5
votes
2answers
163 views

Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$ (although my question may as well be asked for the surface of a polyhedron). Say that $\gamma$ is a shortest halving curve if (a) it partitions the ...
3
votes
0answers
78 views

Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose negative answer I showed by a dimension-increasing map.) Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...
2
votes
1answer
56 views

Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
2
votes
1answer
66 views

Where can I find a translation of Caspar Wessel's “Om directionens analytiske betegning?”

I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...
4
votes
1answer
130 views

Is a smooth union of lines is a subspace?

I apologize if this is a trivial question. Let $X$ be a closed affine subvariety of $\mathbb C^n$ given by the union of lines trough the origin. Does $X$ smooth implies $X$ is a subspace of $\mathbb ...
0
votes
0answers
67 views

Does anyone know of any rapidly converging series for the reciprocal gamma function?

I have seen rapidly converging infinite series for Pi and the such but none for either the gamma or reciprocal gamma function. does anyone know of any that I can use for comparison to mine? Thanks!
4
votes
0answers
85 views

Large subspaces with small basic constants in finite-dimensional Banach spaces

Let $B\in(1,\infty)$. I am interested in estimates for the function $f_B(n)$ defined as the largest $k\in\mathbb{N}$ satisfying the condition: Each $n$-dimensional Banach space contains an ...
1
vote
1answer
117 views

Maximal compact subgroups of a semisimple Lie group are conjugate

I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps: Take one maximal compact ...
2
votes
1answer
181 views

RO(G) grading of Mackey functors

If G is a finite group, I understand that the category of RO(G)-graded spectra, when rationalized, becomes Quillen equivalent to the category of Mackey functors valued in chain complexes of rational ...
2
votes
2answers
188 views

Arbitrarily large number of representations

For a positive integer $k$ let $\gamma_k(n)$ be the number of representations of $n$ as a sum of strictly increasing perfect $k^{\text{th}}$ powers. For example $\gamma_k(2)=0$ for any $k$. Now is the ...
2
votes
1answer
81 views

Generic vs General property of reducedness in a family of projective schemes

Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...
2
votes
0answers
20 views

Implicit/Explicit Time Dependence for Melnikov Functions

My question concerns an article by Koiller and Carvalho found here: http://link.springer.com/article/10.1007/BF01260390 On page 645, they parameterize the time variable $t$ in terms of one of the ...
1
vote
0answers
63 views

Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...
-4
votes
0answers
55 views

If a family of complex analytic space is smooth [on hold]

If I have a deformation in complex analytic space setting. The parametrize space is U, total space is X. If U is smooth and every fiber X(t) is smooth, if X is smooth and the map from X to U is ...
1
vote
2answers
80 views

Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations. Let $\mathcal{P}(X)$ the power set of $X$. Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X. We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
2
votes
0answers
83 views

Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of Are there workable algebraic geometry approaches for the pentagon equation? I've replaced "algebraic geometry" by "numerical" in its content, ...
2
votes
1answer
124 views

Covering by subsets

There is set $A$ with cardinality $2^n$. For every $x \in A$ there is $A_x$ - subset of $A$ with cardinality $2^m$, $x \in A_x$. $M=\{A_x|x \in A \}$. Are there $B \subset A$ with cardinality $\ge ...
-1
votes
0answers
141 views

Research topic selection [on hold]

I am planning to do my PhD in either sheaf theory or generalised functions.pls help me choose one among them by pointing the scope and relevance. thanks in advance.
0
votes
1answer
52 views

A question on parallelizability

Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?
0
votes
0answers
75 views

Weierstrass model for Calabi-Yau fourfolds

The local Weierstrass model for a Calabi-Yau threefold looks like: $$ y^2 = x^3 + f\left(z_i\right)x + g\left(z_i\right) \subset \mathbb{P}^2_{\left[x,y\right]} $$ With $f\left(z_i\right) \in ...
0
votes
0answers
65 views

The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
0
votes
0answers
142 views

Zariski open set of linear forms

Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$. I am looking for a Zariski open ...
0
votes
0answers
7 views

How to characterize elements in the Bruhat open cell? [migrated]

This might be an elementary question. For simplicity, let's assume $G=GL(n,F)$, where $F$ is a local field. Let $U$ be the subgroup of upper triangular unipotents, $A$ the subgroup of diagonal ...
-5
votes
0answers
44 views

Alternating series sum [on hold]

