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https://mathoverflow.net/questions/297186 | 17 | The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of functional analysis
| https://mathoverflow.net/users/86432 | Research topics in distribution theory | While I do not know much about current development of the general theory of distributions, I can say something about the current research topics in a special class of distributions, **the theory of Sobolev spaces**.
Theory of Sobolev spaces was one of the greatest discoveries in
the XXth century mathematics. This the... | 18 | https://mathoverflow.net/users/121665 | 297210 | 130,558 |
https://mathoverflow.net/questions/297015 | 24 | The Ford-Fulkerson algorithm
============================
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex set $V$ and arc set $A$. Fix two distinct vertices $s$ and $t$ of $D$, which we call the... | https://mathoverflow.net/users/2530 | Is the Ford-Fulkerson algorithm a tropical rational function? | Here is an extremely inefficient implementation of Edmonds-Karp (breadth-first search) given as a tropical rational map. (Hopefully I've understood the definition of tropical rational map correctly.)
First, observe that in the usual implementation of Edmonds-Karp, no path is used more than once, since if $p$ is a sh... | 4 | https://mathoverflow.net/users/396 | 297211 | 130,559 |
https://mathoverflow.net/questions/297177 | 8 | Is there any result known about the following generalization of the Erdős-Ko-Rado theorem?
Let $n, k, r, s$ be positive integers. We call a family $\mathcal{F}$ of $k$-element subsets of $\{1,\ldots, n\}$, an *$(r, s)$-intersection family* if among every $r$ elements of $\mathcal{F}$ at least two have an $s$-element ... | https://mathoverflow.net/users/45532 | A generalization of Erdős-Ko-Rado theorem | The case $s=1$ is Erdős hypergraph mathcing conjecture from
Paul Erdős (1965). A problem on independent $r$-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 8 (1965), 93–95.
[users.renyi.hu/~p\_erdos/1965-01.pdf](https://users.renyi.hu/~p_erdos/1965-01.pdf)
A recent paper about it is
Peter Frankl (2017) Proof... | 6 | https://mathoverflow.net/users/12674 | 297214 | 130,560 |
https://mathoverflow.net/questions/297213 | 0 | Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M\_k$ is a sequence of $n \times n$ matrix (**not necessarily symmetric**) such that $M\_k \to O$ where $O$ is the zero matrix. Let $\lambda\_i(H),i=1,...,n$ denote the operator that gets the $i$th largest eigenvalue of $H$ in absolute value. My questio... | https://mathoverflow.net/users/98958 | Limit of eigenvalues of a matrix perturbation sequence | As stated, $H+M\_k$ need not have real eigenvalues; what is the largest then.
If $f\_A(\lambda)$ is the characteristic polynomial of a matrix $A$, then a simple eigenvalue depends locally real analytically on $A$, by the implicit function theorem.
If all eigenvalues are real and you order them by size, then each ei... | 2 | https://mathoverflow.net/users/26935 | 297218 | 130,562 |
https://mathoverflow.net/questions/296903 | 1 | *This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.*
**The problem is**: how to define the bundle $C(M)$ whose sections are all the symmetric affine connections on the manifold $M$ in *terms of natural bundles over... | https://mathoverflow.net/users/22606 | The bundle of symmetric affine connections as quotient of the second-order frame bundle | There *is* an 'identification', i.e., a way to interpret a torsion-free affine connection on $M$ as a section of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})$ in such a way that every (smooth) section of $\check J^2(M,\mathbb{R}^n\_0)/\mathrm{GL}\_n(\mathbb{R})$ over $M$ corresponds to a unique torsion-fre... | 6 | https://mathoverflow.net/users/13972 | 297234 | 130,565 |
https://mathoverflow.net/questions/297236 | 3 | Let $\Box\_{i\in I} X\_i$ denote the [box product](https://en.wikipedia.org/wiki/Box_topology) of the spaces $X\_i$. The box product $\Box\_{n\in\omega}\mathbb{R}$ is not connected, since the collection of bounded sequences is both open and closed.
Is $\Box\_{n\in\omega}[0,1]$ connected?
| https://mathoverflow.net/users/8628 | Is $\Box_{n\in\omega}[0,1]$ connected? | It's not connected.
Let $u=(u\_n)$ be a sequence. For $\ell\in [0,1]$, define $$V\_{u,\ell}=\{(v\_n):\forall n\in \omega:|v\_n-\ell|<\max(2^{-n},2|u\_n-\ell|)\}.$$
Then $u\in V\_{u,\ell}$ and $V\_{u,\ell}$ is open.
Moreover, if $\ell$ is a limit point of $u$ and $v\in V\_{u,\ell}$, then $\ell$ is a limit point of ... | 7 | https://mathoverflow.net/users/14094 | 297237 | 130,566 |
https://mathoverflow.net/questions/297212 | 2 | Let $(x\_{n})\_{n}$ be a sequence in $B\_{L\_{1}}$ that has no weakly Cauchy subsequences. Here $B\_{L\_{1}}$ is the closed unit ball of $L\_{1}$. By Rosenthal's $l\_{1}$-theorem, the sequence $(x\_{n})\_{n}$ has a subsequence that is equivalent to the unit vector basis of $l\_{1}$. But I want a stronger result.
Ques... | https://mathoverflow.net/users/41619 | Bounded sequences in $L_{1}$ that has no weakly Cauchy subsequences | The answer is "No", and can be shown as follows. Let $\{e\_i\}$ be a sequence which is $1$-equivalent to the unit vector basis of $\ell\_1$ and is contained in $L\_1[0,1/2]$ (that is, functions are equal to $0$ on $[1/2,1]$). Let $\{f\_i\}$ be a sequence which is $1$-equivalent to the unit vector basis of $\ell\_2$ and... | 2 | https://mathoverflow.net/users/37822 | 297244 | 130,569 |
https://mathoverflow.net/questions/296192 | 2 | Let $L/K$ be a finite separable field extension and let $\theta$ be a primitive element for $L/K$ with minimal polynomial $\mu(t) \equiv \mu\_{\theta/K}(t) = \sum\_{k=0}^n c\_k t^k$. I am trying to compute the powers $\theta^n,\dots,\theta^{2n-2}$ in terms of $1,\theta,\dots,\theta^{n-1}$. In other words, I am trying t... | https://mathoverflow.net/users/1849 | Formulas for the structure constants of a field extension basis given by a primitive element | I am not entirely sure on the policy of answering one's own question, I am posting this for the sake of completeness and closure for anyone who might be interested in the same question.
Following Ofir Gorodetsky's observation above, let is denote by
$$
h\_k := h\_k(X\_1,\dots,X\_n) := \sum\_{1\leq i\_1\leq i\_2\leq\d... | 1 | https://mathoverflow.net/users/1849 | 297249 | 130,571 |
https://mathoverflow.net/questions/297230 | 3 | Let $X$ be a topological space. $Y$ is a discrete subset of $X$ if it has a discrete topology induced by the topology of $X$. This is equivalent to the fact that for every $y\in Y$ there is an open $U\subset X$ such that $U\cap Y=\{y\}$. Consider a stronger condition: there is a collection $\{U\_y\}\_{y\in Y}$ of mutua... | https://mathoverflow.net/users/53155 | Existence of a discrete subset | The answer to this question is negative and can be obtained with the help of weak P-points in compact spaces with countable cellularity. A non-isolated point $p$ of a topological space $X$ is a *weak $P$-point* if $p$ is not an accumulation point of a countable set $C\subset X\setminus\{p\}$.
If $p$ is a weak P-point... | 5 | https://mathoverflow.net/users/61536 | 297265 | 130,577 |
https://mathoverflow.net/questions/297281 | 41 | $A$ and $B$ take turns to pick integers: $A$ picks one integer and then $B$ picks $k > 1$ integers ($k$ being fixed). A player cannot pick a number that his opponent has picked. If $A$ has $5$ integers that form an arithmetic sequence, then $A$ wins. $B$'s goal is to prevent that from happening. Who has a winning strat... | https://mathoverflow.net/users/115637 | A game on integers | I claim that Player A has a winning strategy in your game, and furthermore, it is a winning strategy for her simply to play the smallest available number.
Let me consider the game along with several natural variations.
**Player A wins the original arithmetic progression game.** To be a little more specific about ... | 65 | https://mathoverflow.net/users/1946 | 297282 | 130,581 |
https://mathoverflow.net/questions/296951 | 6 | For split groups over a $p$-adic field, every irreducible smooth (complex) representation is either infinite-dimensional or one-dimensional. Is it true for quasisplit groups that split over an unramified extension, or quasisplit groups in general? Unfortunately I don't have a good feel for examples, since the books I c... | https://mathoverflow.net/users/122801 | Finite dimensional irreducible representations of quasisplit p-adic groups | See Prop. 3.9 of <http://math.stanford.edu/~conrad/JLseminar/Notes/L2.pdf> for an optimal affirmative answer (no quasi-split condition needed: any connected reductive group over any non-archimedean local field, with the minimal necessary isotropicity hypotheses).
| 7 | https://mathoverflow.net/users/81332 | 297283 | 130,582 |
https://mathoverflow.net/questions/297248 | 9 | Let $X$ be a smooth complex projective Fano threefold. Then the class $c\_1(X)$ can be realised as an effective divisor in $X$. It is it true that the class $c\_2(X)$ can be realised as an effective curve?
(Note that $c\_3$ cannot, since the Euler characteristics of the cubic Fano $3$-fold is negative)
| https://mathoverflow.net/users/13441 | Do all Fano threefolds have effective $c_2$? | By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka `[The Chern classes and Kodaira dimension of an algebraic variety](http://www.mast.queensu.ca/~mikeroth/NotesForSeminars/Miyaoka-Chern-classes-and-Kodaira-dimension.pdf)', 1987), we have that for a Fano variety $X$, $c\_2(X)\cdot H\ge 0$ for any ample divisor $H$. By d... | 10 | https://mathoverflow.net/users/122729 | 297285 | 130,584 |
https://mathoverflow.net/questions/297300 | 5 | For separable metric spaces, three fundamental notions of [dimension](https://en.wikipedia.org/wiki/Inductive_dimension)
are equivalent:
$$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$
Where does the [doubling dimension](https://en.wikipedia.org/wiki/Doubling_space)
fit into the picture?
| https://mathoverflow.net/users/12518 | Doubling dimension vs other metric dimensions | In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and this is discussed in the nice paper below (on the first page):
*Le Donne, Enrico; Rajala, Tapio*, [**Assouad dimensio... | 6 | https://mathoverflow.net/users/11142 | 297316 | 130,597 |
https://mathoverflow.net/questions/297323 | 7 | Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in X$ holds.
