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/400708 | 6 | In mathematics and physics, especially gauge theory, there are many different but related notions of wedge products when discussing vector space- and vector bundle-valued differential forms. For example, if $U,V,W$ are finite-dimensional $\mathbb{R}$-vector spaces and $\mu:U\times V\to W$ is a bilinear map, then we alw... | https://mathoverflow.net/users/259525 | General wedge-product for vector bundle valued forms | The most general definition I know is the following. Every fiberwise bilinear form $\eta: V\_1 \times V\_2 \to W$ of vector bundles $V\_1, V\_2, W$ over $M$ gives rise to the wedge product of vector-bundle-valued differential forms by
$$(\alpha \wedge\_\eta \beta)\_m (X\_1, \dots X\_{p+q}) = \frac{1}{p!q!} \sum\_{\sigm... | 6 | https://mathoverflow.net/users/17047 | 400842 | 164,576 |
https://mathoverflow.net/questions/400813 | 20 | Say that a Diophantine equation is *almost-satisfiable* iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the converse fails - see the discussion [here](https://mathoverflow.net/questions/47442/diophantine-equation-with-no-inte... | https://mathoverflow.net/users/8133 | Is "almost-solvability" of Diophantine equations decidable? | A Diophantine equation is almost-satisfiable iff it is satisfiable over the ring $\widehat{\mathbb Z}$, the profinite completion of $\mathbb Z$ (also called by some the Prüfer ring), by a standard compactness argument (the solutions over each $\mathbb Z/n\mathbb Z$ form an inverse system of finite sets whose inverse li... | 27 | https://mathoverflow.net/users/15934 | 400844 | 164,577 |
https://mathoverflow.net/questions/400840 | 3 | I have seen that some papers talk of computational cost of the network and they measure it in MACs. I didn't find any clear explanation of what it is.
Could anyone explain in clear words the meaning of computational cost and why it should be taken into consideration in a network?
| https://mathoverflow.net/users/178519 | What is the computational cost in a neural network? | Computational cost is simply a measure of the amount of resources the neural network uses in training or inference, which is important so you can know how much time or computing power you'll need to train or use an NN. It can measured in a variety of ways, but common ones are time and number of computations, expressed ... | 6 | https://mathoverflow.net/users/235087 | 400846 | 164,578 |
https://mathoverflow.net/questions/400815 | 3 | Let $\mathcal C, \mathcal D\subseteq 2^\omega$.
Let
$$
\DeclareMathOperator{\Either}{Either}
\Either(\mathcal C,\mathcal D)=\{A\oplus B: \text{either }A\in \mathcal C, B\in\mathcal D\text{, or }B\in \mathcal C, A\in\mathcal D\}
$$
Has this operation been named and studied in the context of Medvedev degrees (i.e., stro... | https://mathoverflow.net/users/4600 | Join-like operation and Medvedev reducibility | Kojiro Higuchi and Takayuki Kihara have studied operations of this flavour in their papers "Inside the Muchnik degrees" I+II ([doi Part 1](https://doi.org/10.1016/j.apal.2014.01.003),[doi Part 2](https://doi.org/10.1016/j.apal.2014.03.001)). It has been a few years since I read those, and I do not remember whether this... | 3 | https://mathoverflow.net/users/15002 | 400847 | 164,579 |
https://mathoverflow.net/questions/400839 | 6 | Let $E$ be a vector bundle on some smooth algebraic variety and $E^\*$ its dual. Suppose $A$ is a sheaf (constructible or a $D$-module) on $E$. Given a linear function $f$ on $E$, we may compute the stalk at $f$ of the Fourier transform of $A$.
Now I’ve heard a slogan along the lines of “the stalk of the Fourier tran... | https://mathoverflow.net/users/101861 | What do nearby/vanishing cycles have to do with Fourier transforms? | You probably want to say $A$ is a constructible sheaf / $D$-module with regular singularities. $D$-modules with irregular singularities (for example, those created by the Fourier transform) will behave very differently, even if $E$ is a vector bundle of rank $1$ on a point.
---
I'm going to pretend you asked the ... | 4 | https://mathoverflow.net/users/18060 | 400848 | 164,580 |
https://mathoverflow.net/questions/400720 | 2 | Let $A$ be a $C^\*$-algebra. Let $M(A)$ be its multiplier $C^\*$-algebras. The strict topology on $M(A)$ is given by
$$x\_\lambda \to x \iff \forall a\in A: (\|x\_\lambda a-xa\| + \|ax\_\lambda - ax\| \to 0 ).$$
We can identify $M(A) \cong \mathcal{L}(A)$ where $\mathcal{L}(A)$ are the adjoinable operators of the rig... | https://mathoverflow.net/users/216007 | Strict topology on the multiplier algebra | Yes, these topologies agree basically by definition once you understand the isomorphism $M(A)\cong \mathcal L(A)$ (since $tθ\_{a,b}=θ\_{t(a),b}$ the canonical isomorphism $A≅ \mathcal K(A)$ takes this element to $t(a)b^\*=t(ab^\*)$). The isomorphism $A \to \mathcal K(A)$ is given by $a \mapsto \theta\_{a\_1,a\_2}$ wher... | 1 | https://mathoverflow.net/users/126109 | 400856 | 164,582 |
https://mathoverflow.net/questions/400818 | 4 | Let $f: Y \to X$ be an etale morphism of schemes.
>
> If $X$ has pseudo-rational singularities then does $Y$ also have pseudo-rational singularities?
>
>
>
For the definition of pseudo-rational see, for example, Definition 9.4 [here](https://arxiv.org/pdf/1703.02269.pdf).
Note that if $X$ has a resolution of... | https://mathoverflow.net/users/519 | Is pseudo-rationality preserved by etale morphisms? | I believe the answer is yes, although I do not know of a reference.
We will use the original definition of pseudo-rationality due to Lipman and Teissier [[Lipman–Teissier 1981](https://doi.org/10.1307/mmj/1029002461), p. 102], and the following characterization:
**Lemma** [[Lipman–Teissier 1981](https://doi.org/10.... | 2 | https://mathoverflow.net/users/33088 | 400859 | 164,583 |
https://mathoverflow.net/questions/400868 | 2 | This is related to one of my previous questions [here](https://mathoverflow.net/questions/398449/k-textth-maxima-of-n-i-i-d-chi-square-random-variables).
Let $(Z\_1, Z\_2, \ldots, Z\_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I\_{n\times n} + \mu \boldsymbol{1}\_n\boldsymbol{1}\_n^\top $. Here $\boldsymbol{1}\_n$ ... | https://mathoverflow.net/users/151115 | Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables | *(To long for a comment.)*
Let $b\_n = \sqrt{\frac{\mu}{n} + \frac{1-\mu}{n^2}} - \frac{\sqrt{1-\mu}}{n}$. Then
$$ \bigl( \sqrt{1-\mu} \, I\_{n\times n} + b\_n \mathbf{1}\_n \mathbf{1}\_n^{\top} \bigr)^2 = (1-\mu) I\_{n\times n} + \mu \mathbf{1}\_n \mathbf{1}\_n^{\top}. $$
In light of this, we can realize $(Z\_i)... | 1 | https://mathoverflow.net/users/15602 | 400870 | 164,585 |
https://mathoverflow.net/questions/400706 | 0 | In *Multiplicative Number Theory - Vol. I* by Montgomery and Vaughan the following result is proved.
**Theorem 7.20** Let $A(x,r)$ denote the number of $n\leq x$ such that $\Omega(n)\leq r \log \log x,$ and let $B(x,r)$ denote the number of $n\leq x$ for which $\Omega(n)\geq r \log \log x.$ If $0<r\leq 1$ and $x\geq ... | https://mathoverflow.net/users/17773 | Proportionality constant in Montgomery-Vaughan Theorem 7.20 | **Edit:** *My only goal is to mark this as answered.*
As explained by Greg Martin in a comment:
Certainly $(,)$
is a decreasing function of $$,
but that doesn't imply that the bound on $(,)$
continues to hold for $\geq 2$.
The reason that $<2$
is required in the given proof is that it proceeds via upper bounds fo... | 0 | https://mathoverflow.net/users/17773 | 400881 | 164,587 |
https://mathoverflow.net/questions/393515 | 3 | I'm currently studying Mumford's *Geometric Invariant Theory*. Unfortunately, I'm stuck understanding a detail in Theorem 1.1.
(Partial) Claim of Theorem 1.1
------------------------------
Let $X = \operatorname{Spec} R$ be an affine scheme over a characteristic-zero field $k$ and consider a reductive group action ... | https://mathoverflow.net/users/237033 | Why is the image of closed invariant subsets closed? Mumford, GIT, Theorem 1.1 | The idea is actually rather simple. Let $W$ be a closed $G$-invariant subset of $X$, and $y$ a closed point that is not in $\phi(W)$. Note that $\phi^{-1}(y)$ is also closed and $G$-invariant. We already know that
$$
\overline{\phi(W\cap\phi^{-1}(y))}=\overline{\phi(W)}\cap\{y\}.
$$
But the LHS is empty which means tha... | 1 | https://mathoverflow.net/users/333428 | 400903 | 164,593 |
https://mathoverflow.net/questions/400832 | 0 | I try to find an upper bound for the mixing time of a random walk $S$ on a connected graph $L=(V,E)$ which has $k<\min\_{v\in V}d(v)$ loops at every vertex. The transition probabilities of this random walk are given by
$$p\_{v,w}=\dfrac{1}{d(v)+k};\qquad p\_{v,v}=\dfrac{k}{d(v)+k}$$
where $d(v)$ is the degree of the ve... | https://mathoverflow.net/users/333230 | Mixing time for random walk on graph with $k$ loops on each vertex | The inequality you are citing should have a power 2 on the Cheeger constant (a.k.a the bottleneck ratio), so the inequality should read:
$$t\_{\rm mix} \le C\log\left(\min\_{v\in V}\dfrac{1}{\pi(v)}\right)\Phi(L)^{-2} \,.$$
This need not hold on a simple graph without loops; e.g. it fails if the graph is bipartite, w... | 1 | https://mathoverflow.net/users/7691 | 400904 | 164,594 |
https://mathoverflow.net/questions/400892 | 2 | If $G$ is a graph with edge set $E$, let $W$ be the $\mathbb{Z}/2$-vector space generated by the elements of $E$. If $A = \{a\_1, \dots, a\_n\} \subset E$, let $\bar{A} = a\_1 + \dots + a\_n \in V$; then $\bar{A}\_1 + \bar{A}\_2 = \overline{A\_1 \Delta A\_2}$, where $\Delta$ indicates symmetric difference.
