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Why is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ said to be incomplete wrt. to the Lebesgue measure?
As I understand a complete measure space contains all subsets of all sets of measure zero from the $\sigma$-Algebra. Any set of measure zero in the Borel $\sigma$-Algebra of a subset of $\mathbb{R}^d$ ... |
Is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ complete? |
Why is the Borel $\sigma$-Algebra on a subset of $\mathbb{R}^d$ said to be incomplete wrt. to the Lebesgue measure?
As I understand a complete measure space contains all subsets of all sets of measure zero from the $\sigma$-Algebra. Any subset of a set with measure zero in the Borel $\sigma$-Algebra of a subset of $... |
How do you show that $\mathrm{Aut}_\mathbb{Q}(\overline{\mathbb{Q}})$ is uncountable ?
Thanks in advance |
Let $G = (V, E)$ a critical graph; i.e. a graph s.t. for any subgraph $H \subseteq G$ we have $\chi(H) < \chi(G)$. I was requested to prove that $\chi(G) \leq \delta + 1$, where $\delta$ is the degree of the vertex (or vertices) of lesser degree in $G$.
I presume that I need to relate a sub-graph $H$ that involves ... |
I recently came across a problem when trying to deal with a set of numbers with n elements.
The problem is as follows:
Starting with a set of n distinct elements, how would one generalize a unique algorithm such that all the possible permutations of the set would appear exactly once, with the following rules:
1. D... |
I've already seen http://math.stackexchange.com/questions/107341 but I'm looking into a similar but distinct problem, one I don't even know how to approach besides simply enumerating all possible results. Ideally I'd like to 'learn how to fish' instead of just be given fish.
As a concrete example, if the two 12-side... |
Let $E/\Bbb{Q}$ be an elliptic curve.
$H^1(G_{\Bbb{Q}},E[2])$ is bijection with the set of pair of torsors and divisor of degree $2$. Let call the latter set $WCD(E/\Bbb{Q})$.
I tried to construct a bijection between them.
Is the following correct ? I'm having difficulty showing its well-definedness, so I my try... |
I have a question regarding a definition in Kanamori's book on page 136. (it's the beginning of the subsection about Random Reals)
Let $\mathcal{B}^{\star} = \{X \in \mathcal{B} \ \lvert \ X \ \text{is not null} \}$ where $\mathcal{B}$ denotes the Borel-Sigma-algebra.
One defines the forcing partial order for $p,q ... |
In [chess][1] (or [9LX][2]), *Armageddon* is a kind of blitz/rapid chess used as tiebreakers: Black has draw odds, but white gets more time.
Eg In the [2022 women's US Chess Championship][3], Jennifer Yu won the coin toss and then chose black against Irina Krush who played white. Jennifer got 4 minutes, and Irina go... |
Consider the following density function (unnormalized)
$$p(y;b)=\int_{0}^{\infty}\dfrac{\exp\left(-\frac{y^2}{2\left(b+\frac 5u\right)}\right)}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
Is it true that $p(y;b)$ is a log-concave function in $b$ when $y$ is considered fixed?
My attempt: First ... |
Is this a metric space? |
>Let $\varphi=\cdots\circ \operatorname{Fr}^{3!}\circ \operatorname{Fr}^{2!}\circ \operatorname{Fr}^{1!}$, where $\operatorname{Fr}$ is the Frobenius endomorphism. Show that $\varphi \in \operatorname{Gal}(\overline{\mathbb F_p}/\mathbb F_p)\setminus\langle\operatorname{Fr}\rangle$.
First of all, I had to show tha... |
The problem asks to sum it in its circle of convergence.
It's radius of convergence is $R=lim_{n \to \infty} \left|{\frac{\frac{1}{2n+1}}{\frac{1}{2n+2}}}\right|$=1. And when $\left|z\right|=1$ it diverges by the Limit Comparison Test with $1/n$. So its circle of convergence is $\left|z\right|<1$, id est, $z \in D(0... |
I recently came across a problem when trying to deal with a set of numbers with n elements.
