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I was working on an assignment and had a good idea to have a water jet model an epitrochoid to create a fountain. While the overall idea was approved by my teachers, I was told that I need to show a proof of the derivation of the parametric equations for the epitrochoid.
the particular Desmos model of what I am mod... |
I am studying a family of random matrices and observe a universal concentration behavior which I would like to understand.
Let $U$ be an $N \times N$ random unitary drawn from the Haar unitary, and $Z = \text{diag}(1,1,1\cdots,-1,-1,-1\cdots)$ (equal ones and minus ones). Define the random variable $\eta(U) := U Z ... |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
Is it always true that if $x_n\to0$, $y_n\to0$ there exist $\epsilon_n\in\{-1,1\}$ such that both $\sum\epsilon_nx_n$and $\sum\epsilon_ny_n$ converge? |
Is this an adequate definition of the natural numbers?
The term *number* is synonymous with *natural number* in this context. By *mutually distinct ordered pair* I intend that the front element is not the back element. The terms *front* and *back* are used in order to avoid using numbers *first* and *second* to de... |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
I am studying a family of random matrices and observe a universal concentration behavior which I would like to understand.
Let $U$ be an $N \times N$ random unitary drawn from the Haar measure, and $Z = \text{diag}(1,1,1\cdots,-1,-1,-1\cdots)$ (equal ones and minus ones). Define the random variable $\eta(U) := U Z ... |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
*Hint for the second question:* work in $\mathbb R^2$ with the sequence $\left(x_n, y_n\right)$ can you say something about $$\mathcal A_k := \left\{n \in \mathbb N : x_n^2 + y_n^2\le \frac1{(k+1)^2}\right\} ?$$ |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
I'm thinking about a question, that is if *E* is a rational elliptic curve, *K* is a cyclotomic field, how can I find the minimal subfield of *K* such that the torsion subgroup over *K* can also be defined over this subfield? I have found some papers for classification of torsion subgroups over **Q**^ab, quadratic fiel... |
*Hint for the second question:* work in $\mathbb R^2$ with the sequence $\left(x_n, y_n\right)$ can you say something about $$\mathcal A_k := \left\{n \in \mathbb N : x_n^2 + y_n^2\le \frac1{(k+1)^4}\right\} ?$$ |
There's a **equation**
$$N(t) = N(t)\frac{P(t,z)}{B}-C\frac{d(P(t,z))}{dz}$$
$$N(t) = A\frac{dP(t,z)}{dt}$$
**Constants:**
B,
C=3.9878*10⁻⁷,
$A=0.11941$,
**Variables:**
N(t) is a function of t and is defined at a point along z, it's a vector quantity.
P(t) is a function of t as well as z
**B... |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
I saw this interesting problem:
Let $x_n$ and $y_n$ be real sequences with $x_n \to 0$ and $y_n \to 0$ as $n \to \infty$.
- Show that there is a sequence $\varepsilon_n $ of signs (i.e. $\varepsilon_n \in \{−1, +1\}$ for all $n$) such
that $\sum \varepsilon_n x_n$ is convergent.
- Must there be a sequence $\v... |
I have a linear transformation $F: U \text{->} V$. Given that the vectors $\vec{v_1}...\vec{v_n} \in V$ are linearly independent I want to show that $F$ is one-to-one.
Also $F(\vec{u_i}) = \vec{v_i}$ and $\vec{u_i} \text{ form the basis of } U$ $\forall i = 1,2,...,n$.
My approach was
$$s_1\vec{v_1}+...+s_n\vec{... |
*Hints for the second question:* work in $\mathbb R^2$ with the sequence $z_n = \left(x_n, y_n\right)$ without loss of generality we can assume that, $\left\|z_n \right\| \le 1$. For $k\in \mathbb N$, let $$\mathcal A_k := \left\{n \in \mathbb N : \frac1{(k+2)^2} < \left\|z_n\right\|\le \frac1{(k+1)^2}\right\}$$ Since ... |
Is this an adequate definition of the natural numbers?
