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https://mathoverflow.net/questions/268391 | 2 | In the context of reverse mathematics $WKL\_0$ is considered equivalent to Gödel's completeness theorem over $RCA\_0$. Does this mean that e.g. $WKL\_0$ plus the consistency statement CON(PA+X) gives a binary tree which somehow is a model of PA+X?
| https://mathoverflow.net/users/37385 | Does $WKL_0$ plus CON(PA+X) give a binary tree model of PA+X? | The main idea is that the Henkin proof of the completeness theorem is esentially a paths-through-tree argument.
If $T$ is any consistent theory in a finite language, let $\tau$ be the tree of attempts to build a complete consistent Henkin theory extending $T$. So we add the Henkin assertions to $T$, and then at each... | 6 | https://mathoverflow.net/users/1946 | 268392 | 120,371 |
https://mathoverflow.net/questions/268393 | 5 | Consider a polynomial in $d$ variables, $p:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Denote by $\mathcal{C}$ its set of zeros, i.e.
$$\mathcal{C}=\{x\in\mathbb{R}^{d}\ |\ p(x)=0\}.$$
>
> **Q.** Is it possible to find finitely many (not necessarily disjoint) manifolds $M\_{1},\dots, M\_{n}\subset\mathbb{R}^{d}$, with po... | https://mathoverflow.net/users/109086 | Zeros of polynomials as a finite union of manifolds | The answer is *yes*.
Your set $\mathcal{C}$ is by definition a [(semi)algebraic subset](https://en.wikipedia.org/wiki/Semialgebraic_set) of $\mathbb{R}^d$, and any such a subset has a [Whitney stratification](https://en.wikipedia.org/wiki/Whitney_conditions) with finitely many (semi)algebraic strata.
See A. Dimca... | 10 | https://mathoverflow.net/users/7460 | 268397 | 120,373 |
https://mathoverflow.net/questions/266260 | 3 | Let $f\_n$ be a sequence of $C^1$-maps on closed manifold $M$. If $f\_n \to f$ in $C^1$-topology. Does $M\_{f\_n}$ converges to $M\_f$ in the Hausdorff distance?
We define $M(f)=\{\bar{x}=(x\_j) \in M^{\mathbb{Z}}\mid f(x\_{j-1})=x\_j\}$. To define the Hausdorff distance in $(M^{\mathbb{Z}},d)$, define
$\bar{d}((x\... | https://mathoverflow.net/users/74049 | Inverse limit space and $C^1$ topology | I think that an example of non-convergence can be constructed as follows:
Let $f\colon [0,1]\to [0,1]$ be a $C^\infty$ map such that
$g:=f\big|\_{[0,1/4]}\colon [0,1/4]\to[0,1/2]$ is an increasing
diffeomorphism; $f\big|\_{[1/4,3/4]}\equiv 1/2$ and
$f\big|\_{[3/4,1]}\colon [3/4,1]\to[1/2,1]$ is an increasing
diffeomo... | 1 | https://mathoverflow.net/users/889 | 268398 | 120,374 |
https://mathoverflow.net/questions/268384 | 3 | Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M\_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B\_k(r)$ is an open ball with radius $r$ in $M\_{k,n}$.
Question: It is true that $H^{n-1}(\partial B(x,r)) \leq H^{n-1}(\partial B\_k(r)) ... | https://mathoverflow.net/users/84068 | Area of a sphere in Alexandrov spaces | Yes, it is true.
In fact there is is a distance non-contracting map $\ell\colon\partial B(x,r)\to \partial B\_k(r)$.
If $x$ is a regular point then $\ell$ is the $k$-logarithm --- it sends point $z$ to the point $\tilde z$ such that geodesics $[xz]$ and $[\tilde x\tilde z]$ go in the same direction for some identi... | 2 | https://mathoverflow.net/users/1441 | 268403 | 120,378 |
https://mathoverflow.net/questions/268199 | 15 | People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask for help in "navigating" in this matter. So, which advantages does each of the model have? Does any "guide" for this su... | https://mathoverflow.net/users/2191 | What are the advantages of various "models" for the motivic stable homotopy category | The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model structure is also monoidal. See Hovey's "Spectra and symmetric spectra in general model categories", where he introduces the pr... | 10 | https://mathoverflow.net/users/11540 | 268405 | 120,379 |
https://mathoverflow.net/questions/268305 | 14 | I don't even know where to begin. There's a discussion of stacks and they talk about $\mathrm{Bun}(G)$. I don't know what it is, or what it's elements are or why it is important. Google and wikipedia don't really help since they pre-suppose.
One [resource](https://web.stanford.edu/~ebwarner/uniformizationofBunG.pdf)... | https://mathoverflow.net/users/1358 | what is $\mathrm{Bun}(G)$? | I am surprised that nobody has mentioned (yet) that an essential point of the algebro-geometric notions of (algebraic) stack and algebraic space is to completely shift the burden of construction problems: one *gives up* on trying to make any kind of actual ringed space at all, and in fact the whole point is to create a... | 37 | https://mathoverflow.net/users/81332 | 268406 | 120,380 |
https://mathoverflow.net/questions/268169 | 14 | By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- what can I say about the spectrum of this matrix?
For example, when the convex body $B$ is just the unit ball by logi... | https://mathoverflow.net/users/108984 | References for reasoning about the spectrum of a convex body? | A very direct result giving characteristic function control for uniform measures on **compact convex sets** and hence spectrum is [Kulikova& Prokhorov].
It sounds to me that all you want to study is the uniform distribution's spectrum on a convex body $K$. With some additional symmetries it is not surprising that [[... | 8 | https://mathoverflow.net/users/25437 | 268420 | 120,383 |
https://mathoverflow.net/questions/268193 | 7 | In a category $\mathcal{C}$, a *generalized element* of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements of $A$.
In a monoidal category $(\mathcal{C},\otimes)$, it seems more natural to consider generalized elements of a... | https://mathoverflow.net/users/43795 | generalized elements in monoidal categories | If I'm not wrong, the following is an answer to 2.
For any objects $A,B,C$ of $\mathcal{C}$, we have the following natural map
\begin{align}
\Phi\colon\mathrm{Hom}(A\otimes B,C) &\to \mathrm{Nat}(h\_A\boxtimes h\_B,h\_C\circ\otimes) \\
f &\mapsto f\circ-
\end{align}
where
* $h\_A\boxtimes h\_B$ denotes the functor ... | 2 | https://mathoverflow.net/users/43795 | 268431 | 120,389 |
https://mathoverflow.net/questions/268396 | 0 | Let $G=(V,E)$ be a graph with minimal degree $\delta(G) \geq 2$. The set of vertices $V(G)$ can be written as a union of cycles.
Question: is there a universal constant $k\in \mathbb{N}$ such that every graph $G$ with $\delta(G) \geq 2$ can be written as a union of a collection ${\cal C}$ of cycles (not necessarily ... | https://mathoverflow.net/users/8628 | Union of cycles | Counterexample: Consider a graph consisting of three paths $u a\_1\cdots a\_{k-1} v$, $u b\_1\cdots b\_{k-1} v$, and $u c\_1\cdots c\_{k-1} v$, and no other vertices or edges. Every cycle contains exactly two of those paths so any two cycles share $k+1$ vertices.
| 5 | https://mathoverflow.net/users/48859 | 268432 | 120,390 |
https://mathoverflow.net/questions/268433 | 9 | Let $F$ be a finite family of non-empty sets such that any two of them intersect. Consider the set $F'$ consisting of all sets that are a subset of at least one element of $F$. Prove $|F'|\geq 2|F|$.
I conjectured this yesterday but have been completely unable to prove it. I tried doing things like inclusion-exclusio... | https://mathoverflow.net/users/24478 | For an intersecting family of $m$ sets there are at least $2m$ sets that are contained in at least one of them | Assume that all the sets are contained in some master set of $n$ elements. Let $G$ be the set of super sets of at least one element of $F$. We apply the Harris-Kleitman inequality:
**Lemma** (Harris-Kleitman): *For $F'$ a downward-closed and $G$ an upward-closed collection of subsets on a set with $n$ elements,* $$|F... | 17 | https://mathoverflow.net/users/18060 | 268436 | 120,392 |
https://mathoverflow.net/questions/268414 | 4 | I was wondering if there is an analogous of the Problem 8.49 from Strom, Modern classical homotopy theory in a Model Category.
>
> There is a bijection between $[X,Y]$ and $<X,Y>$ when Y is a simply connected space.
>
>
>
Where
1. $[X,Y]$ are pointed homotopy classes of maps $X\to Y$;
2. $\langle X,Y\rangle$... | https://mathoverflow.net/users/104432 | Analogous of Strom 8.49 for Model Categories | Say $\mathcal{M}$ is a model category and $A$ is a cofibrant object of $\cal{M}$. Under these conditions, there's an undercategory $\mathcal{M}\_{A/}$ whose objects are pairs of an object $X$ of $\mathcal{M}$ and a map $A \to X$, and whose morphisms are commutative triangles. The undercategory has a model structure, wh... | 4 | https://mathoverflow.net/users/360 | 268437 | 120,393 |
https://mathoverflow.net/questions/267903 | 18 | Given an $\mathbb E\_\infty$-ring (highly structured commutative ring spectrum if you want) $R$, we have the free $R$-algebra (on one generation) $R\{t\}\simeq \bigoplus\_{n\ge 0} R\_{\mathrm h\Sigma\_n}$ vs. the polynomial $R$-algebra $R[t]\simeq \bigoplus\_{n\ge 0} R.$
It is well-known that $R\{t\}\simeq R[t]$ when $... | https://mathoverflow.net/users/39713 | When do the polynomial algebra and free algebra coincide in brave new algebra? | Any morphism of $R$-algebras $\varphi : R\{t\} \to R[t]$ is determined up to homotopy by an element of $\pi\_0(R[t]) \approx \pi\_0(R)[t]$. If $\varphi$ is an equivalence, then this element must be the generator $t$, so we may as well assume $\varphi$ is the canonical map $\varepsilon\_R : R\{t\} \to R[t]$.
**Claim:*... | 7 | https://mathoverflow.net/users/2503 | 268441 | 120,395 |
https://mathoverflow.net/questions/268440 | 1 | Let $H$ be a (multiplicatively written) commutative monoid with identity $1\_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an *ideal* of $H$ is $I = IH$.
The set $\mathcal I^\ast(H)$ of all *non-empty* ideals of $H$ is made into a commutative, reduced monoi... | https://mathoverflow.net/users/16537 | If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$ | Since $\mathcal I^\*(H)$ is reduced, and $IJ = I$ implies $J = I$, the only question is whether $\mathcal I^\*(H)$ has idempotents other than $H$. This is not even true if $H$ is cancellative and reduced:
**Example.** Let $H$ be the multiplicative monoid $\mathbb R\_{\geq 1}$, and let $I = \mathbb R\_{> 1}$. Then $I^... | 2 | https://mathoverflow.net/users/82179 | 268442 | 120,396 |
https://mathoverflow.net/questions/267970 | 2 | Let $\mathfrak{S}\_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}\_{2n}$ has an *$n$-distant pair* if there is some $j\in [2n-1]$ such that $\vert\pi\_j-\pi\_{j+1}\vert=n$.
