instruction stringlengths 12 30k |
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I know i have to use euclidean algorithm somehow, but i have no idea how to apply it in this task, have i try this algorithm with powers of 3? |
Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/xnotes/tc2018.pdf), or we could call it a symmetric monoidal category.
Furthermore, assume that $(\mathcal{C},\otimes)$ ... |
Let $X$ be a compact metric space,$T$ a transformation on $X$ and $f_n$ a sequence of continuous subadditive functions $X \to \mathbb{R}$.
Des $f_n$ converge pointwise for every $x$? By kingsman subadditive Ergodic theorem the convergence is guaranteed for any regular point of an invariant measure. So if every poi... |
I couldn't find an epsilon-delta proof for continuity of $e^x$ so here's my take:
Suppose $|x - x_0| < \delta$ and fix $\epsilon >0$
Consider $|e^x - e^{x_0}| < \epsilon$, then
\begin{gather}
-\epsilon < e^x - e^{x_0} < \epsilon \\
e^{x_0} - \epsilon < e^x < e^{x_0} + \epsilon \\
\ln(e^{x_0} - \epsilon) < x... |
Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/xnotes/tc2018.pdf), or we could call it a symmetric monoidal category.
Furthermore, assume that $(\mathcal{C},\otimes)$ ... |
Here is Theorem 7.32 in the book _Mathematical Analysis - A Modern Approach to Advanced Calculus_ by Tom M. Apostol, 2nd edition:
> Let $\alpha$ be of bounded variation on $[a, b]$ and assume that $f \in R(\alpha)$ on $[a, b]$. Define $F$ by the equation
$$
F(x) = \int_a^x f \, d \alpha, \qquad \mbox{ if } x \in ... |
Theorem 7.32 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: How to establish differentiability? |
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[-1,1]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ?
According to [this][1] answe... |
A(1,-3), B(4,3) are two points on the curve y = x - 4/x. The points on the curve, the tangents at which are parallel to chord AB are?
The answer to the question is (2,0) and (-2,0).
It's very easy to do this by equating slope of curve to slope of AB, But I tried it with complex numbers and without using calculus, but... |
How to do this question using complex numbers and without using differentiation? |
I am currently doing problem 2, Chapter 1, from the "Functional Equations and HOw to Solve Them" book by Christopher G. Small. The problem is as follows:
> Let $f(x)$ be a function that satisfy $f(x + y) = f(xy)$ for all **positive** $x$ and $y$. Proof that $f(x)$ is constant.
Here are my 2 possible solutions:
... |
The set of irrationals in $[0,1] \times [0,1]$ has measure zero in $\Bbb{R}^2$? |
Let $f:(M, g) \rightarrow (N, h)$ be a pseudo-Riemannian immersion, $\overline D$ the linear connection on $f^*(TN)$ induced from the Levi-Civita connection of $h$, and $\nabla$ the connection induced on the normal bundle $NM$ (which is the orthogonal complement of $TM$ in $f^*(TN)$). Consider also the second fundament... |
Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the left. The probability $p_n(t)$ of being at location $n$ at time $t$ is then described by
$$\frac{\mathrm{d}}{\mathrm{d}t}p_n(t)=k_\mathrm{R}p_{n-1}(t)+k_\math... |
Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and differentiable on $\mathbb{R}\setminus{c}$ and that $\lim_{x\to c} f'(x)$ exists. Prove using the mean value theorem that $f$ is differentiable at $c$ and that $\lim_{x\to c}f'(x) = f(c)$.
My thoughts so far:
Consider $f$ restricted to $[a,b]$ with $a<... |
Let $f:(M, g) \rightarrow (N, h)$ be a pseudo-Riemannian immersion, $\overline D$ the linear connection on $f^*(TN)$ induced from the Levi-Civita connection of $h$, and $\nabla$ the connection induced on the normal bundle $NM$ (which is the orthogonal complement of $TM$ in $f^*(TN)$). Consider also the second fundament... |
Let $E/K$ be an elliptic curve over a number field $K$.
Let $L=K(\sqrt{D})/K$ be a quadratic extension.
Let $E_D/K$ be a twist of $E/K$ by $D$.
Let $\text{Sha}(E/K)$ be a Tate-Shafarevich group of $E/K$.
Let $\text{Sha}(E/K)[2]=\{a\in \text{Sha}(E/K)\mid 2a=0\}$.
