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Is the function $(x^2+y^2)\sin{(\frac{1}{x^2+y^2})}$ differentiable at the point $(0,0)$? |
Let $X_1, X_2, ..., X_n$ be a random sample from $N(\theta+2, \theta^2)$. Find the MLE of $\theta$.
I went through some work and I could not solve for $\theta$. Am I doing anything wrong?
$$L( \theta )= \prod _ { i = 1 } ^ { n } \frac { 1 } { \theta \sqrt { 2 \pi } } \exp \bigg[- \frac{( x _ { i } - \theta - 2 )^... |
I have found that the value of the integral below is always $1/3$ for all positive $n$.
$$\int_0^1 (\sqrt[n]{1-x^n}-x)^2 dx$$
Can anyone prove this for me? Thank you. |
I am reading some context from computer graphics about using eikonal equation to compute geodesics on triangular mesh, I find some reference e.g [text](https://www.sciencedirect.com/science/article/abs/pii/S0019357718304294)
The authors state that if $\| \text{grad} \: \textit{f} \ \|_g=1$ then we get $$\nabla_{\tex... |
For a function $f(a)$ defined recursively by $f(a + 1) = f(a) - \frac{1}{f(a)}$,$f(0) = b >0$, if $f(a) \geq 0$, then there exists a constant $c \geq 0$ such that $a \leq cb^2$.
I ran a computer simulation, and it seems that $c = \frac{1}{2}$. However, I have no idea how to prove that. Do you have any ideas? |
Lets consider a set of 3D points $P$. All points in $P$ are distributed in a non-random (but still not known) way over an ellipsoid surface $E_r$ which parameters are unknown.
NOTE: the points in $P$ are obtained in a linear sequence over the ellipsoid surface (as if you were drawing a continuous line over the surfa... |
I am learning Riemann surfaces and one of the books I use is Algebraic curves and Riemann surfaces by Rick Miranda. He mentions if $f(z,w)\in \mathbb{C}[z,w]$ is irreducible the zero locus is connected. He doesn't prove it. Do you guys have tips how to do this or some literature?
Kind regards |
I am learning propositional equality from an [nlab article][1].
The article states that the elimination rules for propositional equality is as follows:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash P(x, y) \; \mathrm{prop}}{\Gamma \vdash \left(\forall x:A.P(x, x)\right) \implies \left(\fo... |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 >0$, if $a_n \geq 0$, then there exists a constant $c \geq 0$ such that $n \leq ca_0^2$.
I ran a computer simulation, and it seems that $c = \frac{1}{2}$. However, I have no idea how to prove that. Do you have any ideas? |
I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \boxtimes v$ for any $v \in V$, and extended as an algebra homomorphism over $T(V)$ (through the universal property of $... |
The set $Ham(M,\omega)$ of Hamiltonian diffeomorphisms is a subgroup of the set of symplectic diffeomorphisms $Symp(M\omega)$ is proven in the following lectures notes https://jumpshare.com/s/tkHDIaSse4wwcxlG3mW1 (proof of proposition 3.42 -pag 110 (=pag 120 of the pdf))
But I have no idea what is going on in som... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
Evaluate the following double summation over all acceptable values of $m$ and $k$:
$$S=\sum_{0 \leq k < m \leq n+1} \binom {n+1}{m}\binom {n}{k}$$
This sum arose in a probability question a friend of mine came across in her homework, in which two people A and B were tossing $(n+1)$ and $n$ coins respectively and ... |
I'm trying to prove the continuity of $\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$ at $(0,0)$. Is the following working correct, and is the way of writing my proof correct?
Let $\delta = \sqrt{\epsilon}$. Then,
$\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$ $\leq$ $\frac{x^2y}{\sqrt{x^2+y^2}}$ $\leq$ $\frac{(x^2+y^2)\sqrt{x^2+y^2}}{\s... |
$A$ and $B$ are playing a tournament consisting of some matches; the first one to win $(n+1)$ matches wins the tournament. If $A$ has probability $p$ of winning a match, find the probability that $A
$ wins the tournament.
