Saturday, March 20, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

$\mathcal{C}_{3}$ generate Borel $\sigma$-algebra on $\mathbb{R}$

Posted: 20 Mar 2021 08:56 PM PDT

I am aware that $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ generate Borel $\sigma$-algebra on $\mathbb{R}$. However, my goal is to show that $\mathcal{C}_{3}$ also generate Borel $\sigma$-algebra on $\mathbb{R}$

$\mathcal{C}_{1}=\{(a, b) : a, b \in \mathbb{R} \}$,

$\mathcal{C}_{2}=\{(-\infty, d) : d\in \mathbb{R}\}$,

$\mathcal{C}_{3}=\{(x, y) : x \in \mathbb{R} , y \in \mathbb{Z} \}$,

Any help is appreciated!

Are the set of direction ratios for a given line unique?

Posted: 20 Mar 2021 08:50 PM PDT

I perfectly understand the concept of direction cosines for a Vector.

Corresponding to an axis, the direction cosine is just the cosine of the angle that the Vector makes with the positive direction of the axis.

But I have problems in uniquely defining the direction cosines of a Line. According to me there is no way to uniquely specify direction cosines of a Line since it has no unique direction. There should always be 2 sets of direction cosines (of opposite sign).

There was a question in my textbook that read: " If a line is parallel to $-18i+12j-4k$ then what are its direction cosines?"

In the solution they have simply calculated the direction cosines of the Vector given, Why isn't the negative counterpart of the direction cosines also given as the solution?

Is there some convention we have to follow?

If $a^3=20a^2+b^2+c^2-a-340$, $b^3=20b^2+c^2+a^2-b-340$, $a^3=20a^2+a^2+b^2-c-340$, what is value of $abc$?

Posted: 20 Mar 2021 08:48 PM PDT

Question : If $a^3=20a^2+b^2+c^2-a-340$, $b^3=20b^2+c^2+a^2-b-340$, $a^3=20a^2+a^2+b^2-c-340$, what is value of $abc$?

I think I should add this and get close to $abc$, but I can't think about this. I know that answer is 19. Please help me

Maximising log-likelihood to train neural network

Posted: 20 Mar 2021 08:45 PM PDT

I'm quite new to neural network, and I'm trying to train my model with the objective function below.

\begin{align} \frac{1}{N} \sum_{n=1}^{N} \log \sigma (f_{\theta}(x) - f_{\theta}(x')) \end{align}

$f_{\theta}(x)$ and $f_{\theta}(x')$ with weight $\theta$ are the outputs of a neural network, and they can be any scalar values. Is it possible to train the model by maximising the equation?

Prove that there is a non-zero distance between any point outside an open set and any point in a closed-bounded subset (Complex Analysis)

Posted: 20 Mar 2021 08:37 PM PDT

Question: If $S \subseteq \mathbb{C}$ is open and $L \subset S$ is closed and bounded, show that there exists a $\delta$ such that $|z-w|>\delta$ for all $z \in L$ and $w \notin S$

Attempt: I am considering the set $P := K^C \cap S$ which is open as $K^C$ is, so for every element $a$ of $P$, we can find an $r>0$ such that this is the radius of an open ball centered at $a$ contained in $P$. I now want to set $\delta$ to be the minimum of all such $r$ such that $r$ is the maximum for a particular point $a$. However, I am not sure how to show that this set even has a minimum element, and I feel that this probably isn't even the case. I can imagine a set in which $P$ kind of bottlenecks and the set of all $r$ wouldn't have a minimum, but an infimum of $0$.

