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Let $n$ be a prime number. Consider $\mathbb U_n$ which is the set of roots of unity, i.e. the solutions of the equation $z^n=1$, more precisely $\mathbb U_n = ${$1,\epsilon , \epsilon^2 , ... , \epsilon^{n-1}$} , where $\epsilon = \cos \frac{2k \pi}{n}+i\sin{2k \pi}{n} ,k={0,1,2,...,n-1}$. My question is if $n$ is pr... |
Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by
$$
x_{n+1}=A x_n
$$
for some starting vector $x_0\neq 0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the Euclidean norm on $\mathbb R^2$. I want to show that this iteration converges linearly, which means that
$$
\lim_n \frac{\ver... |
Is there polynomial $P(x)$ withbinteger coefficiants that satisfies $P(2)=4$ and $P(6)=6$? |
Is there a polynomial $P(x)$ with integer coefficients that satisfies $P(2)=4$ and $P(6)=6$? |
given the distance of each of the four diagonal planes of a tetrahedron from the origin, is there a formula for finding each of the four vertices? |
We know that every finitely generated metabelian group $\Gamma$ has a faithful representation over a finite product of fields.
Denoting $C_n$ the cyclic group of order $n$, the wreath product $C_6\wr C_\infty$ is linear over a product of two fields ( a field of char. 2 and a field of char. 3). May I ask how to const... |
> "How did they calculate it like that?"
For $~p~$ prime, and $~n \in \Bbb{Z^+},~$ let $~V_p(n)~$ denote the largest integer $~\alpha~$ such that $~p^\alpha ~| ~n.~$ This implies that $~p^{\alpha + 1}~$ does not divide $~n.$
Now, let $~S = \{x_1, x_2, \cdots, x_k\}$ be any set of $~k~$ positive integers, and su... |
Suppose $G$ is a balanced directed graph. 'Balanced' means the in-degree equals to the out-degree at each vertex. It can be shown that the directed spanning trees always exist and the number of directed spanning trees is independent of the choice of the root vertex, so it makes sense to say the number of directed spann... |
I'm taking a course on probability and our professor has defined random variables in a way that I'm not so used to, I'm used to the definition of a random varaible X, being a function from the sample space into the real numbers. Instead we are given this:
[![enter image description here][1]][1]
And then we get r... |
Decompose $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$, let $\Omega \subset \mathbb{R}^N$ be a bounded domain and take $x$ in the set
$$\Pi_1(\Omega) = \{ x \in \mathbb{R}^{N_1} : (x,y) \in \Omega \text{ for some } y \in \mathbb{R}^{N_2}\}.
$$
Define the set
$$
A_x := \{y \in \mathbb{R}^{N_2} : (x,y)... |
I recently saw
[this](https://math.stackexchange.com/questions/1686493/finding-submodules-of-a-module-over-a-ring-of-matrices) question and wanted to generalize it to $\mathbb{Z}^n$ and matrices with integer coefficients. I think $k\mathbb{Z}^n$ is a submodule for any integer $k$. Is there a nice way to generalize th... |
In the polar representation of a complex number, is there a conventional term for the $e^{i\theta}$ thing in $z=\rho e^{i\theta}$? I know $|z|=\rho$ is typically called the *modulus*, and $\theta$ is typically called the *argument*. $|z|\cos\theta=\Re z$ is called the *real part,* and $|z|\sin\theta=\Im z$ is called t... |
Is there a conventional term for the $e^{i\theta}$ thing in $z=\rho e^{i\theta}$? |
**Setup**
I'm taking a course on probability and our professor has defined random variables in a way that I'm not so used to, I'm used to the definition of a random varaible X, being a function from the sample space into the real numbers. Instead we are given this:
[![enter image description here][1]][1]
And ... |
Let $t_{0}>0,$ $S_{t_{0}}=S_{0}$ and $\forall t>0\quad \frac{dS_t}{S_t}=\mu dt+\sigma dW_t$.
Are there any regularity results for $(\mu, \sigma)\mapsto S(\mu,\sigma)$? |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
> "How did they calculate it like that?"