$A$ can win in the following ways; wins $(n+1)$ matches in a row, wins $n$ matches, loses $1$ ... |
**Question.** Consider arbitrary distinct elements $k,j \in \mathbb N_{\geqslant 3}$, i.e., $k,j$ are two distinct integers greater or equal than $3$. I am wondering if the inequality
$$ |2^{-k} - 2^{-j} |\geqslant 8^{-k} + 8^{-j} $$
is satisfied.
**My attempt.** For simplicity, I started by considering the c... |
Let $k,j \in \mathbb N_{\geqslant 3}$ such that $k \neq j$. Is it true that $|2^{-k} -2^{-j}| \geqslant 8^{-k} + 8^{-j}$? |
I have the following matrix:
```
A = 1/3 * [14, -14; -14, 14]
```
I know that for finding eigen values I need to solve for |A - lambda * I| = 0.
I found to solutions:
```
lambda_1 = 28/3
lambda_2 = 0
```
And for finding eigen vectors I need to solve for (A - lambda * I)v = 0 where
v = [x, y]
Loo... |
Let A be a $n\times n$ matrix, $s$ be real number and $I$ be the $n\times n$ identity matrix. Why does $A$ have a worse condition number than $A-sI$?
I suspect it is something to do with the singular values being closer together for the matrix $A-sI$ than for $A$. I was wondering how I could mathematically prove th... |
I noticed that modulus of difference of any two odd numbers raised to even powers individually is always divisible by 4 and their sum is always congruent to 2(mod 4).
Example
|77^2 - 59^4| = 12111432 ≡ 0(mod 4)
53^2 + 15^6 = 11393434 ≡ 2(mod 4)
I haven't been able to prove that though. Can anyone provide a ... |
Im working with Newton method, and during I was trying to see why we got this expression, I fall into a problem, I find the same expression but instead of Hessian matrix I find his transpose. I can't figure where Im wrong D:
Here how I found that :
__Quick note of Newton's Method to aproximate zero of a function ... |
The question arose while trying to derive the moment of inertia of solid vs hollow sphere.<br>
In calculating the surface area of sphere, the integral used is
$$\int{2\pi r×Rd\theta}$$
In calculating the volume of sphere the integral is
$$\int{\pi r^2 dx}$$
After reading from some related [questions:(volume... |
Why is volume of sphere by disc method and its surface area by ring method depend differently on arc? |
**Background**
I have a zero-mean dataset $D(\theta)$ composed by $m$ vectors $y_{1}(\theta)$, $y_2(\theta)$, $\dots$, $y_{m}(\theta)\in \mathbb{R}^2$.
Each observation $y_i(\theta)$ depends on a unknown rotation angle $\theta$, which I want to estimate.
Given the dataset, I'm able to compute two vectors $g_N(\the... |
Are there further estimation techniques for this convergent series? |
The question arose while trying to derive the moment of inertia of solid vs hollow sphere.<br>
In calculating the surface area of sphere, the integral used is
$$\int{2\pi r×Rd\theta}$$
In calculating the volume of sphere the integral is
$$\int{\pi r^2 dx}$$
After reading from some related [questions:(volume... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng). We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying the condit... |
Functions that can be expressed using real numbers, $x, e^x, \sin x, \cos x$ along with their inverses, and addition, subtraction, multiplication, division, and function composition are often called "closed-form" functions.
Any closed-form function can be differentiated, and the result will be another closed-form f... |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 >0$, if $a_n \geq 0$, then there exists a constant $c \geq 0$ such that $n \leq ca_0^2$.
I ran a computer simulation, and it seems that $c \approx \frac{1}{2}$. However, I have no idea how to prove that. Do you have any ideas? |
Functions that can be expressed using real numbers, $x, e^x, \sin x, \cos x$ along with their inverses, and addition, subtraction, multiplication, division, and function composition are often called "closed-form" functions.