There is a natural trace map $\text{Sha}(E/L... |
I am stuck trying to solve this equation using Laplace transform:
> $ y′′(t)+2y(t)=\displaystyle 3\int_0^t (t−u)y(u)du $
with the initial conditions: $y(0)=2 , y'(0)=3$.
I got stuck after performing the Laplace transformation and can't get the equation to a form I can do the inverse Laplace transforma... |
Let $f:(M, g) \rightarrow (N, h)$ be a pseudo-Riemannian immersion, $\overline D$ the linear connection on $f^*(TN)$ induced from the Levi-Civita connection of $h$, and $\nabla$ the connection induced on the normal bundle $NM$ (which is the orthogonal complement of $TM$ in $f^*(TN)$). Consider also the second fundament... |
Let $f:(M, g) \rightarrow (N, h)$ be a pseudo-Riemannian immersion, $\overline D$ the linear connection on $f^*(TN)$ induced from the Levi-Civita connection of $h$, and $\nabla$ the connection induced on the normal bundle $NM$ (which is the orthogonal complement of $TM$ in $f^*(TN)$). Consider also the second fundament... |
If $m = \inf \{ f(x) : x \in [a,b] \}$ and $M = \sup \{ f(x) : x \in [a,b] \}$, and $f$ is continuous, what would you conclude about $c$ as $y \to x$? Can you see why it is unnecessary to specify that $c$ is some function of $x$ and $y$? |
Reading about graph theory they said that, for the Laplacian matrix of a graph, let's call it $A$, its adjoint $Adj(A)$ since $A$ is symmetric, ,
**Here is an example of Laplacian matrix $A$ and its adjoint matrix.**
$$A=\begin{pmatrix}2&-1&0&-1\\-1&3&-1&-1\\0&-1&1&0\\-1&-1&0&2 \end{pmatrix} \to Adj(A)=\begin{pma... |
Does symmetric matrix adjoint of $A$ have all its entries equal? |
If $f,g$ are degree-1 monotone maps of the circle, **why** do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$?
I mean, you might say that we have no right to expect an equality. After all, it's not hard to come up with counterexamples. For example, if $f$ is the piecewise-affine map that interpolates $[0,\frac... |
Let $$M=\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}\in \mathbb R^{m\times n}$$ be a block matrix.
**(a)** If $M$ is orthogonal and $C=0$, are $A$ and $B$ orthogonal?
**(b)** Suppose $A$ is orthogonal and $B=cI$, for some $0<c<1$. Is $M$ orthogonal for any square matrix $C$? (Here $I$ denotes the identity matrix.)
... |
This is called a [Riemann surface][1], in the case of functions of one variable.
You are incorrect about the result of stitching those two spaces together. The Riemann surface domain of $\sqrt{\cdot}$ is diffeomorphic to $\mathbb C\setminus\{0\}.$
In general, if $g:U\to\mathbb C$ is holomorphic, then the inverse ... |
>where is $Q$ in that case?
Let $Q'$ be the point you want.
If I understand your question correctly, you can get $Q'$ by rotating $Q$ by $2.27^\circ$ around $R$.
Since
$$\begin{pmatrix}Q'_x-R_x\\Q'_y-R_y\end{pmatrix}=\begin{pmatrix}
\cos(2.27^\circ) & -\sin(2.27^\circ) \\
\sin(2.27^\circ) & \cos(2.27^\circ)... |
>Are $\mathbb N$ with topology $\{1, 2\}, \{3, 4\}, \{5, 6\}, \{7, 8\}, \ldots$ and all unions of these, and $\mathbb N$ with topology $\{1, 3\}, \{2, 4\}, \{5, 7\}, \{6, 8\}, \{9, 11\}, \{10, 12\}, \ldots$ and all unions of these homeomorphic?
Name the topologies $t, t'$ respectively and let the would-be homeomor... |
Consider the usual coupon collector's problem - $n$ coupons with equal probability, and let $T_n$ be the time required to collect all $n$ coupons. It is well known that:
$$
\mathbb{E}[T_n] = nH_n
$$
with $H_n$ denoting the $n$th harmonic number:
$$
H_n = \sum_{k=1}^n \frac{1}{k}
$$
It is easy to show that:
$... |
### Method 1
Let $$\alpha=\frac{m\pi}{2n+1}$$ $m,n \in ℤ$
We have
$$(\tan(\alpha)+i)^{2n+1}=(\frac{\cos(\alpha)-i\sin(\alpha)}{\cos(\alpha)}i)^{2n+1}=i^{2n+1}(\exp(-i(\frac{m\pi}{2n+1})))^{2n+1}/\cos^{2n+1}(\alpha)$$ $$=i(-1)^{m+n}/\cos^{2n+1}(\alpha)$$
Let $(\tan(\alpha)+i)^{2n}=u+iv$ where $u,v \in ℝ$
... |
Let $E/K$ be an elliptic curve over a number field $K$.
