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https://mathoverflow.net/questions/364629 | 1 | Let $ a = (a\_1,a\_2, \ldots,a\_{10})\in \{ 0,1\}^{10}$ be a binary vector of length $10$.
**Question:** *Without using a computer-aided method*, how to prove that there exists binary vectors $x\_{i,j} \in \{ 0,1\}^{10}$, $i\in \{1,2,3,4,5\}$, $j \in \{1,2,3\}$ such that one can recover $a$ from any two rows of the f... | https://mathoverflow.net/users/nan | System of equations - Proof that a solution exists | We can recover $a$ as soon as $\det(X\_{i,k})=1$ over the field $\mathbb{F}\_2:=\{0,1\}$ for all pairs $i<k$ from $\{1,2,3,4,5\}$, where $X\_{i,k}$ is the $6\times 6$ matrix formed by rows $x\_{i,j}$ and $x\_{k,j}$ for $j\in\{1,2,3\}$ excluding the columns indexed by $2i-1,2i,2k-1,2k$. There are total of $10$ such matr... | 1 | https://mathoverflow.net/users/7076 | 364792 | 153,297 |
https://mathoverflow.net/questions/364809 | 3 | Given a set $\mathcal{X}$ and RKHS $\mathcal{H}$ of functions on $\mathcal{X}$, we can recover a (pseudo)metric on $\mathcal{X}$ by $d(x,y)=||\phi\_x-\phi\_y||\_{\mathcal{H}}$, where $\phi\_x=k(x,\cdot)$.
It is straightforward to see that any function $f \in \mathcal{H}$ which has RKHS norm less than $L$ is Lipschitz... | https://mathoverflow.net/users/118735 | RKHS norm of Lipschitz functions | I think that in general $L(.)$ and $\Vert.\Vert\_\mathcal{H}$ measure quite different things.
Writing $L(f)$ for
$$
\inf\{ M>0:|f(x)-f(x')| \leq Md(x,y) \;\forall \;x,x'\in \mathcal{X}\}
$$
let $\mathcal{X}=\mathbb{Z}$ and $\mathcal{H}=l^2$. Then (unless I've made an embarrassing mistake...) setting $f\_n=1\_{[-n... | 5 | https://mathoverflow.net/users/61771 | 364821 | 153,303 |
https://mathoverflow.net/questions/364811 | 4 |
>
> **Theorem**: Assume VP. Let $\mathcal{M}$ be an [accessible model category](https://ncatlab.org/nlab/show/accessible+model+category)
> such that there exists a set of generating cofibrations $I$ and such
> that all objects are fibrant. Then it is combinatorial.
>
>
>
**Proof**: Consider the left determined m... | https://mathoverflow.net/users/24563 | Almost combinatorial accessible model categories | Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).
Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod\_A B \hookrightarrow I\_A B \to B$ a cylinder object, and let $j\_i : B \hookrightarrow I\_A B$ be the first leg inclus... | 7 | https://mathoverflow.net/users/22131 | 364823 | 153,304 |
https://mathoverflow.net/questions/364518 | 5 | I would appreciate some help proving a conjecture related to combinatorics and representation theory.
Given an integer partition $\lambda\vdash n$, define a polynomial in $N$ whose roots are the negatives of the contents of the partition,
$$ [N]\_\lambda=\prod\_{\square \in \lambda}(N+c(\square)).$$ This polynomial i... | https://mathoverflow.net/users/78061 | Sum involving determinants of binomial coefficients, indexed by partitions | Following the OP's comment that the factorization can be obtained using my comments above, I repost them here.
The quantity $E\_{\lambda\nu}(N)$ can be computed using Cauchy-Binet. Specifically see Lemma 9.1 in Yeliussizov's (nice) paper: <https://arxiv.org/pdf/1601.01581.pdf>.
For the quantity $A\_{\lambda\rho}$, ch... | 2 | https://mathoverflow.net/users/nan | 364824 | 153,305 |
https://mathoverflow.net/questions/364825 | 6 | Cross post from [MSE](https://math.stackexchange.com/q/3744292/272127). and sorry if this is an obvious question.
Here is a line of proof of [Theorem 1.15](https://books.google.com/books?id=GoyDAwAAQBAJ&lpg=PP1&pg=PA8) from
*Brendle, Simon*, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Pr... | https://mathoverflow.net/users/90655 | Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$? | Very roughly speaking: Geodesics $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=p$ and $\operatorname{ind}(\gamma)<k$ correspond to critical points of Morse index less than $k$ of the energy functional $E:\Omega\_p(M)\to \mathbb{R}$, where $\Omega\_p(M)$ is the space of loops based at $p$.
The assumption that $\pi\_k(... | 11 | https://mathoverflow.net/users/8103 | 364826 | 153,306 |
https://mathoverflow.net/questions/364793 | 4 | I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5.
Kobayashi is trying to prove that if $E$ is a vector bundle on some manifold $M$, with a flat connection $D$, then it admits a "flat structure" $\{U,s\_U\}$ which consists on an ope... | https://mathoverflow.net/users/143492 | Integrability condition for flat connections | On the manifold $X=U\times \operatorname{GL}\_r$, with points written $x=(m,a)$, each tangent space $T\_x X$ contains a linear subspace $V\_x$ consisting of tangent vectors on which $a^{-1}da=-\omega'$. The problem is to prove that these $V\_x$ spaces form a smooth subbundle $V \subset TX$, and that this subbundle is c... | 1 | https://mathoverflow.net/users/13268 | 364831 | 153,308 |
https://mathoverflow.net/questions/364801 | 1 | Let $\mu\_0$ be the standard Wiener measure on $C[0,T]$. Let $\mu\_n$ be a sequence of measures with $\mu\_n\ll \mu\_0$ for all $n$ and so that the weak$^\ast$ limit of $\mu\_n$ exists, call it $\mu$. Is it true that $\mu \ll \mu\_0$?
I know for general measures this is not true. For example we can have a Gaussian wi... | https://mathoverflow.net/users/nan | Is the weak* limit of Girsanov measures also a Girsanov measure? | This goes as wrong as can be and has nothing to do with any property of Wiener measure.
**Theorem:** Let $X$ be a separable metric space and $\nu$ a measure on $X$. The set of measures absolutely continuous with respect to $\nu$ is dense in the space of measures supported on the support of $\nu$ in the topology of we... | 1 | https://mathoverflow.net/users/35357 | 364833 | 153,310 |
https://mathoverflow.net/questions/364808 | 11 | Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space.
Suppose $f$ preserves $=^\*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects $=^\*$?
That is, if
$$x=^\* y \implies f(x)=^\* f(y)$$
does it follow that:
$$x=^\* y \iff f(x)=^\* f(y)?$$
Using Ax... | https://mathoverflow.net/users/4600 | Homeomorphisms and "mod finite" | Define $f : C \to C$ by the formula
$$ f(x) = x\_0 \cdot (x \oplus \sigma(x)) $$
where $\cdot$ is word concatenation, $\oplus : C \times C \to C$ is coordinatewise xor, and $\sigma(x)\_i = x\_{i+1}$ is the shift.
Clearly this map is continuous and preserves $=^\*$. It is a bijection because you can deduce the preimage ... | 12 | https://mathoverflow.net/users/123634 | 364844 | 153,314 |
https://mathoverflow.net/questions/364859 | 6 | For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w\_1(TM)\neq 0$) with vanishing $w\_1(TM)\cup w\_1(TM)$ and $w\_2(TM)$, i.e.,
$$w\_1(TM)\cup w\_1(TM)=0, ~~~~~ w\_2(TM)=0, ~~~~~w\_1(TM)\neq 0?$$
Here $w\_i(TM)$ is the $i^{\text{th}}$ Stiefel-Whitney class of the tangent bundle of the manif... | https://mathoverflow.net/users/73398 | Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$ | A smooth manifold $M$ admits a pin$^+$ structure if and only if $w\_2(M) = 0$, and a pin$^-$ structure if and only if $w\_1(M)^2 + w\_2(M) = 0$; see [this page](http://www.map.mpim-bonn.mpg.de/Pin_structures) for some information on pin structures. The manifolds you are enquiring about satisfy both conditions and hence... | 7 | https://mathoverflow.net/users/21564 | 364860 | 153,321 |
https://mathoverflow.net/questions/364854 | 1 | I am reading a paper on PDEs and I has been struggling trying to understand some specific identities (that should be very easy). Let me introduce the main notation of the book, which says that each time a sum like $\sum\_\lambda f(\lambda)$ appears, then the $\lambda$-variable must to be understood as a dyadic variable... | https://mathoverflow.net/users/160247 | Understanding an identity for dyadic sums | We write $\mu=2^k \lambda$ with $k \ge -3$. The standing assumption $\mu \ge 1$ becomes $2^k \lambda \ge 1$. The RHS of the sequence of equalities you reproduce is missing a factor $\lambda^s$; is this typo in the original paper?
The sequence of equalities you included should thus be written (adding $L$ on the left) ... | 2 | https://mathoverflow.net/users/7691 | 364862 | 153,322 |
https://mathoverflow.net/questions/364850 | 1 | Let $\mu\_0$ be standard Wiener measure on $C[0,T]$. Let $F\_C(t)=Ct$ and consider the corresponding Girsanov measure
$$\frac{d\mu\_C}{d\mu\_0}=\exp\left(CB(T)-\frac12C^2T\right)$$
What is the weak\* limit of this measure as $C\to \infty$? We know that the space of signed Borel measures on $C[0,T]$ is compact so we h... | https://mathoverflow.net/users/nan | What is the weak* limit of the Girsanov density associated to drift $F(t)=Ct$? | The probability measure $\mu\_C$ does not converge to any measure as $C\to\infty$. One way to see this is to note that
$$f\_C(u):=\int\_{C[0,T]} e^{iu x(T)}\,\mu\_C(dx)=e^{iCTu-Tu^2/2}$$
for all real $u$, because $f\_C$ is the characteristic function of the random variable $B(T)+CT$. So, for any sequence of real number... | 2 | https://mathoverflow.net/users/36721 | 364865 | 153,323 |
https://mathoverflow.net/questions/364851 | 2 | Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$.
Is there a general method to count integral points $(a,b,c)$ with $\max\{|a|, |b|, |c|\} \leq T$ on the affine cubic surface defined by ... | https://mathoverflow.net/users/10898 | Density of integral points on affine cubic surfaces of a certain type | Not in general.
The involutions of the Markov surface such as
$(a,b,c) \leftrightarrow (bc-a,b,c)$ preserve integral points because
$x^2 + y^2 + z^2 - xyz$ is a *monic* quadratic polynomial in each variable.
