parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/289097 | 11 | Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, *Nonnegative functions as squares or sums of squares*, [J. Funct. Anal. **232** (2006) 137-147](http://www.cmls.polytechnique.fr/perso/bony/BBCP_jfa.pdf) (see also [this MO question](https://mathoverflow.ne... | https://mathoverflow.net/users/11211 | Is every non-negative test function the limit of a sequence of sums of squares of test functions? | Given $f\in\mathcal{D}(\mathbb{R}^n)$, $f\ge0$ choose a $g\in\mathcal{D}(\mathbb{R}^n)$, $g\ge0$ such that $\mathrm{supp}(f)\subset \{g>0\}$, and let $\epsilon>0$. Then $\sqrt{f+\epsilon^2 g^2}$ is $C^\infty$: by composition, at any point where $g(x)>0$, and because it locally coincides with $\epsilon g$, at any point ... | 7 | https://mathoverflow.net/users/6101 | 289156 | 127,520 |
https://mathoverflow.net/questions/289133 | 6 | Let $A\to B\to C\to A[1]$ be a distinguished triangle in a (bounded below) derived category of an abelian category.
>
> Is there a necessary and sufficient condition that it splits, namely $B\simeq A\oplus C$ in a way compatible with the morphisms in the triangle?
>
>
>
Remark. A necessary condition is that t... | https://mathoverflow.net/users/16183 | Splitting of exact triangles in derived category | In any triangulated category, the necessary and sufficient condition for a distinguished triangle $A\to B\to C\to A[1]$ to split is that the morphism $C\to A[1]$ in this distinguished triangle vanishes. This morphism is the same thing as "the image of the identity morphism under $Hom(C,C) \to Hom(C,A[1])$" that you men... | 13 | https://mathoverflow.net/users/2106 | 289160 | 127,522 |
https://mathoverflow.net/questions/289164 | 4 |
>
> Let $G$ be a finite group with a central subgroup $Z$ of order 2. Suppose that $Z$ has a direct factor in some (and hence any) 2-Sylow subgroup of $G$. Does this imply that $Z$ has a direct factor in $G$?
>
>
>
I have checked some families of examples I can discard. First, the result is true when $G$ is abel... | https://mathoverflow.net/users/14094 | Splitting of central extension read on 2-Sylow? | You meant to say "suppose that $Z$ has a *complement* in some/any 2-Sylow subgroup", right? Then you're looking for Gaschütz's theorem. It is usually discussed together with the Schur-Zassenhaus theorem and/or group cohomology in many introductory group theory books, for example [Kurzweil-Stellmacher (theorem 3.3.2)](h... | 9 | https://mathoverflow.net/users/3041 | 289165 | 127,523 |
https://mathoverflow.net/questions/289161 | 9 | Let $C$ be a small category: we define its *nerve* $(N(C)\_k)\_k$ as the following simplicial set: $N(C)\_0=Ob(C)$ (the set of objects), $N(C)\_1=Mor(C)$ (the set of all morphisms) and $N(C)\_k$ to be a set of all $k$-tuples of compasable morhpisms $(f\_1,...,f\_k)$. This is equipped with the face maps $d\_i$ defined b... | https://mathoverflow.net/users/24078 | Classifying space as the geometric realization of the nerve of $G$ viewed as a small category | You're spot on with Hatcher's construction. First he constructs the space $EG$ as a $\Delta$-complex, but this can easily be upgraded to a simplicial set $\mathcal{E}G$ such that $EG$ is the geometric realisation of $\mathcal{E}G$:
In fact let $\mathcal{E}G\_n:=G^{n+1}$ and define the face maps as $(g\_0, ..., g\_n) ... | 5 | https://mathoverflow.net/users/3041 | 289168 | 127,525 |
https://mathoverflow.net/questions/288974 | 1 | Let $V$ be a infinite dimensional Banach space over $\mathbb{C}$
Let $\{a\_{m,n} \cdot v\_{m,n}\}\_{m,n \in \mathbb{N}}$ be a double sequence with $a\_{m,n} \in \mathbb{C}$ and $v\_{m,n} \in V$ such that:
$$
\lim\_{m \to \infty}
\sum\_{n=1}^\infty
(a\_{m,n} \cdot v\_{m,n})
=
u
$$
and
$$
\forall n \in \mathbb{N}:
\lim... | https://mathoverflow.net/users/108867 | A double sequence in a Banach space | I think that the answer is "No", and the following
counterexample works in each space $V$ which is at least
two-dimensional. Let $u$ and $v$ be linearly independent vectors
in $V$.
We let $\{v\_{m,n}\}\_{n=1}^\infty$ be the sequence which starts
with $m$ vectors $u+v$, continues with $m$ vectors $u-v$, and all
furthe... | 3 | https://mathoverflow.net/users/37822 | 289172 | 127,527 |
https://mathoverflow.net/questions/289128 | 4 | I'm interseted in blow-ups of toric varieties, unfortunately, I don't understand the construction of a blow-up built by a refinement of a fan, if to be more specific, I didn't find any constructions.
For example, let us consider a fan generated by rays (1,0,0), (0,1,0), (0,0,1), (-1,-1,-1), (1,1,1), (-1,0,0). First f... | https://mathoverflow.net/users/110428 | Blow-ups of toric varieties | The cone spanned by $(1,0,0), (0,1,0), (0,0,1)$ corresponds to a torus fixed point (which is the origin of $\mathbb{C}^3$). Subdividing this cone by adding the ray $(1,1,1)$ (the sum of the ray generators of this cone) corresponds to blowing up this $\mathbb{C}^3$ chart at the origin. Adding the ray $(-1,0,0)$ similarl... | 3 | https://mathoverflow.net/users/51668 | 289173 | 127,528 |
https://mathoverflow.net/questions/289176 | 8 | Let $X$ be a complex projective surface of general type, $K\_X$ be the very ample canonical divisor (which is automatically minimal). Is there an example for such a $X$ that there exist a sequence of negative curves $\{C\_i\}\_{i=1}^{\infty}$ with $C\_i^2$ being fixed, but the geometric genus $g(C\_i)$ going to infinit... | https://mathoverflow.net/users/63996 | Negative curves on surface of general type | Yes. Start with a rational surface $S$ containing infinitely many $(-1)$-curves $E\_n$ (for instance $\mathbb{P}^2$ blown up along 9 general points). Choose a very ample divisor $H$ on $S$, a smooth curve $B$ in the linear system $|2H|$, and consider the double covering $\pi : X\rightarrow S$ branched along $B$. Then $... | 9 | https://mathoverflow.net/users/40297 | 289179 | 127,529 |
https://mathoverflow.net/questions/289163 | 3 | I am interested in the series
$$\sum\_{n\geq 1}I\_n(x)\lambda^n$$
which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\infty$.
If we can not find a closed form for this series, what relevant information can we extract from it?
| https://mathoverflow.net/users/111000 | What can we know about "the half" of the generating series of Bessel function | The modified Bessel functions satisfy the recurrence
$$ I\_n(x) - \frac{2(1+n)}{x} I\_{n+1}(x) - I\_{n+2}(x) = 0 $$
which translates to a first-order differential equation for
$g(\lambda) = \sum\_{n=0}^\infty I\_n(x) \lambda^n$ (note that I'm including $n=0$ in this sum):
$$ 2 \lambda^2 g'(\lambda) + (1-\lambda^2)... | 5 | https://mathoverflow.net/users/13650 | 289184 | 127,532 |
https://mathoverflow.net/questions/289187 | 7 | I am writing a paper on the topological structure of the Golomb space ([defined here](https://mathoverflow.net/questions/286785/)) and arrived to the following question:
**Question 1.** Is it true that for a number $a\in\mathbb N$ the equation $x^2+x=a$ has an integer solution $x$ if and only if for any number $b\in\... | https://mathoverflow.net/users/61536 | Does the equation $x^2+x=a$ have an integer solution? | The answer is yes, and in general if $f(x)\in\mathbb{Z}[x]$ is a monic irreducible polynomial whose Galois group contains a cycle of length $\deg f$ (which is always the case when $\deg f$ is prime, by [Cauchy's theorem](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory))), then for infinitely many primes ... | 18 | https://mathoverflow.net/users/11919 | 289193 | 127,535 |
https://mathoverflow.net/questions/289136 | 2 | For the $A\_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}\_m$. The action of $\mathbb{Z}\_m$ on $\mathbb{C}^2$ is defined as following
$$
\bar{1} \cdot (z,w) = (z e^{\frac{2\pi i}{m}}, w e^{\frac{-2\pi i}{m}}),
$$
where $\bar{1} \in \mathbb{Z}\_m$.
For $m=2$, in other words $(z,w) \... | https://mathoverflow.net/users/118614 | resolution for the du Val's $(A_3)$-singularity | The quotient $\mathbb{C}^2/\mathbb{Z}\_3$ is the hypersurface
$$
xz = y^3
$$
in $\mathbb{A}^3$. To resolve it it is enough to blow up the origin. The resulting variety is a hypersurface in the blowup of $\mathbb{A}^3$ at the origin (this blowup is isomorphic to the total space of $\mathcal{O}(-1)$ on $\mathbb{P}^2$).
... | 2 | https://mathoverflow.net/users/4428 | 289217 | 127,547 |
https://mathoverflow.net/questions/289230 | 3 | I'm working with signed graphs and I don't know the answer to the following question. Also, I couldn't find the answer anywhere.
Question: If we have two signed graphs with the same underlying graph and they are co-spectral, can we conclude they are switching equivalent?
| https://mathoverflow.net/users/111007 | Are cospectral signed graphs with identical underlying graph necessarily switching-equivalent? | No. Let $H$ be an $n\times n$ Hadamard matrix and let $X$ be the signed graph with adjacency matrix
\[
A = \begin{pmatrix}0&H\\ H^T&0\end{pmatrix}.
\]
Then $A^2=nI$ and so the spectrum of $X$ is determined by $n$. However
if $K$ is a second $n\times n$ Hadamard matrix, then the corresponding signed graphs are switchin... | 1 | https://mathoverflow.net/users/1266 | 289241 | 127,555 |
https://mathoverflow.net/questions/289244 | 2 | Is there a complete classification of compact surface $S\subset \mathbb{R}^3$ for which $\Delta \kappa \geq 0$ where $\kappa $ is the Gaussian curvature of $S$.
Does every (compact) $2$ dimensional manifold admit a Riemannian metric with this property?