Any closed-form function can be differentiated, and the result will be another closed-form f... |
Firstly, you need to define an inner product in $T^1(V)$. As far as I am aware this is defined by summing all of the component wise inner products for each tensor power; but this is only guaranteed to converge in the range of the signature thanks to the factorial decay. Ilya's work that you link doesn't define the inne... |
We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic to the symmetric group $S_n$ on $n$ letters with permutation matrices as representatives.
So my question is: How to ... |
How to describe the partition of $GL_3(\mathbb{C})$ by Bruhat decomposition accurately? |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 = k >0$, if $a_n \geq 0$, then $n \leq O(k^2)$.
I ran a computer simulation, and it seems that $c \approx \frac{1}{2}$. However, I have no idea how to prove that. Do you have any ideas? |
Let $f: M\to N$ be a bounded linear map between von Neumann algebras.
Assume that for every net $0 \le x_i\nearrow x$, we have that $f(x_i)\to f(x)$ $\sigma$-strongly. Is it true that $f$ is normal, i.e. $\sigma$-weakly continuous?
Let $\omega \in N_*$. Then we need to check that $\omega \circ f\in M_*$.
Now,... |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 >0$, if $a_n \geq 0$, then $n \in O(a_0^2)$.
I ran a computer simulation, and it seems true.
Moreover, these questions appear similar to this question: https://math.stackexchange.com/questions/29777/closed-form-for-the-sequence-defined-by-a-0-1... |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 = k >0$, if $a_n \geq 0$, then $n \in O(k^2)$.
I ran a computer simulation, and it seems true.
Moreover, these questions appear similar to this question: https://math.stackexchange.com/questions/29777/closed-form-for-the-sequence-defined-by-a-0... |
Quesion(From Sheldon Axler's Linear Algebra Done Right, Section 7F, Exercise 11):\
Let $V$ be a nonzero finite-dimensional inner product space over $\mathbb{C}$, and let $T \in \mathscr{L}(V)$ be a linear map. Prove that for every $\epsilon>0$， there exists a diagonalizable operator $S \in \mathscr{L}(V)$ such that $0... |
What you mentioned is usually called the substitution rule. And it's fine for logic in general. But if you want to get into categorical logic you need something that fits better with categorical notions.
In this case you get that the equality is left adjoint to weakening of variables. Let me explain a bit further.
... |
I would approach this problem by using the Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are
$ r^T Q_c r = 0 $ (for the cone), and
$r^T Q_e r = 1 $ (for the ellipsoid)
And you want to maximize $z = k^T r $, so define
$ G(r, \lam... |
I am looking at the function $\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$ at $(0,0)$, where the function at $(0,0)$ is defined to be $0$. I understand that I need to check that $\lim_{{h \to 0}}$ $\frac{{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - A\mathbf{h}}}{{\|\mathbf{h}\|}}$ = $\lim_{{h \to 0}}$ $\frac{{f(\mathbf{h})}}{{\|... |
What you mentioned is usually called the substitution rule. And it's fine for logic in general. But if you want to get into categorical logic you need something that fits better with categorical notions.
In this case you get that the equality is left adjoint to weakening of variables. Let me explain a bit further.
... |
For a $a_n$ defined recursively by $a_{n+1} = a_n - \frac{1}{a_n}$,$a_0 = k >0$. Prove that if the first $n$ such that $a_n \leq 0$, then $n \in O(k^2)$.
I ran a computer simulation, and it seems true.
Moreover, these questions appear similar to this question: https://math.stackexchange.com/questions/29777/closed... |
What you mentioned is usually called the substitution rule. And it's fine for logic in general. But if you want to get into categorical logic you need something that fits better with categorical notions.
In this case you get that the equality is left adjoint to weakening of variables. Let me explain a bit further.