That works more generally for any polynomial of the form
$Q(x,y,z) = x^2 + y^2 + z^2 - L(x,y,z)$ with $L$ line... | 5 | https://mathoverflow.net/users/14830 | 364869 | 153,325 |
https://mathoverflow.net/questions/364871 | 8 | Let $\Omega \subset \mathbb{R}^m$ be an open subset bounded with a smooth boundary.
**Problem** : Given any bounded continuous function $f:\Omega\to\mathbb{R}$, can we learn it to a given accuracy $\epsilon$? ($\epsilon>0$).
**Definition** : What do you mean by learning a function to a given accuracy $\epsilon$?
... | https://mathoverflow.net/users/14414 | How to learn a continuous function? | The answer is **no** by a Cantor diagonal argument:
Let $\Omega=(0,1)$.
Let $G$ be all functions that can be computed by a finite number of registers with finite precision. It does not matter where $G$ is learnt from.
* The number of states of $n$ registers with precision $m$ is finite, thus the number of functio... | 19 | https://mathoverflow.net/users/125498 | 364873 | 153,326 |
https://mathoverflow.net/questions/364866 | 5 | Suppose $\mathfrak{g}$ is a real Lie algebra integrating to the connected Lie group $G$. One may consider the $G$-equivariant cohomology of $\mathfrak{g}$ ($\mathfrak{g}^\*$) where the $G$-action is the adjoint (coadjoint) representation. That is, the cohomology induced by the cochain complex of $G$-invariant different... | https://mathoverflow.net/users/4730 | Equivariant cohomology of a semisimple Lie algebra | It vanishes as well. The complex of differential forms on $\mathfrak{g}$ is null homotopic through the standard null-homotopy coming from the linear homotopy to the origin. Since the $G$-action preserve this null-homotopy, it is $G$-equivariantly null and so its $G$-invariants subcomplex is null as well.
| 3 | https://mathoverflow.net/users/115052 | 364883 | 153,330 |
https://mathoverflow.net/questions/364877 | 1 | Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x\_\alpha]$ ($\alpha \in \mathbb{Q}$) that have $\mathbb{R}$ as their residue field. There is an injective map from the set of continuous functions $\mathbb{R}\to\mathbb{R}$ to $S$. Is there a locally closed subset $Y\subset X$ such that $Y\cap S$ i... | https://mathoverflow.net/users/nan | Polynomial constraints on the values of continuous functions $\mathbb{R}\to\mathbb{R}$ | It seems to me that the answer is no. Call $I$ the image of this map: I claim that $I$ is Zariski dense. Since $I\subsetneq S$, this is enough to conclude. Let $f\in \mathbb{R}[x\_\alpha]$ be a polynomial which vanishes on $I$. If $f$ is not identically zero then it is a non-constant polynomial in finitely many indeter... | 1 | https://mathoverflow.net/users/116075 | 364884 | 153,331 |
https://mathoverflow.net/questions/364874 | 1 | Can there exist a consistent, recursively axiomatizable theory $T$, such that $\forall \phi, TA\vdash \phi \Rightarrow$ $T\vdash \tau(\phi)$, where $\tau$ is some suitable translation from the language of $TA$ to that of $T$?
---
**Edit:** By "suitable translation" I was thinking about a translation that preserve... | https://mathoverflow.net/users/75935 | Is TA (true arithmetic) interpretable in a recursively axiomatizable theory? | Without further requirements on $\tau$, this is trivial. Let $T$ be any first-order theory and let $\top$ be any tautology. Define $\tau(\phi)= \top$.
If you want $\tau$ to be injective, then let $T$ be any first-order theory in the language of $TA$ and define $\tau(\phi)=\top\vee\phi$.
If you also want equivalence... | 6 | https://mathoverflow.net/users/9825 | 364886 | 153,333 |
https://mathoverflow.net/questions/364880 | -1 | Imagine we have a sequence of finite measures $\nu\_n << \mu\_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have that the Radon-Nikodým derivatives $h\_n$ of $\nu\_n$ wrt. $\mu\_n$ converge to the Radon-Nikodým derivative $h$ of $\nu$ wrt. $\... | https://mathoverflow.net/users/160649 | Convergence of Radon-Nikodým derivative | The answer is negative for a.e. pointwise convergence as well as $L^1$ convergence with respect to $\mu$. Let $\mu\_n=\mu$ be half the Lebesgue measure, color the torus as a checkerboard with $2^n$ cells, and assume $h\_n$ takes value $1$ on black cells and $0$ on white cells. Then $\nu\_n=h\_n\mu \to \mu$ $\*$-weakly,... | 1 | https://mathoverflow.net/users/4961 | 364895 | 153,335 |
https://mathoverflow.net/questions/364902 | 8 | Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.
---
**Question 1:** Is the construction $A \mapsto \Pi(A)$ a functor from von Neumann algebras to complete orthomodular lattices?
For this to make sense, I should sa... | https://mathoverflow.net/users/2362 | Which complete orthomodular lattices arise from von Neumann algebras? | Question 1: Yes, if you take the von Neumann algebra morphisms to be normal $\*$-homomorphisms. Restricting any such map to the projections will preserve sups and orthocomplements.
Question 2: No, this already fails in the commutative case. Look up "abelian AW\*-algebra" or see the discussion of Stonean spaces in vol... | 4 | https://mathoverflow.net/users/23141 | 364905 | 153,338 |
https://mathoverflow.net/questions/364907 | 13 | How many squares can be formed by using n points on a 3 dimensional space?
* Like using 4 points, there is 1 square be formed
* Using 5 points, still 1 square
* Using 6 points, 3 squares can be formed
| https://mathoverflow.net/users/160663 | How many squares can be formed by using n points? | In the plane $n$ points can determine at most $O(n^2)$ squares. This is because any two distinct points can determine up to three squares.
In $R^3$ this argument no longer holds, since two points can form the corners of arbitrarily many squares. As Gerhard points out, $O(n^3)$ is an upper bound (in any dimension) giv... | 16 | https://mathoverflow.net/users/630 | 364915 | 153,340 |
https://mathoverflow.net/questions/364887 | 14 | Can you recover the characteristic of a perfect field from the category of smooth projective geometrically connected varieties over it?
| https://mathoverflow.net/users/nan | Recover the characteristic of $k$ from the category of $k$-varieties | **Correction**. As correctly noted by Remy van Dobben de Bruyn, there is a mistake in Lemma 4. What follows is a corrected argument, with the original (mistaken) post appended below the corrected argument.
Let $k$ be a perfect field. Denote by $\mathbf{V}$ the category of $k$-schemes that are smooth, projective and g... | 12 | https://mathoverflow.net/users/13265 | 364918 | 153,343 |
https://mathoverflow.net/questions/362693 | 3 | Does there exist a topological space $X$ with the following properties?
1. $X$ is connected.
2. The set of irreducible components of $X$ is locally finite.
3. Not every pair of points in $X$ can be "connected by irreducible components", i.e., there exist points $x,y\in X$ such that there does not exist a finite seque... | https://mathoverflow.net/users/11025 | A connected topological space whose points cannot be connected by irreducible components | No such space can exist.
The proof doesn't use very much about irreducible components. That is, suppose $X$ is connected. Let $S$ be any set of closed subsets of $X$ which exhaust $X$ and suppose $S$ is locally finite in the sense that every point $x$ has a neighborhood $U\_x$ intersecting only finitely many sets $Z\... | 2 | https://mathoverflow.net/users/3075 | 364922 | 153,345 |
https://mathoverflow.net/questions/364913 | 5 | By singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20.
Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?
| https://mathoverflow.net/users/4096 | density of singular K3 surfaces | This is a standard argument and there probably exists a reference but it's not hard once you rephrase it in terms of the period domain.
The moduli space of K3 surfaces is locally isomorphic to its period domain.
The period domain is an open subset of the vanishing locus of a quadratic polynomial in $\mathbb P^{21}(... | 13 | https://mathoverflow.net/users/18060 | 364925 | 153,346 |
https://mathoverflow.net/questions/364924 | 2 | Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ is *dense along $x$* if for every prefix $\sigma \prec x$, there is $\tau \succ \sigma$ such that all finite extensions of ... | https://mathoverflow.net/users/136473 | Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$ |
>
> What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member?
>
>
>
An element that is 1-generic *relative to $T$* will not be on $[T]$ unless $[T]$ contains a whole clopen cone $[\sigma]$.
Since "most" 1-generics are 1-generic relative to $T$, I suppose this means the condition to i... | 3 | https://mathoverflow.net/users/4600 | 364928 | 153,347 |
https://mathoverflow.net/questions/364754 | 5 | Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma\_2\subset M$.
Let $\phi:\Sigma\_2\to \Sigma\_2$ be the map of sets that sends $x\in \Sigma\_2$ to the other point $y\in \Sigma\_2$ such that $f(x)=f(y)$.
$\phi... | https://mathoverflow.net/users/99042 | Is identification of double points of an immersion smooth? | I think the answer to the first 2 questions is yes. Most of the details are in the thesis of Ralph Herbert:
*Herbert, Ralph J.*, [**Multiple points of immersed manifolds**](http://dx.doi.org/10.1090/memo/0250), Mem. Am. Math. Soc. 250, 60 p. (1981). [ZBL0493.57012](https://zbmath.org/?q=an:0493.57012)
The important... | 3 | https://mathoverflow.net/users/8103 | 364940 | 153,351 |
https://mathoverflow.net/questions/364313 | 2 | I came across a generalized Black-Scholes equation formulation in [this](https://www.researchgate.net/publication/260912555_A_random_field_LIBOR_market_model) paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion and for a fixed $t$, $W$ is a co... | https://mathoverflow.net/users/51480 | Stochastic integral with respect to a random field | The notations in the question are ambiguous (Bjørn Kjos-Hanssen showed that the other interpretation cannot be correct). I assume that the expression of interest is given by
$$
g(t) = \int\_0^t \Sigma(s)\,dW(s)\;,
$$
where $W$ is a $C(\mathbb{R})$-valued Wiener process with covariance $\hat c$ (at time $1$) and $\Sigma... | 3 | https://mathoverflow.net/users/38566 | 364946 | 153,352 |
https://mathoverflow.net/questions/364948 | 4 | I'm not a professional mathematician, just an undergraduate student. I was reading Introduction to Graph Theory by West, I came over the topic which discuses the methods to find the spanning trees in a general graph. Kirchhoff's Matrix tree theorem was there. Out of curiosity, I thought, could there be any relation bet... | https://mathoverflow.net/users/160688 | Relation between Kirchhoff's Circuital law and Matrix tree Theorem | Chapter II (pages 12 and following) of [Combinatorics of Electrical Networks](http://www.math.uwaterloo.ca/~dgwagner/Networks.pdf) gives a linear algebra derivation of Kirchhoff's theorem from the circuit laws of Ohm and Kirchhoff.
| 3 | https://mathoverflow.net/users/11260 | 364951 | 153,353 |
https://mathoverflow.net/questions/364949 | 1 | I want to understand the following question:
---
Let $\delta$ be the Euclidean metric in $\mathbb{R}^2$. Is there any criteria for smooth function $u$ such that $(\mathbb{R}^2, e^{2u}\delta)$ can be compactified to a compact closed Riemannian surface?
