1
vote
1answer
29 views

generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
0
votes
0answers
32 views

Maximal “Spot It!” card count

This question was triggered by the game Spot It!. The game consists of cards, each having $k$ different symbols from an alphabet of $n>k$ symbols, with the property that any 2 cards have at least ...
1
vote
0answers
13 views

Central limit theorem for independent random variables, with a Gumbel limit

Consider independent random variables $Y_i$, $i>0$, such that $\mathbb{E}(Y_i)\approx \frac{1}{i}$ and $\text{Var}(Y_i)\approx \frac{1}{i^2}$, where $\approx$ means asymptotically equivalent up to ...
2
votes
0answers
20 views

Convergence of Schwartz Kernels

I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...
1
vote
0answers
16 views

Is any finitely generated nilpotent pro-$p$ group necessarily the pro-$p$ completion of some finitely generated nilpotent group?

While thinking about this question, I was led to the following question: My question: Let $G$ be a topologically finitely generated pro-$p$ nilpotent group. Does there exist a finitely generated ...
1
vote
0answers
15 views

Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shift

I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x_i=1\Rightarrow x_{i+1}=0\}$$ such that any element of $G$ ...
2
votes
1answer
89 views

Equivalent definitions of Calabi-Yau manifolds

How do we prove that a compact Kahler manifold whose 1st Chern class vanishes admits a globally defined nowhere vanishing volume form? Thanks.
0
votes
0answers
15 views

Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...
-2
votes
0answers
35 views

Why the square of ideal in Lie algebra is also ideal? [on hold]

Let $L$ be a Lie algebra over field $F$, $I$ - ideal in this algebra. It's stated that $I^2$ (and so any item of central series) is also ideal in $L$. 1) For any $a, b \in I^2: [a,b] \in I^2$. True, ...
0
votes
0answers
19 views

References on law of large numbers, CLT and iterated logarithm laws

Having access to those references, accumulating many results in one domain is always a bless,like Feller's book in probability, Dembo-Zeitoun's large deviation, Grimmett's percolation and recent ...
4
votes
1answer
64 views

Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities: $d_jd_i = d_id_{j−1}$ for $i < j$ $s_jd_i = d_is_{j−1}$ for $i < j$ $s_jd_i = id$ for $i = ...
0
votes
1answer
55 views

A question about sentences in the language of first order ZFC which assert the existence of cardinal numbers

These sentences are usually of two kinds. The first kind are actually theorems of ZFC asserting the existence of various cardinal numbers and their negations are inconsistent with ZFC. The second kind ...
6
votes
2answers
149 views

Changing combination lock

Suppose you have a combination lock (n digits, m symbols) that is unlocked by one specific n-digit key sequence. However, trying a wrong key changes it according to an fixed but unknown function: new ...
0
votes
0answers
28 views

Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
6
votes
2answers
104 views

A looping of algebraic K-theory

Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K_0(\mathcal{C})$. ...
2
votes
1answer
67 views

Bound for a combinatorial sum

I was playing around with a problem and I obtained a certain combinatorial sum. I was wondering if there was a way to simplify or bound it. I have a real valued function $f$, which satisfies $|f(x)| ...
5
votes
2answers
111 views

Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...
1
vote
1answer
80 views

On infinitesimal neighbourhood of a point in a projective scheme

Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$ and $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} ...
-3
votes
0answers
27 views

Radial neuron teaching [on hold]

Hello i have a task to write programm for teaching radial neuron with 3 inputs, i can't find some information about it, i find a lot of info about teaching netowork. I can't undestand what algorithm i ...
0
votes
0answers
37 views

How to join 2 functions into one? [on hold]

is it possible join for example x^2 and (x-2)^2 into one function, so that the graph displays both of them only using one function (relation, to be exact)? Subsequently, is there a general way to ...
3
votes
3answers
89 views

Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?

I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...
1
vote
3answers
82 views

Modular group modulo $N$

Let $N\geq2$ be a positive integer. Is the canonical homomorphism $\pi$ from $SL_2(\mathbb{Z})$ to $SL_2(\mathbb{Z}/N\mathbb{Z})$ surjective? What if we ask the same question for $SL_n$?
-1
votes
0answers
36 views

Proving the solution of one non-linear first order ODE has value 'e-1' at point 1

Consider the following first order non-linear ODE defined on interval $[0,1]$: $$F(x)=f(x)\ln\left(\frac{f^2(x)}{f^2(x)-1}\right)$$ where $f(x)=\frac{\partial F(x)}{\partial x}$, and the initial ...
2
votes
0answers
59 views

Cohomology of elementary Abelian p-group

Let $E=(\mathbb{Z}/p\mathbb{Z})^n$, an elementary Abelian p-group. Let $k$ be an algebraically closed field of characteristic 0. There is a good description of $H^*(E,F^{\times})$ where $F$ is a field ...
0
votes
0answers
39 views

Cohomology operations over general rings [duplicate]

