primitive 7th root of 1, prove that $F/\mathbb{Q}$ is Galois, and descr... |
Are quasicomponents connected in a compact space?
Background:
* [Quasicomponents are connected in a compact Hausdorff space](https://math.stackexchange.com/q/600904/51161).
* [A non-compact locally compact Hausdorff space may not have connected quasicomponents](https://math.stackexchange.com/q/1884546/51161).
... |
Are quasicomponents connected in a compact non-Hausdorff space? |
I am trying to solve this problem but I'm a little stuck:
You and your friends decide to make a lottery with n tickets, where each ticket has a
number between 1 and n, and each ticket is unique. Each ticket is $5, and the lottery
works in the following way. Once all n tickets have been purchased, a number x is s... |
Let $ (x_k)_{k\in \mathbb{N} }$ be an infinite sequence of the real numbers. For $ n\in \mathbb{N} $, define the partial sum $ s_n$ as
$$ s_n = \sum_{k = 1}^n x_k $$
Then we have a new infinite sequence $ (s_n)_{n\in \mathbb{N} }$. Now, suppose this new sequence converges, i.e., there is a $ L \in \mathbb{R}$ suc... |
For a convergent series, is the limit of partial sum is always equal to the infinite sum of the series? |
The modular group $\mbox{SL}_2({\mathbb Z})=\left\lbrace\begin{pmatrix} a & b\\ c & d\end{pmatrix}: a,b,c,d \in {\mathbb Z}, ad-bc=1 \right\rbrace$ acts on the upper half plane ${\mathbb H}=\lbrace z\in {\mathbb C}: \mbox{Im}(z)>0\rbrace $ by $\begin{pmatrix} a & b\\ c & d\end{pmatrix}z=\dfrac{aż+b}{cz+d}$. For a fixed... |
>I need to find all fields $E$ such that $\mathbb{Q} \subset E \subset \mathbb{Q}(\zeta_{40})$ and $[E:\mathbb{Q}]=2$, where $\zeta_{40}$ is a primitive root of unity of order $40$.
I understand that we have a galois extension, and $Gal(\mathbb{Q}(\zeta_{40})/\mathbb Q) = (\mathbb{Z}/40 \mathbb{Z})^{\times} = \math... |
I get different answers when I try to evaluate the following integral when I use partial fraction decomposition and Cauchy's Integral Formula. The integral is,
$$\color{blue}{\int _{-2}^{2}\frac{1}{z^4+1}dz}.$$
I would greatly appreciate any help finding where I go wrong, and what possible misunderstandings I have ... |
Sometimes I see questions like this:
$$\frac{dy}{y}=2dx$$ where $y = 1$ when $x = 0$
My book tells me to do this:
$$\int_1^y\frac{dy}{y}=\int_0^x2dx$$
At first I thought this was a clever way of saying 0=0 (as an integral from $a$ to $a$ is always zero) to solve the problem.
Then I saw this question:
> One meth... |
Method to find a type of equation that can fit a particular shape? |
I have been reading a paper which uses without proof the following result:
>Let $n \in \mathbb N$ and $p$ be a prime divisor of $n$. For a positive integer $m$, by $\zeta_m$ we mean a primitive $m$-th root of unity, so that $\mathbb Q(\zeta_m)$ is just the $m$-th cyclotomic field. Then the Galois group $G:= \text{Ga... |
I'm trying to understand the Zarinski topology on $\text{Spec}(\mathbb{Z})$. I've just learned about this new concept and I wanted to compute this topology for a more concrete example to see how it looks but I dont know how to start.
For now I have that $\text{Spec}(\mathbb{Z})= \{(p); \text{p prime} \}\cup \{0\}$ ... |
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have
$$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}=\{(p) ; \text{p prime and }p|n \} = \{(p_1),...,(p_n)\}$$
Does that helps you visualizing what the... |
I am trying to solve this problem but I'm a little stuck:
You and your friends decide to make a lottery with $n$ tickets, where each ticket has a number between $1$ and $n$, and each ticket is unique. Each ticket is $\$5$, and the lottery works in the following way. Once all $n$ tickets have been purchased, a numb... |
Trouble with 1st order axiomatization of periods, are there reasons to avoid 2-sorted logic? |
Consider the following limit which hypothetically uniformly converges to some real analytic function $\Delta$:
$$\Delta_t(x)=\lim_{r\to \infty} \frac{1}{r} \sum_{n=1}^\infty e^{\frac{t\log n}{\log r \log x}}$$
I'm looking to evaluate this limit. Here $t$ is some real parameter $t>0$.