--------------------------------------------------------------... | https://mathoverflow.net/users/51546 | compatification of $\mathbb{R}^2$ under a conformal metric | As a Riemann surface, the plane can have only one compactification, the sphere.
Introducing a factor to the metric does not change the conformal structure. So whatever factor you put there, the resulting Riemann surface remains conformally equivalent to the plane and thus have one and only one compactification, namely ... | 2 | https://mathoverflow.net/users/25510 | 364954 | 153,354 |
https://mathoverflow.net/questions/364957 | 6 | is there an established name for the property that a square matrix can be made symmetric by permutation of its columns?
Is it possible to recognize those kind of matrices efficiently?
| https://mathoverflow.net/users/31310 | Name for a matrices having a specific property | Here is a suggestion (not an answer) for the second question, at least for real matrices $A$: Suppose that there is a permutation matrix $P$ such that $(AP)^{T} = AP.$
Then $(AP)^{2} = (AP)(AP)^{T} = AA^{T}$, so that $AP$ is a symmetric square root of the positive semidefinite (symmetric) matrix $AA^{T}$. If the non-ze... | 6 | https://mathoverflow.net/users/14450 | 364959 | 153,356 |
https://mathoverflow.net/questions/364527 | 9 | An elliptic curve $E$ over $\overline{\mathbb{Q}}$ is called a *$\mathbb{Q}$-curve* if it is isogenous (over $\overline{\mathbb{Q}}$) to all its Galois conjugates -- see [Are Q-curves now known to be modular?](https://mathoverflow.net/questions/3927/are-q-curves-now-known-to-be-modular) for example.
If I take a finit... | https://mathoverflow.net/users/2481 | Q-curves and twisting | I think there are examples of $\mathbb{Q}$-curves defined over a quadratic field $K$ which are not strong $\mathbb{Q}$-curves over $K$ in Jordi Quer's paper "$\mathbb{Q}$-Curves and Abelian Varieties of $\mathrm{GL}\_2$-Type". He uses the term $\mathbb{Q}$-curves completely defined over $K$ instead of strong $\mathbb{Q... | 1 | https://mathoverflow.net/users/160702 | 364966 | 153,359 |
https://mathoverflow.net/questions/363230 | 0 | A coin $C$ is tossed $n$ times. The coin $C$ is known to have the following properties :
* Let $p\_i$ denote the probability of showing heads in the $i$-th toss, and $q\_i$ denote the probability of showing tails in the $i$ -th toss, so that $p\_i + q\_i = 1$ for all $i$,
* If the $i$ -th toss gives a heads, then $p\... | https://mathoverflow.net/users/109471 | Decaying probabilities | Building on a hint that was provided by prof. Arnab Chakraborty from Indian Statistical Institute, Kolkata, India (where I study) I am writing this answer.
Let us consider a more general problem :
suppose we have a coin $C$ which shows heads with probability $x$ and tails with probability $1-x$, and suppose $y \in [0... | 0 | https://mathoverflow.net/users/109471 | 364971 | 153,361 |
https://mathoverflow.net/questions/364914 | 1 | Let $G$ be a group of isometries acting effectively by isometries on a connected Riemannian manifold. And let $G'\triangleleft G$ be a normal subgroup. I am trying to prove that $\dim \operatorname{St}\_G(p)\leq \dim \operatorname{St}\_{G'}(p)$ for every $p\in M$; where $\operatorname{St}\_X(p)$ stands for the stratum ... | https://mathoverflow.net/users/134552 | Let $G'\triangleleft G<\operatorname{Iso}(M)$ be a normal subgroup. A $G'$-stratum is the union of $G$-strata of lesser dimension | We have to assume that the index $[G':G]$ is finite. In this case:
Let $G'$ be a normal subgroup of $G$ such that the quotient $\Gamma=G/G'$ is finite and acts by isometries in $X'=M/G'$, and $X=X'/\Gamma$. Thus, $(G')^0=G^0$ and, therefore the orbits $G'(p)$ and $G(p)$ have same connected components through $p$. Thi... | 0 | https://mathoverflow.net/users/134552 | 364972 | 153,362 |
https://mathoverflow.net/questions/364772 | 0 | Let $S$ be an infinite set of positive integers. Let us define the following quantities:
* $N\_S(z)$ is the number of elements of $S$, less or equal to $z$
* $r\_S(z)$ if the number of positive integer solutions to $x+y\leq z$,
with $x,y\in S$ and $z$ an integer
* $t\_S(z)$ if the number of positive integer solutions... | https://mathoverflow.net/users/140356 | Paradox in additive combinatorics | I wrote:
*My test power set (see definition in the example section) consists of integers that are even far more rare than pseudo-super-super-primes, yet for them conjecture A works, as expected. Perhaps this is caused by the fact that these integers are far more abundant than pseudo-super-super-primes among the first... | 1 | https://mathoverflow.net/users/140356 | 364973 | 153,363 |
https://mathoverflow.net/questions/364931 | 5 | Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim\_k g$ if at each point in the domain, the determinants of all $k \times k$ minors of the Jacobians of $f$ and $g$ coincide. This is clearly an equivalence relation; and more generally, one could let... | https://mathoverflow.net/users/18263 | Equivalence generated by Jacobian minors | First, let me point out that the OP's suggested generalization to arbitrary target manifolds $Y$ of asking that $f^\*\omega = g^\*\omega$ for *all* $k$-forms on $Y$, is *not* equivalent to the question about equality of $k$-by-$k$ minors.
To see this, consider the simplest case, $X = Y=\mathbb{R}^1$ and $k=1$. The ge... | 4 | https://mathoverflow.net/users/13972 | 364979 | 153,365 |
https://mathoverflow.net/questions/364978 | 10 | Let $C\_m$ be the cycle with $m$ vertices, defined so that $C\_1$ has a self-loop on its unique vertex. Let $p\_m$ be the generating function enumerating the number of ways to choose $k$ vertices in $C\_m$ so that no two are adjacent. Thus the coefficient of $z^k$ in $p\_m(z)$ is the number of independent sets in $C\_m... | https://mathoverflow.net/users/7709 | Is there a bijective proof of an identity enumerating independent sets in cycles? | It seems that I’ve seen this question here before, but I am not sure whether it had a bijective answer. Anyway, here you are.
Enumerate the vertices in two copies of $C\_m$ as $1,2,\dots,m$ and $1’,2’,\dots,m’$, respectively. Take any independent set of size $k<m$ in the union of these cycles (regard it as *marking* ... | 7 | https://mathoverflow.net/users/17581 | 364987 | 153,367 |
https://mathoverflow.net/questions/364982 | 12 | Let $G$ and $H$ be connected Lie groups. A Lie group homomorphism $\rho:G\to H$ is a smooth map of manifolds which is also a group homomorphism.
**Question**: Can we find a smooth (or real-analytic) map $f:G\to H$ which is not homotopic to any Lie group homomorphism?
For example, if $G=H=S^1$, it seems the answer i... | https://mathoverflow.net/users/69190 | Smooth map homotopic to Lie group homomorphism | If $G$ is a compact simply-connected simple Lie group, then any nontrivial homomorphism $G\to G$ is an automorphism (it is injective because $G$ is simple, and any immersion of closed connected manifolds of the same dimension is covering map), and in particular it has degree $\pm 1$. For example, if $f: S^3\to S^3$ is ... | 22 | https://mathoverflow.net/users/1573 | 364988 | 153,368 |
https://mathoverflow.net/questions/364950 | -1 | When are the line graphs of Cayley graphs Cayley?
From [this](https://mathoverflow.net/questions/150744/the-line-graphs-of-complete-graphs-and-cayley-graphs) link we can know when the line graphs of complete graphs are cayley. But, my question pertains to the larger class of Cayley graphs. Are there new results in th... | https://mathoverflow.net/users/100231 | Which line graphs of Cayley graphs are Cayley | One would expect few Cayley graphs to have this property, as their automorphism group is just not big enough. Either way, a complete classification is almost certainly out of reach. Even the example you give of complete graphs is far from trivial (the published proof requires on quite a few group-theoretic results, inc... | 1 | https://mathoverflow.net/users/22377 | 364993 | 153,370 |
https://mathoverflow.net/questions/364996 | 2 | **Notivation.** Recently I attended a little party (adhering to physical distancing and in accordance to other COVID19-related laws). An attendee told me that he chose drink $Y$ since at least half of his acquaintances at the party had drink $X$. I was sober enough to put this into a graph-theoretic property - and that... | https://mathoverflow.net/users/8628 | Non-conformity colorings | If $\kappa$ is an infinite cardinal, there is a graph $G$ of order $\kappa$ with $\chi\_{nc}(G)=\kappa$. If $\kappa$ is regular, $G$ can be a complete graph; if $G$ is singular, $G$ can be a disjoint union of complete graphs.
If $G$ is a finite graph, then $\chi\_{nc}(G)\le3$; just take a $3$-coloring of $G$ which mi... | 4 | https://mathoverflow.net/users/43266 | 365000 | 153,373 |
https://mathoverflow.net/questions/364142 | 3 | In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We consider $2$ different cases:
For our first question we consider $(M,\omega)$ a symplectic manifold. Then $\Omega^0(M)... | https://mathoverflow.net/users/36688 | When is the exterior derivation $d$ a Lie algebra morphism? | On question 1, to expand on what @BK said: If you have a symplectic structure $\omega$ on a manifold $M$, you get a natural Lie bracket on $\Omega^1(M)$ by the following rule:
$$ [\alpha, \beta ] = \omega^\flat([\omega^\sharp (\alpha), \omega^\sharp(\beta)]) $$
Where:
$$ \omega^\sharp \colon \Omega^1(M) \to \mathfrak{X... | 3 | https://mathoverflow.net/users/91000 | 365004 | 153,375 |
https://mathoverflow.net/questions/364491 | 4 | Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that is, there are only finitely many $S$-unit $u$ such that $1-u$ is also an $S$-unit. Therefor for each such $S$ there exi... | https://mathoverflow.net/users/142000 | P-adic distance between solutions to S-unit equation | There cannot be such a uniform bound, unless $p=2$ in which case there are
no solutions at all (because $p \notin S$ but $u$ and $1-u$ cannot both be
$2$-adic units).