Spatial covariance decomposition

Posted: 20 Mar 2021 08:32 PM PDT

$\{\mathbf{Z}_{\mathbf{i}}\}_{\mathbf{i} \in Z^{d}}$ is a strictly stationary process, following an isotropic short memory dependence with the covariance matrix given by $\big\{ C_{k,l}(\lVert \mathbf{i}-\mathbf{j} \lVert) \big\}_{k,l=1}^{p+q} $ where $ \forall \mathbf{i}\neq \mathbf{j}$, $$C_{k,l=1}(\lVert \mathbf{i}-\mathbf{j} \lVert)=Cov(<\mathbf{Z_{i}},e_{k}>, <\mathbf{Z_{j}},e_{l}>) =\sigma_{k}\sigma_{l}\exp{(-an\lVert \mathbf{i}-\mathbf{j} \lVert)} $$.

I assume that : $\forall \mathbf{i_{1}},\mathbf{i_{2}}\neq\mathbf{j_{1}},\mathbf{j_{2}} , \forall k,l,t,u \in \{1,2,...,p+q\}$ $Cov(Z_{\mathbf{i_{1}}}^{(k)}Z_{\mathbf{i_{2}}}^{(u)}, Z_{\mathbf{j_{1}}}^{(l)}Z_{\mathbf{j_{2}}}^{(t)})=Cov(Z_{\mathbf{i_{1}}}^{(k)},Z_{\mathbf{j_{1}}}^{(l)})Cov(Z_{\mathbf{i_{2}}}^{(u)},Z_{\mathbf{j_{2}}}^{(t)}) + Cov(Z_{\mathbf{i_{1}}}^{(k)},Z_{\mathbf{j_{2}}}^{(t)})Cov(Z_{\mathbf{i_{2}}}^{(u)},Z_{\mathbf{j_{1}}}^{(l)}).$

I have two questions:

1 - is it necessary to set this condition to control the convergence of

$n^{-d}\sum\limits_{l,t=1}^{p+q}\sum\limits_{\parallel\mathbf{i-j}\parallel > 0}u^{(l)}u^{(t)}Cov(\textbf{Z}_{\textbf{i}}^{(k)}\textbf{Z}_{\textbf{i}}^{(l)}, \textbf{Z}_{\textbf{j}}^{(k)}\textbf{Z}_{\textbf{j}}^{(t)})$, if yes I would like to have an article that can clarify me.

2- I have always seen this assumption of covariance decomposition on error processes [Fransco and fernandez(2007) ] and I don't know if it is plausible for all processes. if yes I would like to have an example in an article or an example from you.

Thank you for your reply.

$L^p(\mu)$ space relations

Posted: 20 Mar 2021 08:30 PM PDT

Let $(X,M, \mu)$ be a measure space. Suppose $1 \leq q < r \leq \infty$.

Suppose $f \in L^p (\mu) \cap L^r(\mu)$. Show that $f\in L^s(\mu)$ for $p \leq s \leq r$.

Attempt:

Suppose $f \in L^p (\mu) \cap L^r(\mu)$. Then $\int |f| ^p d\mu < \infty$ and $\int |f|^r d\mu < \infty$.

Isn't it obvious that $\int |f|^s d\mu < \infty$ already?

I'm bit confused on what to show.

Any help will be appreciated! Thank you!

Sections on locally ringed space as functions

Posted: 20 Mar 2021 08:30 PM PDT

Notation/Introduction:

Let $(X, \mathcal{O}_X)$ be a locally ringed space, $U \subseteq X$ an open and $p \in U$. Denote by $\mathfrak{m}_p \lhd \mathcal{O}_{X,p}$ the unique maximal ideal and $k_p=\mathcal{O}_{X,p}/\mathfrak{m}_p$ the residual field at $p$. Let $$ ev_p : \mathcal{O}_X(U) \to \mathcal{O}_{X,p} \to k_p \quad , \quad \quad p \in U $$ and denote $f(p) = ev_p(f) \in k_p$. Taking the product we define $$ ev = \mathcal{O}_X(U) \to \prod_{p \in U} \mathcal{O}_{X,p} \to \prod_{p \in U} k_{p} \quad, \quad \quad ev(f) = (f(p))_{p \in U} $$

The question is: when $ev$ is injective?