The motivation to find the... |
I'm trying to understand the Zariski topology on $\text{Spec}(\mathbb{Z})$. I've just learned about this new concept and I wanted to compute this topology for a more concrete example to see how it looks but I dont know how to start.
For now I have that $\text{Spec}(\mathbb{Z})= \{(p); \text{p prime} \}\cup \{0\}$ s... |
>I would like find the Galois group of
$$\mathbb{Q}(\sqrt[3]{3},i\sqrt{3},\zeta_{13})/ \mathbb{Q}.$$
That field extension is Galois because is the splitting field of separable polynomial $(x^3-3)(x^{13}-1)$. We know $[\mathbb{Q}(\sqrt[3]{3},i\sqrt{3}):\mathbb{Q}]=6$ and $[\mathbb{Q}(\zeta_{13}):\mathbb{Q}]=12$, so... |
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have
$$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}=\{(p) ; \text{p prime and }p|n \} = \{(p_1),...,(p_n),(0)\}$$
Does that helps you visualizing what... |
Let's assume we are faced with showing that the following covariance between two functions $g(\cdot)$ and $f(\cdot)$ of the same set of $Z_1,...,Z_K$ i.i.d. standard normal random variables is equal to $0$ (we want to show that they are independent), such that:
$$Cov\left[f(Z_k),g(Z_k)\right]= Cov\left[\sqrt{\hat{\r... |
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have
$$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}\cup \{0\}=\{(p) ; \text{p prime and }p|n \}\cup \{0\} = \{(p_1),...,(p_n)\}\cup \{0\}$$
Does that ... |
Are quasicomponents connected in a compact space?
Background:
* [Quasicomponents are connected in a compact Hausdorff space](https://math.stackexchange.com/q/600904/51161).
* [A non-compact locally compact Hausdorff space may not have connected quasicomponents](https://math.stackexchange.com/q/1884546/51161).
*... |
Consider (finite-dim) Hilbert spaces $\mathcal{H},\mathcal{K}$ and let $A$ denote an operator on $\mathcal{H}\otimes \mathcal{K}$. There are then two ways of defining the ``entanglement" of operator $A$.
1. Hilbert-Schmidt operators satisfy $L^2 (\mathcal{H}\otimes \mathcal{K}) \cong L^2 (\mathcal{H})\otimes L^2 (\... |
To understand the conversion without standard form I read: https://math.stackexchange.com/questions/249413/directly-from-primal-to-dual-when-primal-not-in-standard-form
The posted LP there is:
$$
Primal =\begin{Bmatrix}
max \ \ \ \ 2x_1 - 2x_2 +x_3 + 4x_4 \\
s.t. \ \ \ \ x_1 - x_2 - x_3... |
How to convert a primal LP into the dual LP without usage of the standard form? |
Are quasicomponents connected in a compact space?
Background:
* [Quasicomponents are connected in a compact Hausdorff space](https://math.stackexchange.com/q/600904/51161).
* [A non-compact locally compact Hausdorff space may not have connected quasicomponents](https://math.stackexchange.com/q/1884546/51161).
*... |
In [this quetion][1], it has been proved that an operator is semisimple iff it's diagonalizable over the algebraic closure of the base field.
However, when the minimal polynomial is not separable, it seems that there's something going wrong. Consider operator on $\mathbb F_2(t)$ to be
$$
T=\begin{bmatrix}
&&1&\... |
In the article linked below, no paywall, their Figure 4 shows a Fourier transform of a Kaiser window with Alpha=8. In it, the attenuation is shown to be As ~ 30db.
Below the figure, their Equation 20 is as follows:
Alpha = 0.1102(As − 8.7) , As > 50
Alpha = 0.5842(As − 21)0.4 + 0.07886(As − 21), 21 < As < 50
... |
In courses on general relativity, I was taught to always track the relative order of top and bottom (contravariant and covariant) tensorial indices. For instance, the first index of the Riemann tensor
$$ R^{\mu}_{ \ \ \nu \alpha \beta} $$
can be moved down using the metric
$$ R_{\mu \nu \alpha \beta} = g_{\... |
Why in mathematical texts the relative order of the top and the bottom tensorial indices is rarely considered? |
In the article linked below, no paywall, their Figure 4 shows a Fourier transform of a Kaiser window with Alpha=8. In it, the attenuation is shown to be As ~ 30db.
