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The ring $\mathbb{Z}[\frac{1}{2}]=\{\frac{u}{2^n}\}$ is the localization of $\mathbb{Z}$ in regards to $S=\{2^n:n \in \mathbb{N}\}$. Find all ideals of this ring.
I'm not sure what is meant by this question. The ideals should be of the form $(2^n u)$ since $\mathbb{Z}[\frac{1}{2}]$ has no other elements to form line... |
The hyperbolic geodetic distance in the half plane model is a pure number since it is given by formulae which are ratios of lengths (using either the cross ratio or the arsinh formula). So what is the unit of length? Deriving the length in the arsinnh formula requires specifying x and y cartesian coordinates, so could... |
What kind of isomorphisms preserve matrix group? |
I've always heard that if the value of a function approaches a given limit as the value of the argument approaches a specific value, then the limit exists. But here is an example that has me confused;
Say we have f(x) = x^2
and we want to check: if |x - (-1)| < delta, then |f(x) - L| < epsilon, for every epsilon > ... |
Let $N^3$ be a smooth manifold, and let $u:\Sigma\to N$, $v:\Sigma'\to N$ be two smooth immersions of closed orientable surfaces. (Note that $\Sigma$ and $\Sigma'$ may be of different topology).
Suppose also that $u$ and $v$ are homologous. (To pass to homology, I can triangulate $\Sigma$ and $\Sigma'$ so that $u$ and... |
There's a nice feature I came up with using GeoGebra about four years ago. I have some kind of argument for it but it's not entirely rigorous, I hope to formalize it in the answers.
I would also be happy if anyone could provide a proof of pure Euclidean geometry
Given: $∆ABC$ is a triangle, the equilateral triang... |
I am working on this exercise 1.5.7 of Dixon-Mortimer Permutation Groups: Let $G \leqslant \mathrm{Sym}(\Omega)$ be a transitive group and let $\Gamma$ and $\Delta$ be finite subsets of $\Omega$. Suppose $G_{(\Gamma)}$ and $G_{(\Delta)}$ act primitively on $\Omega\setminus\Gamma$ and $\Omega\setminus\Delta$ respectivel... |
For what it could be worth
\begin{align*}
a_1=\mathrm{i} \quad \quad a_{n+1}=\mathrm{i} + \frac{\mathrm{i}}{a_n}
\end{align*}
leads to
$$a_n=\frac {\mathrm{i}} 2+\frac{\mathrm{i}}{2} \sqrt{1-4\, \mathrm{i}}\,\, \coth \left(n \tanh
^{-1}\left(\sqrt{1-4\, \mathrm{i}}\right)\right)$$ confirms your observations.... |
Consider a standard ERM problem based on quadratic loss where we solve
$$
\hat{f}_n\in \arg min_{f\in \mathcal{F}} R_{tr}(f)
$$
where $R_{tr}(f)=\frac{1}{n}\sum_{i=1}^n (Y_i-f(X_i))^2$, $\{(X_i, Y_i): i=1,\dots, n\}$ is the training dataset, and $$\mathcal{F}:=\{f_\theta: f_\theta(x)=\sum_{i=0}^9 \theta_i x^i , \t... |
What is the conditional expectation $\mathbb{E}[\hat{f}_n(x)|x]-f^*(x)$? |
I recently come by this little problem: https://vnexpress.net/dap-an-bai-toan-dem-so-tam-giac-3310485.html. It shows that we can make $50$ triangles with sides equal to $10$ or less. But it is more of listing than math.
I wonder how many triangles can we make with all sides equal or less than $2n$.
Here my approa... |
Let $V$ be a vector space of odd dimension over a field F and $B:V\times V\rightarrow F$ an skew symmetric bilinear form. Why can we assert that $B$ is degenerate? Is it also true for any skew symmetric bilinear map $C: V\times V\rightarrow W$ where W is other vector space? |
How do I solve the Diophantine equation $a^n+n+b^n=c^n$ for positive integers, where $n>2$?
The reason for giving $n>2$ is because there are infinitely many solutions when $n=2$.
As of,
$3^2+2+5^2=6^2$
$5^2+2+13^2=14^2$
Please note that,
$3^2+4^2=5^2$
$5^2+12^2=13^2$
I'm struggling to find int... |
How do I solve this Diophantine equation for $n>2$? |
Yesterday I asked a [question][1] about the joint law of $(M_T, B_u)$, where $M_T = \max_{t \in [0, T]} B_t$, $B_t$ is the standard Brownian motion and $u < T$.