Define $A=$ set of $\pi\in\mathfrak{S}\_{2n}$ with exactly one $n$-distant pair an... | https://mathoverflow.net/users/66131 | $n$-distant permutations more than not | There is an injective mapping from $B$ to $A$, which can be constructed as follows:
for a permutation $\pi=(p\_1,p\_2,\dots,p\_{2n})\in B$, find the element $p\_k$ that pairs with $p\_1$ (i.e., $|p\_k-p\_1|=n$); clearly we have $k>2$. Map $\pi$ to the permutation
$\pi'=(p\_2,\dots,p\_{k-1},p\_1,p\_k,p\_{k+1},\dots,p\_{... | 3 | https://mathoverflow.net/users/7076 | 268450 | 120,398 |
https://mathoverflow.net/questions/268453 | 3 | Is there a formula in radicals for the eigenvalues, or at least the largest eigenvalue, of an $n\times n$ symmetric matrix, in terms of the entries? For what $n$ is there a formula? There obviously is if $n\leq 4$, and one suspects not if $n\geq 5$. However, the characteristic polynomial of a symmetric matrix is rather... | https://mathoverflow.net/users/19444 | Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix? | If I calculated correctly, $$\begin{pmatrix}2 & 1 & 0 & 0 & 0\newline 1 & 3 & 1 & 0 & 0\newline 0 & 1 & 1 & 1 & 0 \newline 0 & 0 & 1 & 1 & 1 \newline 0 & 0 & 0 & 1 & 1 \end{pmatrix}$$ has characteristic polynomial $-x^5+8x^4-20x^3+15x^2+4x-5$ which is irreducible mod $3$ and has $3$ linear factors and a quadratic facto... | 24 | https://mathoverflow.net/users/2954 | 268454 | 120,399 |
https://mathoverflow.net/questions/268371 | 6 | I'm writing an essay about Ramanujan's conjecture and have some questions:
1 How is Ramanujan's conjecture connected with the Weil conjectures?
2 How could Ramanujan's conjecture be assumed true or deduced when Deligne proved the Weil conjecture?
3 How Is Ramanujan's conjecture connected or equivalent to the Riem... | https://mathoverflow.net/users/nan | The connection between the Weil conjectures and Ramanujan's conjecture | Basically, the coefficients of an holomorphic cusp form are related to the number of points on a certain smooth projective variety over $\mathbb{F}\_p$, and the Weil-Riemann hypothesis gives the neccesary error term for the number of points. This is the content of
* Pierre Deligne, "[Formes modulaires et représentati... | 9 | https://mathoverflow.net/users/43108 | 268457 | 120,401 |
https://mathoverflow.net/questions/233858 | 1 | In the Portilla Simoncelli paper (page 18):
<http://www.cns.nyu.edu/pub/lcv/portilla99-reprint.pdf>
They go about calculating the derivative of the skewness $\eta(x)$ of a distribution (2D matrix in the case of an image: $x$) through the following expression:
$\eta(x) = \frac{\mu\_3(x)}{\mu\_2(x)^{1.5}}$
where $\... | https://mathoverflow.net/users/75690 | Computing skewness derivative in terms of variance | I just read the same paper. Initially I made the same mistake as you, which is how I came across your question. It's been more than a year since your post but, in case you are still curious, here is how to work out the derivation. The key is to properly compute $\frac{\partial \mu\_n}{\partial x\_{ij}}$ for n=2,3, whic... | 3 | https://mathoverflow.net/users/109093 | 268463 | 120,402 |
https://mathoverflow.net/questions/268462 | 11 | Recall that $\kappa$ is a worldly cardinal if $V\_\kappa$ is a model of $\sf ZFC$. While every worldly cardinal is a strong limit cardinal, it is not necessarily regular. The point being that the short cofinal sequence is not first-order definable, so Replacement is not violated.
In particular, the first worldly car... | https://mathoverflow.net/users/7206 | What is the consistency strength of "Every set is a member of a transitive model"? | The answer is no, because I claim that if $\kappa$ is worldly, then $V\_\kappa$ thinks that every set is a member of a transitive model of ZFC.
To see this, note first that every worldly cardinal $\kappa$ is a beth-fixed point $\beth\_\kappa=\kappa$ and furthermore $V\_\kappa=H\_\kappa$, the set of sets whose transi... | 12 | https://mathoverflow.net/users/1946 | 268471 | 120,404 |
https://mathoverflow.net/questions/151963 | 3 | **Q1:** Can we define Fourier series for a function $\mathbb{Z}\_p\to \mathbb{Q}\_p$?
**Q2:** There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:
$$B\_n(\{x\})=-\frac{n!}{(2\pi
i)^n}\mathop{{\sum}}\_{|m|>0}\frac{e^{2\pi imx}}{m^n}\qquad(n\ge1).$$
It means that Bernoulli poly... | https://mathoverflow.net/users/5712 | What is $p$-adic Fourier series? | This is not an answer, but a list of references. (I can't add comments).
At least as I know, the first approach to $p$-adic fourier theory was done by Woodcock in the 70's. See "Fourier analysis for p-adic Lipschitz functions"
(J. London Math. Soc. (2) 7 (1974), 681–693). It is interesting to note that here he define... | 3 | https://mathoverflow.net/users/109085 | 268474 | 120,406 |
https://mathoverflow.net/questions/268491 | 1 | Let $k$ be an algebraically closed field of characteristic zero and let $X$
be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with an open orbit.
Now comes the question: If $X$ is quasi-affine and smooth, is the $k$-algebra $k[X]$ of regular functi... | https://mathoverflow.net/users/108431 | The algebra of regular functions of a quasi-affine toric variety | One can show that for any normal spherical variety $X$ the algebra $k[X]$ is finitely generated. No quasi-affinity or smoothness needed. This holds more generally for $G$-varieties ($G$ connected reductive) where $G$ has an open orbit and a Borel subgroup has an orbit of codimension $\le1$. The latter condition is opti... | 4 | https://mathoverflow.net/users/89948 | 268499 | 120,417 |
https://mathoverflow.net/questions/268503 | 1 | **Definition**: Let $G$ be a finite solvable group and $\Sigma \in \text{H}(G)$, the set of Hall systems of $G$. The normaliser of $\Sigma$ is defined as $$ N\_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for all} H \in \Sigma \}.$$ A system normaliser of $G$ is a subgroup of the form $N\_G(\Sigma)$ for some $\Sigma \in \t... | https://mathoverflow.net/users/92488 | The order of the system normalizer in a finite solvable group | This is straightforward, using the definitions and the lemma. I am going to vote to transfer the question to MSE.
Let $1 =G\_0 < G\_1 < \cdots < G\_n=G$ be a chief series of $G$, and $U$ a system normalizer in $G$. Then $|U| = \prod\_{i=1}^n [U \cap G\_i:U \cap G\_{i-1}]$.
If $G\_i/G\_{i-1}$ is a central chief fact... | 4 | https://mathoverflow.net/users/35840 | 268505 | 120,420 |
https://mathoverflow.net/questions/268494 | 8 | Given two commutative rings $A$ and $B$, any map of rings $A\to B$ will automatically preserve the commutative structure. This is to say, the forgetful functor $\operatorname{CRing}\to \operatorname{Ring}$ is fully faithful.
The analogous situation in brave new algebra is richer: we have $\mathbb E\_k$-rings for all ... | https://mathoverflow.net/users/39713 | Morphisms of $\mathbb E_l$-rings between $\mathbb E_k$-rings for $l<k$ | Let's consider these in characteristic zero, so that we can use differential graded algebras and so that we can model $E\_\infty$ things by strictly commutative things and $E\_1$ things by associative things.
Let $A$ be the commutative DGA $\Bbb Q[x,y]$ where $x$ and $y$ are in degrees $2n$ and $2m$ respectively, wit... | 9 | https://mathoverflow.net/users/360 | 268521 | 120,426 |
https://mathoverflow.net/questions/268451 | 1 | Few days ago I asked about $WKL\_0$ and the role of binary trees to provide for completeness for first order theories, and the question was nicely answered by Joel David Hamkins: [Does $WKL\_0$ plus CON(PA+X) give a binary tree model of PA+X?](https://mathoverflow.net/questions/268391/does-wkl-0-plus-conpax-give-a-bina... | https://mathoverflow.net/users/37385 | How does $RCA_0$ achieve weak completeness? | I also found the theorem confusing at first until I realized what is going on.
Simpson is claiming that if you have a consistent theory $T$ that is closed under deduction, then in $RCA\_0$ you can prove that it has a model.
What is confusing about the theorem is that one ordinarily thinks of the completeness theore... | 3 | https://mathoverflow.net/users/1946 | 268536 | 120,430 |
https://mathoverflow.net/questions/268510 | 1 | I am trying to understand [Kloosterman](https://en.wikipedia.org/wiki/Kloosterman_sum) sums and their estimates (e.g. from [[1](https://people.math.ethz.ch/~kowalski/exponential-sums-icms.pdf)], which does not prove)
$$ \Big| S(m,n;c) \Big| = \Big| \sum\_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, c^{3... | https://mathoverflow.net/users/1358 | group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$ | You can find a concise but self-contained proof of Weil's bound for Kloosterman sums of prime moduli [in these notes](http://www.renyi.hu/~gharcos/weil.pdf). Note that prime moduli constitute the really hard case.
| 1 | https://mathoverflow.net/users/11919 | 268546 | 120,432 |
https://mathoverflow.net/questions/268535 | 6 | I've been reading K. Dajani and C. Kraaikamp's paper [From greedy to lazy expansions and their driving dynamics](http://dutiosc.twi.tudelft.nl/~cork/monthly.pdf). In the last page previous to the references, they mention that the entropy of the map $T\_{\alpha, \beta}(x) = \beta x + \alpha$ is $\log(\beta)$. They menti... | https://mathoverflow.net/users/10518 | Rohklin’s formula and $\beta$ expansions | This is the formula
$$
h\_\mu(f) = \int \log |f'|\,d\mu
$$
for the entropy of an absolutely continuous $f$-invariant measure $\mu$ on the unit interval which goes back to Rokhlin (<http://www.ams.org/mathscinet-getitem?mr=143873>), also see Ledrappier (<http://www.ams.org/mathscinet-getitem?mr=627788>).
| 5 | https://mathoverflow.net/users/8588 | 268553 | 120,434 |
https://mathoverflow.net/questions/268455 | 7 | It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a given boundary point of P as vertex? The optimal constant should be between $\frac{1}{4}$ and $\frac{1}{3}$, but I can't ti... | https://mathoverflow.net/users/90626 | Largest inscribed triangle with a given vertex | The answer is indeed $1/\pi$, equality achieved in the case of $P$ a semicircle and the point its center.
