parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/407823 | 40 | Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander duality plus some sanity-checks on the topology of the curve in question, which is a fine approach but does require a bit... | https://mathoverflow.net/users/61829 | Results with short, advanced proofs or long, elementary proofs | The associativity of the group law on an elliptic curve can be proved in an elementary way by explicitly manipulating algebraic expressions, but this is not very enlightening. By using more advanced geometric ideas, one can [prove associativity more conceptually](https://mathoverflow.net/q/6870).
| 43 | https://mathoverflow.net/users/3106 | 407842 | 167,058 |
https://mathoverflow.net/questions/407839 | 2 | I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the behavior of the following function,
$$f(\theta) = \frac{P^\mu\_{\nu}(\cos \theta)}{\sin^\mu \theta}$$
when $\theta\to 0$ and $... | https://mathoverflow.net/users/68232 | How to compute this limit involving the associated Legendre function? | I find it useful to represent the Legendre function in terms of a hypergeometric function, using a formula from [Wikipedia](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials#Generalization_via_hypergeometric_functions),
$$f(\theta)=\frac{ (1+\cos \theta)^{\mu/2} \, \_2F\_1\left(-\nu,\nu+1;1-\mu;\frac{1}{2... | 3 | https://mathoverflow.net/users/11260 | 407845 | 167,059 |
https://mathoverflow.net/questions/407619 | 7 | It is not clear to me the role of the domain and target in the definition of [prederivators](https://webusers.imj-prg.fr/%7Egeorges.maltsiniotis/ps/m.pdf).
For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself.
Sometimes $\mathit{Dia}$ comes with some [conditions](ht... | https://mathoverflow.net/users/429204 | Indexing categories of derivators | The idea is to have more freedom of expression. No one wants to work with a fixed $Dia$ for ever. In particular, $Dia$ is made more general to allow us to change $Dia$ at will, according to our needs, including to make weird technical choices in some proof (for instance, if you want that (pre)derivators collectively fo... | 5 | https://mathoverflow.net/users/1017 | 407847 | 167,061 |
https://mathoverflow.net/questions/407542 | 1 | Note: This is not intended to be a research level question, but concerns graduate level material.
**Theorem.** *The opposite $\Delta^\mathrm{op}$ of the simplex category $\Delta^\mathrm{op}$ (as usually defined via nondecreasing maps) can be [presented](https://ncatlab.org/nlab/show/presentation+of+a+category+by+gene... | https://mathoverflow.net/users/442261 | Proving that the simplex category is generated by the face and generacy maps | This is not an answer to the question as stated, but an elaboration on my comment (the OP asked me to clarify, but this would be too long for a comment).
Let me start by pointing out that this theorem becomes really easy to prove once one observes that the composition of $\delta$'s is injective, and that of $\sigma$'... | 5 | https://mathoverflow.net/users/102343 | 407858 | 167,065 |
https://mathoverflow.net/questions/407859 | 8 | In [The Fourier Transform of the quartic Gaussian $\exp(-Ax^4)$: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to $\exp(-Ax^{2n})$](https://doi.org/10.1016/j.amc.2014.05.001), Boyd derives the asymptotic of the integral
$$\mathcal{A}(x) = \int\_{-\infty}^\infty... | https://mathoverflow.net/users/152473 | Is there a real-analytic way to derive the asymptotics of $\int_{-\infty}^\infty e^{ikx} e^{-k^4}\,dk$ as $|x|\to\infty$? | A differential equation for ${\cal A} (x) $ can be obtained as follows,
$$
\frac{d^3}{dx^3 } {\cal A} (x) = \int\_{-\infty }^{\infty } dk\, (-ik^3 ) e^{ikx} e^{-k^4 } = \frac{x}{4} \int\_{-\infty }^{\infty } dk\, e^{ikx} e^{-k^4 } = \frac{x}{4} {\cal A} (x)
$$
where integration by parts has been used in the second equa... | 12 | https://mathoverflow.net/users/134299 | 407861 | 167,066 |
https://mathoverflow.net/questions/407867 | 0 | Let $u \in C^\infty\_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\langle u, \varphi \rangle = \frac{1}{\lambda} \langle u, (-\Delta)^s\varphi \rangle = \frac{1}{\lambda} \langle(-\Delta)^s... | https://mathoverflow.net/users/122620 | Iterated integrations by parts using the fractional Laplacian | No, we cannot.
Formally, $\varphi$ is the eigenfunction of the unbounded operator $L\_s$ on $L^2(\Omega)$, defined initially by
$$ L\_s u(x) = (-\Delta)^s u(x) \qquad \text{for } x \in \Omega , $$
where $u \in C\_c^\infty(\Omega)$ (and it is understood that $u(x) = 0$ for $x \notin \Omega$), and then extended to an a... | 0 | https://mathoverflow.net/users/108637 | 407874 | 167,069 |
https://mathoverflow.net/questions/407290 | 2 | $\DeclareMathOperator\wt{wt}$Let $\wt(n)$ be [A000120](https://oeis.org/A000120), number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or... | https://mathoverflow.net/users/231922 | Sequence that sums up to INVERTi transform applied to the ordered Bell numbers | First, we let $P(j,k):=1+f(\lfloor\tfrac{j}{2^k}\rfloor+1)$ and sum over $n$ of fixed weight $\ell:=\mathrm{wt}(n)$ (like in [this answer](https://mathoverflow.net/q/406817)):
\begin{split}
s(n) &= \sum\_{\ell=0}^n \sum\_{t\_1 + \dots + t\_\ell \leq n-\ell}
\sum\_{j=0}^{2^\ell-1} (-1)^{\ell-\mathrm{wt}(j)} \prod\_{k=0}... | 2 | https://mathoverflow.net/users/7076 | 407881 | 167,070 |
https://mathoverflow.net/questions/280223 | 6 | Given an augmented simplicial object $d\_\bullet:X\_\bullet \to \Delta X\_{-1}$, suppose there's a simplicial map $s\_\bullet :\Delta X\_{-1}\to X\_\bullet$ making $d\_\bullet$ a deformation retract, i.e such that $d\_\bullet$ is both a retract and a homotopy-section of $s\_\bullet$. This is equivalent to providing the... | https://mathoverflow.net/users/69037 | Why are simplicial objects monadic over split (contractible) simplicial objects? | Coming across this old question and noticing it still unanswered, here’s an attempt at the “geometric” answer OP wanted.
First note that a split simplicial complex isn’t in general contractible — it’s just *1-truncated*, i.e. each *connected component* is contractible. Concretely, the augmentation indexes the connect... | 3 | https://mathoverflow.net/users/2273 | 407887 | 167,072 |
https://mathoverflow.net/questions/407868 | 0 | There is an array $a\_1,\dotsc,a\_n$ whose elements are pairwise distinct. We define a reverse order pair to be an ordered pair $(a\_i,a\_j)$ such that $i < j$ and $a\_i > a\_j$. Consider the total number of reverse order pairs $N$.
Assume the array is permuted uniformly and randomly, it is well known that $E[N] = \f... | https://mathoverflow.net/users/117620 | The distribution of number of reverse order pairs in a randomly permuted array | First, a quick note on terminology: the standard term for a "reverse order pair" is an *inversion*. Knowing this makes it easier to search for the answer:
The generating function for the number of permutations with $k$ inversions is the $q$-factorial $[n]\_q!$ as shown e.g. [here](https://www.tvhoang.com/articles/202... | 2 | https://mathoverflow.net/users/46140 | 407889 | 167,073 |
https://mathoverflow.net/questions/407773 | 1 | **ASSUMPTION 1**: there exists a continuous random vector $(X,Y,Z)$ such that
$$
\begin{cases}
p\_1=\Pr(X\geq 0, Z\geq 0)\\
p\_2=\Pr(Y\geq 0, Z< 0)\\
p\_3=\Pr(X< 0, Y<0)\\
\end{cases}
$$
where $(p\_1,p\_2,p\_3)\in [0,1]^3$ and $p\_1+p\_2+p\_3=1$. Further, the marginal distribution of each of $X,Y,Z$ are symmetric aroun... | https://mathoverflow.net/users/nan | Construct a random vector as a function of another random vector | I will begin with a reformulation of your question which makes it not only more symmetric, by also (at least for me) more natural and interesting. I will pass from your variables $(W,H,Q)$ to new variables $(X,Y,Z)$ (which are **not** your original $X,Y,Z$) by putting $X=W, Y=-Q, Z=-H$. Then your condition (3) becomes ... | 3 | https://mathoverflow.net/users/8588 | 407890 | 167,074 |
https://mathoverflow.net/questions/407882 | 7 | I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction and model problems (for example, Oda’s *A Note on Ramification of the Galois Representation on the Fundamental Group of an... | https://mathoverflow.net/users/170335 | Reference request: the geometry of vanishing cycle | Beyond the definitions in section 9.2 of Lei Fu's book, you also find a finiteness result for $R^q\psi\mathcal{F}$ for a finite type $S$-scheme and constructible $\mathcal{F}$ (9.4.1). Nearby cycles, as well as their finiteness feature in the proof of finiteness for $R^q f\_{\ast}\mathcal{F}$, where $f$ is a morphism o... | 6 | https://mathoverflow.net/users/69401 | 407891 | 167,075 |
https://mathoverflow.net/questions/407879 | 3 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More precisely, what are examples of representations of $\Spin(n)$ that do not factor through a representation of $\SO(n)$? C... | https://mathoverflow.net/users/176218 | Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$ | $\def\Spin{\text{Spin}}$The cover $\Spin(n) \to SO(n)$ is $2$ to $1$, the nontrivial element kernel is a central element of $\Spin(n)$ which I'll call $z$. Since $z$ is central, it acts on any irrep of $\Spin(n)$ by $\pm 1$; if $z$ acts by $1$, then the irrep factors through $SO(n)$, if $z$ acts by $-1$, then it does n... | 8 | https://mathoverflow.net/users/297 | 407895 | 167,076 |
https://mathoverflow.net/questions/407904 | 5 | Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
* The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^\prime}^{\otimes 2} \simeq \mathcal{L}$ is equivalent to the existence of a simple cyclic cover $Y \rightarrow X$ of... | https://mathoverflow.net/users/158892 | Square root of a line bundle up to a finite surjective morphism | Assume $\mathcal{L}$ is associated with an effective Cartier divisor $D$. Let $D'$ be another Cartier divisor such that $D + D'$ is divisible by 2 in $\mathrm{Pic}(X)$. Let
$$
g \colon X' \to X
$$
be the double covering branched at $D + D'$. Then $g^{-1}(D) = 2R$ for a Cartier divisor $R$ on $X'$, hence $g^\*\mathcal{L... | 7 | https://mathoverflow.net/users/4428 | 407908 | 167,079 |
https://mathoverflow.net/questions/407910 | 0 | I ask the question because of the following statement found in Mark Burgin's paper, "Algorithmic complexity of recursive and inductive algorithms", *Theoretical Computer Science* 317 (2004) 31-60 (pg. 34):
>
> It is usually assumed that any finite set is recursively computable and even decidable. When a finite set ... | https://mathoverflow.net/users/20597 | Can finite sets be non-c.e. depending on how they are presented? | Burgin seems to conflate the notion of computability (existence of an algorithm) with a stronger notion such as our knowing an algorithm or existence of a proof that a particular algorithm agrees with the given presentation.