For $~p~$ prime, and $~n \in \Bbb{Z^+},~$ let $~V_p(n)~$ denote the largest integer $~\alpha~$ such that $~p^\alpha ~| ~n.~$ This implies that $~p^{\alpha + 1}~$ does not divide $~n.$
Now, let $~S = \{x_1, x_2, \cdots, x_k\}$ be any set of $~k~$ positive integers, and su... |
Let $F$ be a field containing all $n^{th}$ roots of unity. An $n$-*Kummer Extension* over $F$ is a Galois extension of the form $E=F[r_1,...,r_m]$ where $r_i^n\in F$ for all $i=1,...,m$.
From Kummer theory, the groups $$\operatorname{kum}\left(E/F\right)=\left(F^{1/n}\right)^\times/F^\times\simeq \operatorname{Gal}... |
According to Wolfram Alpha, this number is not prime, but it does not display prime factors of this number as it normally does for other numbers so I am not so sure about it. Several other factorization calculators I tried also failed to find prime factors of this number. The main question I want to ask is this: Is i... |
Is it possible to find the prime factors of $89!+97$? |
Decompose $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$, let $\Omega \subset \mathbb{R}^N$ be a bounded domain and take $x$ in the set
$$\Pi_1(\Omega) = \{ x \in \mathbb{R}^{N_1} : (x,y) \in \Omega \text{ for some } y \in \mathbb{R}^{N_2}\}.
$$
Define the set
$$
A_x := \{y \in \mathbb{R}^{N_2} : (x,y)... |
It has a bit of physics ring to it, but, at least in a mixed company (mignilling with physics majors) I often call it *the phase factor*. [Wikipedia](https://en.wikipedia.org/wiki/Phase_factor) seems to give me some support.
May be I got used to it when working in telcomm? They do waves! I think I used it before my ... |
It has a bit of physics ring to it, but, at least in a mixed company (mingling with physics majors) I often call it *the phase factor*. [Wikipedia](https://en.wikipedia.org/wiki/Phase_factor) seems to give me some support.
Maybe I got used to it when working in telcomm? They do waves! I think I used it before my sti... |
I got this two function to compare in preparation for a test:
$$
2^n \cdot ln(n^2)
$$
and
$$
2\cdot n!+4\cdot n^2
$$
I know that second function I can count as if it was just n!. But i still don't know with of these functions grow faster...
Can someone help me? Thank you very much. |
It's well known that if $U \subset \mathbb{R}$ is bounded, then there exists a monotone increasing sequence $(x_{n})^{\infty}_{n=1}$ converging to $sup(U)$.
My question is: Let $X$ be a lattice, and let $U \subseteq X$ be a totally ordered subset (chain). Does there exist a monotone increasing net $(x_{\alpha})_{\a... |
Does every set with a supremum contain a monotone net converging to that supremum? |
Let $F$ be a field containing all $n^{th}$ roots of unity. An $n$-*Kummer Extension* over $F$ is a Galois extension of the form $E=F[r_1,...,r_m]$ where $r_i^n\in F$ for all $i=1,...,m$.
From Kummer theory, the groups $$\operatorname{kum}\left(E/F\right):=\left(F^{1/n}\right)^\times/F^\times\simeq \operatorname{Gal... |
In order to make sense of your question one has to specify what do "space" and "cover" mean. Without further adjectives, they usually mean "a topological space" and "a topological covering map." However, it is clear from your comments, this is not what you are actually interested in. One can define the notion of a cove... |
I got this two functions to compare in preparation for a test:
$$
2^n \cdot \ln(n^2)
$$
and
$$
2\cdot n!+4\cdot n^2.
$$
I know that second function I can count as if it was just $n!$. But I still don't know which of these functions grow faster...
Can someone help me? Thank you very much. |
Can you give me examples of infinite-rank partial isometries over B(H)? |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
I am tasked with finding whether the function $()=\tan(/(-1))$ can be developed into a Laurent series around $=1$, and if so what is the radius of convergence and what is the residue?
