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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 $\mathcal{B}$ be an ordered basis for a vector space $V$ of dimension $n$. Denote the basis for the set of anti symmetric tensors ($\Lambda^{p}{V}$) by $\mathcal{B}^{p}$. For $n=3$, let $A=\left( a^{i}_{j} \right )$ be the matrix of the linear map $f: V \to V$ with respect to the basis $\mathcal{B}$. Describe the m... |
I am struggling to prove the following statement:
Let $p, q$ be coprime numbers. For all $n \in \mathbb{N}^*$, we define $t_n$ as the order of $[q]$ in $(\mathbb{Z}/{p^n}\mathbb{Z})^\ast$.
Show that $(t_{n+1}/t_{n})$ is stationary.
I managed to reduce the problem: let $p = \Pi_{i=1}^m p_i ^{\alpha_i}$ (prime fact... |
> Express $(\sin(\frac{\pi }{3})+i \cos(\frac{\pi }{3}))^8$ in $a+bi$
> form.
I used demoivre's, so I got $\sin\left(\frac{8\pi}{3}\right)+i \cos \left(\frac{8\pi}{3}\right)$, does that yield my answer upon evaluation? |
In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal of the curve C).
How do i show that defining divergence in this way also implies that
$Div(\vec{F}) = \partial_1 F... |
I am sorry, but I do not know how to search this. I apologize in advance.
Assume I have a set $X$ and a map $f:X\rightarrow X$; there are $n$ elements $x_1,... x_n\in X$ such that $f(x_k)=x_{k+1}$ and $f(x_n)=x_1$.
Does this property have a name? |
I know that the serie $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging.
I try to evaluate out to 10 terms is 0.861229; 20 terms is 0.863979. Would you please share me a way to solve this question?
Best wishes... |
I meet this when reading a paper about cyclic branched covering spaces of knots. There is no explanation nearby, can anyone tell me how to understand this? |
Ravi Pradeep's solution uses a recurrence relation, I am new to recurrence relations but would like to understand this part of the answer.
https://math.stackexchange.com/questions/364038/expected-number-of-coin-tosses-to-get-five-consecutive-heads
Having looked into recurrence relations I know that the methodolog... |
I know that the series $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging.
I try to evaluate out to $10$ terms is $0.861229$; $20$ terms is $0.863979$. Would you please share me a way to solve this question?
Be... |
For any 2D closed shape, is there always a point and angle for which the billiard movement of a ball fully completes the interior of the shape? In other words, does the path eventually converge to the full figure?
For example, will the moose be completely filled?
[![enter image description here][1]][1]
If not,... |
Can billiards fill any figure? |
Suppose I have two sequences $a_n$ and $b_n$. They both start from a very high number say $25000$ and converges to zero.
If I plot them in one plot, as they start from a very large number, the plot looks like as if $a_n \approx b_n \approx 0$. But if we look closely, the difference between them is around $2$ or $3$... |
How to plot two seperate sequences in one plot such that there difference is clear? |
Let $f : X \to Y$ be a morphism of complex affine varieties. Denote the coordinate rings of $X$ and $Y$ by $\mathbb{C}[X]$ and $\mathbb{C}[Y]$ respectively. Then $f$ corresponds to a ring homomorphism $f^* : \mathbb{C}[Y] \to \mathbb{C}[X]$. It is an easy exercise to show that for every ideal $I \lhd \mathbb{C}[X]$, th... |
What subvariety corresponds to the preimage of an ideal? |
In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal of the curve C).
How do i show that defining divergence in this way also implies that
$Div(\vec{F}) = \partial_1 F... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to have something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorph... |
Let $K$ be a field and $R=K^{2 \times 2}$.
Let $U$ be a subspace of $\mathbb{Q}^2$ and $L=\{A \in R: coim(A) \subset U\}$. Then $L$ is a left ideal of $R$.
