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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/396840 | 1 | Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}\_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, and equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}(r)d\theta^{2},$$ where $dr^{2}$ is the standard metric on... | https://mathoverflow.net/users/163368 | Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold | Some results are known under some restrictions. First, no uniform inequality can hold if $U$ is unrestrained: it could take the shape of (a neighborhood of) a bottle, and the bottleneck will make it possible to have $\Omega$ with large volume and small boundary. A natural restriction is to consider convex $U$ (so that ... | 2 | https://mathoverflow.net/users/4961 | 396900 | 163,875 |
https://mathoverflow.net/questions/396776 | 15 | One of my colleagues gave me the following problem about 15 years ago:
Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (The sides of the squares and the rectangle need not be parallel to each other.)
I couldn't find a solution or even ... | https://mathoverflow.net/users/313754 | Three squares in a rectangle | Since the squares are convex, we can draw lines which separate them. In particular, if two separating lines go from $(b-a,0)$ to $(b,1)$ and from $(c,1)$ to $(c+d,0)$, then we can prove the result in terms of those lines and those variables.
So: let the rectangle go from $(0,0)$ to $(2,1)$. Let $A$ be the leftmost sq... | 5 | https://mathoverflow.net/users/nan | 396903 | 163,877 |
https://mathoverflow.net/questions/396849 | 4 | When dealing with some hash functions that I was trying to speed up, I [toyed](https://mathoverflow.net/questions/394296/hamming-distance-between-ab-and-a-oplus-b-oplus-a-land-b-ll-1) with a binary operation with the goal to "approximate" the addition on $\{0,1\}^\*$ when seen as binary representation of the positive i... | https://mathoverflow.net/users/8628 | Non-associative commutative "group" | The answer to the first question is **yes**:
>
> **Claim.** Let $a, b \in \{0, 1 \}^{\ast}$ and let $n \ge 0$ be the least integer such that $a(i) = b(i) = 0$ for every $i > n$. Then the equation $$a +\_2 x = b$$ has a unique solution $x \in \{0, 1 \}^{\ast}$ which is recursively defined by $x(0) = a(0) + b(0)$, $x... | 5 | https://mathoverflow.net/users/84349 | 396908 | 163,880 |
https://mathoverflow.net/questions/396494 | 6 | If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f\*g\rVert\_p\leq\lVert f\rVert\_1\lVert g\rVert\_p\qquad (f\in L^1(\mathbb{R}^n),\,g\in L^p(\mathbb{R}^n)),$$ for complex $f$ and $g$. But I cannot seem to determine one for $p=\infty$ nor ... | https://mathoverflow.net/users/306090 | When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$? | Here is a fairly simple condition.
It uses the following notion: A family of functions $f\_t\in L^1$ depending on a parameter $t$ in a measure space $X$ is said to *tend to $f$ somewhere* if the essential infimum of $\lVert f\_t-f\rVert\_1$ over $X$ is $0$.
Put $g\_s(t)=g(t-s)$. The condition is that $afg\_s\to\lve... | 1 | https://mathoverflow.net/users/306090 | 396910 | 163,881 |
https://mathoverflow.net/questions/396875 | 4 | Considering the Jacobi theta: $\theta\_3(z) = \sum\_{n\in\mathbb{Z}} q^{n^2}$,
we can invert $\theta\_3-1$ in a small enough neighbourhood of 0.
Routine computation with Lagrange-Burmann inversion gives that the inverse have expansion
starting by:
$\frac{q}{2}-\frac{q^4}{16}+\frac{q^7}{32}-\frac{q^9}{512}-\frac{11 ... | https://mathoverflow.net/users/70925 | Closed formula for reversion of Jacobi theta series | Perhaps, the form given by [Lagrange inversion theorem](https://en.wikipedia.org/wiki/Lagrange_inversion_theorem) cannot be much simplified here. It expresses the $n$-th coefficient of a series reversion as the sum of $n-1$ values of exponential Bell polynomials.
From the practical perspective, since $\theta\_3-1$ co... | 3 | https://mathoverflow.net/users/7076 | 396917 | 163,882 |
https://mathoverflow.net/questions/396922 | 1 | There is a coloured operad $sOp$ such that $sOp$-algebras are single-coloured operads. This operad has a simple description in terms of generators and relations, say, as an operad $F(X)/R$. There is a more concrete description of $sOp$ as the operad $OpTrees$ of trees endowed with additional structure, see [1, section ... | https://mathoverflow.net/users/78299 | Lawvere theory of Lawvere theories | Viewing cartesian operads as algebraic theories, your question may be rephrased as: "How can we present the (multisorted) algebraic theory whose models are (monosorted) algebraic theories?"
This is given by the theory of [abstract clones](https://en.wikipedia.org/wiki/Clone_(algebra)#Abstract_clones). You can check t... | 2 | https://mathoverflow.net/users/152679 | 396926 | 163,886 |
https://mathoverflow.net/questions/396931 | 2 | A graph $G$ is Hamiltonian if there is a [Hamiltonian cycle](https://mathworld.wolfram.com/HamiltonianCycle.html) in $G$.
Suppose $G$ is a [$k$-edge connected](https://en.wikipedia.org/wiki/K-edge-connected_graph) $k$-regular graph with $k>1$.
Does this ensure that $G$ is Hamiltonian?
If not, how about [vertex-transi... | https://mathoverflow.net/users/114739 | Is every $k$-edge connected $k$-regular graph Hamiltonian? | Assume $k \geq 4$, since, for $k=2$, the answer for both questions is **yes**, and for $k=3$, **no**, as there's the [Petersen graph](https://en.wikipedia.org/wiki/Petersen_graph).
The answer to the first question is **no**. To see this, we only need to prove that the constructions by Meredith in [1] give $k$-edge-co... | 5 | https://mathoverflow.net/users/125498 | 396936 | 163,888 |
https://mathoverflow.net/questions/396924 | 0 | given a bipartite graph $G(U,V,E\subseteq U\times V)$ with strictly positive edge-weights; is there an established name for the the task of calculating the lightest spanning subgraph and what is the best known algorithmic complexity?
Put more colloquially: if there are $p$ persons and $t$ tasks with possibly differen... | https://mathoverflow.net/users/31310 | Name for a type of assignment task | This is the [transportation problem](https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)). The people correspond to supply nodes, with a lower bound of 1 and upper bound equal to the number of tasks that person can handle. The tasks correspond to demand nodes, with a lower bound of 1 and upper bound impli... | 1 | https://mathoverflow.net/users/141766 | 396937 | 163,889 |
https://mathoverflow.net/questions/396897 | 11 | Given a closed (perhaps irreducible) 3-manifold $M$ with an embedded surface $S$ and a knot $K$, what conditions allow the two to be isotoped to be disjoint? Obviously a necessary condition is that $[S] \cdot [K] = 0$, but when is this sufficient? [This](https://mathoverflow.net/questions/191585/knots-in-3-manifolds) a... | https://mathoverflow.net/users/314845 | When can a surface in a 3-manifold be isotoped off a knot? | Let’s assume that the manifold $M$ is irreducible and orientable and the surface $S$ is orientable. This is to avoid 1-sided surfaces.
First let’s assume that the surface $S$ is fully [compressible](https://en.wikipedia.org/wiki/Incompressible_surface?wprov=sfti1). That means that there is a sequence of compressions ... | 9 | https://mathoverflow.net/users/1345 | 396939 | 163,891 |
https://mathoverflow.net/questions/396918 | 2 | Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\mathfrak{p} \subseteq R \otimes\_K L$ *good* when $\mathfrak{p} \cap R = \mathfrak{m}\_R$. Notice that good prime ideals corr... | https://mathoverflow.net/users/2841 | Good prime ideals in tensor products of local rings | Let $x$, resp $y$ be the closed point of $X$, resp $Y$. Denote $f : Y \to X$ the given morphism. Then $f(y) \leadsto x$ (specialization). Let $x\_L$ be any point of $X\_L$ mapping to $x$. The morphism $X\_L \to X$ is flat. Hence there is a specialization $z \leadsto x\_L$ in $X\_L$ such that $z$ maps to $f(y)$. Since $... | 2 | https://mathoverflow.net/users/152991 | 396942 | 163,892 |
https://mathoverflow.net/questions/396946 | 2 | Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty.
Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped with their uniform convergence on compact topologies. Does the map:
$$
C(X\times Y,Z)\ni f \to (x\mapsto [y\mapsto f(x,y)... | https://mathoverflow.net/users/176409 | Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$ | In the context of functions from one space to another, a key-word here might by "currying", for the process of converting $x,y\to f(x,y)$ to $x\to (y\to f(x,y)$. In the case of spaces of holomorphic functions with the natural topology, verification is not hard.
In other contexts, isomorphisms $\operatorname{Hom}(X\ot... | 2 | https://mathoverflow.net/users/15629 | 396947 | 163,895 |
https://mathoverflow.net/questions/396954 | 0 | Prove that $\sum\_{k=0}^n {k+1\choose 2}^R + \sum\_{k=0}^{-n-2} {k+1\choose 2}^R = 0.$ This can be shown using Faulhaber's formula but it's very long. Is there a nicer, shorter method? Any thoughts or ideas?
| https://mathoverflow.net/users/265714 | Prove that $\sum_{k=0}^n {k+1\choose 2}^R + \sum_{k=0}^{-n-2} {k+1\choose 2}^R = 0.$ | The "correct" definition of $\sum\_{k=0}^{-n-2} f(k)$ is $-\sum\_{k=-n-1}^{-1}f(k)$. Now ${-a+1\choose 2}={a\choose 2}$. Thus the terms of $\sum\_{k=0}^{-n-2}{k+1\choose 2}^R$ cancel with the terms of $\sum\_{k=0}^n{k+1\choose 2}^R$.
| 4 | https://mathoverflow.net/users/2807 | 396960 | 163,899 |
https://mathoverflow.net/questions/396362 | 2 | For all $k,R \in \mathbb{N}$ fixed, prove that $ \sum\_{i=0}^{2k}( {n+R-1\choose R+i} + (-1)^{i+1}{ n+R+i\choose R+i } )\sum\_{j=0}^i {i\choose j}(-1)^j(i+1-j)^{2k}=0 $. I'm quite sure this is true but I'm not very sharp in seeing how to go about proving it. I was studying some stuff in some Pascal triangles is how I s... | https://mathoverflow.net/users/265714 | Prove that $ \sum_{i=0}^{2k}( {n+R-1\choose R+i} + (-1)^{i+1}{ n+R+i\choose R+i } )\sum_{j=0}^i {i\choose j}(-1)^j(i+1-j)^{2k}=0 $ | ~~First off, there should be $(-1)^{i+1}$ not $(-1)^{R+i}$ (now it's corrected in the question).~~
**UPDATE.** Argument below is simplified and streamlined.
The identity generalizes the [previous question](https://mathoverflow.net/q/395758), which essentially represents the case of $R=1$. In fact, this generalized... | 2 | https://mathoverflow.net/users/7076 | 396963 | 163,900 |
https://mathoverflow.net/questions/396941 | 9 | Might there be a good reference on the interaction of number theory with statistical physics? I am particularly interested in innovations in number theory that have led to breakthroughs in statistical physics.
