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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 central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the freq... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integers $c,d... |
The question as worded is a bit ambiguous, but looking at the answer, we can consider the following problem.
> Imagine $24$ balls organized into $12$ bins; in each bin there is one red ball (representing an international trip) and a green ball (representing a non-international trip). You draw $2$ balls at random, wi... |
Division by zero? |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integers $c,d... |
Suppose $(M_i)_{i \in I}$ is a collection of flat $A$-modules ($A$ is a commutative ring with $1$). Is the tensor product $\bigotimes_i M_i$ still flat?
This is obviously true by induction when $I$ is a finite set, since
$$
(M_1 \otimes \cdots \otimes M_n) \otimes N \cong M_1 \otimes (M_2 \otimes \cdots \otimes M_... |
Is the infinite tensor product of flat modules still flat? |
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 central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the freq... |
Consider the first quadrant of a circle. We can represent the first quadrant of a circle as:
$y_1 = \sqrt{1-x^2},$ such that $0\leq x \leq 1 \\$
and in parametric terms as:
$\left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)$.
Let us compute the derivatives for both. So:
$\frac{dy_1}{dx} = \sqrt{1-x... |
$\lim_{n \to \infty}\prod_{k=1}^{n}{\left ( 1+\frac{1}{2^{2^{k} } } \right )}
$
Let $ a_n={\left ( 1+\frac{1}{2^{2^{k} } } \right )} $ ,
$
\left( 1-\frac{1}{2^2} \right) a_n=\left( 1-\frac{1}{2^2} \right) \left( 1+\frac{1}{2^2} \right) \cdots \left( 1+\frac{1}{2^{2n}} \right) =
1-\frac{1}{2^{2n+1}}$
$a_n=\... |
$\lim_{n \to \infty}\prod_{k=1}^{n}{\left ( 1+\frac{1}{2^{2^{k} } } \right )}
$
Let $ a_n=\prod_{k=1}^{n}{\left ( 1+\frac{1}{2^{2^{k} } } \right )} $ ,
$
\left( 1-\frac{1}{2^2} \right) a_n=\left( 1-\frac{1}{2^2} \right) \left( 1+\frac{1}{2^2} \right) \cdots \left( 1+\frac{1}{2^{2n}} \right) =
1-\frac{1}{2^{... |
So, my question is about proving theorems and the axioms needed to prove the theorem. My understanding is that, in the theorem statement there are certain axioms and one conclusion we have to prove. We have to prove that the conclusion of the theorem statement holds given that **exactly** all of the axioms given in the... |
Extra-axiomatic true statements for proving a theorem? |
> Let $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic function s.t $0$ is a
> zero of order $k\geq 1$. Prove that $f(z)=z^kg(z)$, in which
> $g:\mathbb{D}\to\mathbb{D}$ is holomorphic.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, then by Taylor's expansion we have $$f(z)=\sum\limits_{n=0}^{\infty}c_nz^n,\... |
> Let $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic function s.t $0$ is a
> zero of order $k\geq 1$. Prove that $f(z)=z^kg(z)$, in which
> $g:\mathbb{D}\to\mathbb{D}$ is holomorphic.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, then by Taylor's expansion we have $$f(z)=\sum\limits_{n=0}^{\infty}c_nz^n,\... |
I have two categories $C$ and $D$ and a functor $F: C\rightarrow D$. I select an object $A\in ob(C)$, is there guaranteed to be a homomorphism $Aut(A)\rightarrow Aut(F(A))$? It seems like there should be, but I am having a hard time formalizing the argument, given I'm somewhat new to the subject.
If such a homomorph... |
Let $a_{i}$ and $b_{i}$, $i = 1,\dotsb,n$ be real numbers. Denote $S_{pq}:=\sum_{i=1}^{n}a_{i}^{p}b_{i}^{q}$.