I ask: How can I show that $\varphi$ stays in $X$? It sounds natural and I guess it is true, but I fail to... | https://mathoverflow.net/users/122969 | Flows in Hilbert spaces | Consider $\phi(t) = (1-e^{-1/t^2}, t)\in \mathbb R^2$. It satisfies your assumptions for $X= 0\times \mathbb R$, but it does not stay in $X$.
| 7 | https://mathoverflow.net/users/26935 | 297324 | 130,599 |
https://mathoverflow.net/questions/297277 | 1 | I have two sparse matrices: $A$ of dimension $m \times k$ and $B$ of dimension $k \times n$.
Is there a way to know how many non-zero entries there are in $C = A B$ without computing $A B$?
I can see that the trivial upper bound is $m n$ but can I get a better upper bound?
| https://mathoverflow.net/users/122949 | Upper bound on the number of non-zero entries of the product of sparse matrices | Rephrasing:
>
> Given matrices $\mathrm A\_1 \in \mathbb R^{m \times k}$ and $\mathrm A\_2 \in \mathbb R^{k \times n}$, is there an upper bound on the number of nonzero entries of the $m \times n$ matrix $\mathrm A\_1 \mathrm A\_2$?
>
>
>
Let Boolean matrices $\mathrm B\_1 \in \{0,1\}^{m \times k}$ and $\mathr... | 1 | https://mathoverflow.net/users/91764 | 297332 | 130,604 |
https://mathoverflow.net/questions/297333 | 2 | Let $(A\_i,\mathcal{B}\_i,\mu\_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu\_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\mu\_i\ll \nu\_i$ for any $i$. Under what conditions can we deduce that $\prod\_i \mu\_i \ll \prod\_i \nu\_i$?
| https://mathoverflow.net/users/80490 | Absolute continuity of infinite product of probability measures | This is described by [Kakutani's theorem on product measures](https://mathscinet.ams.org/mathscinet-getitem?mr=23331). There is a very detailed exposition in Chapter III, Section 9 of Shiryaev's [Probability](https://mathscinet.ams.org/mathscinet-getitem?mr=1368405).
| 6 | https://mathoverflow.net/users/8588 | 297334 | 130,605 |
https://mathoverflow.net/questions/297337 | 3 | Let $X$ be a variety and $E$ be a principal $G$ bundles, where $G$ is a semisimple group. Is there a variety $f: \tilde{X}\rightarrow X$ such that $f^\*E$ is trivial $G$ bundle?
| https://mathoverflow.net/users/nan | Global trivialization of a Principal G bundle | Yes, a principal $G$-bundle is trivial if and only if it has a section, and if you take $\tilde{X} = E$, then $f^\*E = E \times\_X E$ has a section given by the diagonal map $E \to E \times\_X E$.
| 6 | https://mathoverflow.net/users/4428 | 297338 | 130,607 |
https://mathoverflow.net/questions/296751 | 5 | Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to (iz,iw)$.
Suppose that there is a smooth function $f(z):\mathbb D\to \mathbb C$ defined on a unit disk $\mathbb D$ su... | https://mathoverflow.net/users/13441 | An estimate on deviation of two smooth tangent $J$-holomorphic curves | Yes, this is true. In fact, a more precise statement holds: Unless $f$ vanishes identically on $\mathbb{D}$, there is an integer $n$ and a nonzero complex number $a$ such that $f(z) = a\,z^n + f\_{n+1}$, where $f\_{n+1}$ is a smooth function on the disk that vanishes to order $n{+}1$ at $z=0$.
The proof follows immed... | 7 | https://mathoverflow.net/users/13972 | 297341 | 130,608 |
https://mathoverflow.net/questions/297342 | 3 | In [Theorem 2.7 in the following notes](http://www.math.nus.edu.sg/~matsr/ProbII/Lec1.pdf), we seem to assume the following statement.
>
> Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection $\phi:(A,\mathcal F |\_A)\to([0,1],\mathcal B([0,1]))$ such t... | https://mathoverflow.net/users/94022 | Measurably-isomorphic subsets of polish spaces and the continuum hypothesis | Every Borel subset (and in fact every analytic subset) of a Polish space either is countable or has a perfect subset. In particular, an uncountable Borel subset in a Polish space has the cardinality of the continuum. This can be found, for example, in Moschovakis's book "Descriptive Set Theory" as Corollary 2C.3. The r... | 9 | https://mathoverflow.net/users/6794 | 297346 | 130,610 |
https://mathoverflow.net/questions/297350 | 2 | For any equivalence $\sim$ on some collection of objects $C$ consider the problem of trying to determine if two arbitrary objects $x$ and $y$ in $C$ are equivalent i.e. if $x\sim y$ now by definition an invarient is just a function $\phi$ satisfying $x\sim y\implies f(x)=f(y)$ while the function $\phi$ is a complete in... | https://mathoverflow.net/users/38626 | Name for "partially complete" invariants in classification problems? | It seems totally fine. Maybe instead of "complete over $C'$" use "complete for $C'$" or "complete invariant of path graphs".
| 2 | https://mathoverflow.net/users/4600 | 297356 | 130,612 |
https://mathoverflow.net/questions/297311 | 2 | I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits\_{|\alpha|=2m}a\_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ and $a\_\alpha$ belong to $C\_0^{\infty}(\Omega)$ ($\Omega$ is open in $\mathbb{R}^2$). It is important that differential operator... | https://mathoverflow.net/users/94631 | Positive form for a homogeneous elliptic pde | Unfortunately uniform ellipticity (as it is written above) does not imply solvability in general. The reason for that comes from the properties of the associated quadratic form $\Phi\_\Omega(u,u)$. For simplicity, I will consider only the case of $m=1$.
The property of uniform ellipticity is used to establish what i... | 0 | https://mathoverflow.net/users/94631 | 297364 | 130,618 |
https://mathoverflow.net/questions/296013 | 10 | I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their relevance to learning about Galois action.
To describe a concrete problem which mimics one considered by them in topolo... | https://mathoverflow.net/users/nan | Arithmetic representation stability and Galois action | I strongly doubt that one can say anything in particular about the Galois action on $X$ from the fact that representation stability holds for the configuration spaces of points on $X$.
Church's original proof of representation stability for configuration spaces of points on oriented manifolds used very little "manif... | 4 | https://mathoverflow.net/users/1310 | 297365 | 130,619 |
https://mathoverflow.net/questions/297371 | 2 | Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define projection matrix $P\_A:=A (A^T A)^{−1}A^T$. Can we say the following?
$$\frac{a^T D^{-1/2}\left(I - P\_{D^{-1/2}C}\right)D^{-1/2}b... | https://mathoverflow.net/users/74156 | An inequality regarding projection | The answer is no. E.g., let $a=-[1,0,0]^T$, $b=[0,1,0]^T$, $D=I\_3$, and $C=[1,1,t]^T$ for $t>0$. Then $a^Tb=0<<1$ and the orthoprojector matrix onto the column space of $A=D^{-1/2}C$ is $P\_A=CC^T/(2+t^2)$, whereas your ratio is $\frac1{1+t^2}\to1$ as $t\to0$, which is not $<<1$.
| 1 | https://mathoverflow.net/users/36721 | 297377 | 130,621 |
https://mathoverflow.net/questions/297376 | 10 | Using the Well-Ordering Principle, which is equivalent to the [Axiom of Choice](https://en.wikipedia.org/wiki/Axiom_of_choice), it can be [proved](https://dominiczypen.wordpress.com/2018/04/09/any-graph-or-its-complement-is-connected/) that
>
>
> >
> > (S): for every simple, undirected graph $G$, finite or infin... | https://mathoverflow.net/users/8628 | Does the axiom of choice follow from the statement "Every simple undirected graph is either connected, or its complement is connected"? | (S) is a theorem of ZF.
*Proof:* Let $G$ be a graph, and let $v$ be a vertex of $G$. Define
$$P\_v = \{w \,:\, \text{there is a path from } v \text{ to } w\}.$$
If $P\_v$ is the vertex set of $G$, then $G$ is connected. If not, then $\overline{G}$ (the complement of $G$) contains the complete bipartite graph on $P\_v... | 27 | https://mathoverflow.net/users/70618 | 297381 | 130,622 |
https://mathoverflow.net/questions/297370 | 1 | Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure on $X$. Then there is a natural $L^2$-inner product on $H^0(X,L)$. Now fix $s\in H^0(X,L)$ of norm $1$. Consider the fol... | https://mathoverflow.net/users/80490 | Bound of the measure of the support of a set of divisors in a fixed linear system | If I am not mistaken, it seems the set you are asking about is almost always $X$ itself (and the argument does not use anything about complex geometry).
For any $p$ in $X$, the vector subspace
$$A\_p := \{ \sigma \in H^0(X,L) : \sigma(p) = 0 \}$$
has codimension at most 1 in $H^0(X,L)$.
On the other hand, for ... | 2 | https://mathoverflow.net/users/122997 | 297384 | 130,623 |
https://mathoverflow.net/questions/296870 | 7 | I am looking for a graph for which $2 d\_{i} < \mu\_{i}$, for some index $i$, where $\mu\_{1} \leq \mu\_{2} \leq \dots\leq \mu\_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d\_{1} \leq d\_{2} \leq \dots \leq d\_{n}$ are the node degrees.
According to the literature and existing upper/lower bounds on ... | https://mathoverflow.net/users/91029 | Lower bound on the eigenvalues of the Laplacian | This is an expanded solution based on the comment of @mostafa.
Such graphs dose not exist.
Let $L$ be the Laplacian of the graph. Suppose that diagonal elements of $L$ are sorted
sequence of degrees $d\_1\leq \ldots \leq d\_n$.
By [Min-Max theorem](https://en.wikipedia.org/wiki/Min-max_theorem#Min-max_theorem), w... | 4 | https://mathoverflow.net/users/53059 | 297391 | 130,624 |
https://mathoverflow.net/questions/292110 | 3 | In the book *Sobolev Spaces with Application* of Maz'ya, $\mathring {L^k\_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||\_{L\_p(\Omega)}$.
