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Can this divisibility be proven in general? |
The following is from Williams' _Probability with Martingales_ section 3.12 (I paraphrase a bit):
> Let $F:\mathbb{R}\to[0,1]$ have properties
> 1. $F$ is non-decreasing.
> 2. $F$ is normalized.
> 3. $F$ is right-continuous.
> We can construct a random variable with distribution function $F$ carried by $([0,1]... |
The Lecturer demonstrate that $S:=\{(x_n)\in l^2:\sum^{\infty}_{n=1}|x_n|^2\leq 1\}\subset l^2$ is closed by showing every convergent sequence $(a_n)$ in $S$ has the limit $a^*\in S$.
Now I have a question :
Is this space itself $l^2:=\{(x)_n\in\mathbb{R}^{\infty}:\sum^{\infty}_{i=1}|x_n|^2<\infty\}$ closed and... |
I have a nonlinear operator $\mathbf{A}(\theta): X \rightarrow Y$, where $X$, $Y$ are Banach space with $\|\cdot \|_{X}$ and $\|\cdot \|_{Y}$. $\theta \in \mathbb{R}^q$ is the parameter of $\mathbf{A}$. Fixed $\theta$, we can define the operator norm of $\mathbf{A}(\theta)$ as $\|\mathbf{A}(\theta)\|_{\text{op}} = \max... |
Let $X$ be a discrete random variable (meaning $\text{im}(X)$ is countable) from a probability triple $(A,\mathcal{A},P)$ to a measurable space $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the Borel $\sigma$-algebra of $\mathbb{R}$. As [Wikipedia explains][1] (bold is mine):
> The probability density function $... |
There are many results for finite Markov chains ,e.g. Demob "Large deviations: Technique and its applications".
However I cannot find examples even for irreducible Markov chains, except for i.i.d. sequence such as binormial distribution.
Could you please give me some examples and the form of its rate functions? |
I was doing Probability: Theory and Examples problem 5.3.8
[![enter image description here][1]][1]
To use Theorem 5.3.8, we need to prove $\mathbb{E}_x[\phi(X_1)]<\phi(x)$, which is just $\mathbb{E}_x[X_1]<x$. After some simple calculations, we have $\mathbb{E}_x[X_1]=px+\lambda$, so we must have $\lambda<x(1-p)$ for... |
I am stuck trying to solve this equation using Laplace transform:
> $$ y''(t)+2y(t)=\displaystyle 3\int_0^t (t−u)y(u)du\\ y(0)=2 , y'(0)=3 $$
I got stuck after performing the Laplace transformation and can't get the equation to a form I can do the inverse Laplace transformation on.
Here's what I got:
$$p^... |
I've read in multiple books that a function $F$ is a cumulative distribution function for some random variable $X$ if and only if it satisfies three conditions:
1. $F$ is non-decreasing.
2. $F$ is normalized.
3. $F$ is right-continuous.
but I've never seen an analogous set of conditions for a probability mass fun... |
Consider four positive numbers a, b, c, d and addition modulo N.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo N
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
Is $f$ a probability mass function for a random variable $X$ if and only if it complies with these two properties? |
Consider four positive numbers a, b, c, d and addition modulo N.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo N
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
**ORIGINAL QUESTION (UPDATED):**
I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function:
$$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 \righ... |
I am reading [this paper](https://www.sciencedirect.com/science/article/abs/pii/S0167668799000268) of Lin and Willmot. I dont understand how they come up with formula $2.7$ and why $\tilde{\phi}(s)=\frac{\tilde{H}(s)}{1+\beta-\tilde{g}(s)}$. Can someone help me? |
I've just started studying linear algebra.
I had a question while studying an example.