For $L$ to be a left ideal the difference of two elements of $L$ has to be in $L$ as well. This should be clear because $U$ is a subspace which has exactly th... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to have something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorph... |
Suppose that $F$ is a local field containing $l$-th roots of unity where $l$ is a prime integer and
$l\neq\mbox{char}\ F$ and that $S$ is a finite set of places $v.$ Given some $t_v\in F_v(c^{1/l})$ where $c\in F$ and $c^{1/l}\notin F_v$ for all $v\in S$. Can we find an element $t\in F(c^{1/l})$ such that $t/t_v$ is ... |
I am given :
$$ I = \frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0} $$
My textbook claims , that as a consequence of the Leibniz integral rule :
$$ I = \int^{b(0)}_{a(0)} x f(x)dx + f(b(0)) \ b(0) \left( \frac{\partial}{\partial \epsil... |
My textbook claims , that as a consequence of the Leibniz integral rule:
$$ \frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0} = \int^{b(0)}_{a(0)} x f(x)dx + f(b(0)) \ b(0) \left( \frac{\partial}{\partial \epsilon} b(\epsilon) \right)_{\epsilon=0} -... |
I am struggling to prove the following statement:
Let $p, q$ be coprime numbers. For all $n \in \mathbb{N}^*$, we define $t_n$ as the order of $[q]$ in $(\mathbb{Z}/{p^n}\mathbb{Z})^\ast$.
Show that $(t_{n+1}/t_{n})$ is stationary.
I managed to reduce the problem: let $p = \Pi_{i=1}^m p_i ^{\alpha_i}$ (prime fact... |
Let
$$I:=\frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0}.$$
My textbook claims that, as a consequence of the Leibniz integral rule:
$$I= \int^{b(0)}_{a(0)} x f(x)dx + f(b(0)) \ b(0) \left( \frac{\partial}{\partial \epsilon} b(\epsilon) \right)_{... |
"Self reference is a semantic notion. Don't warry." But if it has been obtained using self-application? In that case it is a syntactic problem, and perhaps demonstrable in the PR system.
The formula satisfied by all formulas that are not self-applicable, and only by them, is self-applicable if and only if it is not ... |
Does self-reference make the Peano-Russell system inconsistent? |
**Background & Context :** The background of the question is an engineering problem.
I want to efficiently represent a set of integers as rounded real valued functions and quickly be able to calculate their partial sums.
For example fonctions describing the number of bits required to store in my previous question... |
How can one compute rounding preservant integrable functions? |
Previously I had the following coefficients in the program. In this program I want to create a transfer function of butterworth approximation (Figital Filter Fesign - Digital Signal Processing).
import control as ct
import numpy as np
# Coefficients
b1_0 = 0.906... |
How to separate 4th order transfer function (Butterworth Approximation) into 2nd order using Python Control Library? |
Let $w$ be from $\mathbb{C}$ with $|w| = 1$. I need to show that there exists exactly one $t$ from $[0,2\pi)$ such that $w = e^{it}$.
My approach would be to rewrite $w$ as $a+bi$ and $e^{it}$ as $\cos(t) + i \sin(t)$, and then set up two equations so that $a = \cos(t)$ and $b = \sin(t)$ and then argue with the inve... |
I am reading the paper "Lectures on the Langlands program and conformal field theory" by E.Frenkel. In page 28, he computes the one-dimensional continuous representations of $\mathbb{Z}_p$ over $\mathbb{Q}_\ell$, here $\ell \neq p$.\
He chooses any $\ell$-adic number $\mu$ such that $\mu-1 \in \ell \mathbb{Z}_\ell$. H... |
A [Surface of Constant Width][1] is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the surface and paired-planes. I want to construct a trio of parametric equations $\bigl(X(u, v), Y(u, v), Z(u, v)\big... |
A matrix group is defined to be a closed subgroup of $GL_n(\mathbb K)$ where $\mathbb K=\mathbb R,\mathbb C,\mathbb H$
In book *Matrix Group: An Introduction to Lie Group Theory* the author identifies $A\in GL_n(\mathbb R)$ with matrix $diag(A,1)\in GL_{n+1}(\mathbb R)$ and thus proved that $GL_n(\mathbb R)$ is a ma... |
What kind of isomorphisms perserve matrix group? |
Let $G$ be a finite $p$-group and $N$ a maximal subgroup (so $G/N$ has order $p$) such that $Z(N) \leq Z(G)$. III.19.2 in Huppert's Book "Endliche Gruppen I" says that there exists a non-inner automorphism of $G$ of order $p$.
