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https://mathoverflow.net/questions/256105 | 6 | I've been studying projective modules in Rotman, as well as the topic of localization. Now on the [Wikipedia](https://en.wikipedia.org/wiki/Projective_module#Projective_modules_over_commutative_rings) article about them, there's an example of a locally free module that is not projective.
The module in question is $R/I$... | https://mathoverflow.net/users/101892 | If a quotient ring is a projective module then the ideal is principal | Let $\pi:R\to R/I$ be the natural projection map. This is an $R$-module homomorphism (as well as a ring homomorphism). If $R/I$ is projective, then this map splits. Call such a splitting $\varphi:R/I\to R$. So we have $R= \varphi(R/I)\oplus \ker(\pi)$ (as internal direct sums of $R$-modules). But $\ker(\pi)=I$, and thi... | 10 | https://mathoverflow.net/users/3199 | 256111 | 115,728 |
https://mathoverflow.net/questions/256109 | 0 | By the answer of [the question](https://mathoverflow.net/questions/255919/references-request-reflections-in-coxeter-groups), the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$.
Do we have a one to one correspondence between positive roots and reflections in a Coxeter... | https://mathoverflow.net/users/11877 | Do we have a one to one correspondence between positive roots and reflections in a Coxeter group? | I suggest, you really look carefully into the reference "Humphreys, Reflection and Coxeter groups" ([link behind paywall](http://www.ams.org/mathscinet-getitem?mr=1066460)), as you again find the answer there.
Please look into the reference for details below. In case, you have no access to it, you might be lucky search... | 3 | https://mathoverflow.net/users/21291 | 256115 | 115,729 |
https://mathoverflow.net/questions/256086 | 4 | Find all rational solutions of
$$x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1).$$
Clearly the following six solutions hold:
$$(x,y)=(1,1),(-1,-1),(-1,1),(1,-1),(0,1),(0,-1)$$
But how to find all rational solutions?
| https://mathoverflow.net/users/38620 | Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$ | This is no complete answer yet, but may get expanded to one in due course.
First we search for points on the curve, which is isomorphic to
$$C \colon y^2 = 2(x^6 - x^2 + 2);$$
this produces points with $x \in \{-3, -1, -4/7, 0, 4/7, 1, 3\}$ as already
mentioned in the comments. It was also mentioned that the rank of ... | 17 | https://mathoverflow.net/users/21146 | 256119 | 115,730 |
https://mathoverflow.net/questions/255414 | 17 | **Disclaimer:** I know very little about both of the fields in question.
My question is pretty simple:
>
> **What's the relation between differential Galois theory and D-modules
> over algebraic curves?**
>
>
>
Differential galois theory can't subsume D-modules obviously since the latter contains also infor... | https://mathoverflow.net/users/22810 | D-modules over algebraic curves VS differential Galois theory | Consider what happens if you take a $D$-module on an algebraic curve (with field of fractions $K$) and remove all the information on the singularities. You can achieve this by tensoring over the structure sheaf with $K$, obtaining a module for the ring of differential operators on $K$. This ring is generated over $K$ b... | 8 | https://mathoverflow.net/users/18060 | 256124 | 115,732 |
https://mathoverflow.net/questions/256129 | 7 | I feel sure that this question must have been addressed in the literature, but I can't seem to find it - I may be looking in the wrong place.
A graph is planar if it can be drawn on the plane such that no two edges intersect in an interior point. In general, there are infinitely many ways to draw a planar graph, of c... | https://mathoverflow.net/users/17599 | In how many ways can a given planar graph be mapped into the plane? | It is a theorem of Whitney, that a $3$-connected planar graph has two planar embeddings (one being the other flipped over). If a graph is two-connected, then you can flip over some, but not all of the three-connected components (so you get $2^n,$ where $n$ is the number of such). Finally, if the graph is $1$-connected,... | 7 | https://mathoverflow.net/users/11142 | 256131 | 115,733 |
https://mathoverflow.net/questions/256118 | 4 | Consider an $S^3$-bundle over a closed surface $\Sigma$ with genus $g>0$, and call the total space $X$. Suppose there is a smooth map $F: X \rightarrow \Sigma$ such that $F$ induces an isomorphism $F\_{\*} : H\_{2}(X,\mathbb{Z}) \rightarrow H\_{2}(\Sigma,\mathbb{Z})$.
>
> **Question** Can we always homotope $F$ so ... | https://mathoverflow.net/users/99732 | Question about maps of $S^{3}$-bundles | Here is an argument that is essentially the same as Oscar's, but organized a little differently.
First, we have a fibration $S^3\to X\to\Sigma$, giving an exact sequence
$$ \pi\_2(\Sigma) \to \pi\_1(S^3)=0 \to \pi\_1(X)\xrightarrow{\pi\_\*}\pi\_1(\Sigma) \to \pi\_0(S^3) = 0, $$
which proves that $\pi\_\*$ is an isom... | 3 | https://mathoverflow.net/users/10366 | 256141 | 115,739 |
https://mathoverflow.net/questions/256142 | 7 | On page 174 of both copies I can access of Mumford's book "Lectures on Curves on an Algebraic Surface" there is a printing omission (like [this one](https://mathoverflow.net/questions/215487/definition-of-strongly-stable-0-cycle)).
>
> LEMMA 2: Let $A$ be a complete $p$-adic ring where $p$ is not a zero-divisor, su... | https://mathoverflow.net/users/37110 | Printing omission in Mumford's "Lectures on Curves on an Algebraic Surface" | In the Russian edition it is $$(\xi\_0,\xi\_1,\xi\_2,\dots)\leftrightarrow f(\xi\_0)+pf(\xi\_1)+p^2f(\xi\_2)+\dots$$ where $f$ is the Teichmuller map.
| 11 | https://mathoverflow.net/users/39304 | 256152 | 115,742 |
https://mathoverflow.net/questions/256133 | 14 | A *non-principal* [*probability*] *measure* on a set X is a function $\mu$ defined on all subsets of $X$, with values in $[0,1]$, which is finitely additive, satisfies $\mu(X)=1$, and vanishes on singletons.
Can one prove in ZF + DC that the existence of such a measure on $\bf N$ (or on $\omega$), the set of natural... | https://mathoverflow.net/users/58784 | Does the existence of a non-principal measure on ω imply that of a non Lebesgue measurable set? | This is stated as an open conjecture in Pincus' 1974 paper [The Strength of the Hahn-Banach Theorem](http://link.springer.com/chapter/10.1007/BFb0066014), which is a pretty good sign that he didn't prove it in a paper published in 1972. That same paper contains the first proof of the analogous question for the Baire Pr... | 9 | https://mathoverflow.net/users/99234 | 256155 | 115,743 |
https://mathoverflow.net/questions/256026 | 14 | By merging two-two color classes, the Four Color Theorem implies that every planar graph can be two-colored such that each color class induces a triangle-free graph.
Is there a simpler proof for this fact?
| https://mathoverflow.net/users/955 | Is there easy proof for triangle-free two-coloring of planar graphs? | Thomassen does indeed prove the vertex version, but [in a different paper](http://www.sciencedirect.com/science/article/pii/S0095895608000336). In fact this paper proves the stronger statement that you can get a coloring without monochromatic triangles from any 2-list coloring.
A bit of googling revealed that origina... | 7 | https://mathoverflow.net/users/2384 | 256163 | 115,744 |
https://mathoverflow.net/questions/256140 | 8 | Let $M$ be a smooth Riemannian manifold without boundary. Let $X\subset M$ be a closed subset which is a smooth submanifold with boundary, $\dim X=\dim M$. Assume that $X$ is locally convex, i.e. any point of $X$ has a neighborhood $U$ such that for any two points from $X\cap U$ there exists a unique shortest geodesic ... | https://mathoverflow.net/users/16183 | Geometry of convex sets in Riemannian manifolds | Yes it is true.
Consider the signed diststance function $f=\mathrm{dist}\_{\partial X}$.
Note that it is semiconcave in a neigborhood of $X$;
moreover, $f''\le C\cdot |f|$ for some constant $C$ defined locally.
For your $\gamma$, we have $(f\circ\gamma)'=0$ and
$$(f\circ\gamma)''\le C\cdot |f\circ\gamma|.$$
The lat... | 8 | https://mathoverflow.net/users/1441 | 256164 | 115,745 |
https://mathoverflow.net/questions/256143 | 2 | Let $p$ be a prime number, $\psi:G\_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}\_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the weight space representing the homomorphism $\mathbb{Z}\_p^{\times} \rightarrow \mathbb{G}\_m$ and $\epsilon\_p$ denotes... | https://mathoverflow.net/users/46460 | extension of the universal cyclotomic character | Yes, this can be done.
One can realise the universal character $\epsilon^{\mathrm{univ}}$ as an $R$-linear representation $G\_{\mathbf{Q}} \to R^\times$, where $R$ is isomorphic to a direct product of finitely many copies of $\mathbf{Z}\_p[[T]]$. Thus $R$ is Noetherian semilocal, and if $S$ is the finite set of prim... | 3 | https://mathoverflow.net/users/2481 | 256165 | 115,746 |
https://mathoverflow.net/questions/256156 | 10 | Suppose $R\_1,R\_2$ are finite unital commutative rings. Consider Heisenberg groups $H\_3(R\_1)$ and $H\_3(R\_2)$ (upper unitriangular marticies $3 \times 3$).
>
> **Proposition.** If $R\_1 \not\cong R\_2$ (as rings) then $H\_3(R\_1) \not\cong H\_3(R\_2)$ (as multiplicative groups).
>
>
>
Is that true? It se... | https://mathoverflow.net/users/35603 | Non-isomorphic Heisenberg groups over rings | Theorem 1.13 of [this paper](http://www.sciencedirect.com/science/article/pii/0168007294900221) *(The model theory of unitriangular groups, Ann. Pure Appl. Logic
68(3), 1994, 225-261)* by O. Belegradek says that the answer is positive even for infinite commutative rings.
However, Proposition 1.9 in the same paper as... | 11 | https://mathoverflow.net/users/15934 | 256167 | 115,747 |
https://mathoverflow.net/questions/256128 | 5 | Let $A$ be a positive definite, real $n \times n$ matrix. This defines a norm on $\mathbb{R}^n$. Now I have a given point $p \in \mathbb{R}^n$ and I want to find the lattice point $x \in \mathbb{Z}^n$ that is closest to $p$ with respect to this norm. This is commonly known as the **Closest Vector Problem** (CVP) and se... | https://mathoverflow.net/users/36563 | Package for the Closest Vector Problem (CVP)? | `fplll` (available here: <https://github.com/fplll/fplll>), a C++ implementation of a selection of lattice algorithms, has a CVP solver:
>
> It also includes a floating-point implementation of the Kannan-Fincke-Pohst algorithm [K83,FP85] that finds a shortest non-zero lattice vector. For the same task, the GaussSie... | 6 | https://mathoverflow.net/users/20186 | 256168 | 115,748 |
https://mathoverflow.net/questions/256146 | 2 | I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a statement is defined, and he says that he likes to think of $\neg P$ as the new sentence "$P$ is false".