Is there a name for this property in $2$ (or higher dimensions)... | https://mathoverflow.net/users/36688 | Compact surfaces whose Gaussian curvature is a subharmonic function | The Laplacian of any function $f \colon S \to \mathbb{R}$ is $\Delta f$ defined by $\Delta f \, dA = -d(\*df)$, so is exact, and hence has integral zero if $S$ has empty boundary. So if $\Delta f \ge 0$ then $0=\int \Delta f$ forces $\Delta f=0$ everywhere, and so (if $S$ is connected) $f$ is constant.
| 8 | https://mathoverflow.net/users/13268 | 289245 | 127,556 |
https://mathoverflow.net/questions/289253 | 12 | It is well known that an integrable function $u \colon \mathbb R^d \to \mathbb R$ is said to be of [bounded variation](https://en.wikipedia.org/wiki/Bounded_variation#BV_functions_of_several_variables) iff the distributional gradient $Du$ is (representable by) a finite Radon measure, still denoted by $Du$.
Then it i... | https://mathoverflow.net/users/100976 | Structure of the Cantor part of the derivative of a BV function | Certainly not with a single $\alpha$, but it is tempting to decompose $D^cu$ into an integral $\int\_{d-1}^d\mu\_\alpha\ d\nu(\alpha)$ with $\mu\_\alpha$ having a density with respect to $\mathcal H^\alpha$ on an $\alpha$-dimensional set. I don't know if it is always possible.
| 8 | https://mathoverflow.net/users/75422 | 289255 | 127,560 |
https://mathoverflow.net/questions/289268 | 2 | I am studying [Fary-Milnor Theorem](http://mduchin.math.tufts.edu/UCD/116/milnor.pdf) on total curvature of knots and I am stuck in a proof. He is proving on page 9:
>
> The Total curvature of a tame knot cannot equal the curvature of its type
>
>
>
k(C) := total curvature of knot C
So by assuming false he t... | https://mathoverflow.net/users/118902 | Fary-Milnor Theorem : Help following a proof on page 9 | First, the OP's reference is Milnor's paper:
*Milnor, John W.*, [**On the total curvature of knots**](http://dx.doi.org/10.2307/1969467), Ann. Math. (2) 52, 248-257 (1950). [ZBL0037.38904](https://zbmath.org/?q=an:0037.38904).
Secondly, the total curvature of a type is the inf of the curvatures of tame knots of tha... | 3 | https://mathoverflow.net/users/11142 | 289270 | 127,563 |
https://mathoverflow.net/questions/289267 | 4 | Let $v =(r,s,t) \in \mathbb{N}^3$ be a vector such that $\gcd(r,s,t)=1$. We know that there are vectors $x= (x\_1,x\_2,x\_3) \in \mathbb{Z}^3 $ such that $v.x =1$. For each $v$, let $O(v)$ be the smallest size ($L^1$ or $L^\infty$ norm) of such $x$. Now for a natural number $n$ define $f(n)$ to be the maximum of $O(v)$... | https://mathoverflow.net/users/56571 | Smallest size of integral vector with certain inner product | $f(n)$ grows linearly in $n$. It grows at least linearly as seen from the triples $(2,2,2m+1)$. But we always may achieve that all coefficients are $O(n)$. For seeing this, we search a linear representation not of $r,s,t$, but of $r,r+s,r+t$, where we assume $r=\max(r,s,t)$. These numbers, I denote them $A,B,C$, lie be... | 3 | https://mathoverflow.net/users/4312 | 289276 | 127,565 |
https://mathoverflow.net/questions/289279 | 4 | Let $L/\mathbb{Q}$ be a finite Galois extension, and let $\mathcal{O}\_L$ be the ring of integers of $L$.
We have $tr\_{L/\mathbb{Q}}(\mathcal{O}\_L)=d\mathbb{Z}$ for some $d\geq 1.$
**Fact.** $d=1$ if and only if $L/\mathbb{Q}$ is tamely ramified.
**Question 1.** Let $p$ be a prime number. Is it true that $tr\_... | https://mathoverflow.net/users/36683 | Image of the trace map of ring of integers | **Question 1.** Yes, and this follows from the results of Chapter VIII in Weil: Basic Number Theory. See especially Corollary 2 of Proposition 4 in that chapter.
**Question 2.** In general, the exponent of $p$ in $d$ equals $\lceil g\_p/e\_p \rceil$, where $g\_p$ is the exponent of any prime $\mathfrak{p}\mid p$ in t... | 5 | https://mathoverflow.net/users/11919 | 289281 | 127,566 |
https://mathoverflow.net/questions/289293 | 0 | Conjecture:
>
> Any positive integer can be written as the difference between two
> coprime semiprimes.
>
>
>
Tested up to 1,000,000.
See also:
<https://math.stackexchange.com/questions/2579578/the-difference-of-two-coprime-composites>
I'm not surprised that there are heuristics backing up the conjectu... | https://mathoverflow.net/users/57255 | The difference between two coprime semiprimes | It seems to me most likely to be true that every integer can be written in infinitely many ways as the difference between a pair of co-prime semiprimes and that the number of pairs with both members under $x$ is a simpley described fiunction. I say this by analogy with twin primes and their generalizations as described... | 3 | https://mathoverflow.net/users/8008 | 289298 | 127,572 |
https://mathoverflow.net/questions/289259 | 49 | I'm currently a young, not-so-young mathematician, finishing its second postdoc. I developed an interest for rather different topics in the last few years but constantly, slowly converged towards something that has to do with (but at this point I'm quite unsure *is*) category theory and its applications. What motivated... | https://mathoverflow.net/users/118946 | The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me) | Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phenomena play a significant role. Many areas of algebraic topology and algebraic geometry have this property. There are al... | 73 | https://mathoverflow.net/users/51164 | 289311 | 127,575 |
https://mathoverflow.net/questions/289303 | 21 | According to this post [Intuition for group homology](https://mathoverflow.net/questions/10879/intuition-for-group-cohomology), I wonder what is the intuition for Hochschild homology.
>
> The Hochschild homology is defined as the homology of this complex
> chain.
> Given a ring $A$ and a bimodule $M$.
> Define a... | https://mathoverflow.net/users/19072 | intuition for hochschild homology | If you have a right $A$-module $M\_A$ and a left $A$-module $\_AN$, then you can form their tensor product
$$M\otimes\_AN:=\operatorname{coker}(M\otimes\_kA\otimes\_kN\xrightarrow{(m,a,n)\mapsto(ma,n)-(m,an)}M\otimes\_kN).$$
There is also a "derived" version of the tensor product, i.e. a chain complex whose homology gr... | 22 | https://mathoverflow.net/users/35353 | 289315 | 127,576 |
https://mathoverflow.net/questions/289307 | 1 | One of Auslanders famous theorems is that he proved that the global dimension of a semiprimary ring is equal to the maximum of the projective dimensions of the simple modules of the ring. This result can be found in the book of Auslander, Reiten and Smalo or in the module theory book of Lam. But in both books I found n... | https://mathoverflow.net/users/61949 | Reference for a result of Auslander about the global dimension | I think you probably want the following paper.
*Auslander, Maurice*, [**On the dimension of modules and algebras. III. Global dimension**](http://dx.doi.org/10.1017/S0027763000023291), Nagoya Math. J. 9, 67-77 (1955). [ZBL0067.27103](https://zbmath.org/?q=an:0067.27103).
| 5 | https://mathoverflow.net/users/22989 | 289316 | 127,577 |
https://mathoverflow.net/questions/289026 | 4 | Given an operator ideal $\mathfrak{I}$, $\mathfrak{I}^\text{dual}$ is the class of all operators $A:X\to Y$ between Banach spaces $X$ and $Y$ such that $A^\*\in \mathfrak{I}$. Given an operator ideal $\mathfrak{I}$, it is often of interest to know when there is another ideal $\mathfrak{J}$ of independent interest such ... | https://mathoverflow.net/users/nan | Duals of ideals of operators between Banach spaces | Let $\mathcal{K}$, $\mathcal{W}$ and $\mathcal{C}$ denote the compact, weakly compact and completely continuous operators, respectively, and let $\mathcal{W}^{-1}\circ \mathcal{K}$ denote the operators $T:X\to Y$ such that $BT\in\mathcal{K}$ for each $Z$ and each $B\in\mathcal{W}(Y,Z)$.
CLAIM: $\mathcal{C}^{dual} = ... | 2 | https://mathoverflow.net/users/39421 | 289318 | 127,578 |
https://mathoverflow.net/questions/289300 | 8 | I'd like to learn the Newman's Lemma or Diamond Lemma (the one used in abstract rewriting system), can someone recommend me some books where I can read it? I'd appreciate self-contained books with examples. Thank you.
| https://mathoverflow.net/users/47294 | Newman's Lemma or Diamond Lemma | * [Franz Baader and Tobias Nipkow, *Term Rewriting and All That*](https://www21.in.tum.de/~nipkow/TRaAT/) is a book fully devoted to term rewriting; much of it is about applying the diamond lemma.
* [Vincent van Oostrom, *Newman's proof of Newman's lemma*](http://gen.lib.rus.ec/book/index.php?md5=98C760060ED7C8CBA02612... | 16 | https://mathoverflow.net/users/2530 | 289320 | 127,579 |
https://mathoverflow.net/questions/289314 | 1 | Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that
$$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$
Why must $f$ be log-concave? (That is, why must
$$\forall a,b\in\mathbb{R}~\forall t\in(0,1)~~a<b\implies t\log f(a)+(1-t)\log f(b)\leq\log f(ta+(1-t)b)$$
hold?)
I came acros... | https://mathoverflow.net/users/110883 | Inequality satisfied for $t=1/2$ and Measurability implies Log-Concavity | First of all, let's look at $g = -\log f$; then this is really just a question about convex functions. We want to know: if $g$ is measurable and "midpoint convex", i.e.
$$g\left(\frac{a+b}{2}\right) \le \frac{g(a)+g(b)}{2} \tag{MC}$$
does it follow that $g$ is convex, i.e.
$$g((1-t)a+tb) \le (1-t)g(a) + t g(b), \quad ... | 2 | https://mathoverflow.net/users/4832 | 289322 | 127,580 |
https://mathoverflow.net/questions/289331 | 2 | As is known (see [Kadison-Ringrose](http://bookstore.ams.org/gsm-15), 3.4.1) each closed ideal $I$ in the $C^\*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form
$$
I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \}
$$
for some closed subset $S$ in $X$.
>
> Is the same true for the two-sided i... | https://mathoverflow.net/users/18943 | Closed two-sided ideals in $C(X,M_n)$ | I'm assuming your compact space $X$ is Hausdorff. Then the answer is yes.
Suppose $I$ is a closed two-sided ideal in $C(X, M\_n)$.
For each $x \in X$, $I(x) = \{f(x): f \in I\}$ is a two-sided ideal in $M\_n$. But
$M\_n$ is a simple ring: it has no two-sided ideals except itself and $\{0\}$.
