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[![enter image description here][1]][1]
I was seeing this interesting generalization, since it will show the case when the sequence is decreasing, that is, when alpha $= 0$, I understand almost the entire proof except for the last part, I do not understand how to get the fact that $r_n = b_n - s_n$ with the fact t... |
$(y+zq)^2=z^2(1+p^2+q^2)$.
How to find the complete integral of the given pde by using Charpit's equations ?
Charpit's equations are:
$$ \frac{dx}{2pz^2}=\frac{dy}{2qz^2-2z(y+zq)}=\frac{dz}{2p^2z^2-2qz(y+zq)}=\frac{dp}{-2pz(1+p^2+q^2)}=\frac{dq}{2(y+zq)-2qz(1+p^2+q^2)} $$ |
I understand that this is a consequence of a related, more general theorem for non-unital $C^*$-algebras, where we can find approximants of a unit, but for the simple case of a unital $C^*$-algebra $\mathscr{A}$ , what is the simple argument that implies that positive linear functionals $\varphi:\mathscr{A} \rightarrow... |
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological manifold and $E$ is another topological manifold with a surjective continuous map $\pi:E\to M$.Then we say $E$ is a vector ... |
Want to know if there are analytic solutions to two equations around $x_0$ and what method is used to compute them? |
>A real-valued function $f$ defined in $(a, b)$ is said to be convex if $f( tx+ (1 - t)y) \leq tf(x) + (1 - t)f(y)$
whenever $a < x < b, a < y < b, 0 < t < 1$
I want to Prove that every convex function is continuous and I had an idea that can be used to prove this statement but my proof writing skills is not the... |
I know that the sum of powers of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\frac{n(n+1)}{2}$. The idea is that we replicate the set and put it in a rectangle, hence we can do the trick. What is ... |
Fitting / drawing a closed curve of a given length within given dimensions.
I'm trying to fit / draw a given length of a closed curve within given dimensions.
**Example:** Curve **length=941mm**, **width=3mm**, and **height=5mm** and it must fit within the dimensions **x=250mm** **y=210mm**
I was thinking of ... |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
Say I have $ A \succcurlyeq 0 $, $ \begin{bmatrix}A & \vec{b}_1 \\\ \vec{b}_1^T & t_1 \end{bmatrix} \succcurlyeq 0 $, and $ \begin{bmatrix}A & \vec{b}_2 \\\ \vec{b}_2^T & t_2 \end{bmatrix} \succcurlyeq 0 $.
Is $ \begin{bmatrix}A & [\vec{b}_1, \vec{b}_2] \\\ [\vec{b}_1, \vec{b}_2]^T & diag([t_1, t_2]) \end{bmatrix} \... |
Is this combination of positive semidefinite matrices positive semidefinite? |
For any fixed $N \in \mathbb{N}$, let $\mathcal{S}(\mathbb{R}^N)$ be the Schwartz space.
Then, it is well-known that $\mathcal{S}(\mathbb{R}^N)$ is a Frechet space with the seminorms:
\begin{equation}
\lVert f \rVert_{n, \alpha} := \sup_{ x \in \mathbb{R}^n } (1 + \lvert x \rvert^n) \lvert \partial^\alpha f \rvert... |
I have a real-world acoustics problem that I'm trying to solve for some work related R&D. Hopefully you big brained folks can explain down to me. :) I'm good at visualizing math concepts, but I lack any substantial experience, save perhaps g/FooBar. So, sans elegance:
<br>
Let's start with a basic open acoustic spa... |
How do I triangulate a point from 3 segments, rather than from 3 points? |
For any fixed $N \in \mathbb{N}$, let $\mathcal{S}(\mathbb{R}^N)$ be the Schwartz space.