In his example, the finite set $X$ is computable. Indeed, any finite set is computable by a t... | 8 | https://mathoverflow.net/users/6794 | 407912 | 167,081 |
https://mathoverflow.net/questions/407630 | 1 | In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $P\_0$ be a minimal $F$-parabolic subgroup and $M\_0$ a Levi factor of $P\_0$ defined over $F$. Let $W := \mathrm{N}\_G(... | https://mathoverflow.net/users/125617 | Can we compare $K$-spherical representations of $p$-adic groups for varying special maximal subgroups $K$? | If I'm not mistaken, $\pi\_{K,\chi}$ and $\pi\_{K',\chi}$ can be different.
Consider a bipartite bi-regular tree, with degrees $q\_1+1<q\_2+1$. There are $p$-adic unitary groups that their building in such a tree with $q\_2=q\_1^3$, see for example <https://arxiv.org/abs/1005.3504>.
Then there are two maximal compa... | 3 | https://mathoverflow.net/users/450073 | 407919 | 167,082 |
https://mathoverflow.net/questions/407924 | 1 | Let $B(n, k)$ be the number of $k$-rank subspaces of $\mathbb{Z}\_2^n$. One can establish $B(n, k) = {n \choose k}\_2 = \frac{F(n)}{F(k)F(n - k)}$, where $F(x) = \displaystyle\prod\_{i = 1}^x (2^i - 1)$. This expression clearly implies that $B(n, k)$ is odd for all $k, n \in \mathbb{N}\_0$ such that $k \leq n$.
One c... | https://mathoverflow.net/users/106512 | Number of $k$-rank subspaces of $\mathbb{Z}_2^n$ is odd: easy proof? | Consider the involution which complements the ''free'' entries in the reduced row echelon form of the matrix representing a subspace. That is, we leave the pivots as well as things that are necessarily zero unchanged while changing everything else. This involution has a single fixed point, e.g.
$$\begin{bmatrix} 0 & 0 ... | 8 | https://mathoverflow.net/users/51668 | 407927 | 167,088 |
https://mathoverflow.net/questions/407826 | 7 | Let $J\_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J\_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that this isomorphism can be easily achieved by the Hodge-theoretical methods, but they do not give any reference.
Maybe some... | https://mathoverflow.net/users/150116 | Relation between the cohomology group of a curve and the cohomology group of its jacobian | $\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-fold $X$: The Albanese variety $\Alb(X)$ and the Picard variety $\Pic^0(X)$. The Albanese has a natural map $X \to \Alb(X... | 14 | https://mathoverflow.net/users/297 | 407930 | 167,090 |
https://mathoverflow.net/questions/407915 | 6 | As a part of the research with which I am involved, I would like to understand how to compute the effect of the Atkin-Lehner operator/Fricke involution $W\_2 = \begin{pmatrix} 0 & 1 \\ -2 & 0 \end{pmatrix}$ on the $q$-expansion modular forms with $\Gamma\_0(2)$ level structure. I know of the Magma function $\operatorna... | https://mathoverflow.net/users/122371 | Explicit computation of the effect of the Atkin-Lehner operator/Fricke involution's effect on $q$-expansion | In short, there is no simple formula for the $q$-expansion of the transform of a modular form under an Atkin-Lehner operator $W\_Q$, in terms of the $q$-expansion of the original modular form.
The action of $W\_Q$ on modular symbols is simply $\{\alpha,\beta\} \mapsto \{W\_Q \alpha, W\_Q \beta\}$. This is easy to imp... | 6 | https://mathoverflow.net/users/6506 | 407937 | 167,091 |
https://mathoverflow.net/questions/407939 | 4 | Define $A=\sup \limits\_{f} \{m^\*(f[0, 1])\}$, here $f$ are all the functions which satisfy that
$f: [0, 1] \rightarrow [0, 1]$; $\forall x, f(x)-x \in \mathbb{Q}; \forall x, y, x-y \in \mathbb{Q} \rightarrow f(x)=f(y)$.
What is the value of $A$? Obviously, $0<A \leq 1$.
| https://mathoverflow.net/users/345221 | What is the value of this supremum? | Your conditions may be restated as: $f$ induces a choice function for the partition of $[0,1]$ defined by the equivalence $x\sim y$ iff $x-y\in \mathbb Q$. So the image of $f$ is a Vitali set (a choice of representatives for that partition). Therefore $A=1$ because there are Vitali set of outer measure $1$. Here are
co... | 7 | https://mathoverflow.net/users/6101 | 407942 | 167,093 |
https://mathoverflow.net/questions/407918 | 6 | I want to simplify the following expression
$$\sum\_{1\leq a,b\leq n}\mathcal{P}(\gcd(a,b,n)),$$ where $\mathcal{P}(x)$ is the number of partitions of $x$.
It turned out this number is the top betti number of the $C\_n\times C\_n$ covering of the Hilbert scheme of $n$ points of the 2-torus. Simplifying this express... | https://mathoverflow.net/users/111070 | $\sum_{1\leq a,b\leq n}\mathcal{P}(\gcd(a,b,n))$ | Introducing $d:=\gcd(a,b,n)$, we get
$$\sum\_{1\leq a,b\leq n} \mathcal{P}(\gcd(a,b,n)) = \sum\_{d\mid n} \mathcal{P}(d) J\_2(\tfrac{n}d),$$
where $J\_2(\cdot)$ is the [Jordan totient function](https://en.wikipedia.org/wiki/Jordan%27s_totient_function) (see also [OEIS A007434](https://oeis.org/A007434)).
In particula... | 5 | https://mathoverflow.net/users/7076 | 407944 | 167,094 |
https://mathoverflow.net/questions/407940 | 12 | Let $A$ be a subset of $\mathbb{R}^2$ which intersects every straight line in exactly two points.
Is there a such set which is Lebesgue measurable or Borel?
A well-known fact is that there exists such set which is not Lebesgue measurable.
| https://mathoverflow.net/users/345221 | Is there a set that intersects every line twice which is Lebesgue measurable or Borel? | There is such a set which is Lebesgue measurable, and indeed of Lebesgue measure zero. To see this, start with a subset $S$ of $\mathbb R^2$ such that every line intersects it in continuum many points, for instance $C\times\mathbb R\cup\mathbb R\times C$, where $C$ is the Cantor set. Now repeat your favorite transfinit... | 20 | https://mathoverflow.net/users/30186 | 407946 | 167,095 |
https://mathoverflow.net/questions/407943 | 8 | In Voevodsky’s ICM address:
<https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf>
In theorem 4.3 it is claimed that given a symmetric monoidal category $(C, \wedge, 1)$ and an object $ X \in C$ in order for $C[X^{-1}]$ be be symmetric monoidal it is enough fo... | https://mathoverflow.net/users/374433 | Inverting objects in a symmetric monoidal category | To be clear, this claim refers to a very specific construction of $\mathcal{C}[X^{-1}]$, where you copy the construction of the localization of the ring and defines it as the colimits of:
$$ \mathcal{C} \overset{\\_ \otimes X}{\to} \mathcal{C} \overset{\\_ \otimes X}{\to} \mathcal{C} \overset{\\_ \otimes X}{\to} \mat... | 8 | https://mathoverflow.net/users/22131 | 407948 | 167,096 |
https://mathoverflow.net/questions/407872 | 2 | There are well-described methods of generalizing arbitrary functions to matrices in a natural way.
Basically, if $A=PD\_AP^{-1}$ where $D\_A$ is a diagonal matrix, then $f(A)=Pf(D\_A)P^{-1}$, where the function $f$ is applied to the diagonal matrix element-wise.
This is automatized in some CAS systems, such as Math... | https://mathoverflow.net/users/10059 | Is the number of values the sign function can take on a ring ("signedness") of any fundamental importance? Can it be predicted? | Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{R}$-algebra. (This covers split-complex, hyberbolic and a lot of other number "systems".) We want to extend the function $\operatorname{sign}(x)$ to $\mathcal{A}$ in a meaningful way (to be determined) and understand the number of values ... | 2 | https://mathoverflow.net/users/1849 | 407960 | 167,101 |
https://mathoverflow.net/questions/407976 | 1 | Let $\mathcal M$ be the collection of martingle diffusions starting at zero and ending in $\{-1,1\}$. Equivalently, $X\in \mathcal M$ iff there exists a measurable function $a$ s.t. it holds almost surely
$$X\_t=\int\_0^t a(s,X\_s)dW\_s ~\in~ [-1,1],~ \forall t\ge 0 \quad\mbox{and} \quad X\_{\infty}:=\lim\_{t\to\inft... | https://mathoverflow.net/users/261243 | Characterization of martingale diffusions ending in $\{-1,1\}$ | This is an extended comment on the time-homogeneous case, when the coefficient $a(t, x)$ does not depend on $t$, so that it can be written as $a(x)$.
Suppose that $a(\pm 1) = 0$ and that $1/(a(x))^2$ is locally integrable over $(-1, 1)$. We claim that in this case the desired process $X\_t$ exists.
---
Define t... | 2 | https://mathoverflow.net/users/108637 | 407980 | 167,106 |
https://mathoverflow.net/questions/407558 | 2 | Is there a *graph manifold* (<https://en.wikipedia.org/wiki/Graph_manifold>) that doesn't admit an orientation reversing involution? If so, what would be a simple example?
| https://mathoverflow.net/users/13441 | A graph manifold without an orientation reversing involution? | $\mathbb{N}il^3$-manifolds and $\widetilde{SL}$-manifolds are orientable
and do not have orientation-reversing self homotopy equivalences.
This is most easily seen algebraically.