I'll defin... | https://mathoverflow.net/users/202668 | Dimension of circuit space of a matroid | The dimension of the circuit space of a matroid $M$ is the corank of $M$ if and only if $M$ is binary. Here is a proof. Given a basis $B$ and $e \notin B$, we let $C(e,B)$ be the unique circuit contained in $B \cup \{e\}$. We will use the following well-known characterization of binary matroids (Theorem 9.1.2 in Oxley'... | 4 | https://mathoverflow.net/users/2233 | 400906 | 164,596 |
https://mathoverflow.net/questions/400899 | 0 | I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified Bessel function of a second kind is indeed log-concave or not.
| https://mathoverflow.net/users/333425 | Log-concavity of the modified Bessel function of a second kind | Theorem 2(b) in [1] is equivalent to log-convexity of $K\_\nu$ for every $\nu$. This is said to be "well-known", and three references are given.
[1] Árpád Baricz, Saminathan Ponnusamy, Matti Vuorinen, *Functional inequalities for modified Bessel functions*, [DOI:10.1016/j.exmath.2011.07.001](https://doi.org/10.1016/j... | 2 | https://mathoverflow.net/users/108637 | 400908 | 164,597 |
https://mathoverflow.net/questions/400922 | 1 | Let $(X,d)$ be a metric space with finite Assouad dimension $0<C\_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted here by $C\_Y$, should satisfy $C\_Y\leq c C\_X$ (where $c$ is some absolute constant independent of $X$ and of $Y$).
Is ... | https://mathoverflow.net/users/176409 | Monotonicity of doubling dimension | This is Lemma 9.6(i) in J. C. Robinson, *Dimensions, embeddings, and attractors.* Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.
In the proof the author says "it is obvious". I am no longer sure if it is obvious (perhaps it is) since I see a potential issue: if $Y\subset X$, then e... | 2 | https://mathoverflow.net/users/121665 | 400924 | 164,602 |
https://mathoverflow.net/questions/400912 | 10 | $\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack.
Is there is a well-defined notion of the *residue field* of a point $x \in |X|$?
Attempts:
1. [Recall](https://stacks.math.columbia.edu/tag/04XE) that a point on a stack is an equivalence class of morphisms $\Spec k \to X$ from fields $k$. The iss... | https://mathoverflow.net/users/5101 | Residue field of point on an algebraic stack | By definition, a residue field is an equivalence class of morphisms $\operatorname{Spec} k \to X$, i.e. of pairs of a field $k$ and an object in $X(k)$
We can upgrade that equivalence class into a category: Given fields $k$, $L$ and objects $a \in X(k) , b\in X(L)$, a morphism is a map $s \colon k \to L $ together wi... | 5 | https://mathoverflow.net/users/18060 | 400927 | 164,603 |
https://mathoverflow.net/questions/398359 | 8 | Let $f$ be a smooth function defined on the unit disc $D \subset \mathbf{R}^2$ with
\begin{equation}
f \geq 0 \text{ in $D$ and } f(0) = 0.
\end{equation}
This is allowed to have a degenerate minimum at the origin, namely it is allowed that $D^2 f(0) = 0.$
**Question.** When is there $\rho \in (0,1)$ and $u \in C^1(D... | https://mathoverflow.net/users/103792 | When does the eikonal equation $\lvert Du \rvert^2 = f$ admit a local solution? | For $n=1$ or $2$, there is no $u\in C^1(D\_\rho)$ for any $\rho>0$ that satisfies $|\nabla u|^2 = (xy)^{2n}$. (Note that $f=(xy)^2$ has a *degenerate* minimum at $(0,0)$, so $n=1$ should be allowed in this discussion.) Meanwhile, for $n\ge 3$, there do exist $u\in C^1(\mathbb{R}^2)$ that satisfy $|\nabla u|^2 = (xy)^{2... | 8 | https://mathoverflow.net/users/13972 | 400928 | 164,604 |
https://mathoverflow.net/questions/400929 | 0 | Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that it is a Banach space. If $A$ is open in $\mathbb{R}^n$ then there exists a topology on $C(A)$ that makes it a Frechet sp... | https://mathoverflow.net/users/41686 | About the normability of the space of continuous functions | Every real or complex vector space can be equipped with a norm (at least under the axiom of choice): Take a Hamel basis $B$ with coefficient functionals $\varphi\_b$ and define $\|x\|=\sum\limits\_{b\in B} |\varphi\_b(x)|$.
This norm however, is hardly ever of any use. For all your examples, natural additional assump... | 4 | https://mathoverflow.net/users/21051 | 400934 | 164,607 |
https://mathoverflow.net/questions/400926 | 3 | Given an elementary abelian $p$-group $A$ of order $p^n$ for $n\geq 2$ and choose a subgroup $H$ of order $p$ from $Aut(A)\cong GL(n,p)$. We can use semi-direct product $A\rtimes B$ to construct a nonabelian $p$-group containing an elementary abelian maximal subgroup. Is there any other example and a complete character... | https://mathoverflow.net/users/134942 | nonabelian $p$-group contains an elementary abelian maximal subgroup | Not every such group has this form, but they can be classified.
Let $G$ be a $p$-group containing an elementary abelian $p$-group $A$ as a maximal subgroup.
Being maximal, it is an index $p$ normal subgroup. Let $g$ be a generator of the quotient $G/A$. Then $g$ acts by conjugation on $A$ as a matrix $\sigma \in GL... | 6 | https://mathoverflow.net/users/18060 | 400942 | 164,610 |
https://mathoverflow.net/questions/400872 | 0 | Let $M: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$,
$(x,y) \mapsto (p,q)$, with $p,q \in \mathbb{C}[x,y]$
satisfying $\operatorname{Jac}(p,q):=p\_xq\_y-p\_yq\_x \in \mathbb{C}-\{0\}$.
Such a polynomial map is called a Keller map, and the two-dimensional Jacobian Conjecture
says that such a map is injective and surjective.
... | https://mathoverflow.net/users/72288 | Injectivity of Keller maps | This can be achieved after appropriate changes of coordinates of the source and target. More precisely, there are automorphisms $A, B$ of $\mathbb{C}[x,y]$ such that $A \circ M \circ B$ has the property you want.
This follows e.g. from [Orevkov's result](https://www.math.univ-toulouse.fr/~orevkov/jc86.pdf) that if $\... | 1 | https://mathoverflow.net/users/1508 | 400945 | 164,613 |
https://mathoverflow.net/questions/400887 | 1 | Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is the covariance kernel. A random function sampled from such a GP can also be regarded as a member of the RKHS $\mathcal{H... | https://mathoverflow.net/users/81633 | Tail bound on the RKHS norm of a zero-mean Gaussian process | In fact, if the RKHS $\mathcal{H}$ is infinite dimensional, then $\mathbb P(f\in\mathcal{H})=0$ -- see e.g. [Corollary 4.10](https://arxiv.org/abs/1807.02582). So, no inequality of the desired form exists in infinite dimensions.
| 2 | https://mathoverflow.net/users/36721 | 400950 | 164,614 |
https://mathoverflow.net/questions/400947 | 1 | A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union
of translated non-overlapping (but perhaps touching) copies
of $\mathcal C$.
The (translational) packing density of $\mathcal C$ is the maximal proportion of $\mathbb R^d$ occupied by a suitable pac... | https://mathoverflow.net/users/4556 | Worst convex compact set for translational packings of $\mathbb R^d$ | I believe this is not in general known for $d>2$. [Chapter 2](http://www.csun.edu/%7Ectoth/Handbook/chap2.pdf) of the *Handbook of Discrete and Computational Geometry* provides some pretty detailed information about translational packing density, which in their notation is $\delta\_T$. In particular, we have that the t... | 2 | https://mathoverflow.net/users/89672 | 400951 | 164,615 |
https://mathoverflow.net/questions/400915 | 15 | Consider a normal first course on category theory (say up to and including the statement and proof) of the adjoint functor theorem (AFT). What are the minimal assumptions for the definition of a set one needs to make in order that everything works? As far as I understand, up to and including the AFT there is very littl... | https://mathoverflow.net/users/153228 | Minimal set of assumptions for set theory in order to do basic category theory | To complement Tom Leinster's answer, let me try to be specific:
1. To form the product category $\mathcal{C} \times \mathcal{D}$, we need ordered pairs, which we can get from the axiom of **unordered pairs**.
2. It's probably a good idea to have the **empty set** $\emptyset$, so that the initial category exists.
3. M... | 17 | https://mathoverflow.net/users/1176 | 400962 | 164,620 |
https://mathoverflow.net/questions/400959 | 2 | We work in a countable language of finite-order arithmetic, which allows us to quantify over natural numbers, sets of natural numbers, sets of sets of natural numbers, and so on. We measure the complexity of sentences with a generalization of the arithmetical and analytical hierarchies to higher subscripts. We call $\P... | https://mathoverflow.net/users/163672 | Special classes of the arithmetical hierarchy of sentences of finite-order arithmetic | Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for *first-order arithmetic* sentences, so I'm not sure what it mean... | 2 | https://mathoverflow.net/users/8133 | 400963 | 164,621 |
https://mathoverflow.net/questions/400961 | 0 | What is a toric lattice? and how can I construct one in `Macaulay2` and compute its basis? is there any alternative method to make one? Since I went through the whole documentation of the M2 but could not find anything. For example, we know that a toric lattice of $\dim=3$ is an identity matrix of size 3.
| https://mathoverflow.net/users/333602 | What is a toric lattice? | From the [documentation](https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.15/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html) of the `normalToricVariety(*,*)` routine (emphasis mine):
>
> This is the general method for constructing a normal toric variety.
>... | 2 | https://mathoverflow.net/users/103164 | 400970 | 164,624 |
https://mathoverflow.net/questions/400946 | 1 | Let $\mathbb{R}$ be the set of real numbers. Given a subset $S$ of $\mathbb{R}$, let $\mathcal{T}\_S$ be the translation-invariant topology generated by $S$. That is, $\mathcal{T}\_S$ is the topology with a subbasis consisting of all translates of $S$. Suppose $A$ is a subset of $\mathbb{R}$ such that for every nonempt... | https://mathoverflow.net/users/132459 | Subsets of $\mathbb{R}$, every nonempty subset of which generates a disconnected translation-invariant topology | This example is inspired by an example given op page 13 in [J. van Mill, Homogeneous subsets of the real line, Compositio Mathematica, 46 (1982) no. 1, pp. 3-13](http://www.numdam.org/item/?id=CM_1982__46_1_3_0).
Let $H$ be a Hamel base for $\mathbb{R}$ over $\mathbb{Q}$ such that
$1\in H$.
For $x\in\mathbb{R}$ let $... | 3 | https://mathoverflow.net/users/5903 | 400981 | 164,628 |
https://mathoverflow.net/questions/400978 | 5 | Let $S$ be the (say, left) shift operator on $\ell^2(\mathbb{Z})$. For a non-zero vector $x \in \ell^2(\mathbb{Z})$, consider the set
$$X = \{ S^n v \mid n \in \mathbb{Z} \}.$$
Is this always a total set, i.e., is its span dense in $\ell^2(\mathbb{Z})$?
| https://mathoverflow.net/users/16702 | Do powers of the shift operator applied to a non-zero vector always yield a total set? | Such sets are not always total. The shift operator $S$ is unitarily equivalent to multiplication by $z$ on $L^2(S^1)$. From this perspective you can see vectors for which the set you write is not total, for example the characteristic function of an interval.
| 9 | https://mathoverflow.net/users/2085 | 400987 | 164,629 |
https://mathoverflow.net/questions/400993 | 0 | Let $G$ be a cyclic group of order $n$ and $K\leq AutG$ be a subgroup of the automorphism group of $G$. We denote the orbits of the natural action of $K$ on $G$ by $O\_1,\cdots, O\_s$. Let $\underline{X}\_i=\sum\_{x\in O\_i}x$ be the sum of elements in each orbit in the integral group ring $\mathbb{Z}G$. Then the $\mat... | https://mathoverflow.net/users/134942 | generator of a subring of integral group ring | Not when $n=4$ and $G$ is the full automorphism group of $\mathbb Z/4$.