My initial attempt was at developing the expression into something like the following:
$
\tan(\frac{z}{z-1}) =
\tan(1 + \frac{1... |
In a question, I am asked to compute the homology of $\Sigma_g\#S_n$ (via Mayer-Vietoris), and deduce that this space is in fact $S_{n+2g}$. I did so by assuming the classification of compact surfaces, but now I am wondering whether this is all circular: how is the classification of compact (or even triangulable) surfa... |
I am tasked with finding whether the function $()=\tan(/(-1))$ can be developed into a Laurent series around $=1$, and if so what is the radius of convergence and what is the residue?
My initial attempt was at developing the expression into something like the following:
$
\tan(\frac{z}{z-1}) =
\tan(1 + \frac{1... |
I hope you are well.
Could you help me with the following?
I want to build a set of quasi-periodic functions that are also orthonormal. This set will help me with a chaos problem. I tried with $\sin (\sqrt (2) x) + \sin (x)$ style functions and using inner product $< f(x),g(x) > = \int_{- \pi}^{\pi} f(x) \cdot g(... |
I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \boxtimes v$ for any $v \in V$, and extended linearly over $T(V)$. I've encountered the following formula that expresses ... |
I've been studying infinite series recently and believe I came across a counterintuitive (at least to me) result in the past from a textbook that I can't seem to find now. Is it possible to show $$\sum_{n=1}^\infty \frac{\csc(n)}{n}$$ converges? I actually think the result I saw was something along the lines of $\displ... |
I am tasked with finding whether the function $()=\tan(/(-1))$ can be developed into a Laurent series around $=1$, and if so what is the radius of convergence and what is the residue?
My initial attempt was at developing the expression into something like the following:
$
\tan(\frac{z}{z-1}) =
\tan(1 + \frac{1... |
I thought about a way to simply explain the 2nd order approximation of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ around a point $(x_0,y_0)$.
First we take the linear approximation of $f$:
$$f(x,y) \approx f(x_0,y_0) + \frac{\partial f}{\partial x} \cdot \Delta x + \frac{\partial f}{\partial y} \cdot ... |
In a question, I am asked to compute the homology of $\Sigma_g\#S_n$ (via Mayer-Vietoris), and deduce that this space is in fact $S_{n+2g}$. I did so by assuming the classification of compact surfaces, but now I am wondering whether this is all circular: how is the classification of compact (or even triangulable) surfa... |
I'm starting to learn a little basics about smooth manifolds, starting from embedded manifolds.
In a linear space, say $\mathbb{R}^{n}$, I know we can define embedded submanifolds with a locally defined function, which is equivalent to the canonical definition via diffeomorphisms. Following is the definition found i... |
Is the zero set of a smooth function disjoint union of smooth manifold in euclidean space? |
as a part of a model theory assignment I am asked to prove that if a theory T has quantiier elimination, then it can be axiomatized by sentences of the form $\forall\bar{x}\exists{y}\phi(\bar{x},y)$, where $\phi$ is a quantifier-free formula. I have a sketch for the solution, but I am not too sure about it. I would app... |
Consider the following exam quesiton
>Let $\left(X_n: n \in \mathbb{N}\right.$ ) be a sequence of independent random variables with common density function
$$
f(x)=\frac{1}{\pi\left(1+x^2\right)} .
$$
Fix $\alpha \in[0,1]$ and set
$$
Y_n=\operatorname{sgn}\left(X_n\right)\left|X_n\right|^\alpha, \quad S_n=Y_1+... |
[![Original Question][1]][1]
Hello,
I have a question for which I don't understand how the solution came about. In the above triangle, AC = 8, and BE = 6. AD and DE are the equal. Also CN and NB are equal i.e. N is the mid point of BC. The question is what is DN?
The solution starts by stating, create a line... |
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. We can define the partial derivative of f with respect to its first argument using either a two-sided limit ($h\to 0$)
$$\lim_{h\to 0} \frac{f(x_1+h,x_2,\ldots,x_n)-f(x_1,x_2,\ldots,x_n)}{h}$$
or a one-sided limit ($h\to 0^+$)
$$\lim_{h\to 0^+} \frac{f(x_1+h,x_2,\... |
What is the significance of defining the partial derivative as a one-sided limit or a two-sided limit? |
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 $\frac{δz(T,x)}{δA(t)}$ , consi... |
Find all Sylow 3-subgroups of $S_3\times S_3$?