One such result that I am aware of is the Lee-Yang theorem. Marc Kac apparently used insights from Pólya's a... | https://mathoverflow.net/users/56328 | Innovations in number theory leading to breakthroughs in statistical mechanics | 1. The paper “The reasonable and unreasonable effectiveness of number theory in statistical mechanics” by George Andrews comes to mind. It is a nice survey that mentions some of the more striking appearances of number-theoretic results in statistical mechanics, such as the Rogers-Ramanujan identities and their connecti... | 10 | https://mathoverflow.net/users/78525 | 396972 | 163,903 |
https://mathoverflow.net/questions/396952 | 4 | In the paper of Rouquier on the dimension of triangulated categories (found [here](https://www.math.ucla.edu/%7Erouquier/papers/dimension.pdf)) lemma 3.5 says:
**Lemma** Let $\mathcal{T}$ be a triangulated category and let $\mathcal{T}\_1$ and $\mathcal{T}\_2$ be triangulated subcategories of $\mathcal{T}$ such that ... | https://mathoverflow.net/users/171303 | How to prove a lemma of Rouquier on the dimension of triangulated categories? | If $\mathcal{T}\_{1}=\langle M\_{1}\rangle\_{d\_{1}+1}$ and
$\mathcal{T}\_{2}=\langle M\_{2}\rangle\_{d\_{2}+1}$, then
$\mathcal{T}\_{1}\ast\mathcal{T}\_{2}\subseteq\langle M\_{1}\oplus
M\_{2}\rangle\_{d\_{1}+d\_{2}+2}$.
**Sketch proof:** Since $\mathcal{T}\_{i}\subseteq\langle M\_{1}\oplus
M\_{2}\rangle\_{d\_{i}+1}$... | 7 | https://mathoverflow.net/users/22989 | 396987 | 163,905 |
https://mathoverflow.net/questions/396985 | 4 | This is a simple question, just looking for a reference for a formula.
As far I understand the genus of a prime Fano $n$-fold is defined to be the genus of a complete intersection of $n-1$ smooth divisors in the system $|-K\_{X}|$ (see [https://www.math.ens.fr/~debarre/ExposePoitiers2013.pdf](https://www.math.ens.fr/... | https://mathoverflow.net/users/99732 | Formula for genus of a Fano variety | It's not clear from your question if you really want a reference (as per the first line) or the formula itself (as per what is written after "Question"). But the computation of the genus is not hard so let me write it here in case that answers the question.
If $C$ is such a smooth curve in $X$, then the normal bundle... | 6 | https://mathoverflow.net/users/121595 | 396988 | 163,906 |
https://mathoverflow.net/questions/396969 | 5 | Suppose $\pi:\mathcal{X}\rightarrow S$ is a smooth family of complex affine varieties (to make things simpler, we can actually assume it is locally trivial). Let $P$ be a smooth projective variety, and assume there is an embedding $\mathcal{X}\hookrightarrow P\times S$ over $S$. Let $\overline{\pi}: \overline{\mathcal{... | https://mathoverflow.net/users/140928 | When is the fiberwise compactification (not) equal to the compactification of the family? | No.
Let $S = \mathbb{A}^1\_{\mathbb{C}}$ and $\mathcal{X} = \mathbb{A}^1\_S \rightarrow S$ be the constant family. Let $\mathcal{Y}$ be the blow-up of the surface $\mathbb{P}^1\_S$ at the closed point over $0\in S$ lying at infinity. In other words: $\mathbb{P}^1\_S = \mathbb{P}^1\times \mathbb{A}^1$ and $\mathcal{Y}$ ... | 4 | https://mathoverflow.net/users/110362 | 396991 | 163,907 |
https://mathoverflow.net/questions/396883 | 3 | $\DeclareMathOperator\SmallGroup{SmallGroup}$**Definition.** A subset $A$ of a group $G$ is called *product-1-free* if for any sequence of pairwise distinct elements $a\_1,\dots,a\_n$ of $A$ the product $a\_1\cdots a\_n$ is not equal to 1 in $G$.
For a finite group $G$, let $f\_1(G)$ be the largest cardinality of a p... | https://mathoverflow.net/users/61536 | Large product-1-free sets in finite groups | Let me sketch a strategy for proving the lower bound:
**Lemma**: Let $S\_1,\dots, S\_k$ be the composition factors of $G$. Then
$$ f\_1(G)\geq f\_1(S\_1)+\cdots +f\_1(S\_k).$$
**Sketch of proof**:
Take a series:
$$G=G\_0\rhd G\_1 \rhd \cdots \rhd G\_k=\{1\}$$
where $G\_{i-1}/G\_i\cong S\_i$. For each $i$ write $\ell\... | 4 | https://mathoverflow.net/users/801 | 397002 | 163,909 |
https://mathoverflow.net/questions/397005 | 1 | Consider a subspace $V$ of $\mathcal{H}\_A \otimes \mathcal{H}\_B$, with $\mathcal{H}\_A$ and $\mathcal{H}\_B$ finite-dimensional Hilbert spaces, that is $1\_A \otimes U$ invariant for all unitary operators $U$ on $\mathcal{H}\_B$. Is $V = V\_A \otimes \mathcal{H}\_B$ for some subspace $V\_A$ of $\mathcal{H}\_A$?
| https://mathoverflow.net/users/69967 | Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$? | The action of $1\_A\otimes U$ on $V$ defines the action of the unitary group $U(H\_B)$ on $V$. This action induces the action of Lie algebra $\mathfrak{u}(H\_b)$ on $V$. The action has the form $1\otimes T$ for skew-hermitian $T$. Every operator on $H\_B$ has the form $T\_1+iT\_2$ for $T\_1,T\_2$ skew-hermitian, so we ... | 2 | https://mathoverflow.net/users/140292 | 397010 | 163,910 |
https://mathoverflow.net/questions/396950 | 4 | Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed **finite** interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies
$$ \nabla\_{\dot{\gamma}(s)}\dot\gamma(s)=0 \quad \text{and} \quad g(\dot{\gamma}(s),\dot{\gamma}(s))=0\quad \text{for... | https://mathoverflow.net/users/50438 | A question on null geodesics in Lorentzian geometry | We argue by contradiction. Suppose there are infinitely intersections. By reversing time orientation if necessary, there exists a monotonically increasing sequence of times $s\_n \in I$ such that at each $\gamma(s\_n)$ the curves $\gamma$ and $\beta$ intersect. Since $I$ is compact $s\_n$ has a limit $s'\in I$. For con... | 4 | https://mathoverflow.net/users/3948 | 397011 | 163,911 |
https://mathoverflow.net/questions/397006 | 4 | This is a question about marginals of probability measures, which seems unrelated to previous questions.
Let $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ be the unit sphere. Assume that for each $\theta\in \mathbb{S}^{d-1}$ there is an associated probability measure $\mu\_\theta$ on $\mathbb{R}$.
**Question:** Under what... | https://mathoverflow.net/users/8354 | Existence of measures with given 1d marginals | $\newcommand\th\theta\newcommand\vpi\varphi\newcommand\R{\mathbb R}\newcommand\S{\mathbb S^{d-1}}$**Condensed version of the answer:** If $\th\cdot X\sim\mu\_\th$ for all $\th\in\S$, then for each nonzero $t\in\R^d$ the distribution, say $\mu\_t$, of $t\cdot X$ is determined by a simple rescaling of $\mu\_\th$ (the cas... | 5 | https://mathoverflow.net/users/36721 | 397012 | 163,912 |
https://mathoverflow.net/questions/396533 | 3 | I am new to Math Overflow, so I am not really sure whether this question fits the community standards. But, I posted this question in Stack Exchange and recieved no answers. Moreover, nothing even close to an answer to my question can be (easily) found online. So, I decided to post it among researchers to see whether I... | https://mathoverflow.net/users/311366 | About Newton's forward and backward interpolation | *The forward and backward finite differences and the derivative lower the degree of a polynomial by one.*
This property underlies the construction of series expansions of polynomials and, therefore, analytic functions with appropriate convergence properties in terms of diverse polynomial sequences, in particular, She... | 3 | https://mathoverflow.net/users/12178 | 397025 | 163,917 |
https://mathoverflow.net/questions/397014 | 2 | Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
Does this hold also when $K$ is not algebraically closed?
| https://mathoverflow.net/users/14514 | Nef and pseudo-effective divisors over non algebraically closed fields | I believe the answer is yes. I will use Keeler's paper [Kee03] (and the corrigendum [Kee18]) as a reference for these questions.
First, the theorem of the base holds over arbitrary fields by [Kee03, Theorem 3.6; Kee18, Theorem E2.2] (see also [Cut15, Proposition 2.3]), and hence one can talk about the Néron–Severi sp... | 4 | https://mathoverflow.net/users/33088 | 397030 | 163,919 |
https://mathoverflow.net/questions/396951 | 7 | A rep-tile is a shape that can tile larger copies of the same shape.
**Question 1:** Are there any convex pentagons that are also rep-tiles?
**Remarks:** 15 convex pentagonal tiles of the plane are known and none of them *appears* to be a rep-tile. Assuming this observation is right, one can invoke a proof given in... | https://mathoverflow.net/users/142600 | Are there any convex pentagonal rep-tiles? | Question 2 can be answered in the affirmative, at least: there are many triangles with the multi-way rep-tile property.
Every triangle has a simple tiling of itself with $k^2$ copies (just take the affine image of the standard equilateral tiling); these are called *quadratic tilings* by Michael Beeson in his 2010 [pa... | 6 | https://mathoverflow.net/users/89672 | 397032 | 163,920 |
https://mathoverflow.net/questions/397023 | 15 | A pair of distinguished generators of the fundamental group $\pi\_1(\partial(S^3 \setminus K))$ of the boundary torus of a knot complement are usually called the "meridian" and "longitude". However, this terminology has always seemed a bit odd to me: in geography a meridian is a line of longitude, so shouldn't the curv... | https://mathoverflow.net/users/113402 | Why is the thing dual to a "meridian" called a "longitude"? | There is a fundamental asymmetry between latitude and longitude on a sphere, whereas on a torus, there is a symmetry between the two generators. This symmetry could motivate the use of nearly synonymous words.
The terminology may originate with the paper [On the homology invariants of knots](https://www.maths.ed.ac.u... | 13 | https://mathoverflow.net/users/3106 | 397035 | 163,922 |
https://mathoverflow.net/questions/397007 | 4 | Let $A$ and $B$ be Hermitian matrices. Let $[A,B] = AB - BA$ be their commutator and let $[A,B]^+$ be the Moore-Penrose pseudoinverse of $[A,B]$.
Is it then true that $A$ and $B$ both commute with the projector $Q = [A,B] [A,B]^+ = [A,B]^+ [A,B]$ ?