Let $S = \{x \in X: \... | 5 | https://mathoverflow.net/users/13650 | 289333 | 127,586 |
https://mathoverflow.net/questions/289124 | 4 | It is a theorem due to Harer that $H\_k(M\_{g,n},\mathbb{Q})=0$ for $k>C(g,n)$, where $C(0,n)=n-3, C(g,0)=4g-5$ for $g>0$, and $C(g,n)=4g-4+n$ for $g,n>0$. Here $M\_{g,n}$ denotes the coarse moduli space of $n$-pointed genus $g$ curves. I was wondering if an analogous vanishing theorem holds for $H\_{g,n}$, and if yes ... | https://mathoverflow.net/users/118889 | Vanishing of homology for hyperelliptic locus | A cohomological bound can be obtained as follows.
**Proposition** The affine stratification number of $H\_{g,n}$ is $1$ for $n > 0$, $0$ for $n=0$.
*Proof* Since $H\_{g,0}$ is affine we are only interested in the case $n > 0$. Since $H\_{g,n+1} \to H\_{g,n}$ is an affine morphism for $n>0$, it's enough to do the ca... | 3 | https://mathoverflow.net/users/1310 | 289339 | 127,588 |
https://mathoverflow.net/questions/289247 | 4 | **Polya urn model**: At time $0$ an urn initially contains $b$ $\tt{B}$lue balls and $r$ $\tt{R}$ed balls. At time $1$, a ball is drawn uniformly at random (removing it) from the urn, and *two* balls of the drawn color are added to the urn. This procedure is then repeated at times $2,3,4,\ldots$, each time increasing b... | https://mathoverflow.net/users/20307 | Polya urn: Mean number of draws to get a specific sequence of colors? | The crucial thing you need concerns exchageability properties of the Polya urn.
The following two procedures give the same law:
1. Generate a sequence of $B$s and $R$s using Polya's urn as you describe, starting from $b$ blue and $r$ red balls;
2. First randomly draw $p$ from a Beta$(b,r)$ distribution. Now, give... | 8 | https://mathoverflow.net/users/5784 | 289341 | 127,589 |
https://mathoverflow.net/questions/289221 | 6 | Generalizing from 1-category theory, there's a simple definition of a "naive complex" in a stable $\infty$-category. Considering bounded positive graded chain complexes, they are a sequence of maps
$$ A\_n \xrightarrow{d\_n} A\_{n-1} \xrightarrow{d\_{n-1}} \ldots \xrightarrow{d\_1} A\_0 $$
with the property that $d... | https://mathoverflow.net/users/nan | Complexes in stable categories | Here is the problem with the notion of naive complex. Suppose we have a naive complex
$$ \require{AMScd} \begin{CD}
A @>f>> B @>g>> C @>h>> D \end{CD} $$
If we propose to compute the realization iteratively, the first step would be to produce the sequence
$$ \require{AMScd} \begin{CD}
\mathrm{cofib}(f) @>>> C @>h>> D \... | 5 | https://mathoverflow.net/users/nan | 289345 | 127,592 |
https://mathoverflow.net/questions/283751 | 28 | This was originally a question about comparing mana costs in Magic: The Gathering, but it's turned into a question about Minkowski sums of upward-closed convex sets in $\mathbb{N}^k$. The original question is preserved below, if you want to see the original motivation (or have an answer to the original question that do... | https://mathoverflow.net/users/5583 | Are Minkowski sums of upward closed "convex" sets in $\mathbb{N}^k$ still "convex"? (WAS: Comparing mana costs in Magic: The Gathering) | It seems to me that all of the standard counter-examples to the polytope question easily adapt to be counter examples to this question: Take any polytopes in $\mathbb{Z}^{k-1}$ with $(A\_0+B\_0) \cap \mathbb{Z}^{k-1} \neq (A\_0 \cap \mathbb{Z}^{k-1}) + (B\_0 \cap \mathbb{Z}^{k-1})$. Embed $\mathbb{Z}^{k-1}$ into $\math... | 5 | https://mathoverflow.net/users/297 | 289347 | 127,593 |
https://mathoverflow.net/questions/289350 | 3 | I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and Links").
I understand from Thurston's work that if the complement of a non-split link is atoroidal and anannular, then... | https://mathoverflow.net/users/118995 | Classifying links with essential annuli in the complement as torus links | Thurston only claims a classification of knots, not of links. See Corollary 2.5 of Thurston's article "[Three dimensional manifolds, kleinian groups, and hyperbolic geometry](https://projecteuclid.org/euclid.bams/1183548782)".
Cromwell's statement is incorrect, as your examples show. However, your examples are almos... | 3 | https://mathoverflow.net/users/1650 | 289359 | 127,596 |
https://mathoverflow.net/questions/289271 | -2 | The following admits of many (easy) proofs, but I am seeing no purely "bijective" argument:
$$
\sum\_{j=n}^N \binom{j}{n} = \binom{N+1}{n+1}.
$$
Any ideas?
| https://mathoverflow.net/users/11142 | Combinatorial proof of identity | Santa Claus has $N+1$ reindeer whose noses are of varying redness. Every year, Santa needs $n+1$ reindeer to pull his sleigh. The reddest-nosed reindeer always leads the sleigh.
The way Santa chooses the reindeer is as follows. First, Santa chooses one reindeer to lead the sleigh; call it the $(j+1)$th reindeer. Then... | 5 | https://mathoverflow.net/users/88133 | 289361 | 127,597 |
https://mathoverflow.net/questions/289185 | 6 | As stated in the title, I am wondering the main difference between Beilinson conjecture and eTNC. If I read correctly, I can see that there are many literature treating both conjectures in the same line - special values of L-functions. If my impression is completely wrong, I am sorry for this stupid question. But, if t... | https://mathoverflow.net/users/44005 | Difference of Beilinson conjecture and equivariant Tamagawa number conjecture | The key difference between these conjectures is the coefficient ring that is involved. You also are leaving out an important "middle" conjecture -- the (non-equivariant) Tamagawa number conjecture, as formulated in Bloch and Kato's article in the Grothendieck Festschrift -- and knowing what this conjecture says might c... | 8 | https://mathoverflow.net/users/2481 | 289362 | 127,598 |
https://mathoverflow.net/questions/289363 | 5 | What is the chromatic number of the ER graph $G(n,d/n)$, when $d < 1$ (there exist expressions for $d > 1$, but what if the graph is super sparse?). Here $n$ is the number of vertices and $d/n$ is the edge generation probability.
| https://mathoverflow.net/users/91159 | What is the chromatic number of the Erdős–Rényi graph G(n,d/n) when d < 1? | Consider subgraphs consisting of two cycles with an edge in common (i.e. a theta-graph or something more complex). The number of such labelled graphs with $t$ vertices is at most $n^t$, and the probability of each is at most $(d/n)^{t+1}$ since they have at least $t+1$ edges. Summing over $t$ shows that the expected nu... | 12 | https://mathoverflow.net/users/9025 | 289365 | 127,599 |
https://mathoverflow.net/questions/286574 | 3 | Let's fix a $(B,f)$ structure with Thom spectra $MB$. I'd like to know the condition for a not-necessarily-multiplicative generalized cohomology theory $E^\*$ such that for a fibration $X\to Y$ with $(B,f)$ structure with $n$-dimensional fiber to have the pushforward map $E^\*(X)\to E^{\*-n}(Y)$.
When $E$ is multipli... | https://mathoverflow.net/users/5420 | pushforward in non-multiplicative generalized cohomology theory | Yes, and is explained in detail in any textbook. See e.g. Rudyak "On Thom Spectra, Orientability, and Cobordism", Chapter 5, Section 2.
| 2 | https://mathoverflow.net/users/5420 | 289370 | 127,600 |
https://mathoverflow.net/questions/289375 | 10 | Let $A$, $B$, and $C$ be $2\times 2$ complex matrices, with $A$ and $C$ rank $1$ Hermitian. Can we find a real number $a$ and a $2\times 2$ unitary $U$ such that
$$A + BV + V^\*B^\* + V^\*CV$$
is a scalar multiple of the identity, where $V = aU$?
The motivation is that this would answer [this question](https://mathov... | https://mathoverflow.net/users/23141 | $2 \times 2$ matrix question | Yes, one can always do this. In fact, one can assume that $\det(U)=1$, as this follows by a homotopy argument, using the fact that $\pi\_3(S^2)\simeq \mathbb{Z}$.
Here are the details: Assume that $\det(U)=1$ so that $V$ has a non-negative real determinant. Then one can write $V$ uniquely in the form
$$
V(x) = \begin... | 15 | https://mathoverflow.net/users/13972 | 289376 | 127,602 |
https://mathoverflow.net/questions/289379 | 7 | Let $A$ be a Cohen-Macaulay local ring, and let $M$ be an $A$-module that is (S$\_2$) and has depth $\ge n$ for some fixed $n$. Let $\mathfrak{p} \subset A$ be a prime of height $\ge n$. Is it true that $\mathrm{depth}\_{A\_{\mathfrak{p}}} M\_{\mathfrak{p}} \ge n$?
I would be happy with a counterexample or a proof ev... | https://mathoverflow.net/users/63877 | Depth under localization over a Cohen-Macaulay ring | Here is a counter-example when $A$ is regular local of dimension $4$ and $n=3$, the first non-trivial case. Let $A = k[[x,y,z,t]]$, and $P$ be a prime ideal of height $3$, say $P=(x,y,z)$. Let $M=\Omega P$, the syzygy of $P$. Obviously, $M$ is also $\Omega^2 (A/P)$.
The depth of $M$ at various localizations can be co... | 9 | https://mathoverflow.net/users/2083 | 289380 | 127,603 |
https://mathoverflow.net/questions/289367 | 4 | Given a positive definite matrix $Q\in\mathbb{R}^{n \times n}$, I want to find a diagnonal matrix $D$ such that $rank(Q-D) \leq k < n$.
I think this can be regarded as a generalization of eigenvalue problem, which is basically problem of finding a diagonal matrix $\lambda I$ such that $rank(Q-\lambda I) < n$.
Is th... | https://mathoverflow.net/users/119006 | About reducing the rank of a matrix by substract a diagonal matrix | Your problem can be restated as follows: To a given symmetric matxix, can
you add a diagonal matrix so that the result has eigenvalue $0$ with high
multiplicity?
This belongs to the theory which is called Additive Inverse Eigenvalue Problems. See, for example this paper, which seems to treat a very similar problem:
... | 4 | https://mathoverflow.net/users/25510 | 289398 | 127,608 |
https://mathoverflow.net/questions/289390 | 3 | Is every group isomorphic to an inverse limit (that is, projective limit) of perfect groups?