... |
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$ denotes the linear interpolation $x_1 \rightarrow x_2$, then
$$S(\Delta x) = \left(1 , \Delta x, \frac{\Delta x^{\ot... |
I am looking at the function: $$\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$$ at $(0,0)$, where the function at $(0,0)$ is defined to be $0$.
I understand that I need to check that: $$\lim_{{h \to 0}} \frac{{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - A\mathbf{h}}}{{\|\mathbf{h}\|}} = \lim_{{h \to 0}} \frac{{f(\mathbf{h})}}{{... |
I am looking at the function: $$\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$$ at $(0,0)$, where the function at $(0,0)$ is defined to be $0$.
I understand that I need to check that: $$\lim_{{h \to 0}} \frac{{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - A\mathbf{h}}}{{\|\mathbf{h}\|}} = \lim_{{h \to 0}} \frac{{f(\mathbf{h})}}{{... |
I would approach this problem by using the Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are
$ r^T Q_c r = 0 $ (for the cone), and
$r^T Q_e r = 1 $ (for the ellipsoid)
And you want to maximize $z = k^T r $, so define
$ G(r, \lam... |
Let $X \sim f_X(x)$ and $Y = g(X)$. If $g(X)$ is a differentiable, monotonic function with inverse such that $X = g^{-1}(Y)$ then the PDF of $Y$ can be described:
$$
f_Y(y) = f_X(g^{-1}(y)) \bigg| \dfrac{d}{dy}g^{-1}(y) \bigg|
$$
In the multidimensional case, I have seen a very similar expression where the abso... |
Do we really know that we are near a local min when using gradient descent or [Nelder-Mead][1] method?
Suppose that
- I have a function $f$ that is analytic on the open ball $B = B(x, r)\subset R^n$,
- if the coordinates of $x$ are rational, then $f(x)$ and $\nabla f(x)$ are rational,
- I can evaluate $f$ and $... |
Given an approximation of the minimum of f using gradient descent, can I prove I am near a local min? |
I want to prove that if R is a PID and M is a finitely generated R-module, then there exist a free R- module F such that M \cong F \oplus M_{\text{tors}}.
So, in order to prove the above I set M/M_{\text{tors}} = F and since R is a PID then it is also an integral domain, hence M/M_{\text{tors}} is torsion free, ie F... |
Let $n$ be a natural number. When constructing the [Goodstein sequence][1] $(n)_{k}$, we start with $(n)_{1}=n$ written in complete base $2$, and we proceed recursively. That is, if $(n)_{k}$ has been defined and it is not $0$, $(n)_{k+1}$ is obtained from $(n)_{k}$ by bumping the base by $1$ in the complete base $k+1$... |
The below is part of the proof of :
>**Theorem 68.7** Let $ G = G_1 * G_2 $. Let $ N_i $ be a normal subgroup of $ G_i $, for $ i = 1, 2 $. If $ N $ is the least normal subgroup of $ G $ that contains $ N_1 $ and $ N_2 $, then $$ G/N \simeq \left(G_1/N_1\right) * \left(G_2/N_2\right). $$
We show that the extension ... |
>**Theorem 68.7** Let $ G = G_1 * G_2 $. Let $ N_i $ be a normal subgroup of $ G_i $, for $ i = 1, 2 $. If $ N $ is the least normal subgroup of $ G $ that contains $ N_1 $ and $ N_2 $, then $$ G/N \simeq \left(G_1/N_1\right) * \left(G_2/N_2\right). $$
**Proof.** The composite of the inclusion and projection homomor... |
I want to prove that if R is a PID and M is a finitely generated R-module, then there exist a free R- module F such that
$M \cong F \oplus M_{\text{tors}}$
So, in order to prove the above I set $M/M_{\text{tors}} = F$ and since R is a PID then it is also an integral domain, hence $M/M_{\text{tors}}$ is torsion... |
>**Theorem 68.7** Let $ G = G_1 * G_2 $. Let $ N_i $ be a normal subgroup of $ G_i $, for $ i = 1, 2 $. If $ N $ is the least normal subgroup of $ G $ that contains $ N_1 $ and $ N_2 $, then $$ G/N \simeq \left(G_1/N_1\right) * \left(G_2/N_2\right). $$
**Proof.** The composite of the inclusion and projection homomor... |
I am learning about the Levi flat hypersurfaces. Let $M \subseteq \mathbb{C}^n$ be a real smooth hypersurface, i.e. for every point $p$ in $M$ there is an open set $U_p$ in $\mathbb{C}^n$ containing $p$ and a smooth function $r: U_p \to \mathbb{R}$ such that $grad(r)$ does not vanish on $U_p$ and $M \cap U_p=\{z \in U_... |
Are there any unanswered questions regarding math behind Rubik's cubes? |
Do we really know that we are near a local min when using gradient descent or [Nelder-Mead][1] method?
