instruction stringlengths 12 30k |
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Since $z = z$<sub>$0$</sub> is a zero of order $n$ of $f(z)$, we have
- $f(z)$ = $(z-z$<sub>$0$</sub>$)^np(z)$
where $p(z)$ is analytic at $z= z$<sub>$0$</sub> and $p(z$<sub>$0$</sub>) ≠ $0$.
It follows that,
- $(f(z))^2$ = $(z-z$<sub>$0$</sub>$)^{2n}(p(z))^2$
At this point, $z = z$<sub>$0$</sub> would ... |
Let $\{X_i\}_{i\ge 1}$ be a sequence of random variables defined on $(\mathbb{R}, \mathcal{B})$. Assume that $E[X_i]=0$, $E[X_i^2]=1$, and $E[X_iX_j]=0$ for all $i\neq j$ and $i\ge 1$. Let $S_n=\sum_{j=1}^n X_j$. Show that
(1) for $\alpha>1$, $\frac{S_n}{n^\alpha}\to 0$ a.s.
(2)for $\alpha>1/2$, $\frac{S_n}{n^\al... |
Since $z = z$<sub>$0$</sub> is a zero of order $n$ of $f(z)$, we have
- $f(z)$ = $(z-z$<sub>$0$</sub>$)^np(z)$
where $p(z)$ is analytic at $z= z$<sub>$0$</sub> and $p(z$<sub>$0$</sub>) ≠ $0$.
It follows that,
- $(f(z))^2$ = $(z-z$<sub>$0$</sub>$)^{2n}(p(z))^2$
At this point, $z = z$<sub>$0$</sub> would ... |
Let $(X,\mathcal{X})$ be a metric space where assume that $X$ is countable and discrete. Show that probability measures $P_n$ converges to $P$ weakly (i.e., for all bounded continuous functions on $X$, one has $\int f dP_i\to \int f dP$) if and only if $P_n(\{x\})\to P(\{x\})$ as $n\to \infty$.
---
For one direct... |
## Question: Find the number of solutions of equation $f(x)=2^x-x^2+x+\cos x=0$ ##
----------
Given answer is $1$
*My Attempt*
Checked the derivative of it. $$f'(x) = 2^x\ln2-2x+1-\sin x\\f''(x)=2^x(ln2)^2-2-\cos x$$ but $f'(x)$ is both +ve and -ve so I could not make any conclusion. I had projected this g... |
I hope I'm not overbearing in this site. Yes, I'm still struggling.
If you can, I have a question about primary decomposition that still needs help, you can find it in my page.
Now I wanted to find the height of a certain ideal. In the ring $ A = K[x,y,z]/(xy,xz,z-y^2)$ I need to find the height of $I = (x,y,z)$.... |
Computing the height of an ideal...? |
Let two smooth $v_1$ and $v_2$ both satisfy the system
$$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$
$$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$
for some fixed given smooth $f: \bar{U}\times (0,\infty) \rightarrow \mathbb{R}$ and $g: \partial U \times (0,\infty).$ Sh... |
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 <= a + b
2) b <= a + b
3) c <= d
4) a... |
Let two smooth $v_1$ and $v_2$ both satisfy the system
$$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$
$$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$
for some fixed given smooth $f: \bar{U}\times (0,\infty) \rightarrow \mathbb{R}$ and $g: \partial U \times (0,\infty).$ Sh... |
In a general directed graph, does a strongly connected component necessarily has a Hamiltonian path or cycle? I don't think so, and I've tried to come up with compact counter example, but have yet to come up with one. |
Does a strongly connected component necessarily has a Hamiltonian path or cycle? |
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... |
In Mathographics Robert Dixon shows how to draw an Egg-curve (or really half of an egg curve) as an involute (see image 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 type (i.e., 0-D pins, and 0 wi... |
In a general directed graph, does a strongly connected component necessarily have a Hamiltonian path or cycle? I don't think so, and I've tried to come up with compact counter example, but have yet to come up with one. |
Does a strongly connected component necessarily have a Hamiltonian path or cycle? |
Suppose that $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is $L$-Lipschitz continuous (for some $L>0$) and monotone, i.e.,
\begin{align}
(\forall x,y \in \mathbb{R}^{n}) \quad \| F(x) - F(y) \| \leq L \| x-y \|,
\end{align}
and
\begin{align}
(\forall x,y \in \mathbb{R}^{n}) \quad \langle F(x) - F(y), x-y \rangle \geq 0,
... |
Sufficient condition for being zero? |
>Let $\vec{OA}=\vec{a}, \vec{OB}=\vec{b}$ and $\vec{OC}=\vec{c}$ and $|\vec{a}|=3, |\vec{b}|=2$ and $|\vec{c}|=4$. We also know that $\measuredangle(\vec{a},\vec{b})=\measuredangle(\vec{a},\vec{c})=\measuredangle(\vec{b},\vec{c})=\dfrac{\pi}{3}.$ Given the affine coordinate system $O{\vec{a}\vec{b}\vec{c}},$ find the c... |
