# All Questions

**5**

votes

**3**answers

195 views

### A balls and urns model for a hashing problem

Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c ...

**4**

votes

**0**answers

63 views

### Approximation of convex body by polytopes

In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$,
$$\rho_H(U,\mathcal{P}_n)\leq c(U) ...

**1**

vote

**0**answers

23 views

### Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations
$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$
where $\alpha=(\alpha_1,...,\alpha_s)$ and ...

**4**

votes

**1**answer

174 views

### Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...

**0**

votes

**1**answer

108 views

### $u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?

Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u_n ...

**3**

votes

**1**answer

133 views

### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...

**1**

vote

**0**answers

111 views

### Density of numbers whose prime factors all come from a fixed congruence class

Let $q$ be a positive integer greater than one, and let $a$ be an integer such that $\gcd(a,q) = 1$. Define
$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$
Do we know the ...

**-4**

votes

**0**answers

55 views

### The Birthday Paradox [on hold]

I was looking at the birthday paradox, and the many solutions. One of them that came up was the Poisson Distribution. The website I was looking at detailed the process to solve ...

**0**

votes

**0**answers

36 views

### Property of summations [on hold]

Suppose to have the following identity:
$$
\sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j) = \sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j)g(i,j),
$$
for 'good' indexes $i,j$ and some functions $f,g$.
What ...

**1**

vote

**2**answers

138 views

### Bertini theorem for big divisors and klt pairs

Let $X$ be a smooth projective variety and let $D$ be a big $\mathbb Q$-divisor on $X$. Assume that for $m$ large $|mD|$ has no fixed components. Is there a $\mathbb Q$-divisor $D'\equiv D$ so that ...

**8**

votes

**0**answers

120 views

### History of preservation theorems in forcing theory

For my honours thesis, I am studying a general preservation theorem using a framework provided by Shelah. I am mainly concerned about revised countable support iteration of $\dot{S}$-semiproper ...

**1**

vote

**1**answer

58 views

### Multinomial proxy variables: Bound on probability of their sums

Suppose $(X_1,X_2,..X_i,..,X_b)$ as multinomial vector of random variables with $N=\sum_{i=1}^b X_i$ and probabilities $p_i$ to parametrize the $X_i$.
Let us take the following imagination to ...

**2**

votes

**1**answer

74 views

### Coarsely trivial Borel cross section for $G\to G/N$

Let $G$ be a locally compact group, and let $N$ be a closed, normal subgroup, and let $\pi\colon G\to G/N$ be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a ...

**0**

votes

**1**answer

63 views

### Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? [on hold]

Let $G$ be a semisimple and simply connected linear algebraic group over $\mathbb{C}$.
Let $H$ be a connected, Zariski closed and semisimple linear algebraic $\mathbb{Q}$-subgroup of $G$.
Is $H$ a ...

**4**

votes

**1**answer

86 views

### Infinite dimensional Cauchy-Lipschitz theorem [duplicate]

From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation
$$
...

**3**

votes

**1**answer

66 views

### Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$.
Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...

**2**

votes

**1**answer

298 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**2**

votes

**0**answers

49 views

### Potentiality classes and Borel reductions

In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ ...

**8**

votes

**0**answers

182 views

### Neighborly family of coins

Here is a puzzle:
Find 5 identical coins. Can you arrange them so that every coin is touching every other coin?
The solution is here. The hint is: use the third dimension.
My questions are ...

**0**

votes

**0**answers

109 views

### Moving lemma for algebraic curves

Let $X$ be a smooth irreducible projective curve contained in $\mathbb{P}^3$ and $Y$ be another reduced but not necessarily irreducible curve in $\mathbb{P}^3$. Denote by $P$ the Hilbert polynomial of ...

**1**

vote

**0**answers

15 views

### Renormalization for Transport Equations with SBD velocity field

In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in ...

**2**

votes

**1**answer

159 views

### Blowing-up the Grassmannian at a point

Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider $G=Gr(r,n)$ and $V\in G$. I want to understand more explicitly what $Bl_V(G)$ should mean.
Of course for affine ...

**0**

votes

**0**answers

65 views

### Filmed lectures by Hassler Whitney

Are there any filmed lectures by outstanding American mathematician Hassler Whitney, besides the two Einstein Chair lectures below?
Old lectures, from the 1940s onwards, would be particularly ...

**7**

votes

**1**answer

153 views

### On the coherence theorem for bicategories

The coherence theorem for bicategories, as usually stated, reads
Any bicategory $B$ is biequivalent to a (strict) 2-category.
It is possible to give an explicit construction of the ...

**4**

votes

**1**answer

138 views

### Uncountably categorical theories which are interpretable in a strongly minimal

Definition: Let $\lambda$ be a cardinal. An $\mathcal{L}$-theory $T$ is called $\lambda$-categorical whenever every two models of $T$ of cardinality $\lambda$ are isomorphic.
Definition: An ...

