# All Questions

14 views

### Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...
88 views

### A question about tensor product [on hold]

For every $f, g$ in $L^1(G)$, we know the function $[(x,y)↦f(x)g(y)]$ belongs to $L^1(G\times G)$ where here $\times$ means cartesian product. Why are these functions dense in $L^1(G\times G)$?
458 views

### Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$$ It is ...
53 views

### I would like to study Industrial Mathematics but needs to know it importance for project managers and the the development of third world countries [on hold]

Key Importance of Industrial Mathematics in the development of third world countries? Why industrial Mathematics important for Project Managers?
105 views

### Are piecewise linear functions dense in $W^{1,\infty}$?

Are piecewise linear functions dense in $W^{1,\infty}$ ?
347 views
+500

### Separating pure states on the $2\times 2$ matrix algebra

I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help. Let $\mathcal{A}$ be the C*-algebra of ...
60 views

101 views

### Non-finitely generated, non-projective flat module, over a polynomial ring [migrated]

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat ...
138 views

### Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
140 views

### Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
122 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?
260 views

### Recent progress on the busy beaver problem? [on hold]

Has there been any progress on the Busy beaver problem in the last few years? It seems like there hasn't been much work done on the problem since 2010. Is there anything amateurs can do to solve the ...
194 views

### Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
51 views

### Numbers, multiplication and subtraction [on hold]

given a and b find c and d such that bc-ad is least and greater than zero? Also a,b,c,d are integers and all lie inside a given range i.e. [0, n]. For example if n=50, a=48 and b=49, then c=49 and ...
82 views

### Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...
140 views

### Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$. What is its $\zeta$-regularized determinant? This should be ...
358 views

### Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?
17 views

### Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients. ...
106 views

### Permutation equivalence classes with kendall-tau distance

I asked this question on Stack Exchange a few days ago with no help. I am not sure if the question seems too trivial or of not enough general interest to get any attention. Anyway, any sort of ...
440 views

### Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
297 views

### Relation between moduli spaces and classifying spaces

I hope this question is suitable to be posted here on MO. I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...
128 views

### Number of bases of a matroid

I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
44 views

### Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed. In particular it is said that the poles form a deformed lattice that ...
54 views

### Maximal opening angle of a polygon from a point [on hold]

I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point. One ...
51 views

### Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...
46 views

### Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$ [migrated]

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if ...
77 views

### Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads: If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...
27 views

### convergence of empirical distribution of random vectors

Given (a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal entries $A_{ij}\sim \mathcal N(0, 1/n)$; and (b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...
38 views

### When does equality occur in the triangle inequality in metric space? [on hold]

When I think of R^n , n<=3 ; it is very easy given the usual metric. But what if the metric is not usual?
138 views

### Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...
46 views

### Pairwise distance distribution for point clouds (normal distribution) [on hold]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
121 views

### Flat cohomology of an ordinary liftable Calabi-Yau threefold

Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and ...
42 views

### biharmonic equation with L^1 data and Navier Condition

I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
189 views

### Counting function for prime pair with bounded gaps between them [duplicate]

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible. Yitang Zhang breakthrough result established that ...
104 views

### How many different sums of parts of a vector

I hope this question isn't too basic for MO. I also asked it on math.se previously. This mathematical puzzle was given to me by a friend a while ago and I can't work out how to solve it. Does anyone ...
88 views

### Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...