Then, it is well-known that $\mathcal{S}(\mathbb{R}^N)$ is a Fréchet space with the seminorms:
\begin{equation}
\lVert f \rVert_{n, \alpha} := \sup_{ x \in \mathbb{R}^N } (1 + \lvert x \rvert^n) \lvert \partial^\alpha f \rvert... |
First of all, the assumption that $M$ and $E$ are manifolds are too strong: There are many reasons to consider vector bundles over topological spaces which are not manifolds. Second, once you assume that $M$ is a topological manifold, Property 2 will imply that $E$ is a manifold as well. Thus, in what follows, I will n... |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
First of all, the assumption that $M$ and $E$ are manifolds is too strong: There are many reasons to consider vector bundles over topological spaces which are not manifolds. Second, once you assume that $M$ is a topological manifold, Property 2 will imply that $E$ is a manifold as well. Thus, in what follows, I will no... |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
I have been working for this problem for a while:
$ln(x) = 3(1-1/t)$
and by graphing, plugging and chugging values and rigorously doing the math, I can clearly see that one of the values that satisfy this condition is x = 1.
However, when I put this into wolfram alpha and Desmos, I see two answers, one that is... |
How to solve ln(x) = 3(1-1/x)? |
I am only allowed to use the fact that a certain function {${x_i}$} that is not equivalent to the zero sequence is cauchy to prove that the reciprocal {$\frac{1}{x_i}$} is also cauchy.
i.e. we know that $\forall \epsilon > 0$, $\exists\,N_{\epsilon} \in \mathbb N$ s.t. if $i, j > N_{\epsilon}$, then $|x_i - x_j| < \e... |
Can someone verify if my proof is accurate? |
The definition of Fourier Integral seems wrong in my textbook.
The definition in book goes like this,
If $f(x)$ is continuous and $\int_{-l}^{l}\left | f(x) \right |dx$ converges to zero as $l$ approaches to infinity, then $f(x)$ can be represented by $$f(x) = \int_{0}^{\infty} \left [ A(\omega)cos\, \omega x + B(\... |
First of all, the assumption that $M$ and $E$ are manifolds is too strong: There are many reasons to consider vector bundles over topological spaces which are not manifolds. Second, once you assume that $M$ is a topological manifold, Property 2 will imply that $E$ is a manifold as well. Thus, in what follows, I will no... |
The definition of Fourier Integral seems wrong in my textbook.
The definition in book goes like this,
If $f(x)$ is continuous and $\int_{-l}^{l}\left | f(x) \right |dx$ converges to zero as $l$ approaches to infinity, then $f(x)$ can be represented by $$f(x) = \int_{0}^{\infty} \left [ A(\omega)\cos\, \omega x + B(... |
I am only allowed to use the fact that a certain sequence {${x_i}$} that is not equivalent to the zero sequence is cauchy to prove that the reciprocal {$\frac{1}{x_i}$} is also cauchy.
i.e. we know that $\forall \epsilon > 0$, $\exists\,N_{\epsilon} \in \mathbb N$ s.t. if $i, j > N_{\epsilon}$, then $|x_i - x_j| < \e... |
How to create row/column selection matrix from a binary vector? |
Let T be a torus (namely, topology homeomorphic to quotient topology $\mathbb{R}/\mathbb{Z}$). Intuitively it makes sense to say that tangent space of torus at identity (i.e. lie algebra of 1D torus) is $\mathbb{R}$ since we can geometrically think of tangent space as straight line passing through the torus at identity... |
So I have this question here:
>Determine the values of the constant $q$ so that the series $$\sum_{n=3}^{\infty}(-1)^{n-1}\frac{1}{n\sqrt{\ln^q(n)}}$$ is absolutely convergent, conditionally convergent, or divergent.
I assume I can just approach this directly.
For absolute convergence, we have:
$$\sum_{n=3... |
Say I have $ A \succcurlyeq 0 $, $ \begin{bmatrix}A & \vec{b}_1 \\\ \vec{b}_1^T & t_1 \end{bmatrix} \succcurlyeq 0 $, and $ \begin{bmatrix}A & \vec{b}_2 \\\ \vec{b}_2^T & t_2 \end{bmatrix} \succcurlyeq 0 $.
