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Convergence is defined as:
Given the sequence $\{a_n\}$, for any $\epsilon>0$, there exists a positive integer $N$ such that $|a_n - L| <\epsilon$ for n$\geq N$
Now, if I consider the sequence
$\{(-1)^{n+1}\}_{n=1}^{\infty}$\\
I take $\epsilon$=2 and $L$=0 then it satis... |
The class material presents the Existence and uniqueness theorem for ODEs as follows
[![enter image description here][1]][1]
[![enter image description here][2]][2]
What I don't understand is the meaning of $r$. I understand that the solution lies within a subinterval of $(t_0 - a, t_0 + a)$, specifically $(t_0 - ... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/FyHls.png
Prove that $u_{n}=2^{n}-1$ for all positive integers $n$, by induction or what means?
What I did was sub $u_{n}=2^{n}-1$ into the $u_{n+1}$, but that got me nowhere. so what options do i have now. |
Convergence is defined as:
Given the sequence $\{a_n\}$, for any $\epsilon>0$, there exists a positive integer $N$ such that $|a_n - L| <\epsilon$ for n$\geq N$
Now, if I consider the sequence
$\{(-1)^{n+1}\}_{n=1}^{\infty}$\\
I take $\epsilon$=2 and $L$=0 then it satisfies the convergence definition for any $n$... |
Convergence is defined as:
Given the sequence $\{a_n\}$, for any $\epsilon>0$, there exists a positive integer $N$ such that $|a_n - L| <\epsilon$ for n$\geq N$
Now, if I consider the sequence
$\{(-1)^{n+1}\}_{n=1}^{\infty}$ \\
I take $\epsilon$=2 and $L$=0 then it satisfies the convergence definition for any $n... |
I am dealing with the following integral from superconductivity theory
$$\int_{0}^{\infty}\frac{1}{x^2}\left(\frac{1}{\cosh^2x}-\frac{\tanh{x}}{x}\right)dx$$
My attempt to calculate this integral: calculate residues of
$$f(x)=\frac{x-\sinh x\cosh x}{x^3\cosh^2 x},$$
then use Cauchy theorem about residues (integrati... |
> A step starting at a point P in the XY -plane consists of moving by
> one unit from P in one of three directions: directly to the right or
> in the direction of one of the two rays that make the angle of ±120
> with positive X-axis. (An opposite move, i.e. to the
> left/southeast/northeast, is not allowed.) A pat... |
Is this mapping based solution correct? |
I was reading [this notes](https://math.okstate.edu/people/lebl/uw522-s12/lec1.pdf) to learn more about Stone-Weierstrass theorem and there is a remark I do not understand:
[![][1]][1]
It says that Stone-Weierstrass theorem applies but not every continuous function on $[0,\pi]$ has uniformly convergent cosine ser... |
Why not every real-valued continuous function on compact set can be expressed as Fourier cosine series? |
I was reading the Wikipedia page on [Abstract Wiener spaces][1]. I am really confused with their definition of the measure defined on cylindrical sets. Why is this well defined?
[![enter image description here][2]][2]
Thank you!
[1]: https://en.wikipedia.org/wiki/Abstract_Wiener_space
[2]: https://i.stack.i... |
For the beginning I want to clarify the quantifiers (I am assuming that we are always working in $\mathbb{Z}$):
Let's take a look at $\forall x. A$. Where $A$ is a statement which can be either true or false.
The $\forall x.$ quantifier, says that the following statement has to be true **for every** possible $x$.... |
I have a real signal $f(t) $that is periodic on $[0,L]$ where $L=1$. I sampled it uniformly $64$ times and computed FFT (in python). I was asked **to find the frequency where the transformed signal is having the largest magnitude**. I found the index $i$ of the largest value in the transformed signal.
How do I find... |
So I know that if all of your moduli are relatively prime to each other, you can simply apply the Chinese remainder theorem. However, what do you do if the moduli are not relatively prime? |
How do you solve a system of linear congruences with moduli that are not relatively prime to each other? |
I was reading the Wikipedia page on [Abstract Wiener spaces][1] to get some intuition on the Cameron-Martin spaces. I am really confused with their definition of the "measure" defined on cylindrical sets on a Hilbert space $H$.