The term *number* is synonymous with *natural number* in this context. By *mutually distinct ordered pair* I intend that the front element is not the back element. The terms *front* and *back* are used in order to avoid using numbers *first* and *second* to de... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/rrmkS.jpg
So based on the diagram here's what I've attempted:
> Note: $\bar{g}$ is the Earth's gravitational constant and $T$ is the tension force acting on the Blocks.
> $a_1,a_2,a_3$ will be the acceleration for Block A, B, and... |
We're given a random red line and a point $A$ outside of it.
From the red line, we take a random point $B$ such that the circle $\omega = \odot (A,AB)$ cuts the red line again at $C \neq B$.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/RASKL.png
Let $D = \odot(B,BA) \cap \odot ... |
Consider a sequence of polynomials $P(x)=x^n+x^{n-1}+x^{n-2}+...+x^2+x-1, n>2$.
(i) Prove that it has a unique positive real root $x_n$
(ii) Find $$\lim_{n \to ∞} 2^n (x_n - 1/2)$$
The first part was relatively simple; upon differentiating $P(x)$, it is easy to see that for all $x>0$, the function is strict... |
We're given a random red line and a point $A$ outside of it.
From the red line, we take a random point $B$ such that the circle $\omega = \odot (A,AB)$ cuts the red line again at $C \neq B$.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/RASKL.png
Let $D = \odot(B,BA) \cap \odot ... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
My calculus book gives for the derivative of $\operatorname{arcsec}(x)$:
$$\frac{d}{dx} \sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}$$
Then it continues to state that:
> "Some authors prefer to define $\sec^{-1}$ as the inverse of the restriction of $\sec(x)$ to the separated intervals $[0,\pi/2)$ and $[\pi,3\pi/2)$ beca... |
I am solving the differential equation
$$\left(y+y \sqrt{x^{2} y^{4}-1}\right) d x+2 x \ d y=0$$
and I have to use the substitution $u=xy^{2}$ to find the generic solution:
I have expressed $$y'=dy/dx = -\frac{y(1+ \sqrt{x^2y^4-1})}{2x}$$ I have $u'=y^{2}+2xy y'$, hence $u'=-y^{2}\sqrt{u^2-1}$. (I just substituted... |
I am struggling with a logarithmic equation where I am required to find integer solutions for `x` such that `a` is also an integer and greater than 4. The equation is given as:
$a = \log_2\left(\frac{9 \cdot 2^x - 112}{2^x - 13}\right) - 1$
To clarify, I am looking for the values of `x` that make `a` an integer... |
Let $F$ be a free group of rank at least two, $A$ free a factor of $F,$ and $x$ a primitive element in $F \setminus A.$ Suppose that for some/every basis $\mathcal A$ of $A,$ the set $\mathcal A \cup \{x\}$ is not a primitive system in $F$ (does not generate a free factor of $F$). What can be said about the element $x,... |
I am struggling with a logarithmic equation where I am required to find integer solutions for `a` such that `x` is also an integer and greater than 4. The equation is given as:
$a = \log_2\left(\frac{9 \cdot 2^x - 112}{2^x - 13}\right) - 1$
To clarify, I am looking for the values of `x` that make `a` an integer... |
I am solving the differential equation
$$\left(y+y \sqrt{x^{2} y^{4}-1}\right) d x+2 x \ d y=0$$
and I have to use the substitution $u=xy^{2}$ to find the generic solution:
I have expressed $$y'=dy/dx = -\frac{y(1+ \sqrt{x^2y^4-1})}{2x}$$ I have $u'=y^{2}+2xy y'$, hence $u'=-y^{2}\sqrt{u^2-1}$. (I just substituted... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
How to calculate the fundamental group of discrete group like the cyclic group $Z_n$?