I want to show that,
$$ (S_{20}S_{02} - S_{11}^2)S_{00} - (S_{10}S_{02} - S_{01}S_{11})S_{10} + (S_{10}S_{11} - S_{01}S_{20})S_{01} \geq 0,$$
and find the condition on $a$ and $b$ for this inequality to be... |
> Let $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic function s.t $0$ is a
> zero of order $k\geq 1$. Prove that $f(z)=z^kg(z)$, in which
> $g:\mathbb{D}\to\mathbb{D}$ is holomorphic.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, then by Taylor's expansion we have $$f(z)=\sum\limits_{n=0}^{\infty}c_nz^n,\... |
> Let $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic function s.t $0$ is a
> zero of order $k\geq 1$. Prove that $f(z)=z^kg(z)$, in which
> $g:\mathbb{D}\to\mathbb{D}$ is holomorphic.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, then by Taylor's expansion we have $$f(z)=\sum\limits_{n=0}^{\infty}c_nz^n,\... |
I have a real signal $f(t) $that is periodic on $[0,L]$ where $L=1$. I sampled it uniformly $64$ times and computed FFT (in python). I was asked to find the central frequency, namely the frequency with the largest magnitude. I found the index $i$ of the largest value in the transformed signal.
How do I find the freq... |
Your (implicit) claim that $$e_1:=(1,1,1,1),\quad e_2:=(0,1,1,1),\quad e_3:=(0,0,0,1),\quad e_4:=(0,1,0,0)$$ make a basis of $\Bbb R^4$ is correct (how would you prove it?), hence for any choice of four vectors $f_1,f_2,f_3,f_4\in\Bbb R^4$, there is a unique linear transformation $T:\Bbb R^4\to\Bbb R^4$ that maps $e_i$... |
Your (implicit) claim that $$e_1:=(1,1,1,1),\quad e_2:=(0,1,1,1),\quad e_3:=(0,0,0,1),\quad e_4:=(0,1,0,0)$$ make a basis of $\Bbb R^4$ is correct (how would you prove it?), hence for any choice of four vectors $f_1,f_2,f_3,f_4\in\Bbb R^4$, there is a unique linear transformation $T:\Bbb R^4\to\Bbb R^4$ that maps $e_i$... |
From the definition, for a completely regular spaces $X$, a nonempty open set $U\subset X$, and a point $x\in U$, there exists a cts $f: X\to[0,1]$ s.t. $f(X-U)=0$ and $f(x)=1$. Then $x\in f^{-1}((1/2,1])\subset U$. Does this qualify as a proof of the statement in the title?
But aren't we supposed to show there are fi... |
> Let $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic function s.t $0$ is a
> zero of order $k\geq 1$. Prove that $f(z)=z^kg(z)$, in which
> $g:\mathbb{D}\to\mathbb{D}$ is holomorphic.
My attempt:\
Since $f$ is holomorphic on $\mathbb{D}$, then by Taylor's expansion we have $$f(z)=\sum\limits_{n=0}^{\infty}c_nz^n,\... |
Let $(a,b,c)$ be the sides of a triangle and let its circumradius be $1$. Is it true that
$$
a^c + b^c - c^c < \pi
$$
**My progress**: In the special case where at least two of the three sides are equal, I have been able to show that the upper bound is about $3.13861$. if two sides are equal and the third side ... |
Let $(a,b,c)$ be the sides of a triangle and let its circumradius be $1$. Is it true that
$$
a^c + b^c - c^c < \pi
$$
**My progress**: In the special case where at least two of the three sides are equal, I have been able to show that the upper bound is about $3.13861$. if two sides are equal and the third side ... |
Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. The category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class of weak equivalences $W_{\mathcal{J}}$, those natural transformations which are objectwise weak equivalences. If $\mathca... |
Let's do the calculation both ways and see what happens:
Under a binomial model, we have $n = 80000$, and under the assumption that any individual has an equal probability of being born on any one of $365$ days in a year, and that partners in marriage are not any more or less likely to marry someone sharing the same... |
I'm plotting stock market charts with matplotlib in Python, where the x-axis represents dates with 0-indexed numbers, and the y-axis shows varying price scales for different stocks.
