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https://mathoverflow.net/questions/401522 | 2 | I asked this question on Math.StackExchange some time ago and got no responses.
Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type
$$
\Sigma\_A=\{(v\_i)\_{i\in\mathbb{Z}} : A\_{v\_iv\_{i+1}}>0, i\in\mathbb{Z}\}\subset V^{\mathbb{Z}}.
$$
For a stochastic m... | https://mathoverflow.net/users/143604 | Exponential mixing for subshifts | I don't think that it's ever possible to mandate any rate of mixing if you are looking at all measurable sets. Here's a silly example using the full $2$-shift $\{0,1\}^{\mathbb{Z}}$ and the i.i.d. measure $\mu$ with $\mu([0]) = \mu([1]) = 1/2$ (in some sense the "most mixing" SFT and measure).
Define $B$ to be $[0]$,... | 3 | https://mathoverflow.net/users/116357 | 401532 | 164,817 |
https://mathoverflow.net/questions/401339 | 7 | Suppose that $\ell\_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell\_\phi^\*$ is isomorphic to $\ell\_\phi$.
Is $\ell\_\phi$ isomorphic to $\ell\_2$?
| https://mathoverflow.net/users/39421 | Self-dual Orlicz sequence spaces | For a given $1<p$ and $\frac{1}{p}+\frac{1}{q}=1$ you can construct a *universal* Orlicz sequence space $\ell\_M$ so that every Orlicz function $N$ *in between* $p$ and $q$ is equivalent to a function in $E\_M$ which corresponds to the Orlicz subspaces spanned by constant block bases of $\ell\_M$ thus complemented. Suc... | 6 | https://mathoverflow.net/users/3675 | 401534 | 164,818 |
https://mathoverflow.net/questions/401513 | 0 | Let $L>0$ and $\Omega \subset \mathbb{R}^n$ a bounded Lipschitz domain. Define
$$
B\_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|\_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \forall s,t \in [0,1]\}.
$$
I would like to show that, for every fixed $s,t \in [0,1]$, the functional $f \mapsto \int\_\O... | https://mathoverflow.net/users/121671 | Continuity of point evaluation on space of Hölder functions with $L^p$ norm | I believe that I found a way to solve the problem. Let us show that the point evaluation $f \mapsto f\_t$ is a bounded operator from $L^2((0,1) \times \Omega)$ to $L^2(\Omega)$ and the rest follows analogously. Namely, from the given condition, for every $s \in [0,1]$ we have $\|f\_t\|\_{L^2(\Omega)}^2 \leq 2(L^2|t-s| ... | 0 | https://mathoverflow.net/users/121671 | 401536 | 164,819 |
https://mathoverflow.net/questions/401500 | 15 | [This book](https://hott.github.io/book/nightly/hott-online-1287-g1ac9408.pdf) has a section with proofs of the fact $\pi\_1(S^1)=\mathbb Z$ using the univalence axiom. They are a bit too technical for me at the moment to read, but I want to understand the following (vague but conceptual) question:
*What is the role ... | https://mathoverflow.net/users/336697 | Role of univalence in homotopy group calculations | Let's start with an easier question: How do we know that loop is not equal to refl in $\pi\_1(S^1)$?
By the universal property of $S^1$ this is exactly saying that there exists some type $T$ and some $t:T$ and some loop $p:t=t$ which is not trivial. This means I want some type where it's easy to write down paths, but... | 15 | https://mathoverflow.net/users/22 | 401538 | 164,820 |
https://mathoverflow.net/questions/401545 | 4 | Let $(X,\mathcal F,μ,T)$ be a dynamical system (i.e. μ is a probability measure and Τ is μ-preserving) and $\mathcal S\subset\mathcal F$ be a family of sets such that for any $A \in \mathcal F$ and $ε>0$ there exists a $B \in \mathcal S$ with $μ(A\triangle B)\lt ε$.
Assume that $\lim\_{N\to \infty}\frac{1}{N}\sum\_{n... | https://mathoverflow.net/users/336624 | Does the following condition imply ergodicity? | The answer is no. Take $X$ to be a disjoint union $X\_1\sqcup X\_2$ of invariant subsystems of positive measure with $T\restriction\_{X\_1}$ and $T\restriction\_{X\_2}$ ergodic. Define $\mathcal{S}$ as
$\{B\in\mathcal{F}\mid \mu(B\cap X\_1),\mu(B\cap X\_2)>0\}$.
Moreover, assume that both $X\_1$ and $X\_2$ have subsets... | 4 | https://mathoverflow.net/users/128556 | 401549 | 164,827 |
https://mathoverflow.net/questions/401525 | 0 | The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal subgroup of $SU(k)$, so $SU(k)/SO(k)$ may not be a quotient group, or may not be any Lie group. However $SU(k)/SO(k)$ may be ... | https://mathoverflow.net/users/336737 | $SU(k)/SO(k)$ as a manifold, for each positive integer $k$ | $SU(k)/SO(k)$ is the set of real structures on $\mathbb C^k$, i.e. $k$-dimensional real subspaces of $\mathbb C^k$ which generate it as a complex vector space, satisfying two conditions:
(1) The Hermitian form $((z\_1,\dots, z\_n) , (w\_1,\dots, w\_n)) \mapsto z\_1\overline{w}\_1+ z\_2 \overline{w}\_2 + \dots + z\_n ... | 3 | https://mathoverflow.net/users/18060 | 401553 | 164,829 |
https://mathoverflow.net/questions/401544 | 5 | Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k \otimes 1\_H.
$$
| https://mathoverflow.net/users/326091 | Classifying Hopf algebras that admit a single irreducible comodule | The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: [Name for a Hopf algebra whose only grouplike element is the identity?](https://mathoverflow.net/q/400726/85967).
Their classification, is in general an open project:
* ... | 7 | https://mathoverflow.net/users/85967 | 401557 | 164,830 |
https://mathoverflow.net/questions/401470 | 8 | Let $F$ be a smooth rank one foliation on a manifold $M$. Suppose that all leaves of $F$ are compact (that is, circles). Then its leaf space (edit: when additional assumptions are taken) is an orbifold. When the foliation is (transversally) Riemannian, this is proven in the book "Riemannian Foliations" by P. Molino. I ... | https://mathoverflow.net/users/3377 | Smooth rank one foliations with closed leaves | The result is not true without additional assumptions. See [A counterexample to the periodic orbit conjecture](http://www.numdam.org/item/PMIHES_1976__46__5_0/) by Sullivan.
---
**Added later.** The paper by Sullivan linked above exhibits a foliation by circles on a compact manifold of dimension $5$ with non-Haus... | 5 | https://mathoverflow.net/users/605 | 401562 | 164,833 |
https://mathoverflow.net/questions/401512 | 0 | Given a field $k$ with characteristic $p$, let $G$ be a transitive permutation group on $4p$ points. Let $P$ be a Sylow $p$-subgroup of $G$ and $Q\leq P$ is a $p$-subgroup of $P$ of index $p$. Now denote $H:=N\_G(Q)/Q$. Could anyone provide me with an counterexample suth that the dimension of the projective cover $P\_k... | https://mathoverflow.net/users/134942 | Dimension of projective cover of trivial $kG$-module | Here is a Magma calculation that shows that the group ${\rm PSL}(2,11)$ is a counterexample to your question
```
> G := PSL(2,11);
> I := AbsolutelyIrreducibleModules(G,GF(3));
> I;
[
GModule of dimension 1 over GF(3),
GModule of dimension 5 over GF(3),
GModule of dimension 5 over GF(3),
GModule of d... | 4 | https://mathoverflow.net/users/35840 | 401572 | 164,835 |
https://mathoverflow.net/questions/401555 | 5 | Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes\_{\mathbb{Z}},\mathbb{Z})$, semirings are monoids in $(\mathsf{CMon},\otimes\_{\mathbb{N}},\mathbb{N})$). Examples include all rings, but also obje... | https://mathoverflow.net/users/130058 | Examples of $\mathbb{E}_{k}$-semiring spaces | (as a sidenote on terminology, as Jonathan points out in the comments, "spectrum" is not really a good name for your objects precisely because you expect them not to be spectra)
If you look at theorem 8.8 in the GGN paper you cited, you'll see that categories with a nicely behaved tensor product yield examples by tak... | 3 | https://mathoverflow.net/users/102343 | 401588 | 164,840 |
https://mathoverflow.net/questions/401552 | 0 | Let $N$ be the number of degree $d$ monomials in $n$ variables. We can then view each non-zero point in $\mathbb{A}^N\_k$ as a degree $d$ homogeneous form, $k$ an algebraically closed field. Let $X$ be the union of $\mathbf{0}$ and the set of points where the corresponding form is not smooth. I have heard that smoothne... | https://mathoverflow.net/users/84272 | Dimension of the set of singular hypersurfaces | Given a smooth projective variety $X\subset\mathbb{P}^{N-1}$, let $I(X)\subset X\times(\mathbb{P}^{N-1})^\*$ denote the locus of pairs $(x,H)$ where $x\in H$ and $T\_x(X)\subset T\_x(H)$; here we are thinking of $(\mathbb{P}^{N-1})^{\*}$ as the collection of hyperplanes in $\mathbb{P}^{N-1}$.
One can show (using the ... | 2 | https://mathoverflow.net/users/124862 | 401589 | 164,841 |
https://mathoverflow.net/questions/395780 | 5 | I have some problem calculating the value of some specific (but quite common) induced maps I stumbled on while reading some papers on group (co)homology and I would like to know if there are general tricks or routines to avoid calculations over the complexes or, if not, which calculations are customary. But let's first... | https://mathoverflow.net/users/127914 | On induced maps in group homology and Künneth formula | 1. The Kunneth formula is natural for products of homomorphisms. Let $\iota:\{1\}\to H$ be the inclusion of the identity element, and consider $\mathrm{Id}\times \iota:H\times\{1\}\to H\times H$ which can be identified with your map $\alpha$. Applying the Kunneth formula, we see that an element $h\in H\_n(H\times\{1\};... | 2 | https://mathoverflow.net/users/8103 | 401600 | 164,847 |
https://mathoverflow.net/questions/401580 | 4 | **This is a cross-post from [math.stackexchange](https://math.stackexchange.com/questions/4220444/proof-of-derived-tensor-hom-adjunction), since I didn't get any answers there.**
As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-bimodule, we have an isomorph... | https://mathoverflow.net/users/81957 | Proof of derived tensor-hom adjunction |
>
> Does the 'derived adjunction' still hold? If yes, how do you prove it? I would prefer a constructive proof which allows me to understand the map.
>
>
>
Yes. The easiest way to get the derived adjunction is to specialize the ordinary adjunction to appropriate resolutions of the given bimodules.