The fundamental group $\Gamma$ of the $S^1$-bundle over the torus $T$ with Euler class a generator of $H^2(T;\mathbb{Z})$ is a central extens... | 6 | https://mathoverflow.net/users/58488 | 407985 | 167,108 |
https://mathoverflow.net/questions/407933 | 0 | Consider the following Sturm-Liouville problem,
$$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$
where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$
I used maple and got the following explicit solution,
$$Y(\theta) = \sin(\theta)^{1/4} \left(c\_1P^\mu\_{\nu}(\cos \theta)... | https://mathoverflow.net/users/68232 | Asymptotic for eigenvalues for the following ode? | I asked Mathematica about the boundary behavior. First, the $P^{1/4}\_{\nu } $ solution: We have, for $\epsilon \searrow 0$,
$$
(\sin \epsilon )^{1/4} P^{1/4}\_{\nu } (\cos \epsilon ) = \frac{2^{1/4} }{\Gamma(3/4)} + O(\epsilon^{2} )
$$
and
\begin{eqnarray\*}
(\sin (\pi -\epsilon ))^{1/4} P^{1/4}\_{\nu } (\cos (\pi -\e... | 1 | https://mathoverflow.net/users/134299 | 407986 | 167,109 |
https://mathoverflow.net/questions/407977 | 1 | Is the Implicit Function Theorem in the following form correct:
Let $V\_1,V\_2,W$ be Banach spaces, and $Ω⊂V\_1×V\_2$ an open subset containing $(x\_0,y\_0)$. Let consider a continuously differentiable map $f:Ω→W$ with $f(x\_0,y\_0)=0$ and s.t. the derivative on the second component
$D\_2f(x\_0,y\_0):V\_2\ni y↦Df(x... | https://mathoverflow.net/users/451228 | Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)? | **No, it is not correct.** Simple counterexamples can be obtained as follows. Let $V\_1=W$ and $V\_2$ be any Banach spaces such that the identity is continuous but not surjective $V\_2\to W$. Then $f$ defined by $(x,y)\mapsto x+y$ is even smooth with $\partial\_2 f(x\_0,y\_0)$ injective $V\_2\to W$ and there cannot be ... | 1 | https://mathoverflow.net/users/12643 | 407992 | 167,111 |
https://mathoverflow.net/questions/407739 | 3 | Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X\_n\}\_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{E}[d(X\_1,x)^p]<\infty$ for some $p>1$ and some $x\in \mathcal{X}$.
... | https://mathoverflow.net/users/36886 | Wasserstein-type concentration inequalities for empirical measures on polish spaces | Yes, there are various results available in more general settings. The typical route would be to combine an upper bound on the expected distance between the law and the empirical measure (like Theorem 1.1 in [Boissard and Le Gouic](http://www.numdam.org/item/10.1214/12-AIHP517.pdf)) with a concentration estimate around... | 3 | https://mathoverflow.net/users/100163 | 407994 | 167,113 |
https://mathoverflow.net/questions/407987 | 3 | Let $A$ be a $n\times m$ random matrix, whose elements $a\_{ij}$ are independent standard Gaussian random variables.
I am interested in the case $n=\alpha N\,$, $\,m=(1-\alpha)N$ for $\alpha\in(0,1)$ fixed and $N\to\infty$.
Denote by $\sigma\_\max(N)$ the largest singular value of $\frac{1}{\sqrt{N}}A$, that is the s... | https://mathoverflow.net/users/58793 | Top singular value of large random matrices: concentration results | This largest singular value is the norm of the matrix. You can use a net argument to show that there is a $C$ so that $$\mathbb{P}(\| A \|\_{op} \geq C\sqrt{N}(\sqrt{\alpha} + \sqrt{1 - \alpha} + t)) \leq 2 e^{-Nt^2}$$
for all $\alpha, t$. For a reference, this appears as Theorem 4.4.5 in Vershynin's [High Dimensional ... | 1 | https://mathoverflow.net/users/69870 | 408014 | 167,116 |
https://mathoverflow.net/questions/408012 | 5 | Let $\boldsymbol{V}\_{1},\dots,\boldsymbol{V}\_{n}\in\mathbb{R}^{d\times m}$ be $n$ “tall” matrices (where $d\ge m$) with orthonormal columns.
And let $\boldsymbol{P}\_{1},\dots,\boldsymbol{P}\_{n}\in\mathbb{R}^{d\times d}$ be the orthogonal projection matrices defined as $\boldsymbol{P}\_{i}=\boldsymbol{V}\_{i}\bold... | https://mathoverflow.net/users/100796 | Can the concatenation of projection operators be nilpotent with an index k>=3? | I think your example is easily generalisable for any index. For example, let
$$
Q\_1=P\_1\oplus(1),
Q\_2=P\_2\oplus(1),
Q\_3=P\_3\oplus(1)=(1)\oplus P\_1,
Q\_4=(1)\oplus P\_2,
Q\_5=(1)\oplus P\_3.
$$
Then $Q\_5Q\_4Q\_3Q\_2Q\_1$ is nilpotent of index 3, and I think that a similar construction with $2n-1$ matrices of s... | 5 | https://mathoverflow.net/users/1306 | 408016 | 167,117 |
https://mathoverflow.net/questions/407594 | 12 | Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix
$$
M = \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix}
$$
has eigenvalues with positive rea... | https://mathoverflow.net/users/173967 | Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts | A simple brute force method worked (even though I'm not happy with this).
Let $\zeta=\xi+i\eta$ be a non-positive eigenvalue of $M$ and
$\left[\begin{matrix} x & y & z \end{matrix}\right]^T$ be
a corresponding eigenvector.
This gives equations
\begin{align}
Ax + By \qquad &= \zeta x \\
B^Tx + Cy - B^Tz &= \zeta y\\
... | 8 | https://mathoverflow.net/users/7591 | 408017 | 167,118 |
https://mathoverflow.net/questions/408010 | 0 | My problem is to prove
$$
\left(\cos\frac{m}{2n}\pi\right)^{4n}\ge \left(\cos\frac{m+1}{2n}\pi\right)^{2n-1}
\left(\cos\frac{m-1}{2n}\pi\right)^{2n+1}
$$
holds for any positive integer $n$ and $m = 1, 2, \dots, n-1$.
| https://mathoverflow.net/users/451922 | An inequality involving the power of cosine | A bit more general inequality is
\begin{equation\*}
g(h):=g\_x(h):=\ln\frac{\cos ^{1+h}(\pi (x-h)) \cos ^{1-h}(\pi (x+h))}{\cos ^2(\pi x) }\le0 \tag{1}
\end{equation\*}
for $x\in(0,1/2)$ and $h\in(0,\min(x,1/2-x))$.
Suppose that indeed $x\in(0,1/2)$ and $h\in(0,\min(x,1/2-x))$, so that $h\in(0,1/4)$.
Note that
\begi... | 5 | https://mathoverflow.net/users/36721 | 408018 | 167,119 |
https://mathoverflow.net/questions/408011 | 8 | Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$?
Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic to a topological group. Also, as a consequence of a recent paper by Arhangelskii and van Mill (Covering Tychonoff cubes... | https://mathoverflow.net/users/128723 | Topological group locally homeomorphic to the Hilbert cube | The answer is no.
Since the Hilbert cube is compact and locally contractible, such a group would be a locally contractible locally compact group. And every locally contractible locally compact group is Lie (i.e., locally homeomorphic to $\mathbf{R}^d$ for some integer $d<\infty$).
---
For a reference
>
> Szen... | 10 | https://mathoverflow.net/users/14094 | 408026 | 167,121 |
https://mathoverflow.net/questions/408034 | 1 | Let $\bar x \in \mathbb R$. Is there a cut-off function such that $\phi\_\epsilon \in C^\infty(\mathbb R)$, $0 \le \phi \le 1$, and
$$\phi\_\epsilon(x) = \begin{cases}
1 &\text{ if } |x - \bar x| \ge \epsilon\\\
0 &\text{ if }|x-\bar x|\le \epsilon/2
\end{cases}
$$
and
$\phi' \le c\_\epsilon \phi$?
| https://mathoverflow.net/users/122620 | Construct suitable cutoff function | The answer is no.
Indeed, let $f:=\phi=\phi\_\epsilon$. Let $a:=\sup\{x\colon f(x)=0\}$. Then $a$ is real, $f(a)=0$ and $f>0$ on the interval $(a,\infty)$. Without loss of generality, $a=0$, so that $f(0)=0$ and $f>0$ on the interval $(0,\infty)$.
Suppose now that for some real $c>0$ we have $f'\le cf$. Then $(\ln ... | 0 | https://mathoverflow.net/users/36721 | 408038 | 167,122 |
https://mathoverflow.net/questions/408035 | 9 | I know the following facts: $\text{SL}\_2(\mathbb{Z})$ is generated by everyone's favorite matrices
\begin{equation\*}
S =
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\end{equation\*}
and
\begin{equation\*}
T =
\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}
\end{equation\*}
and $\text{SL}\_2(\mathbb{Z})$ acts trans... | https://mathoverflow.net/users/419791 | $\text{SL}_2(\mathbb{Z})$ and continued fractions? | 1. See Remark 2.2 [here](https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf).
2. If you expand $p/q$ into a continued fraction then the successive convergents, as columns of a $2 \times 2$ matrix, have determinant $\pm 1$. Provided $p/q$ is in reduced form and $q > 0$, the last convergent $p\_n/q\_n$ in the ... | 13 | https://mathoverflow.net/users/3272 | 408044 | 167,124 |
https://mathoverflow.net/questions/408043 | 1 | Let $Z\_n=\sum\_{k=1}^n a\_k X\_k$ with $(a\_k)$ a strictly decreasing sequence of positive real numbers that tend to zero. The random variables $X\_k$ are independent and satisfy $P(X\_k=1) =p\_k, P(X\_k=-1)=1-p\_k$. Here $p\_k=\frac{1}{2}$. The normalized series is defined as $Z^\*\_n=(Z\_n-\mbox{E}[Z\_n])/\sqrt{\mbo... | https://mathoverflow.net/users/140356 | Generalized random harmonic series | By the [Berry--Esseen inequality](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands), the limit distribution of $Z\_n^\*$ is the standard normal distribution if $a\_k=1/k^{1/2}$.
---
If $a\_k=1/2^k$, then the limit distribution of $Z\_n^\*$ is uniform on the interval ... | 2 | https://mathoverflow.net/users/36721 | 408046 | 167,125 |
https://mathoverflow.net/questions/408033 | 4 | Let $[n]\_q=\frac{1-q^n}{1-q}$ with $[0]\_q=0$. Recall the $q$-factorials $[n]\_q!=[1]\_q[2]\_q\cdots[n]\_q$ (with $[0]\_q!=1$) and the $q$-binomials
$$\binom{n}k\_q=\frac{[n]\_q!}{[k]\_q!\,[n-k]\_q!}.$$
Now, consider the polynomials
$$W\_n(q):=\frac{1-q^{3n}}{1-q^{2n}}\binom{2n}n\_q.$$
**Examples.** $W\_1(q)=q^2+q+1... | https://mathoverflow.net/users/66131 | A (mild?) question on the number of monomials | Using the fact that $1-q^{3n}=(1-q^{2n})+q^{2n}(1-q^n)$, we can write
$$W\_n(q)=\binom{2n}{n}\_q+q^{2n}\binom{2n-1}{n-1}\_q$$
and then the result follows from the fact that the degree of $W\_n(q)$ is $n^2+n$, together with the fact that $q$-binomial coefficients are polynomials in $q$ with positive coefficients (unimod... | 13 | https://mathoverflow.net/users/2384 | 408049 | 167,126 |
https://mathoverflow.net/questions/408031 | 4 | The following is motivated by a (now-deleted) MSE-question by @aglearner.