I'm gonna explain the problem.
$$\\$$
$$\\$$
Augmented matrix for a linear system in the unknowns x, y and z is
$$
\left[\begin{array}{rrr|r}
1 & 0 & 3 & -1 \\
0 & 1 & -4 & 2 \\
0 & 0 & 0 & 0
\end... |
**Context:**
During a mathematical discussion with a good friend, I was brought to think about the functional equation $q(t)q(1-t) = 1$ for $q$ with some regularity. (To be honest the exact context doesn't really matter, since I am not trying to solve the original problem that this stemmed from). Upon brainstorming a... |
Let $S(\epsilon)$ be the area of the region bounded by $y=e^x$ and $y=x+1+\epsilon$, where $\epsilon$ is a small positive number. When $\epsilon\to0,$ we have $$S(\epsilon)=S_0+\epsilon^\alpha S_1+\dots,\alpha>0$$ Find $S_0, S_1$ and $\alpha$.
For starters, that's how the graphs of $y=e^x$ and $y=x+1$ look like:
... |
>If $\tau=(1\ 2)(3\ 4)(5\ 6)(7\ 8)(9\ 10)$ determine whether there is an $n$-cycle $\sigma$ $(n\ge 10)$ with $\tau=\sigma^k$ for some integer $k$.
This question is an exercise from Dummit and Foote, and it has been asked before [here][1]. The answer suggests that $\sigma^k$ should be a product of $\frac{n}{\text{gcd... |
Is the Young Symmetrizer defined as $c_T=a_T b_T$ or $c_T=b_T a_T$? |
Let $p(X)\in \mathbb Z[X]$ be a given integral polynomial with no real zeros. The roots of $p$ comes in conjugate pairs, so we can write $p(X)=q(X)q^*(X)$ in $\mathbb C[X]$, such that $q$ is the product of roots of $p$ which lies in the upper half plane, and $q^*$ is the conjugate (by coefficient) of $q$.
> Is there a... |
Let $a,b\in\mathbb{N}$.\
**Prove that if $1+ab\equiv 0[24]$ then $a+b\equiv 0[24]$**\
My attempts: \
$1+ab\equiv 0[24]\implies ab\equiv -1[24]$\
so : $ab\equiv 23[24]$\
Then there $k\in\mathbb {N}$ such as $ab=24k$ \
But how can I move to $a+b$?.
|
See Enderton's [Logic](https://www.google.it/books/edition/A_Mathematical_Introduction_to_Logic/dVncCl_EtUkC), page 115: in general $\varphi \vdash \psi$ does not imply $\varphi \vDash_t \psi$.
The counter-example is in the text: we have $∀xPx⊢Pc$ but the second is not tautologically implied by the first one.
Wh... |
Let $n$ be a positive integer with $n\equiv 4\mod 6$ and define $p:=\frac{n^2+n+1}{3}$ (which is in this case a positive integer as well).
> Conjecture : $$p^2\mid n^n+(n+1)^{n+1}$$ for every $n$ of the given form.
The conjecture is true upto $n=10^9$
Trial : We have to show $(n^2+n+1)^2 \mid 9(n^n+(n+1)^{n+1}... |
See Enderton's [Logic](https://www.google.it/books/edition/A_Mathematical_Introduction_to_Logic/dVncCl_EtUkC), page 115: in general $\varphi \vdash \psi$ does not imply $\varphi \vDash_t \psi$.
The counter-example is in the text: we have $∀xPx⊢Pc$ but the second is not tautologically implied by the first one.