1. Why is this well defined?
2. The article claims that if the Hibert space $H$ is den... |
Since the motivation of the question was the lack of online resources, the theorem in question is **Théorème I** in Miquel's paper [*Théorèmes sur les intersections des cercles et des sphères*, Journal de mathématiques pures et appliquées v3 (1838), 517-522.][1]
It's proven via angle chasing. As a reading ... |
Prospective math teacher here. So I am giving a quiz today that I inherited from my Host Teacher. The quiz is over circles, tangents, secants, etc. On the quiz, there is the following question:
Determine whether the converse of the statement is true or false:
If two segments from the same exterior point are tang... |
What is your interpretation of the converse of this statement? |
Prospective math teacher here. So I am giving a quiz today that I inherited from my Host Teacher. The quiz is over circles, tangents, secants, etc. On the quiz, there is the following question:
> Determine whether the converse of the statement is true or false:
>
> If two segments from the same exterior point are... |
What is your interpretation of the converse of "If two segments from the same exterior point are tangent to a circle, then they are congruent"? |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
Problem Statement: Let $f$ be a function with domain $[-1, 1]$ such that the coordinates of each point $(x, y)$ of its graph satisfy $x^2 + y^2 = 1$. What is the total number of points at which $f$ is necessarily continuous?
My confusion: I've been treating the first sentence as a roundabout way of saying that we h... |
We are given a function $f\colon R\to R$ that satisfies $f(x+f(y))=f(x)+f(y)$ for any real $x,y$, and also $f(0)\neq0$. The equation seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equatio... |
The class material presents the Existence and uniqueness theorem for ODEs as follows
[![enter image description here][1]][1]
[![enter image description here][2]][2]
What I don't understand is the meaning of $r$. I understand that the solution lies within a subinterval of $(t_0 - a, t_0 + a)$, specifically $(t_0 - ... |
The system is the following:
$\sqrt{\sin w \sin x}+\sqrt{\cos w\cos x}=1\tag1$
$\sqrt{\sin v\cos x}+\sqrt{\sin x\cos v}=4^{1/3}\big(\sin v\sin x\cos v\cos x\big)^{1/12}\tag2$
$\big(\sin u\sin x\big)^{1/4}-\big(\cos u\cos x\big)^{1/4}=\big(\sin v\sin w\big)^{1/4}-\big(\cos v\cos w\big)^{1/4}\tag3$
$\big(\sin... |
I am examining a certain set of integers, $M$.
Let's say, the amount of integers less or equal than $x$ is $S(x)$.
Now my studies show that the density of primes in this set $M$ is the double density of primes in the naturals.
In formula
$$
\operatorname{d}_M(x) \sim 2\operatorname{d}(x) \text{ for } x\to\infty.... |
Is rearanging allowed when we talk about asymptotic equivalence? |
We are given a function $f\colon R\to R$ that satisfies $f(x+f(y))=f(x)+f(y)$ for any real $x,y$, and also $f(0)\neq0$. The equation seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equatio... |
We are given a function $f\colon R\to R$ that satisfies $f(x+f(y))=f(x)+f(y)$ for any real $x,y$, and also $f(0)\neq0$. The equation seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equatio... |
I understand the solution to this particular question.
However, this example has raised a general concern.
I was asked to find the minimal number of elements that span the entire vector subspace.
In the example it's proven that one of the 3 vector found initialy to span the subspace is a linear combination of t... |
I understand the solution to this particular question.
However, this example has raised a general concern.
I was asked to find the minimal number of elements that span the entire vector subspace.
In the example it's proven that one of the 3 vector found initialy to span the subspace is a linear combination of t... |
My textbook claims , that as a consequence of the Liebniz integral rule:
$$ \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} -... |
We are given a function $f\colon R\to R$ that satisfies $f(x+f(y))=f(x)+f(y)$ for any real $x,y$, and also $f(0)\neq0$. The equation seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equatio... |
I understand the solution to this particular question.
However, this example has raised a general concern.
I was asked to find the minimal number of elements that span the entire vector subspace.