In the proof of Hilfssatz III.19.2, Huppert starts with a homomorphism $\tau: G\to Z(N)$ w... |
Huppert, III.19.2: How to construct a homomorphism from a $p$-group into the center of a maximal subgroup? |
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$ $\Rightarrow$ $N=M$ or $N=I$.
If I choose $M:=I+J=\{a+b,a \in I, b \in J\}$ then the first part shou... |
Suppose that $A \in \mathbb{R}^{n \times n}$ is symmetric positive definite. In this case, solving $Ax = b$ with $x,b \in \mathbb{R}^{n}$ is equivalent to find
\begin{align}
\underset{x \in \mathbb{R}^{n}}{\arg\min}\ f(x) = \frac{1}{2} x^{\mathrm{T}}Ax - b^{\mathrm{T}}x.
\end{align}
So, I can use graident descent (... |
How one can notice when a famous formulas is extended? |
Consider the following ODE:
$$P_n\frac{d f_n(x)}{d x} = \sum_{n' = 1}^NQ_{n,n'}f_{n'}(x),$$
for $n$ and $n' = 1, 2, ..., N$, and in which $P_n$ and $Q_{n,n'}$ are real constant parameters. If we expand this equation for every possible value of $n$ and $n'$, we would get a $N \times N$ system of ODE's that could b... |
Is it possible to solve a system of first-order linear PDE's through a matricial approach? |
Help - How to separate 4th order transfer function (Butterworth Approximation) into 2nd order using Python Control Library? |
Proof- Let H = \langle x_1,...,x_n \rangle and G/H = \langle y_1H,...,y_mH \rangle. Claim that G = \langle x_1,...,x_n,y_1,...,y_m \rangle. Note first that \langle x_1,...,x_n,y_1,...,y_m \rangle \subset G by closure of the group G. Let g\in G. Then gH\in G/H so that we can write gH = (g_1H)^{n_1}\cdot \cdot \cdot (g_... |
This integral: $\int\left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$ seems to be complicated even [Wplfram Alpha][1] can't solve it, but it can be easily solved if one know the formula: $$\tan(a+b+c)= \frac{\tan(a)+ \tan(b)+ \tan(c ) - \tan(a) \tan(b)\tan(c)}{1-\tan(a)\tan(b)-\tan(... |
Let's start from your definition.
1. $\mathbf{GL}_n(\mathbb{R})$ being closed in itself, it is a matrix group; no need to embed it somewhere. Maybe the definition you are quoting isn't the right one?