I guess, the answer is no. In that case: Is there a characterization of the groups that are isomorphic to inverse limits of perfect groups?
| https://mathoverflow.net/users/24916 | Inverse limits of perfect groups | Yes:
>
> Every group is isomorphic to a filtering projective limit of a sequence of perfect groups.
>
>
>
Start from a group $G$ and embed it into a perfect group $P$ (e.g., a symmetric group).
Consider the group $H$ obtained by amalgamating countably many copies $P\_n$ of $P$ over $G$. Then $H$ is perfect.... | 5 | https://mathoverflow.net/users/14094 | 289400 | 127,610 |
https://mathoverflow.net/questions/289391 | 3 | Is every group isomorphic to an inductive colimit (that is, directed colimit, also called inductive limit, or directed limit) of free groups?
I guess, the answer is no. In that case: Is there a characterization of the groups that are isomorphic to inductive colimits of free groups?
| https://mathoverflow.net/users/24916 | Inductive colimits of free groups | Let $G$ be a group. Equivalences:
1. $G$ is locally free (i.e., all its finitely generated subgroups are free)
2. $G$ is a filtering inductive limit of free groups with injective connecting homomorphisms
3. $G$ is a filtering inductive limit of free groups.
The equivalence between (1) and (2) is clear (given that s... | 5 | https://mathoverflow.net/users/14094 | 289401 | 127,611 |
https://mathoverflow.net/questions/289387 | 0 | Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-monotone how does it grow (heuristically $N(a)=O(\log a)$)?
This was my heuristic. [Number of pairs in $[1,a]$ with produc... | https://mathoverflow.net/users/10035 | Elementary description to count of perfect squares - I | As hinted in Alex Kruckman's comment, it is easier to work with
$$N'(a) = \left|\{x\in \{1,\dots,a-1\} : \sqrt{x(a-x)}\in \mathbb{Z}\}\right|,$$
which is the same as $N(a)$ up to a factor of $2$.
It is straightforward to see that $N'(a)$ equals the number of representations
$$a=d(y^2+z^2),\qquad \gcd(y,z)=1,$$
the conn... | 1 | https://mathoverflow.net/users/11919 | 289407 | 127,614 |
https://mathoverflow.net/questions/289368 | 1 | I have a question about convergence of resolvents of Markov processes.
Let $X$, $X^n$ be Markov processes on a locally compact separable metric space $E$.
We denote $\{ R\_{\alpha}\}\_{\alpha>0}$ and $\{ R\_{\alpha}^n\}\_{\alpha>0}$ by the resolvents of $X$ and $X^n$, respectively.
We assume the following:
* fo... | https://mathoverflow.net/users/68463 | Uniform convergence of resolvents of Markov processes | In general: *no*.
Let $X$ be the uniform motion to the right with speed $1$ (so $X\_t = X\_0 + t$), and let $X^n$ be the uniform motion to the right with speed $1$ plus an independent Brownian motion with variance $t/(2 n^2)$. Then $R\_\alpha$ is the convolution operator with kernel $e^{\alpha x} \mathbf{1}\_{(-\inft... | 3 | https://mathoverflow.net/users/108637 | 289410 | 127,616 |
https://mathoverflow.net/questions/289416 | 8 | My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are supposed to be thought of as like derivatives in a wider context than commutative rings, and I don't really understand how.... | https://mathoverflow.net/users/83073 | Semantics of derivations as derivatives | In all of these contexts, derivations are infinitesimal automorphisms, in the sense that $D$ is a derivation on $A$ (an algebra, a Lie algebra, etc.) iff $\exp(Dt)$ is an automorphism of $A \otimes k[t]/t^2$. On a commutative ring automorphisms correspond to automorphisms of the spectrum so derivations correspond to in... | 12 | https://mathoverflow.net/users/290 | 289421 | 127,621 |
https://mathoverflow.net/questions/289414 | 17 | Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, is there a reasonable way to:
(1) given a finite topological space $X$, construct a finite simplicial set $nX$.
(2) ... | https://mathoverflow.net/users/5690 | Are finite spaces a model for finite CW-complexes? | The answer to question (1) is **yes** and it follows from the following theorem by McCord:
>
> **Theorem 1.** (i) For each finite topological space $X$ there exist a finite simplicial complex $K$ and a weak homotopy equivalence $f:|K|\to X$. (ii) For each finite simplicial complex $K$ there exist a finite topologic... | 14 | https://mathoverflow.net/users/43054 | 289427 | 127,624 |
https://mathoverflow.net/questions/289392 | 0 | What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+1$ (corresponding to $x=0$ and $a-x=b^2+1$ or $x=1$ and $a-x=b^2$)?
| https://mathoverflow.net/users/10035 | Elementary description to count of perfect squares - II | *Is $g(a)=1\iff a=b^2+1$ (corresponding to $x=0$ and $a-x=b^2+1$ or $x=1$ and $a-x=b^2$)*
No. It may happen, that, say, $x,a-x$ are perfect squares and $x+1$, $a-x-1$ are twice perfect squares. These Pell type equations have infinitely many solutions, the smallest is $x=1,a-x=9$.
| 1 | https://mathoverflow.net/users/4312 | 289430 | 127,626 |
https://mathoverflow.net/questions/288640 | 2 | Let $P\_1, \ldots, P\_m$, $Q\_1, \ldots, Q\_k \in \mathbb{C}[x\_0,\ldots,x\_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and for every $j$ the polynomial $f$ belongs to the ideal $\langle P\_i, Q\_j \rangle$.
Is it true that the rank of ... | https://mathoverflow.net/users/31356 | Linear homogenous polynomials that generates one quadratic polynomial | We will assume that $f$ is irreducible (if $f$ is not irreducible then in fact the argument of Zach Teitler's answer works).
Consider $M:= f \cap P\_1$ (I mean the intersection of the zeros $f$ and $P\_1$).
This set is the zeros of a quadratic form in plane $P\_1$ of codimension $1$ (it can not be $P\_1$ since $f$ ... | 0 | https://mathoverflow.net/users/31356 | 289435 | 127,629 |
https://mathoverflow.net/questions/289429 | 0 | In my research of linear algebra and optimization, I wish to modify the following well-known problem:
>
> $ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector and we optimize over matrices of bounded rank.
>
>
>
My question is, can there be a way to solve the modified pr... | https://mathoverflow.net/users/69446 | Modification of a known optimization problem | This can be expressed (converted into) as a convex Second Order Cone Problem or SDP, depending on the norms used. CVX or another modeling tool can be used to convert the entered problem into a standard form for a solver. For example,
```
cvx_begin
variable A(m,n)
minimize(norm(x-A*x,p1))
norm(A,p2) <= bound_on_norm
... | 1 | https://mathoverflow.net/users/75420 | 289440 | 127,630 |
https://mathoverflow.net/questions/289385 | 1 | Let $G$ be a regular graph having spanning regular subgraphs $G\_1,\dots, G\_k$
whose edge sets are disjoint and their union is the whole edge set of $G$.
Is it true that the clique number of $G$ is bounded above by the sum of clique numbers
of $G\_i$s? If not, under what conditions the answer is positive.
| https://mathoverflow.net/users/19075 | Clique number of a regular graph with respect to that of a certain edge decomposition | The edges of $K\_{2n+1}$ can be split into $n$ Hamiltonian cycles, whose clique numbers are 2, which gives a counterexample to the initial conjecture. Moreover, these cycles can be merged into larger regular graphs without increasing of the clique number (check, e.g., the case when $2n+1$ is prime!), which provides muc... | 3 | https://mathoverflow.net/users/17581 | 289442 | 127,631 |
https://mathoverflow.net/questions/289402 | 19 | In Wikipedia's page for [Bertrand's postulate](https://en.wikipedia.org/wiki/Bertrand%27s_postulate#Generalizations), it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to the first source, or at least to a previous one?
... | https://mathoverflow.net/users/1234 | Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)? | I have finally found the following papers and results, which predate Nagura's paper of 1952. I cite them from newest to oldest:
1. (Molsen, 1941):
* For $n\geq 118$ there are primes in $(n,\frac43n)$ congruent to 1,5,7,11 modulo 12.
* For $n\geq 199$ there are primes in $(n,\frac87n)$ congruent to 1,2 modulo 3. Thi... | 24 | https://mathoverflow.net/users/1234 | 289448 | 127,632 |
https://mathoverflow.net/questions/289405 | 5 | Consider an elementary class $\mathcal{K}$. It is quite common in model theory that a structure $K$ in $\mathcal K$ comes with a closure operator $$\text{cl}: \mathcal{P}(K) \to \mathcal{P}(K), $$ which establishes a [pregeometry](https://en.wikipedia.org/wiki/Pregeometry_(model_theory)) on $K$.
Any pregeometry yield... | https://mathoverflow.net/users/104432 | Dimension and model theory | Here's an example showing that in general, for pregeometries arising in model theory, you can't characterize the dimension of a union of a chain of models just in terms of the dimensions of the models. In other words, it matters how the models embed into each other.
Consider the theory of a single equivalence relatio... | 2 | https://mathoverflow.net/users/2126 | 289450 | 127,634 |
https://mathoverflow.net/questions/289458 | 10 | I am looking for a proof for the following statement
>
> A smooth projective variety $X$ has maximal Albanese dimension if and
> only if the cotangent bundle of $X$ is generically generated by its
> global sections, that is,$$ H^0(X,\Omega\_X^1)\otimes\mathcal O\_X\to
> \mathcal \Omega\_X^1$$ is surjective at th... | https://mathoverflow.net/users/119052 | Maximal Albanese variety | In characteristic $0$ that follows from Sard's theorem / generic smoothness. In characteristic $p$, I believe that this is false (I still need to check how double point singularities, $\text{Zero}(x\_1^2 + \dots + x\_{n-1}^2+x\_n^2 + x\_{n+1}^p)$, affect the cotangent sheaf).
**Characteristic 0.**In characteristic $... | 10 | https://mathoverflow.net/users/13265 | 289462 | 127,639 |
https://mathoverflow.net/questions/289476 | 10 | The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in the $(\infty, 1)$ setting and can reasonably follow Lurie's arguments in Higher Algebra as to why $Sp$ has such a structur... | https://mathoverflow.net/users/101861 | What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra? | Lurie characterizes the symmetric monoidal structure on $\mathsf{Sp}$ by a universal property (HA.4.8.2.19): it is uniquely determined up to a contractible space of choices by the property that $S^0$ is the unit and $\wedge$ commutes with homotopy colimits in both variables.
I think on first glance this sounds like ... | 32 | https://mathoverflow.net/users/6936 | 289478 | 127,644 |
https://mathoverflow.net/questions/289480 | 0 | If $r\in\Bbb Z\_{\geq0}$ and $m$ is odd then let $2^\ell\mid\binom{m}{2^r}$ and $2^{\ell+1}\nmid\binom{m}{2^r}$.
