instruction stringlengths 12 30k |
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*Given the following Lagrangian function:*
$$L(x, y, λ_1, λ_2)=x-y^2+2y+11-λ_1(3x+2y-4)+λ_2x$$
*And given the first order conditions:*
I. $\frac{∂L}{∂x}=1-3λ_1+1λ_2=0$
II. $\frac{∂L}{∂y}=-2y+2-2λ_1=0$
III. $λ_1(3x+2y-4)=0$
IV. $λ_2x=0$
*Find the optimal solution.*
I've done similar exercises bef... |
Wikipedia states the following (can be found at the bottom of [this section][1]):
> However, differential entropy does not have other desirable
> properties:
>
> It is **not invariant under change of variables**, and is therefore
> most useful with dimensionless variables.
Where differential entropy is as ... |
Why is differential entropy (continuous entropy) not invariant under change of variables? |
What is the probability of drawing a blue ball on the 3rd trial (without replacement)? |
Let $E/K$ be an elliptic curve over a field $K$.
Let $Aut(E)$ be a group of all automorphism of $E$ as algebraic curves.
Then, does $AutE\cong E$ holds ?
There is a natural injection $E\to \text{Aut}(E)$ given by $P\mapsto \{E\to E, Q\mapsto Q+P\}$.
To prove this is surjective, we should prove every automor... |
Why is this integral not invariant under change of variables? (differential entropy function / continuous entropy) |
Without having the book at hand, I am slightly speculating. However, as stated, this construction defines $n$-tuples as functions. So the equality $(x_i) = (y_i)$ is an equality of functions.
More often than not, equality of two functions requires three ingredients: (1) equality of domains, (2) equality of codomain... |
Is the following statement true?
>Let $N_k=3\cdot 2^k+1$. Then $N_k$ is a prime prime if and only if $2^{N_k-1}\equiv 1 \pmod {N_k}$
One implication is simply a Fermat's Little Theorem, so the difficult part is proving or disproving that $2^{N_k-1}\equiv 1 \pmod {N_k}$ implies $N_k$ is a prime. Or in other words ... |
Is $N=3\cdot 2^k+1$ prime if and only if $2^{N-1}\equiv 1 \pmod N$? |
*Given the following Lagrangian function:*
$$L(x, y, λ_1, λ_2)=x-y^2+2y+11-λ_1(3x+2y-4)+λ_2x$$
*And given the first order conditions:*
I. $\frac{∂L}{∂x}=1-3λ_1+1λ_2=0$
II. $\frac{∂L}{∂y}=-2y+2-2λ_1=0$
III. $λ_1(3x+2y-4)=0$
IV. $λ_2x=0$
*Find the optimal solution.*
I've done similar exercises bef... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by V.V. Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satis... |
My research is on radar images and the images are collected in several conical surfaces. These conical surfaces have the same origin, the same maximum length (max flare or max range), but different elevations angles. The images are collected on the surface of the cones only.
I want to determine the velocity field i... |
Need help solving a system of three trigonometric equations in three unknowns. These equations are used to find the coordinates $φ_s$, $θ_s$ and time $t_s$ at the signal source.
($φ_1$, $θ_1$); ($φ_2$, $θ_2$); ($φ_3$, $θ_3$) - coordinates of the three points to which the signal arrived.
$t_1$, $t_2$, $t_3$ - the ti... |
I am looking for a proof that the commutator subgroup of F is simple, I have found lots of articles about the commutator subgroup in general but havent found anything directly involving F and am struggling apllying the general case to F. Does anybody know of any articles that have this proof explicitily? Thanks :) |
I'm trying to understand the following extract from a book.
> Let $coord$ be a mapping from set $D \to D'$. Let's use the notation $D_{d'}$ to indivate the representing block of $d'$, i.e., the subset of $D$ containing the elements of $D$ mapping to $d' \in D'$ through $coord$.