For an arbitrary convex region $P$ and a point $X$ on its boundary, consider the convex hull of $P$ and $2X - P$, and call it $S$. Then any $S$ is clearly symmetric with respect to $X$. Any triangle in $P$ havin... | 5 | https://mathoverflow.net/users/36579 | 268557 | 120,437 |
https://mathoverflow.net/questions/266565 | 9 | Consider a (closed) Riemann surface and let $G(x,y)$ be the Green function of the Laplace-Beltrami operator. We can informally identify $G$ with the two-point correlation function for the Gaussian random field:
$$G(x,y)=\left<\phi(x)\phi(y)\right>=\frac{1}{Z} \int \mathcal{D}\phi\;\phi(x)\phi(y) \exp\left(-\frac{1}{2... | https://mathoverflow.net/users/22773 | Variation of the Green function with respect to the metric | It seems that the naive derivation from the path integral only picks up the term coming from quasiconformal variations. Combining the known result for the quasiconformal variation with the much more well known conformal variation, I arrived at the expression
$$\frac{\delta G(x,y)}{\delta g^{\mu\nu}(z)}=(-\nabla\_{(\m... | 3 | https://mathoverflow.net/users/22773 | 268559 | 120,438 |
https://mathoverflow.net/questions/268382 | 6 | I would like to get a demonstration of the stable homeomorphism conjecture (SHC$\_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in dimension $n$, for $n \neq 4$.
According to several sources, for example the upvoted answer to the post [connectivity of the... | https://mathoverflow.net/users/109080 | Proof of the stable homeomorphism conjecture | For your first question, yes Kirby did prove the n-dimensional annulus conjecture AC$\_n$ for $n>4$ in the 1969 Annals paper that you cite. The only reason this might not be clear from reading the paper is that the original version of the paper proved AC$\_n$ only assuming another statement that he calls HT$\_n$ (namel... | 11 | https://mathoverflow.net/users/23571 | 268575 | 120,444 |
https://mathoverflow.net/questions/268151 | 5 | (This is a follow-up to [The spectral radius of a binary matrix](https://mathoverflow.net/questions/267958/the-spectral-radius-of-a-binary-matrix))
Let $\mathcal B\_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$.
**QUESTION.** Is there a $\delta\in\bigl(0,\frac12\bigr), \alpha\in(0,1)$ and $b>... | https://mathoverflow.net/users/8131 | The spectral radius of a binary matrix - polynomial growth? | A paper that seems to directly address your question is the 1987 paper of Brualdi and Solheid, \*On the minimal spectral radius of matrices of zeros and ones". That paper shows that if the number of 1's is $(\frac12+\delta)n^2$, then the spectral radius is essentially $2\delta n$. The paper also allows you to answer th... | 4 | https://mathoverflow.net/users/11054 | 268580 | 120,446 |
https://mathoverflow.net/questions/268579 | 3 | For a smooth projective curve $Y$ over an algebraically closed field $k$ of characteristic 0, it is known that there exists a one-to-one correspondence between finite \'{e}tale morphisms $f:X\to Y$ of degree $p$ (a prime), and $p$-torsion elements of $\mathrm{Pic}Y$.
Is this correspondence still true when one replace... | https://mathoverflow.net/users/44005 | Etale coverings of non-projective curves | What you say is not correct: when $Y$ is projective, $p$-torsion elements of $\mathrm{Pic}(Y)$ correspond to *cyclic* étale covers of degree $p$. In general, the exact sequence (for the étale topology)
$$1\rightarrow \mu \_p\rightarrow \mathbb{G}\_m\xrightarrow{\ \times p\ }\mathbb{G}\_m \rightarrow 1$$
gives an exact ... | 8 | https://mathoverflow.net/users/40297 | 268582 | 120,447 |
https://mathoverflow.net/questions/268558 | 0 | Let $(\Omega,\mathcal{F},\mathbb{P}):=(M^{\mathbb{N}\_{0}},\mathcal{M}^{\mathbb{N}\_{0}},\mathbb{P})$ be a probability space where $M=\left\{0,1,2,3,4\right\}$, $\mathcal{M}^{\mathbb{N}\_{0}}$ is product $\sigma$-algebra. Note that if we consider the cylinder $I=\left[x\_{0},x\_{1},\ldots,x\_{k}\right]$ defined by
\be... | https://mathoverflow.net/users/109184 | Calculate of Lyapunov Exponents of a sequence of random matrices | Strictly speaking, Lyapunov exponents are only defined for products of non-degenerate matrices. Since you are dealing with diagonal matrices, you are just asking about the exponential growth rates of the diagonal entries, which can be considered separately, so that you don't really need to talk about matrix products at... | 1 | https://mathoverflow.net/users/8588 | 268586 | 120,449 |
https://mathoverflow.net/questions/268425 | 7 | I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of totally geodesic submanifolds. Is there anything one can say about such a manifold?
To be more precise, consider such an $M... | https://mathoverflow.net/users/106940 | Negatively curved manifolds with many totally geodesic submanifolds | Here is a partial answer: Let $(M^{2n},g,J)$ be a compact Riemannian manifold endowed with a *$g$-orthogonal* almost complex structure $J$ with the property that, for every nonzero $v\in T\_pM$ there exists a $J$-holomorphic curve $C\subset M$ passing through $p$ with tangent space spanned by $v$ and $Jv$ that is total... | 7 | https://mathoverflow.net/users/13972 | 268588 | 120,451 |
https://mathoverflow.net/questions/268581 | 5 | Given two real vector spaces $V$ and $U$ with subspaces $A, C \subset V$, $A\cap C \neq \{0\}$ and $B, D \subset U$, $B\cap D \neq \{0\}$, is it true that
$$(A\otimes B)\cap (C\otimes D) = (A\cap C)\otimes (B\cap D)$$
and if so, is there a proof available in a paper or textbook? It is stated to be true in a commen... | https://mathoverflow.net/users/54431 | tensor products and intersections of subspaces | See e.g. Lemma 1.4.5 in the book
S. Dascalescu - C. Nastasescu - S. Raianu: "Hopf Algebras. An Introduction", Pure and Applied Mathematics, 2001
There the statement is proved under the assumption that $C\subseteq A$ and $D\subseteq B$: the general case easily follows from this.
| 8 | https://mathoverflow.net/users/14653 | 268596 | 120,453 |
https://mathoverflow.net/questions/268595 | 5 | Let $G$ be a compact Lie group and let $M$ be a $G$-homogeneous manifold. Suppose that $M$ is endowed with a complex structure invariant by the action of $G$. Denote by $G\_{\mathbb C}$ the complexification of $G$.
Are there known conditions so that the action of $G$ on $M$ can be extended to a holomorphic action of ... | https://mathoverflow.net/users/12233 | Holomorphic extension of an action by a compact Lie group on a complex homogeneous manifold | I think this should work for a connected group $G$. The Lie algebra action extends, by multiplying by $J$, to a complex Lie algebra action. By compactness of $M$, these vector fields are all complete. So some covering group of the complexification acts, for example the universal covering group $\tilde{G}\_{\mathbb{C}}$... | 6 | https://mathoverflow.net/users/13268 | 268597 | 120,454 |
https://mathoverflow.net/questions/268618 | 1 | I am a university student, and I am interested in some pure math areas related to algebra, for example, ring theory like Aliyah & Macdonald (I learned it 60 - 80 percent), algebraic(, or arithmetic) geometry (though I have never learned something like scheme seriously), (algebraic) number theory (I also love other numb... | https://mathoverflow.net/users/109220 | Recommended summer schools for this summer [a university student] | There is an annual summer school for undergraduates in Lisbon, at the Gulbenkian Foundation. The topic this year is algebraic topology. See <https://www.math.tecnico.ulisboa.pt/~ggranja/Talentos/school2017/index.html>. The deadline for applying is May 15th (click on the registration link on the left side of the page).
... | 2 | https://mathoverflow.net/users/3272 | 268624 | 120,460 |
https://mathoverflow.net/questions/268621 | 38 | Is a number whose infinite decimal part is the sequence of even numbers, transcendental? How about a number whose infinite decimal part is the odd numbers? Would the odds be more difficult to prove since they contain almost the entire sequence of primes?
| https://mathoverflow.net/users/nan | Is 0.24681012141618202224... transcendental? | In point of fact, K. Mahler proved in [this paper](https://carma.newcastle.edu.au/mahler/docs/046.pdf) that, if $p(x)$ in a non-constant polynomial such that $p(n) \in \mathbb{N}$ for every $n\in \mathbb{N}$, then the number
$$0.p(1)p(2)p(3)p(4)\ldots,$$
which is formed concatenating after the decimal point the val... | 57 | https://mathoverflow.net/users/1593 | 268632 | 120,464 |
https://mathoverflow.net/questions/268641 | 4 | A friend and I were discussing the properties of continued fractions (as "best" approximations). For fun, we checked the continued fractions of Liouville's constant. The terms in the sequence fit a very clear pattern, see
<http://mathworld.wolfram.com/LiouvillesConstant.html>
Is there an actual reference to show th... | https://mathoverflow.net/users/18974 | Continued fraction of Liouville's constant | This was first done in my paper, [Simple Continued Fractions for Some Irrational Numbers II](http://www.sciencedirect.com/science/article/pii/0022314X82900476), published in J. Number Theory 14 (1982), 228-231.
From the proof there you can deduce the 4 things you listed.
| 8 | https://mathoverflow.net/users/44797 | 268643 | 120,466 |
https://mathoverflow.net/questions/268649 | 3 | Let $\mathfrak{g}$ be a simple Lie algebra with a compact subalgebra $\mathfrak{k}$ such that $(\mathfrak{g},\mathfrak{k})$ corresponds to an irreducible Riemann symmetric space. Denote by $\sigma$ be the involutive automorphism. Suppose that $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ where $\mathfrak{p}$ is the eigenspa... | https://mathoverflow.net/users/56989 | Irreducible Symmetric Pairs | It is true that $\mathfrak p$ is always an irreducible $\mathfrak k$-module. Be aware though that the complexification $\mathfrak p\_{\mathbb C}$ might be reducible as an $\mathfrak k\_{\mathbb C}$-module. This happens precisely in the case that $\mathfrak g$ is of Hermitian type.
**Edit:** If $\tau$ is not a Cartan ... | 5 | https://mathoverflow.net/users/89948 | 268676 | 120,475 |
https://mathoverflow.net/questions/268671 | 8 | Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known?
Any pointers to the literature are welcome. I don't find much on this topic online. Thanks.
| https://mathoverflow.net/users/66288 | Why is the Euler-Mascheroni constant not a Liouville number? | At the present time, we do not even know how to prove that the Euler-Mascheroni constant $\gamma=\lim\_{n\to\infty} \sum\_{k=1}^n\frac{1}{k} - \log n$ is irrational, much less transcendental; although it is conjectured to be transcendental. The reason you won't find a lot about this topic online (or in the research lit... | 18 | https://mathoverflow.net/users/11926 | 268677 | 120,476 |
https://mathoverflow.net/questions/268680 | 11 | Let $M^d$ be a $d$-dimensional orientable spin manifold, and $N^4$ is a closed $4$-dimensional orientable submanifold of $M^d$.