Below the figure, their Equation 20 is as follows:
Alpha = 0.1102(As − 8.7) , As > 50
Alpha = 0.5842(As − 21)0.4 + 0.07886(As − 21), 21 < As < 50
... |
To understand the conversion without standard form I read: https://math.stackexchange.com/questions/249413/directly-from-primal-to-dual-when-primal-not-in-standard-form
The posted LP there is:
$$
Primal =\begin{Bmatrix}
max \ \ \ \ 2x_1 - 2x_2 +x_3 + 4x_4 \\
s.t. \ \ \ \ x_1 - x_2 - x_3... |
I was reading the paper ['A Proposal on Machine Learning via Dynamical Systems'][1], where I came across the following steps:
Consider a system-\
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the solution is of the form $z(t,x)$. \
Then, the following is considered:
> To compute $\frac{δz(T,x)}{δA(t)}$ , c... |
Consider the following limit which hypothetically uniformly converges to some real analytic function $\Delta$:
$$\Delta_t(x)=\lim_{r\to \infty} \frac{1}{r} \sum_{n=1}^\infty \exp\left({\frac{t\log n}{\log r \log x}}\right)$$
I'm looking to evaluate this limit. Here $t > 0$ is some real parameter.
The motivatio... |
Im working with Newton method, and during I was trying to see why we got this expression, I fall into a problem, I find the same expression but instead of Hessian matrix I find his transpose. I can't figure where Im wrong D:
Here how I found that :
__Quick note of Newton's Method to aproximate zero of a function ... |
Newton's methods, do we use Hessian matrix or its transpose? |
Eight tennis players wish to play three doubles matches. Players must have different partners for each match and at the end of the three matches each player must have played in a match involving each of the other seven players at least once. Can this be arranged? |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then two vertices in $A$ are... |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then two vertices in $A$ are... |
A very basic question but given an $R$-module $M$, if $S \subseteq M$ generates $M$ freely, must $S$ necessarily be finite or can it be infinite? |
I have discovered a pretty weird sequence
$$a_0=1$$
$$a_1= 2$$
$$a_n=\frac{a_{n-1}+a_{n-2}}{a_{n-1}-a_{n-2}}$$
The first few terms are:
$$1,\ 2,\ 3,\ 5,\,\ 4,\ -9,\ \frac{5}{13},\ -\frac{56}{61},\ \frac{423}{1033},...$$
The sequence is very unpredictable and looks very random. I wonder if there is a closed-fo... |
>Let $f: \mathbb{C} \to \mathbb{C} $ be an holomorphic function.
>Furthermore, suppose that:
\begin{align*}
\exists \ (a,b) \in \mathbb{R}^2 \setminus \{(0,0)\}:a\mathit{Re}(f)+b\mathit{Im}(f) \ \text{is constant}
\end{align*}
Show that f is constant
By the hypothesis above, we can arrive at the following line... |
First, let me clarify the relationship between transfinite *induction* and *recursion*. Transfinite induction is a proof technique. Transfinite recursion, on the other hand, is a construction technique. You use transfinite recursion to build some mathematical object (usually but not always a function), and you use tran... |
>Let $f: \mathbb{C} \to \mathbb{C} $ be an holomorphic function.
>Furthermore, suppose that:
\begin{align*}
\exists \ (a,b) \in \mathbb{R}^2 \setminus \{(0,0)\}:\quad a\operatorname{Re}(f) + b\operatorname{Im}(f) \text{ is constant}.
\end{align*}
Show that $f$ is constant.
By the hypothesis above, we can arriv... |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then at least two vertices i... |
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have
$$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}\cup \{0\}=\{(p) ; \text{p prime and }p \mid n \}\cup \{0\} = \{(p_1),...,(p_n)\}\cup \{0\}$$
Does ... |
I have kind of an interesting and fun question I encountered today and I'm unsure how the derive the answer.
So let's say I have a mystery box that's worth $120 for 10 items. You are able to get repeats and each item is independent of the next.