One of the reasons I was interested in this law is because I needed to compute $\mathbb{E} \{ \tau B_u \}$, where $\tau$ is the a.s. unique point at which t... |
In this https://web.abo.fi/fak/mnf/mate/kurser/fourieranalys/chap3.pdf Section 3.7: Convolutions (Page 79) lecture notes, it is said that the fourier transform of a convolution of distributions $\mathcal{F}(f*g)$ can be defined if the fourier transform of $g$ is in the space of tempered test functions, i.e. $\mathcal{F... |
I have the coordinates of the vertices of a convex polytope in $R^3$ and no other information. I'd like to count the faces of this polytope. I can determine which vertices are coplanar; this will pick out the faces but it will also include internal sections. Is there a way to distinguish between these two? |
I tried to find galois group of polynomial $p(x) = x^5 + 2x + 2$ and I managed after a while using the fact that it has only $1$ real root and the others are complex so they correspond to transposition and real root to cycle of length $5$. Combining everything gives Galois group is going to be $S_5$.
Now, my questio... |
Is the Galois group of polynomial invariant under iterating? |
In this https://web.abo.fi/fak/mnf/mate/kurser/fourieranalys/chap3.pdf Section 3.7: Convolutions (Page 79) lecture notes, it is said that the fourier transform of a convolution of distributions $\mathcal{F}(f*g)$ can be defined if the fourier transform of $g$ is in the space of tempered test functions, i.e. $\mathcal{F... |
Sorry this is a mathematically illiterate question! I'm coming form a philosophy background here.
Is there a way to find out what 100 divided by 2, a billion times over, would be?
As in 100 / 2 = 50, 50 / 2 = 25 etc for a billion divisions.
I don't know how to begin to do that kind of calculation, as far as I... |
Any resources on the long term dynamics of reaction-diffusion pdes? |
I've always heard that if the value of a function approaches a given limit as the value of the argument approaches a specific value, then the limit exists. But here is an example that has me confused;
Say we have $f(x) = x^2$
and we want to check: if $|x - (-1)| < \delta$, then $|f(x) - L| < \varepsilon$, for every... |
Hint…indicate the point where the dotted line through $D$ meets the $y$ axis as $X$.
Then $XD=L$ and $XF=L\cos\varphi$.
Can you finish it now? |
Consider a complex vector subspace $W \subset V$ such that $\mathrm{dim}W = n$. Suppose I have a set $S$ of $n$ linearly independent complex matrices $S = \{ M_1, \cdots, M_n\}$ that act on $V$.
Let $w$ be some vector in $W$ and consider
$$
w_1' = M_1w, \quad w_2' = M_2 w, \quad \cdots \quad w_n'= M_n w.
$$
Is ... |
*A standard $6$-sided fair die is rolled until the last $3$ rolls are [strictly] ascending. What is probability that the last $3$ digits start with $1,2,3,4$ respectively?*
**My attempt**
We can investigate the $3-$roll (when there are exactly $3$ rolls), the $4-$roll, the $5-$roll, the $n-$roll.
I did that via ... |
A [Surface of Constant Width][1] is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the surface and paired-planes. I want to construct a trio of parametric equations $\bigl(X(u, v), Y(u, v), Z(u, v)\big... |
How can I convert a strict linear (affine) matrix inequality feasibility problem into a non-strict one? |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
While reading a proof of a theorem under the section **The General Solution of the Homogeneous equation** in *Differential Equations and Transform Techniques* by George F. Simmons, it is given that
>We now assume that the Wronskian is identically zero and prove linear
dependence. If $y_1$ is identically zero on $[... |
Problem: Let $(X, || \cdot ||)$ be a normed space. Prove that the following are equivalent:
i) There exists an inner product $< \cdot , \cdot >$ on $X$ such that $||x|| = \sqrt{< x, x >}$ for every $x \in X$.
ii) The $|| \cdot ||$ satisfies the parallelogram law, i.e.,
\begin{align}
||x + y||^2 + ||x - y||^2 ... |
Before solving the problem, let us ponder a little. If the problem statement is true, then you can pick, from each equivalence class of skew symmetric matrices of the same ranks, a representative as a canonical form.
