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The population OLS coefficient for some $X_i \in \mathbb{R}^d, Y \in \mathbb{R}$ is defined as
$$
\beta =\mathbb{E}[X_iX_i']^{-1}\mathbb{E}[X_iY_i]
$$
and if $X$ is a scalar random variable, the commonly shown formula for simple regression is
$$
\beta = Var(X_i)^{-1}Cov(X_i,Y_i)
$$
But from the more general fir... |
Back in $1895$, before there was the notion of topology, Poincaré had written his definition of manifolds in *Analysis Situs*. The definition (in French) can be found [here](https://analysis-situs.math.cnrs.fr/p-1-Premiere-definition-des-varietes.html). Stated with modern symbols, it goes like this:
> ($\star$) A none... |
I have a question that I need help with.
Suppose you have some integers $k,q$ and $N$ and a **fixed** q-coloring on the edges of $K_N$ such that it contains no monochromatic $K_k$.
Is there a nice way to bound the number of possible monochromatic $K_i$ for $i<k$ and a fixed color?
So for $i=2$ we know every two ve... |
The population OLS coefficient for some $X_i \in \mathbb{R}^d, Y \in \mathbb{R}$ for the model $Y = \beta’X + e$ is defined as
$$
\beta =\mathbb{E}[X_iX_i']^{-1}\mathbb{E}[X_iY_i]
$$
and if $X$ is a scalar random variable, the commonly shown formula for simple regression is
$$
\beta = Var(X_i)^{-1}Cov(X_i,Y_i)
$... |
Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space on $\mathbb{R}^n$ and $C^\infty(S^n)$ be the space of smooth functions on $n$-sphere.
Now fix a point $x \in S^n$ and define
\begin{equation}
C^\infty_x(S^n) := \{ f \in C^\infty(S^n) \mid f(x)=0\}
\end{equation}
Then I heard that $\mathcal{S}(\mathbb{R}^... |
Is it really true that $\mathcal{S}(\mathbb{R}^n)$ is identified with smooth functions on $S^n$ vanishing at a fixed point? |
>Suppose that $\rho$ and $\sigma$ are degree $n$ irreducible representations of a group $G$ over $\mathbb{C}$ and that for every $g\in G,$ there is a matrix $T_g\in GL(n,\mathbb{C})$ depending on $g$ so that $\sigma(g) = T_g^{-1} \rho(g) T_g$. Does there exist $T\in GL(n,\mathbb{C})$ so that $\sigma(g) = T^{-1} \rho(g)... |
For any given $j,k\in\mathbb{N}$, with $j\leq k$, how is it possible to prove the following equivalence?
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}
$$
My attempt is:
$$
\frac{2^j(2k-j)!}{(k-j)!}=\sum_{l=j}^k\frac{2^ll(2k-l-1)!}{(k-l)!}\\
2^j\binom{2k-j}{k-j}k!=\sum_{l=j}^k\binom{2k-l-1... |
**Question:** Given $p\sin^{4}{\theta}-q\sin^{4}{\phi}=p$
and
$p\cos^{4}{\theta}-q\cos^{4}{\phi}=q$, find $\theta$ and $\phi$.
Here is my solution (help improving it would be much appreciated):
$p(1-2\cos^{2}{\theta}+\cos^{4}{\theta})-q(1-2\cos^{2}{\phi}+cos^{4}{\phi})=p$
Subtracting this from the origin... |
Is it possible to simplify/improve this result? (Trig identities, simultaneous equations) |
Is $(\mathbb{Q},+)$ elementary equivalent to $(\mathbb{R},+)$? |
There are two issues with your proof. Firstly, apriori to $\lim\sup$ and $\lim\inf$ of a real sequence existing and being equal, the term "$\lim$" does not make sense. So you should say $\{x:\lim\sup I_{F_{n}}\neq 0\}$. Also note that since this is a sequence of $0$s and $1$'s, the limit exists and is $0$ if and only i... |
Is $(\mathbb{Q},<)$ elementary equivalent to $(\mathbb{R},<)$? |
Assuming that matrix A & B are both square and invertible, is there a reliable way to separate out matrix A and B?
