Saturday, May 8, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

If we allow an (improper) uniform distribution over the real numbers, is the Wiener process then differentiable?

Posted: 08 May 2021 07:38 PM PDT

Many mathematical texts assert that the Wiener process (and Brownian motion more generally) is not differentiable. I have seen some heuristic proofs of this, and they tend to rely on the fact that the variance of the rate-of-change of the process over small intervals tends to infinity as the interval gets smaller. Specifically, by definition, the Wiener process has the property that $W(t + \Delta) - W(t) \sim \text{N}(0,|\Delta|)$, so for all $\Delta \neq 0$ we have:

$$Z(\Delta) \equiv \frac{W(t + \Delta) - W(t)}{|\Delta|} \sim \text{N} \Big( 0,\frac{1}{|\Delta|} \Big).$$

This "rate-of-change" quantity obviously exists for all $\Delta \neq 0$. As $\Delta \rightarrow 0$ the variance of this distribution tends to infinity, which does not correspond to any proper distribution. However, if we adopt a conception of probability that allows improper distributions (which requires weakening the standard probability axioms)$^\dagger$, then the above distribution tends to the (improper) uniform distribution on the real numbers. Within this framework, I was wondering if it would then be reasonable to assert that:

$$\frac{dW}{dt} \equiv \lim_{\Delta \rightarrow 0} \frac{W(t + \Delta) - W(t)}{\Delta} \sim \text{U}(\mathbb{R}).$$

In short, if we allow improper distributions, would it then be okay to say that the Wiener process is differentiable, and its derivative at any point is a uniformly distributed real number. The heuristic demonstration above suggests to me that this is plausible, and it also seems to accord reasonably well with intuition --- i.e., that the instantaneous rate-of-change is equally likely to be any real number. I am aware that there may be more technicality involved in this matter. As far as I can see, it essentially comes down to the following question: if the limit of a sequence of distributions tends to an improper distribution, does a sequence of random variables with those distributions tend to a random variable with that improper distribution?

Question: Is it possible to "formalise" the above argument so that the Weiner process can be considered to be differentiable, with its derivative being a uniform real value? Is there any fundamental flaw in the argument?

$^\dagger$ There is quite a bit of existing literature in probability theory that talks about how you can obtain improper distributions like the uniform distribution on the real numbers (for an overview see e.g., Lindqvist and Taraldsen 2018).

Combinations vs permutations with replacement

Posted: 08 May 2021 07:35 PM PDT

The formulas for combinations and permutations with replacement are, respectively:

$$ C = \binom{n + k - 1}{k} \\ P = n^k $$

Where $n$ and $k$ correspond to the number of distinct objects we are to draw $k$ times from with replacement.

My question is I can't wrap my head around why $C * k! \neq P$.

$C$ is the number of ways we can draw $k$ (not necessarily) items from $n$ distinct items with replacement. Here, order does not matter. To account for order mattering, I multiply by $k!$. So why isn't $C * k! = P$?

Splitting field over the field of fractions $\mathbb{Z}_p(x)$

Posted: 08 May 2021 07:23 PM PDT

Let $F=\mathbb{Z}_p(t)$ for some $p$ prime number. Let $E$ be the splitting field of $f(x)=x^p-t$ over $F$.

(a) We need to show that degree of $E$ over $F$ is $p$, i.e. $[E:F]=p$. (b) Also we need to show that $|G|=1$ where $G=Gal_F(E)$.

My work

Since $f'(x)=px^{p-1}=0$ then $f$ is inseparable, so $[E:F] \leq p$ and so if $[E:F]=d<p$ then there should be a minimal irreducible polynomial of degree $d$ but then this polynomial should divides $f$ which cannot happen as $f$ is inseparable, so $[E:F]$ must be $p$. Is that correct and sufficient argument for (a)?

Almost complex structure on a contractible manifold

Posted: 08 May 2021 07:45 PM PDT

Let $M$ be a contractible manifold with an almost complex structure $J:TM\to TM$. Suppose $J':TM\to TM$ is another almost complex structure. Since $M$ is contractible, so is $TM$, hence $J$ and $J'$ are homotopic, say via a homotopy $J_t:TM\to TM$. But we are not guaranteed that each $J_t$ is an almost complex structure. Can we choose a homotopy so that each $J_t$ is an almost complex structure? (actually I'm interested in the simple case $M=\Bbb C^n$)

By the way, is there a notion of "equivalence" between almost complex structures on a fixed manifold?