The problem is as follows:
Starting with a set of n distinct elements, how would one generalize a unique algorithm such that all the possible permutations of the set would appear exactly once, with the following rules:
1. D... |
There are two obvious (smooth left) actions of $\mathrm{SO}(3)$ on $S^2$. There is the standard action by which $\mathrm{SO}(3)$ acts by 3D rotations on the standard embedding of $S^2$ in $\mathbb{R}^3$, and there is the trivial action in which $\mathrm{SO}(3)$ leaves the points of $S^2$ unchanged. It feels like these ... |
Generalization of the action for unbounded systems? |
I'm having a hard time understanding this (translated literally from lecture script):
"let $\rho_1,...,\rho_k $ be a list of irreducible inequivalent unitary representations of a finite group G. $\rho_{\alpha,ij}(g), \alpha = 1,...,k$ denote the Matrix entries of $\rho_{\alpha}(g)$ w.r.t. an orthonormal basis. The f... |
To understand is to see:
Step 1:
$5^5 \times 10^{10} \times 15^{15} \times \dots \times 120^{120} \times 125^{125}=$
$(1\times 5)^5 \times (2\times 5)^{10} \times (3\times 5)^{15}\times ....\times (24\times 5)^{120} \times (25\times 5)^{125}=$
$[5^5\times 5^{10} \times 5^{15} \times ....\times 5^{120} \time... |
Where
li(x) is defined as: https://i.stack.imgur.com/UYuIG.png and Ei(x) is defined as
https://i.stack.imgur.com/vy01Q.png. I can't find any proof of this expression but I did find many articles stating it, could anyone show me how it could be proven? Thank you a lot. |
How can it be proven that li(x)=Ei(logx)? |
How to show that $\operatorname{li}(x)=\operatorname{Ei}(\log x)$?
Here, $\operatorname{li}(x)$ and $\operatorname{Ei}(x)$ are defined as:
$$\operatorname{li}(x)=\int_0^x\frac{\mathrm dt}{\log t}$$
$$\operatorname{Ei}(x)=\int_{-\infty}^x\frac{e^t}t\,\mathrm dt$$
I can't find any proof of this expression but I d... |
How can it be proven that $\operatorname{li}(x)=\operatorname{Ei}(\log x)$? |
Let $(\mathcal{M}, \mathcal{W}, \mathcal{C},\mathcal{F})$ be a model category and $\text{Ho}(\mathcal{M})$ its homotopy category. Now we consider $ \mathcal{M}_{\mathcal{F}}$ the full subcategory of fibrant object. Now, I want to show that we have an equivalence between $\text{Ho}(\mathcal{M})$ and $\text{Ho}(\mathcal{... |
I am confused about conditional expectations. Let $(\Omega, \mathcal{F},P)$ be a probability space. Next, let $X$ and $Y$ be random variables on this space. Next, let $E[X|\sigma(Y)]$ be conditional expectation, which is $\sigma(Y)$-measurable random variable, and, therefore, also $\mathcal{F}$-measurable.
The ques... |
I have a system of differential equations in which a variable converts to several other ones. I'd like to add randomness to the parameters, but I am not sure the best approach.
A minimal example (the real one has more terms) is
$$
\begin{aligned}
\frac{dx}{dt} &= -\Gamma \, x,
&
\Gamma &= \sum_{i = 1}^N \gamma_... |
I have been given this rather simple looking assignment, which is confusing me a lot.
Given is:
$f:[0,5] \rightarrow \mathbb{R}$ where $f(x)=2x+3$.
The first thing I had to do, was to determine the area under the curve, for the given interval $[0,5]$, from the left and right side, for 30 subintervals.
I solve... |
How to find exact Riemann-Sum? |
Theorem: The set of compact operators that map to any Banach space is closed.