Suppose that $X\subset {\mathbb C}^n$ is an affine subvariety, equipped with the classical (Euclidean) topology. Consider the group $G= {\mathbb C}^\times$, and suppose that $G\times X\to X$ is an algebraic action. Assume, in addition, that eac... | https://mathoverflow.net/users/39654 | Group actions on affine varieties with closed orbits | I think the answer to Question 1 is yes, since under your hypothesis the set-theoretical quotient *is* essentially the GIT quotient. This follows from the fact that $G$-invariant regular functions on $X$ separate orbits (when these are closed).
There is an analytic proof of this fact. Let $V$ and $W$ be two disjoint ... | 3 | https://mathoverflow.net/users/173096 | 408055 | 167,127 |
https://mathoverflow.net/questions/408050 | 4 | I have been reading Felix E. Browder's [Convergence Theorems for Sequence of Nonlinear Operators in Banach Space](https://link.springer.com/article/10.1007/BF01109805) and I was hoping I could find answers to a couple of questions I have about the paper. Consider Lemma 6.
>
> Let $F$ be a closed convex subspace of ... | https://mathoverflow.net/users/nan | On some convergence theorems by Felix E. Browder (1967) | **Concerning Lemma 6:** Clearly, this lemma is false if $F=\emptyset$ and, say, $u\_n=nu$ for some nonzero $u\in H$. Assume therefore that $F\ne\emptyset$.
The inequality $\|u\_n\|\le\|u\_1\|+\|f\|$ will not always hold in this setting -- consider e.g. the case when $u\_1=0$ and $u\_n=2f\ne0$ for $n\ge2$.
However, ... | 1 | https://mathoverflow.net/users/36721 | 408060 | 167,131 |
https://mathoverflow.net/questions/408059 | 2 | Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation
$$\partial\_t u -\Delta\_gu=0 \quad \text{on $(0,T)\times M$},$$
subject to initial data $f$ at times $t=0$. Let the map
$$ G: L^2(M)... | https://mathoverflow.net/users/50438 | Compactness for initial-to-final map for heat equation | Parabolic regularity show that $u$ is regular for all positive times; in particular $u(t,\cdot) \in W^{1,2}(M)$ for all $t > 0$. Interior parabolic estimates additionally show that there is a constant $C = C(g,T) > 0$ so that
\begin{equation}
\lvert u(t,\cdot) \rvert\_{W^{1,2}} \leq C \lvert f \rvert\_{L^2}
\quad \text... | 6 | https://mathoverflow.net/users/103792 | 408062 | 167,132 |
https://mathoverflow.net/questions/408008 | 4 | Here is a statement having a proof that involved the CFSG.
Let $p$ be a prime, and $S$ be a nonabelian finite simple group such that $S$ is isomorphic to a subgroup of $S\_p$ with $p\mid |S|$. Then $\mathrm{Out}(S)$ is a $p'$-group (i.e. $p\nmid |\mathrm{Out}(S)|$).
**Question** Is there a CFSG-free proof for this ... | https://mathoverflow.net/users/44312 | Outer automorphism of a finite simple group which is isomorphic to a subgroup of $S_p$ | Let $p$ be a prime, and let $G$ be a non-abelian simple group with $p \mid |G|$. Suppose that $G \leq S\_p$ and that $G$ is transitive.
By a theorem of Burnside a non-solvable transitive group of prime degree is $2$-transitive, so $G$ is $2$-transitive.
Let $B \leq G$ be such that $|B| = p$. Then $G = AB$, where $A... | 8 | https://mathoverflow.net/users/38068 | 408068 | 167,134 |
https://mathoverflow.net/questions/407707 | 1 | In [this book](https://books.google.co.kr/books?id=doLoDwAAQBAJ&pg=PA372&lpg=PA372&dq=every%20graph%20with%20m%20edges%20and%20maximum%20degree%20k%20has%20a%20proper%20(k%20%2B%201)-%20edge%20coloring%20in%20which%20each%20color%20is%20used%20%E2%8C%88%20m%20k%2B1%20%E2%8C%89%20or%20%E2%8C%8A%20m%20k%2B1%20%E2%8C%8B%2... | https://mathoverflow.net/users/384338 | An equitable edge-coloring of bipartite graphs | There is a proof with a very similar principle than in Petrov's proof.
Take an arbitrary coloring. If the property is not respected, you have a vertex $v$ where one of the colors $c\_1$ appear strictly over $⌈\frac{d(v)}{k}⌉$, and another color $c\_2$ appear less or equal than $⌊\frac{d(v)}{k}⌋$ (or one which appear st... | 1 | https://mathoverflow.net/users/381833 | 408076 | 167,136 |
https://mathoverflow.net/questions/408042 | 0 | Suppose that we have the following PDE
$$\partial\_t \mu\_t = \nabla\cdot \left(\nabla \mu\_t - (b\*\mu\_t)\mu\_t\right), \tag{1}$$
with $\mu\_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$ and with the notation $$b\*\mu(x) := \int\_{\mathbb{R}^d} b(x,z)\,\mu(z)\,\mathrm{d}z.$$ For $N \in \mathbb{N}\... | https://mathoverflow.net/users/163454 | Rewriting PDE as "push-forward" | Let's focus on $N=1$ only, the case $N>1$ is just a tensorization of the argument below.
The whole argument is actually unrelated to the specific (aggregation-diffusion) PDE or gradient flow: As soon as you have a curve of probability measures $\mu\_t$ satisfying the continuity equation
$$
\partial\_t\mu\_t+\nabla\cd... | 1 | https://mathoverflow.net/users/33741 | 408080 | 167,137 |
https://mathoverflow.net/questions/408085 | 7 | It was mentioned in a lecture on Faltings’s proof of Mordell Conjecture that there’s some kind of correspondence between Galois representation (of cohomology, or some complex of constructible sheaves?) , but it hadn’t been explained explicitly. Also, I’m studying étale cohomology and Weil conjectures. The trace formula... | https://mathoverflow.net/users/170335 | L-functions and Galois representations: What’s the explicit relation? | The situation is:
One defines an $L$-function associated to a representation $V$ of the Galois group of a number field $K$ as $$L(V,s) = \prod\_{ \mathfrak p } \frac{ 1}{ \det( 1 -|\mathfrak p|^{-s} \operatorname{Frob}\_{\mathfrak p} , V^{I\_{\mathfrak p}} )}$$ with the product taken over the primes of $K$, where $V^... | 18 | https://mathoverflow.net/users/18060 | 408086 | 167,140 |
https://mathoverflow.net/questions/408032 | 3 | I have stumbled upon the following definitions in a paper by Gavin Wraith.
**Definition 1.** Say a ring morphism $A\overset \varphi \to B$ is formally unramified if every $b\in B$ admits:
* $b\_0\in B$ which is a simple root of some $\varphi (p)$ for some monic $p\in A[x]$;
* monics $f,g\in A[x]$ such that $\varphi... | https://mathoverflow.net/users/69037 | Alternative definitions of étale and formally unramified in Wraith | Following a suggestion made in the comments, I post this as an answer.
These are the "local structure theorems" of unramified and étale algebras. They are equivalent to the definitions with liftings and square-zero ideals. The proofs of the equivalence are not straightforward as far as I know: they are elementary but... | 2 | https://mathoverflow.net/users/69401 | 408087 | 167,141 |
https://mathoverflow.net/questions/408091 | 10 | Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence
$$1\to \mathbb{G}\_a^n\times \mathbb{G}\_m^m\to G\to A\to 1,$$
where $A$ is an abelian variety. (See [here](https://icerm.brown.edu/materials/Slides/sp-s12-w3/On_Mordell-Lang_in_algebraic_gr... | https://mathoverflow.net/users/131975 | What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$? | One example is provided by the generalized Jacobian. For a smooth projective curve $C$ and a divisor $D$ on $C$, the generalized Jacobian is defined to be the moduli space parameterizing pairs consisting of a line bundle of degree $0$ on $C$ together with a trivialization of that line bundle over $D$.
This admits a m... | 17 | https://mathoverflow.net/users/18060 | 408092 | 167,142 |
https://mathoverflow.net/questions/408045 | 0 | I wonder if it is possible to solve analytically the following equation
$$
\dot{\alpha}\_t = -\frac{2}{m} \alpha^2\_t + \frac{1}{2m} (\alpha\_t - \alpha\_t^\*)^2
$$
Where $\alpha\_t$ is a complex function, $\alpha\_t^\*$ is its complex conjugate and $\dot{\alpha}\_t$ is the time derivative.
All the best!
| https://mathoverflow.net/users/191299 | Solution of this differential equation | Yes, this can be integrated explicitly. First, notice that, since $m\not=0$, we can write $\alpha(t) = 2m\bigl(x(t)+iy(t)\bigr)$, in which case, the given equation becomes
$$
\dot x + i\,\dot y = -4(x + iy)^2 + (2iy)^2 = -4 x^2 - i\,(8xy),
$$
so $\dot x = -4x^2$ and $\dot y = -8 xy$. Thus, by standard ODE techniques,
$... | 5 | https://mathoverflow.net/users/13972 | 408093 | 167,143 |
https://mathoverflow.net/questions/407780 | 2 | To recall Arrow's theorem:
Suppose we have a finite set $X$ of voters and a finite set $Y$ of candidates.
An **election** is a map $\phi: X \rightarrow T$ where $T$ is the space of total orderings of $Y$. So we are voting via a total ranked list ballot. Let $E$ be the space of all elections.
