3
votes
2answers
129 views

Plane measurable sets and measurable rectangle

Does every measurable set in the plane with positive Lebesgue measure contain a cartesian product of two measurable sets of the real line with positive Lebesgue measures?
1
vote
1answer
150 views

Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...
4
votes
0answers
108 views

Reference to forcing with a sigma ideal $\cong$ Cohen forcing

This is a historical question: Who was the first person to notice the following? If $V \models \kappa$ is measurable and $P$ adds $\kappa$ Cohen reals, then in $V^P$, letting $\hat{I}$ to be the ...
0
votes
0answers
56 views

Curvature of vector bundles associated to holomorphic fibrations

Let $D=U\times \Omega$ in $\mathbb C^m\times\mathbb C^n$ be a pseudoconvex domain and $\phi$ is a strictly plurisubharmonic function on $D$. We suppose that $\phi$ is smooth up to the boundary. Now, ...
-4
votes
0answers
28 views

How are residuals calculated in rugarch package [on hold]

I have a question regarding the rugarch package. I try to fit a ARMA(1,1)+GARCH(1,1) to a time series $x$ using the following command: spec <- ugarchspec(variance.model=list(model="sGARCH", ...
0
votes
0answers
62 views

Sources on evolution of submanifolds subject to Ricci flow

I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome. This old MO post may be ...
0
votes
0answers
149 views

Determine existence of irreducible variety in given homology class

Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$? ...
10
votes
2answers
317 views

Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML. There is a term "Partially Observed Groups" in machine learning theory which has been popularized by ...
4
votes
1answer
73 views

Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing. The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of ...
-1
votes
0answers
88 views

Is the cusp point of the curve $y^2=x^3$ a regular embedding? [on hold]

This question maybe trivial. Let's consider the curve $y^2=x^3$ over a field $k$. It seems to me that the point $(0,0)$ is a regular embedding of codimension $1$ because it is given by the equation ...
2
votes
1answer
59 views

Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [on hold]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...
9
votes
2answers
208 views

Embedding $\mathrm{PGL}(n,q^h)$ in $\mathrm{PGL}(nh,q)$

It is not very hard to see that for each prime power $q$ and natural numbers $n,h$, we have an embedding $$\iota \colon \mathrm{GL}(n,q^h) \hookrightarrow \mathrm{GL}(nh, q),$$ obtained by choosing a ...
20
votes
1answer
898 views

Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather ...
0
votes
0answers
31 views

When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?

Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$? The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective. Any criterions ...
5
votes
0answers
61 views

What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?

The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the ...
-4
votes
0answers
52 views

indecomposable finite groups [on hold]

A finite group $G$ is called indecomposable if it is non trivial and it cannot be decomposed as direct products of two non trivial subgroups, if not it's called decomposable. $\textbf{Examples:}$ ...
-5
votes
0answers
53 views

Is this set compact? [on hold]

Let A denote the set of $m\times n$ matrices whose entries bounded by $1$ in absolute one. Is the set $A$ a compact set in the smooth function space $C^1(R^n,R^m)$ with $C^1$ topology?
1
vote
0answers
24 views

Find coordinate infinity points from unity point using synthetic geometric constructions [migrated]

A common way to put coordinates on $\mathbb P^k\mathbb R$ is to choose $k+2$ points (such that no one of them lies on the hyperplane generated by any $k$ of the others) and interpret them as the ...
-2
votes
0answers
51 views

valuations on residue field of an algebraically closed nonarchimedean field [on hold]

Let $C$ be an algebraically closed nonarchimedean field, and $k$ be its residue field (like $\mathbb{C}_p$ and $\bar{F_p}$). We know that continous valuations on $C$ are 1-1 corresponds to valuations ...
1
vote
0answers
65 views

Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question. Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...
-2
votes
0answers
24 views

Derive a closed formula for the generating function of this recurrence relation [on hold]

This is a given recurrence relation $a_n=19(F_0a_{n-1}+F_1a_{n-2}+...+F_{n-1}a_0)$ where $F_n$ is Fibonacci number and $a_0=9$. Find the generating function $A(x)$ of the sequence $a_n$ I get the ...
-3
votes
0answers
28 views

Packing, Optimization [on hold]

I have a large box of 82cm * 82cm * 120 cm(this is always constant). How many smaller boxes of size 59cm* 34cm *43 cm can I fix in the large box? The size of the smaller boxes are the same and ...
2
votes
0answers
36 views

