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https://mathoverflow.net/questions/304565 | 8 | Let $\mathbb{A}\_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials.
Suppose $V \subseteq \mathbb{A}\_{\mathbb{R}}^n$ is an irreducible affine variety. Let $U$ be an open (with respect to the usual topology) ball $U... | https://mathoverflow.net/users/84272 | Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$ | As noted in [this answer](https://mathoverflow.net/a/297013/82179) to a previous question of yours, for any subvariety $W \subseteq \mathbb A\_\mathbb R^n$, we have
$$\dim\_{\mathbb R} W(\mathbb R) \leq \dim W$$
(where $\dim W$ denotes the dimension in the sense of scheme theory, written there as $\dim\_\mathbb C W(\ma... | 4 | https://mathoverflow.net/users/82179 | 304581 | 133,258 |
https://mathoverflow.net/questions/304586 | 5 | Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ the map $t \mapsto A(t)x$ is continuous.
Consider the initial value problem
$$\varphi'(t)=A(t)\varphi(t)$$
with $\... | https://mathoverflow.net/users/126333 | Invariant subspace in infinite dimensions | In general, the answer is "no" even in the autonomous case, i.e. in the case where $A := A(t)$ does not depend on $t$.
First note that if $A$ generates a $C\_0$-semigroup on $H$, if $X$ is a dense subspace of $H$ and if $X$ is invariant under this $C\_0$-semigroup, then $X$ is even dense in $D(A)$ (which we endow wit... | 5 | https://mathoverflow.net/users/102946 | 304593 | 133,262 |
https://mathoverflow.net/questions/295923 | 5 | Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:
If we rotate $C$ around $p$ by 0 < $\alpha$ < $\pi$ clockwise on the plane, then the created curve $C'$ intersects $C$ at points other than $p$.
... | https://mathoverflow.net/users/101132 | Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations? | Thanks for the efforts have done on this problem specially by @erz ,but It seems that *Mark J.Nielsen* have solved this problem [here](https://link.springer.com/article/10.1007%2FBF00151519) (that I have found it recently), while he was proving this theorem about inscribed triangles in closed simple jordan curves:
**... | 1 | https://mathoverflow.net/users/101132 | 304611 | 133,272 |
https://mathoverflow.net/questions/304525 | 1 | I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right.
Since Singular (the underlying engine) can work with number fields but not $\overline{Q}$, the idea is to convert the base ring into a number field... | https://mathoverflow.net/users/78871 | Is this algorithm for primary decomposition correct? | (Just making an answer out of the above comment, with a small modification.) I think this will have problems if the ideal's generators don't factor at all. For example, for an ideal like
$$
(y^2 - x^3 , 200(x-1)^2 - (y+2)^3 ),
$$
the generators don't factor at all. (Graphing these curves shows that they are cusps. The... | 1 | https://mathoverflow.net/users/88133 | 304622 | 133,276 |
https://mathoverflow.net/questions/304628 | 9 | Sorry if this is a vague question. I remember from my younger days that
before proving his trace formula, Kuznetsov had a pretty result on
orthogonality of Bessel functions. The formulas that I am going to write are
WRONG, but the whole point is that I would like to remember the correct one
(or a pointer to a reference... | https://mathoverflow.net/users/81776 | Kuznetsov trace formula, orthogonality of Bessel functions | The Bessel functions $J\_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights $\ell+1$) of $L^2(\Gamma\backslash H)$. The orthogonal complement of the span of these $J\_\ell$'s is continuously (an... | 13 | https://mathoverflow.net/users/11919 | 304633 | 133,278 |
https://mathoverflow.net/questions/304632 | 4 | Given a real analytic family of Lipschitz continuous functions $f\_t:\overline{U}\rightarrow\mathbb{R}^n$, $t\in\mathbb{R}$, with $U\subset \mathbb{R}^n$ some open and bounded domain. For each $t\_0\in \mathbb{R}$ there exists $\epsilon >0$ and Lipschitz functions $f^k:\overline{U}\rightarrow\mathbb{R}^n$ such that for... | https://mathoverflow.net/users/125729 | Derivative of Lipschitz continuous functions | Let me assume that
$
f(t,x)=\sum\_{k\ge 0} f\_k(x) t^k, \quad \vert x\vert \le 1, \quad \vert t\vert < 1,
$
with $f\_k$ Lipschitz-continuous with an $L^\infty$ norm on $\vert x\vert \le 1$ bounded above by 1 and $\Vert f'\_k\Vert\_{L^\infty}\le C\_0 R^k$.
Then each $f\_k$ is a.e. differentiable, i.e. $\forall k, \exis... | 1 | https://mathoverflow.net/users/21907 | 304634 | 133,279 |
https://mathoverflow.net/questions/304618 | 7 | I think it is true that there is no free-action of $\mathbb{Z}\_p(p\neq 2$) on product of $\mathbb{CP}^n(n$ odd) and $\mathbb{S}^{2m}$. But I don't know how to prove it. Any solution will be helpful.
Thanks in advance.
| https://mathoverflow.net/users/117065 | What about the action on this space? | I think the following will work for $p > 3$. I am not sure if it can be made to work for $p =3$, but maybe it can. I will the appeal to the representation theory of $\mathbb{Z}/p\mathbb{Z}$, but probably this is overkill.
Given a continuous action of $\mathbb{Z}/p\mathbb{Z}$ on your space $X = \mathbb{CP}^n\times S^... | 12 | https://mathoverflow.net/users/98320 | 304641 | 133,281 |
https://mathoverflow.net/questions/304636 | 2 | Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to some controlled error:
The Sobolev space $W^{k,2}(\mathbb R)$ with $k>0$ is defined by all the $L^2$ functions $f$ su... | https://mathoverflow.net/users/126409 | Support of functions in Fourier domain | No. You may ask the same equivalent question for a function in $L^2$. A function $u$ belongs to $L^2$ means measurability and
$$
\int \vert u(x) \vert^2 dx<+\infty,
$$
which implies
$
\lim\_{R\rightarrow +\infty}\int\_{\vert x\vert\ge R} \vert u(x) \vert^2 dx=0,
$
but the rate of convergence can be arbitrarily slow.
| 1 | https://mathoverflow.net/users/21907 | 304642 | 133,282 |
https://mathoverflow.net/questions/304579 | 7 | If $W\_1,W\_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d\_1, d\_2 \leq d$, respectively, then $d\_1 + d\_2 > d$ suffices to guarantee $W\_1 \cap W\_2 \neq \{0\}$. There are similar results for affine subspaces of a spaces of a $k$-vector space, $E\_1,E\_2 \subset V$.
I'm looking for analogou... | https://mathoverflow.net/users/94086 | Intersection of free/affine submodules, comparison with vector spaces | There is a natural generalization of the aforementioned dimension-based reasoning to modules $M$ over commutative domains or commutative Noetherian reduced rings.
Let $M$ be a module over a commutative domain $R$ and let $K$ be the fraction field of $R$. We set $\text{rk}(M) \Doteq \dim\_K(M \otimes\_R K)$.
>
> ... | 8 | https://mathoverflow.net/users/84349 | 304657 | 133,289 |
https://mathoverflow.net/questions/304568 | 34 | A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.
In the theorem as stated, $\mathbb{F}$ is any field and $T\_n(\mathbb{F})$ denotes the algebra of upper triangular $n\t... | https://mathoverflow.net/users/126377 | If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$? | This is false! Let
$$A = \begin{bmatrix}
0&0&0&1 \\
&0&1&0 \\
&&0&0 \\
&&&0 \\
\end{bmatrix}.$$
Imposing that $XA=AX$ for upper triangular $X$ gives linear equations on the $10$ entries of $X$. Solving them, I get that this occurs precisely for $X$ of the form
$$X=\begin{bmatrix}
a&0&\ast&\ast \\
&b&\ast&\ast \\
&&b&0... | 30 | https://mathoverflow.net/users/297 | 304662 | 133,291 |
https://mathoverflow.net/questions/304661 | 1 | Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the interesection of positive operator and trace class with trace no more than 1.
For $A,B\in D(\mathcal{H})$, we say $A\leq B$ if $A-B$ is positive, or (A,B) satis... | https://mathoverflow.net/users/4987 | Completeness of Lowner order in separable Hilbert space | This isn't research level, but if the sequence is increasing and bounded then yes, it has a least upper bound. For each $v \in H$ the sequence $\langle A\_iv,v\rangle$ is bounded and increasing and therefore converges to its least upper bound in $\mathbb{R}$. By polarization it follows that for all $v,w \in H$ the sequ... | 4 | https://mathoverflow.net/users/23141 | 304664 | 133,292 |
https://mathoverflow.net/questions/304663 | 12 | This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up."
Given two Laurent power series $A(t)=\sum\_{k>N}a\_kt^k$ and $B(t)=\sum\_{k>M}b\_kt^k$ for $a\_k,b\_k\in F$ a field, we say that these are expansions at 0 and $\infty$ of a rational function $f$ if i... | https://mathoverflow.net/users/66 | Relations between coefficients of expansions of a rational function at 0 and infinity | One common criterion that is used to algebraically encode whether a power series $\sum\_{n\geq 0} a\_nx^n$ represents a rational function is that the infinite Hankel matrix $(a\_{i+j-1})$ be of finite rank. If this is the kind of criterion you are looking for then for your pair of series $A(t)$ and $B(t)$ satisfy $A(t)... | 16 | https://mathoverflow.net/users/2384 | 304666 | 133,293 |
https://mathoverflow.net/questions/300052 | 4 | Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and injective model structures. Suppose that furthermore $F\colon C^{\mathrm{op}}\times D\to E $ is any bifunctor having a (p... | https://mathoverflow.net/users/80467 | Deriving the functor $ \int_{\Gamma} F(-,-)$ | We eventually found a solution which solved the original issue, though not in the way it is exactly stated here. Namely, we found what would appear to be the most general way that ends can be derived. Specifically,
>
> **Theorem:** Let $\mathscr{C}$ be a model category and $\Gamma$ a category. Regard the functor ca... | 2 | https://mathoverflow.net/users/80467 | 304679 | 133,298 |
https://mathoverflow.net/questions/286640 | 20 | A few weeks ago Jean-Louis Verger-Gaugry announced a proof of Lehmer's conjecture, see <https://arxiv.org/pdf/1709.03771.pdf>. The key result (Theorem 5.28, p. 122) is a Dobrowolski type minoration of the Mahler Measure $M(\beta)$, namely
\begin{align}\label{eq:1} M(\beta) \geq \Lambda\_r\mu\_r-\frac{\Lambda\_r\mu\_r \... | https://mathoverflow.net/users/115255 | Proof of the conjecture of Lehmer: a Dobrowolski type minoration | There are other strange assertions in the paper. For instance p. 131. The author wants to prove (Theorem 7.3) that that a certain meromorphic function $P/f$ has no pole, where the holomorphic function $f$ has (at least) a simple zero at $\omega\in \mathbb C$, and $P$ is a polynomial. The proof is very odd:First of all,... | 7 | https://mathoverflow.net/users/126423 | 304680 | 133,299 |
https://mathoverflow.net/questions/304654 | 3 | This question was posted in <https://math.stackexchange.com> but I got hardly any view. If posting here is an objection please let me know I would delete it immediately.