For nice domain (and correct values of $k,p,q,n$), Sobolev's embedding theory tell us that $\mathring {L^k\_p}(\Omega)\hookrig... | https://mathoverflow.net/users/80191 | Typical elements of the space $\mathring {L^k_p}(\Omega)$ | The answer presented here is copied from the paper [2]. Many similar results (sometimes with more complicated proofs) can be found in [1].
Let the space $L^{k,p}$ be defined by:
$$
L^{k,p}(\mathbb{R}^n)=\{ f\in \mathcal{D}'(\mathbb{R}^n):\, \nabla^kf\in L^p(\mathbb{R}^n)\},
\quad
\Vert f\Vert\_{L^{k,p}}=\Vert \nabla^kf... | 3 | https://mathoverflow.net/users/121665 | 297392 | 130,625 |
https://mathoverflow.net/questions/297191 | 4 | Consider a real analytic $H\_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies what happens to (Lagrangian) tori which are invariant under the Hamiltonian flow $\phi\_{H\_{\epsilon}}$, associated to
... | https://mathoverflow.net/users/47228 | symplectic topology of (perturbed) KAM tori | No. (Answer courtesy of Jacques Féjoz, via email correspondence.) The invariant torus might be translated in the actions,in which case the perturbed torus is not Hamiltonian isotopic to the original. Consider the trivial, integrable perturbation $H\_\epsilon(q,p) = H\_0(p+ \epsilon \Delta p\_0)$, where $\Delta p\_0$ is... | 3 | https://mathoverflow.net/users/2906 | 297401 | 130,626 |
https://mathoverflow.net/questions/297368 | 4 | An operator $T:X\rightarrow Y$ is said to be completely continuous if $T$ maps weakly convergent sequences to norm convergent sequences.
Let $Q: l\_{1}\rightarrow l\_{2}$ be any surjection and $J:l\_{1}\rightarrow Y$ be an isomorphic embedding.
Question. Is there a completely continuous operator $S:Y\rightarrow l... | https://mathoverflow.net/users/41619 | A question on completely continuous operators | By Grothendieck's theorem, $Q$ is absolutely summing, in particular $2$-summing. As such, it has a $2$-summing extension to $Y$, by Pietsch's factorisation theorem. But $2$-summing operators are completely continuous (again by Pietsch's theorem).
| 4 | https://mathoverflow.net/users/24953 | 297402 | 130,627 |
https://mathoverflow.net/questions/236392 | 31 | The existence of a 4-chromatic unit distance graph (e.g., [the Moser spindle](https://en.wikipedia.org/wiki/Moser_spindle)) establishes a lower bound of 4 for the chromatic number of the plane (see the [Nelson-Hadwiger problem](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem)).
Obviously, it would be ni... | https://mathoverflow.net/users/39475 | Has there been a computer search for a 5-chromatic unit distance graph? | As of this morning [there is a paper on the ArXiv](https://arxiv.org/pdf/1804.02385) claiming to show that there exists a 5-chromatic unit distance graph with $1567$ vertices. The paper is written by non-mathematician Aubrey De Grey (of anti-aging fame), but it appears to be a serious paper. Time will tell if it holds ... | 44 | https://mathoverflow.net/users/12128 | 297407 | 130,630 |
https://mathoverflow.net/questions/297389 | 5 | This is along the lines of [this question](https://mathoverflow.net/questions/298437/confusion-in-definition-of-gerbes-in-hitchins-notes)
>
> Gerbes are not just topological objects: we can do differential geometry with them too. We shall next describe what a connection on a gerbe is.
>
>
> To begin with, let’s l... | https://mathoverflow.net/users/118688 | Connection on a Principal bundle and transition functions, as in Hitchin's notes | A principal $G$-bundle $P\to M$ can be described by an open cover $(U\_\alpha)$ of $M$ and a cocycle $g\_{\beta\alpha}: U\_\alpha\cap U\_\beta\to G$. The total space is the quotient of the disjoint union of the spaces $U\_\alpha\times G$ via the equivalence relation
$$ U\_\alpha\times g\ni (x, g')\sim (y, g'')\in U\_... | 4 | https://mathoverflow.net/users/20302 | 297409 | 130,631 |
https://mathoverflow.net/questions/297251 | 2 | In his well-known [paper](http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101110.pdf) Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha\_\beta: \beta,\alpha < \omega\_1\}$ of metric indecomposable continua and retractions. ... | https://mathoverflow.net/users/58082 | Example of an $\omega_1$ decreasing chain of dense semicontinua? |
>
> I am **not** assuming any CH.
>
>
>
Let $\ \Omega\ := \{\alpha: \alpha<\omega\_1\},\ $ and $\ i:\Omega\rightarrow [0;1]\ $ be injective and such that
$\ \Gamma\ :=\ i(\Omega)\ $ is condensed in $\ [0;1].\ $ More generally,
let $\ \Gamma\_\alpha\ :=\ i([0;\alpha)).\ $ Also let $\ M:=[0;1]^2.\ $
Then
$$ S\_\... | 1 | https://mathoverflow.net/users/110389 | 297412 | 130,632 |
https://mathoverflow.net/questions/297410 | 6 | $SU(2)$ can be seen as a subgroup of $SO(5)$ through the following chain of subgroups
$$
SU(2) \subset SO(4) \subset SO(5).
$$
If we identify $SU(2)\cong Sp(1)$, does the inclusion $Sp(1) \to SO(5)$ factor through $Spin(5) \cong Sp(2) \to SO(5)$ as the standard embedding of $Sp(1)$ in $Sp(2)$. I understand, that th... | https://mathoverflow.net/users/118622 | Does $SU(2)\cong Sp(1)\subset SO(5)$ factor through $Spin(5)\cong Sp(2)$ as the standard embedding $Sp(1) \to Sp(2)$? | Yes. The rep of $Sp(2)$ on $\mathbb{C}^5$ that induces the covering map is the complement to the line spanned by the symplectic form in $\bigwedge{}^{2}\mathbb{C}^4$. What you are asking is whether there is another line in this space which is invariant under $Sp(1)$, that is, if the action of $Sp(1)$ on the wedge squar... | 5 | https://mathoverflow.net/users/66 | 297413 | 130,633 |
https://mathoverflow.net/questions/297408 | 2 | Let $S^3=\{(z,w)\in {\mathbb{C}}^2:|z|^2+|w|^2=1\}$,
and $T\_{\pi/4}^2:=\{(e^{i\alpha}/\sqrt{2},e^{i\beta}/\sqrt{2}):\alpha,\beta\in \mathbb{R}\}$.
Is there an isometry $\phi:S^3\rightarrow S^3$ whose fixed point set (i.e.,
$\{p\in S^3:\phi(p)=p\}$) is exactly $T\_{\pi/4}^2$? If yes, then what is an explicit express... | https://mathoverflow.net/users/123014 | Clifford torus as the fixed point set of an isometry from $S^3\rightarrow S^3$? | An isometry of the sphere with respect to the restriction of the Euclidean distance metric (turns out to be) the same as an isometry of the sphere with respect to the [great-circle metric](https://en.wikipedia.org/wiki/Great-circle_distance), which is the induced distance of the round Riemannian metric on the sphere (r... | 5 | https://mathoverflow.net/users/40804 | 297414 | 130,634 |
https://mathoverflow.net/questions/297411 | 3 | This question is inspired by ["Number of collinear ways to fill a grid"](https://mathoverflow.net/questions/297385/number-of-collinear-ways-to-fill-a-grid) by Sebastien Palcoux and the comments of user44191 on [this earlier question of Palcoux's](https://mathoverflow.net/questions/297113/on-the-number-of-eulerian-order... | https://mathoverflow.net/users/353 | Counting "connected" edge orderings (shellings) of the complete graph | Write $\langle m\rangle\_i=m(m+1)\cdots (m+i-1)$. There are $\frac 12
n!$ ways to choose the order in which new vertices are attached (since
at the first step we attach two at once). There are $(n-1)!$ ways to
choose the vertex that each new vertex is attached to. Suppose we have
made these choices. Now there are $(n-2... | 6 | https://mathoverflow.net/users/2807 | 297416 | 130,636 |
https://mathoverflow.net/questions/297400 | 3 | The following question may be a bit imprecise in its formulation, I guess however the problem I have in mind is clear. Although to me it looks like a fairly standard question, I couldn't find any reference approaching it so far and hope someone here can help.
Assume that for every $\epsilon>0$, $\lbrace X^{\epsilo... | https://mathoverflow.net/users/85194 | Family of large deviation principles | First, to see why this is not enough, suppose all variable involved take value in some compact interval. Suppose the sequence $X\_n$ satisfies the LDP, with rate function $J(x)$, and let $X\_n^\epsilon= X\_n+1$ if $n>1/\epsilon$ and $X\_n^\epsilon=X\_n$ if $n<1/\epsilon$.
Then $X\_n^\epsilon\to\_{\epsilon \to 0} X\_n$.... | 2 | https://mathoverflow.net/users/35520 | 297418 | 130,637 |
https://mathoverflow.net/questions/297431 | 4 | I would like to know criteria for a C\*-algebra $A$ to have a positive contraction $a$ with full spectrum, ie $\sigma(a) = [0,1]$. I am particularly interested in the simple case. I believe that if a C\*-algebra is simple, unital, and nonelementary (ie not the compacts) then this should be true. Is there a good referen... | https://mathoverflow.net/users/119857 | Full spectrum positive elements of a $C^*$-algebra | I claim that a C\*-algebra $A$ lacks such an element if and only if every self-adjoint element of $A$ has countable spectrum. Such C\*-algebras are called "scattered"; a good reference is Ghasemi and Koszmider, Noncommutative Cantor-Bendixson derivatives and scattered C\*-algebras. Scattered implies that minimal projec... | 8 | https://mathoverflow.net/users/23141 | 297435 | 130,641 |
https://mathoverflow.net/questions/297437 | 6 | I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the following question:
>
> Let $\gamma:\mathbb{S}^1\to\mathbb{R}^n$, $n\geq 3$, be a closed (rectifiable, piece-wise sm... | https://mathoverflow.net/users/1849 | Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it? | **No**, you cannot define a winding number if $n\geq 3$ since, as pointed out in a comment by Anthony Carapetis, any two curves in $\mathbb{R}^n\setminus\{ p\}$ are homotopic, and a winding number should be invariant under homotopies. You can however, define a linking number between two continuous disjoint images of sp... | 13 | https://mathoverflow.net/users/121665 | 297440 | 130,643 |
https://mathoverflow.net/questions/297432 | 4 | Suppose $X$ is sampled from a symmetric Dirichlet distribution with arbitrary shape and $n$ dimensions. Equivalently we can independently sample $z\_i \sim \text{Gamma}(\alpha, 1)$ and then set $x\_i=\frac{z\_i}{\sum z\_i}$
I want to show that the coordinates $x\_i$ are negatively associated. Intuition tells me that,... | https://mathoverflow.net/users/123034 | Coordinates of Dirichlet Distribution Negatively Associated? | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\o... | 2 | https://mathoverflow.net/users/36721 | 297441 | 130,644 |
https://mathoverflow.net/questions/297443 | 12 | My apologies if this is too elementary, but since I have seen similar questions here I offer it.
