# All Questions

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### Try to prove that a discrete distribution function is a singular distribution function

actually it is the 6th question in the section 3 of chapter 1 in the book named by The course in probability theory, author:Chun Kai Lai. someone asserts that the derivate of the discrete function on ...
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### Compact elements of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field

Let $F$ be a nonarchimedean local field and let $\mathcal{O}$ be its ring of integers. An element $g$ of $\mathrm{GL}_n(F)$ is called compact if the cyclic subgroup that it generates has compact ...
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### Contractibility of space of embeddings of a disc

I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$. The ...
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### Connes on Integers / Primes and Quantum Field Theory / Elementary Particles

I was browsing Matthew Watkins' Number Theory and Physics archive, and came across the following quote by Alain Connes (without context, aside from the fact that it was in a popular book called Dr. ...
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### Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...
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### curvature flow for loops in S^2

Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...
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### co-dimension one minimizing verifolds

It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the ...
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### Online algorithm for nested optimization problem(with locally optimization)

How to construct a sequence ${x_t;\theta_t}$, which is online algorithm for following optimization problem: $\arg\min_\theta \sum_t\min_{x_t}\ell_t(x_t;\theta)$ For simply, we can assume ...
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### Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...
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### same paper was published in the same journal twice

I just realized that the paper "Hitchin's connection and differential operators with values in the determinant bundle" by Xiaotao Suna and I-Hsun Tsai and was published twice in the Journal of ...
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### L-function of twist

I'd like to ask the following easy question, since I can't find a reference. Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...
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### On the local L-parameter of U(3) obtained by the theta lift from U(1)

Let $E/F$ be a quadratic extension of number fields and $v$ a place of $F$. Let $\sigma$ be a automorphic character of $U(1)(\mathbb{A}_F)$ and $\Theta(\sigma)$ the theta lift of $\sigma$ to ...
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### Characterization of a subset of $[0,1]$

Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property: For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
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### Holder continuous analytic function

Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
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### Solution or Reference Request for a Closed Form of the Sum

I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol ...
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### How to distribute groups over activities in rounds [on hold]

This problem started with my sister asking me how she can distribute 12 groups over 6 activities in 6 rounds. She wants to organize a camp for for year students. There are 6 acitvities and in every ...
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### Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...
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### edge transitivity and edge deletion

Let G be a graph which has the following properties: 1) For every $e_1,e_2 \notin E(G)$, $G \cup e_1 \cong G \cup e_2$ 2) For every $e_1,e_2 \in E(G)$, $G\setminus e_1 \cong G\setminus e_2$ i.e. ...
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### Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
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### Square filling self avoiding walk

I want to create an algorithm that fills a square grid with a random Hamiltonian path starting at a particular point. See this example. One approach is to try a free direction as a next step, and ...
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### Learning roadmap in Algebra [on hold]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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### Expanding graphs from iterated zig-zag product

Let $\Gamma \, zz\, G$ denote the zig-zag product of graphs $\Gamma$ and $G$. Reingold, Vadhan, and Wigderson proved that the second largest eigenvalue of the normalized adjacency matrix of the ...
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### How can I prove: If A is a subset of C and B is a subset of D, then the union of A and B is a subset of the union of C and D? [on hold]

How could I write a proof for the above statement, considering I'm studyng the first Enginnering math's course? (in other words, my math level is pretty basic) Thanks in advance
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### How many simultaneous polynomial equations of degree 2 can software solve today?

Consider the following problem: Input: $n$ polynomial equations of degree $2$ in approximately $n$ variables. Each equation contains about $\sqrt{n}$ monomials. We would like to find one ...
Does anyone know of a convergence test for a complex series of the form $$\sum_n a_n \cdot \exp(i \cdot b_n)$$ ? The particular series I need to understand has a_n going to zero as n goes ...
### Expression and growth bound for $r_{p^m,k}(n)$
Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...