# All Questions

98,519 questions

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### Two versions of the Möbius inversion formula

Consider the following versions of Möbius inversion:
Let $(A,+)$ be an abelian group, and let $f$ and $g$ be functions $\mathbb N\rightarrow A$. Then $$\left((\forall n )\;g(n)=\sum_{d|n}f(d)\right)\;...

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38 views

### A conjectural formula for the “minimal degree function”, $k\rightarrow d\min(k)$, attached to recursion, $f\rightarrow A(f)$, in char $3$

THE RECURSION: $f\rightarrow A(f)$
$A: t(\mathbb{Z}/3)[t^3]\rightarrow t(\mathbb{Z}/3)[t^3]$ is the $\mathbb{Z}/3$-linear map with $A(t)=0, A(t^4)=t, A(t^7)=t^4$, and $A((t^9)f)=(2t^9)A(f)+(t^3)A((t^...

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**2**answers

166 views

### Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?

$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...

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**1**answer

138 views

+50

### Nonabelian finite groups with “locally commuting” presentation

Let $G = \left\langle S | R \right\rangle$ be a finitely presented group where S is a set of generators and R is a set of relations. We say that the presentation is "locally commuting" if whenever two ...

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18 views

### Frobenius norm induced distance between two multivariate normals

There is a norm on covariance matrices defined by the Frobenious norm. http://mathworld.wolfram.com/FrobeniusNorm.html
Can this be used to define a valid distance between two multivariate gaussian ...

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**1**answer

71 views

### Definition of $\in_c$ for power objects

On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...

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**1**answer

260 views

### Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...

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43 views

### Combining Fubini theorem and Chebyshev inequality

Let $g$ be a measurable function.
Is it true that
$$\mathcal L^{2d}(\{(x,y) \in \mathbb{R}^d \times \mathbb{R}^d: g(x,y)>\epsilon\}) \le \mathcal L^d\left(\left\{y \in \mathbb{R}^d:\frac{1}{\...

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**1**answer

75 views

### Regularity of John's ellipsoid

Consider a finite dimensional real Banach space $E$, with norm say $|\cdot|$. Let $N$ denote the set of all norms on $E$. Suppose that $\varphi_1, \varphi_2 \in N$ have unit balls $B_1$ and $B_2$, ...

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33 views

### Linear transformation group of paraboloid in $\mathbb{R}^d.$

Suppose $P$ is a paraboloid in $\mathbb{R}^d.$ What is the group of all linear transformations $\mathcal{L}: \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $\mathcal{L}(P)=P?$
Could someone please ...

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132 views

### A refinement of Faltings' lemma

In his proof of the Mordell conjecture, Faltings proved the following important result:
Let $K$ be a number field and $S$ a finite set of primes in $K$. Then for any $g \geq 2$ there exists a number $...

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**1**answer

139 views

### Connected incomparability graph

Let $X$ be a finite set equipped with a partial order. (Not a preorder: $a < b$ implies $b \not< a$.) The corresponding incomparability graph has vertex set $X$ with an edge between two points ...

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69 views

+100

### Some confusion about weights and roots in parabolic root systems

I was reading James Arthur's book An Introduction to the Trace Formula and had a couple of questions. Here $A_0$ is a maximal split torus of a reductive group $G$, $P_0 \supset A_0$ is a minimal ...

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97 views

### Hecke operators that lower level

I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N)...

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199 views

+200

### A strong plus-one hypothesis

To make this more easily readable, I'll start with the question and then give the explanation/motivation.
Question. Is the following principle (or its weakening, with "for every real $r$" replaced ...

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**1**answer

166 views

### Henselianizations over countable index sets

Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...

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**0**answers

127 views

### Ramified covering morphisms

I found the following theorem in the book Revêtements
Étales et Groupe Fondamental$-$Alexander Grothendieck.
Théorème 10.11. — Soit $f : X\longrightarrow Y$ un morphisme quasi-fini séparé. On ...

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**1**answer

149 views

### Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...

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136 views

### The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...

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**4**answers

5k views

### What to do if you notice a substantial improvement to a result in a paper whilst refereeing it?

What would you do/have you done in such a situation?
Hand out the improvement for free in your report
Wait until the result is published and then submit elsewhere
Inform the editor about the ...

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**0**answers

61 views

### Prove that the following formula is valid in LTL (linear temporal logic): □(p→ q) → (◊ p → ◊ q) [on hold]

Can someone help me solve this? Don't know how to solve LTL question.

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150 views

### Foundation of topology based on genus? [on hold]

I have heard the following slogan numerous times - Topology is the study of shapes you can do "whatever you want to" (continously) as long as you never tear them.
I have also heard, that the ...

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**1**answer

187 views

### Quadratic algebras and Koszul algebras

Let $A$ be a quadratic algebra and $B$ the Ext-algebra of $A$.
In case $A$ is a Koszul algebra, we should have that the global dimension of $A$ plus one is equal to the Loewy length of $B$ (is there a ...