2. Being a closed subgroup of something isn't preserved by isomorphism: take any non-closed subgroup $H$ of a grou... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on some reasonble site over $X$, say to have something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorp... |
Consider an expression of the form
$$0=\int_a^b f_0(t) \varphi(t) + f_1(t)\varphi'(t) + \ldots + f_n(t) \varphi^{(n)}(t) \, \mathrm{d}t, \qquad \forall \varphi \in C_c^\infty(a,b)$$
and $f_0, \ldots, f_n \in L^2(a,b)$. Can I conclude that
$$f_i \in W^{i,2}(a,b)$$
where $W^{i,2}(a,b)$ denotes the set of all $f \in ... |
**Background:**
A thin filament of fluid mixes into a flow field in accord with the so-called "compression-diffusion equation" (CDE), which is essentially an advection-diffusion equation with a particular space-time dependent velocity field $u(x,t) = \dot{s}(t)x/s(t)$. The variable $s(t)$ is the time-dependent thick... |
Consider the matrix ode $$\mathrm dx/\mathrm d t=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. it seems that the following formula
$$x(t)=cY(t),Y(t)=I+\int_{0}^tA(t_1)dt_1+\int_0^t[\int_0^{t_2}A(t_1)dt_1]A(t_2)dt_2+\cdots\tag{1}$$
gives a solution. But in the Wikipedia page [wiki]... |
I am working on this exercise 1.5.7 of Dixon-Mortimer Permutation Groups: Let G $\leqslant$ Sym($\Omega$) be a transitive group and let $\Gamma$ and $\Delta$ be finite subsets of $\Omega$. Suppose $G_{(\Gamma)}$ and $G_{(\Delta)}$ act primitively on $\Omega\setminus\Gamma$ and $\Omega\setminus\Delta$ respectively, and ... |
Proof- Let $H = \langle x_1,...,x_n \rangle$ and $G/H = \langle y_1H,...,y_mH \rangle$. Claim that $G = \langle x_1,...,x_n,y_1,...,y_m \rangle$. Note first that $\langle x_1,...,x_n,y_1,...,y_m \rangle \subset G$ by closure of the group G. Let $g\in G$. Then $gH\in G/H$ so that we can write $gH = (g_1H)^{n_1}\dots (g... |
Consider the following ODE:
$$P_n\frac{d f_n(x)}{d x} = \sum_{n' = 1}^NQ_{n,n'}f_{n'}(x),$$
for $n$ and $n' = 1, 2, ..., N$, and in which $P_n$ and $Q_{n,n'}$ are real constant parameters. If we expand this equation for every possible value of $n$ and $n'$, we would get a $N \times N$ system of ODE's that could b... |
Consider the matrix ode $$\frac{\mathrm d x}{\mathrm d t}(t)=x(t)A(t),x(0)=c$$where $x(t)\in\mathbb R^n, A(t) \in M_{n\times n}$ is continuous. it seems that the following formula
$$x(t)=cY(t),Y(t)=I+\int_{0}^tA(t_1)dt_1+\int_0^t[\int_0^{t_2}A(t_1)dt_1]A(t_2)dt_2+\cdots\tag{1}$$
gives a solution. But in the Wikipedia... |
Let $Y$ be a scheme and $\mathcal{G}$ a *presheaf* on any reasonble site over $X$; to have something concrete in mind let's take the small etale site $X_{\text{et}}$. Let furthermore $f:X \to Y$ any morphism of schemes.
There exist a presheaf pullback functor $f^P$ associating to $\mathcal{G}$ a presheaf on $X$ d... |
Prove for this problem If H and K are subgroups. Then [H∨K:H]≥[K:H∩K] using injective map? |
The ans given is C(10,3)+C(10,4)+C(10,5)+C(10,6)+C(10,7)=912. Why can't it be 2^4=16 instead?
I tot that the way to solve the above qns might be similar to the below:
Qns: There are 4 choices {G,U,A,C} for each element. How many 6-element sequence ending with GU can we form?
Ans: 4^4=256
|
Qns: How many bit strings of length 10 contain at least three 1's and at least three 0's? |
> If H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G.
Proof- Let $H = \langle x_1,...,x_n \rangle$ and $G/H = \langle y_1H,...,y_mH \rangle$. Claim that $G = \langle x_1,...,x_n,y_1,...,y_m \rangle$. Note first that $\langle x_1,...,x_n,y_1,...,y_m \rangle \subset G$ by ... |
In the context of the Discrete Fourier Transform, a function in time may be expanded as a Fourier series,
$$ f_m = \frac{1}{T} \sum_{n=0}^{N-1} \tilde{f}_n e^{2\pi inm/N}, $$
where $ f_m $ is the function in the time domain in time step $ m\Delta t $, $ f_m = f(m\Delta t) $, $ T = N \Delta t $ is the period, $ f(... |
From [Wikipedia][1]:
> If $F, G$ are partial functions on the natural numbers, the notation $F\simeq G$ indicates that, for each $n$, either $F(n)$ and $G(n)$ are both defined and equal, or are both undefined.
