instruction stringlengths 12 30k |
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Another partial answer based on numerical evidence. It is easy to note that your numbers can be written as
$$
\frac{a_n}{4^n}
$$
where
$$
a_n=
1,5,22,91,370,1491,5986,23986,96028,384277,1537438,6150406,\dots\tag{*}
$$
After playing a bit with the number one finds:
$$
a_1=1;\quad a_n=4a_{n-1}+\xi_n,\tag1
$... |
I am trying to understand Discrete Fourier Transform (DFT), after only having experience with the continuous transformation. The natural idea was to try to understand DFT on the simplest function, being an exponential, which should transform into an exponential too. To check this in practice, I've drafted up a Python s... |
Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules:
$$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$
I claim that we have a split exact sequence:
$$\{0\}\to\varprojlim_{n\in\Bbb Z}E_n\to\varinjlim_{n\in\Bbb Z}E_n\to\bigoplus_{n\in\B... |
Let $E$ be the splitting field for $\{x^2-p: p>0, \mbox{ a prime }\}$ over $\mathbb{Q}$. The Galois group of $E$ over $\mathbb{Q}$ is an uncountable group and this is one counterexample for failure of bijection between intermediate fields $\mathbb{Q}\subset L \subset E$ and subgroups of $\mbox{Gal}(E/\mathbb{Q})$.
... |
Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules:
$$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$
I claim that we have a split exact sequence:
$$\{0\}\to\varprojlim_{n\in\Bbb Z}E_n\to\varinjlim_{n\in\Bbb Z}E_n\to\bigoplus_{n\in\B... |
I came across a problem related to the denseness of $a\mathbb{Z}+\mathbb{Z}$. Explicitly, the problem asks:
> Let $a$ be irrational.
(a) For any $y \in [0,1]$, prove that there are infinitely many $n \in \mathbb{N}$ so that $\{na\} \in [y-\epsilon, y+\epsilon]$.
>
> (b) Prove that there exists $b \in [0,1]$ such ... |
In Section 6.4, page 116, Shoenfield defines
$$
\beta(a,i)=\mu x_{x<a\stackrel{\cdot}{-}1}\exists y_{y<a}\exists z_{z<a} (a=\text{OP}(y,z)\wedge\text{Div}(y,1+(\text{OP}(x,i)+1)\cdot z))
$$
with $\text{OP}(u,v):=(u+v)\cdot (u+v)+u+1$ and $\text{Div}(u,v):\longleftrightarrow \exists t(tv=u)$.
Furthermore, $\... |
It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that "if A, then B" can be understood as "A *coupled* with B" when it means $\mu(x,y) = T_{norm} (\mu_{A}(x), \mu_{B}(y))$, ins... |
I'm trying to prove the following statement:\
If $\mathcal{F}$ is an algebra, then it is a $\sigma$-algebra if and only if $\{A_n\} \subset \mathcal{F}$ with $A_i \cap A_j = \emptyset$ for all $i \neq j$ implies $\cup_{n=1}^\infty A_n \in \mathcal{F}$.
I did the following proof but i donno if it makes sense: \
**P... |
Is this proof correct (measure theory)? |
We denote the $(A|B):=-{\rm tr}(AB)$ as the inner product of $M_n(\mathbb{K})$, which is easy to prove by the definition. I'm stuck by the problem when I read the Lie algebra of matrix group, finding it's a calculation problem of matrix .
> $A\in{{\rm SU}(2)}$
So $A$ can be written as
> $$A=\left(\begin{array}{... |
How to prove $SUS=\frac{(S|S)}{2}U$ with $S$ and $U$ are skew-hermitian with following conditions? |
In Nakahara (Theorem 7.4, Pg 291) we have for an $r$-form on an $m$-dimensional manifold $$ * *\omega = (-1)^{r(m-r)}\omega$$ when Riemannian and $$ * *\omega = (-1)^{1+r(m-r)}\omega$$ when Lorentzian. So, for $m = 1$, it is possible to have $$ * * = -1$$ for a $0$-form in $1D$ Lorentizan manifold. Hence, my question. ... |
Too long for another comment.