1. Is $N^4$ always spin?
2. If $d=5$, is $N^4$ always spin?
3. If $N^4$ is a boundary in $M^d$, is $N^4$ always spin?
| https://mathoverflow.net/users/17787 | Is a 4-dimensional submanifold of a spin manifold always spin? | Let $i$ denote an immersion $N \to M$. There is an exact sequence of vector bundles on $N$ given by
$$0 \to TN \to i^\*TM \to \nu \to 0$$
where $\nu$ is the normal bundle. As total Stiefel-Whitney classes are multiplicative in short exact sequences (alternatively, $i^\*TM \cong TN\oplus\nu$ smoothly), it follows t... | 19 | https://mathoverflow.net/users/21564 | 268682 | 120,477 |
https://mathoverflow.net/questions/268683 | 18 | Caveat: I am not at all a number theorist, and I randomly came up with the following question while I was hiking. But I already asked two serious number theorists, and since they did not know the answer to my question, I decided to pose it here.
Let $c > 0$ be given. Suppose $a\_1,a\_2,a\_3,\dots$ is a sequence of po... | https://mathoverflow.net/users/14233 | Is every sequence that looks like an AP really an AP? | For each $n$, the differences $a\_{n+1}-a\_n$, $a\_{n+2}-a\_{n+1}$, and $a\_{n+2}-a\_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since
$$
\frac{a\_{n+2}-a\_{n+1}}{a\_{n+2}-a\_n}+\frac{a\_{n+1}-a\_n}{a\_{n+2}-a\_n}=1,
$$
this is a solution to the [S-unit equation](https://en.wikipedi... | 22 | https://mathoverflow.net/users/5263 | 268686 | 120,478 |
https://mathoverflow.net/questions/268573 | 16 | Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
>
> **Question:** Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int\_a^b f(t) \, dt \ ?$$
>
>
>
In other words, is there a Riemann sum with two terms that hits the bull's-eye?
**EDIT:** Prompted by a down... | https://mathoverflow.net/users/1516 | Bull's-eye Riemann sum | Not necessarily. Take small $\delta>0$. Spend time $\frac 23(1-\delta)$ at $(0,1)$. Then, in time $\delta/4$ travel the route $(0,1)\to(0,2)\to(-1,2)\to(-1,-2)$ so that the integral of the vertical coordinate is $0$. Then stay at $(-1,-2)$ for the time $\frac 16(1-\delta)$ and return to $(0,1)$ running the same movemen... | 9 | https://mathoverflow.net/users/1131 | 268696 | 120,481 |
https://mathoverflow.net/questions/268697 | 3 | Let $q(x,y,z) = ax^2 + by^2 + cz^2$ be a non-singular diagonal ternary quadratic form with integer coefficients. The discriminant $\Delta(q)$ of $q$ is then equal to $abc$, and for any positive number $X$ there are finitely many non-singular diagonal quadratic forms of discriminant at most $X$. Indeed, the number of no... | https://mathoverflow.net/users/10898 | The density of diagonal isotropic ternary quadratic forms with respect to discriminant | The density must be zero: for each prime $p$,
the probability that $q$ is *not* $p$-adically isotropic is
$3/2p + O(1/p^2)$ (when one of $a,b,c$ is a multiple of $p$,
and the product of the other two is $-1$ times a quadratic nonresidue);
thus for any finite set $P$ of primes, the probability is
$$
\prod\_{p \in P} ... | 5 | https://mathoverflow.net/users/14830 | 268700 | 120,482 |
https://mathoverflow.net/questions/268702 | 2 | Or polyominoes with no hollow in $\mathbb{R}^3$?
I created this conjecture and tried to make counterexample, but it doesn't work well. Thank you for any answer or correcting question.
| https://mathoverflow.net/users/80691 | Does one pieces of every kind of connected polyominoes P in $\mathbb{R}^2$ which has no hole cover a plane? | I think one can do this by standard inductive trickery.
First take a square grid in the plane. Index the union of the set of cells and the set of polyominoes by natural numbers $i$.
At each stage $i$, we will have placed $n\_i$ connected polyominos with no holes in the plane so that they form a single connected pol... | 9 | https://mathoverflow.net/users/18060 | 268703 | 120,483 |
https://mathoverflow.net/questions/98639 | 16 | **This number is divisible by the order of the subgroup** <http://arxiv.org/abs/1205.2824>.
The proof is short but non-trivial. Is this fact new or is it known for a long time?
| https://mathoverflow.net/users/24165 | The number of group elements whose squares lie in a given subgroup | In 2017, we learned that this fact was [proven by Shiro Iwasaki in
1982](https://doi.org/10.1016/0021-8693(82)90093-X).
| 6 | https://mathoverflow.net/users/24165 | 268714 | 120,484 |
https://mathoverflow.net/questions/268712 | -2 | Let $G=(V,E)$ be a connected simple undirected graph. A *way* in $G$ is a function $w:\{1,\ldots, n\}\to V(G)$ for some positive integer $n$, such that $\text{im}(w) = V(G)$, and for all $k\in \{1,\ldots, n-1\}$ we have $\{w(k), w(k+1)\} \in E$.
We define the *hamiltonicity* of $G$ by $$H(G) = \min\{n\in\mathbb{N}:\t... | https://mathoverflow.net/users/8628 | "Hamiltonicity" of a graph | $H(G)\leq \lvert V(G)\rvert-1$ for every finite connected graph.
Proof. Consider any spanning tree $T$ of $G$, double every edge of $T$, choose any Euler tour $W$ of this auxiliary graph, consider the projection $\mathrm{p}(W)$ of $W$ to $G$. Then $\mathrm{p}(W)$ is a way in your sense having a domain of $2\lvert V(... | 4 | https://mathoverflow.net/users/108556 | 268715 | 120,485 |
https://mathoverflow.net/questions/268691 | 5 | It is an odd and arguably unacceptable situation that $PA$ does not have $\vdash\_{PA}(Pr\_{PA}\ulcorner A\urcorner\to A)$ for false recursive sentences $A$.
However, it is not clear to me that Löb's theorem is already derivable in Robinson arithmetic $Q$, for one cannot assume that the provability predicate of $Q$ ... | https://mathoverflow.net/users/37385 | Can extensions of $Q$ contradict Löb with recursive reflection? | No consistent recursively axiomatized extension $T$ of $Q$ can prove $\mathrm{Pr}\_T\ulcorner\bot\urcorner\to\bot$, that is, $\mathrm{Con}\_T$.
In fact, no consistent r.e. theory $T$ can interpret $Q+\mathrm{Con}\_T$.
In fact, no consistent r.e. theory $T$ can interpret $Q+\{\mathrm{RCon}\_T(\overline n):n\in\mathb... | 6 | https://mathoverflow.net/users/12705 | 268716 | 120,486 |
https://mathoverflow.net/questions/268669 | 2 | Let $L$ be a finite extension of $\mathbb{Q}\_p$. Colmez defines [here](https://webusers.imj-prg.fr/~pierre.colmez/SP-ast.pdf)
the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in [this paper](https://webusers.imj-prg.fr/~pierre.colmez/SP-ast.pdf) he contructs the representa... | https://mathoverflow.net/users/69289 | p-adic representations of $GL_2(\mathbb{Q}_p)$ | Apparently you didn't read the references you quoted terribly carefully, since exactly this question is addressed by the footnote on the bottom of page 4 of the paper you link:
>
> "L'application qui a $\phi \in B(s)$ associe [...] est une isomorphisme de $B(s)$ sur [...], **ce qui munit $B(s)$ d'une structure de ... | 6 | https://mathoverflow.net/users/2481 | 268717 | 120,487 |
https://mathoverflow.net/questions/268710 | 4 | Let $G$ be the subgroup of $\mathrm{GL}\_9(\mathbf{Q})$ defined as follows:
Letting $(a\_1,a\_2,a\_3,a\_4,a\_5,a\_6,a\_7,a\_8,a\_9)$ be the canonical basis of $\mathbf{Q}^9$, $G$ is generated by:
* All permutation matrices permuting the basis elements $(a\_1,\dots,a\_8)$ (fixing $a\_9$)
* The additional transformatio... | https://mathoverflow.net/users/31476 | Identify one group of linear transformations | If I understand your question right, your group $G$ has order $5160960$,
and it has an elementary abelian normal subgroup $N$ of order $2^7$ such
that $G/N \cong {\rm S}\_8$. This can be found with [GAP](https://www.gap-system.org) as follows:
```
gap> A := PermutationMat((1,2),9);;
gap> B := PermutationMat((1,2,3,4... | 13 | https://mathoverflow.net/users/28104 | 268718 | 120,488 |
https://mathoverflow.net/questions/268592 | 2 | "It is known that, for a curvature continuous surface, if the curve $\gamma$ shrinks to the point $v\_i$, the (line) integral converges to the mean curvature $\kappa(v\_i)$ of the surface at the point $v\_i$ times the normal vector $N\_i$ at the same point "
$$\lim\_{\epsilon \to 0} \frac{1}{|\gamma\_\epsilon|}\int\_{... | https://mathoverflow.net/users/109199 | Mean curvature vector approximated for the discrete Laplace Beltrami Operator | In both papers you mention, the curve integral is introduced as a motivation only. Let $S\subset\mathbb R^3$ be a surface and $p\in S$.
We may assume that $p=0$, and that the tangent plane to $S$ at $p$ is the $x$-$y$-plane. We also assume that the $x$- and $y$-axes are the principal curvature directions of $S$ at $p$.... | 3 | https://mathoverflow.net/users/70808 | 268732 | 120,491 |
https://mathoverflow.net/questions/268428 | 19 | For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2's are different.
Since there are $\binom{n+2}{2}$ different triples of nonnegative integers summing up to $n$ such a seque... | https://mathoverflow.net/users/35593 | Sequences with 3 letters | Here is a sketch of a proof that there are no such complete sequences for $n>4$.
Consider the graph where the vertices are the triples of nonnegative integers that sum to $n$ and construct
an edge between two vertices when one can get from one to the other by incrementing one element and decrementing another.
This gr... | 5 | https://mathoverflow.net/users/4422 | 268747 | 120,497 |
https://mathoverflow.net/questions/268733 | 5 | A formula found by Jacobi for $\psi(x)=\sum\_{n=1}^{\infty} e^{-n^2\pi x}$ says
$$\psi(\frac{1}{x})=\sqrt{x} \psi(x)+\frac{\sqrt{x}-1}{2}$$
If we define
$$\phi(x)=\sum\_p e^{-p^2\pi x}$$
where $p$ runs through all prime numbers. Now we want to know the behavior of this function near 0 and $+\infty$, then what can be ... | https://mathoverflow.net/users/41499 | Behavior of the sum $\sum_p e^{-p^2 \pi x}$ around 0 and $+\infty$ | There is no Jacobi-like functional equation for your $\phi(x)$. Nevertheless, with a bit of analysis, we can determine its asymptotic behavior as $x\to\infty$ and as $x\to 0+$.