Suppose that
- I have a function $f$ that is analytic on the open ball $B = B(x, r)\subset R^n$,
- if the coordinates of $x$ are rational, then $f(x)$ and $\nabla f(x)$ are rational,
- I can evaluate $f$ and $... |
For non-negative functions, the definition of $\int f$ is consistent among Stein, Folland and Wikipedia. However, for real-valued functions, there's a difference.
Stein (page 64) first defines integrable functions, and then says:
> If $f$ is Lebesgue integrable, we give a meaning to its integral as follows..."
... |
I have the below function:
$$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot \gamma+ r_0}- x$$
with:
- $x, s_0, s_1, r_0, r_1 \in R^+$
- $\lambda, \gamma \in [0,1]$
- $k \in \{0.25, 4... |
I can't find a definition of what it means that a periodic point of $f$ is isolated, where $f$ is a diffeomorphism. Is this related to the existence of discontinuities of $f$ about the periodic point?
Thanks |
What does it mean when we say that a periodic point is isolated? |
We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic to the symmetric group $S_n$ on $n$ letters with permutation matrices as representatives.
So my question is: How to ... |
Let $X$ be a metric space. I consider elements in $Y=2^X\setminus \emptyset$ and use the Hausdorff metric for $Y$.
Suppose that $A_n \subseteq B_n$ for $A_n,B_n \in Y$ and $A_n \rightarrow A$ and $B_n \rightarrow B$, where $A,B \in Y$ as well.
Is it true that $A \subseteq B$?
|
Is subset relation preserved under limit for Hausdorff metric? |
Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the of $$\frac{d}{dz}E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$
$X$ and $Y(X)$ have no density functions and no joint density.I thought about changing integral and derivative to get $$E[e^X(A_Y(X))\frac{d}{dz}... |
Prove that ordinal multiplication $\alpha \cdot (\beta \cdot \gamma) = (\alpha \cdot \beta) \cdot \gamma$ is associative by using the following facts:
$$\beta \cdot 0=0$$
$$\beta \cdot (\alpha +1)=\beta \cdot \alpha+\beta$$
$$\beta \cdot \alpha=\sup\{ \beta \cdot \gamma \mid \gamma < \alpha \}$$
The second fa... |
I have the following matrix:
```
A = 1/3 * [14, -14; -14, 14]
```
I know that for finding eigen values I need to solve for |A - lambda * I| = 0.
I found to solutions:
```
lambda_1 = 28/3
lambda_2 = 0
```
And for finding eigen vectors I need to solve for (A - lambda * I)v = 0 where
v = [x, y]
Loo... |
Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the of $$\frac{d}{dz}E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$
$X$ and $Y(X)$ have no density functions and no joint density.I thought about changing integral and derivative to get $$E[e^X(A-Y(X))\frac{d}{dz}... |
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:**
$$ \sum_{\hat{x} \in \hat{X}} p(\hat{x}|x) = 1 \quad \te... |
For a complex number $z$, we don't speak about the "absolute value of $z$" but we speak about the **modulus** of $z$ denoted by $|z|$. If the algebraic form of $z$ is $a + ib$, then by definition you set $|z| = \sqrt{a^b + b^2}$. Note that this definition extends the absolute value on $\mathbb R$.