Your metric can be written as
\begin{align}
g_{11}&=g_{22}=\frac{4}{(1+x^2+y^2)^2}\,, &g_{12}&=g_{21}=0\,.
\end{align}
I verified with a symbolic calculator that in this case
the Ricci tensor $R_{ij}$ is indeed equal to $g_{ij}$
and the scalar curvature is two like we get on the ... |
Let $S = \{S_1,\ldots,S_N\}$ be the number of states of a hidden markov process, and let $V = \{V_1,\ldots,V_N\}$ be the alphabet of possible observations one can make.
Given a model $\lambda = (\pi,A,B)$ for a *hidden markov process*, where $\pi$ is the initial distribution, $A$ is the transition-matrix and $B$ is... |
Does there exist a positive integer value of $n$ such that the sum of $1!^n+2!^n+3!^n+\cdots+2024!^n$ is a perfect power?
I know that $n\neq1$, as the last digit of the sum when $n\geq4$ is $3$ and no perfect square ends with $3$, and it cannot be a perfect power since $3^3$ does not divide the sum when $n\geq8$.
... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
> Find the solution to the given derivative of the product
> $$\big(\cos{x}\delta{x}\big)^{(k)}$$
We need to use the differentiation formula for generalized functions:
$$\big(D^\alpha f, \phi)=(-1)^{|\alpha|}(f,D^\alpha\phi)$$
However,I think we need to use also the product formula for generalized function... |
Recently, I have tried to resolve the following problem.
Let periodic $f$ be defined by the following Fourier series,
\begin{align}
f(x) = \sum_{k=1}^\infty \frac{1}{k^{3/2}} e^{ikx}.
\end{align}
Prove that $f$ belongs to Nikol'skii space $H^1_2(-\pi,\pi)$ but
\begin{align}
\frac{1}{2\pi}\int_{-\pi}^\pi \vert ... |
> Find the solution to the given derivative of the product
> $$\big(\cos{x}\delta{x}\big)^{(k)}$$
We need to use the differentiation formula for generalized functions:
$$\big(D^\alpha f, \phi)=(-1)^{|\alpha|}(f,D^\alpha\phi)$$
with $a(x)=cos{x}$, $f=\delta(x)$, and $\alpha=k $
$$\big(D^k(\cos{x},\delta(... |
Let $S = \{S_1,\ldots,S_N\}$ be the number of states of a hidden markov process, and let $V = \{V_1,\ldots,V_N\}$ be the alphabet of possible observations one can make.
Given a model $\lambda = (\pi,A,B)$ for a *hidden markov process*, where $\pi$ is the initial distribution, $A$ is the transition-matrix and $B$ is... |
So, there is apparently a result by Łojasiewicz (in some lecture notes of his: "Ensembles semi-analytiques", published in 1965) that states the following:
> Let $X \subset M$ be a subanalytic subset of an analytic manifold. Then, $X$ is semianalytic of dimension $\leq k$ if and only if **locally** in $M$, there is ... |
Does there exist a positive integer value of $n$ such that the sum of $1!^n+2!^n+3!^n+\cdots+2024!^n$ is a perfect power?
I know that $n\neq1$, as the last digit of the sum when $n\geq4$ is $3$ and no perfect square ends with $3$, and it cannot be a perfect power since $3^3$ does not divide the sum when $n\geq8$.
... |
> Let $A \in \mathbb R^{m \times n }, b \in \mathbb R^m$ and $0 \neq c \in \mathbb R^m$. Prove: Either the system $Ax = c$ or the system $A^T y = 0, c^T y = 1$ has a solution.
Looks a bit like Farkas Lemma but the $c^T y = 1$ confuses me. Can someone give me a hint? |
$$\sum_{n=0}^{\infty}(-1)^n\binom{2n}{n}^5\left(\frac{1+4n}{2^{10n}}\right)=\frac{\Gamma^4(1/4)}{2\pi^4}$$
Is there any way to prove this?
I don't even know where to start with this one.
The following Generating Function is known.