What is the difference between this and $F = G$? If $F, G$ take on exactly the same values for each $n$... |
I'd argue the technical details are necessary but can be avoided. To say you have to study the syntactic details to understand what logic is about is similar to say you have to learn a programming language to understand what an algorithm is. Or you have to know PA to understand arithmetic. Or you have to know how to co... |
The given answer is $C(10,3)+C(10,4)+C(10,5)+C(10,6)+C(10,7)=912$. Why can't it be $2^4=16$ instead?
I thought that the way to solve the above question might be similar to the question below:
There are 4 choices ${G,U,A,C}$ for each element. How many $6$-element sequences ending with $GU$ can we form?
The an... |
How many bit strings of length $10$ contain at least three $1$'s and at least three $0$'s? |
### Context
Suppose I have two measurable spaces $(\mathsf{X}, \mathcal{X})$ and $(\mathsf{Y}, \mathcal{Y})$ on which we have two probability measures $\pi$ and $\nu$ respectively. I form a joint distribution $\mu = \pi\otimes \nu$ on the joint space $(\mathsf{Z}, \mathcal{Z}) := (\mathsf{X}\times\mathsf{Y}, \mathcal{... |
What is the formula or piecewise function that gives how many moves [or half moves (plies)] are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random N-piece chess position if any checkmate exists? (also consider the positions that there is only one checkmate variation) ... |
Consider the following equation:
$$\left (A + \text{sgn}(x)\text{sgn}(x)^T \right )x= y \, x \tag{1} $$
where $A$ is an $n \times n$ **symmetric** matrix. The variables are $y \in \mathbb R, x \in \mathbb R ^n$ with $\| x \|_2=1$. Note this can be equivalently written as
$$Ax +\|x\|_1 \text{sgn}(x) =y \, x.... |
What is the formula or piecewise function that gives how many moves [or half moves (plies)] are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random N-piece chess position if any checkmate exists? (also consider the positions that there is only one checkmate variation) ... |
Suppose that $F$ is a global field containing $l$-th roots of unity where $l$ is a prime integer and
$l\neq\mbox{char}\ F$ and that $S$ is a finite set of places $v.$ Given some $t_v\in F_v(c^{1/l})$ where $c\in F$ and $c^{1/l}\notin F_v$ for all $v\in S$. Can we find an element $t\in F(c^{1/l})$ such that $t/t_v$ is... |
Let $Y$ be a scheme and $\mathcal{G}$ a *presheaf* on any reasonble site over $X$; to have something concrete in mind let's take the small etale site $X_{\text{et}}$. Let furthermore $f:X \to Y$ any morphism of schemes.
There exist a presheaf pullback functor $f^P$ associating to $\mathcal{G}$ a presheaf on $X$ d... |
I have a homework problem that seems wrong. Previous to the problem we showed that the inner product of a vector space $V$ on the field $\mathbb{C}$ defined as
$$ \langle v,w \rangle = a^i(b^j)^*\delta_{ij} $$
where $a^i, b^j$ are the components of the vectors $v = a^i e_{i}$, $w = b^i e_{i}$ satisfy the inner produc... |
The given answer is ${10 \choose 3}+{10 \choose 4}+{10 \choose 5}+{10 \choose 6}+{10 \choose 7}=912$. Why can't it be $2^4=16$ instead?
I thought that the way to solve the above question might be similar to the question below:
There are 4 choices ${G,U,A,C}$ for each element. How many $6$-element sequences endi... |
What is the formula or piecewise function that gives how many moves [or half moves (plies)] are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random $N$-piece chess position if any checkmate exists?