For example,... | 4 | https://mathoverflow.net/users/402 | 401605 | 164,850 |
https://mathoverflow.net/questions/401357 | 2 | Suppose $R$ is a Noetherian ring and $I$ a nontrivial ideal of $R$, and $A\_0\to B\_0$ a finite faithfully flat lci map of smooth $R\_0 := R/I$-algebras.
We fix a smooth $R$-algebra $A$ lifting $A\_0$ and assume $A$ and $A\_0$ integral.
Assume $R$ is $I$-adically complete.
>
> Does there exist a smooth $R$-alge... | https://mathoverflow.net/users/nan | Lifting of flat lci maps | No, I don't think so.
Lemma: Given $R$, $I$, $R\_0 = R/I$, $A$, $A\_0 = A/I$. Assume $2$ is invertible in $R\_0$. If the answer to the question is "yes" then the image of the map $Pic(A) \to Pic(A\_0)$ contains all $2$-torsion of $Pic(A\_0)$.
Proof: Namely, suppose $L\_0$ is an invertible $A\_0$-module of order $2$... | 1 | https://mathoverflow.net/users/152991 | 401618 | 164,854 |
https://mathoverflow.net/questions/401563 | 4 | In [page 197, Equivalents of the Riemann Hypothesis Vol 1](https://doi.org/10.1017/9781108178228), the following statement caught my eye
>
> There is an editorial comment in [102] that includes an observation by
> the GCHQ Problem Solving Group. They contest that one could replace the
> number 2 in inequality (7.94... | https://mathoverflow.net/users/502093 | Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH | **1.** By an inequality due to Robin (see (2.2) in Lagarias's paper),
$$\sigma(n)\leq e^\gamma n\log\log n+\frac{n}{\log\log n},\qquad n\geq 3.$$
By Lemma 3.1 in Lagarias's paper, we also know that
$$\exp(H\_n)\log(H\_n)\geq e^\gamma n\log\log n,\qquad n\geq 3.$$
Combining these two estimates, we infer that
$$\sigma(n)... | 7 | https://mathoverflow.net/users/11919 | 401620 | 164,855 |
https://mathoverflow.net/questions/401575 | 2 | Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product.
>
> Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the case when either $A$ or $B$ is commutative.)
>
>
>
| https://mathoverflow.net/users/129638 | Primitive ideals of minimal tensor product | I expect that this answer is satisfactory, although it isn't a complete answer. This is really the best result one can hope for.
For a $C^\ast$-algebra $A$ let $Prime(A)$ be the prime ideal space (defined exactly as the primitive ideal space, but with prime (two-sided closed) ideals instead). It is well-known that $P... | 5 | https://mathoverflow.net/users/126109 | 401649 | 164,863 |
https://mathoverflow.net/questions/401652 | 1 | Consider an $h \times w$ binary matrix (a matrix with all entries $a\_{ij}$, $1 \le i \le h$, $1 \le j \le w$, equal to $0$ or $1$) with $w$ even and $h \ge 3$. We know that each row has $\frac{w+2}{2}$ entries equal to $1$ and $\frac{w-2}{2}$ entries equal to $0$. All rows are different. For each $j$, $1 \le j \le w$ ... | https://mathoverflow.net/users/136218 | Counting the equivalence classes of some binary matrices | Equivalently, you are counting unlabelled bicolored graphs that have $h$ red vertices ("rows") and $w$ blue vertices ("columns"); such that red vertices have degree exactly $w/2+1$, and blue vertices have degree at most $h-1$ (each column has at least one zero). And finally, each red vertex has different neighborhood (... | 2 | https://mathoverflow.net/users/171662 | 401657 | 164,865 |
https://mathoverflow.net/questions/401659 | 2 | Let $L$ be an ample line bundle in $\mathbb{P}^n$, with at least $n$ global sections. Choose two sets of $n$ linearly independent global sections of $L$, say $S\_1:=\{D\_1,...,D\_n\}$ and $S\_2:=\{E\_1,....,E\_n\}$, where $D\_i$ and $E\_i$ are effective divisors corresponding to global sections on $L$. Does there exist... | https://mathoverflow.net/users/58203 | Cremona transformations and divisors | In general, I think that the answer is negative. In fact, any birational transformation must preserve the geometric genus of divisors, but this is not fixed in the linear system $|L|$.
For instance, take $n=2$ and $L=\mathcal{O}(3)$. Let $D\_1, \, D\_2$ be smooth plane cubics and let $E\_1$ be a nodal cubic. There is... | 4 | https://mathoverflow.net/users/7460 | 401663 | 164,867 |
https://mathoverflow.net/questions/401662 | 10 | Let $M$ and $N$ be topological spaces.
Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$.
It seems to me that the equivalence
$$\operatorname{Sh}(M) \otimes \operatorname{Sh}(N) = \operatorname{Sh}(M \times N)$$
where $\otimes$ is the standard symmetric monoidal structur... | https://mathoverflow.net/users/41259 | Taking the category of sheaves is symmetric monoidal | 1. Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11.
2. Products in $\mathrm{RTop}$ can be computed as tensor products in $\mathrm{Pr^L}$. This is HA Example 4.8.1.19.
... | 12 | https://mathoverflow.net/users/126667 | 401667 | 164,868 |
https://mathoverflow.net/questions/401626 | 4 | Let $X$ be a projective variety and $A$ and $B$ are two vector bundles on $X$. Let $C\_{\bullet}$ denote the complex of sheaves
$$
0\rightarrow A\rightarrow B\rightarrow 0
$$
Then we have a cup product in hypercohomology
$$
\mathbb H^i(C\_{\bullet})\otimes \mathbb H^j(C\_{\bullet})\rightarrow \mathbb H^{i+j}(C\_{... | https://mathoverflow.net/users/nan | Cup product of hypercohomologies | Although I doubt this counts as a "modern" reference, chapter II section 6 of Godement's *Topologie algébrique...* gives the most detailed account that I know for cup products in sheaf cohomology. This includes explicit formulas in terms of Cech cocycles.He doesn't treat products in hypercohomology, but the formulas ar... | 3 | https://mathoverflow.net/users/4144 | 401668 | 164,869 |
https://mathoverflow.net/questions/401661 | 0 | I am recently reading the pde book of Evans's. I am reading the chapter 9.5.2 which is a topic about the radial symmetry for the solution of a elliptic equation. In the book there is a method of moving planes. That is when a function is symmetric for all directions, we can have that it is radial symmetric. I do not kno... | https://mathoverflow.net/users/241460 | Why the symmetry for all directions implies the radial symmetry? | Notations:
1. given $P$ a hyperplane, let $r\_P$ denote the reflection operation $\mathbb{R}^d\to\mathbb{R}^d$ about the plane $P$.
2. given $R$ a co-dimension 2 affine subspace of $\mathbb{R}^d$, denote by $\rho\_{R,\theta}$ the rotation by angle $\theta$ that fixes the "axis" $R$.
Basic linear algebra/affine geom... | 3 | https://mathoverflow.net/users/3948 | 401672 | 164,870 |
https://mathoverflow.net/questions/401669 | 21 | Suppose a proof came out (and was verified by credible peer review) of the following statement:
>
> There is a $T\_0$ such that for all $t>T\_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$
>
>
>
where $T\_0$ is totally ineffective. What interesting consequences would this partial result have?
---
Of... | https://mathoverflow.net/users/6043 | What are the consequences of an ineffective proof of the Riemann Hypothesis? | As a strengthening of what @KConrad commented, it would imply that the density of nontrivial zeros on the critical line is 100% in each horizontal strip of height 1, which is not useless: this is equivalent to the Lindelöf Hypothesis, which states that $\zeta \left( \frac{1}{2} + i t \right) = \mathcal{O}\_{\varepsilon... | 19 | https://mathoverflow.net/users/88679 | 401678 | 164,873 |
https://mathoverflow.net/questions/259136 | 6 | I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm wondering if there is a generalization or analogue for $f: H \mapsto \tilde{H}$, sufficiently nice operators between Hilbe... | https://mathoverflow.net/users/36886 | Reference Request: Vector-Valued Ito Formula | [Curtain and Falb (1970)](https://core.ac.uk/download/pdf/82209342.pdf) treated this case.
| 1 | https://mathoverflow.net/users/60775 | 401680 | 164,874 |
https://mathoverflow.net/questions/401633 | 12 | Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the Theorems and (some) proofs.
As a student, I feel more comfortable with standard algebraic topology and topological manifolds... | https://mathoverflow.net/users/141114 | Roadmap for L-Theory | I apologize for the self promotion -- I hope the content of this answer can be useful anyway...
---
My favourite introduction to L-theory is Lurie's notes on [Algebraic L-theory and surgery](https://www.math.ias.edu/%7Elurie/287x.html) (warning: aggressively modern).
Working from the ideas in this notes my coau... | 11 | https://mathoverflow.net/users/43054 | 401689 | 164,876 |
https://mathoverflow.net/questions/401688 | 6 | I was talking with a non-mathematician the other week at a workshop about the fact that many mathematicians, like myself, are indexed in the [math genealogy database](https://genealogy.math.ndsu.nodak.edu/). We talked a little about how many people tend to have family trees linking back to a few influential/well-known ... | https://mathoverflow.net/users/118731 | Graphs resembling the math genealogy graph must have concentration in a small number of families? | Precisely this question was the starting point of the Galton - Watson theory of branching processes. To quote the opening paragraph of their 1875 paper *On the Probability of the Extinction of Families*:
>
> The decay of the families of men who occupied conspicuous positions in past times has been a subject of freq... | 13 | https://mathoverflow.net/users/8588 | 401694 | 164,879 |
https://mathoverflow.net/questions/401686 | 6 | Say that a logic $\mathcal{L}$ is **directed** iff whenever $\mathfrak{A}\equiv\_\mathcal{L}\mathfrak{B}$ there is some $\mathfrak{C}$ with $\mathcal{L}$-elementary substructures $\mathfrak{A}'\preccurlyeq\_\mathcal{L}\mathfrak{C}$, $\mathfrak{B}'\preccurlyeq\_\mathcal{L}\mathfrak{C}$ with $\mathfrak{A}\cong\mathfrak{A... | https://mathoverflow.net/users/8133 | Failure of "directedness" for second-order logic? | The answer to the question is yes.