$P = 1 - Q$ is effectively a projector onto the subspace where our ... | https://mathoverflow.net/users/174368 | Do any two hermitian matrices A and B commute with the support of their commutator? | To attack this more theoretically: if A and B have a common eigenvector, then that must lie in $\ker Q$ and obviously $[Q,A]$ and $[Q,B]$ act trivially on this guy. Thus, we can take the perp to this eigenvector and ask the same question about the leftover matrices.
Thus, we should assume that $A$ and $B$ have no com... | 4 | https://mathoverflow.net/users/66 | 397036 | 163,923 |
https://mathoverflow.net/questions/396978 | 3 | The classical Busemann-Feller lemma in Euclidean space says the following.
Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then
1. for any point $x\in \mathbb{R}^n$ there exists unique nearest point in $K$; let us denote it by $p(x)$.
2. the map $p\colon \mathbb{R}^n \to K$ does not increase distances, i.e. ... | https://mathoverflow.net/users/16183 | Busemann-Feller lemma in hyperbolic space | In any Hadamard space, projection into convex sets is non-expansive; see Proposition 2.4(4) in *[Metric spaces of non-positive curvature](https://www.springer.com/gp/book/9783540643241)* by Bridson and Haefliger.
| 7 | https://mathoverflow.net/users/68969 | 397045 | 163,924 |
https://mathoverflow.net/questions/397042 | 5 | Let $K$ be a field. Suppose $A$ and $B$ are $K$-algebra and there is a derived equivalence $F:D^b(A)\cong D^b(B)$ between their bounded derived categories. If we assume that $A$ has a tensor decomposition $A\cong A\_1\otimes\_K A\_2$ for two $K$-algebras $A\_1$ and $A\_2$.
Does $B$ also have a tensor decomposition $B... | https://mathoverflow.net/users/134942 | Tensor decomposition under derived equivalence | Not in general.
For an easy example, let $C$ and $D$ be derived equivalent algebras. Then $A=C\times C$ and $B=C\times D$ are derived equivalent, and $A=C\otimes\_K(K\times K)$ has a tensor decomposition, but $B$ will usually not.
There are also connected examples. For example, $KA\_2\otimes\_K KA\_2$ is derived eq... | 5 | https://mathoverflow.net/users/22989 | 397049 | 163,925 |
https://mathoverflow.net/questions/397054 | 0 | Suppose that I have a Jordan curve $J$ parametrized by the function $\phi$. Consider a sequence of parametric functions $\phi\_n$ parametrizing a sequence of Jordan curves $J\_n$, and denote by $H$ and $H\_n$ the mean curvature of $J$ and $J\_n$, respectively.
Suppose also that these Jordan curves are contained in a la... | https://mathoverflow.net/users/249354 | convergence of the mean curvature under $L^\infty$ norm | The hypotheses you impose are not strong enough to ensure convergence of the curvature.
This is ultimately a local problem, so to find a counterexample we work with graphs of functions. Let $(f\_n \mid n \in \mathbf{N})$ be a sequence of functions in $C^2(-1,1)$ say with
$$
\sup \lvert f\_n \rvert + \lvert f\_n' \rve... | 1 | https://mathoverflow.net/users/103792 | 397069 | 163,930 |
https://mathoverflow.net/questions/397072 | 1 | Can someone point out links to Applied Topology/Topological Data Analysis conferences and journals?
Thank you!
| https://mathoverflow.net/users/316155 | Applied Topology/Topological Data Analysis conferences and journals | ATMCS (Algebraic Topology: Methods, Computation, and Science), Oxford University, Oxford, UK, June 20–24, 2022. (I just googled.)
International Conference on Computational Topology and Data Analysis ICCTDA on September 20-21, 2021 in Toronto, Canada
International Conference on Computational Topology and Topological... | 0 | https://mathoverflow.net/users/13268 | 397073 | 163,932 |
https://mathoverflow.net/questions/397076 | 2 | Consider the following mixture model for a univariate density function
$$
(1) \quad f(x)=\int\_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2))
$$
where $D$ is a compact subset of $\mathbb{R}\times \mathbb{R}^+$, $(m, \sigma^2)$ denotes the pair of mean and variance, $g(\cdot; m, \sigma^2)$ is the univariate ... | https://mathoverflow.net/users/42412 | About a mixture | $\newcommand\R{\mathbb R}\newcommand\si{\sigma}\newcommand\ep{\varepsilon}\newcommand\de{\delta}$Any probability distribution on $\R$ can be approximated by a discrete probability distribution on $\R$. Any discrete probability distribution on $\R$ is a mixture of Dirac probability distributions on $\R$. The Dirac proba... | 3 | https://mathoverflow.net/users/36721 | 397079 | 163,935 |
https://mathoverflow.net/questions/397082 | 4 | Let $[n]\_q=1+q+\dots +q^{n-1}$, $ {[n]\_q}! =[1]\_q [2]\_q \dots [n]\_q$ and $\binom{n}{j}\_q = \frac{[n]\_q!}{[j]\_q![n-j]\_q!}$ be the usual $q$-notation.
Consider the polynomials $p\_n(q,r,x)= \sum\_{j=0}^n q^{r\binom{j}{2}}\binom{n}{j}\_qx^j$ and let $d\_{n,r}(q)=\Delta\_{x} p\_n(q,r,x)$ be their [discriminants]... | https://mathoverflow.net/users/5585 | Discriminants of some $q$-analogs of $(1+x)^n$ | This is true.
We have
\begin{align\*}
p\_n (q^{-1}, 1-r, x) &= \sum\_{j=0}^n q^{ (r-1) \binom{j}{2}} \binom{n}{j}\_{q^{-1}} x^j \\
&= \sum\_{j=0}^n q^{ (r-1) \binom{j}{2}} q^{-j (n-j)} \binom{n}{j}\_{q} x^j \\
&=\sum\_{j=0}^n q^{ (r+1) \binom{j}{2}} q^{-j (n-j) - j (j-1)} \binom{n}{j}\_{q} x^j \\
&= \sum\_{j=0}^n q... | 7 | https://mathoverflow.net/users/18060 | 397084 | 163,936 |
https://mathoverflow.net/questions/397024 | 3 | I stumbled upon the following identity, which I have not tried to prove, but seems true:
the function $$f(t):={}\_2F\_1(1/2,2t;1-t;4)$$
is periodic of period 1, and more precisely
$$f(t)=\dfrac{1+2e^{-2\pi it}}{3}\;.$$
This begs several questions:
(1). Is this true? (2). Are there other (infinitely many?) formulas of... | https://mathoverflow.net/users/81776 | Periodic Gauss hypergeometric function | Gauss' contiguous relations provide a basis for finding a linear relationship between three functions of the form ${}\_2F\_1(a+k, b+l, c+m, z)$, henceforth $\mathbf{F}\left(\begin{matrix}a+k, b+l \\ c+m\end{matrix}\right)$. Various papers have been published on calculating these relationships; I'm using the notation of... | 4 | https://mathoverflow.net/users/46140 | 397085 | 163,937 |
https://mathoverflow.net/questions/396990 | 7 | Can the following uniformization statement be proved by $ZF+AD+DC$?
>
> For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable and nonempty})$, then there is a function $f:\mathbb{R}\to \mathbb{R}$ uniformizing $R$.
>
>
>
| https://mathoverflow.net/users/14340 | Uniformization under AD | Here's an argument under the further assumption that $V=L(\mathbb{R})$. Something like this is presumably recorded somewhere. The main point comes from Steel's paper "Scales in $L(\mathbb{R})$", but I don't find it quite explicitly there.
Since $V=L(\mathbb{R})$, we can fix a real $x\_0$ such that our relation $R$ is... | 7 | https://mathoverflow.net/users/160347 | 397096 | 163,941 |
https://mathoverflow.net/questions/396968 | 6 | In section 35.1 of the book "Linear algebraic groups" by Humphreys, it is stated that the quasi-split but not split semisimple groups can only arise when the root system admits a nontrivial graph automorphism.
Moreover, it seems that the relative root system in this case is obtained by adjoining the vertices of Dynki... | https://mathoverflow.net/users/304053 | Quasisplit but not split semisimple groups | The quasi-split forms of a split reductive group $G$ over a field $k$ are classified by the elements of the first Galois cohomology group of $k$ with values in $Out(G)$ (see Theorem 23.51 of Milne's book *Algebraic Groups*). So no outer automorphisms means no nonsplit quasi-split forms. When $G$ is semisimple, the oute... | 5 | https://mathoverflow.net/users/316316 | 397098 | 163,942 |
https://mathoverflow.net/questions/394249 | 4 | Let $X:=X\_{18}$ be an index one smooth prime Fano threefold of degree 18.
Consider its semi-orthogonal decomposition: $D^b(X)=\langle\mathcal{O}\_X,\mathcal{E}^{\vee},\mathcal{A}\_X\rangle=\langle\mathcal{Q}^{\vee},\mathcal{O}\_X,\mathcal{A}\_X\rangle$, where $\mathcal{E},\mathcal{Q}$ are tautological sub and quotie... | https://mathoverflow.net/users/41650 | Auto-equivalences of non-trivial components of derived category of $X_{18}$ | Let me answer the question by myself. After a intensively literature research, I found that the habilitation of Faenzi,Daniele contains everything I need, here is the link <http://dfaenzi.perso.math.cnrs.fr/publis/faenzi.hdr.pdf>
Section 3.2
| 1 | https://mathoverflow.net/users/41650 | 397112 | 163,945 |
https://mathoverflow.net/questions/397113 | 4 | Assuming that for each fixed $k$, $(X\_{n,1},\ldots,X\_{n,k})\Longrightarrow(X\_1,\ldots,X\_k)$ where $X\_1,\ldots,X\_k$ are i.i.d. with mean zero and variance $\sigma^2$, will the array inherit the CLT from its limit? i.e. do I have (if $r\_n\to\infty$):
$\frac{1}{\sqrt{r\_n}}(X\_{n,1}+\cdots+X\_{n,r\_{n}})\Longrighta... | https://mathoverflow.net/users/316497 | CLT for a converging array of random variables | No. Take any sequence $r\_n \to \infty$ and let $X\_{n,k}$ be iid $N(0,1)$ for $k \ne r\_n$, and $X\_{n,r\_n} = r\_n$. The hypothesis is satisfied because $(X\_{n,1}, \dots, X\_{n,k})$ are iid $N(0,1)$ as soon as $n$ is so large that $r\_n > k$. But
$$\frac{1}{\sqrt{r\_n}}(X\_{n,1} + \dots + X\_{n,r\_n}) \sim N(\sqrt{r... | 4 | https://mathoverflow.net/users/4832 | 397118 | 163,948 |
https://mathoverflow.net/questions/397115 | 7 | Recall that a set is **amorphous** iff it is infinite but has no partition into two infinite subsets. I'm interested in the possible structure (in the sense of model theory) which an amorphous set can carry. For example, if $T$ is a complete first-order theory with an amorphous model then $T$ must be strongly minimal; ... | https://mathoverflow.net/users/8133 | Can second-order logic identify "amorphous satisfiability"? | Here's a monadic example. The second-order theory of the vector space $\mathbb F\_2^{\oplus\omega}$ as a vector space over $\mathbb F\_2$ is second-order strongly minimal because given a finite subspace $X$ **of any model of this theory**, any two points outside $X$ are related by an automorphism of the whole space fix... | 6 | https://mathoverflow.net/users/164965 | 397121 | 163,951 |
https://mathoverflow.net/questions/396393 | 4 | Let $\mathfrak{g}$ be a real semisimple Lie algebra with complexification $\mathfrak{g}\_\mathbb{C}$. Recall that a parabolic subalgebra in $\mathfrak{g}\_\mathbb{C}$ is one which contains a Borel subalgebra, and a parabolic subalgebra in $\mathfrak{g}$ is one which complexifies to a parabolic one in $\mathfrak{g}\_\ma... | https://mathoverflow.net/users/126256 | Complexifications of minimal parabolic subalgebras | NB: for brevity I'm going to refer to Borel subalgebras and parabolic subalgebra as Borels and parabolics, respectively
Even for a split (or quasi-split) Lie algebra not all complex Borels are the complexification of real Borels. Otherwise, for example, the real and complex flag manifolds would be identical. Naturall... | 2 | https://mathoverflow.net/users/163024 | 397137 | 163,953 |
https://mathoverflow.net/questions/397089 | 7 | Let $X$ be a topological space.