This question has evolved from watching the video:
<https://www.youtube.com/watch?v=dh5hHpJ79jc>
In the video the speaker talked about white noise... | https://mathoverflow.net/users/71105 | Are Holder Condition and signal to noise ratio (SNR) related? | The Hölder exponent $h$ and the Hurst exponent $H$ (related by $h=H-1$ for a continuous signal) characterise correlations in the noise, not the size of the noise (as quantified by the signal-to-noise ratio). So these are different quantities. This plot illustrates two signals with the same signal-to-noise ratio but dif... | 4 | https://mathoverflow.net/users/11260 | 304688 | 133,302 |
https://mathoverflow.net/questions/304155 | 7 | *I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new content in it. If, however, anyone decides after reading this question that it is a duplicate I apologize in advance*.
... | https://mathoverflow.net/users/22810 | Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions | Let me try: Consider $S^m\_{1,0}$ and $SG^{m,0}$, where the second class of SG-symbols $SG^{m\_\psi,m\_e}$ is defined by the estimates
$$|\partial\_x^\alpha \partial\_\xi^\beta a(x,\xi)| \lesssim\_{\alpha,\beta} \langle x\rangle^{m\_e-|\alpha|} \langle \xi\rangle^{m\_\psi-|\beta|}.$$
Clearly, $SG^{m,0}$ is a subset of ... | 2 | https://mathoverflow.net/users/20155 | 304698 | 133,305 |
https://mathoverflow.net/questions/304660 | 4 | A simplicial pseudo-manifold of dimension $d$ with boundary is a simplicial complex satisfying the following conditions.
1. Every maximal face has dimension $d$
2. Each face of dimension $d-1$ is a face of at most two maximal faces.
3. For each two distinct maximal faces $A$ and $A'$, there is a sequence $A\_1,...,A... | https://mathoverflow.net/users/126419 | Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension | To Question 1: there is no upper bound of the form $f(d) \cdot n$, because [cyclic polytopes](https://en.wikipedia.org/wiki/Cyclic_polytope) have $O(n^{\lfloor\frac{d}{2}\rfloor})$ facets.
Cyclic polytopes maximize the number of faces of any dimension among triangulated spheres with $n$ vertices. This is called the [... | 3 | https://mathoverflow.net/users/98590 | 304703 | 133,307 |
https://mathoverflow.net/questions/304588 | 17 | Let $K$ be a finite extension of $\mathbb{Q}\_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}\_p} / \mathbb{Q}\_p)$ trivial ?
If yes, how can I show it ?
| https://mathoverflow.net/users/125529 | Centraliser of an absolute Galois group | For an extension $L/\mathbf{Q}\_p$, let $G\_L$ denote the absolute Galois group $\mathrm{Gal}(\overline{L}/L)$.
If $\sigma \in G\_{\mathbf{Q}\_p}$ acts centrally on $G\_K$, then it also acts centrally on the subgroup $G\_L \subset G\_K$ for any finite $L/K$. But then it also acts trivially on the abelianization of $G... | 15 | https://mathoverflow.net/users/126439 | 304708 | 133,310 |
https://mathoverflow.net/questions/299056 | 9 | Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in [our paper](https://ijgeometry.com/wp-content/uploads/2018/04/81-86.pdf) equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\sum{(a-b)^2}\end{equation}
where $\sum{(a-b)^2}=(a-b)^... | https://mathoverflow.net/users/122662 | Strengthened version of Isoperimetric inequality with n-polygon | The conjectured inequality, with $k=\tan{\frac{\pi}{n}}-\tan{\frac{\pi}{n+2}}$, is false for $n=3$. More specifically, the constant factor $k=1$ is optimal in the Hadwiger--Finsler inequality: e.g., consider $a=b=1$ and $c\approx0$.
| 2 | https://mathoverflow.net/users/36721 | 304712 | 133,313 |
https://mathoverflow.net/questions/304658 | 3 | Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less than 1.
The set of positive maps consists of all elements that maps positive operator to positive operator. The set of ... | https://mathoverflow.net/users/4987 | Completely Positive Maps and their dual in Separable Hilbert Space | This is true; though I don't know a canonical reference (perhaps someone else can provide this).
But let me sketch a proof. In infinite dimensions, we need to be a little careful about topologies. Let $\newcommand{\mc}{\mathcal}\mc T(H)$ be the trace-class operators on $H$ with norm $\newcommand{\tr}{\operatorname{tr... | 3 | https://mathoverflow.net/users/406 | 304716 | 133,315 |
https://mathoverflow.net/questions/304314 | 24 | Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In [these notes](https://wstein.org/edu/Fall2003/252/lectures/10-17-03/10-17-03.pdf) by William Stein this is stated without proof in remark 1.5.8; it is attributed to personal conversation w... | https://mathoverflow.net/users/119444 | Is every abelian variety a subvariety of a Jacobian? | You can find a detailed proof [here](https://webusers.imj-prg.fr/~anna.cadoret/TC.pdf "here") (theorem 1.2) in the case of principally polarized abelian varieties. One reduces to this case using the Zarhin's trick.
The assumption of $k$ being infinite should not be necessary (see remark 1.3 in the paper)
| 7 | https://mathoverflow.net/users/105092 | 304723 | 133,317 |
https://mathoverflow.net/questions/304693 | 3 | Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P\_n(\cosh(x)) ~
\substack{\huge\rightarrow\\\scriptstyle{n\rightarrow\infty}}~
\frac{1}{\sqrt{\pi n}}\frac{e^{(n+1/2)x}}{\sqrt{2\sinh(x)}} \\
... | https://mathoverflow.net/users/94200 | Asymptotic forms of Legendre functions for large degree | Actually I got hold of Hobson's book and it has the expressions to second order anyway. They are:
$$
P\_n(\cosh x) ~
\substack{\huge\rightarrow\\\scriptstyle{n\rightarrow\infty}}~
\frac{1}{\sqrt{\pi n}}\frac{e^{(n+1/2)x}}{\sqrt{2\sinh x}}\bigg(1-\frac{2-\coth x}{8n}\bigg)\\
Q\_n(\cosh x)~
\substack{\huge\rightarrow\\... | 2 | https://mathoverflow.net/users/94200 | 304732 | 133,321 |
https://mathoverflow.net/questions/304729 | 12 | Is there an example of a finite group $G$ and an action on $M=\mathbb{Z}^n$ such that $H^2(G,M)$ has exponent greater than the exponent of $G$?
(Especially, can we have $G=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and some free $\mathbb{Z}$ module $M$ with $G$ action, such that $H^2(G,M)$ has elements of o... | https://mathoverflow.net/users/nan | Example of group cohomology not annihilated by exponent of $G$? | For each finite group $G$ there is a $G$-module $M$ that is a free abelian group of finite rank such that $H^2(G,M)=\mathbb{Z}/|G|$.
*Proof:* Let $I$ be the augmentation ideal of $\mathbb{Z}G$. Then $H^1(G,I)=\mathbb{Z}/|G|$. $\,\,I$ is a finitely generated $\mathbb{Z}G$-module (it's f.g. even as free abelian group)... | 21 | https://mathoverflow.net/users/18571 | 304734 | 133,322 |
https://mathoverflow.net/questions/304749 | 2 | Cubical sets with connection form a presheaf category on some category $C$. Is $C$ just the full subcategory of the category of posets whose objects are products of the interval $\Delta[1]$?
| https://mathoverflow.net/users/2362 | Is the site for cubical sets with connections equivalent to a full subcategory of posets? | No. Cubical sets with or without connections do not have diagonals. That is, there is no map $\square^1 \to \square^2$ which maps $0$ to $(0,0)$ and $1$ to $(1,1)$.
| 4 | https://mathoverflow.net/users/62782 | 304751 | 133,325 |
https://mathoverflow.net/questions/304691 | 1 | Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \frac{d}{dx}. $$
For sure some conditions on $\mu$ and $\sigma$ are needed. However I’m unsure what conditions are needed... | https://mathoverflow.net/users/115967 | Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$ | First of all, by a suitable change of variables, we can rewrite the equation as $$f''(y) = r a(y) f(y) ,$$
for an appropriate coefficient $a(y)$. Indeed, if $y(x)$ is an increasing solution of $A y(x) = 0$, then
$$ \begin{aligned} A f(y(x)) & = \tfrac{1}{2} \sigma^2 (f''(y) (y')^2 + f'(y) y'') + \mu f'(y) y' \\ & = \tf... | 2 | https://mathoverflow.net/users/108637 | 304754 | 133,327 |
https://mathoverflow.net/questions/304727 | 3 | Let $x\_1, ..., x\_n$ be formal variables. One variant of the Newton-Girard identities expresses
$$\sum\_{\pi \in S\_n} x\_{\pi(1)} x\_{\pi(2)} \cdots x\_{\pi(k)}$$
as a polynomial in the power sums of the $x\_i$-s.
