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https://mathoverflow.net/questions/312284 | 0 | We all know that given a symbol $a(x,\xi) \in S^{\mu,\rho}(\mathbb{R}^n,\mathbb{R}^n)$, a pseudo-differential operator can be defined as
\begin{equation}
Op(a)u(x)=(2\pi)^{-n}\int \int e^{i(x-x')\cdot \xi}a(x,\xi)u(x')dx'd\xi
\end{equation}
for $u$ in Schwartz class of test functions, $\mathcal{S}(\mathbb{R}^n)$. $... | https://mathoverflow.net/users/102092 | Associating a pseudo-differential operator to the symbol in the SG setting | The point is that the term arises if you want to compose pseudo-differential operators (the theorem on p.144). The symbol space is just the space of products of SG-ψdo's. This kind of "quantization" comes from the representation $Op(a)Op(b) = Op(c)$, where $$c(x,\xi) = e^{i<D\_\xi, D\_y>} a(x,\xi) b(y,\eta)|\_{y=x,\eta... | 1 | https://mathoverflow.net/users/20155 | 312414 | 135,869 |
https://mathoverflow.net/questions/312416 | 3 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We call it $T\_0$ if for all $x\neq y \in V$ there is $e\in E$ with $\{x,y\}\not\subseteq E$ and $\{x,y\}\cap e\neq \emptyset$ (i.e., $e$ contains exactly one of $x,y$).
If $H=(V,E)$ is a $T\_0$-hypergraph, it is possible that $|E|<|V|$: Let $... | https://mathoverflow.net/users/8628 | $|V|$ and $|E|$ in hypergraphs with a separation property | There is a map $m: V \to \mathcal P(E)$, picking out the edges a vertex is contained in. Given $x \neq y \in V$, there is an edge $e$ containing precisely one of $x$ or $y$, and so $x$ and $y$ are not contained in precisely the same collection of edges. Therefore $m$ is injective and $|V| \leq |\mathcal P(E)| = 2^{|E|}... | 4 | https://mathoverflow.net/users/40804 | 312423 | 135,871 |
https://mathoverflow.net/questions/312419 | 2 | Can someone please tell me the brief sketch (or any known reference) of the following results?
1. Why $\ell\_2$ is finitely representable in any infinite-dimensional Banach space?
2. Why every Banach space is finitely representable in $c\_0$?
A Banach space $Y$ is said to be finitely representable in some Banach sp... | https://mathoverflow.net/users/76412 | About finite representability of Banach space | 1 is very hard and 2 is very easy. For 2, take an $\varepsilon$-net $(x\_i)\_{i=1}^m$ in the sphere of the $n$ dimensional space ($m$ depends on $n$ and $\varepsilon$), and norming functionals $f\_i$'s. Now check that the map from the $n$-dimensional space into $\ell\_{\infty}^m\subset c\_0$ given by $x\to (f\_1(x), f\... | 8 | https://mathoverflow.net/users/3675 | 312434 | 135,873 |
https://mathoverflow.net/questions/312386 | 3 | Let $X$ be a Noetherian normal scheme. Denote by $B$ the singular locus of $X$. By normality, points in $B$ have codimension strictly greater 1. If $X$ is of finite type over field (or some J-2 ring), then it is known that $B$ is closed.
I am interested in the general situation without the J-2/finite type hypothesis.... | https://mathoverflow.net/users/120280 | Singular Loci in Noetherian normal schemes | Counterexamples surely exist. For instance, let $(R, \mathfrak{m})$ be a Noetherian normal local domain of dimension $3$ whose regular locus is not open; an explicit such $R$ can be found in Example 2.11 of Nishimura's "A few examples of local rings I." By normality, the singular locus $B$ of $\mathrm{Spec}(R)$ interse... | 4 | https://mathoverflow.net/users/5498 | 312451 | 135,876 |
https://mathoverflow.net/questions/312452 | 3 | Consider two massless representations of the connected Poincare group $ISO\_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless representations with helicity s are defined in [Wigner's classification](https://en.wikipedia.org/wiki/Wigner%27s_classificatio... | https://mathoverflow.net/users/56920 | tensor product of massless Poincare representations | I think the answer is given in the paper <https://aip.scitation.org/doi/10.1063/1.1703659> (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
| 4 | https://mathoverflow.net/users/32389 | 312455 | 135,877 |
https://mathoverflow.net/questions/312453 | 1 | Let $X$ be a GO-space with the topology $\tau$ and $\lambda$ be the usual open interval topology on $X$. Put
$$ R= \{x\in X: [x, \rightarrow) \in \tau\setminus \lambda \} \text{ and } L= \{x\in X: (\leftarrow,x] \in \tau\setminus \lambda \}. $$
Define $X^\* \subset X\times \mathbb{Z}$ as follows:
$$X^\*=(X\times \{0\})... | https://mathoverflow.net/users/129635 | A sequence in generalized order spaces | 1. Oddly enough, no: take $x\in R$ and let $a\_n=\langle x,-n\rangle$ for all $n$ (so $x\_n=x$ and $k\_n=-n$) then $\{x\_n\}$ is constant and converges to $x$, but $\{a\_n\}$ does not converge at all.
2. Yes; just consider cases, say if $x\in R\setminus L$ and $\{[x,c\_n):n\in\mathbb{N}\}$ is a local base for $X$ at $x... | 2 | https://mathoverflow.net/users/5903 | 312459 | 135,878 |
https://mathoverflow.net/questions/312454 | 7 | Assume $\pi$ is an unramified representation of $GL\_n(F)$, where $F$ is a p-adic field. And $\phi$ is an unramified vector for $\pi$. Assume $W\_{\phi}$ is a Whittaker function associated to $\phi$. Then whether $|W\_{\phi}(1)|^2=\frac{1}{L(1,\pi,Ad)}$?
I will appreaciate if you can show me a detailed calculation of... | https://mathoverflow.net/users/121163 | Corollary for Casselman-Shalika formula | The answer depends on your normalisation of $W$. Perhaps the most standard normalisation of $W$ is such that $W(1\_n) = 1$, where $1\_n$ is the $n \times n$ identity matrix.
Let $\pi$ be a unitary spherical representation of $\mathrm{GL}\_n(F)$ with Satake parameters $\alpha\_1,\ldots,\alpha\_n$. Let $\widetilde{W}$ ... | 7 | https://mathoverflow.net/users/3803 | 312461 | 135,879 |
https://mathoverflow.net/questions/312439 | 31 | This question isn't related to any specific research. I've been thinking a bit about how existence theorems are generally proven, and I've identified three broad categories: constructive proofs, proofs involving contradiction/contrapositive, and proofs involving the axiom of choice.
I'm convinced that there must be s... | https://mathoverflow.net/users/120414 | "Strange" proofs of existence theorems | There are [probabilistic proofs](https://en.wikipedia.org/wiki/Probabilistic_method) of existence. Do they fall into one of your three categories?
For example, prove the existence of a real number that is normal in all bases: To do it, we show that "almost all" real numbers (according to Lebesgue measure) have that p... | 51 | https://mathoverflow.net/users/454 | 312466 | 135,880 |
https://mathoverflow.net/questions/312465 | 14 | 1. Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
2. What if in Question 1 we restrict the cone angles to be $\leq \pi$?
3. Is it true that, if $M$ is a cone-manifold as in question 2, its volume is at most the simplicial v... | https://mathoverflow.net/users/42912 | Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$? | The results proved in
S. Kojima, "Deformations of hyperbolic 3-cone-manifolds",
J. Differential Geom. 49 (1998), no. 3, 469-516
provide complete answers to questions 1 and 3.
The main theorem of the cited paper ensures that any compact hyperbolic cone metric with cone angles at most π can be continuously deforme... | 14 | https://mathoverflow.net/users/6206 | 312470 | 135,883 |
https://mathoverflow.net/questions/312417 | 4 | Consider an undirected graph $K(n,k,i)$, with the all $k$-element subsets of $\{1,\dots,n\}$ as vertices, and two vertices connected by an edge if their sets intersect in less than $i$ elements.
This [paper](https://cdm.ucalgary.ca/cdm/index.php/cdm/article/download/557/259) claims on page 74 (Theorem 5.1) that the ... | https://mathoverflow.net/users/83212 | Understanding proof about chromatic number | This appears to be a genuine logical error. If $k$ is at least twice $i-1$ and $n$ is large enough that we can cover the sphere in a fine dust, then almost any pair of antipodal points will correspond to hemispheres $H\_1$, $H\_2$ containing at least $k-i+1$ points from sets $S\_1$, $S\_2$ with intersection of size $2(... | 4 | https://mathoverflow.net/users/25485 | 312478 | 135,888 |
https://mathoverflow.net/questions/312227 | 17 | My question is to find the minimum of the following expression:
$$A(x,y) = \sum\_{1\leq i<j\leq n} |x\_i-x\_j|\ |y\_i-y\_j|,$$
over the set of pairs of real vectors $x=(x\_1,\dots,x\_n),y=(y\_1,\dots,y\_n)$ subject to the following conditions:
$$\sum\_{i=1}^n x\_i^2 =\sum\_{i=1}^n y\_i^2 = 1, \sum\_{i=1}^n x\_i=\sum\_{... | https://mathoverflow.net/users/51663 | What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$? | The minimum must occur at vectors $x,y$ where
$x\_i$ and $y\_i$ take only two values each.
This should make it easy to check **Neil Strickland**'s
experimental result (where $x$ and $y$ are indeed of this form).
[*EDIT*: see the answer by **Adam P. Goucher** for this check.]
It is convenient to extend $A$ homogeneous... | 17 | https://mathoverflow.net/users/14830 | 312481 | 135,890 |
https://mathoverflow.net/questions/312479 | 6 | The computer found this, but couldn't prove it.
Let $\psi(n,x)$ denote the polygamma function.
With precision 500 decimal digits we have:
$$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \frac16)) $$
Is it true?