Let $\gamma:[a,b]\to X$ be continuous and injective.
$\gamma$ is said to be "openly extendable" if there is $[a,b]\subset (a',b')$ and a continuous and injective curve $\gamma':(a',b')\to X$ with $\gamma'\vert\_{[a,b]} = \gamma$.
Under what conditions on $X$, all continuous injecti... | https://mathoverflow.net/users/95265 | Extending continuous injective curves both continuously and injectively | ~~**EDIT 1:** Fixed the mistakes and added a full proof below~~
**EDIT 4: Found how to prove the existence of a connected neighborhood.
I am satisfied with this proof, so I'll accept the answer**
**Theorem:** If $X$ is connected, ~~locally connected~~ **EDIT 4: locally path connected**, locally compact and metrizab... | 0 | https://mathoverflow.net/users/95265 | 397139 | 163,955 |
https://mathoverflow.net/questions/397152 | 2 | Recently I [asked a question](https://mathoverflow.net/questions/397115/can-second-order-logic-identify-amorphous-satisfiability) about whether a second-order analogue of strong minimality could correspond to amorphous satisfiability (= having a model whose underlying set cannot be partitioned into two infinite pieces)... | https://mathoverflow.net/users/8133 | Second-order strong minimality and amorphousness, take 2 | I claim that if the language is countable, or merely well-orderable, then the witness property by itself prevents amorphous domains, and even infinite Dedekind-finite domains. Strong minimality has nothing to do with it.
**Theorem.** Any satisfiable complete theory $T$ in a well-ordered language with the witness prop... | 3 | https://mathoverflow.net/users/1946 | 397155 | 163,958 |
https://mathoverflow.net/questions/397140 | 1 | Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum\_{i=1}^3 \lambda\_i\ n\_i\otimes n\_i$$
where $\lambda\_i$, $n\_i$ and $\otimes$ denote the eigenvalues, eigenvectors and dyadic product, respectively. Further, let $a$ be a vector of the fo... | https://mathoverflow.net/users/160345 | Derivative of eigenpair with respect to matrix | If the eigenvalues are distinct you simply fill in the [first order perturbation answer](https://en.wikipedia.org/wiki/Eigenvalue_perturbation) for the eigenvalues and eigenvectors,
$$\frac{\partial\lambda\_i}{\partial A\_{jk}}=(n\_i)\_{j}(n\_i)\_{k}(2-\delta\_{jk}),$$
$$\frac{\partial n\_{i}}{\partial A\_{jk}}=\sum\_{... | 1 | https://mathoverflow.net/users/11260 | 397158 | 163,960 |
https://mathoverflow.net/questions/397141 | 0 | Let us consider a matrix $A \in M\_{2 \times n}(\mathbb C)$ such that $rank A=2$. Let us denote by
$$
a\_1,\ldots,a\_d \in \mathbb C,
$$
where $d:=\binom{n}{2}$, the value of the $2 \times 2$ minors of $A$. We identify these values as a point $P=[a\_1:\ldots:a\_d] \in \mathbb P^{d-1}$, giving a map
$$
\psi: X:=\{A \in ... | https://mathoverflow.net/users/147236 | How many matrices with given minors? | This answer is really just an expansion of Mohan's and Zach Teitler's comments. I have only provided more detail, that's all. I don't care much for the points and if you want, just give the points to Mohan and Zach Teitler, but I thought it may be useful to the OP and to other readers to provide a complete answer.
Le... | 1 | https://mathoverflow.net/users/81645 | 397168 | 163,962 |
https://mathoverflow.net/questions/397162 | 7 | It is well known that (small) coproducts [commute with connected limits](https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#coproducts_commute_with_connected_limits) in $\mathbf{Set}$. With which class of limits do finite coproducts commute?
Ideally, we should furthermore like to know whether the clas... | https://mathoverflow.net/users/152679 | Finite coproducts commute with which limits in Set? | Indeed, let $D$ be a category. The canonical functor $D \to \pi\_0(D)$ is both cofinal and coinitial. Therefore, if finite coproducts commute with $D$-limits in a category $\mathcal C$, then finite coproducts commute with $\pi\_0(D)$-limits. And it is easily seen that the only discrete limit shapes with which finite co... | 11 | https://mathoverflow.net/users/2362 | 397170 | 163,964 |
https://mathoverflow.net/questions/397146 | 2 | Consider the power sum
$$S\_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$
Let $\nu\_3(x)$ denote the $3$-adic valuation of $x$.
>
> **QUESTION 1. (milder)** Is this true?
> $$\nu\_3\left(\frac{S\_a(b)}{S\_a(1)}\right)=0.$$
> **QUESTION 2.** Is this true? $\nu\_3(S\_a(b))=\nu\_3(2a-1)$.
>
>
>
| https://mathoverflow.net/users/66131 | Divisibility of (finite) power sum of integers | Notice that
$$2S\_a(b) \equiv 1^{2b} + 2^{2b} + \dots + (6a-4)^{2b} \pmod{6a-3}.$$
From Faulhaber's formula, we have
$$1^{2b} + 2^{2b} + \dots + (6a-4)^{2b} \equiv B\_{2b} (6a-3) \pmod{6a-3}.$$
It follows that
$$S\_a(b) \equiv \frac32B\_{2b} (2a-1)\pmod{3(2a-1)},$$
where $\nu\_3(\frac32 B\_{2b})=0$ by [von Staudt–Cla... | 6 | https://mathoverflow.net/users/7076 | 397172 | 163,966 |
https://mathoverflow.net/questions/397174 | 2 | Except for $\ p=2\ $ primes split into two disjoint classes, $\ p\equiv1\mod4\ $ and $\ p\equiv3\mod4.\ $ Squares respect this partition, odd prime $\ p=m^2+n^2\ \Leftrightarrow\ p\equiv1\mod4.\ $ On the other hand, *triangles* $\ \binom k2\ $ are oblivious to the $\mod4\ $ classification, as well as to the other class... | https://mathoverflow.net/users/110389 | Equation $\ p=\binom m2+\binom n2$ | $p={m\choose 2}+{n\choose 2}$ is equivalent to $8p+2=(2m-1)^2+(2n-1)^2$. On the other hand, if $8p+2$ is a sum of two squares, these squares must be both odd (this is seen modulo 4). So, applying the criterion for representability as a sums of two squares, we get that *$p$ is a sum of two triangular numbers if and only... | 15 | https://mathoverflow.net/users/4312 | 397176 | 163,967 |
https://mathoverflow.net/questions/397185 | 0 | Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}\_p,a \ne 0$,
define
$E\_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E\_a(\mathbb{F}\_p) -p - 1) \in \{2,-2,B,-B\}$?
| https://mathoverflow.net/users/12481 | Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$ | Look at Section 18.4 in Ireland-Rosen "A classical introduction to modern number theory". Note $p\equiv 1 \pmod{4}$ and $p = \pi\cdot \bar{\pi}$ with $\pi = 1- iu \equiv 1 \pmod{2+2i}$. Let $\lambda: \mathbb{F}\_p \to \langle i \rangle$ be the character of order $4$, which is equal to $(\tfrac{\cdot}{p})\_4$.
Theorem 5... | 7 | https://mathoverflow.net/users/5015 | 397186 | 163,970 |
https://mathoverflow.net/questions/397132 | 4 | Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
>
> If the special fiber of $X\to S$ is a closed immersion, is $X\to S$ a closed immersion?
>
>
>
**remarks**
* My guess is "no", but I'm looking for a counterexample (I can... | https://mathoverflow.net/users/nan | Detecting closed immersions on fibers | I am just posting my comment as an answer, mostly to correct the mistake identified by @LaurentMoret-Bailly.
The property of being universally closed depends only on the underlying reduced scheme. Thus, there are counterexamples coming from a nilradical that is quasi-coherent, yet not coherent.
For one example, let... | 2 | https://mathoverflow.net/users/13265 | 397190 | 163,971 |
https://mathoverflow.net/questions/397187 | 3 | $\newcommand{\sym}{\mathfrak{S}}$
$\newcommand{\rarr}{\rightarrow}$
Given a partition $\lambda$ of $n$ and a commutative ring $R$, writing $\sym\_n$ for the symmetric group, there is a Specht module $S^\lambda$ well-defined as an $R\sym\_n$-module. For example James does this in his book using polytabloids.
By a Spec... | https://mathoverflow.net/users/21848 | Does a finite coresolution with Specht-filtered modules imply a Specht filtration? | No. Working with modules for $\mathbb{F}\_5S\_5$, take the Specht module $S^{(2,1,1,1)}$. Using $S^{(2,1,1,1)} \cong (S^{(4,1)} \otimes \mathrm{sgn})^\star$ and that $S^{(4,1)}$ has top $D^{(4,1)}$ and socle $\mathbb{F}\_5$, we get a short exact sequence
$$0 \rightarrow D^{(4,1)} \otimes \mathrm{sgn} \rightarrow S^{(... | 4 | https://mathoverflow.net/users/7709 | 397191 | 163,972 |
https://mathoverflow.net/questions/397180 | 6 | There is a well-know quotation of Euler in a letter from 1746 to Goldbach:
>
> Letztens habe ich gefunden, dass diese expressio $\sqrt{-1}^{\sqrt{-1}}$ einen valorem realem habe, welcher in fractionibus decimalibus
> $=0,2078795763$, welches mir merkwürdig zu seyn scheinet.
>
>
>
For the principal value of the... | https://mathoverflow.net/users/21051 | How did Euler calculate $i^i$? | [On the numerical value of $i^i$](https://www.jstor.org/stable/2972387%0A) and [Historical notes on the relation $e^{-\pi/2}=i^i$](https://doi.org/10.2307/2972388) describe how these accurate computations can be performed with logarithmic tables.