Being familiar wi... | https://mathoverflow.net/users/3199 | Defining negation | The only problem with defining negation is that there are at least two notions of "negation":
Negation (let's call it $\lnot\_1$) as a syntactic operation is quite clear: you prepend the symbol $\lnot$ to a formula.
Negation (or $\lnot\_2$) is also a unary function on the two-element Boolean algebra $\{\top,\bot\}... | 5 | https://mathoverflow.net/users/14915 | 256169 | 115,749 |
https://mathoverflow.net/questions/256074 | 15 | Consider the ring of polynomials $R:=\mathbb{Z}[x\_1,x\_2,x\_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x\_1,x\_2,x\_3)=f(x\_1-1,x\_2,x\_3)$ and the identity $If=f$.
Let $\mathcal{L}:R\rightarrow R$ be the operator given by
$$\mathcal{L}f=[(x\_1+x\_2)(x\_1+x\_3)E-x\_1^2I]f.$$
Let $1$ stand for the const... | https://mathoverflow.net/users/66131 | An orbit of symmetric polynomials | Suppose $f(x,y,z)$ is symmetric (in the following, *symmetric* tout court always means "symmetric w.r.to the three variables $(x,y,z)$") . Then $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ is already symmetric w.r.to $(y,z)$, so it is symmetric if and only if it is symmetric w.r.to $(x,y)$, that is, after si... | 15 | https://mathoverflow.net/users/6101 | 256176 | 115,753 |
https://mathoverflow.net/questions/256160 | 4 | Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit $(n-1)$-dimensional simplex: $x\_1+\cdots+x\_n=1$. I'm interested in upper (and lower) bounds on the expected absolute ... | https://mathoverflow.net/users/11054 | Determinant of a random row stochastic matrix | The paper to look at is [Bordenave, Caputo, Chafai](https://arxiv.org/abs/0808.1502), which is cited in Nguen's paper.
| 4 | https://mathoverflow.net/users/11142 | 256178 | 115,754 |
https://mathoverflow.net/questions/256172 | 5 | In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}\_p}$ of $\mathbb{Q}\_p$. My question is if this "conjecture" has been proved or disproved before.
Consider the following result proved in "Local Fields" Berlin. (1980) by Serre, J. P.:
... | https://mathoverflow.net/users/47542 | p-adic expansion for elements in algebraic closure of p-adic numbers | The following article discusses p-adic expansions in $\overline{\mathbb{Q}}\_p$
and of $\mathbb{C}\_p$.
Algebraic $p$-adic expansions,
David Lampert,
*Journal of Number Theory* **23** (1986), 279–284.
| 7 | https://mathoverflow.net/users/11926 | 256182 | 115,756 |
https://mathoverflow.net/questions/256180 | 3 | I am not having the best luck getting this, so any advice is appreciated. I apologize in advance if the question is too low-level. Let $G$ be a locally compact group $\sigma$-compact group, and $H\leq G$ a closed subgroup such that $G/H$ carries a finite invariant measure $\mu$. Let $(\pi,\cal H\_\pi)$ be a unitary rep... | https://mathoverflow.net/users/69876 | Induced representation of locally compact groups | Such a $\Phi$ is not going to exist — nor is $H$ going to have nonzero $\mu(H)$ — unless $H$ is *open* as well as closed in $G$. This is the very case treated in Mackey [(1951, Part II)](http://www.ams.org/mathscinet-getitem?mr=42420): then $G/H$ is discrete, $\mu$ can be normalized to be counting measure (i.e. $\mu(eH... | 3 | https://mathoverflow.net/users/19276 | 256187 | 115,759 |
https://mathoverflow.net/questions/256181 | 7 | Let $j, k ,n$ be nonnegative integers such that $0 \leq j, k \leq n \leq k +j $. Pick integer $m$ such that $0 \leq m \leq k + j - n$.
Let $\langle x \rangle\_m$ denote the falling factorial $x(x-1)\ldots (x-m+1)$.
I've stumbled across the need to prove the following equality:
$$\sum\_{i=0}^n \frac{\binom{k}{i}\b... | https://mathoverflow.net/users/78672 | Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$ | It is very probable that what is written below is the simplification of Darij's argument. I use the notation $x^{\underline{n}}=x(x-1)\dots(x-n+1)$ [as in Knuth's books] for the falling factorial, and $[t^n] f(t)$ for the coefficient of $t^n$ in the polynomial $f$.
For a polynomial $f(t)$ of degree at most $j$ we hav... | 10 | https://mathoverflow.net/users/4312 | 256199 | 115,763 |
https://mathoverflow.net/questions/256166 | 3 | Let $$D\_2(n) =\sum\_{pq\leq n} 1,$$
and
$$F\_2(n) =\sum\_{pq\leq n} \frac{1}{pq}$$
where $p,q$ are primes. Similarly define
$$D\_k(n) =\sum\_{p\_1\cdots p\_k\leq n} 1,$$
and
$$F\_k(n) =\sum\_{p\_1\cdots p\_k\leq n} \frac{1}{p\_1\cdots p\_k}.$$
I believe my calculations showing
$$F\_2(n)\sim \frac{(\log\log n)^2}{2}... | https://mathoverflow.net/users/17773 | Estimating $\sum_{p_1\cdots p_k\leq n} \frac{1}{p_1\cdots p_k}$ for various $k$ | For any fixed $k \geq 1$, one has $$D\_k(n) \sim \frac{x}{\log x} \frac{(\log \log x)^{k-1}}{(k-1)!}.$$ This was originally proved by Landau in 1909 using induction of $k$.
For $k \ll \log \log x$, see Theorem 4 of chapter 6 of Tenenbaum's book "Introduction to analytic and probabilistic number theory." This is the ... | 7 | https://mathoverflow.net/users/50426 | 256200 | 115,764 |
https://mathoverflow.net/questions/256201 | 1 | Let $\phi(\cdot)$ be the Euler totient function, and let $n=p\_1^{k\_1}\cdots p\_s^{k\_s}$ be the prime factorization of $n\in \mathbb{N}$. The well-known Euler's product formula states that $\phi(n)=n(1-\frac{1}{p\_1})\cdots(1-\frac{1}{p\_s})$. For some fixed positive integer $a\leq n$, let $\phi(n,a)$ denote the numb... | https://mathoverflow.net/users/75264 | Euler's Totient Function | The formula is given in [this question](https://mathoverflow.net/questions/88777/bound-the-error-in-estimating-a-relative-totient-function), while the answers thereto discuss the accuracy of the obvious approximation (that this is like $a \phi(n)/n.$)
| 1 | https://mathoverflow.net/users/11142 | 256203 | 115,766 |
https://mathoverflow.net/questions/256058 | 4 | For a linear algebraic group over an algebraically closed field of characteristic zero $G$, with unipotent radical $U$, we have that $G/U$ is reductive.
When $G$ is solvable, then Lie's theorem says that irreducible representations are 1-dimensional, so the unipotent radical acts trivially, and so the irreducible rep... | https://mathoverflow.net/users/6059 | Irreducible representations of the reductive quotient | Let $V$ be an irreducible representation of $G$. Consider it as a representation of $U$. Since $U$ is unipotent, $V^U\neq 0$. On the other hand, since $U$ is normal in $G$, $V^U$ is a subrepresentation. But the irreducibility of $V$ implies then that $V^U=V$, which is what you wanted.
| 4 | https://mathoverflow.net/users/41644 | 256205 | 115,767 |
https://mathoverflow.net/questions/256068 | 6 | Does there exist some subring $k \subset \mathbb{C}$ such that the following assertion holds?
* **($k$-Hodge conjecture)** For each nonsingular algebraic variety $X$ over $\mathbb{C}$, and each $q = 0, 1, \ldots, \dim\_{\mathbb{C}}(X)$, each class $\mathfrak{z} \in H^{2q}(X; k) \cap H^{q,q}(X)$ is a $k$-linear combin... | https://mathoverflow.net/users/nan | Status of Hodge conjecture over subrings of $\mathbb{C}$ | The $k$-Hodge conjecture is false for any $k$ containing an irrational real $\alpha$.
Indeed we may clearly assume $\alpha$ is negative. Consider the elliptic curve $E = \mathbb C / \langle 1, \sqrt{\alpha} \rangle$. Let $e\_1$ and $e\_2$ be a basis of $H^1(E,\mathbb Z)$ such that the map to $H^{1,0}(E)$ sends $e\_1$... | 10 | https://mathoverflow.net/users/18060 | 256207 | 115,769 |
https://mathoverflow.net/questions/256196 | 15 | I am trying to understand Shapiro's lemma for $H^2$ in the concrete language of extensions of finite groups.
Let $H$ be a subgroup of a finite group $G$, and let $A$ be an $H$-module. Let ${\rm Ind}\_G^H(A)$ be the induced module (see Serre, Galois Cohomology, Ch. I, 2.5). Shapiro's lemma says that the inclusion $H\h... | https://mathoverflow.net/users/101929 | Shapiro's lemma in the language of group extensions | Shapiro's Lemma boils down to the following isomorphism for a subgroup of finite index: Let us write $G=\bigcup\_{i=1}^t g\_iH$ for left coset representatives of $H$ in $G$. For a $G$-module $P$ we have the following isomorphism $$\Psi:Hom\_H(P,A)\cong Hom\_G(P,Ind\_H^G(A))$$
$$ \Psi(f)(p) = \sum\_{i}g\_i\otimes f(g\_i... | 4 | https://mathoverflow.net/users/41644 | 256208 | 115,770 |
https://mathoverflow.net/questions/256215 | 2 | The following occurred to me while playing [Sokoban](https://en.wikipedia.org/wiki/Sokoban).
Let $G=(V,E)$ be a finite, simple, undirected, and connected graph with $|V|>1$. We call a function $f:V\to V$ a *push function* if $\{x,f(x)\}\in E$ for all $x\in V$. (Note that a push function cannot have fixed points, and ... | https://mathoverflow.net/users/8628 | Push function on simple undirected graphs | Yes. If you have an odd cycle, then you can reduce the size of the image by first pushing two different vertices of the current image to the cycle, and then push them to the same vertex. So a graph is contractible if and only if it is connected and not bipartite (or has only one vertex).
| 3 | https://mathoverflow.net/users/955 | 256216 | 115,773 |
https://mathoverflow.net/questions/256217 | 4 | Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$.