I draw lines between 2 points of varying lengths on each of these charts and calculated their slope.
$$ m = \frac{{y2 - y1}}{{x2 - x... |
How to compare the slope of lines from different charts? |
After searching for a while, I've found the answer in [this paper][1]. $H(1)$ does indeed not exist. And this is not just the case for $H^2=D$ but for any linear operator $H$ that satisfies $H^a = D$ for $a > 1$. A fractional derivative with index law simply doesn't work on constant functions.
[1]: https://www.n... |
Consider the intersection of the curves $x^2+y^2+6x-24y+72=0, x^2-y^2+6x+16y-46=0$. Determine the sum of the distances of their intersection points and $(-3,2)$.
My first thought looking at this problem was homogenization after shifting the origin to $(-3,2)$ but then I realized that homogenization is done with a li... |
I am studying measure theory, topology, and functional analysis in mathematics. Let $X$ be a compact topological space. We assume that there exists a countable family $\{f_n\: X \to \mathbb C: n\in\mathbb{N}\}$ of continuous complex-valued functions on $X$ that {\em separates points} in the following sense: for a... |
The linear system: $Ax=b$
I know that if the system is underdetermined, there will be closed-form solutions. But if $b\sim any~distribution$, are there closed-form solutions?
More detailedly, this is what I am trying to solve:
$$
\arg \min_x \lVert x - X\rVert_2, X\sim any~distribution, X, x \in \mathbb{R}^{n\t... |
From the definition, for a completely regular spaces $X$, a nonempty open set $U\subset X$, and a point $x\in U$, there exists a cts $f: X\to[0,1]$ s.t. $f(X-U)=0$ and $f(x)=1$. Then $x\in f^{-1}((1/2,1])\subset U$. Does this qualify as a proof of the statement in the title?
But aren't we supposed to show there are fi... |
Proving sequence by induction or what means? |
How to show that the determinant of a square matrix is equal to the determinant of its transpose? |
This question arises from [this one](https://math.stackexchange.com/q/4743554/863176), and more generally from many textbook asserting that *« for any stochastic process having the Markov property, the past and the future are independent given the present. »*
Let $X = \{X(t) : t\in \left[0, \infty\right) \}$ be a st... |
Let $(a,b,c)$ be the sides of a triangle and let its circumradius be $1$. Is it true that
$$
a^c + b^c - c^c < \pi
$$
**My progress**: In the special case where at least two of the three sides are equal, I have been able to show that the upper bound is about $3.13861$. If two sides are equal and the third side ... |
In the proof of lemma in https://math.stackexchange.com/questions/1887976/hurwitz-constant-proof-of-lemma-6-1-1-in-esmonde-and-murtys-problems-in-alge#:~:text=There%20is%20a%20constant%20H,%E2%88%92%CE%B2)%7C%3C1.
The Hurwitz constant is not free of elements from the chosen integral basis. So if we consider another in... |
In Algebraic number theory, is the Hurwitz constant remain same if we consider another integral base? |
Atiyah Macdonald Chapter 10. Is a topological group which is isomorphic to a complete topological group also complete? |
In the proof of lemma in https://math.stackexchange.com/questions/1887976/hurwitz-constant-proof-of-lemma-6-1-1-in-esmonde-and-murtys-problems-in-alge#:~:text=There%20is%20a%20constant%20H,%E2%88%92%CE%B2)%7C%3C1.
The Hurwitz constant is not free of elements from the chosen integral basis. So if we consider another in... |
I have a Manager who has no information about the profit from investment into two bonds, other than the fact that they are independently drawn from uniform distribution ${m, 1}$ where, m>0. Profit from 1st and 2nd bond are denoted by a and b respectively.
He asks his partner (who is knowledgeable about profit from ... |
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0& -I_{16}\end{bmatrix} \tag{2}$$
I know that a symmetric matrix has $28 \times 29/2=406$ free parameters and I know that the final a... |
Let $V=\mathbb{C}\backslash \left\{-1\right\}$ and define the special addition $\oplus$ on $V$,for $\forall a,b\in V$
$$a\oplus b=a+b+ab.$$
Then we can check that $V$ satisfies all additive conditions in linear space with the special addition $\oplus$.
