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https://mathoverflow.net/questions/350039 | 1 | Let $K$ be an infinite field positive characteristic and $F(X,Y)\in K[[X,Y]]$. Assume that $F(Z\_1+U\_1,Z\_2+U\_2)=F(Z\_1,Z\_2)+F(U\_1,U\_2)$ where $Z\_1,Z\_2,U\_1,U\_2$ are four indeterminates. Can one assert that $F(X,Y)=G(X)+H(Y)$ with $G,H\in K[[X]]$?
| https://mathoverflow.net/users/33128 | Linear formal series in positive characteristic | Yes. It suffices to consider the case that $F$ is a homogeneous polynomial. Write $F(X,Y) = aX^d + Y^eP(X,Y)$, where $d = \deg(F)$ and $P$ is homogeneous of degree $d - e$, $e \geq 1$.
Claim: if $a \neq 0$, then $d$ is a power of the characteristic $p$ of $K$.
Proof: Otherwise the coefficient of $Z\_1^{d-1}U\_1$ ... | 3 | https://mathoverflow.net/users/1508 | 350046 | 148,137 |
https://mathoverflow.net/questions/350048 | 3 | Let $\mu$ be the Mobius function from $\mathbb{N}$ to $\{-1, 0, 1\}$. It is well known for the frequency of $-1, 1$, and $0$ for the sequence $(\mu(1), \mu(2), \mu(2), \dots, )$.
For any $k\in \mathbb{N}$, it is natural to ask what is the frequency of any given block of $k$-digits in $\{-1, 0, 1\}^{k}$ . I do not kn... | https://mathoverflow.net/users/11966 | frequence of block of digits in Mobius sequence | Terry Tao has a blog post on this [here](https://terrytao.wordpress.com/2015/09/06/sign-patterns-of-the-mobius-and-liouville-functions/)
The Chowla conjecture asserts that all $k-$ blocks are equidistributed.
Matomaki, Radziwill and Tao (MRT) have shown that each of the sign patterns in
$\{-1,0,+1\}^k$
is attaine... | 4 | https://mathoverflow.net/users/17773 | 350054 | 148,139 |
https://mathoverflow.net/questions/350009 | 8 | Let $L: \mathcal C^\to\_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, [$L$ is left exact iff the class of $L$-local morphisms is stable under base change](https://mathoverflow.net/questions/128446/general-theory-of-left-exact-localization) [1].
This appears to be t... | https://mathoverflow.net/users/2362 | When is an $\infty$-categorical localization left exact? | Unless I misunderstand the statement, this is precisely proposition 6.2.1.1 in Higher Topos Theory.
| 4 | https://mathoverflow.net/users/43054 | 350058 | 148,140 |
https://mathoverflow.net/questions/350056 | 2 | *It is clear that one can obtain a discrete dynamical system from a continuous one, but is the converse possible if the system is "nice"?*
Define the discrete-time dynamical system on $\mathbb{R}^d$ by
$$
x\_{n+1} = f(x\_n);\, x\_0\triangleq x
$$
where
$f \in C^2(\mathbb{R}^d;\mathbb{R}^d)$ and $x \in \mathbb{R}^d$.... | https://mathoverflow.net/users/36886 | Continuous-time extension of a discrete dynamical system | I provide a discrete expansion of an iterated function, but a symmetry constraint is also needed to simplify it to a continuous solution. Works for Schroeder's and Abel's Functional Equations. Note that convergence issues make if difficult to provide a complete answer.
Let $f(x)$ and $g(x)$ be functions in Banach spa... | 2 | https://mathoverflow.net/users/nan | 350063 | 148,141 |
https://mathoverflow.net/questions/350057 | 2 | $B\_e=B\_{\text{cris}}^{\phi=1}$, so if a $p$-adic Galois representation $V$ is $B\_e$ admissible, then it is crystalline, so I want to know an example that $V$ is crystalline but not $B\_e$ admissible.
Thanks!
| https://mathoverflow.net/users/nan | An example that a $p$ adic Galois representation is crystalline but not $B_e$ admissible | We have $(V \otimes B\_e)^{G\_K} = D\_{\mathrm{cris}}(V)^{\varphi = 1}$. So any representation which is crystalline, but such that $\mathbf{D}\_{\mathrm{cris}}(V)$ has zero $\varphi$-invariants, is an example of a representation which is $B\_{\mathrm{cris}}$-admissible but not $B\_e$-admissible.
For instance, taking... | 4 | https://mathoverflow.net/users/2481 | 350068 | 148,142 |
https://mathoverflow.net/questions/349922 | 2 | Let $S$ be a linear subspace of $\Bbb R^n$ having dimension $k<n$ and assume $S$ is described by $n-k$ linear equations with integer coefficients. Look at now the intersection $\Lambda=S\cap \Bbb Z^n$ - such an intersection is a lattice in $S$. Theoretically speaking, I already know the existence of a basis $\{v\_1,\do... | https://mathoverflow.net/users/69185 | Basis for a lattice in a subspace of $\Bbb R^n$ | The answer to your question, with a full proof, appears in "An Introduction to the Geometry of Numbers" by J.W.S. Cassels, Corollary 3 in page 14. It is also stated as Theorem 1.28 [here](https://books.google.com/books?id=i5AkDxkrjPcC&pg=PA13&lpg=PA13&source=bl&ots=Pz00ZD6qFZ&sig=ACfU3U0u37_G6qfpiAIyVmLeJUAkSe220g&hl=e... | 3 | https://mathoverflow.net/users/109085 | 350069 | 148,143 |
https://mathoverflow.net/questions/350062 | 1 | I am integrating the following Gaussian over all possible matrix elements $J\_{ij}$:
$$ I=\int \exp{\left\{-a\sum\_{ij}J\_{ij}^2+b\sum\_{ij}J\_{ij}+c\sum\_{ij}J\_{ij}J\_{ji} \right\}} \left (\prod\_{ij}\mathrm{d}J\_{ij} \right)$$
How can I deal with the $\sum\_{ij}J\_{ij}J\_{ji}$ terms? The fact that I am integrating... | https://mathoverflow.net/users/142153 | How can we do a Gaussian integral over matrix elements? | Decompose the sum over $i,j$ as
$$-a\sum\_{ij}J\_{ij}^2+b\sum\_{ij}J\_{ij}+c\sum\_{ij}J\_{ij}J\_{ji}=$$
$$\qquad\qquad=\sum\_{i}\left[(c-a)J\_{ii}^2+bJ\_{ii}\right]+\sum\_{i<j}\left[-a(J\_{ij}^2+J\_{ji}^2)+b(J\_{ij}+J\_{ji})+2cJ\_{ij}J\_{ji}\right]$$
$$\qquad\qquad\equiv\sum\_{i} A\_i+\sum\_{i<j} B\_{ij}.$$
Then perfor... | 3 | https://mathoverflow.net/users/11260 | 350071 | 148,144 |
https://mathoverflow.net/questions/350040 | 3 | I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be [Fröhlich's](https://link.springer.com/article/10.1007/BF01609843) article, which is my main reference at the moment. We co... | https://mathoverflow.net/users/150264 | Mathematical meaning for the (continuous) Sine-Gordon transformation | I'm not sure what exactly the "Sine-Gordon transformation" is (Frölich's article doesn't use that terminology), but I guess your question is specifically about the meaning of the symbol ${:} e^{i \epsilon T(x)} {:}\_V$, when not inside the expectation value $\langle - \rangle\_V$, and what algebraic manipulations are a... | 2 | https://mathoverflow.net/users/2622 | 350072 | 148,145 |
https://mathoverflow.net/questions/350081 | 1 | Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d\_T$ induced by the euclidean metric $d$ on $\mathbb{R}^n$: $$d\_T(x+\Lambda,y+\Lambda):=\min\_{a,b\in\Lambda}(d(x+a,y+b)).$$ Now my... | https://mathoverflow.net/users/36563 | What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$? | Yes, we can. Consider the universal cover $U$ of your torus $T$. One can easily show that $U=R^n$ equipped with a Euclidean metric.
So we have $f:R^n\to T$,
The $f$-preimage of a point is your lattice, up to the shift of the origin (to one point of this preimage) and an orthogonal transformation (an isometry of the E... | 7 | https://mathoverflow.net/users/25510 | 350083 | 148,148 |
https://mathoverflow.net/questions/350089 | 2 | A polynomial formula for the primes (with 26 variables) was presented by Jones, J., Sato, D., Wada, H. and Wiens, D. (1976). Diophantine representation of the set of prime numbers. *American Mathematical Monthly*, *83*, 449-464.
The set of prime numbers is identical with the set of positive values taken on by the pol... | https://mathoverflow.net/users/74668 | Diophantine representation of the set of prime numbers of the form $n²+1$ | Call your polynomial $P$. I propose the following polynomial:
$$
P' = (\xi^2+1)(1 - (\xi^2+1-P)^2)
$$
**Proof** (that the positive values of $P'$ are exactly the primes of the form $N^2+1$):
Let $P\_0$ be one of the values of $P$, and let $\xi\_0$ be any integer.
Case (i). Suppose $P\_0 = \xi\_0^2+1$. Then the va... | 13 | https://mathoverflow.net/users/17907 | 350095 | 148,154 |
https://mathoverflow.net/questions/347256 | 7 | The Weinstein Lagrangian neighborhood theorem says that if $(M,\omega)$ is a symplectic manifold and $L\subset M$ is a Lagrangian submanifold, then there are neighbourhoods $U$ of $L$ in $M$, and $U'$ of the zero-section in $T^\*L$, and a symplectomorphism $U\to U'$ which restricts to the identity on $L$.
>
> **Que... | https://mathoverflow.net/users/90299 | Holomorphic Weinstein Lagrangian neighborhood theorem | A classic example that shows that WLNT doesn't always hold in the holomorphic category is an elliptically fibered $K3$ surface.
A K3 surface $S$ is a compact complex symplectic manifold of complex dimension $2$. Any smooth curve $C\subset S$ is a Lagrangian submanifold. If the Darboux Theorem were true in the sense ... | 9 | https://mathoverflow.net/users/13972 | 350111 | 148,159 |
https://mathoverflow.net/questions/350116 | 2 | In probability theory, the term *subexponential distribution* has historically been used for a distribution whose CDF $F(x)$ satisfies the relation
$$
n(1-F(x)) \sim 1 - F^{\*n}(x)
$$ for any $n \ge 1$ as $x \to \pm \infty$ (depending on the context), where $F^{\*n}(x)$ is the CDF of the $n$-fold additive convolution. ... | https://mathoverflow.net/users/23959 | History of the name "subexponential distribution" in probability | In the known survey [Subexponential distributions](https://mediatum.ub.tum.de/doc/1120625/1120625.pdf) by Goldie and Klüppelberg, we find this:
>
> "The name arises from one of their properties, that their tails decrease more slowly than any exponential tail".
>
>
>
So, the logic behind this term seems to be... | 3 | https://mathoverflow.net/users/36721 | 350119 | 148,164 |
https://mathoverflow.net/questions/350105 | 6 | Is it known whether any **projective geometry statement** that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the **standard projective axiomatics**?
