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Suppose that $M$ is simply connected, compact manifold.
Then I want to show that every closed $1-$ form is exact on $M$.
This is same showing that the first de-Rham cohomology space of $M$ is $0$.
**Question**: Is it possible to answer this without using universal coefficients theorem?
(The reason for not usi... |
Let $X$ be a topological space and $A\subseteq X$ a subset. Suppose I know that $V\subseteq X$ is such that $V\subseteq X\setminus A$ and $V$ is open
Can I conclude $V\subseteq X\setminus\overline{A}$? Why? |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
What numbers can be wrriten uniquely as a sum of two squares? |
And what if $f > g$ ?
I tried to find some counter-example but it's less obvious ! |
What numbers can be written uniquely as a sum of two squares? |
I am confused about what can and cannot be done in a proof in order to avoid having an infinite number of "steps", thus making the argument invalid.
For instance, let $A_{0}, A_{1}, \dotsc$ be subsets of $\mathbb{R}^{n}$, and suppose that I want to show that $A_{0}$ satisfies some desired property *after shrinking ... |
How do I know if I am using infinitely many "steps" in a proof? |
And what if $f > g$ ?
I tried to find some counter-example but it's less obvious !
Feels like we can prove it on segments, no ?
|
I have the following proof in my notes. It is a particular case of the more general rank theorem.
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is inj... |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
>Surely if $X_t$ is nice and smooth, then $\alpha > \frac{1}{2}$, but then $2\alpha > 1$, and wouldn't this degenerate into saying that $\mathbb{X}$ is constant? From the known result on Holder continuous greater than 1 $\implies$ constant?
As mentioned in the comments, the rough path lift $\mathbb{X}_{s,t}$ is n... |
For any two finite subsets $A,B$, of an abelian group, is
$$|A+B|^2 |A-B|^2 \geq |A+A||A-A||B+B||B-B| \ ?$$
I’m interested in finding out if there are sumset inequalities that are sharper than the triangle inequalities and that show that some quantity related to sums and differences is smaller when the sums and ... |
Let $X$ be a topological space and $A\subseteq X$ a subset. Suppose I know that $V\subseteq X$ is such that $V\subseteq X\setminus A$ and $V$ is open
Can I conclude $V\subseteq X\setminus\overline{A}$? Why?
Equivalently, I tried to prove $\overline{A}\subseteq X\setminus V$.
If $x\in A$ then $x\in X\setminus... |
If $|z - i| \le 2$, then $z = 3i$ certainly satisfies this inequality, since
$|3i - i| = |2i| = 2$, yet $|3i| = 3 > 1$. So you clearly have an error. Where is it?
Well, the triangle inequality does say that $$2 = |z-i| = |z+(-i)| \color{red}{\le} |z| + |-i| = |z| + 1. \tag{1}$$ But the problem here is that you wri... |
I have the following proof in my notes. It is a particular case of the more general rank theorem.
Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is inj... |
How do I remove the specific assumptions in the proof of the rank theorem in the special case of injective differential? |
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^\alp... |
Given $f(x) = \frac{x+1}{x^2+2 \sqrt{x}}$, How to prove $\lim_{x \rightarrow 4} f(x)$ using $\varepsilon$-$\delta$ definition? |
Is it possible to simplify the following expression, where $k$ is a given constant? I want to simplify it to something in terms of only $k$.
$$ \sum_{i=0}^{\lfloor \frac{k}{2} \rfloor} 2^{k} \binom{k-i}{i}$$
I was confused on how to deal with the $i$ in the combination. |
Let $e_1,\ldots,e_n$ be the standard basis in $\mathbb{R}^n$. Suppose the scalars $\lambda_1,\ldots,\lambda_n$ satisfy $0< \lambda_1,\ldots,\lambda_n\leq1$ and $\lambda_1^2+\ldots+\lambda_n^2 = m$, where $m$ is a positive integer less than or equal to $n$. Does there always exist a $m$-dimensional subspace $W$ of $\mat... |
Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of prime numbers $p$ such that $a$ is a primitive root modulo $p$ is
$$
\tag{$\star$}
A(a)=\begin{cases}
\left(\displa... |
An upper bound for the [primorial](https://en.wikipedia.org/wiki/Primorial) can be found based on the first [chebyshev function](https://en.wikipedia.org/wiki/Chebyshev_function).