Thanks in advance.

Probability of picking two standard trig functions whose graphs never intersect

Posted: 08 May 2021 07:32 PM PDT

Suppose you are given a set $S$ that contains the six standard trigonometric functions $$S=\{\sin(x), \cos(x), \tan(x), \cot(x), \sec(x), \csc(x)\}$$

What is the probability given the 6 standard trigonometric functions, where defined on the reals, you pick a pair of two that never intersect (assume order does not matter)?

My thought process was that since $\sin(x)$ and $\sec(x)$ never intersect, and $\cos(x)$ and $\csc(x)$ never intersect, then the probability is $2/15$. But the answer ended up being $4/15$.

I thought that order didn't matter, so unless I missed some cases, I'm not sure where I went wrong.

To which field is this quotient isomorphic $\frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle}$?

Posted: 08 May 2021 07:17 PM PDT

I need to show the quotient $\frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle}$ is a field and find to which field is this quotient isommorphic.

I think this: $\frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle}\cong \frac{\mathbb{Z}_5[x]}{\langle 2x-1 \rangle}$. The polynomial $2x-1$ is irreducible in $\mathbb{Z}_5[x]$ thus, the quotient $\frac{\mathbb{Z}_5[x]}{\langle 2x-1 \rangle}$ must to be a field, but I don´t know to which is isomorphic.

How to measure the similarity among vectors

Posted: 08 May 2021 07:16 PM PDT

Question

Suppose there are three vectors $x$, $y$, and the target $t$.

  • The angle between $t$ and $x$ is smaller than that of $t$ and $y$, but
  • The distance between $t$ and $y$ is smaller that that of $t$ and $x$

Which vector $x$ or $y$ should I say "more similar" to $t$ and what formula to use to quantify it?

enter image description here

Background

word2vec or more generally Thought Vectors encodes the similarity of words or concepts into fixed length vectors. Suppose this vector space is $W$ and each vector has $D$ dimensions (features).

Then $queen \approx (king-man) + woman$ can be achieved using the vectors.

Implementing Word2Vec in Tensorflow

enter image description here

However, not sure how this "Similarity" should be defined and measured in the vector space. What would it mean "angle is smaller but distance is larger" or "angle is larger but distance is smaller".

I can think of below but not sure which ones to use.

  1. Distance measured as $|t - x|$
  2. Angle measured as $\frac {t}{|t|} \cdot \frac {x}{|x|} $
  3. Dot product $t \cdot x$

Articles discusses cosine, Word mover's distance, Euclidean distance, etc but it looks there are no solid discussion over the meaning of each method and why use them.

Hence would like some guidances on how to understand "Similar" in a vector space and what formulas to use to quantify it.

Is $\sqrt{n}(\bar X^*_n - \bar X^*_m)$ tight?

Posted: 08 May 2021 07:09 PM PDT

My question is related to this question: Is $\sqrt{n}(\bar X_n - \bar X_m)$ tight?

Suppose $\{(X_i, d_i)\}$ is iid with distribution $P$ with finite fourth moments. Suppose $d_i \in \{0,1\}$, and suppose $d_i$ is independent of $X_i$.

Let $n_d = \sum_{i=1}^n d_i$. In the previous question, I asked if $$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i - \frac{1}{n_d}\sum_{i}^n X_i\cdot d_i\right)= O_p(1)~,$$ which is true. The answer there shows it directly, using a characteristic function approach, but it is also to possible to show it using the multivariate CLT and the Delta method.

Now, I want to know if it is true that $$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X^*_{n,i} - \frac{1}{n^*_d}\sum_{i}^n X^*_{n,i}\cdot d^*_{n,i}\right)= O_p(1)~,$$ where, for each $n$, we draw $\{(X^*_{n,i}, d^*_{n,i})\}$ iid from $\hat P_n$, which is the empirical distribution based on $\{(X_{i}, d_{i})\}_{i=1}^n$ which is drawn iid from $P$. Basically, we use $\hat P_n$ as an approximation of $P$ (as you do in resampling, for example), and I ask if the claim still holds.