>
> Is there a way to find if $\ell$ is even or odd without computing $\binom{m}{2^r}$ (assume $\ell$ odd if $m<2^r$)?
>
>
>
[Lucas theorem](https://en.wikipedia.org/wiki/Lucas%27s_theorem) states ... | https://mathoverflow.net/users/10035 | Divisibility criterion of binomial coefficients | The exponent $\ell$ is the same as the maximal $\ell$ for which $2^\ell$ divides $\lfloor m/2^r\rfloor$. This follows from the criterion with carries or may be proved directly.
| 0 | https://mathoverflow.net/users/4312 | 289482 | 127,645 |
https://mathoverflow.net/questions/288850 | 10 | Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?
Currently, persistent homology is usually done over a field (like $\mathbb{Z}/2$), as the algorithms for producing the barcode only work for a field.
However, working over a field loses a lot ... | https://mathoverflow.net/users/83274 | Persistent homology over the integers | As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modules over the polynomial ring $R[t]$. If (and only if!) $R$ is a field, $R[t]$ is a principal ideal domain and isomorphism... | 10 | https://mathoverflow.net/users/18263 | 289485 | 127,647 |
https://mathoverflow.net/questions/289441 | 10 | Let $F(x,y) = a\_3 x^3 + a\_2x^2 y + a\_1 xy^2 + a\_0 y^3$ be a binary cubic form, say with real coefficients. Put $H(x,y) = H\_F(x,y)$ for the *Hessian covariant* of $F$, defined by
$$\displaystyle H\_F(x,y) = \frac{1}{4} \begin{vmatrix} F\_{xx} & F\_{xy} \\ F\_{xy} & F\_{yy} \end{vmatrix},$$
and put
$$\displays... | https://mathoverflow.net/users/10898 | A curious identity involving a covariant of binary cubic forms | As Abdelmalek notes, if $G$ is to be the cubic covariant $G\_F$ of another form $F$, and has a nonzero discriminant of the expected shape $\Delta(G)=3^6n^3$, then $F$ must divide $G'=G\_G$ - indeed the only candidate (comparing discriminants and exploiting homogeneity) is $F=-G\_G/(3^6n^2)$, which will have rational co... | 6 | https://mathoverflow.net/users/49003 | 289492 | 127,649 |
https://mathoverflow.net/questions/289373 | 7 | Recall that the Silver forcing $\mathbb{P}$ is defined as the set of all partial functions $p\in 2^{\le\omega}$ such that $\omega\setminus dom(p)$ is infinite. As usual, $p\le\_\mathbb{P}q$ if $p$ extends $q$. The Silver model is the model $V$ obtained by the countable support iteration of length $\omega\_2$ of the Sil... | https://mathoverflow.net/users/15860 | Silver forcing and Cichoń's diagram | Answer to question 2: $cof(\mathcal N)=\aleph\_1$, so all cardinals in Cichoń's diagram stay small (and their smallness is witnessed by the set of reals from the ground model).
Proof sketch: For $q\le p$ in Silver forcing, write $q\le\_n p$ if $\omega\setminus dom(p)$ and $\omega\setminus dom(q)$ agree on their firs... | 5 | https://mathoverflow.net/users/14915 | 289499 | 127,652 |
https://mathoverflow.net/questions/289502 | 7 | I'm confused and probably have a thinking error. Exercise [6 on page 420](https://books.google.com/books?id=6iLUBwAAQBAJ&pg=PA420) of Lam's *Lectures on Modules and Rings* says essentially:
>
> Let $R$ be a ring and $C$ a cyclic right $R$-module: $C=R/A$ with some right ideal $A$ in R. Let $(-)^{\*}$ denote the fun... | https://mathoverflow.net/users/61949 | Where is my mistake in calculating duals? | If $C$ is a right module then $C^\*$ is a left module.
In Landrock, $\text{soc}(R)$ is the *right* socle. As he proves, it is a two-sided ideal, but it may not be semisimple as a *left* module (it is not necessarily equal to the left socle). Your example illustrates this: the right socle is spanned by $x$ and $yx$, b... | 13 | https://mathoverflow.net/users/22989 | 289503 | 127,653 |
https://mathoverflow.net/questions/289355 | 7 | Jensen proved that under $\Diamond$ there is a homogeneous Suslin continuum, so the square of a ccc homogeneous space can fail to be ccc. What about ccc topological groups?
>
>
> >
> > Is there a ccc topological group whose square is not ccc?
> >
> >
> >
>
>
>
The obvious thing to try would be the free to... | https://mathoverflow.net/users/11647 | The square of a ccc topological group | Yes. There are such groups after adding a Cohen real (see Theorem $4$ in "Nonpreservation of properties of topological groups on taking their square", Malykhin, 1987) or under RVM (see Theorem $0^c$ in "Some applications of S and L combinatorics", Todorcevic, 1993).
| 4 | https://mathoverflow.net/users/17836 | 289508 | 127,656 |
https://mathoverflow.net/questions/289494 | 10 | Inspired by [this question](https://mathoverflow.net/questions/264513/mathieu-group-m-23-as-an-algebraic-group-via-additive-polynomials), in particular by the indeed elegant description of the Mathieu group $M\_{23}$ it starts with, I am wondering about the following:
Instead of $C$, defined as the multiplicative su... | https://mathoverflow.net/users/29783 | This group is "dual" to the Mathieu group $M_{23}$. Is it known? | This group is the semidirect product $H=C\_{89}\rtimes C\_{11}$. Note that $H\le G$, where $C\_{89}$ is multiplication by elements of order $89$ (and $1$), and $C\_{11}$ is generated by the Frobenius automorphism $x\mapsto x^2$ of $\mathbb F\_{2^{11}}$.
As $89$ is prime, $G$ is a primitive group. Primitive groups of ... | 10 | https://mathoverflow.net/users/18739 | 289522 | 127,663 |
https://mathoverflow.net/questions/289523 | 3 | Maybe this question is not suitable for this platform, I already put that same question in math.stackexchange and I find only vague answers.
I'm studying the book of Rick Miranda; Algebraic Curves and Riemann Surfaces. I'm studying about degree of projective curves and I find a term used very often and that is very ... | https://mathoverflow.net/users/29836 | Meaning of general hyperplane $H$ in $\mathbb{P}^n$ | The sentence:
>
> Let $H$ be a general hyperplane. Then do so and so...
>
>
>
Means:
>
> Pick a hyperplane $H$. Then do so and so, while keeping in mind that the "so and so" might sometimes not work out/be false/be impossible to do.
>
>
>
The use of the word *general* means that the *set of hyperplan... | 8 | https://mathoverflow.net/users/5690 | 289527 | 127,665 |
https://mathoverflow.net/questions/289377 | 6 | Let $X$ be a real Banach space and $Y\subset X$ be a (closed) subspace of $X$. Assume that a sequence $y\_n^\*\in S\_{Y^\*}$ weak\*-converges to some $y^\*\in S\_{Y\*}$. (Here $S\_{Y^\*}$ stands for the dual unit sphere.)
Let $x\_0 \in X \setminus Y$ and define $Z:= \textrm{span} (Y\cup \{x\_0\})$.
Is the following... | https://mathoverflow.net/users/119014 | Extending a weak*-converging sequence onto a superspace | The norm preserving condition, is as Mikhail points out, essential to prove that the such an extension does not exist. The question is stated for Banach spaces, but notice that the same question for normed spaces is equivalent to the question for Banach spaces because a bounded sequence of functionals converges weak$^\... | 6 | https://mathoverflow.net/users/2554 | 289536 | 127,667 |
https://mathoverflow.net/questions/289531 | 2 | Let $K$ be a field, $\alpha, \beta \in \mathrm{Br}(K)$, let $X,Y$ be their Brauer-Severi Varieties, is there a way to calculate $A^\*(X\times Y)$?
For example, if $\alpha,\beta$ both has degree $5$, $2\alpha=\beta$, then $A^\*(X\times Y)$ is a subring (Will two non-rational equivalents cycles become rational equival... | https://mathoverflow.net/users/nan | Chow ring of product of Brauer-Severi Varieties | In general I think it is fairly subtle.
There are some partial results though.
If $\beta$ is in the subgroup of $\text{Br}(K)$ generated by $\alpha$ then the ring $\text{CH}(X\times Y)$ is isomorphic to a direct sum of shifted copies of $\text{CH}(X)$. In this case, the claim is due to the fact the product of Severi-... | 1 | https://mathoverflow.net/users/65919 | 289537 | 127,668 |
https://mathoverflow.net/questions/289520 | 4 | We say (according to <https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+with+duals>) that a symmetric monoidal $(\infty,1)$ category $\mathcal{C}$ *has duals* if its homotopy category $h\mathcal{C}$ is rigid monoidal.
I'm interested in the $\infty$-category $Sp$ of spectra. What is the largest stable $\infty... | https://mathoverflow.net/users/101861 | Dual objects in the $\infty$-category of spectra | As requested, the comments turned into answers:
1. The dualizable objects in spectra are precisely the finite spectra (i.e. spectra of the form $\Sigma^{-k}\Sigma^{\infty}X$ where $X$ is a finite complex.)
2. If you only want the statement 'dualizable objects are finite spectra and their retracts' there is a very for... | 8 | https://mathoverflow.net/users/6936 | 289551 | 127,673 |
https://mathoverflow.net/questions/289534 | 4 | I am looking for a proof of the following statement:
Let $f: X \to B$ be a surjective morphism between smooth projective varieties
such that $-K\_X$ is nef and $B$ is non-uniruled then Kodaira dimension of base $\kappa(B)= 0$. What about when we replace projective varieties with "Kähler manifolds"
| https://mathoverflow.net/users/119052 | Kodaira dimension of the base | I think you are looking for [Q. Zhang, On projective varieties with nef anticanonical divisors, Math. Ann. 332 (2005), 697–703.]
See also [Meng Chen; Qi Zhang. On a question of Demailly-Peternell-Schneider. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1853–1858] for a generalization to tell you that in fact $K\_B\sim\... | 2 | https://mathoverflow.net/users/42636 | 289553 | 127,674 |
https://mathoverflow.net/questions/289547 | 2 | Notation: Let $K$ be a subset of natural numbers $\mathbb{N}$. We set $$\delta(K)=\lim\limits\_{n\rightarrow \infty}\frac{1}{n}|\{k\in K:k\leq n\}|.$$
Question: Let $(a\_{n})\_{n\in \mathbb{N}}$ be a sequence of reals such that $\lim\limits\_{n\rightarrow \infty}a\_{n}=0$. Is there a set $K=\{k\_{j}:j\in \mathbb{N}\}... | https://mathoverflow.net/users/41619 | A question on statistically convergent sequences | No, this does not hold in general. If we simply let $a\_n=\frac1n$, then the given assumptions imply $\delta(K)=0$.