Physically, it seems that $\pi_1(Z_2)$ will be trivial, as there can be no vortex structure for $Z_2$ symmetry. But there can be some "vortex" structure in $Z_n, n\geq3$, so $\pi_1(Z_n), n\geq3$ may be non-trivial. |
**what is exactly is the conditional probability**
I saw this this kind of definitions for conditional probability
> **Definition:**\
Conditional probability is the likelihood
of an event occurring based on the occurrence of a previous event.
and some definitions follows like this
>**Definition:**\
P(E|F... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
**what is exactly is the conditional probability**
I saw this this kind of definitions for conditional probability
> **Definition:**\
Conditional probability is the likelihood
of an event occurring based on the occurrence of a previous event.
and some definitions follows like this
>**Definition:**\
$P(E|... |
Is this an adequate definition of the natural numbers?
The term *number* is synonymous with *natural number* in this context. By *mutually distinct ordered pair* I intend that the front element is not the back element. The terms *front* and *back* are used in order to avoid using numbers *first* and *second* to de... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold:
1. If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \mathbb{D}^2 =V$. So $V$ is a closed subset of $\mathbb{D}^2$.
2. For any $\lambda\in \mathbb{D}^2-V$, there is a b... |
> **Lemma :** Let $u_1,\ldots,u_n \in \mathbb R^2$ then $$\min_{\epsilon \in \left\{-1, 1\right\}^n} \left\|\sum_{i=1}^n \epsilon_i u_i\right\| \le 6\max_{1 \le i \le n}\left\|u_i\right\|.$$
*Proof:* Induction on $n$. The result is trivially true if $n \le 6$. If $n\ge 6$ and if the statement is true for all $k\le n... |
As we all know,
$$
\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)=1
$$
And I encountered many questions where I used this standard limit and it gave the right answer. However in the following questions I am unable to get the right answer by this.
I actually want to solve this limit.
$$
\lim _{x \right... |
My lecture notes define a regular parametric surface as follows:
> Definition 1: Let $D \subset \mathbb{R}^n$ be open such that $\bar{D}$
> is Jordan measurable. Then a map $f: \bar{D} \to \mathbb{R}^{n+k}$ is
> regular if
>
> $$rank(df_x) = x \quad \text{for all } x \in D.$$
>
> We call $f: \bar{D} \to \mat... |
I tried solving this problem for quite some time now but I think there must be a simple solution I am just not seeing so I would appreciate some kind of hint how to proceed.<br>
**Problem:** Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and $f(x+1) = f(x) \ \forall x \in \mathbb{R}$. Show that $\forall m \in \mathb... |
Supposing $X_{t_i}$ is a real-valued stochastic process defined for $i=1,\ldots, N$, where $N=N_n\rightarrow\infty$, and $a_n\rightarrow 0$ as $n\rightarrow\infty$, I am trying to see if the following holds:
\begin{equation}
\sum_{i=1}^{N}(X^2_{t_i}-a_n^2)=o_p(a_n)
\end{equation}
when we have
\begin{equation}
\ma... |
Consider a closed Riemannian manifold $(M,g)$ and the heat equation $\partial_tu = \Delta_g u$ on it. Let $P_t$ be the heat semigroup generated by the equation. Of course the laplacian is symmetric with respect to the $L^2(dvol_g)$ inner product. Is there an easy way to see that also $P_t$ is symmetric? On what space o... |
Let $n$ people hold $n$ dice, throwing all at once. What's the probability that we find exactly $k $ 6's , $k \leq n$?
My approach is this experiment is equivalent to throwing the dice successivly n times. So the probability we have exactly $k$ 6's is :
$${n \choose k}\left(\frac{1}{6}\right)^k \left(\frac{5}... |
If $n$ people throw $n$ dice, what's the probability that we find exactly $k$ 6's? |
Let $V$ be a (possibly infinite dimensional) vector space, and $U$ a non-zero subspace of $V$ such that $U\neq V$. Also, let $F = \{f_i\}$ be the family of linear functionals on $V$ such that $U \subseteq \text{ker}(f_i)$. Then,
1. Is there always a non-zero function in $F$? And one for which $\ker{(f)}=U$?
2. Mu... |
I'm studying logical proposition. And I know that $P$ is a proposition if it's either true (T) or false (F), and if it can't be true and false. I know for exemple that "tomorrow it will be sunny" is not a proposition, because we can't know if it's true or false.