When $x>0$ is large, the leading term $p=2$ dominates, so we have the asymptotics
$$\phi(x)\sim e^{-4\pi x},\qquad x\to\infty.$$
When $x>0$... | 6 | https://mathoverflow.net/users/11919 | 268749 | 120,498 |
https://mathoverflow.net/questions/267954 | 2 | Is there an algorithm which, given a string $s$, generates a sequence of $|s|$ strings, such that it can be proven in some axiomatic system $S$, that the Kolmogorov complexity of each successive term is smaller than the preceeding one?
We further impose a constraint that the terms of the strings themselves, do not de... | https://mathoverflow.net/users/34859 | Generating an arbitrarily long sequence with decreasing Kolmogorov complexity of terms | Suppose there is such an algorithm.
Let $x\_n$ be the first string outputted on input $$s=00\cdots 0=0^n.$$
Then $x\_n$ has complexity at most $\log\_2 n+C$ since I just described it in terms of $n$.
On the other hand, the complexity is at least $n$ by assumption...
which is a contradiction.
| 1 | https://mathoverflow.net/users/4600 | 268758 | 120,505 |
https://mathoverflow.net/questions/268693 | 3 | **Context:** I am reading a physics paper [Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime](https://arxiv.org/abs/gr-qc/0103074) which applies the notion of the wave front set to operator-valued distributions in a globally hyperbolic spacetime. In particular, the paper considers v... | https://mathoverflow.net/users/109255 | Wave front set of vector-valued Dirac delta distribution | (1) A careful reading of the paper will reveal that the notation $\delta(x\_1,\ldots,x\_n)$ is precisely the $\delta$-distribution supported on the diagonal of the $n$-fold Cartesian product $M \times \cdots \times M$. If we look at (a possibly small portion of) the diagonal as covered by the coordinates $(x\_1,\ldots,... | 5 | https://mathoverflow.net/users/2622 | 268766 | 120,508 |
https://mathoverflow.net/questions/268764 | 39 | Do there exist countably many proper holomorphic submersions of complex manifolds $\mathcal{X}\_n \to B\_n$ such that every compact complex manifold appears as a fiber in at least one of the families?
(*Conventions*: Assume that each $B\_n$ is Hausdorff, connected, and paracompact; these conditions imply that $B\_n$... | https://mathoverflow.net/users/2757 | Do compact complex manifolds fall into countably many families? | **Edit.** I spoke with some of the experts in my department. I added one or two references. I reorganized some of the arguments.
This is **true**. This follows from a couple of big theorems and then a separability / second countability argument that ultimately reduces to the Stone-Weierstrass approximation theorem.
... | 22 | https://mathoverflow.net/users/13265 | 268793 | 120,514 |
https://mathoverflow.net/questions/268743 | 3 | Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this [nLab](https://ncatlab.org/nlab/show/pro-object) page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category. The Hom set between $F: D\to \mathcal{C}$ and $G: E\to \mathcal{C}$ is give... | https://mathoverflow.net/users/24965 | Why do we need cofiltered condition on the index category in the definition of pro-categories? | The introduction of $Pro(C)$ is in [Grothendieck's seminar Bourbaki 195, (1960)](http://archive.numdam.org/article/SB_1958-1960__5__369_0.pdf). There you can see the properties that using pro-objects gives you. This is related to Dylan's two comments above. In the classical case of handling inverse systems (pro-objects... | 7 | https://mathoverflow.net/users/3502 | 268815 | 120,520 |
https://mathoverflow.net/questions/268811 | 45 | I'm writing a paper in which I cite a lot of results that appear in Schikhof's *Ultrametric Calculus*. Some of these results are exercises in Schikhof's book. These exercises are not difficult, but are laborious. Thus, if I write the proofs, the article may extend by about two or three pages.
Should I write the proof... | https://mathoverflow.net/users/109085 | Citing exercises in an article | The answer is essentially given in the comments, so let me summarize:
1. It is a frequent situation that one has to cite an exercise.
2. It is legitimate. (Polya-Szego is cited > 1400 times according to Mathscinet)
3. The best thing is to cite a place where the statement is proved, but if you cannot find such a place... | 27 | https://mathoverflow.net/users/25510 | 268818 | 120,521 |
https://mathoverflow.net/questions/268634 | 10 | Is there a notion of "space" satisfying the following requirements?
1. Spaces form (at least) a category; morphisms between spaces are called "continuous maps."
2. Every topological space is a space, and the inclusion of $\mathbf{Top}$ in the category of spaces is fully faithful and injective on objects.
3. The categ... | https://mathoverflow.net/users/26080 | Is there a notion of "space" such that vector bundles can be understood in this way? | As Qiaochu said, probably what you want are topological stacks.
1. Let $T$ be a small full subcategory of $\mathrm{Top}$, with the Grothendieck topology of open covers, and consider the 2-category of stacks of groupoids on $T$. Call its objects "spaces" and its morphisms "continuous maps". You could call its 2-cells ... | 7 | https://mathoverflow.net/users/49 | 268829 | 120,525 |
https://mathoverflow.net/questions/268814 | 7 | The answer to the following question might be obvious but I haven’t found a full proof yet (neither by myself nor in the literature). So my apologies if it is trivial.
Let $X$ be a (for simplicity quasi-projective and non-singular) complex variety $X$ on which a finite group $G$ acts.
Deligne has shown in Hodge III... | https://mathoverflow.net/users/109326 | Two mixed Hodge structures on equivariant cohomology for actions by finite groups | As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1],
$$[X/G]\_n = (G^{n+1}\times X)/G$$
One has a descent spectral sequence
$$E\_1= H^q([X/G]\_p,\mathbb{Q})\cong (G^{p+1}\times H^q(X,\mathbb{Q}))/G$$
abutting to $H^{p+q}([X/G]\_\bullet, \mathbb{Q})=H^{p+q}\_... | 7 | https://mathoverflow.net/users/4144 | 268837 | 120,528 |
https://mathoverflow.net/questions/268841 | 2 | I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1} \right]$.
The matrix is \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix}
I tought of writing my matrix in the following form
$$ \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix} = t \begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}... | https://mathoverflow.net/users/109329 | Powers of small square matrices over the Laurent polynomial ring with integer coefficients | This is not much different from Robert Israel's answer, but here goes:
Let $A(t)=\begin{pmatrix}0&1\\1&t\end{pmatrix}$ and write $A^n(t)=\begin{pmatrix}c\_n&b\_n\\b\_n&a\_n\end{pmatrix}$. Then,
$$\begin{pmatrix}c\_{n+1}&b\_{n+1}\\ b\_{n+1}&a\_{n+1}\end{pmatrix}=A^{n+1}=\begin{pmatrix}c\_n&b\_n\\b\_n&a\_n\end{pmatrix... | 2 | https://mathoverflow.net/users/66131 | 268849 | 120,532 |
https://mathoverflow.net/questions/268853 | 2 | Consider the set $S$, $\{1,2,3\}$. The set I want to find, $P$, is the set of all subsets of $S$ which contain a majority of elements of $S$ - $\{\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$. What are the sufficient conditions for defining $P$? Stated another way: given some subset $Q$ of the powerset of $S$, how can we tell w... | https://mathoverflow.net/users/109349 | What are the sufficient conditions for a set P to be the set of all majority subsets of a set S? | I think this is equivalent (for an odd number of elements of $S$) to:
1. $Q$ is closed under supersets;
2. For any set $A\subseteq S$, exactly one of $A\in Q$, $A^c\in Q$ holds;
3. For any permutation $\pi$ of $S$, $\pi[Q]=Q$.
Because by (1) and (3), $Q$ has to be some collection of all "sufficiently large" subsets... | 3 | https://mathoverflow.net/users/4600 | 268860 | 120,534 |
https://mathoverflow.net/questions/268848 | 3 | Here $\mathbb{F}\_{q}$ means a finite field with $q = p^m$ elements where $p$ is the characteristic of the field in question and $\overline{\mathbb{F}\_{q}}$ means its algebraic closure.
I am studying an article about the existence of a normal basis with a primitive element for finite extensions of $\mathbb{F}\_{q}$.... | https://mathoverflow.net/users/109348 | Viewing $\overline{\mathbb{F}_{q}}$ as a $\mathbb{F}_{q}[X]-$module | By the normal basis theorem, there exists $a \in \mathbb F\_{q^n}$ such that $\{\sigma(a)\mid \sigma \in \operatorname{Gal}(\mathbb F\_{q^n}/\mathbb F\_q)\}$ spans $\mathbb F\_{q^n}$ as $\mathbb F\_q$-vector space. Since $\operatorname{Gal}(\mathbb F\_{q^n}/\mathbb F\_q)$ is generated by Frobenius, this means that $\ma... | 4 | https://mathoverflow.net/users/82179 | 268861 | 120,535 |
https://mathoverflow.net/questions/268859 | 8 | One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x\_1&x\_1^2&\dots&x\_1^{n-1}\\
1&x\_2&x\_2^2&\dots&x\_2^{n-1}\\
\ldots&\ldots&\ldots&\ldots&\ldots\\
1&x\_n&x\_n^2&\dots&x\_n^{n-1}\\
\end{vmatrix}=\prod\limits\_{1\leq i<j\leq n}(x\_j-x\_i).$$
In short, we write $\det(x\_i^{j-1})$. I ... | https://mathoverflow.net/users/66131 | "Almost Hankelized" numerical Vandermonde | Here is a slightly more general version: If $P\_n(x)$ are some polynomials of degree $n-1$ with leading term $a\_nx^{n-1}$ then $$\det(P\_j(x\_i))=\prod\_{j=1}^n a\_n\prod\limits\_{1\leq i<j\leq n}(x\_j-x\_i).$$
The proof is a simple reduction to Vandermonde's determinant. The first column is all constant, and every co... | 13 | https://mathoverflow.net/users/2384 | 268863 | 120,536 |
https://mathoverflow.net/questions/268196 | 18 | *I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory.*
I'm thinking about the beautiful story of "geometrical engineering" by Vafa, Hollowood, Iqbal (<https://arxiv.org/pdf/hep-th/0310272.pdf>) where various types of $\mathcal{N}=2$ SYM... | https://mathoverflow.net/users/105661 | Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory? | The answer is Yes, at least when the ALE space (more precisely the ALF space instead) is of type $A$. Very roughly, one consider Donaldson-Thomas invariants for the same noncompact Calabi-Yau space, but not of rank $1$, of higher ranks instead. (I do not know whether they correspond to Gromov-Witten invariants.) I do n... | 10 | https://mathoverflow.net/users/3837 | 268868 | 120,537 |
https://mathoverflow.net/questions/268883 | 4 | Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with bounded derivative. Define the Nemytskii map $F:H^1(\Omega) \to H^1(\Omega)$ by $F(u)(x) := f(u(x))$. Here $\Omega$ is a bounded smooth domain.
There exists work where we can deduce continuity and differentiability of $F$ under some assumptions on $f$.