From $\vartheta(x) < 1.00028x$, it is clear that:
$$\prod\limits_{p \le x}p \le e^{1.00028x} < (2.72)^x$$
I am looking at the follo... |
From the ["Finite affine planes" section](https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes) of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$:
- each line contains $n$ points,
- each point is contained in $n+1$ lines,
- ther... |
An upper bound for the [primorial](https://en.wikipedia.org/wiki/Primorial) can be found based on the first [chebyshev function](https://en.wikipedia.org/wiki/Chebyshev_function).
From $\vartheta(x) < 1.00028x$, it is clear that:
$$\prod\limits_{p \le x}p \le e^{1.00028x} < (2.72)^x$$
I am looking at the follo... |
>Surely if $X_t$ is nice and smooth, then $\alpha > \frac{1}{2}$, but then $2\alpha > 1$, and wouldn't this degenerate into saying that $\mathbb{X}$ is constant? From the known result on Holder continuous greater than 1 $\implies$ constant?
As mentioned in the comments, the rough path lift $\mathbb{X}_{s,t}$ is n... |
The definition of a Pentagon says (https://en.wikipedia.org/wiki/Pentagon) "five-sided polygon" but let ABCD be a square and E the middle of AB. Is AEBCD a pentagon?
What in the definition of a pentagon "prevents" this from happening? And more generally hexagons, etc ...
Does "five-sided" does not work if the side ... |
I have to test the following Nullspace property: $$ Null(A) \subset Null(BA) $$
where $$ Null(A) = {x\in \mathbb{R}^n/ A x=0} $$
$$ Null(BA) = {y\in \mathbb{R}^n/ AB y=0} $$
For any matrix $A$,$B$ such that they have the correct dimensions to make the product.\
The problem is that when I write the linear ... |
How can I test this property of the nullspace subspace? |
From this [answer][1], it implicitly becomes evident that
$$\sum_{k=1}^\infty \frac{kn^k}{(k+n)!} =\frac{1}{(n-1)!}$$
holds for any integer $n$.
I **guess** that the following general formula holds for any $x>0$:
$$\color{blue}{\sum_{k=1}^\infty \frac{kx^k}{\Gamma(k+1+x)}=\frac{1}{\Gamma(x)}}.$$
I cou... |
Consider the numbers $ax^2$, $c-by^2$, where $x,y=0,1,…,p-1$. There are $2p$ such numbers. Any residue class, except for $0$, can have at most two elements of the form $ax^2$ and at most two elements of the form $c-by^2$ from the ones listed above.
I’m trying to understand the logical deductions happening here. Any ... |
Let $A$ be a compact topological space equipped with the Borel $\sigma$-algebra, and $X=B_b(A)$ be the vector space of bounded measurable functions. Let $Y=\mathcal M(A)$ be the vector space of finite signed measure on $A$. Define the dual pair $<\cdot, \cdot>$ between $(X,Y)$ such that $
<f, \mu >=\int_A f(a)\mu(da... |
Let $\mathcal{O}_{\mathbb{P}^m}(1)$ be the tautological bundle on $\mathbb{P}^m$ and let $T := (\mathbb{C}^\times)^{m+1}$ act on $\mathbb{P}^m$ diagonally, i.e.,
$$
(t_0,\ldots,t_m) \cdot [x_0:\cdots:x_m] := [t_0x_0:\cdots:t_mx_m].
$$
The total space of $\mathcal{O}_{\mathbb{P}^m}(1)$ is given by $\mathbb{C} \tim... |
We consider the sequence $(f_n)_{n \in \mathbb{N}}$ defined by $f_n : [0, 2] \rightarrow \mathbb{R}$ as follows:
$f_n(x) =\begin{cases}
x^n, & \text{if } 0 \leq x \leq 1 \\
1, & \text{if } 1 < x \leq 2
\end{cases}$
for $n \in \mathbb{N}$.
Show that the sequence $(f_n)_{n \in \mathbb{N}} \in (... |
Is it possible to simplify the following expression, where $k$ is a given constant? I want to simplify it to something in terms of only $k$.
$ \sum_{i=0}^{\lfloor \frac{k}{2} \rfloor} 2^{k} \binom{k-i}{i}$.
I was confused on how to deal with the $i$ in the combination. |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
We consider the sequence $(f_n)_{n \in \mathbb{N}}$ defined by $f_n : [0, 2] \rightarrow \mathbb{R}$ as follows:
$f_n(x) =\begin{cases}
x^n, & \text{if } 0 \leq x \leq 1 \\
1, & \text{if } 1 < x \leq 2
\end{cases}$
for $n \in \mathbb{N}$.