This is what I already found:
Since $o(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then $n_3|4$ and $3|(n_3 - 1)$. Hence $n_3$ should be $1$ or $4$. Now how can I find at least one subgroup of order $9$? |
First, let me recall some useful definitions
We recall that if $A=U \Lambda U^*$ is a Hermitian matrix with $U U^*=U^* U=I$ and $\Lambda=\operatorname{diag}\left(\lambda_1\right)$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function then the matrix $f(A)$ is defined by
$$
f(A)=U \operatorname{diag}\lef... |
$f:\mathbb{C^{*}} \to \mathbb{C} \\ \quad \; z \to \frac{1}{z^2} + |z|^2 $
Exploring the possibility of the existence of a non-empty open set $U$ of $\mathbb{C^*}$ such that $f|_{U}$ is holomorphic; I did the following:
Let $z_0 \in \mathbb{C^*}$; $\mathit{C} = \{ z \in \mathbb{C} \mid |z|=|z_0| \}$, then:
$$... |
I'm starting to learn a little basics about smooth manifolds, starting from embedded manifolds.
In a linear space, say $\mathbb{R}^{n}$, I know we can define embedded submanifolds with a locally defined function, which is equivalent to the canonical definition via diffeomorphisms. Following is the definition found i... |
You, Ada, and Carlos are on a game show where the host will ask contestants to wager (or risk) money. Assume that you are all very smart, and that each of you will be willing to wager money if and only if there is a greater than 50% chance you will win that wager. The three of you each roll a 12-sided die (numbered 1 t... |
I am studying Gaussian measures on nuclear spaces, which for concreteness I take to be Schwartz space $\mathcal{S}'(\mathbb{R}^n)$. Just as in finite dimensions, these Gaussian measures can be uniquely characterized by their mean and covariance. However unlike in finite dimensions, the covariance acts on the base space... |
Why is the covariance operator of a Gaussian measure not defined on the dual space? |
Let's rewrite the number as a product to try to understand this better: $\prod_{x=1}^{25} 5x^{5x}$
As you already stated, zeros are formed when $5$ and $2$ and multiplied.
$20+60+80+100+240+240+10^{10+20+30+\dots+120} = 1520$
Since 2's and 5's form 0's, we know there are more 5's than 2's so we can just find t... |
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... |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
Let V be a vector space over a field F and let V1, V2 be subspaces
of V . Prove that the union of V1, V2 is a subspace of V if and only if
V1⊆V2 or V2⊆V1 .
Please someone explain why this holds? |
Now asked on HSM [here][1].
-------------------
>$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \ \ \ \ x>0. $$
Bohr and Mollerup showed that the gamma function is the only positive function $f$ defined on $(0,\infty)$ that has these properties
1. $f(x+1)=xf(x)$
2. $f(1)=1$
3. $f(x)$ is a... |
Tractatus Logico-Philosophicus states simply that
> 6 The general form of the truth function is: $[\bar p, \bar\xi, N(\bar \xi)]$. This is the general form of the sentence.
Wikipedia and other sources say
* $\bar p$ stands for an atomic proposition;
* $\bar \xi$ stands for a set of propositions;
* $N(\xi)$ ... |
What is the logical system of Tractatus Logico-Philosophicus? |
For a different straightforward solution, set up the coordinate geometry with $ E = (0,0) $ (why is this a good choice?) $A = (0, 2b), B= (6, 0), C = (0, 2b-8)$.
Show that $D = (0, b), N= (3, b-4)$ and hence $|DN| = 5$. |
I'm preparing for a course on Topology next semester (so I am a beginner) and in Klaus Jänich's book "Topologie" the question above is asked which I cannot answer.
To give more details, the relevant definition is given as
Let X be a topological space.
1. $A \subset X$ is called closed when $X \setminus A$ is ... |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
I have this recurrence relation above to find a value for. I have attempted to solve this by generating functions and ended up with a closed form solution as:\
$g(x) = \frac{4x}{(1-x)^3} - \frac{4x}{(1-x)^2}$\
I feel like this is correct, though I'm not totally sure. I then rewrote these as infinite sums so I can fin... |
Let $f:[0, +\infty) \to \mathbb{R}$ be continuous and bounded. Define $$F(s) = \int_0^{\infty} f(t) e^{-st} \; \mathrm{d}t$$Consider an arbitrary $s_0>0$. Prove that $F(s)$ is represented by a power series $F(s) = \sum_{n = 0}^{\infty} a_n (s-s_0)^n$ which has a radius of convergence $R \geq s_0$.