Wh... | https://mathoverflow.net/users/103376 | When is a Nemytskii map between Sobolev spaces compact? | I think in your setting $F$ is not compact, unless $f$ is a constant map. Indeed, if $f$ is not a constant map, it coincides on some non-trivial interval $[a,b]$ with a smooth diffeomorphism $g:\mathbb{R}\to\mathbb{R}$ with derivatives $g'$ and $(g^{-1})'$ bounded on $\mathbb{R}$. Thus the corresponding Nemytskii map $... | 4 | https://mathoverflow.net/users/6101 | 268888 | 120,541 |
https://mathoverflow.net/questions/268893 | 4 | Since stochastic analysis in infinitely dimensional spaces has been developed in the past decades, e.g. Hida distributions, Malliavin calculus, just to name a few. However, I have almost never seen that the models from forementioned fields have been treated from a statistician's perspective. To be precise, I mean here,... | https://mathoverflow.net/users/41105 | Application of stochastic analysis in infinitely dimensional spaces in statistics? | (1) Malliavin calculus has been used in mathematical finance to compute sensitivity parameters of option prices, see these [lecture notes](http://www.math.wisc.edu/~kurtz/NualartLectureNotes.pdf) and this [research article.](http://www.tandfonline.com/doi/full/10.1080/00949655.2013.814135?src=recsys) And [Statistical i... | 2 | https://mathoverflow.net/users/11260 | 268895 | 120,543 |
https://mathoverflow.net/questions/268882 | 6 | Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$.
>
> **Question** Is it true that there exists a continuous finite dimensional representation $\pi$ of $O(n)$ such that $\rho$ is a direct summand of the restriction of $\pi$ to $G$?
>
>
>
I am almost s... | https://mathoverflow.net/users/16183 | Imbedding of a representation of a compact subgroup | I don't have a reference, but here is a short argument.
Assume $G<H$ are compact groups (in our example take $H=\text{O}(n)$)
and recall that unitary representations of compact groups are completely reducible. Recall also that by Peter-Weyl every irreducible unitary representation is finite dimensional.
Assume first ... | 13 | https://mathoverflow.net/users/89334 | 268898 | 120,545 |
https://mathoverflow.net/questions/268809 | 3 | Let$^1$
* $T>0$
* $U,H$ be separable $\mathbb R$-Hilbert spaces
* $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
* $(e^n)\_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe^n=\lambda\_ne^n\;\;\;\text{for all }n\in\mathbb N\tag1$$ for some $(\lambda\_n)\_{n... | https://mathoverflow.net/users/91890 | Covariation of the stochastic integral and the Wiener process | Since $$\lambda\_n\left\|\Phi\_s(\omega)e^n\right\|\_H\le\begin{cases}\lambda\_n&\text{, if }\left\|\Phi\_s(\omega)e^n\right\|\_H\le1\\\left\|\Phi\_s(\omega)e\_0^n\right\|\_H^2&\text{, if }\left\|\Phi\_s(\omega)e^n\right\|\_H\ge1\end{cases}\tag7$$ for all $n\in\mathbb N$, $\sum\_{n\in\mathbb N}\lambda\_n=\operatorname{... | 0 | https://mathoverflow.net/users/91890 | 268910 | 120,551 |
https://mathoverflow.net/questions/268890 | 6 | This is a [cross-post](https://math.stackexchange.com/questions/2246157/are-all-symmetries-of-the-dirichlet-functional-isometries) from MSE (no answer there).
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ **compact\***, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $E\_{... | https://mathoverflow.net/users/46290 | Are all symmetries of the Dirichlet functional isometries? | The answer to your question is "yes, every symmetry (in the sense you have specified) is an isometric immersion".
To see why, first note that, if $(M,g)$ is a compact Riemannian $m$-manifold and $h$ is *any* smooth quadratic form on an $n$-manifold $N$, one can always define the Dirichlet energy $E\_{g,h}(f)$ of a sm... | 2 | https://mathoverflow.net/users/13972 | 268912 | 120,552 |
https://mathoverflow.net/questions/268786 | 7 | Let $(M,g)$ be a Riemannian manifold with $LC$ conncection $\nabla$.
Assume that for every three **global** vector fields $X,Y,Z \in \chi^{\infty}(M)$ with $[X,Y]=0$ we have $\nabla\_{X} \nabla\_{Y} Z=\nabla\_{Y} \nabla\_{X} Z$
>
> Is the metric necessarily a flat metric?
>
>
>
| https://mathoverflow.net/users/36688 | A characterization of flat metrics via global vector fields | To see that $g$ is flat, we need enough pairs of commuting global vector fields. To construct these, let $f(r)\colon[0,\infty)\to[0,1)$ be a monotone function with $f(r)=1-\frac1{\log r}$ for $r\gg 1$. Consider the diffeomorphism $\mathbb R^n\to B^n$ given by $\Phi(x)=f(|x|)\,\frac x{|x|}$. We can map constant (and the... | 2 | https://mathoverflow.net/users/70808 | 268913 | 120,553 |
https://mathoverflow.net/questions/268873 | 5 | In the paper ["Schreier split epimorphisms between monoids"](https://link.springer.com/article/10.1007/s00233-014-9571-6) by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among monoids. I am hoping for a simpler proof of one direction.
*... | https://mathoverflow.net/users/69037 | Short proof a monoid is a group iff every splitting is right homogeneous | Frieder Ladisch noticed, that even less of 2. is needed than I thought. By his suggestion I make my comment into an answer.
Given 2. (in fact even less than one-sided homogeneity of one particular splitting suffices), let $A=B\times B$, let $f:B\times B\to B$ be the projection $(b',b)\mapsto b$, and let $s$ be the di... | 8 | https://mathoverflow.net/users/41291 | 268920 | 120,555 |
https://mathoverflow.net/questions/268828 | 8 | Let us say that a set $A \subseteq \mathbb N$ sends a series $\sum\_{n \in \mathbb N}a\_n$ of real numbers *to infinity* if the subseries $\sum\_{n \in A}a\_n$ sums either to $\infty$ or to $-\infty$.
>
> Given four or more conditionally convergent series, is there a single $A \subseteq \mathbb N$ that sends them a... | https://mathoverflow.net/users/70618 | Given four conditionally convergent series, is there a single sequence of naturals such that each corresponding subseries sums to $\pm\infty$? | There is a counterexample with 4 series. Notice that the problem is equivalent to asking if for every sequence of vectors $X\_j$ in $\mathbb R^4$ with lengths tending to $0$ and the infinite sum of absolute values of the projections to each coordinate axis, you can find a sequence of signs $\varepsilon\_j=\pm 1$ such t... | 11 | https://mathoverflow.net/users/1131 | 268932 | 120,560 |
https://mathoverflow.net/questions/268892 | 5 | Let $ k, n, r \geqslant 1 $ be integers. Let $ \lambda $ be a partition of $r$, what we denote by $ \lambda \vdash r $.
I would like a lower and an upper bound for the following quantity, for all $ r \leqslant nk $
$$ b\_r(n, k) := \frac{ \# \left\{ \lambda \vdash r : \lambda \subset (n^k) \right\} }{ \# \left\{ \... | https://mathoverflow.net/users/109373 | Proportion of partitions in a rectangle | For many purposes, you might be satisfied with just the distribution of the largest part, which was studied by Erdős and Lehner, "The distribution of the number of summands in the partitions of a positive integer." Duke Math. J. 8 (1951) 335. Asymptotically, it has a Gumbel distribution.
For the joint distribution o... | 4 | https://mathoverflow.net/users/2954 | 268942 | 120,562 |
https://mathoverflow.net/questions/268947 | 4 | The number of connected labeled graphs with $n$ edges and $n$ nodes are listed in [OEIS A057500](https://oeis.org/A057500).
I suspect the following must be known but I can't find a reference to it.
>
> **Question.** How many among the above graphs are triangle-free?
>
>
>
**NOTE.** This problem is connecte... | https://mathoverflow.net/users/66131 | Triangle-free labeled graphs | Since a connected graph with $n$ vertices and $n$ edges is unicyclic, you just need to subtract the ones whose cycle is a triangle (<http://oeis.org/A053507>) from the total (<http://oeis.org/A057500>).
Starting at $n=4$, I get 3, 72, 1500, 32280, 748440, 18898992, 520107840, 15555704400, 503580654720, 17569154733240... | 9 | https://mathoverflow.net/users/9025 | 268949 | 120,564 |
https://mathoverflow.net/questions/268525 | 21 | I have heard stated the following
**Theorem.** If $\Sigma$ is a (orientable) surface, then $\mathrm b\_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C\_1,\ldots,C\_m$) that you can make on $\Sigma$ without disconnecting it (i.e. with $\Sigma\smallsetminus (C\_1\cup\ldots\cup C\_m)$ still ... | https://mathoverflow.net/users/4721 | Do Betti numbers beyond the first have a "number of cuts" interpretation? | Here is one way that your observation about the first Betti number generalizes to higher-dimensional manifolds.
Suppose that $M$ a compact, orientable smooth manifold and $W$ is a regularly embedded, closed, orientable submanifold of codimension 1. Then if the number of path components of $W$ is greater than the firs... | 11 | https://mathoverflow.net/users/360 | 268951 | 120,565 |
https://mathoverflow.net/questions/268798 | 2 | Denote $[n]=\{1,2,\dots,n\}$. Assume $n\geq2$.
>
> **Question.** Is it true that given any $S\_1,S\_2,\dots,S\_{2n}$ (repetition allowed) subsets of $[2n]$ with $a\in S\_a$ and $\# S\_a=n$ for all $1\leq a\leq 2n$, there exist $i, j, k\in[2n]$ (not all equal) such that
> $$i\in S\_j, \qquad j\in S\_k, \qquad k\in... | https://mathoverflow.net/users/66131 | Existence of 3-distributed subsets | For $n>2$ this is true. Consider a directed graphs with arrows from $a$ to $S\_a\setminus a$. If it has arrows $a\to b$, $b\to a$, take $i=j=a$, $k=b$. If not, consider a vertex $a$ with maximal indegree, it is at least $n-1$ (since all outdegrees are equal to $n-1$). Let $B=\{i\ne a:a\in S\_i\}$, $C=S\_a\setminus a$. ... | 6 | https://mathoverflow.net/users/4312 | 268959 | 120,568 |
https://mathoverflow.net/questions/268339 | 1 | Consider an $\mathbb{F}\_q$-linear map $f:\mathbb{F}\_{q^n}\to \mathbb{F}\_{q^n}$ (so $f$ is a linearized polynomial). Suppose also that $f$ is not $\mathbb{F}\_{q^i}$-linear, where $i>1$.
My question is about the number of values that the function $f(x)/x$ can take, where $x$ is a nonzero element of $\mathbb{F}\_{q^... | https://mathoverflow.net/users/109065 | The number of values of $f(x)/x$ when $f$ is a linearized polynomial | This is an answer to the second part of your question.
The upper bound seems obvious, for the same reason as in the original case. The lower bound, however, is 1.