By **projective geometry statement** I mean any statement it terms of incidence of points and lines, that is a ... | https://mathoverflow.net/users/148443 | Does any real projective plane incidence theorem follow from axioms? | This is false. I don't see how to get a counterexample from Andreas Blass's comment (the only uses for the existence of $\sqrt2$ which are obvious to me require a more flexible notion of incidence statement), so I am posting this as an answer although it is probably more complicated than necessary.
For fixed prime $... | 3 | https://mathoverflow.net/users/75344 | 350121 | 148,165 |
https://mathoverflow.net/questions/349987 | 2 | I am trying to understand the following definition in C. Sabbah's paper
(Quelques remarques sur la géométrie des espaces conormaux), page 186, [Numdam link](http://www.numdam.org/item/AST_1985__130__161_0/).
Let $\phi\colon X\to \mathbb{C}^2$ be a map such that
1. X is irreducible
2. $\phi^{-1}(t)$ is of dimension... | https://mathoverflow.net/users/98788 | Strict transform by normalization | I think I got what it means by strict transform in the paper. In the Stacks Project 31.33, the strict transform is defined as follows
Let $f\colon X\to B$ be a morphism, let $p\colon \hat{B}\to B$ be the blowup of $B$ along a subscheme $Z$, then the strict transform $\hat{X}$ of $X$ by $p$ is just the blowup of $X$ a... | 0 | https://mathoverflow.net/users/98788 | 350127 | 148,169 |
https://mathoverflow.net/questions/350131 | 1 | Let $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\mu)$ be a $\sigma$-finite Borel measure on $d$-dimensional Euclidean space. Can one always construct a sequence of finite *equivalent* measures $\left\{\mu\_n\right\}\_{n \in \mathbb{N}}$ on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ such that
$$
\lim\_{n \rightarrow \infty... | https://mathoverflow.net/users/36886 | Approximation of $\sigma$-finite Borel measures by equivalent finite measures | Since $\mu$ is $\sigma$-finite there is a $\mu$-integrable function $f$ with $0<f\le 1$. For an increasing sequence $A\_n$ with $\bigcup\_{n\in\mathbb N} A\_n=\mathbb R^d$ define densities $f\_n=I\_{A\_n}+fI\_{A\_n^c}$. Then $\mu\_n=f\_n\cdot \mu$ are equivalent finite measures whose densities converge to $1$.
| 3 | https://mathoverflow.net/users/21051 | 350136 | 148,172 |
https://mathoverflow.net/questions/350140 | 0 | Let $n \ge 2$ be an integer, which may be assumed to be **very large**. For $r > 0$, consider the hemi-sphere $H\_n(r) := S\_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where
$$
S\_n(r):= \{x \in \mathbb R^n \mid \|x\|\_2 \le r\}
$$
is the $n$-sphere of radius $r$.
and consider the measure $\lambda\_n$ define on b... | https://mathoverflow.net/users/78539 | Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere | Let $\nu:=\mathcal N(0,\sigma^2 I\_n)$ and $p\in[0,1]$.
Concerning Question 1: For the minimizing $E$, without loss of generality (wlog) we clearly have $E\subseteq H\_n(r)$ and $\lambda\_n(E;r)=p$. It is then clear that, subject to these conditions, wlog the set $E$ must consist of the points in $H\_n(r)$ with the ... | 2 | https://mathoverflow.net/users/36721 | 350151 | 148,175 |
https://mathoverflow.net/questions/350142 | 1 | The 3-dimensional Vlassov -Poisson equation I am studying at university is
$$ \partial\_t f (t,x,v) + v\cdot \nabla\_x f (t,x,v) - \nabla\_x \phi (t,x) \nabla\_v f (t,x,v) =0,$$
where $$\Delta \phi = 4\pi\gamma \rho (t,x,v) \text{ and } \rho (t,x) = \int\_{\mathbb{R}^3} f(t,x,v)\ dv$$
I am trying to prove the momentu... | https://mathoverflow.net/users/137336 | Vlasov Poisson: linear momentum conservation | It helps to consider components of $q$, in order to keep the notations clear. From what you have we can write
$$ q\_i'(t) = \int v\_i \sum\_{j} \nabla\_{x\_j} \phi \nabla\_{v\_j} f ~\mathrm{d}x ~\mathrm{d}v $$
integrate by parts in $v$, and using that $\phi$ is independent of $v$ and $\nabla\_{v\_j} v\_i = \delta\_... | 4 | https://mathoverflow.net/users/3948 | 350156 | 148,177 |
https://mathoverflow.net/questions/350133 | 3 | **Motivation.** My elder son played the following game. He had a bunch of coins, all with heads up, arranged in a circle. He flipped one coin, so that it showed tails, then he moved $1$ position clockwise, flipped that coin, then moved $2$ positions clockwise, flipped that coin, then moved $3$ positions clockwise etc. ... | https://mathoverflow.net/users/8628 | Coin flipping game | Only a partial answer: Assume that $n$ is not a power of $2$. Then there is a prime $p | n$ with $p > 2$. First we consider the case $n = p$. Then for $p = 3$ $\{(k (k+1)/2 \mod 3 \colon k = 0,1,2\} = \{0,1\}$ and $f\_p(k) := k(k+1)/2 \mod p$ is not surjective. For $p > 3$ we have $f\_p(k) \equiv f(p-k-1) \mod p$, in p... | 6 | https://mathoverflow.net/users/100904 | 350159 | 148,180 |
https://mathoverflow.net/questions/350138 | 35 | In the Soviet times there was a famous Encyclopedia of Mathematics. I think it is still familiar to every Russian mathematician maybe except very young ones, and yours truly is in possession of all 5 volumes. Browsing it recently (with no real purpose) I came across a certain peculiarity. In the article "Kervaire invar... | https://mathoverflow.net/users/9833 | Incorrect information in an old article about the Kervaire invariant | I found the following remark in Zhouli Xu's paper ["The strong Kervaire invariant problem in dimension 62"](https://arxiv.org/abs/1410.6199):
>
> In [19], R. J. Milgram claims to show that under the same condition as in Theorem 1.1, one has $θ\_{n+2}$ exists. If this were true, then we would have that $\theta\_6$ e... | 27 | https://mathoverflow.net/users/2384 | 350162 | 148,182 |
https://mathoverflow.net/questions/350114 | 5 | Given some group $G$ that is generated by $a$ and $b$, each of which has infinite order, and some free subgroup $N$ generated by $a^k$ and $b^k$, is there any algorithm that tells me if some $x \in G$ is also in $N$? If not generally, what about for $k=2,3,4$?
| https://mathoverflow.net/users/150179 | The generalized word problem on groups | The answer is "no". Take the free group of rank 2 $F=<a,b>$. Take an infinite recursively enumerable non-recursive set $X=\{u\_1,u\_2,...\}$, a subset of a recursive set $X'$ of words in $a,b$ satisfying the small cancelation condition $C'(1/12)$ and containing neither $a^2$ nor $b^2$. Take another infinite set of word... | 10 | https://mathoverflow.net/users/nan | 350167 | 148,184 |
https://mathoverflow.net/questions/350144 | 0 | Let $f\in C^0([-1,1])$ and $P\_n(f)$ its interpolation polynomial at the Chebyshev nodes.
I would be interested to know about any existing results (positive or negative) about the convergence of $P\_n(f)$ to $f$ in $L^1([-1,1])$ (only assuming that $f$ is continuous).
The negative results I'm aware of (existence o... | https://mathoverflow.net/users/150933 | Convergence of Chebyshev interpolation in L^1 | Here are the details (for the second definition of Chebyshev nodes). We can define the new continuous function $g$ on the circle by $g(z)=f(\frac{z+z^{-1}}2)$ and consider the interpolation by the trigonometric polynomials $Q\_n(z)=\sum\_{k=0}^n b\_k\frac{z^k+z^{-k}}{2}$ (so that $P\_n(\frac{z+z^{-1}}{2})=Q\_n(z)$) on ... | 1 | https://mathoverflow.net/users/1131 | 350171 | 148,186 |
https://mathoverflow.net/questions/350172 | 3 | Let $\mathcal{K}$ be a $2$-category. Is there a special name of those objects $B \in \mathcal{K}$ which have the property that the category $\mathrm{Hom}\_{\mathcal{K}}(B,C)$ is essentially discrete for all $C \in \mathcal{K}$? This means that for every two morphisms $f,g : B \to C$ any $2$-morphism $f \to g$ is an iso... | https://mathoverflow.net/users/2841 | "discrete" objects of a $2$-category | I would call those objects "codiscrete" or "co-0-truncated", since "discrete" and "0-truncated" are used for the dual property, e.g. [here](https://ncatlab.org/nlab/show/discrete+morphism) and [here](https://ncatlab.org/michaelshulman/show/discrete+object). It's equivalent to saying that $B$ is equivalent to the copowe... | 2 | https://mathoverflow.net/users/49 | 350176 | 148,187 |
https://mathoverflow.net/questions/350177 | 6 | Let $R$ be a not necessarily commutative ring, and denote by $\_R\mathrm{lp}\_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, denoted by $\_R\mathrm{Hom}\_R(M,N)$, for any two objects $M,N$.
i) Is $\_R\mathrm{lp}\_R$ a monoidal category? In o... | https://mathoverflow.net/users/125790 | Monoidal categories from the projective modules of a ring | Yes to (i).
It's easier to think about if you weaken/generalize.
Suppose that $M$ is an $(R,S)$-bimodule and $N$ is a left $S$-module. If $N$ is projective then the $R$-module $M\otimes\_SN$ is a summand of a direct sum of copies of $M\otimes\_SS\cong M$, so that it is a projective module if $M$ is.
No to (ii).... | 10 | https://mathoverflow.net/users/6666 | 350186 | 148,190 |
https://mathoverflow.net/questions/350175 | 11 | If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$.
W H Lin, in his 1973 PhD thesis at Northwestern, showed that the only Hopf subalgebras of the mod 2 Steenrod algebra for which this can be done are the algebras $A(n)$ — this is t... | https://mathoverflow.net/users/4194 | W H Lin's thesis and Hopf subalgebras of the Steenrod algebra | Using @CarloBeenakker's answer, our librarian found an electronic version, produced from the microfilm copy of the original: <https://search.proquest.com/docview/302701183> (full text may require accessing through a university library or similar). At a quick glimpse, it looks complete.
| 11 | https://mathoverflow.net/users/4194 | 350191 | 148,193 |
https://mathoverflow.net/questions/338564 | 9 | In his lecture, [*The Categorical Origins of Lebesgue Measure*](https://www.maths.ed.ac.uk/~tl/cambridge_ct14/cambridge_ct14_talk.pdf), Professor Tom Leinster mentions the following theorem:
>
> **Theorem 1:** (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with two distinct basepoints $0$ and $1$, ... | https://mathoverflow.net/users/30211 | On the universal property for interval objects | The obvious thing is to consider the category of unital $C^\*$-algebras $A$ equipped with two distinct characters $\chi\_0,\chi\_1 : A \to \mathbb{C}$ (let me stick to $\mathbb{C}$ since strange things might happen over $\mathbb{R}$, but I am not sure). There is a non-unital monoidal structure $\otimes$ given by pullba... | 2 | https://mathoverflow.net/users/2841 | 350194 | 148,195 |
https://mathoverflow.net/questions/349968 | 2 | This might be a trivial question, but I am trying to prove equi-coerciveness of some family of functions on the space of Probability measures on some space. I could reduce the problem to showing that $$\{\nu:\mathcal{W}\_2^2(\mu, \nu)\le t\}$$ is compact (or is at least contained in some compact subset of $\mathcal{P}(... | https://mathoverflow.net/users/69849 | Is unit ball in 2-Wassestein metric weakly compact? | Yes, it is true. It follows from Prokhorov's theorem that in order to prove (pre-)compactness, it suffices to prove tightness. However, if we define $K$ to be the compact set such that $\mu(\mathbb{R}\setminus K)<\varepsilon$, and $K\_T:=\{x\in \mathbb{R}:\mathrm{dist}(K,x)\leq T\}$, then $\nu(\mathbb{R}\setminus K\_T)... | 3 | https://mathoverflow.net/users/56624 | 350200 | 148,196 |
https://mathoverflow.net/questions/350197 | 7 | I have a problem where I have $n$ commuting matrices $M\_1,\dots,M\_n$. It is a well-known fact that commuting matrices are simultaneously diagonalizable/triangularizable. I need to find the eigenvalues of these matrices, but I need to know the eigenvalues grouped up by the common eigenspaces.
