# All Questions

25 views

162 views

### fixed vector of a generic representation of GL(n,F)

Let $F$ be a locally compact non-archimedean field and $G_{n}$ the locally profinite group $GL(n,F)$. Let $\Gamma_{n,k}$ be the subgroup of $G_{n}$ whose elements are the matrices of the form  ...
21 views

### How to firgure out if a set of vectors represent lines, planes or hyperplanes? [on hold]

if i am given a span of, let's say 3 vectors, what would be a way to determine if they represented a line, plane or a hyperplane? i have reduced siad vectors to reduced row echelon form, but don't see ...
33 views

1k views

### “Circular” domination in ${\mathbb R}^4$

The following problem is related to (and motivated by) the first open case of this MO question. It is difficult to believe that this is a hard problem; and yet, I do not have a solution. For two ...
76 views

### Random circle rotations

Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever ...
89 views

### Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...
33 views

### Algorithm proofs [on hold]

so I've got 2 algorithms in pseudocode: ...
26 views

### How do ideal sheaves behave on the special fibers of the projective line over the integers?

Let $X=\mathbb{P}^1_{\mathbb{Z}}$ and $Y\subset X$ be a local complete intersection of codimension two with Ideal sheaf $I_Y$. (I'm mostly interested in the case where $Y$ is a single point $x$ ...
51 views

### $a_{0}$ such that $0<\lim\sup_{n\to\infty}\frac{p_{n+k}-p_{n}}{H_{k}\log^{a_{0}}(\frac{p_{n+k}+p_{n}}{2})}<\infty$

This question is somehow a follow-up from Would the following conjectures imply Cramer's conjecture? Let $g_{n,k}$ denote the quantity $p_{n+k}-p_{n}$, $s_{n,k}$ denote the quantity ...
125 views

### Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?

From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
45 views

### Menon's identity basics [on hold]

this is the problem I'm having. It's pretty basic as you will see but still I hope you can clear this momentary confusion for me seeing that I'm stuck on this for a few hours. Menon's identity says ...
88 views

### Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$. Why ...
202 views

### Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5). Since the cited ...
89 views

### definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the transformation $(T − \lambda)/(T + \bar \lambda\ )$ with domain and range in the Hilbert space is contractive for ...
28 views

### differentiation of matrix norm [on hold]

Could you please help me how to derive differentiation of the following: d/dW |W - (1/K)a|^2 where || denotes frobenius norm, W denotes M-by-K (nonnegative real) matrix, a = w_1 + ... + w_K (w_k ...
73 views

### Fourier Transform of compactly supported $L^1$ functions

Background Given a (translation bounded) positive definite measure $\gamma$ lets say on $\mathbb R^d$, its Fourier transform as a tempered distribution is a positive measure $\widehat{\gamma}$. I am ...
28 views

### Strength of claims about extensions of partial preorders and orders to linear ones

Consider these two axioms: Every partial order extends to a linear order. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: ...
85 views

### efficient arithmetic with (short) Conway games?

We consider "games" in the sense of ONAG. Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations ...
220 views

### A problem in the domino shuffling algorithm

The domino shuffling algorithm first appeared in the following paper by Propp and Kuperberg: Alternating-sign matrices and domino tilings They used this algorithm to give a fourth proof that the ...
Let A be a (Lebesgue) measurable set in $\mathbb{R}^n$. Consider the 'cone with base A' $A(1) = \{\alpha x \in \mathbb{R}^n : x \in A, \alpha \in (0,1] \}$. Is B Lebesgue measurable? I assume it is, ...