>
> Given a set of maps
>
> $h_... |
I can't simplify the first one, but $(\tan{x}-1)(\tan{x}+1)^{2}=\tan^{3}{x}+\tan^{2}{x}-\tan{x}-1$ so your second expression should (unless I've made an error - it happens) become $((\tan{x}-1)(\tan{x}+1)^{2})^{\frac{1}{3}}$
Does this help? |
Is there any way to expand these two fomulas below:
$\sqrt[3]{\tan x+1}$
$\sqrt[3]{\left(\tan x\right)^{3}+\left(\tan x\right)^{2}-\tan x-1}$? |
anyway to expand $\sqrt[3]{\tan x+1}$ or $\sqrt[3]{\left(\tan x\right)^{3}+\left(\tan x\right)^{2}-\tan x-1}$? |
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 as an algebra homomorphism over $T(V)$ (through the universal property of $... |
Need help solving a system of three trigonometric equations in three unknowns. These equations are used to find the coordinates $φ_s$, $θ_s$ and time $t_s$ at the signal source.
($φ_1$, $θ_1$); ($φ_2$, $θ_2$); ($φ_3$, $θ_3$) - coordinates of the three points to which the signal arrived.
$t_1$, $t_2$, $t_3$ - the ti... |
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \right)}{\pi^2 (1-t)^2 x^2} \right) + \te... |
Let $I$ a non-trivial interval and $f \in D^{2}(I)$ such that $f''(x) = f(x)$ for all $x \in I$. Show that, if there exists some $a \in I$ such that $f(a) = f'(a) = 0$, then $f(x) = 0$ for all $x \in I$.
My attempt:
A reasonable guess I've been pondering is that $f$ could be written as
$$f(x) = be^{x}+ce^{-... |
I'll give a proof here, which uses Hahn-Banach theorem and fills out the holes in the answer of the [post][1] mentioned above.
We consider the collection $\{H_x\}_{x\in I}$ of functionals on $L^\infty$, where each $H_x$ is the Hahn-Banach extension of the functional $G_x: C(I) \to \mathbb{R}$ defined by
$$
G_x(f... |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You have the decision to ca... |
Need help solving a system of three trigonometric equations in three unknowns. These equations are used to find the coordinates $φ_s$, $θ_s$ and time $t_s$ at the signal source.
($φ_1$, $θ_1$); ($φ_2$, $θ_2$); ($φ_3$, $θ_3$) - coordinates of the three points to which the signal arrived.
$t_1$, $t_2$, $t_3$ - the ti... |
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \right)}{\pi^2 (1-t)^2 x^2} \right) + \te... |
If I am not mistaken, the norm operator topology should make the set of bounded operators into a sequential space, since the norm defines a metric. I was wondering if the Weak and Strong Operator topologies also turn the bounded operators into a sequential space, i.e. is a sequentially closed set in those topologies au... |
Are Strong- and weak Operator topologies on separable Hilbert spaces sequential? |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You have the decision to ca... |
Let $E/K$ be an elliptic curve over a field $K$.
Let $Aut(E)$ be a group of all automorphism of $E$ as algebraic curves.
Then, does $AutE\cong E$ holds ?
There is a natural injection $E\to \text{Aut}(E)$ given by $P\mapsto \{E\to E, Q\mapsto Q+P\}$.
To prove this is surjective, we should prove every automor... |
Suppose that there is a sequence defined as below
$$a_0=0,a_{n+1}^3=a_n^2-8$$
How to prove the series $\sum_{n=0}^{+\infty}|a_{n+1}-a_n|$ convergens?<br>
I know that $a_n$ converge to a fixed point,but the question is how to estimate how fast $a_n$ converges to the fixed point. |
How to prove this series converges? |
Suppose that there is a sequence defined as below
$$a_0=0,a_{n+1}^3=a_n^2-8$$
How to prove the series $\sum_{n=0}^{+\infty}|a_{n+1}-a_n|$ convergens?<br>
I know that $a_n$ converge to a fixed point, but the question is how to estimate how fast $a_n$ converges to the fixed point. |
If I understand correctly, the question is how to prove that the family
$$\tag{*}
\left\{\sum_{k=1}^M \beta_k \prod_{j=1}^{N_k} \varphi(w_{jk}^T x+\theta_{jk})\ :\ \beta_k, \theta_{jk}\in\mathbb R, w_{jk}\in\mathbb R^n\right\}$$
is an algebra. That is rather obvious for the following reason. Let me rename
$$\tag{$... |
Suppose that there is a sequence defined as below
$$a_0=0,a_{n+1}^3=a_n^2-8$$
How to prove the series $\sum_{n=0}^{+\infty}|a_{n+1}-a_n|$ converges?
