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/353660 | 4 | What form of the axiom of choice is equivalent (in ZF) to the statement that every distributive lattice is isomorphic to a lattice of sets?
| https://mathoverflow.net/users/20300 | Distributive lattices and axiom of choice | [According to Wikipedia](https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem) the prime ideal theorem for distributive lattices is equivalent to the Boolean prime ideal theorem. (Look in the "Further prime ideal theorems" section.)
| 3 | https://mathoverflow.net/users/23141 | 353664 | 149,421 |
https://mathoverflow.net/questions/280920 | 12 | In Definition 4.1.1 of [$(\infty,2)$-Categories and the Goodwillie Calculus I](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), Lurie defines a *weak $\infty$-bicategory* to be a scaled simplicial set that has the extension property with respect to every scaled anodyne morphism. In Theorem 4.2.7, he defines ... | https://mathoverflow.net/users/25477 | Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory? | A few years later, it has been shown by [Gagna, Harpaz, and Lanari](https://arxiv.org/abs/1911.01905) that the answer is *yes*. Every weak $\infty$-bicategory is an $\infty$-bicategory.
| 8 | https://mathoverflow.net/users/2362 | 353670 | 149,423 |
https://mathoverflow.net/questions/353672 | 3 | Let $S=\{c\_1,\dots,c\_n\}$ be a set of vectors in $\mathbb{R}^M$. Is the below problem studied in literature?
$$\max\limits\_{S'\subset S} \vert S' \vert $$
$$s.t. dim(span(S')) < dim(span(S))$$
which is to find the cardinality of the largest subset which does not span the span of S.
| https://mathoverflow.net/users/124703 | Largest subset not spanning the span | This seems to be a matroid theory question. If you let $S$ be the ground set of the matroid and you let $r$ be the rank of the matroid (i.e. $\dim \mathrm{span}(S)$), then your question amounts to finding the largest flat of rank $r-1$.
For reference, I would recommend Oxley's Matroid Theory.
That said, one general... | 2 | https://mathoverflow.net/users/152900 | 353674 | 149,424 |
https://mathoverflow.net/questions/353621 | 0 | From a delay system, I obtain the following as part of a characteristic equation:
$$f(\lambda) = \lambda - a + be^{-c\lambda},$$
where $a, b,$ and $c$ are positive number and $a<b, ac<1$. **My goal is to find the sign of the real part of the root to $f(\lambda) = 0$**. Taking $\lambda = x + iy$, I obtain:
\begin{align}... | https://mathoverflow.net/users/109419 | Conditions to determine sign of real roots | I think your equation is $f(\lambda) = \lambda - a + be^{-c \lambda}$. Let us consider the parametrized family $f\_{\varepsilon}(\lambda) = \lambda - a + b e^{-\varepsilon c \lambda}$, where $\varepsilon \in [0,1]$. First of all note that the number of roots $f\_{\varepsilon}(\lambda)=0$ in every half-plane $\operatorn... | 2 | https://mathoverflow.net/users/85336 | 353677 | 149,427 |
https://mathoverflow.net/questions/353421 | 1 | Consider the Banach space $\mathcal K=S\_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$. I am looking for an example of two pairs of sequences $\{T^{(i)}\},\{\tilde T^{(j)}\}$ and $\{S^{(i)}\},\{\tilde S^{(j)}\}$ in the unit ball of $\mathcal K$ and an anti-linear operator $\phi:\mathcal K\to \mathcal K$ suc... | https://mathoverflow.net/users/145729 | Sequence of Hilbert Schmidt operators | The limits are always the same.
As $\mathcal K = S\_2(H)$ is a Hilbert space, and as only the Schur product is involved anywhere, we can infact identify $\mathcal K$ with $\ell^2 = \ell^2(\mathbb N)$ and consider the pointwise product of vectors in $\ell^2$.
Let $x=(x\_r)\in\ell^2$ and let $(y^{(i)})$ be a bounded ... | 0 | https://mathoverflow.net/users/406 | 353679 | 149,428 |
https://mathoverflow.net/questions/353673 | 5 | Let $G = \mathrm{PSL}(3,q)$ for $q$ odd. I am trying to understand a question that involves understanding the subgroups that contain a Sylow $2$-subgroup, and in particular, are subgroups of odd index in $G$. I need to find a complete description of the maximal subgroups of odd index in the group $G = \mathrm{PSL}(3,q)... | https://mathoverflow.net/users/92488 | Maximal subgroups of odd index in $\mathrm{PSL}(3,q)$ | The subgroups of ${\rm PSL}\_3(q)$ for odd $q$ were first enumerated by H.H. Mitchell in 1911. (The case $q$ even was done by R.W. Hartley in 1925/6.)
Table 8.3 of the book "The Maximal Subgroups of Low-Dimensional Finite Classical Groups" by Bray, Holt and Roney-Dougal provides a convenient list. Using that it is no... | 8 | https://mathoverflow.net/users/35840 | 353680 | 149,429 |
https://mathoverflow.net/questions/353682 | 19 | Consider the following two-player pebble game. We have finitely
many stones on a finite linear track of squares. We take turns, and
the allowed moves are:
* move any one stone one square to the left, if that square is empty, or
* remove any one stone, or
* remove any two adjacent stones.
Whoever takes the last ston... | https://mathoverflow.net/users/1946 | What is the winning strategy in this pebble game? | The positions which are a win for the second player are those with:
* an even number of pebbles in odd-numbered squares, and
* an even number of pebbles in even-numbered squares.
Indeed, from a position in this set $P$, any move will be to a position not in that set, whereas from a position not in that set one can ... | 32 | https://mathoverflow.net/users/17064 | 353684 | 149,430 |
https://mathoverflow.net/questions/353681 | 7 | For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries.
When doing online search I always land at complex analysis and curves related to conformal maps.
Explicit formulas or recipes for generating such curves would be great; even... | https://mathoverflow.net/users/31310 | Examples of complicated parametric Jordan curves | Some examples and references are mentioned here [Examples of plane algebraic curves](https://mathoverflow.net/questions/352957/examples-of-plane-algebraic-curves). You can find many Jordan curves in the family $e^{it}+re^{int}, 0\leq t\leq 2\pi,$ by choosing parameters properly.
To generalize this, take any polynomia... | 9 | https://mathoverflow.net/users/25510 | 353686 | 149,431 |
https://mathoverflow.net/questions/353668 | 1 | I am trying to understand a part of the proof of an extension of Azuma's inequality, where there is a small failure probability, as it appears in proposition 34 in "Random matrices: universality of local spectral statistics of non-hermitian matrices" by Terence Tao and Van Vu.
Here's the url for Arxiv:
<https://arxi... | https://mathoverflow.net/users/152905 | Proof for an extension of Azuma's inequality | The statement "$Y'$ satisfies the condition of Azuma’s inequality" is incorrect in general if it is supposed to mean, for instance, that $C'\_i\le C\alpha\_i$ for some real constant $C$ not depending on the distribution of $Y$, where $C'\_i=C'\_i(\xi)$ is defined similarly to $C\_i(\xi)$ in definition (4.1) on page 28 ... | 0 | https://mathoverflow.net/users/36721 | 353692 | 149,432 |
https://mathoverflow.net/questions/29054 | 25 | **L1 distance between gaussian measures: Definition**
Let $P\_1$ and $P\_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m\_1,C\_1$ and $m\_0,C\_0$ (I assume matrices have full rank). I know that calculating the L1 distance between $P\_1$ and $P\_0$:
$$d\_1=\int|dP\_1-dP\_0|$$
**The ... | https://mathoverflow.net/users/6531 | L1 distance between gaussian measures | Explicit upper and lower bounds are obtained in Theorem 1.2 and Proposition 2.1 of [The total variation distance between
high-dimensional Gaussians](https://arxiv.org/pdf/1810.08693.pdf).
| 2 | https://mathoverflow.net/users/78539 | 353698 | 149,435 |
https://mathoverflow.net/questions/352980 | 26 | I was explaining to my students the other day why $\cos(2x)$ is not a linear combination of $\sin(x)$ and $\cos(x)$ over $\mathbb{R}$. Besides the canonical method of using special values of sine and cosine, I noticed something interesting. In the following, all vector spaces are over $\mathbb{R}$.
Consider the line... | https://mathoverflow.net/users/40789 | Linear combination of sine and cosine | As noted by the OP we can replace $f$ by $af(bx)$ for suitable $a,b\in\mathbb{R}$ so that wlog we can take $c=1$ and ensure that $\sup f=-\inf f=1$.
Firstly we note that $f(z)$ is infinitely differentiable on $\mathbb{R}$ so we can form the taylor series at 0, $f(z)=\sum\_{i=0}^{\infty}\frac{f^{(i)}(0)}{i!}z^i$. Sinc... | 7 | https://mathoverflow.net/users/7113 | 353712 | 149,440 |
https://mathoverflow.net/questions/353691 | 4 | I want to show that simplicial abelian sheaves are fibrant. For this, I wonder whether a morphism between simplicial sheaves is a fibration iff it has RLP w.r.t. all morphisms like
$$\Lambda^n\_k\times X\to\triangle^n\times X$$
where $X\in Sm/k$. Is this claim true?
Here weak equivalences are stalkwise weak equivalen... | https://mathoverflow.net/users/149491 | Are simplicial abelian sheaves fibrant? | Fibrant in what model structure?
Simplicial abelian sheaves (and presheaves) are fibrant
in the projective model structure because
all simplicial abelian groups are fibrant.
Simplicial abelian sheaves are definitely
not fibrant in the local projective model structure
because sheaf cohomology groups can be nontrivia... | 5 | https://mathoverflow.net/users/402 | 353717 | 149,442 |
https://mathoverflow.net/questions/353287 | 21 | Say $p$ is a polynomial of degree $k$ in $\mathbb C[x]$. Then $p$ can have at most $k$ distinct roots. A somewhat obtuse way to state that is to say that among any set of $k+1$ distinct complex numbers, there must exist a value $a$ for which $p(a)\neq 0$.
The question here has to do with generalizing the above fact t... | https://mathoverflow.net/users/2502 | Smallest $S\subset \mathbb C$ on which no degree $k$ polynomial always vanishes? | This is probably just another way to present Fedor Petrov's solution: Expand
$$\frac{(1-t \alpha\_1) (1-t \alpha\_2) \cdots (1-t \alpha\_{n+k-1})}{(1-t \beta\_1)(1 - t \beta\_2) \cdots (1-t \beta\_n)}$$
as a formal power series in $t$. The coefficient of $t^k$ is a degree $k$ polynomial in the $\beta$'s, which vanishes... | 11 | https://mathoverflow.net/users/297 | 353718 | 149,443 |
https://mathoverflow.net/questions/345586 | 1 | I'm interested in the asympotic behaviour (assuming an exact solution is intractable) of $$I(m,d)=\int\_0^{\infty} \left[ Q(m,x)\right]^d dx$$
for fixed $d \in \mathbb{N}$ (in particular, for $d=3$) and $m\to +\infty$.