In machine readable form:
```
pi^2 == 1/4*(15*psi(1, 1/3) - 3*psi(1, 1/6))
```
| https://mathoverflow.net/users/12481 | $\pi$ in terms of polygamma | Note that
$$
\psi(m,x) =(-1)^{m+1} m! \sum\_{k=0}^{\infty} \frac{1}{(x+k)^{m+1}}.
$$
Therefore
$$
\psi(m,1/6) = (-1)^{m+1} m! \sum\_{k=0}^{\infty} \frac{1}{(k+1/6)^{m+1}} =(-1)^{m+1} m! 6^{m+1} \sum\_{n\equiv 1 \mod 6} \frac{1}{n^{m+1}}.
$$
Writing the condition $n\equiv 1 \mod 6$ as $n\equiv 1 \mod 3$ but not $... | 14 | https://mathoverflow.net/users/38624 | 312501 | 135,893 |
https://mathoverflow.net/questions/312510 | 4 | Let $X$ be a smooth complete algebraic variety over $\mathbb{C}$. Can it happen that the underlying complex manifold is not Kahler? If yes, are there explicit examples? If not - how to prove this?
| https://mathoverflow.net/users/3891 | Kahler manifolds and algebraic varieties | Nonprojective compact algebraic manifolds are never Kähler. Any compact algebraic manifold is Moishezon, and Moishezon's theorem says that a Moishezon manifold is Kähler if and only if it is a projective variety.
| 13 | https://mathoverflow.net/users/1310 | 312511 | 135,896 |
https://mathoverflow.net/questions/312509 | 2 | Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?
| https://mathoverflow.net/users/60732 | Is every prime greater than 5, less than the sum of the two previous primes? | I am to elaborate a bit on how it is that the answer I left in [this previous](https://mathoverflow.net/q/113840/1593) question implies a positive answer to Mr. Recamán's question.
By elementary means (i.e., without resorting to any complex analysis), in his paper [*Mémoire sur les nombres premiers*](http://sites.mat... | 10 | https://mathoverflow.net/users/1593 | 312513 | 135,897 |
https://mathoverflow.net/questions/312515 | -2 | Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f\_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent subsequence, say $f\_{n\_k}$, converging to $f$. Can we say that $f\_n\to f$ strongly?
| https://mathoverflow.net/users/113264 | Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? | Let $(f\_n)$ be any sequence which converges to $f$ weakly but not strongly. Define a new sequence $(g\_k)$ by setting $g\_{2n} = f\_n$ and $g\_{2n+1} = f$. Then $(g\_k)$ shows that the answer to the question is negative.
| 4 | https://mathoverflow.net/users/30721 | 312517 | 135,898 |
https://mathoverflow.net/questions/312520 | 3 | Suppose we have a given action $\varphi : A\rightarrow\text{Aut}(N)$ with $A,N$ abelian groups. Is it possible describe the isomorphism classes of extensions $G$ of $A$ by $N$ realizing $\varphi$ such that the map $G\rightarrow A$ is abelianization (ie, such that $N = G'$)?
(Without the requirement $N = G'$ this is c... | https://mathoverflow.net/users/88840 | Is it possible to classify extensions $G$ of an abelian $A$ by an abelian $N$ such that the map $G\rightarrow A$ is abelianization? | When $A$ acts trivially on $N$, this $H^2(A,N)$ has a canonical map $\Phi$ into the group $\mathrm{Hom}(\Lambda^2A,N)$ ($\Lambda^2A$ being the second exterior power, quotient of $A\otimes\_\mathbf{Z}A$ by the subgroup generated by elements of the form $x\otimes x$ when $x$ ranges over $A$), induced by the commutator ma... | 4 | https://mathoverflow.net/users/14094 | 312521 | 135,899 |
https://mathoverflow.net/questions/312530 | 6 | Suppose $f:X\rightarrow \mathbb{P}\_k^n$ is a proper smooth morphism, where $k$ is an algebraically closed field. If $f$ admits a rational section, can we prove that the fiber of $f$ above any codimension 1 point contains a geometrically integral open subscheme?
| https://mathoverflow.net/users/126961 | To prove the fiber above a codimension 1 point contains a geometrically integral open subscheme | This is true:
Because $f$ is smooth, any fibre $X\_y$ is smooth. If $y$ has codimension $1$, then the rational section $\sigma \colon \mathbb P^n\_k \dashrightarrow X$ is defined at $y$ by the valuative criterion of properness. Thus, $X\_y$ has a rational point $\sigma(y)$. If $U \subseteq X\_y$ is the connected comp... | 7 | https://mathoverflow.net/users/82179 | 312535 | 135,904 |
https://mathoverflow.net/questions/209023 | 16 | [This problem](https://math.stackexchange.com/questions/1278465/maximum-number-of-sylow-subgroups/) (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the problem came from, even though he/she... | https://mathoverflow.net/users/62135 | Origin of group theory problem (bound on number of Sylow subgroups) | Since this question has resurfaced (after 3 years) let me
say something about the origin of this Monthly problem.
---
A few years back a colleague gave me the task of making
up the algebra preliminary exam for our first-year
graduate students. Among the group theory problems that made
my initial list were:
*... | 8 | https://mathoverflow.net/users/75735 | 312537 | 135,905 |
https://mathoverflow.net/questions/312533 | 7 | In a set-theoretic system using first-order logic, every proof could be written as a goal followed by a finite sequence of sentence where each one is justified by an axiom or previously established sentence and the last line is the goal. Computers can easily verify proofs like these using the rules of first-order logic... | https://mathoverflow.net/users/126581 | Are there logical systems where formal proofs are not computer verifiable? | There are logical systems whose formal proofs are not computer verifiable. One such example is [infinitary logic](https://en.wikipedia.org/wiki/Infinitary_logic) in which logical statements can be infinitely long, and a specific statement in a proof may require infinitely many premises to be checked. Such logical syste... | 18 | https://mathoverflow.net/users/1176 | 312546 | 135,907 |
https://mathoverflow.net/questions/312404 | 10 | Let $G$ be a $2$-edge-connected graph drawn in the plane (such that the edges intersect only at the endpoints). I want to orient
the edges of $G$ such that for each vertex $v$, there are no
three consecutive edges (in the clockwise direction) such that all of them
are oriented towards $v$ or all of them are oriented o... | https://mathoverflow.net/users/81107 | Orientations of Planar Graphs | Such an orientation always exists, here is a proof.
Take your 2-edge-connected graph $G$, and consider its dual graph $D$. $D$ has a proper 4-coloring in which each face of $D$ contains at most 3 different colors (add a vertex inside each face of $D$, connect it to all the vertices of the face, and apply the four col... | 10 | https://mathoverflow.net/users/45855 | 312556 | 135,910 |
https://mathoverflow.net/questions/312493 | 5 | Let $N \in \mathbb N$ and suppose that $\phi$ is a submeasure on $[1,N] = \{1,2,\dots,N\}$, by which I mean that $\phi$ is a function $\mathcal P ([1,N]) \rightarrow \mathbb R$ such that
i. $A \subseteq B$ implies $\phi(A) \leq \phi(B)$,
ii. $\phi(A \cup B) \leq \phi(A) + \phi(B)$ for any $A, B \subseteq [1,N]$, an... | https://mathoverflow.net/users/70618 | What is the expected value of the submeasure of a random set? | Let $n$ be chosen arbitrarily and let $S\_k=\{1,2,\ldots,n\}^k$ and let $\mathcal P$ be the collection of all Cartesian products of the form $A\_1\times\ldots\times A\_k$ where $|A\_i|=n-1$. For $Z\subset S\_k$, define $\phi(Z)=\min\{j\colon \exists P\_1,\ldots,P\_j\in \mathcal P:P\_1\cup\ldots\cup P\_k\supset Z\}$.
... | 5 | https://mathoverflow.net/users/11054 | 312557 | 135,911 |
https://mathoverflow.net/questions/312544 | 1 | I have a recurrence,
$$F(n, m) = F(n-1, m) + F(n, m-1) + F(n-1,m-1) $$
$$F(n,1) = 0$$
$$F(1,n) = 2\*(n-1)$$
I would like to compute $F(N,M)$ in terms of $N$ and $M$. The system is defined for $1 \leq n \leq N$ and $1 \leq m \leq M$ where $N$ and $M$ are non-negative integers.
I have solved many linear recurrence... | https://mathoverflow.net/users/129956 | Finding closed form of recurrence relation in two variables | $F(n,m)=D(n,m-1)-D(n-1,m-1)$ where $D(n,m)$ are the [Delannoy numbers](https://en.wikipedia.org/wiki/Delannoy_number).
| 2 | https://mathoverflow.net/users/2480 | 312558 | 135,912 |
https://mathoverflow.net/questions/312551 | 5 | Let $A$ be an Artin algebra of finitistic dimension zero and $B$ an algebra derived equivalent to $A$. Does $B$ also have finitistic dimension zero?
In case this is true, this might generalise the result that selfinjective algebras are closed under derived equivalence. (Since selfinjective algebras have finitistic di... | https://mathoverflow.net/users/61949 | Derived equivalences of Artin algebras with finitistic dimension zero | Let $A$ be the radical square zero algebra whose quiver has two vertices $1$ and $2$, with a loop at each vertex and an arrow from vertex $1$ to vertex $2$. Then the projective (right) modules have structure
$$P\_1=\matrix{&1&\\1&&2}\mbox{$\quad$ and $\quad$}P\_2=\matrix{2\\2},$$
and the left and right finitistic dimen... | 3 | https://mathoverflow.net/users/22989 | 312561 | 135,913 |
https://mathoverflow.net/questions/312499 | 7 | This question was posted on MSE and got very little attention, so I'm also posting it here.
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \mathbf{Set})$ be its category of presheaves regarded as a closed symmetric monoidal category via Day convolution of pr... | https://mathoverflow.net/users/101861 | Dualizable presheaves with respect to Day convolution |
>
> **Lemma:** In a closed symmetric monoidal category where the unit object $1$ is [tiny](https://ncatlab.org/nlab/show/tiny+object) (meaning $\text{Hom}(1, -)$ preserves colimits), every dualizable object is tiny.
>
>
>
*Proof.* If $x$ is dualizable, then $\text{Hom}(x, -) \cong \text{Hom}(1, x^{\ast} \otimes ... | 8 | https://mathoverflow.net/users/290 | 312571 | 135,916 |
https://mathoverflow.net/questions/254453 | 6 | Given a finite dimensional algebra $A$, define $\Omega^{n}(mod-A)$ (modules here are always finite dimensional) to be the full subcategory of projective modules or modules $M$ such that $M \cong \Omega^{n}(N)$ for some other module $N$. How can one check in an easy way to see whether a given module $M$ (we can assume i... | https://mathoverflow.net/users/61949 | How to check whether a module is an n-th syzygy | A method for this is now implemented in QPA as IsNthSyzygy.