I know that $a_n$ converge to a fixed point, but the question is how to estimate how fast $a_n$ converges to the fixed point. |
$a_0=0,a_{n+1}^3=a_n^2-8.$ How to prove the series $\sum |a_{n+1}-a_n|$ converges? |
HINT: being surjective means that $\dim(\operatorname{null}T)=2$ as $\dim(\operatorname{img}T)=\dim V_2=5$. Now, as $T$ is surjective, the map $\tilde T:V_1/\operatorname{null}T \to V_2$ given by $\tilde T(\operatorname{v+null}T)=Tv$ is an isomorphism.
Thus $H:=\tilde T^{-1}(W)$ have dimension $2$ and observe that $... |
In Hartshorne's proof of Bertini's theorem (CH. 2, Theorem 8.18), I am having difficulty in the first part of the proof which tells us that the scheme $H \cap X$ is regular at every point.
I could follow till the statement that $x \in H \cap X \iff \varphi_x(f) \in \mathfrak{m}_x$.
(Q-1) I just want to confirm he... |
I know that the union of countably null sets must be measurable and have measure 0, but what happens when the number of sets is uncountable?
For example, the union measure of all single point sets on [0,1] is 1, so the measure changes after taking the union, but is still measurable. |
Is the union of any set of Lebesgue null test sets a measurable set? |
Let $Cat^{rex}$ be the category whose objects are (small) categories admitting finite colimits and let $Cat^{lex}$ be the category whose objects are (small) categories admitting finite limits.
My first question is: $Cat^{lex}$ and $Cat^{rex}$ are equivalent?
I suspect that the answer is yes and the the equivalence fu... |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You have the decision to ca... |
A and B flip a fair dice until 6 occurs. I have read [this post][1] and tried to work out the expectation of flipping on A wins. But it seems something goes wrong. Following is my work:
A wins iff $6$ occurs in odd round, so the EXP flippings on A wins is therefore:
$$E = \frac{1}{6}*1 + (\frac{5}{6})^2*\frac{1}{6}... |
A and B flip a fair dice until 6 occurs. Why was my calculation wrong? |
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \right)}{\pi^2 (1-t)^2 x^2} \right) + \te... |
The following is considered in Lary Guth's Polynomial Methods in Combinatorics, page 14.
Let $L$ be a set of lines in $\mathbb R^3$. A point $x$ which lies in any three non-co-planar lines of $L$ is called a joint of $L$. Suppose $L$ has $6$ lines. Then, why is it that $L$ has at most $4$ joints?
This has been my ... |
Why is maximum number of joints of 6 lines is 4? |
Greeting,
I have problems understanding the assignment in the field of automata theory and formal languages.
The task says:
Construct a minimal DFA that accepts the following languages:
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[1]: https://i.stack.... |
There is a very simple approach here; just make sure you will assess the special cases, which are very small numbers.
Case $1$: The prime is of the form $p=9k+2$, then:
$$\frac{\frac{4p+1}{9} \times 9-1}{p}.$$
Case $2$: The prime is of the form $p=9k+4$, then:
$$\frac{\frac{2p+1}{9} \times 9-1}{p}.$$
Cas... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
Suppose that there is a sequence defined as below
$$a_0=0,a_{n+1}^3=a_n^2-8$$
How to prove the series $\sum_{n=0}^{+\infty}|a_{n+1}-a_n|$ converges?
I know that $a_n$ converge to a fixed point, but the question is how to estimate how fast $a_n$ converges to the fixed point.
*From comment:* $a_n$ converges to th... |
There is a very simple approach here; just make sure you will assess the special cases, which are very small numbers.
Case $1$: The prime is of the form $p=9k+2$, then:
$$\frac{\frac{4p+1}{9} \times 9-1}{p}.$$
Case $2$: The prime is of the form $p=9k+4$, then:
$$\frac{\frac{2p+1}{9} \times 9-1}{p}.$$
Cas... |
I have the following matrix:
$$
A = \frac{1}{3}
\begin{bmatrix}
14 & -14 \\ -14 & 14
\end{bmatrix}
$$
I know that for finding eigen values I need to solve for $|A-\lambda I|=0$.
I found to solutions: $\lambda_1=\frac{28}{3},\lambda_2=0$.