Here $Q(m,x) = \frac{\Gamma(m,x)}{\Gamma(m)} $ is the upper regularized gamma function.
[Empir... | https://mathoverflow.net/users/5428 | Asympotics of $\int_0^{\infty} \left[ Q(m,x)\right]^d dx$ | Short:
The answer for $d=3$ is $a=\frac{3}{2\sqrt{\pi}}=0.846283 \dots$.
Long:
The integral is splitted in two at $x = m$.
$$
\int\_{0}^{m} \ Q(m,x)^d \ dx + \int\_{m}^{\infty} \ Q(m,x)^d \ dx
$$
For the asymptotic calculation of the integrals two approximations of the reguralized gamma function, $Q(m,x)$, for la... | 3 | https://mathoverflow.net/users/37436 | 353725 | 149,445 |
https://mathoverflow.net/questions/353553 | 2 | Consider dynamical systems $(X,T)$ where $X$ is a compact metric space, $T:X\rightarrow X$ is continuous, the system is minimal and finally, $0<h\_{\rm{top}}(X)<\infty$. I am looking for examples of such systems that do not admit a measure of maximal entropy (mme). Non-minimal topological systems without mme are easy t... | https://mathoverflow.net/users/128556 | Examples of minimal topological systems which are not intrinsically ergodic | The question
[Transitive shifts with multiple fully supported MMEs](https://mathoverflow.net/questions/43564/transitive-shifts-with-multiple-fully-supported-mmes)
that @DanRust mentioned above also discusses MME's, but it is concerned with a different way that a system may fail to be intrinsically ergodic, i.e. havin... | 0 | https://mathoverflow.net/users/128556 | 353727 | 149,447 |
https://mathoverflow.net/questions/353722 | 5 | **Edit:** It turns out that this is equivalent to the RH which gives the idea that this might a *a little* difficult to show. As such we could consider an even simpler case in which the number $n$ is squarefree (all values $k\_j$ are equal to $1$. In previous papers it has been shown that squarefree numbers satisfy Rob... | https://mathoverflow.net/users/120654 | Proving a specific case of Robin's Inequality | By Theorem 1.2 in [this paper](https://jtnb.centre-mersenne.org/item/?id=JTNB_2007__19_2_357_0), Robin's Inequality is true for every odd integer $n>10$. If we knew what the OP wants to prove, then we would also know Robin's Inequality for every integer $n$ whose odd part exceeds $5040$. In particular, we would know Ro... | 6 | https://mathoverflow.net/users/11919 | 353733 | 149,448 |
https://mathoverflow.net/questions/353719 | 3 | Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be *entropy-minimal* if every subshift $Y \subsetneq X$ satisfies that $$h\_{\mathrm{top}}(Y) < h\_{\mathrm{top}}(X).$$ Equivalently, $X$ is entropy-minimal if for every word $\omega \in \mathcal{L}(X)$ the subshift $$ X\_{\omega}... | https://mathoverflow.net/users/10518 | Entropy-minimal subshifts | Let $f$ be a sublinear function that tends to infinity, such as $f(n) = \sqrt{n}$. Define $X \subset \{0,1,2\}^{\mathbb{N}}$ by forbidding all long enough words $w$ with more than $f(|w|)$ occurrences of $2$. Then $X$ has entropy $\log 2$ and is mixing, and properly contains the binary full shift, which likewise has en... | 6 | https://mathoverflow.net/users/66104 | 353765 | 149,459 |
https://mathoverflow.net/questions/353762 | 5 | Let $X$ be the [classifying space](https://en.wikipedia.org/wiki/Classifying_space) of the [Higman group](https://en.wikipedia.org/wiki/Higman_group) $G$. It is well known that $G$ is an acyclic group
$$H\_{\ast}(X;\mathbb{Z})=H\_{\ast}(pt;\mathbb{Z}).$$
Now, suppose that $\mathcal{M}$ is a [local system](https://en... | https://mathoverflow.net/users/141953 | Trivial homology with local system | For $X = BG$ local systems on $X$ can be identified with $G$-modules, and homology with the derived tensor product $-\otimes^L\_{\mathbb ZG}\mathbb Z$, i.e. $H\_i(X;M) \cong \operatorname{Tor}^i\_{\mathbb Z G}(M,\mathbb Z)$. One way to see this is to take the definition $H\_i(X;M):= H\_i(\mathcal S\_\*(\widetilde X)\ot... | 14 | https://mathoverflow.net/users/35687 | 353777 | 149,464 |
https://mathoverflow.net/questions/353778 | 2 | I have a question on a certain property of morphisms between schemes endowed with Galois action. The motivation arises from a comment by Phil Tosteson on [this question](https://mathoverflow.net/questions/353622/application-of-galois-descent).
Phil wrote: "If the map factors through the projection, it factors uniquel... | https://mathoverflow.net/users/108274 | Galois action on morphism between $\overline{k}$ schemes | The first action is the usual one.
Actually, your second action is not well-defined. By definition, $\bar{X}(\bar{k})$ is the set of morphisms $\alpha:\operatorname{Spec}\bar{k}\to \bar{X}$ such that $\pi\_2\circ\alpha$ is the identity on $\operatorname{Spec}\bar{k}$, where $\pi\_2:\bar{X}\to\operatorname{Spec}\bar{k... | 4 | https://mathoverflow.net/users/137902 | 353783 | 149,466 |
https://mathoverflow.net/questions/353788 | 5 | This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the [first part](https://math.stackexchange.com/questions/3562213/karoubi-envelope-idempotent-completion-of-r-mod) we considered the interpretation o... | https://mathoverflow.net/users/108274 | Motivation for Karoubi envelope/ idempotent completion | The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Moreover, over any field the category of numerical motives is abelian semi-simple.
The proof of the latter statement is ... | 3 | https://mathoverflow.net/users/2191 | 353791 | 149,467 |
https://mathoverflow.net/questions/342823 | 3 | Let $\psi$ be an smooth admissible Shearlet with compact support, cand let $\mathcal{M}$ be a bounded region in $\mathbb{R}^2$ and let $m= \chi\_{\mathcal{M}}$ be the characteristic function of $\mathcal{M}$.
Now define
$$
m\_h = \mathcal{F}^{-1}[\hat{m}\chi\_h],
$$
where $h = \{(\xi\_1,\xi\_2)\in \mathbb{R}^2: |\xi... | https://mathoverflow.net/users/114299 | $f \in L^p(\mathbb{R}^2)$ for all $p \geq 1$, and $f$ has zero integral. What can we say about this function's fourier Transform? | Let $f$ be in $L^1(\mathbb R^n)$, then $\hat f$ belongs to $L^\infty(\mathbb R^n)$ and is (uniformly) continuous with $\lim\_{\vert \xi\vert\rightarrow +\infty} \hat f(\xi)=0$: this is the Riemann-Lebesgue Lemma.
If $f$ belongs to $L^p(\mathbb R^n)$, for some $p\in [1,2]$, then $\hat f$ belongs to $L^{p'}(\mathbb R^n)$... | 3 | https://mathoverflow.net/users/21907 | 353792 | 149,468 |
https://mathoverflow.net/questions/353789 | 3 | The following question naturally came up when dealing with 4-rank of certain class groups. In this case I want to inductively deal with some Legendre symbols, and to do so I want my squarefree integers to be "decently" spaced in the sense below.
Is there an absolute constant $C > 0$ such that for all functions $f$ go... | https://mathoverflow.net/users/96891 | Spacing of prime divisors | This is not true for the all $k<r$ problem. Consider random $n$ below $x$, and put $z=\log x$. How many prime factors would a random number have in $[z,z^e]$? This is approximately Poisson with parameter $\sum\_{z <p \le z^e} 1/p \approx 1$. So with positive probability you would find numbers with as many prime factors... | 3 | https://mathoverflow.net/users/38624 | 353794 | 149,470 |
https://mathoverflow.net/questions/353775 | 3 | In 1945 Wiman [W] showed that certain elliptic curves $E$ over $\mathbf Q$ have rank\* at least 4. (It seems this was the highest known rank of an elliptic curve over $\mathbf Q$ until 1974, when Penney--Pomerance found a curve of [rank at least 6](https://web.math.pmf.unizg.hr/~duje/tors/rk6.html).)
The method of h... | https://mathoverflow.net/users/122997 | Wiman's method for bounding the rank of an elliptic curve | I have not read all the details of the article, but most of what I see is just descent by the isogeny $[2]$. The map is the Kummer map $$E(\mathbb{Q})\to \mathbb{Q}^{\times}/\square \times \mathbb{Q}^{\times}/\square \times \mathbb{Q}^{\times}/\square$$ which sends $(x,y)$ to $(x-e\_1,x-e\_2,x-e\_3)$. In modern terms t... | 3 | https://mathoverflow.net/users/5015 | 353796 | 149,471 |
https://mathoverflow.net/questions/353790 | 7 | I apologize in advance if this question is basic.
If $P\_{\bullet}$ is a perfect complex over say a ring $R$ such that
1. $H\_{i}(P\_{\bullet})=0 $ if $i\neq n$
2. $H\_{i}(P\_{\bullet})=E$ if $i=n$
is $E$ a finitely generated $R$-module ?
What can we say about the homology of a generic perfect complex in gen... | https://mathoverflow.net/users/129583 | Homology of perfect complexes | Yes. Let $... 0\to P\_r \to ... \to P\_0 \to 0 ...$ be a complex of projective modules of finite type and denote by $Z\_\*$ the cycles. If $n=0$ it is clear. If not, $0\to Z\_1\to P\_1\to P\_0\to 0$ is exact and so $Z\_1$ is projective and of finite type. Then if $n=1$, $H\_1(P)$ is of finite type. If $n\neq 1$, $0\to ... | 5 | https://mathoverflow.net/users/92322 | 353799 | 149,472 |
https://mathoverflow.net/questions/353319 | 0 | Let $X\_s(\omega)$ be measurable and adapted.
Under what conditions will the process
$$
F\_{t}(\omega) = \int\_0^t X\_s(\omega) \, ds
$$ also be adapted?
To me it seems that adaptedness and measurability should be enough but at the bottom of page 133 in Karatzas and Shreve they say this is not enough. Why?
| https://mathoverflow.net/users/nan | Is the integral of an adapted, measurable process adapted? | Assuming that $\int\_0^t|X\_s(\omega)|\,ds<\infty$ for all $t>0$ and all $\omega$, and that the filtration satisfies the usual conditions, the process $F\_t:=\int\_0^t X\_s\,ds$ is well defined and adapted (even predictable, being continuous). This matter is discussed in the paper "Un exemple de processus mesurable ada... | 2 | https://mathoverflow.net/users/42851 | 353808 | 149,473 |
https://mathoverflow.net/questions/353806 | 6 | The title pretty much sums it up.