Let $U$ be a neighborhood of $\Omega$ such that $\exp\_p(t\nu(p))$ is diffieomorphism, where $p\in\Sigma$ and $\nu$ is the outward unit normal vector field along $\Sigma$. We now consider the distance function $... | https://mathoverflow.net/users/101796 | Riccati equation and principal curvatures | I have found an answer to my question on Mathoverflow.net [here](https://mathoverflow.net/questions/249184/comparison-of-eigenvalue-of-solution-of-riccati-equation?rq=1). @Raziel gave the reference
H. L. Royden, MR 948079 Comparison theorems for the matrix Riccati equation, [Comm. Pure Appl. Math. 41 (1988), no. 5, ... | 2 | https://mathoverflow.net/users/101796 | 256228 | 115,778 |
https://mathoverflow.net/questions/256066 | 11 | Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.
What I would like is a formula for the (normalized) Haar measure integral
$$
\int\_{U(N)} g\_{i\_1 j\_1}\cdots g\_{i\_n j\_n} {\bar{g}}\_{k\_1 l\_1}\c... | https://mathoverflow.net/users/7410 | Formula for $U(N)$ integration wanted | To expand on my comments, this paper <https://arxiv.org/pdf/hep-th/9209083v2.pdf> by Shatashvili deals
with ``correlation functions'' of Haar unitary matrices of the form
$$
\int\_{U(N)}^{} d\mu(U) e\_{}^{tr(UAU\_{}^{-1}B)}
U\_{i\_1j\_1}^{}\bar U\_{k\_1\ell\_1}^{}\ldots
U\_{i\_mj\_m}^{}\bar U\_{k\_m\ell\_m},
$$
and pr... | 6 | https://mathoverflow.net/users/979 | 256237 | 115,779 |
https://mathoverflow.net/questions/256235 | 2 | In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2\_r(f)=\{g\in L^1(0,1):\; \|f-g\|\_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
| https://mathoverflow.net/users/97212 | Separability of $L^1$ in $L^2$ topology | No. Let $f\_a(x)=|x-a|^{-1/2}$. If we had a countable dense set, then (pigeonhole principle) there would be a $g\in L^1$ with $g-f\_a, g-f\_b\in L^2$ for some $a\not= b$, so $f\_b-f\_a\in L^2$, but of course this is false.
| 5 | https://mathoverflow.net/users/48839 | 256239 | 115,780 |
https://mathoverflow.net/questions/256220 | 7 | The notion of [vertex algebra](https://en.wikipedia.org/wiki/Vertex_operator_algebra#Vertex_algebra), like any reasonable algebraic notion, makes sense inside any (sufficiently linear) symmetric monoidal category. The standard pictures of the operator product, however, suggest that symmetry is more than is needed: one ... | https://mathoverflow.net/users/78 | What are braided vertex algebras? | For the case of vector spaces graded by an abelian group (with braiding determined by an abelian 3-cocycle following Joyal-Street), this was done by Dong and Lepowsky in their 1993 book "Generalized Vertex Algebras and Relative Vertex Operators". The object has the name "abelian intertwining algebra", and standard exam... | 8 | https://mathoverflow.net/users/121 | 256241 | 115,782 |
https://mathoverflow.net/questions/256244 | 2 | I am interested in the following graphs: $\Gamma\_n$ has vertex set $V\_n=\{0,\ldots,2^n-1\}\subset\mathbb{Z}$ and two vertices $a,b$ span an edge whenever $|a-b|$ is a power of $2$.
Does the family of graphs $(\Gamma\_n)$ form an expander?
These graphs are not regular, so perhaps one way would be to show that the... | https://mathoverflow.net/users/35269 | Graphs with edges corresponding to powers of 2 | No, the same sets that show that the hypercube is not a vertex expander show that this graph is not a vertex expander.
Suppose $n$ is even. Let $\theta(x)$ be the number of 1's in the binary expansion of $x$ and let
$S=\theta^{-1}([0,n/2])$ be the set of integers between $0$ and $2^n$ whose binary expansion contains ... | 2 | https://mathoverflow.net/users/25051 | 256267 | 115,789 |
https://mathoverflow.net/questions/256268 | 29 | I heard the claim as in the title for a long time, but can not find the precise reference for this claim, what's the reference with proof for this claim? Thanks for the help.
To be more precise, is there a canonical topology structure on the space $\Omega$ of all compact $n$-dim smooth manifolds, such that for any com... | https://mathoverflow.net/users/100486 | Most manifolds are hyperbolic? | The quotes are from Thurston's survey paper *[Three dimensional manifolds, kleinian groups and hyperbolic geometry](http://www.csee.wvu.edu/~xinl/library/papers/math/geometry/thurston1982.pdf)* page 362:
>
> 2.6. THEOREM [Th 1]. Suppose $L \subset M^3$ is a link such that $M — L$ has a hyperbolic structure. Then mo... | 33 | https://mathoverflow.net/users/39082 | 256271 | 115,791 |
https://mathoverflow.net/questions/256236 | 5 | Let $G$ be a quasi-split connected reductive group over a perfect field $F$, with Borel subgroup and maximal torus $B \supseteq T$ defined over $F$. Assume everything splits over a Galois extension $L/F$, with $\Gamma = \textrm{Gal}(L/F)$. The choice of $B$ gives us a set of simple roots $\Delta \subseteq \Phi(G,T)$, a... | https://mathoverflow.net/users/38145 | Splitting field of a root vector $x_{\alpha}: \mathbf{G}_a \rightarrow U_{\alpha}$ | This is explained by the Borel-Tits relative structure theory for connected reductive groups over arbitrary fields. In particular, there is no need to assume $F$ is perfect. The explanation below is long when written out, but the underlying principles are rather natural and clean.
Moreover, we will find a collection of... | 5 | https://mathoverflow.net/users/81332 | 256275 | 115,793 |
https://mathoverflow.net/questions/256276 | 8 | Let $\mathfrak{S}\_n$ be the permutation group on $\{1,\dots,n\}$. Given $\pi=\pi\_1\pi\_2\dots\pi\_n\in\mathfrak{S}\_n$, its [major index](https://en.wikipedia.org/wiki/Major_index) statistic is denoted **maj**$(\pi)$. Define the polynomials
$$Q\_{n,k}(x):=\sum\_{\pi\in\mathfrak{S}\_n}x^{\text{maj}(\pi)+\pi\_n+\pi\_{n... | https://mathoverflow.net/users/66131 | factorization of polynomials wrt the major index stat | For inv instead of maj this is rather clear. I claim that when we fix a $(k+1)$-element subset $K$ of $\{1,2,\ldots,n\}$, the polynomial
$$\sum\_{\pi}x^{\operatorname{inv}(\pi)+\pi\_n+\pi\_{n-1}+\cdots+\pi\_{n-k}} ,$$
where the sum is running over the $\pi\in\mathfrak{S}\_n$ with $\{\pi\_{n-k},\dots,\pi\_n\} = K$,
equa... | 5 | https://mathoverflow.net/users/4312 | 256285 | 115,796 |
https://mathoverflow.net/questions/256284 | 0 | Assume we are given numbers $n,k\in\mathbb{N}$. We want to construct a graph $G=([n],E)$ such that its independence number satisfies $\alpha (G)<k$. Denote $s=|E|$, the number of edges in this graph.
We want to construct a graph $G$ with the minimum possible $s^\*(n,k)$. How can one determine this number? Are there g... | https://mathoverflow.net/users/81429 | Constructing graphs with independence number $\alpha (G)<k$ | This is [Turán's theorem](https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem) (for a complement graph).
| 4 | https://mathoverflow.net/users/4312 | 256286 | 115,797 |
https://mathoverflow.net/questions/156320 | 9 | Let $\mathbb F\_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f\_0(x), \ldots,
f\_k(x) \in \mathbb F\_q[x]$ be polynomials such that
$$e= \deg f\_0 > \deg f\_1 > \cdots > \deg f\_k,$$
and let us define $W:= \langle f\_0, f\_1, \ldots,... | https://mathoverflow.net/users/45664 | On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$ | Generalize your definition slightly - Let $M\_{e,k,q,r}$ be the max over $W$ and $S \subseteq \mathbb F\_q^\times$ of cardinality $q-1-r$ of the number of squarefree monic polynomials over $\mathbb F\_q$ lying in $W$ whose roots all lie in $S$.
Your lower bound clearly becomes
$$M\_{e,k,q,r} \geq {q-1-e-r+k\choose... | 3 | https://mathoverflow.net/users/18060 | 256292 | 115,801 |
https://mathoverflow.net/questions/256306 | 7 | For $n\in\mathbb{Z}^{+}$, consider the polynomials
$$P\_n(x)=\prod\_{k=0}^{n-1}(x^n-x^k).$$
>
> **QUESTION.** Is it possible to find a closed formula for the number of monomials in $P\_n(x)$, after expansion?
>
>
>
Those interested in such enumeration may like to [read this](https://arxiv.org/pdf/0811.3652.pdf... | https://mathoverflow.net/users/66131 | counting monomials and integrality | The claim is correct. We begin with a standard lemma.
**Lemma:** Let $p$ be an odd prime, and $x$ an integer coprime to $p$. Let $k$ be a positive integer and let $o$ be the multiplicative order of $x$ modulo $p$. Then $$v\_p(x^k-1) = \begin{cases} v\_p(x^o-1) + v\_p(\frac{k}{o}) & o \mid k \\ 0 & o \nmid k \end{case... | 4 | https://mathoverflow.net/users/31469 | 256314 | 115,809 |
https://mathoverflow.net/questions/256308 | 1 | A *saturated linear hypergraph* is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ such that
1. $|e|\geq 2$ for all $e\in E$,
2. $|e\_1\cap e\_2| = 1$ for all $e\_1, e\_2\in E$ with $e\_1\neq e\_2$, and
3. $|\{e\in E:v\in e\}| = 2.$
Let $E$ be the set of $n\in\mathbb{N}$ such that it is impossib... | https://mathoverflow.net/users/8628 | Cardinalities of saturated linear hypergraphs | This is similar to an answer I gave before.
For each point $v$, let $L\_v$ denote the edges containing $v$. Then we know that each set $L\_v$ has cardinality 2, and they're distinct.
Moreover, if the edges are $e\_1, e\_2, \ldots , e\_m$, then since every pair of edges intersect, we'll need that the sets $L\_v$ are... | 3 | https://mathoverflow.net/users/22512 | 256317 | 115,811 |
https://mathoverflow.net/questions/256214 | 0 | Let $V,H,V^\*$ be a Gelfand-Triple, $\phi\colon V \to \mathbb{R}$ convex, lower semicontinuous and proper. There exists a so called Moreau-Enveloppe $\phi\_j$, which is Gateâux-differentialable. It's derivative $\phi\_j'\colon V \to V^\*$ is demicontinuous and maps bounded sets to bounded sets in $V^\*$. Now my questio... | https://mathoverflow.net/users/100894 | Moreau-Enveloppe from $L^2(0,T;V) \to L^2(0,T;V^*)$ | Your assumptions are not sufficient to consider this mapping property.