And for finding eigen vectors I need to solve for $(A-\lambda I)v=0... |
The following is considered in Lary Guth's Polynomial Methods in Combinatorics, page 14.
Let $L$ be a set of lines in $\mathbb R^3$. A point $x$ which lies in some set of three non-co-planar lines of $L$ is called a joint of $L$. Suppose $L$ has $6$ lines. Then, why is it that $L$ has at most $4$ joints?
This has ... |
I'm trying to solve exercise 5.2.2 of Weibel's Introduction to homological algebra : if a spectral converging to $H$ has $E_{p,q}^2=0$ expect for $q=0,1$ then there is a long exact sequence
$$\cdots \to H_{p+1} \to E_{p+1,0}^2 \overset{d}{\to} E_{p-1,1}^2 \to H_p \to E_{p,0}^2 \overset{d}{\to} E_{p-2,1}^2 \to H_{p-1} ... |
There is a very simple approach here; just make sure you will assess the special cases, which are very small numbers.
Case $1$: The prime is of the form $p=9k+2$, then:
$$\frac{\frac{4p+1}{9} \times 9-1}{p}.$$
Case $2$: The prime is of the form $p=9k+4$, then:
$$\frac{\frac{2p+1}{9} \times 9-1}{p}.$$
Cas... |
I'm trying to read Kreisel's paper "Analysis of the Cantor-Bendixson Theorem by Means of the Analytic Hierarchy" 1959.
From what is written in the paper, I'm missing something simple. Here is the fragment (in italic) I'm struggling with.
*THEOREM $0$. If $F$ is in $\Pi^1_1$, then $F_p$ is in $\Sigma^1_2$.*
... |
Is $\lim_{n\to\infty}n\cdot\cos\left(\frac{1}{n}\right) = 0$? |
Say that $\{X_n\}$ are none-negative random variables converging almost surely to zero, and $Y_n$ are a sequence of i.i.d random variable, non-negative, with finite $r$-moment for any $r>0$, and whose distribution does not depend on $n$.
Then $X_nY_n\rightarrow 0$ in probability (using Markov inequality), but is it tr... |
Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following:
$$
\left \| AB \right \|_F \leq f(m,r,n)g(A,B) ,
$$
or
$$
\left \| AB \right \|_F \ge f(m,r,n)g(A,B) ,
$$
where $f(m,r,n)$ is a function of the dimension of matrix $m,r,n$, and $g(A,B)$... |
Lets say we are given a list of durations (5s, 10s, 10s, 15s, 15s, 15s, 25s, 30s....) and we want to find a list of **unique durations** that can be created using this list of single durations.
for example if we have the original list of (5, 5, 15, 25) we can create the following durations:
- 5 -> using one 5 e... |
Consider a continuous non-symmetric random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the left. The probability $p_n(t)$ of being at location $n$ at time $t$ is then described by
$$\frac{\mathrm{d}}{\mathrm{d}t}p_n(t)=k_\mathrm{R}p_{n-1}(t)+k_\ma... |
As [rufflewind said](https://math.stackexchange.com/a/4886995/11107), you intuition of inductive proofs as recursive programs is actually a good one and made precise by proof checkers like Coq, Lean or Agda. I won't go quite there, but would like to highlight some concrete things that would be different.
One thing t... |
I can't prove this result:
[Judson - Abstract Algebra, 2022 edition, Corollary 9.18.][1]
[1]: http://abstract.ups.edu/aata/isomorph-section-direct-products.html#hHS
I understand the proof (lcm formula) for case of direct product of two groups, but I can't prove the case with 3 or more groups. I've tried ... |
The lemma 4.26 states the the following:
> Suppose G is a nonempty open subset of $\mathbb{R}$. Then there exists a closed set $F \subset G \setminus \mathbb{Q}$ such that $\left|F\right| > 0$.