For a given indecomposable module $M$, one checks whether $M$ is a direct summand of $\Omega^n(\Omega^{-n}(M)) \oplus P(M)$ when $P(M)$ is the projective cover of $M$.
| 4 | https://mathoverflow.net/users/61949 | 312573 | 135,918 |
https://mathoverflow.net/questions/312577 | -1 | I stuck in determining the expected value of the following product
$E[W\_{t\_i}W\_{t\_{i+1}}^2]$ where $W\_{t\_i}$ and $W\_{t\_{i+1}}$ are Brownian with normal distribution, i.e. $W\_{t\_i}\sim N(0,t\_i)$. I would appreciate if anybody give me a hint about this.
| https://mathoverflow.net/users/129994 | Expected value of $W_{t_i} W^2_{t_{i+1}}$ | Assuming that $t\_{i+1} \ge t\_i$, let $X = W\_{t\_i}$, $Y = W\_{t\_{i+1}} - W\_{t\_i}$. Note that $X,Y$ each have a centered normal distribution and are independent. Then the desired quantity is
$$\begin{align\*}E[X(X+Y)^2] &= E[X^3 + 2 X^2 Y + XY^2] \\
&= E[X^3] + 2 E[X^2] E[Y] + E[X] E[Y^2] && \text{by linearity and... | 0 | https://mathoverflow.net/users/4832 | 312586 | 135,924 |
https://mathoverflow.net/questions/311829 | 7 | One well-known, extremely slick proof of Cayley's tree enumeration theorem is the use of Prüfer sequences. Cayley also proved a version for forests, namely that the number of forests with $n$ labelled vertices that consist of $s$ distinct trees such that $s$ specified vertices belong to distinct trees is $sn^{n-s-1}$; ... | https://mathoverflow.net/users/125803 | Are there Prüfer sequences for rooted forests? | We can prove this using ordinary Prüfer sequences; no generalization is needed.
First I'll describe
Prüfer's correspondence, as a bijection from trees on $\{0,1,2,\ldots,n\}$, where $n\ge1$, to
sequences $a\_1 \ldots a\_{n-1}$ of elements of $\{0,1,2,\ldots,n\}$:
If $n=1$ then $a\_1\ldots a\_{n-1}$ is empty. Otherwi... | 5 | https://mathoverflow.net/users/10744 | 312587 | 135,925 |
https://mathoverflow.net/questions/312566 | 4 | The starting point of this question is a slight reformulation of the [$T\_0$ separation axiom](https://en.wikipedia.org/wiki/Kolmogorov_space): A topological space $(X,\tau)$ is $T\_0$ if for all $x\neq y\in X$ there is a set $U\in \tau$ such that $$\{x,y\}\cap U \neq \emptyset \text{ and } \{x,y\}\not\subseteq U.$$
... | https://mathoverflow.net/users/8628 | Generalizing the $T_0$-axiom | Fleshing out [Will Brian](https://mathoverflow.net/users/70618/will-brian)'s [suggestion](https://mathoverflow.net/questions/312566/generalizing-the-t-0-axiom/312591#comment779594_312566) (and saving him the trouble of writing it up):
If $\kappa$ is finite you topologize $\lambda$ using the base $[\kappa-1,\lambda)$ ... | 3 | https://mathoverflow.net/users/5903 | 312591 | 135,926 |
https://mathoverflow.net/questions/312357 | 10 | From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators
$$\mu\_n := \sum\_{i=0}^{n} (-1)^{n-k}\binom{n}{k} \psi^k$$
However, from the collapse of the Atiyah-Hirzebruch spectral sequence for BU, we ... | https://mathoverflow.net/users/120548 | How to write K-theory Conner-Floyd Chern classes in terms of Adams operations? | I am not sure about the facts you mention, and I don't think I'll quite answer your question, but here are some facts I do know.
First, it is not the case that all $KU$-operations can be written as (even infinite) sums of Adams operations; Adams operations are additive, and general $KU$-operations need not be. So I w... | 10 | https://mathoverflow.net/users/111541 | 312595 | 135,929 |
https://mathoverflow.net/questions/312592 | 9 | Let $\mathcal U$ be a free ultrafilter on a set $X$. For $n\in\mathbb N$ let $\mathcal F$ be a family of $n$-element subsets of $X$ such that $\bigcup\mathcal F\in\mathcal U$.
>
> **Question.** Is there a set $U\in\mathcal U$ and a subfamily $\mathcal E\subset\mathcal F$ such that $\bigcup\mathcal E\in\mathcal U$ a... | https://mathoverflow.net/users/61536 | A property of an ultrafilter | Here's a try for $n=3$ which I think generalizes to any $n$.
Well-order $\mathcal{F}$ as $\{F\_\alpha\}$. By discarding any $F\_\alpha$ which is contained in $\bigcup\_{\beta <\alpha} F\_\beta$, we can ensure that each $F\_\alpha$ contains at least one point which is not in any previous $F\_\beta$, without affecting ... | 7 | https://mathoverflow.net/users/23141 | 312599 | 135,930 |
https://mathoverflow.net/questions/312539 | 1 | Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask for the amount of collisions $X$ in $M$, where a collision is defined as a pair of indices $(i,j)$, $i\neq j$ with $M[... | https://mathoverflow.net/users/89983 | lower bound the probability of at least L collisions | **Main claim**.
$$\Pr[|X - \mathbb{E} X| \geq t] \leq \frac{\mathbb{E} X}{t^2} \approx \frac{L}{t^2}. $$
You can bound the variance and use Chebyshev's Inequality as you suggest, and the calculations are not pretty but it helps if someone has already done them. Here's a sketch. Let's write $m = |M|$ for the number of ... | 2 | https://mathoverflow.net/users/29697 | 312605 | 135,933 |
https://mathoverflow.net/questions/312604 | 7 | Let $\lambda$ be an integer partition, denote the number of [Standard Young Tableaux](https://en.wikipedia.org/wiki/Young_tableau) of shape $\lambda$ by $f\_{\lambda}$. This number is [computed by](https://en.wikipedia.org/wiki/Hook_length_formula) the formula
$$f\_{\lambda}=\frac{n!}{\prod\_{u\in\lambda}h\_u}$$
where ... | https://mathoverflow.net/users/66131 | SYT and contents of a partition | Notice that $f\_{\lambda}$ is the number of standard Young tableaux of shape $\lambda$, whereas $f\_{\lambda}\frac{\prod\_{u\in \lambda} (t+c\_u)}{n!}$ is the number of semistandard Young tableaux of shape $\lambda$ and content $t$. It was mentioned in a [previous question](https://mathoverflow.net/questions/245076/an-... | 6 | https://mathoverflow.net/users/2384 | 312609 | 135,934 |
https://mathoverflow.net/questions/312553 | 24 | I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version.Here's what I'm trying to prove.
>
> let $x\_{1}\ge x\_{2}\ge\cdots\ge x\_{n}\ge 0,y\_{1}\ge y\_{2}\ge\cdots\ge y\_{... | https://mathoverflow.net/users/38620 | show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$ | Let $x:=(x\_1,\dots,x\_n)$ and $y:=(y\_1,\dots,y\_n)$; we identify $x$ and $y$ with the corresponding functions on the set $[n]:=\{1,\dots,n\}$. Take any real $S,T\ge0$ and nonnegative integers $n\_-$ and $n\_+$ such that $n\_-+n\_+\le n$. Let $Z=Z(n,n\_-,n\_+,S,T)$ denote the set of all pairs $(x,y)\in[0,\infty)^n\tim... | 7 | https://mathoverflow.net/users/36721 | 312615 | 135,938 |
https://mathoverflow.net/questions/312596 | 0 | Let $X$ be a smooth, projective variety and $F$ a coherent sheaf on $X$. Let $\{U\_i\}\_{i \in I}$ be an open affine covering of $X$ and $\{f\_{ij}\}\_{i<j}$ with $f\_{ij} \in \Gamma(U\_{ij},F)$ satisfying the cocycle condition. Denote by $\alpha \in H^1(X,F)$ corresponding to the collection $\{f\_{ij}\}\_{i<j}$.
Cho... | https://mathoverflow.net/users/32151 | Naive question in Cech cohomology | Let $U = \coprod\_i U\_i$ and $U'=V \amalg \coprod\_i U\_i$. Since there is $V \to U\_{i\_0}$, there is a map $U' \to U$ over $X$, and hence a map $U'\times\_X U'\to U\times\_X U$ (also over $X$). The original cocycle is a map $U\times\_X U\to F$, and then the new cocycle is the induced map $U'\times\_X U' \to U\times\... | 2 | https://mathoverflow.net/users/4177 | 312616 | 135,939 |
https://mathoverflow.net/questions/312017 | 1 | This is a cross-post from [signal processing forum](https://dsp.stackexchange.com/questions/52272/maximum-magnitude-deviation-between-dft-and-dtft) as it was not conclusive.
Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum\_{n = 0}^{N-1} x[n] e^{-j \om... | https://mathoverflow.net/users/51478 | Maximum Magnitude Deviation between DFT and DTFT | I believe to have found a partial answer in the paper:
Gronwall, T. H. (1921). A sequence of polynomials connected with the $n$th roots of unity. Bulletin of the American Mathematical Society, 27(6), 275–279. <http://doi.org/10.1090/S0002-9904-1921-03411-2>
It actually gives a full construction of a polynomial $F(z... | 1 | https://mathoverflow.net/users/51478 | 312617 | 135,940 |
https://mathoverflow.net/questions/312620 | 6 | Let $K\_1$ the trefoil (left or right hopefully does not matter?) and let $K\_2$ be the figure-eight knot in $S^3 = \partial B^4$. Are there any smooth properly embedded annulus $A$ in $B^4$ with $\partial A = K\_1 \coprod K\_2$?
I'm really interested in knowing some general obstructions for this sort of thing for mo... | https://mathoverflow.net/users/99414 | Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$? | The relationship you're asking for is called **concordance**. Determining if knots are concordant is quite difficult: there are many concordance invariants, but no kind of global picture of what it means.