Hess-Schrader-Uhlenbrock inequality for non-symmetric operators

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some ...
0
votes
0answers
64 views

Pull-back of reflexive sheaves

Let $X$ be a noetherian, projective scheme, $\mathcal{F}$ be a reflexive sheaf on $X$ pure of dimension $\dim(X)$ and $Y \subset X$ be a closed subscheme of $X$. Is it possible that the pull-back of ...
11
votes
2answers
462 views

Obstructions for a group to be the multiplicative group of a field [duplicate]

It is well known that every finite multiplicative subgroup of a field is cyclic. I somehow got interested in a possible reverse implication: Assume we have an abelian group $G$ whose every finite ...
2
votes
0answers
47 views

lower bound on A(k,4,floor(k/2))

A(k,4,r) is the independence number of the Johnson graph J(k,r). What is the best known asymptotic lower bound on A(k,4,floor(k/2)) ? I only obtained ...
1
vote
0answers
27 views

Lattice-isotopic essentialization of arrangements

I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
2
votes
1answer
143 views

Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$. Let $\epsilon\in[\frac{1}2,1)$. Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...
6
votes
6answers
429 views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
1
vote
1answer
75 views

Is $\textbf{FHILB}$ locally regular?

Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this ...
0
votes
0answers
61 views

Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...
1
vote
0answers
75 views

Condition for a finite group scheme to be étale [on hold]

My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a ...
6
votes
2answers
148 views

Models for graphs representing real-life networks

I am interested in basic models of graphs (stochastic or deterministic) that are offered for real-life networks (like social networks, the Internet, neuron networks). I will be thankful for answers ...
3
votes
2answers
101 views

Minimal Hausdorffness reversed

It turns out that not every Hausdorff topology is contained in a minimal Hausdorff topology. Let's put this question on its head: is every non-$T_2$ topology contained in a topology that is maximal ...
2
votes
1answer
119 views

A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question. In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is ...
0
votes
0answers
42 views

compute standard basis in local rings

Let$>'$ be the order in $k[t,x_1,\cdots,x_n]$ as follows: Each semigroup order > on monomial in the $x_i$ extends to a semigroup order >' on monomial in $t,x_1,\cdots,x_n$ in the following way. We ...
1
vote
2answers
96 views

Poset-enrichment of abelian categories

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another ...
-2
votes
0answers
236 views

When do boring objects exist? [closed]

Let's provisionally call an integer boring if it is not the root of a polynomial over $\mathbb{Z}$ with a small number of variables and with small coefficients and arguments (note that this requires ...
6
votes
2answers
870 views

What defines a “short proof”?

I would like to know what the definition of a short proof is. In Lance Fortnow’s article “The Status of the P Versus NP Problem”, Communications of the ACM, Vol. 52 No. 9, he says, If a formula θ ...
-1
votes
0answers
51 views

$GL(2, \mathbb{Z})$ modular form [closed]

Recall that an ordinary modular form (of weight $k$) is a holomorphic function on the upper half plane $\mathbb{H}^+$ satisfying $$ f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{k} \, f(\tau) . ...
1
vote
3answers
256 views

Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...
0
votes
0answers
96 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
0
votes
0answers
67 views

Factoring a polynomial in a specific manner

Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive ...
1
vote
1answer
334 views

Atiyah's vector bundles over an elliptic curve

I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs. Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a ...
1
vote
1answer
62 views

Linear algebra formulation for colored node graph isomorphism

(Please see a few paragraphs below by what I mean by “colored node graph isomorphism”.) Some basic definitions for completeness: Given two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ the graph ...
-2
votes
0answers
73 views

Parallel transport along a geodesic and the related Jacobi field [closed]

Crossposted from: http://math.stackexchange.com/questions/1255018/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found mentioned in ...
0
votes
1answer
87 views

Backward Uniqueness for the wave equation [closed]

Does the wave equation $u_{tt} - \Delta u = 0$ have any backward uniqueness results that are similar to the ones for the heat equation (see for example Theorem 11 page 64 in Evans)? If not, are there ...
3
votes
2answers
212 views

Residue for the generating function of the Euler totient function

Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series $$ f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n $$ Since $0\le \varphi(n)\le n$, I believe this gives a ...
1
vote
0answers
72 views

Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...
-1
votes
1answer
65 views

Algebraic Groups of Type H_3 and H_4 [closed]

By coincidence i stumbled over this page http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html , which was installed for a workshop on algebraic groups in 2012. In the ...

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