For example, if I want to find matrix A and matrix B:
$$
AB =
\begin{pmatrix}
4 & -4 & -1 \\
5 & 1 & 1 \\
-5 & -1 & -1 \\
\end{pmatrix}
$$
and
$$
BA =
\begin{pmatrix}
5 & 4 & 1 \\
-4... |
Is it possible to find matrix A and matrix B if you know AB and BA? |
Assuming that matrix A & B are both square and invertible, is there a reliable way to separate out matrix A and B from AB and BA?
For example, if I want to find matrix A and matrix B:
$$
AB =
\begin{pmatrix}
4 & -4 & -1 \\
5 & 1 & 1 \\
-5 & -1 & -1 \\
\end{pmatrix}
$$
and
$$
BA =
\begin{pmatrix}
5 &... |
The step $\boldsymbol{\beta}^{t+1}$ in solving an optimization problem
is given below.
$$\boldsymbol{\beta}^{t+1}=\mathrm{argmin}_{\alpha}\frac{1}{2}\left\Vert \mathbf{u}^{t}-\boldsymbol{\alpha}^{t+1}+\mathbf{r}-\mathbf{A}\boldsymbol{\beta}\right\Vert _{2}^{2}+\frac{1}{\rho}\left\Vert \boldsymbol{\beta}\right\Vert ... |
The bounds
$$n^{l(n)^2/108-cl(n)} \le {\rm Perf}(\le n) \le n^{l(n)^2/48 + l(n)},$$
where $c$ is a constant and $l(n)= \log_2(n)$,
are proved in [this paper](https://academic.oup.com/jlms/article-abstract/s2-39/1/67/887064?redirectedFrom=fulltext): "Enumerating Perfect Groups" by D. F.
Holt, J. London Math, Soc. 3... |
> Given
$$p\sin^{4}{\theta}-q\sin^{4}{\phi}=p$$
and
$$p\cos^{4}{\theta}-q\cos^{4}{\phi}=q$$
find $\theta$ and $\phi$.
Here is my solution (help improving it would be much appreciated):
$$p(1-2\cos^{2}{\theta}+\cos^{4}{\theta})-q(1-2\cos^{2}{\phi}+cos^{4}{\phi})=p \tag1$$
Subtracting this from the origi... |
Suppose there is a set of arrays $S= [A_1, A_2, ..., A_k] $, where $A_i$ is a finite subset of $\mathbb{Z}$. Given a positive integer $g$, I want to build $g$ sets $G_1,G_2,\dots,G_g$ such that $\forall i\in[k], \exists j\in[g], A_i \subseteq G_j$ and I want the maximum size of the groups minimized, i.e.,
$$\min \max... |
The bounds
$$n^{l(n)^2/108-cl(n)} \le {\rm Perf}(\le n) \le n^{l(n)^2/48 + l(n)},$$
where $c$ is a constant and $l(n)= \log_2(n)$,
are proved in [this paper](https://academic.oup.com/jlms/article-abstract/s2-39/1/67/887064?redirectedFrom=fulltext): "Enumerating Perfect Groups" by D. F.
Holt, J. London Math, Soc. 3... |
To prove that if matrix $U$ is obtained from $A$ by the row operation $R_i = R_i + kR_j$, then $L$ is obtained from the identity matrix $I_n$ by the reverse row operation $R_i = R_i - kR_j$, such that $A = LU$, we can express $A$ as the product of $L$ and $U$, and then show that the given row operations on $A$ and $I_n... |
I started studying differentials on schemes (Hartshorne II.8), and read Eisenbud to gain more intuition and algebra. After vaguely understanding cotangent bundle and conormal sequence, I tried to calculate fiber of $\Omega_{S/k}$, which I expected to be the cotangent space $m/m^2$.
Let $S=k[x_1,\dots,x_r]/I$ where ... |
Assuming that matrix $A$ & $B$ are both square and invertible, is there a reliable way to separate out matrix $A$ and $B$ from $AB$ and $BA$?
For example, if I want to find matrix $A$ and matrix $B$:
$$
AB =
\begin{pmatrix}
4 & -4 & -1 \\
5 & 1 & 1 \\
-5 & -1 & -1 \\
\end{pmatrix}
$$
and
$$
BA =
\beg... |
Is it possible to find matrix $A$ and matrix $B$ if you know $AB$ and $BA$? |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y\tag{I}$$
I cannot pursue it because of the floating point ... |
I found a question in my textbook but I think the answer provided is wrong.