Here, it is easy to show that $\frac{1}{n}\sum_{i=1}^n X^*_{n,i} - \frac{1}{n^*_d}\sum_{i}^n X^*_{n,i}\cdot d^*_{n,i}= o_p(1)$ (using Slutsky for example), and I basically want to know if $\frac{1}{n}\sum_{i=1}^n X^*_{n,i} - \frac{1}{n^*_d}\sum_{i}^n X^*_{n,i}\cdot d^*_{n,i} = O_p(n^{-1/2})$.

It's not feasible to apply CLT directly here, since we are dealing with triangular arrays, but the finite fourth moment here does allow a multivariate version of the Lyapunov CLT to be used. The main challenge seems to be that the independence between $d_i$ and $X_i$ need not hold in the re-sampled samples that are drawn from $\hat P_n$.

Show that $\frac{d}{dt}\langle Y,Z\rangle\Big|_{t=0}=X\langle Y,Z\rangle$

Posted: 08 May 2021 07:35 PM PDT

Let $X,Y,Z \in \mathcal X(\gamma)$ s.t :$X(\gamma(t))=\dot\gamma(t). $

I need to sow this : $$\frac{d}{dt}\langle Y,Z\rangle (\gamma(t))\Big|_{t=0}=X\langle Y(\gamma(t)),Z(\gamma(t))\rangle$$ I could not understand how this happens. Can someone explain this to me please !?

Why do keyhole contours work?

Posted: 08 May 2021 07:01 PM PDT

When evaluating integrals on $(0,\infty)$, we often extend the integral into the complex plane by using a "keyhole contour" with inner radius $\epsilon$ and outer radius $R$: enter image description here

Credit for the image goes to Linda J. Cummings.

My question is - why do we do this? When we take $R\to\infty$ and $\epsilon\to 0$, the contribution from the circular arcs typically vanish, but what we have remaining is two integrals, i.e $\int\limits_0^\infty$ and $\int\limits_\infty^0$. But these will just cancel out right? When we integrate over the same line in the opposite direction, the result is negated, is it not?

I have not seen anyone rigorously justify this. More worrying is a section on Wikipedia where they conclude $$\int\limits_R^\epsilon \frac{\sqrt{z}}{z^2+6z+8}\mathrm{d}z=\int\limits_R^\epsilon \frac{-\sqrt{z}}{z^2+6z+8}\mathrm{d}z=\int\limits_\epsilon^R \frac{\sqrt{z}}{z^2+6z+8}\mathrm{d}z$$ But from this surely the integral is trivially zero?!

Why is this not a completely pointless endeavor? Does it have to do with the fact we are taking a limit as the upper bound of the integral goes $\to\infty$? Or does domain reversal work differently in the complex plane? I'd appreciate if someone could explain this.

Proving that the set $\{\omega: X_t(\omega)=Y_t(\omega), \text{for all t}\}$ is measurable.

Posted: 08 May 2021 07:00 PM PDT

Let $X_t, Y_t$ be stochastic processes on a common probbility space, $(\Omega, \mathcal{A},P)$. As I understand the event:

$\{\omega: X_t(\omega)=Y_t(\omega), \text{for all t}\}$

need not be measurable. However, I read on a mathematical blog that if both $X_t$ and $Y_t$ are jointly measurable, then the set is measurable. Do you know how to prove this?

Attempt:

Since they are jointly measurable we have that the set $K=\{(\omega, t): X_t(\omega)=Y_t(\omega)\}$ is in $\mathcal{A}\otimes \mathcal {B}([0,T]).$ Does that help us? We also have the relation

$$\{\omega: X_t(\omega)=Y_t(\omega), \text{for all t}\}=\{\omega: (\omega,t)\in K \text{ for all t}\}.$$

But is the last set measurable? Any idea on how to solve this?

Question about Wade Analysis Exercise 12.3.8

Posted: 08 May 2021 07:30 PM PDT

Maybe the notation varies so let me just clarify this: if $F\subset\mathbb{R}^n$, we define:

$\overline{\text{Vol}}(F) := inf_{G \text{ is a grid on a rectangle $R\supset F$}}V(F,G)$

(the outer sum $V(F,G)$ is the sum of volumes of rectangles in the grid $G$ which intersect $\overline{E}$)

and if is a Jordan set (boundary has volume zero, that is, $\forall \epsilon > 0$ there is a grid $G$ on a rectangle $R\supset F$ s.t. the outer sum $V(\partial F,G)$ is less than $\epsilon$), then we define:

Vol$(F) := $ $\overline{\text{Vol}}(F)$.