In his proof he constructs such a set by removing from a closed interval $J$ contained in G exponentially decreasing ope... |
Alternative proof to 4.26 in Axler’s MIRA? |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
I can't prove this result:
[Judson - Abstract Algebra, 2022 edition, Corollary 9.18.][1]
[1]: http://abstract.ups.edu/aata/isomorph-section-direct-products.html#hHS
I understand the proof for case of direct product of two groups, but I can't prove the case with 3 or more groups. I've tried to use Inducti... |
I am attending a course on Introduction to Complex Geometry, and the definition they have given me of a holomorphic function between complex manifolds is as follows:
> Let $X,Y$ be complex manifolds. A map $f: X \rightarrow Y$ is $\textbf{holomorphic}$ if 1. f is continuous and 2. $\forall p\in X$, there exist chart... |
In complex geometry, is an holomorphic function continuous by definition? |
In order to remove the question from the unanswered list I will mention the following ($\alpha$-induction means induction up to $\alpha$, also called induction of length $\alpha$):
The fairly common "double induction" is $\omega^{2}$-induction, and it can occur as explained [here][1]. Let's unpack that a bit more t... |
The optimal strategy clearly takes the form of playing until you’ve obtained some threshold value and then cashing out, so we need to determine the threshold. At the last value under the threshold, you know you’re going to cash out if you don’t roll a $5$ or $6$. So the threshold is determined by the condition that a s... |
Assume we meet this problem:
$$
\min_x f(x)\\
s.t. g(x)=0, h(x)\leq 0
$$
Let's assume the solution is $(x^*,\nu^*,\lambda^*)$
Its dual problem is
$$
\max_{\nu,\lambda} l(\nu,\lambda)\\
s.t. \lambda\geq 0
$$
where $l(v,\lambda)=\min_xf(x)+\nu g(x)+\lambda h(x)$. The solution is $\nu^{**},\lambda^{**}$.
... |
Given the ANOVA model $Y_{ij} = \mu_i + \varepsilon_{ij}, \varepsilon_{ij}\sim N(0, \sigma^2)$, $i = 1, 2, \ldots , I, \space j = 1,2, \ldots, n_i$, I am trying to find the MLE's $\hat\mu_1, \hat\mu_2, \ldots , \hat\mu_n, \hat\sigma^2$.
I have that the likelihood function is
$L = \prod_{i=1}^{I}\prod_{j=1}^{n_i} ... |
metabelian groups can be represented by matrices, do we know the exact representation? |
If a sequence of random variables $X_n$ is defined on the space $(\Omega,\mathscr{F},\mathbb{P})$, such that $X_n$ converges to $ X$ a.s. and $X_n$ converges to $ X$ in $L^1$
Is it true that for any sub $\sigma$-field $\ \mathscr{G},\mathbb{E}[X_n|\mathscr{G}]\to\mathbb{E}[X|\mathscr{G}] \quad\mathbb{P}$-a.s. ?
... |
How to show if the assertions are true? |
I can't prove this result:
[Judson - Abstract Algebra, 2022 edition, Corollary 9.18.][1]
[1]: http://abstract.ups.edu/aata/isomorph-section-direct-products.html#hHS
I'll restate the result with my words below, in case the link will stop working in the future:
Result: Let n be a natural number bigger o... |
Prove the following:
Let n be a natural number bigger or equal to 2. Let G1, G2, ..., Gn be groups. Let (g1, g2, ..., gn) be an elemement of G1xG2x ... xGn. The order of (g1, g2, ..., gn) is lcm( ord(g1), ord(g2), ... , ord(gn) )
I understand the proof for case n=2, but I can't prove the case with n larger than... |
Ultimately, the filtration comes from the $E^\infty$ page. So let's attempt to calculate explicitly what this page is. Well, from the $E^2$ page we can only immediately infer the $E^3$ page - so let's focus on that first.
Obviously if $q\neq 0,1$ we have that $E^3_{p,q}=0$; the only subquotient of $0$ is $0$.
If ... |
Prove the following:
Let $n$ be a natural number bigger or equal to $2$. Let $G_1,G_2,\ldots,G_n$ be groups. Let $(g_1,g_2,\ldots,g_n)$ be an elemement of $G_1\times G_2\times\cdots\times G_n$. The order of $(g_1,g_2,\ldots,g_n)$ is $\operatorname{lcm}(\operatorname{ord}(g_1), \operatorname{ord}(g_2),\ldots,\opera... |
I have encountered a PDE
$$K^2\frac{f_{KK}-f_{KKK}}{f_K}+H^2\frac{f_{HH}-f_{HHH}}{f_H}=0$$
where $f$ is homogeneous function of degree 1 in $H$ and $K$.