Then $\mathcal A$ is spanned by $\underline{X}\_1,\underline{X}\_2, \underline{X}\_4$ where $\underline{X}\_i$ is the sum of all elements of order $i$.
Then $\underline{X}\_1=1$ is the identity, $\underline{X}\_2^2 = 1$, $\underline{X}\_2 \unde... | 5 | https://mathoverflow.net/users/18060 | 401001 | 164,635 |
https://mathoverflow.net/questions/400995 | 6 | I want to build a finite CW complex such that $\pi\_1$ is non-abelian and $H\_i$ are zero for $i\geq 2.$
From Hatcher for a given group G, one can create an example of a 2-complex $X\_G$ with $\pi\_1(X\_G)=G.$ I also checked from Mayer-Vietoris that if $G$ is cyclic such complex won't have any higher homology for $i\ge... | https://mathoverflow.net/users/333818 | Finite CW complex with finite non-abelian fundamental group and higher homologies zero | **Theorem.** Let $G$ be a group. There exists a finite 3-complex $X\_G$ with $\pi\_1 X\_G = G$ and $H\_i X\_G = 0$ for $i > 1$ if, and only if, $G$ is finitely presentable and has second group homology $H\_2(G) = 0$.
The more interesting question to me is whether this is possible for a *finite 2-complex*, and Jens Re... | 7 | https://mathoverflow.net/users/40804 | 401008 | 164,638 |
https://mathoverflow.net/questions/401033 | 1 | Let $A,B$ are two $p\times p$ positive definite matrices such that $0<\delta\_0\leq \min\{\lambda\_{\min}(A), \lambda\_{\min}(B)\}\leq \max\{\lambda\_{\max}(A), \lambda\_{\max}(B)\}\leq \delta\_1$. Also assume that $\Vert A-B\Vert\_{op}\leq \varepsilon$. Can we upper bound $\Vert A^{-1} - B^{-1} \Vert\_{op}$ in terms o... | https://mathoverflow.net/users/151115 | If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close | $A^{-1}-B^{-1}=B^{-1}(B-A)A^{-1}$ so
$\|A^{-1}-B^{-1}\| \le \delta\_0^{-2} \epsilon $. This is the best possible bound in terms of the given parameters, as you can see by considering 2 by 2 diagonal matrices: Consider
$A=$diag$(\delta\_0,\delta\_1)$ and $B=$diag$(\delta\_0+\epsilon,\delta\_1+\epsilon)$ where $\delta\_1... | 5 | https://mathoverflow.net/users/7691 | 401035 | 164,643 |
https://mathoverflow.net/questions/401043 | -1 | I am trying to evaluate a fairly simple summation:
$\sum\_{k=1}^n ka^kb^{n-k}$
Which is related to the common identity for $\sum\_{k=1}^n ka^k$ available on Wikipedia.
I've previously seen lengthy lists of obscure summation formulas in the comments but could not find any this time via the search function.
I fou... | https://mathoverflow.net/users/334014 | List of obscure summation identities | As suggested by the OP, I post my comment as an answer:
Try out Wolfram Alpha. The code for your example is <https://www.wolframalpha.com/input/?i=Sum%5B+k+a+%5Ek+b%5E%28n-k%29+%2C+%7B+k%2C+1%2C+n+%7D+%5D>
| 0 | https://mathoverflow.net/users/37436 | 401044 | 164,644 |
https://mathoverflow.net/questions/401057 | 3 | In a paper I was reading, it was mentioned that if $M$ is a closed Riemannian manifold, then by fixing a basis for $L^2(M)$ consisting of eigenfunctions of the Laplacian, the space of smoothing operators on $L^2(M)$ can be identified with the algebra of matrices $a\_{ij}$ such that
$$\sup\_{i,j}i^k j^l |a\_{ij}| <\in... | https://mathoverflow.net/users/78729 | Identification of smooth operators with rapidly decreasing matrices | This is, of course, a long story incorporating many strands but I will try to give a quick overview. Firstly, it is, as so often, convenient to skip to a more general framework. In your case, this would be that of an unbounded self-adjoint operator $T$ on Hilbert space (here that would be the Laplacian--more later).
... | 2 | https://mathoverflow.net/users/317800 | 401061 | 164,646 |
https://mathoverflow.net/questions/401059 | 1 | *EDIT (August 9, 2021):* I would like to ask a more general question. The original question that was fully answered is below the line.
For a positive real number $x$, denote the fractional part $x-[x]$ of $x$ by $\langle x \rangle$.
Let $\ell>0$ be an integer. Is
$$\Phi\_{\ell} := \liminf\_{n>0 \text{ not a } {\e... | https://mathoverflow.net/users/14233 | Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$ | Let $n=m^p+1$ for some large enough $m$. For $0<k<p$ we then have $m^{kp}<n^k<m^{kp}+O(m^{(k-1)p})=m^{kp}(1+O(m^{-p}))$, so $$m^k<n^{k/p}<m^k(1+m^{-p})^{k/p}\leq m^k(1+O(m^{-p}))=m^k+o(1).$$ Hence $\langle n^{k/p}\rangle=o(1)$ as $m\to 0$ for each $0<k<p$, and in particular the $\liminf$ in your question is zero.
The... | 4 | https://mathoverflow.net/users/30186 | 401062 | 164,647 |
https://mathoverflow.net/questions/401020 | 7 | Does one need the axiom of replacement in the [small object argument](https://ncatlab.org/nlab/show/small+object+argument) and in the [transfinite construction of free algebras](https://ncatlab.org/nlab/show/transfinite+construction+of+free+algebras)?
My motivation for the question is that I heard that the axiom of r... | https://mathoverflow.net/users/333306 | Does the small object argument need replacement? | The way it is usually presented, certainly yes. As you point out it usually refers to possibly uncountable regular ordinals, which would usually mean von Neumann ordinals. Once you have the regular ordinal, say $\kappa$, then regardless of whether $\kappa$ is a von Neumann ordinal or just a well ordered set, you need t... | 7 | https://mathoverflow.net/users/30790 | 401065 | 164,649 |
https://mathoverflow.net/questions/401064 | 15 | [A recent algorithm](https://www.maths.ox.ac.uk/node/38304) unknots in quasipolynomial time. But I want to know what happens to the crossing number. Assuming your unknot has $n$ crossings, if I remember correctly it might be necessary to increase $n$ temporarily. But to what? $n+C$? $C\*n$? Even worse? (I don't even kn... | https://mathoverflow.net/users/11504 | Unknot recognition - how tangled does it get? | Joel Hass and Jeff Lagarias proved that one can transform any unknot diagram with $n$ crossings into the standard unknot diagram using not more than $2^{cn}$ Reidemster moves. They were able to obtain the explicit value for $c$ of $c=10^{11}$. See [here](https://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-... | 23 | https://mathoverflow.net/users/127690 | 401066 | 164,650 |
https://mathoverflow.net/questions/400965 | 9 | In [Infinitesimal analysis without the Axiom of Choice](https://doi.org/10.1016/j.apal.2021.102959), Hrbacek and Katz have shown that it is possible to formulate an axiomatic theory which provides a formalisation of calculus procedures which make use of infinitesimals (known as SPOT, an acronym of its axioms).
Elsewh... | https://mathoverflow.net/users/119114 | SPOT as a conservative extension of Zermelo–Fraenkel | In plain terms, the conservativity of SPOT over ZF means that if a particular statement S in the language of ZF is provable in SPOT, then ZF can already prove S (with a possibly different proof). Note that ZF does not include the axiom of choice.
More formally, the conservativity of SPOT over ZF is a statement about ... | 18 | https://mathoverflow.net/users/9269 | 401076 | 164,653 |
https://mathoverflow.net/questions/401048 | 2 | Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B\_t)\_{t\ge 0}$, can we prove
$$\limsup\_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B\_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B\_s\le \limsup\_{n\to\infty}\ell^n(s)],\quad... | https://mathoverflow.net/users/nan | Question concerning an inequality on probabilities of hitting times in a paper | $\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$This is not quite obvious, and it has hardly anything to do with the reverse Fatou lemma.
Indeed, for all $s\in[0,t]$, let
\begin{equation\*}
l\_n(s):=\sup\_{m\colon m\ge n}\ell^m(s),
\end{equation\*}
so that
\begin{equation\*}
\ell^n(s)\le l\_n(s)\downarrow l... | 2 | https://mathoverflow.net/users/36721 | 401084 | 164,657 |
https://mathoverflow.net/questions/401068 | 2 | I'm studying the book *Etale cohomology and the Weil conjecture* by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is the "sheafification" procedure in étale world): see pages 11-13.
Let $A$ be a commutative, unital, Noetherian ring ... | https://mathoverflow.net/users/108274 | Some facts about sheafification functor on étale site | A good trick for answering questions of this type is that there are multiple foundational resources in étale cohomology. If a detail isn't explained in one you're reading, you can quickly check another one - the notation should hopefully be similar enough that you can transfer the proof over. In particular, the Stacks ... | 2 | https://mathoverflow.net/users/18060 | 401093 | 164,661 |
https://mathoverflow.net/questions/400769 | 5 | Say $A\_0$ is an ordinary abelian variety over ${\mathbf{F}}\_q$. Call $\mathcal{A}$ the canonical lift of $A\_0$ over $R := W({\mathbf{F}}\_q)$. It carries a lift of the $q$-th power map on $A\_0$. We call $\phi : \mathcal{A}\to\mathcal{A}$ this lift. It exists by functoriality of the canonical lift.
Call $K = \text... | https://mathoverflow.net/users/nan | Ordinary abelian varieties and Frobenius eigenvalues | Let's examine how $\phi$ acts on the algebraic Dolbeaut cohomology $$H^1(\mathcal A\_K , \mathcal O\_{\mathcal A})+ H^0 ( \mathcal A\_K, \Omega^1\_{\mathcal A}).$$
I claim its eigenvalues on $H^1(\mathcal A\_K , \mathcal O\_{\mathcal A})$ are units and its eigenvalues on $H^0 ( \mathcal A\_K, \Omega^1\_{\mathcal A})$... | 4 | https://mathoverflow.net/users/18060 | 401096 | 164,662 |
https://mathoverflow.net/questions/401095 | -1 | This is inspired by a recent [math.SE question](https://math.stackexchange.com/questions/4215964/does-there-exist-n-m-in-mathbbn-such-that-lvert-left-frac32-ri).
Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide lots... | https://mathoverflow.net/users/29783 | A pathological (?) function involving powers | The answer to both of your questions is no.
Notice that $d(2, 3) = 1$. However, for any $\varepsilon > 0$, there exists a number $2 - \varepsilon < x < 2$ such that $d(x, 3) = 0$ and thus it is not a continuous function. To see this, take $n$ to be some large positive integer and $m$ the unique integer satisfying
$$2... | 1 | https://mathoverflow.net/users/88679 | 401097 | 164,663 |
https://mathoverflow.net/questions/327495 | 7 | I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^\*$-equivariant $K$-theory of flag varieties.
Let $G$ be a semisimple, simply connected algebraic group, with $T \subset B \subset G$ a chosen maximal torus and Borel subgroup. In order to gain geometric infor... | https://mathoverflow.net/users/119460 | Geometric interpretations of nil-Hecke ring and affine Hecke algebra | The subvariety $\overline{Y}\_{s\_i}\subset G/B \times G/B$ is the fiber product $G/B\times\_{G/P\_i}G/B$. The set $\overline{Y}\_{s\_i}$ is the saturation for the diagonal $G$-action of $\{B/B\}\times P\_i/B$, by definition, and of course, that also lies in the fiber product; since they are smooth irreducible varietie... | 4 | https://mathoverflow.net/users/66 | 401106 | 164,666 |
https://mathoverflow.net/questions/401108 | 6 | I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the intersection $\bigcap\{N\trianglelefteq G\mid N\text{ open}\}$ of all open *normal* subgroups is not the trivial subgroup.
... | https://mathoverflow.net/users/5801 | Intersection of all open subgroups vs. the intersection of all open normal subgroups | $S\_\infty$, the group of all permutations of $\mathbb{N}$, has a neighborhood base of the identity of open subgroups. (In fact a Polish group with that property is isomorphic to a closed subgroup of $S\_\infty$).
But without thinking about exactly which ones are open, $S\_\infty$ has a very limited supply of normal ... | 6 | https://mathoverflow.net/users/6342 | 401109 | 164,667 |
https://mathoverflow.net/questions/401036 | 2 | Let $V$ be a Ternary rings of operators(TRO) i.e. closed subspace of $B(H,K)$ such that $xy^\*z \in V$ for all $x,y,z \in V$. A subspace $I$ of $V$ is called a left (right)TRO ideal provided $VV^\*I \subset I$$(IV^\*V \subset V)$.
>
> Sum of closed left and right ideals is closed provided one of the ideal has bound... | https://mathoverflow.net/users/129638 | Looking for an old paper of Kirchberg | I guess you don't need approximate identity assumption to prove it (in any case what does approximate identity mean for a TRO?). I am not sure which paper of Kirchberg you are referring to, but indeed Kirchberg has proved that the sum of a closed left ideal and a closed right ideal is closed (for $C^\*$-algebras) and t... | 1 | https://mathoverflow.net/users/7591 | 401113 | 164,668 |
https://mathoverflow.net/questions/401070 | -3 | This question is a follow-up to [Are there infinitely many L-rigs?](https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs) which is already pretty convoluted.
Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon\_{\varphi,s}:F\mapsto \varphi(F)(s)$ where $F$ is a map from... | https://mathoverflow.net/users/13625 | Structure of the automorphism group of an L-rig | This an answer following an argument from Wojowu: as we require the equality $\epsilon\_{g(\varphi),s}=\epsilon\_{\varphi,s}$ to hold for all $(g,\varphi,s)$, and thus for all $s$, this means that $g(\varphi)=\varphi$ for all $\varphi$, so that $g$ is the identity. So the automorphism group of $G\_{\mathcal{L}}$ is tri... | 0 | https://mathoverflow.net/users/13625 | 401118 | 164,669 |
https://mathoverflow.net/questions/401125 | 4 | In his paper "Paul Levy's Isoperimetric Inequality" (published as appendix C in *Metric Structures for Riemannian and Non-riemannian Spaces*), Gromov claims that if $H$ is a minimal $n$-dimensional hypersurface dividing a Riemannian into two pieces of fixed volume, $v$ is any point and $h \in H$ satisfies $dist(H,v)=di... | https://mathoverflow.net/users/106263 | Nearest point is always regular for isoperimetric hypersurfaces | I think you're inadvertently opening a big can of worms. The question can be answered by a combination of two facts: the absence of branch points in (almost-)minimising hypersurfaces and Allard's regularity theorem.
Specifically, the tangent cones to $H$ at $h$ must be multiples of an $n$-dimensional hyperplane $P$ s... | 2 | https://mathoverflow.net/users/103792 | 401127 | 164,672 |
https://mathoverflow.net/questions/397619 | 15 | I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself.
Let $f(z)$ be a power-series and $[z^n]\{-\}$ denote the $n$'th coefficient. Show that the following holds, whenever $[z^0]{f(z)}=1$:
$$
\exp... | https://mathoverflow.net/users/109370 | Comparing two power-series | I decided to summarize the two main proofs that I liked. The first one was motivated by the answer by esg. The second one can be found in the answer of Alex Gavrilov and is made more explicit. I am very grateful for their help.
**1. proof:**
Using Gessel [(2.4.4)](https://arxiv.org/abs/1609.05988) and the unique solu... | 1 | https://mathoverflow.net/users/109370 | 401129 | 164,673 |
https://mathoverflow.net/questions/401130 | 4 | Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $E$.
My questions are derived from this fact:
1. Let $\{D\_t\}\_{t\in [0,1]}$ be a smooth family of Dolbeault operato... | https://mathoverflow.net/users/41200 | Family of Dolbeault operators on complex vector bundles over $\mathbb{CP}^1$ | The answer to the first question is in fact no: consider the family of Dolbeaut operators on the complex vectorbundle of degree 0 and rank 2 underlying $$V=\mathcal O(-1)\oplus\mathcal O(1).$$ Consider the family of operators $$\bar\partial^t=\begin{pmatrix}\bar\partial^{\mathcal O(-1)} & t\, \gamma \\0& \bar\partial^{... | 2 | https://mathoverflow.net/users/4572 | 401134 | 164,674 |
https://mathoverflow.net/questions/400967 | 15 | I have a vague memory of an infinite game due to Ernst Specker with
the following properties:
(1) It is a two-person perfect information game, where the players
move alternately.
(2) The possible moves depend only on the current position.
(3) There is no winning strategy where each move is based only on
the curre... | https://mathoverflow.net/users/2807 | An infinite game possibly due to Ernst Specker | I don't know about the game attributed to Specker, but here is a
simple game with your desired features.
Let us call it the **Chocolatier's game**. There are two players,
the Chocolatier and the Glutton. To begin play, the Chocolatier
serves up finitely many unique and exquisite chocolate creations on
a platter, and ... | 23 | https://mathoverflow.net/users/1946 | 401136 | 164,675 |
https://mathoverflow.net/questions/401140 | 2 | Let $\epsilon\_1,...,\epsilon\_n$ be i.i.d. random signs, $\mathbf{u}\_1,...,\mathbf{u}\_n$ i.i.d. uniform random vectors on the unit sphere $\mathbb{S}^{d-1}$, assuming $d$ even, and $\mathbf{v}\_1,...,\mathbf{v}\_n$ be their half-truncations, that is $\mathbf{v}\_i[j] = \mathbf{u}\_i[j]$ for all $j \in \left \{1,...,... | https://mathoverflow.net/users/334327 | Lower bounds on random process | $\newcommand\ep\epsilon\newcommand\v{\mathbf v}$Conditioning on the $\ep\_i$'s and using Jensen's and [Szarek's](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-smv58i1p13bwm) inequalities, we have
$$E\Big|\sum\_1^n\ep\_i\|\v\_i\|\_2^2\Big|
\ge E\Big|\sum\_1^n\ep\_ic\_i\Big|\ge\frac1{\sqrt2}\sq... | 3 | https://mathoverflow.net/users/36721 | 401142 | 164,676 |
https://mathoverflow.net/questions/401149 | 3 | Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.
I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically hyperbolic group is either finite or virtually cyclic but couldn't find a reference. I wonder if someone knows where it is and ... | https://mathoverflow.net/users/104837 | The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic | I do not know a reference where this statement exactly is proved, but Theorem 1.1 from Osin's article *Acylindrically hyperbolic groups* does most of the work.
It implies that, if the stabiliser $H$ of a pair of points at infinity $\alpha,\omega \in \partial S$ is not virtually infinite cyclic, then it has bounded or... | 2 | https://mathoverflow.net/users/122026 | 401155 | 164,679 |
https://mathoverflow.net/questions/401147 | -1 | Consider the projective plane $\mathbb{P}^2\_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$.
Take the point $p\_0 = [0:1:0]\in \mathbb{P}^2\_{\overline{\mathbb{C}(t)}}$ and eight more general points $p\_1,\dots,p\_8\in \mathbb{P}^2\_{\overline{\mathbb{C}(t)}}$ which are n... | https://mathoverflow.net/users/14514 | Coefficients of elliptic curves over function fields | Are you asking if, for all tuples $p\_1,\dots, p\_8$, there exists such a $C'$ with $A\_9$ of degree one? This is false, assuming $A\_9=0$ does not count as degree one.
We can for example choose one of the $p\_i$ to equal $[0: 0 : 1]$ for $t=1$ and and one to equal $[0,0,1]$ for $t=2$. Then regardless of which $C'$ w... | 1 | https://mathoverflow.net/users/18060 | 401164 | 164,685 |
https://mathoverflow.net/questions/401157 | 7 | Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$$
be the functor sending a space $X$ to the topos of sheaves on $X$. Does this functor have a lef... | https://mathoverflow.net/users/333306 | Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint? | In this answer, Topos is interpreted as a 2-category.
(As a side remark, the 1-category of toposes does not make sense
until one picks a specific model for toposes and geometric morphisms, and different models
need not be equivalent as 1-categories.
For the 1-categorical framework to make sense, at the very least
one n... | 9 | https://mathoverflow.net/users/402 | 401166 | 164,687 |
https://mathoverflow.net/questions/401103 | 2 | Let $v(x, t) = \mathbb E [f(x + W\_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$:
$$\partial\_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W\_t) W\_t ].$$
I am interested in $u'(x)$ with $u$ defined by
$$u(x) = \mathbb E \int\_0^T f(x + W\_t) dt$$
for some function $f$. First we... | https://mathoverflow.net/users/5656 | Existence of the derivative of functionals of Brownian motion | $\newcommand{\ep}{\varepsilon}\newcommand{\R}{\mathbb R}\newcommand{\vpi}{\varphi}\newcommand{\De}{\Delta}\newcommand{\Om}{\Omega}$The derivative $u'(x)$ exists and the equality
\begin{equation\*}
u'(x) =\int\_0^T \frac{dt}t\, Ef(x+W\_t)W\_t \tag{1}
\end{equation\*}
holds under very mild restrictions on $f$, just a bi... | 2 | https://mathoverflow.net/users/36721 | 401167 | 164,688 |
https://mathoverflow.net/questions/401034 | 14 | I am trying to bound a function that includes $\sum\limits\_{\substack{d < n^{1/3} \\ d \mid n}} 1$.