Show that the sequence $(f_n)_{n \in \mathbb{N}} \in (... |
I'd like to know if the following statement is true:
For any $f \in \mathcal{C}^\infty(\mathbb{R})$ such that $f(0) = 0$, then $f(x)/x$ is smooth at $0$.
- if $f$ is analytic, then this is clear by simply Taylor-expanding it.
- hence, my question is for when $f$ is non-analytic. I am hoping that, maybe the fac... |
$f$ smooth non-analytic, $f(0) = 0$. Is $f(x)/x$ smooth at $0$? |
In most of the cases I have studied, I haven't seen what it means for a form "to be" on the manifold. We usually let $M$ to be a manifold, and we let $\omega\in \bigwedge ^kT^*_pM$ in the exterior algebra of the dual tangent space, we then say $\omega$ is of the form $\sum_{I}f_Idx^I$ (I is a multiindex) and we don't g... |
Given the following relation, which properties does it fulfill?
Is my reasoning correct or am I missing something?
$(m, n) ∈ R \iff m > n$
***1.) reflexivity***
no, $1 > 1$ (trivially false)
***2.) symmetry***
no, since $x > y\not\rightarrow y > x$, since $2 > 3 \not\rightarrow 3 > 2$
***3.) antisy... |
I'd like to know if the following statement is true:
For any $f \in \mathcal{C}^\infty(\mathbb{R})$ such that $f(0) = 0$, then $f(x)/x$ is smooth at $0$.
- if $f$ is analytic, then this is clear by simply Taylor-expanding it.
- hence, my question is for when $f$ is non-analytic. I am hoping that, maybe the fac... |
From this [answer][1], it implicitly becomes evident that
$$\sum_{k=1}^\infty \frac{kn^k}{(k+n)!} =\frac{1}{(n-1)!}$$
holds for any integer $n$.
I **guess** that the following general formula holds for any $x>0$:
$$\color{blue}{\sum_{k=1}^\infty \frac{kx^k}{\Gamma(k+1+x)}=\frac{1}{\Gamma(x)}}.$$
I cou... |
Suppose $S$ is an invertible symmetric matrix with the following property:
For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number then $S_{ij} \neq 0$.
Is it possible to prove that the inverse matrix $S^{-1}$ also has the above property? |
I was working on another problem, and the last step to prove it would be finding a function like this. So to be clear, I am looking for a function $f(x)=\sum_{n=0}^\infty a_nx^n$ such that $0=f(1)=f'(1)=f''(1)=\ldots$ , and $a_n\to 0$. I might be just too tired, but I haven't been able to think of one. Does a function ... |
I am currently working through [this][1] paper by Denef, and I have run into a little bit of a snafu with Lemma 3.1.
The setup is as follows: let $K$ be any field of characteristic zero, and let $E_0$ be the elliptic curve defined by the equation
$$Y^2 = X^3 + aX + b$$
Identify $T$ with the rational function $... |
I am working on this question:
> Consider $\mathcal{P}_2(\mathbb{R})$ be all polynomials of degree at most 2 with real coefficients. Let $\mathcal{S}$ be a subspace defined by $\mathcal{S} = \{p \in \mathcal{P}_2(\mathbb{R}): p(0) = p(1) \}$. Let T: $\mathcal{S} \rightarrow \mathcal{S}$ be a linear operator defined by... |
Consider two random variables $X$ and $Y$ such that $P(X \le Y) = 1$ but $P(X < Y) < 1$, that is $P(X < Y) > 0$. Then, how do we prove that $E[X] < E[Y]$? |
How to prove that E[X] < E[Y]? |
> What numbers can be written uniquely as a sum of two squares?
I was looking at sequence [A125022](https://oeis.org/A125022), which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$... |
Consider two random variables $X$ and $Y$ such that $P(X \le Y) = 1$ but $P(X < Y) < 1$, that is $P(X < Y) > 0$. Then, how do we prove that $E[X] < E[Y]$?
My attempt is as below:
Let $A = \{x \in \mathbb{R}: x = X(w) < Y(w) = y \}$,
then
$$
E[Y]
=
\int_{\mathbb{R}} x \, dF_{Y}(x)
=
\int_{\mathbb{R}/A... |
In the paper [Magnetic Bloch analysis and Bochner Laplacians](https://www.academia.edu/65733916/Magnetic_Bloch_analysis_and_Bochner_Laplacians) of Ruedi Seiler, this expression appears in this context:
[![enter image description here][1]][1]
The conventional notation for a fiber bundle would be $\pi: b\ \to\ M$.... |
What kind of notation is $\pi: b\ \varnothing\ M$? |
So let's say we have two vectors $a \in \mathbb R^n$ and $b \in \mathbb R^m$. Let $L=\text{lcm}(n,m)$.