In the example it's proven that one of the 3 vector found initialy to span the subspace is a linear combination of t... |
Here's the problem, as far as I understand it. For given measure spaces:
$$(\mathcal{X}_0, B(\mathcal{X}_0), \mathbb{P}_0)$$
$$(\mathcal{X}_1, B(\mathcal{X}_1), \text{don't-care-about-this-measure-yet?})$$
Chopin and Papaspiliopoulos, in their 2021 Sequential Monte Carlo book, introduce the idea of a probabi... |
My textbook claims , that as a consequence of the Leibniz integral rule:
$$ \frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0} = \int^{b(0)}_{a(0)} x f(x)dx + f(b(0)) \ b(0) \left( \frac{\partial}{\partial \epsilon} b(\epsilon) \right)_{\epsilon=0} -... |
I am trying to prove that the set of equivalence classes of measurable functions with the metric $\rho(f,g)=\int_{0}^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|} \, d\mu$ is a vector topological space.
By definition, a vector topological space is a linear space, with a topology in which addition and multiplication by a sca... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
>Solve $7\sqrt{x-8}-\sqrt{21x+12}=2\sqrt{3}$
$\Rightarrow 7\sqrt{x-8}-\sqrt{3}\cdot\sqrt{7x+4}=2\sqrt{3} \tag{1}$
$\Rightarrow 7\sqrt{x-8}=\sqrt{3}\cdot(2+\sqrt{7x+4}) \tag{2}$
$\Rightarrow [7\sqrt{x-8}]^2=[\sqrt{3}\cdot(2+\sqrt{7x+4})]^2 \tag{3}$
$\Rightarrow 49\cdot(x-8)=3\cdot[4+(7x+4)+4\sqrt{7x+4}] \tag... |
Let $R \subset S$ an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$.
In [this answer][1] Haran elaborated that if $R \subset S$ a faithfully flat extension, then the intersection of ... |
Let $f(n) = \left\{\begin{matrix}
a_1 + b_1 \cdot c_1^n n \equiv 1 \pmod{2}
\\ a_2 + b_2 \cdot c_2^n \equiv 0 \pmod{2}
\end{matrix}\right.$ where $a_1, a_2, b_1, b_2, c_1, c_2 \in \mathbb{N}$
Is it possible to find a general formula that fits both even and odd $n$? |
How to merge subsequences with a closed form to a universal closed form? |
Notice that from $(8)$ to $(9)$ You completely lost $4\times144$
So It should've been
$$
784x^2 + 12480-24304x=0
$$
Also maybe you made an error while copying the problem? Because $$7\sqrt{11-8}-\sqrt{231+12}\neq 2\sqrt{3}$$
$$7\sqrt{3}-9\sqrt{3}\neq 2\sqrt{3}$$
But as @PinkRabbit pointed out $-2\sqrt{3}\neq 2\s... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(-1, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing... |
1.
$\frac{8!*3^8*12!*2^{12}}{3*2*2}$ This is the solution, do youDo know a paper in which is it explained, why this is the formula?
Do you know a paper in which this formula is explained, why we calculate like this?
2.
Then I heard that the numerator of the fraction would be the result if one did not consider the... |
Why do we divide by 12 when calculating the number of different instances of the Rubik's cube? |
I have a smooth function $f : P \rightarrow R$, where $P$ is the positive orthant of $R^3$. I know that $\partial f / \partial x,\; \partial f / \partial y$ are both $> 0$ everywhere on $P$ and that $\partial f / \partial z < 0$, also everywhere on $P$. Finally, I know that all three partials are functions only of $z$... |
1.
$\frac{8!*3^8*12!*2^{12}}{3*2*2}$ This is the solution, do you know a paper in which this formula is explained, why we calculate like this?
2.
Then I heard that the numerator of the fraction would be the result if one did not consider the mechanics of the cube, but could simply assume that all 54 faces of the c... |
Let $f_1:\mathbb{N}_{odd} \to \mathbb{R}$
$$f_1(n) = a_1 + b_1 \cdot c_1^n$$
and $f_2:\mathbb{N}_{even} \to \mathbb{R}$
$$f_2(n) = a_2 + b_2 \cdot c_2^n$$
Is it possible to find a general formula $f:\mathbb{N} \to \mathbb{R}$ that fits both even and odd $n$? |
I have a smooth function $f : P \rightarrow R$, where $P$ is the positive orthant of $R^3$. I know that $\partial f / \partial x,\; \partial f / \partial y$ are both $> 0$ everywhere on $P$ and that $\partial f / \partial z < 0$, also everywhere on $P$. Finally, I know that all three partials are functions only of $z$... |
1.