Wh... |
$\newcommand{\Gal}{\text{Gal}}$
>Let $K_1/K$ and $K_2/K$ be two Galois extensions. I have a theorem that says that $K_1K_2/K$ is a Galois extension and if $K_1\cap K_2=K$ then $$\Gal(K_1K_2/K)\cong \Gal(K_1/K)\times \Gal(K_2/K).$$
I would like to generalize it for three extension fields, i.e.,
>Let $K_1/K,K_2/K$... |
Was reading up my textbook about the identification of onto and into functions and one of the method given is as follows
Solve $f(x) = y$ for $x$, say $x = g(y)$ Now if $g(y)$ is defined for each $y \in \text{ codomain }$ and $g(y) \in \text{ domain of } f \text{ for all } y \in { codomain}, then f(x) \text{ is onto... |
For a cylinder, the volume formula is given by $V = \pi r^2 H $. To find the differential \( dV \), we differentiate \( V \) with respect to \( r \) and \( H \), yielding \( dV = 2 \pi r H dr + \pi r^2 dH \). Upon integrating \( dV \) again, we obtain $( 2 \pi r^2 H )$. However, I'm confused about where the factor of t... |
why cant i integrate the derivative of a cylinders volume to find it again? |
Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=\mathrm{Gal}(\bar{k}/k)$ be an Abelian group (may be infinite). If $K,L$ are intermediate fields, denote $G_K=\mathrm{Gal}(\bar{k}/K), G_L=\mathrm{Gal}(\bar{k}/L)$. Is it true $G_K G_L=G_{L\cap K}$? Generally, is it true $\langle G_{K_i}\mid i\in I\rangle=G_... |
Let V be the set of all contimuous functions $f : [0,1] \to \mathbb{R} $, differentiable on $(0,1)$,with the property that $f(0) = 0$ and $f(1)= 1$. Determine all $a\in \mathbb{R}$ such that for every $f\in V$,there exists some $\xi\in (0,1)$ such that
$$f(\xi)+a=f'(\xi)$$
I have tried some functions like $x$ and $x^... |
In order to remove the question from the unanswered list I will mention the following ($\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... |
Isnt raising curve equations to powers "dangerous" in general? Take
$$ x=y $$
if you square that, you get
$$x^2 = y^2$$
The new equation contains the points described by $ x=y $, and $ x=-y $ combined, ie, 2 lines, slopes 1 & -1, unlike the original equation that only described a single line.
I'm self-stu... |
When is it OK to take powers of the equation of a curve? |
**Problem:**
Evaluate in terms of bessel functions:
$$\int_{}^{} x^{-2} J_2(x) dx $$
**Attempt**
Rewriting the integrasl as: $\int_{}^{} x^{-1}*x^{-1}J_2(x) dx $ then using Integration By Parts to get:
$$\int_{}^{}x^{-2} J_2(x) = -\frac{1}{x^2}J_1 -\int_{}^{} \frac{1}{x^3} J_1 dx $$
Integration by p... |
Can essentially primitive idempotents be defined both as $e_\lambda=s_\lambda a_\lambda$ and $e_\lambda=a_\lambda s_\lambda$? |
Are idempotents in the group algebra of $S_n$ equivalent to Specht modules? |
**Background**
I am trying to understand different notions of aperiodicity in graphs, and how these concepts relate to <a href="https://en.wikipedia.org/wiki/Aperiodic_tiling">aperiodic tilings</a>. There is a different description of aperiodicity in graphs <a href="https://en.wikipedia.org/wiki/Aperiodic_tiling">o... |
I am a beginner in Koopman operators，and I want to apply it to the robotics. But I think the Koopman operator approximated by EDMD is more like a fitting tool. Because we always choose some dictionary functions $\Phi=\left \{ \phi_1(x),\phi_2(x),\dots,\phi_n(x) \right \} $, and the approximation of koopman operator i... |
This question arises in the context of Cosmology, where the smoothing of a vector field $f(\vec{x})$ is defined through the following integration:
$$f_s(\vec{x})=\int d^3x'\ W(|\vec{x}-\vec{x}'|)f(\vec{x}')\tag{1}$$
with $W$ being a function called window function, usually a Gaussian. While reading research paper... |
If $A$ is normal with $\sigma(A)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$? |
Let $$L=\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}=0$$
given that $a,b \in \mathbb R$ and are finite.