Well, let $f \in \mathcal{O}_X(U)$ such that $f(p) = 0$, $\forall p \in U$, them $f_p = \mathrm{stalk}_p(f) \in \mathfrak{m}_p = \mathcal{O}_{p, X} \setminus \mathcal{O}_{p, X}^{\times}$. Then $f|_V \notin \mathcal{O}_X(V)^{\times}$ for all open $V \subseteq U$. Then what?

In schemes I know that $X$ be reduced is a sufficient condition, but what about the general case of locally ringed spaces? The case of schemes is reduced to affine schemes and the proof relies on commutative algebra (the intersection of all primes is the nil radical), so I can't use those ideas now (I guess).

In some sense, this condition of $ev$ being injective is about understanding the abstract sheaf $\mathcal{O}_X$ as a sheaf of rings of functions (what sounds like a really reasonable question to me).

Trouble finding my error showing the limit of $e^{x^{e^{-x}}}$

Posted: 20 Mar 2021 08:46 PM PDT

I have found a way to proof this, but I would really like to know where the error in my initial approach is, since I am only comfortable when I can use different methods. So I don't need a solution to this problem but a correction of my (false) way. The limit is: $$\lim_{x \to \infty} (e^{x^{e^{-x}}}) = e $$

I tried: $$ a = e^{x^{e^{-x}}}\\ ln(a) = ln (e^{x^{e^{-x}}}) \\ ln(a) = e^{-x} ln(e^x) \\ e^{ln(a)} = e^{e^{-x}ln(e^x)} \\ a = e^{e^{-x}x}  \\ a = e^{\lim_{x \to \infty}\frac{x}{e^x}} \ \ \ (l'hopital) \\ a = e^{\lim_{x \to \infty}\frac{1}{e^x}} \\ a = e^0 = 1$$ I know it's not written down flawlessly, but I hope you get the idea of what I was trying to do..Why is my result 1 and not e?

Show that $v_1$ is the projection of $C$ onto the direction $u_1$ scaled by $1/\sigma_1$.

Posted: 20 Mar 2021 08:28 PM PDT

Let $U, \Sigma, V^T$ be the SVD of the matrix $C$ where the diagonal entries of $Sigma$ are arranged from largest to smallest. Let $U = \begin{bmatrix} u_1 & u_2 & \cdots & u_m\end{bmatrix}$ be the $m$ columns of $U$ and $V = \begin{bmatrix} v_1 & v_2 & \cdots & v_n\end{bmatrix}$ be the $n$ columns of $V$. Show that $v_1$ is the projection of $C$ onto the direction $u_1$ scaled by $1/\sigma_1$.

Proof: By definition, $C = U\Sigma V^T$. Note that $C$ may be rewritten as $$ C = \sum\limits_{i=1}^n \sigma_i u_i v_i^T$$ where $\sigma_i$ are the eigenvalues along the diagonal of $\Sigma$. I want to show that $proj_{u_1} C = \dfrac{1}{\sigma_1}v_1$.

It's been a minute since I have done linear algebra. How would I find such projection for this problem?

Intuition about non-existence of non-integrable functions.

Posted: 20 Mar 2021 08:35 PM PDT

Recently in the class of integration our teacher told us that some functions like $e^{x^2}$ are non integrable , so it is not possible to obtain a function which is it's anti-derivative. But we know that integration is area under the curve so for a smooth function like $e^{x^2}$ I have an intuitive feeling that the graph of area under the curve be another smooth curve. Also every curve on the graph represents a function so isn't there a contradiction ?

Why we assume henselian ring is local?

Posted: 20 Mar 2021 08:32 PM PDT

Henselian ring is defined as local ring in which hensel lemma holds. Why do we assume local ring?

What is wrong with the definition that 'henselian ring is defined as hensel lemma holds' ?