1) What about *"yesterday it was sunny"* ? I guess th... |
Is "yesterday it was sunny" a proposition? |
Let, *K* be a set of subgroups of *S*<sub>5</sub> (symmetric group of 5 elements) that are isomorphic to the non-cyclic group of order 4. How many conjugacy classes are there in *K*?
I know that a non-cyclic group of order 4 is isomorphic to *K*<sub>4</sub>. Now, do I have to find manually all the subgroups isomorph... |
The statement of the Calderon-Zygmund decomposition as given in Duoandikoetxea's book is the following.
For $f$ integrable on $\mathbb{R}^n$ and non-negative and given $\lambda>0$, there exist a sequence $\{Q_j\}$ of disjoint dyadic cubes such that
1) $f(x) \leq \lambda$ for almost every $x \notin \cup Q_j$
2)... |
I've come across different proofs for one of the limit laws, namely,
$$lim_{x\to c} [f - g](x) = lim_{x \rightarrow c}\ f(x) - lim_{x \rightarrow c}\ g(x) .$$
Now, some authors end their proof like so,
$$\begin{aligned}|[f-g](x)-(L-M)| & =|(f(x)-L)-(g(x)-M)| \\ & \leq|f(x)-L|+\mid g(x)-M \\ & <\frac{\epsilon}{2... |
This is a German Olympiad exercise from 2015.
>Call a number $n\in\mathbb{N}$ _smooth_ if there are $a_1,...,a_n\in\mathbb{Z}$ such that
$$
a_1+a_2+...+a_n=a_1\cdot a_2\cdot ... \cdot a_n=n.
$$
Find all smooth numbers.
________
So I think that it is evident that every composite number can be written as
$$
... |
It's not clear to me if this is the "reasoning" you are hoping is explained, but here goes.
When you prove the existence of a limit using the $\epsilon$-$\delta$ definition, you are proving a statement of the form "for every $\epsilon>0$, there exists a $\delta >$ such that..." This means that the first line of your... |
I'm studying logical proposition. And I know that $P$ is a proposition if it's either true (T) or false (F), and if it can't be true and false. I know for exemple that *"tomorrow it will be sunny"* is not a proposition, because we can't know if it's true or false.
1) What about *"yesterday it was sunny"* ? I guess ... |
In my course on Nonlinear PDEs, we investigate the integral functional
$\int_{t_0}^{t_1} L(\dot{\gamma}(t), \gamma(t))dt$, defined on $q$-absolutely continuous curves $\gamma: I \rightarrow \mathbb{R} $ with some interval $I \subset \mathbb{R}.$ They form a Banach space, denoted as $AC^q(I)$ and defined as in the Wiki... |
>Given $U_1, U_2, \ldots, U_n$ i.i.d $\sim \text{Unif}(-1, 1)$, what's the probability that $U_1^2 + U_2^2 + \ldots + U_n^2 \le 1$?
I tried to reduce it to $U_1^2 + U_2^2 \le 1$ and interpret the result geometrically but I'm not able to reason about it properly.
$U_1^2 + U_2^2 \le 1$ represents the circle with ... |
Your question has two parts.
We fix a finite group $G$ of order $n$.
**(1) Does $G$ act transitively on a set $\Omega$ of size $n$?**
The answer is **yes**, and the reason is already found in the standard proof of Cayley's theorem, which considers the the action of $G$ on itself (so $\Omega = G$) via
$$g \bul... |
This is one of exercises from my uni's Discrete math problems. (Non-mandatory) And the job is to either found a counterexample or prove it.