The question says:
Let $S$ be a relation defined over $\mathbb R$ such that $(a,b) \in S \iff ab≥0$. Is $S$ equivalence?
<br> Book says that it is. <br>But let $x=4, y=0$ and $z=-6$ be three real numbers. Then clearly $(x,y),(y,z)\in S$ but ... |
<h6>Problem Clarification - Solving a Simpler Related Problem</h6>
Given ( with $f^{-1}=\frac{1}{f}$ ) as $f^{-1}$ is the fraction (and not the inverse, as there are two possible conventions for $f^{-1}$ ):
$$f^{-1} = f^{'}
\tag{ Eq. 1}$$
Then solve for $f$.
<h6>Solution in the case that $f^{-1}=\frac{1}{f... |
> Given
$$p\sin^{4}{\theta}-q\sin^{4}{\phi}=p$$
and
$$p\cos^{4}{\theta}-q\cos^{4}{\phi}=q$$
find $\theta$ and $\phi$.
Here is my solution (help improving it would be much appreciated):
$$p(1-2\cos^{2}{\theta}+\cos^{4}{\theta})-q(1-2\cos^{2}{\phi}+\cos^{4}{\phi})=p \tag1$$
Subtracting this from the orig... |
Consider the integral
$$I(t) = \int_0^{1/2} \frac{t dx}{x^{-t} - x^t}$$
in the vicinity of $t=0$.
When $t$ is viewed as a real variable, $I(t)$ is very well-behaved. For example, it's infinitely differentiable.
On the other hand, when $t$ is on the imaginary axis, the integrand is very poorly behaved as a ... |
I need to study the Lyapunov stability of these matrices.
Is there a criterion that does not need to calculate the eigenvalues?
For example if the trace of a matrix is > 0 then it is unstable.
$$
\begin{bmatrix}
-1 & 2 & 0 & -5 & 3 \\
0 & -1 & 1 & 6 & -4 \\
0 & 0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0 & 2 \\
0 & 0 & ... |
Counting the numbers of quantifiers, how are there 4? |
Consider the integral
$$I(t) = \int_0^{1/2} \frac{t dx}{x^{-t} - x^t}$$
in the vicinity of $t=0$.
When $t$ is viewed as a real variable, $I(t)$ is very well-behaved. For example, it's infinitely differentiable.
On the other hand, when $t$ is on the imaginary axis, the integrand is very poorly behaved as a ... |
Let $\mathbb{F}$ be an arbitrary field of characteristic $\mathrm{char}(\mathbb{F})\neq 2$ and $V$ a $\mathbb{F}$-vector space. The definition of a quadratic form I am used to is a map $\varphi\colon V\to\mathbb{F}$ with the property $\varphi(\lambda v)=\lambda^{2}\varphi(v)$ such that
$$\beta_{\varphi}(v,w):=\frac{1... |
Consider a category with two objects $X, Y$. Let
$$\begin{aligned}
\hom(X, X) = \hom(Y, Y) &= \mathbb Z \times \{\text{even numbers}\} \\
\hom(X, Y) = \hom(Y, X) &= \mathbb Z \times \{\text{odd numbers}\}
\end{aligned}$$
Let composition be
$$(d, r) \circ (d', r') = (d + d', r + r').$$
Now construct an involutive... |
This is probably rather trivial, but anything that's combinatorics related messes me up. Just to check: are the following summations identical?
$$\sum_{k_1+k_2+k_3 = n} a^{k_1} b^{k_2} c^{k_3} \quad \text{and} \quad \sum_{0\leq i \leq j \leq n} a^i b^{j-i} c^{n-j}$$
where $k_1,k_2,k_3 \geq 0$. I believe both summ... |
Let $s\in (0, 1)$ and $f\in H^s(\mathbb R^n)$. By definition, $f\in L^2(\mathbb R^n)$ and $[f]_s<+\infty$, where $[\cdot]_s$ denotes the Gragliardo seminorm.