Proof: Let $\{\mathcal{T}_n\}$ be a sequence of compact operators from $\mathcal{X}_1$ to
$\mathcal{X}_2$, where $\mathcal{X}_2$ is a Banach space. Our goal is to show that if $\|\mathcal{T}_n - \mathcal{T}\| \to 0$
for some $\mathcal{T}... |
I am confused about conditional expectations. Let $(\Omega, \mathcal{F},P)$ be a probability space. Next, let $X$ and $Y$ be random variables on this space. Next, let $E[X|\sigma(Y)]$ be conditional expectation, which is $\sigma(Y)$-measurable random variable, and, therefore, also $\mathcal{F}$-measurable.
The ques... |
In Weibel's definition of a homology spectral sequence collapsing at $E^r$ he says that $H_n$ is the unique non zero $E_{pq}$ such that $p+q=n$. I am trying to prove it but so far, I haven't been able to go very far, I am trying to use the fact that $E_{pq}^\infty\cong F_pH_n/F_{p-1}H_n$ but I haven't been really succe... |
The problem asks to sum it in its circle of convergence.
It's radius of convergence is $R=lim_{n \to \infty} \left|{\frac{\frac{1}{2n+1}}{\frac{1}{2n+2}}}\right|$=1. And when $\left|z\right|=1$ it diverges by the Limit Comparison Test with $1/n$. So its circle of convergence is $\left|z\right|<1$, id est, $z \in D(0... |
I am following Roger'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$, with even and odd p... |
Determine the value of $\lambda$ such that the vectors $5\vec{a}+6\vec{b}+7\vec{c},7\vec{a}+\lambda\vec{b}+9\vec{c},3\vec{a}+20\vec{b}+5\vec{c}$ are coplanar given that $\vec{a},\vec{b},\vec{c}$ are non coplanar.
For the vectors to be coplanar,there must exist scalars $x_1,x_2,x_3$ not all $0$ such that their linear c... |
I am following Roger'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$, with even and odd p... |
What books could serve as a modern equivalent to classics like *Higher Algebra* by Hall and Knight and *Higher Algebra* by Barnard and Child? Basically, what would be a modern rigorous, and comprehensive book on high school algebra? |
I am studying topology through the Topology without Tears book, and on the first Unit the finite-closed topology is presented with the following definition:
> 1.3.1
Definition.
Let X be any non-empty set. A topology
τ on X is
called the finite-closed topology or the cofinite topology if the closed subsets
of X ar... |
I am trying to prove that vector field $X$ is symplectic iff $\scr{L}$$_{X}\omega=0$
$(M,\omega)$ symplectic manifold (compact, smooth and connected if this is necessary)
If X is symplectic, then by definition $i_X\omega$ is closed, ie. $d(i_X\omega)=0$
Using Cartan's magic formula $\scr{L}$$ _X\omega=i_Xd\o... |
Determine the value of $\lambda$ such that the vectors $$5\vec{a}+6\vec{b}+7\vec{c},7\vec{a}+\lambda\vec{b}+9\vec{c},3\vec{a}+20\vec{b}+5\vec{c}$$ are coplanar given that $\vec{a},\vec{b},\vec{c}$ are non coplanar.
For the vectors to be coplanar,there must exist scalars $x_1,x_2,x_3$ not all $0$ such that their line... |
Let's clear the integrand after the change of variable $z = e^{i\theta}$, which can be rewritten as
$$
\int_{-\pi}^\pi \frac{\sin\theta}{a-\sin\theta} \,\mathrm{d}\theta = - \oint_C \frac{z^2-1}{z^2-2iaz-1} \frac{\mathrm{d}z}{iz},
$$
where $C$ is the unit circle centered at the origin. A simple pole is located at ... |
Consider the following density function (unnormalized)
$$p(y;b)=\int_{0}^{\infty}\dfrac{\exp\left(-\frac{y^2}{2\left(b+\frac 5u\right)}\right)}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
Is it true that $p(y;b)$ is a log-concave function in $b$ when $y$ is considered fixed?