I considered approaching this problem using proof by contradiction – "Connected graph G contains exactly one cut vertex and there exists a longest path such that it doesn't con... |
Let $C_i\sim C$ and $R_i \sim R$ denote the durations of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$ the weathe... |
> Consider a linear operator as defined below
$$S_N[u(x)] : = \int_\Bbb R \chi_{[-N,N]}(\xi)\hat u(\xi)e^{ix\xi}d\xi, \ u \in L^1(\Bbb R)$$
Prove that $S_n:L^1(\Bbb R) \to C^0_b(\Bbb R)$ is well defined and bounded - $C^0_b(\Bbb R)$ is the space of continuos bounded fuction -
I'm having some trouble understa... |
I saw this problem : Suppose $f$ is a real function defined on $\mathbb{R}^1$ which satisfies $\lim\limits_{h\to0 } f(x+h)- f(x-h)= 0$ for every $x \in \mathbb{R}^1$ • Does this imply that $f$ is continuous?
I tried for an hour to prove that $f$ is continuous and failed to (I was very upset because I thought that t... |
Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
**what is exactly is the conditional probability**
I saw this this kind of definitions for conditional probability
> **Definition:**\
Conditional probability is the likelihood
of an event occurring based on the occurrence of a previous event.
and some definitions follows like this
>**Definition:**\
$P(E|... |
Given the Lindeberg-Lévy Central Limit Theroem for i.i.d $X_i$'s,
\begin{equation}
\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}X_i-\mu\right)\overset{d}\rightarrow N(0,\sigma^2)
\end{equation}
I understand this implies
\begin{equation}
\frac{1}{n}\sum_{i=1}^{n}X_i-\mu = O_p\left(\frac{1}{\sqrt{n}}\right)
\end{... |
If you mean the [ Weierstrass M-test ][1], you should be careful about the differences between the theorem's statement and your own problem as in your case we are not dealing with sums.
However, please note that for every $n \in \mathbb N$, $0<\frac{1}{n^{\frac{3}{2}}} \leq 1$ and:
$$|S_n(\frac{1}{n^{\frac{3}{2}... |
Note that either all $|x|+|y|-|z|$, $|y|+|z|-|x|$, and $|z|+|x|-|y|$ are positive or just one of them is negative. Hence,
$$(|x|+|y|-|z|)(|y|+|z|-|x|)(|z|+|x|-|y|)>0 \iff |x|,|y|,|z| \text{ are sides of a triangle.}$$
So, $f^*(x,y,z)=(|x|+|y|-|z|)(|y|+|z|-|x|)(|z|+|x|-|y|)$ can work as a source for generating... |
I saw this problem : Suppose $f$ is a real function defined on $\mathbb{R}^1$ which satisfies $\lim\limits_{h\to0 } f(x+h)- f(x-h)= 0$ for every $x \in \mathbb{R}^1$ Does this imply that $f$ is continuous?
I tried for an hour to prove that $f$ is continuous and failed to (I was very upset because I thought that th... |
Suppose I have two matrices, given as:
$A= \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$, $B= \begin{pmatrix}
e & f \\
g & h
\end{pmatrix}$
Then,the Kronecker product of $A$ and $B$ is
$A \otimes B=\begin{pmatrix}
ae & af & be & bf\\
ag & ah & bg & bh\\
ce & cf & de & df\\
cg & ch & dg & dh
\end{... |
Question: How many ways are there to form a sequence of 10 letters from four 'a's, four 'b's, four 'c's, and four 'd's if each letter must appear at least twice?
My Approach: I started out with the exponential generating function where:
- XA represents the number of 'A's,
- XB represents the number of 'B's,
- XC ... |
Context:
1. Continuous maps between topological spaces are structure preserving in the following sense:
Given two topological spaces $(X,\tau_X),(Y,\tau_Y)$ (where $(X,\tau_X)$ is the topological space of set $X$ endowed with a topology $\tau_X$; similarly $(Y,\tau_Y)$), a map $f:(X,\tau_X)\to (Y,\tau_Y)$ is con... |
What structure do Smooth Maps and Diffeomorphism? |
You can do it with basic combinatorial methods:
What you have are the letters aabbccdd?? with ?? representing wild card letters. There are $\frac{8!}{2!2!2!2!}$. There are 4 possibilities where the two ?? are the same, and $\binom{8}{2}-4=24$ where they are different. There are $\frac{10!}{2!2!4!2!}$ in the first ca... |
> Can I do it directly using permutation with limited repetition?