After years doing almost exclusively classical complex geometry, using analysis and topology, I am trying to learn schemes, starting from the “red book”, which I have browsed for years, but vowing now not to skip anythin... | https://mathoverflow.net/users/9449 | etale topology of local schemes | Let me describe how to answer the etale cohomology questions for a slightly different ring, which is the ring of algebraic elements in the ring of formal power series, also known as the etale local ring / Henselization of the algebraic local ring (although these definitions become different in the greater generality of... | 5 | https://mathoverflow.net/users/18060 | 297481 | 130,657 |
https://mathoverflow.net/questions/297470 | 14 | I have [asked this question on Math.StackExchange](https://math.stackexchange.com/questions/2729335/is-the-quotient-of-a-toric-variety-by-a-finite-group-still-toric), but haven't got any reply.
Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $\... | https://mathoverflow.net/users/87910 | Is the quotient of a toric variety by a finite group still toric | No. If $X$ is the 47-dimensional torus and $G=\mathbb{Z}/47\mathbb{Z}$ acting by the permutation representation, then $X/G$ is not a rational variety (R. Swan, Inv. math. 7, 148-158 (1969)), therefore not toric.
| 36 | https://mathoverflow.net/users/7666 | 297482 | 130,658 |
https://mathoverflow.net/questions/296400 | 2 | In the literature, is there any paper or research investigating [the invariant subspace problem](https://en.wikipedia.org/wiki/Invariant_subspace_problem) with consideration of differential operators acting on an appropriate Sobolev space?In particular is there any research investigating the "Invariant Subspace Conject... | https://mathoverflow.net/users/36688 | A possible dynamical approach to the "Invariant Subspace Problem" | The following papers are related to your question:
1. A. Atzmon, A model for operators with cyclic adjoint, *Integral
equations and Operator theory*, 10(1987), 153-163.
2. A. Atzmon, Nuclear Frechet spaces of entire functions with transitive
differentiation, *J. Analyse Math.* 60(1993), 1-19.
A.Atzmon
| 3 | https://mathoverflow.net/users/123063 | 297494 | 130,664 |
https://mathoverflow.net/questions/296851 | 0 | Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E\_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E\_\lambda $, then why $BX$ is positive and self-adjoint?
I am struggling in dealing with unbounded operators...
see page 48, line +6 (j... | https://mathoverflow.net/users/91769 | For $B=\int \lambda d E_\lambda $ and $X$ commutes with every $E_\lambda $, why $BX$ is positive and self-adjoint? | If $B$ is positive self-adjoint then $B=A^2$ with $A$ positive self-adjoint. If $X$ is bounded non-negative and commutes with $B$, it commutes as well with a function of $B$ such as $A=\sqrt B$. Then we have
$$
XB=BX=AAX=AXA\ge 0.
$$
The domain of $BX$ can be taken as the domain of $B$ with $XB u$ well-defined for $u\... | 2 | https://mathoverflow.net/users/21907 | 297496 | 130,665 |
https://mathoverflow.net/questions/297330 | 5 | Sorry if this is elementary.
Let $C$ be a category, $X$ an object in $C$ and let $S(X)$ denote the poset of subobjects of $X$.
According to the [nlab entry](https://ncatlab.org/nlab/show/subobject#the_poset_of_subobjects), if $C$ has all limits and co-limits, so does $S(X)$. It's a little unclear to me why this is ... | https://mathoverflow.net/users/74739 | How do (co)limits in posets of subobjects relate to (co)limits in ambient category? | To summarize what's been said in comments:
* If $C$ has small limits, then so does $S(X)$: they are inherited from $C/X$, which inherits them from $C$ by adding $X$ as a terminal object in the diagram being taken a limit of.
* In particular, therefore, if $C$ is well-powered, so that $S(X)$ is small, then it is a sma... | 4 | https://mathoverflow.net/users/49 | 297499 | 130,666 |
https://mathoverflow.net/questions/297467 | 6 | There is a notion of combinatorial curvature due to Forman, see [here (published paper)](https://link.springer.com/article/10.1007/s00454-002-0743-x) or [here (preprint).](http://math.rice.edu/~forman/ric.ps) I checked for a couple of small triangulations of $\mathbb{RP}^2$ (6-vertex, 7-vertex, 9-vertex) and the 1-curv... | https://mathoverflow.net/users/50846 | Combinatorial curvature of real projective plane | Take any triangulation and any pair of adjacent faces $ABC$ and $BCD$. Now subdivide the shared edge $BC$ and both faces into three edges $BE$, $EF$, $FC$ and six faces $ABE$, $AEF$, $AFC$, $BED$, $EFD$, $FCD$. The edge $EF$ has $1$-curvature $2+2-2=2$.
| 2 | https://mathoverflow.net/users/112641 | 297503 | 130,667 |
https://mathoverflow.net/questions/297502 | 0 | Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows:
\begin{align}
f(x)\triangleq \sum\_{i=1}^K \frac{\alpha\_i \gamma\_i \sin(x-\theta\_i)}{1+\gamma\_i[1+\cos(x-\theta\_i) ]},
\end{align}
where anything except $x$ is a given parameter and we have $\alpha\_i >0, \forall i$ and $\gamma\_i >0, \forall i$. I a... | https://mathoverflow.net/users/123067 | Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$ | Even for the case $K=2$, closed form solutions seem hopeless. Counting the number of solutions should be possible, though. Expand the sines and cosines and put everything over a common denominator: the numerator will be a trigonometric polynomial $P(\sin(x), \cos(x))$. Let $R(s)$ be the resultant of $P(s,c)$ and $s^2 +... | 2 | https://mathoverflow.net/users/13650 | 297506 | 130,668 |
https://mathoverflow.net/questions/297501 | -1 | A topological space $X$ is called Artinian if the descending chain condition holds for open subsets of $X$. If the descending chain condition holds for open basis subsets of a Hausdorff space $X$ with the property that the intersection of two open basis is an open basis, can we prove that $X$ is Artinian?
Also, is th... | https://mathoverflow.net/users/123066 | A condition for Artinian topological spaces | If $X$ is Artinian, then for every $x \in X$ there is a minimal neighbourhood $U\_x$ such that if $O$ is open with $x \in O$, then $U\_x \subseteq O$.
This is a simple application of Zorn's lemma on the poset of open neighbourhoods of $x$, ordered by reverse inclusion. The Artianness of $X$ implies that all chains in... | 1 | https://mathoverflow.net/users/2060 | 297529 | 130,676 |
https://mathoverflow.net/questions/297534 | 3 | It is known that $L^1(\mathbb{R}) \ast f$ is dense in $L^1(\mathbb{R})$ for some $f\in L^1(\mathbb{R})$.
So for such $f$ the closure of $L^1(\mathbb{R}) \ast f$ in the $L^1$ norm is $L^1(\mathbb{R})$.
But apparently
(1)$\quad\quad L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$ *for every*
$f\in L^1(\mathbb{R})$.
I... | https://mathoverflow.net/users/95282 | Proof of $L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$ | I'm guessing $\*$ means convolution (since this is a math forum) and not pointwise multiplication (since this is not a computer forum).
Some steps to try ... suppose $L^1 \* f = L^1$
$\widehat{f} \ne 0$ a.e.
There is $g \in L^1$ so that $g \* f = f$
$ \widehat{g} \widehat{f} = \widehat{f}$
$\widehat{g} = 1$ ... | 6 | https://mathoverflow.net/users/454 | 297535 | 130,678 |
https://mathoverflow.net/questions/297525 | 5 | It has bug me for a while that I don't have a good understanding of the theory of Hecke operators. For elliptic modular forms, it was explained in Koblitz's book that they arose from viewing the modular forms as function on modular points (lattices in $\mathbb{C}$, possibly with additional structures) but I feel this i... | https://mathoverflow.net/users/14725 | Hecke operators for hermitian modular forms of general level | It's best to think of these things adelically. Suppose you have a reductive group $G$ over a number field $F$, and you set
$$\mathcal{G} = G(\mathbf{A}\_{F, f}) = \sideset{}{'}\prod\_{\text{$v$ finite place of $F$}} G(F\_v).$$
For any open compact $U \subset \mathcal{G}$ you can form the Hecke algebra $\mathbf{C}[U \b... | 4 | https://mathoverflow.net/users/2481 | 297556 | 130,683 |
https://mathoverflow.net/questions/297477 | 3 | Let $\Omega=\mathbb{D}\cap\{ (x,y)\, \vert\, y>0\}$, $I=(-1,1)\times \{0\}$ and $A=\partial\Omega\setminus I$. Let $Q\in L^1(\Omega)$, and $R\in C^\infty\_{loc}(I)$.
I am looking to the following problem
$$
\left\{
\begin{aligned}
\Delta \psi = Q & \hbox{ in } \Omega \\
\psi = 0 & \hbox{ on } A \\
\partial\_{\... | https://mathoverflow.net/users/9253 | Existence en regularity of elliptic PDE with mixed boundary | The continuity of $\psi$ up to $\partial \Omega$ is false without more control on $R$.
Consider for example the harmonic function that is $1$ on the upper half-circle and $-1$ on the lower half-circle. A model for the behavior near e.g. the lower left corner is the (zero-homogeneous) angle function $\frac{2}{\pi} \I... | 3 | https://mathoverflow.net/users/16659 | 297563 | 130,686 |
https://mathoverflow.net/questions/297479 | 1 | I am having trouble with a summation notation in a paper I am reading (talking about Semi Markov Processes), I am not how to use it.