Is $ \begin{bmatrix}A & [\vec{b}_1, \vec{b}_2] \\\ [\vec{b}_1, \vec{b}_2]^T & \operatorname{diag}([t_1, t_2]) ... |
Say I have $A \succcurlyeq 0$, $ \begin{bmatrix}A & \vec{b}_1 \\\ \vec{b}_1^T & t_1 \end{bmatrix} \succcurlyeq 0 $, and $ \begin{bmatrix}A & \vec{b}_2 \\\ \vec{b}_2^T & t_2 \end{bmatrix} \succcurlyeq 0 $.
Is $ \begin{bmatrix}A & [\vec{b}_1, \vec{b}_2] \\\ [\vec{b}_1, \vec{b}_2]^T & \operatorname{diag}([t_1, t_2]) \e... |
Let $X$ be an $n \times n$ random matrix with *i.i.d. standard Gaussian entries*, i.e., the entries $X_{ij}$ are independent standard normal random variables. Given any deterministic unit vectors $u, v \in \mathbb R^n$, calculate the *mean and variance* of $Z_n := u^\top X^\top Xu + \sqrt n v^\top Xv$, and prove the ce... |
OP is asking for clarification about the contents of some text books , hence it is a valid query.
The hitch here is that these are not mainstream text books with mainstream thinking : hence there are not many comparable text books where we can get clarifications.
With that , these are **my equally non-mainstrea... |
Let $N$ be a von Neumann subalgebra of the von Neumann algebra $M$. Is the type of $N$ at most the type of $M$? |
I know this has probably been asked 100 times, but I couldn't find anything resembling this and I've been on this easy number for 2 hours... My work is below.
We want to show that $\forall \epsilon >0$, $\exists N$ such that $\forall n > N$, then $|x_n-2|<\epsilon$.
$$
|\frac{4n^2-3n}{2n^2−1}-2|=|\frac{4n^2-3n-4n^... |
I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$.
Can someone help me?
$$
K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 \sin^2t}}
$$
Results so far:
$$
K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 \sin^2t}} = \int^{\frac{\pi}{2}}_0 (1-k^2 \sin^2t)^{-\frac{1}{2}}dt... |
Without loss of generality, we can consider that $n\geq 2$.
Consequently, one has that
\begin{align*}
\left|\frac{4n^{2} - 3n}{2n^{2} - 1} - 2\right| & = \left|\frac{3n - 2}{2n^{2} - 1}\right| \leq \frac{3n + 2}{2n^{2}} \leq \frac{3n + n}{2n^{2}} = \frac{2}{n} < \frac{2}{n_{\varepsilon}} < \varepsilon
\end{alig... |
Without loss of generality, we can consider that $n\geq 2$.
Consequently, one has that
\begin{align*}
\left|\frac{4n^{2} - 3n}{2n^{2} - 1} - 2\right| & = \left|\frac{3n - 2}{2n^{2} - 1}\right| \leq \frac{3n + 2}{n^{2}} \leq \frac{3n + n}{n^{2}} = \frac{4}{n} < \frac{4}{n_{\varepsilon}} < \varepsilon
\end{align*... |
Here's the problem I am trying to solve:
[![enter image description here][1]][1]
I get that the second column would have an independent variable (as it would be a pivotal column) and the third and first columns would have nonpivotal columns (as they could be expressed as free variables).
But what's next? How to ... |
Given general solution of a matrix, how to reconstruct its RREF form? |
If we have a $2$-state model (i.e. the simplest non-trivial example) in a HMM, and some generated observation-data $\mathcal{O}$ from the algorithm for generating observations, is it always the case that the maximum of a given family $\Theta = \{\theta_i\}_{i \in I}$ of parameters, where $I$ is some index-set, if the t... |
In Garfako's *Modern Fourier Analysis* I came across the following inequality:
$$
\left| \left| f \right|-\underset{Q}{\mathrm{Avg}}\left| f \right| \right|\le \left| f-\underset{Q}{\mathrm{Avg}}f \right|+\underset{Q}{\mathrm{Avg}}\left| f-\underset{Q}{\mathrm{Avg}}f \right|,
$$
where
$$
\underset{Q}{\mathrm{Av... |
Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\sqrt{r-1}}$ for $r>1$, where $M^s$ denotes the amplification of the $\mathrm{II}_1$ factor $M$. This notation is consis... |
If we have a $2$-state model (i.e. the simplest non-trivial example) in a HMM, and some generated observation-data $\mathcal{O}$ from the algorithm for generating observations, is it always the case that the maximum of a given family $\Theta = \{\theta_i\}_{i \in I}$ of parameters, where $I$ is some index-set, if the t... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
Let's say I have $f: \mathbf{R}^{2} \to \mathbf{R}$, and I want to know if $\lim_{(x,y) \to (0,0)} f(x,y)$ exists.