Let $L=K(\sqrt{D})/K$ be a quadratic extension.
Let $E_D/K$ be a twist of $E/K$ by $D$.
Let $\text{Sha}(E/K)$ be a Tate-Shafarevich group of $E/K$.
Let $\text{Sha}(E/K)[2]=\{a\in \text{Sha}(E/K)\mid 2a=0\}$.
There is a natural trace map $\text{Sha}(E/L... |
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[-1,1]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ?
According to [this][1] answe... |
Recall an orthogonal matrix is a matrix $M$ where $M^TM=MM^T$. So let's try on your case.
$$M^T=\begin{pmatrix}A^T & 0\\
C^T & B^T\end{pmatrix}=\begin{pmatrix}A^T & 0\\
C^T & cI\end{pmatrix}$$
So we have $$M^TM=\begin{pmatrix}A^T & 0\\
C^T & cI\end{pmatrix}\begin{pmatrix}A & C\\
0 & cI\end{pmatrix}=\begin{pmatr... |
I am taking a first course on group theory.
I understand why $\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_6$.
How can I use this fact to show that
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3 \cong \mathbb{Z}_2\times\mathbb{Z}_6$ ?
More in general, it looks like (but I am not sure if this is true) t... |
Let $\mathfrak{g}$ be a simplex complex Lie algebra. Looking at this [question][1] we see that the weights of the non-zero weight vectors of an irreducible $\frak{g}$-module $V$ have weights strictly less than that of the highest weight vector of $V$. However this does not seem to work for the vector space representati... |
I am going through question in my text book regarding proof. I have done that but I think I am making mistake of proving that $u_1=u_2$ and $w_1=w_2$ by using this same statement that I have to prove. Am I wrong or I have done following proof correctly. Kindly help
**Question:**
If $U$ and $W$ are subspaces of vect... |
In Lemmon's Beginning Logic, page 41 exercise 1g says
to prove that from $\neg (P \cap Q)$ then $\neg P \cup \neg Q$ follows.
1 (1) $\neg (P \cap Q)$ (A)
2 (2) $\neg (\neg P \cup \neg Q)$ (A)
3 (3) $\neg P$ (A)
3 (4) $\neg P \cup \neg Q$ (3 vI)
2,3 (5) $(\neg P \cup \neg Q) \cap \neg(\n... |
Here is the standard definition of vector space:
> Let $F$ be a field, and let $X$ be a non-empty set such that
> (A0) for each pair $x, y$ of elements of $X$, there exists a unique element $x + y$ in $X$;
> (M0) for each element $\alpha \in F$ and for each element $x \in X$, there exists a unique element $... |
Which vector space axiom(s) is (are) going to fail if we take $\mathbb{C}$ as our set of vectors and $\mathbb{Z}$ as our set of scalars? |
I am interested in understanding the connection between homology and homotopy. In particular, I know that Hurewicz theorem states that for path connected topological spaces we can always find a homomorphism between $H_n(G)$ and $\pi_n(G)$, with $G$ a path connected topological space. If we restrict to $n = 1$, I also k... |
Connection between homology and homotopy invariants? |
Consider the usual coupon collector's problem - $n$ coupons with equal probability, and let $T_n$ be the time required to collect all $n$ coupons. It is well known that:
$$
\mathbb{E}[T_n] = nH_n
$$
with $H_n$ denoting the $n$th harmonic number:
$$
H_n = \sum_{k=1}^n \frac{1}{k}
$$
It is easy to show that:
$... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
For me you proof is very confusing and it is wrong. It is wrong because the equation $|f(x)-f(2)| < \frac{\delta}{\sqrt{3}+\sqrt{2}}$ does not hold. $|f(2.1)-f(2)|\approx 0.034$, but $\frac{0.1}{\sqrt3+\sqrt2}\approx 0.032$.
But now to the confusion. You state
> Givens: $\varepsilon > 0$ and $\delta > 0$ and $|x... |
Let $\{e_n\}_{n \in \Bbb{N}}$ be an orthonormal basis for separable Hilbert space $H$. Suppose $\{f_n\}_{n \in \Bbb{N}}$ is an orthonormal set such that
$$\sum_n \vert \vert e_n - f_n \vert \vert<1.$$
Prove $\{f_n\}$ is a basis for $H$.