There are 9 common items, 7 rare items, and 4 epic items in total.
... |
There is a combinatorial way of doing this, for those who know basic graph theory, that uses far less number theory as well. Indeed, we need just the following:
**Lemma 1:** *Let $C$ be a cycle and let $A$ be a subset of the vertices of $C$ s.t. $A$ is more than half the vertices of $C$. Then at least two vertices i... |
**The following is an excercise where we try to prove Lemma 5.3.1. Oksendal "Stochastic Differential Equations"**
Suppose that $b(t,x)$ and $\sigma(t,x)$ satisfy the hypotheses of the existence/(strong) uniqueness theorem. Suppose that $X_t,\hat{X}_t$ are adapted processes, both of which are continuous in mean squar... |
Use the fact that: for $x\in\mathbb R$ [(cosx)][1],
we have
$$1-\frac{x^2}{2}\le \cos(x)\le1-\frac{x^2}{2}+\frac{x^4}{24},$$
so
$$0\leq\cos(x)-1+\frac{x^2}{2}\leq\frac{x^4}{24},\quad\forall x\in\mathbb R.$$
Hence
$$0\leq\frac{\cos x-1+x^2/2}{x^2+y^2}\leq\frac{x^2}{24}\to0,\quad (x,y)\to(0,0).$$
[1]: https... |
Let $ (x_k)_{k\in \mathbb{N} }$ be an infinite sequence of the real numbers. For $ n\in \mathbb{N} $, define the partial sum $ s_n$ as
$$ s_n = \sum_{k = 1}^n x_k $$
Then we have a new infinite sequence $ (s_n)_{n\in \mathbb{N} }$. Now, suppose this new sequence converges, i.e., there is a $ L \in \mathbb{R}$ suc... |
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\cap P_k$. Given $\varphi \in C^\infty_0(\Omega)$, how to prove directly that there exist a positive constant $C$ such that
... |
Let $X_1, \dots, X_n$ be i.i.d. random variables from a categorical distribution with parameter vector $p = (p_1, \dots, p_k)$.
Suppose we’re interested in the estimating $\vartheta := f(p)$ for some function $f : \mathbb R^k \to \mathbb R$, e.g. the true Gini-Simpson index of this distribution $\vartheta = f(p) = 1... |
Consider an operator $\eta$ on the (tensor) product of Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_2$ such that its range is of the form $V_1\otimes V_2$. Does this imply that $\eta$ must be a product operator, i.e., of the form $\eta = A\otimes B$? |
HINT
First, get some precise definitions going, so you can refer to those in your proof.
So, relative to any expression or function $\phi$ built up from $\neg$, $\lor$, and $\land$, define the expression $\phi^D$ to be the result of changing all $\land$'s into $\lor$'s and vice versa. Formally and recursively:
... |
From an interview book, where the answer is not so clear I believe.
You keep generating $\mathcal U_{[0,1]}$ iid random variables until their sum exceeds 1, then compute the expected value of the last random variable, i.e. the one responsible for letting the sum of rvs overflow 1.
My idea (not working):
The $i$-... |
From an interview book, where the answer is not so clear I believe.
You keep generating $\mathcal U_{[0,1]}$ iid random variables until their sum exceeds 1, then compute the expected value of the last random variable, i.e. the one responsible for letting the sum of rvs overflow 1.
My idea (not working):
The $i$-... |
[![Demostration of what I'm talking about][1]][1]
Given a sigmoid equation `L / ( 1 + e ^ (-kx - m))`
How can I determine "k" and "m" given inputs "x1" and "x2" (see picture)?
Where "x1" corresponds to the start of the "sigmoid curve" section, and "x1" corresponds to its end.
I'm just looking for approxima... |
In response to the question "Can we prove [the '101'] pattern holds?", here is a visual proof in Wolfram Language. The triangular numbers can be defined as the sequence of (*duplicate*) successive odd numbers, where each odd number is scaled by $n$ or $n + 1$ depending on its parity in the sequence. These are listed in... |
I looked today again on the question, have also considered from the beginning the same (periodic) series approach, which is just an other way to write the operation $\star$, and saw no constructive progress. However, it may be of interest to have also this way to look at the problems, so here is an attempt of an answer... |
>**Mission**: To find a minimum-length walk from "base" to a point _g_ distance units away and back for 1 person alone, given:
>
>1. An unlimited supply of rations provided at base, to be consumed continuously while walking.