Note that each $k\_j\ge j^3$, hence on any interval $[1,n]$ with $j^3\le n< (j+1)^3$ there are at most $j$ many elements of $K$. So for all $n$,
$$\frac1n|\{k\le n:k\in K\}|\le \frac{j}n\le\frac{\sqrt[3... | 2 | https://mathoverflow.net/users/4600 | 289558 | 127,675 |
https://mathoverflow.net/questions/289559 | 11 | Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L\_{A/\kappa}^{\operatorname{alg}}$ is known to vanish, due to the Frobenious automorphism having simultaneously to induce on the cotangent complex an automorphism and ... | https://mathoverflow.net/users/39713 | Cotangent complex of perfect algebra over a perfect field | Let me explain why the $E\_\infty$-cotangent complex $L\_{B/A}$ vanishes for any map $A \to B$ of perfect rings over $\mathbf{F}\_p$. (I do not know the answer to the more general question at the end.)
The proof uses formal properties of the cotangent complex (Kunneth formula, transitivity triangle) and relies on the... | 7 | https://mathoverflow.net/users/117273 | 289575 | 127,679 |
https://mathoverflow.net/questions/289560 | 17 | *There's already a [question](https://mathoverflow.net/questions/177367/reference-request-for-instantons) about the same topic but I think its aim is different.*
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely differential-geometric terms (where the m... | https://mathoverflow.net/users/22810 | What is an "Instanton" in classical gauge theory? (to a mathematician) | A linguistic remark: "Instantons" are the same mathematically to "solitons", particle-like solutions of classical field theories (explaining the suffix "on"). Unlike solitons, instantons are structures in time (explaining the prefix "instant").
A mathematical remark (using Donaldson's book on Yang-Mills Floer homolog... | 12 | https://mathoverflow.net/users/12310 | 289594 | 127,688 |
https://mathoverflow.net/questions/289590 | 1 | In my research on linear algebra and optimization, I have come across the following problem repeatedly:
>
> Given constant matrices $C\in\mathbb{R}^{k \times k}$ and $X\in\mathbb{R}^{n \times n}$, $$\min\_{A\in\mathbb{R}^{n\times k}, B\in\mathbb{R}^{k \times n}} \| X - A C B X \|\_F$$
>
>
> where $C$ may be singu... | https://mathoverflow.net/users/69446 | Possible analytical way to solve or approximate a specific optimization problem's solution | $ACB$ ranges over all the matrices with rank smaller or equal to the rank of $C$, so this is equivalent to a problem with $C=I$ (and possibly with a smaller $k$). That said, it is not clear to me how you planned to solve the problem with $C=I$. It's not the standard setup of low-rank approximation with SVD (Eckart-Youn... | 1 | https://mathoverflow.net/users/1898 | 289595 | 127,689 |
https://mathoverflow.net/questions/289571 | 1 | In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}\_{m\times n}$ and $q\in\mathbb{R}^n$. Define
$$x^\* := \arg\min\_x\,\tfrac{1}{2} x^\mathsf{T} P x - q^\mathsf{T}x$$
and
\begin{align}
x\_c^\* &:= \arg\min\_x \, \tf... | https://mathoverflow.net/users/108236 | Norm of solution of quadratic program | We can show more, namely that if $K$ is a closed convex set (such as $\ker A$) containing the origin and
\begin{align\*}
x\_c^\* &= \operatorname\*{argmin}\_x \,\frac{1}{2}x^\mathsf{T} P x - q^\mathsf{T}x\\
&\quad\,\,\operatorname{subj.to}\,\,x\in K
\end{align\*}
then $\|x\_c^\*\|\_P \leq \|x^\*\|\_P$. To see this, not... | 1 | https://mathoverflow.net/users/108236 | 289601 | 127,690 |
https://mathoverflow.net/questions/289240 | 2 | There seems to be many ways to obtain a 1-category out of a 2-category:
1. Dumb truncation. $\delta: 2\text{-Cat} \to \text{Cat}$ sends a 2-category $\cal K$ into the 1-category obtained forgetting the 2-cells. Only works with strict 2-categories.
2. Core truncation. $c : 2\text{-Cat} \to \text{Cat}$ sends a 2-catego... | https://mathoverflow.net/users/7952 | Co/completeness of truncated 2-category | 1. Every strict conical 2-limit in $K$ is also a 1-limit in $K^\delta$. So if $K$ has all of those, then $K^\delta$ is complete. (This is a special case of a general fact about enriched categories, since $K^\delta$ is the underlying ordinary category of the $\mathrm{Cat}$-enriched category $K$ in the sense of enriched ... | 2 | https://mathoverflow.net/users/49 | 289602 | 127,691 |
https://mathoverflow.net/questions/289369 | 16 | The following problem (call it THEOREMS) belongs to class NP.
* **Input:** Mathematical statement $S$ (written in some formal system such as ZFC) and positive integer $n$ written in unary.
* **Output:** "Yes" if $S$ has a formal proof of length at most $n$. "No" otherwise.
It is known that there is an algorithm whi... | https://mathoverflow.net/users/31472 | Representing mathematical statements as SAT instances | Improving on Brumleve's answer, I have a method for encoding a length-$n$ proof with a quasilinear $\tilde {O} (n)$-bit 3SAT instance.
In most formal systems, proofs and objects that appear in proofs (propositions, formulas, etc.) have a tree-like structure in that each such object can be built from other such object... | 6 | https://mathoverflow.net/users/41947 | 289612 | 127,696 |
https://mathoverflow.net/questions/289564 | 6 | A $n-1$ dimensional submanifold $N\subset \mathbb{R}^n$ is called a convex submanifold if for every $x\in N$ ,ther is a neighborhood $W$ of $x$ in $N$ such that $W$ entirly lies at one side of $T\_x N$. A (local) diffeomorphism $\phi$ on $\mathbb{R}^n$ is called a convex diffeomorphism if $\phi$ and its inverse preserv... | https://mathoverflow.net/users/36688 | The concept of convex foliation | If $n>1$, and a smooth diffeomorphism $f:U\to V$ (where $U$ and $V$ are, say, convex, open subsets of $\mathbb{R}^n$) carries convex sets to convex sets, then it is easy to show that it must carry each intersection $U\cap H$, where $H\subset\mathbb{R}^n$ is a hyperplane, to an intersection $V\cap H'$, where $H'\subset\... | 7 | https://mathoverflow.net/users/13972 | 289621 | 127,697 |
https://mathoverflow.net/questions/289615 | 4 | Kunen showed that there is no nontrivial $j: V \rightarrow\_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.
Let's denote the universe by $U$. We aren't assuming that the atoms form a set, or even that there are no more atoms than pure sets. Of course, if there at least two atoms, there will be nont... | https://mathoverflow.net/users/91635 | Kunen inconsistency with atoms | **Theorem.** The Kunen inconsistency works over ZFC with atoms.
That is, in this theory, there is no non-identity elementary
embedding $j:V\to V$ that fixes every atom.
**Proof.** Suppose that $j:V\to V$ is an elementary embedding
fixing every atom. If $j$ is not the identity embedding, then I
claim that $j$ must mov... | 11 | https://mathoverflow.net/users/1946 | 289625 | 127,700 |
https://mathoverflow.net/questions/289443 | 3 | Let $X$ be a real $n\times n$ positive semidefinite matrix of rank $m\le n$ and let $Y\in\mathbb{R}^{m\times n}$ be the unique matrix satisfying (i) $X=Y^\top Y$, and (ii) $Y\, [I\, |\, 0]^\top = L$ with $L\in\mathbb{R}^{m\times m}$ being upper triangular with positive diagonal entries. (Notice that when $n=m$, $Y$ coi... | https://mathoverflow.net/users/62673 | Closed-form expression for differential of matrix function | It suffices to consider the case $n=2$, $m=1$. Namely, write $Y=[Y\_1|Y\_2]$ etc, then $L=Y\_1$ is upper triangular with positives on the diagonal, and
\begin{align\*}
X &= Y^\top Y = \begin{pmatrix} Y\_1^\top Y\_1 & Y\_1^\top Y\_2 \\ Y\_2^\top Y\_1 & Y\_2^\top Y\_2\end{pmatrix}\,,
\\
dX &= \begin{pmatrix} (dY\_1)^\to... | 4 | https://mathoverflow.net/users/26935 | 289626 | 127,701 |
https://mathoverflow.net/questions/289630 | 3 | Let $X$ be a topological space with a dense subset $D\subseteq X$. Suppose that every open cover of $X$ has a finite subfamily which covers $D$. Can I conclude that $X$ itself is compact?
The answer is clearly negative in general: one can take any space $X\_0$ and form a new space $X:=X\_0\cup\{\infty\}$ by adjoining... | https://mathoverflow.net/users/27013 | Checking finite subcover property on dense subset | Regularity is enough. Given an open cover $\mathcal{A}$ of $X$ we can use regularity to build another cover $\mathcal{B}$ such that for any $U \in \mathcal{B}$ there is a $V \in \mathcal{A}$ with $\overline U \subseteq V$. Now a finite subset of $\mathcal{B}$ covering $D$ naturally provides a finite subset of $\mathcal... | 6 | https://mathoverflow.net/users/17836 | 289636 | 127,704 |
https://mathoverflow.net/questions/289628 | 0 | Suppose we define "finite set" as a set on which there can exist a cyclic path that passes through all of its elements. (The details of the formulation are present below).
I have two questions:
1. Is the above definition equivalent to a known definition of "finite set"?
2. If the answer to 1. is yes, then to which... | https://mathoverflow.net/users/95347 | To which of the known definitions of 'finite set' this graphical definition is equivalent? | It seems to me that your definition is equivalent to the usual definition of finite set in set theory (a set is finite if it has $n$ elements for some natural number $n$).
It is clear that any finite set satisfies your definition, since we can easily build cycles on an $n$-element set.
Conversely, if we have such... | 0 | https://mathoverflow.net/users/1946 | 289637 | 127,705 |
https://mathoverflow.net/questions/289643 | 2 | When atomhood is definable, the answer is clearly yes, so essentially the question is interesting primarily when we are working in ZF with extensionality weakened to apply only to inhabited sets. In this case, we may as well include the empty set with the atoms. So, if we have an elementary embedding from $V\_0$ to $V\... | https://mathoverflow.net/users/90758 | Does every elementary embedding $j:V \to V$ in ZFA arise from a self-injection on the class of atoms? | In ZFCA, the answer is yes, every elementary embedding $j:V\to V$
is the unique extension to $V$ of an injection on the atoms. If the
class of atoms is a set, then it must be a permutation of the
atoms.