My Thoughts,
By Besssels inequality, one has
$$\sum_n \vert \langle... |
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 $... |
$
\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... |
When is the initial form of a principal ideal generated by the initial form of the former's generator? |
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... |
Let $\{e_n\}_{n \in \Bbb{N}}$ be an orthonormal basis for separable Hilbert space $H$. Suppose $\{f_n\}_{n \in \Bbb{N}}$ is an orthonormal set such that
$$\sum_n \vert \vert e_n - f_n \vert \vert<1.$$
Prove $\{f_n\}$ is a basis for $H$.
My Thoughts,
By Besssels inequality, one has
$$\sum_n \vert \langle... |
There are no problems, except for that fact that $\mathbb Z$ is not a field.
That is, there are no problems *until* you start attempting to apply theorems which require the scalars to form a field.
For example, here is a theorem about vector spaces:
> For any vector space $X$ over any field $F$, and for any $... |
Working on Divisors Sum Efficient calulcation topic. Accidentaly discovered one interesting relation which is accurate up to $10^{17}$ order.
$$\sum_{i=1}^{\infty}{\frac{\sigma(i)}{e^{i}}}\approx\frac{\pi^2}{6}-\frac{1}{2}+\frac{1}{24}$$
To get things more clear look at the below numbers:
$$\sum_{i=1}^{\infty}{\... |
Let $z=\alpha+i\beta$ and $s=x+iy$ be complex numbers.
Consider the calculation of the Laplace transform of $e^{-zt}$.
$$\mathcal{L}(e^{-zt})=\int_{0^-}^\infty e^{-(x+\alpha+i(\beta+y))t}dt$$
Suppose $x+\alpha=0$. That is $x=-\alpha$ which means $\text{Re}(s)=-\text{Re}(z)$.
Then the integral becomes
$$\... |
What is the technical reason that $\mathcal{L}(e^{-zt})=\int_{0^-}^\infty e^{-(x+\alpha+i(\beta+y))t}dt$ diverges when $x+\alpha=0$? |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
A, B, C are flipping a coin independently until they got a head (same experiment "to get a head" is repeated by these 3 people). Denote `X, Y, Z` stand for the number of flips they need, respectively. Find the probability `P(X<Y<Z)`.
My solution is that they form a permutation like `XYZ` or `XZY` etc. Each combinati... |
The first solution does not account for ties. What if they all found a head on the first try? |
I recently had an exam in my number theory class and one of the questions was to compute the last seven digits of $7^{5^{2024}}$. Usually, this is a pretty standard problem since we can just use FLT and CRT (see below for my approach). However, that we need to compute the first *seven* digits makes this method seem com... |
I wrote a proof on [ProofWiki](https://proofwiki.org/wiki/User:Sandbox/Du_Bois-Reymond_Constants/Example/First) of this problem:
>Prove $\displaystyle \int_0^\infty \left|\dfrac d {d t} \left(\dfrac {\sin t} t\right)^n\right|\mathrm d t$ diverges for $n=1$.
To fill the gaps in the proof, I need to prove the follo... |
Let $\sigma:\mathbb{N}^*\longrightarrow\mathbb{N}^*$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=1}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[-1,1]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ?
According to [this][1] a... |
I recently had an exam in my number theory class and one of the questions was to compute the last seven digits of $7^{5^{2024}}$. Usually, this is a pretty standard problem since we can just use FLT and CRT (see below for my approach). However, that we need to compute the last *seven* digits makes this method seem comp... |
I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed.
The question at hand asks to prove that the rings $k[x,y]/(xy^2,x^2y)$ localized in $(x)$ is (or isn't) isomorphic to th... |
Let $D$ be a bounded convex domain in $\mathbb R^2$, if $\xi\in\partial D$ is it true that, there exists an $R>0$ and a convex function $\gamma:\mathbb R\to \mathbb R$ such that upto suitable choice of coordinates we have
$$D\cap B(\xi,R):= \{\ (x,y)\in B(\xi,R)\ :\ y>\gamma(x)\ \}?$$ And for strictly convex domain w... |
**One-Step**
I'm reviewing answers to this [code golf][1] challenge. Here's a target cell marked by `X` that was chosen using `rand()*512` ([rand()][2] in C) ensuring that each cell has an equal chance ($\frac{1}{512}$) of being chosen for modification.