A **social choice fun... | https://mathoverflow.net/users/126543 | Is there a version of Arrow's theorem without unrestricted domain? | There are two possible directions one can take this. One is to look at weaker conditions than full domain that still allow one to obtain ArrowÄs theorem, or one can look at restrictions that allow for more positive results. A classical criterion in the first category is the following:
**Chain Property:** We say $E'$ ... | 4 | https://mathoverflow.net/users/35357 | 408094 | 167,144 |
https://mathoverflow.net/questions/408099 | 3 | Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial\_t$ with smooth coefficient. Suppose I have $u\_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving
\begin{equation}
\left\{\begin{aligned}
&L u\_1 -\partial\_t u\_1= 0& \hspace{10pt} &\text{for $(x,t) \in (0,1) \times (0,T]$}
;\\
&... | https://mathoverflow.net/users/87922 | Gluing of two solutions to the same parabolic equation | Absolutely not! Taking the difference $v=u\_1-u\_2$, you see that $v(x,t)$ solves
$$
\begin{cases}
(L-\partial\_t) v=0 & \mbox{for }(x,t)\in(0,1/2)\times(0,T];
\\
v(0,t)=f(t) & \mbox{for }x=0,\,t\in(0,T]\\
v(1/2,t)=0 & \mbox{for }x=1/2,\,t\in(0,T]\\
v(x,0) =0 & \mbox{for }x\in (0,1/2),\,t=0
\end{cases}
$$
for some left... | 5 | https://mathoverflow.net/users/33741 | 408103 | 167,145 |
https://mathoverflow.net/questions/408121 | 3 | Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads
$$\sum\_{n\geq0}p(n)\,x^n=\prod\_{k\geq1}\frac1{1-x^k}.$$
There are many congruences for $p(n)$ including those due to Ramanujan: $p(5n+4)\equiv\_50, p(7n+5)\equiv\_70$ and $p(11n+6)\equiv\_{11}0$. Here, I would like to as... | https://mathoverflow.net/users/66131 | Congruence residues of integer partitions | Yes, this is true. In fact something even stronger is known:
For arbitrary positive integers $r,j$, and prime $\ell\geq 5$, there are infinitely many values of $n$ for which
$$p(n)=r\pmod{\ell^j}.$$
This is a special case of a conjecture of Newman and is proved in the paper "[Coefficients of half-integral weight modu... | 5 | https://mathoverflow.net/users/2384 | 408124 | 167,147 |
https://mathoverflow.net/questions/408117 | 9 | Is there a "nice" way to compute the signature of a smooth toric manifold of even complex dimension in terms of the moment polytope? By signature I mean in the sense of topology (see <https://en.wikipedia.org/wiki/Signature_(topology)>).
If one goes through all the machinery it is clear that the signature is encoded.... | https://mathoverflow.net/users/99732 | "Nice" way to compute the signature of a toric manifold? | The Hodge index theorem for smooth projective varieties (or compact Kähler manifolds) says that the signature is $$\sum\_{p,q} (-1)^p h^{p,q}(X)$$
For toric varieties, $h^{p,q}=0$ unless $p=q$, and equals the Betti number $h^{2p}(X)$, so the signature is $\sum\_p (-1)^p h^{2p}(X)$.
The Betti number $h^{2p}$ is equa... | 12 | https://mathoverflow.net/users/18060 | 408134 | 167,148 |
https://mathoverflow.net/questions/408138 | 8 | A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi\_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi\_1$-equivalent if there is $π\_1$-equivalence between them.
Let $X, Y$ be CW-complexes
1. Is it true that if $f \colon X \... | https://mathoverflow.net/users/148161 | Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism? | No and no. For an explicit counterexample to 1. (which is also a counterexample to 2.) take the map $\mathbb{R}P^2\to \mathbb{R}P^{\infty}$.
| 24 | https://mathoverflow.net/users/39747 | 408142 | 167,149 |
https://mathoverflow.net/questions/408021 | 3 | Consider the fractional Sobolev space
$$
W^{k,2}(\mathbb R^n):=\big\{f \in \mathcal S'\,\big|\,(1+\|\xi\|)^k\hat f(\xi)\in L^2(\mathbb R^n)\big\}
$$
for some $k\in\mathbb R$, and let $\mathcal M$ denote the space of Lebesgue-measurable functions on $\mathbb R^n$ (equivalence classes of functions, where two functions ar... | https://mathoverflow.net/users/5690 | Sobolev embedding into measurable functions | My question has been answered in this MathStackexchange post: <https://math.stackexchange.com/questions/4033589/sobolev-space-with-negative-index>
For every $k<0$, there exist a measure $\mu\_k$ which is singular with respect to Lebesgue measure, and such that $\mu\_k\in W^{k,2}(\mathbb R^n)$.
So $W^{k,2}(\mathbb R... | 0 | https://mathoverflow.net/users/5690 | 408156 | 167,154 |
https://mathoverflow.net/questions/407832 | 1 | Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C\_{d,r}(X)$. The Chow variety can have many topologically connected components (The number of connected components can grow to infinity as $d$ goes to infinity). Each connected component can be a... | https://mathoverflow.net/users/127776 | Singularities of Chow varieties | I am just posting my comments as an answer.
**Question 1 is false.** Here is a variant of my comment. Consider the dense open subscheme of the Chow variety parameterizing degree-$3$ curves in projective $n$-space that are (geometrically) irreducible and reduced. This has one irreducible component that is a dense open... | 2 | https://mathoverflow.net/users/13265 | 408162 | 167,157 |
https://mathoverflow.net/questions/408151 | 0 | Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H\_{\*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus\_{p+q=\*}(H\_{p}(X; \mathbb{Z}) \otimes H\_{q}(Y; \mathbb{Z}))$ by the Künneth theorem, which can be assumed if $H\_{i}(Y; \mathbb{Z})$ has n... | https://mathoverflow.net/users/41200 | Künneth formula and induced map in homologies | Here is an example which will not make you very happy. There is a degree one map $f:S^2\times S^1 \to S^3$ which just collapses the complement of an embedded open disk. Take $a\in H\_2(S^2;\mathbb{Z})$ and $b\in H\_1(S^1;\mathbb{Z})$ to be generators. Then $f\_\*(a\times b)\in H\_3(S^3;\mathbb{Z})$ is a generator, whil... | 6 | https://mathoverflow.net/users/8103 | 408163 | 167,158 |
https://mathoverflow.net/questions/408169 | 4 | I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of isolated double points over the original. By smoothable, I mean when does there exist a flat family containing the singularity ... | https://mathoverflow.net/users/103594 | Are Du Val singularities smoothable? | Du Val singularities are indeed smoothable and, in fact, more is true: they are *of class* $T$, namely, they are quotient singularities admitting a $1$-parameter $\mathbb{Q}$-Gorenstein smoothing.
See Definition 1.1 and Proposition 1.2 in the paper
M. Manetti: [On the moduli space of diffeomorphic algebraic surface... | 8 | https://mathoverflow.net/users/7460 | 408172 | 167,161 |
https://mathoverflow.net/questions/408175 | 8 | One of the standard examples of a forcing which is distributive but not proper (equiv. not closed) is a club shooting into a stationary co-stationary set.
But that forcing is $S$-proper for the stationary set into which we shoot the club. So it's at least a little bit proper. I want more. I am looking for an example ... | https://mathoverflow.net/users/7206 | Example of a distributive forcing which is entirely improper | $\mathbb P$ being $S$-proper means that for all countable $M \prec H\_\theta$ with $M \cap \omega\_1 \in S$, every $p \in \mathbb P \cap M$ can be extended to an $M$-generic condition.
A positive answer to your question is given by [work of Gitik](https://mathscinet.ams.org/mathscinet-getitem?mr=2648158), who showed ... | 9 | https://mathoverflow.net/users/11145 | 408180 | 167,164 |
https://mathoverflow.net/questions/408170 | 3 | Optimizing the spectral norm of some positive semidefinite matrix $A(x) \in S^{n}$, w.r.t. a list of variables $x \in \mathbb{R}^d$ and semidefinite constraints is, in general, a nonconvex problem (ref.: [Can one maximize the spectral norm of a matrix via semidefinite programming?](https://mathoverflow.net/questions/11... | https://mathoverflow.net/users/455887 | Relaxations for the spectral norm maximization problem | Minimizing a concave function subject to convex constraints is Concave Programming.
If the constraints of a Concave Programming problem are compact, as in your example, there must be a global optimum at an extreme of the constraints. In this example, if any semidefinite constraints are ignored, the extreme point of t... | 1 | https://mathoverflow.net/users/75420 | 408183 | 167,165 |
https://mathoverflow.net/questions/408174 | 2 | Let $X$ be a scheme. For $Y$ a scheme over $X$, the representable presheaf $h\_Y : U\mapsto \mathrm{Hom}\_X(U,Y)$ on the small étale site $X\_{et}$ is actually a sheaf, and by the Yoneda lemma the mapping $Y\to h\_Y$ is fully faithful when restricted to the category of schemes étale over $X$. Is there a bigger subcateg... | https://mathoverflow.net/users/138396 | Extending the domain of the yoneda embedding map from étale schemes to the small étale topos so that it is still fully faithful | The good setting to answer this question really is that of algebraic spaces. Then I believe that the answer is no, basically because every sheaf $F$ on the small étale site is representable by an étale $X$-algebraic space. The algebraic space that represents $F$ is constructed as follows: consider the collection of pai... | 4 | https://mathoverflow.net/users/17988 | 408186 | 167,167 |
https://mathoverflow.net/questions/408184 | 2 | Consider the following wave-type equation,
$$u\_{tt}-\frac{2}{t}u\_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u\_{t})=(0,0)$ at $t=\epsilon.$ My goal is to estimate the energy $E(u\_t) = \int (u\_{tt})^2 + |\nabla u\_t|^2.$ I already know tha... | https://mathoverflow.net/users/68232 | How to estimate higher order regularity for wave type equation with time dependant coefficients? | $$ \tilde{u}\_{tt} - \frac{2}{t}\tilde{u}\_{t}-\Delta \tilde{u} = g\_t -\frac{2}{t^2}\tilde{u} $$
$$ \dot{E} = 2 \int \tilde{u}\_t (2 t^{-1} \tilde{u}\_t + g\_t - 2 t^{-2} \tilde{u} ) $$
$$ \dot{E} = 2 \int \tilde{u}\_t (2 t^{-1} \tilde{u}\_t + g\_t) - 2 t^{-2} \frac{d}{dt} \int \tilde{u}^2 $$
$$ \dot{E} + \frac{... | 2 | https://mathoverflow.net/users/3948 | 408189 | 167,168 |
https://mathoverflow.net/questions/408204 | 9 | Let $L$ be any differential operator (not necessarily linear).
Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:
Given a boundary value problem on some domain $\Omega$. If $L$, the initial condition, and the boundary are of a certain form, then separa... | https://mathoverflow.net/users/114299 | General validity of separation of variables | The short answer is No. A major problem is that there is no single universal definition for what it means for an equation to be solvable by separation of variables. This defect in the theory, as well as the lack of general statements of the form that you would like to see, was highlighted a while ago in this BAMS book ... | 10 | https://mathoverflow.net/users/2622 | 408206 | 167,170 |
https://mathoverflow.net/questions/408136 | 3 | Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}\_{\*}$ (the category of point spaces; all continuous maps and homotopies preserve the chosen base points). We define $π'\_1(X) $ as the set of homotopy classes of prod... | https://mathoverflow.net/users/148161 | Can the loops in the definition of the fundamental group be considered injective? | In the [Griffiths Twin Cone](https://wildtopology.com/2014/06/28/the-griffiths-twin-cone/) (or double cone over the shrinking wedge of circles), $G\subseteq \mathbb{R}^3$, all injective loops are null-homotopic yet $\pi\_1(G)$ is uncountable. Hence, injective loops don't generate any of the fundamental group. However, ... | 4 | https://mathoverflow.net/users/5801 | 408211 | 167,173 |
https://mathoverflow.net/questions/408181 | 0 | It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words,
if
$$M(X,T) := \{ \mu\,\, \text{probability measure} : \mu= T\_\*\mu \} $$
is the set of invariant measures for $T$, then
$$h\_\text... | https://mathoverflow.net/users/297294 | Metric entropy and topological entropy | It's true for a very simple reason: the entropy of a dynamical system with respect to a (not necessarily ergodic) invariant measure is the average of the entropies of its ergodic components.
| 2 | https://mathoverflow.net/users/8588 | 408213 | 167,174 |
https://mathoverflow.net/questions/407731 | 12 | The Airy differential equation
$$y''(x)\ = \ xy(x)$$
is one of the simplest *irregular* differential equations (so not determined by its monodromy data, there is more structure, the Stokes data). Associated to this, Katz constructs an *Airy sheaf* $\text{Ai}$ on the affine line $\mathbf{A}^1\_{\mathbf{F}\_q}$; $\text{A... | https://mathoverflow.net/users/119012 | Katz's $\ell$-adic Airy sheaf | (1.) The part of the Stokes data we can see is the restriction of the differential equation to the field of formal Laurent series at each point. There is a classification of differential equations over this field, and a classification of Galois representations of a field of formal Laurent series over a finite field, an... | 6 | https://mathoverflow.net/users/18060 | 408214 | 167,175 |
https://mathoverflow.net/questions/408131 | 2 | Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak\* topologies.
Let $C$ be a weak\* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of minimisers of $\langle x, \cdot \rangle$ on $C$. Each $M(x)$ is non-empty, convex, and closed.
My question is: What... | https://mathoverflow.net/users/96899 | How to choose minimisers in a continuous way | I think the answer is: only in the trivial case when each $M(x)$ is a singleton $\{f(x)\}$. Indeed, if $M(x)$ has more than one point, it has at least $2$ extremal points. An extremal point of a convex compact set $C$ of a LCTVS (here, $X’$ with its weak\* topology) is the unique minimiser on $C$ for some continuous li... | 2 | https://mathoverflow.net/users/6101 | 408216 | 167,177 |
https://mathoverflow.net/questions/408194 | 13 | The sequence that is addressed here is resourced from the most useful site OEIS, [listed as A014153](https://oeis.org/A014153), with a generating function
$$\frac1{(1-x)^2}\prod\_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look at these two interpretations mentioned there.
$a(n)=$ Number of partitions of $n$ with t... | https://mathoverflow.net/users/66131 | Two interpretations of a sequence: an opportunity for combinatorics | Here is a bijective proof that equates both $a(n)$ and $b(n+1)$ to the quantity
$$p(n)+2p(n-1)+\cdots+np(1)+(n+1)p(0) \tag1$$
where $p(n)$ is the number of partitions of $n$.
For $a(n)$: each partition with three types of $1$'s corresponds to a partition of $k$ together with $(n-k)$ parts from $\{1',1''\}$. This seco... | 18 | https://mathoverflow.net/users/2384 | 408219 | 167,178 |
https://mathoverflow.net/questions/408146 | 4 | I want
\begin{equation}
\int\_a^b f(g(x))dg=\int\_{g(a)}^{g(b)} f(t)dt,\tag{$\heartsuit$}\label{heart}
\end{equation}
where $g$ is a continuous but not necessarily monotone (or bounded variation) function of $[a,b]$, both integrals are Riemann–Stieltjes (that is, $\int\_a^b h(x)dg(x)$ is the limit of Riemann–Stieltjes ... | https://mathoverflow.net/users/4312 | Change of variables in Riemann–Stieltjes integral | Looks like the answer is yes, they are equal, and perhaps you do not really need to assume the existence of the integral in the right-hand side, that is, $\int\_{g(a)}^{g(b)} f(t) dt$. This is proved in the context of Henstock–Kurzweil integrals as Theorem 6.1 in [1].
The key lemma is that if $g(a) = g(b)$, then $\in... | 5 | https://mathoverflow.net/users/108637 | 408220 | 167,179 |
https://mathoverflow.net/questions/408193 | 6 | Disclaimer: This question was originally posted in [math.stackexchange.com](https://math.stackexchange.com/questions/4272335/are-there-a-in-depth-classification-of-branch-points-in-complex-analysis) and, after 30 days with no answers, I followed the instructions of [this topic](https://meta.mathoverflow.net/questions/2... | https://mathoverflow.net/users/424703 | Is there a, in depth, classification of branch points in complex analysis? | Yes, there is a classification. An isolated branch point can be algebraic or logarithmic. If the branch point is at 0, algebraic means that $f(z^n)$ has a pole or removable singularity at 0. It can also have an essential singularity, but this does not have an accepted name. In the case of a logarithmic point
$f(e^z)$ i... | 8 | https://mathoverflow.net/users/25510 | 408224 | 167,181 |
https://mathoverflow.net/questions/408221 | 5 | This is a question inspired by T. Amdeberhan's [recent question](https://mathoverflow.net/questions/408194/two-interpretations-of-a-sequence-an-opportunity-for-combinatorics), as well as [another previos MO question](https://mathoverflow.net/questions/127000/partitions-sum-of-divisors-identity).
For an integer partit... | https://mathoverflow.net/users/25028 | Coefficients obtained from ratio with partition number generating function | The bijection described by Mark Wildon in the second linked question can be adapted to your generalization. Indeed, $|\lambda|\_k$ counts the number of ways of selecting a box $B$ of $\lambda$ such that, if the row containing $B$ has length $m$, then there are at most $k-1$ other rows of length $m$ above it. By erasing... | 9 | https://mathoverflow.net/users/2384 | 408228 | 167,182 |
https://mathoverflow.net/questions/408201 | 9 | Let $S$ be a set of $2n$ points in $\mathbb{R}^2$. Which is the maximum number of different bi-partitions of $S$ generated by a straight line?
More precisely, which is the maximum number of partitions $S=S\_1 \sqcup S\_2$ such that $|S\_1|=n$ and there exists a line $\ell$ such that $S\_1$ and $S\_2$ and are the inte... | https://mathoverflow.net/users/167834 | Bi-partitioning $2n$ points on the plane with a straight line | These are called halving lines, and we don't know the exact order of their magnitude, just that it is between $\Omega(n2^{\sqrt\log n})$ and $O(n^{4/3})$.
For more information, see <https://jeffe.cs.illinois.edu/open/ksets.html>.
| 11 | https://mathoverflow.net/users/955 | 408233 | 167,184 |
https://mathoverflow.net/questions/408237 | 4 | Let $G$ be a reductive group and $X$ a smooth $G$-variety. Then the fixed point subvariety $X^G$ is also smooth (this is theorem 13.1 of Milne's book on algebraic groups). Suppose in addition that the canonical bundle $\omega\_X$ is trivial. Then is $\omega\_{X^G}$ also trivial?
One idea I had was to use the adjuncti... | https://mathoverflow.net/users/334560 | If $X$ is a smooth $G$-variety with trivial canonical bundle, then does $X^G$ also have trivial canonical bundle? | No. Let $C \subset \mathbb{P}^2$ be a smooth sextic curve, let $X$ be the double covering of $\mathbb{P}^2$ ramified over $C$, and let $G = \mathbb{Z}/2$ acting on $X$ be the involution of the double covering. Then $X$ is a K3 surface, hence its canonical bundle is trivial. But $X^G = C$ is a curve of genus $10$, its c... | 14 | https://mathoverflow.net/users/4428 | 408238 | 167,185 |
https://mathoverflow.net/questions/408208 | 8 | [This list](http://shpilrain.ccny.cuny.edu/gworld/problems/probhyp.html) of open problems from <http://grouptheory.info/> includes the question:
*"Is every biautomatic group which does not contain any $\mathbb{Z} \times \mathbb{Z}$ subgroups, hyperbolic?"*
It is credited to Gersten but I don't see any mention of it... | https://mathoverflow.net/users/133733 | When are biautomatic groups hyperbolic? | $\DeclareMathOperator\BS{BS}$Since this question goes in several directions, I hope a discursive answer is appropriate.
The question fits into an important family of questions in geometric group theory which look to provide some sort of "algebraic" characterisation of hyperbolic groups. The search for this characteri... | 13 | https://mathoverflow.net/users/1463 | 408239 | 167,186 |
https://mathoverflow.net/questions/408243 | 9 | I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection.
Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ equipped with some connection $D$, I want to find an orthogonal bundle $W$ with a *flat* connection $D'$ such that $V\subset... | https://mathoverflow.net/users/142627 | Embedding of a bundle with connection into a bundle with flat connection? | The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate grassmanian by a map. Since the canonical connection... | 12 | https://mathoverflow.net/users/12605 | 408258 | 167,197 |
https://mathoverflow.net/questions/408247 | 5 | In pseudo-Riemannian geometry it is well known that every three-dimensional Einstein manifold has constant curvature. A proof of this is sketched [here](https://math.stackexchange.com/a/2194463/242708).
**Question**. Does anyone know where in the literature I can find a proof of such result?
| https://mathoverflow.net/users/74033 | Proof that every three-dimensional Einstein manifold has constant curvature | This is precisely Proposition 1.120 on p.49 in Besse's "Einstein Manifolds" (I am using the reprint of the 1987 edition, so the numbering may be different in the older edition):
>
> A 3-dimensional pseudo-Riemannian manifold is Einstein iff it has constant (sectional) curvature.
>
>
>
Note that the proof is be... | 6 | https://mathoverflow.net/users/1849 | 408259 | 167,198 |
https://mathoverflow.net/questions/408168 | 7 | The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}\_2(\mathbb{C})$-Eisenstein series:
\begin{align\*}
\int\_{-\infty}^{\infty}\left(\int\_{-\infty}^{\infty} \frac{(x\_1+ i x\_2)^{\ell}}{(x\_1^2+x\_2^2+a^2)^s} e^{ i x\_1\zeta\_1}dx\_1 \right... | https://mathoverflow.net/users/11765 | On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series | Following @Abdelmalek's great advice in the comments above it is enough to compute the following triple integral:
\begin{align\*}
I=\frac{\pi^s}{\Gamma(s)}\int\_{0}^{\infty}\left(\int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}
(x\_1+x\_2 i)^{\ell}\cdot e^{-\pi t(x\_1^2+x\_2^2+a^2)} e^{2\pi i (x\_1\zeta\_1+x\_2\zeta\_... | 4 | https://mathoverflow.net/users/11765 | 408260 | 167,199 |
https://mathoverflow.net/questions/408261 | 2 | Given a finite simple graph $G$ with $n$ vertices, for each 0-1 colouring $\alpha \in \mathbb{Z}\_2^n$ of its vertices consider the subgraph $G\_\alpha$, whose vertices are the $1$-coloured vertices in $\alpha$, created as follows: if two vertices with labels $1$ are adjacent, then add the edge between them.