Consider $V = \mathbb{R}$ and the mapping $f(x) = x^3$. It maps bounded sets to bounded sets, but its Nemytskii operator $y \mapsto y^3$ is not well defined from $L^2(0,T)$ into itself, only from $L^{3p}(0,T)$ into $L^p(0,T)$, $p \ge 1$.
| 0 | https://mathoverflow.net/users/32507 | 256320 | 115,813 |
https://mathoverflow.net/questions/256323 | 2 | Consider a multiset $S=\{a\_1,a\_2,...,a\_{2n}\}$ of natural numbers. There are $2n$ elements (not necessarily unique since $S$ is a multiset) in $S$. All elements of $S$ belong to a set of natural numbers $R$ with size $N$ (i.e |R|=N). The probability that a number $i$ is in the multiset $S$ is $P\_i$. Each element $a... | https://mathoverflow.net/users/64387 | Average number of rows to fit all elements in a multiset of natural numbers | You are selecting iid random variables $a\_1, \ldots, a\_{2n}$ from some distribution. And you want to estimate how often the most common value appears.
Let $p\_{0}$ be the highest probability. If this is much larger than the rest of the values, then $p\_{0} 2n$ is a good estimate (always a lower bound for what you w... | 2 | https://mathoverflow.net/users/22512 | 256330 | 115,818 |
https://mathoverflow.net/questions/256161 | 0 | I like to ask a simple question: **how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.**
Let us take a nontrivial 2-cocycle $\omega\_3^G(g\_a, g\_b) \in H^2(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwo... | https://mathoverflow.net/users/44768 | Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$ | The given particular 2-cocycle $\omega\_2 \in H^2(G,\mathbb{R}/\mathbb{Z})$ of $G=(\mathbb{Z}\_2)^2$ is
$$
\omega\_2(g\_a , g\_b)=(-1)^{[g\_{a\_1}]\_2 [g\_{b\_2}]\_2 }, \;\;\;\;
$$
where $g\_a=(g\_{a\_1},g\_{a\_2}) \in G=(\mathbb{Z}\_2)^2$, and similarly for $g\_b$.
Below I aim to show that both $Q\_8 \to (\mathbb{Z}\_... | 4 | https://mathoverflow.net/users/27004 | 256336 | 115,819 |
https://mathoverflow.net/questions/256338 | 2 |
>
> Suppose $f:I\rightarrow\mathbb R$ has antiderivatives on an interval $I\subset\mathbb R$. Then $f$ can be decomposed as $f=g+h$, where both $g,h:I\rightarrow\mathbb R$ have antiderivatives. In addition, $g$ is bounded below and $h$ is bounded above.
>
>
>
I have been long thinking about this statement, but i... | https://mathoverflow.net/users/101992 | Decompose a function having antiderivatives into bounded components | In general, $f$ does not admit such a decomposition. If it does, $f=g+h=(g+m)+(h-m)$, so we can assume $g\ge0$ and $h\le0$, that is, any antiderivative $F$ of $f$ is bounded variation. But in general this is not the case for an everywhere derivable function , e.g $F(x)=x^2\sin( 1/x^2)$ for $x\neq0$, $F(0)=0$.
| 2 | https://mathoverflow.net/users/6101 | 256339 | 115,820 |
https://mathoverflow.net/questions/256342 | 23 | Endow $S^7$ with a structure of an $H$-space induced from multiplication in octonions $\mathbb{O}=\mathbb{R}^8$. It is not associative as octonion multiplication is not associative.
Is it associative up to homotopy, i.e. are maps $m(m(-,-),-):S^7\times S^7\times S^7\to S^7$ and $m(-,m(-,-)):S^7\times S^7\times S^7\to... | https://mathoverflow.net/users/39304 | Is the $H$-space structure on $S^7$ associative up to homotopy? | It is not. See Theorem 1.4 of [this paper by I.M. James](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0090812-8/) (Trans. AMS 84 (1957), 545-558).
In particular, there exists *no* homotopy associative multiplication on $S^n$ unless $n=1$ or $n=3$.
| 30 | https://mathoverflow.net/users/11546 | 256343 | 115,821 |
https://mathoverflow.net/questions/256299 | 1 | We can join two $S^d$'s via a connected sum to obtain $S^d\#S^d=S^d$. We can also deform a single $S^d$ via a self-connected sum (ie add a handle) to obtain $\#\_\text{self}S^d=S^{d-1}\times S^1$.
When $d=2$, the self-connected sums can generate all 2-dimensional closed oriented manifolds from $S^2$. Do we have a si... | https://mathoverflow.net/users/17787 | Constructing manifolds via generalized connected sums or fiber sums | Obtaining a manifold from $S^d$ by iterative connected sum is not always possible. The homology groups of the $d$-manifold $M$ change in a controlled way when taking such a connected sum with itself to obtain a manifold $M'$: we have
$$
H\_k(M') = \begin{cases}
H\_k(M) &\text{if }k \neq 1,d-1\\
H\_k(M) \oplus \Bbb Z &\... | 5 | https://mathoverflow.net/users/360 | 256357 | 115,824 |
https://mathoverflow.net/questions/255975 | 4 | Given classes $H$, $R$, can we find a class $G$ such that the following statement is provable in ZFC?
If $R\subseteq V\times V$ and $H$ is a function from $V$ into $Ord$ such that for all sets $x,y$, $y\ R\ x$ implies that $H(y)<H(x)$, then $G$ is a function from $V$ into $V$ such that for all sets $x$, $G(x)=\{G(y)\... | https://mathoverflow.net/users/101817 | The existence of the Mostowski collapses for a non-set-like well-founded relation from which there is a homomorphism to $Ord$ | After 4 days, I solved my own question. The answer is negative, that is, we can not prove in ZFC the existence of the required $G$ for all classes $H$, $R$. In what follows, we construct special classes $H$ and $R$ such that if there is a required $G$ then we can prove in ZFC the consistency of itself.
Consider in ZF... | 1 | https://mathoverflow.net/users/101817 | 256364 | 115,829 |
https://mathoverflow.net/questions/256365 | 3 | I have a system of $n \times 1$ equations
$$
0 = A\,vec(xx^t) + B x + C
$$
where
* $x$ is a $n \times 1$ vector of unknowns
* $x^t$ means transpose
* $vec$ means $xx^t$ has been vectorized so has dimension $n^2 \times 1$
* $A$ is a known matrix with dimensions $n \times n^2$
* $B$ is a known matrix with dimensions $n... | https://mathoverflow.net/users/102005 | Solving a vector of quadratic equations | Shameless advertisement to a paper of mine: <http://www.sciencedirect.com/science/article/pii/S0024379511004484> *Quadratic vector equations*, in Linear Algebra and its Applications, volume 438, 2013. Arxiv version on <https://arxiv.org/abs/1004.1500>.
I studied this exact problem in the case in which $A,B,C$ all con... | 5 | https://mathoverflow.net/users/1898 | 256368 | 115,830 |
https://mathoverflow.net/questions/255275 | 7 | I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.)
An easy example is the real projective space $RP^n$ in the case $n \equiv 1 \ (\operatorname{mod} 4)$. But unfortunately, I don't kn... | https://mathoverflow.net/users/13356 | Second Stiefel-Whitney class is a square | At least there are quite a lot of such manifolds: up to multiplying by powers of 2, any oriented bordism class contains such a manifold.
Proof: Let $f: X \to BSO$ be the universal map such that $w\_2$ pulled back to $X$ is a square of a class $x$. I.e. $X$ is the fiber of a map $w\_2 - x^2: BSO \times K(\mathbb{F}\_2... | 14 | https://mathoverflow.net/users/80296 | 256369 | 115,831 |
https://mathoverflow.net/questions/256301 | 3 | A finite group G is called a 2-Frobenius group if it has a normal series $1\unlhd H\unlhd K\unlhd G$, where $K$ and $G/H$
are Frobenius groups with kernels $H$ and $K/H$, respectively.
We consider the relationship between $A=\{2\text{-Frobenius groups}\}$ and $B=\{\text{Frobenius groups}\}$:
>
> **QUESTION.** Is ... | https://mathoverflow.net/users/99750 | Is there a finite group that is both 2-Frobenius and Frobenius? | Attempt at correction ( or more precisely, justification that original attempt was basically correct): let $G$ be a Frobenius group with kernel $L$. Then $L \leq F(G)$ by Thompson's theorem, so $Z(F(G)) \leq C\_{G}(L) \leq L,$ as $L$ is a Frobenius kernel. Then $F(G) \leq C\_{G}(Z(F(G)) \leq L,$ so in fact $F(G) = L.$
... | 6 | https://mathoverflow.net/users/14450 | 256371 | 115,832 |
https://mathoverflow.net/questions/256353 | 2 | It is known that
$$\int\_ 0^{\infty}\frac {e^{-x - \frac {1} {x}}} {x} dx=2 K\_0(2),$$ but now I want to get the closed form approximate result of
$$\int\_ 0^a\frac {e^{-x - \frac {1} {x}}} {x} dx.$$
I have searched the classic Table of Integrals, Series, and Products, but there is no pattern match this situation.
Is ... | https://mathoverflow.net/users/64177 | Is there a way to get the closed form approximate result of $\int_ 0^a\frac {e^{-x - \frac {1} {x}}} {x}\, dx$ | 
The blue curve is the desired integral $\int\_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K\_0(2)a^3(1+a^3)^{-1}$.
You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-... | 5 | https://mathoverflow.net/users/11260 | 256376 | 115,833 |
https://mathoverflow.net/questions/256224 | 5 | Consider the following result (which is Lemma 2.8 in Mitchell and Steel's paper on Fine Structure and Iteration Trees):
**Lemma 2.8** Let $\pi \colon \mathcal{H} \to \mathcal{M}$ be generalized $r \Sigma\_{k}$
elementary, where $\mathcal M$ is a ppm (not of type III) and $1
\leq k < \omega$. Suppose that $\rho\_{k}... | https://mathoverflow.net/users/57114 | Mitchell, Steel. FSIT. Lemma 2.8: Is $k$-solidity actually needed? | In my question I already verified that (assuming 1. and 2.) item 3. is provable without assuming that $\pi(r)$ is $k$-solid over $(\mathcal{M},q)$. To see that we can actually drop $k$-solidity in this lemma, it hence suffices to see that 1. and 2. also don't require that $k$-solidity of $\pi(r)$.