What canonical form would have a nice structure? Diagonal matrix would be nice, but if a matrix is ... |
Let $\mathfrak{g}$ a Lie algebra and $B:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathbb{K}$ be a non-degenerate invariant symmetric bilinear form. Then there exists a unique $C_{B} \in \mathfrak{U(g)}$, called the Casimir element of $\mathfrak{g}$ corresponding to $B$, such that $$\sum_{i=1}^{n} e_{i}(B^{\sharp}e... |
Why $B^{\sharp}e_{1}^{*},...,B^{\sharp}e_{n}^{*}$ is a basis of $\mathfrak{g}$? |
Sorry this is a mathematically illiterate question! I'm coming from a philosophy background here.
Is there a way to find out what $100$ divided by $2$, a billion times over, would be?
As in $\frac{100} { 2} = 50, \frac{50} { 2} = 25$ etc for a billion divisions.
I don't know how to begin to do that kind of ... |
Here is the question I am trying to see the solution of part (a) in it:
Fix $\mathbb F_p$ a field with $p$ elements for $p$ prime. Consider $f(X) = X^{p^n} - X + 1$ over $\mathbb F_p$ and let $E$ be a splitting field of $f.$
$(a)$ Prove that $E$ has a subfield isomorphic to $\mathbb F_{p^n}.$
$(b)$ Prove that $[... |
Let $\mathfrak{g}$ be a finite dimensional Lie algebra and $B:\mathfrak{g} \times \mathfrak{g} \rightarrow \mathbb{K}$ be a non-degenerate invariant symmetric bilinear form. Then there exists a unique $C_{B} \in \mathfrak{U(g)}$, called the Casimir element of $\mathfrak{g}$ corresponding to $B$, such that $$\sum_{i=1}^... |
Consider a complex vector subspace $W \subset V$ such that $\mathrm{dim}W = n$. Suppose I have a set $S$ of $n$ linearly independent complex matrices $S = \{ M_1, \cdots, M_n\}$ that act on $V$.
Let $w$ be some non-zero vector in $W$ and consider
$$
w_1' = M_1w, \quad w_2' = M_2 w, \quad \cdots \quad w_n'= M_n w. ... |
I am not sure if my solution to the following problem is correct
1. Let x be an infinite set. for x $\in$ X and y $\in$ X we define:
\begin{equation}
d(x,y) = \left\{
\begin{array}{ll}
1 & x \neq y \\
0 & x=y
\end{array}
\right.
\end{equation}
Prove that d is metric, Which subsets of X are o... |
My attempt: $P$ is projective. Take $0 \to B \to I_0 \to I_1 \to 0$ is an injective resolution of $B$. Apply $Hom(P,\_)$, left exactness of covariant functor implies exactness of the sequence: $0 \to Hom(P,B) \to Hom(P,I_0) \to Hom(P,I_1)$. Note $Hom(P,I_0) \to Hom(P,I_1)$ is onto. Then $Ext^1(P,B) = \frac{ker(Hom(P,I_... |
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:
$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{1}{1+\cdots x}}}}\right)-C=0... |
Yes, these are indeed two different definitions of the limit of a function at a point. In my experience, your observations can be generalized: most introductory textbooks use the first definition (where the function is required to be defined in a punctured neighborhood of the point, and thus in particular one-sided lim... |
If we have a conic with equation: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ or S =0, and a point $P(x_1, y_1)$.
Then the equation of the chord with P as its midpoint is given by T = $S_1$.
$S_1$ is obtained by plugging point P in S. ($S_1$ = $ax_1^2 + 2hx_1y_1 + by_1^2 + 2gx_1 + 2fy_1 + c = 0$)
>The 'T' form... |
Why does T = S1 work for conics? |
g : R −→ R, g(x) := (2x + 1 for x < 0, e^x for x>= 0)
I am trying to understand the properties of this function.
I already know that it is continuous, as its' components are continuous functions(because 2x+1 is linear and because e^x is exponential).
But I don't understand why it is not differentiable, because ... |
Why is this continuous function not differentiable? |
g : R −→ R, g(x) := (2x + 1 for x < 0, e^x for x>= 0)
I am trying to understand the properties of this function.
I already know that it is continuous, as its' components are continuous functions(because 2x+1 is linear and e^x is exponential).