Is there a... | https://mathoverflow.net/users/16507 | number of components of subgraphs | Something to start with.
We have $$\Theta(G)=\sum\_{\alpha} (-1)^{\alpha}\#G\_\alpha=\sum\_{\alpha}(-1)^\alpha\sum\_U \mathbf{1}(U\,\text{is a component of}\,G\_{\alpha}),$$
where $U$ runs over all induced connected subgraphs of $G$. Changing the order of summation, we get
$$
\Theta(G)=\sum\_U \sum\_{\alpha:\, U\,\te... | 2 | https://mathoverflow.net/users/4312 | 408263 | 167,200 |
https://mathoverflow.net/questions/408250 | 8 | I wonder whether there is a website or a survey collecting all known NP-complete or NP-hard problems on graph theory?
| https://mathoverflow.net/users/375270 | Is there a website or a survey collecting all NP-complete problems on graph theory? | Here is a [section on graph theory](https://www.csc.kth.se/%7Eviggo/wwwcompendium/node8.html) in *A compendium of NP optimization problems* by P. Crescenzi and V. Kann.
| 7 | https://mathoverflow.net/users/7076 | 408269 | 167,201 |
https://mathoverflow.net/questions/408264 | -1 | Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int\_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$
or any reasonable Euclidean norm such that bounded functions have a finite norm.
Can we construct a continuous function mapping from $S$ to the space of distributions... | https://mathoverflow.net/users/8737 | Fundamental of a signal | Make a sequence of functions of period $2\pi$ whose limit is of period $\pi$. Your map is discontinuous.
| 1 | https://mathoverflow.net/users/13268 | 408271 | 167,202 |
https://mathoverflow.net/questions/408262 | 11 | If $0^\sharp$ exists then the $L$-indiscernibles form a proper class of ordinals without any infinite constructible subset, as $0^\sharp$ can be defined from any infinite increasing sequence $\langle \kappa\_i\mid i<\omega\rangle$ of them as
$$0^\sharp = \{\varphi(v\_0,\dots, v\_n)\mid L\_{\sup\_{i<\omega}\kappa\_i}\mo... | https://mathoverflow.net/users/125703 | A proper class of ordinals without an infinite constructible subset |
>
> *Stanley, M. C.*, [**A cardinal preserving immune partition of the ordinals**](http://dx.doi.org/10.4064/fm-148-3-199-221), Fundam. Math. 148, No. 3, 199-221 (1995). [ZBL0843.03028](https://zbmath.org/?q=an:0843.03028).
>
>
>
An infinite set (or class) of ordinals is said to be immune if it neither contains ... | 14 | https://mathoverflow.net/users/11115 | 408272 | 167,203 |
https://mathoverflow.net/questions/408276 | 2 | I posed this question on Math.Stackexchange (see [here](https://math.stackexchange.com/q/4301068/75923)) but until now there was no response. This made me decide to give it a try here.
---
Let $k\subseteq F$ denote an algebraic field extension and let $\alpha\in F$ having $f\in k[x]$ as its minimal polynomial. Fu... | https://mathoverflow.net/users/40263 | If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$? | Here is a different interpretation of the question, which is hopefully closer to OP's intent:
>
> Let $F/k$ be an algebraic field extension, and let $\alpha\in F$. Does $Aut\_k(F)$ act transitively on the conjugates of $\alpha$ *which are contained in $F$*?
>
>
>
The answer to this question is no in general. F... | 7 | https://mathoverflow.net/users/30186 | 408283 | 167,207 |
https://mathoverflow.net/questions/408061 | 1 | I am looking for a reference or derivation for the following question:
Consider a cycle graph $G$ with $N$ vertices (see example [here](https://en.wikipedia.org/wiki/Cycle_graph)). Let two independent continuous-time random walkers$^\star$ start on node $i$ and node $j$. Let $T$ be the time when the two walkers meet ... | https://mathoverflow.net/users/420641 | Reference: probability distribution of first meeting time of two random walks on a cycle graph | Assume that $i<j.$ You can reduce this to the following simpler-looking question: Consider a continuous-time RW on the integers, moving at rate two, started at $k:=j-i$. Let $T$ be the hitting time of $\{0,N\}$ by this walk, also known as as the exit time from $[1,N-1]$. Let $\tau$ denote the hitting time of $\{0,N\}$ ... | 0 | https://mathoverflow.net/users/7691 | 408303 | 167,211 |
https://mathoverflow.net/questions/408301 | 20 | Are there arbitrarily large sets $\mathcal S=\{a\_1,\ldots,a\_n\}$ of strictly positive integers such that all sums $a\_i+a\_j$ of two distinct elements in $\mathcal S$ are squares?
Considering subsets in $\mathbb Z$ should essentially give the same answer since such a set can contain at most one negative integer.
... | https://mathoverflow.net/users/4556 | Size of set of integers with all sums of two distinct elements giving squares | The size of such sets is bounded by some (unknown) constant, assuming a big conjecture in arithmetic geometry.
The Bombieri-Lang conjecture (non-trivially via the Uniformity Conjecture, see Stanley Yao Xiao's comment) implies that for any $f(x)\in \mathbb{Z}[x]$ of degree $5$, with no repeated roots, there are at mos... | 24 | https://mathoverflow.net/users/385 | 408306 | 167,212 |
https://mathoverflow.net/questions/408287 | 1 | Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi\_k\}\_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of Dirichlet eigenfunctions of $-\Delta\_g$ on $M$.
Now let us denote by $X$ the collection of all pairs $(f,g)$ with $f$ a smo... | https://mathoverflow.net/users/50438 | A property for generic pairs of functions and metrics | I believe that a combination of the following two facts essentially confirms the desired result. I phrased the second point in $C^2(M)$—working directly in $C^\infty(M)$ is a bit awkward because it is not a Banach space—but I think it should hold more widely.
* The generic simplicity of the Laplacian eigenvalues is a... | 3 | https://mathoverflow.net/users/103792 | 408312 | 167,214 |
https://mathoverflow.net/questions/406784 | 5 | Remember, that an incomplete r.e. set $A$ is cuppable if there is an incomplete r.e. set $B$ such that $A\oplus B \equiv\_T 0'$. It's relatively easy to build a low cuppable set but my question is whether for every cuppable set $A$ there is a low r.e. set $B$ such that $A \oplus B \equiv 0'$
I'm pretty sure this must... | https://mathoverflow.net/users/23648 | Does every cuppable r.e. set cup with a low set? | It seems that a short comment is not sufficient.
***The answer is negative. I.e. there is a cuppable r.e. set which is no low cuppable.***
By the results from Soare's book, low cuppability is equivalent to prompt simplicity which is equivalent to noncappbability.
Now the fact is that there is an r.e. degree which... | 3 | https://mathoverflow.net/users/14340 | 408313 | 167,215 |
https://mathoverflow.net/questions/408230 | 0 | I'm looking for some continuous functions $\{f\_i(x,t)\}$, here $x=(x\_1,x\_2..., x\_n)$, such that:
* $f\_i(x,t):R^n\times [0, +\infty)\rightarrow R ~~\text{is continuous for each}~~i $
* $f\_i(x,0)=x\_i$
* $\Sigma\_1^n f\_i^2(x,t)\rightarrow \infty ~~\text{as}~~\|(x,t)\|^2=\Sigma\_1^n x\_i+t^2 \rightarrow \infty$
... | https://mathoverflow.net/users/166368 | Looking for some special functions | I assume you want squares on $x\_i$ in the expression for the norm: $\|(x,t)\|^2=x\_1^2+\dots+x\_n^2+t^2$.
Then I believe there are no such functions for topological reasons. Consider the restriction of $f=(f\_1,\dots,f\_n)$ to a half-sphere
$$X=\{(x,t);\,x\_1^2+\dots+x\_n^2+t^2=C,\,t\geq 0\}.$$
Suppose that $f\neq 0... | 2 | https://mathoverflow.net/users/10846 | 408319 | 167,217 |
https://mathoverflow.net/questions/322043 | 12 | On Saturday 4 September 1999, [Vaughan Jones](https://en.wikipedia.org/wiki/Vaughan_Jones) put on arXiv a paper entitled [*Planar algebras, I*](https://arxiv.org/abs/math/9909027).
Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the mention "to appear in New Zealand ... | https://mathoverflow.net/users/34538 | Why is Planar algebras I (by Vaughan Jones) not published? | This paper is now published in New Zealand Journal of Mathematics Vol. 52 (2021).
<https://doi.org/10.53733/172>
[pdf file](https://nzjmath.org/index.php/NZJMATH/article/view/172/61)
| 11 | https://mathoverflow.net/users/164194 | 408320 | 167,218 |
https://mathoverflow.net/questions/408266 | 2 | Is it true the following statement?
>
> Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$, there exists a Polish space $Z$ and a Borel function $g:X \rightarrow Y\times Z$ with closed graph such that $f(x) \ = \pi\_Y(g(x))$ for all $x \in X$.
>
>
>
In case it was true, do you have any hint ... | https://mathoverflow.net/users/141146 | Is every Borel function a projection of a Borel function with closed graph? | Yes, this is true: by Exercise 13.5 in [Kechris' "Classical Descriptive Set Theory"](https://link.springer.com/book/10.1007/978-1-4612-4190-4), for any Borel function $f:X\to Y$ between Polish spaces there exists a continuous bijective map $i:Z\to X$ from a Polish space $Z$ such that the function $f\circ i:Z\to Y$ is c... | 6 | https://mathoverflow.net/users/61536 | 408325 | 167,220 |
https://mathoverflow.net/questions/408298 | 0 | For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\ -x^T & C & -A^T \end{matrix} \right),
\end{align}
where $x, y$ are $1 \times n$ matrices, $A,B,C$ are $n \times n$ matrices. Are there... | https://mathoverflow.net/users/11877 | Matrix representations of Lie groups of type $B_n$ | The quadratic form whose matrix is $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & I\_n \\ 0 & I\_n & 0 \end{pmatrix}$ gives an embedding of $\operatorname{SO}(2n + 1, \mathbb C)$ in $\operatorname{GL}(2n + 1, \mathbb C)$ whose derivative is your specified embedding $\mathfrak{so}(2n + 1, \mathbb C) \to \mathfrak{gl}(2n + 1, \ma... | 1 | https://mathoverflow.net/users/2383 | 408326 | 167,221 |
https://mathoverflow.net/questions/408268 | 1 | Consider the integral
$$\mathcal{I}=\int\_0^t\left(\frac{Ae^{-\lambda t}-Ae^{-\lambda s}}{2}\right)^{2m+1}e^{-\epsilon(t-s)}ds,\tag{1}$$
for constants $A,\lambda,\epsilon,t\in\mathbb{R}$ and $m\in\mathbb{Z}^+$.