After using the Log map, as defined in this paper [Riemannian approaches in Brain-Computer Interfaces: a
review][1] (Section III. A page 2&3), to project points from the manifold onto the tangent space $T_P(M)$, do we use the metric defined by the ambient Euclidean space or the Riemannian metric to measure distances ... |
Let two smooth $v_1$ and $v_2$ both satisfy the system
$$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$
$$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$
for some fixed given smooth $f: \bar{U}\times (0,\infty) \rightarrow \mathbb{R}$ and $g: \partial U \times (0,\infty).$ $U... |
Let $Q(x_1,\ldots,x_n)$ be a positive definite quadratic form.
I would like to show that there exists $C>0$ such that $Q(x_1,\ldots,x_n)\geq C(x_1^2+x_2^2+\ldots x_n^2)$.
Proof:
Let $\vec{x}=\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}$. The quadratic form can be expressed as $Q(x_1,\ldots,x_n)=\vec{x}^\... |
**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... |
I have to show that Torus defined as the quotient space obtained from the square $Q$ using the word $aba^{-1}b^{-1}$ to identificate the sides, is a topological 2-variety.
I've already proved that is locally 2-euclidean. Now I have to show that it is an Hausdorff space.
I want to use the following result.
**Th... |
Suppose $f\in L^{1}(\mathbb{R}^{n}\rightarrow \mathbb{C})$. What do we know about
$$F(y_{1},\cdots,y_{n-1},x):=\int_{\mathbb{R}}
e^{\dot{\imath} xy_{n}} f(y_{1},\cdots,y_{n-1},y_{n})\,dy_{n}
$$
Can we still say $x\mapsto F(y_{1},\cdots,y_{n-1},x)$ is uniformly continuous and
$$\|F(y_{1},\cdots,y_{n-1},x)\|_{... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
There are no problems, except for that fact that $\mathbb Z$ is not a field.
That is, there are no problems *until* you start attempting to apply theorems which require the scalars to form a field.
For example, here is a theorem about vector spaces:
> For any vector space $X$ over any field $F$, and for any $... |
Assume that $G$ is a group such that $|G|=55$ and $G$ has exactly four elements with order $5$. Prove that center of $G$ is not trivial and $G$ is a cyclic group.
My try:
We know that $|h_1|=|h_2|=|h_3|=|h_4|=5$ and there does not exist $h_5$ such that $|h_5|=5$. Hence we know that $h_1^5=h_2^5=h_3^5=h_4^5=1$. If... |
**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... |
I have to show that Torus defined as the quotient space obtained from the square $Q$ using the word $aba^{-1}b^{-1}$ to identificate the sides, is a topological 2-variety.
I've already proved that is locally 2-euclidean. Now I have to show that it is an Hausdorff space.
I want to use the following result.
**Th... |
$
\DeclareMathOperator{\init}{in}
\DeclareMathOperator{\gr}{gr}
\newcommand{\calO}{\mathcal{O}}
$
Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\gr_{\mathfrak{m}}(R)=\bigoplus_{i=0}^\infty\mathfrak{m}^i/\mathfrak{m}^{i+1}$ be the associated graded ring. For $f\in R$, the initial form of $f$ is $\init(f)=f... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
Can someone help me to see where the problem of the functional derivative below is?
**Minimize the functional:**
$$ F[p(\hat{x}|x)] = I(X; \hat{X}) + \beta \sum_{x \in X}\sum_{\hat{x} \in \hat{X}} p(x,\hat{x})d(x,\hat{x}) $$
**Subject to the constraint:**
$$ \sum_{\hat{x} \in \hat{X}} p(\hat{x}|x) = 1 \quad \te... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
**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... |
**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... |
I'd suggest not "by hand" but using some simple programming language (an interpreter language will do, so you do not need to compile your routine before running it). It should offer trace options or simple output to a cmd window (console). Myself I use REXX (I am a mainframe dinosaur). I am not up to date what else is ... |
**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 $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/wiki/Modular_lambda_function).
