**0**

votes

**0**answers

13 views

### Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?

**-2**

votes

**0**answers

16 views

### Does the language suggest hard average cases?

\begin{equation*}
\begin{aligned}
\ \\
L & = \{ D \, | \, permuted \, C \, on \, its \, submatrices \, C_{i} \, \}
\ \\
\ \\
C & = [\,C_{1}\, C_{2}\, ...\, C_{k-1} \, C_{k} \, C_{k+1} \, ... ...

**0**

votes

**0**answers

23 views

### Davenport constant of class group

Denote $\mathsf{C(\Delta)}$ where $\mathsf{\Delta=b^2-4ac<0}$ with $\mathsf{gcd(a,b,c)=1}$ be class group of all equivalence classes of integral quadratic forms with discriminant $\mathsf{\Delta}$. ...

**-3**

votes

**0**answers

29 views

### Why should “small” P be preferred?

In contrast, of course, is the approach of finding an NP language of super-polynomial complexity.
But why the overwhelming, obvious yet implicit favoritism?
Has it anything to do with our ...

**0**

votes

**1**answer

68 views

### An exercise in the Kaplansky's book

I saw the following exercise in the Kaplansky's book that is due to D. Lizard. Where can i find the main text for the proof of this exercise?
Let $P$ be a prime ideal of $R$, $I$ the ideal generated ...

**0**

votes

**0**answers

36 views

### A quadrant of residues

Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$
...

**3**

votes

**1**answer

46 views

### The growth of a subset of a group

Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it ...

**0**

votes

**0**answers

74 views

### Abelian varieties in DAG

This is (hopefully) a pretty simple question. There is the notion of a derived scheme, etc., in derived algebraic geometry. What is the analogue of an abelian variety in derived algebraic geometry? I ...

**-2**

votes

**0**answers

89 views

### $\mathsf{GCD}$ in arithmetic progression

Given $\mathsf{M\in\Bbb N}$, pick $\mathsf{r,s,A,B\in\Bbb N}$ randomly with $\mathsf{0<r<s<A<B<M}$ satisfying $\mathsf{gcd(A,B)=1}$.
Given $\mathsf{c\geq1}$, what is the probability ...

**4**

votes

**0**answers

78 views

### Derived global functions on (derived) stacks $BG$ and $G/G$

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...

**3**

votes

**0**answers

56 views

### Fell topology vs. convergence of matrix coefficients

My question is partially inspired by the following discussion:
Topology on the Unitary Dual
Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let ...

**0**

votes

**0**answers

46 views

### Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...

**1**

vote

**0**answers

33 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

**4**

votes

**0**answers

97 views

### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

**1**

vote

**1**answer

37 views

### Standard name / symbol for intersection in Brouwerian lattices

A Brouwerian lattice has a lower adjoint $\cdot - B$ to $B\lor\cdot$. It is called pseudodifference. The main reference is http://www.jstor.org/stable/1969038
Once you have pseudodifference, you can ...

**0**

votes

**0**answers

18 views

### Is there an official name for the intersection of the join-irreducible representations of two lattice elements?

Given a lattice provided with a join-irreducible representation of its elements, there is a natural "intersection" operator $A \mathbin{\dot\cap} B$ that returns the join of the setwise intersection ...

**0**

votes

**0**answers

115 views

### Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$ [migrated]

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...

**11**

votes

**0**answers

155 views

### Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...

**0**

votes

**0**answers

19 views

### Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...

**4**

votes

**0**answers

40 views

### $AXB$ sort of decomposition? [migrated]

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d ...

**-3**

votes

**0**answers

56 views

### Connection between Haar measure of locally compact group G and Haar measure compact subgroup of it [on hold]

Is there a connection between the Haar measure of the locally compact group G and the Haar measure of a compact subgroup?

**4**

votes

**0**answers

122 views

### Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...

**2**

votes

**0**answers

81 views

### Can one complete a morphism of commutative triangles to a “commutative cube” in a triangulated category?

This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?.
I am deeply grateful for the contributions there; they roughly say that ...

**3**

votes

**0**answers

31 views

### Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...

**4**

votes

**0**answers

65 views

### Constructing a simple $A$-module

Let $n \ge 2$, and let $A$ be the (unital and asociative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is the ...

**-2**

votes

**0**answers

34 views

### How would one describe the rules of Pascal's triangle (which gives us the “normative curve” of probability) in cellular automata terms? [on hold]

If cellular automata simple rules can create complex structures, then how pascal's triangle can be explain as these rules as they are so symmetric ??
For example, elementary cellular automata rule ...

**0**

votes

**0**answers

35 views

### Interpolating a Polynomial Given Multiplier of each $y_i$

We have polynomial $P(x)=(x-\beta)\cdot g(x)$, where degree of $P(x)$ is fixed n-1, $\beta$ chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get ...