In exact arithmetic, th... | https://mathoverflow.net/users/143176 | Numerical method for simultaneous computation of eigenvalues of a family of commuting matrices | For small Hermitian (or real symmetric) matrices, yes, but really this is a hard problem not fully solved. See [1,2] for algorithms. The Cardoso paper [2] looks at the non-commuting case, but in the commuting case should minimize the off diagonal errors with respect to the Frobeneius norm.
I don't know about about ge... | 5 | https://mathoverflow.net/users/6133 | 350205 | 148,197 |
https://mathoverflow.net/questions/350202 | 7 | Suppose that $G$ is a reductive algebraic group acting on a smooth variety $X$, and that the action has finite stabilizers. When is the action of $G$ on $X$ proper? What is an example where the action is not proper?
I am aware of a similar statement which is Proposition 0.8 in Mumford's GIT, which says that the actio... | https://mathoverflow.net/users/113933 | Properness of reductive group actions on smooth varieties | Actions of reductive groups with finite stabilizers on quasi-projective varieties are often not proper. The simplest example I know is given by he action of $\mathrm{PGL}\_2$ on the projective space $\mathbb P^4$ of effective divisors of degree $4$ on $\mathbb P^1$. Consider the open subset $X \subseteq \mathbb P^4$ of... | 7 | https://mathoverflow.net/users/4790 | 350209 | 148,198 |
https://mathoverflow.net/questions/350132 | 4 | Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be sufficiently large integers.
Is it true that, for any irreducible hypersurface $X$ of bi-degree $(d,1)$ in $\mathbb{P}^r\times\mathbb{P}^r$, the Picard group $\mathrm{Pic}(X)$ or the divisor class group $\mathrm{Cl}(X)$ equals to $\mathbb{Z}\o... | https://mathoverflow.net/users/nan | Picard group of hypersurfaces in $\mathbb{P}^r\times\mathbb{P}^s$ | Ok, this follows from Lazarsfeld positivity I, Example 3.1.25.
| 0 | https://mathoverflow.net/users/nan | 350217 | 148,200 |
https://mathoverflow.net/questions/350223 | 3 | Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Does there exist a countable set of finite measures $\{\mu\_n\}\_{n \in \mathbb{N}}$ on $(X,\Sigma)$ such that $L^1\_{\mu}(\Sigma)$ can be written as the projective-limit in the category of LCS
$$
L^1\_{\mu}(\Sigma) = \projlim\, L^1\_{\mu\_n}(\Sigma),
$$
for some... | https://mathoverflow.net/users/36886 | $L^1_{\mu}$ as limit | There is a strictly positive integrable function $f\in L^1\_\mu(\Sigma)$, hence $\nu=f\cdot \mu$ is a finite measure and $\Phi:L^1\_\nu(\Sigma)\to L^1\_\mu(\Sigma)$, $g\mapsto gf$ is an isomorphism. In particular, $L^1\_\mu(\Sigma)$ is *isomorphic* to a projective limit of $L^1\_{\mu\_n}(\Sigma)$ with finite measures (... | 7 | https://mathoverflow.net/users/21051 | 350225 | 148,203 |
https://mathoverflow.net/questions/350154 | 1 | I did a JavaScript interactive picture of the Malfatti circles of a triangle. The user can drag the vertices of the triangle and the Malfatti circles are updated accordingly.
Now, I would like to restrict the transformations of the original triangle in order that they preserve the outer Soddy circle of the Malfatti c... | https://mathoverflow.net/users/21339 | Triangles with a given outer Soddy circle of the Malfatti circles | At the request of the poster, I am expanding my comment to an answer. We are given a circle $\cal C$ which we take to be the unit one with centre at the origin of the coordinate plane. Now if $PQR$ is a triangle in the plane, there is a unique triangle $ABC$ with $A=(0,0)$, $B=(1,0)$ and $C=(p,q)$ to which it is direct... | 1 | https://mathoverflow.net/users/131781 | 350232 | 148,205 |
https://mathoverflow.net/questions/350098 | 13 | Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.
Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^\*\_{\rm sing}(X\_{\mathbb C}, \mathbb{Z}\_p)$ using a varian... | https://mathoverflow.net/users/131945 | Is there a version of algebraic de Rham cohomology that can be used to calculate torsion classes? | You should read the introduction to Bhargav Bhatt's lecture notes on prismatic cohomology: [available here](http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/lecture1-overview.pdf). This is a new cohomology theory introduced by Bhatt-Scholze (closely related to prior work by Bhatt-Morrow-Scholze) for ex... | 9 | https://mathoverflow.net/users/56878 | 350242 | 148,210 |
https://mathoverflow.net/questions/350245 | 1 | I reading Yuri Manin's famous paper on "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS" and struggle with his definition for so called *pseudo-abelian completion* given on page 453 by a reason I would like to explain below. first of all the setting:
Definition 1: An additive category $\mathscr{D}$ is called pse... | https://mathoverflow.net/users/108274 | Pseudo-Abelian Completion in the constrution of Motifs (by Y. Manin) | This is an equivalent way of describing the same thing. To see that, notice that for any $f$ such that $q\circ f=f\circ p$, $$( f\circ p - q\circ f\circ p)\circ p =q\circ( f\circ p - q\circ f\circ p)=0$$ Therefore $[f\circ p]=[q\circ f\circ p]$ in our homomorphism group. Similarly, $[f]=[f\circ p]$ in our homomorphism ... | 3 | https://mathoverflow.net/users/131196 | 350248 | 148,212 |
https://mathoverflow.net/questions/350255 | 2 | If a real function $f:ℝ→ℝ$ is twice differentiable at a point $x$, then the first derivative must be continuous at $x$, and assuming $f′(x)>0$, then there exist $δ>0$ such that $f′(y)>0 $ for all $y∈(x−δ,x+δ)$, then on this interval $f$ must be increasing. Repeating this process for all $x$, we conclude that $δ$ is a f... | https://mathoverflow.net/users/74668 | Is $δ=δ(x)$ a continuous function | $\delta(x)$ (defined as the maximum of appropriate $\delta$'s) is even 1-Lip (for any $f$). Indeed, if $|x-y|=a$, then $\delta(y)\geqslant \delta(x)-a$, since $(y-c,y+c)\subset (x-\delta(x),x+\delta(x))$ for $c=\max(\delta(x)-a,0)$. Analogously $\delta(x)\geqslant \delta(y)-a$ and therefore $|\delta(x)-\delta(y)|\leqsl... | 9 | https://mathoverflow.net/users/4312 | 350256 | 148,214 |
https://mathoverflow.net/questions/350230 | 9 | Let $\mathbb{F}$ be a local non-Archimedean field. Let $\mathcal{O}\subset \mathbb{F}$ be its ring of integers. Let $GL\_n(\mathcal{O})$ be the (compact) group of $n\times n$ invertible matrices with entries in $\mathcal{O}$ such the inverse matrix also has entries in $\mathcal{O}$. Let $Gr\_{i,n}$ be the Grassmannnian... | https://mathoverflow.net/users/16183 | Is the representation of $GL_n(\mathcal{O})$ in functions on Grassmannian multiplicity free? | Yes, this is due to Hill:
*Hill, Gregory*, [**On the nilpotent representations of (GL\_ n({\mathcal O}))**](http://dx.doi.org/10.1007/BF02567703), Manuscr. Math. 82, No. 3-4, 293-311 (1994). See especially Corollary 3.2.
This was generalised and extended by Bader and Onn in
*Bader, Uri; Onn, Uri*, [**Geometric r... | 9 | https://mathoverflow.net/users/2381 | 350263 | 148,217 |
https://mathoverflow.net/questions/350102 | 20 | here it's a question that I've posted [in MSE](https://math.stackexchange.com/questions/3501788/) but unfortunately got no answers:
Let $A$ and $B$ be matrices of finite order with integer coefficients.
Let $n\in\mathbb{N}$ and let $G\_A=\mathbb{Z}\ltimes\_A \mathbb{Z}^n$ be the semidirect product, where the action... | https://mathoverflow.net/users/150901 | Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$ | I believe now that David Speyer's example can be adapted to provide a counterexample to the original question. (So I retract my earlier comment on the question and will delete it soon.)
In David's example, $A$ is a degree $\phi(m)$ matrix of order $m$ defining the action by multiplication of $\zeta\_m$ on the ideal $... | 13 | https://mathoverflow.net/users/35840 | 350265 | 148,218 |
https://mathoverflow.net/questions/350264 | 0 | Am trying to show that $\lim\_{n \rightarrow \infty} \frac{1}{2^{2n}} \sum\_{k=1}^n \sum\_{i=0}^{k-1} \binom{n}{k} \binom{n}{i} =0.5.$
I think that the above result is true but am not sure how to prove this. Any help on this will be greatly appreciated. Thanks in advance!
| https://mathoverflow.net/users/151009 | A combinatorics question: $\lim\limits_{n \to \infty} \frac1{2^{2n}} \sum\limits_{k=1}^n \sum\limits_{i=0}^{k-1} \binom nk \binom ni = \frac12$ | We can rewrite the sum as
\begin{align\*}
\sum\_{k=1}^n \sum\_{i=0}^{k-1} \binom{n}{k} \binom{n}{i}
&=\sum\_{0\leq i<k\leq n}\binom{n}{k} \binom{n}{i}\\
&=\frac{1}{2}\sum\_{\substack{0\leq i,k\leq n\\i\neq k}}\binom{n}{k}\binom{n}{i}\\
&=\frac{1}{2}\left(\sum\_{0\leq i\leq n}\binom{n}{i}\right)^2-
\frac{1}{2}\sum\_{0\l... | 5 | https://mathoverflow.net/users/11919 | 350266 | 148,219 |
https://mathoverflow.net/questions/350268 | 2 | In [On the Question of Absolute Undecidability](http://logic.harvard.edu/koellner/QAU_reprint.pdf), Peter Koellner investigates whether it is possible to prove or disprove [$V = L$](https://en.wikipedia.org/wiki/Axiom_of_constructibility) using (EDIT: both first and second-order) reflection principles, ie. statements o... | https://mathoverflow.net/users/116605 | Can the axiom of choice or its weaker versions be (dis)proved using reflection principles? | The question seems to me asking if sufficiently large cardinals defined by indescribability properties will prove the axiom of choice or its weak variants hold below such cardinals. (Disproving is moot since these are consistent with $V=L$.)
