5
votes
1answer
139 views

Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
0
votes
0answers
23 views

How to prove prime avoidance for graded cases?

Let $R$ be a nonnegatively graded ring such that $R_0$ is local with infinite residue field. Let $I,J_1,...,J_s$ be a homogeneous ideals of $R$ such that $I$ is not contained in $J_i$. Please prove ...
1
vote
1answer
105 views

Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...
0
votes
0answers
58 views

inverse of operator [migrated]

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial ...
1
vote
1answer
84 views

The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers. Consider the exponential Diophantine equation $$ a^n-b^m=x^3+y^3 \qquad (*) $$ Nontrivial solution ...
3
votes
1answer
125 views

4-th order diophantine equation

I met in many places the equation $(a^4-b^4)(c^4-d^4)=\square$ It is well known that this was investigated by Euler. But I was unable to find the general solution of this equation. Could you please ...
4
votes
1answer
155 views

A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following: Let $G=K_{\Bbb C}$ be a ...
-1
votes
0answers
44 views

General Solution of Quadratic Diophantine Equation System

I am looking for the general solution of: $x^2-y^2=\square$ $x^2-z^2=\square$ $y^2-z^2=\square$ I have found a solution in tpiezas site. But not really sure if this is the general solution.
0
votes
0answers
25 views

Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if $$\sigma(\sigma(n)) = 2n.$$ A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$ Here is my question: Is ...
1
vote
1answer
121 views

Analogue of fundamental theorem of real surfaces for complex surfaces

Is there an analogue of the fundamental theorem of surfaces for complex surfaces? If I know only differentiable functions $E,F,G,e,f,g$ (coefficients of the first and second fundamental forms) where ...
0
votes
0answers
35 views

how to measure a bidrectional relationship effect on third variable [on hold]

Sorry that my question was unclear: I decide to determine if there is a relationship between two variables (gross national income, X and enrollment, Y) in Country A, between 2000-2007 My results ...
-3
votes
0answers
20 views

List comparison [on hold]

What is the best way to compare two lists? For example, how far is the list 2284, 1161, 493, 227, 116, 43, 13, 7 from being a list where each number is half the ...
3
votes
0answers
57 views

Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots: Maucher, Fabian, and Paul ...
4
votes
1answer
108 views

“Small” simplicial complex with torsion trees

I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...
7
votes
0answers
126 views

Are there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any ...
2
votes
0answers
56 views

Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...
0
votes
0answers
93 views

Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free. ...
2
votes
0answers
82 views

First Chern class and second Chern class in Quantizable Kaehler manifolds

Assume that $(X,\omega)$ is a K\"ahler manifold and $L\to X$ be a pre-quantum line bundle, then is there any relation between first Chern class and second chern class?
4
votes
1answer
140 views

Schwartz-Zippel lemma for an algebraic variety

Let $X $ be a smooth affine subvariety of $(\overline{\mathbb{F}_q})^n$ defined by a prime ideal $I$. Let $f$ $\in \mathbb{F}_q[x_1,\ldots,x_n]$ be a polynomial such that $f \notin I$. Let $r_1, ...
3
votes
1answer
96 views

Alternative parallel paths

There are $n$ non-intersecting strings (with ends $x_1,\dots, x_n$ and $y_1,\dots, y_n$). An additional string intersects the first $n$ strings somehow. All the intersections are simple (vertices of ...
0
votes
0answers
6 views

What's the definition of multivariate mode? [migrated]

In the case of grouped data where a frequency curve have been constructed to fit the data, the mode will be the value (or values) of x corresponding to the maximum point (or points) on the curve. From ...
1
vote
0answers
83 views

List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...
5
votes
2answers
91 views

References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
-1
votes
0answers
18 views

Using Headpose Vector and 2D Point to Compute Distances [on hold]

I have a frame taken from a video. The frame contains a face and I have the (x, y) locations of the features (corners of lips, edge of eyebrows, etc.) and the headpose vector (pitch, yaw, roll), which ...
0
votes
0answers
61 views

Fastest algorithm to compute isogeny

Let $E/GF(p)$ and $E'/GF(p')$ are two isogenous elliptic curves($\#E=\#E'$). We know that there exist the map $$\psi : E \to E'$$ Suppose that we haven't any information about degree of $\psi$. ...
2
votes
1answer
52 views

Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
2
votes
0answers
67 views

Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
1
vote
1answer
82 views

Formula for negative polylogarithms

Theorem. We have that $\displaystyle \underbrace{x\left(\dfrac{d}{dx}\left(\cdots x \left(\dfrac{d}{dx} \left( \dfrac{x}{1-x}\right)\right)\cdots\right)\right)}_{\text{$x \frac{d}{dx}$ $m$ ...
1
vote
0answers
87 views

Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...
0
votes
0answers
67 views

How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields. I dont know ...
19
votes
0answers
334 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...
1
vote
1answer
72 views

Size of connected components of a graph via its spectrum

I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is: Is there a way to understand the size of each connected ...
5
votes
3answers
289 views

Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true: Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
5
votes
1answer
95 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...
1
vote
0answers
95 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
5
votes
1answer
53 views

$G$-invariant holomorphic vs. polynomial functions

Let $X\subseteq\Bbb C^n$ be a smooth affine variety and $G$ a complex reductive group acting algebraically on $X$. Let $x_0\in X$. If there is a non-constant $G$-invariant holomorphic function ...
9
votes
2answers
262 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
2
votes
0answers
62 views

Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?

Recall that a locally compact group $G$ is said to be an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure; an $[SIN]$ group, if each neighborhood of the identity includes a ...
0
votes
0answers
28 views

Integrable modules and comodules

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...
12
votes
0answers
218 views

The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in ...
0
votes
0answers
64 views

Use of algebraic topology in geometry(differential and complex analytic) [on hold]

I'm curious in seeing connections of algebraic topology with other areas of mathematics. As far as I know, it's not really used in scheme-theoretic algebraic geometry and arithmetics(aside from ...
-2
votes
0answers
79 views

proof that ${\rm SL}_n (R)=E_n(R)$ in a local ring? [on hold]

I have to prove that ${\rm SL}_n (R)=E_n(R)$ and I need some help. $R =R_1\cdot R_2\cdots R_n$ , and every $R_i$ is a local ring . $E_n(R)$ is the elementary group and ${\rm SL}_n(R)$ is the special ...
1
vote
0answers
70 views

How many real-analytic forms exist on a real-analytic manifold?

I hope my question isn't too vague: Let $M$ be a real-analytic compact manifold (say surface for simplicity). How big is $\Omega^k(M)$ (space of global real-analytic $k$-forms)? Let me explain ...
-4
votes
0answers
26 views

Calculus limit question [on hold]

enter image description here this formula, why this must be the first item 2x ^ 3 multiplied by 2 One is -5x ^ 5 / (3x ^ 2) (4x ^ 3) another is a -5x ^ 5 / (3x ^ 2) (2x ^ 3) all can be reduction, why ...
0
votes
1answer
27 views

Existence of stationary tangent cones

My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42 So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density ...
0
votes
0answers
18 views

Dennis More' Superlinear Convergence_refrences request

Why in the proof of superlinear convergence of restricted broyden class (for the unconstrained optimization) we need the bounded deterioration condition for the approximation of all the true hessian ...
3
votes
1answer
105 views

Two point function of a free scalar field in Euclidean space-time

This question was previously asked here http://physics.stackexchange.com/questions/251927/two-point-function-of-a-free-massless-scalar-field-in-euclidean-space-time though I did not get there an ...
-1
votes
0answers
34 views

Calculus limit question, why in the (p (x) / (px) frist) X (p (x) fris) formula, the non-moving trend must be a 1? [on hold]

For example, this formula,enter image description here why this must be the first item 2x ^ 3 multiplied by 2 One is -5x ^ 5 / (3x ^ 2) (4x ^ 3) another is a -5x ^ 5 / (3x ^ 2) (2x ^ 3) all can be ...
2
votes
1answer
87 views

Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [on hold]

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
7
votes
1answer
258 views

Choosing two-colorable subgraph in a triangulation

Consider a planar graph $G$ which is a triangulation. Is it possible to find a two-colorable subgraph $H$ of $G$ which has a common edge with every face of $G$? It is known that it is not always ...

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