# All Questions

**5**

votes

**0**answers

63 views

### Why is there a $\sqrt{5}$ in Hurwitz's Theorem?

Hurwitz's theorem is an extension of Minkowski's Theorem and deals with rational approximations to irrational numbers. The theorem states:
For every irrational number $\alpha$, there are infinitely ...

**1**

vote

**0**answers

31 views

### How do we respond to the question, “What is mathematics?”? [on hold]

The mathematical community has offered various responses, depending on who was asking, and depending on what century it was, and other variables. The Wikipedia article "Definitions of mathematics" ...

**1**

vote

**0**answers

43 views

### Diagonalizing against a non stationary set of functions

Suppose $\kappa$ is regular uncountable (assume $2^{< \kappa} = \kappa$ and $\kappa$ is weakly Mahlo if needed). Is the following true?
For every sequence $\langle f_i: i \to 2 \mid i \in A ...

**0**

votes

**2**answers

48 views

### Operator on a Sobolev space

I'm studying Sobolev spaces using Evans' PDE book.
I can't figure out this simple fact.
Let $L$ be an operator in this form:
$$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$
I can't understand why ...

**0**

votes

**0**answers

3 views

### More precise statement about lower bounds on the cover time of general graphs

Uriel Feige has shown in 1995 in his paper "A Tight Lower Bound on the Cover Time for Random Walks on Graphs", the following result:
For any graph $G$ on $n$ vertices, and any starting vertex $u$ ...

**0**

votes

**0**answers

18 views

### Period of closed phase curve [on hold]

I'm currently reading Arnolds "Mathematical Methods of Classical Mechanics" and I'm having a hard time solving some of the problems in Chapter 2.
I think that the following problem is fairly simple ...

**1**

vote

**0**answers

43 views

### Actions of the unit circle on complex matrices

Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle.
How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on ...

**3**

votes

**2**answers

84 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

**0**

votes

**0**answers

6 views

### lattice basis reduction of the orbit of a rational vector on the torus

LEt $v=(p_1/q,...,p_n/q)$ be a vector of the torus $\mathbb{T}^n$, such that for any $i$, $p_i$ and $q$ are relatively prime. Let $L= \{ kv \mod \mathbb{T}^n , k=0,...,q-1 \}$.
What is the lattice ...

**0**

votes

**0**answers

34 views

### Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...

**0**

votes

**0**answers

42 views

### Well-ordered reference

I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.)
Let $(S,<)$ ...

**0**

votes

**0**answers

21 views

### Enumeration of simple graphs with given degree distribution/sequence

Is there any exact formula for asymptotic/exact enumeration of simple graphs with given degree sequence? I just found some results about it, but the formula is hold on for some conditions, for example ...

**0**

votes

**0**answers

16 views

### Motion tracking formula by tracking information [on hold]

I have tracking information of a short clip that demonstrates a first person view flight.
The red dot is a x,y tracking information of a specific point in the clip.
Additionally, we have the angle, ...

**0**

votes

**0**answers

37 views

### What does the Plancherel theorem say about positive-definite distributions?

I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...

**2**

votes

**0**answers

58 views

### Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block.
Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...

**3**

votes

**1**answer

336 views

### Is a non-trivial finite perfect group of order 4n?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.
Question: Is a non-trivial finite perfect group of ...

**0**

votes

**0**answers

13 views

### Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point.
Let us assume that $C_1$ and $C_2$ are two ...

**2**

votes

**0**answers

12 views

### Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial.
proof: First if ...

**-1**

votes

**0**answers

36 views

### examples of manifolds having two ends with $Ric \geqslant -(n-1)$ but not split

Let $M$ be an $n$-dim complete, non-compact Riemannian manifold with two ends. Then we can construct a line on $M$. If we assume that $Ric \geqslant 0$, by Cheeger-Gromoll's splitting theorem, the ...

**-1**

votes

**0**answers

19 views

### Solve Higher ODE with boundary condition at infinity [on hold]

Given below is a second-order linear differential,
$y''+Ay'+By=0$
The boundary conditions are:
(a) $y(t=0)=1+H\frac{dy}{dt}$
(b) $y(t=\inf)=0$
Does a solution exist for this problem?

**6**

votes

**2**answers

87 views

### Moments of a random variable and of its conditional expectation

Let $X$ be a bounded random variable with $\mathbb{E}X=0$. Since $X$ is bounded, all its moments exist. Let $\mathcal{G}$ be any $\sigma$-field and let $Y:=\mathbb{E}[X|\mathcal{G}].$ I am interested ...

**8**

votes

**1**answer

154 views

### Is an explicit $c$ known to lead to a noncomputable Julia set?

Braverman & Yampolsky have shown that there exist noncomputable Julia sets,
i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$
is not computable.
"A set is ...