Let $\alpha\_0<\alpha\_1$ be the least ordinals (in the reverse lex order, say) such that $V\_{\alpha\_0}$ and $V\_{\alpha\_1}$ have the same second order theory $T$. Assume towards a contradiction that $V\_{\alpha\_0}$ and $V\_{\alpha\_1}$ are elementarily embeddable into a common ... | 8 | https://mathoverflow.net/users/102684 | 401696 | 164,880 |
https://mathoverflow.net/questions/401675 | 2 | Let $R$ be a commutative ring with identity and $A$ and $B$ be two proper ideals of $R$ such that $A+B=R$ and for each $r^2=r\in R$ we have either $r\not\in A$ or $r-1\not\in B$. How can we prove the existance of two maximal ideals $m\_1$ and $m\_2$ of $R$ such that $A\subseteq m\_1$, $B\subseteq m\_2$ and $\{r\in m\_1... | https://mathoverflow.net/users/338309 | The existence of two maximal ideals with the same set of idempotents | Sketch: First, if $e$ is an idempotent in $A$, show that $B$ can be replaced with $B+Re$, and the hypotheses still holds. Use this to reduce to the case that $A$ and $B$ contain the same idempotents.
Second, if $e$ is an idempotent of $R$ with $e,1-e\notin A$ (and hence also not in $B$) show that we can replace $A$ a... | 2 | https://mathoverflow.net/users/3199 | 401704 | 164,881 |
https://mathoverflow.net/questions/346647 | 1 | Is there an algorithmic way to map the natural numbers to unique k-ary trees?
I am familiar with the work of Tychonievich who created a mapping from integers to binary trees. <https://www.cs.virginia.edu/~lat7h/blog/posts/434.html>
Is there something similar for k-ary trees?
| https://mathoverflow.net/users/148964 | mapping integers to k-ary trees | If you search for "ranking $k$-ary trees" or "ranking $t$-ary trees" you will find several published papers on this. For example:
[This](https://epubs.siam.org/doi/abs/10.1137/0207039?casa_token=L7KsyEFGnGMAAAAA%3AtRJdKQBkcAnPHnV1GQaFhmsMHA1znPEnW942T513tvnhIhfLFposVAp5vi_XK2JmlbSVWBGlDKeS&)
[This](https://epubs.si... | 3 | https://mathoverflow.net/users/9025 | 401710 | 164,882 |
https://mathoverflow.net/questions/401705 | 2 | Let $X$ and $Y$ be smooth algebraic varieties over $\mathbb{C}$. Let $f$ be a continuous map from complex points of $X$ to $Y$. Are there Zariski opens $U$ and $V$ inside $X\times \mathbb{A}^1$ and $Y\times \mathbb{A}^1$ respectively such that $U$ contains $X\times\{0\}$ and $X\times \{1\}$ (similarly $V$ contains $Y\t... | https://mathoverflow.net/users/127776 | Are continuous maps generically homotopic to a regular map? | I think that the answer is negative.
For instance, let $X=Y$ be a smooth curve of genus $g\geq 3$ with $\operatorname{Aut}(X)=\{\mathrm{id} \}$, and take as $f \colon X \to X$ an isotopically non-trivial diffeomorphism.
| 6 | https://mathoverflow.net/users/7460 | 401711 | 164,883 |
https://mathoverflow.net/questions/401656 | 3 | Let $P\subseteq \mathbb{N}$ be the set of primes, and for any integer $n>1$ let $L(n) = \max\{p \in P: p \mid n\}$ be the largest prime divisor of $n$. Moreover, for $n \in \mathbb{N}$ with $n>1$ we let $M(n)$ to be the **median** of the set $$\{L(m)/m : m\in \mathbb{N} \land 1 < m \leq n\}.$$
Does $\lim\_{n\to\infty... | https://mathoverflow.net/users/8628 | Behavior of biggest prime divisor of $n$ as $n$ grows large | The number of prime divisors of $n$ grows typically as $\log \log n$. Suppose $n$ has $k$ prime factors. Now $n/L(n)$ has only $k-1$ prime factors, so
$$
k-1 \approx \log \log \frac{n}{L(n)} = \log \log n + \log\left( 1-\frac{\log L(n)}{\log n}\right) = k + \log\left( 1-\frac{\log L(n)}{\log n}\right)
$$
and hence
$$
\... | 3 | https://mathoverflow.net/users/120914 | 401713 | 164,884 |
https://mathoverflow.net/questions/401683 | 13 | Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup\_{x \in M} \|\text{Rm}(x) \| = \infty$$
I'm looking for these examples because they might provide a counter-example to a problem of mine, but I can't think of any. Obviously such a manifold can't be compact (and therefore it can't be complete ... | https://mathoverflow.net/users/119418 | Are there examples of Einstein manifolds with unbounded curvature? | If you don't care about completeness, here's a fairly simple way to construct such examples: Start with a compact Einstein manifold $(M^n,g)$ with Einstein constant $1$ (i.e., $\mathrm{Ric}(g) = (n{-}1)\,g$) that is not conformally flat. Now take the sine-cone, i.e., $\bigl(M\times(0,\pi),h\bigr)$ where $h = \mathrm{d}... | 12 | https://mathoverflow.net/users/13972 | 401715 | 164,886 |
https://mathoverflow.net/questions/401712 | 0 | For a convergent sequence $(a\_n)\_n \rightarrow a$ consider the exponential series
\begin{equation\*}
\exp\_{(a\_n)\_n}(-x) := \sum\_{n=0}^{\infty} \frac{(-x)^n a\_n}{n!}.
\end{equation\*}
Can there be anything said about the resulting expression, more than convergence? I presume that, since e.g. changing only $a\_0... | https://mathoverflow.net/users/338748 | Exponential Series with a sequence | The Appell Sheffer polynomial formalism can be used to deal with these types of polynomials for $a\_0 = b\_{0,0} = 1$.
In umbral notation for which, e.g., $(c.)^n = c\_n$,
$$ e^{c.t} \; e^{xt} = e^{(c.+ x)t} = e^{t \; p.(x)},$$
with the Appell Sheffer polynomials
$$p\_n(x) = (c.+x)^n = \sum\_{k=0}^n \; \binom{n... | 0 | https://mathoverflow.net/users/12178 | 401718 | 164,887 |
https://mathoverflow.net/questions/401703 | 6 | Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model.
Assume throughout that $\mathcal{F}$ is non-compact.
I am interested in properties of $G$, and how these properties are connected with Poincaré's theorem... | https://mathoverflow.net/users/338619 | Non-compact Dirichlet fundamental domains and free Fuchsian groups | For (1) the answer is "yes". Since the surface has finitely generated fundamental group, there is a finite collection of disjoint embedded bi-infinite geodesics that cut the surface into a collection of ideal (or hyperideal) polygons, each with at most one cone point in its interior. (There is a special case when the c... | 2 | https://mathoverflow.net/users/1650 | 401725 | 164,889 |
https://mathoverflow.net/questions/401726 | 3 | Let $M\_1(\mathbf{x})$ and $M\_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x\_1, \ldots, x\_n]$.
I would like to show that
$$
\{ \mathbf{x} \in \mathbb{A}^n\_{\mathbb{C}}:\dim (\ker M\_1(\mathbf{x}) \cap \ker M\_2(\mathbf{x})) \geq C \},$$
for any $C > 0$,
is 1) an affine ... | https://mathoverflow.net/users/84272 | How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety? | Evidently, $\mathrm{Ker} M\_1\cap \mathrm{Ker} M\_2=\mathrm{Ker}(M\_1,M\_2)$,
where $(M\_1,M\_2)=:M$ is the $2m\times m$ matrix obtained by putting $M\_1,M\_2$
together ($k$-th column of $M$ consists the of the $k$-th column of $M\_1$ followed by the $k$-th column of $M\_2$).
Then $\mathrm{dim}\,\mathrm{Ker} M=2m-r$,... | 6 | https://mathoverflow.net/users/25510 | 401728 | 164,890 |
https://mathoverflow.net/questions/401721 | 0 | Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^\*$-module
$E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that
$$q: E \to E: (f,g) \mapsto (f-g, 0)$$
is not adjointable. This is claimed in Lance's book on Hilbert $C^\*$-modules, p22.
Here is what I tr... | https://mathoverflow.net/users/216007 | Why is $q(f,g) = (f-g,0)$ not adjointable? | This is just a calculation. Continuning your argument, $q^\*(s,t) = q^\*(s,0) = (s\_1,-s\_2)$ for some $s\_1\in A, s\_2\in J$ (I add the minus sign for convenience later). Then
$$ \overline{f} s - \overline{g}s = \langle (f,g), (s\_1,-s\_2) \rangle
= \overline{f} s\_1 - \overline{g}s\_2, $$
for all $f\in A, g\in J$.
Se... | 4 | https://mathoverflow.net/users/406 | 401730 | 164,892 |
https://mathoverflow.net/questions/401732 | 1 | Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi\_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, so $\pi\_1(\widetilde{G}) = 0$.
>
> Question: What are the necessary and sufficient conditions to derive that
>... | https://mathoverflow.net/users/336737 | Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space | Lifting to covers is completely understood; the necessary and sufficient requirement is that the image of the fundamental group lies in the subgroup associated to the cover. So, for example, you are correct that to lift to the universal cover the map on fundamental groups must be trivial. For a reference, see chapter 1... | 3 | https://mathoverflow.net/users/134512 | 401738 | 164,894 |
https://mathoverflow.net/questions/401736 | 7 | Let $\mathcal E$ be a topos and $\varphi$ a statement formulating a property of toposes. There are two ways of checking whether $\mathcal E$ satisfies $\varphi$:
1. Consider the first-order language $L$ of a category. Each topos can be considered as an $L$-structure. So, using standard model-theoretic notions, one ca... | https://mathoverflow.net/users/338895 | Comparing Kripke-Joyal semantics of toposes to model-theoretic satisfaction | As you observe yourself, the question does not quite make sense as $\phi$ in 1. is a formula in the first order language of a category and in 2. $\phi$ is a formula in higher order logic (something like the Mitchel-Benabou language).
The only framework I can think of where this question makes perfect sense is if you ... | 7 | https://mathoverflow.net/users/22131 | 401742 | 164,896 |
https://mathoverflow.net/questions/401761 | 0 | Let $X$ be a separable space and let $x^{\*\*}\in X^{\*\*}$. If $x^{\*\*}(x^{\*}\_{n})\rightarrow 0$ for each weak\*-null sequence $(x^{\*}\_{n})\_{n}$ in $X^{\*}$, is $x^{\*\*}$ in $X$ ?
Thank you!
| https://mathoverflow.net/users/41619 | weak*-null sequences in the dual space of a separable space | Yes, because $(B\_{X^\*},w^\*)$ is compact metrizable. So $x^{\*\*}$ is $w^\*$-continuous at any $x^\*\in B\_{X^\*}$.
| 2 | https://mathoverflow.net/users/76412 | 401762 | 164,897 |
https://mathoverflow.net/questions/401755 | 3 | Functional version of the counting hierarchy is $FCH$. It is an open problem whether there a sequence of $poly(log(n))$ number of $+,\times$ operations utilizing the assistance of $O(1)$ number of constants and arbitrary number of integer variables to compute $n!$.