Assume that there is a constant $K>0$ such that $$[f]_s\le K.$$
There is any way to use this information to prove that there exists $c>0$, possibly dependin... |
If $[f]_s\le K$, is $||f||_{L^2}$ bounded by a constant depending on $K$? |
<h6>Problem Clarification - Solving a Simpler Related Problem</h6>
Given ( with $f^{-1}=\frac{1}{f}$ ) as $f^{-1}$ is the fraction (and not the inverse, as there are two possible conventions for $f^{-1}$ ):
$$f^{-1} = f^{'}
\tag{ Eq. 1}$$
Then solve for $f$.
<h6>Solution in the case that $f^{-1}=\frac{1}{f... |
I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. So the branching effect on the "log" is that its imaginary part is $\pi$ on the right and above the point $ai$ on the p... |
This is a question about first countable spaces. Topology of such spaces can be defined in terms of convergent sequences, and many topological properties of such spaces can be expressed in terms of sequences by taking general theorems about nets and replacing the term "net" with the term "sequence". For example: A set ... |
From the book "Gentle Introduction to Art of Mathematics":
>How many quantifiers does this have and what kind?
>
> "Everybody has some friend that thinks they know everything about a sport.”
The answer given is "four", but no explanation has been provided.
I think that "everybody" implies Universal and "some... |
Consider a category with two objects $X, Y$. Let
$$\begin{aligned}
\hom(X, X) = \hom(Y, Y) &= \mathbb Z \times \{\text{even numbers}\} \\
\hom(X, Y) = \hom(Y, X) &= \mathbb Z \times \{\text{odd numbers}\}
\end{aligned}$$
Let composition be
$$(d, r) \circ (d', r') = (d + d', r + r').$$
Now construct an involutive... |
Let $s\in (1/2, 1)$ and $f\in H^s(\mathbb R^n)$. By definition, $f\in L^2(\mathbb R^n)$ and $[f]_s<+\infty$, where $[\cdot]_s$ denotes the Gragliardo seminorm.
Assume that there is a constant $K>0$ such that $$[f]_s\le K.$$
There is any way to use this information to prove that there exists $c>0$, possibly depend... |
If $s\in (1/2, 1)$ and $[f]_s\le K$, does $||f||_{L^2}$ happen to be bounded by a constant depending on $K$? |
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by the basis $B = \{ X_d : d \mid p_n\#\}$ and $X_d = \{ x \in \Bbb{Z}/p_n\#: x^2 \neq 1 \pmod q, \forall \text{ primes } q \mid d\}$. Then clearly $X_c \cap X_d = X_{\text{lcm}(c,d)}$.
The full space $\Bbb{Z}/p_n\#$ is given vacuously by $X_1 = \{... |
Why is this minimizer obvious? |
Let $\mathbb{H}_+\subseteq \mathbb{C}$ be the upper half plane,and $\bar{\mathbb{H}}_+=\mathbb{H}_+\cup \mathbb{R}$. Then consider the following equation:
$$z=-\frac{1}{m}-m+\frac{a^2}{ca^2m+1}.$$
Show that for each fixed $a,c\in \mathbb{R}_+$ and $z\in \bar{\mathbb{H}}_+$, there is a unique $m=m(z)\in \bar{\mathbb... |
So I have a graph-related problem, where I need to determine **the number of paths** of length $L$ from Node $N_A$ to node $N_B$, in a **complete graph** containing $N$ nodes. Nodes $N_A$ and $N_B$ can point to the same node.
An example application of this problem is finding the number of words of length $L$, starti... |
I'm trying to find base of
$span\{(1,0,1), (2,1,0)\} \ \ \cap \ \ \ \{(x,y,z) \in \mathbb{R}^{3}: x-y+z=0 \} \subset \mathbb{R}^{3} $
What I've done is:
1. Found the subspace/base of the $x-y+z=0$. That is vectors e.g. $(0,1,1), (-1,0,1), (2, -1, -1)$. Checked that they are linearly independent:
$$\begin{bma... |
Why are column having no leading entry are called free variable? |
Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$
I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$
I use $a+b+c\ge \sqrt{3(ab+bc+ca)}=3$ and we need to prove $$a^2b+b^2c+c^2a+abc\ge 4.$$
But it is wrong when $c=0,b=3,a=1.$
I got stuc... |
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by the basis $B = \{ X_d : d \mid p_n\#\}$ and $X_d = \{ x \in \Bbb{Z}/p_n\#: x^2 \neq 1 \pmod q, \forall \text{ primes } q \mid d\}$. Then clearly $X_c \cap X_d = X_{\text{lcm}(c,d)}$.