1. Prove by induction that for the case when $n$ is a multiple of $3,$ that $n+1$ can be achieved, where $n> 8$

Posted: 20 Mar 2021 08:25 PM PDT

I understand the first part of the induction like for n=8 blah blah but I'm confused on the proving part of the question.. any help would be appreicated

Is this an equivalent definition for $(\subseteq \omega)-$presentations?

Posted: 20 Mar 2021 08:44 PM PDT

(I am looking at the first part of Antonio Montalban's manuscript on Computable Structure Theory, for those that want specifics or to read along.)

In computable structure theory it is common to take the union of finite structures to get an infinite one. A nested sequence of strings with characters in $\Bbb N$, $\sigma_0 \subseteq \sigma_1 \subseteq ...$, can be unioned over, $\bigcup\limits_{i \in \Bbb N} \sigma_i$, to get a possibly infinite string of these characters, and this could be seen as a function. For a Turing functional or c.e. operator with oracle $f$, $\Phi^f_e$, encoding them (their domains, $W^f_e$) is done similarly, writing $W^\sigma_e = \{ n \in \Bbb N \vert \langle \sigma , n \rangle \in W_e \}$, we have $W^f_e = \bigcup\limits_{\sigma \subset f}W^\sigma_e$.

An $\omega-$presentation of a structure $\mathcal{A}$ (or of a copy of $\mathcal{A}$) is just a structure whose domain is $\Bbb N$, and we say it is computable when the set (or string)

$\tau^{\mathcal{M}} := \bigoplus\limits_{i \in I_R}R_i^{\mathcal{M}}\oplus \bigoplus\limits_{i \in I_F}F_i^{\mathcal{M}}\oplus\bigoplus\limits_{i \in I_C}\{ c_i^\mathcal{M}\}$

is computable. The atomic diagram of an $\omega-$presentation $\mathcal M$ is the infinite binary string $D(\mathcal{M}) \in 2^{\Bbb N}$ defined as:

$D(\mathcal{M})(i) :=$ $\begin{cases} 1 \text{ if } \mathcal{M} \models \phi_i^{at}[x_j \mapsto j : j \in \Bbb N] \\ 0 \text{ otherwise,} \end{cases}$

where $\phi_i^{at}$ is the $i^{th}$ atomic $\tau$-formula in an effective enumeration of all the atomic $\tau$-formulas of the set $\{ x_0, x_1, ... \}$.

More generally, for subsets of $\Bbb N$, we can talk about $( \subseteq \omega) -$presentations, which can be used to present finite structures as well as infinite sequences of finite structures.

This being established, the same sort of "finite approximation" approach with atomic diagrams is applied, where the sequence $\{ \mathcal{M}_s : s \in \Bbb N \}$ is identified with the sequence of codes $\{ D(\mathcal{M}_s) : s \in \Bbb N \}$.For these finite approximations, we have

$D(\mathcal{M_0}) \subseteq D(\mathcal{M_1}) \subseteq D(\mathcal{M_2}) \subseteq \cdots \text{ so } D(\mathcal{M}) = \bigcup\limits_{s \in \Bbb N} D(\mathcal{M}_s)$.

I imagine this union sort of as circles based at $0$ including more and more characters of the atomic diagram's string, these characters arranged on a line going to the right. Bigger circles capture more of the structure. I was wondering if this analogy could be "rephrased" in the following sense. So just focusing on the last bit of the string, i.e., ignoring the segment going all the way to $D(\mathcal{M}_{i-1}$), let's call this $D'(\mathcal{M}_i)$. It seems to me you could get the same string by just taking the disjoint union over all these $D'(\mathcal{M}_i), \coprod\limits_{i \in \Bbb N} D'(\mathcal{M}_i)$, and it would just look like the string, but with each of these primed segments "marked" in a way.

Q. Is this notion equivalent, or does something go wrong?

Relations between differential geometry and algebraic geometry

Posted: 20 Mar 2021 08:53 PM PDT

I am currently an undergrad student looking to study some algebraic geometry, I have heard that differential geometry is useful for intuition in algebraic geometry, but I have no background in that (DG).

