# All Questions

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### 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 ...

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2 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 ...

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14 views

### Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition 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 ...

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**2**answers

29 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 $K_0(\mathcal{C})$.
...

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**1**answer

44 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)| ...

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**1**answer

59 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 $\varphi(x,y)$ ...

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**1**answer

36 views

### On infinitesimal neighbourhood of a point in a projective scheme

Let $X, Y$ be projective schemes over $\mathbb{C}$ and suppose that $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} ...

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22 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 ...

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29 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 ...

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**2**answers

44 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 ...

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**3**answers

70 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 $SL_n$?

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28 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 ...

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32 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 ...

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35 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 = ...

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7 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 ...

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34 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 ...

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**1**answer

100 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 ...

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28 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 ...

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18 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 ...

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93 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 ...

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**1**answer

86 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 ...

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**1**answer

50 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 $\mathbf{R}^k$ ?

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**1**answer

58 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 ...

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20 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 ...

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76 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 ...

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42 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 ...

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59 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 ...

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26 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 ?

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55 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 ...

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67 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 ...

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38 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 ...

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102 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 $H\cong G$? ...

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**1**answer

73 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 ...

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39 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 ...

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**1**answer

69 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 ...

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76 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?

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**1**answer

42 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 ...

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207 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 ...

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**1**answer

268 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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9 views

### Bigraded analogue of Ratliff-Rush closure filtration

Consider the filtration $\lbrace{I^rJ^s}\rbrace_{r,s\in\mathbb{Z}}.$
What will be the bigraded analogue of Ratliff-Rush closure filtration $\tilde{{I}^n}=\cup_{k\geq1}({I}^{n+k}:{I}^k)$?
Will it be ...

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36 views

### How do I prove a matrix A is self adjoint to an inner product? . [on hold]

In the source question
B is an element of $M_n(R)$ and is a symmetic matrix
such that $v^tBv>0$.
Also $<.|.>$ is an inner product on $R^n$ called the $B$-inner product.
we are asked to ...

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**2**answers

107 views

### Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...

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42 views

### A limit of a sum related to integer lattice and power series

I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit
$$\lim_{z \to (1,\ldots,1)^-} (\sum_{v ...

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37 views

### An inequality involving Bessel functions of imaginary index

The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J_{ik}(\tau)$, I found in http://link.springer.com/article/10.1134%2F1.558677 ...

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125 views

### “topological” Ochanine genus?

The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus I am aware of a lift to a "spin orientation of Tate K-theory", namely to ...

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29 views

### If $f(x) = Ax$, show that for all $t \in \mathbb{R}$, the extreme $x_n = x_n(t)$ of polygon converges to $e^{At} x_0$ [on hold]

Let $f$ is a vector field in $\mathbb{R}^n$, $x_0 \in \mathbb{R}^n$ and $x_{k+1} = x_k + f(x_k)\Delta t$, $k= 0,1,...,n-1$, where $\Delta t = \frac{t}{n}$. A polygon whose points are the $x_i$ ...

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184 views

### A Special Pair of Formulas

Consider the first order language $\mathcal{L}=\{\in,\subseteq\}$ and $\{\in\}$-theory $\text{ZFC}$. Is there a formula $\psi (x,y) \in \{\subseteq\}-Form$ with the following ...

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73 views

### A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to ...

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118 views

### Riemann Hypothesis and Kahr, Moore and Wang

Is there an expression of the Riemann Hypothesis in the First-Order Logic?
Is there a conversion of this expression to the Kahr, Moore, Wang AEA reduction class for satisfiability?

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63 views

### Riemannian metric on complexification of Lie group

Let $G$ be a compact linear group and $G^c$ be its complexification. Then there is a diffeomorphism $f: G^c \to G \times Lie(G) $ given by $$ x e^{iA} \to (x,A).$$
Let $h$ be the pull back metric of ...