$\frac{8!*3^8*12!*2^{12}}{3*2*2}$ This is the solution, do you know a paper in which this formula is explained, why we calculate like this?
2.
Then I heard that the numerator of the fraction would be the result if one did not consider the mechanics of the cube, but could simply assume that all 54 faces of the c... |
**Problem:**
Is the function $f\colon \mathbb R \to \mathbb R^2$, defined by $f(x)=(x, \cos x)$, $M$-bilipschitz for some $M\geq 1$? (Using the standard metric)
**So far:**
By the MVT we have $\lvert \cos x -\cos y \rvert \leq \lvert x-y\rvert$ for all $x,y\in \mathbb R$, so we easily get that $$d(f(x),f(y))=\... |
1. $\frac{8!\cdot3^8\cdot12!\cdot2^{12}}{3\cdot2\cdot2}$ This is the solution, do you know a paper in which this formula is explained, why we calculate like this?
2. Then I heard that the numerator of the fraction would be the result if one did not consider the mechanics of the cube, but could simply assume that all... |
How to integrate $$\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
My try to integrate
$$\text{I}=\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
\begin{aligned}
&= -\frac{1}{2} \int_{0}^{1} \frac{\ln(x) \ln(1 + x^2)}{1 - x} \, dx \\
&= -\frac{1}{2} \int_{0}^{1} \frac{\ln... |
Let $R \subset S$ an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$.
In [this answer][1] Haran elaborated that if $R \subset S$ a faithfully flat extension, then the intersection of ... |
I recently came across the following claim in a paper:
$$\langle X^i - \alpha \rangle = \langle X - \alpha^{i^{-1}} \rangle$$
as the generators have the same roots in (in some extension of the base field $\mathbb{Z}/\mathbb{Z}p$), where $X^i - \alpha$ is irreducible modulo $p$ for prime $p$.
I can see that the... |
I explained in a comment why your approach cannot work; here’s a start on one that does.
Let $D_d$ be the set of points at which the monotonically increasing function $f$ is discontinuous, let $D_c$ be a countable dense subset of $\Bbb R\setminus D_d$, and let $D=D_d\cup D_c$. Note that we may assume that $n,-n\in D... |
I noticed some books define the Riemann curvature tensor with opposite signs on all $3$ terms. I wonder what happens to the Ricci curvature, the scalar curvature, and the Einstein tensor when using one or other definition? Would they also change sign? And if I change the signs of the metric from mostly plus to mostly ... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
What are the weakest conditions to assure that $A$ is already discrete valuation ring too?
... |
I am interested in formulating a discrete finite time constrained LQR in CVXPY.
\begin{align}
\text{minimize } J = & \sum_{k=0}^N x'(k)Qx(k) + u'(k)Ru(k) \\\\
\text{subject to } & x(k+1) = Ax(k) + Bu(k) \\\\
& Cx(k) + Du(k) \leq e
\end{align}
I came across the *batch-approach* [1] in a paper solving a similar... |
I'm trying to better understand the concept of a linear transformation acting as an isomorphism on a specific subspace of the vector space, even though it may not be an isomorphism on the entire vector space.
To elaborate, I am referring to a situation where a linear transformation acts like an isomorphism but only... |
I recently came across the following claim in a paper concerning polyonomials in $\mathcal{R}_p = \mathbb{Z}_p[X]/\langle X^d + 1\rangle$ (where $\mathbb{Z}_p = \mathbb{Z}/\mathbb{Z}p$) and their automorphism $\sigma_i: X \mapsto X^i$.
$$\sigma_i \langle X - \alpha \rangle = \langle X^i - \alpha \rangle = \langle X ... |
Finite map of local rings $R \to A$ with $R$ DVR. When is Target $A$ DVR too? |
Let $R \subset S$ be an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$.
In [this answer][1] Haran elaborated that if $R \subset S$ a faithfully flat extension, then the intersection o... |
I'm trying to better understand the concept of a linear transformation acting as an isomorphism on a specific subspace of the vector space, even though it may not be an isomorphism on the entire vector space.
To elaborate, I am referring to a situation where a linear transformation acts like an isomorphism but only... |
While self-studying an elementary geometry book, I found the following claim:
> If $r$ and $s$ are lines contained in the plane $P$, then for every pair of points $A\in r$ and $B\in S$, with $A\neq B$, it holds that $\overline{AB}\setminus\{A,B\}\subseteq SP^B_r\cap SP^A_s$.
where $\overline{AB}$ is the segment t... |
So I am trying to prove that
$f: ℝ X ℝ -> ℝ$ be the function $f(x, y) = x^3 - xy^2$
is surjective and injective.