Euler described how he arrived at the identity in a paper read at the B... | 10 | https://mathoverflow.net/users/11260 | 397193 | 163,974 |
https://mathoverflow.net/questions/397178 | 8 | Assume $X$ is a smooth projective variety over $\overline{\mathbf{F}}\_p$ and fix a prime $\ell\neq p$.
Let $F\_i$ be the geometric Frobenius on $\ell$-adic cohomology
$$H^i\_{\rm ét}(X,\overline{\mathbf{Q}}\_{\ell})$$
for fixed $i\ge 0$.
>
> * What relation is expected to hold between the minimal polynomial $m... | https://mathoverflow.net/users/nan | Minimal vs characteristic polynomial of geometric Frobenius | It's conjectured -- see e.g. [this question](https://mathoverflow.net/questions/104102/semisimplicity-of-frobenius-operation-on-etale-cohomology?noredirect=1&lq=1) -- that the Frobenius is always semisimple, so its minimal polynomial is the radical of its characteristic polynomial (the product of its distinct linear fa... | 6 | https://mathoverflow.net/users/2481 | 397195 | 163,975 |
https://mathoverflow.net/questions/397150 | 9 | Let $L/K$ be a (abelian, Galois) quadratic extension of number fields with $\text{Gal}(L/K)$ generated by $\sigma$ and $\mathfrak{p} = \alpha\mathcal{O}\_K$ a principal prime ideal of $\mathcal{O}\_K$. Assume $\mathfrak{p}$ splits as $\mathfrak{P} \sigma(\mathfrak{P})$ in $\mathcal{O}\_L$ and that $\alpha = \beta \sigm... | https://mathoverflow.net/users/106850 | Is it true that this ideal must be principal? (proof verification) | I claim that under the given circumstances $\mathfrak{P}$ is not necessarily principal, i.e., the statement claimed in the question is wrong.
Here is a counterexample. Consider $K = \mathbb{Q}$ and $L = \mathbb{Q}[\sqrt{-47}]$. Let $\alpha = -3$ and $\beta = \frac{1}{4}(1 \pm \sqrt{-47})$. Then the minimal polynomial... | 10 | https://mathoverflow.net/users/148992 | 397200 | 163,976 |
https://mathoverflow.net/questions/397078 | 3 | In some urn model with parameter $p$, the generating function
$$
f\_p(z) \;=\; \frac{1+p\,z}{1-(1-p)\,z\,(1+p\,z)}
$$
is such that $[z^n]f\_p(z)$ is the probability that an $n$-urn configuration has a particular property.
It is known that this probability tends to zero if $p\gg n^{-1/2}$ and tends to one if $p\ll n^{... | https://mathoverflow.net/users/34341 | Using singularity analysis for probability at a threshold? | There are a few things working in this particular question that allow for that to occur. This is a rational function and the denominator is a quadratic in $z$ meaning that we can write $$[z^n] f\_p(z) = c\_1 \rho\_1^{-n} + c\_2 \rho\_2^{-n}$$ where $\rho\_j$ are the roots of the denominator and $c\_j$ are functions of ... | 2 | https://mathoverflow.net/users/69870 | 397205 | 163,978 |
https://mathoverflow.net/questions/396867 | 9 |
>
> Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$?
>
>
>
I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?
| https://mathoverflow.net/users/309203 | Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$? | As I suspected, the statement that the bundle $\mathrm{SO}(8)\to S^7$ is a product bundle, i.e., that
$$
\mathrm{SO}(8)\simeq S^7\times\mathrm{SO}(7)\tag1
$$
as bundles over $S^7$ is in N. Steenrod's [The topology of fibre bundles](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&... | 6 | https://mathoverflow.net/users/13972 | 397209 | 163,979 |
https://mathoverflow.net/questions/397210 | 22 | Bertrand Russell, I believe, somewhere presents a joke (if I remember correctly). Someone is shown the boat of another, and the first says: "I thought that your boat is longer than it is." The owner replies: "No, my boat is not longer than it is."
Does someone know the reference?
| https://mathoverflow.net/users/37385 | "The boat is not longer than it is." | Bertrand Russel, [**On denoting**](http://web.mnstate.edu/gracyk/courses/web%20publishing/russell_on_denoting.htm)
*Mind*, October 1905, pages 479-493.
>
> When we say: "George IV wished to know whether so-and-so", or when we
> say "So-and-so is surprising" or "So-and-so is true", etc., the
> "so-and-so" must be... | 19 | https://mathoverflow.net/users/11260 | 397211 | 163,980 |
https://mathoverflow.net/questions/397060 | 3 | I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found [here](http://www.mi.uni-koeln.de/%7Egeiges/contact05.pdf), and am trying to figure out the effect of a half Lutz twist on the Euler class
of t... | https://mathoverflow.net/users/316096 | Effect of a Lutz twist on Euler number | I have actually solved this by now, his statement is indeed correct but
the proof is a bit misleading. I will only discuss the case where $K$ is a knot.
Instead of choosing a field which does not vanish in a tubular neighbourhood of $K$, one simply chooses a generic vector field which is given by the radial vector fi... | 1 | https://mathoverflow.net/users/316096 | 397216 | 163,981 |
https://mathoverflow.net/questions/397043 | 3 | I am currently trying to understand the properties of Deligne-Lusztig induction, following Carter's *Finite groups of Lie type* and Digne-Michel's *Representations of finite groups of Lie type*. I am reasonably satisfied with the construction, but I am having difficulty understanding the proofs of the properties of the... | https://mathoverflow.net/users/175051 | Frobenius reciprocity for Deligne-Lusztig induction/restriction | **Answer to Questions 1 and 2:** At least in Digne--Michel the functor ${}^\ast R\_{T \subseteq B}^G$ is rigorously defined. The reference is [Digne--Michel, p.47]. What they do is the following: If $G$, $H$ are any two finite groups and $M$ is a $G$-module-$H^{\rm opp}$, then we have the functor $R\_H^G \colon E \maps... | 3 | https://mathoverflow.net/users/148992 | 397222 | 163,982 |
https://mathoverflow.net/questions/397201 | 2 | Let $E$ be the elliptic curve over $\mathbf{Q}\_3$ with Weierstrass equation $y^2 = x^3-x$.
It has complex multiplication by $\mathbf{Z}[\sqrt{-1}]$, with $\sqrt{-1}$ acting as $(x,y)\mapsto(-x,iy)$.
Call $R:=\mathbf{Z}\_3[\sqrt{-1}] = W(\mathbf{F}\_9)$.
Let $\mathcal{E}$ be the Néron model over $\mathbf{Z}\_3$, an... | https://mathoverflow.net/users/nan | An example of Serre on the cohomology of some CM elliptic curves | Yes. There exists a "nice" cohomology theory for $p$-adic varieties, taking values in vector spaces over $\mathbb Q$, defined by fixing an embedding $\mathbb Q\_p \to \mathbb C$, base-changing along this embedding, and taking singular cohomology. Every morphism of varieties over $\mathbb Q\_p$ induces a map on cohomolo... | 2 | https://mathoverflow.net/users/18060 | 397225 | 163,984 |
https://mathoverflow.net/questions/397217 | 0 | The Dirichlet inverse of the Euler totient function is:
$$\varphi^{-1}(n) = \sum\_{d \mid n} \mu(d)d \tag{1}$$
and the von Mangoldt function can be expanded/computed as:
$$\Lambda(n) = \sum\limits\_{k=1}^{\infty}\frac{\varphi^{-1}(\gcd(n,k))}{k} \tag{2}$$
Consider the sequences:
$$a(n)=\sum \_{k=1}^{j} \frac{\varphi^{-... | https://mathoverflow.net/users/25104 | Correlating the von Mangoldt function with periodic sequences | TL;DR: This is not a full answer. I compute asymptotically the correlation of $b(n)$ and $\Lambda (n)$ (it turns out to be $0$) and the correlation of $b(n)$ and $a(n)$ in a relatively closed form sum (I don't think there's really a simpler expression). Although I do not show when the correlation of $b(n)$ and $a(n)$ i... | 1 | https://mathoverflow.net/users/88679 | 397226 | 163,985 |
https://mathoverflow.net/questions/397154 | 3 | Let $G$ be a reductive group over the finite field $\mathbb{F}\_q$. Then a *regular* embedding of $G$ is an $\mathbb{F}\_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group over $\mathbb{F}\_q$, such that $\iota$ maps the derived group of $G$ isomorphically onto that of $G'$ and the cent... | https://mathoverflow.net/users/148992 | Regular embeddings of a reductive groups with induced center | The answer is **Yes**.
Let $G\hookrightarrow G'$ be a smooth regular embedding. We write $Z(G')$ for the center of $G'$, which is an $F$-torus, where $F={\Bbb F}\_q$.
We construct a regular embedding $G'\hookrightarrow G''$ such that $Z'':=Z(G'')$ is an induced torus. We write $S=(G,G)=(G',G')$ for the derived group ... | 2 | https://mathoverflow.net/users/4149 | 397229 | 163,987 |
https://mathoverflow.net/questions/397221 | 3 | Suppose $Y$ is an $(n-1)$-connected space, $n>2$, so we have Hurewicz isomorphisms $\pi\_n(Y)\cong H\_n(Y)$ and $\pi\_{n-1}(\Omega Y)\cong H\_{n-1}(\Omega Y)$. Let a map $\alpha\colon X\to\Omega Y$ be given. Naturally it induces a map $\beta\colon X\times S^1\to Y$. I want to show the following diagram is commutative:
... | https://mathoverflow.net/users/310606 | Induced map in homology for a map to a loop space | The general case does follow from the case $Y=S^n$.
Without loss of generality $X=\Omega Y$ and $\alpha$ is the identity. Now the statement comes down to the assertion that the composition
$$
\pi\_{n-1}\Omega Y\to H\_{n-1}\Omega Y\to H\_n(\Omega Y\times S^1)\to H\_n Y
$$
corresponds to the Hurewicz map $\pi\_nY\to H\... | 7 | https://mathoverflow.net/users/6666 | 397231 | 163,988 |
https://mathoverflow.net/questions/396816 | 3 | If one wanted to obtain a fan for a toric variety of dimension $ n>1 $ whose Cox ring is $ \mathbb{Z}^{2} $ graded with weights $ \{(a\_{i},b\_{i})\}\_{i=1}^{n+2} $, then one could let $ B $ be the $ n \times (n+2) $ matrix whose $ (i,j) $-th entry is $ \delta(i,j) $ if $ 1 \le i, j \le n $, $ a\_{i} $ if $ j $ is equa... | https://mathoverflow.net/users/113893 | How to create a toric variety whose Cox ring has a specific grading? | I realized an answer to this. Let $ B $ be the $ n \times (n+2) $-matrix with entries $ \delta(i,j) $ if $ 1 \le i,j \le n $, $ a\_{i} $ if $ j $ is equal to $ n+1 $ and $ b\_{i} $ if $ j $ is equal to $ n+2 $. Now perform row reduction (with operations strictly in the integers) on the matrix $ B $ until all entries of... | 1 | https://mathoverflow.net/users/113893 | 397240 | 163,990 |
https://mathoverflow.net/questions/397245 | 2 | Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that the map $H^1(k,\mathrm{Pic}^0(X\_{\bar{k}})) \rightarrow H^1(k,\mathrm{Pic}(X\_{\bar{k}}))$ is surjective. Indeed, applyi... | https://mathoverflow.net/users/172132 | Computing $H^1$ with coefficients in a torsion-free abelian group | I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X\_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to your question.