>2. The ability to walk up to 1 distance unit per unit of rations consumed.
>3. The ability... |
Is this proof vigorous enough? (Feel free to provide any suggestions for improvement as well).
(A centre of a connected graph $G$ is a vertex $v$ with the property that the maximum of the distances between $v$ and all other vertices of $G$ is as small as possible.)
Note: I will not prove the tree properties liste... |
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have
$$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}=\{(p) ; \text{p prime and }p \mid n \} = \{(p_1),...,(p_n)\}$$
Does that helps you visualizing wha... |
Given two matrices $A,B \in M_n(\mathbb{C})$, is there any way of telling if there exists a single $P \in M_n(\mathbb{C})$ such that $PAP^{-1}$ and $PBP^{-1}$ are conjugate to some integer matrix?
This [post][1] describes the necessary and sufficient conditions for any single matrix to be conjugate to an integer mat... |
Choosing the lattice $L=[1, \sqrt{-2}]$, what is the value of $\wp(0)$?
The definition of the Weistrass $\wp$ function for the lattice $L$ is
$$
\wp(z;L) = \frac{1}{z^2} + \sum_{\omega \in L - \{ 0 \}} \bigg(\frac{1}{(z - \omega)^2} - \frac{1}{\omega}\bigg) \tag{w} \label{w}
.$$
#### My attempts
I tried to ... |
Let's say $X \sim \mathcal{N}(\mu, \Sigma)$, where $\mu \in \mathbb{R}^{3}$. If I have two half spaces: $H_{1} = \{x: a^{T}x \geq 0\}, H_{2} = \{x: b^{T}x \geq 0\}$, is there a way I can compute the total probability mass in the area $H_{1} \cap H_{2}$? |
Can I analytically integrate a high dimensional Normal distribution? |
Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous.
**Attempt:**
Note
$$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$
As $C_c(\Bbb{R}) \subset L^1(\Bbb{R})$ is dense, there exists $g_n \in C_c(\Bbb{R}$) such that
$$\... |
I deal with a lot of multi-sigma functions, which I assign compact formulas through a step-wise "pull-back". It is best explained through a well-known example:
$$\begin{align} \sum_{i_1=1}^n \sum_{i_2=1}^{i_1}\cdots \sum_{i_k=1}^{i_{k-1}} 1 &= \sum_{i_1=1}^n \sum_{i_2=1}^{i_1}\cdots \sum_{i_{k-1}=1}^{i_{k-2}} i_{k... |
I'm currently reading Conway and Sloane's *Sphere Packings, Lattices and Groups*. It is a great read for facts, but they only sketch proofs at a high level.
Within it, they define the lattice $D_n$ to be the set of all points with even coordinates. Another similar collection of points is
$$D_n^+ = D_n \cup \left\{... |
Choosing the lattice $L=[1, \sqrt{-2}]$, what is the value of $\wp(0)$?
The definition of the Weierstrass $\wp$ function for the lattice $L$ is
$$
\wp(z;L) = \frac{1}{z^2} + \sum_{\omega \in L - \{ 0 \}} \bigg(\frac{1}{(z - \omega)^2} - \frac{1}{\omega}\bigg) \tag{w} \label{w}
.$$
#### My attempts
I tried t... |
Given two matrices $A,B \in M_n(\mathbb{C})$, is there any way of telling if there exists a single $P \in M_n(\mathbb{C})$ such that $PAP^{-1}$ and $PBP^{-1}$ are integer matrices?
This [post][1] describes the necessary and sufficient conditions for any single matrix to be conjugate to an integer matrix, so I would ... |
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\cap P_k$. Given $\varphi \in C^\infty_0(\Omega)$, how to prove directly that there exist a positive constant $C$ such that
... |
Assume a heavy chain (constant mass per unit length) takes the shape of a plane curve $\mathcal C$ after being suspended by its two ends from the same height. Let $s$ be its arc length starting from its lowest point, let $T$ be the tension in the chain, let $\varphi$ be the angle between the tangent line and the horizo... |
In response to the question "Can we prove [the '101'] pattern holds?", here is a visual proof in Wolfram Language. The triangular numbers can be defined as the sequence of (*duplicate*) successive odd numbers, where each odd number is scaled by $n$ or $n + 1$ depending on its parity in the sequence. These are listed in... |
This kind of result is often most easily proved by ‘element-chasing’: let $x$ be any element of the set on the lefthand side, and show that $x$ must be an element of the set on the righthand side. (If you were proving an equality, instead of an inclusion, you’d then repeat the process in the other direction.)