I have been trying to solve:

Let $E\subset \mathbb{R}^2$ be nonempty and Jordan and let $f: E\to [0,\infty)$ be integrable (on $E$). Prove that the volume of $\Omega:= \{(x,y,z)\in \mathbb{R}^3 \mid (x,y)\in E, 0\leq z\leq f(x,y)\}$ satifsifes: Vol$(\Omega) = \int\int_EfdA$.

Once I can show that (i) $\Omega$ is Jordan, (ii) $A:= \{(x,y,z)\in \mathbb{R}^3 \mid (x,y)\in \overline{E}, 0\leq z\leq f(x,y)\}$ is Jordan and equal to $\overline{\Omega}$, and (iii) Vol$(\Omega) = $ Vol$(A)$,

and I assume WLOG that f is continuous on $E$ (*) (the set of points of discontinuity has measure zero and I think this implies volume zero and I guess that implies Jordan measure or Vol$(-)$ of zero)

then I am really done ($A$ is certainly projectiable under the assumption (*), so by theorem 12.39 we will have: $\int_E f = \int\int_E(\int_0^{f(x,y)}1dz)dydx = \int_A f = Vol(A) = $ Vol$(\overline{\Omega}) = $ Vol$(\Omega) + $ Vol$(\partial \Omega) = $ Vol$(\Omega) + 0 = $Vol $(\Omega)$ (the first equality holds due to the FTC from single variable calculus).

I guess I can prove (i), (ii), (iii) assuming (*) (maybe some of them, especially (i) without) but I feel like this is kind of sloppy and making the argument more convoluted than it needs to be (we haven't proved my class for instance that if the set differences between $U$ and $V$ in $\mathbb{R}^2$ have measure zero (definition 12.27), then if one of $U$ or $V$ is Jordan then I guess they both are and if $f:C\supset U,V\to \mathbb{R}$ is integrable then $\int_Uf = \int_Vf$, but I feel like this is all getting really convoluted and long, and there is probably an easier way. (I think I have more or less proven that $A = \overline{\Omega}$ if we assume $f$ is continuous but still...). It just seems like I am doing it wrong just because it so complicated at this point (I am up to like 6 pages of scratch work now).

Also, no I cannot use the change of variables theorem (**) here (idk if that would be helpful anyway but let's just assume not, that isn't until the section in the book after this exercise anyway, so apparently it is feasible without it I just don't know how).

It really kind of looks like I should be using Fubini somewhere but I don't think $\Omega$ needs to be a product of rectangles (it seems like far from it, possibly besides using that theorem (**) I mentioned that I can't use for this). Maybe I should be using Caravelli Principle somewhere but I am not really sure how either (I was trying to use that corollary of Caravelli that involves projective sets as you can see, the problem is really just that $f$ need not be continuous and/or I don't know how to rigorously justify saying WLOG assume f is continuous on $E$)

Prove that $2n\choose n$ divides $LCM(1,2,3,...,2n)$

Posted: 08 May 2021 07:15 PM PDT

What I did:

L = $LCM(1,2,...,2n)$

M = $2n\choose n$

If we can show that the exponent of any prime in the prime factorization of M is less than that in L, the problem is solved.

So, $e_p(M) \leq e_p(L)$ for all prime p.

$e_p(M) = e_p((2n)!) - 2e_p(n!) = \sum_{r\geq1}\lfloor \frac{2n}{p^r}\rfloor - 2\sum_{r\geq1} \lfloor \frac{n}{p^r}\rfloor$

$e_p(L) =$ largest $m$ so that $p^m \leq 2n$ or $m = \lfloor\frac{\log 2n}{\log p}\rfloor$

Problem is, how do I proceed?