Fix $p$ and $e$. We shall construct a set $S$ of uniformly bounded size
and $S$-units $u\_1,u\_2$ that are congruent modulo $p^e$.
By a theorem of Ch... | 5 | https://mathoverflow.net/users/14830 | 365013 | 153,378 |
https://mathoverflow.net/questions/364991 | 2 | A variety $ Z $ over a field $ k $ of characteristic zero is ruled if there is a variety $ M $ and a dominant, birational map $ \phi: M \times \mathbb{P}^{1}\_{k} \dashrightarrow Z $. A variety $ Z $ over a field $ k $ of characteristic zero is uniruled if there is a variety $ M $ and a dominant, rational map $ \phi: M... | https://mathoverflow.net/users/113893 | Example of a projective variety over a field of characteristic zero which is uniruled but not ruled | I think you just need to know that there exists a threefold $X$ which is unirational but not rational (e.g. the cubic threefold). If $X$ is birational to $S\times \mathbb{P}^1$, there is a dominant rational map $X -\!-\!\!\!> S$, thus $S$ is unirational, hence rational by Castelnuovo's theorem. Therefore $X$ is rationa... | 6 | https://mathoverflow.net/users/40297 | 365018 | 153,379 |
https://mathoverflow.net/questions/365015 | 4 | $h\_{d\_1}, h\_{d\_2}$ and $h\_{d\_1d\_2}$ are class number of $\Bbb Q(\sqrt{d\_1}),\Bbb Q(\sqrt{d\_2}),\ and \ \Bbb Q(\sqrt{d\_1d\_2})$ respectively.
| https://mathoverflow.net/users/131448 | Is there any relation between $h_{d_1},h_{d_2}$ and $h_{d_1d_2}$? | There is a relation between the class numbers of those quadratic fields *and* the class number of the biquadratic field $\mathbf Q(\sqrt{d\_1},\sqrt{d\_2})$. The key term to look for is "Brauer relation", which has a wider scope than just this example. A reference for this topic has already been mentioned in an earlier... | 5 | https://mathoverflow.net/users/3272 | 365023 | 153,381 |
https://mathoverflow.net/questions/365020 | 0 | I wonder if the limit below $$\lim\_{x\rightarrow +\infty} e^{-x}\sum\_{j=0}^{\infty}\frac{x^{j+a}}{\Gamma(j+a+1)}$$
equals 1, for real constant $a>0$, and how shall we get this result?
| https://mathoverflow.net/users/146060 | A limit calculation | If we take $$\chi(x)=\sum\_{j=0}^{\infty} \frac{x^{j+a}}{\Gamma(j+a+1)}$$ then we get, $$\chi'(x)-\chi(x)=\frac{x^{a-1}}{\Gamma(a)}$$
Hence, the solution gives $$\psi(x)=\chi(x)e^{-x}=\frac{1}{\Gamma(a)}{\int e^{-x} x^{a-1} dx}+c$$
This means $$\lim \limits\_{x \to \infty} \psi(x) -\psi(0)=1$$
As, $$\color{grey}{... | 4 | https://mathoverflow.net/users/156029 | 365027 | 153,382 |
https://mathoverflow.net/questions/364867 | 9 | It is known that the connected totally geodesic complex submanifolds of a projective space ${\rm P}V$ equipped with a Fubini-Study metric are precisely the projective subspaces ${\rm P}Z$, where $Z \subseteq V$ is a complex subspace. Direct implication is obvious, and if for the converse we assume that $N\subseteq {\rm... | https://mathoverflow.net/users/54656 | Totally geodesic submanifolds of complex Grassmannians | There is an extensive literature on totally geodesic submanifolds of symmetric spaces. A good place to start to read about this (and references to the preceeding literature) would be, for example,
Bang-yen Chen and Tadashi Nagano, *Totally geodesic submanifolds of symmetric spaces, I*, Duke Math Journal **44** (1977)... | 4 | https://mathoverflow.net/users/13972 | 365033 | 153,383 |
https://mathoverflow.net/questions/365043 | 1 | I am looking for a **reference** for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". i.e. for any $x\_1,x\_2>0, \alpha \in [0,1]$ satisfying $\alpha x\_1 + (1- \alpha)x\_2 =c$, we have
$$
\phi(c)=\phi\left(... | https://mathoverflow.net/users/46290 | Convexity at a point and Jensen inequality | For any real numbers $u,v,c$ such that $u\le c\le v$, let $\mu\_{c;u,v}$ denote the unique probability distribution on the set $\{u,v\}$ with mean $c$.
Your generalization of Jensen's inequality follows immediately from the well-known fact that any probability distribution $\mu$ on $\mathbb R$ with a given mean $c\in... | 4 | https://mathoverflow.net/users/36721 | 365045 | 153,389 |
https://mathoverflow.net/questions/365046 | 12 | What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^\*(P,\mathbb{Q})=H^\*(G,\mathbb{Q})\otimes H^\*(M,\mathbb{Q})$$
| https://mathoverflow.net/users/36688 | A nontrivial principal bundle which satisfies Leray-Hirsch theorem | Let $P$ be any $SU(2)$-bundle on $X$ with vanishing second Chern class $c\_2(P)$. The hypotheses of the Leray-Hirsch theorem are satisfied if there is a class in $H^3(P)$ which restricts to the generator of $H^3(SU(2))$. This happens if and only if in the Leray spectral sequence, the map $d\_3: H^0(X,H^3(SU(2))) \to H^... | 10 | https://mathoverflow.net/users/125523 | 365047 | 153,390 |
https://mathoverflow.net/questions/363713 | 6 | In page 79 of Bott-Tu, "Differential Forms in Algebraic Topology", they define the relative de Rham theory as follows:
Let $f:S\to M$ be a smooth map. Define the complex $\Omega^\*(f)$ by
$$\Omega^k(f):=\Omega^k(M)\oplus\Omega^{k-1}(S)$$ $$\underline{\mathrm{d}}(\alpha,\beta)=(\mathrm{d}\alpha,f^\*\alpha-\mathrm{d}\b... | https://mathoverflow.net/users/107013 | De Rham's theorem for top-forms in manifolds with boundary | It is indeed true that $H^n(M)=0$ if $M$ is a compact manifold with boundary. In particular, $H^n(M,\partial M)\cong\mathbb{R}$ by Lefschetz duality (as Chris Gerig mentioned) and the integral (1) is an isomorphism.
The only reference I have found that states this results is:
Differential forms: theory and practice... | 4 | https://mathoverflow.net/users/107013 | 365048 | 153,391 |
https://mathoverflow.net/questions/365052 | 13 | I wanted to ask for your intuition about ordinal fixed points $\alpha = \aleph\_\alpha$, where $\aleph\_\alpha$ stands for the $\alpha$-th Aleph number in the Aleph sequence of cardinalities.
For background why I am asking this. I was surprised when I first learned $|\mathbb{Q}| = |\mathbb{N}|$ and $|{\cal P}(A)|>|A|... | https://mathoverflow.net/users/156936 | Intuition about ordinal fixed points $\alpha = \aleph_\alpha$ | Your intuition is finitary, and therefore wrong. Compare, for example, the two sequences:
1. $\alpha\_n=n$, and
2. $\beta\_n=2^n$.
It is easy to see that $\alpha\_n<\beta\_n$ for all $n$. We even know from elementary calculus that the rate of change between them is growing very fast as well, so there is no possible... | 14 | https://mathoverflow.net/users/7206 | 365056 | 153,393 |
https://mathoverflow.net/questions/365057 | 2 | A homogeneous structure is a countable first order structure $M$ over a relational language such that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$.
Lachlan proved that if $M$ is any stable countably homogeneous structure over a finite relational language $\mathcal{L}$,... | https://mathoverflow.net/users/120374 | Question on countably homogeneous structures | Let's first observe that any stable countable homogeneous structure in a finite relational language is $\aleph\_0$-categorical and $\aleph\_0$-stable. This is explained in *A survey of homogeneous structures* by Macpherson: $\aleph\_0$-categoricity is Corollary 3.1.3 on p. 17, and $\aleph\_0$-stability is in the paragr... | 6 | https://mathoverflow.net/users/2126 | 365059 | 153,394 |
https://mathoverflow.net/questions/365024 | 5 | Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, [On some functions connected with $\varphi(n)$](https://doi.org/10.1090/S0002-9904-1929-04799-2), Bull. Amer. Math. Soc. 35 (1929), 832–836). I'm interested to know about bounds on solution... | https://mathoverflow.net/users/51189 | Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound? | **UPD.** Bound simplified.
Here is a constructive bound for the number of solutions to $\phi(x)=m$.
Let $\varphi(a) = m$. If $p^k\mid a$ for some $k\geq 1$, then $p^{k-1}(p-1)\mid m$, and thus $k\leq 1+\frac{\log(m)}{\log(p)}\leq 1+\frac{\log(m)}{\log(2)}$.
Then the number of such $a$ is bounded by
$$\prod\_{d\mid ... | 8 | https://mathoverflow.net/users/7076 | 365067 | 153,396 |
https://mathoverflow.net/questions/365061 | 5 | Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result:
>
> **Theorem** (Lomonosov): Every nonscalar $T \in B(H)$ which commutes with a nonzero compact operator $K$ has a nontrivial hyperinvariant subspace.
>
>
>
It has been shown that... | https://mathoverflow.net/users/160051 | An extension of Lomonosov Theorem | Looking again at Solution 12.4 in Kubrusly's book, I have noticed that the proof can be used even to prove the statement above, with some small changes. I will briefly sketch such small modifications: instead of considering the operator $TS-ST$, we will consider $C:=TS-\alpha ST$ ($\alpha \neq 0$). The claim:
>
> I... | 1 | https://mathoverflow.net/users/160051 | 365069 | 153,398 |
https://mathoverflow.net/questions/364909 | 1 | Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions:
* $N\_S(z)$ is asymptotic continuous version of the function counting the number of elements in $S$ less or equal to $z$.
* $N'\_S(z)$, the derivative of $N\_S(z)$, is the "probability" for... | https://mathoverflow.net/users/140356 | Curious inversion formula in additive combinatorics | This is not an answer to the question, but an explanation as to how I came up with the formula for $w(z)$. We assume here that $S$ is a random set. That is, let us consider $X\_z$ as a Bernouilli random variable of parameter $N'\_S(z)$. A positive integer $z$ belongs to $S$ if and only if $X\_z = 1$. Thus $P(z\in S) = ... | 1 | https://mathoverflow.net/users/140356 | 365080 | 153,403 |
https://mathoverflow.net/questions/365070 | 17 | In *A survey of homogeneous structures* by Macpherson (Discrete Mathematics, vol. 311, 2011), a stable or unstable theory is defined as (Definition 3.3.1):
>
> A complete theory $T$ is *unstable* if there is a formula $\varphi(\overline{x}, \overline{y})$ (where $\ell(\overline{x}) = r$ and $\ell(\overline{y}) = s)... | https://mathoverflow.net/users/120374 | Intuition behind stability and instability in model theory | I am going to explain some motivation by relating this definition to stability to other definitions, and discussing some examples. For simplicity I am going to assume we are working in a countable language.