But I don't understand why it is not differentiable, because I think ... |
Here is the question I am trying to see the solution of part (a) in it:
Fix $\mathbb F_p$ a field with $p$ elements for $p$ prime. Consider $f(X) = X^{p^n} - X + 1$ over $\mathbb F_p$ and let $E$ be a splitting field of $f.$
$(a)$ Prove that $E$ has a subfield isomorphic to $\mathbb F_{p^n}.$
$(b)$ Prove that ... |
Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(- \frac{(x-y)^2}{2t} \right)$ and let $u^\alpha(x,y)$ be the potential density of $B_t$, i.e. $u^\alpha(x,y) = \int_0^\i... |
The use of $F\simeq G$ as in your quote from Wikipedia is, as you said, hard to justify, but the closely related notation $F(n)\simeq G(n)$ is actually quite reasonable, and may have suggested the $F\simeq G$ notation. Let me explain how $F(n)=G(n)$ and $F(n)\simeq G(n)$ came to have different meanings.
The idea, as... |
In order to make sense of your question one has to specify what does "space" and "cover" mean. Without further adjectives, they usually mean "a topological space" and "a topological covering map." However, it is clear from your comments, this is not what you are actually interested in. One can define the notion of a co... |
Let $X$ and $X'$ be two birationnal algebraic curves over $\mathbb{Q}$.
Is it true (or not) that $X(\mathbb{Q})$ is Zariski-dense iif $X'(\mathbb{Q})$ is Zariski-dense ?
If no, where can I find a counter-example?
If yes, what about the higher dimensional varieties? |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
Given some unitary matrix $U$, we can possibly find many unitary square root matrices $S_i$ so that $S_i^2=U$. Let's assume we have an additional (complex) matrix $T$ so that it commutes with $U$:
$$ [U,T]=0. $$
What can we say about the commutator $[S_i,T]$? Do these matrices necessarily commute? If not, is ther... |
Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by
$$
x_{n+1}=A x_n
$$
with some starting vector $x_0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the Euclidean norm on $\mathbb R^n$. I want to show that
$$
\lim_n \frac{\vert x_{n+1}\vert}{\vert x_n\vert}\neq 0.
$$
*Some obser... |
Let us define a π-system on a set $\Omega$ as a collection $\mathscr A$ of certain subsets of $\Omega$ having two properties: $\mathscr A$ being non-empty and being closed under finite intersections (meaning that if $A, B \in \mathscr A$, then $A \bigcap B \in \mathscr A$ as well).
However, the π-system page in Wiki... |
Why is there no empty set in a π-system? |
I'm studying the existence and the uniqueness of the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ but i don't understand why $\mathfrak{U(g)}$ is generated by $i(\mathfrak{g})$, where $i$ is the following morphism: $$i: \mathfrak{g} \rightarrow \mathfrak{U}_{L}.$$
Is there anyone who can help me to u... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
**Question**
>The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first and fourth quadrant is?
___________
The path given is [![this the path on which the particle has moved][1]][1]
... |
Let $X$ and $X'$ be two birational algebraic curves over $\mathbb{Q}$.
Is it true (or not) that $X(\mathbb{Q})$ is Zariski-dense iif $X'(\mathbb{Q})$ is Zariski-dense ?
If no, where can I find a counter-example?
If yes, what about the higher dimensional varieties? |
Let $X$ and $X'$ be two birational algebraic curves over $\mathbb{Q}$.
Is it true (or not) that $X(\mathbb{Q})$ is Zariski dense iif $X'(\mathbb{Q})$ is Zariski dense ?
If no, where can I find a counter-example?
If yes, what about the higher dimensional varieties? |
I had a question I was hoping someone could answer. I'm working on this homework problem, and when graphed, the volume looks like it would be the same revolving around the x-axis or on the axis of $y=5$, but it isn't. Why is this?
Problem:
Find the volume of region R when region R is revolved about the line $y = 4... |
Golden Ratio in a functional equation? |
If we have three random variables $X,Y,Z$, which are jointly normal, how can it be shown that $I(X;Y) = I(X;Z)= 0 \implies I(X;Y;Z) = 0$?
I know that for jointly normal distributions $X,Y,Z$: $I(X;Y) = \frac{1}{2} \log \frac{\det(\Sigma_X) \det(\Sigma_Y)}{\det(\Sigma_{XY})}$. Knowing this much, how can I move on fro... |
Is this right for converting an infinite sum into an integral with the limit $[0; \infty]$? |
I've done the following to get from an infinite sum to an integral, the question is now, did i do everything properly and didn't made any mistakes, because i coulnd't find anything like that when searching for "how to convert inifinite sum to integral?"
$$
\int_{a}^{b}f(x) dx = \lim_{n \rightarrow \infty} \sum_{i =... |
In an exam there are $20$ questions, one can obtain either $-1,0$ or $4$ in each question based on marking scheme. How many different totals can he obtained in the test?