I know that the L... |
I'm preparing for a course on Topology next semester (so I am a beginner) and in Klaus Jänich's book "Topologie" the question above is asked which I cannot answer.
To give more details, the relevant definition is given as
Let X be a topological space.
1. $A \subset X$ is called closed when $X \setminus A$ is ... |
From the equation $u(x) = x+g(x)c$, we see that there can be no further eigenvectors. Suppose $u(x)=\lambda x$ for some scalar $\lambda$. Then $(\lambda-1)x = g(x)c$. Either $\lambda=1$ and $g(x)=0$ (we cannot have $c=0$ since $u\ne 1_E$) or $x = \frac{g(x)}{\lambda-1} c$, in which case $x\in H$ and $g(x)=0$, so $x=0$.... |
Given the right triangle with the point M at (x,y).
There's an ellipse that touches the point M, with the centre at 0,0. No matter where you place M.
The formula for it would be:
$x^2/a^2 + y^2/b^2 = 1$
According to Pythagoras theorem:
$(a+b)^2 = (x + b * cos ABC)^2 + (y + a * sin ABC)^2$
Can we obta... |
I was reading a computer vision paper for 3D reconstruction and was a bit confused by two notations: https://www.researchgate.net/publication/327805907_3D_Image_Reconstruction_from_Multi-Focus_Microscope_Axial_Super-Resolution_and_Multiple-Frame_Processing
1) $f \in R^{{N_x}\times{N_y}\times{N_z}}$, which denotes... |
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 easily solved 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 $$\int\... |
To my understanding (from reading Wikipedia) the general form of homogeneous first order ODE is
$$
h(x,y)\frac{dy}{dx}=g(x,y)
$$
where $h$ and $g$ are both homogeneous functions of the same degree, for example, the following equation that I just made up
$$
x^2\frac{dy}{dx}-\frac{y^4}{xy+x^2}=0
$$
would have bot... |
What's the general form of a homogeneous ODE (linear or not, of any order)? |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
I know there are [various ways][1] to compute eigenvalues, and they are generally not very efficient for large matrices.
Graph Laplacians are symmetric and positive (except on the diagonal, which is negative). I see that [this question and answer][2] gives one way to decompose a symmetric matrix into an easier Eige... |
How to compute eigenvalues of graph laplacian? |
**Setup**
I'm taking a course on probability and our professor has defined random variables in a way that I'm not so used to, I'm used to the definition of a random variable X, being a function from the sample space into the real numbers. Instead we are given this:
[![enter image description here][1]][1]
And ... |
Solve a simple exponential recursive equation? |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
I have a research project on building an object detection model for detecting animals in thermal imageries. I already have a baseline model, and want to do some uncertainty quantification on my predictions.
My idea was to do bootstrap sampling. Given X number of images, create 30 bootstrapped samples each with X im... |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
>More precisely, if $m\mid n$, can we always find a group $G$ of order $n$ and a subgroup $H$ of $G$ of order $m$?
Yes.
For each $n$, there exists a cyclic group of order $n$. In the cyclic group of order $n$, for each $d\mid n$, there exists a subgroup of order $d$.
---
*Note that this assumes $m\mid n$,... |
After searching on the web without success, I'm asking for help here. I wasn't taught statistics, and I get lost in a lot of formulas that I often find hard to understand. I'm also not used to posting on a forum, let me know if I need to change anything, or give more details!
I'm following a protocol to get the perf... |
I have this recurrence relation
$$h_{n} = h_{n-1} + 4(n-1) \ \text{for} \ n \geq 1 \ \text{with} \ h_{0} = 0, h_{1} = 0$$
I am asked to find $h_{9}$.