The full space $\Bbb{Z}/p_n\#$ is given vacuously by $X_1 = \{... |
> Found the subspace/base of the $x-y+z=0$. That is vectors e.g. $(0,1,1), (-1,0,1), (2, -1, -1)$. Checked that they are linearly independent:
$$\begin{bmatrix}0 & 1 & 1\\-1 & 0 & 1\\ 2 & -1 & -1\end{bmatrix} = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}$$ so I guess that is the base (and subspace tha... |
Proof using infinite descent or otherwise? |
Let $\mathbb{H}_+\subseteq \mathbb{C}$ be the upper half plane,and $\bar{\mathbb{H}}_+=\mathbb{H}_+\cup \mathbb{R}$. Then consider the following equation:
$$z=-\frac{1}{m}-m+\frac{a^2}{ca^2m+1}.$$
Show that for each fixed $a,c\in \mathbb{R}_+$ and $z\in \bar{\mathbb{H}}_+$, there is a unique $m=m(z)\in \bar{\mathbb... |
I am curious what the object you obtain is when you cut the $d$-dimensional flat torus $\mathbb{T}^d :=\left(\mathbb{R} / \mathbb{Z}\right)^d$ along certain hyper-planes. Specifically, if you identify it with $[0,1)^d$ and consider a typical element $x=(x_1,\dots,x_d)$, I would like to cut it along the planes $x_i=x_j,... |
Cutting the $d$-dimensional torus? |
I am wonder why we say $E$ is subspace of $E^{**}$，do the elements of $E$ really belong to $E^{**}$？
I know the definition of canonical injection $J:E\to E^{**}$，$J$ is linear and $J$ is an isometry.
Another question: if $Y$ belongs to $E^{**}$ and $Y=J(X) $, can we say that $Y$ belongs to $E$? |
Let's call a map closed if it takes any closed subset to a closed subset.
I'm wondering if there are some "standard" examples of closed (in this sense) linear maps on a Banach space, other than the zero map or a linear homeomorphism. More precisely:
**Question:** What is an example of a bounded linear map $T\colo... |
Solve $3a^2 + 3ab + b^2 - p^3 = 0$ using infinite descent or otherwise? |
It is well known that if $X$ is a Kahler manifold, then $$\bigoplus_{p+q=k}\mathcal{H}^{p,q}=\mathcal{H}^{k}(X,g)_{\mathbb{C}}=\mathcal{H}^{k}_{\overline{\partial}}(X,g)=\mathcal{H}_{\partial}^{k}(X,g)$$
If one loosens the restriction that $X$ is Kahler to simply being Hermitian, are there any quick counter examples t... |
Let $H$ be a Hilbert space, and consider some linear operators $ A, B: H\rightarrow H $. In functional analysis, I knew that if $A$ and $B$ are both bounded, then the composition $AB$ is bounded.
I'm curious about the situation when $A$ and $B$ are unbounded. Now, is $AB$ also unbounded or not? I tried to prove it ... |
The composition of two unbounded linear operator is also unbounded or not? |
Let $H$ be a Hilbert space, and consider some linear operators $ A, B: H\rightarrow H $. In functional analysis, I knew that if $A$ and $B$ are both bounded, then the composition $AB$ is bounded.
I'm curious about the situation when $A$ and $B$ are unbounded and linear. Now, is $AB$ also unbounded or not? I tried t... |
I've been unable to get over the hurdle of understanding this concept. However, I'm willing to bet that once I finally understand it will all seem obvious.
I do understand that the real part of the curve divides the complexification of the curve into two halves.
What I've been unable to observe is that the two c... |
$f(x)=f(x+T),T\notin \mathbb Q,\int_0^T f(x)\mathrm dx=-T$. Does ther exist a function $f$ such that $sup \sum_{k_1<i\le k_2}f(x+i)=+\infty$? |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
Any assured method for finding the derivative of p Euclidian norms? |
The main thing is
> The remainder when $97$ is divided by $7$ should be same as the remainder when $a$ is divided by $7$.