One thing I observed different that even without repetition there is repetition as different permutation can obtain same number.
I tried several... |
How many different totals can be obtained in a test with $20$ questions if one can obtain $-1$, $0$, or $4$ for a question? |
I attemped to solve [this problem](https://math.stackexchange.com/questions/1463787/probability-of-winning-in-a-die-rolling-game-with-six-players?rq=1) using a method that is slightly different from the solutions provided (but results in a correct answer), and I'd like to check if the logic is correct (it seems incorre... |
6 Player Dice Game (is my method "accidentally" correct?) |
**Context.** This question is a follow-up of [the following one.][1]
Let $n\geq 1$ be an integer, let $d\mid n$, and let $H=\langle\sigma, \tau \rangle\subset S_n,$ where $ \sigma=(1 \ 2 \cdots n), \tau=( 1 \ d)$.
According to an exercise of Serre's finite groups, after correction of some typo, one should have $... |
I'm trying to show that |f(r)| $\rightarrow \infty$ as r $\rightarrow 1^-$. Here, f(z) = $\sum_{n\geq 0}z^{2^n}$.
Then I need to show by induction that if z = $re^{i\theta 2^m}$ = 1 for a certain m in $\mathbb{N}$ then |f($re^{i\theta}$)| $\rightarrow \infty$ if r$\rightarrow 1^-$.
For this one, I imagine the c... |
I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an arbirtrary norm), at a point at the boundary $\partial B$. Recalling that the normal cone to a convex set $C$ at a point $x$ in... |
Finding functions $h$ satisfying $h(h(n))+h(n+1) = n+2$ for all natural $n$. Why does the Golden Ratio appear? |
I tried to find galois group of polynomial $p(x) = x^5 + 2x + 2$ and I managed after a while using the fact that it has only $1$ real root and the others are complex so they correspond to transposition and real root to cycle of length $5$. Combining everything gives Galois group is going to be $S_5$.
Now, my questio... |
I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an arbirtrary norm), at a point at the boundary $\partial B$. Recalling that the normal cone to a convex set $C$ at a point $x$ in... |
I'm trying to show that $|f(r)| \rightarrow \infty$ as $r \rightarrow 1^-$. Here, $f(z) = \sum_{n\geq 0}z^{2^n}$.
Then I need to show by induction that if $z = re^{i\theta 2^m} = 1$ for a certain $m$ in $\mathbb{N}$ then $|f(re^{i\theta})| \rightarrow \infty$ if $r\rightarrow 1^-$.
For this one, I imagine the c... |
I am considering $2^n\times 2^n$ complex matrices and a general anti-involution $A^{**}=A$, $(AB)^*=B^* A^*$. To my knowledge all anti-involutions are a combination of transpose, field involution and similarity transform (is that correct?).
What are the possible dimensions of the space created by all matrices which ... |
What are the possible dimensions of self-anti-involution matrices? |
I have a conceptual problem when we extend the correlation to time series. I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer som... |
I attemped to solve [this problem](https://math.stackexchange.com/questions/1463787/probability-of-winning-in-a-die-rolling-game-with-six-players?rq=1) using a method that is slightly different from the solutions provided (but results in a correct answer), and I'd like to check if the logic is correct (it seems incorre... |
I'm trying to show that $|f(r)| \rightarrow \infty$ as $r \rightarrow 1^-$. Here, $f(z) = \sum_{n\geq 0}z^{2^n}$.
Then I need to show by induction that if $e^{i\theta 2^m} = 1$ for a certain $m$ in $\mathbb{N}$ then $|f(re^{i\theta})| \rightarrow \infty$ if $r\rightarrow 1^-$.
For this one, I imagine the case $... |
So I am working on a problem and I am required to convert multiple summations of the following form into a single summation,
\begin{align}
\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty} \alpha_{ijk} \,X^{i+j+k} = \sum_{n=0}^{\infty} \beta_{n} \,X^{n}
\end{align}
So now I want to compare terms of the ... |
I am considering $2^n\times 2^n$ complex matrices and a general anti-involution $A^{**}=A$, $(AB)^*=B^* A^*$. To my knowledge all anti-involutions are a combination of transpose, field involution and similarity transform (is that correct?).