One concordance invariant is the knot signature. The trefoil has signature $-2$ and the figure-8 knot has signatu... | 11 | https://mathoverflow.net/users/317 | 312623 | 135,942 |
https://mathoverflow.net/questions/312618 | 10 | The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
1. What is the significance of the meanings of "Anderson dual" to the sphere spectrum? (For example, I understand the significant meanings of Pon... | https://mathoverflow.net/users/27004 | Sphere spectrum, Character dual and Anderson dual | The Anderson dualizing spectrum $I\_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi\_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded abelian groups. Since $\mathbf{Q/Z}$ is an injective $\mathbf{Z}$-module, this functor is representable by a spectrum $I\_\... | 16 | https://mathoverflow.net/users/102390 | 312628 | 135,944 |
https://mathoverflow.net/questions/312625 | 8 | **Background:** I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property holds, but upon closer inspection it holds for some primes and not others and now I am wondering if this "elementary... | https://mathoverflow.net/users/10930 | For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.) | It's easier to work with the inverse permutation. Starting from $1$ and working backwards we end up with a sequence $1=2^0\to \pm 2^1\to \pm 2^2\to \cdots \to\pm 2^{\frac{p-1}{2}}=1$. Where the signs are uniquely determined so that $\pm 2^r\in \{1,2,\dots,\frac{p-1}{2}\}$. This cycle covers all numbers in $\{1,2,\dots,... | 10 | https://mathoverflow.net/users/2384 | 312632 | 135,946 |
https://mathoverflow.net/questions/312584 | 5 | Let $\varphi$ be a plurisubharmonic function in the unit ball $B\_1\subset \mathbb{C}^n$ with $\varphi\le 0$. Suppose that the Lelong number $\nu(\varphi,0)<k$ for some $k>0$. Does it follow that there exists $\alpha>0$, possibly depending on $k$, such that $\int\_{B\_{\frac{1}{2}}}e^{-\alpha\varphi}dvol(z)\le C$?
No... | https://mathoverflow.net/users/130000 | Lelong numbers and integrability of psh functions | **The answer is yes** :
If the Lelong number of a plurisubharmonic function
$\varphi$ at a point $a$ satisfies the condition $\nu(\varphi,a) < 2$, then the function $e^{-\varphi}$ is locally
integrable with respect to the Lebesgue measure in a neighbourhood of $a$.
It follows from a result of Skoda, Proposition 7.1... | 2 | https://mathoverflow.net/users/89429 | 312648 | 135,951 |
https://mathoverflow.net/questions/312638 | 9 | Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every finite simplicial set $K$ we have an isomorphism of the form: $$\mathcal{H}(K,X) \cong \mathcal{H}(K,Y)$$ induced by postc... | https://mathoverflow.net/users/57280 | Do finite simplicial sets jointly detect isomorphisms in the homotopy category? | The answer is no. Otherwise, it would follow from Brown's representability theorem (and here I mean very specifically Theorem 2.8 from Brown's 1965 paper *Abstract Homotopy Theory*) that every "half-exact" functor on the homotopy category of unbased simplicial sets is representable. That is however false by my answer [... | 7 | https://mathoverflow.net/users/12547 | 312649 | 135,952 |
https://mathoverflow.net/questions/312650 | 5 | I am trying to understand direct limits in the category of $C^\*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
>
> Let $(A\_n,f\_n)$ be a direct sequence of $C^\*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim\_{\rightarrow} M\_... | https://mathoverflow.net/users/129638 | Behaviour of direct limit with matrices | You’re asking whether the functor $M\_2$ on Banach spaces preserves colimits of direct sequences.
(In case you’re not familiar with the categorical terminology I’m using here, don’t be put off — it’s not very deep, it’s just useful packaging-up of the kind of properties you’re discussing in the question, and hopefull... | 10 | https://mathoverflow.net/users/2273 | 312651 | 135,953 |
https://mathoverflow.net/questions/312652 | 13 | It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group, there exists a unique probability over $K$ that is left-invariant (Haar measure). The same problem as above still mak... | https://mathoverflow.net/users/8799 | Probability of commutation in a compact group | Edit: let me reformulate my answer since I apparently didn't answer the right question.
(a) if $K$ is a compact Lie group, the commuting probability is positive iff $K\_0$ is abelian. As you noticed, $\Leftarrow$ is trivial.
Conversely, assume that $K\_0$ is not abelian. We can view $K$ as Zariski-closed in matrix ... | 13 | https://mathoverflow.net/users/14094 | 312655 | 135,954 |
https://mathoverflow.net/questions/312656 | 6 | **Short version of question.** Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph\_0}$ such that all points of $S$ have distinct pairwise distances?
**Formal version of question.** If $X$ is a set, let $[X]^2=\big\{\{x,y\}:x\neq y\in X\big\}$. Let $(X,d)$ be a metric space. Let $d\_{\text{set}}:[X]^2\to \mathbb{R}... | https://mathoverflow.net/users/8628 | Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances? | Consider $\Bbb R$ as a vector space over $\Bbb Q$, choose (Zorn) a basis $S$ consisting (after rescaling with rational scalars) of elements in $(0,1)$. Then we have no relation of the shape $\pm(s\_1-s\_2)=(s\_3-s\_4)$ for $s\_1,s\_2,s\_3,s\_4\in S$. The cardinality of the basis is also the required one.
| 13 | https://mathoverflow.net/users/122945 | 312658 | 135,955 |
https://mathoverflow.net/questions/291158 | 152 | Are there examples of originally widely accepted proofs that were later discovered to be wrong by attempting to formalize them using a proof assistant (e.g. Coq, Agda, Lean, Isabelle, HOL, Metamath, Mizar)?
| https://mathoverflow.net/users/119858 | Proofs shown to be wrong after formalization with proof assistant | Since this question was asked in January there have been some developments. I would like to argue that the scenario raised in the question has now actually happened. Indeed, Sébastien Gouëzel, when formalising Vladimir Shchur's work on bounds for the Morse Lemma for Gromov-hyperbolic spaces, found an actual inequality ... | 145 | https://mathoverflow.net/users/1384 | 312661 | 135,957 |
https://mathoverflow.net/questions/312664 | 4 | I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}\_{\geq 0}$, and for each 1-cochain $\xi: G \rightarrow GL\_r\left(L\right)$, we define $W\_{\xi}$ to be $L^r$ with equipped with the f... | https://mathoverflow.net/users/120479 | Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism | Your mistake is in the meaning of $\xi$ and $\xi'$ being cohomologous. The condition should be
$\xi\_\sigma =f^\sigma \xi'\_\sigma f^{-1}$, for some $f\in GL\_r(L)$. This is the correct definition of cohomologous cocycles in the non-abelian cituation, and then $f$ just give you the isomorphism.
In general one shoul... | 7 | https://mathoverflow.net/users/115052 | 312669 | 135,958 |
https://mathoverflow.net/questions/312663 | 4 | W. Veech on **[Teichmüller curves in moduli space, Eisenstein series and applications to triangular billiards](https://eudml.org/doc/143714)** says on the second paragraph of page 579:
"Thurston's original construction [8] corresponds to the case $r=s=1$." In this context [8] refers to a preprint of Thurston. A refe... | https://mathoverflow.net/users/892 | Thurston's preprint: "On the geometry and dynamics of diffeomorphisms of surfaces" | As you know, the preprint was published in the Bulletin of the AMS [here](http://www.ams.org/journals/bull/1988-19-02/S0273-0979-1988-15685-6/). There, at the end of the preface, Thurston writes:
>
> There would be no simple stopping point if I began to incorporate the more recent developments in the original paper... | 5 | https://mathoverflow.net/users/353 | 312672 | 135,959 |
https://mathoverflow.net/questions/312645 | 9 | Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a\_{k})\in \ell^{p}(\mathbb{Z})$ and $(b\_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]) that
$$\sum\_{j\neq k}\frac{a\_{j}b\_{k}}{|j-k|^{\lambda}}\leq C\_{p,q} \parallel a \parallel\_{p}\parallel b \parallel\_{q}.$... | https://mathoverflow.net/users/130023 | The discrete Hardy-Littlewood-Sobolev inequality | Write $$\sum\_{j\ne k,\,j,k=1,\dots,N} \frac{|a\_kb\_j|}{|j-k|^{\alpha}}=\sum\_{d=1}^{N}\frac{1}{d^{\alpha}} \Big(\sum\_{l=1}^{N-d} |a\_lb\_{l+d}| + \sum\_{l=1}^{N-d} |a\_{l+d}b\_{l}|\Big).$$
By Holder's inequality one has
$$\sum\_{l=1}^{N-d} |a\_lb\_{l+d}|\le \Big(\sum\_{l=1}^{N-d} |a\_l|^p \Big)^{1/p}\Big(\sum\_{l=... | 3 | https://mathoverflow.net/users/127070 | 312681 | 135,963 |
https://mathoverflow.net/questions/312679 | 7 | Find the number of ways of selecting k cells from a $(2\times n)$-board such that no two selected cells share a side (non-adjacent).
For $n=3$ and $k=2$, the answer is $8$; for $n=5$ and $k=3$, the answer is $38$.
Counting manually works for small numbers.
>
> **Question.** Is there any formula that can be d... | https://mathoverflow.net/users/130047 | Counting configurations on a 2xn board under restrictions | $\dots$ continued from above: to find the coefficient of $t^nq^k$, rewrite the generating function as
\begin{align}G(t,q):=\frac{t+(2t+t^2)q}{1-t-(t+t^2)q}
&=\frac{t+(2t+t^2)q}{1-t}\cdot\frac1{1-\left(\frac{t+t^2}{1-t}\right)q} \\
&=\frac{t+(2t+t^2)q}{1-t}\sum\_{j\geq0}\left(\frac{t+t^2}{1-t}\right)^jq^j \\
&=\sum\_{k\... | 9 | https://mathoverflow.net/users/66131 | 312688 | 135,968 |
https://mathoverflow.net/questions/306929 | 5 | Let $u$ be a solution of the heat equation $$u\_t - u\_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u\_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly the heat kernel)
$$\sup\_{t>0}\int\_{\mathbb{R}} t^\alpha |u\_x|^2 \ dx,$$
in terms of the initial data? Could you point out... | https://mathoverflow.net/users/122620 | Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$ | I'm sorry for the late answer, but joined MathOverflow just this week. The Fourier Splitting method, developed by María Elena Schonbek in the 80's asserts that "decay is determined by the low frequencies of the solutions" for many dissipative linear and nonlinear equations (heat, fractional heat, Navier-Stokes, dissipa... | 4 | https://mathoverflow.net/users/129968 | 312695 | 135,971 |
https://mathoverflow.net/questions/312685 | 1 | I am interested in the closed form solution to the following problem:
$\int\_a^b\int\_0^{cx+d}(x+e)f(x)f(y)dydx$, where $f(.)$ is the pdf of the lognormal distribution with mean $0$ and variance $\sigma^2$. $a,b,c,d,e$ are constants. Say $F(.)$ is the corresponding cdf.