With $K$ as the Complete Elliptic Integral of the First Kind.
$$K^2\left(\... |
I have conjectured that for all functions $$ g(t) = \sum_{n=0}^{\infty}\frac{a_n}{n!}t^n, $$ it holds that $$ \int_0^{\infty}\frac{(g(tx)-g(0))^j}{j!}e^{-t} dt = \frac{1}{j!a_0a_1...a_j}\prod_{r=1}^j(\int_0^{\infty}g(rt_rx)e^{-t_r}dt_r - g(0)), $$
provided that the product of coefficients only includes the non-negativ... |
Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain and $n>1$ and let $T_p\colon W^{1,p}(\Omega)\to L^p(\partial\Omega)$ denote the trace theorem for $p\in[1,\infty]$.
Suppose, we are given $u\in W^{1,2}(\Omega)\cap L^q(\Omega)$ for some $q\in(2,\infty]$. Of course, we have $T_2u\in L^2(\partial\Omega)$ with... |
Is it possible to exploit higher integrability in the trace theorem? |
We denote the $(A|B):=-{\rm tr}(AB)$ as the inner product of the skew-hermitian matrix, which is easy to prove by the definition. I'm stuck by the problem when I read the Lie algebra of matrix group, finding it's a calculation problem of matrix .
> $A\in{{\rm SU}(2)}$
So $A$ can be written as
> $$A=\left(\begin... |
Suppose we have a pairwise distinct vector of reals $[x_1,x_2,\ldots,x_n]$ and for any such vector, we create a matrix $A^n$ such that
$$a_{ij} = \frac{x_i}{x_i-x_j}$$ where $i \neq j$ and $$a_{ii} = \sum_{j\neq i} \frac{x_i}{x_i-x_j} $$
What can be said about the set of eigenvalues and eigenvectors of any $A... |
Does this iterative process always reach $1$? |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
I read Poincaré's paper called [Analysis Situs](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf). And here's the thing about chain complex.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equal t... |
I am currently working on problem 76 (section 11.2) in Stewart's Calculus textbook.
It asks what the area of a series of inscribed circles in an equilateral triangle with side lengths of 1 is. The circles are said to tend towards the vertices of the triangle, so I believe the center of the first circle is the median... |
Geometric series(?) problem with circle center at the median of a triangle |
Historical data show that approximately 70 000 vehicles cross Älvsborgsbron every day. What is the probability that more than 7 500 000 vehicles cross Älvsborgsbron in 100 days? For full points you must motivate and explain all assumptions and approximations. |
Assuming the points to be $a_{1},a_{2}\cdots,a_{7}$ we can put them into the vertices and the centre of a regular hexagon, where we can find $12$ distances which are equal and the remaining $9$ distances are non equal. But how can I show that this is the maximum number and it cannot exceed $12$? |
Geometric series(?) problem with circle center at the centroid of a triangle |
For an example of $2$, you took the usual metric on $d_2$ and made is so that a sequence $x_n$ which is going to $\infty$ becomes Cauchy by shrinking the distances. This is exactly the right idea. The function $f:(-1,1)\to \mathbb{R}$ given by $$f(x)=\frac{x}{1-|x|}$$ also works, and has inverse $$f^{-1}(x)=\frac{x}{1+... |
I am reading Wong's book on "Asymptotic Approximations of Integrals". On page 497, the book recalls (without proof) the following estimate: for all $\delta>0$ and $\nu>1$,
$$
\int_\delta^\infty t^m e^{-\nu t^2} dt \le K_\delta e^{-\nu \delta^2},
$$
where $K_\delta$ is a constant independent of $\nu$.
May I kno... |
What conclusions can be drawn about the ideal generated by these polynomials? |
> Let $$X = \{ x \in \mathbb R^n: 1/2 \leq x_1 \leq 1 \text{ and } x_{i-1} \leq x_i \leq 2x_{i-1}\forall i=2,\ldots,n \}.$$ Prove that $X$ is a polyhedron and determine all of its extreme points.
**My attempt:**
First, we'll get $X$ in the form $X = P(A, b) = \{ x \in \mathbb R^n \mid Ax \leq b\}$. In order to do... |
I’m studying representation theory of semisimple Lie algebras through highest weights. We have seen that finite dimensional irreducible representations are in 1 to 1 correspondence with dominant weights.