I am looking for a variant which does the following. For every $1 \le j \le m$, let $x^{(j)}\_1, ..., ... | https://mathoverflow.net/users/31469 | A generalization of Newton-Girard Identities | Just a quick answer for now (will try to give more details later):
This is easy to do by expanding the exponential of a logarithm as in the usual symmetric function case. Basically, you are asking for an explicit formula for the multisymmetric elementary polynomials in terms of the multisymmetric power sums. You can fi... | 7 | https://mathoverflow.net/users/7410 | 304763 | 133,332 |
https://mathoverflow.net/questions/304699 | 5 | Let $f,g:\mathbb{C} \to \mathbb{C}$ be holomorphic and have the property $f^{-1}(S)=g^{-1}(S)$ where S is the unit circle centered at 0. What can be said about $f$ and $g$.
| https://mathoverflow.net/users/51203 | Holomorphic functions with equal inverse images of unit circle | An exhaustive study of this question is contained in the paper
MR0825840
Stephenson, Kenneth
Analytic functions sharing level curves and tracts.
Ann. of Math. (2) 123 (1986), no. 1, 107–144 (freely available online).
| 5 | https://mathoverflow.net/users/25510 | 304770 | 133,335 |
https://mathoverflow.net/questions/304791 | 6 | The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is this. Should one count only the singularities that are rational points or all the singularities over the complex field?... | https://mathoverflow.net/users/4274 | Faltings theorem and number of singularities | The definition of the geometric genus in terms of (d-1)(d-2)/2 minus the contributions of the singularities is not a great one. It's better to give a more intrinsic definition, as the dimension of the space of global section of the canonical line bundle of the normalization (or the first sheaf cohomology of the normali... | 13 | https://mathoverflow.net/users/18060 | 304794 | 133,341 |
https://mathoverflow.net/questions/304669 | 1 | This is a cross-post to the [question](https://math.stackexchange.com/questions/2816308/gluing-locally-defined-continous-functions-over-complex-domain) I asked at MSE over almost a month ago.
Suppose $n, l, m \in \mathbb N$ and $n \ge l > m$. Let $T: \mathbb C \to \mathcal M(n \times l; \mathbb C)$ be continuous and ... | https://mathoverflow.net/users/103704 | Gluing locally defined continous functions over complex domain | I am not sure whether this gluing method will work, but as I mentioned in the comment, this problem is equivalent to "whether every rank $m$ (typo in the comment, sorry) complex vector bundle over $\mathbb C$ is trivial". To see this, first construct the trivial bundle of rank $n$ over $\mathbb C$, that is, $\mathbb C\... | 1 | https://mathoverflow.net/users/37103 | 304820 | 133,352 |
https://mathoverflow.net/questions/304788 | 5 | Consider a system of $n$ "bounded" Poisson processes over the integers, $X\_1, \ldots X\_n$, all incrementing at rate $\lambda$. Initially all the processes begin at $0$. The process $X\_i$ is inactive until $X\_{i+1} - X\_i > 1$, at which point it begins incrementing itself at rate $\lambda$. Whenever $X\_{i+1} - X\_i... | https://mathoverflow.net/users/126491 | Expected time of distinguishability of a series of Poisson processes bounded by each other | The process you are looking at is called TASEP (totally asymmetric simple exclusion process); though your initial conditions are unusual. A more conventional version of your question would be to consider the TASEP with step initial conditions (one particle at every negative integer) and ask for a typical time that the ... | 3 | https://mathoverflow.net/users/56624 | 304825 | 133,356 |
https://mathoverflow.net/questions/304798 | 5 | Product CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb D^n\times\mathbb D^m=\mathbb D^{m+n}$. I want to define products in a synthetic way, only manipulating objetcs up to ho... | https://mathoverflow.net/users/126328 | CW Product via Whitehead map | The attaching map of the product of cells is sometimes described as an *exterior join construction*.
Let $F:D^n\to Cf\subseteq X$ be the characteristic map of an $n$-cell of $X$ with attaching map $f:S^{n-1}\to X\_{n-1}$, and let $G:D^m\to Cg\subseteq Y$ be the characteristic map of an $m$-cell of $Y$ with attaching ... | 7 | https://mathoverflow.net/users/8103 | 304827 | 133,357 |
https://mathoverflow.net/questions/304833 | 1 | Let $X$ be set. Let $f$ be a function from $X$ into $X$. For a given set $E\subseteq X$, we say $E$ determines a $U$-part of $f$ if $f(E)\subseteq E$ and the restriction $f:E\to E$ is a bijection.
I am looking for a commuting pair of ***injective*** functions $f:X\to X$ and $g:X\to X$ (I mean $fg=gf$) such that both... | https://mathoverflow.net/users/84390 | A commuting pair of functions with no common bijective part | Let $X=\mathbb{N}$, with
$f(2n)=2n$, $f(2n+1)=2n+3$ and $g(2n)=2n+2$, $g(2n+1)=2n+1$.
| 2 | https://mathoverflow.net/users/22989 | 304835 | 133,359 |
https://mathoverflow.net/questions/304535 | 5 | Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)\_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u\_n+u\_m$) ?
**Remark** : This question has been [asked previously on math.SE](https://math.stackexchange.com/questions/2838191/additive-basis-... | https://mathoverflow.net/users/97798 | Additive basis of order 2 | For $1<\alpha<\frac32$, $(\lfloor n^{\alpha}\rfloor)\_{n\geqslant0}$ is an asymptotic basis of order 2. I finally found these two articles:
J-M. Deshouillers, [**Un problème binaire en théorie additive**](http://matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2549.pdf), Acta Arith. 25 (1974), 393-403
S.V. Konyagin, [**An addit... | 4 | https://mathoverflow.net/users/97798 | 304844 | 133,363 |
https://mathoverflow.net/questions/37498 | 8 | Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distributed" in a way which is due to the continued fraction form of φ, making the constant "as difficult as possible" to approximate using rational values (... | https://mathoverflow.net/users/4336 | Sequences of evenly-distributed points in a product of intervals | **Part 1. Equidistribution.**
As mentioned in the comments, the equidistribution theorem states that any irrational value will produce an equidistributed sequence. That is, in the limit as $n \rightarrow \infty$, all finite subintervals of $(0,1)$ are equally likely.
However, as you have mentioned, equidistributio... | 3 | https://mathoverflow.net/users/126485 | 304860 | 133,367 |
https://mathoverflow.net/questions/304866 | 1 | Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this may be a rather open question, does anyone have any examples of calculations whose result is the computation of some part... | https://mathoverflow.net/users/10007 | Computing a factorization of a monad | Your last question, the one you call "a question all its own," is essentially the subject of [this paper](https://link.springer.com/article/10.1007%2Fs10485-016-9473-8). (Free version from the author's website [here](http://www.math.wayne.edu/~asalch/papers/kleisli6c.pdf).)
Here is a relevant excerpt from the introdu... | 5 | https://mathoverflow.net/users/nan | 304877 | 133,371 |
https://mathoverflow.net/questions/304850 | 3 | Suppose I have groups $A,B$ and $C$ for which the following information is given:
1) The group $G\_{AB}$ is a central extension of $B$ by $A$, where the abelian group $B$ acts trivially
2) The group $G\_{BC}$ is an extension of $C$ by $B$, where $C$ does *not* act trivially, and the action is given by $\theta: C \r... | https://mathoverflow.net/users/125997 | When is a sequence of group extensions associative? | This situation was studied by Eilenberg and MacLane in a series of papers in the 1940s. They studied exact sequences
$$1 \to A \to G\_{AB} \stackrel{\alpha}{\to} G\_{BC} \stackrel{\beta}{\to} C \to 1$$
in which $A$ maps into the centre of $G\_{AB}$, where we are given an action of $G\_{BC}$ on $G\_{AB}$ that restricts ... | 6 | https://mathoverflow.net/users/35840 | 304880 | 133,372 |
https://mathoverflow.net/questions/304832 | 8 | We say $S$ is a Suslin forest if adding a minimum to $S$ we have a Suslin tree. So a Suslin Forest is essentially a Suslin tree $S$ in which we drop the requirement for $S$ to have a single root.
Notice that every uncountable subset of a Suslin tree is a Suslin forest.
Given a Suslin forest $S$, I would like to know ... | https://mathoverflow.net/users/121875 | Is there a function from a Suslin tree to itself which send compatible elements to incompatible elements? | Yes, this can happen, and indeed it happens in a subforest of any given Souslin tree.
Let's start with an illustrative case. Sometimes people consider Suslin trees that are not necessarily
normal, and where for example, a sequence converging to a limit level
can have more than one bounding node at that level.
Let u... | 5 | https://mathoverflow.net/users/1946 | 304882 | 133,373 |
https://mathoverflow.net/questions/303971 | 1 | It is an ancient result of Jensen that
$$(s-1)\zeta(s)=\frac{\pi}{2} \int\_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$
where $\zeta$ denotes the Riemann zeta function.
Is there a generalisation of this formula valid for all Dirichlet L-functions?
| https://mathoverflow.net/users/480516 | On a certain integral representation for Dirichlet L-functions | As I suspected in my comment above, there is an analog of Jensen's 'ancient' result for the Hurwitz zeta function: For $a>1/2$,
$$
\zeta(s,a)=\frac{\pi}{2(s-1)}\int\_{-\infty}^\infty\frac{(a-1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
This is (23) on p. 92 of *Series Associated with Zeta and Related Functions*, by Srivasta... | 3 | https://mathoverflow.net/users/6756 | 304893 | 133,377 |
https://mathoverflow.net/questions/304884 | -4 | EDIT: it appears that my original question has some confusion between auto-morphisms and elementary embeddings as it is obvious from the answer below, therefore I'll clarify here what I exactly want.
Can we have the following?
A transitive model $M$ of ZF-Regularity such that we have an external injective function ... | https://mathoverflow.net/users/95347 | Can there be such an elementary embedding? | EDIT: Based on the comments, I think it's worth clarifying a bit of the nature of the Boffa-Jensen construction of a model of NFU (which appears to be part of the motivation for this question).
In this construction, we do not have an elementary embedding; rather, we begin with a model $M$ of ZFC and an *automorphism*... | 2 | https://mathoverflow.net/users/8133 | 304900 | 133,380 |
https://mathoverflow.net/questions/304498 | 6 | Let $G = \operatorname{GL}\_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi\_1 \otimes \cdots \otimes \chi\_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is to say that $\chi\_i \neq \chi\_j$ for $i \neq j$. Equivalently, $\chi \neq w.\chi$ for all $w \in W(T,G)$.