I got as far as
$\int\_a^b\int\_0^{cx+d}(x+e... | https://mathoverflow.net/users/130049 | Solving integral over the space of two lognormally distributed variables | For $c=0$ there is a lengthy closed form,
$$\int\_a^b\int\_0^{d}(x+e)f(x)f(y)dydx=\frac{1}{4} \left(\text{erf}\left(\frac{\ln d}{\sqrt{2} \sigma}\right)+1\right) $$
$$\qquad\qquad\times\left(e \left(\text{erf}\left(\frac{\ln b}{\sqrt{2} \sigma}\right)-\text{erf}\left(\frac{\ln a}{\sqrt{2} \sigma}\right)\right)+e^{\sigm... | 0 | https://mathoverflow.net/users/11260 | 312700 | 135,975 |
https://mathoverflow.net/questions/312636 | 3 | A sequence $a\_0,a\_1,\dots,a\_n$ of real numbers is called *concave* if $a\_{0}=0$, and for each $0<i<n$, we have $a\_i\geq\dfrac{a\_{i-1}+a\_{i+1}}{2}$.
Find the largest $c(n)$ such that for every concave sequence $a\_0,a\_1,\dots,a\_n$ of non-negative real numbers, we have
$$c(n)\sum\_{i=1}^{n}a^2\_{i}\le \left(\s... | https://mathoverflow.net/users/38620 | it's convex sequence inequality | $\newcommand{\R}{\mathbb{R}}$
The constant $c(n)$ can be improved from $\frac{n-1}2$ to the optimal value
\begin{equation}
c\_\*(n):=\frac{3n(n-1)}{2(2n-1)}
\end{equation}
for $n\ge2$.
Indeed, for $i\in[n]:=\{1,\dots,n\}$, let
\begin{equation}
g\_i:=a\_i-a\_{i-1},\quad h\_i:=\frac12\,(g\_i-g\_{i-1})=\frac12\,(a... | 5 | https://mathoverflow.net/users/36721 | 312701 | 135,976 |
https://mathoverflow.net/questions/312691 | 45 | Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$?
Remark. I am aware of examples (due to Mike Davis) of compact $n$-manifolds whose universal covering spaces are fake $R^n$'s i.e. are contractile but not homeomorphic to $R^n$.
| https://mathoverflow.net/users/39654 | Exotic $R^4$ as the universal covering space | This is problem 4.79(A) of Kirby's 1995 problem list, contributed by Gompf. It is my impression that it is still open.
There is some small progress: Remark 7.2 in [this article](https://arxiv.org/pdf/math/9712292.pdf) observes that their constructions imply that a specific countable set of examples of exotic $\Bbb R... | 40 | https://mathoverflow.net/users/40804 | 312711 | 135,980 |
https://mathoverflow.net/questions/312708 | 8 | Let $\lambda$ be an integer partition: $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq0)$. Denote its [conjugate partition](http://mathworld.wolfram.com/ConjugatePartition.html) by $\lambda'$. For example, if $\lambda=(4,3,1)$ then $\lambda'=(3,2,2,1)$.
Recall also the notation for the content of a cell $u=(i,j)$ in ... | https://mathoverflow.net/users/66131 | A link between hooks, contents and parts of a partition | Another approach is to notice that $$\sum\_{u\in\lambda}h\_u=\sum\_{u}d\_u$$
where $d\_u=i+j-1$ for $u=(i,j)$.
**Proof:** The easiest way to see this is that both sides count the number of pairs $\{(i\_1,j\_1),(i\_2,j\_2)\}\in\lambda$ such that either $i\_1=i\_2$ and $j\_1\le j\_2$ or $j\_1=j\_2$ and $i\_1\le i\_2$.... | 8 | https://mathoverflow.net/users/2384 | 312717 | 135,984 |
https://mathoverflow.net/questions/312706 | 7 | Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E\_2$-algebras: <https://arxiv.org/pdf/1309.2820.pdf>
I was wondering if anybody knows of a nice relationship between the categories of left modules over a Hopf Algebra and left modules over the underlying e1 structure of an $E\_2$ alg... | https://mathoverflow.net/users/124286 | Modules over Hopf Algebras and $E_2$-algebras | Let $A$ be a brace algebra and $B$ the Koszul dual bialgebra. There is a natural adjunction
$$
\Omega\colon \mathrm{CoMod}\_B\rightleftarrows \mathrm{LMod}\_A
$$
where, for instance, the functor $\mathrm{LMod}\_A\rightarrow\mathrm{CoMod}\_B$ sends $M\mapsto M\otimes B$ equipped with some standard differential.
For $\... | 6 | https://mathoverflow.net/users/18512 | 312725 | 135,987 |
https://mathoverflow.net/questions/312731 | 3 | It is easy to see that for any finite Abelian group $G$ and any numbers $a,b$ with $|G|=ab$ there exist a subgroup $A\subset G$ and a subset $B\subset G$ such that $|A|=a$, $|B|=b$ and $G=A+B$, where $A+B=\{a+b:a\in A,\;b\in B\}$.
>
> **Problem.** Is it true that for any finite abelian group $G$ and numbers $a,b$ w... | https://mathoverflow.net/users/61536 | On decomposition of finite Abelian groups | I don't think it's true for $G=\mathbb{F}\_2^3$ and $a=b=3$.
If there were such sets $A$ and $B$, they must have exactly three elements each.
By applying a translation and a group automorphism, we may as well take
$$A=\{(0,0,0),(1,0,0),(0,1,0)\}.$$
Then $B$ either has at most one element with third coordinate ze... | 8 | https://mathoverflow.net/users/22989 | 312734 | 135,992 |
https://mathoverflow.net/questions/312726 | 3 | Let $Z\_1 \rightarrow Z\_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (<http://pi.math.cornell.edu/~hatcher/AT/AT.pdf>, section 4.F, last lines) it's stated that the natural map $$\underset{\rightarrow}{\lim} \ \... | https://mathoverflow.net/users/118786 | Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$ | First note that Hatcher's exercise says "where direct limits mean mapping telescopes", so he is defining $\underset{\rightarrow}{\lim}$ to mean the telescope. I disapprove of that quite strongly. The telescope is the same as the homotopy colimit, and the standard notation for that is $\underset{\rightarrow}{\text{holim... | 12 | https://mathoverflow.net/users/10366 | 312736 | 135,993 |
https://mathoverflow.net/questions/312716 | 12 | The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}\_1=A\operatorname{-mod}$ and $\mathcal{C}\_2=B\operatorname{-mod}$, then $\mathcal{C}\_1\boxtimes\mathcal{C}\_2 =A\otimes B\operatorname{-mod}$ is independent of the choice of $A$ and $... | https://mathoverflow.net/users/66 | Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$? | I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references:
* Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, *Reflexivity and dual... | 11 | https://mathoverflow.net/users/2926 | 312738 | 135,994 |
https://mathoverflow.net/questions/312467 | 9 | Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem for Laplace operator associate to data $f:\Omega\to\mathbb{R}$ and $g: \partial\Omega \to \mathbb{R}$ (measurable functions... | https://mathoverflow.net/users/112207 | Global regularity for Neumann problem | Proposition 7.7 of chapter 5 in the first volume of "Partial Differential Equations" by M. E. Taylor (Springer 1996) proves the regularity result for $C^\infty$ boundaries. (Notice that any two solutions differ at most by an additive constant.) The argument of the proof should go through for boundaries with $C^{2,1}$ r... | 6 | https://mathoverflow.net/users/nan | 312740 | 135,995 |
https://mathoverflow.net/questions/312704 | 5 | I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a nonsingular complex matrix $A \in \text{GL}\_n(\mathbb{C})$ is an expression $A = UP$, where $U \in \text{GL}\_n(\mathbb{C}... | https://mathoverflow.net/users/30211 | Generalizing Polar Decomposition of Matrices | The basic answer to the question here is "Yes, there is a strong analogy via the Iwasawa decomposition for a semisimple Lie group". If I were trying to study this kind of question, I'd probably start with a MathSciNet search, which of course does require access. For example, searching for "lie group" and "polar decompo... | 4 | https://mathoverflow.net/users/4231 | 312746 | 135,997 |
https://mathoverflow.net/questions/312748 | 4 | Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h\_L$ denotes the Hilbert polynomial.
>
>
> >
> > Is it true that, if $L$ and $L'$ are ample line bundles which are equal in the Neron-Severi group, then $h\_L = h\_{L'}$?
> >
> >
> > Does this imply, together with finite g... | https://mathoverflow.net/users/130079 | Does a projective variety have only finitely many associated Hilbert polynomials? | Yes for the first question, by Riemann--Roch.
No for the second --- even in the simplest case of a projective line, the polynomial $td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).
| 7 | https://mathoverflow.net/users/4428 | 312749 | 135,998 |
https://mathoverflow.net/questions/312411 | 7 | Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite products and coproducts (a.k.a meets and joins in lattice-speak); we may or may not require 2-morphisms in $\mathbf{Lat}$ ... | https://mathoverflow.net/users/128639 | Products, coproducts and equalizers in category of lattices | I'm not sure about sheaf theory, but limits and colimits in categories of lattices are routine to construct. You just have to be clear about your categorical setup. Consider the following categories:
* $sLat^{\vee}$, the category of posets with $(\vee,\bot)$ and morphisms which preserve these.
* $Lat$, the category o... | 4 | https://mathoverflow.net/users/2362 | 312760 | 136,003 |
https://mathoverflow.net/questions/312752 | 22 | This question is inspired by [Probability of commutation in a compact group](https://mathoverflow.net/questions/312652/probability-of-commutation-in-a-compact-group), which asked whether $P(xyx^{-1}y^{-1} = 1)$ could take values strictly between $0$ and $1$ on a compact connected group. That question turned out to have... | https://mathoverflow.net/users/20598 | Probability of satisfying a word in a compact group | The negative answer follows easily from a very useful fact that should be better known, so I am writing it explicitly.
**Fact (Tannaka, Chevalley):**
Every (connected) compact Lie group is isomorphic as a topological group to the group of real points of a (connected) reductive affine real algebraic group.