Weak solution of nonlinear PDE

Posted: 08 May 2021 06:56 PM PDT

Let $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary.Prove there exists a positive constant $\epsilon_0$ so that for all real number $\epsilon<\epsilon_0$ snd $f\in L^2(\Omega)$,there exists a unique $u\in H_0^1(\Omega)$ so that

\begin{equation} -\Delta u+\epsilon \sin u=f \end{equation} in the sense of distribution.

For the uniqueness part we may use energy method, just subtract two equations of different weak solutions and then multiply both sides $\varphi_\epsilon *u$.I am not sure if it works.

I have no idea about the existence part, I know some functional analysis methods such as Riesz representation theorem can deal with linear equations, but how to solve nonlinear equations? Is there any good book given an introduction to nonlinear equations?(I'm familiar with real analysis and functional analysis)

Study the convergence the improper integral $\int_1^\infty \frac1{\ln^7(x)}dx$

Posted: 08 May 2021 07:24 PM PDT

I know that $\int_1^\infty \frac1xdx$ diverges. I can probe that $\int_1^\infty \frac1{\ln(x)}dx$ diverges, as $\forall x>1:\frac1x<\frac1{\ln(x)}$

I also know that $\int_1^\infty \frac1{x^2}dx$ converges $ \therefore \int_1^\infty \frac1{x^7}dx$ also converges.

I know that $\forall x>1:x>\ln(x) \rightarrow x^7>\ln^7(x)\rightarrow \frac1{x^7}<\frac1{\ln^7(x)}$ but this doesn't help me much.

Prove: If $f,g \in \Bbb Z[x]$ then $C(fg) = C(f)C(g)$

Posted: 08 May 2021 07:31 PM PDT

If $f,g \in \Bbb Z[x]$ then $C(fg) = C(f)C(g)$. C is the content of a polynomial (greatest common divisor of the coefficients). The proof states that proving: "For any prime, $p$ we have $p|C(fg)$ iff $p|C(f)$ or $p|C(g)$." implies the result. I'm not sure why this is. Can someone explain; is it just because of the prime factorisation? Explanations/examples would be great!

equality involving sums

Posted: 08 May 2021 07:17 PM PDT

Let $n\in\mathbb{Z}^+.$ Prove that for $a_{i,j}\in\mathbb{R}$ for $i,j = 1,\cdots, n,$ $$\left(\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\right)^2 + n^2\sum_{i=1}^n\sum_{j=1}^n a_{i,j}^2 - n\sum_{i=1}^n \left(\sum_{j=1}^n a_{i,j}\right)^2 - n\sum_{j=1}^n\left(\sum_{i=1}^n a_{i,j}\right)^2 \\= \frac{1}4\sum_{i,j,k,l=1}^n (a_{i,j} + a_{k,l} - a_{i,l} - a_{k,j})^2.$$

I tried some small cases (e.g. $n=2$ and $n=1$), but I'm not really sure how to generalize the result. Clearly, this is a problem involving summations and it might be useful to use various summation properties, but I'm not sure which properties to use. Expanding the squares and using the fact that different summation indices are "independent" (e.g. $\sum_{i,j,k=1}^n a_{i,j} = n\sum_{i,j=1}^n a_{i,j}$) gives that

\begin{align*} &\left(\sum_{i,j=1}^n a_{i,j}\right)^2 + n^2\sum_{i, j=1}^n a_{i,j}^2 - n\sum_{i=1}^n\left(\sum_{j=1}^na_{i,j}\right)^2 - n\sum_{j=1}^n \left(\sum_{i=1}^n a_{i,j}\right)^2 \\ &=\sum_{i,j,k,l=1}^n a_{i,j}a_{k,l} + \sum_{i, j,k,l=1}^n a_{i,j}^2 - \sum_{i,k=1}^n\sum_{j,l=1}^na_{i,j}a_{i,l} - \sum_{j,l=1}^n \sum_{i,k=1}^n a_{i,j}a_{k,j}\\ &=\frac{1}4\sum_{i,j,k,l=1}^n[(2a_{i,j}a_{k,l} + 2a_{i,l}a_{k,j}) + (a_{i,j}^2+a_{k,l}^2 + a_{i,l}^2 + a_{k,j}^2)- (2a_{i,j}a_{i,l}+2a_{k,l}a_{k,j}) - (2a_{i,j}a_{k,j} + 2a_{k,l}a_{i,l})]\\ &=\frac{1}4\sum_{i,j,k,l=1}^n(a_{i,j}+a_{k,l}-a_{i,l}-a_{k,j})^2. \end{align*}

Is this incorrect?