My attempt: First ... |
I heard of urelements, which don't contain elements; I've also heard of proper classes which can't be elements of sets, has anyone ever discussed objects that can't be elements in a set or be themselves sets? |
Are there objects that can't be elements of sets, or be sets themselves? |
I need help with trying to understand the solution to $\operatorname{y}'' + \operatorname{y} = 5x{\rm e}^{2x}$:
- Clearly the auxillary equation is just $m^2 + 1 = 0$ which can be solved by $m= \pm {\rm i}$, thereby giving us $A\cos\left(x\right) + B\sin\left(x\right)$ for our complementary function.
- The issue co... |
Using local Lipschitz continuity of $f(\cdot)$:
\begin{align}
f(\mathbf{a}) &\leq f(\mathbf{b})+ L_0 \lVert{\mathbf{a}-\mathbf{b}}\rVert
\end{align}
In the FedProx Paper (https://arxiv.org/pdf/1812.06127.pdf, Equation 9 of the Theorem 4 proof (page no. 13), the authors have used a relation between local Lip... |
How do I find the corner shape of the bounding box of a smooth curve of constant width? |
This problem was in a competition for a job. It seems simple BUT the challenge is you cannot use trigonometry.\
*Let there be 3 squares with side length of l arranged in such a way that it forms a rectangle with length 3l and width l.\
So ABCD,CEHD and EFGH are squares*(this is notation because it's easier to tell yo... |
In Weibel's definition of a homology spectral sequence collapsing at $E^r$ he says that if it converges to $H_*$ then $H_n$ is the unique non zero $E_{pq}^r$ such that $p+q=n$. I am trying to prove it but so far, I haven't been able to go very far, I am trying to use the fact that $E_{pq}^\infty\cong F_pH_n/F_{p-1}H_n$... |
**Question.**
I am having difficulty with both intuitive understanding and necessity of, a technicality in a proof of the Vapnik-Chervonenkis inequality in the notes of [Robert Nowak (2009)](https://nowak.ece.wisc.edu/SLT09/lecture19.pdf#page=2). The proof strategy is taken from the yellow book by [Luc Devroye, Lász... |
This is not a full answer. I found a different perspective to view the problem in, which turns the question from a complicated integral to a complicated discrete summation.
---
Consider a Poisson process, with unit density, on the positive real line. This means that for any $0<a<b$, the number of arrivals in $[a... |
Suppose I had to choose $n$ people from $p$ of one type, $q$ of another type and $r$ of the last type, given $p,q,r>n$. Then how many ways are there to choose $n$ people with at least one from each category?
How could I generalise this for more, say $m$, categories? |
Let's consider the following deterministic optimal control problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables:
\begin{align}
J(t) = \inf_{c(t)} \ &\int_0^\infty e^{-\delta u} \left( x^2(u) + \lambda y^2(u) \right) du
\end{align}
where the dynamics for the states are:
\begin{align}... |
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{{\displaystyle #1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}... |
In mathematics we do not consider questions like "are there objects that are XYZ?", which is a rather philosophical question. In mathematics we consider questions like "are there objects that are XYZ under ABC axioms?". Everything in maths is relative to some axiomatic theory.
In particular the answer to this questi... |
We aim to compute $$I(p,q)=\int_{0}^1e^{ia/(1-\tau)+ib/\tau}\frac{d\tau}{\sqrt{\tau^{2p+1}(1-\tau)^{2q+1}}}$$
where $p$ and $q$ are nonegative integers. This is an improper integral if $p$ or $q$ is $>0.$
An interesting point of this calculation is the fact that $I(0,0)$ cannot be made explicit.
Felipe menti... |
Functions that can be expressed using real numbers, $x, e^x, \sin x, \cos x$ along with their inverses, and addition, subtraction, multiplication, division, and function composition are often called "closed-form" functions.
Any closed-form function can be differentiated, and the result will be another closed-form f... |
In mathematics we do not consider questions like "are there objects that are XYZ?", which is a rather philosophical question. In mathematics we consider questions like "are there objects that are XYZ under ABC axioms?". Everything in maths is relative to some axiomatic theory.
In particular the answer to this questi... |
Find the length of the curve defined by the equation:
$$\sqrt{x}+\sqrt{y}=\sqrt{a}$$
from point $(0, a)$ to point $(a, 0)$.