In the case of $sl(2;\mathbb{C})$ we have only one simple root $\alpha_1$, with fundamental dominant weight $w_1=\... |
I am currently working on a project where I model the motion of a snowboarder using a 4th-degree polynomial. The polynomial is currently defined in abstract units along the x-axis and y-axis. I have encountered a challenge when attempting to convert these abstract units into real-world time (seconds on x-axis and meter... |
> Is it possible to apply "standard symmetrization and density
> arguments" in a compact Riemannian manifold to prove something at
> first in a smaller subdomain of it?
So suppose we have to show finiteness of some integral, say ($f:\mathbb{R}^n\to\mathbb{R}$ is some arbitrary function)$$\int_\Omega f(x)\,dx$$ ove... |
Consider this particular instance for the $0/0$ case of the Stolz-Cesaro Theorem: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0$$ with $a_n$ and $b_n$ two sequences of real numbers, $b_n$ strictly monotone, whereby $$\lim_{n \to \infty} \frac{a_{n+1} - a_{n}}{b_{n+1} - b_{n}} = 0$$ then also $$\lim_{n \to \i... |
Calculate the spectral radius of
$\begin{equation}A=
\begin{pmatrix}
1 &1\\ 2 &1
\end{pmatrix}
\end{equation}
$
My understanding is spectral radius is maximum modulus of the largest eigenvalue of $A$
eigenvalues are $1+\sqrt2$ and $1-\sqrt2$
So, $\rho(A)=1+\sqrt2$
But in one research article $\rho... |
I failed this question at an exam and would like help in explaining how to solve it: Historical data show that approximately 70 000 vehicles cross Älvsborgsbron every day. What is the probability that more than 7 500 000 vehicles cross Älvsborgsbron in 100 days? For full points you must motivate and explain all assumpt... |
Let's say we have $n$ urns (numbered $1$ through $n$) and the first $k$ urns have a ball in them (for some $k$ unknown to us) and the remaining urns are empty. Our goal is to determine $k$ by looking into as few urns as possible.
One method would be to use a binary search - look into the urn with index $\left\lfloo... |
Is it possible to find something better than binary search for this problem? |
Consider the following $3\times3$ real matrix:
$$
A=\begin{pmatrix}
1-a_1-a_2 & b_1 & c_1 \\
a_1 & 1-b_1-b_2 & c_2\\
a_2 & b_2 & 1-c_1-c_2\\
\end{pmatrix}
$$
Note that
$$
I_3-A=
\begin{pmatrix}
a_1+a_2 & -b_1 & -c_1\\
-a_1 & b_1+b_2 & -c_2\\
-a_2 & -b_2 & c_1+c_2\\
\... |
This question is possibly related to [my previous question][1] and it's answer.
Suppose $a_n>0\ \forall\ n\in\mathbb{N}\ $ and $\displaystyle\sum_{n=1}^{\infty} a_n = a.$
Does there exist a subsequence $(a_{k_n})_{n\in\mathbb{N}}$ of $(a_n)_{n\in\mathbb{N}}$ such that the sequence defined by: $b_1 = a_{k_1}=a_1;\... |
Does every positive real sequence whose series converges to $a$ have a "straight line" convex subsequence that also converges to $a?$ |
Suppose we have a pairwise distinct vector of reals $[x_1,x_2,\ldots,x_n]$ and for any such vector, we create a matrix $A_n$ such that
$$a_{ij} = \frac{x_i}{x_i-x_j}$$ where $i \neq j$ and $$a_{ii} = \sum_{j\neq i} \frac{x_i}{x_i-x_j} $$
What can be said about the set of eigenvalues and eigenvectors of any $A... |
This question is possibly related to [my previous question][1] and it's answer.