If $X$ is a topological space and $R$ is a commutative ring, then the singular cohomology groups $H^*(X,R)$ support cohomology operations coming from the homology of symmetric groups. If $R = ...
0
votes
0answers
11 views

The use of wavelets in time series modelling

I have been working on modelling a time series using wavelets for a long time. I am quite familiar with the wavelet theory and all...However, I have a big understanding issue and really appreciate it ...
1
vote
0answers
38 views

symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory. I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...
1
vote
1answer
118 views

Number theoretic functions that have an irregular behaviour at primes

Usually, number theoretic functions have "trivial" (or at least easily defined) values for primes. In this thread, I am rather asking for functions which are only defined on primes (well, this ...
3
votes
0answers
35 views

Petersson product of newforms to different level

Let $\text{S}_k^{new}(\Gamma_0(N),\chi)$ be the space of newforms. We call $f\in\text{S}_k^{new}(\Gamma_0(N))$ a newform if $f$ is a Hecke eigenform i.e $\text{T}_nf=\lambda_nf$ ($\text{T}_n$ hecke ...
1
vote
0answers
25 views

Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...
-3
votes
0answers
99 views

Math Instructor [on hold]

How do you obtain a disjoint family from an arbitrary family of sets? This is mentioned in Kelley's book, p. 201, Theorem 35. It's also been mentioned in this site. (Arbitrary union of meager open ...
3
votes
2answers
127 views

“Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ? Is there some kind ...
1
vote
1answer
60 views

About embeddings of connected sums

Let $M_1$ and $M_2$ be two soomth manifolds who're already embedded in $\mathbf{R}^k$. Can one prove that the connected sum of $M_1$ and $M_2$ can also be embedded into $\mathbf{R}^k$ ?
3
votes
1answer
75 views

On Neron-Severi group of normal projective surfaces and blow up

Let $X$ be a normal projective surface with at most rational singularites (in finitely many points). Let $\pi:\tilde{X} \to X$ be the blow up of $X$ at finitely many singular points. The question is ...
1
vote
0answers
25 views

Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course. I've seen a reasonable amount of literature about ...
7
votes
0answers
82 views

“abstract” description of geometric fixed points functor

I'm sure this must be well known, but I could not find any references. My basic question is: Are there "abstract" descriptions of the geometric fixed point functors in equivariant stable homotopy ...
4
votes
1answer
56 views

Diagonalization for sums of Hermitian matrices

I found an interesting question about diagonalizable matrices, Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$. Do there exist complex numbers $u\neq v$, such that $A+uB$ and ...
4
votes
0answers
65 views

Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...
2
votes
0answers
29 views

Augmentation ideal of the cohomology of an elemntary abelian 2-group [on hold]

Let V be an elemntary abelian 2-group and $R=H^{*}V$ its cohomology. What is the Augmentation ideal of R and what is the quotient of R by its augmentation ideal ?
1
vote
0answers
60 views

Is the “Hilbert scheme of curves” in $\mathbb C^3$ a degeneracy locus?

It is known that the Hilbert scheme of $n$ points in $\mathbb C^2$ is expressible as a degeneracy locus, i.e. the zero locus of $\textrm{d}f$, where $f$ is some regular function on a smooth variety. A ...
2
votes
0answers
70 views

Local Systems on Function fields over $\mathbb{F}_p$

Suppose $X$ is a smooth proper curve over $\mathbb{F}_p$ for some prime number $p$. Let $l\neq p$ be a prime, and suppose $L$ is a rank 2 local system over $X$ with coefficients in $\mathbb{Z}_l$ such ...
2
votes
0answers
39 views

Explicit descriptions of self-replicating pro-$p$ groups

A group $G$ is called self-replicating, if there exists a finite index subgroup $H$, such that $H\cong G\times\dots\times G$. Maybe the most famous example of a self-replicating group is a subgroup ...
4
votes
1answer
133 views

Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...
1
vote
1answer
85 views

Groups in which lower central series and upper central series coincide

Let $G$ a finite two-generated $p$-group in which lower and upper central series coincide. Clearly we obtain that the upper central series become strongly central, we have also that at least half of ...
0
votes
1answer
41 views

Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?

I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet: Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset ...
1
vote
1answer
84 views

$\Gamma$-action on maximal tori in Borel-Tits

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...
6
votes
0answers
84 views

What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
1
vote
1answer
44 views

Original sources for two theorems by Bass, Matlis, Papp,

It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...
1
vote
1answer
261 views

spicing up riemann surfaces course

I am a master's student planning to write master's thesis in riemann surfaces.It plan to study forster's riemann surfaces.What side topics could one study to spice up the thesis.I am particularly ...
3
votes
1answer
279 views

When does a cohomology class induce an isomorphism between homotopy groups?

A representative $\alpha$ of a cohomology class $[\alpha]\in H^n(X;\pi_n(X))$ is equivalent to a map from $X$ into an Eilenberg-Mac Lane space as $\alpha:X\to K(\pi_n(X),n)$. After applying a ...

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