Well, I decided to plot it as I didn't have intuition about how the graph actually looked. Let's take $a=3$:
[![enter image description here][1]][1]
Note that the graph doesn't actually... |
In Sharpe's book on Cartan geometry, he mentions in kind of an offhand way that Cartan atlases extend to unique maximal Cartan atlases, but I'm feeling skeptical. For a Klein pair $(G,H)$ with Lie algebras $(\mathfrak{g},\mathfrak{h})$, a Cartan atlas is defined as a family of $\mathfrak{g}$-valued $1$ forms on an open... |
I'm a little embarrassed to ask this but I couldn't answer it myself.
I am looking the groups that contains $D_4$ and larger than $D_4$. Here is what I think:
We cannot say $D_4 \subseteq D_n, n\ge 5$ because their group operation are different. But what else? $D_4$ doesn't lie in $S_4$.
Any help is appreciat... |
What are the groups containing dihedral group $D_4$ of order $8$? |
Let $X,Y$ be real symmetric matrices. We say $X \geq Y$ if $X-Y$ is PSD.
**Question:** What is a basic ($2\times 2$) example of matrices $X,Y$ s.t.:
$$
X \leq Y
$$
but:
$$
X^2 > Y^2
$$
e.g.$X^2 - Y^2$ is strictly positive definite? All I know is that $X,Y$ cannot commute but I am having difficulty coming up... |
So I was looking at the integral of $\int{f(x)\cdot\frac{d}{dx}(\frac{df(x)}{dx})dx}$ and I got the following using the DI method (for those who don't know it search DI bprp on youtube):
choosing f(x) for D and $\frac{d}{dx}(\frac{df(x)}{dx})$ for I we get on the first row $\frac{df(x)}{dx}$ for both D and I, and co... |
Where did I go wrong? Integration by parts |
I'm taking a course in functional analysis that is using Rudin's book. I want to ask if there is a relationship between the characteristic polynomials and their elliptic linear operators?
I understand that we call $L$ an elliptic linear operator of $n$ variables of order $N$ if its characteristic polynomial $p(x,... |
I have the following problem:
Let $B$ be a set $B=(x_1 , x_2) \in \mathbb{R}^2 | x_1^2 + x_2^2 \leq 1$ and $x_1 \geq 0$. [1]
A function $f:B \rightarrow \mathbb{R}$ is given by
$\begin{equation*}
f(x_1 ,x_2 )=x_1^2 x_2^2 +3x_1 x_2 +x_2 -4.
\end{equation*}$
State the image of $f$.
To find the image, I w... |
**Question.**
I am having difficulty with both intuitive understanding and necessity of, a technicality in a proof of the Vapnik-Chervonenkis inequality in the notes of [Robert Nowak (2009)](https://nowak.ece.wisc.edu/SLT09/lecture19.pdf#page=2). The proof strategy is taken from the yellow book by [Luc Devroye, Lász... |
I do realize that there were several questions regarding the expected value of the product of two discrete variables. However, I have not found any *natural* derivation of the covariance terms that appear in the expectation of the product. Hence,
*Out of mathematical curiosity*...
We can define the expected val... |
Let $p_1,p_2,p_3$ be distinct prime numbers and consider $n \in \mathbb N$ . Find the number of functions $f : \{ 1,2,...,n \} \rightarrow \{p_1,p_2,p_3 \} $ for which the number $f(1)f(2)...f(n)$ is a perfect square.
My attempt : If $n$ is an odd number, it is obvious that there are no functions, because it is esse... |
Suppose I had to choose 5 people from 8 engineers, 7 scientists and 6 mathematicians. How many ways can I choose at least one person from each profession?