What is the growth rate of the values of higher order derivatives of $f(z)$ for a fixed $... |
I came across this interesting task. Fischer's inequality is well known and states the following:
> Let $(X, B)$ be a $2$-design with parameters $(v, k, λ_2)$, where $v > k$. Then $|B| ≥ |X|$, which implies $b ≥ v$.
Our task is to characterize those $2$-designs where Fisher's inequality is achieved, i.e., $b = v... |
The inequality holds for $$n\ge2.$$
I used Wolfram numerical integration to show:
for $n=1$, it [becomes][1]: 0.761578
for $n=2$, it [becomes][2]: 0.980789
for $n=3$, it [becomes][3]: 1.09101.
In the links, I used
$$\Phi(x)=\frac{1}{2}+\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}} \right).$$
whe... |
About the title, I've read [here] (https://math.stackexchange.com/questions/4452753/expected-length-of-longest-piece-when-a-stick-is-broken-into-three-pieces) that the answer is 11/18.
I've tried with a simple algorithm:
```
def cut(first_len:int) -> List[Union[float,float]]:
c1 = random.uniform(0,first_len)
r... |
Actually, regarding connected sums, it is much more subtle than you think. On one hand, if $M_1, M_2$ are 3-dimensional manifolds each of which embeds in $\mathbb R^4$, then their connected sum also embeds. But the converse is false. For instance, lens spaces $L(p,q), p>1$, do not embed smoothly in $\mathbb R^4$, whil... |
I understand the logic behind the polynomial rootfinding algorithms that use companion matrices. I checked the [numpy implementation][1] and it's pretty straightforward but I don't understand [this comment][2]:
> rotated companion matrix reduces error
Why is this true mathematically? I don't even understand why w... |
About the title, I've read [here](https://math.stackexchange.com/questions/4452753/expected-length-of-longest-piece-when-a-stick-is-broken-into-three-pieces) that the answer is 11/18.
I've tried with a simple algorithm:
```
def cut(first_len:int) -> List[Union[float,float]]:
c1 = random.uniform(0,first_len)
re... |
I need to find out if the limit exists.
$$\lim_{(x,y)\to(0,0)}\frac{-x^6y^1(x^2+1)}{(x^6+y^2)\sqrt{x^2+y^2}}$$
First, I approached the limit from $y=0$, and the result was $\frac{0}{x^7}$. Then, I approached from $x = 0$, and the result was $\frac{0}{y^3}$. This made me assume that the limit does not exist. Howe... |
I'm teaching a real analysis course and I'm trying to find a video or picture or app that illustrates the Lebesgue integral of a nonnegative function as the "supremum of integrals of smaller simple functions."
I know https://math.stackexchange.com/questions/1721065/lebesgue-integral-graphical-concept has some stuff... |
Let four random variables form the Markov chain
$${\displaystyle \raise{1.5ex}W\overset{\longleftarrow}{\searrow\lower{1.5ex} X\swarrow}\raise{1.5ex} Y\searrow\lower{1.5ex} Z}$$
such that the conditional distribution of $X$ depends only on $W$ and $Y$ and is conditionally independent of $Z$ given $Y$. What is th... |
Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I am looking for the gradient bound of $u$ explicitly on the behavior of the coefficients.
I found that in Gi... |
**Proposition 5.4.9** (The non-negative reals are closed). Let $a_1, a_2, a_3, \ldots $ be a Cauchy sequence of non-negative rational numbers. Then $\text{LIM}_{n \to \infty}a_n$ is a non-negative real number.
**Tao's Proof.**
We argue by contradiction, and suppose that the real number $x:= \text{LIM}_{n \to \inf... |
**Question:** \
Let f be a function f: A → B, and let A1, A2 ⊆ A. Then f(A1 ∩ A2) = f(A1) ∩ f(A2).