I have attempted to solve this by generating functions and ended up with a closed form solution as:\
$g(x) = \frac{4x}{(1-x)^3} - \frac{4x}{(1-x)^2}$\
I feel ... |
[Resolution][1]
"The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)\). Shouldn't it be (x = 2 cos(t))?
[Resolution][2]
[1]: https://i.stack.imgur.com/11p5z.png
[2]: https://i.stack.imgur.com/2wWma.png |
[Resolution][1]
"The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)). Shouldn't it be (x = 2 cos(t))?
[Resolution][2]
[1]: https://i.stack.imgur.com/11p5z.png
[2]: https://i.stack.imgur.com/2wWma.png |
Let four random variables form the Markov chain
$${\displaystyle W\rightarrow X\leftarrow Y\rightarrow Z},$$
such that the conditional distribution of $X$ depends only on $W$ and $Y$ and is conditionally independent of $Z$ given $Y$. What is the relationship between the mutual informations I(W;X|Y) and I(W;X|Z) ... |
Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a *finite type* quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (more formally, such that $\mathcal{O}_X/\mathcal{J}$ has support $Z$?)
In the affine case the answer is yes: $V(\m... |
I came across the following problem: Let $p$ be a prime, prove that $$(p-1)^np^{(n^2-n)/2}\mid \prod_{k=1}^n(p^n-p^{k-1})$$ for any $n\in \mathbb{N}$. It is quite hard to prove it using elementary number theory. However, the problem becomes trivial if we notice that $|\operatorname{GL}_n(\mathbb{F}_p)|=\prod_{k=1}^n(p^... |
Let $M$ a compact and connected smooth manifold.
Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for all $t$.
**I know that**:
Given a symplectic form $\omega$ on $M$, we denote by $Symp(M, \omega)$ the space of... |
Let $M$ a compact and connected smooth manifold.
Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for all $t$.
**I know that**:
Given a symplectic form $\omega$ on $M$, we denote by $Symp(M, \omega)$ the space of... |
[Resolution][1]
"The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)). Shouldn't it be (x = 2 cos(t))?
[Resolution][2]
The theorem in the book is this. From what I understand, you calculate both sides of the equality in the theorem and obtain the same value.Rot is ... |
Given a variety $X$, is the constant sheaf given by the function field $K=k(X)$ a quasicoherent sheaf?
I think so because to each affine open we assign K, and then for a localization $A \to A_f$ the map $K \otimes_A A_f \to K$ is an isomorphism.
But I don’t find this stated anywhere. |
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
... |
We have that a walk regular graph is one in which $A^k$ is a constant diagonal matrix for all $k \geq 0$.
Given this, do operations on walk regular graphs also result in walk regular graphs? For example do the cartesian product, direct product or complement of walk regular graphs also result in walk regular graphs? ... |
[Resolution][1]
"The question is a problem from Leithold's calculus book. I didn't understand the ($x = 5 \cos(t)$). Shouldn't it be ($x = 2 \cos(t)$)?
[Resolution][2]
The theorem in the book is this. From what I understand, you calculate both sides of the equality in the theorem and obtain the same value.Ro... |
Let $M$ be a smooth manifold, and $\Gamma(TM)$ the space of all vector fields on $M$. [Wikipedia says][1] that an affine connection on $M$ is a map $\nabla:\Gamma(TM)^2\rightarrow \Gamma(TM)$ that satisfies:
$$\nabla(X,Y)=\nabla_XY$$
$$f(\nabla_XY)=\nabla_{f(X)}Y$$
$$\nabla_Xf(Y)=\partial_Xf(Y)+f(\nabla_XY)$$
for a... |
In the problem section, there is a problem asking the reader to prove
$$|(f,g)|\le \|f\|\cdot\|g\|$$
where $|(f,g)|$ is the inner product and $\|f\|$ is the norm by defining the function $F(\alpha) = \|f+\alpha g\|^2$ and (I assume this is how they intended) showing how there is at most one root for $\alpha$ and ... |
I approached this problem the following (apparently incorrect) way:
In $K_{9,9}$, there are 18 vertices, so 18 choices for a starting point. From there, since the graph is bipartite, whichever "side" we start on, we must pick one of the 9 points on the other side. Then, we must pick one of the 8 remaining points on ou... |
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