Here $\mathrm{Pic}^0(X\_{\bar{k}})$ is trivial so it is certainly *not true* in general that the map $... | 4 | https://mathoverflow.net/users/5101 | 397249 | 163,996 |
https://mathoverflow.net/questions/397257 | 2 | Let $k=m+\sum^{m+1}\_{j=1} a\_j$ such that $a,m,k\in\mathbb{N}$ and $a\_1$ or $a\_{m+1}\geq 0$ with all other $a\geq1$. Note that we assume natural numbers start from $0$ and we have the restriction that $\sum^{m+1}\_{j=1} a\_j\geq m-1$. Why do there exist $F\_{k+2}$ solution sets for values of $m$ and $a\_\zeta$, $\fo... | https://mathoverflow.net/users/167822 | Is this case of a generalised partition equivalent to Fibonacci numbers? | Without mistake on my behalf, a proof can be given as follows:
Denote the set of all solutions (for a given value of $k$)
by
$\mathcal F\_k$. Every element of $\mathcal F\_k$ ending
with a last coefficient $\geq 1$ corresponds to an element
of $\mathcal F\_{k-1}$ after decreasing its last element
(of the correspondin... | 3 | https://mathoverflow.net/users/4556 | 397262 | 163,998 |
https://mathoverflow.net/questions/396342 | 11 | In Lurie's `Higher Algebra`, construction 4.4.2.7 presents a Bar construction in the setting of $\infty$-categories. The construction in 4.4.2.7 takes as input an $\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ and a suitable pair of bi-modules in $\mathcal{C}^\otimes$ and gives a... | https://mathoverflow.net/users/nan | Making the ($\infty$-categorical) Bar construction valued in (bi)-modules | In Lurie's construction of the relative tensor product, there's an $\infty$-operad $\mathrm{Tens}\_{[2]}^{\otimes}$ whose algebras in a monoidal $\infty$-category $\mathcal{C}$ are given by 3 associative algebras (say A,B,C) and two bimodules (say M for (A,B) and N for (B,C)). The simplicial bar construction is obtaine... | 5 | https://mathoverflow.net/users/1100 | 397265 | 164,000 |
https://mathoverflow.net/questions/397233 | 5 | In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this [question](https://mathoverflow.net/q/397132).
Let $R$ be a discrete valuation ring, $X$ and $S$ two $R$-flat and universally closed $R$-schemes.
Let $f : ... | https://mathoverflow.net/users/nan | On universally closed morphisms of reduced schemes | Let $R$ be $\mathbb Z\_p$ (or any other dvr).
Let $S$ be obtained by gluing two copies of $\mathbb P^1\_{\mathbb Z\_p}$ away from the $0$-point in the special fiber, i.e. away from the vanishing locus of the ideal $(p,x)$ in local coordinates away from $\infty$.
Let $X$ be obtained by gluing two copies of $\mathbb ... | 2 | https://mathoverflow.net/users/18060 | 397266 | 164,001 |
https://mathoverflow.net/questions/397237 | 0 | Consider the space $\mathcal{M}\_{loc} (\mathbb{R}^d)$ of locally finite signed Radon measures, equipped with the weak\* topology in duality with $C\_b (\mathbb{R}^d)$. It is known that this is space is not metrizable, nor first countable (although I believe it is a Souslin space?).
On the other hand, in practice one... | https://mathoverflow.net/users/100163 | Reference request: sequential weak* topology on the space of signed Radon measures | This is not an answer since I cannot give the references you are requesting for reasons that will soon be apparent but am not entitled to comment. Firstly, there is something fundamentally wrong with the framework of your question--there is no duality between spaces of bounded, continuous functions on euclidean space a... | 6 | https://mathoverflow.net/users/317800 | 397268 | 164,002 |
https://mathoverflow.net/questions/397232 | 3 | Let $C\_n=\frac1{n+1}\binom{2n}n$ be the well-known Catalan numbers. Here is a curiosity.
>
> **QUESTION.** Are there infinitely many $C\_n$ that are "radical", i.e. that are square-free?
>
>
>
| https://mathoverflow.net/users/66131 | "Radical" Catalan numbers? | If $s(n)$ is the largest integer such that $s(n)^2 \mid \binom{2n}{n}$, then the main result of Sárközy, A. (1985). On divisors of binomial coefficients, I. Journal of Number Theory, 20(1), 70-80 is
>
> If $\varepsilon > 0$, $n > n\_0(\varepsilon)$ then we have $$e^{(c-\varepsilon) \sqrt{n}} < s(n)^2 < e^{(c+\varep... | 3 | https://mathoverflow.net/users/46140 | 397270 | 164,003 |
https://mathoverflow.net/questions/395393 | 4 | Let $R=\mathbb{Z}/1024\mathbb{Z}$ and $G=GL(3,R)$. Let $H$ be the subgroup of $G$ consisting of all matrices with determinant $1$ which are congruent to the identity matrix modulo the ideal $4R$. Let
$A=\begin{pmatrix} 1 &0&512\\ 0& 1&0\\0&0 &1\end{pmatrix}$ and $B=\begin{pmatrix} 1 &4&2\\ 8& 1&4\\16 &8 &1\end{pmatri... | https://mathoverflow.net/users/104638 | A question about the possibilities of GAP | It is contained. Here is the way I did the calculation with GAP. It ought to work nicer, but there is a stupid technical issue in the way that makes it hard to implement a membership test in a subgroup. (The issue is basically that we cannot guarantee that elements will always lie in one big parent group.)
Thus, one ... | 5 | https://mathoverflow.net/users/59303 | 397274 | 164,004 |
https://mathoverflow.net/questions/397286 | 23 | Why do we have two theorems one for the density of $C^{\infty}\_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ and one for the density of $C^{\infty}\_c(\Omega)$ in $L^p(\Omega)$? with $\Omega$ an open subset of $\mathbb{R}^n$.
Why not just the second one?
I was asked by the prof what is the difference between the density... | https://mathoverflow.net/users/144902 | Why do we have two theorems when one implies the other? | Some mathematicians seem to agree with you, and strive only to state and prove the most general versions of their theorems. I've had co-authors express that view. And I've sometimes had referee reports on my papers state this philosophical perspective explicitly, objecting to a warm-up theorem that I stated and proved ... | 135 | https://mathoverflow.net/users/1946 | 397289 | 164,008 |
https://mathoverflow.net/questions/396953 | 10 | Let $G$ be a reductive algebraic group with choices of Borel subgroup and maximal torus $B \supseteq T$ and unipotent radical $U$ over an algebraically closed field $k$ of characteristic zero. Is the affine closure $\overline{G/U} = \operatorname{Spec}(A)$ Cohen–Macaulay? If so, is there a reference to this fact? If no... | https://mathoverflow.net/users/104690 | Is the affine closure of the basic affine space of a reductive algebraic group Cohen–Macaulay? | Update: It seems that every affine closure of such a quotient $G/U$ *is* Cohen-Macaulay. In the same list of properties as above, the author states that, by a theorem of Brion, the ring of functions of $\overline{G/U}$ has rational singularities. Immediately before this statement, the author also cites [Elkik - Singula... | 4 | https://mathoverflow.net/users/104690 | 397293 | 164,010 |
https://mathoverflow.net/questions/293118 | 3 | I'm looking for the PDF version/scan of Henry P McKean Jr.'s paper on propagation of chaos. The reference is as follows -
**Propagation of chaos for a class of non-linear parabolic equations., In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) (1967): 41-5... | https://mathoverflow.net/users/106281 | Looking for access to McKean's original paper? | I am not sure whether there is still a quest for McKean's original paper, but as a nice substitute, you can take a look at the very recent [review paper](https://arxiv.org/pdf/2106.14812.pdf) on "propagation of chaos". Theorem 5.1 there is a result due to McKean and the original proof of him is presented (starting from... | 1 | https://mathoverflow.net/users/163454 | 397295 | 164,012 |
https://mathoverflow.net/questions/397285 | 4 | I'd like to prove the following conjecture.
Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$).
Then
$f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$
is irrational if $x$ is not an integer multiple of $\frac{\pi}{8}$. Is this true? Is the other direction true too?
I've tried ... | https://mathoverflow.net/users/317106 | Irrationality of this trigonometric function | **Lemma.** Let $n$ be a positive integer such that each number in the open interval $(n/4,3n/4)$ is not coprime with $n$. Then $n\in \{1,4,6\}$.
**Proof.**
* If $n=2m+1$ is odd, and $m\geqslant 1$, then $m\in (n/4,3n/4)$.
* If $n=4m+2$ and $m>1$, then $2m-1\in (n/4,3n/4)$
* If $n=2$, then $1\in (n/4,3n/4)$
* If $n=... | 6 | https://mathoverflow.net/users/4312 | 397304 | 164,017 |
https://mathoverflow.net/questions/397316 | 4 | In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x\_1, ..., x\_s) = k$ for $(x\_1, ..., x\_s) \in \mathbb{Z}^s$, where $F$ is some indefinite integral quadratic form, and $k\neq 0$ is a fixed integer.
An old p... | https://mathoverflow.net/users/318125 | Hardy-Littlewood circle method for non-diagonal quadratic forms | The "best" way to deal with quadratic forms using the circle method is via Heath-Brown's delta symbol method.