An alternate definition of stability, for a complete theory $T$ is as follows.
>
> **Definition.** $T$ is *... | 25 | https://mathoverflow.net/users/38253 | 365081 | 153,404 |
https://mathoverflow.net/questions/364635 | 1 | Given a linear diophantine equation $$x\_1+\dots+x\_n=m\leq nn'$$ how many solutions does it have with each $x\_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $n'$ over different ranges for both situations
1. $x\_1\leq\dots\leq x\_n$ and
2. unordered.
| https://mathoverflow.net/users/136553 | Integer partitions into restricted parts | Since $m$ is a dummy variable (*i.e.* a bound variable) and $n,n'$ are "real" variables (*i.e.* they are free) perhaps we should rewrite the problem accordingly as
$``$compute the following
$$ f(y,z) = \#\left\lbrace (x\_1,... , x\_y )\mid x\_1 + ... + x\_y = m,\ x\_i \in \mathbb{N},\ m \leq yz ,\ i < j \implies x\_i \... | 3 | https://mathoverflow.net/users/157298 | 365094 | 153,407 |
https://mathoverflow.net/questions/365122 | 2 | Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some constant $c \geq 0$, where $\left| \! \left| \cdot \right| \! \right|$ is the operator norm,
\begin{align\*}
\left| \! \left... | https://mathoverflow.net/users/nan | A norm inequality for operators | $$
\|\sqrt{A+B}Cx\|^2=(\sqrt{A+B}Cx,\sqrt{A+B}Cx)=\\
((A+B)Cx,Cx)=(ACx,Cx)+(BCx,Cx)=\|\sqrt{A}Cx\|^2+\|\sqrt{B}Cx\|^2,
$$
taking the supremum over unit vectors $x$ we get
$$
\|\sqrt{A+B}C\|^2\leqslant \|\sqrt{A}C\|^2+\|\sqrt{B}C\|^2\leqslant (\|\sqrt{A}C\|+\|\sqrt{B}C\|)^2.
$$
| 9 | https://mathoverflow.net/users/4312 | 365123 | 153,415 |
https://mathoverflow.net/questions/365135 | 2 | Let me first explain the setup:
>
> Let $(X\_t)\_{t \geq 0}$ be a stochastic process on some probability space $(\Omega,\mathcal{F},P)$ with values in a complete and separable metric space $E$ (e.g. $E = \mathbb{R}$) and let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. The conditional probability distrib... | https://mathoverflow.net/users/157982 | Continuous version of conditional probability distributions $( \mathcal{L}(X_t | \mathcal{G}) )_{t \geq 0}$ if $(X_t)_{t \geq 0}$ is continuous? | For simplicity take $E=\Bbb R$ and the time interval to be $[0,1]$, and think of $X=(X\_t)\_{0\le t\le 1}$ as a random element of $C=C([0,1]\to\Bbb R)$, a Polish space. We then have a regular conditional distribution of $X$ given $\mathcal G$, call it $Q=Q(\omega,B)$, $\omega\in\Omega, B\in\mathcal B(C)$. And the induc... | 3 | https://mathoverflow.net/users/42851 | 365138 | 153,419 |
https://mathoverflow.net/questions/364470 | 5 | As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of this term provides important simplifications to the theory and we can give meaning to the theory when this term is not inc... | https://mathoverflow.net/users/150264 | A set of questions on continuous Gaussian Free Fields (GFF) | Essentially, what is asked is the continuation of my previous MO answer
[Reformulation - Construction of thermodynamic limit for GFF](https://mathoverflow.net/questions/362040/reformulation-construction-of-thermodynamic-limit-for-gff/362118#362118)
and the solution of the exercise I mentioned at the end of that ans... | 5 | https://mathoverflow.net/users/7410 | 365144 | 153,421 |
https://mathoverflow.net/questions/365141 | 0 | I don't know how to prove that the definition
\begin{equation}
\lambda\_r = \frac{1}{r} \sum\_{j=0}^{r-1} (-1)^j {r - 1 \choose j} E[X\_{r-j:r}]
\end{equation}
where
\begin{equation}
E[X\_{r:n}] = \frac{n!}{(r - 1)! \: (n - r)!} \int\_{0}^{1} x(u) \: u^{r-1} \:(1-u)^{n-r} \: du
\end{equation}
is consistent wi... | https://mathoverflow.net/users/160827 | Consistent of the two definitions of L-moments | We have
\begin{equation}
EX\_{r-j:r}=\frac{r!}{(r-j-1)!\,j!} \int\_0^1 du\,x(u)\,u^{r-j-1}\,(1-u)^j \\
=r\binom{r-1}j \int\_0^1 du\,x(u)\,u^{r-j-1}\,(1-u)^j
\end{equation}
and
\begin{equation}
(1-u)^j=\sum\_{i=0}^j(-1)^i \binom ji u^i.
\end{equation}
So,
\begin{equation}
\lambda\_r=\int\_0^1 du\,x(u)\,p\_r(u),
\e... | 2 | https://mathoverflow.net/users/36721 | 365154 | 153,427 |
https://mathoverflow.net/questions/365082 | 10 | With the usual topology on $\Bbb R$, a compactification $\mathrm{id}\_{\Bbb R}:\Bbb R\to v\Bbb R$ can have a remainder $v\Bbb R \setminus \Bbb R$ of cardinality $1,2, 2^{\aleph\_0}=\mathfrak c,$ or $2^{\mathfrak c}.$ The only possibilities less than $\mathfrak c$ are $1,2.$
Suppose $\mathfrak c^+<2^{\mathfrak c}.$ Wh... | https://mathoverflow.net/users/81583 | Possible cardinalities of the remainders of compactifications of $\Bbb R$ | Every connected compact Hausdorff space of weight $\aleph\_1$ is the remainder $v \mathbb R \setminus \mathbb R$ of some compactification of $\mathbb R$. In particular, $[0,1]^{\aleph\_1}$ is the remainder of a compactification of $\mathbb R$, and therefore $\mathbb R$ has a compactification with remainder of cardinali... | 8 | https://mathoverflow.net/users/70618 | 365158 | 153,430 |
https://mathoverflow.net/questions/365101 | 1 | I'm a young researcher and I'm pretty new in this field. I want to work on packing problem "L's in Tans" and "L's in L's" as presented on <https://erich-friedman.github.io/packing> . On link can be found some known optimal packings, but I can't find any papers related to this problem.
Can someone show me some papers ... | https://mathoverflow.net/users/155474 | Packing L's in Tans and L's in L's | It might be hard to find literature on the specific families of packing problems you mentioned. However, they are special cases of a more general pattern of asking for the optimal packing of $N$ congruent copies of an object inside another object. You can find many papers tackling this type of problem, see what kind of... | 2 | https://mathoverflow.net/users/20186 | 365168 | 153,433 |
https://mathoverflow.net/questions/365163 | 1 | Suppose you roll a dice 100 times, How many times would you expect the most common number to show up.
I.e. roll a dice 100 times and document the frequency of each value, then repeat this process infinitely many times and take the mean of the highest frequency from each trial.
Is there a way to derive a formula or ... | https://mathoverflow.net/users/160834 | Dice roll expectation question | According to the multinomial probability mass function formula, the expected maximum frequency in $n$ rolls of a fair die is
$$e\_n=\frac1{6^n}\sum\frac{n!}{x\_1!\cdots x\_6!}\,\max(x\_1,\dots,x\_6),$$
where the sum is taken over all $n$-tuples $(x\_1,\dots,x\_6)$ of nonnegative integers such that $x\_1+\dots+x\_6=n$. ... | 2 | https://mathoverflow.net/users/36721 | 365169 | 153,434 |
https://mathoverflow.net/questions/365166 | 0 | I have a Poisson process where new elements arrive to a set with Poisson intensity $\lambda$. Initially, there are $N\_0$ elements in the set. The probability that there are $N\_0 + M$ elements in the set at time $t$ is $Pr[N(t) = N\_0+M] = \frac{(\lambda t)^M}{M!} e^{-\lambda t}$.
I'm interested in the expected numb... | https://mathoverflow.net/users/51134 | Expected size of binomial coefficient with Poisson arrivals? | Suppose that a real $c:=\lambda t>0$ and a natural $k$ are fixed, whereas $n:=N\_0\to\infty$.
Take any real $m>0$. Then
$$f(k,t)=g\_m(k,t)+h\_m(k,t),$$
where
$$g\_m(k,t):=e^{-c}\sum\_{0\le j<m}\frac{c^j}{j!}\,\binom{n+j}k,$$
$$h\_m(k,t):=e^{-c}\sum\_{j\ge m}\frac{c^j}{j!}\,\binom{n+j}k.$$
For each $j$, $\binom{n+j}k\si... | 2 | https://mathoverflow.net/users/36721 | 365174 | 153,435 |
https://mathoverflow.net/questions/365175 | 9 | It is a standard consequence of the Brown Representability Theorem for $\operatorname{Ho}(\operatorname{Top}\_\*)$ that the category of generalized cohomology theories for spaces (pointed CW complexes, more specifically) is equivalent to the stable homotopy category $\operatorname{SHC}$ (defined as the homotopy categor... | https://mathoverflow.net/users/158123 | Cohomology theories for spaces vs cohomology theories for spectra | The category of cohomology theories on pointed CW-complexes is *not* equivalent to the stable homotopy category. The latter projects onto the former, and this projection induces a bijection on isomorphism classes, but there is a kernel, containing superphantom maps, see [Christensen, J.Daniel. “Ideals in Triangulated C... | 14 | https://mathoverflow.net/users/12166 | 365180 | 153,437 |
https://mathoverflow.net/questions/365140 | 5 | Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the *operadic suspension* $\mathfrak{s}\mathcal{O}$ is defined in several different ways depending on the author. Some standard refere... | https://mathoverflow.net/users/144957 | Is operadic desuspension inverse to operadic suspension? | What you really need to show is that
$$f(a\circ\_ib)=(-1)^{(n-1)(m-1)}f(a)\circ\_if(b).$$
Here, $n$ is the arity of $a$, $m$ is the arity of $b$, and $\circ\_i$ is the infinitesimal composition in $\mathcal{O}$ (once you twist the definition of the infinitesimal composition by your sign, you get the usual equation for ... | 6 | https://mathoverflow.net/users/12166 | 365182 | 153,438 |
https://mathoverflow.net/questions/364890 | 1 | My question is that how much information we can get form integer moments of a complex random variable?