More in detail. Let $C$, $D$ and $E$ be categories, let $F:C\to D$ and $G:C\to E$ be functors, and let $P:C^{op}\to \mathrm{Set}$ be a presheaf. The colimit of $F$ in $D$ satisfies
$$
D(\mathrm{colim} \,F, d) \cong [C,D](F, d)
$$
for each object $d$ of $D$, where in the right-hand s... | https://mathoverflow.net/users/30366 | Is there such a thing as a weighted Kan extension? | Yes. Given $F:C\to D$ and a [profunctor](https://ncatlab.org/nlab/show/profunctor) $H:E$ ⇸ $C$, i.e. a functor $H : C^{\rm op}\times E\to \rm Set$ (or to the enriching category $V$), the $H$-weighted colimit of $F$ is the functor $L : E \to D$ such that each value $L(e)$ is the $W(-,e)$-weighted colimit of $F$ (in a co... | 6 | https://mathoverflow.net/users/49 | 353817 | 149,477 |
https://mathoverflow.net/questions/353804 | 7 | This is an off-shot from [my previous post](https://mathoverflow.net/questions/353715/an-identity-for-polynomials-over-partitions) on MO.
Given an integer partition $\lambda=(\lambda\_1,\dots,\lambda\_{\ell(\lambda)})$ of $n$, denote $\ell(\lambda)$ to be the length of $\lambda$.
Let $r\_2(n)$ denote the number of ... | https://mathoverflow.net/users/66131 | Sum of squares and partitions | Start by checking that the following formal product can be expanded as a sum over partitions
$$\prod\_{i\geq 1}\left(1+\sum\_{r\geq 1}a\_r(x\_1x\_2\cdots x\_i)^r\right)=\sum\_{\lambda}\left(\prod\_{j\geq 1}a\_{\lambda\_j-\lambda\_{j+1}}\right)\left(\prod\_{j\geq 1}x\_j^{\lambda\_j}\right)$$
with the convention that $a\... | 15 | https://mathoverflow.net/users/2384 | 353823 | 149,480 |
https://mathoverflow.net/questions/353828 | -1 | Coming from a non-group theory background, I noticed that the finite groups I was dealing with seem to all have the following property. Let $G$ be a finite group, $H$ a subgroup. Then the normalizer $N\_G(N\_G(H))$ of the normalizer of $H$ is just $N\_G(H)$. It seems to be an exercise in almost any group theory book th... | https://mathoverflow.net/users/142072 | Normalizer of a Normalizer of a subgroup of a finite group with no elements of order $p^2$ | No, take the wreath product $G$ of $C\_3$ and $C\_2$, two cyclic groups of orders 3 and 2. It has order $18$, and an abelian normal subgroup $A=C\_3\times C\_3$. Let $H$ be one of the factors $C\_3$ in $A$. Then $N(H)=A, N(N(H))=G$ and $G$ does not have elements of order $p^2$ for any $p>1$.
| 3 | https://mathoverflow.net/users/nan | 353830 | 149,482 |
https://mathoverflow.net/questions/353849 | 4 | Let $x\_0, x\_1, \ldots x\_{n-1}$ be arbitrary vectors in a complex Hilbert space. Define the $n \times n$ symmetric real matrix $M$ by $M\_{ij} = \lvert \langle x\_i, x\_j \rangle \rvert^2$. Must $M$ be positive semidefinite?
| https://mathoverflow.net/users/60487 | Positive-semidefiniteness of a Gram-like matrix | Yes. The matrix $A$ with $a\_{i,j} = \langle x\_i,x\_j \rangle$ is a Gram matrix and thus positive semidefinite, so $A^T = \overline{A}$ is positive semidefinite too. It then follows from the [Schur product theorem](https://en.wikipedia.org/wiki/Schur_product_theorem) that your matrix $M = A \circ \overline{A}$ (where ... | 8 | https://mathoverflow.net/users/11236 | 353850 | 149,486 |
https://mathoverflow.net/questions/353760 | 1 | Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) The *score* of each vertex $x$ is the minimum number of edges that need to be deleted so that $x$ cannot reach some other vertex $y$. (So... | https://mathoverflow.net/users/83212 | Size of minimum cut in random graph | Yes, the limit goes to $1$.
An observation: by the Chernoff bound, with (very) high probability all vertices have degree between $0.49n$ and $0.51n$, so let's assume this holds in the following.
First, let's see that the number of edges needed to remove all directed paths from a given $x$ to a given $y$ is either t... | 1 | https://mathoverflow.net/users/36212 | 353854 | 149,488 |
https://mathoverflow.net/questions/353860 | 4 | Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}\_{i=1}^n$ be normal coordinates centered around $p$.
Using Jacobi field, one can show that the metric $g$ has the following Taylor expansion
\begin{align}
g\_{ij}(x)&=\delta\_{ij}-\frac{1}{3}R\_{ipqj}(p)x^px^q-\frac{1}{6}\nabla\_r... | https://mathoverflow.net/users/137708 | Taylor expansion of determinant of Riemannian metric in normal coordinates up to higher order | After dropping the first order terms using the normal coordinate condition,
$$\partial^4\_{ijkl} \det(g) = \partial^3\_{ijk} (g^{-1} \partial\_l g) = g^{-1} \partial^4\_{ijkl} g + ( \partial^2\_{ij} g^{-1} \partial^2\_{kl} g + \partial^2\_{ik} g^{-1} \partial^2\_{jl} + \partial^2\_{jk} g^{-1} \partial^2\_{il} g)$$
... | 4 | https://mathoverflow.net/users/3948 | 353862 | 149,492 |
https://mathoverflow.net/questions/353715 | 3 | Given an integer partition $\lambda=(\lambda\_1,\dots,\lambda\_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its [conjugate partition](http://mathworld.wolfram.com/ConjugatePartition.html) $\lambda'$. Denote by $\lambda''=\lambda',0$ found by appending one extra zero at the right ... | https://mathoverflow.net/users/66131 | An identity for polynomials over partitions | Yes, your identity $(1)$ is true. We can give a proof as follows:
Let's denote the left hand side of your identity $(1)$ by $A\_n(q)$. Starting with the identity
$$\prod\_{i\geq 1}\left(1+\sum\_{r\geq 1}a\_r(x\_1x\_2\cdots x\_i)^r\right)=\sum\_{\lambda}\left(\prod\_{j\geq 1}a\_{\lambda\_j-\lambda\_{j+1}}\right)\left(... | 5 | https://mathoverflow.net/users/2384 | 353865 | 149,494 |
https://mathoverflow.net/questions/353801 | 2 | This question basically follows [this earlier question of mine](https://mathoverflow.net/q/353184/8133) but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA\_0$. For $X$ a Turing ideal we get the map $c\_X$ on $2^\omega$ given by $c\_X(x)=[b[x]]\cap X$ where $b$ is some computable... | https://mathoverflow.net/users/8133 | Detecting comprehension topologically | Statements about existence of $\omega$-models can be topologically detected.
Specifically, fix $X$ a Turing ideal. For $t\in X$ say that $t$ *enumerates a family of sets* if:
* Exactly one $p\in c\_X(t)\cap X$ has $c\_X(p)=X$.
* For every other $q\in c\_X(t)$ we have $c\_X(q)=\{a\}$ for some $a\in y$.
* For each $a... | 0 | https://mathoverflow.net/users/8133 | 353870 | 149,497 |
https://mathoverflow.net/questions/353878 | 14 | Let's consider square matrices $A\_{n \times n}$, $B\_{n \times n}$ and $X\_{n \times n}$ with elements from $\mathbb{R}$. Could you tell me please, what would be the necessary conditions for the existence of solution (may be not unique) of Sylvester equation:
$$
AX=XB.
$$
As I know, sufficient condition looks like (bu... | https://mathoverflow.net/users/152731 | Necessary conditions for the existence of solution of Sylvester equation AX=XB | This equation always has *a* solution: $X = O$. I'll assume throughout this answer that you're interested in a non-zero solution.
The equation $AX = XB$ is equivalent to $(A \otimes I - I \otimes B^T)\mathbf{x} = \mathbf{0}$, where $\otimes$ denotes the Kronecker product and $\mathbf{x}$ is the vectorization of $X$. ... | 32 | https://mathoverflow.net/users/11236 | 353880 | 149,498 |
https://mathoverflow.net/questions/353874 | 5 |
>
> I am looking for a smooth **closed** 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is fixed.
>
>
>
*A naive attempt:*
If we consider the action given by $\mathbb{S}^1$ on $\mathbb{S}^3\subset \... | https://mathoverflow.net/users/99042 | Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it | Such a closed $4$-manifold does not exist, and this follows from:
Church, P., & Lamotke, K. (1974). Almost free actions on manifolds. Bulletin of the Australian Mathematical Society, 10(2), 177-196
Let me present the argument anyway. The answer breaks down into a local and global part. The local question is to unde... | 15 | https://mathoverflow.net/users/66405 | 353884 | 149,500 |
https://mathoverflow.net/questions/353832 | 3 | Let $M=B \times\_f F$ be a warped product of two pseudo-Riemannian manifolds. If $X, Y, Z \in L(B)$ and $U, V, W \in L(F)$, (with $L(B)$ and $ L(F)$ I mean the set of all horizontal and vertical lift to M), then we have:
(1) $R(V, W)U = {R(V, W)U}^F +
\frac{\langle \nabla f, \nabla f \rangle }{f^2}(\langle V, U \ran... | https://mathoverflow.net/users/111304 | From Riemannian curvature to Ricci curvature in warped product manifold | Using (1), the relevant trace is the following, where $e\_1, \dots, e\_k$ is an orthonormal frame on $F$:
\begin{align\*}
\mathrm{Ric}(V,W) &= \cdots - \sum\_{i=1}^k \langle e\_i, R(V,e\_i)W\rangle\\
&=
\cdots - \sum\_{i=1}^k\frac{\langle\nabla f,\nabla f\rangle}{f^2}(\langle e\_i,e\_i\rangle\langle V,W\rangle - \langl... | 2 | https://mathoverflow.net/users/613 | 353885 | 149,501 |
https://mathoverflow.net/questions/353888 | 11 | Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a functor from local Artin CDGA's to homotopy types satisfying a certain sheaf condition. If the commutativity condition is weak... | https://mathoverflow.net/users/7108 | Connectedness, loops and formal moduli problems | The presentation of the formal moduli problems story in Gaitsgory-Rosenblyum *A Study in Derived Algebraic Geometry, Vol 2* may be what you are looking for. We review it here (in the case over $\mathrm{Spec}\, k$ for a field $k$ of characteristic zero, that the question concerns):
**1. Looping/delooping equivalence i... | 11 | https://mathoverflow.net/users/39713 | 353897 | 149,506 |
https://mathoverflow.net/questions/353844 | 3 | Suppose we have a bounded linear operator $A = A(\gamma):H\_1\to H\_2$ where $H\_1$ and $H\_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to
$$
(I-A)x = y.
$$
If $\|A\|<1$ we can use a Neumann series expansion and get a series representation:
\begin{align}
x & = (I-A)^{-1... | https://mathoverflow.net/users/152373 | Are there any techniques that can be used in the case when a Neumann series doesn't converge? | If the spectrum of $A$ is contained in a disk $\{z: |z - a| \le r\}$ where $|1-a| > r$, then the series $\sum\_{n=0}^\infty (1-a)^{-1-n} (A - a I)^n$ converges to $(I-A)^{-1}$.
| 4 | https://mathoverflow.net/users/13650 | 353899 | 149,507 |
https://mathoverflow.net/questions/353853 | 4 | This question is a reference request for the following result or two results, which I believe are rather easy to prove.
Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\subset\mathcal K$ be a coreflective full subcategory. Assume that the coreflector $C\colon\mathcal K\to \mathcal A$ is an ac... | https://mathoverflow.net/users/2106 | Coreflective subcategories in Grothendieck/locally presentable categories | Since a coreflective full subcategory is the category of coalgebras for the induced idempotent comonad, 1. is answered in [presentability rank of categories of coalgebras](https://mathoverflow.net/questions/350351/presentability-rank-of-categories-of-coalgebras) (the corresponding comonad is accessible).
| 5 | https://mathoverflow.net/users/73388 | 353903 | 149,510 |
https://mathoverflow.net/questions/353882 | 1 | Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{\mathbb{C}^{\times}})) \cong (\mathbb{C}^{\times})^{2g}$, where $\underline{\mathbb{C}^{\times}}$ is the constant sheaf on... | https://mathoverflow.net/users/94696 | Map from local systems to holomorphic line bundles on a curve | I think the following theorem answers your question.
>
> **Theorem:** Let $X$ be a smooth, proper connected curve over $\mathbf C$ with a line bundle $\mathscr L$. Then $\mathscr L$ admits a flat connection $\nabla$ if and only if $c\_1(\mathcal L)=0$.
>
>
>
**Remark:** A more general statement would be that ... | 4 | https://mathoverflow.net/users/115211 | 353907 | 149,512 |
https://mathoverflow.net/questions/353906 | 3 | Let $X$ be a very general surface of degree $\ge 5$ in $\mathbb{P}^3$ and $ Y$ is arbitrary irreducible cubic hypersurface. Is $X \cap Y$ reduced ?
| https://mathoverflow.net/users/130022 | Reducedness of complete intersection | Yes (of course, $Y$ should be reduced). Since $X$ is very general we may assume $\operatorname{Pic}(X)=\mathbb{Z}\cdot [\mathscr{O}\_X(1)]$. If the divisor $Y\_{|X}$ on $X$ is not reduced, it is of the form $2H+H'$, where $H$ and $H'$ are hyperplane sections of $X$ (possibly equal). But since the restriction map $H^0(\... | 8 | https://mathoverflow.net/users/40297 | 353908 | 149,513 |
https://mathoverflow.net/questions/353915 | -2 | Before the current problem I work on, I proved the following:
>
> Let $q$ be a polynomial with $\deg(q) \le n$. If $q(x)=o(x^n)$ for $x \to 0$, then $q$ is the zero polynomial.
>
>
>
I **have** to use the above for the problem I'm working on currently now, which is
>
> Let $f:I \to \mathbb{R}$ be a $C^n$ ... | https://mathoverflow.net/users/153006 | Proof of: If $f(x)=p(x)+o(x^n)$ for $x \to 0$, then $b_{k}=\frac{f^{(k)}(0)}{k !} $ for $ k=0,1, \ldots, n$ | Let $T$ be the [Taylor polynomial](https://en.wikipedia.org/wiki/Taylor%27s_theorem#Statement_of_the_theorem) for $f$ of order $n$ at $0$, so that $f(x)=T(x)+o(x^n)$ (as $x\to0$). Comparing this with the condition $f(x)=p(x)+o(x^n)$, we see that $(p-T)(x)=o(x^n)$. Using now what you have proved, we see that $p=T$, and ... | 2 | https://mathoverflow.net/users/36721 | 353919 | 149,517 |
https://mathoverflow.net/questions/353916 | -1 | In my recent work I stumbled across a problem of this type:
G with two partial oders $\leq$ and $\preceq$ on every set, i.e. for every $n \in \mathbb{N}$ $A\_n \subset A\_{n+1}$ and $(A\_n, \leq) $ and $(A\_n, \preceq)$ are partially ordered sets (partial orders do not depend on $n$).
The question now is whether th... | https://mathoverflow.net/users/153033 | (maximal) antichains with respect to two different partial orders on the same set | Almost a trivial counterexample:
Let $A\_n$ be $\{0,\dots,n+2\}$ with one order being the usual $\leq$, that is a linear order, and another being $=$, that is the discrete order.
Note that all the $A\_n$ have at least two elements. But the only sets which are antichains in *any* $A\_n$ are singletons, which are nev... | 0 | https://mathoverflow.net/users/7206 | 353924 | 149,518 |
https://mathoverflow.net/questions/353891 | 3 | Suppose $f$ is a normalized cuspidal eigenform of level $p^2N$ ($p\nmid N$) and trivial character, such that the corresponding representation at $p$ is supercuspidal. Now suppose $\chi$ is primitive Hecke character of conductor $p$. We can apply the usual twisting operator by $\chi$ or $\chi^{-1}$ to $f$ to obtain norm... | https://mathoverflow.net/users/108548 | Atkin-Lehner operator on supercuspidals | What you are asking for is a formula for the local epsilon-factors $\varepsilon(\pi \otimes \chi)$ where $\pi = \pi\_{f, p}$ is the local component of $f$ at $p$. This is a deep question: it has to be, in some sense, since you can recover $\pi$ uniquely if you know the epsilon-factors of all its twists (Jacquet's local... | 4 | https://mathoverflow.net/users/2481 | 353928 | 149,520 |
https://mathoverflow.net/questions/353342 | 6 | In a [recent question on MSE](https://math.stackexchange.com/questions/3537191/inner-automorphisms-of-group-algebras-vs-inner-automorphisms-of-the-group) I asked about conditions under which the canonical morphism $Out(G) \to Out(k[G])$ is injective.
>
> Is it true that this morphism is indeed injective if $G$ is f... | https://mathoverflow.net/users/3041 | Inner automorphisms of group algebras vs. inner automorphisms of the group | The question for finite $G$ and $k = \mathbb{Z}$ is the **normalizer problem**, see [1, Section 1]. By a result of Jan Krempa, the kernel of the cannonical morphism is in that case always an elementary abelian $2$-group. As far as I know, there is basically only one example known where the kernel is non-trivial [1, The... | 4 | https://mathoverflow.net/users/153043 | 353934 | 149,524 |
https://mathoverflow.net/questions/353773 | 1 | Let $D$ be a linear differential operator on $\mathcal{C}^\infty(\mathbb{R})$, and let $\mathcal{E}\_\lambda=\{f\in\mathcal{C}^\infty(\mathbb{R})|Df=\lambda f\}$ be the space of eigenfunctions of $D$ to the eigenvalue $\lambda$. It is easy to see that $\bigcup\_{\lambda}\mathcal{E}\_\lambda$ can be characterized by the... | https://mathoverflow.net/users/45250 | Differential equation satisfied by linear combinations of eigenfunctions of linear differential operator | If $f$ is a linear combination of at most $N$ eigenfunctions, then $f$,$Df$,$D^2f$,...,$D^Nf$ are linearly dependent. Hence $W(f,Df,...,D^Nf)=0$, where $W$ denotes the Wronskian.
| 3 | https://mathoverflow.net/users/12120 | 353940 | 149,527 |
https://mathoverflow.net/questions/353905 | 27 | Let $G = \{ g\_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is
$$ G! = [ \prod\_i g\_{\sigma(i)} | \sigma \in S\_n] $$.
I'm interested in knowing how $G!$ behaves as a set, and also how often does every elemen... | https://mathoverflow.net/users/94076 | Multiplying all the elements in a group | Yes, your $G!$ (as a set) is always either $[G,G]$ (if the order of $G$ is odd, or its Sylow $2$-subgroup is non-cyclic) or $z[G,G]$ if $G$ has cyclic Sylow $2$-subgroup, where $z$ is the involution in the Sylow $2$-subgroup. This was apparently a question/conjecture of [Golomb](https://www.ams.org/journals/bull/1970-7... | 41 | https://mathoverflow.net/users/1392 | 353954 | 149,529 |
https://mathoverflow.net/questions/353965 | 0 | Let ${\frak C} \subseteq {\cal P}({\cal P}(\omega))$ be the collection of all covers of $\omega$ (that is, ${\cal C} \in {\frak C}$ iff $\bigcup {\cal C} = \omega$.)
We define the following binary relation on ${\frak C} = $: For ${\cal A}, {\cal B} \in {\frak R}$ we say ${\cal A} \leq\_\text{r} {\cal B}$ if ${\cal A}... | https://mathoverflow.net/users/8628 | Is this ordering on the set of all covers of $\omega$ a (complete) lattice? | Yes. The l.u.b. of $\mathcal A$ and $\mathcal B$ is $\mathcal A \cup \mathcal B$. The g.l.b. of $\mathcal A$ and $\mathcal B$ is $\{A\cap B: A\in\mathcal A , B\in\mathcal B\}$.
We can even generalize by replacing $\subseteq$ by an arbitrary meet-semilattice preordering, in which case
* The l.u.b. of $\mathcal A$ a... | 3 | https://mathoverflow.net/users/4600 | 353967 | 149,534 |
https://mathoverflow.net/questions/353608 | 5 | Set $\mathcal{F}:=\{ A \in \text{SL}\_2(\mathbb{R}) \, | \, Ae\_1 \in \operatorname{span}(e\_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and
$\mathcal{NC}:=\{ A \in M\_2(\mathbb{R}) \, | \det A \ge 0 \, \,\text{ and } \, A \text{ is not conformal} \,\}$.
By a non-conformal matrix, I mean a matr... | https://mathoverflow.net/users/46290 | Is this subset of matrices contractible inside the space of non-conformal matrices? | **Edited.** In the first version of the answer I was assuming that the space in which the contraction was taking place was not $\cal NC$ but the complement to non-conformal matrices in $SL(2,\mathbb R)$. I'll suggest a fix for this now.
Note, that we have a natural continuous map $u: {\cal NC}\to S^1=\mathbb RP^1$. N... | 2 | https://mathoverflow.net/users/943 | 353972 | 149,538 |
https://mathoverflow.net/questions/353960 | 5 | $\mathcal{O}$ notation describes an onto function $f:\mathcal{O} \rightarrow \omega\_{CK}$. In calculating all values $n \in \mathbb{N}$ such that $f(n)=\alpha$, when $\alpha$ is a limit, all indexes $e$ of ordinary programs are considered such that $\phi\_e(i)=n\_i$ (with $i \in \mathbb{N}$). The values $n\_i$ must sa... | https://mathoverflow.net/users/112385 | How far does this restricted definition on $\mathcal{O}$ goes? | For Q2, the answer is $\omega^2$, for both recursive and primitive recursive notations. It's not hard to see that every ordinal below $\omega^2$ can be reached.