Suppose I substitute $x$ and $y$ for their polar forms $r\cos(\theta )$, $r\sin(\theta )$.
Then, when analysing $\lim_{r \to 0} f(r, \theta)$, I get some $\theta$-dependent trigonometric function (... |
I need to find the volume of $K$ bounded by:
$0\leq z\leq 10-x^2-y^2 $, $x+1-y^2 \geq 0$ and $x+y^2-1 \leq 0$. I dont really understand how to go about this problem. What is the projection on the $xy$-plane?. My first idea was:
$$\int \int \int_K dzdydz = \int \int_D ( \int_0^{10-x^2-y^2}dz)dxdy = \int \int_D(10-x^2-... |
I need to find the volume of $K$ bounded by:
$0\leq z\leq 10-x^2-y^2 $, $x+1-y^2 \geq 0$ and $x+y^2-1 \leq 0$. I dont really understand how to go about this problem. What is the projection on the $xy$-plane?. My first idea was:
$$\iiint_K dzdydz = \iint_D ( \int_0^{10-x^2-y^2}dz)dxdy = \iint_D(10-x^2-y^2)dxdy.$$ But ... |
Set $$P/L=L/B$$.
Assume this is rational. If so, we can find coprime integers $P$ and $L$ to satisfy this equation, so that $P/L$ is a fraction in lowest terms
Recall that
$$2L+B=P$$
Subtracting $2L$ from both sides, we get
$$B=P-2L$$
Substituting into the equation above:
$$\frac P L=\frac{L}{P-2L... |
Let $m^*$ be an outer measure on $\mathbb{R}$, i.e. for $A\subseteq\mathbb{R}$ then:
$$
m^*(A):=\inf\left\{\sum_{n=1}^{\infty}\mathscr{l}(I_n): I_n\text{ is an open interval and } A\subseteq \bigcup_{n=1}^{\infty}\ I_n\right\}.
$$
Define a semi-metric $d: \mathscr{P}(\mathbb{R}) \times \mathscr{P}(\mathbb{R})... |
By using the Maclaurin series $K(k)==\frac{\pi}2\sum_{n=0}^\infty c_n ^2 k^{2n}$ where $c_{n}={2n\choose n}2^{-2n}$ we have
$$\int_0^{\frac{\pi}2}K(\sin t)dt\\
=\frac{\pi}2\sum_{n=0}^\infty c_n ^2\int_0^{\frac{\pi}2}\sin^{2n}t\,dt\\
=\frac{\pi^2}4\sum_{n=0}^\infty c_n ^3.
$$
But, how can I show that $$\int_0^{\fra... |
Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\sqrt{r-1}}$ for $r>1$, where $M^s$ denotes the amplification of the $\mathrm{II}_1$ factor $M$. This notation is consis... |
When the equivalent relaton $x\sim y \iff f(x)=f(y)$ gives a foliation? |
Let me take an arbitrary function f which is continuous and differentiable wherever defined. <br> According to the first principle, $f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$.
Also, $f'(x)=\lim_{h\to 0} \frac{f(x)-f(x-h)}{h}$. <br>
However, if I were to replace $x$ with $x+h$ and say $f'(x+h)=\lim_{h\to 0} \frac{... |
For each ${n}$, let ${(V_n,E_n)}$ be an Erdös-Renyi graph on ${n}$ vertices with parameter ${1/2}$ (we do not require the graphs to be independent of each other).