Is an overring of a UFD a normal domain?

Posted: 08 May 2021 07:35 PM PDT

Old question: Let $R$ be a UFD, and $Q(R)$ its field of fractions.

Let $S$ be an overring of $R$, namely, a ring such that $R \subseteq S \subseteq Q(R)$.

Question: Is it true that $S$ is a UFD?

Notice that $R=k[x^2]$, $S=k[x^2,x^3]$ is not a counterexample, since $S$ is not contained in the field of fractions of $R$, $Q(R)=k(x^2)$.

Please see this similar question: Here we require that $S \subseteq Q(R)$ (and remove the requirement for simplicity); the answer there presents rings $R \subseteq S$ with $S \not\subseteq Q(R)$.

Thank you very much!

New question: After receiving several comments, including the following counterexample $k[x,y] \subset k[x,y,y/x] \subset k(x,y)$, I would like to change my question to the following one:

New question: Is $S$ normal, namely, integrally closed in its field of fractions $Q(S)$?

This and this paper are relevant. If I am not wrong, $(k[x],k(x))$ is a normal pair, namely, every ring $k[x] \subseteq S \subseteq k(x)$ is integrally closed in $k(x)$, so for $R=k[x]$ the new question has a positive answer.

More generally, if I am not wrong, if $R$ is a Dedekind domain, then $(R,Q(R))$ is a normal pair, hence answers the new question positively.

What about $R=k[x,y]$? Is $(k[x,y],k(x,y))$ a normal pair?

Form of line elements in curvilinear

Posted: 08 May 2021 07:32 PM PDT

I've just started studying this book. On page 5 of it, the following sentence has written:

Generally speaking, when using curvilinear (e.g. polar, cylindrical, spherical) coordinates $(y^1,...,y^n)$ for the Euclidean $\mathbb{R}^n$ we should expect the line elements to take the form of $$ds^2 = \sum_{ij}g_{ij}dy^idy^j$$

I want to know why our expectations should be in this form? What is the intuition behind it?

A question about determining the distance between a point on the edge of a circle to a point on a equilateral triangle

Posted: 08 May 2021 07:44 PM PDT

Equilateral triangle ABC has side length 6. Let D be the point on segment BC such that BD=4. The circle passing through points A, B, C intersects line AD at A and at another point E. The length of DE can be expressed in simplest radical form as (A rad B)/C, where A, B, and C are positive integers. What is A+B+C?

I keep on getting 15, but the answer key says the answer is 18. enter image description here

I don't know if the link to the image will work, but what I have tried so far is R-r (6rad3/3 - 6rad3/6). However, I realize that R is calculated assuming that you are going from the center of the circle inscribed in the equilateral triangle to a point on the outside circle, and in this problem the line going through points AD doesn't go through the middle of an inscribed circle. My question is what other methods should I try to find the length of DE where the answer can be expressed in terms of (AradB)/c.

Cubic and Conic curves has an linear factor in common?

Posted: 08 May 2021 07:20 PM PDT

Let $V(f)$ be a cubic projective curve and a $V(g)$ be a conic projective curve, with $f,g \in \mathbb{C}[X,Y,Z]$. If $V(f)\cap V(g)$ has more than six points, then $f$ and $g$ has a linear factor in common?

A friend of mine said that it's false, but I couldn't find any conterexample for this. Just need one conterexample, can you help me? Thanks.

Simple Linear Regression with Dummy Variable: Testing Gender Differences

Posted: 08 May 2021 06:56 PM PDT

If we have data on sample who report their wage $W$ and gender $D$ (w/ $D=1$ male): $$ \ln W_{i}=\beta_{1}+\beta_{2} D_{i}+\varepsilon_{i} $$ The OLS estimators of the regression coefficients are $$ \begin{array}{c}\hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(1-D_{i}\right) \ln W_{i}}{\sum_{i=1}^{n}\left(1-D_{i}\right)}=\ln \bar{W}_{f} \\ \hat{\beta}_{2}=\frac{\sum_{i=1}^{n} D_{i} \ln W_{i}}{\sum_{i=1}^{n} D_{i}}-\frac{\sum_{i=1}^{n}\left(1-D_{i}\right) \ln W_{i}}{\sum_{i=1}^{n}\left(1-D_{i}\right)}=\ln \bar{W}_{m}-\ln \bar{W}_{f}\end{array} $$ with $$ \begin{array}{l}\bar{W}_{f}=\prod_{i=1}^{n} W_{i}^{\frac{1-D_{i}}{n_{f}}} \\ \bar{W}_{m}=\prod_{i=1}^{n} W_{i}^{\frac{D_{i}}{n_{m}}}\end{array} $$

