# All Questions

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6 views

### What techniques are available for constructing D-modules over smooth projective varieties?

I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved ...

**1**

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7 views

### Covering Number of a Positive Semidefinite Cone (Approximate the Objective of a SDP)

I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program
\begin{align}
\max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~...

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38 views

### Research topics in Curves and Surfaces

I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...

**-3**

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**0**answers

71 views

### Questions about the numbers 1-6 [on hold]

I have no background in math or topology. I am a writer and have developed a storytelling technique I'm starting to find out may have strong roots in math.
What is the pattern 1,2,3,4,5,6..2,1,4,3,6,...

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42 views

### Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...

**2**

votes

**2**answers

66 views

### Combinatorial identity involving number of cycles (of any length) in a permutation

I am going through Phil Hanlon's paper and on page 127, right after the first paragraph, "It is well known that.."
which boils down to the following identity:
$$
\prod_{i=0}^{n-1}(\beta-i) = \sum_{\...

**1**

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32 views

### About Composition diamond lemma

Composition Diamond lemma for Lie algebra over a field $F$ has already been investigated in several papers :
L.Bokut and Y.Q.Chen Groebner-Shirshov bases for Lie algebras
and
A.I Shirshov, ...

**3**

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**1**answer

129 views

### ($\oplus$, $\otimes$) is a semiring. If $\otimes$ = +, what are the possible operators $\oplus$?

Assume that ($\oplus$, $\otimes$) is a semiring over the non-negative reals.
If $\otimes$ is +, what are the possible operators for $\oplus$?
So far I have proven that ...

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**0**answers

33 views

### Reference request for a well-known lemma in Parabolic Vector Bundle

In the paper- "Moduli Space of parabolic vector bundles on a curve" - Usha N Bhosle, Indranil Biswas-Beitr Algebra Geom (2012), 53:437-449, DOI: 10.1007/s13366-011-0053-7, Lemma $2.1$ is being ...

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67 views

### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem:
THEOREM. Let $M$ be a ...

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**1**answer

42 views

### Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...

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**0**answers

22 views

### Moduli space of Parabolic Vector Bundles with arbitrary parabolic weights

I have just posted another question related to Moduli Space of Parabolic Vector Bundles on Curve. The questions came up when I was trying to read the paper (Desingularisation of the Moduli Varieties ...

**1**

vote

**1**answer

43 views

### Isofibrations and Diagonal Functors

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor.
Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...

**-3**

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41 views

### Fredholm operators: how to calculate Coker and Ker [on hold]

Exercise:
Let $1\leq p \leq \infty$. For each $n\in\mathbb{Z}$ construct a Fredholm operator $F:l^p\to l^p$ whose index is $n$.
Solution (given in the lecture classe):
$F_n(x_i):=(0,\ldots, 0,x_1,...

**-1**

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**0**answers

29 views

### Direct product between joins of subgroups [on hold]

Suppose that $G$ is a finite group with $A, B, H, K \leq G$. Suppose that $H\times A \leq G$ and $K\times B \leq G$. I want to show that $\langle H,K \rangle \times \langle A, B \rangle = \langle H \...

**2**

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**1**answer

63 views

### Linear independency and compactness of the set of pure states of a $C^*$-algebra

Let $\mathcal{A}$ be a noncommutative $C^*$-algebra and $PS(\mathcal{A})$ be the set of its pure states.
Question 1. Is $PS(\mathcal{A})$ linearly independent (as vectors over $\mathbb{R}$)? (If $\...

**5**

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**0**answers

106 views

### An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...

**-1**

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**0**answers

19 views

### Limits of an indeterminate form $\lim_{t\to\infty} (a+b(-m)^t)/(c+d(-m)^{t-1})$ [on hold]

I'm trying to solve the limit of the following indeterminate form:
$$\lim_{t\to\infty} \frac{a+b(-m)^t}{c+d(-m)^{t-1}}$$
where $t=1, 2, 3, \cdots$ denotes time and all the coefficients are positive ...

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25 views

### Extending homomorphisms into Hahn groups

Let $\Omega$ be a linearly (i.e. fully) ordered set, and let $\Lambda_{\Omega}$ be the ordered abelian group consisting of
those $(\lambda_\omega)_{\omega\in\Omega}\in\mathbb{R}^{\Omega}$ with well-...

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**1**answer

205 views

### Is a one-dimensional compact complex analytic space necessarily projective?

Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...

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60 views

### Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation:
There are two approaches in defining Homology with local coefficients of a ...