Let $\mathcal{Z}$ be a complex value random variable, and assume that we can compute $$\int \mathcal{Z}^k d\mu,$$
For $k \in \mathbb{N}$ and $\mu$ be a measure.
I also am looking for an example of a complex random... | https://mathoverflow.net/users/18950 | Moments of complex random variables | This is to rewrite the excellent example by Mateusz Wasilewski in a more conventional form.
Let $Z:=XU$, where $X$ and $U$ are independent random variables (r.v.'s); $P(X>0)=1$; $X$ is unbounded; $EX^k<\infty$ for all natural $k$; $U=e^{iT}$; $T$ is a r.v. with values in the interval $[0,2\pi)$ and pdf $p$ given by t... | 2 | https://mathoverflow.net/users/36721 | 365197 | 153,444 |
https://mathoverflow.net/questions/362604 | 3 | Is there a generalization of Segal's theorem that the inclusion of $X\_1$ into $\Omega|X\_\*|$ is a weak equivalence for a $\Gamma$-space $X\_\*$ if $X\_1$ is group like? Specifically, I am looking for a result something like $\hat{X} \rightarrow \Omega ^n |X\_{\*,\dots,\*}|$ is a weak equivalence where $X\_{\*,\dots,\... | https://mathoverflow.net/users/134512 | Multi-simplicial generalization of $\Gamma$-spaces | The paper "Iterated monoidal categories" by Balteanu, Fiedorowicz, Schwanzl, Vogt gives an analogous result for functors $(\Delta ^{op})^n \rightarrow Top$ and n-fold loop spaces.
| 1 | https://mathoverflow.net/users/134512 | 365198 | 153,445 |
https://mathoverflow.net/questions/365199 | 4 | Ultimately, I'm trying to figure out whether or not the full subcategory in $\mathbf{sSet}$ spanned by Kan complexes is finitely complete (as a $1$-category).
Since fibrations are stable under pullback in general, I know that Kan complexes are closed under finite products, so the question boils down to whether the pull... | https://mathoverflow.net/users/160838 | ($1$-)pullbacks of Kan complexes | Take any simplicial set $X$ which is not a Kan complex. Let $K$ be a Kan replacement of $X$, and let $L$ be a Kan replacement of the pushout $K\amalg\_X K$. Then the two maps $K\to L$ are levelwise injective, and the pullback $K\times\_L K$ is precisely $X$.
| 8 | https://mathoverflow.net/users/39747 | 365204 | 153,446 |
https://mathoverflow.net/questions/360629 | 4 | Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$
satisfying the following two conditions:
**(i)** $\operatorname{Jac}(p,q):=p\_xq\_y-p\_yq\_x \in \mathbb{C}-\{0\}$.
(Generally, $\operatorname{Jac}(p,q) \in \mathbb{C}(x,y)$).
**(ii)** One of $\{p,q\}$ can be written as $\fra... | https://mathoverflow.net/users/72288 | Certain endomorphisms of $\mathbb{C}(x,y)$ | The answer is **no**.
Take $p=\frac{x^2}{2}$, take $q=\frac{y}{x}$. The Jacobian matrix is $\begin{pmatrix} x& -\frac{y}{x^2} \\ 0 & \frac{1}{x}\end{pmatrix}$ whose determinant is equal to $1$. However, $f$ is definitely not an automorphism of $\mathbb{C}(x,y)$.
More generally, take any polynomial $p\in\mathbb{C}[x... | 5 | https://mathoverflow.net/users/23758 | 365208 | 153,447 |
https://mathoverflow.net/questions/365210 | 7 | Let $D$ be the unit disk in the complex plane, and assume that $g$ is a Riemannian metric on $D$ which is complete and conformal to the standard Euclidean metric. Can it be the case that the Gaussian curvature of $g$ approaches zero as we approach $\partial D$?
| https://mathoverflow.net/users/160856 | Curvature of complete conformal metrics on the open unit disk | Yes. Take the metric with length element $\rho(z)|dz|$ where $\rho(z)=(1-|z|)^{-2}$.
It is complete since $\int^1\rho(t)dt=\infty$, and the curvature
$$-\rho^{-2}\Delta\log\rho=\rho^{-4}({\rho'}^2-\rho\rho'')=-2(1-r)^2\to 0,$$
where $r=|z|$ and the primes indicate differentiation with respect to $r$.
| 7 | https://mathoverflow.net/users/25510 | 365226 | 153,452 |
https://mathoverflow.net/questions/365219 | 3 | I recently have begun reading about DF-spaces and its clear to me that $C(K)$ is a DF-space for any compact subset (non-empty) $K$ of some $\mathbb{R}^D$ for finite D, since $C(K)$ is Banach. However, what if we relace $K$ by all of $\mathbb{R}^D$ itself? Is $C(\mathbb{R}^D)$ a DF-space when it is equipped with the usu... | https://mathoverflow.net/users/36886 | Is $C(\mathbb{R}^n)$ is a DF-Space? | No. With the topology of uniform convergence on compact sets, $C(\mathbb R^d)$ is a Fréchet space (for the sequence of semi-norms $p\_n(f)=\sup\{|f(x)|: |x|\le n\}$) and the intersection of the classes of Fréchet and DF-spaces consists precisely of all Banach spaces.
Here is a proof of this well-known fact: By defini... | 8 | https://mathoverflow.net/users/21051 | 365232 | 153,455 |
https://mathoverflow.net/questions/365114 | 11 | I asked this [same question](https://math.stackexchange.com/questions/3728239/how-to-understand-the-effect-of-adjoint-functors) on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good grasp of all differe... | https://mathoverflow.net/users/54507 | How to understand adjoint functors? | Nice question Bumblebee. So, let us start with some "metaphysics of adjointness":
>
> THE LEFT AND RIGHT ADJOINTS TO A FUNCTOR
>
>
> $ \mathcal{F}:\mathcal{C}\hookrightarrow\mathcal{D}$
>
>
> ARE THE FREE (LEFT) AND CO-FREE (RIGHT) WAYS TO GO BACK FROM $D$ TO
> $C$.
>
>
>
If you choose some easy examples, ... | 7 | https://mathoverflow.net/users/15293 | 365237 | 153,456 |
https://mathoverflow.net/questions/365173 | 15 | The question [here](https://mathoverflow.net/questions/364902/which-complete-orthomodular-lattices-arise-from-von-neumann-algebras) inspires my present question.
Reyes proves [here](https://arxiv.org/abs/1101.2239) that the contravariant functor Spec from the category of commutative rings to the category of sets cann... | https://mathoverflow.net/users/6269 | Can one associate a "nice" topos to a von Neumann algebra? | (I'm going to be a bit informal to be able to go to the point relatively directly, but if you want more details on some specific aspect. I can try to add them)
Toposes are closely related to topological groupoids, in fact, they can be seen as a special type of localic groupoids or localics stack, the "étale-complete ... | 12 | https://mathoverflow.net/users/22131 | 365242 | 153,457 |
https://mathoverflow.net/questions/365218 | 8 | Say I pick $n$ i.i.d. random standard normal points in $\mathbb{R}^d$. Roughly, as long as $n$ is much smaller than exponential in $d$, with high probability all points will be on the convex hull. This is because with high probability they will all be near the radius $\sqrt{d}$ sphere and all almost orthogonal, and thu... | https://mathoverflow.net/users/22930 | Probability that random high dimensional vectors are all on the convex hull | It's not too bad to see that the probability is at most $2n^2 e^{-d/2e}$. Let $x\_1,\ldots,x\_n$ be the points. We will use a union bound, so it is sufficient to examine the probability that $x\_1$ is in the convex hull of $x\_2,\ldots,x\_n$. This happens if and only if there are $\lambda\_j \in [0,1]$ with $\sum \lamb... | 6 | https://mathoverflow.net/users/69870 | 365249 | 153,460 |
https://mathoverflow.net/questions/364926 | 1 |
>
> If we let $\omega\_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $\Pr\_{n\in\mathbb{N}}[\omega\_Q(n)=k]$ as $Q\to\infty$ if $k$ remains fixed (or perhaps very small with respect to n)?
>
>
>
I am asking this question since my study led... | https://mathoverflow.net/users/159298 | Asymptotic for the probability that a number has $k$ prime factors less than $Q$ | As pointed out in the question, we have that
$$\prod\_{p<Q}\left(\frac{x-1}{p}+1\right)=\sum\_{k=0}^{\pi(Q)}\Pr\_{n\in\mathbb{N}}[\omega\_Q(n)=k]x^k$$
which can be derived by showing that on both the RHS and the LHS the coefficient of $x^k$ is equal to
$$\sum\_{\substack{S\subseteq \{p<Q\} \\ |S|=k}} \left(\prod\... | 2 | https://mathoverflow.net/users/159298 | 365250 | 153,461 |
https://mathoverflow.net/questions/365244 | 6 | Fix a smooth function $f:\mathbb{R}\to\mathbb{R}$. Do there exist real numbers $a<b$, an infinite set $S\subset (a, b)$ and an analytic function $g$ defined on $(a-\epsilon, b+\epsilon)$ for some $\epsilon>0$ such that $f|\_S=g|\_S$?
If $g$ is only required to be defined on $(a, b)$ the question has a positive answer... | https://mathoverflow.net/users/nan | A smooth function $\mathbb{R}\to\mathbb{R}$ agrees with an analytic function on a bounded infinite set | The answer is **no**, not necessarily.
Let $f$ be any smooth function such that the Taylor series of $f$ about any point $p$ has zero radius of convergence; see [this MO answer](https://mathoverflow.net/a/81479/108637) for an explicit example.