If ${|T_n|}$ is the number of triangles in ${(V_n,E_n)}$, (i.e. the set of unordered triples ${\{i,j,k\}}$ in ${V_n}$ such that ${\{i,j\}, \{i,k\}, \{j,k\}... |
Jacobi and Madden found infinitely many primitive solutions to,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4$$
using an elliptic curve. We will use a different approach that, like the method for,
$$a^4+b^4+c^4 = d^4$$
discussed in [this post][1], also yields **pairs** of solutions.
---
**I. The system**
Define,... |
For each ${n}$, let ${(V_n,E_n)}$ be an Erdös-Renyi graph on ${n}$ vertices with parameter ${1/2}$ (we do not require the graphs to be independent of each other).
If ${|T_n|}$ is the number of triangles in ${(V_n,E_n)}$, (i.e. the set of unordered triples ${\{i,j,k\}}$ in ${V_n}$ such that ${\{i,j\}, \{i,k\}, \{j,k\}... |
Consider a bookstore that has an inventory of used books $U = \{\text{'To Kill a Mockingbird', '1984'}, \ldots\}$, and an inventory of new books $N = \{\text{'Project Hail Mary', 'The Midnight Library'}, \ldots\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $U'$... |
What is $\sin$? What is $\sin(57.94°)$? According to the idea to which OP refers$$\sin(57.94°):=\frac{MH'}{OM}=\frac{OH}{OM}\simeq \frac{8.5}{10}=0.85$$
[![enter image description here][1]][1]
But as the drawing suggests, his idea is just a *restriction* of a more general idea of what $\sin$ really is. For example, $... |
Consider an asymmetric random walk $(X_n)$ in which the initial point is one ($X_0 = 1$). It increases by $a$ with a probability of 0.3, remains the same with a probability of 0.4, and decreases by $b$ with a probability of 0.3, where $a<b$ and $a,b \in \mathbb{R}$. Also, suppose this process stops once $X_n \geq \lamb... |
With
$g(x) = {\large\int} \ \sin(x) \cos(x) f(x) \ dx$
or
$g(x) = {\large\int}_0^\infty \sin(x) \cos(x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(x) \cos(x)}{f(x) \ + \ a} dx$
or
${\large\int}_0^\infty \frac{\sin(x) \cos(x)}{f(x) \ + \ a} dx$
that may contain... |
What is $\sin$? What is $\sin(57.94°)$? According to the idea to which OP refers$$\sin(57.94°):=\frac{MH'}{OM}=\frac{OH}{OM}\simeq \frac{8.5}{10}=0.85$$
[![enter image description here][1]][1]
But as the drawing suggests, his idea is just a *restriction* of more general ideas of what $\sin$ really is. For example, $$... |
Consider a bookstore that has an inventory of used books $U' = \{\text{`To Kill a Mockingbird', `1984'}, \ldots\}$, and an inventory of new books $N' = \{\text{`Project Hail Mary', `The Midnight Library'}, \ldots\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $U... |
Consider a bookstore that has an inventory of used books $U' = \{\text{`To Kill a Mockingbird', `1984'}, \ldots\}$, and an inventory of new books $N' = \{\text{`Project Hail Mary', `The Midnight Library'}, \ldots\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $U... |
I'm working with Magma to do computations with Jacobians of genus 2 hyperelliptic curves, and I find some behavior for elements that I don't understand. Specifically, let us take the prime $p = 431$ and the finite field $\mathbb{F}_p$, and the curve
$$C : y^2 = x^6 + 305x^5 + 64x^4 + 430x^3 + 367x^2 + 305x + 430.$$
... |
Why does magma give a cubic as the second coordinate of a Mumford representation for genus 2 curves? |
Consider a bookstore that has an inventory of used books $U' = \{\text{`To Kill a Mockingbird', `1984'}, \ldots\}$, and an inventory of new books $N' = \{\text{`Project Hail Mary', `The Midnight Library'}, \ldots\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $U... |
As per daw's comment, the answer is to use Besicovitch space, [described here][1]. This space contains all periodic functions, which can be seen for example by using a Fourier series.