To show that $\omega^2$ cannot be reached, the argument is the same for both primitive recursive and full recursive. For each $a$, we construct a $b$ ensurin... | 5 | https://mathoverflow.net/users/32178 | 353981 | 149,540 |
https://mathoverflow.net/questions/353959 | 2 | Consider a random vector X with the coordinate distribution is uniformly distributed in the set $\{\sqrt{n}e\_i : i = 1,..., n\}$, where $e\_i$ denotes the n-element set of the canonical basis vectors in $R^n$. Show that $ \parallel X \parallel\_{ \psi 2}\asymp \sqrt{\frac{n}{{ log n}}}.$
By the definition of the sub... | https://mathoverflow.net/users/153056 | Show the coordinate distribution has a very large sub-gaussian norm | The subgaussian norm of a real-valued random variable $Y$ is
$$\|Y\|:=\|Y\|\_{\psi\_2}:=\inf\{t>0\colon Ee^{Y^2/t^2}\le2\}.$$
If $Y$ is such that $P(Y=0)<1$ and $Ee^{Y^2/t^2}<\infty$ for all real $t>0$, then
$Ee^{Y^2/t^2}$ continuously decreases in real $t>0$ from $\infty$ to $0$, so that $\|Y\|$ is the unique positi... | 0 | https://mathoverflow.net/users/36721 | 353983 | 149,541 |
https://mathoverflow.net/questions/353950 | 13 | Aleksandrov [A], proved a remarkable property of convex functions.
>
> **Theorem.** If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and a symmetric $(n\times n)$ matrix $D^2f(x)$ such that
> $$
> \lim\_{y\to x}
> \frac{|f(y)-f(x)-Df(x)(y-x)-\fra... | https://mathoverflow.net/users/121665 | Aleksandrov's proof of the second order differentiability of convex functions | The paper [On the second differentiability of convex surfaces](https://link.springer.com/article/10.1007/BF00150866) by Bianchi, Colesanti, and Pucci (*Geometriae Dedicata* volume 60, pages 39–48 (1996)) concerns the proof of the Busemann-Feller-Alexandroff Theorem on the second order differentiability of convex functi... | 11 | https://mathoverflow.net/users/3948 | 353984 | 149,542 |
https://mathoverflow.net/questions/354005 | 1 | This question comes from Huybrechts' lecture notes on K3 surfaces, more specifically, chapter 2.
Let $ X $ be a K3 surface (over an algebraically closed field $ k $) and $ L $ a line bundle on $ X $. The base locus of the linear system $ |L| $ is defined as a closed subscheme of $ X $ by $$ \text{Bs} (L) := \cap\_{s ... | https://mathoverflow.net/users/152391 | Fixed part of a line bundle on a K3 surface | (1) I guess, as a scheme, the base locus might have embedded points. But the fixed part is defined as the pure 1-dimensional part of the base locus scheme.
(2) If $F$ is the fixed part, it means that every divisor in the linear system can be written as
$$
D = D' + F.
$$
One can also assume that $F$ has no common comp... | 5 | https://mathoverflow.net/users/4428 | 354006 | 149,547 |
https://mathoverflow.net/questions/352717 | 3 | A topological group $G$ is defined to be
$\bullet$ *precompact* if for any neighborhood $U\subseteq G$ of the unit there exists a finite subset $F\subseteq G$ such that $G=UF$;
$\bullet$ *narrow* if for any neighborhood $U\subseteq G$ of the unit there exists a countable subset $S\subseteq G$ such that $G=US$;
$\... | https://mathoverflow.net/users/61536 | Is each preseparable topological group narrow? | Jan Pachl has informed me that the answer to this problem is affirmative and can be derived from the following helpful fact, proved in Lemma 3.31 of his book ["Uniform spaces and measures"](https://www.springer.com/gp/book/9781461450573). I also remember that a similar theorem was proved in the book "Topologies on grou... | 1 | https://mathoverflow.net/users/61536 | 354007 | 149,548 |
https://mathoverflow.net/questions/353998 | 1 | This is my first question here, so if I am doing things incorrectly, please let me know. Now on to the question:
The forcing notion $Fn(\kappa,2)$, which constists of partial functions from $\kappa$ to $2$ with finite support, ordered by reverse inclusion, is known to add $\kappa$ new cohen reals.
Are there similar... | https://mathoverflow.net/users/118455 | Are there forcing notions adding $\kappa$ random, sacks, prikry, or Mathias reals? | The reason $Fn(\kappa,2)$ works for adding $\kappa$ new cohen reals is, because there exists a bijection between $\kappa$ and $\kappa\times\omega$, therefore the forcing $Fn(\kappa,2)$ is isomorphic (as a partial order) to $Fn(\kappa\times\omega,2)$ which itself is isomorphic to a Finite Support iteration of Length $\k... | 1 | https://mathoverflow.net/users/138274 | 354010 | 149,549 |
https://mathoverflow.net/questions/353962 | 3 | Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $p(n)$ denote the probability that there is only one vertex with the maximum out-degree. What is $\lim\_{n\rightarrow\infty}p(n)$?
... | https://mathoverflow.net/users/83212 | Unique maximum degree in tournament | I can't find a published proof of this known result, but here is a close miss.
In [this paper](https://core.ac.uk/reader/82426260), page 256, is a short proof that a random undirected graph has a unique vertex of maximum degree almost surely. If you replace "graph" by "tournament" and "degree" by "out-degree", the ex... | 3 | https://mathoverflow.net/users/9025 | 354016 | 149,552 |
https://mathoverflow.net/questions/354018 | 1 | Let $E$ be an elliptic curve over $\mathbb{Q}$. For a prime $p$, let $\mathcal{E}\_p$ denote its Neron model over $\mathbb{Z}\_p$. Also, let $\Phi\_p(E)$ denote the component group of $\mathcal{E}\_p$.
The structure of $\Phi\_p(E)$ is well-known, and I want to study it when $E$ has multiplicative reduction at $p$. F... | https://mathoverflow.net/users/116950 | Elliptic curves and its Neron model | The Galois group of $L/{\mathbb Q\_p}$ acts on the component group over $L$ and the component group over $\mathbb Q\_p$ should be the subgroup of elements fixed by the Galois group.
| 1 | https://mathoverflow.net/users/153093 | 354019 | 149,553 |
https://mathoverflow.net/questions/354017 | 4 | A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $\mathbb{R}$.
Assume that $p(x,y)\in \mathbb{C}[x,y]$ has all its roots in a real half space $K$. Is it true that all criti... | https://mathoverflow.net/users/36688 | A possible generalization of Gauss Lucas theorem to higher dimension | There is a very pretty multivariate extension of Gauss-Lucas proved by Marek Kanter [here](https://arxiv.org/abs/1203.6426) As far as I can tell, the paper has not been published, but the proof of the main theorem is half a page, so...
| 4 | https://mathoverflow.net/users/11142 | 354020 | 149,554 |
https://mathoverflow.net/questions/354025 | 2 | Let $q$ be a prime power. Let $\mathbb{F}\_q$ be the finite field with $q$ elements. Then $\mathbb{F}\_{q^n}$ is a field extension of $\mathbb{F}\_q$ of degree $n$ and can be considered as an $n$-dimensional vector space $V$ over $\mathbb{F}\_q$. Now consider the action of $GL(V)$ on $V$. Any element $x$ in $\mathbb{F}... | https://mathoverflow.net/users/74343 | General linear group action on extensions of finite fields | Let $F$ be any field and $F<E$ a finite field extension.
Fix $x\in E^\*$ and consider the multiplication operator $g\_x\in \text{GL}\_F(E)$,
the group of invertible $F$-linear transformations of $E$.
The centralizer of $g\_x$ could be naturally identified with the subgroup $\text{GL}\_{F[x]}(E)<\text{GL}\_F(E)$, where ... | 2 | https://mathoverflow.net/users/89334 | 354035 | 149,559 |
https://mathoverflow.net/questions/353943 | 2 | I first posted this on [mathematics](https://math.stackexchange.com/questions/3555306/weak-tauberian-theorem). However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought.
Karamata's Tauberian theorem states the following. Let $A(z)=\sum a\_nz^n$ be a power series wi... | https://mathoverflow.net/users/111917 | Weak version of Karamata's Tauberian theorem | This seems to follow easily from *de Haan–Stadtmüller Theorem*; see Theorem 2.10.2 in the Bingham–Goldie-Teugels book:
>
> **Theorem:** Let $U$ be non-decreasing, and vanish on $(-\infty, 0)$. The following are equivalent:
>
>
> (*i*) $U \in OR$;
>
>
> (*ii*) $\hat{U}(1/\cdot) \in OR$;
>
>
> (*iii*) $\hat{U}(... | 2 | https://mathoverflow.net/users/108637 | 354036 | 149,560 |
https://mathoverflow.net/questions/354027 | 3 | Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-zero) $\ell$-torsion? This appears to be closely related to the presense of torsion in the etale cohomology group $H^2\_{et... | https://mathoverflow.net/users/2191 | Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion? | It seems that existence of $\ell$-torsion is possible. Here's one example.
Let $k$ be an algebraically closed field of characteristic $p$. In the paper "An Example of Unirational Surfaces in Characteristic p." (<https://eudml.org/doc/162649>) Shioda proves that a hypersurface $X\_n = \{x\_1^n + x\_2^n + x\_3^n + x\_4... | 5 | https://mathoverflow.net/users/14440 | 354038 | 149,561 |
https://mathoverflow.net/questions/354044 | 6 | It is known that a finite dimensional basic algebra over an algebraically closed field is isomorphic to the path algebra of a finite quiver modulo an admissible ideal.
Question 1: Is the same true for algebras over $\mathbb{Q}$? If not, are there suitable further assumptions that would guarantee this?
Question 2: A... | https://mathoverflow.net/users/142444 | Finite dimensional algebras over $\mathbb{Q}$ | Question 1: No, take any finite field extension of $\mathbb{Q}$. It is basic but has a simple module that is not 1-dimensional and thus it is not of the form $KQ/I$ for $I$ admissible (since all simple modules are 1-dimensional for algebras of the form $KQ/I$).