Then $I... | https://mathoverflow.net/users/38145 | Irreducibility of the unramified principal series | Suppose $F$ is a non-archimedean local field. This regularity assumption is not needed, nor is the assumption that $\chi$ is unramified. The sufficient part for irreducibility is part of Theorem 4.2 of [Bernstein and Zelevinsky - Induced representations of reductive $\mathfrak p$-adic groups. I](http://www.math.tau.ac.... | 5 | https://mathoverflow.net/users/31327 | 304901 | 133,381 |
https://mathoverflow.net/questions/304897 | 0 | Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ will cut out some divisor $C\_s \subset X$.
Need it be the case that, for generic $s$, the curve $C\_s$ is irreducible?
Th... | https://mathoverflow.net/users/126543 | Do line bundles with enough sections on surfaces have generic divisors which are irreducible? | Not in general. Take a high genus hyperellitpic curve $C$ with a degree 2 map $f:C\to\mathbb{P}^1$. Take your surface to be $C\times D$ for some smooth curve $D$ and let $L$ be the pull back of $\mathcal{O}\_{\mathbb{P}^1}(1)$ by the obvious map. Then $H^0(L)=2$ which is globally generated and every section vanishes al... | 3 | https://mathoverflow.net/users/9502 | 304902 | 133,382 |
https://mathoverflow.net/questions/304898 | 22 | I am writing an ODEs textbook for second year students and I would like to get inspirations on general good designs on undergraduate textbooks taught in the first two years (i.e. calculus, linear algebra, real analysis and ODEs ) that enhance student understanding .
>
> Q: Can you recommend some design principles ... | https://mathoverflow.net/users/99863 | Design principles for good undergraduate textbooks to enhance student understanding | I've published a number of undergraduate and graduate science books, some heavy in mathematics, but no true mathematics textbooks. I've thought long and hard about how to design and craft them, and have several professional calligraphers, type designers, book designers in my immediate family, and they (and of course my... | 11 | https://mathoverflow.net/users/89654 | 304904 | 133,383 |
https://mathoverflow.net/questions/304908 | 10 | I'm playing around with products $M = \Bbb S^{n\_1} \times \Bbb S^{n\_2}$, and a quick computation using the Künneth formula tells us that if $(n\_1,n\_2)$ is not $(1,1)$ or $(2,4)$, $M$ is not symplectic (WLOG $1 \leq n\_1 \leq n\_2$, of course). The $(1,1)$ case is obviously symplectic, but I couldn't decide about th... | https://mathoverflow.net/users/54656 | Is $\Bbb S^2 \times \Bbb S^4$ symplectic? | No. Note that $H^2(S^2\times S^4,\mathbb R)$ is one dimensional, spanned by $\pi^\*\alpha$, where $\pi:S^2\times S^4\to S^2$ is the projection, and $\alpha$ is a volume form on $S^2$. Suppose $\omega$ is a symplectic form on $S^2\times S^4$. Then $[\omega]=c[\pi^\*\alpha]$ for some $c\in\mathbb R^\times$. Then $[\omega... | 28 | https://mathoverflow.net/users/37103 | 304911 | 133,387 |
https://mathoverflow.net/questions/304918 | 2 | Let $k$ be a field, $D$ be a crossed product division algebra over $k$, namely $D$ has a maximal subfield which is Galois over $k$. Is it possible $D$ contain some other maximal subfield which is non-Galois over $k$
| https://mathoverflow.net/users/nan | Crossed product division algebra | Take a prime number $p>2$ and a field $k$ of characteristic zero which does not contain a $p$-th root of unity. Assume that $K/k$ is a Galois extension of order $p$ with a Galois group $\langle g\rangle$ and that $D = K\*\_{\alpha}\langle g \rangle$ where $[\alpha]\in H^2(\langle g\rangle,K^{\times})$ is non-trivial. T... | 2 | https://mathoverflow.net/users/41644 | 304924 | 133,390 |
https://mathoverflow.net/questions/304922 | 1 | We know predual of a von Neumann algebra $M$ as a Banach space is independent of Hilbert space where the $M$ is represented. Now the question is if we represent $M$ in $B(\mathcal{H})$, where $M$ has separating vector, then is the predual is same as weak operator dual, finally boiled down the question is weak\*-topolog... | https://mathoverflow.net/users/125816 | Clarification on predual on existence of separating vector | So I *think* you are asking:
>
> Let $(M,H)$ be a von Neumann algebra with a separating vector $\xi\in H$. Is the weak operator topology on $M$ the same as the weak$^\*$-topology coming from $M\_\*$?
>
>
>
The answer is of course "yes". As you allude to, if $(M,H)$ has a separating vector, then every $\omega\i... | 3 | https://mathoverflow.net/users/406 | 304930 | 133,391 |
https://mathoverflow.net/questions/255472 | 4 | In [the article](https://en.wikipedia.org/wiki/Free_Lie_algebra), the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X.
It is said that the graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle al... | https://mathoverflow.net/users/11877 | How to show that the graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra | All that has been said in the excellent comments by Darij is true for modules (the set of scalars being a ring) except maybe Radford's theorem (see below). I complete here what has been said, too long for a comment though.
The shuffle product is called such because it "shuffles" the letters of two words considered a... | 3 | https://mathoverflow.net/users/25256 | 304931 | 133,392 |
https://mathoverflow.net/questions/304923 | 0 | Let $K$ be a function field of positive characteristic $p$.
Let $b$, $u$, $u'$ be in $K\setminus K^p$.
I would like to show that there are only finitely many natural numbers $r$ prime to $p$ such that for some natural number $n\_r$ we have $u'=(b^r)(u^{p^{n\_r}})$.
But I don't even know whether this assertion holds.
... | https://mathoverflow.net/users/4948 | A question in a function field of positive characteristic $p$ | That is not true. Let $K$ be $\mathbb{F}\_p(u)$, a purely transcendental extension of degree $1$. Let $u'$ equal $u$. Let $b$ equal $u^{1-p}$. None of these is in $K^p$. For every integer $n\geq 0$, for the positive natural number $r=(p^n-1)/(p-1)$, then $b^ru^{p^n}$ equals $u'$. Of course $r$ is congruent to $1$ modul... | 2 | https://mathoverflow.net/users/13265 | 304932 | 133,393 |
https://mathoverflow.net/questions/304759 | 1 | It is hard to find any reference which contains a proof of the following statement: slope stability is an open condition in a flat family.
There is one I have found, '65 paper of Narasimhan and Seshadri. Is this the unique reference which proves the statement?
There are some references which prove the openness of Gies... | https://mathoverflow.net/users/75699 | reference required: slope stability is an open condition | In the second edition of "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, the arguments in Proposition 2.3.1 which imply (for instance) that Gieseker-semistability is an open condition in families also imply the same for $\mu$-stability (this is pointed out just after Definition 3.A.1 on p. 82).
| 2 | https://mathoverflow.net/users/5496 | 304933 | 133,394 |
https://mathoverflow.net/questions/304772 | 5 | I am aware that if an elliptic surface contains multiple fibers, then it has no section. Is the converse false?
In particular, I am looking for an example of a projective, properly elliptic surface (Kodaira dimension 1), fibered over $\mathbb{P}^1$, with no multiple fibers and no section.
| https://mathoverflow.net/users/126479 | Properly elliptic surface with no multiple fibers and without a section | Please confer Corollary 2.2 of the following with $d$ equal to $3$ and with $n$ equal to $2$.
Jason Starr
A pencil of Enriques surfaces of index 1 with no section
<https://arxiv.org/pdf/math/0602639.pdf>
This proves that for every integer $e\geq 2$, for a very general hypersurface $X$ in $\mathbb{P}^2\times... | 5 | https://mathoverflow.net/users/13265 | 304934 | 133,395 |
https://mathoverflow.net/questions/304946 | 9 | Let $\Gamma$ be a finite graph, then $H^1(\Gamma,\mathbb{Z})\cong \mathbb{Z}^{g(\Gamma)}$ can be viewed as a $\mathrm{Aut}(\Gamma)$ module.
Conversely, given a finite group $G$, and a $G$-module $\mathbb{Z}^n$, does there always exist a finite graph $\Gamma$ such that the $G$ module $\mathbb{Z}^n$ arises as $G\overs... | https://mathoverflow.net/users/nan | Finite group representation as $\mathrm{Aut}(\Gamma)$ action $H^1(\Gamma,\mathbb{Z})$ of graph? | No. An action of a group $G$ on a graph $\Gamma$ induces a homomorphism $G\to \mathrm{Out}(\pi\_1(\Gamma))=\mathrm{Out}(F\_n)$. So a representation $G\to \mathrm{GL}\_n({\mathbb Z})$ can come from an action on a graph only if it lifts to a homomorphism $G\to \mathrm{Out}(F\_n)$ over the quotient homomorphism $\mathrm{O... | 10 | https://mathoverflow.net/users/6668 | 304959 | 133,401 |
https://mathoverflow.net/questions/304892 | 7 | Equation
$$(2x^2+1)(2y^2+1)=4z^2+1$$
has no solutions in the positive integers. Its true?
| https://mathoverflow.net/users/126541 | $(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions? | By popular demand, I turn my comment to an answer:
There are no solutions according to Theorem 6 in Kashihara: Explicit complete solution in integers of a class of equations $(ax^2−b)(ay^2−b)=z^2−c$, Manuscripta Math. 80 (1993), 373-392.
| 14 | https://mathoverflow.net/users/11919 | 304962 | 133,402 |
https://mathoverflow.net/questions/304957 | 2 | I am studying pell conics and the source I am using (Franz Lemmermeyer: Conics - A Poor Man's Elliptic Curves) defines its discriminant as follows:
For equations of the form $X^2 + XY + \frac{1-d}{4}Y^2$ = 1, the discriminant is $d$, ($d\equiv 1$ mod $4$.)