The intention of evaluating $\mathcal{I}$, is to find
\begin{equation}
\begin{split}
\mathcal{S}&=\in... | https://mathoverflow.net/users/167822 | Integral of $J_1\left(Ae^{-\lambda t}-Ae^{-\lambda s}\right)e^{-\epsilon(t-s)}$ with respect to $s$? | If $\lambda>0$ and $\lambda>\epsilon$ one has
$$\lim\_{t\rightarrow\infty}e^{\epsilon t}\mathcal{S}=\int\_0^\infty J\_1\left(-Ae^{-\lambda s}\right)e^{\epsilon s}ds$$
$$\qquad=\frac{A/2}{\epsilon-\lambda} \, \_1F\_2\left(\frac{1}{2}-\frac{\epsilon}{2 \lambda};2,\frac{3}{2}-\frac{\epsilon}{2 \lambda};-A^2/4\right).$$
Th... | 3 | https://mathoverflow.net/users/11260 | 408335 | 167,223 |
https://mathoverflow.net/questions/408308 | 5 | In P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations.
pg.118, they go over some special cases of nonlinear SDEs $dX\_t=\alpha(t,X\_t)\,dt+\sigma(t,X\_t)\,dB\_t$ that have exact solutions.
I am just wondering if there are any more large lists somewhere that you came across. Than... | https://mathoverflow.net/users/99863 | Long list of exactly solvable nonlinear SDEs | A follow-up of Kloeden and Platen is C.H. Skiadas, [Exact Solutions of Stochastic Differential Equations](https://link.springer.com/article/10.1007/s11009-009-9145-3) (2010).
| 2 | https://mathoverflow.net/users/11260 | 408341 | 167,225 |
https://mathoverflow.net/questions/408305 | 3 | Do there exist non-rational algebraic numbers that belong to $\mathbb Q\_p$ for all prime $p$? If yes, can one characterize them?
I spent several days for the first question, and I found nothing. The second one looks even more diffuclt and surely out of my skills.
| https://mathoverflow.net/users/33128 | Algebraic numbers in all $\mathbb Q_p$ | (cw answer based on Wojowu's link)
>
> The only algebraic extension of $\mathbf{Q}$ that embeds into $\mathbf{Q}\_p$ for all $p$ (or even for a density 1 set of primes) is $\mathbf{Q}$ itself.
>
>
>
For if $P\in\mathbf{Q}[t]$ is an irreducible polynomial of degree $\ge 2$, the set of primes $p$ for which $P$ h... | 9 | https://mathoverflow.net/users/14094 | 408343 | 167,226 |
https://mathoverflow.net/questions/408344 | 11 | I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly theoretical unfortunately).
Right now I'm interested in the cohomology of quiver and character varieties and their links with... | https://mathoverflow.net/users/146464 | Reference for combinatorics with view towards representation theory/algebraic geometry | M. Haiman "Notes on Macdonald polynomials and the geometry of the Hilbert scheme of points on $\mathbb{P}^2$". By one of the greatest specialists of interactions between combinatorics and algebraic geometry.
| 12 | https://mathoverflow.net/users/37214 | 408347 | 167,227 |
https://mathoverflow.net/questions/408364 | 3 | Consider the [binary partitions](https://oeis.org/A000123) of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function
$$\sum\_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod\_{k\geq0}\frac1{1-x^{2^n}}.$$
A result of De Bruijn shows the asymptotic growth
$$\log b(2n)\sim \frac1{\log 4}\log^2\left(\frac{n}{\log n}\ri... | https://mathoverflow.net/users/66131 | Asymptotic growth of ternary partitions of integers $3n$ | It's right in the beginning of De Bruijin's paper. More generally,
$\log p(rn) \sim \frac 1 {2 \log r} \log^2 \frac n {\log n}$
where $p(rn)$ is the number of partitions of $rn$ into powers of $r$.
The result is attributed to Kurt Mahler's paper "On a special functional equation", published in Journ. London Math.... | 4 | https://mathoverflow.net/users/114143 | 408367 | 167,231 |
https://mathoverflow.net/questions/407905 | 0 | [EDIT] The older exposition of this theory was proved inconsistent by EmilJeřábek (see comments). Here, this is a possible salvage. (the new information over the older post shall be put in square brackets)
Working in mono-sorted FOL with equality and membership, add the following axioms:
Define: $set(y) \iff \exist... | https://mathoverflow.net/users/95347 | Is this theory equivalent to MK? | This theory is consistent since NBG proves this form of reflection and the rest of axioms are either axioms or theorems of MK, so this theory is a subtheory of MK. Now to establish the other direction of equivalence with MK, we note that Extensionality, Pairing, Boolean union, Set Union, Power, Infinity and Separation ... | 0 | https://mathoverflow.net/users/95347 | 408368 | 167,232 |
https://mathoverflow.net/questions/408294 | 6 | A subset of a linear space $X$ is called *infinite-dimensional* if it is not contained in a finite-dimensional linear subspace of $X$.
>
> **Problem.** Let $L$ be an infinite-dimensional subset of the linear space $\mathbb R^\omega$. Is there an infinite set $I\subseteq\omega$ such that for every infinite set $J\su... | https://mathoverflow.net/users/61536 | Infinite-dimensional projections of linearly independent sets | For $n :=\{0,1,\dots,n-1\}\in\omega$ let’s denote $P\_n:\mathbb{R}^\omega\to \mathbb{R} ^n$ the projection, which is the restriction map $f\mapsto f\_{|n}$. The dimensions of the subspaces $P\_n(L)\subset \mathbb{R} ^n$ can’t be stationary, because that would mean that for some $n\_0$, every function $f\in P\_{n\_0}(L)... | 4 | https://mathoverflow.net/users/6101 | 408375 | 167,235 |
https://mathoverflow.net/questions/408374 | 21 | Let $u(i,j)$ denote the number of lattice paths from the origin to a fixed terminal point $(i,j)$ subject only to the condition that each successive lattice point on the path is closer to $(i,j)$ than its predecessor. For example, $u(1,1) = 5$ counts the one-step path $(0,0) \to (1,1)$ and the 4 two-step paths with lon... | https://mathoverflow.net/users/29500 | Why are the numbers counting "ever-closer" lattice paths so round? | For any $k\in\mathbb N$ let $a\_k$ be the number of points on the circle of radius $\sqrt{k}$ (this number may be zero). For any path as in question and for any $k$ between $1$ and $i^2+j^2-1$, there is going to be either none or exactly one of the points on the circle of radius $\sqrt{k}$ around $(i,j)$. As the sequen... | 41 | https://mathoverflow.net/users/30186 | 408377 | 167,236 |
https://mathoverflow.net/questions/408362 | 2 | Let $[x\_1,x\_2,x\_3,x\_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I\_Z=(x\_1,x\_2,x\_3^2) \subset \mathbb{C}[x\_1,x\_2,x\_3,x\_4]$$ i.e. $Z$ is a double point supported on the line $L:(x\_1=x\_2=0)$.
I want to consider the blowup of $\mathbb{P}^3$ along $Z$, i... | https://mathoverflow.net/users/146431 | Equivalence of sequences of blowups of $\mathbb{P}^3$ | These are definitely not the same. $Z$ is a complete intersection subscheme, so if you blow up $Z$, then the exceptional divisor (=set) is a single $\mathbb P^2$, while in the second case it's a union of a copy of $\mathbb P^2$ and a copy of a $\mathbb P^2$ blown up at a point (as you point out it is easy to see that i... | 1 | https://mathoverflow.net/users/10076 | 408393 | 167,243 |
https://mathoverflow.net/questions/408396 | 8 | It is not known if there are infinitely many prime Fibonacci numbers. But can one assert that there is no Fibonacci number >2 that is also highly composite (<https://en.wikipedia.org/wiki/Highly_composite_number>) - or that there are only finitely many such numbers?
**Remarks:** As given in <http://www.maths.surrey.a... | https://mathoverflow.net/users/142600 | On Fibonacci numbers that are also highly composite | The largest highly composite Fibonacci number is $F\_{3} = 2$.
If $p$ is a prime number, then either $p \mid F\_{p-1}$ (if $p \equiv \pm 1 \pmod{5}$), $p \mid F\_{p}$ (if $p = 5$), or $p \mid F\_{p+1}$ (if $p \equiv \pm 2 \pmod{5}$). It follows that if $n > 12$ and $p$ is a prime that divides $F\_{n}$ and no previous... | 21 | https://mathoverflow.net/users/48142 | 408421 | 167,250 |
https://mathoverflow.net/questions/408409 | 8 | What do people in homotopy theory mean when they say "... is a model for spaces / $(\infty,1)$-categories"? And why does one need models?
Is this related to the notion of a model category?
| https://mathoverflow.net/users/442261 | "Models" in homotopy theory | Yes, this is absolutely related to model categories (although the concept of models for an homotopy theory is much more general): the notion of model category was introduced by Quillen exactly to express the idea of models for homotopy types. I quote Quillen's *Homotopical Algebra* (Chapter I, page 0.3):
>
> The te... | 13 | https://mathoverflow.net/users/1017 | 408424 | 167,251 |
https://mathoverflow.net/questions/408351 | 2 | Let $ G $ be a group of $ n \times n $ matrices. Suppose that some subset $ \{ g\_j: 1 \leq j \leq n^2 \} $ of $ G $ is a basis for the space of all $ n \times n $ matrices. Furthermore suppose that the set
$$
\{ \overline{g\_i}: 1 \leq j \leq n^2 \}
$$
is a group in $ \operatorname{PGL}\_n $. What can we say about the... | https://mathoverflow.net/users/387190 | Bound on the size of group related to a matrix basis | I consider the situation over $\mathbb{C}$. We can replace $G$ by the subgroup generated by the $g\_j$'s, so assume that $G$ is generated by the $g\_j$'s. That $G$ contains a spanning set of the space of all matrices yields that the center of $G$ only contains scalar matrices, and also that the inclusion $G \hookrighta... | 5 | https://mathoverflow.net/users/10266 | 408434 | 167,256 |
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