Also consider the positions that there is only one checkmate variat... |
Just for a University reference, from the [Cambridge Dictionary Definition of a Prime][1]:
> a number that cannot be divided by any other number except itself and
> the number
Clearly, if negative numbers were allowed as factors, there would be no primes. So the number under consideration to be a prime itself c... |
In the context of the Discrete Fourier Transform, a function in time may be expanded as a Fourier series,
$$ f_m = \frac{1}{T} \sum_{n=0}^{N-1} \tilde{f}_n e^{2\pi inm/N}, $$
where $ f_m $ is the function in the time domain in time step $ m\Delta t $, $ f_m = f(m\Delta t) $, $ T = N \Delta t $ is the period, $ f(... |
If you take a random messy Cauchy problem like
$$y'=y^3+xy+e^x,\quad y(0)=1$$
Chances are it will explode (if it does) at a some $x$ values that are nothing special, like $-2.8172...$ and $1.2234...$. Recently [I posted][1] about an ODE that seemed hard to solve, at least at my level, because it involved some Bessel ... |
What are examples of surprisingly well behaved Cauchy problems? |
A disjoint union of open balls is of course disconnected. [Here](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/spaces-which-cannot-be-written-as-a-countable-disjoint-union-of-closed-subsets/D24FAD263DF49239740F1A548826CA1B) it is proved that a locally compact, connected, Hausdorff space... |
Must a countable disjoint union of closed balls in $\mathbb{R}^n$ with positive radius be disconnected? |
Let $K$ be a field and $R=K^{2 \times 2}$.
Let $U$ be a subspace of $\mathbb{Q}^2$ and $L=\{A \in R: coim(A) \subset U\}$. Then $L$ is a left ideal of $R$.
For $L$ to be a left ideal the difference of two elements of $L$ has to be in $L$ as well. This should be clear because $U$ is a subspace which has exactly th... |
Take the partial sums
$$ \sum_{n=0}^N\frac{(-1)^{n+1}}{4n+1}=-\sum_{n=0}^N\int_0^1 (-x^4)^ndx=-\int_0^1\frac{1-(-x^4)^{N+1}}{1+x^4}dx. $$
Now,
$$ \left|\int_0^1\frac{(-x^4)^{N+1}}{1+x^4}dx\right|\leqslant\int_0^1 x^{4N+4}dx=\frac{1}{4N+5}\underset{N\rightarrow +\infty}{\longrightarrow}0 $$
so, letting $N\rightarrow... |
Let $G$ be a (complex, compact, commutative) Lie group. Apparently the endomorphism ring $\textrm{End}(G)$ of $G$ (i.e., holomorphic group homomorphisms) acts on the cotangent space $T^\ast_eG$ at the identity $e \in G$.
If $\phi \in \textrm{End}(G)$ and $\omega \in T^\ast_eG$ then
\begin{align}
\phi \cdot \omega ... |
Let $\theta_i$ be a Bernoulli random variable of success probability $p$.
Given two fixed sets $A = \{a_1, a_2, ..., a_N\}$ and $B = \{b_1, b_2, ..., b_N\}$ where $a_i$ and $b_i$ are constants.
For any realisations $Z = \{\theta_1, \theta_2,..., \theta_N\}$, wee define two random variables $X$ and $Y$ such that:
... |
"Self reference is a semantic notion. Don't warry." But if it has been obtained using self-application? In that case it is a syntactic problem, and perhaps demonstrable in the PR system.
The formula satisfied by all formulas that are not self-applicable, and only by them, is self-applicable if and only if it is not ... |
I have the coordinates of the vertices of a polytope in $R^3$ and no other information. I'd like to count the faces of this polytope. I can determine which vertices are coplanar; this will pick out the faces but it will also include internal sections. Is there a way to distinguish between the two? |
I recently come by this little problem: https://vnexpress.net/dap-an-bai-toan-dem-so-tam-giac-3310485.html. It shows that we can make 50 triangles with sides equal to 10 or less. But it is more of listing than math.
I wonder how many triangles can we make with all sides equal or less than 2n.
Here my approach:
... |
Given an edge set $T$ of a complete bipartite graph $K_{n,n}$ such that $|T|=n+2$ for $n\ge 4$, can we find a proper edge coloring of $K_{n,n}$ (if two edges are incident, then they receive different colors) using $n$ colors such that any color class intersects with $T$ in at most two edges? |
There's a nice feature I came up with using GeoGebra about four years ago. I have some kind of argument for it but it's not entirely rigorous, I hope to formalize it in the answers.