A thing that I overlooked is that the space of all functions for which the mean value of squares
$$
\lim_{T\to\infty}\frac{1}{\math... |
I am trying to find out the Eisenstein integers that have norm values of 3,5,7....
I want to see if there is some pattern.
For the norm value of 3, I was able to find six Eisenstein integers.
For the norm value of 7, I was able to find twelve Eisenstein integers.
I am unable to find even a single Eisenstein in... |
When reading a proof of the sub-additivity property of the outer measure on $\mathbb{R}$, that is:
If $\lbrace A_n \rbrace_{n=1}$ in $\mathscr{P}(\mathbb{R})$ then
$$
m^*\left( \bigcup_{n=1}^{\infty}\ A_n \right) \le \sum_{n=1}^{\infty}\ m^*(A_n)
$$
I see the following sentence at the beginning of the proof:... |
I'm making a game and need a bit of mathematics help.
If I have a circle with a radius of $x$ and there is an equilateral triangle in the center made up of $3$ equal chords. How can I work out the length of the sides of the triangle?
I actually want to split the equilateral triangle into $3$ different isosceles t... |
Bob and Hilal fall in a prison. If both/none of them confess that they stole the money, they will both stay 11 months in prison. If one of them confesses but the other does not, the one who confesses will be free, the one who doesn't will stay for 12 months.
What is the Nash equilibrium in this game?
My thinking:... |
Let $n>1$ be a positive integer, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be continuously differentiable on the interval $[1,n]$. I want to calculate an integral of the form
$$
\int _1^n\lfloor u \rfloor f'(u) \mathrm d u
$$
I realise that I can set upper and lower bounds on the integral by writing $\lfloo... |
# 1D basis functions
In 1D space, I'm looking for a set of [basis functions](https://en.wikipedia.org/wiki/Basis_function) like below. They would be equal to one at certain points and zero at others.
[![Schematic 1D][1]][1]
## 2D schematics
In 2D space, the basis function could be something below, taken fro... |
Let $F$ be a multiplicative generalized cohomology theory.
Let $\sigma^0 \in F^0(pt) \simeq \tilde{F}^0(S^0)$, and let for any $n\in \mathbb{Z}$, $\sigma^n \in \tilde{F}^n(S^n)$ be the element associated to $\sigma^0$ under the suspension isomorphism.
In the famous paper of Atiyah, Bott and Shapiro (Clifford M... |
For each ${n}$, let ${(V_n,E_n)}$ be an Erdös-Renyi graph on ${n}$ vertices with parameter ${1/2}$ (we do not require the graphs to be independent of each other).
If ${|T_n|}$ is the number of triangles in ${(V_n,E_n)}$, (i.e. the set of unordered triples ${\{i,j,k\}}$ in ${V_n}$ such that ${\{i,j\}, \{i,k\}, \{j,k\}... |
I don't get why commutative property is not valid under subtraction, because I find that it is
for example:
$5 - 3 = 2 = -3 + 5$ or rather $5 + (-3) = 2 = -3 + 5$
So how does it not hold true for negative numbers?
|
How does commutative property not hold for subtraction? |
Do these two equations have analytical solutions, and if so, what methods can be employed to find them? |
> People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives there are no two people with the same birthday, but when person X arrives there i... |
## Part 1
The differential forms approach is indeed very powerful. What Hestenes points out in his *From Clifford Algebra to Geometric Calculus* is that to give a complete treatment of differential geometry of manifolds, you need various structures. In the book, you will find an alternative. The starting point (as w... |
Given a finite dimensional real vector space, we can view all the complex structures on it as a subspace of ${\rm GL}(V)$. I wonder if it is a submanifold of ${\rm GL}(V)$. And moreover, given a symplectic form $\omega$, will the $\omega$-compatible (for simplicity, I discard the requirement that $\omega(v,Jv)>0$
fo... |
$\def\fra{\mathfrak{a}}
\def\c{\mathrm{c}}$I found the answer to be quite subtle, for the question brings up the issue that $\fra$-adic completions may not be $\fra$-adically nor $\hat{\fra}$-adically complete (see the containments \eqref{contain}, which in general are strict). You mention in your first “thought” two ... |
This [question][1] states a PMF using combination notation, and the expression is actually equal to a permutation:
$$\frac{\binom{365}{k}k!}{365^k}$$
I have been trying to write permutations using latex, and I find that it is quite awkward.
