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https://mathoverflow.net/questions/227522 | 3 | In a [question](https://math.stackexchange.com/questions/781461/extensions-of-the-hermite-bielher-and-hermite-kakeya-theorem) asked by Bobby Ocean, the following theorem is cited:
***Hermite-Kakeya Theorem(for polynomials)*** - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros for... | https://mathoverflow.net/users/33672 | Hermite-Kakeya Theorem for entire functions | Hermite's theorem indeed generalizes to entire functions but your statement for entire functions is incorrect.
$$2\cos2z-\cos z=4\cos^2z-1-\cos z$$
has all zeros real, but the zeros of $2\cos 2z$ and $\cos z$ do not interlace.
Substitute the square root if you want zeros to be on a ray.
The correct statement: If $f$... | 3 | https://mathoverflow.net/users/25510 | 227528 | 106,170 |
https://mathoverflow.net/questions/227504 | 3 | Here a signed graph is one where each edge is signed either odd or even. A cycle is odd or even according to the sum of the signs of its edges. For a given signed graph, a resigning may be performed by flipping the signs of all edges in the cutset of some vertex subset. Clearly this does not change the sign of any cycl... | https://mathoverflow.net/users/31016 | Is there a version of Robertson-Seymour's graph minor theorem known to apply to signed graphs? | Yes. This is one of the results of the [Matroid Minors Project](http://www.math.uwaterloo.ca/~jfgeelen/Research/research.html) of Geelen, Gerards and Whittle as part of their proof of [Rota's Conjecture](https://www.wikiwand.com/en/Rota's_conjecture). In fact, they prove that for any finite abelian group $\Gamma$, the ... | 11 | https://mathoverflow.net/users/2233 | 227534 | 106,171 |
https://mathoverflow.net/questions/227543 | 12 | It’s obvious that [Givens rotation](https://en.wikipedia.org/wiki/Givens_rotation) works better with sparse matrices. But I don’t know why [Householder reflection](https://en.wikipedia.org/wiki/Householder_transformation) is better for dense matrices. Does it require less computations? Or it’s numerically more stable t... | https://mathoverflow.net/users/50309 | Why Householder reflection is better than Givens rotation in dense linear algebra? | Implementing the QR factorization with Householder rotations is cheaper ($2n^2m$ vs $3n^2m$ for a $m\times n$ matrix), and equally accurate in practice. See Section 19.6 of Higham's *Accuracy and Stability of Numerical Algorithms*, or Golub-Van Loan for more explicit algorithms.
Moreover, in a Householder-based imple... | 14 | https://mathoverflow.net/users/1898 | 227545 | 106,176 |
https://mathoverflow.net/questions/227539 | 9 | Let $a\_1, a\_2, \dots, a\_n$ and $b\_1, b\_2, \dots, b\_n$ be real numbers, and assume that $\{a\_i\} \neq \{b\_i\}$. Can the equation
$$ e^{a\_1 x} + e^{a\_2 x} + \dots + e^{a\_n x} = e^{b\_1 x} + e^{b\_2 x} + \dots + e^{b\_n x}$$
have more than $n$ real roots including $0$ and counting multiplicities?
There are s... | https://mathoverflow.net/users/36579 | Number of real roots of an exponential polynomial | Seems that it can not have more than $n$ roots. Let us use the following generalization of Descartes rule for signed measures. Namely, let $\mu$ be a Borel signed measure on a real line, with compact support (this condition may be of course weakened). We say that $\mu$ has at most $k$ changes of sign if there exist poi... | 13 | https://mathoverflow.net/users/4312 | 227547 | 106,177 |
https://mathoverflow.net/questions/227429 | 4 | Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
| https://mathoverflow.net/users/84390 | Is the Fourier-Stieltjes algebra of a locally compact group semi-simple? | (The following was originally a comment, but I've moved it to an answer. Please don't TeXify this.)