For any field $K$, a basic algebra is isomorphic to a qu... | 7 | https://mathoverflow.net/users/61949 | 354051 | 149,562 |
https://mathoverflow.net/questions/354048 | 1 | Let $A$ be a von Neumann algebra. I want to understand the precise meaning of the $\sigma$-weak topology on $A$. What I understand so far is the following: The $\sigma$-weak topology, which we will denote by $\tau$, is the weak$^\*$-topology on $A$ (which can be defined since every von Neumann algebra has a (unique) pr... | https://mathoverflow.net/users/153115 | Explanation of $\sigma$-weak topology von a von Neumann algebra | This is not really research level, and is probably better suited to math.stackexchange, but here's an answer anyway. I will take as given that you know the notation $\sigma(E^\*,E)$ for the weak-\* topology on $E^\*$, or more generally $\sigma(F,E)$ on a Banach space $F$ isomorphic to the dual space of a Banach space $... | 1 | https://mathoverflow.net/users/61785 | 354056 | 149,564 |
https://mathoverflow.net/questions/354004 | -1 | I ran into the following question; let $x,y$ be two points in $\mathbb{R}^d$. Let $(\psi\_t)\_{t\geq 0}$ be the mapping from $\mathbb{R}^{2d}$ to $\mathbb{R}^{2d}$ defined, for all $t\geq 0$, by
$$
\psi\_t(x,y) = \Big(xe^{-t}+\sqrt{1-e^{-2t}}y, -\sqrt{1-e^{-2t}}x+ye^{-t}\Big).
$$
Re-parameterizing this family of mappin... | https://mathoverflow.net/users/nan | Reparameterization and group structure | I am not sure precisely what you want, but I am intrigued that in the two cases that you display you replace an apparently arbitrary parametrisation by one of a very special kind, namely where the $y$-coordinate is a primitive of the $x$ one. (There are many reasons why this is useful but there is no point in my going ... | 1 | https://mathoverflow.net/users/131781 | 354064 | 149,565 |
https://mathoverflow.net/questions/354081 | 13 | I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes ([OEIS A034962](https://oeis.org/A034962)), that is for instance:
$$5+7+11=23$$
$$7+11+13=31$$
$$11+13+17=41$$
$$17+19+23=59$$
$$19+23+29=71$$
$$23+29+31=83$$
$$29+31+37=97$$
$$...$$
The number of such triplets, till ... | https://mathoverflow.net/users/150698 | About the number of primes which are the sum of 3 consecutive primes (OEIS A034962) | The question asks how many primes $p\_n \le x$ are there such that $p\_n + p\_{n+1}+p\_{n+2}$ is also prime. This is beyond our reach to answer, but one can use Hardy-Littlewood type heuristics to attack this. Since $p\_n + p\_{n+1}+ p\_{n+2}$ is roughly of size $x$, it has about $1/\log x$ chance of being prime, and s... | 24 | https://mathoverflow.net/users/38624 | 354090 | 149,571 |
https://mathoverflow.net/questions/354082 | 13 | I would like to ask about (old\* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical models that are used to predict the spread, mathematical studies of effectiveness of various strategies, etc.
\*(Add... | https://mathoverflow.net/users/1532 | Relevant mathematics to the recent coronavirus outbreak | There is the whole discipline of math models of epidemics.
See, for example, Fitzgibbon, William E.(1-HST); Morgan, Jeffery J.(1-HST); Webb, Glenn F.(1-VDB); Wu, Yixiang(1-VDB)
Spatial models of vector-host epidemics with directed movement of vectors over long distances. (English summary)
Math. Biosci. 312 (2019), 77... | 14 | https://mathoverflow.net/users/nan | 354093 | 149,572 |
https://mathoverflow.net/questions/354057 | 6 | Let $\mathbf{Cat}\_\Delta$ denote the category of simplicially enriched categories (ie simplicial objects in $\mathbf{Cat}$ with all face and degeneracy maps bijective on objects). There is a canonical functor $\mathfrak{C}:\Delta\rightarrow\mathbf{Cat\_\Delta}$ that takes $[n]$ to the simplicial category $\mathfrak{C}... | https://mathoverflow.net/users/nan | Understanding the adjunction $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightleftharpoons \mathbf{Cat}_\Delta:\mathcal{N}$ | To a fairly crude approximation: **$\newcommand{\C}{\mathfrak{C}}$think of the functor $\C(-)$ as like geometric realisation, but realising the basic simplices as the categories $\C[n]$ instead of as the topological simplices.** Lots of functors out of $\hat{\Delta}$ are defined analogously by left Kan extension of som... | 2 | https://mathoverflow.net/users/2273 | 354099 | 149,575 |
https://mathoverflow.net/questions/354078 | 2 | Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so that:
$$ \lvert f(x) - f(y)\rvert \leq Ld\_M(x,y)\quad \forall x, y \in M. $$
**My question is:** What is either a **nece... | https://mathoverflow.net/users/35936 | Riemannian submanifolds of Euclidean space admitting Lipschitz extension of Lipschitz functions, and converse statement | Say $M$ admits universal Lipschitz extension if, for any Lipschitz $f : M \to \mathbb R$, there exists Lipschitz$F : \mathbb R^n \to \mathbb R$ such that $F|\_M = f$. $M$ admits universal Lipschitz extension if and only if there exists $C$ such that $d\_M(x,y) \leq C \|x-y\|$ for all $x,y \in M$ (note that it always ho... | 5 | https://mathoverflow.net/users/91418 | 354101 | 149,576 |
https://mathoverflow.net/questions/354085 | 3 | Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the smoothing effect of the heat flow, I expect that the following estimate holds true:
\begin{equation}\label{tre}
(1)\qquad \left... | https://mathoverflow.net/users/54552 | Smoothing-Strichartz estimates for the heat-Schrodinger evolution | I think the estimate you want does not really require Strichartz.
First, your estimate is equivalent to the following, which is a bit easier for me to think about: let $u$ be the solution to the equation
$$ \partial\_t u - (1 + i) \triangle u = F \tag{\*}$$
with initial data
$$ u(0,x) \equiv 0 $$
Then the desired es... | 4 | https://mathoverflow.net/users/3948 | 354109 | 149,578 |
https://mathoverflow.net/questions/354106 | 2 | Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|\_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i.d. standard normal random variables $N\_1,\dots,N\_d$ such that
$$
X \stackrel{d}{=}(N\_1/N,\cdots,N\_d/N),
$$
where $... | https://mathoverflow.net/users/127150 | Volume computation using probabilistic approach | This approach is of course well known. Clearly, it just says that
$$P(X\in A)=P((N\_1,\dots,N\_d)\in C\_A),$$
where $A$ is a Borel subset of the unit sphere $S^{d-1}$ and $C\_A:=\mathbb R\_+A$ is the corresponding cone.
The hard part is to compute the Gaussian measure, $P((N\_1,\dots,N\_d)\in C\_A)$, of the cone $C... | 3 | https://mathoverflow.net/users/36721 | 354113 | 149,580 |
https://mathoverflow.net/questions/295129 | 9 | I'm a condensed matter physicist who tries to understand the details of deformation quantization.
In my self-made training, I've found two huge pieces of work, namely
>
> Fedosov, B. V. (1994). "A simple geometrical construction of deformation quantization". Journal of Differential Geometry, 40 : 213–238.
>
>
... | https://mathoverflow.net/users/37254 | Fedosov vs. Kontsevich deformation quantization : a beginner survey |
>
> Fedosov's work seems to be also available with details in a book Deformation quantization and index theory. Are the two references overlapping ?
>
>
>
Yes, indeed. The book contains strictly more than the contents of the paper from 1994.
>
> I've found several documents from Kontsevich having similar ti... | 7 | https://mathoverflow.net/users/7031 | 354141 | 149,588 |
https://mathoverflow.net/questions/353807 | 0 | Are there any(other than the full complex/1-case)?
Is there a name for this ($k$-edge-regular I call it)?
Thanks.
| https://mathoverflow.net/users/142777 | Examples for simplicial complexes in which every k-edge is contained in exactly $d$ $k+1$-edges | There are many such examples. If d=2 (plus some connectivity) those are called pseudomanifolds, so there are many of those, and there are many examples for larger values of d. When every set of size k is a k-edge these are designs.
| 2 | https://mathoverflow.net/users/1532 | 354158 | 149,592 |
https://mathoverflow.net/questions/349102 | 1 | Almost 5 years ago (time flies), I asked in [Rankin-Selberg convolution and product of degrees](https://mathoverflow.net/questions/194770/rankin-selberg-convolution-and-product-of-degrees?r=SearchResults) whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{GL}\_{n}(\... | https://mathoverflow.net/users/13625 | Rankin-Selberg convolution and product of degrees as of Christmas 2019 | [Newton and Thorne](https://arxiv.org/abs/1912.11261) proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}\_2(\mathbb{A}\_{\mathbb{Q}})$ corresponding with a holomorphic cuspidal newform of even integral weight $k\geq 2$, squarefree level, and trivial central character, then for each $n\geq 1$,... | 6 | https://mathoverflow.net/users/111215 | 354161 | 149,594 |
https://mathoverflow.net/questions/354077 | 5 | Let $M$ be a manifold, $p:E\to M$ a rank $d$ vector bundle. Suppose that $U \subset E$ is an open subset such that $U \cap p^{-1}(x)$ is nonempty and convex for all $x \in M$. Is it true that $U \to M$ is a fiber bundle with fiber $\mathbb R^d$? And that $U \cong E$ as fiber bundles? We may assume with no loss of gener... | https://mathoverflow.net/users/1310 | Shrinking and stretching of vector bundles | Since there are no references so far, let me give a sketch proof along the lines of my comment. I'll assume that $M$ is compact.
0. Let's show first that there is a smooth section of $E$ lying in $U$. Indeed, for any point $x\in M$ there is a neighbourhood $U\_x$ with a section $s\_x$. Take a finite cover $U\_i$ of $... | 3 | https://mathoverflow.net/users/943 | 354162 | 149,595 |
https://mathoverflow.net/questions/354145 | 7 | Let $Y\_1, \ldots, Y\_n$ and $X\_1, \ldots, X\_n$ be i.i.d. $p$-Bernoulli random variables and let $T \in \{0, \ldots, n\}$ be a [stopping time](https://en.wikipedia.org/wiki/Stopping_time) for the process. From [Wald's equation](https://en.wikipedia.org/wiki/Wald%27s_equation), we know
$$
E\left[\sum\_{i=1}^T Y\_i \ri... | https://mathoverflow.net/users/153090 | Chernoff-type bounds for a stopped sum of independent random variables | The desired statement will not hold. E.g., suppose that $n\ge2$; $X\_1,\dots,X\_n,Y\_1,\dots,Y\_n$ are independent; $p=1/2$; $T=1\_{X\_1\ne Y\_1}+n1\_{X\_1=Y\_1}$; and $\delta=1/2$. Then $\mu:=p\,ET>n/4\to\infty$ (as $n\to\infty$), so that $1-\exp(-c\delta^2\mu )\to1$ for any fixed $c>0$. However,
$$P\Big(\sum\_{i=1}^... | 6 | https://mathoverflow.net/users/36721 | 354173 | 149,598 |
https://mathoverflow.net/questions/354168 | 3 | Let $(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$,
where $\rho(x)$ denotes the spectral radius of an element $x\in A$.