If $\rho\_{k}^{\mat... | 3 | https://mathoverflow.net/users/57114 | 256381 | 115,836 |
https://mathoverflow.net/questions/256033 | 13 | Let $\Phi$ be a (crystallographic) root system and $\Phi^{+}$ a choice of positive roots, with $\Delta$ the corresponding choice of simple roots. So the root lattice of $\Phi$ is just $\mathbb{Z}\Delta$ ($=\mathbb{Z}\Phi^{+}=\mathbb{Z}\Phi$).
Then $\rho := \frac{1}{2}\sum\_{\alpha \in \Phi^{+}}\alpha$ (the "half-sum ... | https://mathoverflow.net/users/25028 | Significance of half-sum of positive roots belonging to root lattice? | There is no single answer to this question. Here is one observation.
The element $z\_G=e^{2\pi i\rho^\vee}$, where $\rho^\vee$ is one-half the sum of the positive co-roots, is a canonical (independent of the choice of positive co-roots) element of $G$, fixed by every automorphism of $G$. If $G$ is simply connected $z... | 3 | https://mathoverflow.net/users/6030 | 256388 | 115,838 |
https://mathoverflow.net/questions/235636 | 5 | Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a decomposition $TE\_x=TE\_x^h \oplus TE\_x^v$, $TE^v=\ker d... | https://mathoverflow.net/users/89961 | Perburb the Monodromy of Lefschetz fibration over a disk | The answer to your question is yes, given that the Hamiltonian perturbation indeed is sufficiently small. Conceptually the idea of the construction is better formulated as follows: fix the symplectic form $\Omega$ and instead deform the *fibration* by a smooth (in general non-symplectic) isotopy in order to obtain a di... | 1 | https://mathoverflow.net/users/48067 | 256396 | 115,841 |
https://mathoverflow.net/questions/256300 | 2 | I wish to study the following linear program
$$\begin{array}{ll} \text{minimize} & \mathrm c^{\top} \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\\ & \mathrm x \geq 0\end{array}$$
where
* $\mathrm A$ is an *infinite* matrix with a *finite* number of nonzero elements in each row. In other words,... | https://mathoverflow.net/users/94774 | Linear programming with infinitely many constraints | * H. Edwin Romeijn, Robert L. Smith, [Shadow Prices in Infinite-Dimensional Linear Programming](https://www.researchgate.net/publication/266425025_Shadow_Prices_in_Infinite-Dimensional_Linear_Programming), Mathematics of Operations Research, Vol. 23, No. 1, February 1998.
>
> We consider the class of linear program... | 1 | https://mathoverflow.net/users/91764 | 256404 | 115,843 |
https://mathoverflow.net/questions/256055 | 8 | In [N=2 Quantum Field Theories and Their BPS Quivers](https://arxiv.org/abs/1112.3984v1) by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair $(C,\phi)$ consisting of a Riemann surface $C$ and a meromorphic quadratic differential $\phi$ on $C$ a (non-compact) Cal... | https://mathoverflow.net/users/5323 | How do you get the spectral curve from a Calabi-Yau? | In general there is no way to extract a spectral curve from a Calabi-Yau threefold.
In the study of strings on Calabi-Yaus, one object of interest is the periods, i.e. integrals of the holomorphic top form. We are particularly interested in the behavior of these period as the complex structure of the Calabi-Yau is va... | 7 | https://mathoverflow.net/users/5124 | 256411 | 115,847 |
https://mathoverflow.net/questions/256277 | 8 | Using the internal logic of a topos it's often possible to derive newer theorems about sheaves from earlier ones about simpler objects, assuming that you can prove the earlier ones constructively. In addition, Barr's Theorem allows one to directly reuse classical results when they have a geometric statement relative to... | https://mathoverflow.net/users/99234 | Can the algebraic geometry of schemes be developed internally in topoi? | The notes of Ingo Blechschmidt, [*Using the internal language of toposes in algebraic geometry*](http://rawgit.com/iblech/internal-methods/master/notes.pdf) cover this. See also [his lecture at Topos à l'IHÉS](https://www.youtube.com/watch?v=7S8--bIKaWQ) of the same name.
His work very much extends Hakim's thesis.
| 9 | https://mathoverflow.net/users/4177 | 256414 | 115,848 |
https://mathoverflow.net/questions/256382 | 13 |
>
> **Definition.** Call an object $X$ of a category $\mathbf{C}$ *nearly initial* iff firstly, it is weakly initial, and secondly, for all objects $Y$ and all morphisms $f,g : X \rightarrow Y$, there exists an automorphism $\alpha$ of $X$ such that $g = f \circ \alpha$.
>
>
>
It's straightforward to show that a... | https://mathoverflow.net/users/26080 | What are "nearly initial" objects really called? | In Definition 4.1 of their paper [*Shapely monads and analytic functors*](https://arxiv.org/pdf/1512.05980.pdf), Richard Garner and Tom Hirschowitz call such an object a "Galois object".
| 14 | https://mathoverflow.net/users/57405 | 256415 | 115,849 |
https://mathoverflow.net/questions/256408 | 5 | Could I get some help with proving this identity?
$$\sum\_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}.$$
It has been checked in Matlab for various small $n,m$ and $p$. I have a proof for $m=0$ that involves Pascal's rule to split it into two sums that mostly cancel, but this does not work for... | https://mathoverflow.net/users/94200 | Binomial ID $\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}$ | Such identities are often reduced to the [Chu--Vandermonde's identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity) $\sum\_{i+j=\ell} \binom{x}i\binom{y}j=\binom{x+y}\ell$ by using reflection formulae $\binom{x}k=\binom{x}{x-k}$, $\binom{x}k=(-1)^k\binom{k-x-1}k$.
In your case you may write
$$
\sum\_{k=m}^... | 5 | https://mathoverflow.net/users/4312 | 256424 | 115,853 |
https://mathoverflow.net/questions/256422 | 14 | Fix $n \ge 2$. Let $V\_n$ be the $\binom{n}{2}$-dimensional vector space (over $\mathbb{C}$) generated by a set of vectors $\{w\_{ij} : 1 \le i < j \le n\}$. Let $\bigwedge^\* V\_n$ be the exterior algebra on $V\_n$. Note that $\bigwedge^\* V\_n$ is graded as an algebra (e.g. as a vector space) by $\bigwedge^\* V\_n = ... | https://mathoverflow.net/users/102036 | Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result? | This turns out to be a completely general phenomenon for configuration spaces on *any* open manifold, though we did not know this at the time; it came in the later paper "[FI-modules and stability for representations of symmetric groups](https://arxiv.org/abs/1204.4533)" by Church, Ellenberg, Farb (all references are t... | 21 | https://mathoverflow.net/users/250 | 256425 | 115,854 |
https://mathoverflow.net/questions/256391 | 2 | Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}\_\mathcal{C}X$-modules (via the Yoneda functor). My question: are there some "niceness" conditions we can impose on the pair $(\mathcal{C}, X)$ w... | https://mathoverflow.net/users/7108 | Is the property of being a dg generator open? | Let me give an example (which you are probably aware of) that shows how a generator may become a non-generator after a deformation.
Let $\mathcal C$ be the derived category of $\mathbb{P}^1$ and $X = \mathcal{O} \oplus \mathcal{O}(-1)[1]$. This is clearly a generator. On the other hand, we can deform the sum to an e... | 5 | https://mathoverflow.net/users/4428 | 256428 | 115,855 |
https://mathoverflow.net/questions/256435 | 15 | Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t\_0,t\_1]\rightarrow M$ is a smooth curve and is a geodesic, then in local coordinates we have $$ \ddot{\gamma}^\mu(t)+\Gamma^\mu\_{\alpha\beta}(\gamma(t))\dot{\gamma}^\alpha(t)\d... | https://mathoverflow.net/users/85500 | When is a flow geodesic and how to construct the connection from it | **Note:** I've decided that this answer should be rearranged a bit
so that it clearly separates the discussion of the basic properties of
the tangent bundle from the discussion of the formulae associated to a connection.
The content is the same, but I hope it's clearer.
The standard way to discuss the geometry of co... | 19 | https://mathoverflow.net/users/13972 | 256437 | 115,860 |
https://mathoverflow.net/questions/256438 | 3 | Working on a problem I encounter the following sum:
$\sum\_{r<R}\frac{\mu(r)^{2}\tau\_{k}(r)^{2}}{\varphi(r)}$ that I have to estimate on the above; here $\tau\_{k}(r)$ is the number of ways of writing $r$ as a product of $k$ natural integers. I now that $\sum\_{r<R}\tau\_{k}(r)=RP\_{k-1}(R)+E(R)$ with $P\_{k-1}(x)$ ... | https://mathoverflow.net/users/85782 | Particular sum on product partitions into k parts | * If you only need an upper bound of the right order of magnitude, I strongly recommend the use of Rankin's trick, which amounts to write
$$
\sum\_{r \leq R} f(r) \leq \prod\_{p \leq R} \left( \sum\_{\ell \geq 0} f(p^{\ell}) \right)
$$
for a nonnegative multiplicative function $f$. For $f(r) = \frac{\mu^2(r) \tau\_k(r)... | 7 | https://mathoverflow.net/users/21724 | 256439 | 115,861 |
https://mathoverflow.net/questions/255795 | 5 | (Cross-posted from <https://math.stackexchange.com/questions/2029407/ray-class-groups-through-binary-quadratic-forms>)
If $d$ is the discriminant of a quadratic number field, then the primitive classes of binary quadratic forms of discriminant $d$ form a group isomorphic to the narrow ideal class group of $\mathbb{Q}... | https://mathoverflow.net/users/5373 | Ray class groups through binary quadratic forms | I guess that the answer is no. The binary forms with discriminant
$\Delta \cdot f^2$ describe ring class fields modulo $f$ (see Cox's book), so by taking the limit as $f \to \infty$ something like the idelic version of the full ring class group will result, and this is what I expect you will get using your weak equiva... | 4 | https://mathoverflow.net/users/3503 | 256449 | 115,862 |
https://mathoverflow.net/questions/256451 | 2 | Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184)=p(2^{6}\cdot 3^{4})=24,\qquad
p(65536)=p(2^{16})=16.$$
Is $p(n) = O(\log^{k}(n))$ for some constant $k$? Thanks in advance!
| https://mathoverflow.net/users/102059 | Upper bound for product of exponents of prime factorization | No, $p(n)$ is not bounded by a power of $\log(n)$. For example, if $$n=p\_1^2\dots p\_r^2,$$
where $p\_1<p\_2<\dots$ is the sequence of prime numbers, then by the Prime Number Theorem we have
$$ \log n=2(\log p\_1+\dots +\log p\_r)=(2+o(1))p\_r=(2+o(1))r\log r, $$
so that $r=(1/2+o(1))\log n/\log\log n$, and therefore... | 6 | https://mathoverflow.net/users/11919 | 256452 | 115,863 |
https://mathoverflow.net/questions/256454 | 6 | I [asked the following question on StackExchange](https://math.stackexchange.com/questions/2041770/relative-version-of-whitney-immersion-theorem) but received no response:
Let $M$ be a smooth $n$-dimensional compact manifold with boundary. Let $U$ be an open neighborhood of $\partial M$. Assume that we have a fixed i... | https://mathoverflow.net/users/82724 | Relative version of Whitney Immersion Theorem | No, you can't generally do this even with the added assumptions. The bundle $\tau = TS^{n-1} \to S^{n-1}$ is non-trivial for $n-1 \neq 1,3,7$, but it is stable after (one) trivialisation. Hence $\tau \oplus TD^n\vert\_{S^{n-1}}$ is a trivial bundle over $S^{n-1}$.