Is there an upper bound known for this sum, either in general or in terms of $\sum\limits\_{\substack{d \mid n}} 1$? Or in general is there a bound for $\sum\limits\_{\substack{d < n^{1/k} \\ d \mid n}} 1$? Any help i... | https://mathoverflow.net/users/333969 | How many divisors of $n$ are below $n^{1/3}$? | One thing you asked for is a lower bound.
Following FusRoDah, I will let $d\_k(n)$ be the number of divisors of $n$ of size less than $n^{1/k}$, and $d(n)$ be the number of divisors of $n$.
Then I claim $$ d\_1(n) \leq d\_3(n) (d\_3(n)+5),$$ giving an explicit lower bound of size roughly $d\_1(n)^{1/2}$.
Proof: F... | 11 | https://mathoverflow.net/users/18060 | 401172 | 164,691 |
https://mathoverflow.net/questions/401154 | 1 | Let $(W\_t)\_{t\ge 0}$ be a standard Brownian motion. For each $t\in [0,1]$, it is known that, e.g. from Burkholder-Davis-Gundy's inequality
$$\mathbb E\big[\sup\_{s\in [t,t+\Delta t]}|W\_s-W\_t|^p\big]=O(\Delta t^{p/2}),\quad \forall p\ge 1,$$
where $O$ refers to "of order". Do we have an estimate of
$$\mathbb E... | https://mathoverflow.net/users/261243 | On the "uniform continuity" of Brownian motion under expectation | For $n\in\mathbb{Z}\_{\geq 0}$ and $0\leq i< 2^n$, denote
$$
X\_{n,i}=\sup\_{t\in[i2^{-n}, (i+1)2^{-n}]}{|W\_t-W\_{i2^n}|}.
$$
Let $n$ be such that $2^{-{n}}<|\Delta t|\leq 2^{-{n+1}}$. Then,
$$\sup\_{s,t\in[0,1],~ |s-t|\le\Delta t}|W\_s-W\_t|^p\leq 4\sup\_{i}|X\_{i,n}|^p,$$ because such $s$ and $t$ must belong to... | 1 | https://mathoverflow.net/users/56624 | 401174 | 164,692 |
https://mathoverflow.net/questions/400976 | 6 | Voisin uses the fact "If $X$ is a K3 surface with an ample line bundle $\mathcal L$ such that $\mathcal L$ generates $\mathop{\mathrm{Pic}}(X)$ and $(\mathcal L^2) = 4t - 2$, then every smooth curve $C \in \lvert\mathcal L\rvert$ satisfies $K\_{t, 1}(C, K\_C) = 0$." to prove the Green conjecture holds for generic curve... | https://mathoverflow.net/users/129738 | Existence of curves of arbitrary genus on some K3 surface | This should be a consequence of the surjectivity of the period map for K3 surfaces. I believe with this in mind the reasoning is somewhat standard, but it's useful to try and make it explicit. The underlying strategy is as follows: 1) identify a non-empty locus $\mathcal{W}$ in the period domain to which a K3 surface $... | 4 | https://mathoverflow.net/users/76148 | 401185 | 164,696 |
https://mathoverflow.net/questions/365820 | 7 | Given a (fibrant) simplicially enriched category $\mathcal{C}$, I'm interested in the possibility of replacing it with a weakly equivalent one (in Bergner model structure) such that all the mapping spaces are minimal Kan complexes, that I read about for example in $\textit{Higher Topos Theory}$, section 2.3.3.
The naiv... | https://mathoverflow.net/users/134438 | Locally minimal simplicial categories | It's not possible in general to ensure that all the hom-spaces in a simplicial category are minimally fibrant. Here's a counterexample [inspired by Isbell](https://mathoverflow.net/a/128629/2362).
Consider $Set$ with its cartesian monoidal structure (the same approach, *mutatis mutandis*, will work with the cocartesi... | 4 | https://mathoverflow.net/users/2362 | 401186 | 164,697 |
https://mathoverflow.net/questions/159554 | 5 | I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a good reference?
| https://mathoverflow.net/users/22055 | Reference for statement that almost every $n$-element partial order has trivial automorphism group | Prömel (1987) proves a more general statement of rigidity for many classes of structures. In particular he has:
Corollary 2.3. Let $P^u(n)$ denote the number of unlabeled partial orders
on an $n$-element set. Then there exists a constant $s$ such that for all $n$
$$
P^u(n) \le \frac{P(n)}{n!} \left(1 + \frac{s}{2^{n/... | 3 | https://mathoverflow.net/users/171662 | 401199 | 164,701 |
https://mathoverflow.net/questions/401025 | 15 | The *Cantor bijection* given by
$$(x,y)\longmapsto {x+y\choose 2}-{x\choose 1}+1$$
is a bijection from $\{1,2,3,\dotsc\}^2$ onto $\{1,2,3,\dotsc\}$.
It can be generalized to bijections $\varphi\_d:\{1,2,3,\dotsc\}^d
\longrightarrow \{1,2,3,\dotsc\}$ given by
$$(x\_1,\dotsc,x\_d)\longmapsto (d+1\bmod 2)+(-1)^d\sum\_{k... | https://mathoverflow.net/users/4556 | Are there exotic polynomial bijections from $\mathbb N^d$ onto $\mathbb N$? | [Wikipedia says](https://en.wikipedia.org/wiki/Fueter%E2%80%93P%C3%B3lya_theorem) "The generalization of the Cantor polynomial in higher dimensions" is $$(x\_1,\ldots,x\_n) \mapsto x\_1+\binom{x\_1+x\_2+1}{2}+\cdots+\binom{x\_1+\cdots +x\_n+n-1}{n}$$ Note that this is not equivalent to your generalisation $$(x\_1,\ldot... | 14 | https://mathoverflow.net/users/46140 | 401201 | 164,702 |
https://mathoverflow.net/questions/401198 | 3 | Let $V^n$ be a Stein space(or Stein manifold) in $\mathbb{C}^N$. I want to construct a Stein space(or Stein manifold) $W^{n+1}$ such that $H\_i(V;\mathbb{Z})=H\_{i+1}(W; \mathbb{Z}).$
If we take the suspension of $V,$ is it a Stein space? Or can I get a Stein space $W^{n+1}$ which has the same homotopy as the suspens... | https://mathoverflow.net/users/333818 | How to get a Stein space which has homotopy type of suspension of another Stein space | I'm not sure what you mean by "suspension" of $V$ here. The notion of suspension I have in mind (doubling the cone of $V$ over its base) doesn't yield a manifold, and even if it did it would give an odd-dimensional manifold, so the answer to your question would be no.
About the homotopy type, the answer is yes. Since... | 6 | https://mathoverflow.net/users/13119 | 401205 | 164,705 |
https://mathoverflow.net/questions/401210 | 2 | For a sufficient large field $k$ with characteristic 2, $S\_3$ and $D\_{10}$ both do have the property that the trivial module over their group algebra has nontrivial self-extension. Puig's work on nilpotent blocks shows that being p-nilpotent for the considered characteristic is sufficient to guarantee the trivial mod... | https://mathoverflow.net/users/134942 | Nontrivial self-extension of trivial $kG$-module | $\mathrm{Ext}^1\_{kG}(k,k)$ classifies these extensions, and it's not too hard to show that $\mathrm{Ext}^1\_{kG}(k,k) \cong H^1(G;k) \cong \hom(G, k)\cong \bigoplus\_{i\in I}\hom(G^{ab},\mathbb F\_p)$ where $|I| = \dim\_{\mathbb F\_p}(k)$.
In particular, this is nonzero if and only if $\hom(G^{ab},\mathbb F\_p)\neq ... | 8 | https://mathoverflow.net/users/102343 | 401212 | 164,706 |
https://mathoverflow.net/questions/359594 | 6 | Write $n$ iid draws of $(x,y)$ as $(x^n, y^n)$. Fix $R\in (0,H(x))$. What is the min of $n^{-1}H(y^n|f(x^n))$ over maps $f$ with range $\lbrace 1,\dots,\exp nR\}$, taking $n\to \infty$?
| https://mathoverflow.net/users/10668 | Can information be extracted more precisely using more random trials? | The characterization is given in terms of a so-called auxiliary random variable. It is as explicit of an answer as you'll get, unless you consider very special cases (like jointly Gaussian $X,Y$, or binary-valued $X,Y$). Namely, you have
$$
\lim\_{n\to\infty}\min\_{f: x^n \mapsto f(x^n)\in \{1,\dots,2^{nR}\}} \frac{1... | 1 | https://mathoverflow.net/users/99418 | 401219 | 164,708 |
https://mathoverflow.net/questions/401151 | 26 | I have a question about the Chocolatier's game, which I had
introduced in [my recent answer to a question of Richard
Stanley](https://mathoverflow.net/a/401136/1946).
To recap the game quickly, the Chocolatier offers up at each stage
a finite assortment of chocolates, and the Glutton chooses one to
eat. At each stage... | https://mathoverflow.net/users/1946 | The Chocolatier's game: can the Glutton win with a restricted form of strategy? | *(Not an answer; promoted from a comment on another answer)*
If we modify the game so that the glutton can remember (only) the last chocolate they ate, they have a winning strategy as follows:
Well-order $X$. At each step, let $c\in X$ be the last chocolate we ate. If, among the chocolates offered to us, there is a... | 13 | https://mathoverflow.net/users/64294 | 401221 | 164,709 |
https://mathoverflow.net/questions/401218 | 5 | Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions?
Say now $p = 1$, Does its Fourier Series converge in the metric space $L^1[a,b]$? Does the series converge pointwise? And under which condi... | https://mathoverflow.net/users/334508 | Convergence of Fourier series | 1. **Convergence in $L^p$, $p>1$.**
True, by [M. Riesz's Theorem](https://www.jstor.org/stable/1993749). This is a standard topic in every harmonic analysis course, with several readable proofs.