---
Recall that for commutative unital algebras, the Jacobson radical is exactly the set of quasi-nilpotent elements. By definition (see e.g. [Eymard's original 1964 article](http://www.numdam.org/item?id=BSMF_1964... | 1 | https://mathoverflow.net/users/763 | 227565 | 106,179 |
https://mathoverflow.net/questions/227458 | 7 | After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma\_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma\_x(6)=0 \bmod 6$ for $x$ odd and 2 mod 6 if $x$ is even
**Edit:01** :${\sigma}\_x(n) =\sum\_{d|n} d^x$ is the sum divisor function
**... | https://mathoverflow.net/users/74330 | Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? | Let $r=\gcd(k,e+1)$, and $p$ a prime. Then $\sigma\_k(p^e) \equiv r\frac{p^{e+1}-1}{p^r -1} \bmod \sigma(p^e)$. Also, $r=1$ if and only if $\sigma(p^e)$ divides $\sigma\_k(p^e)$. Thus for $k$ coprime to $\tau(n)$, we have $\sigma(n)$ divides $\sigma\_k(n)$. The relation also suggests that for a given $n$ the sequence $... | 5 | https://mathoverflow.net/users/3402 | 227569 | 106,182 |
https://mathoverflow.net/questions/227568 | 3 | Assume I have a chain of real numbers, s.th.
$x\_0 < y\_0 < x\_1<y\_1<x\_2<\dots <x\_n<y\_n$.
I'm trying to explicitely solve the expression
$$ \sum\_{i=0}^n \frac{\prod\_{j=0}^n(x\_j-y\_i)}{\prod\_{j=0, j\neq i}^{n}(y\_j-y\_i)}$$
Calculation until $n=3$ it seems the expression is equal to $\sum\_{j=0}^n x\_j - \... | https://mathoverflow.net/users/75382 | equality of expressions for sum | Consider polynomial $f(t)=\prod\_{i=0}^{n-1}(t-x\_i)$, interpolate it in points $y\_0,\dots,y\_n$:
$$
f(t)=\sum\_{i=0}^n f(y\_i)\frac{\prod\_{j\ne i}(t-y\_j)}{\prod\_{j\ne i}(y\_i-y\_j)}.
$$
Now compare coefficients of $t^{n}$.
| 10 | https://mathoverflow.net/users/4312 | 227571 | 106,183 |
https://mathoverflow.net/questions/227173 | 10 | The invention of intrinsic differential geometry is usually attributed to Gauss in the context of his *theorema egregium* but the notion of the curvature of an embedded surface existed before. Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures?
In <http:... | https://mathoverflow.net/users/28128 | Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures? | Let me start with your first quotation:
"In 1763, Euler started a thorough study of curvature of embedded surfaces. In 1767, he found an expression of the curvature in terms of the product of principal curvatures."
Karin Reich's article in *Leonhard Euler: Life, Work and Legacy* (p. 482 in particular) explains what... | 6 | https://mathoverflow.net/users/3503 | 227572 | 106,184 |
https://mathoverflow.net/questions/227555 | 2 |
>
> Is there a nontrivial commutative Hausdorff topological group that is extremely amenable?
>
>
>
Recall that a topological group is called *extremely amenable* if any continuous action on a compact Hausdorff topological space has a fixed point. For instance, it is known that no nontrivial locally compact grou... | https://mathoverflow.net/users/83382 | Abelian extremely amenable group? | The very first examples of extremely amenable Polish groups were abelian, if I remember correctly (the so-called "exotic" groups of Herer and Christensen, Math. Ann. 213 (1975), 203-210).