By Smale--Hirsch this means there is an immersion $i... | 3 | https://mathoverflow.net/users/318 | 256462 | 115,866 |
https://mathoverflow.net/questions/256426 | 7 | If $T$ is a complete first-order theory and $\kappa$ is a cardinal, let $\mathrm{Mod}\_\kappa(T)$ be (a skeleton of) the category of $\kappa$-small models of $T$ (i.e. of cardinality $<\kappa$), with elementary embeddings as morphisms. What are the possible cardinalities of (the set of morphisms of) $\mathrm{Mod}\_\kap... | https://mathoverflow.net/users/2362 | How many elementary embeddings can there be? | It is a fact (following from the Ehrenfeucht–Mostowski theorem) that for every complete theory $T$ and for every $\lambda \geq |T|$, there is $M \models T$ with $|M| = \lambda$ and $M$ having $2^\lambda$-many automorphisms (assuming $T$ has infinite models). So if I'm understanding your question correctly then what you... | 4 | https://mathoverflow.net/users/26705 | 256464 | 115,868 |
https://mathoverflow.net/questions/256467 | 8 | Consider two topological spaces $X$ and $Y$.
The notion of homotopy equivalence between $X$ and $Y$ is defined as a pair of continuous maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g$ and $g\circ f$ are homotopic to the identities.
But what if we weaken the definition a little bit, by considering the following... | https://mathoverflow.net/users/100552 | On a weaker version of homotopy equivalence between topological spaces | Saying that there are maps $f \colon X \to Y$ and $g \colon Y \to X$ such that $g f$ is homotopic to $\mathrm{id}\_X$ means that $X$ is a homotopy retract of $Y$. (By the way, we say that maps are *homotopic* not *homotopy equivalent*.) In your situation you have two spaces that are homotopy retracts of each other and ... | 16 | https://mathoverflow.net/users/12547 | 256468 | 115,870 |
https://mathoverflow.net/questions/256474 | 1 | Let $(X\_i)$ be a countable collection of bounded metric spaces, pre-compact in the Gromov-Hausdorff metric. It is well-known that for any choice of non-principal ultrafilter $U$ on $\mathbb N$, the metric ultra-limit of $(X\_i)$-s with respect to $U$ is a Gromov-Hausdorff limit of some subsequence $(X\_{i\_k})$.
Let... | https://mathoverflow.net/users/2234 | can every Gromov-Hausdorff limit be realized as a metric ultralimit? | Yes, any ultrafilter which contains $\{i\_j\}\_j$ will do.
| 5 | https://mathoverflow.net/users/1441 | 256477 | 115,873 |
https://mathoverflow.net/questions/256469 | 21 | This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order intuitionistic logic, because most such topoi aren't boolean. But, I have an extremely hard time accepting that $\mathbf{S... | https://mathoverflow.net/users/26080 | Why would the category of sets be intuitionistic? | You wrote:
>
> Suppose our intuition for the phrase "subset of $X$" comes from the idea of having an effective total function $X \rightarrow \{0,1\}$ that returns an answer in a finite amount of time. In this case, the subsets of $X$ ought to form a Boolean algebra.
>
>
>
Unfortunately, this is not a workable ... | 23 | https://mathoverflow.net/users/1176 | 256480 | 115,874 |
https://mathoverflow.net/questions/256472 | 2 | Consider a set of positive matrices $(P\_k)\_{k\in K}$ in $\mathbb R^{p\times p}\_{++}$ ($P\_k$ is positive in the sense that all entries of $P\_k$ are positive).
Let $X\_0\in\mathbb R\_{++}^p$ and define for every $n\ge0$ $X\_{n+1}=\max\_{k\in K}\{ P\_kX\_n\}$, where the max operator has to be understood element-wis... | https://mathoverflow.net/users/102063 | Perron-Frobenius for a "maximum" of matrices | The answer is no, as the map you define is not linear.
For example consider the matrices
$$ P\_1 = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end {bmatrix},\quad P\_2 = \begin{bmatrix} 1 & 2 \\ 1 & 2 \end {bmatrix}.$$
For simplicity, I'll call the map in question $F(x) := \max\{P\_1x,P\_2x\}$, where as you say the maximum is t... | 3 | https://mathoverflow.net/users/85570 | 256481 | 115,875 |
https://mathoverflow.net/questions/256482 | 5 | Let $f: X \to Y$ be a **finite map of projective varieties**.
I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Aut(X/Y)$.
>
> 1. **When is $f$ isomorphic to the quotient map by an action $X \to X^G$?**
>
>
>
If $X... | https://mathoverflow.net/users/22810 | When are finite maps quotients by finite groups? |
>
> If $X$ is normal and $G$ acts on $k(X)/k(Y)$ then the $G$ acts also on $X/Y$ (in a way consistent with its action on $k(X)$).
>
>
>
Because $X$ is integral, its ring of functions on each affine open embeds into $k(X)$, so the action of $G$ on an affine open is determined by the action on the field of fractio... | 9 | https://mathoverflow.net/users/18060 | 256486 | 115,877 |
https://mathoverflow.net/questions/256473 | 1 | Let $X\neq \emptyset$ be a set, let $\text{Part}(X)$ be the set of partitions of $X$ (where we require that $\emptyset \notin P$ whenever $P\in\text{Part}(X)$).
For $P, Q\in \text{Part}(X)$ we say that $P$ *covers* $X$ *more efficiently* than $Q$ if $$\text{card}(P\setminus Q) < \text{card}(Q\setminus P), $$ and we w... | https://mathoverflow.net/users/8628 | Is the covering efficiency relation transitive? | I believe the answer is yes. Let $X\neq \emptyset$ and let $P,Q,R\in {\rm Part}(X)$.
First, note that if $|P|<|Q|$, then $P<\_{\rm eff}Q$, since $|P-Q|\leq |P|<|Q|=|Q-P|$.
Now we are ready for the proof. Assume $P<\_{\rm eff}Q<\_{\rm eff}R$. Without loss of generality, we may remove $P\cap Q\cap R$ from all the set... | 3 | https://mathoverflow.net/users/3199 | 256495 | 115,880 |
https://mathoverflow.net/questions/256497 | 15 | I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me (being an "abstract algebraist") so that the student would make a talk on this subject. Any ideas? I don't want this pi... | https://mathoverflow.net/users/2191 | What areas of algebra could be interesting to probability theorists? | I am firstly an algebraist and later shifted to probability somehow, so I think I can answer your question from my own experience. I am a algebraist from the bottom of my heart though...A natural thought is you can teach them some basic random matrices, yet you may think it as "linear algebra" so:
**(1) Boolean algeb... | 21 | https://mathoverflow.net/users/25437 | 256502 | 115,881 |
https://mathoverflow.net/questions/256510 | 6 | Since$$\prod\_{p \leq n} \left(1-\frac{1}{p}\right) =\frac{ e^{-\gamma}+o(1)}{ \log n},$$ by Mertens theorem, the density of integers in $$(X^{\theta},X],$$ which aren't divisible by primes $$p \leq X^{\theta}$$
is $$\rho(X)\sim \frac{ e^{-\gamma}}{\theta \log X}.$$
How small a subinterval in this interval can inheri... | https://mathoverflow.net/users/17773 | Local density of numbers not divisible by small primes | Actually, the density you indicate is false, because $X^\theta$ is too large for the primes $p\leq X^\theta$ to behave sufficiently independently. For example, for $\theta=1/2$ the sifted set consists precisely of the primes in $(X^{1/2},X]$ whose density is $1/\log X$ (instead of $2e^{-\gamma}/\log X$) by the Prime Nu... | 13 | https://mathoverflow.net/users/11919 | 256511 | 115,883 |
https://mathoverflow.net/questions/256526 | 15 | Is the Fourier transform of the function
$$ f(\xi) = e^{-t|\xi|^{2m}}$$
positive for $t>0$ and $m \in \mathbb{N}\_0$?
| https://mathoverflow.net/users/16702 | Positivity of certain Fourier transform | it is positive for $m=1$, but not for $m=2$, see this Mathematica output:
| 16 | https://mathoverflow.net/users/11260 | 256528 | 115,887 |
https://mathoverflow.net/questions/256527 | -3 | Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum\_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\alpha-\beta|}.$
Now assume that $B=UAU^\*$ and $U$ is unitary on $\ell^2$ with
$(Bu)(\alpha) = \sum\_{\beta}B(\alpha,\bet... | https://mathoverflow.net/users/101813 | Exponential decay of kernel | Consider $\ell^2({\mathbb N\_0})$. Let $A(\alpha,\beta)=e^{-|\alpha-\beta|}$. Let $\sigma$ be a permutation of ${\mathbb N}\_0$ such that $\sigma(0)=0$. Let $U(u)(\alpha)=u(\sigma^{-1}\,(\alpha))$. Then the estimate $|B(\alpha,\beta)|\le Ce^{|\alpha-\beta|}$ gives for $\beta=0$ that $e^{-\sigma^{-1}(\beta)}\le Ce^{-\be... | 0 | https://mathoverflow.net/users/nan | 256529 | 115,888 |
https://mathoverflow.net/questions/256475 | 2 | In my research problem, I'm arrived at the following simple looking but highly non-linear pde which is related to the von Karman equations for plates with incompatible elastic strain (<http://rspa.royalsocietypublishing.org/content/467/2126/402>).
A sufficiently smooth (possibly analytic) function $w:X\to\mathbb{R}$ ... | https://mathoverflow.net/users/25516 | On the solution of a Monge-Ampere type non-linear partial differential equation | When you write *the* solution, you must have some other conditions in mind, since one generally does not have unique solutions. For example $\zeta = 0$ and $\zeta = 2 w$ both satisfy your equation, so there is no uniqueness without further assumptions, such as boundardy conditions or initial conditions.