2. **Convergence pointwise almost everywhere, $p>1$.**
True, by the [Carleson-Hunt Theorem](https://terrytao.wordpress.... | 25 | https://mathoverflow.net/users/142740 | 401224 | 164,710 |
https://mathoverflow.net/questions/401196 | 2 | Let $\Omega \subset \mathbb{R}^n$ be an open open subset. Let $u,v\colon \Omega\to \mathbb{R}$ be two functions such that at least one of them is compactly supported. Assume each of $u$ and $v$ can be presented as a difference of two bounded subharmonic functions in $\Omega$. Thus in particular the distributional Lapla... | https://mathoverflow.net/users/16183 | Comparing integrals of bounded subharmonic functions | Without loss of generality, $u$ has compact support $K\subset\Omega$. Therefore the (signed) measure $\Delta u$ is supported in $K$ as well. Let $(\phi\_k)$ be a sequence of smooth (radial) mollifiers such that $\phi\_k\*u$ is supported in $K^\delta$ (the closed $\delta$ neighborhood of $K$, with $\delta>0$ so small th... | 0 | https://mathoverflow.net/users/42851 | 401231 | 164,714 |
https://mathoverflow.net/questions/401237 | 0 | Prove that this sum holds for all positive integers $k$. I'm quite sure this is right but I can't see immediately how to go about proving it. This will help resolve a problem regarding sums of binomial coefficients that I'm working on. Any ideas?
| https://mathoverflow.net/users/265714 | Prove for all $k \in \mathbb{N}$, that $\sum_{j=0}^{2k+1} {n+j-1\choose j} + \sum_{j=0}^{2k+1}(-1)^j{n+2k+2\choose j} = 0$ | It's known that $\frac1{(1-x)^n}=\sum\_{i\geq0}\binom{n+j-1}jx^j$. Combined with the relation $\frac1{1-x}\frac1{(1-x)^n}=\frac1{(1-x)^{n+1}}$, one finds that
$\sum\_{k\geq0}x^k\sum\_{j=0}^k\binom{n+j-1}j=\sum\_{k\geq0}x^k\binom{n+k}k$. In particular, one gathers that
$$\sum\_{j=0}^{2k+1}\binom{n+j-1}j=\binom{n+2k+1}{2... | 2 | https://mathoverflow.net/users/66131 | 401243 | 164,719 |
https://mathoverflow.net/questions/401181 | 10 | In the $\infty$-world, [connective spectra play the role of abelian groups](https://ncatlab.org/nlab/show/abelian+infinity-group), while [$\mathbb{E}\_\infty$-spaces play that of commutative monoids](https://ncatlab.org/nlab/show/E-infinity+space). This may be rephrased by saying that we may identify the $\infty$-categ... | https://mathoverflow.net/users/130058 | Tensor products of $\mathbb{E}_\infty$-spaces | The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E\_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful point of view that does not appear in loc. cit. is that this tensor product comes from the Lawvere theory of commutative m... | 10 | https://mathoverflow.net/users/20233 | 401245 | 164,720 |
https://mathoverflow.net/questions/401244 | 10 | $\newcommand{\K}{\mathrm{K}}$The abelian group completion functor $\K\_0\colon\mathsf{CMon}\to\mathsf{Ab}$ satisfies
$$
\K\_0(A)
\cong
\mathbb{Z}\otimes\_{\mathbb{N}}A,
$$
naturally in $A\in\mathrm{Obj}(\mathsf{CMon})$, where
* $\mathbb{Z}$ is the additive monoid of integers (i.e. $\K\_0(\mathbb{N})$, the group compl... | https://mathoverflow.net/users/130058 | Group completion of $\mathbb{E}_{\infty}$-monoids via tensor products | Yes, for the same reason. Let me sketch a proof.
1- $QS^0\otimes X$ is group-complete. Indeed, its $\pi\_0$ is $\mathbb Z\otimes \pi\_0(X)$, and that's a group for the usual reasons. Another way to prove it is to prove that the shear map for $X\otimes Y$ is (the shear map of $X)\otimes Y$, which can be seen by noting... | 9 | https://mathoverflow.net/users/102343 | 401247 | 164,722 |
https://mathoverflow.net/questions/397778 | 16 | I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction?
$$\begin{equation\*} \zeta (3)=\dfrac{6}{5+\overset{\infty }{\underset{n=1}{\mathbb{K}}}\dfrac{-n^{6}}{34n^{3}+51n^{2}+27n+5}}\end{equation\*}$$
Fu... | https://mathoverflow.net/users/174578 | Extending Apéry's proof to Catalan's constant? | **Summary**:
* The continued fraction, the recurrence and the explicit form of the sequence are interchangeable and for the Apéry numbers, we don't know what come first. This extend to other constructions for other constants.
* The approximation for the Catalan's constant $G$ fails because it doesn't converge too fas... | 16 | https://mathoverflow.net/users/302667 | 401250 | 164,723 |
https://mathoverflow.net/questions/401233 | 2 | I'm looking for a simple example of the following (so that I can get better intuition for it). If possible, I'd like a 2 dimensional rational example.
$X$ is an irreducible projective variety, with an open subset $U$ that is (isomorphic to) the total space of a line bundle over an irreducible variety - in other words... | https://mathoverflow.net/users/3077 | Example sought: disconnected closures of fibers of line bundles | Take the blowup $X = \text{Bl}\_p(\mathbb{P}^1\times \mathbb{P}^1)$ with exceptional $E$ and $l,l' \subset X$ the proper transforms of the rulings through $p \in \mathbb{P}^1\times \mathbb{P}^1$. Then $U := X\setminus (E\cup l)$ is the trivial line bundle over $\mathbb{P}^1$ but $Z$ will be the disconnected set $l\cup ... | 4 | https://mathoverflow.net/users/76148 | 401253 | 164,724 |
https://mathoverflow.net/questions/401252 | 10 | I paraphrase part of the wikipedia article on the Weyl character formula: [Weyl character formula](https://en.wikipedia.org/wiki/Weyl_character_formula).
If $\pi$ is an irreducible finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$ is a choice of Cartan subalgebra ... | https://mathoverflow.net/users/81645 | Can the numerator in Weyl's character formula be written as a determinant? | The classical definition of the Schur polynomials, which considerably predates the Weyl character formula, is as a ratio of two determinants (a so-called "bialternant"): see, e.g., [https://en.wikipedia.org/wiki/Schur\_polynomial#Definition\_(Jacobi's\_bialternant\_formula)](https://en.wikipedia.org/wiki/Schur_polynomi... | 14 | https://mathoverflow.net/users/25028 | 401254 | 164,725 |
https://mathoverflow.net/questions/401268 | 2 | **Problem:**
>
> let $P\_1(x), P\_2(x), Q\_1(y), Q\_2(y)$ be some polynomials of degree $d$ in $\mathbb{F}\_p$. Let
> \begin{equation}
> A := \{ (x, y) \in \mathbb{F}\_p^2 : P\_1(x) = Q\_1(y) \},\\
> B := \{ (x, y) \in \mathbb{F}\_p^2 : P\_2(x) = Q\_2(y) \}.
> \end{equation}
> Let us also assume that equations $P\_... | https://mathoverflow.net/users/334675 | Common roots to "independent" equations $P_1(x) = Q_1(y)$ and $P_2(x) = Q_2(y)$ in $\mathbb{F}_p \times \mathbb{F}_p$ | The bound $d^2$ is, indeed, correct. It follows from Bézout's theorem, see [here](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem#Plane_curves). In your case, affine plane curves are exactly $A$ and $B$, the condition on common divisor is the lack of common component. You can take $F=\mathbb F\_p$ and $E=\overlin... | 2 | https://mathoverflow.net/users/101078 | 401270 | 164,730 |
https://mathoverflow.net/questions/398544 | 26 | Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
>
> **Question 1.** Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ and $\left(1+b\right)\left(1+a\right)$ are invertible. Does it follow that $1+a$ and $1+b$ are invertible as w... | https://mathoverflow.net/users/2530 | Subtraction-free identities that hold for rings but not for semirings? | Tim Campion's [idea](https://mathoverflow.net/a/401257) works, though his example needs a little fixing. As in Tim's answer, we will find a rig with two elements $X$ and $Y$ such that $X+Y=1$ but $XY \neq YX$.
Let $(M,+,0)$ be any commutative monoid. Let $R$ be the set of endomorphisms of $M$ obeying $\phi(x+y)=\phi(... | 11 | https://mathoverflow.net/users/297 | 401273 | 164,732 |
https://mathoverflow.net/questions/401278 | 1 | If I start with a, say, 3-CW complex $X$ which can be embedded in $\mathbb{R}^5$, I can get a neighbourhood $U$ of $X$ which has the same homotopy type of $X$. Then $U$ is a $5-$ dimensional open manifold. Can I get a close manifold (compact without a boundary) $M$, of dimension $6$ (or some higher dimension) such that... | https://mathoverflow.net/users/333818 | How can I construct a closed manifold from a finite CW complex? | Take $X=S^3$. Then no closed manifold of dimension at least 6 has the same homotopy type.
| 9 | https://mathoverflow.net/users/334338 | 401280 | 164,736 |
https://mathoverflow.net/questions/401283 | 1 | We consider the sequence $n\longmapsto {n\choose k}+1$
for $k\geq 1$ a fixed integer. For $k\geq 3$ odd,
this sequence seems to contain surprisingly few prime numbers
while there are many primes (perhaps roughly the expected
amount) among the first terms of this sequence for even $k\geq 2$.
Is there an explanation fo... | https://mathoverflow.net/users/4556 | There seem to be only few primes of the form ${n\choose k}+1$ if $k\geq 3$ is odd | When $k$ is odd, writing $f\_k(n)={n\choose k}+1$ you have $f\_k(-1)=0$ as a polynomial evaluation, and removing the denominator gives you the extra $k!$ in integers.
I.e., $5!f\_5(n)=n(n-1)(n-2)(n-3)(n-4)+120$, so $$5!f\_5(-1)=(-1)(-2)(-3)(-4)(-5)+120=-120+120=0.$$ As $-1$ is a root of $f\_k(n)$, then $n+1$ divides ... | 7 | https://mathoverflow.net/users/334725 | 401284 | 164,737 |
https://mathoverflow.net/questions/401282 | 5 | Let $Q=(Q\_0,Q\_1)$ be the following quiver, $Q\_0$ consist of 2 vertices, denoted by 1,2. $Q\_1$ consist a loop at 1 called $\gamma$, an arrow $\alpha$ from 1 to 2 and an arrow $\beta$ from 2 to 1. The relation $\rho$ is $\{\beta\alpha, \beta\gamma, \gamma\alpha, \gamma^m\}$ for some integer $m\geq 2$. Given a field $... | https://mathoverflow.net/users/134942 | Is this quiver with relations of finite representation type | An easy way to see that the algebra is of infinite representation type (for any $m \geq 2$) is to observe that it is a [string algebra](http://www.math.uni-bonn.de/people/schroer/fd-atlas-files/FD-BiserialAlgebras.pdf), and that you have strings of arbitrary length, each corresponding to an indecomposable module.
For... | 8 | https://mathoverflow.net/users/18756 | 401294 | 164,741 |
https://mathoverflow.net/questions/401289 | 29 | I came across this problem while doing some simplifications.
So, I like to ask
>
> **QUESTION.** Is there a closed formula for the evaluation of this series?
> $$\sum\_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{a^2b^2}$$
> where the sum runs over all pairs of positive integers that are relatively prime.
>
>
>
... | https://mathoverflow.net/users/66131 | Closed formula for a certain infinite series | Apply Möbius summation, the formula for $\sum\_{n>=1}\cos(2\pi n x)/n^2$ to obtain:
$$11/4-45\zeta(3)/\pi^3=1.00543...\;$$
| 49 | https://mathoverflow.net/users/81776 | 401304 | 164,743 |
https://mathoverflow.net/questions/401302 | 3 | Let $R$ be a dvr and $U\to \text{Spec}(R)$ an affine smooth $R$-scheme with non-empty special fiber $U\_0$.