The equation is as follow:
$$MTTSF\_{\phi}=\sum\_{i - 1}\frac{1}{\xi\_{i}}\sum\_{j}x\_i.\theta\_{ij}$$
Which give you the Mean Time to Security Failure of your system.
Here are t... | https://mathoverflow.net/users/109680 | Summation unknown notation | Actually I have identified the problem. In the paper, authors example used an SMP where the attacker can chose between two paths, both with the same probability $\frac{1}{2}$. Because the start state is not taken into account in the system fromalized control flow, I belive that they made a mistake which resides here.
... | 2 | https://mathoverflow.net/users/109680 | 297572 | 130,689 |
https://mathoverflow.net/questions/297328 | 24 | **Warning**: non-specialist writing, some rubbish possible.
The formula $h^\*(BG)\cong h^\*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal torus $T$ and Weyl group $W$ suggests that it might come from some sort of equivalence like $G\sim T///W$ where "///... | https://mathoverflow.net/users/41291 | Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori? | Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain generalized cohomology theories. If you replace the role of the maximal torus with "arbitrary abelian subgroups", this "looking... | 14 | https://mathoverflow.net/users/437 | 297582 | 130,693 |
https://mathoverflow.net/questions/297555 | 2 | After asking [this question](https://mathoverflow.net/questions/295158/generalizations-of-abhyankar-moh-theorem-embeddings-of-the-line-in-the-plane/295575?noredirect=1#comment739839_295575), I figured out that I am also interested in the following related question:
>
> Is [Abhyankar-Moh theorem 1.6](https://eudml.o... | https://mathoverflow.net/users/72288 | Abhyankar-Moh embedding theorem without algebraic closedness | In the van den Essen's paper, a polynomial map $t\mapsto (x(t),y(t))$ is called an embedding if there is a polynomial $F$ such that $F(x(t),y(t))=t$. There is no problem with his proof: Theorem 1 is true over any field if we interpret embedding in this sense.
The reason why he considered $\mathbb{C}$ rather then an ... | 3 | https://mathoverflow.net/users/9833 | 297591 | 130,696 |
https://mathoverflow.net/questions/297526 | 5 | If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute Galois groups have finite abelianizations)?
| https://mathoverflow.net/users/83073 | Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed? | Here is another example, which also answers the "followup question":
Let $K$ be the field of Laurent series over $\mathbb{R}$. Its absolute Galois group is the infinite profinite dihedral group $\hat{\mathbb{Z}}\rtimes(\mathbb{Z}/2\mathbb{Z})$, where the action is by inversion. This group is the free profinite product ... | 4 | https://mathoverflow.net/users/101929 | 297597 | 130,698 |
https://mathoverflow.net/questions/297602 | 1 | Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.
Are there any general results which can be applied to sets like this (Cantor set times a plane set) to see if they do, or do not, embed into the plane?
Is there a... | https://mathoverflow.net/users/95718 | Trouble with plane embedding | I can answer the last question.
Let $X$ be a "tripod" in $\mathbb{R}^2$. For concreteness, let
$$ X = ([-1,1]\times \{0\}) \cup (\{0\}\times [0,1]).$$
Then $X$ is a $1$-dimensional plane continuum such that $\mathbb{R}^2 \setminus X$ is path-connected.
But $C\times X$ does not embed into the plane: only countably... | 5 | https://mathoverflow.net/users/123122 | 297604 | 130,701 |
https://mathoverflow.net/questions/297571 | 3 | Let $X$ be a random variable with $E[X] = \mu < \infty$.
For $n=1,2,\dots$, construct a triangular array of random variables as
\begin{equation}
Y\_{n,i} = X\_i \frac{\sqrt{\mu}}{\sqrt{\sum\_{j=1}^n X\_j/n}}.
\end{equation}
Then, does the following hold?
\begin{equation}
\frac{1}{n} \sum\_{i=1}^n Y\_{n,i} \overse... | https://mathoverflow.net/users/80302 | Weak law of large numbers for triangular arrays | $\newcommand{\al}{\alpha}
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\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}... | 2 | https://mathoverflow.net/users/36721 | 297606 | 130,702 |
https://mathoverflow.net/questions/297618 | 6 | This is admittedly, probably an easy question for the right person here, but I cannot seem to track down an answer. The question itself may not be hard, but the answer is crucial to a math paper I am writing (and I don't know enough number theory).
>
> True or false: There are at least exp$(r)$ irreducible polynomi... | https://mathoverflow.net/users/122188 | Number of irreducible polynomials of degree $r$ in $F_2[x]$ | Sloane's OEIS sequence [A001037](https://oeis.org/A001037) counts ($n=r$ in your definition):
Number of degree-$n$ irreducible polynomials over $GF(2)$;
number of $n$-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period $n$;
number of binary Lyndon words of length $n$... | 12 | https://mathoverflow.net/users/17773 | 297620 | 130,705 |
https://mathoverflow.net/questions/297549 | 2 | This is a follow-up on an [older question](https://mathoverflow.net/questions/297236/is-box-n-in-omega0-1-connected).
Let $\Box\_{i\in I} X\_i$ denote the [box product](https://en.wikipedia.org/wiki/Box_topology) of the spaces $X\_i$. Is there a Hausdorff space $(X,\tau)$ with $|X|>1$ such that $\Box\_{n\in\omega}X$ ... | https://mathoverflow.net/users/8628 | Connected box products of Hausdorff spaces | If $X$ is the [Irrational slope topology](https://topology.pi-base.org/spaces/S000067) then the closures of any two non-empty open sets must intersect. It easily follows that $\Box\_{n\in\omega}X$ is connected. Note that $X$ is Hausdorff but not regular. It seems (see the comments to the OP) that there are no $T\_3$ ex... | 2 | https://mathoverflow.net/users/17836 | 297623 | 130,706 |
https://mathoverflow.net/questions/297583 | 10 | In Lemma 5.2(a) of Z. Fiedorowicz, *Classifying Spaces of Topological Monoids and Categories* American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 301-350 the author proves the following.
>
> **Lemma 5.2(a)** Suppose that $\{M\_i\}\_{i\in I}$ is a collection of monoids with a common submonoid $W$ such ... | https://mathoverflow.net/users/15934 | A flatness result of Fiedorwicz for amalgamated free products of monoids in connection with classifying spaces of monoids | This appears to be false to me (unless maybe there are commutativity hypotheses), unless I've made a mistake below. (This is based on an example that I saw Andrew Ranicki give, a number of years ago, about non-exactness of Cohn localization.)
Consider the diagram of monoids
$$
\Bbb Z \leftarrow \Bbb N \rightarrow \Bb... | 6 | https://mathoverflow.net/users/360 | 297628 | 130,709 |
https://mathoverflow.net/questions/254116 | 4 | I am trying to show an estimate of the following form: Given any $p(x)$ such that $1<p^-\leq p(x) \leq p^+ <\infty$ and $p(\cdot)$ is log-Holder continuous, does there exists an $R\_0$ (depending only on $p(\cdot)$, $n$ and log-Holder continuity of $p(x)$) such that
$$ \int\_{B\_R} M\_{<{2R}} (|f|)^{p(x)}(x) \ dx \leq ... | https://mathoverflow.net/users/100801 | Integral form of maximal function estimate on variable exponent spaces | The answer to this question can be found in Theorem 4.8 and Corollary 4.9 of arxiv.org/abs/1707.02535 where a bound of the above form is proved with a size restriction on $\int\_{B\_{2R}} |f(x)|^{p(x)} dx$.
| 0 | https://mathoverflow.net/users/100801 | 297635 | 130,712 |
https://mathoverflow.net/questions/297616 | 4 | Maybe this is a question to naive for the MO community! For a projective smooth variety $X$ defined over a field $F$, and for simplicity let's assume $F$ is a number field. One way to define motivic cohomology is through $K$-theory
\begin{equation}
H^i\_M(X,\mathbb{Q}(j))=K\_{2j-i}(X)^{(j)}\_{\mathbb{Q}}
\end{equation}... | https://mathoverflow.net/users/87910 | Absoluteness of motivic cohomology and restriction of scalars | By the work of Voevodsky, A^1-homotopy theory, or Cisinski-Déglise, Triangulated categories of mixed motives, we have motivic cohomology of arbitrary schemes (maybe noetherian and finite dimensional, but this is the case here) with integral coefficients. For regular schemes, motivic cohomology with Q-coefficients is is... | 1 | https://mathoverflow.net/users/6506 | 297640 | 130,713 |
https://mathoverflow.net/questions/297651 | 1 | What are all the totally geodesic submanifolds of $\mathbb{R}P^N$?
If I were to guess the answer, I would have thought that it is all $\mathbb{R}P^n$, where $1\leq n\leq N$, based on the analogy with what happens with the sphere in $\mathbb{R}^{N+1}$. Is this true, and is there an easy way to see this?
| https://mathoverflow.net/users/123154 | Determining all totally geodesic submanifolds of $\mathbb{R}P^N$ | I think Lemma 3, Section 3 of this paper of Wolf provides a positive answer to your question(s):
* J.A. Wolf. Elliptic spaces in Grassmann manifolds. Illinois J. Math.
Volume 7, Issue 3 (1963), 447-462. [(link to journal website)](https://projecteuclid.org/euclid.ijm/1255644952)
| 1 | https://mathoverflow.net/users/50846 | 297653 | 130,720 |
https://mathoverflow.net/questions/297585 | 9 | Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer is no if $K=\overline{\bf F}\_p$ (in that case ${\rm End}(A)\_{\bf Q}$ is a CM field). My question is for other fields.
| https://mathoverflow.net/users/17308 | Endomorphism ring of simple ordinary abelian variety | Let $D$ be a non-split quaternion algebra over $\mathbb{Q}$, split at $p$ and $\infty$. Let $\mathcal{O}$ be a maximal order (I think these are all conjugate). Then $\mathcal{O}^1$, the multiplicative group of norm 1 elements, embeds into $SL\_2(\mathbb{R})$ by picking an isomorphism $\iota:D\otimes \mathbb{R}\cong M\_... | 6 | https://mathoverflow.net/users/791 | 297663 | 130,724 |
https://mathoverflow.net/questions/297658 | 4 | Let $X$ be a locally compact Hausdorff space and suppose that $X$ can be written as the disjoint union of countably many non-empty closed subsets. Is at least one of the subsets clopen?
| https://mathoverflow.net/users/121269 | closed decomposition of locally compact Hausdorff space | Let $X$ be a countable compact Hausdorff space consisting of a unique accumulation point $0$ with discrete complement $X'$. Let $j$ be a bijection $\mathbf{N}\to X'\times\mathbf{N}$.