What are some specific examples of concepts that are common in both and would help understand the other subject? I am aware "manifold -> variety", but what else?

Limit of derivative given specific function.

Posted: 20 Mar 2021 08:40 PM PDT

Given function $f(x)$ ,$f :R^+ \rightarrow R^+$ that is smooth, strictly decreasing, and $lim_{x \rightarrow \infty} f(x) =0$ .

Can one show that $lim_{x \rightarrow \infty} f'(x) = 0$ ?

Understanding the technique of finding a basis of the row space of $A$ is to find a basis of column space of $A$ transposed

Posted: 20 Mar 2021 08:45 PM PDT

I understand the row space of $A$ is equivalent to the column space of $A^T.$ For example :

$$ \begin{align} A &= \begin{bmatrix} 1 & 1 & 2 & 2 & 1 \\ 2 & 2 & 1 & 1 & 1 \\ 3 & 3 & 3 & 3 & 2 \\ 1 & 1 & -1 & -1 & 0 \\ \end{bmatrix} \\ \\ rref(A) &= \begin{bmatrix} 1 & 1 & 0 & 0 & 1/3 \\ 0 & 0 & 1 & 1 & 1/3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \end{align} $$

So the first and the second row of $rref(A)$ form a basis in the row space of $A.$

The second way to find the basis in the row space of $A$ is :

$$ \begin{align} A^T &= \begin{bmatrix} 1 & 2 & 3 & 1 \\ 1 & 2 & 3 & 1 \\ 2 & 1 & 3 & -1 \\ 2 & 1 & 3 & -1 \\ 1 & 1 & 2 & 0 \\ \end{bmatrix} \\ \\ rref(A^T) &= \begin{bmatrix} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \\ \end{align} $$ We pick the first and the second column of $rref(A^T)$, and their corresponding columns in $A^T$ (which are the first and the second rows of $A$) form a basis in the column space of $A^T.$

My question basically is why the second approach works. Why does it just so happen that after row operations on $A^T,$ the first and the second columns are the pivot columns of $rref(A^T)$? Is it true that if the kth row of the $rref(A)$ has pivot then the kth column of $rref(A^T)$ has pivot too ?

The probability of $7 7$’s in $17$ digit random number?

Posted: 20 Mar 2021 08:50 PM PDT

$$205-7895777-7786742$$

What is the probability that the number $7$ appears $7$ times in a $17$ digit random number?

Taylor's Theorem, question on $(x - n)^k$ term

Posted: 20 Mar 2021 08:50 PM PDT

I read this post which explains the factorials in Taylor's Theorem. Does anyone know how the $(x-x_0)^k$ terms contribute? Are these terms a specific weight placed on the functions of $f(x)$?

For instance a $k$ times differentiable function: $$ f(x) = f_0 + f_1 x + f_2 x^2 + ... + f_n x^k $$

where the derivatives centered at $x = 0$ are: $$ f'(0) = 0, f''(0) = 2! f_2, f'''(0) = 3! f_3 $$

and

$$ f_0 = f(0), f_1 = \frac {f'(0)} {1!}, f_2 = \frac {f''(0)} {2!}, f_3 = \frac {f'''(0)} {3!} $$

which results in: $$ f_T(x) = f_0 + f_1(x - x_0) + f_2(x - x_0)^2 + f_3(x-x_0)^3 + ... + f_n(x-x_0)^n $$

Wiki says:

$$f(x) \approx f(a) + f'(a)(x - a)$$

but I'm having trouble making the connection between the two.

Proving that if $a^{2020}+b^{2020} = a^{2022}+b^{2022}$ then $a+b\le 2$ for $a,b\in \mathbb{R}$

Posted: 20 Mar 2021 08:27 PM PDT

The task obviously amounts to simply expressing the relationship between $a$ and $b$ but I could not come up with anything of interest besides a few reformulations of the equation. Are there any useful methods for solving this? Perhaps the general case as well?