....................X...........
..................... |
I've been trying to work on a solution for this equation
$$
(1+x{ ²})y''+y=0
$$
using power series around $x_{0}=0$. So far, I have reached the following recursion relation:
$$
a_{n+2}=\frac{-a_{n}(n^{2}-n+1)}{(n+2)(n+1)}
$$
but I have been struggling to find a generic form for the coefficients. $a_{0}$... |
$
\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... |
The title pretty much says it all.
A directed graph $G = (V, E)$ is two-terminal series-parallel, with terminals $s_G$ and $t_G$, if it can be produced by a sequence of the following operations:
1. Create a new graph, consisting of a single directed edge directed from $s_G$ to $t_G$.
2. Given two two-terminal ser... |
In a directed series-parallel graph, is every pair of non-terminal nodes connected by at most one path? |
Introduction:
Find a such family of polynomial then we can always use Newton's method provided we have a procedure to start.
We introduce a rational function where $b_i$ are real numbers:
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac... |
Antie, a smart ant living on a smooth surface $S$ of $\mathbb{R}^3$, would like to evaluate the Gauss curvature $K$ at a certain point $P\in S$. Antie is aware of Gauss Theorema Egregium, according to which Gauss curvature may be evaluated using the first fundamental
form of the surface. So, Antie grabs a ruler and... |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed.
The question at hand asks to prove that the rings $k[x,y]/(xy^2,x^2y)$ localized in $(x)$ is (or isn't) isomorphic to th... |
I was watching a video on ["system identification"][1] in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many contexts, related to control problems or nonlinear odes, etc. I think I am reacting to these kinds of blanket statements, and I ... |
What is meant when mathematicians or engineers say we cannot solve nonlinear systems? |
Antie, a smart ant living on a smooth surface $S$ of $\mathbb{R}^3$, would like to evaluate the Gauss curvature $K$ at a certain point $P\in S$. Antie is aware of Gauss Theorema Egregium, according to which Gauss curvature may be evaluated using the first fundamental
form of the surface. So, Antie grabs a ruler and... |
For $n=1,\dots,7$, clearly $S_7$ contains $n$-cycles: each of then is isomorphic to $\mathbb Z_n$. For $n=10$, for example the subgroup $\langle (12)(34567)\rangle$ is isomorphic to $\mathbb Z_{10}$. Likewise, for $n=12$, for example the subgroup $\langle (123)(4567)\rangle$ is isomorphic to $\mathbb Z_{12}$. Can you s... |
When is the initial form of a principal ideal generated by the initial form of the original ideal's generator? |
Suppose I have a quasiconvex function $g:\mathbb{R} \rightarrow \mathbb{R}$ and a step-function $f(x)$ defined as: \begin{align} f(x) = \begin{cases} a \mbox{ if } x\geq \alpha \\ b \mbox{ if } x< \alpha \end{cases}, \end{align} for some $a,b,\alpha \in \mathbb{R}$.
Suppose I take the pointwise minimum $h(x)=\min\... |
According to https://en.wikipedia.org/wiki/Condition_number, the conditional number for the matrix $M$ is $||A|| \times ||A^{-1}|| = ||A A^{-1}|| = 1$. In another word, is $||A||$ the absolute value of the determinant $|A|$ in the above definition?
I have manually tested a few $||A|| \times ||A^{-1}||$ including He... |
The ODE $xy' + y = 0$ has no strong solution over $\mathbb R$ but has solution $y(x) = \begin{cases} c_1/x & x<0 \\ c_2/x & x>0 \end{cases}$ over $\mathbb R^*$, which may equivalently be written as $y(x) = \frac ax + \frac b{|x|}, x\neq 0$. In Friedlander's "Introduction to the theory of distributions" exercise 2.3 pa... |
I have a nonlinear system of ODEs with **known** constant coefficients $A, B, C, D, E, M$:
\begin{align}
&\dot{n}(t)=-An(t)+Bm(t)n(t)+C \\
&\dot{m}(t)=-Bm(t)n(t) + (M-m(t))R_0 \sin{\omega t} +Dm(t)+E,
\end{align}
with driving amplitude $R_0$. I know from numerical simulations that solutions of $n(t)$ and $m(t)$ ... |
$A(1,-3), B(4,3)$ are two points on the curve $y = x - \frac 4x$. The points on the curve, the tangents at which are parallel to chord $AB$ are?
The answer to the question is $(2,0)$ and $(-2,0)$.