For equations of the form $X^2 - dY^2 = 1$, the discriminan... | https://mathoverflow.net/users/126566 | Definition of a Discriminant in Three Variables | Given $n$ homogeneous polynomials $F\_i(x\_1,\ldots,x\_n)$ in $n$ variables with respective degrees $d\_i$, there is a unique polynomial ${\rm Res}(F\_1,\ldots,F\_n)$ in the coefficients of the $F\_i$ called the multidimensional resultant which satifies:
1. It is irreducible.
2. It is equal to 1 when $F\_i=x\_i^{d\_i... | 7 | https://mathoverflow.net/users/7410 | 304964 | 133,404 |
https://mathoverflow.net/questions/302877 | 4 | Suppose we have a logic for counterfactuals as with David Lewis. I here use $\Rrightarrow$ for the counterfactual conditional. So suppose we have:
Rules:
(1) If $A$ and $A\rightarrow B$ are theorems, then $B$ is a theorem.
(2) If $(B\_1\wedge ...)\rightarrow C$ is a theorem, then so is $((A\Rrightarrow
B\_1)\we... | https://mathoverflow.net/users/37385 | Axioms for modal logics based upon counterfactuals | In ["Completeness and decidability..."](http://www.andrewmbailey.com/dkl/Completeness_and_Decidability.pdf), Lewis shows the system C1 to be decidable and complete with respect to the semantics of corresponding canonical $\alpha$-models (81-4). It seems that K holds in all relevant filtrations of standard $\alpha$-mode... | 2 | https://mathoverflow.net/users/124813 | 304968 | 133,406 |
https://mathoverflow.net/questions/304973 | 24 | The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible in the language of set theory. (One may think of $\sigma$ as a family condition/decision in the real life, say having kids o... | https://mathoverflow.net/users/82843 | Forcing and Family Contentions: Who wins the disputes? | I like this question a lot. It provides an interesting way of talking about
some of the ideas connected with the maximality principle and the
modal logic of forcing.
Let me make several observations.
First, Alice can clearly win, in one move, with any forceably
necessary statement $\sigma$, which is a statement for... | 20 | https://mathoverflow.net/users/1946 | 304976 | 133,412 |
https://mathoverflow.net/questions/304965 | 5 | A recent question [On a certain integral representation for Dirichlet L-functions](https://mathoverflow.net/questions/303971/on-a-certain-integral-representation-for-dirichlet-l-functions) referenced an integral representation of $\zeta(s)$ due to Jensen that was new to me:
$$
(s-1)\zeta(s)=\frac{\pi}{2(s-1)}\int\_{-\i... | https://mathoverflow.net/users/6756 | On a certain integral representation for Hurwitz zeta functions | All these are special cases of the Abel--Plana formula: under simple regularity
and growth condition which are easily given, for any $a$ with $\Re(a)>-1/2$ we have
$$\sum\_{n\ge 0}f'(n+a)=-\dfrac{\pi}{2}\int\_{-\infty}^\infty \dfrac{f(a-1/2+it)}{\cosh^2(\pi t)}\,dt$$
plus some correction terms due to the possible poles... | 7 | https://mathoverflow.net/users/81776 | 304977 | 133,413 |
https://mathoverflow.net/questions/306017 | 6 | Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page:
<https://en.wikipedia.org/wiki/Symmetric_space>
In the Berger classifiction of holonomy groups
<https://en.wikipedia.org/wiki/Holonomy#The_Berger_classifica... | https://mathoverflow.net/users/126606 | Holonomy groups of compact Riemannian symmetric spaces | At the request of the OP I put my comment as an answer: in general, the holonomy group and the isotropy group have the same identity component (this is a theorem of E. Cartan). So if you assume that $M$ is simply-connected, they are equal. You can see a proof (for instance) in section 10.79 of Arthur Besse's *Einstein ... | 6 | https://mathoverflow.net/users/40297 | 306024 | 133,435 |
https://mathoverflow.net/questions/306037 | 4 | Is the following claim correct (Chapter 13 before Theorem 87 of Todorcevic's book: Notes on forcing axioms):
Let $\alpha$ be an infinite countable indecomposable ordinal and $U$ be an uniform ultrafilter on $\alpha$ (namely elements in $U$ have order type $\alpha$). Then for any collection $\{B\_i\in U: i<\mathfrak{m}\... | https://mathoverflow.net/users/119731 | Indecomposable ordinals and pseudointersection | I believe the claim is wrong:
If the claim is right I claim I can show $\alpha\to (\alpha)^2\_2$ which is obviously wrong for countable ordinal $\alpha\geq \omega+2$.
Given a coloring $f: [\alpha]^2\to 2$, for each $\beta\in \alpha$, let $A\_\beta^i=\{\gamma<\alpha: f(\beta,\gamma)=i\}$ for $i<2$. Let $g: \alpha\... | 3 | https://mathoverflow.net/users/23835 | 306038 | 133,438 |
https://mathoverflow.net/questions/304999 | 1 | A hyperbolic link is one whose complement admits a hyperbolic metric.
Hyperbolic links, and especially hyperbolic knots, are quite popular these days. However, I am currently interested in links whose complement admits a flat (i.e. locally euclidean) metric. If I got it right, the major difference from the hyperbolic ... | https://mathoverflow.net/users/9833 | "Flat links", a reference request | The canonical reference for this sort of thing is:
*Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P.*, [**Three-dimensional orbifolds and cone-manifolds**](http://dx.doi.org/10.2969/msjmemoirs/005010000), MSJ Memoirs. 5. Tokyo: Mathematical Society of Japan (MSJ). ix, 170 p. (2000). [ZBL0955.57014](https://zbma... | 0 | https://mathoverflow.net/users/11142 | 306043 | 133,439 |
https://mathoverflow.net/questions/306040 | 2 | I am somewhat a beginner in the field of operator algebras and was wondering about the following:
Let $T$ be a linear map between the space of bounded operators $B(H)$ on some Hilbert space and $S$ a map between the space of trace-class operators that we denote by $N(H)$ in the sequel.
Then, one defines maps $T\_n:... | https://mathoverflow.net/users/126620 | Description of (completely) bounded operator | Okay, I think the most important thing to start with is that there are different ways of defining tensor products of C\*-algebras. There's a brief introduction [here](https://ncatlab.org/nlab/show/spatial+tensor+product). So when you talk about "the space $B(H) \otimes B(H)$" this could mean different things. The von N... | 1 | https://mathoverflow.net/users/23141 | 306045 | 133,440 |
https://mathoverflow.net/questions/306044 | 1 | Take a category $C$, and take all endofunctors of $C$, so the set $E= \{ M| M: C \rightarrow C \}$. $E$ forms the objects of a category with morphisms given by all natural transformations $\mu : M \rightarrow N$ for $M,N \in E$. Let $\mathcal{C}$ be the endofunctor category as defined. What are the internal categories ... | https://mathoverflow.net/users/10007 | What are the internal categories in an endofunctor category | (I assume here that by “internal category” you mean the usual sense defined in any category with finite limits.)
If $\newcommand{\C}{\textbf{C}}\C$ has finite limits, then for any $\newcommand{\D}{\textbf{D}}\D$, $[\D,\C]$ will have finite limits, constructed as pointwise limits, and so internal categories in $[\D,\C... | 4 | https://mathoverflow.net/users/2273 | 306053 | 133,442 |
https://mathoverflow.net/questions/306049 | 3 | Let $\mathcal{B}\_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu\_L$ be the Lebesgue measure on $\mathbb{R}$.
Define a new $\sigma$-algebra $\mathcal{B}\_0$ as follows:
$$\mathcal{B}\_0=\{A\in \mathcal{B}\_{\mathbb{R}}:\mu\_L(A)=0\ \text{or}\ \mu\_L(A^c)=0\}.$$
I want to prove that the family o... | https://mathoverflow.net/users/58096 | Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$? | It is consistent that such a set $E$ exists, but I do not know if ZFC proves that such a set exists.
A set $S$ is called a Sierpiński set if $S$ is uncountable, but $S\cap N$ is countable for every (Borel) null set $N$.
If $S$ is a Sierpiński set, then $S$ satisfies your requirement: $S$ is not measurable, and ea... | 7 | https://mathoverflow.net/users/14915 | 306055 | 133,443 |
https://mathoverflow.net/questions/304937 | 5 | On the Wikipedia page for restricted representations
<https://en.wikipedia.org/wiki/Restricted_representation>
there is presented a number of explicit "branching rules". In particular, there is the Weyl's branching rule from U(N) to U(N-1) given in terms of signatures $f\_1 \geq \cdots \geq f\_N$, for $f\_i \in \m... | https://mathoverflow.net/users/125941 | Weyl's Branching Rule for $SU(N)$-Setting | The question is answered on page 385 of the classical Zhelobenko book
***Compact Lie groups and their representations***
for the more general case of $SU(n+m)/SU(n) \times SU(m)$.
| 4 | https://mathoverflow.net/users/125941 | 306074 | 133,450 |
https://mathoverflow.net/questions/306078 | 6 | It is well known that there is a correspondence between homomorphism of rings $A\to B$ and morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$.
**Question:** (1) In analogy, is there anything similar between homomorphism of adic rings and morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{S... | https://mathoverflow.net/users/69190 | about morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$ | See EGA I, sec. 10. For the specific question you mention, see paragraph 10.2. There is further issues on the cohomology of formal schemes in EGA III, sect 3.4. Also, the existence theorem is treated in the volume "Fundamental Algebraic Geometry: Grothendieck’s FGA explained", Mathematical Surveys and Monographs 123 (2... | 6 | https://mathoverflow.net/users/6348 | 306082 | 133,452 |
https://mathoverflow.net/questions/306058 | 5 | It is basic that the norm map $N:\mathbf{F}\_{q^n}^\* \to \mathbf{F}\_q^\*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?
A basic example is an intermediate extension $\mathbf{F}\_{q^d}$. On $\mathbf{F}\_{q^d}^\*$ we have
$$N(x) = \left(x^{(q^... | https://mathoverflow.net/users/20598 | Surjectivity of norm map on subspaces of finite fields | The number of elements with norm $a$ is $$\frac{1}{q-1} \sum\_{\chi: \mathbb F\_q^\times \to \mathbb C^\times} \sum\_{x \in V} \chi(Nx) \overline{\chi(a)}$$ The summand vanishes unless $\chi$ has order dividing $n$ so there are at most $gcd(n,q-1)$ terms. One of the terms has size $q^{\dim V}$ so it is sufficient that ... | 5 | https://mathoverflow.net/users/18060 | 306085 | 133,454 |
https://mathoverflow.net/questions/306046 | 16 | As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.
| https://mathoverflow.net/users/126532 | How to visualize a Witt vector? | You can view the spectrum of the ring Witt vectors, in the sense of scheme theory, pretty reasonably.
