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Bayesian Update: Poisson Distribution
Consider a hypothetical bin containing countably infinite number of balls. A machine draws balls randomly according to Poisson distribution with parameter $\lambda>0$, namely the probability the ...
- reputation score 1361
Score of 2
1 answer
67 views
Prove that $0 < a < b$ implies $0 < a^n < a^{n-1}b < \cdots < ab^{n-1} < b^n$ and $0 < a^{1/n} < b^{1/n}$
I encountered the fact that $a^2 < a^2 + b^2$ implies $$\sqrt{a^2} \leq \sqrt{a^2 + b^2}$$ in Baby Rudin's proof of Theorem 1.13(d), which states that for complex $z$, $|\text{Re }z| \le |z|$.
...
- reputation score 21
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1 answer
33 views
Proving smooth algebraic varieties remain smooth after base change by any field extension from first principles
Let $X$ be a smooth algebraic variety over a field $k$, and let $K/k$ be any field extension. I want to prove that
$$
X_K:=X\times_{\operatorname{Spec}k}\operatorname{Spec}K
$$
is smooth over $K$.
I ...
- reputation score 43
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45 views
How can I systematically count all integer pairs $(a,b)$ satisfying $(a+1)(b+1)=−2015$ and $a+b≥0$?
I am trying to solve the following problem:
Find the number of integer pairs (a,b) such that
$(a+1)(b+1)=−2015$,
where
$−2015=(−1)⋅5⋅13⋅31$,
and the additional condition
$∣a+b∣$ $=$ $a+b$
holds (...
- reputation score 1078
Score of 6
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46 views
What is the tangent space of $C^2(S^1,S^2)$?
Let $S^1$ is 1-sphere, $S^2$ is 2-sphere. They all have smooth differential structures and metric. Denote $C^2(S^1,S^2)$ as the set of 2 times continuously differentiable map from $S^1$ to $S^2$.
From ...
- reputation score 7480
Score of -1
1 answer
52 views
Sum of two squares is equal to square of a prime, then the solution is unique [duplicate]
It is different from “sum of two squares is a prime” case.
$a^2 + b^2 = p^2$ where $p$ is a prime, then $(a,b)$ is unique.
Any proof or links?
Thanks.
- reputation score 9
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92 views
Triangle ABC is valid if and only if what?
Is there any set of conditions for the side lengths of a triangle which we can call conditions $X$ such that the statement below is true?
Given a triangle ABC w/ side lengths $a, b, c$ (corresponding ...
- reputation score 1043
Score of 2
0 answers
29 views
Is the constant of the convolution with the Riesz potential attainable?
Suppose that $N \in \mathbb{N}$; $0 < \lambda < N$; $p, r > 1$ and
$\frac{1}{p} + \frac{\lambda}{N} = 1 + \frac{1}{r}$. The Hardy-Littlewood-Sobolev inequality states that convolution with ...
- reputation score 766
Score of 2
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31 views
Euler Number of Bundles on a Grassmannian
Consider the bundle $\mathcal{V}:=\bigoplus^{4n} \wedge^2 \mathcal{S}\to G(2,2n+2)$, where $G(2,2n+2)$ is the Grassmannian parametrizing $2$-subspaces of $(2n+2)$-space and $\mathcal{S}$ is the ...
- reputation score 646
Score of 3
1 answer
115 views
What distribution does this data fall into? It represents discrete count data over a time interval, but doesn't fit anything I've checked.
I'm working with a dataset of count data, specifically consumer complaints of businesses submitted to a third party platform within the last 3 years. Any given business has a non-negative integer ...
- reputation score 666
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32 views
Is it correct to think that the plane with the maximum metric “curves” around the point from which distances are measured?
I am trying to develop a geometric intuition for the metric space
$$(\mathbb{R}^2,d_\infty),$$
where
$$d_\infty((x_1,y_1),(x_2,y_2))
=
\max\{|x_1-x_2|,|y_1-y_2|\}.$$
When this space is drawn using the ...
- reputation score 471
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38 views
Does the theory of symmetric functions show up in statistics?
Given an I.I.D sample $X_1,...,X_n$ and parameter $\theta$, a point estimator $\hat{\theta}$ of $\theta$ is a function of the sample:
$$\hat{\theta}=g(X_1,...,X_n)$$
It follows from independence that ...
- reputation score 21
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44 views
On the dynamic invariant of $6n \pm 1$ twin-track arithmetic lattice and its consecutive prime structures [closed]
I am an independent researcher investigating the arithmetic and structural properties of prime distributions formulated within the twin-track lattice of $6n \pm 1$.
I would like to inquire about a ...
- reputation score 1
Score of 1
1 answer
148 views
If the sum of a sequence of positive numbers is infinite, is every point in the real plane accessible relative to the sequence?
This is a generalization of a question I asked previously, here: Sufficient condition for a point in the plane to be accessible. Let $x_n$ be an infinite sequence of positive real numbers. Let $P$ be ...
- reputation score 26016
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Bijective holomorphic maps have non-singular jacobian
I have some difficulties to understand the proposed proof at page 19 of Principles of Algebraic Geometry by J. Harris and P. Griffiths for the following result:
If $f : U \to V $ is a one-to-one ...
- reputation score 1122