Sure. Every letter has to appear twice. That means $8$ out of the $10$ positions are already occupied and we have only $2$ positions left to fill. These $2$ positions can be filled using the same letter or two different letters. Let's consider these c... |
[![enter image description here][1]][1]With respect to the graph
Why is the blue vertices have degree sequence {3, 3, 4} and the
yellow vertices have degree sequence {3, 4, 4, 4} ?.
[1]: https://i.stack.imgur.com/unfds.png |
Question: How many ways are there to form a sequence of 10 letters from four 'a's, four 'b's, four 'c's, and four 'd's if each letter must appear at least twice?
My Approach: I started out with the exponential generating function where:
- XA represents the number of 'A's,
- XB represents the number of 'B's,
- XC ... |
In Calculus on Manifolds Spivak gives the following definition of a manifold:
[![enter image description here][1]][1]
But $\Bbb{R}^k \times \{0\}$ is a set of $(k+1)$-tuples and $V$ is a set of $n$-tuples, so
$$V \cap (\Bbb{R}^k \times \{0\}) = \emptyset $$ if $k+1 \neq n$. This doesn't add up with what Spivak st... |
Let $T$ be a, possibly unbounded, self-adjoint operator with a dense domain $D(T)$ in a Hilbert space $H$. If $T=0$ on $D(T)$, can $T$ be different from the $0$ operator, i.e. $Tv=0$ for all $v\in H$? |
**Exercise:**
>$X$ is a normed linear space and $\Vert \quad \Vert$ is a norm on $X$. $x$ is a non-zero element in $X$. Then for all $y\in X$, there is a real number $a$ such that $\Vert y-ax \Vert=\inf \{ \Vert y-bx \Vert: \text{$b$ is real number}\}$.
**My Attempt**
I tried to work it out but failed. I haven... |
**Exercise:**
>$X$ is a normed linear space and $\Vert \quad \Vert$ is a norm on $X$. $x$ is a non-zero element in $X$. Then for all $y\in X$, there is a real number $a$ such that $\Vert y-ax \Vert=\inf \{ \Vert y-bx \Vert: b\in\Bbb R\}$.
**My Attempt**
I tried to work it out but failed. I haven't a clue about... |
I recently watched [this video](https://youtu.be/NmM9HA2MQGI) by Computerphile, where Mike explains the mathematics of the Diffie Hellman exchange. I've been wondering, since as explained $g$, $g^a$ and $g^b$ is public, can't you just do this equation
$$
\frac{1}{log_{(g^a)}(g)} = a
$$
and $$
\frac{1}{log_{(g^b)}(... |
Wouldn't it be easy to find private keys with the Diffie Hellman exchange? |
I am a neophyte undergraduate student reviewing multivariate analysis. I am learning from *Mathematical Tools for Applied Multivariate Analysis* (J.D. Caroll, P.E. Green).
I covered the **co-factor expansion method** to yield the determinant of a matrix. In my textbook, it is said that for a 3x3 matrix, d(A) is ba... |
**Question:** How many ways are there to form a sequence of 10 letters from four 'a's, four 'b's, four 'c's, and four 'd's if each letter must appear at least twice?
**My Approach:** I started out with the exponential generating function where:
- XA represents the number of 'A's,
- XB represents the number of 'B's... |
Examine if
$\{(x, y)\in \Bbb R^2 \mid 2x+3y≤6,2x+3y≥6, x≥0, y≥0\}$
is a convex set
After solving the constraints, we come to the conclusion that the set is basically a line segment joining points (3, 0) and (0, 2) and since a line segment is convex the given set is a convex set. Or we can also use the theor... |
[![enter image description here][1]][1]With respect to the graph
Another concept central to an understanding of fractional isomorphism is that of the iterated degree
sequence of a graph. Recall that the degree of a vertex $v\in G$ is the cardinality of its neighbor set:
$d(v) = |N(v)|$. The degree sequence... |
This is problem 9-1 from John Lee's Introduction to Smooth Manifolds. Suppose $M$ is a smooth manifold $X$ is a smooth vector field on $M$ and $\gamma$ is a maximal integral curve of $X$.