| 5 | https://mathoverflow.net/users/84847 | 227577 | 106,188 |
https://mathoverflow.net/questions/227556 | 9 | The Azuma inequality states that if we have a martingale $X\_1,\ldots,X\_N$ that satisfies a bounded difference condition:
$$|X\_k - X\_{k-1}| \leq c\_k$$
Then:
$$\Pr\left[X\_N - X\_0 \geq \sqrt{2\sum\_kc\_k^2 \ln(1/\delta)}\right] \leq \delta$$
My question is:
>
> Does the same inequality hold if the constants $... | https://mathoverflow.net/users/3027 | Adaptive version of the Azuma–Hoeffding inequality | There is no such inequality even if we further restrict $c\_k$ to be in $\{0,1\}$ and weaken the inequality to include a constant factor. (I think it is natural to add the condition that the $c\_k$ values are uniformly bounded.) Suppose $c\_k \in \{0,1\}$ and without loss of generality no $1$ follows a $0$. The choice ... | 3 | https://mathoverflow.net/users/2954 | 227579 | 106,190 |
https://mathoverflow.net/questions/227581 | 3 | If $P$ is the transition matrix of a reversible Markov chain, and $\pi$ is its stationary distribution, and let $R$ be defined by:
$$R\_{ij} = \sqrt{\frac{\pi\_i}{\pi\_j}}P\_{ij}~.$$
By reversibility, it follows that $R$ is a symmetric matrix arising from a similarity transformation of $P$ and hence, both $P$ and $... | https://mathoverflow.net/users/7576 | Similarity transformation of transition matrix of reversible Markov chain (reference request) | If $P$ is the Markov transition matrix of a graph, the matrix $A=P\pi^{-1}$ is called the affinity matrix and $R=\pi^{1/2}A\pi^{1/2}=\pi^{1/2}P\pi^{-1/2}$ is called the *normalized affinity matrix*, see for example section 3 of [this paper.](http://papers.nips.cc/paper/2665-hierarchical-eigensolver-for-transition-matri... | 2 | https://mathoverflow.net/users/11260 | 227585 | 106,192 |
https://mathoverflow.net/questions/227548 | 3 | If I'm not mistaken, there is a bicategory $\mathsf{Monad}$ given as follows:
1. Start with the associative operad.
2. Deloop it to obtain a multicategory.
3. Adjoin objects and morphisms as necessary to obtain a monoidal category.
4. Deloop *that* to obtain a bicategory.
So basically, $\mathsf{Monad}$ is the "doub... | https://mathoverflow.net/users/26080 | Does every bicategory have a "delaxing object"? | I think that the theory of lax functors of $(\infty,\infty)$-categories is not sufficiently developed to answer that version of your question. But for your main question, yes.
Given a bicategory $\mathfrak A$, you certainly have a set of objects and for each pair of objects $a,b\in\mathfrak A$ a category $\mathfrak ... | 4 | https://mathoverflow.net/users/78 | 227590 | 106,194 |
https://mathoverflow.net/questions/227541 | 7 | Am looking for a name for:
$$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$
$$=2.529477472079152648180116154253954242$$
[Wolfram|Alpha](http://www.wolframalpha.com/input/?i=product%20%28n%21%29%2F%28n%21-1%29)
Expanding the formula gives:
$$(1+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\frac{1}{3!}+\frac{1}{3!^2}+\dots)(1+\frac... | https://mathoverflow.net/users/70355 | Factorial-based constant | I don't know about a name, but it does have a history. Knopfmacher, Odlyzko, Pittel, Richmond, Stark [D., not H.], Szekeres, and Wormald, The asymptotic number of set partitions with unequal block sizes, available [here](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJC/Volume_6/PDF/v6i1r2.pdf), find that it is the res... | 15 | https://mathoverflow.net/users/3684 | 227596 | 106,196 |
https://mathoverflow.net/questions/227523 | 6 | I have a simple linear algebra question regarding the definition of dual of a lattice; it was asked by someone else [here](https://math.stackexchange.com/questions/1450162/dual-lattice-is-a-lattice-for-the-dual-tori) three months ago on mathstackexchange but got no answer and few views, so forgive me for asking simple ... | https://mathoverflow.net/users/80739 | Why is dual lattice a lattice, in the context of complex tori | As explained in my comments, it is the same to treat the question where the notation $L^{\ast}$ is defined with $V^{\ast}$ taken to be the $\mathbf{C}$-dual (rather than the conjugate dual), so we do that. That is, $L^{\ast}$ now consists of the $\mathbf{C}$-linear (rather than conjugate-linear) forms on $V$ whose imag... | 5 | https://mathoverflow.net/users/81332 | 227599 | 106,197 |
https://mathoverflow.net/questions/227496 | 5 | This is cross-posted on MSE: <https://math.stackexchange.com/q/1596565/9464>
In the *Partial Differential Equations* by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1\_0(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and $H^1(\Omega)=W^{1,2}(\Omega)$, $H^1\_0(\Omega)=W^{1,2}\_0... | https://mathoverflow.net/users/nan | Difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$ | In general, $X$ does not embed into $Y$. Indeed, suppose that $\partial \Omega$ is sufficiently smooth. Then there is the trace map $\gamma\_0 \colon H^1(\Omega) \to H^{1/2}(\partial\Omega)$, $u\mapsto u\bigr|\_{\partial\Omega}$, which fits into a short exact sequence
$$
0 \longrightarrow H\_0^1(\Omega) \longrightarrow... | 6 | https://mathoverflow.net/users/69194 | 227615 | 106,201 |
https://mathoverflow.net/questions/227642 | 3 | $$e^x=\sum \limits\_{k=0}^\infty \frac{x^k}{k!}$$
We can rewrite the equation as
$$e^x=\sum \limits\_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)} \tag{1}$$
because $x!=\Gamma(x+1)$ where $x$ is non-negative integer.