**0**

votes

**1**answer

40 views

### Solutions of an nonlinear evolution problem

We consider the following continuous-time nonlinear evolution problem
\begin{equation}
\begin{cases} \dot{y}(t)=Ay(t)+f(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}
\end{equation}
where ...

**2**

votes

**1**answer

64 views

### Set of regular points in an Alexandrov space with curvature bounded below

Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$.
...

**-2**

votes

**0**answers

40 views

### Two rational and one irrational root of a cubic? [on hold]

Let $p(x)=a_3x^3+a_2x^2+a_1x+a_0$, with $a_i\in\mathbb{Q}$. Is it true that if two of the roots of $p(x)$ are in $\mathbb{Q}$, then the third is as well?

**8**

votes

**1**answer

690 views

### Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor

In their paper, Darmon, Diamond and Taylor remarked the following :
(the previous paragraph of Section 2.2 (p. 55), https://www.math.wisc.edu/~boston/ddt.pdf)
If $\rho : G \rightarrow ...

**3**

votes

**2**answers

196 views

### Lower bounding the multiplicative order of 2 modulo p

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.
Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?
...

**-1**

votes

**0**answers

17 views

### classification open problems by complexity [on hold]

i am looking for a standard for classification of open problems by complexity,is there any standard that tells us certain problem is in first class or 3th class of hard open problems?
thanks.

**-3**

votes

**0**answers

50 views

### One question about group algebra [on hold]

Let G be an locally compact group and H is closed normal subgroup of it. If f belong to L^1(G), Is restriction of f to H belong to L^1(H)? conversely, can we extend every member of L^1(H) to some ...

**4**

votes

**0**answers

76 views

### Intuition for the tensor algebra? [on hold]

As the question suggests, can someone give me their intuitions for working with the tensor algebra? Thanks in advance.
Here is my intuition/understanding for the tensor algebra. Given a ring $A$ ...

**4**

votes

**1**answer

94 views

### Zero divisors with support of size 3 in group algebras of finite groups

Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$?
Recall that the support of ...

**0**

votes

**0**answers

14 views

### Lebesgue-integrability of piecewise function with random variable [on hold]

This function is Lebesgue-integrable:$$\chi(x)= \left\{
\begin{array}{ll}
1 & \text{if}~x~\text{is rational}\\
0 & \text{if}~x~\text{is irrational}.
\end{array}
...

**0**

votes

**2**answers

105 views

### Making idempotent element by a relation [on hold]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?

**-6**

votes

**0**answers

58 views

### |(a,b)| = |R| ? [on hold]

I want to prove that any open interval (a,b) has the same cardinality of the real numbers (|(a,b)| = |R|).
Do I have to find an function to prove it? or is there a theorem to prove it easier? or any ...

**1**

vote

**0**answers

85 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**0**

votes

**0**answers

53 views

### Cyclic faithfully flat modules

Iam looking for an example of a cyclic faithfully flat R-module but not projective. Could someone help me?

**0**

votes

**2**answers

58 views

### Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...

**1**

vote

**0**answers

51 views

### largest subgroup of $Out(\hat{F_2})$ which preserves the Nielsen invariant

Let $x,y$ be generators for the free group $F_2$. It's known that $Aut(F_2)$, and hence $Out(F_2)$ preserves the conjugacy class of the subgroup $\langle[x,y]\rangle$ generated by $[x,y]$ (This ...

**17**

votes

**7**answers

850 views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

**2**

votes

**0**answers

74 views

### Interchanging the tensor product with infinite product

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the ...

**29**

votes

**1**answer

625 views

### Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...

**0**

votes

**0**answers

43 views

### What is number of faces in a k-ary n-dim cube? [on hold]

What is the number of $(n-r)$ dim faces for a $k$-ary $n$-dim cube ?
Definition of k-ary cube: In a $k$- ary $n$- cube , each node is identified by an $n$-bit base-$k$ address $b_{n − ...

**2**

votes

**0**answers

79 views

### A factorial related statement

Is statement $\mathsf{S}$ below in $\mathsf{NP}$ or in $\mathsf{coNP}$?
$$\mathsf{S}:\mathsf{Given}\mbox{ }n,a,s,c\in\Bbb N,\mbox{ }\mathsf{with}\mbox{ }n\mbox{ }\mathsf{a}\mbox{ }\mathsf{prime}\mbox{ ...

**-2**

votes

**0**answers

41 views

### Prime ideals decomposition [on hold]

How to prove the decomposition law for prime ideals in finite separable extensions of number fields? How we use the "conductor condition"?

**1**

vote

**0**answers

38 views

### Doubling theorem for Alexandrov spaces

Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem?
The only reference I am aware of is the original ...