Consider the natural inclusions $a,b \mapsto \mathbb R^L$, which simply continually concatenate $a$ or $b$ to itself until the dimension matches $L$, which of course is possible due to $L$ being the least common mu... |
Is this summation over Fourier coefficients a meaningful way to compute dot product despite its weaknesses? Or what is the better way to define it? |
From the ["Finite affine planes" section](https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes) of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$:
- each line contains $n$ points,
- each point is contained in $n+1$ lines,
- ther... |
Consider two random variables $X$ and $Y$ such that $P(X \le Y) = 1$ but $P(X = Y) < 1$, that is $P(X < Y) > 0$. Then, how do we prove that $E[X] < E[Y]$?
My attempt is as below:
Let $A = \{x \in \mathbb{R}: x = X(w) < Y(w) = y \}$,
then
$$
E[Y]
=
\int_{\mathbb{R}} x \, dF_{Y}(x)
=
\int_{\mathbb{R}/A... |
From the ["Finite affine planes" section](https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes) of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$:
- each line contains $n$ points,
- each point is contained in $n+1$ lines,
- ther... |
I am currently reading Range's *Holomorphic functions and integral representations in several complex variables* and would like some clarification on the following definition.
> Let $M$ be a complex submanifold of $\mathbb{C}^n$ and let $p \in M$. Then a function $f : M \to \mathbb{C}$ is *holomorphic* at $p$ if $f ... |
The other solution uses class field theory. I will avoid that here.
Your question is the same as asking when $\mathbf Q(\sqrt{n}) \subset \mathbf Q(\omega_m)$.
By replacing $n$ by is biggest squarefree factor, we can assume $n$ is *squarefree*. And surely you are not interested in $n$ being a square, so we take... |
I was working on another problem, and the last step to prove it would be finding a function like this. So to be clear, I am looking for a function $f(x)=\sum_{n=0}^\infty a_nx^n$ such that $0=f(1)=f'(1)=f''(1)=\ldots$ , and $a_n\to 0$, and there exists a nonzero coefficient. I might be just too tired, but I haven't bee... |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3. :
N... |
Does anyone have an intuitive explanation for this result? My textbook uses a proof by contradiction. But it just doesn't come logically to me why this holds. What would go wrong if, for example, |a|=3 and |b|=4, where b is the largest order element? |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
How to show that gcd of two quasiperiods is a quasiperiod? |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3. :
N... |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now... |
Definition $3.1.1$ in page $29$ of [this book][1] is the definition of quasiperod and Proposition $3.1.3.$ shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition $3.1.3.... |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now... |
Definition $3.1.1$ in page $29$ of [this book][1] is the definition of quasiperod and Proposition $3.1.3.$ shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition $3.1.3.... |
Definition $3.1.1$ in page $29$ of [this book][1] is the definition of quasiperod and Proposition $3.1.3.$ shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition $3.1.3.... |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
Suppose we have a vector field $\vec{A}$ in $\mathbb{R}^n.$ The following “vector quantity”:
$$(d\vec{r} \cdot \nabla)\vec{A}$$
looks strange perhaps, but when expanded is seen to be a simple quantity:
$$(d\vec{r} \cdot \nabla)\vec{A} = (dx \frac{\partial}{\partial x} + dy \frac{\partial}{\partial y} )\vec{A}= \begi... |
Let $f \colon [0, 1] \to \mathbb{R}$ be defined by
$$
f(x) = \sum_{n=1}^\infty \frac{a_n}{n} \sin(n\pi x),
$$
for some square-summable real sequence $a_n$.
Is it true that $f$ is absolutely continuous? I tried a basic estimate,
$$
\sum_{k=1}^K |f(u_k) - f(v_k)| \leq \sum_{n \geq 1} \sum_{k \leq K} \frac{|a_... |
Is the function $f(x) = \sum_{n \geq 1} \tfrac{a_n}{n} \sin(n\pi x)$ absolutely continuous on $[0, 1]$? |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now... |
Definition $3.1.1$ in page $29$ of [this book][1] is the definition of quasiperod and Proposition $3.1.3.$ shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition $3.1.3.... |
Definition 3.1.1 in page 29 of [this book][1] is the definition of quasiperod and Proposition 3.1.3. shows that gcd of two quasiperiods is a quasiperiod. The whole proof is clear except for the part about CRT.