**Answer:**
$$ f(A_1 \cap A_2) \subseteq f(A_1) \cap f(A_2) :$$ Let $ y $ be an arbitrary element in $ f(A_1 \cap A_2) $. This means there exists an $ x \in A_1 \cap A_2 $ such that $ f(x) = y $. Since $ x $ is in bo... |
Actually, regarding connected sums, it is much more subtle than you think. On one hand, if $M_1, M_2$ are 3-dimensional manifolds each of which embeds in $\mathbb R^4$, then their connected sum also embeds. But the converse is false. For instance, lens spaces $L(p,q), p>1$, do not embed smoothly in $\mathbb R^4$, whil... |
From what I understand (and I am not an expert in probability), a "point process" on a set $X$ is a probability measure on $2^X$, the set of subsets of $X$. I only want to consider the case when $X$ is finite. In particular, I have been interested recently in determinantal processes, where probabilities are given by th... |
This is exercise 46 from Piskunov Chapter 14
Find the area of that part of the surface of the sphere $x^2+y^2+z^2=1$ cut out by the cilinder $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 \text{ , } a>b $
First, I parametrized the sphere using the function:
$g(x,y)=(x,y,\sqrt{1-x^2-y^2})$
and then I used the next s... |
Well of course $0^\sharp$ cannot be added by forcing. What I mean is the following. Suppose $0^\sharp$ exists. Let $\mathbb{P}=\mathrm{Add}(\omega,1)$ be the Cohen poset, and $L^\mathbb{P}$ be the class of $\mathbb{P}$-names in $L$. Let $W=\{\sigma[0^\sharp]:\sigma\in L^\mathbb{P}\}$ where $\sigma[0^\sharp]$ is the int... |
I'm reading van Bentham's *Modal Logic for Open Minds* (section 5.7), where he proves the (weak) completeness of K. I'm trying to work out each step, but I seem to need to use an intuitionistically invalid inference, which I'm not sure if necessary for this proof. I'm also not sure how to extend the same proof to prove... |
Assume $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (let's denote this as $\mathcal{C}$) is a tensor-category in the sense of Deligne/J.S. Milne (https://www.jmilne.org/math/xnotes/tc2018.pdf).
Assume furthmore that for each object $X$ in $\mathcal{C}$, we have an adjunction $- \otimes X \dashv \underline{\text{Hom}... |
This is exercise 46 from Piskunov Chapter 14
Find the area of that part of the surface of the sphere $x^2+y^2+z^2=a^2$ cut out by the cilinder $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 \text{ , } a>b $
First, I parametrized the sphere using the function:
$g(x,y)=(x,y,\sqrt{1-x^2-y^2})$
and then I used the next... |
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
1. why "the area of the polygon $abcelef$
.... represents the number of ways the three points can be taken, so that the circle circumscribing the triangle will lie wholly within the given polygon." It looks li... |
I am writing this question to try to unconfuse myself in two different directions.
Throughout we work over $\mathbb{C}$ and the choice of topology should not matter, but say we are using the etale topology. Let $G$ be an algebraic group, $X$ an algebraic variety over which $G$ acts. I want to study equivariant Vecto... |
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
1. why "the area of the polygon $abcelef$
.... represents the number of ways the three points can be taken, so that the circle circumscribing the triangle will lie wholly within the given polygon." It looks li... |
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
1. why "the area of the polygon $abcelef$
.... represents the number of ways the three points can be taken, so that the circle circumscribing the triangle will lie wholly within the given polygon." It looks li... |
Let $\mathcal{H},\mathcal{K}$ be Hilbert spaces ($\cong \mathbb{C}^2$ if necessary) and let $h_1,h_2$ denote semi-positive $\ge 0$ operators on $\mathcal{H}\otimes \mathcal{K}$. If $h_1,h_2$ are entangled, i.e., not a trivial product operator, then is this also true for $h_1 +h_2$? (not including the trivial case where... |
Assume $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (let's denote this as $\mathcal{C}$) is a tensor-category in the sense of Deligne/J.S. Milne (https://www.jmilne.org/math/xnotes/tc2018.pdf).