I'm grateful for any help.

Can the infinite jungle gym surface be expressed by an exhaustion of compact surfaces with one boundary component?

Posted: 20 Mar 2021 08:44 PM PDT

Is it possible to write the infinite jungle gym surface as the increasing union of compact surfaces whose boundaries consist of only one simple closed curve? I believe that this is true. This surface has only one end. Therefore the result follows from the existence of canonical exhaustions (made of compact surfaces whose components of the complement have one boundary component) (as explained in Ahlfors-Sario book). If anyone could help me, I would like to "see" such a curve. Thanks.

Question about limit of integrals

Posted: 20 Mar 2021 08:34 PM PDT

I want to prove the following:

$$ \lim_{n \rightarrow \infty} \int_{0}^{b_{n}} f_{n}(x) dx= \int_{0}^{1} f(x) dx $$ provided that $f_{n}$ is Riemann-integrable on $[0,1]$ and $f_{n} \rightarrow f$ uniformly on $[0,1]$ and $b_{1} \leq b_{2}...\leq b_{n}\le \dots$ and $b_{n} \rightarrow 1$.

I started the proof by saying that since $f_{n} \rightarrow f$ uniformly so we can exchange the limit and the integral. But, that seems to be making it almost trivial. Am I missing something very important?

Finding the marginal distributions

Posted: 20 Mar 2021 08:24 PM PDT

I got these 2 models for hourly wage for 2 periods:

The hourly wage for period 1 is normally distributed with mean $µ$ and variance $σ^2$ so $Y_1 \sim N(\mu,\sigma^2)$.

And the hourly wage for period 2 is given by: $$Y_2=\alpha+\beta Y_1+U$$ where $Y_1$ and $U$ are independent and $U \sim N(0,v^2)$. Then we assume that $\beta \neq 0$ and let $\mu = 350$ and $\sigma^2=12365$ and $\alpha=350\cdot(1-\beta)$ and $v^2=12365 \cdot (1-\beta^2)$.

Now I have to find the marginal distributions of $Y_1$ and $Y_2$. I have found on Wikipedia that the marginal probability is $𝑝_𝑋(𝑥)=𝐸_𝑌[𝑃_{𝑋|𝑌}(𝑥|𝑦)]$ and I have in a previous task found that $𝐸(𝑌_2|𝑌_1)=𝛼+𝛽⋅𝑌_1$. Is that the same value (I'm not totally sure on the notation will be the same) and how can I use it to find the marginal distributions of $Y_1$ and $Y_2$?

sample variance divided by variance is Chi Square

Posted: 20 Mar 2021 08:36 PM PDT

To estimate the sample variance, the following relation is often used:

$$\frac{(n-1)s^2}{\sigma^2} \sim \chi^2(n-1) $$

With $(n-1)$ being the degrees of freedom.

Could someone provide me a formal proof and some intuition for this relation?

Probability of rolling a 6 immediately after a 1 is rolled

Posted: 20 Mar 2021 08:29 PM PDT

Question

Ann and Bob take turns to roll a fair six-sided die. The winner is the first person to roll a six immediately after the other person has rolled a one. Ann will go first. Find the probability that Ann will win.

Answer

$\mathbb{P} (\mathrm {Ann\ wins}) = \frac {36} {73}$

I have thought long and hard about this question but I am unable to even start. I have tried considering cases, but got stuck along the way. For example, it is trivial to calculate the probability that Ann wins if there are only three rolls (which is the minimum number of rolls needed for Ann to win). However, the problem easily becomes very complicated when we consider five rolls and more.

The suggested solution by my professor uses first-step decomposition, but it is a new concept to me and I am struggling to understand it.

If anyone can provide a detailed and intuitive explanation as to how this problem should be solved, that will be greatly appreciated!

Do overlapping left and right coset spaces have a name?