>
> In terms of output size $n!$ is not even in po... | https://mathoverflow.net/users/10035 | Is factorial computation known to be in a class smaller than $FEXP$? | Yes, $n!\bmod p$ is computable in FCH. More generally, if $f$ is a polynomial-time computable function, then given $n$ and $m$ in binary, we can compute
$$\prod\_{i<n}f(i)\bmod m\tag1$$
in FCH. This follows from the fact that if we are given in *unary* $n$, $m$, and a sequence of numbers $a\_0,\dots,a\_{n-1}$, then we ... | 3 | https://mathoverflow.net/users/12705 | 401767 | 164,899 |
https://mathoverflow.net/questions/238060 | 7 | It is known that there are characterizations of weak compactness in most of classical non-reflexive spaces (e.g. $L\_{1}$-spaces and $C(K)$-spaces). I wonder whether there are characterizations of weak compactness in James space $J$ or its dual $J^{\*}$. Can we establish a criterion for it if there is no? Thank you!
| https://mathoverflow.net/users/41619 | Weak compactness in the James space and its dual | James space is a commutative Banach algebra with pointwise operations. $J^{\ast\ast} = J\oplus\mathbb{C}$ is just $J$ with a unit attached [https://doi.org/10.4153/CJM-1980-083-7].
Second, since $J$ contains no copy of $\ell^1$, every bounded sequence $(x\_n)$ in $J$ has a weakly Cauchy subsequence, say $(u\_n)$, tha... | 0 | https://mathoverflow.net/users/164350 | 401770 | 164,901 |
https://mathoverflow.net/questions/401775 | 12 | Let $G$ be a real Lie group.
What conditions must $G$ satisfy so that the following is true:
>
> For any finite group $\Gamma$ there exist finitely many conjugacy classes of subgroups of $G$ that are isomorphic to $\Gamma$.
>
>
>
I believe that for $G=GL(n,\mathbb{R})$ this is true because:
subgroups of $GL(n,... | https://mathoverflow.net/users/164084 | Which Lie groups have finitely many conjugacy classes of subgroups of fixed isomorphism type? | A natural condition is that $G$ has finitely many connected components.
One can easily reduce this case to the connected group case, and then to the compact group case, as all [maximal compact subgroups](https://en.wikipedia.org/wiki/Maximal_compact_subgroup) in a connected Lie group are conjugated.
Then the representa... | 13 | https://mathoverflow.net/users/89334 | 401777 | 164,904 |
https://mathoverflow.net/questions/401774 | 0 | **Motivation.** My eldest son starts school tomorrow. His class is split in two groups of $10$ students each. From time to time, the groups are rearranged. I wondered how many rearrangements are needed until every student has been in the same group with every other student at least once. This led to a more general ques... | https://mathoverflow.net/users/8628 | Putting $\omega$ in two boxes | Let $n=3$ and partition $\omega$ into three infinite sets, $X\_0, X\_1, X\_2$. The functions $f\_0,f\_1,f\_2$ defined by $$f\_i(x) = \begin{cases} 1 & \mbox{if $x$ belongs to } X\_i \\ 0 & \mbox{otherwise} \end{cases}$$ are fair functions, and for all $a, b \in \omega$ there is $i \in \{0,1,2\}$ such that $f\_i(a) = f\... | 6 | https://mathoverflow.net/users/8049 | 401781 | 164,905 |
https://mathoverflow.net/questions/401571 | 5 | Let $X$ be a compact complex manifold in Fujiki class $\mathcal C$, that is bimeromorphic to a compact Kähler manifold, let $T$ be a Kähler current of $X$, then we have the De Rham class $[T]\in H^{1,1}(X,\mathbb R)$, pick a smooth form $\tau$ in the same class as $[T]$, then does the wedge map $\tau^q\wedge :H^0(X,\Om... | https://mathoverflow.net/users/99826 | Hard Lefschetz theorem for non-Kähler manifolds | I think that the answer to your question is no.
In order to construct a counter-example, the idea is the following: let $T$ be a Kahler current, and $E$ a $d$-closed positive current on a compact complex $3$-fold $X$ such that $T^3>0$ and $E^3<0$. Then, for $t>0$, the current $S\_t:=T+tE$ is a Kahler current. Under t... | 1 | https://mathoverflow.net/users/48958 | 401799 | 164,909 |
https://mathoverflow.net/questions/401794 | 6 | I have been reading about Puiseux series in the context of the Newton–Puiseux algorithm for resolution of singularities of algebraic curves in $\mathbb{C}^2$. Given a curve $f(x,y)=0$ with $f$ a convergent power series and such that the curve has a singularity at the origin, the algorithm produces convergent Puiseux se... | https://mathoverflow.net/users/206706 | How to treat Puiseux series as functions? | One standard way to bring actual functions in the picture is the following formulation of the existence + convergence results of Puiseux roots: "Given an irreducible power series $f \in \mathbb{C}[[x, y]]$ which is a [Weirstrass polynomial](https://en.wikipedia.org/wiki/Weierstrass_preparation_theorem) in $y$ of degree... | 10 | https://mathoverflow.net/users/1508 | 401800 | 164,910 |
https://mathoverflow.net/questions/401789 | 7 | A well-known conjecture of Berge and Fulkerson says that every bridgeless cubic graph has a collection of six perfect matchings that together cover every edge exactly twice. Is this still open for bridgeless cubic planar graphs?
| https://mathoverflow.net/users/148974 | Berge-Fulkerson conjecture --- the planar case | The Berge-Fulkerson conjecture holds for planar graphs. Here is a proof.
Let $G$ be a bridgeless cubic planar graph. The dual graph $G^\*$ is a triangulation. By the Four Colour Theorem, $G^\*$ has a 4-colouring $c$. We will use $\mathbb{Z}\_2 \times \mathbb{Z}\_2$ as the set of colours for $c$. Now for each edge $e ... | 12 | https://mathoverflow.net/users/2233 | 401810 | 164,913 |
https://mathoverflow.net/questions/401606 | 2 | Let $G$ be an algebraic group acting on $X$ (a finite type scheme on $k$).
A $G$-invariant $k$-morphism $f : X \rightarrow S$ is a map such that the following commute:
$\require{AMScd}$
\begin{CD}
G \times\_k X @>\rho>> X\\
@V \pi\_2 V V @VV f V\\
X @>>f> S
\end{CD}
Where $\rho$ is the action map and $\pi\_2$ is th... | https://mathoverflow.net/users/175944 | $G$-invariant morphism and coarse moduli spaces | This is not true without the assumption that $G$ is reduced. Here is a counterexample.
Fix a prime number $p$ and any field $k$ of characteristic $p$. We define the nonreduced algebraic subgroup $\alpha\_p \subset \mathbb{G}\_a$ by $\alpha\_p = Spec(k[t]/(t^p))$ (here $\mathbb{G}\_a$ denotes the additive group over $... | 3 | https://mathoverflow.net/users/339730 | 401811 | 164,914 |
https://mathoverflow.net/questions/401004 | 7 | Let $n \geq 1$ be an integer and consider the symmetric function
$$D\_n = \sum\_{d|n} p\_d^{n/d},$$
where $p\_{d}$ are the power-sum symmetric functions.
It can be checked up to $n=35$ that the symmetric function $D\_n$ is Schur-positive.
The multiplicity of the Schur function $s\_n$ in $D\_n$ is given by the n... | https://mathoverflow.net/users/10881 | About the sum of rectangular power sums | Talking to Sheila Sundram, as suggested in the comments, was a good idea. After some conversation, the following proof became apparent. I don't know of any proof already in the literature.
Let $g$ be an $n$-cycle in $S\_n$. Since the inverse Frobenius characteristic of the given symmetric function is supported only o... | 3 | https://mathoverflow.net/users/36466 | 401815 | 164,916 |
https://mathoverflow.net/questions/401822 | 5 | Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all $x \in (a, b)$. Denote by $\mathcal{Z} = \{x \in \mathbb{R}, f(x) = 0\}$ the zero point set of $f$. Assume $\mathcal{Z}... | https://mathoverflow.net/users/114951 | Zero points of a smooth function on $\mathbb{R}$ | The answer is yes.
We can make a smooth function whose zero set is the Cantor set. Simply place a smooth function on each middle third segment as you build the Cantor set, so that on each of these segments, the function arches from 0 up to a height that vanishes quickly as the segments become small, and then back dow... | 11 | https://mathoverflow.net/users/1946 | 401823 | 164,918 |
https://mathoverflow.net/questions/400948 | 9 | In the section 3.2 of *Sheaves in Topology* by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f\_!$ has finite cohomological dimension, then the following holds:
>
> **(Verdier duality, ... | https://mathoverflow.net/users/131975 | Verdier duality under more general conditions | The status of the following answer is a bit speculative, unfortunately. To be precise, I believe that all of what I say below is true; I also believe that there is no proof in the literature of some of the things I state below, and I am not enough of an expert to supply those proofs. Nevertheless I'm putting it out the... | 8 | https://mathoverflow.net/users/1310 | 401863 | 164,933 |
https://mathoverflow.net/questions/401851 | 4 | This is a follow-up question to an [older question](https://mathoverflow.net/questions/401774/putting-omega-in-two-boxes).
Let $\alpha \in \big(\omega\cup\{\omega\}\big) \setminus \{0,1\}$ be an ordinal. We say that a function $f: \omega \to \alpha$ is *fair* if $$|f^{-1}(\{j\})| = \aleph\_0$$ for all $j\in\alpha$.
... | https://mathoverflow.net/users/8628 | Putting $\omega$ into $\alpha$ boxes where $\alpha \in \big(\omega\cup\{\omega\}\big)\setminus\{0,1\}$ | $B\_\alpha = 3$ for every $\alpha \in (\omega \cup \{\omega\}) \setminus \{0,1\}$. Here is a construction that shows $B\_\alpha \le 3$.
1. Let $g : \mathbb{N} \to \alpha$ be any function such that $g^{-1}(\{j\})$ is infinite for all $j \in \alpha$.
2. Let $h$ be any bijection from $\omega$ to $\{0,1,2\} \times \mathb... | 7 | https://mathoverflow.net/users/8049 | 401874 | 164,938 |
https://mathoverflow.net/questions/351800 | 8 | Consider the closure $K \subset \overline{\mathcal{M}}\_{0,n}(\mathbb{P}^1, n)$ in the stable maps space of the locus $K\_0$ of maps $(f : C \to \mathbb{P}^1, p\_1, \ldots, p\_n)$ where $C \cong \mathbb{P}^1$ is smooth and the set of marked points is exactly the preimage of $\infty \in \mathbb{P}^1$; $\{p\_1, \ldots, p... | https://mathoverflow.net/users/12402 | The closure of a locus in $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$ | It turns out that the answer is yes and these conditions are sufficient.