Where did the expression for $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ come from? I'm familiar with the standard formulas: $\begin{array}{c}\hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}} \\ \hat{\beta}_{0}=\bar{y}-\hat{\beta_{1}} \bar{x}\end{array}$

Can you explain this relation between finite fields and circles?

Posted: 08 May 2021 07:20 PM PDT

Let $p$ be a prime such that $p \bmod 4 = 1$, so there exists some $i=\sqrt{-1}$ in $\mathbb{F}_p$. Furthermore, let $r \in \mathbb{N}$ be the radius of a circle such that there are $p-1$ lattice points on it. (The sums of squares function allows to compute such $r$.)

I think that for every lattice point $(x,y)$ there is a unique $z = x + iy$ in $\mathbb{F}_p$. Also, there is a generator $g$ in $\mathbb{F}_p$ such that $g^\alpha$ traverses the lattice points in the order given by their angle on the circle.

Here's an example with $g=2$:

enter image description here

(The line segments in the picture above point to $(x \bmod p)$ and $(y \bmod p)$. They describe a circle of radius $(r \bmod p)$ in the finite plane $\mathbb{F}_p \times \mathbb{F}_p$.)

Is there any literature about that phenomenon?

Can you prove the order of points fits the order given by $g^\alpha$?

Here are questions related to that topic:

Calculating Angle Between Two Circles With Same Center Point - Mercury Retrogrades

Posted: 08 May 2021 07:05 PM PDT

I want to calculate the angle between points on two circles with the same center point. Each point is a planet in its orbit. I have the degrees of each point in the circle. What I want to know is how to calculate the angle between the two. I'm using this to calculate the retrogrades of Mercury, and have been working on this project for about three weeks. Maybe you know a better way to do it?

Mercury: 329  Earth  : 77

Where the value for Mercury and Earth are in degrees.

I'm learning the math as I go, so if you could explain things that would be great!

I have the ratio of Mercury's orbit to Earth. This changes based on the date.

Mercury: 0.431094585293355    Earth: 0.985210404350114

Try to calculate the angle between the gray dot and the blue dot, from the perspective of the black dot. Angle ME When the numbers are given in degrees.

enter image description here

why does the recurrence $a_{n+1} = a_n - 2 a_n^{3/2}$ look like $1/n^2$ for large n? [closed]

Posted: 08 May 2021 07:05 PM PDT

The following question has come up in my research:

Consider the recurrence $a_{n+1} = a_n - 2 a_n^{3/2}$, with initial condition $a_0 \in (0, \frac{1}{4})$.

I know that $a_n \approx 1/n^2$ for large $n$, but I don't understand why that is true. I'd appreciate any insights!

Does knowing the nth digit of pi help in finding the next digit?

Posted: 08 May 2021 07:33 PM PDT

Obviously, we can generate as many digits of $\pi$ as we please, but is there any way to make use of knowing digit $n$ when seeking next digit of $\pi$?

What is the minimal number of different symbols in the game "Dobble"?

Posted: 08 May 2021 07:19 PM PDT

There is a game called "Dobble" (or "Spot It!" in some countries) which implies an interesting problem I couldn't solve. The game consists of some amount of cards $c$, which have $s$ distinguishable symbols on them. For every two cards it is guaranteed that they have exactly one symbol in common. Several other questions have been asked about this game (see https://stackoverflow.com/questions/6240113/what-are-the-mathematical-computational-principles-behind-this-game), but I haven't found my question anywhere. I asked myself:

Given the total number of cards $c$ and the number of symbols $s$ per card, what is the minimum number of distinguishable symbols one has to use in order to fulfill the abovementioned condition?