Let $Z\subset U$ be a closed subset. Assume the intersection of $Z$ with $U\_0$ is empty.
>
> Is $Z$ empty?
>
>
>
If $U\to \text{Spec}(R)$ was proper then the answer would be yes because the image of $... | https://mathoverflow.net/users/nan | Non-empty closed subsets with empty special fiber | $R=\mathbb{Z}\_p, U=\mathrm{Spec}\:\mathbb{Z}\_p\times \mathbb{Q}\_p, Z=\mathrm{Spec}\:\mathbb{Q}\_p\neq \emptyset$
| 3 | https://mathoverflow.net/users/334839 | 401305 | 164,744 |
https://mathoverflow.net/questions/401241 | 3 | Let $G=(V,E)$ be a simple, undirected graph, finite or infinite, with $V \neq \emptyset$. Is the following statement true?
>
> There is a cardinal $\kappa \leq |V|$ and an injective map $\psi : V \to {\cal P}(V)$ such that for $v\neq w\in V$ we have: $$\{v,w\} \in E \; \text{ if and only if }\; \big|\big(\psi(v) \s... | https://mathoverflow.net/users/8628 | Representing graphs by sets of small symmetric difference | Statement fails for $G = K\_{2, 3}$. Proof is either with computer search, or by case analysis (an attempt follows).
Let the parts of $G$ be $v\_0, v\_1$ and $u\_0, u\_1, u\_2$ respectively. Consider $\Delta = |\psi(u\_0) \triangle \psi(u\_1)| + |\psi(u\_0) \triangle \psi(u\_2)| + |\psi(u\_1) \triangle \psi(u\_2)|$. ... | 2 | https://mathoverflow.net/users/106512 | 401317 | 164,750 |
https://mathoverflow.net/questions/401291 | 3 | Given a $C^\*$-algebra $A$, we write $\Omega(A)$ for its space of characters, i.e. its non-zero algebra homomorphisms $A \to \mathbb{C}$. If $X$ is a compact Hausdorff space, it is well-known that
$$X \to \Omega(C(X)): x \mapsto \operatorname{ev}\_x$$
is a homeomorphism of topological spaces.
---
Let $X,Y$ be com... | https://mathoverflow.net/users/216007 | A $*$-homomorphism $C(X) \to C(Y)$ gives a continuous map $Y \to X$ | Given a maximal ideal $\mathfrak{m} \subseteq C(X)$, the corresponding point $x \in X$ or rather its singleton $\{x\}$ is the intersection of all zero sets $Z(f)$ of all functions $f \in \mathfrak{m}$.
So the map $\varphi : Y \to X$ associated to $\pi : C(X) \to C(Y)$ is defined by
$$\{\varphi(y)\} = \bigcap\_{f \in ... | 5 | https://mathoverflow.net/users/2841 | 401319 | 164,751 |
https://mathoverflow.net/questions/401321 | 7 | Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are **parametrically equivalent** iff every primitive relation/function in one is definable (with parameters) in the other. For example, every group $\mathfrak{G}=(G;\*,{}^{-1})$ is parametrically equivalent to its "torsor reduct" $\mathfrak{T... | https://mathoverflow.net/users/8133 | Is $\mathbb{Q}$ "equivalent" to a structure with transitive automorphism group action? | Consider $(\mathbb{Q},R)$, where $R$ is the 6-ary relation defined by $$R(a,b,c,d,e,f) \iff (a-b)(c-d)=(e-f)$$
The automorphism group of this structure includes at least the translations $x\to x+h$. Since those translations include $x \to x+(b-a)$, there is always a translation taking $a$ to $b$, so the group of auto... | 6 | https://mathoverflow.net/users/nan | 401326 | 164,753 |
https://mathoverflow.net/questions/401348 | 3 | Let $X$ be a subset of $\mathbb R^2$ consisting of $n$ distinct points. Let $d\_1(X)$ be the number of pairs of points of $X$ on distance $1$ from each other. Define
$$d\_1(n)=\sup\_{X\subset \mathbb R^2|, |X|=n}d\_1(X).$$
In particular $d\_1(1)=0$, $d\_1(2)=1$, $d\_1(3)=3$, $d\_1(4)=5$, etc.
**Question.** I wond... | https://mathoverflow.net/users/13441 | Planar subsets with many pairs of points on distance $1$ | This is the so-called "Erdős unit distances problem"; see for instance the [related Wikipedia entry](https://en.wikipedia.org/wiki/Unit_distance_graph#Number_of_edges) or this [recent survey by Szemerédi](https://doi.org/10.1007/978-3-319-32162-2_15).
As you might expect, a good deal is known, but the problem is by n... | 9 | https://mathoverflow.net/users/25028 | 401351 | 164,762 |
https://mathoverflow.net/questions/401342 | 2 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(X\_n:\Omega\rightarrow \mathbb{R}^m)\_n$ be a sequence of i.i.d. random variables and let $L:\mathbb{R}^m\rightarrow [0,\infty)$ be Lipschitz. Let $\mu\_n:=\frac1{n} \sum\_{k=1}^n \delta\_{X\_k}$. Are there conditions under which:
$$
\mathbb{P}\left(|\math... | https://mathoverflow.net/users/298030 | Concentration Inequality for Bounding Lipschitz Empirical Lass | Your inequality is trivial and useless as written. On its left-hand side we have a probability which is $\le1$ and goes to $0$ as $t\to\infty$, whereas on the right-hand side we have an expression which is $\ge1$ and goes to $\infty$ as $t\to\infty$, because for any good rate function $I$ on $[0,\infty)$ we have $I(t)\... | 6 | https://mathoverflow.net/users/36721 | 401355 | 164,764 |
https://mathoverflow.net/questions/401345 | 6 | Let $\mu$ be a probability measure with finite support on integers or the real line with the property that $\mu( 0) \le p$ for a fixed $0<p <1$. Let $S\_n$ denote the random walk starting at $0$, where each step has distribution $\mu$. Denote by $p\_n$ the probability that $S\_n=0$, so $p=p\_1$.
Is there a function $... | https://mathoverflow.net/users/3635 | Uniform upper bounds for the return probability of random walks on $ \mathbb{R}$ | The actual (negative) answer was given in fedja's comment.
Since fedja said he would wait for somebody to come up with a reference for the relaxed statement, here we go:
Let $X$ be a random variable (r.v.) with distribution $\mu$. By (say) inequality (2.5) of Chapter III in [Petrov's book](https://link.springer.com... | 3 | https://mathoverflow.net/users/36721 | 401360 | 164,765 |
https://mathoverflow.net/questions/401364 | 6 | The [1991 paper](https://www.sciencedirect.com/science/article/pii/0022404991900306#!) of Lewis, “*Is there a convenient category of spectra?*” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:
1. There is a symmetric monoidal smash product $\wedge$;
2. We have an a... | https://mathoverflow.net/users/130058 | Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces | For $E\_\infty$ spaces, homotopy-theoretically there is a functor $L: \mathcal{S} \to E\_\infty \mathcal{S}$ with a right adjoint $R$. The only property on this list that really needs replacing on this list is property (5): the unit
$$
X \to RL(X)
$$
should be homotopy equivalent to the natural inclusion
$$
X \to Free\... | 9 | https://mathoverflow.net/users/360 | 401367 | 164,768 |
https://mathoverflow.net/questions/398427 | 8 | This is a follow-up question to this [MO question](https://mathoverflow.net/questions/374180/function-of-x-1-x-2-x-3-x-4-that-factors-in-two-ways-as-phi-1-x-1-x-2/374197#374197), which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x\_1, \dots, x\_n)$ be a functio... | https://mathoverflow.net/users/2233 | Functions over monoids which factor in two different ways | Take $S$ to be the monoid on the set $\{0,2,3,4,5,6\}$ with operation $x\oplus y=\min(x+y,6).$
Let $g:\{0,1\}^3\to S$ be the function $(x,y,z)\mapsto \min(x+y+z+4,6).$ Using any surjective $h:S\to \{0,1\}$ this can be converted to $f(x,y,z)=g(h(x),h(y),h(z)).$ I'll just work with $g.$
$g$ factors with respect to $1... | 4 | https://mathoverflow.net/users/164965 | 401397 | 164,775 |
https://mathoverflow.net/questions/401385 | 1 | Is there a $P$ time definable sequence of **succinct polynomial sized representation of balanced bipartite graphs** whose number of perfect matchings is a primorial?
For factorial a complete bipartite graph suffices.
Motivations:
1. The speed of growth of the function defining the sequence might capture prime gap... | https://mathoverflow.net/users/10035 | Succinct polynomial sized representation of balanced bipartite graphs whose perfect matching count is a primorial | Consider the graph $G\_k$ with vertex set $$\{u\_1, \ldots, u\_k, v\_1, \ldots, v\_k\}$$ and edges $$\{(u\_1, v\_1), \ldots, (u\_1, v\_k)\} \cup \{(u\_2, v\_1), \ldots, (u\_k, v\_{k-1})\} \cup \{(u\_2, v\_k), \ldots, (u\_k, v\_k)\}$$ It has $k$ perfect matchings, because once $u\_1$ is assigned to $v\_i$ this forces th... | 4 | https://mathoverflow.net/users/46140 | 401399 | 164,776 |
https://mathoverflow.net/questions/401398 | 2 | At 1st we consider some weak statement of Chevalley–Warning theorem for any finite field: If $f$ is a homogeneous polynomial of degree $d$ with $n$ independent variables over a finite field $F$. Then if $ n > d $ then there is a non trivial solution of this homogeneous polynomial in $ F^{n} / \{0,0,...,0\} $.
Now if ... | https://mathoverflow.net/users/215016 | Chevalley–Warning theorem for rational field $\mathbb{Q} $ | The condition that $d$ is odd just implies that there is no real obstruction to the existence of rational points, but there could still be $p$-adic obstructions for some prime $p$.
As an example, let $p$ be a prime and $a$ an integer which is coprime to $p$ for which $a \bmod p$ is not a cube (this necessarily implie... | 4 | https://mathoverflow.net/users/5101 | 401403 | 164,777 |
https://mathoverflow.net/questions/401415 | 0 | Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k\_1> k\_2>0$. Does there necessarily exist a diffeomorphism $\phi^{k,\alpha}\in C(\Omega,\Omega)$ satisfying:
$$
\lVert\phi-1\_{\Omega}\rVert\_{k,\alpha}= k\_1 ... | https://mathoverflow.net/users/176409 | Existence of a Hölder homeomorphism satisfying prescribed norm constraints | As it is the answer is no, by the following counter-example
$$.$$
| 1 | https://mathoverflow.net/users/6101 | 401420 | 164,780 |
https://mathoverflow.net/questions/401417 | 3 | The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$.