**0**

votes

**0**answers

36 views

### Embedding rational simple algebras in the real quaternions [duplicate]

Is there any way to embed a rational division algebra of dimension higher than 4 over its center in the real quaternions ?
I think not, but I cannot prove it.

**0**

votes

**1**answer

50 views

### On a reciprocal of Ostrowski theorem on Newton polytopes and factorization

$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...

**0**

votes

**1**answer

85 views

### A bijection between Lusztig series induced by inflation

Context:
Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F ...

**0**

votes

**0**answers

27 views

### Unitary derivative and countable set

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...

**0**

votes

**1**answer

29 views

### Degree of the negative part of a divisor

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for more requirement). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree $n$ with ...

**1**

vote

**1**answer

41 views

### Graph of bounded continous functions with distance 1

Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$
We set $E = \{\{f,g\}: f,g \in ...

**2**

votes

**2**answers

84 views

### Completion of a single totally ordered down-set

This is a follow-up question to Complete sets of incompatible totally ordered down-set in a partially ordered set.
Let $(P,\leq)$ be a partially ordered set such that for every $p\in P$ the set ...

**3**

votes

**0**answers

188 views

### Galois correspondence subgroups/subsystems

In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
By applying this result to finite groups, we get a Galois correspondence ...

**1**

vote

**0**answers

32 views

### How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...

**-1**

votes

**0**answers

41 views

### Combinatorics: Identical objects and distinct groups [on hold]

I'm confused between the following 2 formulae:
1) Number of ways to put n identical objects into r distinct boxes, such that the ordering is NOT important is:
(n+ r - 1) C r
2) Number of ways to ...

**1**

vote

**0**answers

28 views

### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where ...

**0**

votes

**0**answers

53 views

### A combinatorial question on ranks

Denote
$$\mathscr{C}[r]=\{Q\in\Bbb Z_{\geq 0,\leq 1}^{n\times n}:\mathsf{rk}(Q)= r\}.$$
$$\mathscr{D}[A,t]=\{B\in\mathscr{C}[\mathsf{rk}(A)]:\mathsf{dim}(col(A)\cap col(B))\geq t\}.$$
Given ...

**2**

votes

**0**answers

52 views

### the choosing of an independent set in a specific $k$-partite graph

Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class ...

**5**

votes

**2**answers

286 views

### contractible configuration spaces

Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$.
My question: is $F(S^\infty,k)$ ...

**1**

vote

**0**answers

28 views

### Skein theory: How axiomatizing a 2-box space?

Let $(A,+,\times, *)$ with an adjoint operation compatible with $+$, $\times$ and $*$, such that $(A,+,\times)$ and $(A,+,*)$ are finite dimensional ${\rm C}^{*}$-algebras.
What are the axioms on ...

**5**

votes

**0**answers

93 views

### How do you categorify the cycle index series?

Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be
$$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$
It was observed by Baez and Dolan in their paper ...

**3**

votes

**1**answer

146 views

### Question about of comeager set

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...

**-3**

votes

**0**answers

51 views

### Limit of (n^2+1)^(1/n) [on hold]

I am struggling to figure out $\lim\limits_{n \to inf} \sqrt[n]{n^2+1} $.
I've tried manipulating the inside of the square root but I cannot seem to figure out a simplification that helps me find the ...

**-4**

votes

**0**answers

49 views

### Is a vector space with two identical vectors a vector space with one or two vectors? [on hold]

I'm new this, and cannot find any answers by searching. If a vector space has 2 identical vectors, in particular the zero vector, is it a vector space with 2 vectors or since they are linearly ...

**7**

votes

**1**answer

316 views

### Regularized sums of Mobius sequence

Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$?
Experimentally this seems plausible (up through ...

**1**

vote

**0**answers

103 views

### Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$ [on hold]

Let $q=p^n$ for $p>3$.I want to know whether the group $G_2(q)$ has a subgroup of order $q^3.(q^3-1)$. First I look the paper of Peter Kleidman. In this paper Kleidman determined all maximal ...

**0**

votes

**0**answers

64 views

### Help in finding the distribution and pdf

Considering a set of $n$ points that are $d$ dimensional and are independently and uniformly distributed on a surface. The points are homogeneous poisson point process.
Considering nearest neighbor ...

**0**

votes

**0**answers

19 views

### exit time of a non degenerate diffusion

Let $n, d \geq 1$, $b : \mathbb{R}^n \to \mathbb{R}^n$ and $\sigma: \mathbb{R}^n \to \mathbb{R}^{n \times d}$ two Lipschitz functions.
We assume that
\begin{equation}
\exists \mu >0, \xi^T ...

**0**

votes

**0**answers

37 views

### Non Normal operator [on hold]

Standard example for non normal operator is the shift operator. It is continous but the image of the left shift is not dense. Can we have an example of a non normal operator $A$ which is continuous ...

**5**

votes

**1**answer

294 views

### Group laws in class field theory

In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.
In the case of a ...