On the one hand, every injection $\pi:A\to A$ on the class $A$ of
atoms extends naturally to a map defined on all o... | 4 | https://mathoverflow.net/users/1946 | 289648 | 127,711 |
https://mathoverflow.net/questions/289635 | -1 | let $ S = \sum\_{k=0}^\infty (-1)^k (k!)a\_k $ a divergent series such that $b\_k=(-1)^k (k!)a\_k >0 $ for $k>1$ , and $b\_k$ signed [this](https://oeis.org/A214645/list) from $k=1$ to $20$ ,The asymptotic of the titled series is :$$ A(x) = x + x^2/2! + 3\*x^3/3! + 16\*x^4/4! + 126\*x^5/5! + 1333\*x^6/6!+\cdots$$
> ... | https://mathoverflow.net/users/nan | Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence? | **Note**:This is not an answer but probably helping you to get the answer of your question.just to show some properties related to your formel series .
we denote by $f$ your divergent power series
,The formel series you have is non analytic function and it is smooth ,it's has a positive radius of convergence.Then, it e... | -1 | https://mathoverflow.net/users/51189 | 289651 | 127,712 |
https://mathoverflow.net/questions/289620 | 4 | Let $k=\mathbb{F}\_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1\_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-zero.
Let $K = k(t)$ be the function field of $\mathbb{P}^1\_k$. It is a global field.
Is the $K/\mathbb{F}\_q$-tr... | https://mathoverflow.net/users/119125 | On families of supersingular abelian surfaces over the projective line | I will answer the question for the specific family of abelian surfaces as constructed by Moret-Bailly [MB]. There might be other types of examples for which the answer is different (?). We will recall the construction here:
**Lemma** [MB]. *There exists a non-isotrivial family $\mathscr A \to \mathbb P^1\_{\mathbb F\... | 4 | https://mathoverflow.net/users/82179 | 289654 | 127,714 |
https://mathoverflow.net/questions/289669 | 1 | Playing around with [this series](https://mathoverflow.net/questions/289662/series-involving-gamma-function) for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum\_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{(a+c)!(b+c)!}$$ and obviously the factorial... | https://mathoverflow.net/users/29783 | Series involving factorials | The sum
$$\sum\_{k=0}^\infty \frac{(a+k)!\,(b+k)!}{k!\,(a+b+c+k+1)!}z^k.$$
is not only a generalized hypergeometric series; it's [the original ungeneralized Gauss hypergeometric series](https://en.wikipedia.org/wiki/Hypergeometric_function),
$$\frac{\Gamma(a+1)\,\Gamma(b+1)}{\Gamma(a+b+c+2)}{}\_2F\_1\left({a+1,b+1\atop... | 14 | https://mathoverflow.net/users/10744 | 289677 | 127,717 |
https://mathoverflow.net/questions/289650 | 11 | In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is
given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior algebra of a manifold. For what it's worth, the Lagrangian in question is
$$L = \frac{1}{2} g\_{ij}(\phi) \dot{\phi}\_i \dot... | https://mathoverflow.net/users/119133 | supersymmetry and the de Rham complex | I think there is a typo in the references to "Supersymmetry and Morse theory", [21] should be replaced by [22]="Constraints on supersymmetry breaking". The quantization of non-linear sigma models and its relation with the de Rham complex is discussed in Section 10 of this paper.
In addition to the big "Mirror symmetr... | 10 | https://mathoverflow.net/users/25309 | 289678 | 127,718 |
https://mathoverflow.net/questions/289683 | 15 | Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent.
Must there exist a nonzero real linear combination $aA + bB$ which has a repeated eigenvalue?
| https://mathoverflow.net/users/23141 | Existence of double eigenvalue | The answer is 'no'. The generic pair $A$ and $B$ of $4$-by-$4$ Hermitian symmetric matrices will not have any nonzero real linear combination that has a double eigenvalue.
For a specific example, take
$$
A = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&-2&0\\0&0&0&2\end{pmatrix}
\quad \text{and}\quad
B = \begin{pmatrix}0&i... | 23 | https://mathoverflow.net/users/13972 | 289687 | 127,720 |
https://mathoverflow.net/questions/289679 | 6 | Let $A$ be a selfinjective algebra and for an indecomposable module $M$ define $\psi\_M:= \inf \{ i \geq 1 | Ext\_A^i(M,M) \neq 0 \}$.
Questions:
1. In case $A$ is symmetric, do we have $\psi\_M \leq max \{ \psi\_S | S $ is simple $\}$ for each indecomposable non-projective module $M$?
This should be true in case $... | https://mathoverflow.net/users/61949 | Ext in symmetric algebras and group algebras | I think this example answers both questions.
Let $k$ have characteristic $3$, and let $G=C\_3\times S\_3$.
Then $kG$ has two simple modules, both one-dimensional, and for each simple module $S$, $\text{Ext}^1(S,S)$ is one-dimensional.
But if $M=kC\_3$, with $S\_3$ acting trivially, then $\text{Ext}^i(M,M)=0$ for ... | 5 | https://mathoverflow.net/users/22989 | 289695 | 127,725 |
https://mathoverflow.net/questions/289689 | 6 |
>
> Every discrete central subgroup of a connected Lie group is finitely generated.
>
>
>
This result was alluded to without comment in a book I was reading (Lie Group Actions in Complex Analysis by D. Akhiezer Proposition on page 38).
Assuming it was a trivial result, I posted to math.stackexchange where YCor... | https://mathoverflow.net/users/105628 | Discrete central subgroup of a connected Lie group is finitely generated | An old result of Iwasawa is that in any connected Lie group $G$, every compact subgroup is contained in a maximal compact subgroup, and all maximal compact subgroups are conjugate.
Let $G$ be a connected Lie group and let $Z$ be a discrete central subgroup. Then $Z$ has an infinite torsion quotient $Z'=Z/B$ (lemma be... | 4 | https://mathoverflow.net/users/14094 | 289700 | 127,729 |
https://mathoverflow.net/questions/289711 | 35 | From a [recent answer](https://mathoverflow.net/questions/289259/the-derived-drift-is-pretty-unsatisfying-and-dangerous-to-category-theory-or/289609#289609) by Mike Shulman, I read:
>
> "HoTT is (among other things) a foundational theory, on roughly the same ontological level as ZFC, whose basic objects can be rega... | https://mathoverflow.net/users/5690 | Defining $SU(n)$ in HoTT | This isn't easy to do, and the reason it isn't easy is because of the step "$\infty$-groupoids are the same thing as spaces." Of course the homotopy hypothesis tells you that any $\infty$-groupoid is equivalent to the fundamental $\infty$-groupoid of a space, but that doesn't mean that they're literally the same thing.... | 26 | https://mathoverflow.net/users/22 | 289713 | 127,734 |
https://mathoverflow.net/questions/289646 | 7 | This seems like a really basic question, but I somehow don't know and haven't been able to find the answer.
I suspect that (at least under suitable assumptions) there should be a relation between the following two constructions, but I'm looking for a precise theorem.
If I have a family $X$ over a disk, which I'm ... | https://mathoverflow.net/users/84144 | intersection cohomology and nearby cycles | Let me give a quick and lazy answer. Throughout all statements are up to shift in the derived category.
Let us assume that $X$ and $f : X \to \eta$ is smooth. Then the nearby cycles sheaf $\Psi\_f \mathbb{Q}\_X$ is generically a constant sheaf (generically the "nearby cycles" are points, in other words generically th... | 7 | https://mathoverflow.net/users/919 | 289722 | 127,738 |
https://mathoverflow.net/questions/288936 | 1 | Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very degenerate, because all the energy levels with close energy values do not contain periodic trajectories: a really weird pi... | https://mathoverflow.net/users/106583 | Isolated periodic trajectories of Hamiltonian systems | If we take $H=I + \frac13(p^3 -q^3) + I^2(p-q)$ with $\omega =dI\wedge d\theta + dp\wedge dq$,
then
$\dot q = p^2 + I^2$
$\dot p = q^2 + I^2$
$\dot I = 0$
$\dot\theta = 1 + 2I(p-q)$
and the only periodic orbit is for $q=p=I=0$.
| 2 | https://mathoverflow.net/users/85369 | 289725 | 127,739 |
https://mathoverflow.net/questions/289721 | 4 | The following seems to be useful, and probably well-known, but I can't find a reference for it. If anyone can point me to a textbook or paper which states it, then I'd be grateful.
Consider a partition of $(A, B)$ of a finite set $X$. That is, $X = A \cup B$ and $A \cap B = \emptyset$. Suppose further that the size o... | https://mathoverflow.net/users/57425 | Partitions of finite sets and their behavior under permutations of the set | (Not an answer, but too long for a comment.) What if $X = \{1,2,3,4,5,6\}$, $A=\{1,2,3\}$, and $B=\{4,5,6\}$; and $\sigma\_1=(3,6)$, $\sigma\_2=(2,6)\sigma\_1^{-1} = (2,6,3)$. Anyway $\sigma\_2 \sigma\_1 = (2,6)$.
Now
* $X\_A = A = \{1,2,3\}$,
* $X\_{AA} = X\_A \cap \sigma\_1^{-1}(A) = A \cap \{1,2,6\} = \{1,2\}$, ... | 2 | https://mathoverflow.net/users/88133 | 289727 | 127,740 |
https://mathoverflow.net/questions/289733 | 5 | I wonder if there is an example of a knot $K$ in the 3-sphere which can be realized as cables of two distinct (up to isotopy) knots $K\_1 \neq K\_2$.
It is known that if a knot $K$ is the $(p,q)$-cable of another knot $K'$, then there is a unique annulus in its exterior $X\_K$ with slope $pq$ on $\partial X\_K$. This... | https://mathoverflow.net/users/88357 | Can a knot be cables of two different knots? | Yes, $K'$ is uniquely determined by $K$ and so, no, a knot cannot be a cable in two different ways. This follows from the [Gordon-Luecke theorem](https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem) and from a result of [Feustel and Whitten](https://cms.math.ca/openaccess/cjm/v30/cjm1978v30.1284-1295.pdf). See ... | 8 | https://mathoverflow.net/users/1650 | 289742 | 127,744 |
https://mathoverflow.net/questions/91371 | 23 | Let $\Delta(s\_1,s\_2,\ldots,s\_n) := \prod\_{i<j}(s\_i-s\_j)^2$. Is there a standard way to estimate the decay of the Selberg-type integral
$$ I\_n:= \frac{1}{n!^2}\int\_0^1 \int\_0^1\cdots\int\_0^1 \frac{\Delta(s\_1,s\_2,\ldots,s\_n) \Delta(t\_1,t\_2,\ldots,t\_n)}{\prod\_{i,j}(1-s\_i t\_j)^2} d s\_1\ldots d s\_n d t\... | https://mathoverflow.net/users/7831 | Asymptotics of a Selberg-type integral | You can follow large deviation-type estimates from random matrix theory, starting with Ben Arous & Guionnet's paper: <https://link.springer.com/article/10.1007/s004400050119>
You will eventually obtain that your Selberg-type integral behaves like
$$
\frac{1}{(n!)^2}e^{-n^2 E\_\*}
$$
where $E\_\*$ is the minimum of t... | 8 | https://mathoverflow.net/users/15517 | 289746 | 127,745 |
https://mathoverflow.net/questions/289696 | 1 | If $G=(V,E)$ is a loopless finite directed graph and $v\in V$, we set $\text{In}(v) = \{(w,v): w\in V \land (w,v) \in E\}$.