I tried the following approach,
We know, $\lim\limits_{x\to 0}{\frac{\sin 3x}{3x}}=1$
$$\therefore L=\lim\limits_{x\to 0}{\frac{\sin 3x}{3x}\cdot \frac{3}{x^2}+\frac{a}{x^2}+b}$$
substitu... |
If the limit $\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}$ exists and equals $0$ then what can $a$ and $b$ be? |
Say $e^{m_1x}$ and $e^{m_2x}$ are the solutions of the differential equation
$$y''+py'+qy=0\tag{1}$$
where $p$ and $q$ are constants.
And the values of $m_1$ and $m_2$ are found by solving the auxiliary equation
$$m^2+mp+q=0\tag{2}$$
We get complex roots if $p^2-4q>0$ and each of the roots $(m_1,m_2)$ can be ... |
suppose $f:D \subset \mathbb{R} \to \mathbb{R},f \ is \ continuous \ and \ f \ satisfies \ the\ Lipschitz \ condition,that \ is \ : \ \exists M>0, \forall x,y\in D,\,we \ have \ |f(x)-f(y)|\leq M|x-y|$
I want to know whether $f'(x) \ exists \ for \ all \ x\in D,and \ whether \ f'(x) \ is \
Riemannian \ i... |
I think my question is very easy but I need to understand. The problem is,
I have a ticket with 2 numbers from 1 to 10. The first number cannot be greather than the second number. How many valid ticket do I have?
Is there any formula to solve that? Because I have tried with permutation formula but it doesn't take car... |
Let $B$ be an infinte set.
$∀ b∈ B, ∃!n∈ N, f(b)=n$
$∀ n∈ N, ∃!b∈ B, f(b)=n$
Does this ensure B is bijective ? |
Does the following conditions ensure the infinite set B is bijective? |
Let $B$ be an infinte set.
$∀ b∈ B, ∃!n∈ N, f(b)=n$
$∀ n∈ N, ∃!b∈ B, f(b)=n$
Does this ensure B is enumerable ? |
Does the following conditions ensure the infinite set B is enumerable? |
We need to show that the map
$$\theta:\mathbb{C}[z_0,\ldots,z_n]_k\to H^0(\mathbb{P}^n,\mathcal{O}(k)) $$
Is an isomorphism.
Huybrecht's describes the map $\theta$ right before the proposition.
***This is what I understand from the description*:**
Given a homogeneous polynomial of degree $k$ we can ... |
In how many ways can we distribute 6 white and 6 black ball in 10 different boxes such each box has atleast 1 ball? |
Let $B$ be an infinte set.
$∀\ b∈ B, ∃!\ n∈ \mathbb{N}, f(b)=n$
$∀\ n∈ \mathbb{N}, ∃!\ b∈ B, f(b)=n$
Does this ensure $B$ is enumerable ? |
Consider four positive numbers a, b, c, d and addition modulo N.
I would like to determine the minimum number of constraints needed to express that:
1) $a + b + c = d$
2) $a + b + c$ does not wrap around the modulo N
I think I need at least 5 constraints, for instance:
1) $a \le a + b$
2) $b \le a + b$
3) ... |
Say I have a matrix $A \in R^{p \times p}$ which is symmetric and with non-negative diagonal entries (i.e. $a_{ii} \geq 0 \forall i\in \{1, \ldots, p\}$). However, $A$ is not positive (semi-)definite.
I want to somehow "perturb" the diagonal elements of $A$ so that it becomes positive (semi-)definite. The obvious wa... |
For $u,v\in \mathcal{S}(\mathbb R^d)$, $s,s_1,s_2\in\mathbb R$,
$$
\|uv\|_{H^s}\lesssim_{d,s,s
_1}\|u\|_{H^{s_1}}\|v\|_{H^{s_2}}.
$$
where $s_1+s_2=s+d/2>0$.
**My attempts:** We work in a direct estimate in Fourier side except introducing Bony's decomposition. Indeed, $$
|(\hat{u}\ast \hat{v})(\xi)|\le \left... |
Do you want this kind of curve modelizing a string rolled/unrolled along a convex polygon (here the particular case of a regular octogon) where the position of the extremity of the string can be seen as a kind of spiral modelizing rather well a half-egg-shape ?