Suppose $a_n>0\ \forall\ n\in\mathbb{N}\ $ and $\displaystyle\sum_{n=1}^{\infty} a_n = a.$
Does there exist a subsequence $(a_{k_n})_{n\in\mathbb{N}}$ of $(a_n)_{n\in\mathbb{N}}$ such that the sequence defined by:
$b_1 = a_{k_1}=... |
Hi everyone I'm doing an exercise of logic from the Science of programming book of David Gries, one of the exercises says:
> Prove that from $\neg q$ follows q $\Rightarrow (p \land $ $\neg p)$
I proved that in the following way:
From $\neg q$ infer $q \Rightarrow (p \land $ $\neg p)$
1. $\neg q$ Pr... |
Let's consider a quantity denoted by $a$. This could represent various measures such as an area (e.g., the area under a curve).
Initially, we estimate the value of $a$. As we refine our approximation, **we find that we can make it arbitrarily close to a number $\boldsymbol\ell$.** This suggests that $a$ equals $\ell... |
Is the argument for convergence of an approximated quantity to its true value sound? |
Let $U$ be the universe of a multiple choice questionnaire. We can say $|U| = 1$. We have $k \geq 3$ sets of answers in $U$ such that for all $k$, we have $ S= \sum_{u_i \in U} u_i > 1$. The answers are in percentages.
We are interested in the quality of a certain set, say $u_c'$, such that we want to know the perce... |
Consider a quantity, $a$, which could represent various measures such as the area under a curve. Initially, we estimate the value of $a$. As we refine our approximation, we find that we can make it arbitrarily close to a number $\ell$. This suggests that $a$ equals $\ell$. But why is this the case?
Here's the ration... |
Is the argument for approaching of an approximated quantity to its true value sound? |
Use the fact that every root $\alpha$ is conjugated to a simple root $\alpha_i \in \Delta$ by a product of simple reflections $\sigma = \prod_i \sigma_{\alpha_i} \in W$ where $W$ is the Weyl group of $\Phi$. That means $\sigma(\alpha) = \alpha_i \in \Delta$.
Now define $\langle \lambda, \alpha \rangle:= \frac{2 (\l... |
I have the impression you're looking for the single one solution. You do realise there are two solutions, do you?
[![enter image description here][1]][1]
P.s. I realise very good this should be a comment and not an answer, but there's no way to add an image to a comment.
[1]: https://i.stack.imgur.com/BUma... |
This question is possibly related to [my previous question][1] and it's answer.
Suppose $a_n>0\ \forall\ n\in\mathbb{N}\ $ and $\displaystyle\sum_{n=1}^{\infty} a_n = a.$
Does there exist a subsequence $(a_{k_n})_{n\in\mathbb{N}}$ of $(a_n)_{n\in\mathbb{N}}$ such that the sequence defined by:
$b_1 = a_{k_1}=... |
I have seen this equation from a link named [Asisten and German Academy][1], (it is a video of Facebook) where there is a complicate solution (I invite to watch it) for
$$(2n)^{\log 2}=(5n)^{\log 5}$$
I have adopted, instead, this approach:
$$(2)^{\log 2}(n)^{\log 2}=(5)^{\log 5}(n)^{\log 5} \iff 2n^{\log 2}=5... |
Does the inner semidirect product of Lie groups need these two subgroups both be closed? |
Let T be a positive compact operator on a Hilbert space then we know that an operator equation $X^2 = T$ have a solution.
What would be an example of an operator $T$ which have uncountable many solutions? |
The exterior product of two vectors doesn't represent an simple area, but an _oriented_ area. It's just that in two dimensions, this becomes a scalar since it only has one vector component (but it can still be considered oriented because if the sign of the component).
In three dimensions, an oriented are is equivale... |
I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex plane, when $A$ has an accumulation point in $D$.
Now let us assume I have a function of two complex variables $f(z... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
How to answer this question in GTM167? |
I apologise that this is similar to older questions, but I’ve seen no definitive answer yet.
If $a,b,c$ are $1$-forms and $b\otimes c$ is the tensor product, is the wedge product $a\wedge ( b\otimes c)$ defined or is it just meaningless?