By brute force total number of selection possible $n$:
$$n = \binom{8}{1}\binom{7}{1}\binom{6}{3}+\binom{8}{1}\binom{7}{3}\binom{6}{1}+\binom{8}{3}\binom{7}{1}\... |
Let's assume we are faced with showing that the following covariance between two functions $g(\cdot)$ and $f(\cdot)$ of the same set of $Z_1,...,Z_K$ i.i.d. standard normal random variables is equal to $0$ (we want to show that they are independent), such that:
$$Cov\left[f(Z_k),g(Z_k)\right]= Cov\left[\sqrt{\hat{\r... |
I'm kind of surprised that I've been unable to find a closed form for the expected number of intersections of $n$ independent standard Brownian motions up to time $t > 0$. I'm well aware of the Karlin-McGregor formula which gives the asymptotics of non-intersecting Brownian motions as $t\to\infty$. I have been looking ... |
Suppose I had to choose 5 people from 8 engineers, 7 scientists and 6 mathematicians. How many ways can I choose at least one person from each profession?
By brute force total number of selection possible $n$:
$$n = \binom{8}{1}\binom{7}{1}\binom{6}{3}+\binom{8}{1}\binom{7}{3}\binom{6}{1}+\binom{8}{3}\binom{7}{1}\... |
There is "Sufficient Condition Theorem" to test if a function's Laplace transform exists.
**If a function $f$ is
piecewise continuous on $[0, \infty)$ and is of exponential order $\alpha$, then the Laplace transform of $f$ exists
for $s>\alpha$.**
I read a document stating that this sufficient condition may not... |
If it is correct to claim that the function $\frac{1}{\sqrt t}$ is not continuous? |
I know there is a fairly elementary way to factor $n = pq$ (distinct primes) given the value of the totient function, but is there an efficient way to find this if we know $n = pqr$ and we know $\phi(n) = (p-1)(q-1)(r-1)$? |
They can't be abelian.
What about products? For any group $G,$ $$G×D_4,G*D_4.$$
Direct and free, respectively. Don't think you can always do a semi-direct or a wreath; but sometimes.
Also, many dihedral groups contain embedded $D_4$'s. |
Let $p_1,p_2,p_3$ be distinct prime numbers and consider $n \in \mathbb N$ . Find the number of functions $f : \{ 1,2,...,n \} \rightarrow \{p_1,p_2,p_3 \} $ for which the number $f(1)f(2)...f(n)$ is a perfect square.
My attempt : If $n$ is an odd number, it is obvious that there are no functions, because it is esse... |
Assume we have a subset of measure zero $N$ in the Borel-$\sigma$-field $\mathcal{B}([a,b])$.
I think that $[a,b]\setminus N$ is a dense set. But how to prove that?
My attempt:
let $x \in [a,b]$ and $\varepsilon>0$ arbitrary then $\lbrace z \in [a,b]: |x-z|<\varepsilon\rbrace \cap ([a,b]\setminus N)\neq \em... |
Complement of a set of measure zero is dense in $[a,b]$? |
Let $(M,g)$ be a Riemannian manifold and $X\in\Gamma(TM)$ a vector field. We choose local coordinates $x^{i}$ on $M$. My question:
> Does the notation $X_{;ij}^{k}$ mean the components of $\nabla_{\partial_{j}}\nabla_{\partial_{i}}X$ or of the double covariant derivatice $\nabla^{2}_{\partial_{j},\partial_{i}}X:=\na... |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
**********************
*Intro__________*
On a [previous question](https://math.stackexchange.com/q/4838311/909869) I found that the following approximation could be used:
$$\dfrac{\ln\left(1+x^{\ln(2)}\right)}{\... |
How can I convince my teacher that he is wrong in this task? We were given the following task and my teacher says I have to use a z-test. In my opinion I have to use a t-test.The task is as follows:
A company installs a nuclear power blocker into the power grid to improve the health of its employees. Before installa... |
They can't be abelian.
What about products? For any group $G,$ $$G×D_4,G*D_4.$$
Direct and free, respectively. Don't think you can always do a semi-direct or a wreath; but sometimes.
Also, many dihedral groups contain embedded $D_4$'s. As does $S_n,n\ge4.$ |
In [this quetion][1], it has been proved that an operator is semisimple iff it's diagonalizable over the algebraic closure of the base field.