For examle, the zero element is $0$, the additive inverse... |
I have a second order diff eqn that is in the form of
\begin{equation}
y^{\prime\prime} + \Big(\frac{3}{x} - \frac{3x^{-4}}{2[x^{-3} + B]}\Big)y^{\prime} - \frac{3}{2}\frac{1}{x^5[x^{-3} + B]}y = \mathcal{F}\,.
\end{equation}
where $B$ and $\mathcal{F}$ are **known** constant. I have two **initial condition... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
Let $V=\mathbb{C}\backslash \left\{-1\right\}$ and define the special addition $\oplus$ on $V$, for $\forall a,b\in V$
$$a\oplus b=a+b+ab.$$
Then we can check that $V$ satisfies all additive conditions in linear space with the special addition $\oplus$.
For example, the zero element is $0$, the additive inver... |
I have a second order diff eqn that is in the form of
\begin{equation}
y^{\prime\prime} + \Big(\frac{3}{x} - \frac{3x^{-4}}{2[x^{-3} + B]}\Big)y^{\prime} - \frac{3}{2}\frac{1}{x^5[x^{-3} + B]}y = \mathcal{F}\,.
\end{equation}
where $B$ and $\mathcal{F}$ are **known** constant. I have two **initial condition... |
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.
Does there exist an example of the following?
* A commutative ring with unity $R$
* A mapping $\operatorname{d}: R\operatorname{-Mod} \to \overline{\mathbb{R}}_{\geq 0}$ with the following properties:
* If $A \cong B$, $\operatorname{... |
I've been reading "On a small cancellation theorem of Gromov" by Yann Ollivier (https://projecteuclid.org/journalArticle/Download?urlId=10.36045%2Fbbms%2F1148059334 ) recently and these definitions aren't really clear for me since the author never really described what he meant by combinatorial 2-complex. I am also not... |
Can someone explain these Definitions about Cell Complexes for me? |
I just came into contact with matroid theory through James Oxley's book 'Matroid Theory (2011)', and I was stuck on a problem while doing the exercises in the first section of Chapter 1:
> Deduce that, as $n\rightarrow\infty$,the proportions of non-isomorphic $n$-element matroids that are graphic or are $GF(2)$-repr... |
I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry:
> If seven vertices of a hexahedron lie on a sphere, then so does the eighth vertex.
Is it suffice to prove: If seven vertices of a hexahedron lie on a [quadric surface](https://en.wikipedia.org/wiki/Quadric), th... |
I have doubts for what concerns the end of this exercise. I have the function $f(x, y, z) = x$ and I have to find min and max subject to the constraints $x^2+y^2+z^2 = \frac{5}{2}$ and $y + z = 1$.
I calculated the Jacobian matrix of the constraints and I found it has maximum rank of $2$, unless $x = y = z = 0$ thou... |
As we know from 1st year Calculus, for function $f(x) = x^n, x \in \mathbb{R}, n \in \mathbb{N}$, if $n$ is odd, the $f(x)$ is increasing; if $n$ is even $f(x)$ is increasing when $x \geq 0$ and decreasing when $x<0$. But here we only have the cases where $n$ is a positive integer. We don't know how it goes if $n$ is r... |
Can you please check whether this is a correct solution by using $\mathcal{F}[\operatorname{sgn}(t)]=\frac{2}{i\omega}$? From the FT of sign function, we get
$\mathcal{F}^{-1}[\frac1\omega]=\frac{i\operatorname{sgn}(t)}{2}$. Now we are close to the target but still need some shifting. We will continue
\begin{ali... |
I am reading Dummit & Foote for studying undergraduate level field theory, and here is what I see on Page 555:
[![enter image description here][1]][1]
I followed the words of writers and understood the proof for a statement
$$\mathbb{Q}(\zeta_8) = \mathbb{Q}(i, \sqrt{2})$$
where $\zeta_8$ denotes the primit... |
This question is motivated by the fact that we can write any rational function $f$ on $\mathbb{C}$ as
$$f(z) = g(z)+\dfrac{p(z)}{q(z)}$$
where $g,p,q$ are polynomials with $\mathrm{deg}(p)\leq \mathrm{deg}(q)$. Now, we can develop $\dfrac{p}{q}$ via partial fraction decomposition as
$$\dfrac{p(z)}{q(z)}=\sum_{i}\d... |
Let $f,g:\mathbb{R}\to \mathbb{R}$. Let $a\in\mathbb{R}$. Let $\lim_{x\to a}f(x)=0=\lim_{x\to a}g(x)$. Now I want to say which function goes to zero faster as x approaches zero. I have seen in a post that the accepted definition is: if $\lim_{x\to a}\frac{f(x)}{g(x)}=0$ then we say f goes to zero faster than g as x app... |
I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry:
> If seven vertices of a (quadrilaterally-faced) hexahedron lie on a sphere, then so does the eighth vertex.
