0
votes
0answers
3 views

Intrinsic definition of the weigth 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 ...
0
votes
0answers
11 views

Standard name / symbol for intersection in Browverian lattices

A Browverian 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
7 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
39 views

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

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 ...
6
votes
0answers
68 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
6 views

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

Let us define polynomial $T(x)=(x-\beta)\cdot g(x)$. So $T(x)$ is product of two polynomials namely $x-\beta$ and $g(x)$. We define $T(x)$ over finite field $\mathbb{Z}_p$. Degree of $T$ is at most ...
4
votes
1answer
32 views

$AXB$ sort of decomposition?

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
40 views

Calculate the trajectory of an object around earth [on hold]

I am making a small project for fun in Unity3D Engine now I am trying to find out how to calculate the trajectory of an object. The know info is: Weight of object Speed of object Altitude of object ...
-3
votes
0answers
43 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?
5
votes
0answers
86 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 $n$ ...
0
votes
0answers
38 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 ...
2
votes
0answers
19 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
44 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
23 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
28 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
0answers
23 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
56 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
33 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?
7
votes
1answer
386 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
130 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
15 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
44 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 ...
5
votes
0answers
72 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
83 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
82 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
46 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
79 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
50 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
47 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
39 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 ...
14
votes
7answers
677 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
69 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 ...
22
votes
1answer
388 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
42 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
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
37 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 ...
4
votes
0answers
55 views

Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...
4
votes
1answer
76 views

Time for brownian motion to cross a coordinate plane

Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the ...
1
vote
0answers
41 views

Approximating a divergent integral with modified Bessel functions of the first and second kinds

I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds $I = \int_0^{\infty} \frac{\cos(ax)}{x} ...
6
votes
1answer
176 views

Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...
-1
votes
0answers
25 views

Any polynomial-time algorithm for hypergraph bisection? [on hold]

I work with hypergraph partitioning. I want to divide a complete weighted hypergraph into 2 parts using cut-net metric, a sum of all edges cut, and connectivity metric. Is there a polynomial-time ...
5
votes
0answers
120 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
0
votes
0answers
120 views

A finite group with O_{p}(G)=1

Let $G$ be a finite group of order $p(p^2-1)/2$, where $p$ is prime number. If $O_{p}(G)=1$, then what is the number of Sylow $p$-subgroups G?
1
vote
1answer
123 views

Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly. For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...
0
votes
0answers
27 views

Algorithm for 2-coloring classes of 3-uniform hypergraphs

Hi all and thanks in advance for your efforts. I'm interested in 2-coloring 3-uniform hypergraphs. I know that in general, the problem of deciding if a 3-uniform hypergraph is 2-colorable is ...
1
vote
0answers
53 views

What characteristic of a graph depend on the vertex labeling?

Different labeling on a graph produces class of isomorphic graphs. Two isomorphic graphs possess similar characteristic such as connectivity, degree distribution of vertices, equality of spectrum and ...
1
vote
1answer
60 views

Estimate $\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$

Let $\mu$ be a singular Borel probability measure on $[0, 1)$, and $f\in L^2(\mu)$. Estimate $$\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$$ where $\alpha_n\in\mathbb R$ and ...
-1
votes
0answers
46 views

About perturbation of spectral radius of a matrix because of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (you can assume the diagonal matrices to be such that ...
4
votes
2answers
201 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...

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