Suppose that $g$ is a smooth function such that for some sequence $(x\_n... | 3 | https://mathoverflow.net/users/108637 | 365259 | 153,463 |
https://mathoverflow.net/questions/364800 | 14 | Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$? Note that we are not merely aski... | https://mathoverflow.net/users/nan | Non-algebraic holomorphic maps between algebraic curves | Just turning my comments into an answer:
Following the OP, let $V$ be a smooth projective connected curve with negative Euler characteristic (i.e., genus at least two) over $\mathbb{C}$. Then $V$ is hyperbolic in the sense that Kobayashi's pseudometric is a metric. In particular, by a theorem of Kwack, it is "Borel h... | 3 | https://mathoverflow.net/users/4333 | 365273 | 153,465 |
https://mathoverflow.net/questions/365171 | 5 | Suppose that we have a compact Kaehler manifold $X$ with big and nef canonical class $c\_1(K\_{X})$, does it imply that $X$ is projective? By the base point free theorem, big and nef implies semi ample but it is for projective algebraic manifolds. So it seems to suggest that big and nef does not necessarily imply proje... | https://mathoverflow.net/users/142966 | Does big and nef imply projectivity? | If $X$ has a big line bundle $L$ then for an appropriate natural number $m$, sections of $L^m$ define a meromorphic map $\varphi: X \dashrightarrow \mathbf P^N$ which is bimeromorphic onto its image. Therefore $X$ is bimeromorphic to the projective variety $\overline{\varphi(X)}$, hence it is Moishezon. But Moishezon p... | 7 | https://mathoverflow.net/users/121595 | 365291 | 153,468 |
https://mathoverflow.net/questions/365286 | 5 | Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$
$$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator!
Then, let $l$ be the left-shift and $r$ the right-shift on $\ell^2(\mathbb Z).$
We consider the compact operator on $\ell^2(\mathbb Z;\mathbb C^2)$ defined... | https://mathoverflow.net/users/108483 | Compact operator without eigenvalues? | Note that $$T^2 = \begin{pmatrix}lMrM&0\\0&rMlM\end{pmatrix},$$ and hence the eigenvectors of $T^2$ are $$v\_j = (e\_j, 0) , \qquad w\_j = (0, e\_j),$$ with corresponding eigenvalues $$\lambda\_j = \frac{1}{(1 + |j|) (1 + |j+1|)} \, , \qquad \mu\_j = \frac{1}{(1 + |j|) (1 + |j-1|)} \, ,$$ respectively. In particular, t... | 9 | https://mathoverflow.net/users/108637 | 365293 | 153,469 |
https://mathoverflow.net/questions/365303 | 3 | Let $G$ be a finite group. Let $p$ be a prime.
Let $O\_p(G)$ be the $p$-core of $G$.
>
> Are there any theorems known saying something like
>
>
> $O\_p(G)$ is trivial, if and only if ... and
>
>
> $O\_p(G)$ is non-trivial, if and only if ..., respectively ?
>
>
>
I am especially interested in the case $p... | https://mathoverflow.net/users/46117 | Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial | There are many such theorems. By the way, I would say that your definition of $p$-core may be non-standard if you are using it to denote the largest normal $p$-subgroup of $G$. I think many people would use $p$-core of $G$ to be $O\_{p^{\prime}}(G)$, the largest normal subgroup of $G$ of order co prime to $p$. Certainl... | 5 | https://mathoverflow.net/users/14450 | 365306 | 153,474 |
https://mathoverflow.net/questions/363214 | 2 | Let $(X\_t,Y\_t)$ be a pair of stochastic processes such that
$$
\begin{aligned}
dX\_t =& A\_t X\_t dt + C\_t dW\_t,\\
dY\_t = & H\_t X\_t dt + K\_tdB\_t
\end{aligned}
$$
for some non-random matrix-valued functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear ... | https://mathoverflow.net/users/36886 | Kalman filter distribution of observation process | If we assume $A$ constant $$\frac{d}{dt}\mathbb{E}(X\_t )=A \mathbb{E}(X\_t ) $$so $\mathbb{E}(X\_t)=e^{tA}X\_0$ and $\mathbb{E}Y\_t = Y\_0 + \int\_0^t
H\_s e^{sA}X\_0ds$.
For the variance, we can assume $X\_0=0$ and $Y\_0=0$. And we have $$\frac{d}{dt}\mathbb{E}(X\_tX\_t^T )=A\mathbb{E}(X\_tX\_t^T )+\mathbb{E}(X\_tX... | 2 | https://mathoverflow.net/users/99045 | 365316 | 153,476 |
https://mathoverflow.net/questions/365310 | 4 | *DISCLAIMER: I [posted](https://math.stackexchange.com/q/3743628) the same question a week ago on Mathematics Stack Exchange.*
We know by an abstract argument that there exist Banach spaces $E$, $F$, $G$ and maps $E \to F \hookrightarrow G$ such that $E \to F$ is non-nuclear, $F \hookrightarrow G$ is an isometry (met... | https://mathoverflow.net/users/120251 | Concrete example of non-nuclear operator $E \to F$ and isometry $F \hookrightarrow G$ so that the composition $E \to F \hookrightarrow G$ is nuclear | Take any sequence $a\_n$ of scalars that is square summable but not summable. That is the "hard" (in the technical sense) part of the argument. The rest is "soft". Let $T$ be the diagonal operator on $\ell\_2$ with diagonal $a\_n$. So $T$ is $2$-summing (Hilbert-Schmidt) but not nuclear (trace class). Let $S: \ell\_2 \... | 6 | https://mathoverflow.net/users/2554 | 365325 | 153,478 |
https://mathoverflow.net/questions/365278 | 3 | For any set $A\neq\varnothing$ let $\text{End}(A)$ denote the endomorphism monoid, consisting of all functions $f:A\to A$, together with composition. If $A, B\neq \varnothing$ are sets and $g:B\to A$ is a surjection, is there a surjective monoid homomorphism $\varphi:\text{End}(B)\to \text{End}(A)$?
| https://mathoverflow.net/users/8628 | Surjective monoid homomorphism $\text{End}(B)\to \text{End}(A)$ given surjection $g:B\to A$ | This is a cleaner rewrite of my original answer. The answer is no (assuming the surjection is not injective and the smaller set does not have cardinality $1$).
Let $T\_A$ be the full transformation monoid on the set $A$. Then the set $C\_A$ of constant maps is the unique minimal two-sided ideal of $T\_A$. Since $C\_A... | 7 | https://mathoverflow.net/users/15934 | 365329 | 153,479 |
https://mathoverflow.net/questions/365125 | 3 | Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\subseteq \mathbb P^1$, with $p\in\Delta\subseteq U$, such that $H^1(f^{-1}(U),{{\mathcal O}^{\rm an}}^\*)\to
H^1(f^{-1}(... | https://mathoverflow.net/users/88058 | Analytic vs Zariski neighbourhood of a fibre | I don't think the modified question works either. Let $E$ be a general elliptic curve. Take $X$ to be the quotient of $E\times \mathbb{P}^{1}$ by an involution which is a translation by a point of order 2 on $E$ and is $z \to 1/z$ on $\mathbb{P}^{1}$. Take the map $f : X \to \mathbb{P}^{1}$ that corresponds to the proj... | 1 | https://mathoverflow.net/users/439 | 365333 | 153,482 |
https://mathoverflow.net/questions/365324 | 3 | I’m mostly interested in this is the case where $A, \hat{A}$ are r.e. but the general case seems worth asking too.
Suppose I have sets $A >\_T \hat{A}$ with $A' \equiv\_T \hat{A}'$. Does this imply that if $C$ is r.e. in $A$ then there is a $\hat{C}$ r.e. in $\hat{A}$ such that $A \oplus C \equiv\_T \hat{A} \oplus \h... | https://mathoverflow.net/users/23648 | Does degree of jump determine degrees of relatively r.e. sets? | The answer is no. Every properly n-REA set for n < 3 (I believe Peter Cholak and I have shown this fails at 3 but could always fall apart in write-up) can be extended to a properly n+1 REA set by adding a relative r.e. set. Now apply this result to a low r.e. set. You can find that result in a paper by Peter Cholak and... | 2 | https://mathoverflow.net/users/23648 | 365334 | 153,483 |
https://mathoverflow.net/questions/365317 | 8 | We know that if $A$ is a separable $C^{\*}$-algebra then $K\_1(A)$ is countable.
Can anybody give an example of a C\*-algebra for which $K\_1(A)$ is uncountable?
| https://mathoverflow.net/users/137242 | Example of a C*-algebra whose $K_1$ is uncountable | There must be tons of ways to do this, but a simple one is to start with an uncountable set $X$, equipped with the discrete topology, and consider $c\_0(X)$. There are uncountably many pairwise inequivalent minimal projections in this algebra, so its $K\_0$ group is uncountable. Now use $K\_0(c\_0(X)) \cong K\_1(Sc\_0(... | 9 | https://mathoverflow.net/users/23141 | 365341 | 153,485 |
https://mathoverflow.net/questions/365339 | 3 | Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k > 0$?
Of course, this is a special case of $\Delta (f \wedge g)$. I would expect the formula to involve $\Delta f, \De... | https://mathoverflow.net/users/104461 | Product formula for Laplace de-Rham operator | Yes, and you can find it and its proof as a special case of Proposition 2.5 in the book of Berline-Getzler-Vergne: let $E$ be a hermitian vector bundle with unitary connection $\nabla,$ then the induced Laplacian $\Delta=\Delta^\nabla$ satisfies
$$\Delta (f s)=(\Delta f)s+f\Delta^\nabla s-2\langle\operatorname{grad} ... | 4 | https://mathoverflow.net/users/4572 | 365343 | 153,486 |
https://mathoverflow.net/questions/365344 | 7 | I want to understand if there is an intuition approchable with
most possible 'elementary geometrical' knowledge for
$n$-(co)skeleta of simplicial sets?
Formally sketleton & coskeleton functions arise as follows: For $\Delta$ the simplex category write $\Delta\_{\leq n}$ for its full subcategory on the objects
$[0],[1... | https://mathoverflow.net/users/108274 | Visualize (co)sketeton of a simplicial set (geometrical intuition) | For $k \le n$, the $k$-simplices in $\mathrm{cosk}\_n(X)$ are the same as in $X$. For larger $k$, there is a unique $k$-simplex for every $n$-skeleton of a $k$-simplex you find in $X$, that is, $(\mathrm{cosk}\_n(X))\_k \cong \mathrm{Hom}(\mathrm{sk}\_n \Delta^k, X)$.
You can also think inductively: again, for $k \le... | 8 | https://mathoverflow.net/users/644 | 365347 | 153,487 |
https://mathoverflow.net/questions/365322 | 1 | I don't know Swedish and I'm not finding the article *"G. Blom and C. E. Froberg, On money changing"* translated into English... so I tried to read the original (Swedish) with the help of Google Translate but, from the results, I fear that some semantics could get lost...
The article prove a denumerant's upper and l... | https://mathoverflow.net/users/160935 | (Translation request) Hypotheses of the Blom-Fredberg bounds on denumerants? | The article is [Blom, G. and Fröberg, C-E., *Om Myntväxling*, Nordisk Matematisk Tidskrift, 1962, Vol. 10, No. 1/2 (1962), pp. 55-69] for anyone who wishes to sing along.