I ask because I was trying to derive the associative law for the wedge produ... |
Let $I$ be an injective $\mathcal{O}_X$-module. For every open set $U \subset X$ and $\mathcal{O}_U$-module $\mathcal{F}$, define an $\mathcal{O}_X$-module $j_!(\mathcal{F})$ by
$$
j_!(\mathcal{F})(V) = \begin{cases}
\mathcal{F}(V), & \text{if } V \subset U \\
0, & \text{if } V \not\subset U.
\end{cases}
$$
... |
If any of the following exposition is unclear, please write a comment.
In essence, I am looking at the graph $G$ that is generated by the polynomial $q(x) = x^2+ax+b$ ($a,b \in \mathbb{Z}$) via the edge set
$$\{(n, q(n) \text{ mod } p) : n =0,\ldots,p-1\}$$
for some prime $p \in \mathbb{P}$. After some thought, it... |
Question from an old exam:
>In a full binary tree of height $n$, each vertex is randomly colored either white or black. A path is considered good if it goes from any vertex to a leaf and all vertices on this path are of the same color. Let $X$ denote the number of good paths. For example, for a tree of height $n = 3$ ... |
$X_n \xrightarrow[]{d} X$, $Y_n \xrightarrow[]{d} Y$ where $X \sim N(\mu_x, \sigma_x)$ and $Y \sim N(\mu_y, \sigma_y)$, but $X_n \not\!\perp\!\!\!\perp Y_n$. What do we need to analyze $(X_n, Y_n) \xrightarrow[]{d} \: ?$.
In other words, we have two dependent random variables that converge in distribution and we a... |
Let $T: V \rightarrow W$ be a continuous/bounded linear map between Banach spaces. Let $J \subseteq \ker(T)$ be a closed subspace. Then I know that this induces a unique linear map on the quotient $\bar{T}:V/J \rightarrow W$ s.t. $\bar{T}\circ \pi=T$. I also know from topology that $\bar{T}$ is continuous w.r.t. the qu... |
Let us define the complete elliptic integral of the first kind as follows:
$$
K(k)=\int_{0}^{1} \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2t^2} }
\text{d}t,
$$
where we restrict the modulus $k$ to $|k|<1$ with its complementary $K^\prime(k)=K\left ( \sqrt{1-k^2} \right )$. They are shortened as $K,K^\prime$ respectively.
... |
Let $$ (\log x_1)^2+ (\log x_2)^2 + \cdot\cdot\cdot + (\log x_n)^2= r $$
For $r=0,1,2,…$
The goal is to count the number of lattice points enclosed by the curve/surface for $r$. In this way the problem is a generalization of the Gauss circle problem.
> What is the exact solution or at least an approximate so... |
Let $$P = 3^{\frac13}\times3^{\frac29}\times3^{\frac3{27}}\dots\infty$$
What would be the value of $P^{1/3}?$
$$P = 3^{1/3\ +\ 2/9 +\ 3/27\dots\infty}$$ $$Let\ S = \frac13 + \frac29 + \frac3{27}\dots\infty$$
How would I evaluate $S?$ I see 2 series in it, the one in the numerator, i.e, $1+2+3+4\dots$ and the o... |
Question from an old exam:
>In a full binary tree of height $n$, each vertex is randomly colored either white or black. A path is considered good if it goes from any vertex to a leaf and all vertices on this path are of the same color. Let $X$ denote the number of good paths. For example, for a tree of height $n = 3$ ... |
The theorem below is from the book “Elementary Number Theory” by “David Burton”
“Let n be a composite square-free integer, say, n = p1 p2 · · · pr , where the pi are distinct primes. If pi−1|n−1 for all i=1,2,...,r, then n is an absolute pseudoprime.”
The proof in the book considers only the case where gcd(a, n) ... |
I'm currently reading some stuff on the existence of a measurable inverse, or measurable choice theorems from [Bogachev's Measure Theory book, Volume II][1], where I'm trying to connect these two theorems on measurable choice:
The first one is (see P.34): see below, the third screenshot for the definition of Souslin... |
Let $X$ be an infinite set. Then the space of all bounded functions $\mathcal{B}(X,\mathbb{R})$ is disconnected (where $\mathbb{R}$ has the usual metric and $\mathcal{B}(X,\mathbb{R})$ has the topology induced by the sup metric).