If one writes... | 4 | https://mathoverflow.net/users/13972 | 256530 | 115,889 |
https://mathoverflow.net/questions/256541 | 6 | It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.
| https://mathoverflow.net/users/41105 | measurable linear functionals are also continuous on separable Banach spaces? | If "measurable" means "Borel" or even "Baire measurable", then this is true. It's a special case of a more general result that any Borel homomorphism of Polish groups is continuous. See for instance Kechris, *Classical Descriptive Set Theory*, Theorem 9.10.
Also, you can drop the word "separable". Suppose $X$ is an a... | 9 | https://mathoverflow.net/users/4832 | 256544 | 115,891 |
https://mathoverflow.net/questions/256523 | 4 |
>
> **QUESTION.** How do I show that if $T$ is a tempered distribution that is harmonic, then $T$ is a polynomial?
>
>
>
Any help is greatly appreciated.
| https://mathoverflow.net/users/102092 | $T$ is tempered distribution that is harmonic,then $T$ is polynomial | The Fourier transform is well-defined over the space ${\mathcal S}'(R^d)$ of tempered distributions into itself. If $T$ is harmonic, that is $\Delta T=0$, then $|\xi|^2\hat T=0$. This tells you that the support of $\hat T$ is $\{0\}$ (unless $T=0$ of course). Using the topology of ${\mathcal S}'(R^d)$, you know that th... | 8 | https://mathoverflow.net/users/8799 | 256545 | 115,892 |
https://mathoverflow.net/questions/256384 | 3 | Let $\epsilon\_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G\_\mathbb{Q}$ of finite image and with conductor $N$ prime to $p$, and $F$ be the set of the primes of $F$ above $p$.
Is the dimension of $H^1\_f(G\_{F,S}, \b... | https://mathoverflow.net/users/46460 | Crystalline extension the $p$-adic cyclotomic character | Yes, this is known.
By Shapiro's lemma we have
$$H^1\_f(G\_{F, S}, \psi \epsilon^k) = H^1\_f(G\_{\mathbf{Q}, S}, \psi \epsilon^k) \oplus H^1\_f(G\_{\mathbf{Q}, S}, \chi\_F \psi \epsilon^k),$$
where $\chi\_F$ is the quadratic character attached to $F$. There are standard formulae for the dimensions of both terms on th... | 2 | https://mathoverflow.net/users/2481 | 256565 | 115,899 |
https://mathoverflow.net/questions/256410 | 3 | Let $C/S$ be a semistable curve, and $G$ a finite group acting faithfully on $C/S$, whose order is invertible on $S$.
Let $P\in C$ be a geometric point with image $s\in S$, such that $P$ is a node in $C\_s$. For short, let $A := \widehat{\mathcal{O}}\_s$, and $R := \widehat{\mathcal{O}}\_P$. Then, at $P$, we may fix ... | https://mathoverflow.net/users/88840 | local picture of ramification at a node | Okay! Finally I see how to prove this.
First note that any $r\in R$ has a unique expansion as:
$$r = r\_0 + \sum\_{i\ge 1}a\_ix^i + \sum\_{i\ge 1}b\_iy^i\qquad r\_0,a\_i,b\_i\in A$$
Since $R$ is a local ring (with maximal ideal $m\_R := (x,y,m\_A)$), the quotient $\varphi : R\rightarrow R/m\_R \cong k$ is a local rin... | 1 | https://mathoverflow.net/users/88840 | 256573 | 115,901 |
https://mathoverflow.net/questions/256572 | 4 | Denote $(x)\_t = x(x-1)(x-2)\cdots(x-t+1)$ and fix some $t\_1,\dots,t\_n\in\mathbb{N}$. Now consider the polynomials
$$f\_n(x)=\sum\_{\pi\in L[n]}(-1)^{\vert\pi\vert-1}(\vert\pi\vert-1)!\prod\_{A\in\pi}(x)\_{\sigma(A)}$$
where the sum extends over all non-empty set-partitions $L[n]$ of $[n]:=\{1,\dots,n\}$ and $\sigma(... | https://mathoverflow.net/users/66131 | degree of a polynomial over set-partitions | Yes, this is true. Let $M\_1,\dots,M\_n$ be disjoint sets, $|M\_i|=t\_i$, $M=\cup M\_i$. Fix $s\in \{0,1,\dots,n-1\}$. Consider a permutation $w$ of the set $M$ which has exactly $|M|-s$ cycles. Assume that $w$ fixes each of the sets $\cup\_{i\in A} M\_i$, where $A$ is a part of a certain partition $\pi$ of the index s... | 7 | https://mathoverflow.net/users/4312 | 256577 | 115,903 |
https://mathoverflow.net/questions/256575 | 4 | My question is really easy to state, but I'm having trouble hitting the final nail in the coffin in a proof of the result. The question concerns fractional iterations of holomorphic functions, for clarity I'll define what I mean.
Suppose we have a holomorphic function $f(\xi)$ which takes the unit disk $D$ to itself.... | https://mathoverflow.net/users/nan | Super attracting fixed points have no fractional iteration | I suppose this result (non-existence of fractional iterates) is known and very old. From continuity $z\mapsto f(z,\zeta)$
follows that the fixed point at $0$ must be independent of $z$.
Suppose, for example that $f(\zeta)=a\zeta^2+\ldots$
Taking $z=1/2$ and denoting $g(\zeta)=f(1/2,\zeta)$ we obtain $f=g\circ g$.
Th... | 6 | https://mathoverflow.net/users/25510 | 256582 | 115,905 |
https://mathoverflow.net/questions/256580 | 6 | Often, large cardinals imply that many definable inner models and transitive sets have theories which are absolute under forcing. For example, assuming a proper class of Woodin cardinals, the theory of $L(\mathbb{R})$ is absolute under forcing in the following sense: if $V[G]$ is a set-generic extension of $V$, then $H... | https://mathoverflow.net/users/8133 | Absoluteness for the Chang model | Corollary 3.1.7 in Larson's Stationary Tower notes states that if $\delta$ is a Woodin limit of Woodin cardinals, then no forcing in $V\_\delta$ can change the theory of the Chang model, even with real parameters. Corollary 3.1.10 states the same thing when $\delta$ is strongly compact.
| 8 | https://mathoverflow.net/users/99234 | 256583 | 115,906 |
https://mathoverflow.net/questions/246864 | 4 | The Joyal Tierney model structure on $Cat$ (sometimes called the ``canonical" model structure) is (as mentioned in <https://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/>) proper, cartesian, simplicial, combinatorial, and every object is both fibrant and cofibrant. Is there a reference for th... | https://mathoverflow.net/users/84563 | Simplicial Enrichment of Joyal-Tierney Model Structure on $Cat$ | I asked Chris Schommer-Pries (the author of the blog post) about this. It's discussed in an unpublished note by Charles Rezk (either scroll down [here](http://www.math.uiuc.edu/~rezk/papers.html) to the note "A model category for categories", or else use [this](http://www.math.uiuc.edu/~rezk/cat-ho.dvi) direct link. On... | 5 | https://mathoverflow.net/users/2362 | 256587 | 115,909 |
https://mathoverflow.net/questions/256588 | 2 | Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x\_1,y\_1)\le f(x\_2,y\_2)$ whenever
$x\_1\le x\_2$ and $y\_1\le y\_2$. It is known that the partial derivatives
$\partial f/\partial x$ and $\partial f/\partial y$ exist almost everywhere, but are they Lebesg... | https://mathoverflow.net/users/102106 | Are the partial derivatives of a function increasing in both variables measurable? | Yes. As Nik Weaver's comment points out, your function is automatically measurable. Therefore you can apply Theorem 1 of [Measurability of partial derivatives](http://www.ams.org/journals/proc/1977-063-02/S0002-9939-1977-0437696-1/) by Moshe and Mizel.
| 1 | https://mathoverflow.net/users/99234 | 256591 | 115,911 |
https://mathoverflow.net/questions/256584 | 10 | Let $$p\_k(x\_1,\ldots,x\_n)=x\_1^k+\cdots+x\_n^k$$ and let
$$e\_k(x\_1,\ldots,x\_n)=\sum\_{1\le i\_1<i\_2<\ldots<i\_k\le n}x\_{i\_1}\cdots x\_{i\_k} $$
be the $k$'th symmetric polynomial. It is well known that there exists a unique polynomial
$$ F\_k(y\_1,\ldots,y\_k)\in\mathbb Z[y\_1,\ldots,y\_k] $$
such that
$$
p\_... | https://mathoverflow.net/users/11926 | Coefficients of polynomials appearing in Newton identities relating symmetric polynomials to power sums | OEIS A00041 gives the number of monomials. The absolute values of the coefficients of the polynomials sum to $2^k-1$. See A263916 for more notes and references on these Faber polynomials.
| 6 | https://mathoverflow.net/users/12178 | 256592 | 115,912 |
https://mathoverflow.net/questions/256598 | 13 | It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which is the automorphism group of the *oriented* projective line over the field with seven elements. (More details are on [W... | https://mathoverflow.net/users/4177 | The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin? | V. Dotsenko's construction, on math.stackexchange:
<https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450>
may fit your requirement "combinatorial mapping of these geometries that induces an isomorphism".
| 7 | https://mathoverflow.net/users/4600 | 256602 | 115,914 |
https://mathoverflow.net/questions/256610 | 5 | Let $A$ be an algebra over $k$, $\operatorname{tr\_A}(x, y):=\operatorname{tr}(m\_{xy})$ be a trace form on $A$, and $V\_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \mapsto V\_A$ gives a bijection
$$\left\{ \begin{array}{cc} \text{central simple algebras over } k \\ \text{ of dimens... | https://mathoverflow.net/users/43639 | Reference request: correspondence between central simple algebras and quadratic forms | Everything is in Lam's book *Introduction to Quadratic Forms over Fields*. Theorem III 5.1 says:
All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\right)$.
Note that as quadratic space, $A = \langle 1, -a, -b, ab\rangle$ and $V\_A = \langle -a, -b, ab\rangle$.
T... | 10 | https://mathoverflow.net/users/41094 | 256611 | 115,917 |
https://mathoverflow.net/questions/256525 | 17 | Let $G$ be a finite group. Is $\mathbb{Z}G\otimes\_{Z(\mathbb{Z}G)}\mathbb{Z}G$ free as a $\mathbb{Z}$-module, where $Z$ denotes the centre?
| https://mathoverflow.net/users/82435 | Freeness of tensor product | There's a distinctly non-zero chance that my calculations are wrong, but I think it has a lot of $2$-torsion for $G=A\_5$, the alternating group of degree $5$. I'm afraid the method I've used is a little indirect, and I haven't extracted an explicit torsion element.