It's very easy to do this by equating slope of curve to slope of AB, But I tried it with complex numbers and without usin... |
I want to understand, how to deal with following task:
*Which of the following extensions are normal?*
1. $\mathbb{Q}(i \sqrt[6]{3}) / \mathbb{Q}$;
2. $\mathbb{C}(t) / \mathbb{C}\left(t^4\right)$;
3. $\mathbb{R}(t) / \mathbb{R}\left(t^4\right)$;
The first one is pretty obvious, if i am right. I know, that exte... |
Which of the following extensions are normal? |
I am trying to set up a probabilistic framework that can be easily switched between a frequentist and a Bayesian approach.
Assume the measurable spaces $(\mathcal{X}, \mathfrak{X}), (\mathcal{T}, \mathfrak{T})$ and $(\Omega, \mathfrak{F})$, where $\Omega = \mathcal{X} \times \mathcal{T}$ denotes the sample space equ... |
For $n=1,\dots,7$, clearly $S_7$ contains $n$-cycles: each of them is isomorphic to $\mathbb Z_n$. For $n=10$, for example the subgroup $\langle (12)(34567)\rangle$ is isomorphic to $\mathbb Z_{10}$. Likewise, for $n=12$, for example the subgroup $\langle (123)(4567)\rangle$ is isomorphic to $\mathbb Z_{12}$. Can you s... |
Let $X$ be a random variable from a probability triple $(A,\mathcal{A},P)$ to $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the Borel $\sigma$-algebra of $\mathbb{R}$. As [Wikipedia explains][1] (bold is mine):
> The probability density function $f$ of $X$ with respect to the counting measure, **if it exists,** ... |
Is every probability mass function $f_X$ of a random variable $X$ the Radon-Nikodym derivative of $X_*P$ with respect to the counting measure? |
I want to prove that there exists $x>0$ such that $x^x = c$ for $c>0$, and that it is unique for $c>1$. I have no idea how to begin, and I want to try to prove it without using the intermediate value theorem, as I do not how to show that $x^x$ is continuous for $x>0$, and I also want to see whether there is proof that ... |
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[0,2]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ?
According to [this][1] answer... |
I wish to compute an analytic solution to
$\int_{-\infty}^{\infty}dx\, e^{-x^2} \operatorname{erf}(ax +b)$
I gave integration by parts a go but essentially ended up with the same integral. Anyone know if this can be solved? |
This is a second part of a two-part question.
In the first part I was asked to prove that any group of order $5784 = 2^3 \cdot 3 \cdot 241$ has subgroups of the following *indexes*: $3, 6, 8, 12,$ and $24$.
I proved that part by showing:
- G has subgroups of orders $2, 2^2, 2^3, 3, 241$ (by Sylow theorem).
... |
I have a nonlinear system of ODEs with **known** constant coefficients $A, B, C, D, E, F, M$:
\begin{align}
&\dot{n}(t)=-An(t)+Bm(t)n(t)+Cm(t) \\
&\dot{m}(t)=-Bm(t)n(t) + (M-m(t))R_0 \sin{\omega t} +Dm(t)+En(t)+F,
\end{align}
with driving amplitude $R_0$. I know from numerical simulations that solutions of $n(t)... |
There is another similar question in the forum but it requires the compactness https://math.stackexchange.com/q/2488691
So I was wondering if instead of compact we ask for locally compact, would be still true?
As one can see the only thing left to prove is the continuity of the inversion mapping $x\to x^{-1}$.
... |
I want to prove that there exists $x>0$ such that $x^x = c$ for $c>1$, and that it is unique. I have no idea how to begin, and I want to try to prove it without using the intermediate value theorem, as I do not how to show that $x^x$ is continuous for $x>0$, and I also want to see whether there is proof that solely rel... |
I'm grappling with a problem involving seating arrangements in a room that features two circles of chairs. One circle consists of 11 chairs, and the other has 7, making a total of 18 chairs. The twist is considering scenarios with 20 people, focusing on the case where the chairs are indistinguishable. The challenge I f... |
I want to understand, how to deal with following task:
*Which of the following extensions are normal?*
1. $\mathbb{Q}(i \sqrt[6]{3}) / \mathbb{Q}$;
2. $\mathbb{C}(t) / \mathbb{C}\left(t^4\right)$;
3. $\mathbb{R}(t) / \mathbb{R}\left(t^4\right)$;
The first one is pretty obvious, if i am right. I know, that exte... |
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