The answer is negative, but more complicated. First of all, we can violate a... | 3 | https://mathoverflow.net/users/7206 | 350271 | 148,220 |
https://mathoverflow.net/questions/345620 | 3 | For any positive integer $n$, let $[n]:=\{1,\ldots,n\}$. Let $S\_n$ denote the set of permutations (bijections) $\pi:[n]\to [n]$. For any $n>1$ and $\pi\in S\_n$ we let the *minimal neighbor distance* be defined by $$\text{md}(\pi) = \min \big(\{
|\pi(k) - \pi(k+1)|: k\in [n-1]\}\cup \{|\pi(n)-\pi(1)|\}\big).$$
For $n>... | https://mathoverflow.net/users/8628 | Minimal neighbor distance in permutations | Here is a quick argument showing that $P(\mathbf{md}(\pi)>m)\le Ce^{-cm}$ though I'll not try to make the bounds sharp. Let us consider $n$ independent random variables $X\_k$ uniformly distributed on $[0,1]$. The rearrangement $\pi$ will be determined from that model as $\pi(k)=\#\{i\in[n]:X\_i\le X\_k\}$. Clearly, we... | 3 | https://mathoverflow.net/users/1131 | 350274 | 148,221 |
https://mathoverflow.net/questions/349983 | 4 | For sets $\cal X$ and $\cal Y$, let $a:{\cal X}\times{\cal X}\rightarrow \mathbb{R}$ and $a:{\cal Y}\times{\cal Y}\rightarrow \mathbb{R}$ be positive definite symmetric kernels. Define the tensor product $a\otimes b$ as usual via
$$ (a\otimes b)((x,x'),(y,y')) = a(x,x')a(y,y') $$
for all $x,x'\in \cal X$ and $y,y'\in \... | https://mathoverflow.net/users/141657 | Relation between Gaussian processes and RKHSs with tensor product kernels | For simplicity, I assume that the Gaussian process $g$ has mean zero such that it is completly specified by its covariance function $k$.
Note that the support of a (Borel) measure depends on the chosen topology.
So I cannot give a general answer, which is independet of the chosen topology.
For non-Borel measures, the... | 1 | https://mathoverflow.net/users/25523 | 350285 | 148,224 |
https://mathoverflow.net/questions/350060 | 29 | The following families of polytopes have received a lot of attention:
* [permutahedra](https://en.wikipedia.org/wiki/Permutohedron),
* [associahedra](https://en.wikipedia.org/wiki/Associahedron),
* [cyclohedra](https://en.wikipedia.org/wiki/Cyclohedron),
* ...
>
> My question is simple: **Why?**
>
>
>
As I u... | https://mathoverflow.net/users/108884 | Why are we interested in permutahedra, associahedra, cyclohedra, ...? | Philosophical questions deserve philosophical answers, so I am afraid no amount of references and specific results will probably satisfy you. Let me try to explain it in a somewhat generic way.
Think about it this way - why care about sequences like $\{n!\}$, Fibonacci or [Catalan numbers](https://www.math.ucla.edu/~... | 22 | https://mathoverflow.net/users/4040 | 350286 | 148,225 |
https://mathoverflow.net/questions/350298 | 1 | The simplex category has for objects totally ordered sets $[n]$ , and for morphisms order-preserving functions between those sets.
We can see the totally ordered set $[n]$ of size $n$ of the simplex category as a very simple form of category (skeletal), for which between 2 elements, there is **at most** one arrow, wh... | https://mathoverflow.net/users/44206 | Symetrical simplex category | You haven't specified what the *morphisms* in your "symmetric simplex category" are supposed to be, and there are two natural choices:
* Functors, or
* Natural isomorphism classes of functors.
In the first case, you get a category equivalent to [FinSet](https://ncatlab.org/nlab/show/FinSet), and the presheaves on i... | 3 | https://mathoverflow.net/users/27013 | 350302 | 148,231 |
https://mathoverflow.net/questions/350222 | 4 | I learned from Wolfram MathWorld about [hypersine](http://mathworld.wolfram.com/Hypersine.html), as being a dimensional analog trig function for hypersolid angles. There it is being defined by
>
> The hypersine ($n$-dimensional sine function) is a function of a vertex angle of an $n$-dimensional parallelotope or si... | https://mathoverflow.net/users/118679 | addition theorems for hypersine | From the paper mentioned in the other answer it looks to me, that addition theorems are kind out of near reach.
But at least we could come up with a result for the **crosspolytopes (orthoplexes)** none the less. This can be done by using the already mentioned dissection of the vertex corner angle into according subsi... | 0 | https://mathoverflow.net/users/118679 | 350308 | 148,232 |
https://mathoverflow.net/questions/350297 | 1 | Are any non-trivial properties known about the constant 0.2357111317192329... that is obtained by catenating the digits of sequence of prime numbers in base 10 or in other bases, especially whether it is normal?
| https://mathoverflow.net/users/31310 | Prime analogue of Champernowne's constant | In base 10 this is called the *[Copeland–Erdős constant](https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant);* it is normal. See [this article](https://projecteuclid.org/euclid.bams/1183509721).
| 2 | https://mathoverflow.net/users/11025 | 350319 | 148,235 |
https://mathoverflow.net/questions/285742 | 2 | Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\mathbb P$-semimartingale and $H$ a predictable process $\mathbb P$-integrable with respect to $X$.
By Girsanov-Meyer theo... | https://mathoverflow.net/users/85330 | Is the stochastic integral invariant under equivalent change of probability? | The measure-invariance of stochastic integrals with (locally) bounded integrands is shown in Meyer (1976, VI.26) or Dellacherie & Meyer (1978, VIII.12). Analogous statement is available for general integrands, e.g. Protter (2004, Theorem II.14).
*Meyer, P. A.*, [**Un cours sur les integrales stochastiques**](http://w... | 3 | https://mathoverflow.net/users/113782 | 350327 | 148,236 |
https://mathoverflow.net/questions/350258 | 3 | Rings are supposed to be associative and unital, but not necessarily commutative.
Some definitions:
* (Bass) A ring $R$ is said to have stable range $1$ if for all $a,b \in R$, whenever $Ra+Rb=R$, then there exists $x\in R$ with $a+xb$ being a unit.
* (Warfield) A ring $R$ is said to be an exchange ring if it has th... | https://mathoverflow.net/users/151007 | Example of an associative unital ring R with stable range 1 and Jac(R)=0 that is not an exchange ring | I know at least two:
1. The integral closure of $\mathbb Z$ in $\mathbb C$
2. The ring of holomorphic functions on $\mathbb C$.
Each nontrivial ideal of an exchange ring with Jacobson radical zero must contain a nonzero idempotent, but since these are both domains, this is clearly not the case for them.
I found t... | 4 | https://mathoverflow.net/users/19965 | 350342 | 148,240 |
https://mathoverflow.net/questions/350324 | 1 | Let $\varphi$ be an $\tau$-sentence, we define the generalized spectrum of $\varphi$ as the class of its finite models, $$\text{GenSpec}(\varphi):=\{\mathcal{A}; \mathcal{A} \models \varphi, \lvert A\rvert < \aleph\_0\}$$
and the spectrum of $\varphi$ as the set of cardinalities of finite models
$$\text{Spec}(\varphi):... | https://mathoverflow.net/users/123891 | Category of finite models of a $\tau$-sentence | A correction, to start: "$\mathcal{C}$ is discrete" does *not* mean "$\text{Mor}(\mathcal{C}) = \emptyset$". Instead, a discrete category has *only identity* arrows. And even with that correction, your question as written has a somewhat trivial negative answer: The class $\text{GenSpec}(\varphi)$ is always closed under... | 3 | https://mathoverflow.net/users/2126 | 350343 | 148,241 |
https://mathoverflow.net/questions/350337 | 6 | Playing around with Matlab I noticed something very peculiar:
Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by
$$A\_{ij}= i \delta\_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$
Here $\delta\_{ij}$ is the Kronecker delta.
We first note that this matrix is not diagonally dominant if $n$ i... | https://mathoverflow.net/users/119875 | Phase transition in matrix | The claim is true with $\epsilon=\frac6{\pi^2}\,$.
To see this, remark that by changing variable $x\_i=y\_i\sqrt i\,$, this is equivalent to proving that
$$\epsilon\left(\left(\frac1{ij}\right)\right)\_{1\le i,j}\le I\_\infty.$$
The first (infinite) matrix is $V\otimes V$ with $V=(1,\frac12\,,\ldots,\frac1n\,,\ldots... | 7 | https://mathoverflow.net/users/8799 | 350350 | 148,242 |
https://mathoverflow.net/questions/350292 | 3 | In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^\*(i): SH^\*(M)\rightarrow SH^\*(N).$$
In particular, given an exact Lagrangian $i\_L: L\hookrightarrow M$, its Weinstein neighborhood $T^\*L$ is a Liouville su... | https://mathoverflow.net/users/114985 | Viterbo restriction map surjective on Weinstein neighbourhood | This is almost never true in general, although it's obviously true for boundary connected sums. For example, it is usually the case that Weinstein handle attachment will kill (non-trivial) invertible elements in $\mathit{SH}^0(T^\ast L)$, so the corresponding Viterbo restriction map can never be surjective. The counter... | 1 | https://mathoverflow.net/users/43423 | 350356 | 148,244 |
https://mathoverflow.net/questions/333086 | 24 | This is a slight generalization of a [question](https://math.stackexchange.com/questions/3203167/which-root-lattices-have-a-theta-series-with-this-property) I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this site.
... | https://mathoverflow.net/users/115044 | Which even lattices have a theta series with this property? | After revisiting my question, I think I have managed to find a proof that there are no other examples of lattices with the requested property. I'm posting it as a self-answer in case someone is interested. Here is the outline:
* The case of even dimension $\mathrm{dim}\: \Lambda \ge 4$ was already described$^\dagger$... | 1 | https://mathoverflow.net/users/115044 | 350359 | 148,245 |
https://mathoverflow.net/questions/350335 | 4 | Let $X$ be a compact metric space, and let $B$ be a convex balanced bounded set in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$.
Let $M=\{u\in C(X),~ uf\in B,~\forall f\in B\}$ and let $N=\{u\in C(X),~ uf\in \overline{B},~\forall f\in \overline{B}\}$.
Since multiplication $(f,g)\to fg$ is ... | https://mathoverflow.net/users/53155 | Approximation of multipliers by multipliers of a smaller set | Let $X=[0,1]$, let $p\_n(x)=x^n$ for $n\in{\mathbb N}$, let $a(x) =e^{-x}$.
Let $V\_0 = \operatorname{lin}\{p\_n \colon n\in {\mathbb N}\}$ and let $V = {\mathbb C}a + V\_0$.