Show that the image of $\gamma$ is an immersed submanifold of $M$, diffeomorphic to $S^1$ if $\gamma$ is periodic and nonconstan... |
I'm studying logical proposition. And I know that $P$ is a proposition if it's either true (T) or false (F), there are no other possibilities than being (T) or (F) and it can't be true and false. I know for exemple that *"tomorrow it will be sunny"* is not a proposition, because we can't know if it's true or false.
... |
A road is constructed with n lines (n>2) sharing exactly one point in the center. There are n cars, each at a unique endpoint.
* Cars can move forwards and backwards, each considered 1 move.
* Cars can't pass each other (road to narrow)
* Cars want to finish at a different endpoint facing away from the center.
... |
Suppose function $f$ and sets $X$ and $Y$ such that $f : X \rightarrow Y$.
What does this notation exactly mean, out of the following options?
- $\text{domain(} f) \subseteq X \wedge \text{range(} f) \subseteq Y$
- $\text{domain(} f) = X \wedge \text{range(} f) \subseteq Y$
- $\text{domain(} f) = X \wedge ... |
Let $\mathcal{H}$ be a Hilbert space and $(\Omega, \mathcal{F}, P)$ a probability space with $L^2$-space $L^2(\Omega)$. I am looking at the space of square integrable $\mathcal{H}$-valed random variables defined as
$$L^2(\Omega,\mathcal{H})=\{X:\Omega \to \mathcal{H} \text{ measurable} : \mathbb{E}[\|X\|^2]<\infty\}... |
I have a simple question. Curve in ${R}^{1,1}$ The 2 dimensional Minkowski space.
\begin{equation}
\gamma:[0,1] \longrightarrow R^{1,1},
\end{equation}
How can I calculate the extrinsic curvature of this curve seen as a 1 dimensional surface embedded in $R^{1,1}$?
Thanks! |
> Can I do it directly using permutation with limited repetition?
Sure. Every letter has to appear twice. That means $8$ out of the $10$ positions are already occupied and we have only $2$ positions left to fill. These $2$ positions can be filled using the same letter or two different letters. Let's consider these c... |
I'm attempting to unpack the following paragraph from a proof of the Model Existence Theorem (p.44 on [this set of notes][1]):
> Suppose we have a consistent $S$ in the language $L = L(Ω, Π)$. Extend $S$ to a consistent $S_1$ such that $p ∈ S_1$ or $(¬p) ∈ S$ for each sentence $p ∈ L$ (by applying Zorn’s lemma to g... |
A road is constructed with n lines (n>2) sharing exactly one point in the center. There are n cars, each at a unique endpoint.
* Cars starts facing the center
* Cars can move forwards and backwards, each considered 1 move.
* Cars can't pass each other (road to narrow)
* Cars want to finish at a different endpoi... |
I am trying to understand the meaning of Peirce's Law, namely:
[(P→Q)→P]→P
What does this mean? I am trying to find a practical example of P and Q where the truth of P would follow from the truth of P→Q. Any suggestion?
|
Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
I have been trying to solve a problem that came to me randomly about a week ago and I am very close to solving it but have reached a roadblock when re-arranging the final equation to solve it. I do not believe the problem is necessary but here is the equation (and the cosign function is using degrees not radians)
((... |
I have been trying to solve a problem that came to me randomly about a week ago and I am very close to solving it but have reached a roadblock when re-arranging the final equation to solve it. I do not believe the problem is necessary but here is the equation (and the cosign function is using degrees not radians)
((... |
Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
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