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**1**answer

95 views

### Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.
Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.

**4**

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**0**answers

96 views

### Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g.
A question about ordinal definable real numbers
.
A negative answer was obtained in Archive for ...

**3**

votes

**1**answer

186 views

### A generalization of Erdős–Mordell inequality [on hold]

I proposed my conjecture generalization of Erdős–Mordell inequality as following:
Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...

**-1**

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**0**answers

21 views

### Hyperbola application [on hold]

A curved mirror is placed in a store for a wide angle view of the room. the right hand branch of x squared over one minus y squared over three equals one models the curvature of the mirror. a small ...

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19 views

### Algorithm for Longest Common Subtour

For a new kind of heuristic for the TSP I need to calculate the longest subtour, that is common to a set $T_1,\ ...,\ T_m$ of tours, that are "good" approximations of the optimal tour $T_{opt}$.
By a ...

**-1**

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**0**answers

50 views

### How many number of finite points exists inside the circle? [on hold]

I am doing project on Image processing dealing with circular images. So I need an approximate number of pixels present inside circle image of radius R and Circle center of (x,y). Please give me the ...

**0**

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**1**answer

52 views

### Computing canonical forms from orbit partitions

Suppose we know the orbit partition of the vertices of a graph (due to the action of its automorphism group). Is it easy (as in "polynomial time") to generate a canonical form (aka "canonical labeling"...

**1**

vote

**1**answer

31 views

### Proving that an integral of several cdf and pdf functions is increasing in a certain parameter

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy
variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the
usual general CDF and PDF (no specified distribution here)....

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69 views

### Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...

**3**

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**0**answers

50 views

### Intersection multiplicity of limit linear spaces

Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...

**5**

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**1**answer

158 views

### The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic?
Thank you!

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23 views

### On balanced bipartite graphs

Of the $2^{n^2}$ balanced bipartite graphs on $2n$ vertices how many of them have $i$ perfect matchings where $i\in\Bbb N\cup\{0\}$ and $0\leq i\leq n!$ holds?

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37 views

### A priori estimates for elliptic operators

Suppose $L : L^{m,p}(M)\rightarrow L^p(M)$ is some elliptic operator of order $m$, and $(M,g)$ is a compact Riemannian manifold. Then it is known that there exists a constant $C$ such that we have the ...

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309 views

### Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...

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34 views

### Sums of unit vectors has a binary span after constrained permutations

Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, f_{2(m_1+m_2)}\}...

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104 views

### A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...

**0**

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79 views

### Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many equivalent formulations of the Birch and ...

**3**

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**1**answer

77 views

### Isoperimetric inequality via Crofton's formula

I have seen various assertions that one can derive the isoperimetric inequality in the plane from Crofton's formula in geometric probability. Unfortunately, I have not managed to figure out such a ...

**4**

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**1**answer

120 views

### Lie subalgebra of $\chi^{\infty}(M)$ of codimension one

Assume that $M$ is an arbitrary manifold.
Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
If not, what is a counter ...

**-1**

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**1**answer

119 views

### number of partitions from 0 to n^2 [on hold]

You are given the numbers 0 to n^2. You must use n numbers with no number greater than n to form all the partitions of the numbers 0 to n^2. For example with n=4 you want to find the partitions of 7:...

**-3**

votes

**1**answer

144 views

### Does one need an external, peer-reviewed grant to become tenured faculty in this field? [on hold]

As a secondary question, how important is it to be awarded grants to remain employed as a PhD mathematician at an academic research institution? Is it common or uncommon for PhD mathematicians to have ...

**2**

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**0**answers

60 views

### Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is
ample. then there is a Kahler metric $\omega_E$ such that
$Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then
$h=\frac{1}{\...

**3**

votes

**1**answer

109 views

### Elements in the group von Neumann algebra which are not summable

Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...

**0**

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**1**answer

46 views

### Count Functional digraph [on hold]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...

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34 views

### Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that
$$S^°=\{s\in S| f\text{ is ...

**5**

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**1**answer

172 views

### How to compute $[CP^2, G/PL]$?

Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...

**3**

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60 views

### Gauss Bonnett on a flat surface with border

I'm reading the article "Euler Characteristics of Teichmuller Curves in genus two" by Matt Bainbridge and there's a point I don't understand in the proof of theorem 5.5. Maybe you can help me clarify ...

**2**

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69 views

### Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...

**2**

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86 views

### Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.
I have been searching for similar reflexive $...