What are the possible dimensions of the space created by all matrices which ... |
Can someone teach me how to prove the statement $(A\Rightarrow B)\Leftrightarrow C$?
My idea is:
1. For $\Rightarrow $ part, we assume that A and B are true then show C is true
2. For $\Leftarrow $ part, suppose that C and A are true then show B is true
Am I right? Thanks for your help! |
How to prove $(A\Longrightarrow B)\iff C$? |
I’m trying to understand equation (1.8) on p. 4 of Chow et al.’s “The Ricci flow: techniques and applications”. There, the authors say that, using indices, the equation
$$ -2\operatorname{Rc}(g) = \epsilon g + \mathcal L_X g$$
becomes
$$ -2R_{ij} = \nabla_i X_j + \nabla_j X_i + \epsilon g_{ij}. \tag{1.8}$$
... |
**Context.** This question is a follow-up of [the following one.][1]
Let $n\geq 1$ be an integer, let $d\mid n$, and let $H=\langle\sigma, \tau \rangle\subset S_n,$ where $ \sigma=(1 \ 2 \cdots n), \tau=( 1 \ d)$.
According to an exercise of Serre's finite groups, after correction of some typo, one should have $... |
> Is it the closure $\overline{f(Z)}$?
Yes indeed. Remember that we think of elements $g \in \mathbb C[Y]$ as functions $g: Y \to \mathbb C$. The map $f^*$ is then just composition $f^*g = g \circ f: X \to Y \to \mathbb C$. Now such a function $g$ vanishes on $f(X)$ if and only if $f^*g$ vanishes on $X$, i.e. $f^*g ... |
Consider a complex vector subspace $W \subset V$ such that $\mathrm{dim}W = n$. Suppose I have a set $S$ of $n$ linearly independent complex matrices $S = \{ M_1, \cdots, M_n\}$ that act on $V$.
Let $w$ be some non-zero vector in $W$ and consider
$$
w_1' = M_1w, \quad w_2' = M_2 w, \quad \cdots \quad w_n'= M_n w. ... |
Let $BA$ be the set of axioms of Boolean algebras.
Let $Th(BA)$ be the set of first-order sentences that are true when conjoined with the axioms in $BA$.
I believe that $Th(BA) = Th(\langle \{0,1\}, \land, \lor, \lnot \rangle)$ where $\land, \lor,\lnot$ are the usual propositional operations interpreted over $\{... |
Is the theory of Boolean algebras equal to the theory of Boole's algebra? |
Let $g : \mathbb{R} \to \mathbb{R}$, such that
\begin{equation}
g(x)=\begin{cases}
2x + 1 & \text{ if } x < 0,\\
e^x & \text{ otherwise}.
\end{cases}
\end{equation}
I am trying to understand the properties of this function.
I already know that it is continuous, as its' components are continuous ... |
The proof that the automorphism group of $Z_n$ is isomorphic to $(\mathbb Z/n\mathbb Z)^\times$ in Dummit and Foote uses the fact that if $\varphi \in \operatorname{Aut}(Z_n)$ then $\varphi(x) = x^a$ for some integer $a \in \mathbb Z$.
I can understand that this is the case because the isomorphism must send generato... |
Why must an automorphism of $Z_n$ be of the form $x \mapsto x^a$? |
I am considering $2^n\times 2^n$ complex matrices and a general anti-involution $A^{**}=A$, $(AB)^*=B^* A^*$. To my knowledge all anti-involutions are a combination of transpose, field involution and similarity transform (is that correct?).
What are the possible dimensions of the space created by all matrices which ... |
**Context.** This question is a follow-up of [the following one.][1]
Let $n\geq 1$ be an integer, let $d\mid n$, and let $H=\langle\sigma, \tau \rangle\subset S_n,$ where $ \sigma=(1 \ 2 \cdots n), \tau=( 1 \ d)$.
According to an exercise of Serre's finite groups, after correction of some typo, one should have $... |
The answer is, technically, yes. As I hinted at in my comment, making $h$ constant satisfies the conditions. Given any $g$, and any constant function $h$ (which is convex), $h \circ g$ is constant and hence convex.
However, in the realm of [tag:convex-optimization], this is likely unhelpful. The unstated spirit of t... |
This is one of my extension sheet questions and I was really stumped on how to approach it.
$\sqrt{99} + \sqrt{101} < 2\sqrt{100}$
First I had approached it by looking at smaller and larger square roots, however the numbers are much too small to be practical in an exam. I next have done the following:
Let $n =... |
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