However, when the minimal polynomial is not separable, it seems that there's something going wrong. Consider operator on $\mathbb F_2(t)$ to be
$$
T=\begin{bmatrix}
&&1&\... |
A semisimple matrix that is not diagonalizable over algebraic closure? |
In [this answer][1] the point is made that since a unitary and symmetric matrix $U$ commutes with all $U^j$ (I assume those are the matrix powers $U^2$, $U^3$ etc?) it must also commute with $S=\sqrt{U}$, where $\sqrt{\cdot}$ denotes the matrix squre root. I do not understand this argument. The matrix square root is no... |
This is the subtraction counterpart to my previous universal algebra question on addition with a nonzero constant, here: https://math.stackexchange.com/questions/1648926/no-simplifying-identities-for-any-single-nonzero-number-under-addition. I know that the structure $(\mathbb{R};-)$ of the binary operation of subtract... |
Are there any new identities when we go from subtraction to subtraction with a nonzero constant? |
I spent some time learning about the standard proof of Goodstein theorem using ordinals, and its strength. In order to remove the question from the unanswered list I will give an answer. I would appreciate feedback if my understanding of the concepts is erroneous.
The answer to the first questions is yes, it can be ... |
If we defined $f(x)= \sum_{m=0}^{\infty} x^{2^m} = \lim_{n\to\infty}f_n(x)= \lim_{n\to\infty}\sum_{m=0}^{n} x^{2^m}$.
It's easy to see that $|{f(x)}|$ converge in the interval (-1,1). (We can easily born it with the geometric series)
Supose I want to look at the limit :
$$\lim_{x\to1^-}|f(x)|$$
**How could a p... |
[Statement 1.7 from Fractal Geometry by Falconer][1]
[1]: https://i.stack.imgur.com/SHAKJ.png
I am reading through Fractal Geometry Mathematical Foundations and Analysis by Falconer and I am not very familiar with measure theory.
While reading through equation 1.7, I have been stuck trying to figure out if... |
Let $p_1,p_2,p_3$ be distinct prime numbers and consider $n \in \mathbb N$ . Find the number of functions $f : \{ 1,2,...,n \} \rightarrow \{p_1,p_2,p_3 \} $ for which the number $f(1)f(2)...f(n)$ is a perfect square.
*My attempt* : If $n$ is an odd number, it is obvious that there are no functions, because it is es... |
They can't be abelian.
What about products? For any group $G,$ $$G×D_4,G*D_4.$$
Direct and free, respectively. Don't think you can always do a semi-direct or a wreath; but sometimes.
Also, many dihedral groups contain embedded $D_4$'s (some contain more than one copy of $D_4$).
$S_n,n\ge4$ contai... |
This problem was in a competition for a job. It seems simple BUT the challenge is you cannot use trigonometry.\
*Let there be 3 squares with side length of l arranged in such a way that it forms a rectangle with length 3l and width l.\
So ABCD,CEHD and EFGH are squares*(this is notation because it's easier to tell yo... |
You could use ordinary generating functions, and find the coefficient of $x^5$ in the expansion of
$\left[\binom81x+\binom82x^2+\binom83x^3\right]\left[\binom71x + \binom72x^2+\binom73x^3\right]+\left[\binom61x+\binom62x^2 + \binom63x^3\right]$
which [evaluates][1] to the expected $14140$
Note that we have re... |
This problem was in a competition for a job. It seems simple BUT the challenge is you cannot use trigonometry.
> Let there be 3 squares with side length of $\ell$ arranged in such a way that it forms a rectangle with length $3\ell$ and width $\ell$. So $ABCD$, $CEHD$, and $EFGH$ are squares. (This is notation becaus... |
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... |
Sometimes I see questions like this:
$$\frac{dy}{y}=2dx$$ where $y = 1$ when $x = 0$
My book tells me to do this:
$$\int_1^y\frac{dy}{y}=\int_0^x2dx$$
At first I thought this was a clever way of saying 0=0 (as an integral from $a$ to $a$ is always zero) to solve the problem.
Then I saw this question:
> One meth... |
How can you set bounds algebraically for integral equations? |
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