If $R$ is $p$-torsion free, then $\mathrm{Spec}(W\_n(R))$ is $n+1$ copies (or $n$ if you use the traditional indexing) of $\mathrm{Spec}(R)$ glued together in a suitable way along their fibers over $p$. But there are... | 12 | https://mathoverflow.net/users/1114 | 306086 | 133,455 |
https://mathoverflow.net/questions/306083 | 5 | $\require{AMScd}$
**Background:** This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar constructions on ncatlab can be found at [[1]](https://ncatlab.org/nlab/show/bar+and+cobar+construction). I will also mention ... | https://mathoverflow.net/users/123015 | Homology of bar complex vs homology of indecomposables | The functor of indecomposables is the left adjoint of a Quillen adjunction between dg-algebras and dg-modules. (For a general reference, see Section 12.1.3 of the book *Algebraic Operads* by Loday and Vallette, though this was certainly known before the book – I just happen to have it on my desk.) As such it preserves ... | 4 | https://mathoverflow.net/users/36146 | 306103 | 133,459 |
https://mathoverflow.net/questions/306117 | 3 | Given an infinite cardinal $\kappa$, is there a graph on $\kappa$ vertices that contains $2^\kappa$ pairwise non-isomorphic [induced subgraphs](https://en.wikipedia.org/wiki/Induced_subgraph)?
| https://mathoverflow.net/users/8628 | Infinite graph with lots of non-isomorphic induced subgraphs | You don't need something as complicated as the Rado graph.
Take the disjoint union of $K\_n$ (the complete graph on $n$ vertices) for all $n$. This has countably many vertices. For any subset of the natural numbers $S \subseteq \mathbb{N}$ consider the induced subgraph consisting of $K\_i$ for $i \in S$. For distinct... | 5 | https://mathoverflow.net/users/18606 | 306128 | 133,462 |
https://mathoverflow.net/questions/306104 | 6 | The binary product of two $\lambda$-presentable objects (in a locally presentable category) is $\mu$-presentable for some regular cardinal $\mu \geq \lambda$ (because all objects are $\mu$-presentable for some regular cardinal $\mu$). I don't see any reason for $\mu$ to be equal to $\lambda$. Even if $\lambda^2=\lambda... | https://mathoverflow.net/users/24563 | The binary product of two presentable objects | Unfortunately, no example for the first question is coming to mind at the moment.
For the second question, let $\lambda$ be such that the product functor $\times: \mathcal C \times \mathcal C \to \mathcal C$ is $\lambda$-accessible [1]. Let $\mu \rhd \lambda$ be such that the binary product of $\lambda$-presentable o... | 3 | https://mathoverflow.net/users/2362 | 306129 | 133,463 |
https://mathoverflow.net/questions/306118 | 3 | I was able to conclude, numerically, the following:
$$s(n) = \sum\_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum\_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$
for $x\in[0,1]$. For example
\begin{array}{ccc}
n& max(s(n)) & n+1 \\
0& 1.00& 1\\
1& 1.53& 2\\
2& 2.07& 3\\
3& 2.60& 4\\
4& 3.13& 5\\
\en... | https://mathoverflow.net/users/126661 | Inductive proof of $s(n)≤n+1$ | The alternative form can be expressed as
$$ \frac{1}{2} - \frac{n+1}{2} \cos (n+1)t
+ \frac{1}{2}\bigl( \cos nt + \cdots + \cos t \bigr).$$
The third summand has period $2\pi$ and takes its maximum value of $n/2$ when $t=0$. So just from the triangle inequality we get
$$s(n) \le \frac{1}{2} + \frac{n+1}{2} + \fr... | 3 | https://mathoverflow.net/users/7709 | 306131 | 133,465 |
https://mathoverflow.net/questions/306134 | 13 | Who first chose the names Alice and Bob for the players (or observers) A and B?
| https://mathoverflow.net/users/30395 | Who first chose the names Alice and Bob for players A and B? | Quoted from [Wikipedia](https://en.wikipedia.org/wiki/Alice_and_Bob):
>
> *The Alice and Bob characters were invented by Ron Rivest, Adi Shamir, and Leonard Adleman in their 1978 paper "A method for obtaining digital signatures and public-key cryptosystems".*
>
>
>
*Rivest, R. L.; Shamir, A.; Adleman, L.*, [**... | 28 | https://mathoverflow.net/users/82843 | 306136 | 133,468 |
https://mathoverflow.net/questions/306138 | 8 | Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over characteristic zero.has anyone used Manin ideas to prove theorem for positive characteristic ?
| https://mathoverflow.net/users/65846 | Mordel's conjecture for function fields in positive characteristic | Implicitly, my proof (Inv. Math. 104 (1991) 643-646) is an extension of Manin's approach to char p. I first extended Manin's ideas to char p for elliptic curves (Comp. Math. 74 (1990) 247-258) and realized the connection with p-descent so, for the Mordell conjecture, I just used the p-descent directly. See also my pape... | 11 | https://mathoverflow.net/users/2290 | 306143 | 133,469 |
https://mathoverflow.net/questions/306122 | 2 | Let $\{x\_n\}\_{n=1}^{\infty}$ be a monotone decreasing sequence of positive real numbers such that $\sum\_{n=1}^{\infty} x\_n$ diverges. Also let $\{k\_n\}\_{n=1}^{\infty}$ be a strictly increasing sequence of positive integers such that $\sum\_{n=1}^{\infty} \frac{1}{k\_n}$ diverges.
Can $\sum\_{n=1}^{\infty} x\_{k\... | https://mathoverflow.net/users/109471 | Sum of a terms in a divergent series taken along indices the sum of whose reciprocal diverges. Can the sum converge? | Yes, this can happen. E.g., let $m\_1,m\_2,\dots$ be natural numbers such that
$$m\_r\sim\ln r$$
(all asymptotic relations here are for $r\to\infty$). Let $k\_r:=m\_1+\dots+m\_r$, so that $k\_r\sim r\ln r$ and hence $\sum\_r1/k\_r=\infty$. Let
\begin{equation}
x\_n:=y\_r\sim1/(r\ln^2 r)\quad\text{if}\quad k\_{r-1}+1... | 4 | https://mathoverflow.net/users/36721 | 306148 | 133,472 |
https://mathoverflow.net/questions/306141 | 3 | The following lemma is in [Bosch](http://www.math.purdue.edu/~tongliu/seminar/rigid/Bosch.pdf)'s book "Lectures on Formal and rigid geometry" p198.
**Lemma** Let $K$ be a non-archimedean field and $R$ its valuation ring. Let $X= \mathrm{Spf}A$ be an affine admissible formal $R$ scheme. Then there are canonical biject... | https://mathoverflow.net/users/69190 | A translation between formal and rigid geometry | Using your notation: it corresponds, like in the "classic" case of polynomials over a field, to the ideal generated by $\zeta\_1-x\_1,\dots,\zeta\_n-x\_n$, where $x=(x\_1,\dots,x\_n)\in \mathbb B^n(K)$. In another terms, it is as before the kernel of the evaluation map $A\to R$ given sending $f$ to $f(x)$ (which conver... | 1 | https://mathoverflow.net/users/24442 | 306150 | 133,473 |
https://mathoverflow.net/questions/306156 | 2 | This is a technical question coming out of my research.
Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that
$$
\angle(u, v) \le \sum\_{\Omega} \angle(u\_{\Omega}, v\_{\Omega})?
$$
Here the $\Omega$ indexes length-two subvectors. My ... | https://mathoverflow.net/users/19098 | Controlling angles between vectors using sum of subvector angles? | In the positive orthant you may use the following argument.
We have $\sin \angle(u,v)=\frac{\|u\times v\|}{\|u\|\cdot \|v\|}$, where $\times$ denotes the vector product. Note that the coordinates of $u\times v$ are exactly $\pm \|u\_{\Omega}\times v\_{\Omega}\|$. Combined with Pythagoras and obvious inequalities like... | 2 | https://mathoverflow.net/users/4312 | 306160 | 133,475 |
https://mathoverflow.net/questions/306097 | 3 | Let $0<q<1$ and consider the entire function $f(z)=\displaystyle \sum\_{k=0}^\infty q^{k^2}z^k$. For $a>1,$ denote $m\_j=f(a^j),\; j=0,1,2,\dots.$
Question: Does there exist an entire function $g(z)=\displaystyle \sum\_{k=0}^\infty c\_k z^k$ with $c\_k>0$ (strictly) and a number $b\neq a$ so that $g(b^j)=m\_j$ for al... | https://mathoverflow.net/users/78726 | On the values of an entire function | The answer is **yes**, at least in the special case when $a > q^{-4}$ and $b = \sqrt{a}$.
---
Denote by $h(z)$ the [q-Pochhammer symbol](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol) with parameter $\tfrac{1}{b}$:
$$ h(z) = \prod\_{k = 0}^\infty \biggl(1 - \frac{z}{b^k}\biggr) = \biggl(z; \frac{1}{b}\biggr)\... | 1 | https://mathoverflow.net/users/108637 | 306163 | 133,476 |
https://mathoverflow.net/questions/298502 | 1 | I'm trying understand the article "[Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem](https://arxiv.org/pdf/0908.2682.pdf)" by Ben Andrews and Paul Bryan and they stated on the final of the article (precisely, on the proof of corollary 4) that
>
> The un-norm... | https://mathoverflow.net/users/114870 | A change of parameters used on Curve Shortening Flow |
>
> Just fixing notation,
>
>
> $\gamma\_1$ is the normalized curve;
>
>
> $\gamma$ is the curve which is a solution for the Curve Shortening Flow (CSF);
>
>
> $t\_1$ is the time parameter of $\gamma\_1$;
>
>
> $t$ is the time parameter of $\gamma$;
>
>
> $k\_1(p,t\_1)$ is the curvature of $\gamma\_1 ( \cdo... | 1 | https://mathoverflow.net/users/114870 | 306164 | 133,477 |
https://mathoverflow.net/questions/304849 | 4 | Combine my [first previous question](https://mathoverflow.net/questions/303013/a-muirhead-like-inequality) and [second previous question](https://mathoverflow.net/questions/302956/an-inequality-related-to-power-sum-and-elementary-symmetric-polynomial-and-major) with the [Muirhead inequality](https://en.wikipedia.org/wi... | https://mathoverflow.net/users/122662 | Combination power elementary symmetric polynomial inequality | These inequalities are true when $m=1$, and they both follow from Karamata's inequality, since $x^a$ is concave when $0\le a\le 1$, and $a^x$ is convex. However they are both *false* for every $m\geq 2$.
