0
votes
0answers
12 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: ...
1
vote
0answers
12 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 $E$ ...
0
votes
0answers
12 views

Lp harmonic and harmonic Lp; is it the same?

In Harmonic function theory there is a notion of Harmonic Bergman space. For an open set $\Omega$ in $\mathbb{R}^{n}$ and $1\leq p<\infty$ we denote by $b^{p}\left(\Omega\right)$ a set of all ...
1
vote
0answers
31 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
0answers
20 views

Hilbert polynomials and a moving lemma type question

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
0answers
5 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 the authors prove a chain rule for vector fields $B\in SBD(R^d)$ with ...
1
vote
1answer
43 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
0answers
27 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 ...
5
votes
1answer
47 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 ...
1
vote
0answers
17 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 ...
1
vote
0answers
46 views

alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
0
votes
0answers
32 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
1answer
23 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
1answer
42 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
0answers
10 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
1answer
18 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
1answer
27 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
2answers
50 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 ...
0
votes
0answers
60 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
0answers
20 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 ...
0
votes
0answers
32 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
0answers
16 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
0answers
49 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 ...
1
vote
0answers
23 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 ...
2
votes
0answers
86 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
0answers
26 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
0answers
62 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 ...
2
votes
0answers
79 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
0answers
46 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
0answers
42 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 ...
5
votes
1answer
225 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
0answers
93 views

Does SL(3,q) have a subgroup of order $q^3.(q^3-1)$

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
0answers
32 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
0answers
16 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
0answers
36 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 ...
3
votes
0answers
162 views

Group laws in class field theory

In the case of quadratic 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 local field, ...
1
vote
1answer
92 views

Questions about a possible way of representing construcive ordinal numbers

Let $K$ be the set of all total recursive functions of non-negative integers having only non-negative integers as values. Let $L$ be any well-ordered subset of $K$ in which the ordering $<$ is ...
4
votes
0answers
33 views

Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
2
votes
1answer
124 views

What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
6
votes
1answer
317 views

Adding sets not containing arithmetic progressions of length three by forcing

Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where: 1) $s\in 2^{<\omega}$, 2) $N\in \mathbb{N}$, 3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...
0
votes
0answers
39 views

separating parameters in generalized quadratic Gauss sum

The normalized generalized quadratic Gauss sum is defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$. Under what conditions on $c$ can we ...
5
votes
0answers
87 views

Tree property and singular strong limit cardinals

I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question: Theorem. It is consistent, relative to the existence of large cardinals, that ...
1
vote
0answers
89 views

When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space. Let $d$ a word metric on $\Gamma$ coming from some finite set of generators. My question is: Does there exist a ...
6
votes
0answers
95 views

Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups ...
1
vote
2answers
206 views

Tensor calculus on the frame bundle

Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its ...
3
votes
0answers
96 views

Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
1
vote
1answer
132 views

About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$. Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$. Let ...
-1
votes
0answers
50 views

Minimum rank non-negative matrix summations

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$. What is minimum $k$ such that $$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...
0
votes
1answer
44 views

Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where $\rho,s>1$, ...
-5
votes
0answers
55 views

I don't get how -6cos3xsin3x becomes -3sin6x in the later part [on hold]

y = cos²3x dy/dx = 2cosx(-sin3x)(3) = -6cos3xsin3x = -3sin6x I found this answer key in my guidebook but I can't find any trigonometric function's or differentiation formula ...

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