I was thinking of finding a subset of B (different from B) and showing that that set is... |
How to prove that the space of all bounded functions $\mathcal{B}(X,\mathbb{R})$ is disconnected? |
Denote by $\omega$ any non-trivial cube root of unity. If $\mathbb Q_p(\sqrt[3]{2})$ was galois then it contains all cube roots of $2$ i.e. $\sqrt[3]{2}$, $\omega\sqrt[3]{2}$ and $\omega^2\sqrt[3]{2}$ hence it contains $\omega$. But can $\omega \in \mathbb Q_p(\sqrt[3]{2}) - \mathbb Q_p$? If this was true then $\omega$... |
If $\mathbb Q_p(\sqrt[3]{2})$ is galois over $\mathbb Q_p$ then does $\mathbb Q_p$ necessarily contain a cube root of unity? |
I'm trying to build a 3d visual to illustrate a concept. Imagine a [circular helix][1]. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We have a big helix made of a small one. What i'm trying to do is to repeat that process and use each new helix as the... |
I'm studying the following theorem in (Schindler, 2014: Set Theory
Exploring Independence and Truth), p. 178-180:
**Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue measurable.**
Proof (sketch):\
Let $\epsilon_n = \frac{1}{2n+1}$ for n < $\omega$. By Lemma 9.14 *(A non-t... |
[In Wikipedia](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula#Generalizations) the following generalisation of the Möbius inversion formula is given (and proved):
>Suppose $F(x)$ and $G(x)$ are complex-valued functions defined on the interval $[1, ∞)$ such that
$$G(x)=\sum_{1\le n\le x} F\left(\frac xn\r... |
Which is the error in this application of Möbius inversion formula? |
**A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.**
I have tried solving it by cutting the original triangle into several triangles but each time I end either with all equilate... |
As far as my studies have brought me, I've only see so far the definition of a "pole" for a complex valued function $f:\Omega \rightarrow \mathbb C$ if we assume that the function is holomorphic in a neighbourhood around a singularity $z_0$.
Does it make sense to talk about poles without this condition, i.e. without... |
Can a non-holomorphic function have a pole? |
If I sample $N+1$ uniform $k$-bit integers $x, x_1, \ldots, x_N$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{x_i\}$? Upper/lower bounds and/or asymptotic approximations would also be of interest.
For prime $p$ and $x, x_1, \ldots, x_N$ as above, say $x$ is $p$*-uniqu... |
While working on a project, I came across a paper that includes this sum:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0 (g(ξ), g(\bar{ξ}))
$$
where we have
- "$$ being the tetrahedral group", "a discrete subgroup of isometries $ ⊂ O(3)$";
- $r_0(t) = -\frac{t^6 - 15 t^4 + 15 t^2 - 1}{16 (t^2 + 1)^3} × S +... |
I'm currently reading some stuff on the existence of a measurable inverse, or measurable choice theorems from [Bogachev's Measure Theory book, Volume II][1], where I'm trying to connect these two theorems on measurable choice:
The first one is (see P.34): see below, the third screenshot for the definition of Souslin... |
> For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation}
My solution: Write $n$ as a product of distinct prime powers, i.e. $n=p_1^{\alpha_1}\times\cdots\times p_r^{\alpha_r}$. Then ... |
If you only want to show the first equality in 2., i.e. $$\varphi(0) = \int_{\mathbb R^d} - \phi_x \Delta\varphi \mathrm d \lambda,$$
it might be helpful to consider a ball $\Omega = B(0,2r)$ of radius $2r$, such that $\operatorname{supp}(\varphi) \subseteq B(0,r)$ and set $u = \varphi$. Now $u$ vanishes on a neighb... |
In his book "Differential Geometry" Loring. Tu writes the derivative of a vector field $X$ along a curve $c: [a,b] \to \mathbb{R}^n$ as $\frac{dV(t)}{dt}=\sum \frac{dv^i(t)}{dt} \partial_i |_{c(t)}$. It then goes on to say that such a derivation is only defined in the $\mathbb{R}^n$ manifold and not in an arbitrary man... |
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