**Claim 1:** $V\_0 = \{ f \in V \colon f(0)=0\}$.
Proof: the LHS is obviously contained in the RHS; for the converse inclusion, note that $... | 4 | https://mathoverflow.net/users/763 | 350361 | 148,246 |
https://mathoverflow.net/questions/350345 | 5 | Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ denotes its (right and left) derived functor.
Under which as... | https://mathoverflow.net/users/66686 | When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful? | I suspect that it is fully faithful if and only if $\mathcal F$ preserves $Ext^n$-s between objects. The "only if" part is clearly necessary (see below). I am slightly hesitant about the "if" part, especially for the unbounded categories.
On the other hand, you can write many explicit sufficient conditions for that f... | 3 | https://mathoverflow.net/users/5301 | 350366 | 148,249 |
https://mathoverflow.net/questions/350364 | 15 |
>
> Does $\mathbb{Q}$ embed into a finitely generated solvable group?
>
>
>
I've checked that $\mathbb{Q}$ is not a subgroup of any finitely generated metabelian group. I don't know how to show this (or whether it is true) for solvable groups of higher step.
| https://mathoverflow.net/users/123459 | Does $\mathbb{Q}$ embed into a finitely generated solvable group? | Yes, it's due to Ph. Hall. It embeds into a f.g. 3-step solvable group.
Let $s:\mathbf{Z}\to\mathbf{Q}^\*$ be a map (thought as an bi-infinite word) such that every finite sequence of nonzero rational numbers occurs as subword. Define two automorphisms $u,v$ of $\mathbf{Q}^{(\mathbf{Z})}$ (vector space over $\mathbf{... | 28 | https://mathoverflow.net/users/14094 | 350367 | 148,250 |
https://mathoverflow.net/questions/350336 | 1 | Is there tight estimates for the following logarithmic summation ($\gamma\in(0,1)$)
$$\ln\Bigg(\sum\_{t=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum\_{\ell=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum\_{k=0}^t\binom{\ell}{k}\binom{n-\ell}{t-k}\Bigg)?$$
Is it roughly bounded a... | https://mathoverflow.net/users/136553 | Tight estimates for binomial summation | By [Vandermonde's identity](https://en.wikipedia.org/wiki/Vandermonde%27s_identity), $$\sum\_{k=0}^t\binom{\ell}{k}\binom{n-\ell}{t-k}=\binom{n}{t},$$
so the triple sum reduces to
$$
\sum\_{t=n/2-\gamma n^\gamma}^{n/2+\gamma n^\gamma}\sum\_{\ell=n/2-\gamma n^\gamma}^{n/2+\gamma n^\gamma}\binom{n}{t}
=\sum\_{t=n/2-\gamm... | 1 | https://mathoverflow.net/users/141766 | 350372 | 148,252 |
https://mathoverflow.net/questions/350303 | 7 | I'll write "$\mathcal{L}\_\alpha$" for the fragment $\mathcal{L}\_{\infty,\omega}\cap L\_\alpha$.
---
Say that a countable admissible $\alpha$ is *Robinsonian* if there is some sentence $\varphi\in\mathcal{L}\_\alpha$ such that $L\_\alpha\models\varphi$ and there is no $T\subseteq\mathcal{L}\_\alpha$ which is con... | https://mathoverflow.net/users/8133 | "Robinson arithmetic" for (some) levels of $L$? | *EDIT: to my chagrin, the notion of "$n$-admissibility" is not what I thought it was! What I wanted was $\Sigma\_n$-admissibility. You can find the definition of $n$-admissibles [here](https://link.springer.com/chapter/10.1007/BFb0059534); they are vastly smaller than their $\Sigma\_n$ counterparts, and indeed for each... | 2 | https://mathoverflow.net/users/8133 | 350378 | 148,254 |
https://mathoverflow.net/questions/350391 | 0 | Let $(X,\tau)$ be a non-metrizable topological space which is not first-countable and let $\emptyset \neq Y\subset X$ be a proper dense subset. Is it possible for $(Y,\tau\_Y)$ *(where $\tau\_Y$ is the relativisation of $\tau$ to $Y$)* be metrizable?
| https://mathoverflow.net/users/36886 | Restriction of non-metrizable topology to dense subset is non-metrizable | Yes: the order topology on $\omega\_1+1$ (the first uncountable successor ordinal) is an example. It is not first countable, because it's "top" point $\omega\_1$ has no countable neighborhood base. But the set of all isolated points of this space is dense in it, and the relative topology on this set is discrete (hence ... | 2 | https://mathoverflow.net/users/70618 | 350402 | 148,261 |
https://mathoverflow.net/questions/350394 | 7 | If $\Phi\_1,\Phi\_2$ are convex polyhedra in $\mathbb{R}^3$ such that the sets of outer normals to facets coincide, but $\Phi\_1$ is not a translate of $\Phi\_2$, then there exist two corresponding facets $F\_1,F\_2$ (with the same outer normal) such that one of them is a translate of a proper subset of another.
This... | https://mathoverflow.net/users/4312 | Alexandrov's rigidity in higher dimensions | Here is a counterexample in dimension four.
Consider positive numbers $x\_1,x\_2,x\_3,x\_4\in\Bbb R$ with $x\_1<x\_2$ and $x\_3<x\_4$ and construct the two 4-orthotopes (cartesian products of intervals)
\begin{align}
O\_1:=[0,x\_1]\times [0,x\_2]\times[0,x\_3]\times[0,x\_4]\\
O\_2:=[0,x\_2]\times [0,x\_1]\times[0,x... | 3 | https://mathoverflow.net/users/108884 | 350410 | 148,265 |
https://mathoverflow.net/questions/350409 | 2 | Is there a vector field $X$ on $\operatorname{M}\_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition:
$$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\cdot \operatorname{Det}=-\operatorname{trace} \end{cases}$$
where $\operatorname{Det}$ is determinant?
| https://mathoverflow.net/users/36688 | A vector field $X$ on $\mathrm{GL}(n,\mathbb{R})$ with $\begin{cases} X.\mathrm{trace}=\mathrm{Det} \\X.\mathrm{Det}=-\mathrm{trace} \end{cases}$ | $\DeclareMathOperator\trace{trace}\DeclareMathOperator\Det{Det}$**No**. Let $\sum\_{i,j}x\_{ij}(M)\frac\partial{\partial m\_{ij}}$ be this vector field. These conditions amount to writing
$$\sum\_ix\_{ii}(M)=\det M,\qquad\sum\_{ij}x\_{ij}(M)\hat m\_{ij}=-{\rm Tr}M,$$
where $\hat M$ is the cofactor matrix. Take $M=a I\_... | 5 | https://mathoverflow.net/users/8799 | 350416 | 148,266 |
https://mathoverflow.net/questions/275406 | 9 | Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the category of commutative [pseudomonoids](https://ncatlab.org/nlab/show/pseudomonoid) $\mathrm{CMon}(\mathcal{C})$ has bica... | https://mathoverflow.net/users/2841 | Pushouts of commutative pseudomonoids | To summarize some of the comments:
I don't know a short answer for why a bicategorical coequalizer doesn't work. If you try to give the bicategorical coequalizer the structure and universal property, you'll find that it just doesn't work somewhere. The intuition is that in higher categories, when you have more cohere... | 2 | https://mathoverflow.net/users/49 | 350419 | 148,267 |
https://mathoverflow.net/questions/350415 | 1 | Consider the function
$$f\_{n}(x)=e^{-x^2}x^n.$$
My goal is to show that
$$ G(y):=\frac{(f\_2\*f\_0)(y)}{(f\_0\*f\_0)(y)}- \left(\frac{(f\_1\*f\_0)(y) }{(f\_0\*f\_0)(y)}\right)^2$$
is log-concave.
Let us first observe that indeed $G(y) \ge 0.$
This just follows from a Cauchy-Schwarz
$$(f\_1\*f\_0)(y) \... | https://mathoverflow.net/users/108483 | Log-concavity of function | Direct calculations show that
$$(f\_2\*f\_0)(y)=\frac{1}{4} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} \left(y^2+1\right),
$$
$$(f\_1\*f\_0)(y)=\frac{1}{2} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} y,
$$
$$(f\_0\*f\_0)(y)=\sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}}
$$
for all real $y$,
so that $G$ is the constant $1/4$ and... | 6 | https://mathoverflow.net/users/36721 | 350421 | 148,268 |
https://mathoverflow.net/questions/350414 | 9 | For which $n$ is there a unique perfect group of order $n$? Are there infinitely many such $n$?
Some guesses for infinite sequences of such $n$: $|\mathrm{PSL}(2,p)|$, $|\mathrm{SL}(2,p)|$, $|A\_m|$, $|S\_m|$.
If you replace "perfect" with "simple", a complete understanding follows from the Classification. Apart fr... | https://mathoverflow.net/users/20598 | When is there a unique perfect group of order $n$? | For the case $n = p(p-1)(p+1)$ for $p >3$ a prime, a theorem of M. Herzog (to be found in the article "Finite groups with a large cyclic Sylow subgroup" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=338163)) in the conference proceedings "Finite Simple Groups" (Oxford, editors M. Powell and G. Higman, publish... | 13 | https://mathoverflow.net/users/14450 | 350424 | 148,269 |
https://mathoverflow.net/questions/350439 | 0 | I am reading the paper: ``ON THE DISTRIBUTION OF FIRST HITS FOR THE SYMMETRIC STABLE PROCESSES" by Blumenthal, Getoor and Ray, (Trans. Amer. Math. Soc. 99 (1961), 540-554).
On page 546, the authors talk about the idea of Riesz regarding spherical inversions. In particular they say, for the sphere $\{u:|x-u| =r\}$, in... | https://mathoverflow.net/users/151110 | Sphere inversion in Riesz potential | Well, you may like to have a look at the original M. Riesz's 1938 paper: it is a *fantastic* read!
---
In the language of these papers, a "potential" of exponent $\alpha$ is a function $f$ of the form
$$ f(x) = \int\_{\mathbb{R}^N} |y - x|^{\alpha - N} \mu(dy) , $$
where $\mu$ is a non-negative measure. M. Riesz ... | 3 | https://mathoverflow.net/users/108637 | 350443 | 148,277 |
https://mathoverflow.net/questions/350449 | 3 | Does there exists two non-constant continuous functions $f,g:[0,1]\rightarrow \mathbb{R}$ so that they are in independent (in probability sense) when viewed as random variables over the measure space $([0,1],\mathrm{Lebesgue},\mathcal{B}\_{[0,1]})$, where $\mathcal{B}\_{[0,1]}$ is the Borel sigma field of $[0,1]$?