For $n\in\mathbf{N}$, write $F\_n=\{(0,n),j(n)\}$. This is a 2-element subset of $X\times\mathbf{N}$. Then $(F\_n)\_{n\in\mathbf{N}}$ ... | 6 | https://mathoverflow.net/users/14094 | 297666 | 130,725 |
https://mathoverflow.net/questions/297669 | 2 | It is well-known that Sklyanin algebras are Koszul, but, is it known an explicit description of the dual algebra Ext\_A(k,k)? (I mean in terms of generators and relations)
| https://mathoverflow.net/users/98863 | Description of Koszul dual of Sklyanin algebras | Section 10 of the following paper spells out this example.
*Smith, S. Paul*, **Some finite dimensional algebras related to elliptic curves**, Bautista, Raymundo (ed.) et al., Representation theory of algebras and related topics. Proceedings of the workshop, Mexico City, Mexico, August 16-20, 1994. Providence, RI: Ame... | 3 | https://mathoverflow.net/users/6263 | 297673 | 130,727 |
https://mathoverflow.net/questions/297570 | 3 | First, a bit of background on orbifolds:
>
> Let $X$ be a connected (effective) orbifold. To every point $x \in X$, we associated a group $G\_x$ called the *isotropy group*. The singular locus $\Sigma X$ is the set of points $x$ for which $G\_x \neq 1$, and these points are called *singular points*. Non-singular po... | https://mathoverflow.net/users/51599 | Do regular points of an orbifold form a connected set? | Part of the reason that people consider orbifolds whose singular points have codimension $\ge 2$ is because they are restricting their attention to oriented orbifolds. For example, if you are working with $n$-orbifolds over the complex numbers $\mathbb{C}$ then each element of each isotropy group is a holomorphic trans... | 9 | https://mathoverflow.net/users/20787 | 297684 | 130,732 |
https://mathoverflow.net/questions/297690 | 8 | Let $N$ be a countably infinite set and let $\mathcal P$ denote power set.
I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$.
But what can be said about automorphisms of $\mathcal P(N)$ mod finite? That is, mod out by the equivalence relation $A\sim B\iff A$ and $B$ di... | https://mathoverflow.net/users/4600 | Automorphisms of power set lattice mod finite | This is known as Rudin-Shelah problem. Note that, by Stone duality, this is equivalent to determine the self-homeomorphism group of the Stone-Cech boundary of $N$. Notably, consider the group induced by bijections between two cofinite subsets of $N$ (modulo cofinite coincidence). It maps homomorphically injectively int... | 12 | https://mathoverflow.net/users/14094 | 297691 | 130,734 |
https://mathoverflow.net/questions/297692 | 2 | It appears that there are two different definitions of category. Some authors require the Hom-sets to be pairwise disjoint. Eilenberg and Mac Lane in their original definition require each identity morphism of the category to uniquely determine an object of the category. But some authors (e.g. Kashiwara & Schapira) do ... | https://mathoverflow.net/users/16046 | Different definitions of category | **The difference is only cosmetic, not serious.** Given a category $\newcommand{\C}{\mathbf{C}}\C$ with not-necessarily-disjoint homsets, we can easily make its homsets disjoint. Precisely, we can define a new isomorphic category $\C'$, with the same objects as $\C$ but with $\hom\_{\C'}(x,y) := \hom\_{\C}(x,y) \times ... | 13 | https://mathoverflow.net/users/2273 | 297693 | 130,735 |
https://mathoverflow.net/questions/297650 | 5 | I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows.
Consider a positive integer $x \in \{1,2,3,...\}$ and a real number $q \in [1,x]$. Prove that
\begin{equation}
\sum\_{m=1}^{x} \frac{m}{\mi... | https://mathoverflow.net/users/123150 | An inequality involving a sum of power terms | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}... | 6 | https://mathoverflow.net/users/36721 | 297696 | 130,737 |
https://mathoverflow.net/questions/297617 | 10 | It is widely known that
$$ \frac{1}{n!}\sum\_{\pi\in S\_n}\chi\_\lambda(\pi)\chi\_\mu(\pi)=\delta\_{\lambda,\mu},$$
where $S\_n$ is the permutation group and $\chi$ are its irreducible characters.
In exercise 7.63 of his classic book *Enumerative Combinatorics*, Richard Stanley computes explicitly the value of
$$\su... | https://mathoverflow.net/users/78061 | sum of character product over derangements | Using standard symmetric function notation, we have
\begin{eqnarray\*} \sum\_{n\geq 0}\sum\_{\lambda,\mu\vdash n}
\frac{1}{n!}\left(\sum\_{\pi\in D\_n}\chi\_\lambda(\pi)\chi\_\mu(\pi)\right)
s\_\lambda(x)s\_\mu(y) & = & \sum\_{n\geq 0}\frac{1}{n!}
\sum\_{\pi\in S\_n}\left.p\_{\rho(\pi)}(x)p\_{\rho(\pi)}(y)\right|\_... | 8 | https://mathoverflow.net/users/2807 | 297708 | 130,743 |
https://mathoverflow.net/questions/297672 | 1 | I have what is in essence a basic analysis question.
To make working out a certain example a bit easier I found that I need to find existence of a function $f\in C^\infty(\mathbb{R})$ with the following properties:
1. $f$ is increasing
2. $f(x)=0$ for all $x\leq 0$
3. $f(x)=1$ for all $x\geq 5$
4. $\frac{f(x)}{x}... | https://mathoverflow.net/users/27224 | Argument for differentiability of a certain quotient of smooth functions | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}... | 1 | https://mathoverflow.net/users/36721 | 297710 | 130,744 |
https://mathoverflow.net/questions/297707 | 5 | Let $X = (X\_1, \ldots, X\_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|\_{\infty}^k \right)$$ for $k=1, \ldots, 6$ ? It is easy to see for example for $k=1$, the upper bounded is of the order $\sqrt{2\log(2d)}$. In particular for... | https://mathoverflow.net/users/16976 | Moments of maximum of independent Gaussian random variables | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$
... | 7 | https://mathoverflow.net/users/36721 | 297711 | 130,745 |
https://mathoverflow.net/questions/297705 | 14 | Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (<http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf>) by Mike Hutchings and these incomplete notes by Ralph Cohen (<http://math.stanford.edu/~ralph/morsecourse/biglectures.pdf>)
[..I really want a ... | https://mathoverflow.net/users/89451 | What are good Morse Theory lecture notes and books? | If you are looking for the classical approach to Morse theory, I feel nothing beats Milnor's book on the subject:
>
> Milnor, J.
> Morse theory.
> Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963
>
>
>
For the Morse homological approach, i.e. counting flowlines, I reall... | 16 | https://mathoverflow.net/users/12156 | 297732 | 130,751 |
https://mathoverflow.net/questions/297743 | 6 | Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$.
If $T$ is an $R$-torsor, it yields by base change a $k$-torsor $T\_k$.
Apparently, we have the following theorem:
**Thm.** If $G$ is smooth and $T,T'$ are two $R$-torsors, ... | https://mathoverflow.net/users/36683 | Torsors over complete local fields | I am rewriting my comment as an answer. That is **false** in characteristic $p$ for torsors for the finite, flat group scheme $\mu\_p=\text{Spec}\ \mathbb{Z}[t]/\langle t^p -1 \rangle$ with the usual group multiplication, $$\mathbb{Z}[t]/\langle t^p-1 \rangle \to \mathbb{Z}[t\_1,t\_2]/\langle t\_1^p-1,t\_2^p-1\rangle, ... | 5 | https://mathoverflow.net/users/13265 | 297747 | 130,755 |
https://mathoverflow.net/questions/297738 | 24 | Let me summarize what I think I understand about constructivism:
"Constructive mathematics" is generally understood to mean a variety of theories formulated in intuitionist logic (i.e., not assuming the law of excluded middle, $\neg\neg A\Rightarrow A$ (LEM)) so that, broadly speaking, in order to prove $A \lor B$ on... | https://mathoverflow.net/users/17064 | In what ways is ZF (without Choice) "somewhat constructive" | As I have said in a comment, Levy proves a weak form of the existential property for $ZF$ and $\Pi\_2$ sentences. He also proves that his results are best possible. Let me state a simple fact that is how I like to think about this existential property. If $ZF$ proves a $\Pi\_2$ sentence $\forall x\exists y A$ then $ZF$... | 14 | https://mathoverflow.net/users/9825 | 297759 | 130,759 |
https://mathoverflow.net/questions/297765 | 6 | Assuming the axiom of choice I can write for any cardinal number $\kappa$ and any simple graph $G$ that a function $f$ is a $\kappa\text{-coloring}$ of $G$ if and only if the cardinality of the image of $f$ is equal to $\kappa$ and that:
$$\forall u,v\in V(G)\left[\{u,v\}\in E(G)\implies f(u)\neq f(v)\right]$$
Now ... | https://mathoverflow.net/users/38626 | Does the existence of a unique chromatic (possibly transfinite) number for every (possibly non-finite) simple graph imply the axiom of choice? | It seems that your question has a positive answer, as shown by Galvin and Komjáth in their paper
>
> *Galvin, F.; Komjáth, P.*, [**Graph colorings and the axiom of choice**](http://dx.doi.org/10.1007/BF02309111), Period. Math. Hung. 22, No.1, 71-75 (1991). [ZBL0748.05056](https://zbmath.org/?q=an:0748.05056).