After reading through the article, **no assumption is made on the coprimality of the $a\_i$**, in the sense that no added assumptions appear to be ... | 1 | https://mathoverflow.net/users/120914 | 365350 | 153,490 |
https://mathoverflow.net/questions/365342 | 4 | The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.
The [measurable Riemann mapping theorem](https://en.wikipedia.org/wiki/Measurable_Riemann_mapping_theorem) asserts the existence and uniqueness of a quasiconformal homeomorphism $f$ sa... | https://mathoverflow.net/users/88498 | The (measurable) Riemann mapping theorem | You misstated Riemann's (original) theorem:
a crucial assumption is that your open subset
is simply connected.
Both theorems can be considered as classification theorems
of Riemann surfaces. The Riemann original theorem says that every simply connected domain in the sphere, whose complement contains at least 2 points... | 18 | https://mathoverflow.net/users/25510 | 365353 | 153,491 |
https://mathoverflow.net/questions/365352 | 5 | Let $\chi\_\lambda(\mu)$ be the usual characters of the irreducible representations of the permutation group $S\_n$. The normalized character is the quotient $\chi\_\lambda(\mu)/f^\lambda$, where $f^\lambda=\chi\_\lambda(1)$ is the dimension of the representation.
Can I hope for a nice formula expressing their sum
$$... | https://mathoverflow.net/users/78061 | Summing over normalized characters of the permutation group | The quantity you are asking about is in fact a well-known expression: When multiplied by $n!$, it is the number of ordered pairs $\sigma, \tau \in S\_{n}$ such that
$[\sigma, \tau] = \mu$, where $[\sigma, \tau] = \sigma^{-1}\tau^{-1}\sigma \tau$ is the commutator of $\sigma$ and $\tau$. However, I do not know how to re... | 7 | https://mathoverflow.net/users/14450 | 365354 | 153,492 |
https://mathoverflow.net/questions/365328 | 9 | Let $k=\mathbb F\_q(T)$. Can one prove (or disprove) that the series $\sum\_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no clue to attack this question. I thought about Dwork-Polya-Bertrandias theorem, but I did not find a several variables ve... | https://mathoverflow.net/users/33128 | A series that is rational? | If you set $T=0$ or $X=0$ then you get the series $\sum\_{n\geq 0} Y^{q^n}$. This cannot be rational because a rational power series in one variable that is not a polynomial cannot have arbitrarily long sequences of 0 coefficients (since the coefficients satisfy a linear recurrence relation with constant coefficients).... | 23 | https://mathoverflow.net/users/2807 | 365358 | 153,494 |
https://mathoverflow.net/questions/365323 | 18 | It is well known that:
$$\zeta(s):=\prod\_{n=1}^{\infty} \frac{1}{1-p\_n^{-s}} \qquad \Re(s) \gt 1$$
with $p\_n =$ the $n$-th prime. It also known that:
$$\zeta(2n):= \frac{(-1)^{n+1} B\_{2n}(2\pi)^{2n}}{2(2n)!}$$
where $B\_{2n}$ is the $2n$-th Bernoulli number.
Now define the function:
$$f(k,N,x):= \zeta(2... | https://mathoverflow.net/users/12489 | Could computing the next prime in a finite Euler product be made rigorous? | $2k=1+p\_N$ works for $N>1$, but $2k\le 0.56 \, p\_N$ will fail if
$p\_{N+2}=p\_{N+1}+2$.
With $q=p\_{N+1}$, we have
$$
\frac{1}{1-q^{-2k}} < \frac{1}{1-x^{-2k}} = \frac{1}{1-q^{-2k}} \prod\_{p>q} \frac{1}{1-p^{-2k}} .
$$
It follows that
$$
q^{-2k} < x^{-2k} < q^{-2k} + \sum\_{j\ge 2} (q+j)^{-2k} < q^{-2k} +\frac{1}{... | 15 | https://mathoverflow.net/users/12947 | 365362 | 153,495 |
https://mathoverflow.net/questions/365357 | 3 | Is there a simple construction of a confomral mapping of the half-plane onto a "circular trianagle", i.e. a domain whose sides are the arcs of three circles.
| https://mathoverflow.net/users/124426 | Conformal mapping | Yes, it is (how simple, is a matter of opinion). You may always assume that $0,1,\infty$ on the boundary of upper half-plane
are preimages of the vertices. Now suppose that the inner angles of your triangle
are $\pi\alpha\_j,$ and let us assume that $\sum\alpha\_j$ is not an odd integer (Euclidean triangles must be con... | 10 | https://mathoverflow.net/users/25510 | 365365 | 153,496 |
https://mathoverflow.net/questions/365346 | 6 | **Disclaimer** : I asked this question on Maths.StackExchange 20 days ago (and started a bounty) [here](https://math.stackexchange.com/questions/3723218/perfect-mathbb-z-ell-modules) but got no answer, so I'm asking it here now (with no modification).
$\newcommand{\l}{\ell} \newcommand{\Z}{\mathbb Z}$
I'm trying to u... | https://mathoverflow.net/users/102343 | Perfect $\mathbb Z_\ell$-modules | Over a PID, you can check perfectness on homology. So the claim is that the homology of an $\ell$-complete $\mathbb{Z}\_\ell$-module is finitely generated if and only if it is so mod $\ell$. The homology groups are $\ell$-complete, and so this follows from the long exact sequence of mod $\ell$-reduction.
| 4 | https://mathoverflow.net/users/39747 | 365377 | 153,500 |
https://mathoverflow.net/questions/365363 | 1 | I found a lemma in [this paper](https://www.jstor.org/stable/24899147) of Constantin and Wu, stated with no proof:
>
> **Lemma
> 3.2.** Let $b=\chi\_{D}$ be the characteristic function of a bounded domain $D\subset\mathbb R^2$ whose boundary has box-counting (fractal) dimension not larger than $d<2:$
> $
> d\_{F}(\... | https://mathoverflow.net/users/70388 | Box counting dimension and Besov spaces on $\mathbb R^2$ | The modulus of continuity in direction $v$ is $\omega\_p(t,v) := \lVert 1\_D - 1\_D(\cdot-tv)\rVert\_p$. Since $\lvert 1\_D(x) - 1\_D(x-tv)\rvert \le 1\_{N(\partial D,t)}$, where $N(\partial D,t)$ is a $t$-neighborhood of $\partial D$, then by the assumption on the dimension of the boundary
$$
\omega\_p(t,v) \le C t^\f... | 2 | https://mathoverflow.net/users/90189 | 365383 | 153,502 |
https://mathoverflow.net/questions/365375 | 1 | Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular.
Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both?
Thanks in advance.
| https://mathoverflow.net/users/125843 | Lower bound for $\vert \det A \vert $ for the adjacency matrix of regular graphs | $k$ is a lower bound, since $k$ is an eigenvalue and the product of other eigenvalues is integer (they are roots of the monic polynomial with integer coefficients). I doubt that it may be improved without further assumptions.
| 4 | https://mathoverflow.net/users/4312 | 365389 | 153,505 |
https://mathoverflow.net/questions/268723 | 9 | In [this](http://www.jstor.org/stable/2269620?origin=JSTOR-pdf&seq=1#page_scan_tab_contents) article, Takeuti has introduced a theory of ordinal numbers, which in his own words, is intended to be a first order theory:
>
> The theory of ordinal numbers we are to develop is based on the first
> order predicate calcul... | https://mathoverflow.net/users/76416 | First order axioms for primitive recursion in Takeuti's theory of ordinal numbers | I’m not familiar with this paper, but what is wrong with just writing out the first-order definitions of the inner functions? That is, $f(a)=g(\mu x\_{x<l(a)}f^x(f^a(h(a))))$ becomes
$$\exists x\,(f(a)=g(x)\land\phi(a,x)),$$
where
$$\begin{align\*}
\phi(a,x)&\iff(x<l(a)\land\psi(a,x,0)\land\forall y\,(y<x\to\neg\psi(a,... | 1 | https://mathoverflow.net/users/12705 | 365402 | 153,508 |
https://mathoverflow.net/questions/365248 | 3 | Question is the following:
>
> Is the functor $H^n\_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
>
>
>
More generally, are cohomology functors sheaves in general (in any reasonably non trivial Grothendieck topology)?
I am also interested in cohomology fu... | https://mathoverflow.net/users/118688 | Are cohomology functors sheaves? |
>
> Is the functor H^n\_dR:Man→Set a sheaf with respect to open cover topology on Man?
>
>
>
As already pointed out in the comments, the answer is no for n>0, yes for n=0.
>
> "in what way is cohomology a sheaf" leads one to notions like ∞-topoi etc.
>
>
>
In the context of this question,
the assignment... | 6 | https://mathoverflow.net/users/402 | 365444 | 153,520 |
https://mathoverflow.net/questions/365448 | 3 | Suppose that $v=(v\_1,\ldots, v\_d)\in \mathbb{R}^d$ lies in the linear subspace $v\_1+\cdots +v\_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\mathcal{S}\_d;v)=Conv(\mathcal{S}\_d\cdot v) \end{equation} is the convex hull of $v$ under the symmetric group action on... | https://mathoverflow.net/users/53199 | Are cyclic orbitopes of permutahedra necessarily simplicies? | Let $M$ be the circulant matrix whose rows are given by cyclic shifts of $(v\_1,\dots v\_d)$ and let $P(x)=v\_1+v\_2x+\cdots+v\_dx^{d-1}$ be the associated polynomial. Moreover, let $s$ be the degree of $\gcd(P(x),x^{d}-1)$.Then the rank of $M$ is equal to $d-s$, so it is possible to come up with examples of vectors $v... | 5 | https://mathoverflow.net/users/2384 | 365450 | 153,523 |
https://mathoverflow.net/questions/365455 | -2 | Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e.
$$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$
Is there a way to bound the product $f(x,y)\cdot f(x,y')^T \in\mathbb{R}^{d\times d}$?
| https://mathoverflow.net/users/156139 | Bounding the product of lipschitz function | Say $f(x,y)=y$. Then $f(x,y)^2=y^2$, which is not Lipschitz in $y$. So the answer is negative.
| 2 | https://mathoverflow.net/users/12518 | 365456 | 153,524 |
https://mathoverflow.net/questions/365361 | 20 | So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory and symplectic geometry.
They have been considered by mathematical physicists for a time, but without a rigorous defi... | https://mathoverflow.net/users/160378 | A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories | As for prerequisite for three papers, I recommend
Chriss-Ginzburg, [Representation Theory and Complex Geometry](https://www.springer.com/gp/book/9780817637927), and Victor Ginzburg, [Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups](https://arxiv.org/abs/math/9802004).
One also needs ... | 20 | https://mathoverflow.net/users/3837 | 365458 | 153,525 |
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