We want to solve $a \equiv 97 (\mod 7)$. Equivalently, $7 \mid (a - 97)$. As you see, $7 \nmid 97$, so $7$ can not divide $a$ (think why?). So $a$ will be in the form of $7k + q$, where $k\in... |
The main thing is
> The remainder when $97$ is divided by $7$ should be same as the remainder when $a$ is divided by $7$.
We want to solve $a \equiv 97 (\mod 7)$. Equivalently, $7 \mid (a - 97)$. As you see, $7 \nmid 97$, so $7$ can not divide $a$ (think why?). So $a$ will be in the form of $7k + q$, where $k\in... |
It is well-known ghe notion of Joyal's combinatorial species. In his treatment, Joyal assumes finiteness. In fact he works with the groupoid of finite sets. The hnderluing idea is to capture gge notkon of a structure on s finite ground set and to work "combinatorially" with. Ok, finiteness is essential for the Joyal's ... |
Let $s\in (1/2, 1)$ and $f\in H^s(\mathbb R^n)$. By definition, $f\in L^2(\mathbb R^n)$ and $[f]_s<+\infty$, where $[\cdot]_s$ denotes the Gagliardo seminorm.
Assume that there is a constant $K>0$ such that $$[f]_s\le K.$$
Is there any way to use this information to prove that there exists $c>0$, possibly dependi... |
Consider the following energy function $$-\sum_{i,j\in[n]}\cos(\theta_i-\theta_j)$$
At any point $\theta$, move it to the point $\theta^*$ such that for any $i,j\in[n]$, $$\theta^*_i-\theta^*_j=t(\theta_i-\theta_j)$$
where $t\in(0,1]$. It is clear that such $\theta^*$ are not unique. Thus, the direction from point ... |
How to verify whether a direction is the steepest descent, in multi-variables case? |
This question is much easier to solve if you use monoids instead of automata.
**Proposition**. The language $L'$ is recognized by the [syntactic monoid](https://fr.wikipedia.org/wiki/Monoïde_syntaxique)${}^*$ of $L$.
It follows immediately that $L'$ is regular, since a language is regular if and only if its syn... |
My answer:
Suppose that f: ℝ $\rightarrow$ ℝ is an even function, that is differentiable everywhere. If f is an even function, then we have that f(x)= f(-x). We now take the derivative using the chain rule
$$f'(x)= \frac{df}{dx}$$
$$=\frac{d}{dx}[f(x)]$$
$$=\frac{d}{dx}[f(-x)]$$
$$= f'(-x)\cdot (-x)'$$
$$= f'(... |
I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent:
> That is, given $a,b,c \sim \operatorname{N}\left(0,1\right)$, what is (a numerical approximation to 5 digits of) $\operatorname{Pr}\l... |
My question is if in a normed space (maybe complete if necessary), a convex (non empty) set with convex complement has non-empty interior.
I cannot think of a counterexample, but neither how to prove it.
Any ideas? |
Convex set with convex complement has non empty interior? |
Consider the following energy function $$-\sum_{i<j\in[n]}\cos(\theta_i-\theta_j)$$ where $\theta_i\in\mathbb{R}$, for $\forall i\in[n]$.
At any vector $\vec\theta$ that is not all-equal vector ($\theta_i=\theta_j,\forall i<j\in[n]$), move it to the point $\theta^*$ such that for any $i,j\in[n]$, $$\theta^*_i-\theta... |
Let $M$ be a von Neumann algebra. If $N_1$ and $N_2$ are two subalgebras of $M$ such that $N_1$ is $*$-isomorphic to $N_2$.
What is the relationship between $A_1:=N_1'\cap M$ and $A_2:=N_2'\cap M$? Can we conclude that $A_1$ is $*$-isomorphic to $A_2$? |
The main thing is
> The remainder when $97$ is divided by $7$ should be same as the remainder when $a$ is divided by $7$.
We want to solve $a \equiv 97 \pmod 7$. Equivalently, $7 \mid (a - 97)$. As you see, $7 \nmid 97$, so $7$ can not divide $a$ (think why?). So $a$ will be in the form of $7k + q$, where $k\in ... |
Consider the polynomial $f(x) = x$, which is certainly a real polynomial. Is there a real polynomial $\bar{f}$ such that $\bar{f}(x) = \bar{x}$? More generally, if both $f$ and $\bar{f}$ satisfying $\bar{f}(x) = \overline{f(x)}$ also individually satisfy the Cauchy-Riemann equations then they have to be constants.