Is it suffice to prove: If seven vertices of a (quadrilaterally-faced) hexahedron lie on a [quadric sur... |
> Consider $\mathbb{R}$ equipped with the excluded point topology $\tau_0 = \{U \subseteq \mathbb{R} : 0 \notin U \text{ or } U = \mathbb{R}\}$. Identify the limit points of the sequence $(1+\frac{1}{n})_{n \in \mathbb{N}}$ in $(\mathbb{R}, \tau_0)$.
So, I've gotten quite far with this, I think I have shown that $0$... |
In proving the Jonhson-Lindenstrauss Lemma with $\mathrm{Uniform[-1,1]}$ random projections I used the following fact which I have been trying to prove:
$$ \mathbb{E}[e^{\lambda(3Z(w)^2-1)}] \leq e^{2\lambda^2}, \quad \forall \ \lambda \leq \frac{1}{4}$$
Where $Z(w)$ is defined as: $Z(w) = \sum_{j=1}^p a_j w_j$,... |
I am working on the following questions:
> Let $X$ be a compact Riemann surface of genus $3$ with two points $p\neq q$.
>
> - Find a non-constant meromorphic function on $X$ with at least a double zero at $p$ and holomorphic everywhere except possibly at $q$.
>
> - What is the smallest possible pole under at $... |
I need to solve polynomial equation and I know an interval with a root. I try to use Newton method, but sometimes it fails due to overshooting. So instead I try to adopt an interval method, to keep myself in interval - it works, but not always.
Let's see a simple example function: y=x2+2x-30 and interval [a,b]
Deriv... |
The Fredholm determinant of $1-T$ with a trace-class integral operator $T$ with kernel $K$ is ("informally") defined by
\begin{equation}
\text{det}_F\big(1-T\big)=\sum_{n=0}^{\infty}\frac{1}{n!}\int_{x_1\dots x_n}\det\big(K(x_j,x_k)\big)\ ,
\end{equation}
where $1\leq j,k\leq n$.
In quantum field theory one often ... |
Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are perpendicular lines on the plane $y=\sqrt{2}$.\
$D,E$ are the image of $A,B$ under the perspective projection centered at $C$ onto $xy$-plane.\
Why are the lines $DF,EF$ perpendicular?
[![enter image description here][1]][1]
... |
There is no other set that contains the same $A$.
Start with a set $Y$ and form
$$
S=\{p\in Y\colon p\not\in p\}
$$
What can we say about $S$?
1. Is $S\not\in Y$? Possibly.
2. Is $S\in Y$ and $S\not\in S$? No because then by definition of $S$, $S\in S$.
3. Is $S\in Y$ and $S\in S$? No because then by ... |
Let $S$ be some subset of $\mathbb{R}^3$ and $\text{Conv}(S)$ be its convex hull. Say $S$ has property $\mathcal{P}$ if each point $x \notin S$ lies on some line contained in $\mathbb{R}^3 \setminus S$ such that there is a ruled surface in $\mathbb{R}^3 \setminus S$ giving a homotopy to a line in $\mathbb{R}^3 \setminu... |
What does it mean $\frac{n-1}{2}K_{2}$?