There is some linear map that sends the chosen subspace to 0, but is not itself 0. This then gives slope 0 on that entire subspace, giving an answer of 1... | 1 | https://mathoverflow.net/users/44191 | 268964 | 120,570 |
https://mathoverflow.net/questions/268782 | 5 | The Question
============
Consider the trace of an $n \times n$ unitary matrix with determinant 1
\begin{align}
f: SU(n) &\rightarrow \mathbb{C}\\
U \mapsto \text{tr}\, U &= \sum\limits\_{i=1}^{n-1} z\_i + \frac{1}{z\_1 \cdots z\_{n-1}}
\end{align}
where the $z\_i$ are the eigenvalues of $U$ and we have used $\de... | https://mathoverflow.net/users/68269 | Boundary of the image of a compact manifold in the complex plane | Here is one way to prove the claims about the image of the map $\mathrm{tr}:\mathrm{SU}(n)\to\mathbb{C}$. I'll just outline the steps. For simpicity, I'll always assume $n\ge 3$. (For completeness, observe that $\mathrm{tr}\bigl(\mathrm{SU}(2)\bigr)$ consists of the interval $[-2,2]\subset\mathbb{R}\subset\mathbb{C}$.)... | 2 | https://mathoverflow.net/users/13972 | 268972 | 120,571 |
https://mathoverflow.net/questions/268836 | 12 | Sheaves of sets on a space are somehow "parametrized sets". This is the philosophy by which one can do mathematics internal to a sheaf topos (of which theory I admit I know essentially nothing), with the possibility of defining all the usual mathematical concepts, except that in this case -if I understand correctly- th... | https://mathoverflow.net/users/4721 | Is the theory of vector bundles just linear algebra done in a suitable topos? | I'll gather all the snippets from the various comments here (and mark this post as community wiki).
The answer to the question "Is the theory of vector bundles just linear algebra done in a suitable topos?" is a resounding **yes**.
One needs to fix a ring object $\mathcal{O}\_X$ in the topos $\mathrm{Sh}(X)$ of set... | 14 | https://mathoverflow.net/users/31233 | 268974 | 120,572 |
https://mathoverflow.net/questions/268978 | 5 | While I was working on a variational problem, I met this equation as its Euler-Lagrange equation, but I cannot solve it:
$ x= \frac{af'(x)}{\sqrt{1+af'(x)^{2}}} + \frac{bf'(x)}{\sqrt{1+bf'(x)^{2}}} \ (a\neq b) $.
| https://mathoverflow.net/users/83772 | An ordinary differential equation | You can solve it parametrically as follows: Write
$$
x(t) = \frac{at}{\sqrt{1+at^2}}+\frac{bt}{\sqrt{1+bt^2}}
$$
and
$$
y(t) = c - \frac{\sqrt{1+at^2}+\sqrt{1+bt^2}}{\sqrt{1+at^2}\sqrt{1+bt^2}}
$$
where $c$ is a constant. Then this gives the general solution as $c$ varies.
If you want an explicit relation between $x... | 7 | https://mathoverflow.net/users/13972 | 268984 | 120,577 |
https://mathoverflow.net/questions/268982 | 6 | Is it possible to have an $\omega\_2$-length sequence of ($\omega\_1$-)Suslin trees such that if one builds the product of finitely many trees in that sequence, one ends up with a Suslin tree again?
The existence of such a sequence of length $\omega$ follows from $\diamondsuit$, as was shown by Jensen. By Shelah and ... | https://mathoverflow.net/users/4753 | $\omega_2$-sequence of Suslin trees | The answer is yes, and indeed, one can even have that any countable
number of the Suslin trees join to a Suslin tree.
To see this, simply force with countable support to add $\omega\_2$
many Suslin trees. The forcing to add one Suslin tree has
conditions consisting of countable normal $\alpha$-tree, for
$\alpha<\omeg... | 5 | https://mathoverflow.net/users/1946 | 268986 | 120,579 |
https://mathoverflow.net/questions/268989 | 8 | I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to [this MO question](https://mathoverflow.net/questions/106660/kroneckers-jugendtraum-for-real-quadratic-fields/) suggests that [the Stark conjectures](https://en.wiki... | https://mathoverflow.net/users/29873 | How to compute with the Stark conjectures? | The main reference here is the very useful User's Guide,
* C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "[User's Guide to PARI / GP](http://math.mit.edu/~brubaker/PARI/PARIusers.pdf)" (2003)
Particularly the sections about **bnrstark** (pp. 108), **quadhilbert** (pp. 87) and **quadray** (pp. 88).
For ... | 14 | https://mathoverflow.net/users/43108 | 268995 | 120,584 |
https://mathoverflow.net/questions/268981 | 1 | Consider the following integral
$$
\int\_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx
$$
where $a,b>0$. The integral can be solved exactly but I am not interested in that. I want to perform the expansion of the integral around $y=0$, maybe using something analogous to Leibnitz's rule for computing derivative... | https://mathoverflow.net/users/42072 | Expansions in terms of a variable in the integration limit in a point where the integration limit is not differentiable | $$\int\_{-\infty}^{-(-a+\frac{b}{y})^{1/2}}\frac{e^{-x^2}}{x}dx=\tfrac{1}{2}{\rm Ei}\,(a-b/y)=$$
$$\qquad\qquad=\tfrac{1}{2}(a-b/y)^{-1}\exp(a-b/y)\sum\_{n=0}^\infty\frac{n!}{(a-b/y)^n}$$
$$\qquad\qquad=-\frac{y }{2 b}e^{a-\frac{b}{y}}\sum\_{n=0}^\infty (-1)^{n}n!(1-a)^n(y/b)^n$$
| 3 | https://mathoverflow.net/users/11260 | 268996 | 120,585 |
https://mathoverflow.net/questions/268905 | 9 | Let $p$ be a prime, let $\zeta\_p=e^{2\pi i/p}$,
let $g\in{\bf F}\_p$ be a non-square and
let $\chi:{\bf F}\_p^\*\rightarrow{\bf C}^\*$ be a non-trivial character.
Then the complex numbers
$$
\chi(n)\sum\_{r\in{\bf F}\_p}\chi(r^2-g)\zeta\_p^{nr},\quad(\hbox{$n\in{\bf F}\_p^\*$})
$$
have the same arguments modulo $\pi... | https://mathoverflow.net/users/18645 | Arguments of exponential sums | There are at least 3 ways of seeing this - an elementary way, via the determinants of $\ell$-adic sheaves, and via Katz's calculus of finite field hypergeometric functions.
---
Elementary:
Subtraction inside a multipliciative character is too difficult. Let's add a new variable
$$=\chi(n) \sum\_{r,t \in \math... | 13 | https://mathoverflow.net/users/18060 | 268997 | 120,586 |
https://mathoverflow.net/questions/268170 | 5 | An operator ideal $\mathcal{I}$ possesses the $\sum\_{p}$-property, say $1<p<\infty$, if for arbitrary collections of Banach spaces $E\_{m}, F\_{n} (m,n=1,2,\ldots)$ the following holds:
if $T\in \mathcal{L}((\sum\_{m} E\_{m})\_{p},(\sum\_{n} F\_{n})\_{p})$ and $Q\_{n}TJ\_{m} \in \mathcal{I}(E\_{m},F\_{n}) \mbox { fo... | https://mathoverflow.net/users/85647 | the sigma-p property | The answer to your first question is no. To see this, one can consider for every ordinal $\xi$ the class $\mathcal{I}=\textbf{Sz}\_\xi$ of operators whose Szlenk index does not exceed $\omega^\xi$. Then $\text{Space}(\mathcal{I})$ contains all $\ell\_p$ spaces for any $\xi>0$, $1<p<\infty$ (these spaces are asymptotica... | 3 | https://mathoverflow.net/users/109440 | 269000 | 120,587 |
https://mathoverflow.net/questions/269001 | 3 | Consider a polynomial $\sum\limits\_{k=0}^n a\_kx^k$ with $a\_k\geq 0$ and $x\geq 0$.
In this [comment](https://mathoverflow.net/questions/229028/log-concave-polynomial-is-a-log-concave-function), [Richard Stanley](https://mathoverflow.net/users/2807/richard-stanley) mentions that polynomials with only real roots are l... | https://mathoverflow.net/users/91545 | Are polynomials with only real zeros log concave functions? | This fact is trivial. Your assumptions imply that $$f(x)=ax^m\prod(1+x/x\_k),$$
where $a>0$ and $x\_k>0$. All zeros $-x\_k$ are negative because you assume that
$a\_k\geq 0$, so there are no positive zeros. Now every factor in this product is log-concave, therefore the product is log-concave.
| 7 | https://mathoverflow.net/users/25510 | 269005 | 120,588 |
https://mathoverflow.net/questions/269002 | 10 | The following is known as *deletion-contraction formula*:
>
> Assume $\Gamma$ is a connectted graph with edge $\rho$ then
> $$t(\Gamma)=t(\Gamma\backslash\rho)+t(\Gamma/\rho),$$
> where $\Gamma\backslash\rho$ is $\Gamma$ with the edge $\rho$ removed and $\Gamma/\rho$ obtained from $\Gamma$ by contracting $\rho$ a... | https://mathoverflow.net/users/1441 | History of deletion-contraction formula | This seems to have first been observed in
*Brooks, R.L.; Smith, C.A.B.; Stone, A.H.; Tutte, W.T.*, [**The dissection of rectangles into squares**](http://dx.doi.org/10.1215/S0012-7094-40-00718-9), Duke Math. J. 7, 312-340 (1940). [ZBL0024.16501](https://zbmath.org/?q=an:0024.16501).
See equation 3.12 and Theorem 3.... | 15 | https://mathoverflow.net/users/1345 | 269011 | 120,590 |
https://mathoverflow.net/questions/268937 | 4 | Let $X$ be a locally connected topological space, which is covered by open sets $\{U\_{\alpha},\alpha\in A\}$ and let $C$ be an arc in $X$, i.e. a homeomorphic image of an interval.
Is it always possible to choose open sets $V\_1,...,V\_n$ that cover $C$ and such that
* Each $V\_i$ is contained in some $U\_{\alpha}... | https://mathoverflow.net/users/53155 | Inscribing a "chain" into an open cover | The answer is ``yes'' if $X$ is Hausdorff.
We can identify the arc $C$ with the unit interval $[0,1]$ and assume that $[0,1]$ is contained in $X$ and is covered by a family $\mathcal U$ of open subsets of $X$. By the compactness of $[0,1]$, there exists an increasing sequence $0=a\_0<a\_1<\dots<a\_n=1$ of real numbe... | 4 | https://mathoverflow.net/users/61536 | 269013 | 120,591 |
https://mathoverflow.net/questions/267618 | 2 | So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$?
Using the Chernoff bound we know that $\frac{1}{\epsilon^2}\log(1/\delta)$ is enough. And I know that this is also a... | https://mathoverflow.net/users/148158 | Lower bound on number of samples for an epsilon delta approximation matching the Chernoff bound | **This argument based on modification of [2].**
**(1)** Let A be the event that $\frac{1}{n}\sum\_{i=1}^{n}X\_{i}\geq\frac{1}{2}$ , let $X\_{i}=0$ mean heads of coin and $X\_{i}=1$ mean tails.