Also, if $z = i$, ... |
My question originates from a short proof in Strang's LA book that shows when
$A\pmb{x}=\pmb{b}$
every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in the column space of $A$. The proof is discussed [in this thread](https://math.stackexchange.com/questions/3585142/ho... |
Can someone help me to see where the problem of the functional derivative below is?
**Minimize the functional:**
$$ 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... |
For a complex number $z$, we don't speak about the "absolute value of $z$" but we speak about the **modulus** of $z$ denoted by $|z|$. If the algebraic form of $z$ is $a + ib$, then by definition you set $|z| = \sqrt{a^2 + b^2}$. Note that this definition extends the absolute value on $\mathbb R$.
Also, if $z = i$, ... |
I've been given this exercise: Given $g:\mathbb{R}\longrightarrow\mathbb{R}$ is differentiable, and $0\leq g'(x)\leq 1$ for all $x\in \mathbb{R}$, prove that there exists some point $0\leq c\leq 1$ such that $g'(c)=3c^2$
Now, here's what I tried:
I want to apply Darboux's theorem, so I want to find a suitable $c$... |
I'm trying to prove the following theorem involving Krull dimension:
> **Theorem** If $A$ is a noetherian ring then $$\dim(A[x_1,x_2, \dots , x_n]) = \dim(A) + n$$ where $\dim$ stands for the Krull dimension of the
> rings. Thus, $\dim(K[x_1,x_2, \dots , x_n]) = n$ for any field $K$.
I know how to prove it by ... |
I'm not just curious; it refers to general relativity.
Specifically, would the hole in the torus' center look to us like a sphere, one you cannot enter because you always slip across the side and go around it instead of through? |
If you lived in a 4-torus, what would the doughnut hole look like from the inside? |
I've read somewhere an MSE that we can understand the normalized valuation of a polynomial in the function field of a curve at a smooth point as the first non-vanishing coefficient (or exponent) of a corresponding laurent series, meaning that (using Silvermans notation) for the function $f \in \overline{K}(C)$ it's suf... |
What is the connection between the valuation of a polynomial of a function filed of a curve and the corresponding laurent series? |
I am currently trying to understand the Levenberg-Marquardt implementation that [(Fetzer et al. 2020)](https://ieeexplore.ieee.org/document/9093299) used in their solution.
*(The paper is publicly accessible [here](https://openaccess.thecvf.com/content_WACV_2020/papers/Fetzer_Stable_Intrinsic_Auto-Calibration_from_Fun... |
What is truncated Levenberg-Marquardt and how does it work? |
I have a coupled system of 2nd order ODEs as follows :
$$r^2 f_{rr} + 2r f_r - \frac{r}{2n+1}h_r - (n+1)(n+2)f = 0$$
$$r^2 g_{rr} + 2r g_r + \frac{r}{2n+1}h_r - n(n-1)g + \frac{n}{2n+1}h = 0$$
$$(n+1)r f_r - nrg_r + (n+1)(n+2)f + n(n-1)g = 0$$
Here, $f, g, h$ are functions of r, taking finite values as $r \righ... |
I am trying to understand how to think about the elements of $H^1(X,\mathcal{O}^*_X)$ I know one way is to think about it as the Picard group via the isomorphism. But when I try to "see" the elements of $H^1(X,\mathcal{O}^*_X)$ I get confused.
We have
$$ 0\to \mathbb{Z}\to \mathcal{O}_X\to \mathcal{O}^*_X\to 0$$... |
I am a beginner to invariant theory and I am reading the Wikipedia page on the subject. There it says the following. Suppose $G = SL_n$ the group of $n*n$ matrices with complex coefficients, and $M_n$ is the vector space of $n*n$ matrices. Let the action of $G$ on $V$ be the normal by one by left multiplication. Then t... |
Why should $|x|=\sqrt{x^2}$? That's a *consequence* of the definition. Not the definition itself. If $x\in \mathbb R$ then $x = x + 0i$ and $Re(x) = x$ and $Im(0) = 0$. And $|x| = \sqrt{Re(x)^2 + Im(x)^2} = \sqrt{x^2 + 0^2} = \sqrt{x^2}$.