In general which graph am I referring to if I write $l\times K_{2}$?
To add a little bit of contest, we are talking about undirected graph on n-1 vertices with adjacency matrix A, Denote the eigenvalues of A by $\lambda_{1}\geq \lambda_{2}\geq…\geq\lambda_{n-1}$.
Then “it is w... |
Consider a non-zero positive element $a$ of a simple C*-algebra $A$. Suppose it has a sub-projection which is Murray-vN equivalent to an infinite projection, i.e. there exists an infinite projection $h$ s.t. $h\precsim a$. Then is $a$ properly infinite?
Here are the steps and I don't see why exactly each one is true... |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Assume that $\mathcal{H}$ is a Hilbert space with inner product $\langle \cdot, \cdot \rangle$ and an orthonormal basis $\{e_j\}_{j=1}^{\infty}$ and $C: \mathcal{H} \to \mathcal{H}$ is a rank one Hilbert-Schmidt operator. I came across the statement that for such an operator it holds that for all $i<j$,
$$\det \begin{... |
Consider the problem
$$
\left\lbrace
\begin{gathered}
-\Delta u = 1 \quad\textrm{in}\quad S\\
u=0\quad\textrm{on}\quad \partial S
\end{gathered}
\right.
$$
where $S$ is the infinity strip $\mathbb{R}\times (0,1)$ or the semi-infinity strip $\mathbb{R}_+\times (0,1)$.
What extra conditions should be imposed ... |
Assume that $\mathcal{H}$ is a Hilbert space with inner product $\langle \cdot, \cdot \rangle$ and an orthonormal basis $\{e_j\}_{j=1}^{\infty}$ and $C: \mathcal{H} \to \mathcal{H}$ is a rank one Hilbert-Schmidt operator. I came across the statement that for such an operator it holds that for all $i<j$,
$$\det \begin{... |
I've learned about differentiability of several variables. I know that the requirement of differentiable is that its partial derivatives are continuous. As for counterexample,
$xy\sin(1/(x^2+y^2))$ is not differentiable but its partial derivatives are continuous.
And if it is differentiable then we can use chain r... |
Assume that $\mathcal{H}$ is a Hilbert space with inner product $\langle \cdot, \cdot \rangle$ and an orthonormal basis $\{e_j\}_{j=1}^{\infty}$ and $C: \mathcal{H} \to \mathcal{H}$ is a rank one Hilbert-Schmidt operator. I came across the statement that for such an operator it holds that for all $i<j$,
$$\det \begin{... |
Assuming $\pi: \tilde M\rightarrow M$ be a universal covering of a complete Riemannian manifold $M$.
$f:\tilde M \rightarrow \tilde M$ is a covering transformation. If $f$ has fixed point, whether $f$ must be the identity? |
Whether covering transformation of universal covering have not fixed point? |
A random sample X1, . . . , Xn of size n is taken from the Poisson distribution with parameter θ.
Let $X = (X_1, . . . , X_n)^T$ and $x = (x_1, . . . , x_n)^T$.
It is proposed now to estimate the estimate the function of θ, g(θ), where
$g(θ) = P(X = k) = exp(−θ)θ^k/k!$
for a specified integer k greater than zero.\\... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
> Of the three properties that a set of sentence may have: "consistent", "complete", "is a theory", which pairs imply the third?
It is Exercise 1.4.42 in Hinman's *Fundamentals of Mathematical Logic*. I guess if a set of sentence $\Delta$ is a complete propositional theory, then it is consistent. As it is similar ... |
Of the three properties that a set of sentence may have: "consistent", "complete", "is a theory", which pairs imply the third? |
A random sample $X_1, . . . , X_n$ of size n is taken from the Poisson distribution with parameter $\theta$.