Let $T=(V,E)$ be a [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) such that for every $v\in V$ the set $\text{In}(v)$ contains at least $2$ elements. Is there a map $c: V... | https://mathoverflow.net/users/8628 | Sum-coloring a tournament | No, there isn't. For a counterexample, let $T$ be the tournament with vertex
set $V=\left\{ 1,2,3,4,5\right\} $ and arc set
\begin{align\*}
E & =\left\{ \left( 1,2\right) ,\left( 1,3\right) ,\left( 2,3\right)
,\left( 2,4\right) ,\left( 3,4\right) ,\left( 3,5\right) ,\left(
4,5\right) ,\left( 4,1\right) ,\left( 5,1\righ... | 4 | https://mathoverflow.net/users/2530 | 289753 | 127,748 |
https://mathoverflow.net/questions/289748 | 5 | For the diffusion equation $\frac{\partial} {\partial t} P\_t(x)=D \frac{\partial^2} {\partial x^2} P\_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x} P\_t(x=0)=0$.
What is the mathematical way of setting the condition that whenever a particle reaches the origin it stays ... | https://mathoverflow.net/users/37545 | How to define (and solve) the diffusion equation with a sticky boundary at the origin? | I would just take an absorbing boundary condition and then add the absorbed density as a delta function at the sticking point. For convenience, translate the origin so that the sticking point is $x\_a>0$ and the particle starts from $x=0$ at $t=0$. The solution then is
$$P(x,t)=f(x,t)-f(2x\_a-x,t)+N(t)\delta(x-x\_a)$... | 4 | https://mathoverflow.net/users/11260 | 289755 | 127,749 |
https://mathoverflow.net/questions/289744 | 0 | Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi\_{t} : M\rightarrow M $ be its flow. Question: Is $\varphi\_{t}$ a biholomorphism? It is a diffeomorphism but is it holomorphic?
| https://mathoverflow.net/users/119173 | flow of holomorphic vector field | The question is local, so assume that $M$ is open in $\mathbb C^n$, so that we do not have to deal with the second tangent bundle. Then:
\begin{align\*}
\partial\_t \phi\_t &= X\circ \phi\_t
\\
T(\partial\_t \phi\_t) &= \partial\_t T(\phi\_t) = TX\circ T(\phi\_t)
\\
\partial\_t J\circ T(\phi\_t) &= J\circ\partial\_t T(... | 1 | https://mathoverflow.net/users/26935 | 289756 | 127,750 |
https://mathoverflow.net/questions/289619 | 6 | Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom\_{Rng}(A,R)$ of ring homomorphisms, where the group operations (multiplication, inverse, unit) are given on this set from the co-operations of the ... | https://mathoverflow.net/users/14443 | Group schemes over ring of Witt vectors and their representing algebras | The Hopf algebra of $G\circ W\_m$ does have an explicit construction. It's appeared in a few of my papers and zillions of Buium's papers, and no doubt many others which aren't coming to mind right now. Some people call $G\circ W\_m$ the `order $m$ arithmetic jet space of $G$' and denote it $J^mG$. The reason is that th... | 4 | https://mathoverflow.net/users/1114 | 289758 | 127,752 |
https://mathoverflow.net/questions/289761 | 0 | Consider a sphere without two poles $U^2$. Will Borsuk–Ulam theorem still work, i.e. $\forall$ continuous functions $f:U^2 \rightarrow \mathbb{R}^2 ~\exists x \in U^2$ such as $f(-x)=f(x)$?
| https://mathoverflow.net/users/112492 | Borsuk–Ulam theorem on the sphere with expluded poles | No, it fails as soon as you remove one point: the stereographic projection is a bijection between $\mathbb{R}^2$ and a sphere minus a point.
| 1 | https://mathoverflow.net/users/35609 | 289765 | 127,755 |
https://mathoverflow.net/questions/289779 | 6 | The answers to this M.O. [question](https://mathoverflow.net/questions/116559/where-do-the-k%C3%A4hler-identities-first-appear) give a history of the [Kaehler identities](http://mathworld.wolfram.com/KaehlerIdentities.html). The identities can be extended to the vector bundle-valued setting, and play a central role in ... | https://mathoverflow.net/users/3072 | Where do the (Akizuki)-Nakano Identities First Appear | * [Curvature and Betti
Numbers](https://www.jstor.org/stable/1969287), Salomon Bochner (1948).
* [On a differential-geometric
method in the theory of analytic stacks](http://www.jstor.org/stable/89226), Kunihiko Kodaira (1953).
* [On complex analytic vector bundles](https://projecteuclid.org/euclid.jmsj/1261414950), Sh... | 4 | https://mathoverflow.net/users/11260 | 289780 | 127,758 |
https://mathoverflow.net/questions/289781 | 8 | Suppose $BS(1,n)$ is the Baumslag-Solitar group and $S\_m$ is the
symmetric group. If $\Phi: BS(1,n) \to S\_m$ is a homomorphism, must the
image of $\Phi$ be abelian?
| https://mathoverflow.net/users/14644 | Do actions of BS(1,n) on finite sets factor through abelian quotients? | No, these groups have many finite non-abelian quotients. Recall that the Baumslag-Solitar group $B(1,n)$ has a presentation
$$
B(1,n)=\langle t,a\,|\,tat^{-1}=a^n\rangle.
$$
A homomorphism $\Phi: B(1,n)\to S\_m$ amounts to a pair $(\tau,\alpha)$ of
permutations, where $\Phi(a)=\alpha$ and $\Phi(t)=\tau$, such that $\t... | 7 | https://mathoverflow.net/users/5740 | 289785 | 127,760 |
https://mathoverflow.net/questions/134449 | 11 | I'm reading the new [HoTT book](http://homotopytypetheory.org/book/) and I'm wondering about a potential equivalent form of the Univalence Axiom: $(A \simeq B) \simeq (A = B)$.
For simplicity, I'm tacitly working in a fixed universe. It is known that the univalence axiom implies function extensionality $$\mathsf{fune... | https://mathoverflow.net/users/2000 | Equivalent form of the Univalence Axiom | Still not an answer to the second question, but I wanted to add something else that's missing: in fact the bare statement $(A=B)\simeq (A\simeq B)$ is not known to be a correct form of the univalence axiom. The correct statement is that *the canonical map* $(A=B) \to (A\simeq B)$ is an equivalence. The statement $(A=B)... | 4 | https://mathoverflow.net/users/49 | 289786 | 127,761 |
https://mathoverflow.net/questions/289763 | 2 | Let $\lambda\_n = n + \delta\_n $ for all $n \in \mathbb{Z}$ where $\delta\_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. How can one show that the sequence $(x\_n)\_{n \in \mathbb{Z}} = (e^{i \lambda\_n t})\_{n \in \mathbb{Z}}$ is minimal in $L^2([-\pi, \pi])$, in the sense that
\begin{equation}
\forall ... | https://mathoverflow.net/users/115381 | A minimal sequence in $L^2([-\pi, \pi])$ | First, you need to say what $\delta\_n$'s are.
I am assuming you meant $0\le \delta\_n\le 1$. This and somewhat more general statement follows from a formula of Carleman: if $\liminf \frac{n}{\lambda\_n}>\frac{A}{\pi}$, then the system is complete in $L\_2[-A,A]$. For detailed proof see the chapter 3 of [Young's boo... | 4 | https://mathoverflow.net/users/3675 | 289788 | 127,763 |
https://mathoverflow.net/questions/289708 | 21 | The [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number) $C\_n$ count both
1. the Dyck paths of length $2n$, and
2. the ways to associate $n$ repeated applications of a binary operation.
We call the latter *magma expressions*; we will explain below.
**Dyck paths, and their lattice structure**
A *Dyc... | https://mathoverflow.net/users/2811 | Is the order on repeated exponentiation the Dyck order? | **EDIT:** I can complete half of the proof, showing that the magma order refines the Dyck order.
---
Following Martin Rubey's comment, there is a standard bijection between association orders and Dyck paths that uses [reverse Polish notation](https://en.wikipedia.org/wiki/Reverse_Polish_notation) (RPN). For $n=3$... | 7 | https://mathoverflow.net/users/3106 | 289802 | 127,769 |
https://mathoverflow.net/questions/289766 | 5 | Let $k$ be a field with algebraic closure $\overline{k}$. Let $f\colon X\to k$ be a smooth projective variety(geometrically connected) over $k$.
Is the base change map $$\phi\_i\colon \mathrm{CH}^i(X)\to\mathrm{CH}^{i}(X\_{\overline{k}})$$ always injective?
(If $i=1$, $\mathrm{CH}^1(X)=\mathrm{Pic}(X)$, the Hochsc... | https://mathoverflow.net/users/nan | Chow group and base change | No, this is not true in general.
A counterexample occurs already for Severi-Brauer varieties. Since the Chow group $\text{CH}(\mathbf{P}^n)$ is torsion free, it's enough to show there are Severi-Brauer varieties with torsion in their Chow groups. This was (I think) first observed in:
>
> Merkurjev, A. S. Certain ... | 5 | https://mathoverflow.net/users/65919 | 289803 | 127,770 |
https://mathoverflow.net/questions/289806 | 3 | This question concerns reduced scheme structure on locally complete intersection, and I guess the answer is related to the number of generators of a radical ideal.
I am confused about the following proposition in Hartshorne's Algebraic Geometry, Chapter 2:
**Proposition 8.23** Let $Y$ be a locally complete intersec... | https://mathoverflow.net/users/119189 | Reduced scheme structure on locally complete intersection | Localization of a regular local ring is still regular. So, if $Y$ is regular in codimension one, then it is also regular in codimension zero, in particular reduced. So, Hartshorne is correct. Your question has a negative answer. For a standard example, take the curve given by $(t^3, t^4, t^5)\subset\mathbb{A}^3, t\in\m... | 6 | https://mathoverflow.net/users/9502 | 289808 | 127,771 |
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