You can read about this in detail in the paper:
Heath-Brown - A New Form of the Circle Method, and its application to Quadratic Forms
Theorem 4 in particular gives an asymptotic formula for the problem you ... | 4 | https://mathoverflow.net/users/5101 | 397318 | 164,022 |
https://mathoverflow.net/questions/397314 | 4 | Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu\_n \rightarrow \mathbb{G}\_m \rightarrow \mathbb{G}\_m \rightarrow 1.$$
**Question 0.** Applying the etale cohomology functor $H\_{et}^i(k,-)$, I know that $H\_{et}^i(k,\mathbb{G}\_m) =... | https://mathoverflow.net/users/172132 | A Kummer exact sequence involving $\mu_\infty$ | Regarding you question 1, it suffices to take the short exact sequence
$$1\to \mu\_\infty\to \mathbb{G}\_m\to \mathbb{G}\_m\otimes\_{\mathbb{Z}}\mathbb{Q}\to 1\,.$$
This sequence is exact in the étale topos of any scheme over $\mathbb{Q}$. Indeed to show surjectivity it is enough to show that for every ring $R$, any $x... | 2 | https://mathoverflow.net/users/43054 | 397335 | 164,024 |
https://mathoverflow.net/questions/397136 | 3 | Let $U\_q(\frak{sl}\_2)$ denote the quantum universal enveloping algebra of $\frak{sl}\_2$, and consider the adjoint action
$$
\mathrm{ad}\_X: U\_q({\frak sl}\_2) \to U\_q({\frak sl}\_2), ~~ Y \mapsto S(X\_{(1)})YX\_{(2)},
$$
where we have used sumless Sweedler notation. This gives $U\_q({$\frak sl}\_2)$ the structure ... | https://mathoverflow.net/users/153228 | The adjoint representation of $U_q({\frak sl}_2)$ on itself | Unlike what happens in the classical case, it is not locally finite dimensional, basically because it has invertible elements. However its ad-locally finite part $U'$ is very large and it decomposes as a direct sum
$$U'=\bigoplus\_{V} V^\* \otimes V$$
where the sum is over the irreducible finite dimensional modules (no... | 8 | https://mathoverflow.net/users/13552 | 397337 | 164,025 |
https://mathoverflow.net/questions/397330 | 22 | I am a PhD student in algebraic / arithmetic geometry and I never took a formal course in algebraic topology, even though I have some basic knowledge.
In algebraic geometry we deal exclusively with sheaf cohomology since we care about non-constant sheaves. But I feel, maybe in my naivety, that a lot of the important ... | https://mathoverflow.net/users/131975 | Why should an algebraic geometer care about singular / simplicial (co)homology? | Sheaf cohomology is a powerful tool, but it isn't a replacement for all of basic algebraic topology. For example, fundamental groups and homology some topics that would get lost. And these topics are certainly relevant to algebraic geometry. Also, as pointed out in the comments, you would lose valuable intuition if you... | 24 | https://mathoverflow.net/users/4144 | 397339 | 164,027 |
https://mathoverflow.net/questions/396687 | 1 | I posted [this question](https://math.stackexchange.com/questions/4187132/reference-request-on-the-relationship-between-inscribed-polytopes-and-shadows-of) thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this on... | https://mathoverflow.net/users/6316 | The relationship between facets of an inscribed polytope and those facets' shadows | The following is a detailed description of the counterexample given in the comments. The construction is the same, but with slightly more notation and detail for clarity. Also, the construction there gave two triangles with different areas but equal-area shadows; this answer constructs one polytope of which both of the... | 2 | https://mathoverflow.net/users/75344 | 397343 | 164,028 |
https://mathoverflow.net/questions/397332 | 2 | Are there any reference for the classification of orientable disk bundle over a closed surface? I am particularly interested in the case if the surface is $S^2,RP^2,T^2$ or the Klein bottle.
Many thanks!
| https://mathoverflow.net/users/280895 | Classification of disk bundle over surfaces | Fix a base space $B$. Taking boundaries gives an equivalence from the category of (isomorphisms of topological) disk bundles over $B$ to the category of (isomorphisms of topological) circle bundles over $B$. When $B$ is a surface the latter are also called “Seifert fibered spaces”. These are described in many different... | 5 | https://mathoverflow.net/users/1650 | 397350 | 164,030 |
https://mathoverflow.net/questions/397250 | 5 | Alice and Bob have $N\_A$ and $N\_B$ warriors under their command, numbered $1$~$N\_A$ and $1$~$N\_B$ respectively. Alice has $1$ fighting power at her disposal, and Bob has $b$ ($b\gt 0$). Before the game, they privately distribute their power between their warriors. When the game begins, both send their warrior #1 to... | https://mathoverflow.net/users/75935 | Is there a dominant strategy for this game? | This answer expands on my previous answer. It is too long for a comment on comment.
The outcome of the game to a player is 1 if he/she wins and 0 if he/she lose. Every pair of strategies induces an expected outcome, which is a number between 0 and 1 - the probability that the player wins. The min-max value of the gam... | 1 | https://mathoverflow.net/users/64609 | 397359 | 164,036 |
https://mathoverflow.net/questions/397317 | 8 | Are there smooth closed manifolds $M^n$ in every dimension $n \geq 3$ with trivial mapping class groups and with $H^1(M^n;\mathbb{Z}/2\mathbb{Z})$ arbitrarily large?
I am under the impression that "generically" a manifold will have no mapping class group (but maybe I am totally mistaken). There are presumably lots of... | https://mathoverflow.net/users/99414 | Manifolds with trivial mapping class group and large $H^1$? | I think that the construction in Belolipetsky--Lubotzky, Finite Groups and Hyperbolic Manifolds (<https://arxiv.org/abs/math/0406607>) provides a more algebraic construction of such manifolds for any $n$.
One of the results in this paper (Theorem 3.1) is the following : given groups $\Gamma \triangleright \Delta \tri... | 5 | https://mathoverflow.net/users/32210 | 397362 | 164,037 |
https://mathoverflow.net/questions/397356 | 3 | Let $M$ be a smooth manifold. A Lie algebroid over $M$ is a vector bundle $E\rightarrow M$ over $M$, with a Lie bracket on $\Gamma(M,E)$, a morphism of vector bundles $\rho:E\rightarrow TM$, such that, the following conditions are satisfied:
1. the map $\rho:E\rightarrow TM$ induce a morphism of Lie algebras $\Gamma(... | https://mathoverflow.net/users/118688 | Use of theory of Lie algebroids in (better) understanding of generalised complex structures | The compatibility conditions that you mention in the definition of a generalized complex structure are equivalent to the statement that the $+i$-eigenbundle $L$ of $J$ is a complex Dirac structure:
1. compatibility of $J$ with the inner product is equivalent to $L$ being 'isotropic', and
2. compatibility of $J$ with ... | 3 | https://mathoverflow.net/users/69713 | 397364 | 164,039 |
https://mathoverflow.net/questions/397373 | 5 | The classical Artin-Rees lemma tells the following. Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Let $M$ be a finitely generated $R$-module and $N\subset M$ be a submodule. Then there exists an integer $m\ge0$ such that for all $n\ge0$ the following equality of two submodules in $N$ holds:
$$
... | https://mathoverflow.net/users/2106 | Artin-Rees lemma for multiplicative subsets? | This is straightforward . Define the submodule $P$ as $N\subset P\subset M$, the set of all elements $m\in M$ such that $sm\in N$ for some $s\in S$. Then, choose $t\in S$ such that $tP\subset N$.
Now for what you need, clearly the right hand side is contained in the left. So, let $x\in stM\cap N$ for $s\in S$. Then, ... | 3 | https://mathoverflow.net/users/9502 | 397382 | 164,042 |
https://mathoverflow.net/questions/397380 | 1 | Let $G$ be a simple graph (finite or infinite), $[n]\mathrel{:=}\{1,...,n\}$. Define the function:
$$\varepsilon\_n(G)\mathrel{:=}\min\_\phi{\lvert{\operatorname{dom} (\phi)}\rvert},$$
where $\phi$ is the partial function $\phi:V(G)\to[n]$ such that $\forall x,y\in[n]$, $x\neq y$, $\exists u,v\in V(G)$: $\phi(u)=x$, $\... | https://mathoverflow.net/users/175589 | What is this invariant graph? | Rephrasing, given $G$ and $n$ you're looking for the smallest subset $V' \subseteq V$ of the vertices of $G$ such that the subgraph $G'$ induced by $V'$ has a vertex colouring $\phi$ on $n$ colours for which every pair of colours $c \neq d$ has an edge between those colours and there is no edge between two vertices of ... | 2 | https://mathoverflow.net/users/46140 | 397383 | 164,043 |
https://mathoverflow.net/questions/397381 | 1 | I'm reading a book and have encountered a relation which seems to me to be impossible to prove, I would like to be sure if this is the case. The author gives a probability function as
$$p\_n = \frac{e^{-c\_1 n - c\_2/n}}{Z},$$
where $c\_1$ and $c\_2$ are constants and Z is a normalization factor and $n \geq 3$. Then by... | https://mathoverflow.net/users/nan | To prove a relation involving a probability distribution | The equality $\alpha + p\_6 = 1$ can be rewritten as
$$\sum\_{n=3}^\infty g(n)p\_n=1,$$
where
$$g(n):=(n-6)^2+1(n=6)\ge1(n\in\{5,6,7\})+4\times1(n\notin\{5,6,7\}).$$
Therefore and because $p\_n>0$ for all $n\ge3$, we have
$$\sum\_{n=3}^\infty g(n)p\_n>\sum\_{n=3}^\infty p\_n=1,$$
so that the equality $\alpha + p\_6 = 1... | 2 | https://mathoverflow.net/users/36721 | 397384 | 164,044 |
https://mathoverflow.net/questions/397368 | 10 | I came upon this statement in a stack answer.
Statement :
If $f\_n$ is a sequence of real valued functions (not necessarily continuous or measurable) on $[0,1]$ such that $f\_n$ converges point-wise to $0$, then there exists an infinite subset of $[0,1]$ where the convergence is uniform.
I couldn't prove it. I beli... | https://mathoverflow.net/users/318305 | Pointwise convergence imples uniform convergence in an infinite subset | For every sequence $(F\_n)\_{n \in \omega}$ with $F\_n:[0,1] \rightarrow \mathbb{R}$ converging pointwise to $0$, we can associate to every $x \in [0,1]$ an $f\_x \in \mathbb{\omega}^\omega$ in the following way: Set $f\_x(m):= \min\{n \in \mathbb{\omega}\,\, \colon \,\, \forall n' \geq n \,\,\,\, \vert F\_{n'}(x) \ver... | 6 | https://mathoverflow.net/users/134910 | 397385 | 164,045 |
https://mathoverflow.net/questions/397135 | 6 | I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim\_{|y|\to \infty}u(y) = 0.
\end{split}
\end{equation}
Here $\alpha$ is a constant and $r > 0$ and $\delta(y)$ is the Dirac delta function. Is it possible to solve this ODE? I e... | https://mathoverflow.net/users/114951 | How to solve the following ODE with a parameter? | This is essentially an elliptic second-order ODE in $(0,\infty)$:
$$ (1 + y^2)^{r/2} u''(y) = (-\alpha) u(y) , $$
with boundary conditions $u'(0) = -1$ and $u(\infty) = 0$. This kind of problems are well-studied, the corresponding theory is known as *Krein's spectral theory of strings*.
The value $h(\alpha)$ of $u$ a... | 7 | https://mathoverflow.net/users/108637 | 397387 | 164,046 |
https://mathoverflow.net/questions/397371 | 3 | Suppose $(M,g)$ is a three dimensional smooth compact simply connected Riemannian manifold with boundary and suppose that $\Sigma$ is a smooth simply connected hypersurface in $M$ with a smooth boundary $\partial \Sigma \subset \partial M$. Let $D\_X Y$ denote the Levi-Civita connection on $(M,g)$. Let $Z$ be a smooth ... | https://mathoverflow.net/users/50438 | A question on Levi-Civita connection and a fixed hyper surface | Not a full answer, but there definitely should be some integrability constraint.