Certainly $\mathbb{Z}G\otimes\_{Z(\mathbb{Z}G)}\mat... | 8 | https://mathoverflow.net/users/22989 | 256615 | 115,919 |
https://mathoverflow.net/questions/256623 | 31 | I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something like this to be true if $E$ (and therefore also $F$) is separable, but I am not totally sure. So, also this special case ... | https://mathoverflow.net/users/101778 | Is a normed space which is homeomorphic to a Banach space complete? | Let $\bar{E}$ be the norm completion of $E$, which is a Banach space. Then we can consider $E$ as a dense linear subspace of $\bar{E}$, where the subspace topology and the norm topology on $E$ coincide. In particular, since this topology is homeomorphic to $F$, it is completely metrizable, so $E$ is a $G\_\delta$ in $\... | 32 | https://mathoverflow.net/users/4832 | 256627 | 115,921 |
https://mathoverflow.net/questions/256624 | 4 | Let $X$ be an infinite-dimensional Banach space and $\beta$ denote Istrățescu's spreading measure of noncompactness, i.e. $$\beta(M) = \sup \{ \varepsilon > 0 \colon \exists\_{(x\_n)^{\mathbb N} \in X^{\mathbb N}} \forall\_{m \ne n} \colon \|x\_n-x\_m\| > \varepsilon \}$$ for any $M \subset X$.
If we denote the (clos... | https://mathoverflow.net/users/11512 | Geometric implications of $\beta(B_X) = 2$ | Edit
Tsirelson space admits spreading models 1-equivalent to the unit vector basis of $\ell\_1$. See the paper
>
> Odell, E.; Schlumprecht, Th. *A problem on spreading models*. J. Funct. Anal. 153 (1998), no. 2, 249--261. [MR1614578](http://www.ams.org/mathscinet-getitem?mr=1614578)
>
>
>
So $\beta =2$ and... | 2 | https://mathoverflow.net/users/3675 | 256653 | 115,931 |
https://mathoverflow.net/questions/256576 | 27 | Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r\_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r\_{Q}(n) = a\_{E}(n) + a\_{C}(n)$ is broken into the Eisenstein series piece, and the cusp form piece. Knowing what numbers... | https://mathoverflow.net/users/48142 | Which quaternary quadratic form represents $n$ the greatest number of times? | **Theorem.** Let $Q(x\_1,\dots,x\_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x\_1,\dots,x\_k)=n$ satisfies
$$r\_Q(n)\ll\_{k,\epsilon}n^{k/2-1+\epsilon}.$$
The implied constant depends only on $k$ and $\epsilon$, so it is independent of the... | 20 | https://mathoverflow.net/users/11919 | 256662 | 115,935 |
https://mathoverflow.net/questions/256666 | 1 | An oriented graph is a digraph without any self-loops, multiple arcs, or 2-cycles. What is the smallest minimum outdegree of an oriented graph on $n$ vertices that ensures there will always be a cycle of length at least $k$? Can the bound be improved if one includes the minimum indegree in the bound?
| https://mathoverflow.net/users/51384 | Lower bound on outdegree/indegree in oriented graph to guarantee cycle of length at least $k$ | Out-degree $k-2$ is sufficient to force a directed cycle of length at least $k$. To see this, consider a longest directed path $P:=v\_1v\_2 \dots v\_\ell$. Since $P$ is a longest path and there are no $2$-cycles, all out-neighbours of $v\_\ell$ are among $\{v\_1, \dots, v\_{\ell-2}\}$. Since $v\_\ell$ has at least $k-2... | 1 | https://mathoverflow.net/users/2233 | 256672 | 115,937 |
https://mathoverflow.net/questions/256370 | 3 | I have a convex optimization problem
\begin{align\*}
\max\_{P\, =\, (p\_{ij})} &\sum\_{ij} p\_{ij} \big(a\_{ij}-\log(p\_{ij})\big)\\
\text{st}\quad &p\_{ij}\in\mathcal{P}\_{n, m} \; \text{and} \sum\_{i=1}^n p\_{ij} (b\_i-1) = 0 \quad \forall j\\
\text{where,}\; &\mathcal{P}\_{n, m} := \{(p\_{ij}) \in \mathbb{R}\_+^... | https://mathoverflow.net/users/99932 | Proximal map for a convex optimization problem | (This is my second attempt at a solution. My initial attempt can be found below.)
Your optimization problem can be expressed as
$$
\operatorname{minimize}\_{p \in \mathbb R^{n \times m}} \quad F(p) + G(p)
$$
where
$$
F(p) = \sum\_{i,j} p\_{ij}(\log(p\_{ij}) - a\_{ij}) + I\_\Omega(p)
$$
and $I\_\Omega$ is the indicato... | 2 | https://mathoverflow.net/users/38114 | 256684 | 115,940 |
https://mathoverflow.net/questions/256638 | 2 | At first, I want to explain why did I say the $n$th power of a matrix by [*companion matrix*](https://en.wikipedia.org/wiki/Companion_matrix). Suppose that $A$ is a matrix
of order $d$ over an ordinary field. There are several methods to obtain a closed-form expression for the $n$th power of the matrix $A$.
* **Firs... | https://mathoverflow.net/users/64181 | The $n$th power of a matrix by Companion matrix | Note: for the notation I'm used to, the 1s for $C$ are subdiagonal, as in Wikipedia, not superdiagonal, as in your question.
Under the assumption that $A$ is conjugate to a companion matrix:
If you are willing to accept a probabilistic answer, then there is a very efficient algorithm. Choose $\vec{v} \in \mathbb{R}... | 7 | https://mathoverflow.net/users/44191 | 256687 | 115,942 |
https://mathoverflow.net/questions/256652 | 3 | Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.
I'm reading a paper [[1](http://projecteuclid.org/download/pdfview_1/euclid.aoas/1267453933 "[1]")], which uses a notation for stochastic processes that doesn't seem to be standard.
>
> Now we generalize the squared covar... | https://mathoverflow.net/users/49465 | Stochastic Process Notation | $U(X)$ just means the stochastic process at a **random time** $X$. So you have two different random things, the stochastic process (collection of random variables) $\{U(t)\}$, and the random time $X$ picks out a particular one $U(X)$.
For instance, imagine you pick a random book from the shelf and open it on a random... | 2 | https://mathoverflow.net/users/4600 | 256688 | 115,943 |
https://mathoverflow.net/questions/256657 | 2 | Let $\mathbf{A}=[a\_{ij}]$ be a matrix of real numbers $a\_{ij}>0$ for all $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n\}$ and let $v$ be a nonnegative real number. The coefficient of the matrix are not *all* greater than $1$ nor *all* less than one.
The problem is to find a subset $S$ of the columns (of $\{1,\ldots,n\... | https://mathoverflow.net/users/82536 | How to find the set of vectors which are as nearly orthogonal as possible? | Wanted to post it as a comment, but I realize it feels more like an answer.
If I am not mistaken, after some simplifications (column rescaling) the problem can be transformed into: given the semidefinite matrix $B=A^\top A$, find an index set $I$ such that $\det B(I,I)$ is maximum. I don't think it is easy to solve. ... | 3 | https://mathoverflow.net/users/1898 | 256695 | 115,944 |
https://mathoverflow.net/questions/256699 | 10 | For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P\_1,\dots,P\_n$ are closed points, then the open curve $C=\overline{C}\setminus\{P\_1,\dots,P\_n\}$ is hyperbolic if $2-2g-\sum \deg P\_i<0$.
The ge... | https://mathoverflow.net/users/50846 | Is there a notion of hyperbolicity for number rings? | My intuition is that there will not be a precise definition of a hyperbolic number field. However, there there may be some number fields you can confidently say are hyperbolic.
Consider the following characterization of a hyperbolic complete curve in the function field case - a degree $d$ cover of $\mathbb P^1$ is hy... | 6 | https://mathoverflow.net/users/18060 | 256701 | 115,946 |
https://mathoverflow.net/questions/256680 | 2 | Consider a non-constant harmonic function $f$ on $\mathbb{Z}^d$ (meaning this that $f(x)$ if the average of the $2d$ values $f(y)$ such that the distance between $x$ and $y$ is one). Let $M\_n$ denote the maximum of the absolute values $|f(x)|$ for all $x$ such that $||x||\_{\mathbf{L}^1} \leq n$ (where $n$ is a positi... | https://mathoverflow.net/users/102178 | On lower bounds for harmonic functions on $\mathbb{Z}^d$ | The way your statement is formulated, it is wrong. The simplest counterexample is the function $f(x,y)=xy$ on $\mathbb Z^2$ which vanishes on the ball of radius 1. In Heilbronn's pioneeriing 1949 paper it is, in particular, proved that any polynomial growth harmonic function for the simple random walk on $\mathbb Z^d$ ... | 1 | https://mathoverflow.net/users/8588 | 256703 | 115,947 |
https://mathoverflow.net/questions/256697 | 1 | I've recently found myself doing some work on local rings,
and I found the following quantity keeps popping up-
Let $A$ be a local commutative unital ring, with maximal ideal $\newcommand{\mfr}{\mathbf} \mfr{m}$. For any $n\in\mathbb{N}$,
put $\beta(n)=\dim\_{A/\mfr{m}}\mfr{m}^n/\mfr{m}^{n+1}$.
It seems quite pla... | https://mathoverflow.net/users/14443 | Is there a commonly used name and notation for $\beta(n)=\dim_{A/\mathbf{m}}\mathbf{m}^{n}/\mathbf{m}^{n+1}$, where $(A,\mathbf{m})$ is a local ring? | Per HeinrichD's comment, I am rewriting my comment as an answer.
The **Hilbert-Samuel function** is $\alpha(n) = \text{length}(A/\mathbf{m}^n)$. The function $\beta(n)$ is $\alpha(n)-\alpha(n-1)$, the first difference of the Hilbert-Samuel function.
If $A$ is Noetherian, then there exists a unique numerical polyn... | 3 | https://mathoverflow.net/users/13265 | 256706 | 115,948 |
https://mathoverflow.net/questions/256702 | 1 | I have read a wikipedia article on Pisot number or PV number: <https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number>
I define a Pisot number $\alpha$ is "small" iff $\alpha<2$.(It is not a qualified definition; is there another reasonable bound for the problem below?)
Is there a small Pisot number $\al... | https://mathoverflow.net/users/102206 | small Pisot numbers with real conjugates | The following theorem of Kronecker seems to (almost) answer your question.
*Let $\beta$ be an algebraic integer such that all its algebraic conjugates are real (i.e. $\beta$ is totally real). Then either $\beta$ is of the form $\beta=2\cos(\pi q)$ with rational $q$, or some algebraic conjugate of $\beta$ has absolut... | 4 | https://mathoverflow.net/users/17581 | 256716 | 115,951 |
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