I... | https://mathoverflow.net/users/151115 | Two independent function when considered as random variable over $([0,1],\mathrm{Lebesgue},\mathcal{B}_{[0,1]})$ | We can take a Peano curve $\varphi: [0,1]\to [0,1]^2$. More precisely, we want a construction like the one [here](https://en.wikipedia.org/wiki/Space-filling_curve#Outline_of_the_construction_of_a_space-filling_curve) that spends its fair share of the time in each dyadic square when traced out at unit speed. This will ... | 7 | https://mathoverflow.net/users/48839 | 350452 | 148,279 |
https://mathoverflow.net/questions/350447 | 4 | Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert\_{L^2[0,1]}=1.$
Does there exist a constant $c>0$ such that any such function satisfies
$$ \Vert f-1 \Vert\_{H^1}>c?$$
I was thinking that the Fourier series could help to prove or disprove something l... | https://mathoverflow.net/users/150549 | Approximate constant function | For all $x$ and $y$ in $[0,1]^2$
$$f(x,y)=
\left\{
\begin{aligned}
\int\_x^y f\_y(x,z)\,dz&\text{ if }x\le y, \\
-\int\_y^x f\_y(x,z)\,dz&\text{ if }x\ge y,
\end{aligned}
\right.
$$
where $f\_y(x,z):=\frac{\partial f(x,y)}{\partial y}|\_{y=z}$, so that
$$|f(x,y)|\le\int\_0^1|f\_y(x,z)|\,dz\le\sqrt{\int\_0^1|f\_y(x,... | 8 | https://mathoverflow.net/users/36721 | 350454 | 148,280 |
https://mathoverflow.net/questions/290511 | 3 | Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum\_{j\in\mathbb Z}\hat\Psi(2^j\xi)=1$ for all $\xi\ne0$, we can define the Littlewood-Paley operator $$\Delta\_j(f) := \left(\hat\Psi(2^{-j}\xi)\hat{f... | https://mathoverflow.net/users/94022 | Is $\mathscr{S}_h'$ a complementary subspace for $\mathscr{S}'/\mathscr{P}$, the space of tempered distributions modulo polynomials? | (This is rather late, but I'll answer the question anyways.) No, it is not true that $\mathscr S\_h'$ and $\mathscr P$ are (topological) complements in $\mathscr S'$ (in fact, they are not even linear complements). Suppose it were the case that every tempered distribution $f$ admitted a (unique) decomposition of the fo... | 4 | https://mathoverflow.net/users/141993 | 350458 | 148,282 |
https://mathoverflow.net/questions/350473 | 12 | Let $G$ be a finite group, and $\rho\_1, \rho\_2: G\to GL\_n(\mathbb C)$ be two representations. Suppose that $\rho\_1$ and $\rho\_2$ are equivalent (i.e. conjugate over $\mathbb C$), and suppose that both groups $\rho\_1(G)$, $ \rho\_2(G)$ belong to $GL(n,\mathbb Z)$. Is it true that these two groups are conjugate in ... | https://mathoverflow.net/users/13441 | Two equivalent irreducible representations given by integer matrices | The smallest counterexample involving irreducible representations of symmetric groups is the $2$-dimensional irreducible module for $\mathbb{C}S\_3$. It can be defined over the integers as the submodule $U = \langle e\_2-e\_1, e\_3-e\_1\rangle\_\mathbb{Z}$ of the natural integral permutation module $\langle e\_1, e\_2,... | 23 | https://mathoverflow.net/users/7709 | 350479 | 148,291 |
https://mathoverflow.net/questions/350478 | 8 | It is a classical result of Quinn that for a simply-connected
closed $4$-manifold $X$ the isometries of its intersection form
are in one-to-one correspondence with
$\pi\_0 \text{Homeo}(X)$. (Isotopy of 4-manifolds, 1986)
Let $X$ have some simple fundamental group, say $\mathbb{Z}\_2$,
and let $h\colon X \to X$ be... | https://mathoverflow.net/users/150186 | Topological mapping class groups of 4-manifolds | Let $X = (S^2\times S^2)/\mathbb{Z}\_2$ where the $\mathbb{Z}\_2$ action is generated by $(x, y) \mapsto (-x, -y)$. Note that $H\_2(X; \mathbb{Z}) \cong \mathbb{Z}\_2$, so every diffeomorphism acts trivially.
Consider the diffeomorphism $f : S^2\times S^2 \to S^2\times S^2$ given by $(x, y) \mapsto (x, -y)$. This de... | 9 | https://mathoverflow.net/users/21564 | 350480 | 148,292 |
https://mathoverflow.net/questions/350467 | 2 | Let $f\_n: \mathbb R^2 \rightarrow \mathbb R$ be a family of probability distributions with the property that they vanish on the diagonal $f\_n(x,x)=0.$
I would like to know: Can we show that a function like this can never converge to a standard Gaussian $f(x,y) = \frac{1}{2\pi} e^{- \frac{\vert x \vert^2+ \vert y \v... | https://mathoverflow.net/users/151010 | Non-convergence to a Gaussian | Your conjecture is true.
Indeed, let $g\_n:=\sqrt{f\_n}$ and $g:=\sqrt{f}$. Let
$$v:=\|g\|,
$$
where $\|h\|:=\|\,h|\_J\,\|\_{L^2(J)}$ for $h\in L^2(\mathbb R^2)$, $J:=I^2$, $I:=[-u,u]$, and $u\in(0,1/20)$ is small enough so that
$$v>u/10;
$$
such a number $u$ exists, because $g(0,0)^2=1/(2\pi)>1/400$ and $g$ is con... | 1 | https://mathoverflow.net/users/36721 | 350492 | 148,297 |
https://mathoverflow.net/questions/350469 | 8 | In this post, *irrep* and *dim* mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) two irreps of dim $5$ (denoted $5\_1$ and $5\_2$) and (at least) two irreps of dim $7$ (denoted $7\_1$ and $7\_2$) ... | https://mathoverflow.net/users/34538 | Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation | I think that there is indeed no such finite group $G$, whether simple or otherwise. Note first that the representation $5\_{1}$ can be assumed to be faithful ( for if $K$ is its kernel, then the group $G/K$ has the same property), so from now on, we assume it faithful.
Note next that $Z(G) = 1$, since if $5\_{1}$ lie... | 11 | https://mathoverflow.net/users/14450 | 350493 | 148,298 |
https://mathoverflow.net/questions/350502 | 3 | Suppose $P\_1,\ldots,P\_n$ are homogeneous polynomials in $\mathbb C[x\_0,\ldots,x\_n]$ of degrees $d\_1,\ldots,d\_n\ge 1$. These define hypersurfaces $H\_1,\ldots,H\_n\subset\mathbb P^n$. **Is there a nonzero polynomial in the coefficients of $P\_1,\ldots,P\_n$ which vanishes whenever when the common intersection $H\_... | https://mathoverflow.net/users/110236 | Resultant in many variables | Let $F\_i$ be a homogeneous polynomial of degree $d\_i$ defining $H\_i$. Let $U(x)$ be a linear form. Then consider the multivariate resultant ${\rm Res}(F\_1,\ldots,F\_n,U)$ which, for fixed $F\_i$'s is a homogeneous polynomial of degree $d\_1\cdots d\_n$ in the coefficients of $U$. You condition is equivalent to the ... | 5 | https://mathoverflow.net/users/7410 | 350507 | 148,302 |
https://mathoverflow.net/questions/350510 | 10 | The quadratic reciprocity law states that for $p\_1\ne p\_2$ prime, the product $\left(\frac{p\_1}{p\_2}\right)\left(\frac{p\_2}{p\_1}\right)$ takes values $1$ or $-1$ depending on whether $p\_1$ and $p\_2$ satisfy some set of restrictions mod $4$.
Is there a "quadratic reciprocity law for three primes"? I suspect th... | https://mathoverflow.net/users/9924 | Quadratic reciprocity for three primes? | Turns out the details are easy so I worked them out myself :) The highlighted statement is true.
Let me assume $4\mid M$. Pick $p\_2,p\_3$ arbitrary satisfying the congruence modulo $M$ (they exist by Dirichlet). Take any $p\_1$ which is congruent to $r\_1\pmod M$, congruent to $p\_2^{-1}\pmod{p\_3}$, and such that
$... | 13 | https://mathoverflow.net/users/30186 | 350512 | 148,303 |
https://mathoverflow.net/questions/350511 | 4 | Let us fix a constant $r>1$. Let $d(x,y)$ denote the distance between points $x,y\in \mathbb R^2$. Suppose we have a discreet subset $X\subset \mathbb R^2$ such that
1) For any two points $x,x'\in X$ we have $d(x,x')\ge 1$.
2) For any point $y\in \mathbb R^2$ there is $x\in X$ such that $d(x,y)\le r$.
**Question... | https://mathoverflow.net/users/13441 | Finding a not too slim triangulation with prescribed vertices on $\mathbb R^2$ | Yes, the Delaunay triangulation will have this property. From 2) the circumradius of any triangle will be at most $r$, so diameter will be at least $2r$. Also a too small angle would either imply an edge of length less than $1$ or a circumradius larger than $r$.
| 4 | https://mathoverflow.net/users/112954 | 350513 | 148,304 |
https://mathoverflow.net/questions/350464 | 1 | I want to prove the following.
>
> For every $\Pi^0\_1$ statement $\forall x\phi(x)$, where $\phi(x)$ is a $\Delta^0\_1$ formula, there is $e\in\mathbb{N}$ such that $\forall x\phi(x)$ implies $W\_e=PA$\* and $PA+\text{$W\_e$ is consistent}\vdash\forall x\phi(x)$.
>
>
>
The argument is inspire by an argument o... | https://mathoverflow.net/users/18879 | Is every true $\Pi^0_1$ statement entailed from a consistency statement of $PA$? | The statement you want to prove is true, and your argument works - but there's a simpler one: drop all reference to the recursion theorem and Godelian incompleteness, and just change the second clause in your definition of $\varphi\_e(n)$ to "$\sigma(n)\wedge\exists x(x\not=x)$." That is, introduce an inconsistency *di... | 2 | https://mathoverflow.net/users/8133 | 350518 | 148,307 |
https://mathoverflow.net/questions/350525 | 0 | Let $f(x,y)$ be a real-valued function on the unit square $[0,1]^2$. Suppose that $f(x,y)$ is Riemann integrable along each straight line. Does this imply that $f$ Riemann integrable on the square? Does the answer change if we only suppose that $f$ is integrable along all vertical lines $x=c$ and integrable along all h... | https://mathoverflow.net/users/99186 | If a real-valued bivariate function on the unit square is integrable along each line, is it integrable on the square? | No, consider the function
$$g(x,y) = \cases {0,&if $(x,y)=(0,0)$\\
xy^2/(x^2+y^6) &otherwise}$$
taken from Exercise 4.7 of Baby Rudin (3ed) via [this Math.SE post](https://math.stackexchange.com/questions/174816/discontinuous-functions-that-are-continuous-on-every-line-in-bf-r2). Along every line it is continuous and ... | 3 | https://mathoverflow.net/users/4832 | 350527 | 148,311 |
https://mathoverflow.net/questions/350514 | 6 | I am reading Hatcher's treatment of the Adam's spectral sequence. <http://pi.math.cornell.edu/~hatcher/SSAT/SSch2.pdf>
On page 20, he states "Thus for each $i$ the groups $\pi\_i(Z^k)$ are zero for all sufficiently large $k$. The same is true for the groups $\pi^Y\_i (Z^k)$ when $Y$ is a finite spectrum, since a map
... | https://mathoverflow.net/users/41616 | moving from sphere spectrum to finite spectrum | Let $\{Y,Z\}$ denote the homotopy group of maps between a spectrum $Y$ and a spectrum $Z$. We are assuming that one has a sequence of spectra $Z^1, Z^2, \dots$ such that for any fixed $i$, $\{S^i,Z^k\} = 0$ for $k>>0$, and we want to show that, if $Y$ is a finite spectrum, then $\{Y,Z^k\} = 0$ for $k>>0$.