The Wiener process is a stochastic process whose definition can be found in any textbook. In any text, the stochastic integrals or stochastic differential... | https://mathoverflow.net/users/56524 | What exactly is the relation between the Wiener process and Wiener measure? | The Wiener measure $w$ is the distribution of the Wiener process/random function $W$ on $C[0,1]$; that is,
$$P(W\in A)=w(A)$$
for all Borel sets $A\subseteq C[0,1]$. Here "Borel sets" can be replaced by "open sets" or "closed sets".
Equivalently,
$$Ef(W)=\int\_{C[0,1]}f\,dw$$
for all (say) nonnegative Borel-measurabl... | 4 | https://mathoverflow.net/users/36721 | 401422 | 164,781 |
https://mathoverflow.net/questions/401414 | 3 | I am trying to upper bound the variance of a centered tree and I would like to get an upper bound which would look like : $$\sum\limits\_{\substack{ (l\_1, ..., l\_d) \neq (k\_1, ..., k\_d), \\ \sum\_{j=1}^d l\_{j} = \sum\_{j=1}^d k\_{j} = k } } \frac{k!}{k\_{1}! ... k\_{d} !} \frac{k!}{l\_{1}!... l\_{d} !} \left(\frac... | https://mathoverflow.net/users/335858 | Upper bound for the crossed-terms of a sum of multinomial coefficients | I think that you cannot hope for $\epsilon(k)$ going to zero. Using the identity
$$(x\_1+\ldots+x\_d)^k=\sum\_{0\leq k\_1,k\_2,\ldots,k\_d\leq k\_1+\ldots+k\_d=k}\frac{k!x\_1^{k\_1}\cdots x\_d^{k\_d}}{k\_1!\cdots k\_d!}$$
we get the lower bound
$$2^{-d}d^{2k}-\sum\_{0\leq k\_1,\ldots,k\_d\leq k\_1+\ldots+k\_d=d}\left(\... | 1 | https://mathoverflow.net/users/4556 | 401430 | 164,783 |
https://mathoverflow.net/questions/401234 | 3 | I want to prove the following: (Here, $W^{2,2}$ is a Sobolev space as defined in [Evans](https://bookstore.ams.org/gsm-19-r), chapter 5; $S$ is a Schwartz space; and if $A$ is a distribution and $a$ a function, then $\langle A, a\rangle$ means $A(a)$).
>
> **Theorem.** Let $\newcommand{\C}{\mathbb C}\newcommand{\R}... | https://mathoverflow.net/users/129831 | How to rigorously differentiate the convolution of a distribution and a $L^2$ function? | The last step is formally justified by **15.8, differentiation property** of José Sebastião e Silva's "Integrals and orders of growth of distributions." (The paper is currently available [here](http://jss100.campus.ciencias.ulisboa.pt/Publicacoes/Artigos-de-Investigacao/Inv-JSS/Integrals%20and%20orders%20of%20growth%20... | 1 | https://mathoverflow.net/users/129831 | 401437 | 164,787 |
https://mathoverflow.net/questions/401439 | 0 | Let $x=(x\_1,\ldots,x\_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
>
> **Question.**
> What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x\_1$ ?
>
>
>
| https://mathoverflow.net/users/78539 | Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$ | Let $g \sim N(0, (1/d)I\_d)$ be independent of $x$. Then $g\_1 \overset{\mathcal L}{=} \|g\| x\_1$, so $(x\_1, \|g\|x\_1)$ is a coupling between the marginal distribution of $x\_1$ and $N(0, 1/d)$.
The norm $\|g\|$ is sharply concentrated around $1$, with fluctuations of order $1/\sqrt{d}$, so the Wasserstein distanc... | 4 | https://mathoverflow.net/users/37014 | 401444 | 164,789 |
https://mathoverflow.net/questions/401441 | 30 | ### Summary
Someone claims $\mathbb{R}$ can be constructed as the following intriguing quotient, which is related to Gromov's bounded cohomology. I want to find out if it is true.
$$\frac{\bigl\{f:\mathbb{Z} \to \mathbb{Z} \mathrel| \mbox{ the set } \{f(m+n)-f(m)-f(n) \mathrel| m, n \in \mathbb{Z}\} \mbox{ is bound... | https://mathoverflow.net/users/124549 | A natural construction of real numbers? | So here is my attempt to reconstruct the construction...
Suppose $f\colon\mathbb Z\to\mathbb Z$ satisfies $|f(m+n)-f(m)-f(n)|\le M$ as $m,n$ run over $\mathbb Z$. Then setting $m=n=2^k$, we see
$|f(2^{k+1})-2f(2^k)|\le M$, from which it follows that $f(2^k)/2^k$ is a Cauchy sequence, and so converges to some $\alpha\... | 27 | https://mathoverflow.net/users/11054 | 401445 | 164,790 |
https://mathoverflow.net/questions/401432 | 2 | This is a reference request/nomenclature question. Let $A \subseteq \mathbb{P}^n$ be a finite set of points not contained in a hyperplane (over some field), and let $\sigma\_r(A)$ be the $r$-th secant variety to $A$. This secant variety forms a *subspace arrangement*, i.e., a finite union of linear subspaces of $\mathb... | https://mathoverflow.net/users/150898 | Secant variety to a zero-dimensional projective variety | I believe this would be a dual arrangement of a star arrangement.
A star arrangement is a union of subspaces defined as follows. Let $H\_1,\dotsc,H\_d$ be a collection of hyperplanes and fix an integer $c$. The codimension $c$ star arrangement $X\_c$ is the union of intersections of $c$ of the $H\_i$, over all size $... | 2 | https://mathoverflow.net/users/88133 | 401462 | 164,796 |
https://mathoverflow.net/questions/401460 | 8 | Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post [When do chain complexes decompose as a direct sum?](https://mathoverflow.net/questions/32854/when-do-chain-complexes-decompose-as-a-direct-sum)) that any chain complex over $R$ is equivalent to its cohomology as an o... | https://mathoverflow.net/users/59235 | Chain complexes split in the derived category over rings of global dimension 1 | One reference is H. Krause, "Derived categories, resolutions, and Brown representability", Contemporary Math. vol.436, AMS, 2007, p.101-139 or <https://arxiv.org/abs/math/0511047> , Section 1.6.
Another possible reference is L. Positselski, O.M. Schnürer, "Unbounded derived categories of small and big modules: Is the... | 8 | https://mathoverflow.net/users/2106 | 401464 | 164,797 |
https://mathoverflow.net/questions/401459 | 4 | Given a positive integer $d$, does there exist an integer $n$ that depends only on $d$ (or perhaps also on the dimension of $X$), such that for any degree $d$ finite étale covering $\pi: \widetilde X \to X$ of projective varieties and very ample line bundle $\mathcal L$ on $X$, $\pi^\ast(\mathcal L)^{\otimes n}$ is ver... | https://mathoverflow.net/users/129738 | Pullback of very ample line bundles under finite étale covering | The answer is no. Take for $X$ a (smooth) plane curve of degree $2p+3$. There exists a line bundle $M$ on $X$ with $M^{2}=K\_X$ and $h^0(M)=0$. Then $\eta :=M(-p)$ is a line bundle of order 2 in $JX$, giving rise to a double étale covering $\pi :\tilde{X}\rightarrow X $. Put $\mathscr{L}=\mathscr{O}\_X(1)$. Then $$H^0(... | 8 | https://mathoverflow.net/users/40297 | 401465 | 164,798 |
https://mathoverflow.net/questions/401476 | 4 | Say that an algebra $\mathfrak{A}$ (in the sense of universal algebra) is **point-transitive** iff for every $a,b\in\mathfrak{A}$ there is a $\pi\in Aut(\mathfrak{A})$ with $\pi(a)=b$. While genuinely point-transitive algebras are somewhat rare, many naturally-occurring algebras yield *the same clone as* a point-transi... | https://mathoverflow.net/users/8133 | Do almost-point-transitive algebras generate almost-point-transitive varieties? | Let me distinguish between **clone** and **polynomial clone**.
The former is the smallest composition-closed
collection of operations on $A$
containing the primitive operations of $\mathbb A$ and the projections,
while the latter is the smallest composition-closed
collection of operations on $A$
containing the primitiv... | 4 | https://mathoverflow.net/users/75735 | 401506 | 164,805 |
https://mathoverflow.net/questions/401497 | 0 | Let $(x\_n)\_{n\in\mathbb{N}}$ be a non-increasing sequence in [0,1], (i.e. $x\_n\ge x\_{n+1},n\in\mathbb{N} $), such that $x\_n\ge\frac{1}{n},n\in\mathbb{N} $.
If we fix $k\in\mathbb{N}$ is there necessarily a lower bound c>0 for the fractions $\frac{x\_{kn}}{x\_n}$ for all $n\in\mathbb{N}$?
I was thinking that if... | https://mathoverflow.net/users/336624 | Lower bound for $\frac{x_{kn}}{x_n}$, where $(x_n)_{n\in\mathbb{N}}$ is a non-increasing sequence in [0,1] with $x_n\ge\frac{1}{n}$ | Such a lower bound does not exist in general.
E.g., for natural $j$ and natural $n\in((j-1)!,j!]$, let
$x\_n:=1/(j-1)!$, with $x\_1:=1$, so that $x\_n\ge1/n$ for all natural $n$. Also, for any fixed natural $k\ge2$ and all natural $j\ge k$, we have $kj!\in(j!,(j+1)!]$ and hence
$$\frac{x\_{kj!}}{x\_{j!}}=\frac{(j-1)!... | 1 | https://mathoverflow.net/users/36721 | 401510 | 164,808 |
https://mathoverflow.net/questions/401483 | 3 | In addition to classic two-valued logic, there are *many* many-valued logics, including Łukasiewicz's and Kleene's three-valued logics, Gödel's many-valued logic $G\_k$, and infinite-valued fuzzy logic and probability logic.
I wonder: Was ever the case that some kind of "0-valued" and "1-valued" logics came up in a n... | https://mathoverflow.net/users/244671 | 0-valued and 1-valued logics? | There is an article by C. L. Hamblin titled *One-valued Logic,* The Philosophical Quarterly, 66 (1967), 38-45, that you may find interesting.
| 2 | https://mathoverflow.net/users/8027 | 401515 | 164,811 |
https://mathoverflow.net/questions/401498 | 1 | I have the following problem: I'm given a linear bounded operator $P\in \mathcal{L}(L^2([a,b]))$, $a,b\in \mathbb{R}$ and I want to find a sequence of approximating linear bounded operators $(P\_n)\_{n\geq 1}$ satisfying the following conditions:
1. $P\_n \to P$ in $\mathcal{L}(L^2([a,b]))$ as $n\to \infty$ (i.e. in ... | https://mathoverflow.net/users/248499 | Approximating linear bounded operator on $L^2([a,b])$ | Here's a more explicitly worked out version of my comment above. Consider $P(\sin nx)=\sin 2^n x$. This is an isometry, so in particular bounded in $L^2(0,\pi)$. It can not be approximated in the desired way.
Indeed, suppose we had an operator $Q$ with $\|P-Q\|<\epsilon$ that is also bounded on $H^1$. Since $\|\sin n... | 2 | https://mathoverflow.net/users/48839 | 401528 | 164,816 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.