This question was answered in greater generality by Gathmann ([Absolute and relative Gromov-Witten invariants of very ample hypersurfaces.](https://arxiv.org/abs/math/9908054) *Duke
Math. J.*, 2002, Proposition 1.14) building off of earlier work... | 1 | https://mathoverflow.net/users/12402 | 401888 | 164,942 |
https://mathoverflow.net/questions/401899 | 10 | GCH for alephs means the statement that, for any aleph $\kappa$, there are no cardinals $\mathfrak{r}$ such that $\kappa<\mathfrak{r}<2^\kappa$.
Does GCH for alephs imply the axiom of choice?
Remark. Lindenbaum and Tarski assert in ``Communication sur les recherches de la théorie des ensembles'' without proof that ... | https://mathoverflow.net/users/101817 | Does GCH for alephs imply the axiom of choice? | The answer is positive, yes.
Note that $2^\kappa\leq 2^{\kappa^+}$, and therefore $\kappa^+\leq\kappa^++2^\kappa\leq 2^{\kappa^+}$. So either $2^\kappa=2^{\kappa^+}$ or $2^\kappa=\kappa^+$.
In the first case $\kappa^+$, $\kappa<\kappa^+<2^\kappa$ is impossible. So the latter case holds.
Therefore the power set of... | 9 | https://mathoverflow.net/users/7206 | 401902 | 164,948 |
https://mathoverflow.net/questions/401909 | 2 | Working with the level set method introduced by *Osher & Sethian* in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the zero level set is continuous with respect to the level set function. The continuity of the area is true (see here [Lebesgue... | https://mathoverflow.net/users/61629 | Continuity of the perimeter of level sets w.r.t. level function | As it is written it's not true, by trivial reasons: take $\Omega\subset \mathbb{R}^2$ the unit open disk, $\phi(x):=1-|x|^2$ and $\phi\_n(x):=\phi(x)+\frac1n$ for $x\in\overline\Omega$. So $\{x\in\overline\Omega:\phi(x)=0\}$ is the unit circle and $\{x\in\overline\Omega:\phi\_n(x)=0\}$ is empty. The measure of *interio... | 6 | https://mathoverflow.net/users/6101 | 401916 | 164,953 |
https://mathoverflow.net/questions/401760 | 14 | The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^\* = x- \partial\_x \text{ and }a = x+\partial\_x.$$
Fix two numbers $\alpha,\beta \in \mathbb R.$
Can we explicitly compute the spectrum of
$$H = \begin{pmatrix} aa^\* + \alpha^2 & \alpha a+ \... | https://mathoverflow.net/users/150549 | Spectrum of matrix involving quantum harmonic oscillator | The Hamiltonian
$$H=\begin{pmatrix}
\alpha^2+a^\ast a&\alpha a+\beta a^\ast\\
\alpha a^\ast+\beta a&\beta^2+a^\ast a
\end{pmatrix}
$$
is known in the physics literature as the *anisotropic Rabi Hamiltonian*. (In the most general case there is an additional term $\Delta\sigma\_z$.) I give some pointers to the literature... | 14 | https://mathoverflow.net/users/11260 | 401927 | 164,955 |
https://mathoverflow.net/questions/401921 | 3 | I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
>
> Consider the following ordinary differential equation $$\dot{\theta}\_1=\omega\_1 \ \ \ \dot{\theta}\_2=\omega\_2 \ \ \ \ \theta\_i\in (0,2\pi] \ \ \forall i\in\{ 1,... | https://mathoverflow.net/users/13904 | Searching for the proof of a certain claim in Arnold's ODE book from 1992 | See at p. 163 of this PDF link:
[https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations\_Red.pdf](https://eclass.uoa.gr/modules/document/file.php/PHYS289/%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CE%B1/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_R... | 7 | https://mathoverflow.net/users/167834 | 401928 | 164,956 |
https://mathoverflow.net/questions/401269 | 15 | I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.
E.g.
* "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.
* "Every set $S$ of natural number... | https://mathoverflow.net/users/170446 | Open problem in analysis with just one quantifier? | The
[Littlewood conjecture](https://en.wikipedia.org/wiki/Littlewood_conjecture) is an example that meets all my requirements.
It is easy to state and obviously $\Pi^1\_1$. Furthermore [this comment by Christian Reiher](https://gowers.wordpress.com/2009/11/17/problems-related-to-littlewoods-conjecture-2/#comment-4395... | 4 | https://mathoverflow.net/users/170446 | 401930 | 164,957 |
https://mathoverflow.net/questions/401924 | 17 | $\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \neq e$ there is an irreducible character $\chi$ such that $\chi(g) = 0$.
I have been led to consider such character tab... | https://mathoverflow.net/users/34538 | The finite groups with a zero entry in each column of its character table (except the first one) | Partial answer: the finite group $G$ is clearly in this class if it has a $p$-block of defect zero for every prime $p$ which divides $|G|$. This is a sufficient condition which may not be necessary. No non-trivial finite solvable group satisfies the sufficient condition. Most (but not all) finite simple groups satisfy ... | 15 | https://mathoverflow.net/users/14450 | 401932 | 164,959 |
https://mathoverflow.net/questions/401925 | 1 | This question is related to [this one](https://mathoverflow.net/questions/401910/effect-of-snowflaking-on-metric-capacity). Let $(X,d)$ be a metric space, let $\epsilon\in [0,1)$ and consider the snowflake $(X,d^{1-\epsilon})$. Suppose that $(X,d)$ has a finite doubling constant, where the doubling constant $\lambda\_{... | https://mathoverflow.net/users/36886 | Effect of snowflaking on doubling constants | Suppose $(X,d)$ has doubling constant $\lambda$. This means that for every $r$, every $d$-ball of radius $2r$ can be covered by $\lambda$ many $d$-balls of radius $r$. With respect to the metric $d^\alpha$, this means that every $d^\alpha$-ball of radius $(2r)^\alpha$ can be covered by $\lambda$ many $d^\alpha$-balls o... | 4 | https://mathoverflow.net/users/23141 | 401937 | 164,962 |
https://mathoverflow.net/questions/401890 | 0 | In the paper [Dissimilarity in Graph-Based Semi-Supervised Classification](http://pages.cs.wisc.edu/%7Eswright/papers/dissim_final.pdf), there are few things I could not understand.
Given that $x\_1, x\_2,..., x\_n$ are the vector representation of $n$ items, $f : X \rightarrow \mathbb{R}$ is the discriminant function,... | https://mathoverflow.net/users/168789 | Why is this function convex? | Call the sum $S(f)$. Let $0<b<1$, and let $f$ and $g$ be two functions mapping $X$ to $\Bbb R$. Then
$$
S(bf+(1-b)g)=b^2S(f)+(1-b)^2S(g)+b(1-b)\sum\_{i,j}w\_{i,j} [f(x\_i)-f(x\_j)]\cdot[g(x\_i-g(x\_j)].
$$
By Cauchy-Schwarz,
$$
\sum\_{i,j}w\_{i,j} [f(x\_i)-f(x\_j)]\cdot[g(x\_i-g(x\_j)]\le 2\sqrt{S(f)S(g)}.
$$
(The non-... | 1 | https://mathoverflow.net/users/42851 | 401939 | 164,963 |
https://mathoverflow.net/questions/401934 | 15 | Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function
$$
\Phi: \quad G \to (0,\infty), \quad x \mapsto \mu(U x U).
$$
My question is:
>
> Can one give a natural characterization of the groups for which this function ... | https://mathoverflow.net/users/59219 | For what LCH groups is the Haar measure $\mu(U x U)$ bounded? | Your conjecture is correct.
Suppose that we have a compact unit neighbourhood $U$ such that $\mu(UxU) \ll 1$ for all $x$. As you have already noted, we can take $\mu$ to be unimodular, and the choice of neighbourhood is not relevant, so we may assume without loss of generality that $U$ is symmetric: $U^{-1} = U$. (Th... | 14 | https://mathoverflow.net/users/766 | 401943 | 164,964 |
https://mathoverflow.net/questions/401933 | 5 | Is there an abelian variety $A/\mathbb R$ of dimension $n$ such that $End\_{\mathbb R}(A)\otimes \mathbb Q$ contains a field $K$ of degree $[K:\mathbb Q]=2n$? ($End\_{\mathbb R}(A)$ is the ring of $\mathbb R$-endomorphisms of $A$)
| https://mathoverflow.net/users/197736 | Abelian variety with CM defined over real numbers | No.
Assume for contradiction that such an $A$ exists. First look at the singular cohomology $H^1(A\_{\mathbb C}, \mathbb Q)$, which admits an action of $K$ and so is a $K$-vector space. It has dimension $2n$ over $\mathbb Q$ and so is a 1-dimensional $K$-vector space.
Tensoring with $\mathbb C$, we see that $H^1(A\... | 10 | https://mathoverflow.net/users/18060 | 401945 | 164,965 |
https://mathoverflow.net/questions/401904 | 8 | $\DeclareMathOperator\SO{SO}$At the very begining of [Akbulut and Kalafat - Algebraic topology of $G\_2$ manifolds](https://arxiv.org/abs/1308.2263v2), the authors stated that there is a "canonical fibration" for $G\_2$ of the form
$$G\_2\to \SO(7)\to \mathbb{R}P^7,$$
where the map $G\_2\to \SO(7)$ is obtained by regar... | https://mathoverflow.net/users/100553 | The "canonical fibration" for the Lie group $G_2$ | In *Spinors and Calibrations* by F. Reese Harvey, you can find proof (p. 283) of $S^7 \simeq Spin(7)/G\_2.$ It takes the same approach as Bryant's notes mentioned in the comments but it is much more detailed in this case.
The main idea is to write down the spinor representation of $Spin(7)$ using octonionic multiplic... | 5 | https://mathoverflow.net/users/6818 | 401947 | 164,966 |
https://mathoverflow.net/questions/401948 | 2 | Let $(I,\leq)$ be a directed set, that is $\leq$ is reflexive and transitive and for every $a,b\in I$ we find $c\in I$ such that $a,b\leq c$. Now consider the set $M$ consisting of all maps $\sigma:I\longrightarrow I$ such that $a \leq b$ implies $\sigma a≤ \sigma b$. We define a reflexive and transitive order on $M$ a... | https://mathoverflow.net/users/145920 | Is the set of "endomorphisms" of a directed set again a directed set? | I claim that the answer is no. This answer is a generalization of the answers to [this previous question](https://mathoverflow.net/q/190464/22277).