Moreover, ... | 20 | https://mathoverflow.net/users/89334 | 312763 | 136,004 |
https://mathoverflow.net/questions/312759 | 5 | For a finite graph with undirected, unweighted edges, a well-known result is that the dimension of the null space of the Laplacian matrix gives the number of connected components. Does this result apply to infinite graphs as well?
The infinite graphs I'm interested are locally finite. That is, the degree of each node... | https://mathoverflow.net/users/94774 | Laplacian of an infinite graph and connected components | For infinite (locally finite) graphs it is not true in general that the dimension of the kernel of the Laplacian is the number of connected components.
The simplest example to have in mind is the standard graph structure on $\mathbb{Z}$, which is connected, where the kernel of the Laplacian consists exactly of all ar... | 10 | https://mathoverflow.net/users/89334 | 312773 | 136,007 |
https://mathoverflow.net/questions/312771 | 8 | This is a question in the spirit of [an earlier problem](https://mathoverflow.net/questions/312708/a-link-between-hooks-contents-and-parts-of-a-partition).
Let $\lambda$ be an integer partition: $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq0)$.
Recall also the notation for the content of a cell $u=(i,j)$ in a p... | https://mathoverflow.net/users/66131 | A link between hooks and contents: Part II | You can also prove this inductively by adding boxes to outside corners of $\lambda$.
Suppose the result is true for $\lambda\vdash n$ and then we add a new box $(i,j)$ in an outside corner.
The RHS increases by $(n+1)^2-n^2 + (i-j)^2=2n+1+(i-j)^2$
How much does the LHS increase by? Let me use $h\_u$ for the hook... | 6 | https://mathoverflow.net/users/25028 | 312775 | 136,009 |
https://mathoverflow.net/questions/312772 | 1 | Let $\pi:X \to \mbox{Spec}(\mathbb{C}[t]/(t^2))$ be a smooth, projective morphism and $L$ be an invertible sheaf on $X$. Denote by $L\_0$ the restriction of $L$ to the closed fiber, say $X\_0$ of $\pi$. Suppose that the natural morphism $H^0(L) \to H^0(L\_0)$ is surjective. Can we conclude that $\pi\_\*L$ is flat over ... | https://mathoverflow.net/users/38832 | Flatness of direct image sheaf over local artinian ring | This follows easily from the theory of modules over $R = \mathbb C[t]/(t^2)$. Indeed, we have a short exact sequence
$$0 \to H^0(\mathscr L\_0) \to H^0(\mathscr L) \to H^0(\mathscr L\_0) \to 0,$$
induced by the observation that $(t) \cong R/(t)$ and your assumption that the map $H^0(\mathscr L) \to H^0(\mathscr L\_0)$ ... | 3 | https://mathoverflow.net/users/82179 | 312786 | 136,015 |
https://mathoverflow.net/questions/312590 | 16 | I'm interested in
$$\sup\_{x, y} \left\| A x + B y\right\|\_2$$ subject to
$$\left\|x\right\|\_2 = \left\|y\right\|\_2 = 1$$
where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of compatible sizes. (Generalizations to more than two terms are also interesting)
Does this problem have a name in... | https://mathoverflow.net/users/30352 | $\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$ | The sets $\{Ax : \|x\|=1\}$ and $\{By : \|y\|=1\}$ are ellipsoids. Hence the set $\{Ax+By : \|x\|=\|y\|=1\}$ is the [Minkowski sum](https://en.wikipedia.org/wiki/Minkowski_addition) of two ellipsoids. Googling for these terms returned this paper which may be interesting to you (although browsing through it I did not fi... | 8 | https://mathoverflow.net/users/1898 | 312816 | 136,030 |
https://mathoverflow.net/questions/312768 | 3 | I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and my problem is that I obtain different results according to the specific matrix representation that I choose. To be more ... | https://mathoverflow.net/users/101308 | Casimir operator of a given Lie algebra and relation with its matrix representation | I think that the reason for this discrepancy is that in the first basis all three vectors have the same norm (with respect to the Killing form) while in the second they do not. You should see the $a\_i$s as elements in the dual of your Lie algebra and so the last step should be mapping these elements to the *duals* of ... | 4 | https://mathoverflow.net/users/6818 | 312818 | 136,031 |
https://mathoverflow.net/questions/312817 | 2 | * $f(n,k) = 2g(n-2,k-1)+f(n-1,k)$
* $g(n,k) = g(n-1,k-1)+f(n,k)$
* when $n\le0$ or $k\le0: \quad f(n,k) = 0$
* when $n < k:\quad f(n,k) = 0$
* when $n-k<-1:\quad g(n,k) = 0$
* when $k=0:\quad g(n,k) = 1$
* $g(1,1) = 3$
Solve the above recurrence relation for two variable and two equations. More formally, How can we ... | https://mathoverflow.net/users/130104 | Find formula for recurrence relation with two function and two variables | A quick way is to generate the first few values and guess the solution, then check that it's correct. Using FriCAS, for example:
```
(1) -> )se fu ca all
In general, interpreter functions will cache all values.
(1) -> f(n,k) == (if n < k or n <= 0 or k <= 0 then 0 else 2*g(n-2,k-1)+f(n-1,k))
... | 4 | https://mathoverflow.net/users/3032 | 312827 | 136,035 |
https://mathoverflow.net/questions/312830 | 9 | Let us call a positive integer $n\in\mathbb{N}$ *consecutively summable* if there are positive integers $m, k < n$ such that $$n=\sum\_{i=0}^k (m+i).$$For $A\subseteq \mathbb{N}$ we set the *lower density* of $A$ to be $$\text{ld}(A)=\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
If $N$ is the set of po... | https://mathoverflow.net/users/8628 | Lower density of numbers not summable by consecutive integers | It is known that the only numbers not "consecutive summable" are the powers of $2$.
This is easy to prove: If $m+m+1+\ldots m+k=2^a$ then $2^a=\frac{(k+1)(2m+k)}{2}$.
This means that $2^{a+1}=(k+1)(2m+k)$ which is impossible since the differences form the two parentheses is an odd number. (And they should be both... | 12 | https://mathoverflow.net/users/38851 | 312831 | 136,037 |
https://mathoverflow.net/questions/312837 | 8 | Let $\lambda$ be an integer partition: $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq0)$. Further, let $h\_u$ denote the [hook-length](https://en.wikipedia.org/wiki/Hook_length_formula#Example) of the cell $u$.
We call $\lambda$ a $t$-core partition if none of its hooks $h\_u$ equals $t$. Define $c\_t(n)$ to be the ... | https://mathoverflow.net/users/66131 | Generating function for $3$-core partitions | The set of $3$-core partitions can be described explicitly.
**Theorem** The partition $\lambda=\{\lambda\_1,\lambda\_2,\dots\}$ of length $k$ (that is, $\lambda\_k > 0$ but $\lambda\_{k+1} = \lambda\_{k+2} = \cdots = 0$) is a $3$-core if and only if the sequence of differences $\{\lambda\_1-\lambda\_2,\lambda\_2-\la... | 13 | https://mathoverflow.net/users/2384 | 312840 | 136,039 |
https://mathoverflow.net/questions/312756 | 0 | Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
$$\min\_{Y}||D-S||\_1$$
| https://mathoverflow.net/users/130071 | How to solve this optimization problem efficiently? | You can model this as an integer quadratic program. Let $x\_i$ be a nonnegative integer variable indicating the number of times index $i$ is selected ($i=1,\dots,2^M$). Rewrite the equation defining $S$ as $$S=\frac{1}{T}\sum\_{i=1}^{2^M} V(i,:)x\_i$$and add one more constraint:$$\sum\_{i=1}^{2^M} x\_i = T.$$
Whether... | 1 | https://mathoverflow.net/users/130122 | 312844 | 136,040 |
https://mathoverflow.net/questions/312803 | 1 | I have a question concerning generalized order spaces.
Is there a generalized order space $X$ with countable tightness which is not first countable?
thanks a lot!
| https://mathoverflow.net/users/129635 | Is there a generalized order space $X$ with countable tightness which is not first countable? | No, in general tightness and character are equal for GO-spaces:
if $x$ is in the closure of $(\gets,x)$ then there is a subset $L$ of cardinality at most the tightness of $(\gets,x)$ that has $x$ in its closure; likewise if $x\in\operatorname{cl}(x,\to)$ there is $R\subseteq(x,\to)$ of cardinality at most the tightness... | 2 | https://mathoverflow.net/users/5903 | 312850 | 136,042 |
https://mathoverflow.net/questions/312710 | 5 | Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F\_n=F[E\_{p^n}]$ and $F\_{\infty}=F[E\_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F\_\infty/F)$ is an open subgroup of $GL\_2(\mathbb{Z}\_p)$.
Consider the natural norm maps on $E\_{p^\infty}(F\_m)$... | https://mathoverflow.net/users/69289 | Elliptic curves and $GL(2)$ Iwasawa theory | For $n$ large enough the Galois group of $F\_n/F\_{n-1}$ identifies with the kernel $G$ of $\operatorname{GL}\_2(\mathbb{Z}/p^{n+1}\mathbb{Z}) \to \operatorname{GL}\_2(\mathbb{Z}/p^{n}\mathbb{Z})$. Let $P$ and $Q$ be a basis of $E\_{p^{n+1}}$. The norm of $P$ is written as a sum over $a$, $b$, $c$, $d$ modulo $p$ as
\b... | 4 | https://mathoverflow.net/users/5015 | 312851 | 136,043 |
https://mathoverflow.net/questions/312846 | 13 | Matsumoto proved in his PhD thesis that if $F$ is a field then $$K\_2(F)=(F^\*\otimes F^\*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks not really nice to me and one guy told me that there is another proof of this fact that uses "sheaves of groups on ... | https://mathoverflow.net/users/88385 | Reference for the algebro-geometric proof of Matsumoto theorem | Apparently, what this one guy meant is Merkurjev's proof of Merkurjev-Suslin theorem, which was writen down by [A. R. Wadsworth](http://www.ams.org/books/conm/055.2/) and later by [W. van der Kallen](https://doi.org/10.1007/BFb0074799). The latter paper starts with the statement of Matsumoto theorem and the proof does ... | 10 | https://mathoverflow.net/users/5018 | 312855 | 136,045 |
https://mathoverflow.net/questions/312847 | 6 | Let $k$ be a global field, and let $G = \mathbf G(\mathbb A\_k)$ for a connected, reductive group $\mathbf G$ over $k$. In [these](https://services.math.duke.edu/~hahn/Chapter3.pdf) notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ is a Hilbert space $V$ together with a continuous homomorphism $\p... | https://mathoverflow.net/users/38145 | Definition of unitary representation of $\mathbf G(\mathbb A_k)$ | It is absolutely essential that the space of (bounded/continuous) operators be given the "strong" operator topology (strictly weaker than the norm topology), and the map $G\times V\to V$ to be jointly continuous.