I would appreciate a simple explanation of the following claim from the proof of Proposition 3.1.3:
Now... |
This question was inspired by the solution to this problem: https://math.stackexchange.com/q/913780.
**Question:**
Suppose you have the real-valued piecewise function:
\begin{equation*}
f(x) =
\begin{cases}
f_1(x), & \text{if } x < a \\
f_2(x), & \text{if } x \geq a
\end{cases},
\end{equation*}
where... |
Is this Proof on 1-Form with Compact Support Correct? |
Let $\mathcal{O}_{\mathbb{P}^m}(1)$ be the tautological bundle on $\mathbb{P}^m$ and let $T := (\mathbb{C}^\times)^{m+1}$ act on $\mathbb{P}^m$ diagonally, i.e.,
$$
(t_0,\ldots,t_m) \cdot [x_0:\cdots:x_m] := [t_0x_0:\cdots:t_mx_m].
$$
The total space of $\mathcal{O}_{\mathbb{P}^m}(-1)$ is given by $\mathbb{C} \ti... |
I'm learning about types of function in discrete maths. I like to learn by exploring concepts through computer code. I came across the following code purporting to determine whether a function is injective:
```
def is_injective(function):
# Check if the function is injective (one-to-one)
codomain_set = ... |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
We know that
$$\sin x = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots = \sum_{n\ge 0} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
But how could we derive this without calculus?
There were some approach using $e^{ix} = \cos x + i \sin x$, pls notice I also would like to avoid such definition as to prove $e^{ix} = \cos x ... |
I have to test the following Nullspace property: $$ Null(A) \subset Null(BA) $$
where $$ Null(A) = {x\in \mathbb{R}^n/ A x=0} $$
$$ Null(BA) = {y\in \mathbb{R}^n/ BA y=0} $$
For any matrix $A$,$B$ such that they have the correct dimensions to make the product.\
The problem is that when I write the linear ... |
I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$.<br>
Or to be more general we can also replace $0$ with constants. <br>
- We do know the max values of $x$ and $y$ can achieve
- We are allowed to shift... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not integrable on unbounded domain but its dirivative is integrable in that unbounded domain.
Consid... |
Let $u,v \in \mathbb{C}[x]$.
Assume that $\mathbb{C}(u,v)=\mathbb{C}(x)$.
Then by [this answer][1], there exists $\alpha,\beta,\gamma \in \mathbb{C}$
such that
$\langle u-\alpha,v-\beta \rangle = \langle x-\gamma \rangle$,
where $\langle * \rangle$ is the ideal generated by $*$.
> Is this result can be genera... |
I am trying to figure out how to produce this exact integral sign (in LaTeX, if possible; if not, in Microsoft Word?). Would anyone know how to? I know that it's really specific, but it could really make the difference for me in a class. I've also attached an image of this integral with an integrand. Thank you!
[The i... |
Does anyone know what font this is? |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain.
... |
I am trying to figure out how to produce this exact integral sign (in pdfLaTeX, if possible; if not, in Microsoft Word?). Would anyone know how to? I know that it's really specific, but it could really make the difference for me in a class. I've also attached an image of this integral with an integrand. Thank you!
[Th... |
I am new to measure theory. I am confused by the following claim from *Measure Theory* by Donald Cohn (Section 1.5 Completeness and Regularity):
> **Claim**$\quad$ *The restriction of Lebesgue measure to the $\sigma$-algebra of Borel subsets of $\mathbb{R}$ is not complete.*
In this post, I will denote the Lebesgue... |
Is it necessary to go back to the definition with epsilons or can you help me find two sub-suites that don't have the same limit? |
How can you show that the absolute value of sinx diverges? |
I have prices $p_i$, sizes $s_i$, with average weighted price $A=\frac{\sum(p_i s_i)}{\sum(s_i)}$
I want to calculate $s'_i$, such that
$\sum(s'_i) = \sum(s_i)$
to give a desired new average weighted price
$B = \frac{\sum(p_i s'_i)} { \sum(s'_i)}$
What is the best way to do this?
Would it be possible... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
Use tensor product to show that $ \mathscr{T}^n=\sum_{j \geq 1} \lambda_j^n e_j \otimes e_j $? |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
Give an example of a sequence of function in $C^1(\mathbb{R})$ to show that the Poincaré inequality does not hold on a general non-compact domain.
I tried to find the sequence in $C^1(\mathbb{R})$ such that it is not belong to $L^2$ on unbounded domain in but its dirivative is in $L^2$ in that unbounded domain.
... |
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