An easy wa... | 8 | https://mathoverflow.net/users/102519 | 350529 | 148,312 |
https://mathoverflow.net/questions/350466 | 2 | CW-complexes are defined by attaching cell with *increasing dimension*: you start with a set of points, then attach 1-cells, then 2-cells and so on. Why are defined so? My question is: why is it necessary to attach cells ordered by dimension and not to attach a 2-cell, then a 1-cell, then a 3-cell...? The proofs I have... | https://mathoverflow.net/users/137622 | CW complexes obtained by attaching cells not with increasing dimension | There is a name for the kind of space you are describing: a **cell complex.**
A CW complex is a cell complex which has cell attachments in the increasing order of dimension.
The main advantage of having a CW complex over a mere cell complex is that the filtration by skeleta defines a finite chain complex model for it... | 3 | https://mathoverflow.net/users/8032 | 350550 | 148,319 |
https://mathoverflow.net/questions/350562 | 1 | Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi\_{x})\_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued *global function*, because it depends on every entry of $\varphi$. The renormalization group setup goes like this: first, for a fixed $N... | https://mathoverflow.net/users/150264 | Gaussian Property of the Renormalization Group | The following (between the horizontal lines) is a statement of what one may call the abstract change of variable theorem. It is taken from some exercise I gave my students a while ago:
---
Let $(X,\mathcal{M},\mu)$ be a measure space and let $(Y,\mathcal{N})$ be a measurable space.
Let $f:X\rightarrow Y$ be an $(... | 2 | https://mathoverflow.net/users/7410 | 350564 | 148,322 |
https://mathoverflow.net/questions/350567 | 9 | Let $C$ be a smooth connected algebraic curve over $\mathbb C$. Assume that $C$ admits no automorphisms. Let $p\_1, p\_2 \in C$ be two distinct points.
**Question:** Are the $\mathbb A^1$ homotopy types of $C - p\_1$ and $C- p\_2$ the same?
I think that the answer is "no", but I do not know how to construct an inv... | https://mathoverflow.net/users/131945 | Is the $\mathbb A^1$ homotopy type of a punctured curve independent of the choice of puncture? | If I understand it correctly, this follows from the results of Severitt's [master's thesis](https://www.math.uni-bielefeld.de/~mseverit/mseverittse.pdf). (I am not an expert on this, so it is possible I misread one of the statements.)
Indeed, Lemma 9.1.1 shows that smooth projective curves of genus $> 0$ are $\mathbf... | 7 | https://mathoverflow.net/users/82179 | 350577 | 148,327 |
https://mathoverflow.net/questions/350542 | 6 | This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H\_{\omega\_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle H\_{\omega\_2},\in\rangle$, consisting of sets of hereditary
size at most $\aleph\_1$. These are both models of
Zermel... | https://mathoverflow.net/users/1946 | Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy? | A theorem of Harrington (Theorem B of his paper "[Long projective wellorders](https://doi.org/10.1016/0003-4843(77)90004-3)") says $\text{MA} + \neg\text{CH}$ is consistent with a projective wellorder of the reals, hence a wellorder of $H\_{\omega\_1}$
definable over $H\_{\omega\_1}$. Since $\text{MA}\_{\omega\_1}$ im... | 8 | https://mathoverflow.net/users/102684 | 350585 | 148,332 |
https://mathoverflow.net/questions/350563 | 9 | Let $X$ be a space with fundamental group $G$. Recall that the de Rham fundamental group of $X$ is the inverse limit of the Malcev completions of the nilpotent truncations of $G$. This has a Lie algebra, which I will denote by $\mathfrak{g}$. The Lie algebra $\mathfrak{g}$ has a natural filtration, and thus has an asso... | https://mathoverflow.net/users/149707 | Different definitions of formality for groups | There is a notion of **$q$-equivalence** between cdgas: it is a chain of morphisms each being isomorphism on cohomology in degrees $\leq q$ and monomorphism in degree $q+1$. Now, we call something **$q$-formal** if it is $q$-equivalent to its cohomology. Obviously, a space $X$ is $1$-formal if and only if $K(\pi\_1(X),... | 5 | https://mathoverflow.net/users/81055 | 350588 | 148,334 |
https://mathoverflow.net/questions/350586 | 2 | I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The set of all functions $\varphi : \Lambda \to \mathbb{R}$ is isomorphic to $\mathbb{R}^{|\Lambda|}$, so that these functio... | https://mathoverflow.net/users/150264 | Imprecise Definition of a $\sigma$-algebra | $\newcommand\vpi{\varphi}$
$\newcommand\La{\Lambda}$
$\newcommand\R{\mathbb R}$
You misunderstood the notes: For each $j\in\{0,\dots,N\}$ and each $x\in\La$, $\vpi\_j(x)$ is (not a real number but) a real-valued random variable (r.v.).
Indeed, (i) formula (2.17) on page 25 in the notes implies that $\vpi\_j=\sum\_{k... | 1 | https://mathoverflow.net/users/36721 | 350602 | 148,340 |
https://mathoverflow.net/questions/350579 | 3 | Sample $m$ times from unknown probability distribution $p=(p\_1,p\_2,\cdots,p\_n)$, we can construct a probability distribution $q=(q\_1.q\_2,\cdots,q\_n)$.
How large $m$ should be to achieve that the probability of $||p-q||\_2<\epsilon$ is at least $1-\delta$? We are particularly interested in the case of $\delta=1/... | https://mathoverflow.net/users/4987 | Reconstructing probability distribution with high probability | $\newcommand\ep{\epsilon}$
$\newcommand\ch{\operatorname{ch}}$
This answer is based on [Theorem 3 and formula (2)](https://epubs.siam.org/doi/10.1137/1130013), which yield
$$P(|S\_m|\ge x)
\le2\exp\Big(-tx+\sum\_{i=1}^m E(e^{t|X\_i|}-t|X\_i|-1)\Big), \tag{0}
$$
where $x$ and $t$ are any nonnegative real numbers, $S\_... | 3 | https://mathoverflow.net/users/36721 | 350607 | 148,344 |
https://mathoverflow.net/questions/350609 | 7 | In an Oberwolfach report from 2016 [1, page 2] it is said that $K\_8(\mathbb{Z})$ has recently been computed. Does anyone know a reference for the computation?
[1] <https://orbilu.uni.lu/bitstream/10993/29499/1/preliminary_OWR_2016_52.pdf>
| https://mathoverflow.net/users/17734 | Reference for computation of $K_8(\mathbb{Z})$ | There are two preprints available:
<https://arxiv.org/abs/1910.11598>
<https://www.utsc.utoronto.ca/people/kupers/wp-content/uploads/sites/50/2021/01/k8zshorter.pdf>
| 9 | https://mathoverflow.net/users/16785 | 350612 | 148,346 |
https://mathoverflow.net/questions/293128 | 1 | Consider a Markov chain $X\_n$ taking values in finite countable set $\mathcal{X}$ with transition matrix $P$. Consider a function $f:\mathcal{X}\to\mathcal{Y}$ inducing a partition $\mathcal{Y}=\{\mathcal{X}\_{[i]}\}$. Then, the process $Y\_n = f(X\_n)$ is Markov if, given any two elements of the partition $\mathcal{X... | https://mathoverflow.net/users/81434 | Comprehensive reference for lumped or projected markov chains | The question of what properties are preserved has attracted a lot of attention in dynamics. I think the state of the art is by Mark Piraino, see 'projection of Gibbs states for Hölder potentials.'
If you care only about starting with a Markov system then probably work of Chazottes and Ugalde is more direct.
Sorry f... | 1 | https://mathoverflow.net/users/24586 | 350618 | 148,349 |
https://mathoverflow.net/questions/350627 | 5 | Suppose that $\kappa$ has the property that for every family $A\subseteq\omega^{\omega}$, if $|A|<\kappa$, then there exists some $g\in\omega^{\omega}$ such that for any $f\in A, \exists^{\infty}n\;f(n)\neq g(n)$. Does it then follow that for any family $A\subseteq\omega^{\omega}$ of size $<\kappa$, there exists a $g\i... | https://mathoverflow.net/users/138274 | Relationship between "infinitely unequal" and "eventually different" | It does not follow. In fact, it is not true in the Cohen model. Let $X$ be a set of $\aleph\_2$ mutually generic Cohen reals over $V$. (Recall: ``mutually generic'' means that if $x \in X$ and $Y \subseteq X$, then $x \in V[Y]$ if and only if $x \in Y$, and otherwise $x$ is Cohen-generic over $V[Y]$.)
In $V[X]$, if $... | 5 | https://mathoverflow.net/users/70618 | 350630 | 148,351 |
https://mathoverflow.net/questions/350615 | 2 | My question: is it true that we can define a spectrum of $\mathbb{C}$-algebra $A$ in such a way that it becomes a complex manifold with the algebra of holomorphic functions $A$?
Maybe it will work if we restrict to algebras that already are an algebra of holomorphic functions for some manifold?
I'm aware of the const... | https://mathoverflow.net/users/143549 | Holomorphic Spectrum | In some cases the spectrum of a commutative Banach algebra $\mathcal A$ may contain "analytic disks" on which the Gelfand transforms of members of $\mathcal A$ correspond to analytic functions. You might look at section 1.5 of Andrew Browder, ["Introduction to Function Algebras"](https://books.google.ca/books/about/Int... | 3 | https://mathoverflow.net/users/13650 | 350631 | 148,352 |
https://mathoverflow.net/questions/350552 | 1 | Let $G$ be a finite solvable group of Fitting length $n$ with upper Fitting series
$1= U\_0G \leq U\_1G \leq \cdots \leq U\_nG = G$.
Is it true that every normal subgroup of Fitting length $i$ is contained in $U\_iG$? (for $i=1$ it is true because the Fitting subgroup contains all nilpotent normal subgroups)
| https://mathoverflow.net/users/126942 | Normal subgroup of Fitting length i contained in i-th term of upper Fitting series? | You have already done the case, when $i = 1$.