This is not a pathology: even in very simple cases, such as $G=\mathbb R$ acting on $V=L^2(\mathbb R)$, t... | 7 | https://mathoverflow.net/users/15629 | 312856 | 136,046 |
https://mathoverflow.net/questions/312839 | 5 | I came across the following alternating sum
$$
\sum\_{k=0}^n (-1)^k \binom{2n}{k} (n-k)^r,\quad 1\leq r < n.
$$
It seems that when $r$ is an even integer the sum is $0$ and when $r$ is an odd integer the sum is not zero (regardless of the parity of $n$).
[Edited] The case when $r$ is even is easy by symmetry as Dari... | https://mathoverflow.net/users/48609 | partial alternating sum involving binomial coefficients | In [a post on Math Stack Exchange, MSE 2827591](https://math.stackexchange.com/questions/2827591/alternate-proof-for-weighted-alternating-shifted-central-binomial-sum-relation), I prove the following:
$$\sum\_{k=1}^n (-1)^{k+1} \binom{2n}{n+k} k^s =
\binom{2n}{n} \sin(\pi s/2) \int\_0^\infty \frac{dx \, \,x^s}{\sinh{... | 4 | https://mathoverflow.net/users/121836 | 312875 | 136,053 |
https://mathoverflow.net/questions/312859 | 15 | Let $g \geq 2$. Let $S = \langle a\_1,b\_2,...,a\_g,b\_g | [a\_1,b\_1] \cdots [a\_g,b\_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F\_g$ be a free group with $g$ generators. Given two surjections $f\_1,f\_2 : S \to F\_g$ is there a way to determine if there are automophisms $\phi: S \to S$ and ... | https://mathoverflow.net/users/99414 | Equivalence of surjections from a surface group to a free group | This is true, and it is written up in lemma 2.2 of "The co-rank conjecture for 3--manifold groups" by C. Leininger and A. Reid [<https://arxiv.org/abs/math/0202261>](https://arxiv.org/abs/math/0202261). They state the result in slightly different language, that is they prove that any such epimorphism is induced by choo... | 9 | https://mathoverflow.net/users/32210 | 312880 | 136,056 |
https://mathoverflow.net/questions/312881 | 3 | Let $\lambda$ be an integer partition: $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq0)$. Further, let $h\_u$ denote the [hook-length](https://en.wikipedia.org/wiki/Hook_length_formula#Example) of the cell $u$.
We call $\lambda$ a $t$-core partition if none of its hooks $h\_u$ equals $t$. Define $c\_t(n)$ to be the ... | https://mathoverflow.net/users/66131 | Generating function for 3 -core partitions: Part II | Suppose the partition with $k$ parts $\lambda=\{\lambda\_1\geq \lambda\_2\geq\dots\geq\lambda\_k\geq \lambda\_{k+1}=0\}$ is a partition with only odd parts. Then we have $\lambda\_i-\lambda\_{i+1}$ is even for all $i\leq k-1$ and odd for $i=k$. Using the same characterization I mentioned in the [previous answer](https:... | 9 | https://mathoverflow.net/users/2384 | 312882 | 136,057 |
https://mathoverflow.net/questions/312886 | 3 | Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that
$$ \sum\_{n \in \mathbb{Z}} w(x+n) = 0$$
for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume that $w$ is continuous?
| https://mathoverflow.net/users/16702 | Constant "periodization" of a function | No, the **Haar wavelet** is a counter example (i.e. $w(x) = \chi\_{[0,1[}(x) - \chi\_{[1,2[}(x)$). Decay is arbitrarily fast and mollifying by convolution gives a smooth counterexample.
| 6 | https://mathoverflow.net/users/9652 | 312887 | 136,059 |
https://mathoverflow.net/questions/312086 | 3 | I have found some applications of the **Frame Theory** in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.
As we know, any frame in a Hilbert space is itself a Bessel sequence.
There is a spectrum of weakening sequences betwee... | https://mathoverflow.net/users/84390 | Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames? | To answer the first question, frame sequences are just countable frames. A somewhat more standard name for upper semi-frames are *Bessel sequences*, that is countable sequences $\{f\_n\}$ in a Hilbert space $\mathcal{H}$ which satisfy
$ \sum\limits\_{n=1}^\infty |\langle f, f\_{n} \rangle |^2 \leq B \|f\|^2 \hspace{... | 4 | https://mathoverflow.net/users/118731 | 312888 | 136,060 |
https://mathoverflow.net/questions/312866 | 2 | Let $F=\mathbb Q\_l$ and $E$ be a finite extension of $\mathbb Q\_p$ with residue field $k \cong O\_E/m\_E$ ($p \neq l$). Let $\bar r: \Gamma\_F=Gal(\bar F/F) \to GL\_2(k)$ be a continuous representation. Let $\mathrm{CLN}\_{O\_E}$ be the category of complete local notherian $O\_E$-algebras and let $\mathrm{Lift}\_{\ba... | https://mathoverflow.net/users/112340 | Steinberg components of local deformation rings | If I understand correctly, I believe that you want $r:\Gamma\longrightarrow\operatorname{GL}\_2(R)$ to factor through $R^{\operatorname{St}}$ if $r$ is a *non-trivial* extension of $\beta$ by $\alpha$ with $\alpha/\beta=\kappa$ (only these correspond to Steinberg representation by the Local Langlands Correspondence). W... | 1 | https://mathoverflow.net/users/2284 | 312893 | 136,063 |
https://mathoverflow.net/questions/312848 | 4 | Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}\_2(\mathbb{A}\_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ and $\mathcal{O}$ the completion of the ring of integers of the field of Fourier coefficients of $\pi$. Does there alw... | https://mathoverflow.net/users/nan | Are there Galois representations associated with any regular algebraic cuspidal automorphic representation? | In *Motifs et formes automorphes: applications du principe de fonctorialité* by Laurent Clozel (in *Automorphic forms, Shimura varieties and $L$-functions* Volume I (1990)), it is asked in 4.3.2 whether the category of algebraic automorphic representation of $\operatorname{GL}\_n(\mathbb A\_F)$ defined over $\bar{\math... | 2 | https://mathoverflow.net/users/2284 | 312897 | 136,065 |
https://mathoverflow.net/questions/312867 | 7 | I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand the Fourier transform is a Hilbert space isometry
$F:L^2(\mathbb R^d) \rightarrow L^2(\mathbb R^d).$
In Quantum Mecha... | https://mathoverflow.net/users/130130 | Making the Fourier transform quantitative | Notice that $\|f1\_{B(0,\delta)}\|\_{L^2}\to 0$ as $\delta\to 0$. So the answer to the question is no because it should reasonably include the requirement $\varepsilon'<1$. -- Rapid decrease of the Fourier transform $F(f)$ implies smoothness of $f$ but, without further assumptions, it does not imply localization.
| 4 | https://mathoverflow.net/users/nan | 312904 | 136,067 |
https://mathoverflow.net/questions/312737 | 5 | Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that
$$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$
Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$
The answer is affirmative when $f:\mathbb{R}\to\mathbb{R}$ is a real function, and the proof is not difficult. The above generalization seems... | https://mathoverflow.net/users/38620 | A variant of Cauchy-type functional equation conjecture | counterexample
$$f(x)=1-e^{\Re(x) i}$$
| 13 | https://mathoverflow.net/users/38620 | 312917 | 136,069 |
https://mathoverflow.net/questions/312914 | -2 | sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i.e. $B$ as positive definite matrix, has contribution to the growth of spectrum?
remark:
sorry my question may be ... | https://mathoverflow.net/users/124254 | sum of positive definite matrix | Yes, this follows from the Maximin characterization of eigenvalues of symmetric matrices. If $A$ is an $n\times n$ symmetric matrix, form the Rayleigh ratio
$R(x)=(x,Ax)/(x,x),$ where $(.,.)$ is the standard dot product. Then the $k$-th eigenvalue is
$$\lambda\_k=\max\_a\min\_{ax=0} R(x)$$
where the inner minimum is un... | 1 | https://mathoverflow.net/users/25510 | 312918 | 136,070 |
https://mathoverflow.net/questions/312753 | 0 | or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?
The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutter paper, "Generalizations of the Kunen inconsistency", *Annals of Pure and Applied Logic*, 163 (2012) pp. 1873-1876 (Sec... | https://mathoverflow.net/users/20597 | What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$ | Here is the strategy behind McCallum's attempted proof that there is no nontrivial elementary embedding $j :V\to V$.
Step 1 is to cite the theorem that if a $j :V\to V$ is consistent, then a $j : V\to V$ is consistent with $V\_\lambda\vDash \text{ZFC}$ where $\lambda = \sup\_{n < \omega} j^n(\text{crt}(j))$. This is ... | 10 | https://mathoverflow.net/users/102684 | 312921 | 136,072 |
https://mathoverflow.net/questions/312931 | 7 | Let $k$ be an algebraically closed field, it is well known that $\mathbb P^1$ is simply connected, but how about smooth projective surfaces $X$ with a smooth morphism to $\Bbb P^1$?
Except the case $X$ is a product of curves or a projective bundle like Hirzebruch surfaces, could we classify all such $X \rightarrow \... | https://mathoverflow.net/users/102104 | Classification of smooth algebraic surfaces with a smooth morphism to $\Bbb P^1$ | Let $k$ be an algebraically closed field.
Let $f:X\to \mathbb{P}^1$ be a smooth proper morphism with fibres of dimension one. Note that the fibres of $f$ are geometrically connected by Stein factorization and the fact that $\mathbb{P}^1$ is simply connected (Riemann-Hurwitz).
>
>
> >
> > **Theorem.** The morphi... | 14 | https://mathoverflow.net/users/4333 | 312935 | 136,075 |
https://mathoverflow.net/questions/312781 | 2 | What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H\_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any symplectic structure?
**Added:** As it is indicated in the comments, this post is not a duplicated post.
| https://mathoverflow.net/users/36688 | An orientable compact even dimensional manifolds whose all even cohomologies do not vanish but it does not admit any symplectic structure | For your $n=2k$, $\mathbb{C} P^n\#\mathbb{C} P^n$ does not even admit an almost complex structure, so it cannot be symplectic.