**a) Surjectivity:**
To prove surjectivity, we need to show that for every \( z \) in the codomain R, there exists at least one pair (x, y) in the domain R X R such that $f(x, y) = z$
Give... |
Am I correct? Subjective and Injective logic (my reasoning) |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" cases appear where ... |
This may be quite an elementary question but I can't quite find a proof for this. So assume i have a morphism $f: X \mapsto Y$ where $X,Y$ are smooth affine varieties over $\mathbb{C}$. Furthermore assume there exist points $x,y$ such that $df_x:T_xX \mapsto T_yY$ is an isomorphism. Is it then true that the induced map... |
I'd like to know how to calculate the resultant of two bivariate polynomials. Searching through the web, I read that there is an Euclid-like algorithm for univariate polynomials, and that one can use it for bivariate polynomials by considering them as univariate polynomials with polynomial coefficients.
The only ver... |
Suppose $X$ is a (smooth) manifold with boundary and $f: [0,\infty)\times \partial X \rightarrow X$ is an open embedding such that $f(0,x) = x (\forall x \in \partial X) $ i.e., a collar.
Is $f([0,1] \times \partial X)$ closed in $X$ ?
It seems to be true but I have some problems: Suppose $f(t_n,x_n) \to y\in X$, ... |
Does collar restrict to a closed neighbourhood? |
While self-studying an elementary geometry book, I found the following claim:
> If $r$ and $s$ are lines contained in the plane $P$, then for every pair of points $A\in r$ and $B\in s$, with $A\neq B$, it holds that $\overline{AB}\setminus\{A,B\}\subseteq SP^B_r\cap SP^A_s$.
where $\overline{AB}$ is the segment t... |
So I am trying to prove:
> The function
$f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$, defined by $f(x, y) = x^3 - xy^2$ is surjective and injective.
##### Surjectivity:
To prove surjectivity, we need to show that for every $z$ in the codomain $\mathbb{R}$, there exists at least one pair $(x, y)$ in the d... |
Consider the graph with $\mathbb{Z}^2$ as a vertex set, and with an edge between two vertices iff their $L^{\infty}$-distance is $1$. Imagine we perform a random walk on this graph, starting at the origin.
Equivalently, imagine a chess king moving randomly on an infinite chessboard.
Does anyone know whether thi... |
How do you solve this limit $ \lim _{x\to 0}\:\:\frac{x^m-\sin^nx}{x^{n+2}} $? |
>I need to solve without l'Hôpital's rule.
$$n,m,r \in \mathbb{N}$$
$$\lim_{x\to 0} \frac{\tan ^nx - \sin ^m x}{x^r}=\ ?$$
MY TRY :
$$\lim_{x\to 0} \frac{\tan ^nx - \sin ^m x}{x^r}=\\\lim_{x\to 0}\frac{\tan^n x}{x^r}-\frac{\sin^m}{x^r}=?$$
|
In Wikipedia's [proof of Bertrand's Postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate), in the second lemma, it is concluded that:
$$R = R(p,{{2n}\choose{n}}) \le \log_p 2n$$
where $R(p,n)$ is the [p-adic order](https://en.wikipedia.org/wiki/P-adic_valuation) of ${2n}\choose{n}$
Later in... |
How to get the vector in the same direction in local coodinate system as the global? |
How to get the vector in the same direction in local coordinate system as the global? |
I am interested in the smallest coordinate of a vector. Given a positive real vector $x=(x_1,...,x_n)$, I need your help to determine if the function $f(x)= min_{1\leq i \leq n}x_i$ is Lipschitz. If not, can we for example restrict $f$ on some subspace of $\mathbb{R}^n$ so that we get Lipschitzity? (Eg say $0\leq x_i... |
While self-studying an elementary geometry book, I found the following claim:
> If $r$ and $s$ are lines contained in the plane $P$, then for every pair of points $A\in r$ and $B\in s$, with $A\neq B$, it holds that $\overline{AB}\setminus\{A,B\}\subseteq SP^B_r\cap SP^A_s$.
where $\overline{AB}$ is the segment t... |
It turns out that the familiar result that subjective monotones $\mathbb R\to\mathbb R$ are continuous extends to general LOTS (linearly ordered topological spaces):
> **Theorem**. If $X$, $Y$ are LOTS with $Y$ being dense and complete, then any surjective monotone $X\to Y$ is continuous.