$\Gamma(x)$ ([Gamma function](https://en.wikipedia.org/wiki/Gamma_function)) also has undefined property ... | https://mathoverflow.net/users/20994 | Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$? | Set $a:=1+n\in \mathbb C$ and assume $a\notin\mathbb Z$; we use the variable $z:=1/x$. The question has already been answered in the comments: the power series $$S(z):=\sum\_{k=0}^\infty \frac{z^k}{\Gamma(a-k)}$$ is divergent (null radius of convergence). Yet a more research-level angle can be reached with the help of ... | 9 | https://mathoverflow.net/users/24309 | 227649 | 106,214 |
https://mathoverflow.net/questions/227045 | 2 | In [this paper](http://arxiv.org/pdf/math/0511710v2.pdf), the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a 2-space by considering only identity morphisms) by:
$$t\_j \bar{t\_i}(x,f)=(x... | https://mathoverflow.net/users/2597 | A question on 2-bundles | They don't say that $f$ is an object of $F$, so it might as well also be a morphism of $F$. In other words, the same equation defines both the object and the morphism parts of $g\_{i j}(x)$, with $f$ interpreted as either an object or a morphism respectively.
| 1 | https://mathoverflow.net/users/49 | 227653 | 106,217 |
https://mathoverflow.net/questions/227588 | 4 | Let $A$ be a bounded subset of a Banach space $X$. Set: $wk\_{X}(A)=\inf\{\epsilon>0:\overline{A}^{w^{\*}}\subset X+\epsilon B\_{X^{\*\*}}\}$, where $\overline{A}^{w^{\*}}$ denotes the $weak^{\*}$ closure of $A$ in $X^{\*\*}$. Since $A$ also can be considered to be a bounded subset of $X^{\*\*}$, my question is:$wk\_{X... | https://mathoverflow.net/users/41619 | A quantity measuring weak non-compactness | Suppose that $A$ is a bounded subset of $X$ and $X$ is a subspace of $Y$. Then
\begin{equation}
wk\_Y(A) \le wk\_X(A) \le 2 wk\_Y(A),\ \ \ (\#)
\end{equation}
and $2$ is the best constant in the right inequality. The choice $Y:=X^{\*\*}$ gives the inequality you want.
First, notice that $wk\_X(A)$ is the supremu... | 2 | https://mathoverflow.net/users/2554 | 227655 | 106,218 |
https://mathoverflow.net/questions/227630 | -2 | Suppose you are given two graphs $G\_1$ and $G\_2$ and are promised that both are twin free. Is the problem of determining if they are isomorphic graph isomorphism hard? I am curious for the cases of simple as well as non-simple directed graphs.
| https://mathoverflow.net/users/41283 | Graph isomorphism for twin free graphs | Twins are easy to detect. Equivalence classes of twins can be replaced by single vertices with colours (or attached gadgets if you don't like colours) that encode the multiplicity. So the general case is no harder than the twin-free case.
| 5 | https://mathoverflow.net/users/9025 | 227661 | 106,221 |
https://mathoverflow.net/questions/227608 | 10 | First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution equations, I had to prove the following lemma
**Lemma:** Let $Z=\{z\_n\}\_{n\in \mathbb{N}}$ be a set of indeterminates,... | https://mathoverflow.net/users/25256 | Algebraic independence of exponentials | First of all, I am not technically answering your question because you ask specifically for a reference. However, here is a nice algebraic proof.
The result should follow immediately from the following lemma, assumed known for $k = \mathbb C$.
**Lemma 1.** Let $k$ be a field of characteristic $0$. Then the elements... | 9 | https://mathoverflow.net/users/82179 | 227663 | 106,223 |
https://mathoverflow.net/questions/227637 | 5 | Consider the numbers of the form: $$A\_n = \prod\_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$
where, the product in taken oven all $2^n$ terms with variations in sign. We know such numbers are integers (even perfect-squares).