See also:
1) Goertsches-Konstantis' paper *"Almost complex structures on connected sums of complex
projective spaces"* which answers the following MO question
2) [Does $\Bbb{CP}^{2n} \# \B... | 6 | https://mathoverflow.net/users/12310 | 312951 | 136,079 |
https://mathoverflow.net/questions/312751 | 2 | **Setup**
---------
Suppose that $U$ is a subset of $L^{\infty}\_{\mu}(\mathscr{F})\cap L^1\_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}\_{\mu}(\mathscr{F})\cap L^1\_{\mu}(\mathscr{F})
,
$$
for a fixed convex and lower-semicontinuous function $g$
. Here $\mu$ ... | https://mathoverflow.net/users/36886 | Echange of Infimum Integral with Pointwise Infimum | If $U$ is non-empty then the value of either side is $m \mu(\mathcal{X})$ where $m= \inf \{y^2 ~|~ g(y)<M\}$. If $\mu$ is non-finite then $m=0$.
To see this note first that for $f\_1,f\_2 \in U$ the function $f$ defined by
$$
f(x)=\begin{cases}
f\_{1}(x) & f\_{1}^{2}(x)\le f\_{2}^{2}(x)\\
f\_{2}(x) & f\_{2}^{2}(x)<... | 1 | https://mathoverflow.net/users/123897 | 312976 | 136,085 |
https://mathoverflow.net/questions/312745 | 3 | How can I prove this: Let $K$ be a real abelian number field, $K\_1$ be the Hilbert Class Field of $K$, and $J=K\_1\cap K(\zeta\_b)$. If a prime $p$ divided $[J:K]$ but did not divide $[K:\mathbb{Q}]$, then there would be an unramified extension of $\mathbb{Q}$ of degree $p$. ?
| https://mathoverflow.net/users/130073 | About real abelian number fields | Let $J^{(p)}\subseteq J$ be the subfield fixed by the $p$-Sylow subgroup of $\operatorname{Gal}(J/K)$ which is also the $p$-Sylow subgroup of the (abelian!) group $\operatorname{Gal}(J/\mathbb{Q})$ since, by assumption, $p\nmid [K:\mathbb{Q}]$. Then, by the structure theorem of abelian groups, we find a direct-product ... | 0 | https://mathoverflow.net/users/18238 | 312978 | 136,086 |
https://mathoverflow.net/questions/260303 | 5 | Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a *covering* of $X$ if $\bigcup U = X$. We call a covering *proper* if for $a\neq b\in U$ we have $a\not\subseteq b$.
Let $\text{Cov}(X)$ denote the collection of all coverings of $X$. For $A, B\in \text{Cov}(X)$ we set $A\... | https://mathoverflow.net/users/8628 | Proper coverings as a quotient of the coverings | **Are $(\text{Cov}(X)/\simeq, \leq\_q)$ and
$(\text{PropCov}(X),\leq)$ isomorphic as posets?**
If $X$ is finite, then $(\text{Cov}(X)/\simeq, \leq\_q)$
and $(\text{PropCov}(X),\leq)$
are isomorphic, but if $X$ is infinite they are not.
To see why they are isomorphic when $X$ is finite it is enough to
note that, in th... | 3 | https://mathoverflow.net/users/75735 | 312980 | 136,087 |
https://mathoverflow.net/questions/312941 | 2 | Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2\_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad x(0)=x\_0\in \mathbb{R}.$$
It is known that for every $x\_0$ and $R$, where $\|u\|\_2\le R$, there is a time $T$ such... | https://mathoverflow.net/users/113264 | Counter example about blow-up solution of DEs | The implicit function theorem usually helps to arrive at such conclusions. You translate the (differential equation) to an operator on function spaces, in your example e.g. $$E \colon W^{1,2}(0,T) \times L^2(0,T) \to L^2(0,T)$$
with $$E(x,u) = ax + f(x) + u - \dot x.$$
(One can also set this up differently, for instanc... | 3 | https://mathoverflow.net/users/85906 | 312985 | 136,088 |
https://mathoverflow.net/questions/312916 | 9 | It is well known in narrow circles that the homological dimension (in the sense of relative Banach homology) of $c\_0$-module $\ell\_\infty$ is 2. As the corollary, this module is not projective. This proof is rather involved, its main ingridient is a lack of a right inverse for the mapping:
$$
\Delta:c\_0\;\hat{\otime... | https://mathoverflow.net/users/19593 | Explicit proof that $c_0$-module $\ell_\infty$ is not projective | I believe the following works...
Notice that $c=c\_0 + \mathbb C1$ and so
$$\newcommand{\proten}{\widehat\otimes} c\proten\ell^\infty = c\_0\proten\ell^\infty + 1 \otimes \ell^\infty.$$
This is an isomorphism, maybe not isometric. Suppose, towards a contradiction, that there is a right inverse $T:\ell^\infty \rightar... | 5 | https://mathoverflow.net/users/406 | 312990 | 136,090 |
https://mathoverflow.net/questions/312983 | 11 | Does there exist an uncountable $P \subset \mathcal{P}(\mathbb{N}) $ with the property that for any distinct $x,y \in P$, $|x \cap y|$ is prime?
A more general, but likely harder, question: is it possible to characterize the set $\mathcal{A}$ of all subsets $S \subset \mathbb{N}$ with the following property: there i... | https://mathoverflow.net/users/130192 | Cardinality of families of subsets of $\mathbb{N}$ whose intersections are finite | I guess this is a variant of Noah's construction:
Let $S$ be an infinite subset of $\mathbf{N}=\{0,1,2,\dots\}$. Define a leafless rooted tree $V\_S$, starting from a root at level 0, such that given a vertex $v$ of level $n\ge 0$, $v$ has exactly 1 successor if $n\notin S$ and exactly 2 successors if $n\in S$.
Let... | 11 | https://mathoverflow.net/users/14094 | 312992 | 136,091 |
https://mathoverflow.net/questions/312928 | 5 | Let $X$ be a compact complex analytic manifold, $D\subset X$ an irreducible smooth divisor, given as zeroes of a global meromorphic function $f\in {\mathfrak M} (X)$. Are there enough other meromorphic functions defining $D$?
Here is a precise question: **can one find, for each $x\in X$, a meromorphic function $g\in ... | https://mathoverflow.net/users/5301 | Are there enough meromorphic functions on a compact analytic manifold? | First, let's clarify some possible confusion. A compact complex manifold does not admit non-constant holomorphic functions, so, assuming that $X$ admits non-constant meromorphic functions, you actually want the divisor of such a function to include poles, not just zeros. Infinity is thus a legitimate value. The points ... | 10 | https://mathoverflow.net/users/14493 | 313002 | 136,096 |
https://mathoverflow.net/questions/312993 | 6 | Let $G$ be an $n \times n$ grid graph. Is there anything known about the asymptotic growth rate of the number of self avoiding paths from $(0,0)$ to $(n,b)$ (from the lower left corner to some arbitrary point on the right hand side)?
For example, is there anything known about the $b$ that maximizes this number? A li... | https://mathoverflow.net/users/41873 | Number of self avoiding paths on a grid graph? | **UPD:** the answer below is in fact completely wrong - it deals with counting walks $\gamma$ weighted by $\mu^{-\text{length}(\gamma)}$. It is clear that without restricting or penalizing for the lengths of the walks, the number of walks will grow exponentially with the *volume* (i. e., as $\eta^{N^2}$ for some $\eta>... | 10 | https://mathoverflow.net/users/56624 | 313003 | 136,097 |
https://mathoverflow.net/questions/312896 | 8 | Let $(a\_1,a\_2\ldots,a\_n)$ and $(b\_1,b\_2,\ldots,b\_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a\_1b\_1,a\_2b\_2,\dots,a\_nb\_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2$, and there are [exa... | https://mathoverflow.net/users/83212 | Product of arithmetic progressions | First, it's easy to see that *real* solutions can be rescaled to be rational and therefore also integer.
For $n=7$ there are no solutions.
For $n=6$ there are essentially 4 primitive integral solutions, with only one consisting of natural numbers (as requested):
$\big(-35+12(1,2,4,3,6,5)\big) \times \big(-5+2(5,6... | 6 | https://mathoverflow.net/users/2480 | 313008 | 136,099 |
https://mathoverflow.net/questions/312986 | 3 | Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^\*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder if this is statement is correct and whether it is written down in some book/article.
| https://mathoverflow.net/users/13441 | The ample cone of a surface with an algebraic $\mathbb C^*$ action | This is true, and follows from Corollary 3.29 of <https://arxiv.org/pdf/0811.0517.pdf>
| 4 | https://mathoverflow.net/users/130201 | 313009 | 136,100 |
https://mathoverflow.net/questions/313027 | 2 | Looking at @Lucia's answer to [this question](https://mathoverflow.net/questions/164874/is-it-possible-to-show-that-sum-n-1-infty-frac-mun-sqrtn-diverg) it appears $\sum\_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I misunderstood ... | https://mathoverflow.net/users/123507 | Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$? | The convergence of $\sum\mu(n)/n^s$ for $\Re(s)>1/2$ is equivalent to the [Riemann hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis). First, by Theorems 1.1 and 1.3 in Montgomery-Vaughan: Multiplicative number theory I, this convergence implies that $1/\zeta(s)$ is holomorphic in $\Re(s)>1/2$ (which is clea... | 15 | https://mathoverflow.net/users/11919 | 313028 | 136,103 |
https://mathoverflow.net/questions/313037 | 2 | In [this](https://arxiv.org/abs/1211.5964) archiv paper which is continuation of following:
*Borodzik, Maciej; Némethi, András; Ranicki, Andrew*, [**Morse theory for manifolds with boundary**](http://dx.doi.org/10.2140/agt.2016.16.971), Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). [ZBL1342.57018](https://zbmath.o... | https://mathoverflow.net/users/nan | Embedding problem for 3-manifolds attacked via 4-manifolds | Since you've asked for an opinion, my answer is also an opinion. I would say that the embedding in $R^5$ is not going to be helpful. You are seeking to take $M = \partial N\_i$ and `improve' the $N\_i$ to make their union into a 4-sphere. This is a very reasonable approach; maybe you can do surgery on the $N\_i$ to giv... | 7 | https://mathoverflow.net/users/3460 | 313063 | 136,112 |
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