$Pr(A)\leq Pr\left\{ \left|\frac{1}{n}\sum\_{i=1}^{n}X\_{i}-\left(\frac{1}{2}-\epsilon\right)\right|\geq\epsilon\right\} $
... | 2 | https://mathoverflow.net/users/25437 | 269027 | 120,597 |
https://mathoverflow.net/questions/268991 | 6 | Let $\mathbf{Sch}\_k$ be the category of $k$-schemes of finite type, and let $K\_0(\mathbf{Sch}\_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the virtual Lefschetz motive $\mathbb{L} := [\mathbb{A}^1\_k]$. Is every element of $\mathbb{Z}[\mathbb{L}]$ the class of a ... | https://mathoverflow.net/users/12884 | Virtual mixed Tate motives | Here is a proof of the fact that over an infinite field $f(\mathbb{L})$ is a class of a scheme if and only if the leading term of $f$ is positive, as suggested by Will Sawin.
**Example.** For $f(t)=a\_nt^n+a\_{n-1}t^{n-1}+\dots+a\_0$ with $a\_n>0$ throw away from affine space of dimension $n$ planes $\mathbb{A}^i$ in... | 6 | https://mathoverflow.net/users/39304 | 269029 | 120,598 |
https://mathoverflow.net/questions/269032 | 7 | Assuming $\text{PA}$ is consistent. Then $\text{PA} + \neg\text{Con}(\text{PA})$ is consistent and have a model, say $M$. We know $M$ must be nonstandard, in which case, there is a nonstandard proof of $0=1$ from ($M$'s version of) $\text{PA}$ in $M$. The fact that $M$'s version of $\text{PA}$ is different from the "re... | https://mathoverflow.net/users/18879 | Can the "real" Peano Arithmetic be inconsistent? | It seems that the Feferman-style description of PA will exhibit your requirements.
Specifically, consider the theory $P$ defined as follows. Begin to enumerate the usual PA axioms, but include the next axiom in $P$ only if doing so keeps $P$ consistent. Never add an axiom to $P$ that would make it inconsistent.
S... | 7 | https://mathoverflow.net/users/1946 | 269033 | 120,599 |
https://mathoverflow.net/questions/269031 | 4 | For $m \in \mathbb{N}$, let $E\_m \colon S^1 \to S^1$ be multiplication map $x \mapsto mx$. Also, let $R\_\alpha$ be the map $x \mapsto x+\alpha$.
Now, consider $E\_m \times R\_\alpha \colon S^1 \times S^1 \to \colon S^1 \times S^1$ , and an invariant measure $\mu$ with respect to this map. Then, is $\mu$ necessarily... | https://mathoverflow.net/users/109453 | What are invariant measures of $E_m \times R_\alpha$ on $S^1 \times S^1$? Are they necessarily product measures? | No. Define a function $f\colon S^1\to\{0,1\}$ by $f(x)=1$ if $x\in [0,\alpha)$ and 0 otherwise. Now define $\pi\colon S^1\to S^1$ by $\pi(x)=\sum\_{n=0}^\infty f(R\_\alpha^n x)m^{-(n+1)}$. It is easy to see that $\pi(R\_\alpha x)=E\_m(\pi(x))$. Now the measure defined on rectangles by $\mu(A\times B)=\lambda(\pi^{-1}(A... | 7 | https://mathoverflow.net/users/11054 | 269034 | 120,600 |
https://mathoverflow.net/questions/268919 | 4 | Physicists frequently talk about symmetries of a theory, and them being generated by Killing vectors. While this is clear to me in the context of gravity, where a Killing field $\xi$ is defined by $\mathcal{L}\_\xi g = 0$, I am confused by the concept in the context of Yang-Mills. Below are my thoughts about the matter... | https://mathoverflow.net/users/104213 | Killing fields for Yang-Mills | An infinitesimal diffeomorphism, generated by $\xi^a$, acts on the metric as $g\_{ab} \mapsto g\_{ab} + 2 \nabla\_{(a} \xi\_{b)}$. The last term is zero precisely when $\nabla\_{(a} \xi\_{b)} = 0$, that is, when $\xi^a$ is a Killing vector.
An infinitesimal gauge transformation on a principal bundle, generated by a $... | 3 | https://mathoverflow.net/users/2622 | 269050 | 120,602 |
https://mathoverflow.net/questions/268284 | 2 | Let $G$ be a semisimple connected complex Lie group, compact real Lie group or linear algebraic group. Let $\chi$ be the character of a finite dimensional irreducible representation of $G$ (I am particularly interested in the adjoint representation, or perhaps the fundamental ones, but general statements are useful too... | https://mathoverflow.net/users/17064 | Critical points of characters on semisimple groups | The question "does every semisimple $\chi$-critical $g\in G$ necessarily have finite order?" (with $\chi$ being the character of the adjoint representation) has a negative answer, as shown by the following fairly trivial counterexample:
* Let $g \in \mathit{SO}\_6$ be the rotation that is the orthogonal direct sum (i... | 0 | https://mathoverflow.net/users/17064 | 269051 | 120,603 |
https://mathoverflow.net/questions/268998 | 8 | The space of $(d+1)$-dimensional antisymmetric matrices has the same dimension as the space of $d$-dimensional symmetric matrices, $\frac12d(d+1)$. There are isomorphisms between the two spaces, e.g. for $d=2$
$$
\begin{pmatrix}
a & b\\ b & c
\end{pmatrix}
\leftrightarrow
\begin{pmatrix}
0 & a & b\\ -a & 0 & c\\ -b &... | https://mathoverflow.net/users/81448 | Coordinate free isomorphism between $d+1$-dimensional antisymmetric rank $2$ tensors and $d$-dimensional symmetric rank $2$ tensors | Here is a **revised** partial answer and some comments:
It seems that you are asking for some kind of isomorphism between $S^2(\mathbb{F}^d)$ and $\Lambda^2(\mathbb{F}^{d+1})$ that would 'have the greatest symmetry', in the sense that it would commute with some group action on $\mathbb{F}^d$ and $\mathbb{F}^{d+1}$, t... | 7 | https://mathoverflow.net/users/13972 | 269056 | 120,607 |
https://mathoverflow.net/questions/269035 | 9 | What is the largest prime number $p$ for which one knows examples of nonisogenous elliptic curves $E\_1$ and $E\_2$ over $\mathbb{Q}$ with isomorphic mod $p$ Galois representations: $E\_1[p] \cong E\_2[p]$? I do not require the isomorphism to preserve the Weil pairing.
| https://mathoverflow.net/users/63877 | Elliptic curves with the same mod $p$ representation | The largest known is for $p = 17$. This can be found by searching Cremona's tables. In particular, in [this paper](https://www.dpmms.cam.ac.uk/~taf1000/papers/highercongr.pdf), Tom Fisher mentions the examples of
curves 3675b1 and 47775b1, which are
$E\_{1} : y^{2} + xy + y = x^{3} + x^{2} - 393x - 9654$
and
$E\_... | 15 | https://mathoverflow.net/users/48142 | 269057 | 120,608 |
https://mathoverflow.net/questions/269030 | 4 | First, an observation: Here are two examples of when different notions of weak equivalence in a category turn out to be the "same":
* (Whitehead's Theorem) Two CW complexes are homotopy equivalent if and only if they are weakly homotopy equivalent (i.e. there is a map between them which induces an isomorphism on homo... | https://mathoverflow.net/users/108901 | When are two kinds of weak equivalence 'the same'? | The notions of weak equivalence are not 'the same' in your examples, rather they coincide on a certain class of objects, namely those that are 'cofibrant-fibrant' as mentioned in Arun's comment. Perhaps there is a more general question hidden in yours, however, as in both examples you have one notion of homotopy equiva... | 4 | https://mathoverflow.net/users/3502 | 269066 | 120,610 |
https://mathoverflow.net/questions/268770 | 8 | For $G = PSU(3)$, it is known that $\dim I(G;\mathbb Q) / I(G;\mathbb Q)^2 = 3$, while $H^{\*\*}(BG;\mathbb Q)$ is obviously a power series ring in two indeterminates since $G$ has rank 2. This would would be fine except that there is an elementary argument the natural map should induce a bijection on spaces of indecom... | https://mathoverflow.net/users/5792 | Doesn't completion of a representation ring preserve its indecomposables? | In fact $\text{dim }I(G)/I(G)^2=2$. Let $x=[V\_{3L\_1}]$, $y=[V\_{2L\_1+L\_2}]$ and $z=[V\_{3L\_1+3L\_2}]$. Then
\begin{eqnarray}R(PSU(3))\cong\mathbb{Z}[x, y, z]/(y^3-y^2-xz-2y(x+z)-x-y-z).\end{eqnarray}
Note that $\text{dim }V\_{3L\_1}=\text{dim }V\_{3L\_1+3L\_2}=10$ and $\text{dim }V\_{2L\_1+L\_2}=8$. If $\overli... | 2 | https://mathoverflow.net/users/2306 | 269070 | 120,614 |
https://mathoverflow.net/questions/269071 | 2 | If $A,B,C$ are acute angles then is it always true :
$$\sum\_{cyc} \frac{ \sin A}{ \sin B} \le \sum\_{cyc} \frac{A}{B} \le \sum\_{cyc} \frac{ \tan A}{ \tan B}$$
where
$\sum\_{cyc}$ denotes the cyclic sum.
| https://mathoverflow.net/users/109471 | if A,B,C are acute then is it true : cyclic sum ( ((sin x)/(sin y)) le ((x)/(y)) le ((tan x)/(tan y)) )? | Yes. This follows from the following
Lemma. If $x,y,z>0$ and $xyz=XYZ=1$ and $\min(x,y,z)\leqslant \min(X,Y,Z)$, $\max(x,y,z)\geqslant \max(X,Y,Z)$, then $x + y + z\geqslant X + Y + Z$.
Proof. Move maximal and minimal elements in $\{x,y,z\}$ making them closer to each other with fixed product. The sum decreases an... | 7 | https://mathoverflow.net/users/4312 | 269078 | 120,616 |
https://mathoverflow.net/questions/268956 | 6 | Consider Prikry's forcing for changing the cofinality of a measurable cardinal into $\omega.$ The forcing has the Prikry property and one can prove this either directly or using Rowbottom's theorem which is a Ramsey type theorem.
Now consider Magidor's forcing for changing the cofinality of a large cardinal into $\om... | https://mathoverflow.net/users/11115 | Ramsey type theorems and Magidor's forcing | There is an analog of Rowbottom's theorem which can be extracted from the proof of the Magidor forcing:
Suppose that $u$ is a sequence of ultrafilters on $\kappa$ of length $\lambda<\kappa$ and $f\colon [\kappa]^{<\omega}\to 2$. Then there are sets $A\_\nu\in u(\nu)$ for $\nu<\lambda$ such that whenever $\langle \nu\... | 4 | https://mathoverflow.net/users/109444 | 269081 | 120,618 |
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