I can write a program plotting this kind of curve in th... |
Say I have a matrix $A \in R^{p \times p}$ which is symmetric and with non-negative diagonal entries (i.e. $a_{ii} \geq 0 \forall i\in \{1, \ldots, p\}$). However, $A$ is not positive (semi-)definite.
I want to somehow "perturb" the diagonal elements of $A$ so that it becomes positive (semi-)definite. The obvious wa... |
I am currently reading *Measure, Integration, and Real Analysis* by Sheldon Axler and I am in the middle of the proof for the following lemma:
> Suppose $B ⊆ R$ is a Borel set. Then for every $ε > 0$, there exists a closed set
$F ⊆ B$ such that $|B\setminus F| < ε$.
The proof begins thus:
> Let $L$ be the set... |
In this proof, why does definition of Lebesgue outer measure imply existence of open superset that is arbitrarily larger? |
Adele owns one T-shirt and one skirt in each of the following seven colors: Red, Blue, Green, Yellow, Black, Orange, and Purple. She always wants to wear a T-shirt and a skirt in different colors. If she wears a red clothing item, she also wants the second clothing item to be yellow. How many such outfits can Adele put... |
How many outfits are possible in this problem? |
In Mathographics Robert Dixon shows how to draw an Egg-curve (or really half of an egg curve) as an involute:
[![enter image description here][1]][1]
This allows you to draw a smoother version of eggs created from the 'arcs of a circle method.'
I want to know the formula for idealized involute curves of this t... |
>The radius of the quarter circle is $6\sqrt 5$ and we assume that $OA= 5$ and $OC=10$. What is the maximum area of the blue triangle?
>
>[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/mdJv3m.png
Interpreting the problem statement, I believe that points $A$ and $C$ are fixed and poin... |
Let $B$ be an infinte set.
Suppose we take input set as $B$ and output set as $\mathbb{N}$
(1) $∀\ b∈ B, ∃!\ n∈ \mathbb{N}, \text{output $(b)$ is $n$}$
This ensures $f:B→N$ exist.
(2) $∀\ n∈ \mathbb{N}, ∃!\ b∈ B, f(b)=n$
Does this ensure $B$ is enumerable ? |
In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that sigma(x) = {z:|z|≤||x||}
But the proof is given only for one side inclusion..that is.,
The spectrum of an element x in a Complex Banach Algebra is a subset of the closed disk with centre 0 and radius ||x||.
Is the reverse i... |
Is the spectrum of an element in an algebra ,sigma(x) = {z in C: |z|≤ ||z||}? |
In some combinatorics book I found the following remarks about multiplicative groups of finite fields:
Let p be an odd prime number and $q=p^2$. Then $F_p^*$ can be identified with the elements $x$ of $F_q^*$ that satisfy $x^{p-1}=1$.
Furthermore, for fixed $y \in F_p^*$ the polynomial $y=x^{p+1}$ has exactly p+1 sol... |
Why the polynomial $x^{p+1}=y$ has exactly p+1 solutions in $F_{p^2}^*$ for $y /in F_p^*$?$ |
Is the spectrum of an element in an algebra ,sigma(x) = {z in C: |z|≤ ||x||}? |
While studying for my functional analysis course I encountered [bounded inverse theorem][1] which states that given a bijective bounded linear operator $T:X\rightarrow Y$, then $T^{-1}:Y\rightarrow X$ is bounded too. (Actually at lesson it has been introduced as a corollary to open map theorem.) In particular, I was tr... |
>The radius of the quarter circle is $6\sqrt 5$ and we assume that $OA= 5$ and $OC=10$. What is the maximum area of the blue triangle?
>
>[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/mdJv3m.png
Interpreting the problem statement, I believe that points $A$ and $C$ are fixed and poin... |
Why the polynomial $x^{p+1}=y$ has exactly p+1 solutions in $F_{p^2}^*$ for $y \in F_p^*$? |
Let $B$ be an infinite set.