Assume furthmore that for each object $X$ in $\mathcal{C}$, we have an adjunction $- \otimes X \dashv \underline{\text{Hom}... |
Let one Hermitian matrix $H$ acts on $\mathbb{C}^d\otimes \mathbb{C}^d\otimes\mathbb{C}^d$. If I know this Hermitian matrix is invariant under the permutation of subsystems, what I can say about the structure of this Hermitian matrix $H$? For example, all linear combinations of tensor product of Hermitian matrices will... |
Structure of Hermitian matrices that commutes with all permutation of subsystems? |
Define $P_\phi := e^{-iP\phi}$, where $P$ is a [Pauli matrix][1] with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed that if $P'P = -PP'$ i.e. we have two anticommuting Pauli matrices $P, P'$, then
$$P_{\frac{\pi}{4}}P'_{\phi} = (iPP')_{\phi}P_{\frac{\pi}{4}}.$$
Note that $iPP'$ is also a Pauli ma... |
Can anyone tell me if my proof is correct or if there is something lacking in my argument? I am not too confident about it. I used some hints to construct my argument but I took a different route to that of the book. If anyone has some advice I would really appreciate it. The problem states the following:
Prove that... |
Define $P_\phi := e^{-iP\phi}$, where $P$ is a [Pauli matrix][1] with some overall phase factor and $\phi\in[0,2\pi)$. It is [claimed](https://arxiv.org/pdf/1808.02892.pdf) that if $P'P = -PP'$ i.e. we have two anticommuting Pauli matrices $P, P'$, then
$$P_{\frac{\pi}{4}}P'_{\phi} = (iPP')_{\phi}P_{\frac{\pi}{4}}.$... |
I am currently writing a small paper regarding the analogy between Fourier analysis and linear algebra for my mathematics class. Unfortunately, I am stuck at proving that $\mathrm{e}^{2\pi\mathrm{t}\omega t}$ forms an orthonormal basis in the $L^2(\mathbb{R})$ for all $\omega\in\mathbb{R}$.
In my paper I already sho... |
I have a question about the standard notation for representing properties of permutations. This is best illustrated with a concrete example: let's take a permutation on 6 elements with cycle representation $\sigma=(134)(2)(56)$. We know that the cycle representation is unique. In the problem I'm working on, I need an a... |
I was reading the book *Introduction to the topological groups* by Taqdir Husain, he gives a proof for this theorem, but I have a problem with understanding some part. The proof goes like this
It remains to show that the inversion mapping $x\to x^{-1}$ is continuous. So let $U$ be a open neighborhood of $e$. We wish... |
I need to minimize the following integral by varying parameter $\lambda$:
$$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$
The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy $f(0)=g(0,\lambda)$ and $f(\infty)=g(\infty,\lambda)=0$. And the integral is convergent.
What is the corresponding Euler-Lagrange... |
How to use variational principle to find the best value for parameter $\lambda$? |
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
Let $\M,\N$ be Riemannian manifolds, $f:\M \to \N$ smooth. Let $\bigwedge^k df:\Lambda_k(\TM) \to \Lambda_k(f^*{\TN})$ be the induced map on the exterior powers.
Define
... |
Could someone help me with this proof, please?<br>
I am aware that topologically equivalent means that the set of open sets are the same for both metrics.
Let $f : (X,d) \to (Y,m)$ be a continuous function between metric spaces. If defined
$$d^*(x,y) := d(x,y) + m(f(x),f(y)),$$
show that $d$, $d^*$ are topologica... |
I am reading through Mathematical Logic by Chiswell and Hodges and just finished section 5.3, which introduces LR, the Language of Relations, for first-order logic. In this section the following notions are defined:
1. A first-order signature
2. Variables
3. A parsing tree for terms of LR($\sigma$)
4. A parsing... |
Why are variables not considered part of the lexicon of LR($\sigma$)? |
I can't understand the red line well. Seemly, if $\tilde \gamma_1\cap \tilde \gamma_2$ has at least two points, it will contradict with the simple connectivity. I want to know why it is ?
**PS(2024-3-26)**: As the hint of Moishe Kohan, the right way is to use the Hadamard Theorem.
**Hadamard Theorem**: Let $M$ b... |
Have you seen this (sort of thing)?