I would also be happy if anyone could provide a proof of pure Euclidean geometry
Given: $∆ABC$ is a triangle, the equilateral triang... |
Take the partial sums
$$ \sum_{n=0}^N\frac{(-1)^{n+1}}{4n+1}=-\sum_{n=0}^N\int_0^1 (-x^4)^ndx=-\int_0^1\frac{1-(-x^4)^{N+1}}{1+x^4}dx. $$
Now,
$$ \left|\int_0^1\frac{(-x^4)^{N+1}}{1+x^4}dx\right|\leqslant\int_0^1 x^{4N+4}dx=\frac{1}{4N+5}\underset{N\rightarrow +\infty}{\longrightarrow}0 $$
so, letting $N\rightarrow... |
I have the coordinates of the vertices of a polytope in $R^3$ and no other information. I'd like to count the faces of this polytope. I can determine which vertices are coplanar; this will pick out the faces but it will also include internal sections. Is there a way to distinguish between these two? |
How to distinguish faces of a polytope from internal sections? |
I have the following stochastic process represented by
$$S_t = 1+\int_0^t \exp\left(\sigma B_s - \sigma^2\frac{s}{2}\right)\,dB_s\,,$$
where $B_t$ is a standard Brownian motion.
I would like to compute its expectations. I thought I could do it in two ways: $(1)$ using the fact that expectation of Ito integral is ... |
Consider the first-order non-linear difference equation:
\begin{equation}
k_{t+1} = sf(k_t) + (1-\delta)k_t
\end{equation}
where $s$ and $\delta$ are two constants and $s,\delta \in (0,1)$.
I assume that $f: \mathbb{R}_+ \to \mathbb{R}_+$ is twice continuously differentiable and $f'(k)>0$ and $f''(k)<0$ for all $k... |
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle lies wholly within the polygon.
**Solution**:
Let $ABCDEF$... be the polygon, $PGR$ the triangle formed by joining th... |
My book<sup>1</sup> says: Let K be the set of real numbers that are sums of the series of the form $\sum_{k=1}^{\infty} \frac{a_k}{3^k}$ with ${a_k} \in \{0, 2\}$.
I can see that sum where ${a_k}=0$ goes to $0$: $$\sum_{k=1}^{\infty} \frac{0}{3^k} = \frac{0}{3^1}+\frac{0}{3^2}+\frac{0}{3^3}+\frac{0}{3^4}.... = 0$$ ... |
I tried solving this problem and I got into a nice way, sadly I think I might be wrong.
What I thought of doing is separating (0/1)H:MM and 2H:MM from each other while calculating
While calculating (0/1)H:MM I came to this calculation: 2 * 9 * 4 * 7 = 588
2 - for two options
9 - for one less option (0 to 9... |
Can someone help me to see where the problem of the functional derivative below is?
**Minimize the functional:**
$$ F[p(\hat{x}|x)] = I(X; \hat{X}) + \beta \sum_{x \in X}\sum_{\hat{x} \in \hat{X}} p(x,\hat{x})d(x,\hat{x}) $$
**Subject to the constraint:**
$$ \sum_{\hat{x} \in \hat{X}} p(\hat{x}|x) = 1 \quad \te... |
I am attending a introductory model theory course this semester. The professor started talking about formulas and sentences without paying much attention to the formalism of the syntax. However, as the course progressed, I began to notice that we need to put the formulas in a set because Zorn's lemma is needed to prove... |
Consider the first-order non-linear difference equation:
\begin{equation}
k_{t+1} = sf(k_t) + (1-\delta)k_t
\end{equation}
where $s$ and $\delta$ are two constants and $s,\delta \in (0,1)$.
I assume that $f: \mathbb{R}_+ \to \mathbb{R}_+$ is twice continuously differentiable and $f'(k)>0$ and $f''(k)<0$ for all $k... |
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