Suppose we take input set as $B$ and output set as $\mathbb{N}$
(1) $∀\ b∈ B, ∃!\ n∈ \mathbb{N}, \text{output $(b)$ is $n$}$
This ensures $f:B→N$ exist.
(2) $∀\ n∈ \mathbb{N}, ∃!\ b∈ B, f(b)=n$
Does this ensure $B$ is enumerable ? |
I’m confused by the notation in a lemma from my textbook:
Lemma: Let F \subset B \subset E be a tower of fields with B/F the splitting field of some polynomial f \in F[x]. If σ \in Gal(E/F), then σ|B \in Gal(B/F).
What does σ|B even mean? I’ve never seen this notation before. Thanks! |
In a algorithm book i'm reading ther's this paragraph (it concerns a recursive algorithm for integer multiplication):
*In general, a number $x$ with an even number $n$ of digits can be expressed in terms of two $n/2$- digit numbers, its first half $a$ and second half $b$:*
$x=10^{n/2} * a + b$
I'm not sure how... |
I have a series question which I can not solve for two days.
$$\sum_{n=1}^\infty\frac{1}{(n+1)(n+2)(n+3)\dots(n+p)} = \frac{1}{p\cdot p!}$$
How can I prove and solve this equation? |
In a algorithm book i'm reading ther's this paragraph (it concerns a recursive algorithm for integer multiplication):
*In general, a number $x$ with an even number $n$ of digits can be expressed in terms of two $n/2$- digit numbers, its first half $a$ and second half $b$:*
$x=10^{n/2} * a + b$
I'm not sure how... |
According to the generic theory of linear diffusion processes, i.e time homogeneous Markov processes with independent increments
<cite authors="Itô, Kiyosi; McKean, Henry P. jun.">_Itô, Kiyosi; McKean, Henry P. jun._, Diffusion processes and their sample paths., Classics in Mathematics. Berlin: Springer-Verlag. xvi... |
I'm calculating the problem descripted in the title, and found it a little bit hard, here is the problem:
Suppose $X_i\sim\mathcal{N}(0,1)$ is standard normal random variables, now we need to calculate the expectation of linear combination of the square of standard normal variables, that is $$\mathbb{E}\left[\sum_{i... |
Denote the set $\mathcal S$ all Hermitian matrices act on $\mathbb{C}^d\otimes \mathbb{C}^d\otimes\mathbb{C}^d$. If for any $H\in\mathcal S$, we have $H$ is invariant under the permutation of subsystems, how can I characterize $\mathcal S$? For example, all linear combinations of tensor product of Hermitian matrices wi... |
Imagine you're at a casino. The game you're playing consists of the following:
There are 52 cards, half of them are red, and the other half are black. Croupier mixes the cards and he draws two. If both of them are red, you get a point, if they are of different color, they get discarded and nobody gets a point, and i... |
There is an elemination game with 10 unkown players.
There are 8 rounds.
At the end of each of the first 7 rounds, 1 player is eliminated.
In the last round, the winner will be decided between the remaining 3 players.
After every round (of the first 7 rounds) gamblers can bet on the eventual winner.
Also, after ... |
Fix $u$ and $x$
Let $u\leq F(x)$. Then as $X^{-}(u)=\inf\{x:F(x)\geq u\}$ we have that $X^{-}(u)\leq x$ as $x$ satisfies that $F(x)\geq u$.
Now let $X^{-}(u)< x$ . Then by property of infimum, there exists $a$ such that $F(a)\geq u$ and $a<x$
Thus by monotonicity, $F(x)\geq F(a)\geq u$ .
But you also have... |
I’m confused by the notation in a lemma from my textbook:
Lemma: Let $F \subset B \subset E$ be a tower of fields with $B/F$ the splitting field of some polynomial $f \in F[x]$. If $σ \in Gal(E/F)$, then $σ|B \in Gal(B/F)$.