$(134)$ can be "described" by: $$\begin {pmatrix}1&2&3&4&5&6\\3&2&4&1&5&6\end {pmatrix}.$$
Because $1\to3,2\to2,3\to4,4\to1,5\to5,6\to6.$ |
In Hui-Hsiung Kuo's Introduction to Stochastic Integration, lemma 4.3.3, equations (4.3.11) says that since $f$ is assumed to be bounded, we have
\begin{equation}
\int_a^b |f(t,\cdot)-f(t-n^{-1}\tau,\cdot)|^2\ dt\to 0,\quad\text{almost surely},
\end{equation}
as $n\to\infty$. I have question about how this converge... |
I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq R^d$
,we have two Monge problem:$$min_{T\in L(P_1,Q_1)}\int \fr... |
I was reading the book *Introduction to the topological groups* by Taqdir Husain, he gives a proof for this theorem, but I have a problem with understanding some part. The proof goes like this
It remains to show that the inversion mapping $x\to x^{-1}$ is continuous. So let $U$ be a open neighborhood of $e$. We wish... |
Imagine a two-dimensional version of you lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you could conduct to determine the orientability and genus of the surface? |
If the cut locus of surface is a segment, the surface is the topologic sphere? |
I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq R^d$
,we have two Monge problem:$$min_{T\in L(P_1,Q_1)}\int \fr... |
I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq R^d$
,we have two Monge problem:$$\min_{T\in L(P_1,Q_1)}\int \f... |
I have recently been studying the theory of computational optimal transport and am very interested in how to describe the similarity between two optimal transport plans. Specifically, suppose $P_1,Q_1,P_2,Q_2$ are probability measures on $\Omega \subseteq R^d$
,we have two Monge problem:$$\min_{T\in L(P_1,Q_1)}\int \f... |
I didn't use Glasser's Master Theorem because it only works on even functions, and $\sin(e^x)$ is definitely not an even function.
Here's what I did.
$$\int_{-\infty}^{\infty}\sin(e^{x-\frac{1}{x}})dx=\int_{-\infty}^{0}\sin(e^{x-\frac{1}{x}})dx+\int_{0}^{\infty}\sin(e^{x-\frac{1}{x}})dx$$
For the first term, $u=-x$... |
Is $\int_{-\infty}^{\infty}\sin(e^{x-\frac{1}{x}})dx=\frac{\pi}{2}$? |
> because $\cos(y\ln(n))+i\sin(y\ln(n))$ could be equal to 0
Why would this be true? $n$ is a positive number, so $y\ln(n)$ is a real number, and for any real number $r$, $\cos(r)+i\sin(r)$ has magnitude $1$. Ergo, we have:
$$
\begin{align*}
& |e^{x\ln(n)}(\cos(y\ln(n))+i\sin(y\ln(n)))| \\
= & |e^{x\ln(n)}| ... |
[Proof][1]
[1]: https://i.stack.imgur.com/5VVok.jpg
In step No. 6, it assumes that the angles are supplementary although it was never specified, is there a reason for this? |
Why in step 6 it tells that the angles are supp?(read desc) |
To the best of our knowledge,in two-dimensional [Complex space][1], there are N vectors in arbitrary directions. Using "Method One," by fixing one vector among the N vectors and independently rotating the remaining N-1 vectors, it is possible to maximize the magnitude of the composite vector obtained by rotating and su... |
a alternative way to get the maximum value of the magnitude of the composite vector? |
[Proof][1]
[1]: https://i.stack.imgur.com/5VVok.jpg
In step No. 6, it assumes that B, C, D are collinear although it was never specified, is there a reason for this? |
To the best of our knowledge,in two-dimensional [Complex space][1], there are N vectors in arbitrary directions. Using "Method One," by fixing one vector among the N vectors and independently rotating the remaining N-1 vectors, it is possible to maximize the magnitude of the composite vector obtained by rotating and su... |
I am currently doing Problem 2, Chapter 1, from the "Functional Equations and How to Solve Them" book by Christopher G. Small.
It is as follows:
> Let $f(x)$ be a function that satisfy $f(x + y) = f(xy)$ for all **positive** $x$ and $y$.
> Prove that $f(x)$ is a constant.
Here are my 2 possible solution... |
show that the set of 2x2 matix of the form
$$\begin{matrix}
a & b \\
c & 0 \\
\end{matrix}
$$
is a vector space with the standard operations
can i incorporate the idea that this certain matrix can be connected (a subspace) to the M_22 matrix? or not since it has not been stated in the problem and prove thi... |
Let $H$ be a Hilbert space. Working on locally compact quantum groups, I have met with operator of the form $\iota \otimes \omega_{\xi, \eta}$ with $\xi,\eta \in H$ and $\omega_{\xi,\eta}(T) = (T\xi,\eta)$. These operators should be defined on $B(H \otimes H)$ but I can't find their definitions.
First, I thought we... |
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