The idea is that $|z|$ is a *norm* which can be thought of as "the distan... |
This will depend on the operation, but essentially you could have (at least) two approaches.
One is to describe how the operation works on the adjacency matrix $A$ (for the complement) or the two adjacency matrices $A, B$ (for products); and then try to algebraically manipulate the expression for the resulting matri... |
I am an engineering student and when I was doing some work on data visualisation I stumbled across an integer sequence after watching a video about sequences that produce interesting graphs. I submitted the sequence to the On-Line Encyclopedia of Integer Sequences [A328225](https://oeis.org/A328225).
The sequence is... |
Your **idea can work** as follows. Here, I provided both distribution and expectation of $X_N$.
We can show that for $\color{blue}{0<t<1}$:
$$\color{blue}{F_{X_{N}|N=i}(t)=1-\left ( 1+ \frac{1}{i-1} \right ) t +\frac{1}{i-1} (1-t)^{i}} \tag{1}.$$
This gives
$$\color{blue}{\mathbb E[X_{N}|N=i]}=\int_0^1 \le... |
I am following Rogers's *Supermanifolds: Theory and Applications* and I might be getting something wrong, because I reach a definition that, as I understand it, doesn't imply what the author states.
Letting $\mathbb{R}_{\infty}$ be a real Grassmann algebra generated by anticommuting $\{\xi_j\}_{j\in\mathbb{N}}$, wit... |
This will depend on the operation, but essentially you could have (at least) two approaches.
One is to describe how the operation works on the adjacency matrix $A$ (for the complement) or the two adjacency matrices $A, B$ (for products); and then try to algebraically manipulate the expression for the resulting matri... |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
Is the polynomial $x^p-u$ irreducible over $\mathbb{Q}_p$?
|
I came across the following coupled system of 2nd order ODEs from the appendix of [this paper][1] :
$$r^2 f_{rr} + 2r f_r - \frac{r}{2n+1}h_r - (n+1)(n+2)f = 0$$
$$r^2 g_{rr} + 2r g_r + \frac{r}{2n+1}h_r - n(n-1)g + \frac{n}{2n+1}h = 0$$
$$(n+1)r f_r - nrg_r + (n+1)(n+2)f + n(n-1)g = 0$$
Here, $f, g, h$ are fun... |
Consider the following 2-category $\mathcal K$:
- objects are finite length abelian categories (i.e., abelian categories where every object has finite length)
- morphisms are exact functors (preserving finite length)
- 2-morphisms are all natural transformations.
For every finite length abelian category $\ma... |
Is “semisimplification” a 2-functor? |
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
Is the polynomial $f=x^p-u$ irreducible over $\mathbb{Q}_p$?
I tried Newton polygons but they are not conclusive in this case. |
Suppose I have a function $f:\mathbb{R}_+ \to [0,1]$ and another similar function $g$.
I consider $g$ to be a fixed function, and I want to know how close $f$ is to $g$.
For this purpose, I define a `residual operator' $R$ which calculates the difference between the two functions, but *as a function of $f$*. C... |
This is question 1a in section 6.3.6 of Zorich's Analysis I.
I went about it as follows:
$
\lim_{n->\infty}[ \frac{n}{(n+1)^2} + ... + \frac{n}{(2n)^2}] =
\lim_{n->\infty}[ \frac{n}{[n(1+1/n)]^2} + ... + \frac{n}{[n(1+n/n)]^2}] =
\lim_{n->\infty}[ \frac{n}{n^2}\frac{1}{(1+1/n)^2} + ... + \frac{n}{n^2}\frac{1}{... |
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