Take the simplest case where $M = \mathbb{R}^3$ (the boundary is unimportant for the discussion here) and $\Sigma$ is the $x$-$y$ plane.
If $\phi$ is a defining function of $\Sigma$, then restricted to $\Sigma$ we have $\nabla \phi|\_{... | 4 | https://mathoverflow.net/users/3948 | 397392 | 164,048 |
https://mathoverflow.net/questions/395866 | 16 | The following statement is a direct consequence of the Continuum Hypothesis:
>
> There exists a sequence $\langle f\_\alpha:\omega\_1\rightarrow\omega\_1 ~ \vert ~ \alpha<\omega\_1\rangle$ of functions such that there is no function $f:\omega\_1\rightarrow\omega\_1$ with the property that the sets $\{\xi<\omega\_1 ... | https://mathoverflow.net/users/90412 | Diagonalizing against $\omega_1$-sequences of functions mod finite | The arguments in Section 6 of my paper "The nonstationary ideal in the $\mathbb{P}\_{\mathrm{max}}$ extension" show that there is a proper forcing adding a function from $\omega\_{1}$ to $\omega\_{1}$ which agrees with each such ground model function in only finitely many places. I say there that Todorcevic had done so... | 6 | https://mathoverflow.net/users/31807 | 397394 | 164,049 |
https://mathoverflow.net/questions/397395 | -6 | Indeed,
**Conjecture** Let $\ p\ $ be a prime, $\ n>1\, $ -- a natural number, and let the largest prime divisor $\ q\ $ of $\ s(p^n) :=\sum\_{k=0}^n\,p^k\ $ satisfy $ q<p.\ $ Then $\ p^n\ =\ 7^3.$
The unique(?) exception would be
$$ s(7^3)\ =\ 400\ = 2^4\cdot5^2 $$
>
>
**EDIT** A micro-observation: when... | https://mathoverflow.net/users/110389 | (not so) Spectacular $\ 7^3$ | There's also
$$s(67^2) = 1 + 67 + 67^2 = 4557 = 3 \cdot 7^2 \cdot 31 \tag{1}\label{eq1A}$$
In this case, $q = 31 \lt 67$ meets your criteria. I haven't checked further, but I'm fairly certain there are additional exceptions.
| 4 | https://mathoverflow.net/users/129887 | 397397 | 164,050 |
https://mathoverflow.net/questions/397347 | 9 |
>
> Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group?
>
>
>
It is well-known from the theory of special cube complexes that a subgroup $H$ of a right-angled Coxeter group $G$ contains a finite-index subgroup isomorphic to a sub... | https://mathoverflow.net/users/122026 | Subgroups of RAAGs vs. subgroups of RACGs | The fundamental group of the (non-orientable) closed surface of Euler characteristic -1 provides a counterexample.
On the one hand, it’s a subgroup of index 4 in the reflection group on the right-angled pentagon, so it embeds in a RACG, and of course it’s torsion-free.
On the other hand, [Crisp—Wiest](https://arxiv... | 11 | https://mathoverflow.net/users/1463 | 397400 | 164,051 |
https://mathoverflow.net/questions/397396 | 4 | Let $A$ and $B$ denote two *countably infinite* sets of ordinals.
Let $W\_A$ denote the supremum of ordinals *writable* by Ordinal Turing Machines with the set $A$ given as the source of parameters. That is, for any element $\tau$ of $A$, a $\tau$-th cell of the input tape is marked with $1$ before the start of a com... | https://mathoverflow.net/users/122796 | Countably infinite sets of ordinals as parameters for Ordinal Turing Machines | The answer is yes. In fact, you can make the difference enormous.
First, let me construct a certain set of ordinals.
**Lemma.** There is a countable set $X$ of ordinals such that:
1. Every OTM program that halts with $X$ as input, halts strictly before $\sup X$.
2. If a program does not halt with $X$ as input, th... | 6 | https://mathoverflow.net/users/1946 | 397406 | 164,053 |
https://mathoverflow.net/questions/397404 | 2 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ is *sane* if $V$ is finite, $E \neq \emptyset$, and $\emptyset \notin E$, and $e\not\subseteq e'$ whenever $e\neq e' \in E$. Moreover, we call $H$ *summable* if there is a map $f:V\to \mathbb{Z}\_{\geq 0}$ such that for all $e, e' \in E$ we have $$\sum\... | https://mathoverflow.net/users/8628 | Summable hypergraphs | No, not every sane hypergraph is summable. To see this, let $G$ be a star with five leaves $\ell\_1, \dots, \ell\_5$ all adjacent to a vertex $u$. Then turn $G$ into a sane hypergraph $H$ by adding the hyperedges $\{\ell\_1,\ell\_2\}$ and $\{\ell\_3, \ell\_4, \ell\_5\}$. Towards a contradiction, suppose that $f: V(H) \... | 2 | https://mathoverflow.net/users/2233 | 397411 | 164,055 |
https://mathoverflow.net/questions/397370 | 1 | What is the expression for the modular S-matrix of (p,q) minimal model? The Wiki <https://en.wikipedia.org/wiki/Minimal_model_(physics)> does not provide S-matrix
| https://mathoverflow.net/users/17787 | Modular S-matrix of (p,q) minimal model | The modular S-matrix appears in Section 10.6 of the Big Yellow Book by di Francesco, Mathieu and Sénéchal. For a freely available reference, there are my lecture notes <https://arxiv.org/abs/1609.09523> , eq. (A.28).
The Wikipedia article on minimal models was mostly written by someone who does not like the modular b... | 2 | https://mathoverflow.net/users/12873 | 397421 | 164,059 |
https://mathoverflow.net/questions/397283 | 2 | Let $P(n)$ be an irreducible polynomial of degree $2$ over the positive integers. Do there exist infinitely many positive integers $n$ such that $P(n)$ divides $n!$?
Edit: motivation by examples:
A) $p(n)=n^2+1$ (true, $21^2+1$ divides $21!$).
B) $p(n)=n^2+n+1$ (true, $74^2+74+1$ divides $74!$).
| https://mathoverflow.net/users/174530 | Polynomial whose values divide $n!$ | Here is a completely elementary proof, inspired by Pasten's comments.
Let $P(n)=an^2+bn+c$.
Take $n=a^5x^4+2a^3(ab+2a+1)x^3+a(2a^3c+a^2b^2+6a^2b+3ab+6a^2+5a+1)x^2+(ab+2a+1)(2a^2c+2ab+b+2a+1)x+a^3c^2+2a^2bc+abc+2a^2c+ac+c+ab^2+b^2+2ab+2b+a+1$
Then
$P(n)=P\_1(x)P\_2(x)P\_3(x)$
where
$$\begin{align\*}
P\_1(x)=... | 8 | https://mathoverflow.net/users/2480 | 397434 | 164,063 |
https://mathoverflow.net/questions/397407 | 1 | Let $X$ be a compact subset of the Euclidean space. Also let $Y$ be a separable Frechet space with the seminorms $\lVert \cdot \rVert\_n$'s.
Then let $C(X,Y)$ be the space of continuous mappings from $X$ into $Y$. It can be given the topology induced from the seminorms $\mid f \mid\_n := \sup\_{x \in X}\lVert f(x) \r... | https://mathoverflow.net/users/56524 | Showing that $C(X,Y)$ is separable when $X$ is compact Hausdorff but $Y$ is just a separable Frechet space | If $d$ is a translation-invariant metric on $Y$ then
$$
D(f,g)=\sup\{d(f(x),g(x)):x\in X\}
$$
defines a translation-invariant metric on $C(X,Y)$, and $D$ is complete if $d$ is complete.
The topology induced by $D$ is the compact-open topology ([Arens, A topology for spaces of transformations, Ann. Math. 47 (1946), 480-... | 2 | https://mathoverflow.net/users/5903 | 397436 | 164,064 |
https://mathoverflow.net/questions/397435 | 36 | We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number pair $(a,b)$. The corresponding complex field operations are expressible entirely within the real field.
Meanwhile, man... | https://mathoverflow.net/users/1946 | Can one show that the real field is not interpretable in the complex field without the axiom of choice? | An interpretation of $(\mathbb R,+,\cdot)$ in $(\mathbb C,+,\cdot)$ in particular provides an interpretation of $\DeclareMathOperator\Th{Th}\Th(\mathbb R,+,\cdot)$ in $\Th(\mathbb C,+,\cdot)$. To see that the latter cannot exist in ZF:
* The completeness of the theory $\def\rcf{\mathrm{RCF}}\rcf$ of real-closed field... | 34 | https://mathoverflow.net/users/12705 | 397444 | 164,067 |
https://mathoverflow.net/questions/397445 | 7 | Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(\cdot,0)=id$ and $F(\cdot, 1)=q \in M$.
Can we find an open neighborhood $U$ of $K$ such that $U$ is homeomorphic to ... | https://mathoverflow.net/users/280895 | Contractible set in a manifold | This is not true. Let K be one component (it doesn't matter which one) of the [Whitehead link](https://en.wikipedia.org/wiki/Whitehead_link), which has two components. Then K is contractible in the complement M of the other component. But K is not contained in a 3-ball in M. This can be seen in many ways; for instance ... | 9 | https://mathoverflow.net/users/3460 | 397447 | 164,068 |
https://mathoverflow.net/questions/397424 | 2 | I have the following question:
Let $u$ be a smooth subharmonic function on the unit disc $\mathbb{D}:=\left\{ z\in\mathbb{C}:\left|z\right|<1\right\} $.
Assume that $u=0$ on the boundary of $\mathbb{D}$ and
$$
\int\_{\mathbb{D}}\Delta u=1.
$$
Here $\Delta u$ is the Riesz measure of $u$.
Is there any chance to show ... | https://mathoverflow.net/users/318721 | A question on subharmonic functions on the unit disc | The first comment already shows that you cannot have $M$ independent of $u$.
But in fact you can construct a function smooth in the closed unit disk for
which $\Delta u$ has no uniform bound as $|z|\to 1$ whatsoever.
Just take
$$u\_1(z)=\sum\delta\_j\left(\log|z-a\_j|-\log|1-a\_jz|\right)=-\sum\delta\_jG(z,a\_j),$$
whe... | 2 | https://mathoverflow.net/users/25510 | 397449 | 164,070 |
https://mathoverflow.net/questions/149867 | 10 | Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by [Mostow-Palais theorem](http://en.wikipedia.org/wiki/Mostow-Palais_theorem), there exists an equivariant smooth embedding $M\to {\mathbb R}^n$ (for some $n$) sending $G$ to a subgroup of $O(n)$. I am ... | https://mathoverflow.net/users/21684 | Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups | There cannot be such an equivariant embedding in general.
For a non-compact semisimple Lie group $G$ one can always find a cocompact lattice $\Gamma<G$, so $M=G/\Gamma$ will be a compact real analytic manifold on which $G$ acts real analytically.
However, any homomorphism $G\to \text{PGL}\_n(\mathbb{R})$ is algebraic, ... | 2 | https://mathoverflow.net/users/89334 | 397458 | 164,073 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.