How to find such alternating series sum? \begin{equation} \sum_{n = 0}^\infty \frac {(-1)^{n-1}}{3^{n-1}} \end{equation}
1
vote
0answers
85 views

Open covering of the Hilbert functor of points

Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then ...
-2
votes
0answers
62 views

Is der a relation between the product of 2 numbers and their sum? [on hold]

a * b = x a + b = y Is there a way to obtain y from x without using a,b ? PS: a,b are prime-numbers. so there is only one possible y.
1
vote
0answers
179 views

Is Independent University of Moscow recognized? [on hold]

What graduate schools recognize the degree from Independent University of Moscow? It is not a university strictly speaking and their degree doesn't have any official status in Russia, but they claim ...
9
votes
0answers
267 views

Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...
3
votes
0answers
60 views

Software tools for medium-scale systems of polynomial equations

I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...
-4
votes
0answers
56 views

Symmetric group representation from alternating groups [on hold]

Is the symmetric group $S_n$ isomorphic to a direct sum or tensor product of alternating groups $A_m$? Can any combination of alternating groups ($(A_m \otimes A_m)\oplus A_m$ .. etc) yield a valid ...
0
votes
0answers
155 views

Non-Abelian Fourier Analysis

I'm currently thinking to generalize a known result on abelian groups to non-abelian groups. This is the problem. Fix an abelian group $G$. We know that: $\mathbb{E}_{\chi\in ...
2
votes
2answers
140 views

An inequality concerning convexity and expectation

Assume $f$ and $g$ are nonnegative with $$\int_0^\infty f(x)dx=1=\int_0^\infty g(x)dx $$ and $$\int_0^\infty xf(x)dx<\infty > \int_0^\infty xg(x)dx $$ Is it true for nonnegative numbers $p$, $q$ ...
-1
votes
0answers
41 views

Geometry of the triangle: from the centroid to the orthocenter and beyond [on hold]

Let A', B', C' be points on the perpendicular bisectors of the sides of a triangle ABC so that the angles A'BC, B'CA and C'AB are equal. The lines AA', BB' and CC' concur. Which is the locus? Do you ...
3
votes
2answers
109 views

Pairwise dependent random walk recurrent

Let $\{D_i\}_{i=0,1,2,\dots }$ be independent $\exp(1)$ random variables. We use the collection $\{D_i\}$ to define a random walk on $\mathbb Z$ by $S_0 = 0$ and $S_n = \sum_1^n X_i$ with $X_i \in ...
-2
votes
0answers
20 views

Asymptotic expansion on 3 nonlinear ordinary differential equations [on hold]

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
7
votes
1answer
199 views

Compute an arbitrary decimal place of $\pi$

Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?
0
votes
3answers
101 views

How do I show that a separable isogeny is central?

I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading: Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: ...
0
votes
0answers
35 views

Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and ...
8
votes
1answer
225 views

Representation of finite groups in a compact Lie group

Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set ...
4
votes
0answers
91 views

Reference for analyticity of $K$-theory

This is a follow-up to my last question, Homotopy excision for structured ring spectra -- reference?. The immediate reason why I care about Blakers-Massey theorems for ring spectra is to prove that ...
4
votes
1answer
122 views

Most Regularity of a Polygon

Conseider $n$ electrons in an empty sphere. What structure do they make? This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of ...
3
votes
1answer
144 views

Carlsson's spectrum BG^-V

In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the ...
0
votes
0answers
69 views

Albanese map on a universal jacobian associated to a curve on an abelian surface

Given an abelian surface $A$ and a curve $C$ on it, consider the component $\{C\}$ of the Hilbert scheme of curves with the same cohomology class of $C$. I would say that it has a fibration structure ...
1
vote
0answers
93 views

Error in trace of operator

Imagine you have a Schroedinger operator $H:=-\frac{d^2}{dx^2}+V$ with $V \in C[a,b]$ on some compact interval $[a,b]$. The boundary conditions are supposed to be taken in such a way that this ...
-5
votes
0answers
43 views

What do rank-2 tensor entries represent? What's their geometrical meaning? [on hold]

given a vector (with, say, 3 dimensions), if I multiply that with a second-rank tensor, I can make it change in both direction and magnitude (for that 2nd rank tensors are involved in vector-vector ...
0
votes
0answers
109 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...
2
votes
0answers
72 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: ...
1
vote
1answer
121 views

Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...
3
votes
1answer
101 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
-1
votes
1answer
76 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...

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