If $X,Y$ are posets, then let $\text{Hom}(X,Y)$ denote the set of all mappings $f:X\rightarrow Y$ where if $x\_{1},x\_{2}\in X$ and $x\_{1}\leq x\_{2}$, then $f(x\_{1})\l... | 3 | https://mathoverflow.net/users/22277 | 401953 | 164,968 |
https://mathoverflow.net/questions/401960 | 7 | I would like to know if anyone has an electronic copy of the following paper:
>
> **"Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Översigt Vetensk. Akad. Handlingar 58, 91–105 (1901)"**
>
>
>
In my search, the best result I found was the (possible) statement of the main result of... | https://mathoverflow.net/users/115618 | Looking for an electronic copy of Holmgren's old paper | The full text of the article can be found scanned [here](https://www.biodiversitylibrary.org/page/32299061#page/103/mode/1up).
| 12 | https://mathoverflow.net/users/120914 | 401963 | 164,971 |
https://mathoverflow.net/questions/401957 | 0 | In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C\_n$ depending only on $n$ such that $R \leq C\_n \|\text{Rm}\|$. It's not clear whether this inequality should actually be $R(t) \leq C\_n \|\text{Rm}(... | https://mathoverflow.net/users/119418 | Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow | **Remark**
As @OthisChodosh and @WillieWong have pointed out, the existence of a constant $C\_n$ that depends only on the dimension can be proved using only elementary linear algebra. I might as well provide the details. Although I like my first answer, it was overkill.
**Simpler answer**
First, recall that if $V... | 6 | https://mathoverflow.net/users/613 | 401973 | 164,975 |
https://mathoverflow.net/questions/401971 | 6 | On the nlab [page](https://ncatlab.org/nlab/show/Chevalley-Eilenberg+algebra#DefForLieAlg) for Chevalley–Eilenberg algebras, it defines $\operatorname{CE}(\mathfrak g)$ for $\mathfrak g$ finite dimensional, and then says "This has a more or less evident generalization to infinite-dimensional Lie algebras", and provides... | https://mathoverflow.net/users/163483 | CE(g) for g infinite dimensional | A definition that always works and does agree with that one in the finite-dimensional case is the following: put
$$
C^k(\mathfrak{g})=({\Lambda}^k\mathfrak{g})^\*=\operatorname{Hom}({\Lambda}^k\mathfrak{g}, \mathbb{F}).
$$
(Here $\mathbb{F}$ is the ground field, of course.)
The differential is given by the formula
$$
... | 6 | https://mathoverflow.net/users/1306 | 401979 | 164,977 |
https://mathoverflow.net/questions/401962 | 3 | I was reading this paper ([arXiv link](https://arxiv.org/abs/0912.3506))
>
> On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations
> Guy Barles (LMPT), Alessio Porretta, Thierry Wilfried Tabet Tchamba (LMPT)
>
>
>
The authors claimed that in general... | https://mathoverflow.net/users/124759 | A regularity estimate for second-derivative | The main tools are the following elliptic regularity results: assume that $u \in W^{2,p}\_{loc}$ and let $f=\Delta u$.
a) If $f \in L^q\_{loc}$ with $q>p$, then $u \in W^{2,q}\_{loc}$;
b) If $f \in W^{1,p}\_{loc}$, then $u \in W^{3,p}\_{loc}$.
With this in mind, first note that $|Du|^m \in W^{1,p}\_{loc}$, becaus... | 1 | https://mathoverflow.net/users/150653 | 401982 | 164,978 |
https://mathoverflow.net/questions/401530 | 3 | Jim Lawrence has a very important paper on the topic of valuations on polyhedra called "*Rational-function-valued valuations on polyhedra*", published in the DIMACS volume *Discrete and computational geometry* of the AMS.
**Does anyone have a scanned copy of this article that can be shared?**
Google books has a pre... | https://mathoverflow.net/users/109085 | Request for an article by Jim Lawrence | I have a scan of the chapter and can email it, just let me know the address.
The reference list (missing from the Google books preview) is here:
| 4 | https://mathoverflow.net/users/11260 | 401990 | 164,981 |
https://mathoverflow.net/questions/401954 | 2 | The following is an excerpt from Marco Gualtieri's [thesis](https://arxiv.org/abs/math/0401221)
>
> A central theme of this thesis is that classical geometrical
> structures which appear, at first glance, to be completely different
> in nature, may actually be special cases of a more general unifying
> structure. O... | https://mathoverflow.net/users/118688 | Does "integrability condition" have a specific meaning or is it used in a casual way? | It turns out there is a well defined notion of integrability of a G-structure on a manifold.
Thanks to the user Thomas Rot who has given the reference [Linear $G$-structures by examples](https://www.few.vu.nl/%7Epasquott/course16.pdf)
Definition $2.1$ is that of $G$-structure on a manifold M.
It defines the notio... | 2 | https://mathoverflow.net/users/118688 | 401994 | 164,982 |
https://mathoverflow.net/questions/401952 | 3 | Let $M$ be a smooth manifold and $E \to M$ a vector bundle.
I'm reading a text which says:
>
> Recall that there is a one-to-one correspondence between:
>
>
> 1. linear vector fields on $E$, and
> 2. linear operators $D : \Gamma(E) \to \Gamma(E)$ such that there exists a vector field $X$ on $M$ such that
> $$D(... | https://mathoverflow.net/users/341298 | Linear vector fields $\leftrightarrow$ certain differential operators | Using google (linear vector field on vecot bundle), I found the following reasonable definition for a linear vector field $\hat X$ on a vector bundle $E\to M:$ it is a vector field $\hat X\in\mathcal X(E)$ which is a vector bundle morphism $\hat X\colon E\to TE$ along a map $X\colon M\to TM$ given by a vector field on ... | 4 | https://mathoverflow.net/users/4572 | 401999 | 164,983 |
https://mathoverflow.net/questions/402006 | 2 | We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as
$$
\mathbb G(k,V):=\{W \subset V : \dim W=k\}.
$$
Then consider a non-degenerate quadratic form on $V$, we write $q: V \times V \to \mathbb C$ if it is symmetric, $\omega: V \times V \to \mathbb ... | https://mathoverflow.net/users/147236 | Tangent bundle for orthogonal and isotropic Grassmannians | The tangent bundle to the orthogonal Grassmannian fits into an exact sequence
$$
0 \to T\_{\mathrm{OG}(k,V)} \to \mathcal{S}^\vee \otimes \mathcal{Q} \to S^2\mathcal{S}^\vee \to 0.
$$
Taking into account an exact sequence
$$
0 \to \mathcal{S}^\perp/\mathcal{S} \to \mathcal{Q} \to \mathcal{S}^\vee \to 0
$$
one can obtai... | 4 | https://mathoverflow.net/users/4428 | 402007 | 164,986 |
https://mathoverflow.net/questions/402010 | 16 | In this question [Sizes of bases of vector spaces without the axiom of choice](https://mathoverflow.net/questions/93242/sizes-of-bases-of-vector-spaces-without-the-axiom-of-choice?newreg=1a6b5b1d756a41139ae970cb60c32e6a) it is said that "It is consistent [with ZF] that there are vector spaces that have two bases with c... | https://mathoverflow.net/users/341908 | Examples of vector spaces with bases of different cardinalities | This is not a very thoroughly studied problem. So to start from the end, there is no standard procedure for this sort of construction. We know of one, it can maybe be adapted slightly to get a mildly more general result, but it's not something like "let's add a new vector space without a basis" or "let's add an amorpho... | 15 | https://mathoverflow.net/users/7206 | 402012 | 164,989 |
https://mathoverflow.net/questions/402002 | 5 | Denote $A$ $-$ set of positive numbers with only prime factors of the form $4k+1$ and
$B$ $-$ set of positive numbers that can be represented as sum of two squares. $A$ is a subset of $B$ and [there is](https://math.stackexchange.com/questions/2376879/does-the-interval-x-x-10-x1-4-contain-a-sum-of-two-squares) upper bo... | https://mathoverflow.net/users/37289 | Upper bound for maximal gap between consecutive numbers consisting only $4k+1$ primes | One can prove a bound comparable to the $O(n^{1/4})$ bound for $B$. To derive it, notice that any odd number of the form $a^2+b^2$ with $(a,b)=1$ lies in $A$. Now, for a given large number $N$, choose the largest even $a$ such that $a^2<N$. Then, of course $N-a^2=O(\sqrt{N})$. We now want to choose $b$ coprime to $a$ s... | 6 | https://mathoverflow.net/users/101078 | 402013 | 164,990 |
https://mathoverflow.net/questions/396809 | 0 | Problem
=======
I was wondering if there are any theoretical results that tackle the following problem:
>
> Construct the following matrices $\mathbf{\mathcal{S}\_{1}},\mathbf{\mathcal{S}\_{2}},\ldots,\mathbf{\mathcal{S}\_{p}}$ all of size $m\times n$ with $m\geqslant n$. The goal is to compute:
> $$
> \mathbf{\m... | https://mathoverflow.net/users/313939 | Using QR or SVD to sum up finite number of matrices | If your matrices happen to be low-rank, you might want to consider a simultaneous low-rank approximation of your matrices (as in [Inoue and Urahama - Equivalence of Non-Iterative Algorithms for Simultaneous Low Rank Approximations of Matrices](https://doi.org/10.1109/CVPR.2006.112)), i.e. you compute
$$
\min\_{A,M\_i,B... | 0 | https://mathoverflow.net/users/135810 | 402014 | 164,991 |
https://mathoverflow.net/questions/401985 | 11 | Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (potentially cellular) map)?
If this setting is to broad, I'm specifically interested in the case, where $M$ is a surfac... | https://mathoverflow.net/users/89741 | Can we embed a closed manifold into a homotopy equivalent CW complex? | Pick a torus, and add two discs along a meridian and a longitude. You get a 2-complex homotopic to a sphere that does not contain a sphere. This generalises easily to any genus by picking a genus-$g$ surface.