So sup... |
The Hodge dual (or formal adjoint) to the exterior derivative $d: \Omega^k(M) \to \Omega^{k+1}(M)$ on a smooth manifold $M$ is the codifferential $ d^* $, a linear map $$ d^*: \Omega^k(M) \to \Omega^{k-1}(M),$$ defined by
$ d^* = (-1)^{n(k+1)+1} * d * $
Could someone kindly provide a detailed explanation of... |
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\cap P_k$. Given $\varphi \in C^\infty_0(\Omega)$, how to prove directly that there exist a positive constant $C$ such that
... |
I have this question: P → ¬¬Q ⊢ ¬Q → ¬P
I have these steps for proof:
1. (1) P → ¬¬Q A
2. (2) ¬Q A
1,2. (3) ¬P 1,2 Modus Tollens
1. (4) ¬Q → ¬P 2,3 CP (conditional proof)
Is this correct or do I need to do a double negation at some point?
The only applicable rules we're allowed to use are mo... |
I have this question: P → ¬¬Q ⊢ ¬Q → ¬P
I have these steps for proof:
1. P → ¬¬Q A
2. ¬Q A
1,2. ¬P 1,2 Modus Tollens
1. ¬Q → ¬P 2,3 CP (conditional proof)
Is this correct or do I need to do a double negation at some point?
The only applicable rules we're allowed to use are modus tollens, mod... |
I have this question: P → ¬¬Q ⊢ ¬Q → ¬P
I have these steps for proof:
1. P → ¬¬Q
2. ¬Q
1,2. ¬P Modus Tollens
1. ¬Q → ¬P CP (conditional proof)
Is this correct or do I need to do a double negation at some point?
The only applicable rules we're allowed to use are modus tollens, modus ... |
I have this question: P → ¬¬Q ⊢ ¬Q → ¬P
I have these steps for proof:
1. (1) P → ¬¬Q Premise
2. (2) ¬Q Premise
1,2. (3) ¬P (1,2 Modus Tollens)
1. (4) ¬Q → ¬P (2,3 CP)
Is this correct or do I need to do a double negation at some point?
The only applicable rules ... |
I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about [partitions](http://en.wikipedia.org/wiki/Partition_%28number_theory%29)).
The asymptotic formula always seems to be written as,
$$ p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi \sqrt{\frac{2n}{3}}}, $$
however I nee... |
Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous.
**Attempt:**
Note
$$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$
As $C_c(\Bbb{R}) \subset L^1(\Bbb{R})$ is dense, there exists $g_n \in C_c(\Bbb{R}$) such that
$$\... |
As the title says.
Let's consider some category $C$ and the contravariant functor $Sub : C^{op} \rightarrow Set$ that for each object gives its set of subobjects.
The introductory book by Leinster says that by the Yoneda lemma, $Sub$ being representable amounts to there being an object $\Omega$ and an element $t:... |
Choosing the lattice $L=[1, \sqrt{-2}]$, what is the value of $\wp(0)$?
The definition of the Weierstrass $\wp$ function for the lattice $L$ is
$$
\wp(z;L) = \frac{1}{z^2} + \sum_{\omega \in L - \{ 0 \}} \bigg(\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}\bigg) \tag{w} \label{w}
.$$
#### My attempts
I tried... |
Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\cap P_k$. Given $\varphi \in C^\infty_0(\Omega)$, how to prove directly that there exist a positive constant $C$ such that
... |
In order to set up the apparatus for some work I'm doing, I'd like to construct a field with certain properties. I'm hoping this is a standard construction, though I've been unable to find it in the literature, and I hope someone can just point me to a good reference for it.
For the complex numbers $\mathbb{C}$ "al... |
I'm currently working on a project where my objective is to sample from a target distribution $p(x)$ using Markov Chain Monte Carlo (MCMC) techniques. The primary tool at my disposal is an unconditional proposal distribution $q(x)$ that is easy to draw independent samples from. The standard approach I'm considering inv... |
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