So my question is: is the math in the question written in combination nota... |
I just want to clarify a point that I haven't seen made explicitly made:
there is no globally defined tautological one-form or symplectic
potential on a closed symplectic manifold (i.e. a compact manifold without
boundary). If there were some $\theta$ (in local coordinates, expressable as $p_i dq^i$) such that $\ome... |
This [question][1] states a PMF using combination notation, and the numerator of the expression is actually equal to a permutation:
$$\frac{\binom{365}{k}k!}{365^k}$$
I have been trying to write permutations using latex, and I find that it is quite awkward.
Question: Is the math written in combinations because tha... |
The domain of consideration is the set of whole numbers, $\mathbb{N}_0$.
The following theorem (see facsimile below) appears in [The Number System, by H.A. Thurston][1]:
>*3. **Theorem:** There is just one operation $\theta$ such that, for every $x$ and $y,$*
\begin{align*}
(i)\text{ }\theta\left(x,0\right)=&x\... |
> People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives there are no two people with the same birthday, but when person X arrives there i... |
There are n! permutations that can be inserted into an initially empty binary tree, but only Catalan(n) different binary trees. A possible method (surely very ineffective, but I'm an old FORTRAN guy, meaning that I automatically hate classes and recursion) to weed out duplicates would be the following: If a left child ... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with $\nabla g = \|\nabla g\|\hat n$. Now let every other point in the pl... |
There are n! permutations that can be inserted into an initially empty binary tree, but only Catalan(n) different binary trees. A possible method (surely very ineffective, but I'm an old FORTRAN guy, meaning that I automatically hate classes and recursion) to weed out duplicates would be the following: If a left child ... |
I have the standard [bump function](https://en.wikipedia.org/wiki/Bump_function) below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
[![Bump function][1]][1]
How can I customize it to be like below:
[![Schematics of desired customization][2]][2]
# Translate and scale
I can translate and sca... |
I have the standard [bump function](https://en.wikipedia.org/wiki/Bump_function) below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
[![Bump function][1]][1]
How can I customize it to be like below:
[![Schematics of desired customization][2]][2]
# Translate and scale
I can translate and sca... |
In p.76 of this book, an example 4 says "Take a strictly decreasing sequence $F_1 \supset F_2 \supset \cdots $ of closed sets in $X$ ".
[Atiyah-Macdonald, Introduction to commutative algebra, p.76][1]
I don't understand if there exists such the sequence.
I think the following statement may be true in general top... |
In p.76 of this book, an example 4 says "Take a strictly decreasing sequence $F_1 \supset F_2 \supset \cdots $ of closed sets in $X$ ".
https://i.stack.imgur.com/3NWcY.jpg
I don't understand if there exists such the sequence.
I think the following statement may be true in general topology.
> Let $(X, \mathcal... |
I am given the following hint: Express $ q $ as a quotient of integers $ m/n $ where $ m,n $ are mutually prime, and show that $ m^2 = 2n^2 $ leads to a contradiction.
Proof solution:
$2$ divides the right-hand side of this equation, hence $2\vert m^2$ ($2$ divides $m^2$). Since $2$ is prime, we must have $2\vert ... |
In p.76 of this book, an example 4 says "Take a strictly decreasing sequence $F_1 \supset F_2 \supset \cdots $ of closed sets in $X$ ".
I don't understand if there exists such the sequence.
I think the following statement may be true in general topology.
> Let $(X, \mathcal{O})$ be an infinite Hausdorff topologi... |
For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite? |
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