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**0**answers

145 views

### Is there a Seifert–van Kampen theorem for etale fondemental group?

Is there a Seifert–van Kampen theorem for etale fondemental group? (for example for varieties over a non-algebraically closed field) Any example is welcome.

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**1**answer

80 views

### 3 dice combination [on hold]

It there any formula for the calculating dice combinations. For example for hex-dice, when I roll one dice it is possible $6$ results. If I do the same with $2$ disces there are $21$ results instead ...

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**3**answers

661 views

### Are inclusions “canonical” injections?

[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly ...

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**1**answer

132 views

### Spectrum of a first-order elliptic differential operator

Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold.
I ...

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51 views

### Counterexample in Kolmogorov theorem about existence of almost surely continuous modification

I want to understand this Kolmogorov theorem about existence of almost surely continuous modification:
A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...

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**1**answer

52 views

### The extension of a pluri-sub-harmonic Functions

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. Proposition 2.9 of the paper is: If $\Psi$ is a pluri-sub-harmonic on the punctured ball $B_n^{*}$ ...

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**3**answers

238 views

### Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$?
Same question, but this time $G$ is a finite group with at most $c$...

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**1**answer

43 views

### On extensions of holomorphic mappings with image in a projective algebraic variety

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that:
Let $N$ be a complex manifold, $S\...

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167 views

### Moishezon manifold vs proper complex variety

Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...

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**1**answer

207 views

### Maps from a scheme over the dual numbers to constant schemes

Let $X$ be a smooth scheme over $k[t]/(t^2),$ where $k$ is a field of characteristic 0 (the case when $X$ is a projective curve is already interesting). Let $X_{0} \to X$ denote $X$ with the reduced ...

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68 views

### Extending permutation models

We know that within ZFC any structure may be extended to a rigid structure. My question is whether this holds also for models. I mean: can a permutation model be extended to a standard model where the ...

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**0**answers

122 views

### Reference to a particular result of Scholl and Faltings

Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...

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**2**answers

71 views

### Definition of the weight lattice for a nonreduced root system

Let $(V,\Phi)$ be a root system with dual root system $(V^{\ast},\Phi^{\vee})$. Let $\Delta = \{\alpha_1, ... , \alpha_n\}$ be a set of simple roots for $V$, and let $\Delta^{\vee} = \{\alpha_1^{\vee}...

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772 views

### Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?

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**1**answer

65 views

### Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...

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231 views

### Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture

I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures:
...

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56 views

### Graph with 5 nodes max that fulfils the following [on hold]

i) It should contain exactly four cycles and these should all have length $4$;
ii) Graph should contain a node which has degree $3$;
iii) Graph should contain a subgraph which is a tree that ...

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126 views

### Giving mod $p$ on $O_{\overline{\mathbb{Q}}}$

Let $O_{\overline{\mathbf Q}}$ be the ring of all algebraic integers. Can I give an equivalence relation $\bmod p$ on $O_{\overline{\mathbf{Q}}}$ satisfying the following conditions (where $p$ is a ...

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**1**answer

68 views

### Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...

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69 views

### An Ehrhart positivity question related to Schur polynomials

Consider the Schur polynomial $s_\lambda(x_1,\dotsc,x_k)$.
It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function
$$
n \to s_{n \lambda}(1,...

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9 views

### How to Create Point-Optimal Objective Functions

Here is a problem that has originated from some IP research i'm working on.
You are given a polyhedron $P$ in standard matrix inequality form of $Ax \le b$, $x \in \mathbb{R}^n$ as well as a point $...

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102 views

### Topology of abstract varieties over $\mathbb{C}$

What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think
they have to have non-zero $b_2$ ...

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**1**answer

136 views

### Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

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46 views

### Asymptotic behavior of an exponential of an integral

Let $G(x)$ be the following function
$$
G(x) = \sum_{i \in I} \alpha_i (t+x)^{\lambda_i} x^{1-\lambda_i}
$$
where we know that the coefficients sum up to one ($\sum_{i \in I} \alpha_i = 1$ ) and that ...

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**1**answer

68 views

### Finding $P$ points among $N$ to approximate a probability density function?

Let $f$ be a probability density function (positive such that $\int_{\mathbb{R}} f(x) \mathrm{d} x = 1$) and $X_0 = \{x_n\}_{1\leq n \leq N}$ be $N$ given real points. We also fix $1 \leq P \leq N$ ...

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60 views

### Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

Let a group $G=G_1 \times G_2$, where
$G_1$ is a discrete group (can be finite or infinite),
$G_2$ be any compact Lie group or finite group.
Question: Is there some simple result that we can ...

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25 views

### About Extension group and weights in $\mathcal{O}^\mathfrak{p}$

Denote $M_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$.
Suppose $\text{Ext}_{\mathcal{O}^\...