(The same proof from $\m... |
I was just reading the proof that a graph is a tree iff it is simply connected in Hatcher, on pg.85, which is given below:
The quotient map $X \to X/T$ is a homotopy equivalence by Proposition $0.17.$
The quotient $X/T$ is a graph with only one vertex, hence is a wedge sum of circles,
whose fundamental group we s... |
Ok so based on your questions, let's try to clear something up first:
When people talk about the number of permutations of a Rubik's Cube, they do NOT include rotating the whole cube. So, for example, let's say you had a solved cube. Hold it with the white side on top. Now rotate the whole cube so that the blue s... |
Compact asymmetric form in the plane? |
Does Euclid's Lemma generalize from primes to any integer? |
The posterior mean prediction of a Gaussian Process is given by
$$\mu(x_*) = \sum_{i=1}^n\alpha_i k(\mathbf{x}_i,\mathbf{x}_*) $$ where
$$\alpha = (K + \sigma_n^2I)^{-1} \mathbf{y}$$
Can we show that $\|\alpha \| $ decreases with more training data? i.e.
$$\| \alpha_{n + 1} \| < \| \alpha_{n} \|$$
|
Let $g:\mathbb{R}^n\to\mathbb{R}^m$ be linear defined by $g(x) = Ax$.
I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have that $h:\mathbb{R}^n\to\mathbb{R}^m$... |
I have the following question that starts from this expression:
$$
u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) .
$$
If $h \in C^2(\bar{B})$ is harmonic and Green function $G(x, y)=\Gamma(y-x)-h(y)$, the above integral le... |
Let $Z_1,...,Z_k$ be independent mean zero random vectors in $R^n$. Show that
\begin{equation*}
E\left\| \sum_{j=1}^{k} Z_j \right\|_2^2 = \sum_{j=1}^{k} E \left\| Z_j \right\|_2^2
\end{equation*}
Proof:
Let $X=Z_1+...+Z_k$ and $X^i,Z_j^i$ the ith component of $X,Z_j$.
\begin{equation}
E\left\| \sum_{j=1}^{k} Z... |
Recently, I tried to calculate this double sum:
$$ F(s) = \sum_{(a,b) \in \mathbb{Z}^2 \backslash (0,0) } \frac{1}{(a^2 + b^2)^s}$$
For $ s \in \mathbb{C}, Re(s) > 1$
I think i found the value of this function with the following reasoning. Indeed, $F(s)$ is :
$$ F(s) = \sum_{n = 1}^{\infty} \frac{r_2(n)}{n^... |
I'm working on a task where I have matrices $A$ and $B$ respectively. I need to find a matrix $B'$ such that $AB = AB'$ yet $\|{B}'\|<\|B\|$. This is mainly because currently $\|{B}\|$ is very large and which hurts the task I'm solving.
Anyone has any idea on how to solve this?
Thanks! |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. The latter example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already ... |
> I mean, why $b/M$ ensures that the solution stays inside the rectangle?
Hint: **if** the solution exists:
$$\dot x(t) = F(t,x(t))\implies |\dot x(t)|\le M \implies -M\le\dot x(t)\le M.$$
Integrating:
$$-M(t - t_0)\le x(t) - x(t_0)\le M(t - t_0).$$
Can you continue? |
This is a bit of a soft question, but I was just wondering, in the real analysis epsilon-delta definitions of limits and continuity, can epsilon be a very big positive number, like a googol, or a googolplex? And if it can be, why don't textbooks restrict epsilon to be a small positive real number, like, "For any positi... |
Can epsilon be a big positive number? |
The Cramér random model for the primes is a random subset ${{\mathcal P}}$ of the natural numbers with ${1 \not \in {\mathcal P}}, {2 \in {\mathcal P}}$, and the events ${n \in {\mathcal P}}$ for ${n=3,4,\dots}$ being jointly independent with ${{\bf P}(n \in {\mathcal P}) = \frac{1}{\log n}}$ (the restriction to ${n \g... |
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