I wrote an algorithm which can answer this question in a reasonable time for $c < 6$ and $s < 6$ in Java, it can be found here. I'm basically using brute-force to check all possibilities until I can be sure that I've found the optimal one. I used some if-clauses to bail out as early as possible and not generate useless possibiilities, but nevertheless the algorithm isn't fast enough for bigger numbers. If anyone is interested, I would explain the code in more detail (the comments I made are in German).

Here is a table with the values I got so far (extended table can be found here):

+---+---+---+----+----+----+----+  |   | 1 | 2 |  3 |  4 |  5 |  6 |  <- c  +---+---+---+----+----+----+----+  | 1 | 1 | 1 |  1 |  1 |  1 |  1 |  | 2 | 2 | 3 |  3 |  5 |  6 |  7 |  | 3 | 3 | 5 |  6 |  6 |  7 |  7 |  | 4 | 4 | 7 |  9 | 10 | 10 | 11 |  | 5 | 5 | 9 | 12 | 14 | 15 | 15 |  +---+---+---+----+----+----+----+    ^    |    s

If $m(c, s)$ denotes the minimal number of distinguishable symbols, we can write down some (trivial) things:

  • $m(1, s) = s$
  • $m(2, s) = 2s - 1$
  • $m(c, 1) = 1$
  • $m(c, s) \le c * (s - 1) + 1$

The last inequality comes from the fact that one can always create the following configuration to meat the criterion:

1 1 1 1 ...  2 4 6 8 ...  3 5 7 9 ...

One column represents one card. This pattern can be applied to any $c$ and $s$ and the amount of different symbols is always $c * (s - 1) + 1$.

Furthermore the columns seem to develop according to $m(c, s + 1) = m(c, s) + c$ for larger $s$ and the rows form according to $m(c + 1, s) = m(c, s) + s - 1$ for larger $c$.

I've also searched the Online Encyclopedia of Integer Sequences for rows and columns of this sequence, but I couldn't find any promising results.

Nevertheless, I have no idea how to develop a formula for such a problem and would be thankful for your help.

Brownian bridge sde

Posted: 08 May 2021 07:01 PM PDT

The SDE for the Brownian bridge is the following:

$$dX_t = \dfrac{b-X_t}{1-t} \, dt+dB_t$$

with the solution

$$X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}.$$

The expectation and covariance are:

$$\mathbb{E}(X_t) = a+(b-a)t$$

$$\operatorname{Cov}(X_s,X_t) = \min(s,t)-st$$

Now I want to have a look at what happens as $t\rightarrow 1$.

For the expectation and covariance I get

$$\mathbb{E}(X_1) = b,$$

$$\operatorname{Cov}(X_s,X_1) = \min(s,1)-s$$

But I'm having trouble to see what happens with $X_t$. The first two summands clearly go to b, and the last summand should go to 0 as Brownian bridge expression for a Brownian motion suggests. The prove in the last comment using Doob's maximal inequality and Borel-Cantelli is quite short and I don't understand, what's exactly happening there, especially not, where the last equation comes from. Would be great if someone could explain it more exact how I get $$\lim_{t \rightarrow 1} (1-t)\int_0^t \frac{dB_s}{1-s} = 0 \text{ a.s.} $$

Thanks in advance!

Prove that isomorphic rings have the same characteristic

Posted: 08 May 2021 07:16 PM PDT

If $\phi: A \to B$ is a ring isomorphism, I have to proof that $\operatorname{char} A= \operatorname{char} B$.

I know that an isomorphism $\phi$ is bijective and: $$\phi(x+y)=\phi(x)+ \phi(y)$$ $$\phi(x*y)=\phi(x)*\phi(y)$$ $$\phi(1)=1$$

I have supposed that $\operatorname{char} A=n$, so: $n1=\underset{n\text{ times}}{\underbrace{1+1+\ldots+1}}=0$ in $A$.

As $\phi$ is surjective, $n1\in A$ exists, where $x=\phi(n1)$.

Now, $\phi(n1)=\phi(0)=\phi(1+1+\ldots+1)=\phi(1)+\ldots+\phi(1)=n*\phi(1)=n$.

I don't know if this is correct and if this is how to finish the proof. Could you help me please?
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