**Counterexample to Inequality 1** Set $a\_1=1$ and $a\_i=0$ for $i>1$. Then your inequality now reads $$\sum\_{i=... | 7 | https://mathoverflow.net/users/2384 | 306167 | 133,479 |
https://mathoverflow.net/questions/306172 | 17 | I know of three homotopy theories of colored operads.
* The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak equivalences given by strict maps $O\to O'$ which induce weak equivalences on spaces of operations.
* The "dendroida... | https://mathoverflow.net/users/7108 | Homotopy theories of operads | The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826.
In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial operads", arXiv:1109.1004) proved a Quillen equivalence between simplicial operads and dendroidal sets. In the paper of... | 15 | https://mathoverflow.net/users/1310 | 306177 | 133,480 |
https://mathoverflow.net/questions/306088 | 5 | I got stuck on the following problem while thinking about [this question](https://mathoverflow.net/questions/305003/does-vizings-conjecture-hold-for-the-infinite-graphs).
Let $G$ be an infinite graph. Say that a set of vertices $S$ of $G$ *has a dominating pair* if there exist $v,w \in S$ such that every other vertex... | https://mathoverflow.net/users/23141 | Domination numbers of infinite graphs | I think I have a counterexample. I will construct the *complement* of the graph $G$ as the union of an infinite sequence of finite graphs $H\_n.$
Let $H\_0$ be a graph with three vertices and no edges.
Suppose the finite graph $H\_n$ has been constructed with vertex set $V\_n=V(H\_n).$ Then $H\_{n+1}$ is a supergra... | 1 | https://mathoverflow.net/users/43266 | 306187 | 133,483 |
https://mathoverflow.net/questions/306161 | 8 | $\require{AMScd}$In *Algebraic homotopy*, Baues defines the notion of homotopy pushout in a cofibration category in the following way: a commutative diagram
\begin{CD}
A @>k>> C \\
@AfAA @AAhA\\
B @>g>> D
\end{CD}
is a *homotopy pushout* if for one factorization $B\hookrightarrow W\stackrel{\sim}\to A$ the induced ... | https://mathoverflow.net/users/21326 | Homotopy pushout independent of factorization and symmetric in cofibration category | I don't remember how Baues does this exactly, but all facts of this sort follow from the Gluing Lemma (see Lemma 1.4.1 in [this paper](https://arxiv.org/abs/math/0610009)) and "Brown type factorization". By this I mean the following construction. Given a morphism $X
\to Y$ and two factorizations $X \to Z\_0 \to Y$ and... | 6 | https://mathoverflow.net/users/12547 | 306191 | 133,487 |
https://mathoverflow.net/questions/306165 | 4 | Let $f(n)=\sum\limits\_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
**EDT 1:** A possible answer is [Analysis of the subtractive algorithm for greatest common divisors](http://www.pnas.org/content/pnas/72/12/4720.full.pdf) by Knuth D. ... | https://mathoverflow.net/users/5712 | Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$ | Let $n=p\_1^{r\_1}\cdots p\_k^{r\_k}$ be a prime factorization of $n$. We need to estimate
$$
\sum\_{\substack{{e\_1\leq r\_1}\\{\cdots}\\{e\_k\leq r\_k}}}\frac{e\_1\log p\_1+\cdots +e\_k \log p\_k}{p\_1^{e\_1}\cdots p\_k^{e\_k}}
$$
We break these up and consider
$$
\sum\_{\substack{{e\_1\leq r\_1}\\{\cdots}\\{e\_k\l... | 5 | https://mathoverflow.net/users/21090 | 306195 | 133,489 |
https://mathoverflow.net/questions/306182 | 1 | The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u\_1+iu\_2&u\_3+iu\_4\\-u\_3+iu\_4&u\_1-iu\_2\end{bmatrix}\in\mathbb{C}^{2\times2}$$
with
$$u\_1^2+u\_2^2+u\_3^2+u\_4^2=1, u\_j\in\mathbb{R} \ \forall j\in\{1,...,4\}$$
... | https://mathoverflow.net/users/126683 | Conditions to obtain a real logarithm of a unitary unimodular complex matrix? | In terms of [Pauli matrices](https://en.wikipedia.org/wiki/Pauli_matrices):
$$U=u\_1I+iu\_2\sigma\_3+iu\_3\sigma\_2+iu\_4\sigma\_1,\;\;u\_1^2+u\_2^2+u\_3^2+u\_4^2=1,$$
$$V=\alpha (n\_1\sigma\_1+n\_2\sigma\_2+n\_3\sigma\_3),\;\;n\_1^2+n\_2^2+n\_3^2=1,$$
$$\exp(iV)=I\cos\alpha + i(n\_1\sigma\_1+n\_2\sigma\_2+n\_3\sigma... | 2 | https://mathoverflow.net/users/11260 | 306206 | 133,492 |
https://mathoverflow.net/questions/305000 | 4 | **Short version of the question.** If $(P,\leq)$ is a partially ordered set (poset), a topology denoted by $\tau\_o(P)$ can be defined (see below). There is also another notion of convergence, called *order-convergence*. My question is: if ${\cal F}$ is a set-filter on $P$, is it true that ${\cal F}$ converges to some ... | https://mathoverflow.net/users/8628 | Order convergence vs topological convergence in partially ordered sets | The answer to this question is negative.
As a counterexample, consider the one-point extension $P:=2^{<\omega}\cup\{\infty\}$ of the binary tree $2^{<\omega}=\bigcup\_{n\in\omega}2^n$. Here $2$ is the ordinal $\{0,1\}$ and $2^n$ is the set of binary sequences of length $n$.
The partial order on $P$ is defined by t... | 4 | https://mathoverflow.net/users/61536 | 306211 | 133,494 |
https://mathoverflow.net/questions/306197 | 14 | The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with duals upon a single object.
The proof sketch by Lurie uses Morse theory extensively, and has since been formalised and ext... | https://mathoverflow.net/users/13767 | Is there a PL, or topological, bordism hypothesis? | This is addressed in Remark 2.4.30 of [Jacob's paper](http://www.math.harvard.edu/~lurie/papers/cobordism.pdf). The PL case has a very nice description but the topological case does not. In particular, there's no difference between *framed* bordisms in the PL and smooth case. So the framed part of the cobordism hypothe... | 11 | https://mathoverflow.net/users/22 | 306215 | 133,495 |
https://mathoverflow.net/questions/306213 | 10 | The transfinite *tower* of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G\_{\alpha+1}=Aut(G\_{\alpha})$ for each ordinal $\alpha$ and the direct limit is taken at the limit stages:
$G\rightarrow Aut(G)\rightarrow Aut(Aut(G))\rightarrow\cdots\rightarrow G\_{\... | https://mathoverflow.net/users/82843 | The Tall Tale of Terminating Transfinite Towers | Here is a relevant paper,
>
> *Kaplan, Itay; Shelah, Saharon*, [**The automorphism tower of a centerless group without choice**](http://dx.doi.org/10.1007/s00153-009-0154-2), Arch. Math. Logic 48, No. 8, 799-815 (2009). [ZBL1192.03026](https://zbmath.org/?q=an:1192.03026).
>
>
>
At least for a centerless group... | 12 | https://mathoverflow.net/users/7206 | 306217 | 133,497 |
https://mathoverflow.net/questions/304176 | 3 | Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1\_Y$. Assume that there exists a map $h:X\to Y$ which induces an isomorphism on fundamental groups, where $X$ is a finite wedge of circles.
Is $\pi\_2 (M\_{h},X)$ a projective (or ... | https://mathoverflow.net/users/114580 | A projective (or free) $\mathbb{Z}\pi_1$-module | Based on your comments to this [answer](https://mathoverflow.net/a/303345/94123), you seem to be in the situation where $\pi\_2(Z)$ is a free $\mathbb{Z}\pi\_1(X)$-module, and where the map $\pi\_1(X)\to \pi\_1(Y)$ is an iso. Under those assumptions, the answer should be a consequence of the following diagram, coming f... | 2 | https://mathoverflow.net/users/94123 | 306228 | 133,500 |
https://mathoverflow.net/questions/306110 | 3 | I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen.
<https://projecteuclid.org/euclid.jdg/1090349447>
I am wondering what versions of comparison principle for viscosity solution was used in Lemma 3.4 to derive the sup n... | https://mathoverflow.net/users/87922 | Comparison principle for viscosity solution | The two conditions in the user's guide are (3.13) and (3.14). The latter holds for your equation, but not the former, which is based on the existence of a zeroth order term $\gamma u$ in the PDE. Assumption (3.13) is only used to perturb a sub (or super) solution into a strict sub (or super) solution, which is the esse... | 4 | https://mathoverflow.net/users/18406 | 306232 | 133,501 |
https://mathoverflow.net/questions/306204 | 0 | I improve [my previous question](https://mathoverflow.net/questions/304849/combination-power-elementary-symmetric-polynomial-inequality). Because this conjecture is exactly natural development of [A Muirhead Like Inequality](https://mathoverflow.net/questions/303013/a-muirhead-like-inequality) and [Muirhead's Inequalit... | https://mathoverflow.net/users/122662 | A symmetric polynomial inequality | Both inequalities are true and can be deduced from their $m=n$ special cases.
**Inequality 1** You can prove that for any choice of $\vec{p}=(p\_1,p\_2,\dots,p\_m)$ we have
$$\sum\_{\text{sym}}x\_{i\_1}^{a\_{p\_1}}\cdots x\_{i\_m}^{a\_{p\_m}}\le \sum\_{\text{sym}}y\_{i\_1}^{a\_{p\_1}}\cdots y\_{i\_m}^{a\_{p\_m}}.$$
T... | 6 | https://mathoverflow.net/users/2384 | 306239 | 133,504 |
https://mathoverflow.net/questions/306236 | 0 | Let $M$ and $N$ are two von Neumann algebras such that their preduals $M\_{∗}$ and $N\_{∗}$ are isomorphic in the sense of Banach spaces, does it imply M and N are $∗$-isomorphic or not??
| https://mathoverflow.net/users/125816 | Isomorphism of preduals implies isomorphism of the $W^*$-algebras or not? | No. Take $M={\bf C}^{\oplus 4}$ (i.e. $\ell\_\infty$ on a 4-element set) and take $N=M\_2({\bf C})$.