What does $σ|B$ even mean? I’ve never seen this notation before. Thanks! |
The First method is feasible. The second method however, According to me, has some issues with it.
Issue 1:
Say you have a set $(S)$ of vectors in the Argand plane,
$$S=\left\{A_{k}e^{i\theta_{k}},\ \forall\ k∊\left\{1,2,3...n\right\}\right\}$$
And denoting
$$C_{all}=\sum_{k=1}^{n}A_{k}e^{i\theta_{k}}$$
Your ... |
First, we divide all patterns of how balls can be distributed into two groups:
- group of patterns A, when an urn has $3$ balls in it,
- and group if patterns B, when two urns have two balls in it.
Examine the group A, there are following patterns in it:
- $3$ white balls in a “big” urn,
- $2$ white, $1$... |
I wish to answer the following question
>Let $C[0,1]$ be the space of real continuous functions on the interval $[0,1]$. A mapping $L: C[0,1] \rightarrow C[0,1]$ is said to be positive if $L(f) \geqslant 0$ for each $f \in C[0,1]$ with $f \geqslant 0$, and linear if $L(a f+b g)=a L(f)+b L(g)$ for all functions $f, g... |
In doing some analysis in finite dimensional space, the following question arose about the link between columns of a matrix under different basis.
More precisely, consider $(e_i)$ the canonical basis in $\mathbb{R}^{d}$ and $(\tilde{e}_i))$ another basis. We denote by $H$ the matrice from $(e_i)$ to $(\tilde{e}_i)$... |
If $A=(0,1)$ then the closure of $A$ in $N$ is equal to $N$ (so $A$ is dense in $N$ ), but can we say $A$ is closed? |
The Fourier transform of a function is given by $$f(ξ) = \int_{∞}^{−∞}
f(x)
e^{-2πixξ}dx$$
the paper I was reading from says that the test function $e^{2πixξ}$ is a periodic with period $2\pi/ξ$ and it continues by saying ("so integrating $f$ against this test function gives
information about the extent to which... |
How the Fourier transform show the frequency extent of $f(x)$? |
I see that for $n = 0, 4, 144, 4900, 166464$, etc. we get a perfect square, but is it possible to obtain an analytical solution? |
For which integers $n$ $2n^2+1$ is a perfect square? |
I am trying to get some practice in computing tensors in Riemannian manifolds. Is there somewhere that I can check my results? For example write now I am computing the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, Einstein and Weyl tensor and the expressions that solve the Killing equations of the pseud... |
The First method is feasible. The second method however, According to me, has some issues with it.
Issue 1:
Say you have a set $(S)$ of vectors in the Argand plane,
$$S=\left\{A_{k}e^{i\theta_{k}},\ \forall\ k∊\left\{1,2,3...n\right\}\right\}$$
And denoting
$$C_{all}=\sum_{k=1}^{n}A_{k}e^{i\theta_{k}}$$
Your ... |
I am trying to get some practice in computing tensors in Riemannian manifolds. Is there somewhere that I can check my results? For example right now I am computing the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, Einstein and Weyl tensor and the expressions that solve the Killing equations of the pseud... |
I have a question about the following definition from Gilkey's book, "Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem" (p. 31):
(Let $M$ be a closed manifold.)
**Definition:** An atlas $\mathcal{U}$ is a collection $\{U_i,h_i,\phi_i\}$ where the $U_i$ are a finite open cover of $M$, $(U_... |
Is it possible to find
$$\int_{-\pi}^\pi d\theta\,e^{i \theta+a e^{i \theta} + b e^{-i \theta}}$$
in terms of Bessel functions? |
It is well-known that a category that has products for all class-indexed families must be thin (i.e. posetal). Does a similar result hold for coproducts?
MY ATTEMPT:
Working with coproduct, I only have that the opposite category is thin. May I infer that also the starting category is thin? |
Coproducts for non-small indexed families of morphisms? |
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