>
> ... | 14 | https://mathoverflow.net/users/7206 | 297768 | 130,762 |
https://mathoverflow.net/questions/297756 | 8 | It is well known that the intersection of two models of ZFC does not have to be a model of ZFC (or even ZF). Now what if we restrict ourselves to models $M[G]$, $M[H]$ which are generic over $M$ for the same poset $\mathbb{P}$? What about their intersection. It is not too difficult to show that if $G \times H$ is $\mat... | https://mathoverflow.net/users/103802 | Intersection of two generic extensions | Let $M$ be a countable transitive model. Fix $a\subseteq \omega$ such that $a$ is not generic over $M$. We plan to construct a pair of $M$-generic Cohen reals $G\_0$ and $G\_1$ such that $P(\omega)\cap M[G\_0]\cap M[G\_1]$ encodes $a$ and hence does not belong to $M[G\_0]$ or $M[G\_1]$ let alone $M[G\_0]\cap M[G\_1]$.
... | 13 | https://mathoverflow.net/users/102684 | 297769 | 130,763 |
https://mathoverflow.net/questions/297777 | 1 | $\textbf{Question}$: Is there a translation from $\textbf{S5}$ modal logic to $\textbf{S4}$ such that
$$\text{If} \hspace{0.3cm} \textbf{S5} \vdash F \hspace{0.3cm} \text{then } \hspace{0.3cm} \textbf{S4} \vdash F'$$
where $F'$ is formed from $F$ by applying a suitable translation $\hspace{0.2cm}'$?
I take it tha... | https://mathoverflow.net/users/122435 | Translations between S4 and S5 modal logics | As written, there is a trivial such translation: just put $F'=\top$ for all formulas $F$.
Assuming you actually wanted to formulate the condition as “if and only if” rather than just “if”, a simple such translation is provided by
$$\mathrm{S5}\vdash A\iff\mathrm{S4}\vdash\Diamond\Box A.$$
| 4 | https://mathoverflow.net/users/12705 | 297779 | 130,767 |
https://mathoverflow.net/questions/297731 | 2 | Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. My question is, what is the dual $E'$ of $E$? Is there an identification as in the case of $L^{\infty}(\Omega,\mu)$ and ... | https://mathoverflow.net/users/123197 | Duality of Bochner $L^{\infty}$ space | I take it from your question and comments that it is important that one considers the Banach space of bounded meaurable functions, not equivalence classes thereof. In the scalar case, the dual is the space of finitely additive bounded measures. In the vector valued case, the dual is, at least for the case of a separabl... | 0 | https://mathoverflow.net/users/122526 | 297781 | 130,768 |
https://mathoverflow.net/questions/297715 | 8 | Let $ X, Y $ be smooth affine varieties over $ \mathbb C $. Let $ T : X \rightarrow Y $ be a dominant quasi-finite morphism and let $ T^\# : \mathbb C[Y] \rightarrow \mathbb C[X] $ be the resulting map on coordinate rings.
($ T $ being quasi-finite means that it can be factored as an open embedding followed by a fin... | https://mathoverflow.net/users/438 | Differential operators and quasi-finite morphisms | Edit: I think skipped a step in the original argument - it is not immediately clear to me that $d$ commutes with $\mathbb C[Y]$ in $D(X)$ unless $d$ is a derivation. I have added an inductive argument for this below.
---
I believe this is true. Here is a sketch of an argument.
We will prove this by induction on... | 5 | https://mathoverflow.net/users/7762 | 297786 | 130,769 |
https://mathoverflow.net/questions/297780 | 5 | I would like to know if there is a closed form formula for the homotopy type of $\widehat{(\mathbb{C^{\ast}})^n}$? For example, it is not difficult to see that $\widehat{\mathbb{C^{\ast}}}$ has the homotopy type of $S^1\vee S^2$.
My guess is that the formula, for general $n$, should look something like this
$$ \bigve... | https://mathoverflow.net/users/7494 | One point compactification of $(\mathbb{C}^{\ast})^n$ | If $\widehat X$ is the 1-point compactification of $X$, then there is a homeomorphism (for, say, locally compact Hausdorff spaces)
$$
\widehat{X \times Y} \cong \widehat X \wedge \widehat Y
$$
with the smash product. Moreover, the smash product preserves homotopy equivalences for well-pointed spaces, which $\widehat{\B... | 12 | https://mathoverflow.net/users/360 | 297788 | 130,770 |
https://mathoverflow.net/questions/297787 | 5 | It is possible to define the determinant of a tensor.
We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map.
If $T:\{1,....,n\}^m\to \mathbb C$ then one can define
$$\operatorname{Det}T:=\sum\_{\sigma\_2,...,\sigma\_m\in S\_n}\left(\left[\prod\_{i=2}^m\operatorn... | https://mathoverflow.net/users/123218 | Looking for a tractable algorithm or formula for the determinant of a tensor | This is the Cayley hyperdeterminant, see [1,2], which is believed to be an NP-hard computation [3].
1. F. Gherardelli, Osservazioni sugli iperdeterminanti, Istit.
Lombardo Accad. Sci. Lett. Rend. A 127, 107 (1993).
2. [The Cayley
Determinant of the Determinant Tensor and the Alon Tarsi
Conjecture](https://core.ac.uk/... | 3 | https://mathoverflow.net/users/11260 | 297795 | 130,772 |
https://mathoverflow.net/questions/297797 | 14 | In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the diagonal and integer entries below embeds as a subgroup of $C^1(S^1)$.
<http://www.math.uchicago.edu/~farb/papers/nilpo... | https://mathoverflow.net/users/123220 | Proving convergence of sum over $\mathbb{Z}^n$ | The sum diverges already for $n=4$. To see this, let $L\geq K$ be a dyadic parameter, and consider the contribution of $q\_1\asymp L^{1/8}$, $q\_2\asymp L^{1/6}$, $q\_3\asymp L^{1/4}$, $q\_4\asymp L^{1/2}$. If the implied constants are sufficiently close to each other, these ranges are pairwise disjoint. However the co... | 8 | https://mathoverflow.net/users/11919 | 297803 | 130,775 |
https://mathoverflow.net/questions/297794 | 0 | Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do not see a simple argument for that. Any help is welcome.
| https://mathoverflow.net/users/892 | Hyperbolic structures on infinite type surfaces | This is a classical theorem: hyperbolic Riemann surfaces of finite hyperbolic area
are compact surfaces with finitely many punctures. Tsuji (Theorem XI.12) credits this to Siegel (1945). The proof is a simple computation of the area of the fundamental polygon of the uniformizing Fuchsian group.
Tsuji, Potential theor... | 3 | https://mathoverflow.net/users/25510 | 297814 | 130,781 |
https://mathoverflow.net/questions/297816 | 8 | 1. Suppose $(x\_n)$ and $(y\_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x\_n), (y\_n)\}=X$. Can we always pass to subsequences $(x\_{n\_k})$ and $(y\_{n\_k})$ such that $\overline{span}\{(x\_{n\_k}), (y\_{n\_k})\}\neq X$?
I had the impression that this must be "obviously ... | https://mathoverflow.net/users/7872 | Two questions about basic sequences | I would suggest to get a positive answer to the second question as follows: Let $\{x\_i\}$ be a basic sequence in $X$ (existing by Mazur's result) and let $\{z\_i\}$ be a rapidly converging to zero sequence in $X$ with dense linear span. Let $\{y\_i\}$ be given by $y\_i=x\_i+z\_i$. The sequence $\{y\_i\}$ is basic by t... | 10 | https://mathoverflow.net/users/37822 | 297823 | 130,786 |
https://mathoverflow.net/questions/297796 | 1 | Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m\_1}, \dots, \lambda^{m\_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m\_1, \dots, m\_n$. The generalized characteristic polynomial of a $k \times k$ matrix $\mathbf{A}$ is then:
$$ p(\lambda) = det(\mathbf{D}(\lambda) - \mathbf{A})$$
Th... | https://mathoverflow.net/users/51478 | Polynomial Eigenvalue Problem with few non-zero coefficients | You may want to have a look at the research by Bini and coauthors on the Ehrlich-Aberth method, e.g., <https://arxiv.org/abs/1207.6292>. That method is sort-of "black box", i.e., you only need a way to evaluate and factor the matrix polynomial.
| 1 | https://mathoverflow.net/users/1898 | 297832 | 130,790 |
https://mathoverflow.net/questions/297833 | 5 | Can Omega limit sets of dynamical systems be connected but not road connected?
In the process of reading Wiggins, we have encountered the definition and properties of Omega limit sets for autonomous equations in a finite dimensional space. The connectivity of Omega limit set of a point is described in nature, but its p... | https://mathoverflow.net/users/123243 | Dynamical system and omega limit set | Yes, the omega-limit set of a dynamical system can be connected but not path-connected. In fact this is the case for hyperbolic flows with an attractor, e.g. the [Lorenz attractor](https://en.wikipedia.org/wiki/Lorenz_system), the Plykin attractor,
some [uniformly hyperbolic attractors](http://www.scholarpedia.org/arti... | 10 | https://mathoverflow.net/users/6129 | 297836 | 130,792 |
https://mathoverflow.net/questions/297808 | 3 |
>
> Let $A:D(A)\subseteq E \to E$ be a closed operator on a complex Banach lattice $E.$ Then $A$ is said to be **real** if $x+iy \in D(A) \implies x,y \in D(A)$ for all $x,y \in E\_{\mathbb R}$ and $A(D(A) \cap E\_{\mathbb R}) \subseteq E\_{\mathbb R}.$ Here $E$ is the complexification of $E\_{\mathbb R}.$
>
>
>
... | https://mathoverflow.net/users/119514 | Reference request: Spectral properties of real operators | **A few preliminary remarks:**
1) Complexifications of Banach lattices are in fact a special case of the more general concept of *complexifications of real Banach spaces*.
2) Most books and articles about complex Banach lattices which contain spectral theoretic results focus on *positive* operators (or, say, genera... | 6 | https://mathoverflow.net/users/102946 | 297838 | 130,793 |
https://mathoverflow.net/questions/297826 | 1 | Let $G$ be a finite simple graph on the vertex set $\{x\_1, \ldots,
x\_n\}$ and $I(G) := (\{x\_ix\_j \mid \{i,j\} \in E(G)\}) \subset R=K[x\_1,
\ldots, x\_n]$ be the edge ideal corresponding to the graph $G$,
where $K$ is a field.
The Castelnuovo–Mumford regularity (or simply, regularity) $reg(I(G))$ of $I(G)$ is
defin... | https://mathoverflow.net/users/68302 | Shedding vertex | In the case of a shedding vertex $v$, $\operatorname{reg} I(G) = \max \{ \operatorname{reg} I(G \setminus N[v]) + 1, \operatorname{reg} I(G\setminus v) \}$, by a theorem of myself and Tài Hà. So your inequality is true. See Theorem 1.5 of the following.