Now suppose it i true for $i$. Let's prove it for $i+1$. Suppose $H \triangleleft G$ has Fitting length $i+1$. That means $\frac{H}{U\_iH}$ is nilpotent. Now let's define $\phi\_i$ as a natural homomorphism between $G$ and $\frac{G}{U\_iG}$. Then $\phi\_i(H)$ is a nilpote... | 1 | https://mathoverflow.net/users/110691 | 350661 | 148,357 |
https://mathoverflow.net/questions/350619 | 2 | For all $k \in \mathbb N$ let $a\_k$ be strictly positive bounded weights, i.e. there are constants $C\_1$ and $C\_2$ such that $0<C\_1\le a\_k \le C\_2$. Now a real valued sequence $(x\_k)\_{k \in \mathbb N}$ satisfies $\lim\_{n \to \infty}\frac{1}{n} \sum\_{k=1}^n (a\_k)^m e^{2\pi i x\_km}=0$ for all non-zero integer... | https://mathoverflow.net/users/128116 | Is this a criterion for uniform distribution modulo one? | I provide a sequence $(x\_k)\_k$ that is not uniformly distributed but satisfies $\frac{1}{N}\sum\_{k \le N} a\_k^m e^{2\pi i x\_k m} \to 0$ as $N \to \infty$ for each $m \ge 1$, where $a\_k = 1$ if $k$ is odd and $a\_k = 3$ if $k$ is even. Let $x\_k = 0$ if $k$ is odd. Then, for any $m \ge 1$, $\sum\_{\substack{k \le ... | 4 | https://mathoverflow.net/users/129185 | 350663 | 148,358 |
https://mathoverflow.net/questions/350653 | 6 | In synthetic differential geometry, an object $M$ verifies the *Wraith axiom* if for all functions $\tau:D\times D\to M$ which are constant on the axes $\tau(d,0)=\tau(0,d)=\tau(0,0)$ for all $d\in D$, there's a unique factorization through the multiplication map, i.e there's a unique function $t:D\to M$ such that $t(d... | https://mathoverflow.net/users/69037 | Intuition and analogue of Wraith axiom from synthetic differential geometry | Axiom W is about the behaviour of the second tangent bundle - it ensures that the vertical bundle of the tangent bundle, $V(M) \subseteq T\circ T(M)$, where $V(M) = T(p)^{-1}(0)$, decomposes as the pullback of the projection $p\_M: T(M) \to M$ along itself. The map $[\bullet, M]:[D,M] \to [D \times D, M]$ would be writ... | 5 | https://mathoverflow.net/users/75783 | 350668 | 148,362 |
https://mathoverflow.net/questions/350633 | 3 | There is a well known fact:
>
> If $G$ is a finitely generated group. Then $|G’| < \infty$ iff $[G:Z(G)]<\infty$.
>
>
>
Suppose $\mathfrak{U}$ is a group variety. Let’s denote the corresponding verbal subgroup as a $V\_{\mathfrak{U}}(G)$ and the corresponding [marginal subgroup](https://groupprops.subwiki.org/... | https://mathoverflow.net/users/110691 | Is finite verbal subgroup equivalent to finite index of marginal subgroup? | It is not true. A counterexample was constructed in Ashmanov, I. S.; Olʹshanskiĭ, A. Yu. Abelian and central extensions of aspherical groups. Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 48–60, 85. They even contsructed a noetherian group $G$ and a word $v$ with finite verbal subgroup $v(G)$ and marginal subgroup $v^\... | 6 | https://mathoverflow.net/users/nan | 350673 | 148,363 |
https://mathoverflow.net/questions/350674 | 1 | I am currently reading Jesse Peterson's [lecture notes on von Neumann algebras](https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf). I'm confused by lemma 4.4.2. In particular, it seems to me that Hahn Banach theorem here can only conclude the map $\phi$ is in the X\*\* not in X. Without th... | https://mathoverflow.net/users/151245 | Proof of uniqueness of predual of von Neumann algebra | The proof is using the version of the Hahn-Banach theorem which works for locally convex spaces. We have a $C^\*$-algebra $A$ and a Banach space $X$ with $X^\*=A$. We then equip $A$ with the weak$^\*$-topology, that is, $\sigma(A,X)$. As the proof notes, $(A)\_1 \cap A\_+$ is $\sigma(A,X)$ compact, so if $a<0$ we can f... | 5 | https://mathoverflow.net/users/406 | 350681 | 148,364 |
https://mathoverflow.net/questions/350648 | 1 | Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph $G$, what is the smallest number of "curves" in the plane drawn from $u$ to $v$ such that no $u$--$v$ path in $G$ intersects each curve at a point other than $u$ or $v$?
(any finite bound is good as well)
Also, can anyone recommend ... | https://mathoverflow.net/users/56833 | Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph, what is the smallest number of "curves"? | Unfortunately, there is no finite bound, even if both vertices have bounded degree. To see this, consider a large grid graph $G$ with $u$ a degree-$4$ vertex in the 'left half' of the grid and $v$ a degree-$4$ vertex in the 'right half' of the grid. Let $\mathcal{I}$ be a family of $u$--$v$ curves such that there is no... | 1 | https://mathoverflow.net/users/2233 | 350688 | 148,365 |
https://mathoverflow.net/questions/350675 | 3 | I originally asked this question on [math stackexchange](https://math.stackexchange.com/questions/3510940/can-matrix-factorizations-be-nonsquare), but got no attention. As such, I am asking here as well.
$\textbf{Background}$
On page 49 of a 1980 [paper by Eisenbud](https://www.ndsu.edu/pubweb/~ssatherw/fa11/790-3/... | https://mathoverflow.net/users/142858 | Can matrix factorizations be nonsquare? | Note that the equality of traces yields $|m-n|x=0$, and in particular this is impossible for $|m-n|=1$.
Still it is possible for $(m,n)=(1,3)$. Let $R$ be the ring $\mathbf{F}\_2[a\_1,a\_2,a\_3,b\_1,b\_2,b\_3]/J$, where the ideal $J$ is generated by all $a\_ib\_j$ for $i\neq j$ and $a\_1b\_1-a\_2b\_2$, $a\_2b\_2-a\_3... | 3 | https://mathoverflow.net/users/14094 | 350699 | 148,368 |
https://mathoverflow.net/questions/350698 | 2 | Let $z$ be a number in the unit disc. I would like to know a lower bound of the form
$$\frac{1}{N}\sum\_{k=1}^N|z-\omega^k|\geq |z|+c$$
when $\omega$ is a primitive $N$'th root and $N>3$, where $c$ is independent of $z$ and $N$. The lower bound $|z|$ follows from the triangle inequality.
| https://mathoverflow.net/users/nan | Average distance to roots of unity | Such bound exists for certain $c>0$, but I do not know which $c$ is optimal.
Denote $$f\_n(z):=\frac{1}{n}\sum\_{k=1}^n|z-\omega^k|,$$
then $f$ is 1-Lip as the average of 1-Lip functions.
Denote $y=z/|z|$ (or $y=1$ if $z=0$), then $|y|=1$, $|z-y|=1-|z|$ and $f\_n(z)-|z|\geqslant f\_n(y)-|z-y|-|z|=f\_n(y)-|y|$, th... | 3 | https://mathoverflow.net/users/4312 | 350710 | 148,372 |
https://mathoverflow.net/questions/350712 | 3 | Can there be a set whose Hausdorff dimension gradually changes?
For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, and changes without jumps?
| https://mathoverflow.net/users/10059 | A set whose Hausdorff dimension gradually changes? | I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A\_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of dimension $1/2$ of $[1/2,1]$.To obtain $A\_2$, we then make our sets larger on $[1/4,1/2]$, $[3/4,1]$ and again, we make th... | 19 | https://mathoverflow.net/users/48839 | 350714 | 148,373 |
https://mathoverflow.net/questions/350713 | 8 | On p. 459 of "Modular elliptic curves and Fermat's last theorem", proof of Prop. 1.1, where it says "Since $H^2(G,\mu\_{p^r}) \rightarrow H^2(G,\mu\_{p^s})$ is injective for $r \leq s$...", is there any chance that that's a typo and he really means to say $H^1$ both times rather than $H^2$?
He uses it to conclude tha... | https://mathoverflow.net/users/15482 | Proof of Prop 1.1 in Wiles' "Modular elliptic curves and Fermat's last theorem" | Well, although *there is a typo* (Wiles forgot to close his parenthesis, and wrote $H^2(G,\mu\_{p^r}\to H^2(G,\mu\_{p^s})$ in his proof), his claim is correct. Let, as *ibid.* $F$ be the finite extension of $\mathbb{Q}\_p$ fixed by $G$, so that your arrow can be written $H^2(F,\mu\_{p^r})\to H^2(F,\mu\_{p^s})$. Conside... | 23 | https://mathoverflow.net/users/18238 | 350715 | 148,374 |
https://mathoverflow.net/questions/350711 | 3 | I wanted to know are there any problems in Functional Analysis (FA) that can possibly be successfully tackled by someone like me who does not have any expertise in this area but is only familiar with a few basic topics that you would find in most undergraduate level courses?
I wanted to mention that I did look around... | https://mathoverflow.net/users/151270 | What are some problems for research in functional analysis that can possibly be solved by someone with basic knowledge of the subject? | Since your aim is to show your abilities and get the attention of a professor that would take you in a PhD program, the best and most natural thing would be, getting the problems from the professors themselves. Read their papers, find their open problems that you may like, and then write to them (with discretion) for c... | 5 | https://mathoverflow.net/users/6101 | 350717 | 148,375 |
https://mathoverflow.net/questions/350671 | 2 | $\mu=1+\epsilon$ where $\epsilon>0$ holds.
>
> 1.Is there a good bound for $$T=\frac{\sum\_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
>
>
>
This quantity can be interpreted as $$\sum\_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}\mathbb P(\frac n2+i)$$
where ... | https://mathoverflow.net/users/136553 | Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution | Your main conjecture is not quite correct.
Indeed, for each natural $j$, let $B\_j$ be a random variable (r.v.) with the binomial distribution with parameters $j$ and $1/2$, and let $C\_j$ be an independent copy of $B\_j$. Let also $u:=\sqrt{\mu n\ln n}$. Then for all even natural $n$
$$T/2^n=U\_n:=\sum\_{k\colon\,|... | 2 | https://mathoverflow.net/users/36721 | 350720 | 148,376 |
https://mathoverflow.net/questions/350729 | 1 | Let $G \leq S\_n$ be $2$-transitive other than $A\_n$ and $S\_n$. Is it possible that there exists $N\lhd G$ with $N\neq G$, $N$ transitive and $G/N$ cyclic?
I am interested mostly in the answer when $n$ is large and also when the group $G$ is $3$-transitive.
| https://mathoverflow.net/users/2042 | Is there a 2 transitive finite group with transitive normal subgroup having a cyclic quotient other than $A_n$ and $S_n$? | For $G = \operatorname{Aut}(M\_{22})$ and $N = M\_{22}$, with the action of $M\_{22}$ on $22$ points you have $N \triangleleft G < S\_{22}$. Here both $N$ and $G$ are $3$-transitive, and $G/N \cong C\_2$.
| 4 | https://mathoverflow.net/users/38068 | 350736 | 148,380 |
https://mathoverflow.net/questions/350716 | 1 | This is similar in spirit to [Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution](https://mathoverflow.net/questions/350671/sum-of-squares-of-middle-binomial-sums-or-truncated-mean-of-binomial-coefficie) but gives some total estimates. Though the other one wa... | https://mathoverflow.net/users/136553 | A conjecture on 'truncated joint moments' of binomial coefficients under binomial distribution | The OP has changed the original question. This change invalidates my previous answer. The answer below is to the changed question.
For any natural $m$ and $n$ such that $m/n$ is also natural, the expression now to be bounded is
\begin{equation}
L:=\frac mn\,\log\_2 T,
\end{equation}
where $T$ is just as in the ques... | 1 | https://mathoverflow.net/users/36721 | 350738 | 148,381 |
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