It is evident that $f=aH+bK$ is a solution of the above equation. But how can I solve it explicitly or can show that $f=aH+bK$ is the only solution.
I have ... |
Ultimately, the filtration comes from the $E^\infty$ page. So let's attempt to calculate explicitly what this page is. Well, from the $E^2$ page we can only immediately infer the $E^3$ page - so let's focus on that first.
Obviously if $q\neq 0,1$ we have that $E^3_{p,q}=0$; the only subquotient of $0$ is $0$.
If ... |
I'm reviewing answers to this [code golf][1] challenge:
> The goal of this challenge is to generate and output a $16 \times 32$ (height
> × width) grid where all cells contain a decoy character, except for a
> random single cell that contains the target character. The input will
> be a pair of ASCII ($32-126$) ... |
I've found [this][1] demonstration (for [this][2] problem taken from "*Mathematical methods of Classical Mechanics*" by V. I. Arnol'd), and I could not decode this particular step:
> A $k$-dimensional plane can be uniquely identified by the value of its
> symplectic form. The symplectic form takes values in the int... |
The question is "Use Warshall’s Algorithm to find the transitive closure of the relation {(a, c), (b, d), (c, a), (d, b), (e, d)} on the set {a, b, c, d, e}."
My matrix at the end is:
( 1 0 1 0 0 )
( 0 1 0 1 0 )
( 1 0 1 0 0 )
( 0 1 0 1 0 )
( 0 1 0 1 0 )
However, I did some code to do the Warshall's Algorithm... |
Let $\tau$ be the most coarse topology on $\mathbb{R}$ that contains the euclidean topology and all subsets of $\mathbb{Q}$. We already know that $\tau_{Eucl} \cup P(\mathbb{Q}$ is a base for $\tau$.
We also have a metric $d$ on $\mathbb{R}$ defined as the following. Let $(r_k)_{k\in \mathbb{Z}_{>0}}$ be a sequence of... |
I am following a first coruse on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$
But, how can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is ... |
I am taking a first course on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$
But, how can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is tru... |
I am taking a first course on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$
How can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is true) th... |
I am taking a first course on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$
How can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is true) th... |
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \right)}{\pi^2 (1-t)^2 x^2} \right) + \te... |
The question is "Use Warshall’s Algorithm to find the transitive closure of the relation {(a, c), (b, d), (c, a), (d, b), (e, d)} on the set {a, b, c, d, e}."
My matrix at the end is:
( 1 0 1 0 0 )
( 0 1 0 1 0 )
( 1 0 1 0 0 )
( 0 1 0 1 0 )
( 0 1 0 1 0 )
However, I did some code to do the Warshall's Algorithm... |
Let $X$ be a randomm variable with CDF $F_X$. If $F_X$ is continuous, is $\text{im}(X)$ uncountable? |
Suppose there is a **d-regular** **graph** with **n** nodes. Now we randomly **delete k nodes** out of them. What is the **probability** of the graph getting **disconnected**? |
Consider a continuous biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the left. The probability $p_n(t)$ of being at location $n$ at time $t$ is then described by
$$\frac{\mathrm{d}}{\mathrm{d}t}p_n(t)=k_\mathrm{R}p_{n-1}(t)+k_\mathrm{L}... |
Limit[-x^6y^1(x^2+1)/(x^6+y^2)Sqrt[x^2+y^2],(x,y)->(0,0)].
[image of the limit to be clear][1]
I need find if the limit exists or not. First i approached the limit from y=0 and the result was 0/x^7. Then i approached from x = 0 and the result was 0/y^3. This made me assume that the limit does not exist. However wo... |
Let $X$ be a random variable with cdf $F_X$. If $F_X$ is continuous, is $\text{im}(X)$ uncountable? |
I am working on my homework and I solved this problem and I just wanted someone to double check to see if my answer makes sense.
5.4.3: Lining up club members for a photo.
Ten members of a club are lining up in a row for a photograph. The club has one president, one VP, one secretary, and one treasurer.
(c)
H... |
This problem should be simple to solve for you guys, but not for me.
My Gun Club holds an annual raffle. There are 15 prizes, you can only win one prize per ticket. I purchased 2 tickets and there were 3500 tickets sold. What are the odds of winning?
This is the most tickets they have ever printed. My argument is tha... |
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