*Hà, Huy Tài; Woodroofe, Russ*, [**Results on the regularity of ... | 3 | https://mathoverflow.net/users/19729 | 297843 | 130,795 |
https://mathoverflow.net/questions/297847 | 4 | Let $X$ be an algebraic variety, $M \in Mod(\mathcal{D}\_X)$. I am studying the characteristic variety associated to this module, and I am trying to understand why all the different definitions coincide. Fix $F$ a good filtration of $M$, we define:
$Ch(M) = supp \left( \widetilde{gr^{F}(M)} := \mathcal{O}\_{T^{\*}X} ... | https://mathoverflow.net/users/91572 | Characteristic variety of a D-module | Although your cautionary statement about tensor products is true sometimes, in this case it’s not. This is because $\pi$ is *affine*, so over your affine open $U$, everything commutes with taking global sections.
| 4 | https://mathoverflow.net/users/36720 | 297857 | 130,800 |
https://mathoverflow.net/questions/297809 | 2 | I'm pretty clear in my understanding of scalar-valued differential $(p, q)$-forms (resp. holomorphic $(p,0)$-forms) on a complex manifold $M$ and the related Hodge theory. What I'm not sure about is whether there is a matrix-valued analogue of such differential forms on a complex manifold in the math literature. If the... | https://mathoverflow.net/users/86315 | matrix-valued differential forms on complex manifolds | Actually, matrix-valued differential forms are used a lot in hypercomplex analysis/hypercomplex geometry, which, as the name suggests, includes certain complex manifolds. There is a nice account of such differential forms in *Rocha-Chavez, Shapiro, and Sommen "Integral Theorems for Functions and Differential Forms in $... | 2 | https://mathoverflow.net/users/1849 | 297865 | 130,803 |
https://mathoverflow.net/questions/297850 | 9 | There is a theorem of Whitehead that lens spaces $L(p,q)$ and $L(p,q')$ are of the same homotopy type iff $\pm qq'≡ m^2 (\mathrm{mod}\ p)$ for some $m$. As a consequence, for a given $p$, there is only one homotopy type of $L(p,q)$ if $p=4k+3$ is prime, and two if $p=4k+1$ is prime (see Rolfsen's book *"Knots and Links... | https://mathoverflow.net/users/108829 | How many homotopy types of lens spaces L(p,q) if the given integer p is not prime? | No; in fact, there can be arbitrarily many homotopy types.
The theorem you quote says that the number of homotopy types, for a given $p$, is the same as the size of the following quotient group: $(\Bbb Z/p\Bbb Z)^\times$, the group of units in the ring $\Bbb Z/p\Bbb Z$, modded out by the subgroup generated by its squ... | 17 | https://mathoverflow.net/users/5091 | 297868 | 130,804 |
https://mathoverflow.net/questions/297871 | 1 | I was wondering : given a full rank lattice $\Lambda$ of $R^n$ (a discrete subgroup spanning $R^n$) the successive minima of $\Lambda$ are for $1\leqslant i \leqslant n$ $\lambda\_i= \min\{r>0 \mid \text{exists i linearly independants vectors of $\Lambda$ un the ball centered in 0 of radius r}\}$. Every text talking ab... | https://mathoverflow.net/users/123269 | Existence of linearly indépendants vectors reaching each minima of a lattice | Define the linearly independent vectors $u\_1,\dots,u\_n\in\Lambda$ recursively as follows. If $u\_1,\dots,u\_{m-1}$ has already been defined, then let $u\_m$ be the shortest lattice vector that is linearly independent of $u\_1,\dots,u\_{m-1}$. By the definition of $\lambda\_m$, it is clear that $|u\_m|\leq\lambda\_m$.... | 1 | https://mathoverflow.net/users/11919 | 297874 | 130,807 |
https://mathoverflow.net/questions/168608 | 6 | Does the system of congruence equations
\begin{eqnarray}
A\_{17k}&\equiv& 0 \pmod {17^2}, \nonumber \\
A\_{17k+1}&\equiv& 0 \pmod {17^2}, \tag{1}
\end{eqnarray}
has solutions other than $k=3$? Here $A\_n$ are Apery numbers:
$$A\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$
Thanks to the recurrence relation... | https://mathoverflow.net/users/32389 | Congruence equation for Apery numbers | Results of Gessel can be used to solve this question. Gessel showed, in [Theorem 1 here](https://www.sciencedirect.com/science/article/pii/0022314X82900713), that for any prime $p$, if $n=\sum d\_i p^i$ is the base-$p$ expansion of $n$, then
$$(\*) A\_n \equiv \prod A\_{d\_i} \bmod p.$$
In [Theorem 4 here](https://www.... | 6 | https://mathoverflow.net/users/31469 | 297901 | 130,817 |
https://mathoverflow.net/questions/297905 | 14 | The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F\_4, G\_2, E\_6, E\_7$ and $E\_8$.
I am trying to find out whether the classification was discovered first (attributed to Wilhelm Killing [1888-1890]), or whether some/all of th... | https://mathoverflow.net/users/103150 | Historically, which came first: the Lie algebras or their classification? | The classification came first. As Killing says in his [introduction](//archive.org/stream/mathematischean64behngoog#page/n12) (translation by Coleman ([1989](//mathscinet.ams.org/mathscinet-getitem?mr=1007036))):
>
> For each $l$ there are four structures supplemented for $l = 2, 4, 6, 7, 8$ by exceptional simple g... | 20 | https://mathoverflow.net/users/19276 | 297912 | 130,820 |
https://mathoverflow.net/questions/297900 | 1 | Consider a sum $$\sum\_{k=0}^{n-1}\sum\_{j=0}^{m}A\_{j,m}(n-k)^jk^j$$
which returns an odd power $n^{2m+1}$ of $n$, for $\ m=0,1,2,...$ given fixed $A\_{0,m}, \ A\_{1,m}, \ ..., \ A\_{m,m}$. The coefficients $A\_{0,m}, \ A\_{1,m},....$ are solutions of system of equations (refer to [.txt-file](https://kolosovpetro.gith... | https://mathoverflow.net/users/113033 | Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}, \ m=1,2,....$ | **EDIT 2018-04-16**: Formulae are corrected.
I'm not sure about connection with $\beta\_{mv}$, but we can obtain a recurrence formula for $A\_{j,m}$ as follows.
First let us fix the unused values $A\_{j,m}=0$ for $j<0$ or $j>m$, so we won't need to care about the summation range for $j$.
Expanding $(n-k)^j$ and u... | 4 | https://mathoverflow.net/users/7076 | 297916 | 130,821 |
https://mathoverflow.net/questions/297746 | 9 | Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}\_2(K) \to \text{GL}\_3(K)$ be the adjoint representation, obtained by $\text{GL}\_2(K)$ acting on $2 \times 2$ matrices with trace $0$ by conjugation. Suppose $\rho\_1, \rho\_2 : G \to \text{GL}\_2(K)$ are representations of a group $G$ s... | https://mathoverflow.net/users/123203 | Do two dimensional representations with the same adjoint representation differ by a character? | $\DeclareMathOperator\ad{ad}$This is true. The way I am viewing it, one has to make two cases, the representation $\ad V$ (and hence $\ad W$) is irreducible, or both are dihedral.
Part I: Assume that $\ad V$ and $\ad W$ are irreducible ($\rho\_1 : G \rightarrow GL(V)$ and $\rho\_2:G \rightarrow GL(W)$. Since $V$, $W... | 4 | https://mathoverflow.net/users/23291 | 297919 | 130,823 |
https://mathoverflow.net/questions/297902 | 1 | Assume that we are given a weighted, undirected graph $G = (V; E)$ where each edge $e \in E$ is assigned weight $w(e) \geq 0$. The goal is to remove a set of edges $D \subseteq E$ with minimum weight such that the remaining graph $G = (V; E\setminus D)$ has no triangles.
How can we formulate this problem as a set cov... | https://mathoverflow.net/users/nan | Removing the minimum number of edges to make a graph triangle-free (using set cover) | In order to formulate the given question as a set cover problem, let's assume that $T^{\*}$ to be the set of all triangles in the graph.
$T^{\*}=\{(e\_{i}, e\_{j}, e\_{k})\in E^{3}: e\_{i}, e\_{j}, e\_{k}$ form a triangle$\}$
For each $e$, let $T\_{e}$ denote the set of all triangles containing $e$. We define the ... | 0 | https://mathoverflow.net/users/nan | 297928 | 130,826 |
https://mathoverflow.net/questions/297926 | 2 | This question follows Noah Schweber's excellent answer to a corresponding question regarding second-order $ZFC$ and the continuum hypothesis: <https://mathoverflow.net/a/78083/24611>
Simply put, it seems that $ZFC\_2$ "decides" $CH$ in a certain sense that can be made formally precise, although second-order logic is ... | https://mathoverflow.net/users/24611 | A question about Second-Order ZF and the Axiom of Choice | Let me start by observing that we have to be a bit careful when talking about ZF$\_2$. Specifically, there is a subtle distinction between set models and class models which needs to be highlighted. In one sense, ZF proves that $V$ is a class model of ZF$\_2$ - in another sense, it can't even express this claim appropri... | 4 | https://mathoverflow.net/users/8133 | 297931 | 130,827 |
https://mathoverflow.net/questions/282781 | 4 | Let $X$ be a projective variety over an imperfect (hence infinite and char(k)=p>0) field $k$. If the local rings of $X$ are all regular, then can we say that a general hyperplane section $H$ is also regular? If it helps, you can assume any combination of the following hypothesis on $k$:
$k$ contains a perfect (infinite... | https://mathoverflow.net/users/80473 | Bertini's type theorems over imperfect fields | I needed to know the answer to this myself, so here is a good reference:
>
> Hubert Flenner, Liam O’Carroll, and Wolfgang Vogel, *Joins and
> intersections,* Springer Monographs in Mathematics, Springer-Verlag,
> Berlin, 1999. MR [1724388](https://mathscinet.ams.org/mathscinet-getitem?mr=1724388) DOI [10.1007/978... | 5 | https://mathoverflow.net/users/33088 | 297934 | 130,829 |
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