**Is $(A,\,^\ast,\lVert\cdot\rVert)$ then a $C^\ast$-algebra?**
| https://mathoverflow.net/users/58125 | Can C*-algebras be characterized among Banach *-algebras by the spectral radius? | [Updated to include Nik Weaver's correction / improvement from the comments.]
I think this follows from the spectral radius formula:
$$\left\| a \right\|^2 = \rho(a^\*a) = \lim\_{k \to \infty} \left\| (a^\*a)^k \right\|^{1/k} \leq \left\| a^\*a \right\| \leq \left\| a^\* \right\| \left\| a \right\|$$
This gives $... | 2 | https://mathoverflow.net/users/4362 | 354175 | 149,599 |
https://mathoverflow.net/questions/354184 | 2 | Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem
$$L\phi=\lambda\phi$$
where $\phi$ is in $L^2(\mathbb{R})$ and vanishes at infinity.
We want to study the distribution of the eigenvalues (approximately)?
Classical e... | https://mathoverflow.net/users/62005 | Eigenvalue problem of Schrodinger equation with polynomial growth potential on the real line | If $L=-\Delta +|x|^s$, $s>0$, in $R^n$, then $N(\lambda)$, the number of eigenvalues less than $\lambda$, behaves like $\lambda^{N(1/2+1/s)}$ as $\lambda \to \infty$. This is in Titchmarsh "Eigenfunction expansions...Part II, Sect. 17.8 or in Reed Simon Section XIII.
| 3 | https://mathoverflow.net/users/150653 | 354186 | 149,601 |
https://mathoverflow.net/questions/354180 | 4 | Let $A$ and $B$ be two $C^\*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $\*$-homomorphism which is injective on a dense $\*$-sub-algebra of $A$. Can such a map have non-trivial kernel, and if so, is it possible that the $K$-theory groups of $A$ and $B$ can be non-isomorphic?
| https://mathoverflow.net/users/128876 | $K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras | Yes to both.$\newcommand{\Cst}{{\rm C}^\*}$
The standard example for the first is: take a discrete group $G$ and let $A$ be its full $\Cst$-algebra, $B$ its reduced $\Cst$-algebra. There is a canonical homomorphism $q:A\to B$ which is injective when restricted to $\ell^1(G)$; but $q$ is injective if and only if $G$ is... | 10 | https://mathoverflow.net/users/763 | 354188 | 149,602 |
https://mathoverflow.net/questions/353987 | 4 | We are searching for rank $8$ elliptic curves with the torsion subgroup $\mathbb{Z}/6$ using newly discovered families similar to Kihara's as described in
>
> A. Dujella, J.C. Peral, P. Tadić, *Elliptic curves with torsion group $\mathbb{Z}/6\mathbb{Z}$*, Glas. Mat. Ser. III 51 (2016), 321-333 doi:[10.3336/gm.51.2... | https://mathoverflow.net/users/95511 | A generator needed for a Z/6 elliptic curve | In this case, it pays to work on the curve $E'$ that is 2-isogenous to $E$, which is given by the equation
$$
y^2 = x^3 + 404100192598226941365253x^2+ 1470175712258164849983363482095324897635296971x.
$$
It is relatively easy to find the 7 independent points
```
(110776963853866550724016 : -805058694686300892101313... | 9 | https://mathoverflow.net/users/151977 | 354195 | 149,603 |
https://mathoverflow.net/questions/353542 | 1 | Suppose that $b:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz and of polynomial growth. Suppose further that there are constants $C\_1,C\_2>0$ such that $(x-y)(b(x)-b(y))\leq C\_1-C\_2(x-y)^2$ for all $x\in\mathbb{R}$. The ODE
$$
\dot{x}(t)=b(x(t))+d(t)+u(t),\quad x(0)=x\_0,\qquad t\in[0,1],
$$
is then seen to have a u... | https://mathoverflow.net/users/124450 | Steering an ODE out of a ball | *Do you have an easy argument...*
Sorry for the late reply, but it is, actually, a rather simple story. Let $R=1$ and suppose that we have declared some $M$ and $\delta$. Then the adversary starts at some $x\_0\in(-\frac 12,\frac 12)$ and he has some guaranteed time $T=T(b,M)>0$ during which the solution stays in $(-... | 4 | https://mathoverflow.net/users/1131 | 354196 | 149,604 |
https://mathoverflow.net/questions/354059 | 2 | We assume that for every real $x$, $L[x]$ only contains countably many reals.
Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that
1. For any reals $x\in I$ and $y$, $y\in L[x]$ implies $y\in I$; and
2. For any finite $F\subseteq X$, there is a real $z\in I$ so that $F... | https://mathoverflow.net/users/14340 | The measure of ideals generated by random reals | The question has a negative answer. The technique is essentially due to Jockusch and Posner.
>
> **Proof**: Let $x$ be a real in which every constructible real is recursive. Now $$A=\{r\mid r\mbox{ is Martin-L\" of random relative to }x \wedge x\oplus r\geq\_T x',\mbox{ the Turing jump of }x.\}$$ Then $A$ is null, ... | 1 | https://mathoverflow.net/users/14340 | 354205 | 149,609 |
https://mathoverflow.net/questions/354040 | 3 | Given a natural number $n$ (of **unknown** factorization) and an arbitrary number $c \in \mathbb{Z}^\*\_n$ (the set of natural numbers smaller than $n$ and coprime to it), is there an **efficient** algorithm which outputs numbers $a \in \{2,\ldots,n-1\}$ and $b \in \mathbb{Z}^\*\_n$ such that:
$$b ^ a \equiv c \pmod ... | https://mathoverflow.net/users/9544 | Given $n, c$ find $a>1,b$ such that $b ^ a \equiv c \pmod n$ | After much struggling, I found out that this is currently deemed a hard problem; so there is no known efficient algorithm to solve it.
From the [Encyclopedia of Cryptography and Security (2011)](https://link.springer.com/referencework/10.1007/978-1-4419-5906-5):
>
> **Strong RSA Assumption**
>
>
> The Strong RS... | 4 | https://mathoverflow.net/users/9544 | 354216 | 149,611 |
https://mathoverflow.net/questions/354217 | 5 | Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K\_{0}(C)=0$, what can be said about the higher groups $K\_{n}(C)=0$ ? Is there a non-trivial example of such exact subcategory $C$ ?
| https://mathoverflow.net/users/17895 | Exact subcategory with trivial Grothendieck group: what are the consequences and examples | This is a long comment more than an answer.
If you think of $K\_0$ as a universal domain for all kinds of functions that associate a "dimension" to a module, then $K\_0(\mathscr{C})=0$ means that at least some of the modules in $\mathscr{C}$ are "the worst kind of infinite dimensional". For example they are not of fi... | 6 | https://mathoverflow.net/users/3041 | 354224 | 149,612 |
https://mathoverflow.net/questions/354218 | 1 | I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of the conditional expectation of the random effect, i.e.
$$ \mathbb{E}[X | Y] = \int f\_{X|Y} (x|y)\: dx = \int \frac{f\_{Y... | https://mathoverflow.net/users/153213 | Conditional density for random effects prediction in GLMM | This has nothing to do with GLMM's per se. All what is done here is using the definition
$$f\_{Y|X}(y|x):=\frac{f\_{X,Y}(x,y)}{f\_X(x)}$$
(if $f\_X(x)\ne0$) to write
$$f\_{Y|X}(y|x)f\_X(x)=f\_{X,Y}(x,y),$$
so that
$$\int f\_{Y|X}(y|x)f\_X(x)\,dx=\int f\_{X,Y}(x,y)\,dx=f\_Y(y)$$
and hence
$$\frac{f\_{Y|X}(y|x)f\_{... | 1 | https://mathoverflow.net/users/36721 | 354227 | 149,614 |
https://mathoverflow.net/questions/354201 | 6 | Let $\langle M,E\rangle$ be a model of $\mathsf{ZFC}$.
Does there exist a $d\in M$ such that, for all $a\in M$, $a\mathrel{E}d$ if and only if $a$ is definable in $\langle M,E\rangle$ without parameters?
A result of J.D. Hamkins, etc. (cf. [Pointwise definable models of set theory](https://doi.org/10.2178/jsl.78010... | https://mathoverflow.net/users/101817 | A question about definable elements in a model of ZFC | In the paper you mention in the original post, we mention several of the possibilities as follows. Item (v) includes the particular situation you asked about.
*Hamkins, Joel David; Linetsky, David; Reitz, Jonas*, [**Pointwise definable models of set theory**](http://dx.doi.org/10.2178/jsl.7801090), J. Symb. Log. 78, ... | 7 | https://mathoverflow.net/users/1946 | 354234 | 149,618 |
https://mathoverflow.net/questions/354215 | 3 | Can it be shown that
>
> There are finitely many positive integers $n$ that can't be expressed as
>
>
> $$n=a+b$$
>
>
> for any composite integers $a$ and $b$ relatively prime to each other?
>
>
>
<http://oeis.org/A096076> is the sequence, *"Not the sum of two relatively prime composite numbers"*.
It's giv... | https://mathoverflow.net/users/149083 | Not the sum of two relatively prime composite numbers | Following fedja's comment, the number of decomposition $n=a+b$ with $\gcd(a,b)=1$ equals $\varphi(n)$. Among these, there are at most $2\pi(n)$ decompositions in which $a$ or $b$ is prime, hence $n$ has a suitable decomposition when $\varphi(n)>2\pi(n)$. Now the well-known explicit lower bounds for $\varphi(n)$ and upp... | 10 | https://mathoverflow.net/users/11919 | 354239 | 149,621 |
https://mathoverflow.net/questions/354222 | 1 | Let $n=3$ and $u$ be the solution to Klein-Gordon equation
\begin{equation}
\begin{cases}\ddot{u}-\Delta u +u=u^3 \\
u(0)=u\_0, \partial\_t u(0)=u\_1,
\end{cases}
\end{equation}
where $(u\_0,u\_1) \in H^1 \times L^2$. If we assume that $u$ exists globally and scatters to a solution $v$ of a free Klein-Gordon equatio... | https://mathoverflow.net/users/137915 | the energy of scattering solution of cubic Klein-Gordon equation in $n=3$? | Step 1: assuming scattering, there exists a solution $v$ to the linear Klein-Gordon equation such that $u-v \to 0$ in $H^1(\mathbb{R}^3)$ as $t \to \infty$. By Sobolev embedding this means that for any $p\in [2,6]$ you also have $u-v \to 0$ in $L^p(\mathbb{R}^3)$.