More generally, a finite 2-complex contains finitely many surfaces, and there are some moves (like the Matvee... | 15 | https://mathoverflow.net/users/6205 | 402027 | 164,997 |
https://mathoverflow.net/questions/402034 | 8 | Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum\_{ 1 \leq i < j \leq n} a\_ia\_j = 0 \mod p$$ where $a\_1, a\_2, \dots a\_n$ are chosen uniformly from the set $S = \{-1, 1\}$. Does this sum equidistribute mod $p$ as $n$ goes to infinity? What would be the ... | https://mathoverflow.net/users/7894 | Question about estimating random symmetric sums modulo p | The condition
$$\sum\_{ 1 \leq i < j \leq n} a\_ia\_j \equiv 0 \pmod p$$
is equivalent to
$$\left(\sum\_{ 1 \leq i\leq n} a\_i\right)^2 \equiv n \pmod p.$$
So a necessary condition is that $n$ is a quadratic residue modulo $p$ (including the zero residue). If $n$ is divisible by $p$, then the above condition says that ... | 14 | https://mathoverflow.net/users/11919 | 402038 | 165,002 |
https://mathoverflow.net/questions/402043 | 3 | $\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\Gamma$-spaces and spectra are quite close in spirit:
* **$\Gamma$-spaces** are pointed functors $X\colon(\Gamma^\op,\l0\r)... | https://mathoverflow.net/users/130058 | Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces |
>
> $\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$can we view nonconnectivity as arising from enlarging Segal's category $\Gamma^{\mathsf{op}}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathsf{FinSets}\_\*)$ of finite pointed sets... | 1 | https://mathoverflow.net/users/402 | 402050 | 165,004 |
https://mathoverflow.net/questions/401967 | 12 | This question is about logical complexity of sentences in third order arithmetic. See [Wikipedia](https://en.wikipedia.org/wiki/Arithmetical_hierarchy) for the basic concepts.
Recall that the Continuum Hypothesis is a $\Sigma^2\_1$ sentence. Furthermore (loosely speaking) it can't be reduced to a $\Pi^2\_1$ sentence,... | https://mathoverflow.net/users/170446 | Example of a $\Pi^2_2$ sentence? | The Suslin hypothesis is $\Pi^2\_2,$ and $T = ZFC + GCH + LC$ (LC an arbitrary large cardinal axiom) does not prove it to be equivalent to any $\Sigma^2\_2$ sentence. Suppose toward contradiction $T$ proves SH to be equivalent to $\exists A \subset \mathbb{R} \varphi(A),$ where $\varphi$ is $\Pi^2\_1.$ Assume $V \model... | 9 | https://mathoverflow.net/users/109573 | 402053 | 165,006 |
https://mathoverflow.net/questions/402056 | 3 | Assume $\Omega$ is a smoothly bounded open domain in $ \mathbb R^n$ and $B$ is an open ball of equal volume. Let $\Phi:\partial\Omega\to \partial B$ be a diffeomorphism. Is it true that there exists a volume preserving diffeomorphism $\tilde{\Phi}:\overline{\Omega}\to\overline{B}$ such that $\tilde{\Phi}\vert\_{\partia... | https://mathoverflow.net/users/79956 | Extension of volume preserving diffeomorphism | Let $\Omega$ be a submanifold of $\mathbb R^n$ with boundary $\partial\Omega$ diffeomorphic to $\partial B$. As far as I know, it is not obvious that $\overline\Omega$ should be diffeomorphic to $\overline B$, and my guess would be that this is actually false in all generality. From what I understand, though, it is tru... | 5 | https://mathoverflow.net/users/129074 | 402058 | 165,008 |
https://mathoverflow.net/questions/332091 | 4 | Let $(A,A^+)$ be a sheafy Tate-Huber pair, and let $X=\operatorname{Spa}(A,A^+)$. It is well-known that $H^i(X,\mathcal{O}\_X)=0$ for $i>0$. I assume it is generally not true that $H^i(X,\mathcal{O}\_X^+)=0$ for $i>0$, but I don't think that I have seen an explicit counterexample. Is there a simple example of a nonzero... | https://mathoverflow.net/users/93798 | Failure of Tate acyclicity for integral structure sheaves | Below is an example of a formal scheme $\mathfrak{X}$ such that its rigid analytic generic fiber $X$ is affinoid, while its special fiber contains a complete elliptic curve $E$. Because $H^1(E,\mathcal{O}\_E) \ne 0$, $H^1(X,\mathcal{O}\_X^+) \ne 0$ as well.
Let $p$ be an odd prime, and let $X$ be the rigid analytic e... | 3 | https://mathoverflow.net/users/93798 | 402070 | 165,012 |
https://mathoverflow.net/questions/402071 | 5 | **Motivation.** (Please skip if you are not in the mood for "chitchat".) Last night I listening to a classical radio station, and for the umpteenth time, they played Mendelssohn's [Psalm 42](https://en.wikipedia.org/wiki/Psalm_42_(Mendelssohn)), a composition that I like very much. Luckily, a week ago, when they played... | https://mathoverflow.net/users/8628 | Radio-playing sequence | I think you're just asking for a de Bruijn sequence of order $2$ on $n$ symbols, in which case the answer is $n^2$ because the non-simple digraph on $n$ vertices where $u \to v$ for every $u, v$ (including $u=v$) and the edges $u \to v$ and $v \to u$ are considered distinct unless $u=v$ is Eulerian.
| 10 | https://mathoverflow.net/users/46140 | 402072 | 165,013 |
https://mathoverflow.net/questions/402076 | 12 | I would like to know (as part of an attempt to streamline some calculations in the cohomology of a Morava stabiliser group) whether $1170\sqrt{-3}\sqrt{5}\sqrt{-7}-19110$ is a square in $\mathbb{Q}(\sqrt{-3},\sqrt{5},\sqrt{-7})$. What is an efficient method for this kind of question?
| https://mathoverflow.net/users/10366 | Squares in a triquadratic field | The field $L=\mathbb Q(\sqrt{-3},\sqrt{5},\sqrt{-7})$ has a lot of intermediate fields which we can exploit. Pick one of its index two subextensions, say $K=\mathbb Q(\sqrt{-3},\sqrt{5})$. The element $\alpha=1170\sqrt{-3}\sqrt{5}\sqrt{-7}-19110$ of $L$ has as its only conjugate over $K$ the element $-1170\sqrt{-3}\sqr... | 15 | https://mathoverflow.net/users/30186 | 402079 | 165,015 |
https://mathoverflow.net/questions/402075 | 3 |
Say you have a 2D broken line you move along, but only some directions are allowed (I give you the angles relative to the usual cartesian plane):
1. (Up-Left): $]\pi, \dfrac{\pi}{2}[$
2. (Down-Left): $]-\pi, \dfrac{-\pi}{2}[$
3. (Down-Right): $ ]\dfrac{-\pi}{2}, 0[$
An additional rule is that you cannot go ... | https://mathoverflow.net/users/342793 | Broken line that can go in specific directions: can it end up on its starting point? | There has to be a self-intersection; the path cannot be a simple polygon.
Suppose the path were indeed simple. A vertex with maximal $y$ coordinate must connect edges that go up and then down. According to the rules, the only way this can happen is first up-left, then down-left. [For a simple polygon](https://en.wiki... | 3 | https://mathoverflow.net/users/1227 | 402089 | 165,018 |
https://mathoverflow.net/questions/402099 | 5 | In the Wikipedia article (<https://en.wikipedia.org/wiki/Quantifier_elimination#cite_note-4>) it is said that every abelian group has quantifier elimination property and a long old paper of W. Szmielew (1955) is given as a reference. But as far as I know, this is true for special classes of abelian groups (like divisib... | https://mathoverflow.net/users/44949 | Quantifier elimination for abelian groups | Abelian groups are the same thing as $\mathbb Z$-modules. In general, for any ring $R$, the theory of left $R$-modules has quantifier elimination down to Boolean combinations of primitive positive formulas and certain sentences (expressing so-called Baur–Monk invariants). This is the Baur–Monk quantifier elimination th... | 10 | https://mathoverflow.net/users/12705 | 402100 | 165,020 |
https://mathoverflow.net/questions/402060 | 3 | Just for fun I was trying to find a formula that calculates the value of the sum of the Riemann zeta non trivial roots raised to a power $n$, $Z(n)$.
$$Z(n) = \sum\_{\rho} ' \frac{1}{\rho ^n}$$
I managed to find one monster of an equation after a while and it seems to work fine for $n=1$ but when I try $n=3$ the su... | https://mathoverflow.net/users/342532 | Proof of the sum of the reciprocal non trivial zeros cubed | This comes directly from the Hadamard product given in the Wolfram page you
refer to by taking logarithmic derivatives and identifying powers of $s$.
However, following Harold Stark, the classical formula given in Wolfram should
be replaced by the much simpler formula
$s(s-1)\Lambda(s)=\prod\_{\rho}(1-s/\rho)$, where t... | 5 | https://mathoverflow.net/users/81776 | 402104 | 165,021 |
https://mathoverflow.net/questions/363297 | 19 | In his PhD thesis, [Categorical Structure of Continuation Passing Style](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.9.3717&rep=rep1&type=pdf), Thielecke studies the ⊗¬-categories, which are premonoidal categories with structure (namely, a functor ¬ which is adjoint to itself), in order to model the semant... | https://mathoverflow.net/users/142280 | How to pronounce "⊗¬-category"? | I shared an office with Hayo Thielecke in the late 1990s.
The pronunciation he used was "Tensor-NOT-category".
| 17 | https://mathoverflow.net/users/170446 | 402107 | 165,023 |
https://mathoverflow.net/questions/402051 | 5 | Fix a finite set of integers $S$ and a prime number $p$. Let $(a\_1, a\_2, \dotsc, a\_n)$, $(b\_1, b\_2, b\_3, \dotsc, b\_n)$ be two sequences of integers where the numbers $a\_i$ and $b\_i$ are chosen uniformly from the set $S$ (In the case I care about S = {-2,-1,1,2 }). I would like to understand what is the probabi... | https://mathoverflow.net/users/7894 | Distribution of some sums modulo p | Using the [Newton identities](https://en.wikipedia.org/wiki/Newton%27s_identities), one can (in the high characteristic regime $p>k$) express the elementary symmetric polynomial $\sum\_{i\_1 < \dots < i\_k} a\_{i\_1} \dots a\_{i\_k}$ in terms of the moments $\sum\_{i=1}^k a\_i^j$ for $j=1,\dots,k$. The question then bo... | 7 | https://mathoverflow.net/users/766 | 402109 | 165,024 |
https://mathoverflow.net/questions/319745 | 8 | Recall that the permanent of an $n\times n$ matrix $A=[a\_{i,j}]\_{1\le i,j\le n}$ is defined by
$$\operatorname{per}A=\sum\_{\sigma\in S\_n}\prod\_{i=1}^n a\_{i,\sigma(i)}.$$
In 2004, R. Chapman [Acta Arith. 115(2004), 231-244] determined the value of
$$\det\left[\left(\frac{i+j-1}p\right)\right]\_{1\le i,j\le(p+1)/... | https://mathoverflow.net/users/124654 | Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive? | This is the sequence A322898 in OEIS.
