# All Questions

**2**

votes

**0**answers

35 views

### A List-Like Frobenius Monad

Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and hence a Frobenius monad. I was reading a paper that, I think, suggested zipper as an example. I think ...

**-1**

votes

**0**answers

47 views

### Closure property of completely monotone functions [on hold]

A $C^{\infty}$ function $f(x_1, \dots, x_n)$ defined on $(0,\infty)^n$ is said to be completely monotone if
$$
(-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0
$$
...

**1**

vote

**0**answers

109 views

### The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...

**-1**

votes

**0**answers

29 views

### Relation between Independent variables in an Equation [on hold]

Description: We define index as an indicator, sign, or measure of something.
Let, $A_{i}$ is an index, that measures the benefits of choosing a network station $i$ among other existing network ...

**4**

votes

**1**answer

65 views

### Restricted Lie algebras with a $p$-nilpotent basis

Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p>0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k}=0$ for some positive integer $k$. If $L$ is ...

**3**

votes

**0**answers

28 views

### continuous injective extension of a map defined on a hemisphere

Let $S^2 = \{x\in \mathbf R^3\colon |x|=1\}$ be the unit sphere, $S^2_+ = S^2 \cap \{x_3 \ge 0\}$ be the upper hemisphere and $S^1 = S^2 \cap \{x_3 = 0\}$ be the unit circle. Let $u\colon S^2_+\to \...

**3**

votes

**0**answers

83 views

### How can I describe explicitely a nonsingular model of the elliptic surface?

Consider the surface $\mathcal{E} = Q_1 \cap Q_2 \subset \mathbb{P}_k^3\times \mathbb{P}_k^1$ with homogenous coordinates $x$, $x^\prime$, $y$, $z$ and $t$, $s$ respectively and a field $k$ of even ...

**8**

votes

**0**answers

94 views

### Trouble with Stable Equivariant Profinite Homotopy Theory

I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra ...

**-1**

votes

**0**answers

12 views

### Rigorious formulation of approximation of integral as area of a square and its radius of convergence [migrated]

We know that the taylor expansion of
$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...

**0**

votes

**0**answers

92 views

### Join of $G$-CW-Complexes

I want to understand the CW-structure on the join of $G$- CW complexes for my master's thesis.
Let $G$ be a discrete group and $X$ and $Y$ $G$-CW-complexes. Furtheremore, let $X*Y$ denote the join
$$[...

**1**

vote

**1**answer

123 views

### Lift Lie group action on a small neighborhood

Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$.
Q: Why in a small neighborhood of $N$, $G$ also action ...

**4**

votes

**1**answer

87 views

### Is there a known criterion for a compact complex analytic space to be projective?

It is known when a compact complex analytic space $X$ is the analytification of a complex projective variety? If $X$ is a manifold, then Kodaira's embedding theorem and Chow's theorem says that $X$ is ...

**5**

votes

**1**answer

161 views

### A two-point inequality

Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...

**0**

votes

**0**answers

48 views

### interpolation inequalities and embeddings

When $Z$ is an interpolation space between two Banach spaces $X$ and $Y$ (say real / complex method), we have a norm inequality
$$
\| x \|_Z \le C \| x \|_X^\theta \| x \|_Y^{1-\theta}
$$
My question ...

**4**

votes

**0**answers

61 views

### Is this map representable? what is the fiber?

Consider the following map of stacks:
Let $S$ be the stack whose $A$ points are diagrams of the form
\begin{array}{ccccc}
{} & {} & U & {} \\
{} & {} & \downarrow & \searrow \\...

**-1**

votes

**0**answers

58 views

### Applications of computing the averages of arithmetical functions

I often read many papers in which the authors compute the average of certain arithmetical functions like 'On the distribution of the Euler function of shifted smooth numbers' of Shparlinski and al. ...

**1**

vote

**1**answer

102 views

### Cluster algebra structure on the coordinate ring of $Mat_3$

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...

**4**

votes

**1**answer

92 views

### Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...

**6**

votes

**0**answers

71 views

### Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....

**1**

vote

**1**answer

41 views

### Fredholm operator and automorphism of unit disk

Recently, I came across the following question while studying Fredholm operator. Recall an operator $S$ on a Hilbert space $\mathcal H$ is said to be Fredholm if $Range(S)$ is closed along with both $...

**10**

votes

**2**answers

193 views

### Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...

**1**

vote

**0**answers

49 views

### prove that $\min\{|z_{j} - w_{1}|,|z_{j} - w_{2}|\}\leq 1$ holds [on hold]

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $$ (z-z_{1})(z-z_{2})+(z-z_{2})(z-z_{3})+(z-z_{3})(z-z_{1})=0$$ ...

**2**

votes

**1**answer

78 views

### On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category.
Question: What about the converse, i.e., can we characterize every unitary modular tensor ...

**0**

votes

**0**answers

48 views

### Formulation of constraints

I would like to relax the following binary SDP problem
$$ \max_{A,x_i,y_i} \ trace(A) $$
$\quad \qquad \qquad $ subject to:
$$ A + \sum_{i=1}^N x_i D_i + \beta \sum_{i=1}^M y_i D_i \preceq \alpha .I ...

**0**

votes

**0**answers

4 views

### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

**0**

votes

**1**answer

103 views

### Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...

**0**

votes

**1**answer

82 views

### An inequality of real continuous function with f'>0 and f''>0

I proposed my conjecture as follows:
Let $f(x)$ is a real continuous function on $[m, M]$ and $f'>0, f''>0$ on $[m, M]$, let $m \le x_i \le M$, for $i=1, 2,..., n$. Then
$$\frac{f(x_1)+f(x_2)+...

**2**

votes

**0**answers

127 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 ...

**2**

votes

**0**answers

24 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}~...

**3**

votes

**0**answers

126 views

### Research topics in Curves and Surfaces [on hold]

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 ...

**6**

votes

**0**answers

133 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 $G$ (...

**3**

votes

**2**answers

143 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**

vote

**0**answers

52 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, ...

**4**

votes

**3**answers

269 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 ...

**0**

votes

**0**answers

60 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 ...

**3**

votes

**1**answer

136 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 ...

**0**

votes

**1**answer

70 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 ...

**0**

votes

**0**answers

32 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

70 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**

votes

**0**answers

53 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**

votes

**0**answers

40 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 \...

**3**

votes

**1**answer

94 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**

votes

**0**answers

139 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 ...

**3**

votes

**0**answers

64 views

### Extending homomorphisms between ordered abelian 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-...

**11**

votes

**1**answer

456 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 ...

**6**

votes

**1**answer

150 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 ...

**0**

votes

**1**answer

117 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.

**6**

votes

**0**answers

228 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 ...

**4**

votes

**1**answer

232 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**

votes

**0**answers

24 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 ...