0
votes
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
7answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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 ...

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