I used a program in PARI and calculated the values of a(26) to a(34).
```
a(26) = -7000008163328
a(27) = 22712032822272
a(28) = 2244036651776
a(29) = 4363027965018112
a(30) = 30229121955004416
a(31) = -46693326700068864
a(32) = -23328907207088128
a(33) = 3173005987716005888
a(34... | 11 | https://mathoverflow.net/users/43683 | 402122 | 165,030 |
https://mathoverflow.net/questions/402102 | 2 | On p.112-113 of volume 10-1 of Gauss's werke, which contain an unpublished fragment dated to 1805, Gauss states some results on cubic and biquadratic "Gaussian periods" in a trigonometric form. While the result on the cubic period was already published by Gauss in a footnote to article 358 of his Disquisitions Arithmet... | https://mathoverflow.net/users/118562 | Explanation of two interrelated identities of Gauss about cubic and biquadratic periods | Let $p \equiv 1 \bmod 3$ be a prime number, let $g$ be a be a primitive root
modulo $p$, and $\zeta$ a primitive $p$-th root of unity. The three
cubic periods are
\begin{align\*}
\eta\_0 & = \zeta + \zeta^{g^3} + \zeta^{g^6} + \ldots + \zeta^{g^{p-4}}, \\
\eta\_1 & = \zeta^g + \zeta^{g^4} + \zeta^{g^7} + \ldots + \ze... | 2 | https://mathoverflow.net/users/3503 | 402129 | 165,033 |
https://mathoverflow.net/questions/401651 | 11 | Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are denoted $J^rY$ ($r\in\mathbb N$, $J^0Y=Y$), and the infinite jet bundle $J^\infty Y$ is - as a topological space - the p... | https://mathoverflow.net/users/85500 | Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations) | The following remarks are based on having previously gone through the literature that you've mentioned also for the purposes of figuring out these differences. It has been a while since then, but the state of our understanding that we reached with Urs Schreiber at the time was recorded in the following work, to be more... | 3 | https://mathoverflow.net/users/2622 | 402134 | 165,035 |
https://mathoverflow.net/questions/402120 | 3 | Any knot or link can be written in braid notation (with implied closure of strands). Some natural questions:
1. Assume I don't allow inverses - only overcrossings are used as generators. Anything known about the braid index then (might as well be $\infty$, i.e., this representation doesn't always exist)?
2. Assume I ... | https://mathoverflow.net/users/11504 | "Effectivity" of braid notation for knots | (1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.
(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to rotate ... | 2 | https://mathoverflow.net/users/1650 | 402144 | 165,038 |
https://mathoverflow.net/questions/402131 | 1 | Let $f\_n\colon \mathbb{C} \to \mathbb{C}$ be a sequence of entire functions, such that $f\_n$ converges to the zero function on an open dense subset $U$ of $\mathbb{C}$ pointwise (or equivalently normally). Then, does $f\_n$ genuinely converge to the zero function on $\mathbb{C}$?
It seems to me that it does converg... | https://mathoverflow.net/users/343456 | Convergence of a sequence of entire functions on an open dense subset | It is not true. Consider the compact sets
$$K\_n=\{ z:|z|\leq n, |\arg z|\geq 1/n\}\cup\{0\}.$$
By Runge's approximation theorem, there exist polynomials $f\_n$,
such that $|f\_n(z)|<1/n,\; z\in K\_n,$ and $f\_n(1)=n.$
This sequence of polynomials evidently converges uniformly on compact subsets of the dense open set $... | 2 | https://mathoverflow.net/users/25510 | 402145 | 165,039 |
https://mathoverflow.net/questions/401484 | 6 | Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal definition seems to be given in this (very long) introduction.
It is just mentioned that $A$ is Serre-smooth if $A$ has finite global dimensi... | https://mathoverflow.net/users/61949 | What is a Serre-smooth algebra? | No, there is no such reference. The introduction to that book is based on a couple of lectures I gave in Luminy and there I had to distinguish between several notions of 'smoothness', formal smoothness a la Kontsevich-Rosenberg, Cayley-smoothness, and Artin-Schelter- or Auslander-Gorenstein-regularity as used by people... | 7 | https://mathoverflow.net/users/2275 | 402149 | 165,040 |
https://mathoverflow.net/questions/356458 | 0 | Claim:
------
let $a,b,c>0$ and $p\geq 1$ then we have :
$$\left(\frac{a^3}{13a^2+5b^2}\right)^p+\left(\frac{b^3}{13b^2+5c^2}\right)^p+\left(\frac{c^3}{13c^2+5a^2}\right)^p\geq 3\left(\frac{a+b+c}{54}\right)^p$$
---
The case $p=1$ have been proved by user RiverLi and I offer a partial proof too in this case (... | https://mathoverflow.net/users/147649 | Olympiad inequality as a generalizing result due at the origin to Vasile Cirtaoje | The case $p>1$ easily follows from the case $p=1$. Indeed, let $M\_p(x,y,z) := \left(\frac{x^p+y^p+z^p}3\right)^{1/p}$. Then for $p>1$ we have
$$M\_p\big( \frac{a^3}{13a^2+5b^2}, \frac{b^3}{13b^2+5c^2}, \frac{c^3}{13c^2+5a^2}\big) \stackrel{(1)}{\geq} M\_1\big( \frac{a^3}{13a^2+5b^2}, \frac{b^3}{13b^2+5c^2}, \frac{c^3}... | 1 | https://mathoverflow.net/users/7076 | 402151 | 165,041 |
https://mathoverflow.net/questions/401635 | 3 | For any group $G$, the *universal example* for proper $G$-actions, $\underline{E}G$, is a proper $G$-space such that for any other proper $G$-space $X$, there exists a map (unique up to $G$-equivariant homotopy)
$$X\to\underline{E}G.$$
The quotient $\underline{B}G$ may be called the *classifying space* for proper $G$-a... | https://mathoverflow.net/users/78729 | Reference request: functoriality of $\underline{E}$ and $\underline{B}$ | Here is model that is obviously functorial: take for $\underline{E}G$ the simplicial complex with vertex set the finite subsets of $G$ and simplices the finite chains of sets ordered by inclusion. A homomorphism $f:G\rightarrow H$ induces a function from the finite subsets of $G$ to the finite subsets of $H$. You can a... | 2 | https://mathoverflow.net/users/124004 | 402157 | 165,044 |
https://mathoverflow.net/questions/402161 | 2 | This is related to [Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups](https://mathoverflow.net/questions/401996/symplectic-group-over-mathbbz-p-mathbbz-is-generated-by-its-root-subgroup). There I was told that in general, the symplectic group $\text{Sp}\_{2n}(R)$ is not generated by its... | https://mathoverflow.net/users/nan | When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus? | Since $\operatorname{Sp}\_{2\ell}$ is simply-connected, there is no need for the maximal torus. So the question is about the triviality of $\operatorname{K\_1}(\mathsf{C}\_\ell, R) = \operatorname{Sp}(2\ell,R)/\operatorname{Ep}(2\ell,R)$, where $\operatorname{Ep}(2\ell,R)$ is the elementrary symplectic group, that is, ... | 2 | https://mathoverflow.net/users/5018 | 402166 | 165,050 |
https://mathoverflow.net/questions/401602 | 8 | Let $f: \mathbb R \to \mathbb R$ be a $C^1$ function.
We say a point $c \in \mathbb R$ is a *mean value point* of $f$ if there exists an open interval $(a,b)$ containing $c$ such that $f’(c) = \frac{f(b) - f(a)}{b-a}$.
>
> **Question:** Is it true that (Lebesgue) almost every point in $\mathbb R$ is a mean value ... | https://mathoverflow.net/users/173490 | Converse of mean value theorem almost everywhere? | Let $U$ be an open and dense subset of $\mathbb{R}$ with finite measure. Let $g: \mathbb{R} \to \mathbb{R}^{\ge 0}$ be a continuous function with $\{g = 0\} = U^c$. Then define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = g(0)+\int\_0^x g(t)dt$. Then $f \in C^1$, and $f' \equiv g$ means $f$ is strictly increasing (since a... | 7 | https://mathoverflow.net/users/129185 | 402171 | 165,052 |
https://mathoverflow.net/questions/402108 | 4 | Based on my previous answer and your help
[Is there a procedure for extracting first integer $q\_0$ from $\sum\limits\_{k=0}^{\infty}\frac{1}{q\_k^z}$, all $0<q\_0<q\_1<...$ integers, $z$ complex?](https://mathoverflow.net/q/300543/113386)
This formula derived in the question
$$ \ln(p\_n)=-\lim\limits\_{\operator... | https://mathoverflow.net/users/nan | Recursive formula for n-th prime derived from a previous question | Based on
1. [Keller - A recursion equation for prime
numbers](http://arxiv.org/abs/0711.3940)
2. [Kawalec - The recurrence
formulas for primes and non-trivial zeros of the Riemann zeta
function](http://arxiv.org/abs/2009.02640)
it is obvious that they meant to use or even used real values. The novelty here is that ... | 0 | https://mathoverflow.net/users/nan | 402180 | 165,053 |
https://mathoverflow.net/questions/402177 | 1 | Let $A$ be a $C^\*$-algebra and $E$ be a (right) Hilbert $C^\*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally complemented, i.e. we have $F^\perp \oplus F^{\perp \perp} = E.$ Can we conclude that $F$ is orthogonally complemented, and i... | https://mathoverflow.net/users/216007 | Complemented submodules of a Hilbert C*-module | If $F^\perp =\{0\}$ then $F^\perp \oplus F^{\perp\perp} = E$ is automatic, so all you need is an example of this where $F \neq E$. For instance, $C[0,1]$ as a Hilbert module over itself with $F =$ the functions which vanish at 0.
| 2 | https://mathoverflow.net/users/23141 | 402183 | 165,055 |
https://mathoverflow.net/questions/400879 | 7 | Given a partition $\lambda$ and its Young diagram $\pmb{Y}\_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}\_{\lambda}$. We now recall a conjecture (now a theorem) of D. Armstrong which states "the total number $(s,t)$-core partitions is $\frac1{s+... | https://mathoverflow.net/users/66131 | Fibonacci embedded in Catalan? | The order ideal corresponding to a core partition with distinct parts cannot contain an element $x \geq s+2$, otherwise it must also contain $x-s, x-s-1$, resulting in two equal parts. For the same reason, it also can't contain two consecutive elements in $\{1,2,\dots, s-1\}$. Therefore the poset in questions is simply... | 6 | https://mathoverflow.net/users/2384 | 402188 | 165,057 |
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