Posted: 20 Mar 2021 08:52 PM PDT

I have been studying they symmetries of the square, D₄. I would like to use the subgroups that are not normal to make a coset space, but since they are not normal, the right coset space is different than the left coset space. However, I did notice that the left and right coset space of pairs of subgroups in the same conjugacy classes overlap. Is there a name for this whole system?

This picture is one of the two conjugacy systems. There are four overlapping coset spaces in the form of the two rows and columns.

Linked system of left and right coset spaces

Best way to play 20 questions

Posted: 20 Mar 2021 08:37 PM PDT

Background

You and I are going to play a game. To start off with I play a measurable function $f_1$ and you respond with a real number $y_1$ (possibly infinite). We repeat this some fixed number $N$ of times, to obtain a collection $\{(f_i,y_i)\}_{i=1}^N$. Define

$\Gamma=\{\mu: \mu $ is a probability measure, and $\int f_i(x)d\mu(x)=y_i \forall i\}$

(We'll assume that $\Gamma$ is not empty, meaning you cannot give inconsistent answers. For example, if I play $f_1(x)=x^2$, then you could not play $y_1=-1$).

Next, I play a measure $P$ which must satisfy $P\in \Gamma$. You then play a measure $Q\in \Gamma$. The game is scored such that your reward is equal to $H(Q|P)$ (the cross entropy) while my reward is equal to $-H(Q|P)$. In other words, I am rewarded to the extent that I can guess your distribution, while you are rewarded to the extent that you can foil my guess.

What should my strategy be for playing this game?

Motivation

This game can be considered as a general model of scientist designing experiments (the functions $f_i$) and using the results of those experiments to develop a theory (the measure $P$). The other player is a stand-in for "Nature", which we assume acts against us at every turn.

Further, the game is a generalization of the setting considered by Grunwald and Dawid in this paper. They consider what amounts to a special case of this game in which the set $\Gamma$ is assumed to be specified ahead of time (so each participant only specifies their distribution). Very interestingly, they show that in this case the optimal strategy is to play the distribution $\text{arg max}_{P\in\Gamma} H(P)$, where $H$ denotes Shannon entropy. This amounts to a passive model of scientific inference, in which the scientist has access only to observational data in the forms of constraints $\int f_i(x)d\mu(x)=y_i$, but has no control over what the actual functions $f_i$ are. Thus I am interested in what would happen if their setting is extended to an "active" one, in which the scientist can control which statistics $f_i$ to measure in the first place.

Morse-Kelley Class Comprehension and Russell's Paradox

Posted: 20 Mar 2021 08:33 PM PDT

As I understand the ZFC solution to Russell's paradox, since $\{x\mid x\notin x\}$ must be $\{x\mid x\notin x\}\cap S$ for some set $S$, the paradox goes away, but in Morse-Kelley, although again there must be some $M$ such that $\{x\mid x\notin x\}\cap M$, this $M$ may be a proper class, which no longer is as limiting as the ZFC version, and hence no longer gives the same solution. So MK must handle Russell's Paradox in a different way. How? I would be grateful for enlightenment. Thanks.

Proof that every repeating decimal is rational

Posted: 20 Mar 2021 08:21 PM PDT

Wikipedia claims that every repeating decimal represents a rational number.

According to the following definition, how can we prove that fact?

Definition: A number is rational if it can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Largest integer that can't be represented as a non-negative linear combination of $m, n = mn - m - n$? Why?

Posted: 20 Mar 2021 08:29 PM PDT

This seemingly simple question has really stumped me:

How do I prove that the largest integer that can't be represented with a non-negative linear combination of the integers $m, n$ is $mn - m - n$, assuming $m,n$ are coprime?

I got as far as this, but now I can't figure out where to go:

$mx + ny = k$, where $k = mn - m - n + c$, for some $c > 0$

$\Rightarrow m(x + 1) + n(y + 1) = mn + c$

If I could only prove this must have a non-negative solution for $x$ and $y$, I'd be done... but I'm kind of stuck.

Any ideas?
Posted by at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)