# All Questions

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### Two questions on the Schur multiplier of groups of order $p^4$

I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$. The case in which $p=2$ is well known, see e.g. Table 1 at ...
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### pushing out families of curves

Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...
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### Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? If $M$ is symmetric ...
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### Nontrivial norm values in rings of integers

Let $L=Q(\sqrt{d})$ for some SQF integer $d\equiv_4 3$ (the same can be asked for $d\equiv_4 1$). In this case the ring of integers of $L$ is $O_L=\{x+\sqrt{d}y \mid x,y\in Z\}$ so the norms of $O_L$ ...
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### Distinct Numbers

Let $x_{ij}\in\{0,1\}$ for ${i=1}$ to ${m}$ and for ${j=1}$ to $n$. How many different values does $$\prod_{i=1}^m\sum_{j=1}^nx_{ij}$$ cover? Is there an $a_{ijk}\in\Bbb R$ (there is a $a_{ijk}\neq0$ ...
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### Generalized Dedekind Sum Reciprocity Law

Is there a reciprocity law for generalized Dedekind sums of the form: $$S(a,b;x,y;c)=\sum_{k \mod c}\tilde{B}_1\left(\frac{ak+x}{c}\right)\tilde{B}_1\left(\frac{bk+y}{c}\right)$$ such that the other ...
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### Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...
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### Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple Lie groups, ...
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### Which known theorems of Lie algebras are still valid for Leibniz algebras?

Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras ...
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### Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...
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### Retractions of Yoneda are retractors, i.e., left adjoints?

Background It is fairly well known that if a full subcategory $i: C \hookrightarrow D$ has a left adjoint $F: D \to C$, then the canonical counit $F i(c) \to c$ is an isomorphism. (A classical ...
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### Fundamental group of connected sum for non-orientable manifolds

For orientable manifolds, $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$, fundamental group of connected sum is free product of fundamental groups. As far as I understand, for non-orientable manifolds connected sum ...
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### Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)

What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation? By co-rank, I mean the ...
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### Stable Household Formation

I want to model the problem of household formation by a finite number of individuals, each of whom has preferences over sets of housemates. A collection of households is unstable if there is a set ...
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### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
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### Cubic-exponential enumerative combinatorics

There are many quantities in enumerative combinatorics that grow roughly exponentially, like the Fibonacci numbers, the Catalan numbers, and the factorials; indeed, most of the functions that arise in ...
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### Bounded functions dense in Sobolev Spaces

Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
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### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem: Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation: $Z_{k+1}-Z_k=P_k(1-2Z_k)$ where $P_k=0$ with probability ...
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### Codim r strata in a scheme

My question is what is the definition of codimension r strata for a scheme $X$, with NCD $D$. I heard the term in a lecture but I could not find it nowhere.
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### Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning: For some $\alpha \in H^n(k,\mu_p)$ ...
### Complex structures on $\Bbb R^4$
Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...