In particular, this means that
$$ E(u,\dot{u})(t) -... | 1 | https://mathoverflow.net/users/3948 | 354241 | 149,622 |
https://mathoverflow.net/questions/354139 | 2 | Will the fundamental representation $\pi\_n$ of type $C\_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be calculated? If so this would make $C\_n$ the only non-exceptional type for which the first and last Dynkin node did not give ... | https://mathoverflow.net/users/143172 | Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$ | The LiE software does these calculations, and is available on line:
<http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html>
| 3 | https://mathoverflow.net/users/6030 | 354245 | 149,623 |
https://mathoverflow.net/questions/354244 | 2 | Coming from a problem in game theory, I arose at some dubious monotonicity like property for matrices of the following art. Let $H=\lbrace h\in\mathbb{R}^{n}\colon h\_{1}+\dots+h\_{n}=0\rbrace$. I'm interested in matrices $A\in\mathbb{R}^{n\times n}$ which satisfy for all $h\in H$ and $i\ne j$
\begin{equation}
(h\_{i}-... | https://mathoverflow.net/users/85194 | Dubious matrix monotonicity | Such a matrix has the form $\theta I\_n+ew^T+ve^T$ where $T$ denotes transposition and $e$ is the vector $(1,\ldots,1)^T$. The parameter $\theta$ is $\le0$.
Here is the proof when $n\ge4$. By continuity, your assumption implies that
$$(e\cdot h\quad\hbox{and}\quad h\_i=h\_j)\Longrightarrow((Ah)\_i=(Ah)\_j).$$
Denotin... | 2 | https://mathoverflow.net/users/8799 | 354252 | 149,626 |
https://mathoverflow.net/questions/354248 | 3 | Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A\_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product
$$
\langle \cdot,\cdot\rangle: A\_0 \times A\_0 \to \mathbb{C}, ~~~~~ (a,b) \mapsto h(a^\*b).
$$
Let $L^2(A\_0)$ be the completi... | https://mathoverflow.net/users/153228 | Reduced compact quantum group and left and right multiplication | I think there are some subtle points here about what the "right action" even means.
For a general $\*$-algebra $A\_0$ and a functional $\phi:A\_0\rightarrow\mathbb C$, we first of all have to decide what "positive" means for $\phi$. We could take this as being $\phi(a^\*a)\geq0$ for all $a$. Then Cauchy-Schwarz holds... | 3 | https://mathoverflow.net/users/406 | 354253 | 149,627 |
https://mathoverflow.net/questions/354157 | 1 | This is a follow up from [my earlier MO question](https://mathoverflow.net/questions/353715/an-identity-for-polynomials-over-partitions).
Given an integer partition $\lambda=(\lambda\_1,\dots,\lambda\_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its [conjugate partition](http:/... | https://mathoverflow.net/users/66131 | Divisibility of polynomials over partitions | Both are true and these follow routinely from Euler's Pentagonal Theorem (PT). We have
\begin{align}
A:&=1+\sum\_{n=1}^\infty (q-1)f\_n(q)x^n\\
&=\prod\_{j=1}^\infty(1+(q-1)x^j+q(q-1)x^{2j}+q^2(q-1)x^{3j}+\ldots) \\
&=
\prod\_{j=1}^\infty\frac{1-x^j}{1-qx^j}.
\end{align}
Consider it modulo small powers of $q$. Modulo $... | 2 | https://mathoverflow.net/users/4312 | 354259 | 149,629 |
https://mathoverflow.net/questions/354250 | 13 | **Remark:** All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.
Motivation:
-----------
I recently had an interesting exchange with s... | https://mathoverflow.net/users/56328 | Mathematical physics without partial derivatives |
>
> Is it possible to accurately simulate any non-trivial physics without computing partial derivatives?
>
>
>
Yes. An example is the nuclear shell model as formulated by Maria Goeppert Mayer in the 1950's. (The same would also apply to, for example, the [interacting boson model](https://en.wikipedia.org/wiki/In... | 8 | https://mathoverflow.net/users/nan | 354265 | 149,632 |
https://mathoverflow.net/questions/354271 | 0 | Consider two sequences of (not necessarily independent) Bernoulli random variables $X\_1, X\_2, \ldots, X\_n$ and $Y\_1, Y\_2, \ldots, Y\_n$. Suppose that for any $i$, we have $\Pr[X\_i = 1] = \Pr[Y\_i = 1] = p\_i$, but the actual value of $p\_i$ is determined only after observing $\{X\_1, \ldots, X\_{i-1}, Y\_1, \ldot... | https://mathoverflow.net/users/153090 | Sum of sequences of random variables, with variable success probabilities | Define the martingale $M\_n = \sum\_{i = 1}^n (X\_i - Y\_i)$ with the filtration $\mathcal{F}\_n = \sigma( \{X\_j,Y\_j\}\_{j=1}^n )$. Then $|M\_{n + 1} - M\_n| = |X\_{n+1}-Y\_{n+1}| \leq 1$ and it is indeed a martingale since $$\mathbb{E}[M\_{n+1}\,|\,\mathcal{F}\_n]= \mathbb{E}[X\_{n+1} - Y\_{n+1}\,|\,\mathcal{F}\_n] ... | 1 | https://mathoverflow.net/users/69870 | 354273 | 149,634 |
https://mathoverflow.net/questions/280666 | 9 | In Bott and Tu's *Differential forms in algebraic topology* there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this:
>
> **Theorem** (Leray-Hirsch): Let $E$ be a fiber bundle over $M$ with fiber $F$. Suppose that $M$
> has a finite good cover. If there are global cohomology classes $e\_1,
... | https://mathoverflow.net/users/12233 | Leray-Hirsch theorem for Dolbeault cohomology | There's a simples proof of Leray-Hirsch theorem for Dolbeault cohomology in this paper:
<https://arxiv.org/pdf/1806.11435.pdf>
| 1 | https://mathoverflow.net/users/12233 | 354281 | 149,635 |
https://mathoverflow.net/questions/354285 | 2 | The Siegel-Walfisz theorem is stated in <https://en.m.wikipedia.org/wiki/Siegel>–Walfisz\_theorem.
I want to know if it can be extended unconditionally to a modulus $q$ such that any factorization $q=\prod\_{i}q\_{i}$ fulfills $(q\_{i},q\_{j})=(q\_{i}q\_{j})^{\delta\_{ij}/2}$ (hence for $q$ being the product of disti... | https://mathoverflow.net/users/13625 | About a possible extension of Siegel-Walfisz theorem | For large $q$ such a result does not hold. This follows from the results in the paper "Limitations to the Equi-Distribution of Primes I" by Friedlander and Granville. Specifically, their Proposition 1 implies that for any constant $B>1$ and arbitrarily large $Q$ (assuming GRH, all sufficiently large $Q$ work), if $q$ i... | 7 | https://mathoverflow.net/users/30186 | 354290 | 149,639 |
https://mathoverflow.net/questions/354293 | 9 | A subset $E$ of $\mathbb R$ is said to contain arbitrarily long arithmetic progressions, if for every natural $n$, there exists $a, d \in R, d$ nonzero, such that $a + kd$ is in $E$ for all natural $k \leq n$.
Does every measurable subset of $\mathbb R$ of non zero Lebesgue measure contain arbitrarily long arithmetic... | https://mathoverflow.net/users/132446 | Does every measurable subset of $\mathbb R$ of non zero Lebesgue measure contain arbitrarily long arithmetic progressions? | Yes. For fixed $n$, we approximate our set $E$ from above by an open set $U=\sqcup \Delta\_i$ ($\Delta\_i$ are disjoint intervals) with such accuracy that one of intervals $\Delta\_i$ satisfies $|E\cap \Delta\_i|>(1-\frac1{n+1})|\Delta\_i|$, where $|\cdot|$ denotes Lebesgue measure. Now if $\Delta\_i=(a,a+(n+1)t)$, we ... | 10 | https://mathoverflow.net/users/4312 | 354298 | 149,641 |
https://mathoverflow.net/questions/354292 | 4 | Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this imply that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{F}\_{2}$ as well? Here $\mathbb{F}\_{2}$ denotes the free gro... | https://mathoverflow.net/users/64444 | Copies of $\mathbb{Z}\oplus \mathbb{F}_2$ in non-affine, irreducible Coxeter groups | Consider the free product $W = \tilde A\_2 \* A\_1 \* A\_1 \* \ldots \* A\_1$. This Coxeter group is not affine, and it has a copy of $\mathbb Z^2$ within the $\tilde A\_2$ part, so it satisfies your condition.
Now assume $W$ contains a copy of $\mathbb F\_2 \oplus \mathbb Z$, such that $u,v$ generate $\mathbb F\_2$ ... | 4 | https://mathoverflow.net/users/135257 | 354308 | 149,642 |
https://mathoverflow.net/questions/354316 | 1 | Let $T$ be a bounded operator on a Banach space $X$ and suppose that there is a non-constant polynomial $p$ such that $p(T) = 0$. It seems to be well known that the spectrum of such an operator coincides with the point spectrum and consists of finite order poles of the resolvent, only. But I can not find any citable re... | https://mathoverflow.net/users/91108 | Spectral properties of operators mapped to zero by some polynomial | Take the annihilating polynomial and shift its argument by $\lambda I$. That is, define the polynomials $q$ and $q\_0$ by the identity $$(T-\lambda I) q(\lambda,T-\lambda I) - q\_0(\lambda) I = p((T-\lambda I) + \lambda I).$$ Then your hypothesis $p(T)=0$ implies the following formula for the resolvent: $$(T-\lambda I)... | 5 | https://mathoverflow.net/users/2622 | 354325 | 149,644 |
https://mathoverflow.net/questions/354092 | 9 | Let $X$ be a compact metrizable space and let $\mathcal{K}\_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric for any compatible metric on $X$).
>
> **Question:** When does there exist a continuous function $f: \mathcal{K}\_{... | https://mathoverflow.net/users/83901 | Which compact metrizable spaces have continuous choice functions for non-empty closed sets? | It's an old (1981) theorem by Jan van Mill and Evert Wattel (see [this paper](https://www.jstor.org/stable/2044129?seq=1)) that a compact space has a continuous selection iff it is orderable. (So has a linear order whose order topology is the topology on $X$). $F \to \min(F)$ and $F \to \max(F)$ are then the two only c... | 7 | https://mathoverflow.net/users/2060 | 354347 | 149,660 |
https://mathoverflow.net/questions/354304 | 2 | Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold *[with boundary](https://en.wikipedia.org/wiki/Manifold#Manifold_with_boundary)*. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the topology of [uniform convergence on compacta]... | https://mathoverflow.net/users/36886 | Density of continuous functions to interior in set of all continuous functions | A boundary of a paracompact manifold has a collar neighborhood, i.e. $U\subset N$ that includes $\partial N$ and is homeomorphic to $\partial N\times [0,1)$ via a map $\psi$ that maps $\partial N$ onto $\partial N\times \{0\}$. Therefore, I will be talking about the points in $U$ as if they were in $\partial N\times [0... | 1 | https://mathoverflow.net/users/53155 | 354358 | 149,662 |
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