Things should get more interesting if you require stricter bounds on the norm of the Banach space isomorphism between the preduals. For instance, isometric isomorphism of the preduals as Banach spaces immediately impl... | 7 | https://mathoverflow.net/users/763 | 306249 | 133,508 |
https://mathoverflow.net/questions/306241 | 6 | First, some quick notation: for any series $\sum\_{n=1}^\infty a\_n$ whose terms are positive real numbers, and for any subset $M = \{m\_1, m\_2,...\} \subseteq \mathbb{N}$, we write $\sum\_M a\_n$ to mean $a\_{m\_1} + a\_{m\_2} + ....$.
Then, for each series, one can associate a topology on $\mathbb{N}$ by declaring... | https://mathoverflow.net/users/1708 | Reference request: A collection of topologies on $\mathbb{N}$ formed via series | What you've described as a topology, is actually a family of analytic P-ideals on $P(\mathbb{N})$ called the "summable ideals." They are studied for different reasons, and the general search terms to use are "summable ideal" or more generally "analytic P-ideal".
| 4 | https://mathoverflow.net/users/8843 | 306254 | 133,510 |
https://mathoverflow.net/questions/306270 | 2 | We have a group of $n$ people who must make a journey of length $d$. They are to start together, and their goal is to arrive at the destination at same time. They have a single bicycle, which they ride in turns. Each time a rider dismounts he leaves the bike by the side of the road, and walks on, while one of the other... | https://mathoverflow.net/users/8628 | Is this cycling problem computable? | I assume that moving backwards or stopping is forbidden, otherwise the answer is "Yes." for trivial reasons.
Clearly, the time it takes the $i$-th person to reach the end of the track depends only on the
distance $l\_i$ that $i$-th person has spent riding the bike. Moreover, this time is
$t\_i := \dfrac{l\_i}{b\_i}... | 3 | https://mathoverflow.net/users/126017 | 306276 | 133,516 |
https://mathoverflow.net/questions/306179 | 1 | For fixed $m = 0, 1, 2, ...$
$$f\_m(k) = \prod\_{j=1}^{m}(k+j).$$
Some examples of $f\_m(k)$ are as following:
$$f\_0(k) = 1, \quad f\_1(k) = (k+1), \quad f\_2(k) = (k+1)(k+2).$$
The $s\_m(n)$ is defined as following:
$$s\_m(n) = \sin\left(\frac{t}{2}\right)\sum\_{k=0}^nf\_m(k)\sin(k+0.5)t,\qquad t\in[0,\pi].$$
The... | https://mathoverflow.net/users/126661 | Proof of an inequality $s_m(n) \le f_m(n)$ | For the people who are interested, inspired from [this answer](https://mathoverflow.net/a/306131/126661),
$$
\begin{align}
s\_m(n)
&= \sin\left(\frac{t}{2}\right)\sum\_{k=0}^nf\_m(k)\sin(k+0.5)t\\
&=\frac{1}{2}\sum\_{k=0}^nf\_m(k)2\sin(k+0.5)t\sin\left(\frac{t}{2}\right)\\
&=\frac{1}{2}\sum\_{k=0}^nf\_m(k)[\cos kt -\c... | 1 | https://mathoverflow.net/users/126661 | 306282 | 133,518 |
https://mathoverflow.net/questions/303298 | 8 | While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum.
Here is a special case. Let $G$ be a group, with $H$ a subgroup of index $2$. Choose $s \in G$ which is not in $H$. For $(\pi,V)$ a representation of $H$, define a repre... | https://mathoverflow.net/users/38145 | Reference request: tensor induction | An introduction to tensor induction is given in §13 of
*Curtis, Charles W.; Reiner, Irving*, Methods of representation theory with applications to finite groups and orders. Volume 1., Wiley Classics Library; New York et al.: John Wiley & Sons. (1990). [ZBL0698.20001](https://zbmath.org/?q=an:0698.20001).
Some pape... | 7 | https://mathoverflow.net/users/10266 | 306288 | 133,522 |
https://mathoverflow.net/questions/306302 | 0 | Assume that $A = U \* S$ for $U$ orthogonal and $S$ diagonal, ordered and positive.
If I only know $A$, is it possible to obtain $U$ and $S$?
My first guess would be taking the singular value decomposition, since $U \* D \* I = A$ by construction, but, due to the non-uniqueness of the SVD, I don't know how to enfor... | https://mathoverflow.net/users/126743 | How to decompose a matrix into its orthogonal and diagonal parts (assuming it has that form)? | To turn Darsh Rajan's comment into an answer:
if $A = US$ with $U$ orthogonal, $S$ diagonal, then $A^TA = S^TU^TUS = S^2$, so we know $S^2$ (and this gives us a condition on which matrices can be expressed in this way). If we further assume $S$ is positive, then we can find $S$ uniquely, and so find $U = AS^{-1}$.
| 1 | https://mathoverflow.net/users/44191 | 306316 | 133,532 |
https://mathoverflow.net/questions/306324 | 3 | In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively.
This notation has been used by several other authors including Hovey in Model Categories and Dwyer and Spalinski in Homotopy Theories and model categories.
Personally ... | https://mathoverflow.net/users/117088 | Best notation for fibrant/cofibrant replacement | Quillen's notations $Q$ and $R$ are by far the most commonly used in my experience (apart from ad hoc constructions like "let $A' \to A$ be a cofibrant replacement"). In general it is a good idea to follow standard notation unless you have a very compelling reason to deviate. The target audience for your paper will oft... | 5 | https://mathoverflow.net/users/1310 | 306330 | 133,539 |
https://mathoverflow.net/questions/306333 | 11 | In category theory, a notion of monoidal category in which every sequence $X\_1, \ldots , X\_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 of Tom Leinster's [book](https://arxiv.org/pdf/math/0305049.pdf) for more details).
Going down one category level, ... | https://mathoverflow.net/users/5690 | Unbiased Hopf algebras | I'll just consider a finite-dimensional bicommutative Hopf algebra $H$ over a field $k$.
* For every map $A\xrightarrow{p} B$ of finite sets, we have maps $\mu\_p\colon H^{\otimes A}\to H^{\otimes B}$ and $\psi\_p\colon H^{\otimes B}\to H^{\otimes A}$
* These have the obvious kind of functoriality for composites $A\... | 4 | https://mathoverflow.net/users/10366 | 306336 | 133,542 |
https://mathoverflow.net/questions/306291 | 4 | Let $P\_1$ be a set of 4 points in the Euclidean plane. Formally, $P\_1$ determines a set $L\_1$ of 6 lines, which then determine only 3 points not already in $P\_1.$ Let $P\_2$ be the set of 7 points thus far determined. Formally, $P\_2$ determines only 3 lines not already in $L\_1.$ Let $L\_2$ be the set of 9 lines t... | https://mathoverflow.net/users/61426 | Counting points and lines in a plane | The question is discussed in Cooper, Joshua; Walters, Mark; Iterated point-line configurations grow doubly-exponentially, Discrete Comput. Geom. 43 (2010), no. 3, 554–562, MR2587837 (2011f:51016). I quote from the review by Tamas Szonyi:
Consider a quadrangle as the starting configuration (stage one). In each stage ... | 7 | https://mathoverflow.net/users/3684 | 306339 | 133,544 |
https://mathoverflow.net/questions/298986 | 0 | Is there a method for solving the following system of generalized Abel's integral equation:?
$(x^2 -1)\int\_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int\_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt =g\_1 (x),\\ x^3 \int\_0^x \frac{u(t)}{(x-t)^{\frac{1}{4}}}\; dt + (1-x)\int\_0^x \frac{v(t)}{(x-t)^{\frac{1}{5}}}\;... | https://mathoverflow.net/users/114476 | A system of generalized Abel's integral equation | The system in question is solved this article: *A new operational method to solve Abel’s and generalized Abel’s integral equations*.[see here](https://www.researchgate.net/publication/319910584_A_new_operational_method_to_solve_Abel's_and_generalized_Abel's_integral_equations)
| 0 | https://mathoverflow.net/users/nan | 306340 | 133,545 |
https://mathoverflow.net/questions/306353 | 8 | Elliptic curves have a group structure over the rational points. Why is this impossible for curves having genus greater than 1? I read somewhere that this impossibility is implied by Faltings theorem, which states that the number of rationals is finite. I don't see the implication. The group over the rational points co... | https://mathoverflow.net/users/4274 | Group over algebraic curves having genus greater than 1 | Let $G/k$ be a proper smooth connected non-trivial group variety. For $1 \neq x \in G$, the translation by $x$ has no fixed point, so by the Lefschetz fixed point formula for $\ell$-adic cohomology (using $G$ connected), the Euler characteristic of $G$ is $0$. If $G/k$ is $1$-dimensional, this implies that $0 = \chi(G)... | 18 | https://mathoverflow.net/users/nan | 306357 | 133,553 |
https://mathoverflow.net/questions/306363 | 5 | Let $G$ be a finite group of order $240$.
If $G\cong C\_4\times A\_5$ or $C\_2\times C\_2\times A\_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are
$
[1,1,1,1,~3,3,3,3,3,3,3,3, ~4,4,4,4,~5,5,5,5 ].
$
Conversely,
Suppose that $G$ is non-solvable, and the all degrees of irreducible $\mathbb{C}... | https://mathoverflow.net/users/99750 | $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$? | $G$ is non-solvable, so must have $A\_{5}$ as a composition factor (as no other non-Abelian simple group has less than $168$). Hence $F(G)$ can have order at most $4$.
If $G$ has no component, then $F(G) = F^{\ast}(G)$ has order $4$, and is centralized by all elements of order $5$, contrary to $C\_{G}(F^{\ast}(G)) \l... | 8 | https://mathoverflow.net/users/14450 | 306372 | 133,559 |
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