Let $X = (X_1, . . . , X_n)^T$ and $x = (x_1, . . . , x_n)^T$.
It is proposed now to estimate the estimate the function of $\theta$, $g(\theta)$, where
$g(\theta) = P(X = k) = exp(−\theta)\theta^k/k!$
for a ... |
> We call a field $F$ called separably closed if the only separable algebraic extension $F\subset E$ is the trivial extension, that is $E=F$. A separable closure of a field $K$ is a separable algebraic extension $K \subset K_s$, with $K_s$ separably closed. \
Q: Show that every field $K$ has a separable extension; furth... |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Let $\sigma(n)$ be the sum of all divisors (includin 1 and $n$) of $n$, and define the abundacy index of $n$ as $I(n) = \sigma(n)/n $. For example: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. Thus all perfect numbers have an abundacy index of 2. I wondered what the average abundacy of all numbers was, and it tur... |
Let $M$ be a spacetime (i.e., a differentiable manifold equipped with a Lorentzian metric) that admits a choice of time orientation.
For any region $R\subset M$, we define the causal complement of $R$ as $R' = M\setminus J(R)$, where $J(R) = J^+(R)\cup J^-(R)$ is the union of the causal future and causal past of $R... |
Is causal convexity equivalent to causal completeness? |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
*Often*, when people use "either ... or ..", they are trying to stress that they are expressing an exclusive disjunction.
But, not always.
Consider. I say "I want to be either rich or happy" ... I think that is a perfectly good usage of English ... but it is also clear that I mean this in an inclusive sense: it... |
In my set theory course, we introduced the Axiom of Union.
> For any set $x$ there exists a set $y$ s.t. all elements of $y$ are elements of some $z\in x$. Notation for this set is $y= \bigcup x$.
>
> $\forall x\ \exists y \ \forall z (z\in y \leftrightarrow \exists w \ (z\in w \land w\in x))$
My professor then ... |
Recursive definition of union seemingly informal? |
Let $\sigma(n)$ be the sum of all divisors (including 1 and $n$) of $n$, and define the abundancy index of $n$ as $I(n) = \sigma(n)/n $. For example: $I(6)= \frac{1+2+3+6}{6} = 1/1+1/2 +1/3 +1/6 = 2$. Thus all perfect numbers have an abundancy index of 2. I wondered what the average abundancy of all numbers was, and it... |
Assuming $M$ is a compact Riemannian manifold. $\gamma_j:[0,1]\rightarrow M$ are equicontinuous curves, namely, there is constant $L$ such that
$$
d(\gamma_j(t_1),\gamma_j(t_2))\le L(t_2-t_1)
~~~~~~
\forall t_1\le t_2 \in [0,1],~~\forall \gamma_j\in\{\gamma_j\}
$$
And the length of $\gamma_j$ converge, assuming ... |
Equicontinuous curves on compact Riemannian manifold have uniform convergent subsequence? |
Let $X$ be a Polish space. Recall that the [Suslin operation](https://en.wikipedia.org/wiki/Suslin_operation) is the operation $\mathcal{A}$ such that for any *Suslin scheme* $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have:
$$
\mathcal{A}\{A_s : s \in \omega^{<\omega}\} := \bigcup_{a \in \omega^\omega} ... |
(I am reading Huybrechts's Complex Geometry)
I also looked some classic sources, but I didn't find anything addressing the below.
---
**Proposition $B.0.35$**
Let $$0\longrightarrow \mathcal{F}^0\longrightarrow \mathcal{F}^1\longrightarrow \mathcal{F}^2\longrightarrow 0$$
be a short exact sequence of she... |
**Expression:**
[![enter image description here][1]][1]
**My attempt:**
1. $(xy)' + (yz)$
2. $(x'+y') + (yz) \quad \textit{After applying de Morgan's Axiom}$
3. $x' + (y' + yz) \quad \textit{After applying 1st Distributive Axiom}$
4. $x'+ (y' + y) + (y'z) \quad \textit{Rewriting}$
5. $x' + 1 + (y'z... |
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