... |
Consider the following energy function $$-\sum_{i<j\in[n]}\cos(\theta_i-\theta_j)$$ where $\theta_i\in\mathbb{R}$, for $\forall i\in[n]$.
At any vector $\vec\theta$ that is not all-equal vector ($\theta_i=\theta_j,\forall i<j\in[n]$), move it to the point $\theta^*$ such that for any $i,j\in[n]$, $$\theta^*_i-\theta... |
See here https://mathoverflow.net/questions/138518/generalization-of-analytic-functors where I was wondering the same
As for why, I don't know the exact answer, but after all this time I know that the features of the category of bijections of finite sets (e.g. being traced) are essential for some of the theory -as a... |
Find the incongruent roots modulo 13 of $x^3$ + 12. By trial and error, I know x=1,3, and 9. But how can I provide a solution where I apply various theorems? I hope someone can help me. |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
Let $\mathbf{N}$ be the natural numbers with the discrete topology (or really, any countable set), and consider the space of natural number sequences $\prod_{\mathbf{N}} \mathbf{N}$ with the product topology (sometimes called "the Baire space"). Is there an "explicit" example of a continuous function $\prod_{\mathbf{N}... |
Essentially infinite continuous maps $\prod_{\mathbf{N}} \mathbf{N} \rightarrow \mathbf{N}$? |
Let $\Lambda$ be a finite-dimensional algebra. The classical Auslander-Reiten translation $\tau$ is defined as $\tau= \mathrm{D}\mathrm{Tr}$ where $\mathrm{Tr}$ is a duality between $\underline{\mathrm{mod}}\;\Lambda$ and $\underline{\mathrm{mod}}\;\Lambda^{\mathrm{op}}$ and $\mathrm{D}$ is a duality between $\mathrm{m... |
Consider $f(x)=\log(x^2)$. Clearly, the domain of $f$ is $D_f=\mathbb{R}-\{0\}$, since $x^2>0$ for any $x<0$. However, by the fundamental properties of logarithms:
$$
f(x)=\log(x^2)=2\log(x)\equiv g(x)
$$
where I abuse notation to give separate names $f$ and $g$ to two equivalent (but different in appearance) e... |
Does the domain of the real logarithm need to be "extended" when dealing with even exponents in the argument? |
Consider the following energy function $$-\sum_{i<j\in[n]}\cos(\theta_i-\theta_j)$$ where $\theta_i\in\mathbb{R}$, for $\forall i\in[n]$.
At any vector $\vec\theta$ that is not all-equal vector ($\theta_i=\theta_j,\forall i<j\in[n]$), move it to the point $\theta^*$ such that for any $i,j\in[n]$, $$\theta^*_i-\theta... |
- There are `16` possible characters (hexidecimal).
- The length of each string is `40`.
- total number of possible combinations or unique strings:
- $16^{40} = 1461501637330902918203684832716283019655932542976$
∴ the probability of generating a unique hash is $1/(16^{40})$ ?
## For a bit more context ... |
For example, modulo $30$ we have that $\gcd(5, 30) = 5$, but $5, 5^2, 5^3 = 5, 5^2, \dots$ goes the list, so both repeat.
We know it's true when $\gcd(a,n) = 1$ because a cyclic group is formed in $\Bbb{Z}/n^{\times}$, s it's sufficient to prove the case when $\gcd(a, n) = d \neq 1$.
Another example:
$$
n... |
For any square-free $n \geq 1$ and $a \in \Bbb{Z}/n$ including $\gcd(a,n) \neq 1$, then in the list $a, a^2, a^3, \dots$ either $a$ or $a^2$ repeats? |
From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them that. Here are some examples:
- The functor $U: \mathcal{Grp} \rightarrow \mathcal{Set}$ that sends each group to its underlying set is usua... |
Are right-adjoints of a forgetful functor reflectors? |
$I,J$ are proper ideals in $R$, an integral domain with unity, and $I$ or $J$ is finitely generated. Can we say $IJ\subsetneq I?$ I would also like to know if we could more generally say something like this for modules over $R$.
The only ideals I could find where the equality $IJ=I$ was achieved were in non-domani... |
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