**5**

votes

**2**answers

67 views

### Differential operators between modules, $\mathcal{D}_A(M, M)$ necessarily a filtered, almost commutative ring?

See here. Does it follow immediately that $\mathcal{D}_A(M, M)$ as defined in the link is a filtered, almost commutative ring? How can I visualize this geometrically?

**13**

votes

**2**answers

321 views

### What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?

Over the years, advances in machine learning has allowed us to communicate and interact, using the same natural language, more and more semantically with computers, e.g. Google, Siri, Watson, etc. On ...

**0**

votes

**0**answers

46 views

### higher direct images of relative canonical sheaf plus a fractional divisor

For a map $f: Y \rightarrow X$ branched over simple normal crossing divisor $B=\sum_iB_i$, do we know of similar local freeness property for higher direct image of relative canonical sheaf plus a ...

**-5**

votes

**0**answers

32 views

### profit and loss related aptitude qustion? [on hold]

A sold a watch to B at a gain of 5% and B sold it to C at a gain of 4%. If C paid Rs. 1902 for it, the price paid by A is?

**2**

votes

**3**answers

123 views

### Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...

**0**

votes

**0**answers

9 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**0**

votes

**0**answers

48 views

### Intersections of ideals and nilpotence

Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...

**2**

votes

**0**answers

62 views

### Infinite-dimensional admissible representations of SL(2,C)

I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...

**0**

votes

**0**answers

50 views

### Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...

**1**

vote

**2**answers

88 views

### Embedding finite lattices into the lattice of partitions of a finite set

For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.
Let $L$ be a ...

**3**

votes

**0**answers

53 views

### Relations between modular functions of certain $q$-continued fractions

Given the j-function $j:=j(\tau)$, and $q=e^{2\pi i\tau} = \exp(2\pi i\tau)$ where, for convenience, we set $\tau=\sqrt{-n}$.
I. $\frac{A_2(q)}{A_1(q)} = \text{q-cfrac}:\;$ Icosahedral group
...

**7**

votes

**2**answers

382 views

### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a ...

**14**

votes

**2**answers

311 views

### Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...

**-1**

votes

**0**answers

27 views

### Stochastic integration with respect to Fractional Brownian Motion

I would like to know what can be said about integral process
$X_t = \int_0^t e^{-sr} dB_s^H,t\in[0,\infty)$, where $B^H_t$ is Fractional Brownian Motion with Hurst parameter $H>\frac{1}{2}$, ...

**13**

votes

**2**answers

603 views

### Counterexamples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...

**-4**

votes

**0**answers

66 views

### Is Zsigmondy's Theorem utilized in Sociology to the extent that a lay person might be familiar with its use? [on hold]

Is Zsigmondy's theorem somehow useful in sociological studies? Has it in some way been co-opted as a sociological term with a corresponding sociological theory, no matter how far off base from the ...

**2**

votes

**2**answers

609 views

### About Abhyankar's conjecture

I just came to this conjecture (proved by M.Raynaud and D.Harbater in 1994) last weekend, in Fresnel and v.d.Put's book Rigid Geometry and Its Applications. It claims that all quasi $p$-group $G$ ...

**2**

votes

**1**answer

104 views

### Proving inequation with ceilings in Finite Field of characteristic $p$

Take $ui = pt_i +j_i$ where $p$ is a prime number and $u(p-r) \equiv 1 $ $(\mbox{mod p})$ for positive integers $1 \le i, r, j_i\le p-1$ and $t_i \ge 0$. How can I prove that:
\begin{equation}
...

**2**

votes

**0**answers

44 views

### Expected absolute value of the determinant of an $n$ by $n$ Toeplitz $(0,1)$ matrix

If $A$ is chosen uniformly over all $n$ by $n$ $(0,1)$-Toeplitz matrices, what is the expected absolute value of the determinant?
Alternatively, what if anything is known about the determinant of ...

**2**

votes

**1**answer

53 views

### A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...

**11**

votes

**1**answer

236 views

### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties ...

**-3**

votes

**0**answers

76 views

### Non-flat $R \subseteq S$, which is integral, separable, $R$ is a noetherian (not integrally closed) integral domain

On ramification theory in noetherian rings,
of Auslander and Buchsbaum say:
"Chapter 4 is devoted to showing that under various conditions if $S$ is unramified over $R$, then $S$ is $R$-projective. ...

**14**

votes

**1**answer

317 views

### Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem ...

**2**

votes

**0**answers

88 views

### Are these two $q$-continued fractions equivalent?

In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24,
$$U(q) = ...

**8**

votes

**2**answers

162 views

### Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...

**3**

votes

**0**answers

86 views

### Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...

**0**

votes

**0**answers

68 views

### What is the significance of the eigendecomposition of a Cayley table? [on hold]

Treating the Cayley table of a group $G$ as a matrix $M_g$, one notices interesting things about its eigendecomposition.
For instance, for the symmetric groups $\{S_n\}$, the rank of the Cayley table ...

**4**

votes

**1**answer

88 views

### Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form
$P( \Vert X \Vert_\infty < r ),$
where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...