>
> Is there a way to investigate the asymptotics of the largest prim... | https://mathoverflow.net/users/62680 | Estimating size of greatest prime divisor of a sequence of integers | The sequence grows too rapidly and I doubt that one can get meaningful asymptotics for the largest prime factor. However, one can show that the largest prime factor is bounded by $(n\sqrt{n})^{2^{\pi(n)}}$ where $\pi(n)$ is the number of primes up to $n$, and since $\pi(n) \sim n/\log n$, this implies a stronger bound ... | 7 | https://mathoverflow.net/users/38624 | 227676 | 106,230 |
https://mathoverflow.net/questions/227319 | 1 | Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C\_0(\Omega)^\*$ , dual of the commutative C\*-algebra $C\_0(\Omega)$, is just the space of complex Radon measures $M(\Omega)$.
$$\gamma: M(\Omega)\simeq C\_0(\Omega)^\* : \gamma(\mu)(f)=\int f d\mu$$
Let... | https://mathoverflow.net/users/84390 | The total variation of a complex measure | The answer is directly obtained by the following interesting fact:
Theorem. Let $M$ be a von neumann algebra and $f$ be a normal functional on $M$.
Assume $a$ is in the unit ball of $M$ with $||af||=|f||$ and $af$ is positive. Then $af$ is just the absolute value of $|f|$. (see theorem 3.2 in Order ideals in a C\*-... | 1 | https://mathoverflow.net/users/84390 | 227680 | 106,231 |
https://mathoverflow.net/questions/227677 | 5 | On [page 52 of this paper](http://projecteuclid.org/euclid.jmsj/1260976291), Iwasawa considered the bilinear symmetric non-degenerate pairing $\Phi\_n \times \Phi\_n \rightarrow \mathbb{Q}\_p/\mathbb{Z}\_p$ defined by
$$\langle \alpha, \beta \rangle\_n := \text{the class of } T\_n(\alpha\beta) \text{ in } \mathbb{Q}\_p... | https://mathoverflow.net/users/14725 | Why is $(A^\perp)^\perp = A$? | The trace form is a perfect pairing in the sense of harmonic analysis, i.e. it identifies $\Phi\_n$ with its Pontryagin dual, see Weil's "Basic Number Theory" or Tate's thesis. Therefore, your statement follows from the Pontryagin duality, for example Proposition 3.6.1 in the book "Principles of Harmonic Analysis" by D... | 9 | https://mathoverflow.net/users/nan | 227686 | 106,234 |
https://mathoverflow.net/questions/227487 | 4 | Let $G$ be a finite group $n = |G|$. Let $\sigma : G \rightarrow GL(n,\mathbb{C})$ be the regular representation. Hence every element of $G$ can be seen as a permutation matrix. Let $\mathbb{Q}[x\_1,...,x\_n]^G = \mathbb{Q}[f\_1,...,f\_m]$ be the invariant ring. Let $s\_1,...,s\_r$ (this might be empty) be the algebrai... | https://mathoverflow.net/users/nan | Common zero of invariants of finite groups | Venkataramana's answer really leaves little to add. But let me nevertheless add that the assertion is true for all reductive groups. It is known as the "surjectivity of the categorical quotient." This can be found in probably all textbooks on invariant theory. Out of personal bias, let me quote one here: Derksen and Ke... | 1 | https://mathoverflow.net/users/82616 | 227726 | 106,250 |
https://mathoverflow.net/questions/227161 | 12 | I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew of any direction on how to move on:
Define the functions $t\_k(x) = \sum\_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ f... | https://mathoverflow.net/users/nan | Connection between cyclic group and exponential function | The natural generalization to a product of cyclic group is having generating functions on several variables. Take the Klein four group as an example. Define
$$t\_{k\_1,k\_2}(x\_1,x\_2) = \sum\_{n\_1,n\_2=0}^\infty \frac{x\_1^{2n\_1+k\_1} x\_2^{2n\_2+k\_2}}{(2n\_1+k\_1)!(2n\_2+k\_2)!}.$$
These functions satisfy
$$
\begi... | 8 | https://mathoverflow.net/users/7732 | 227728 | 106,251 |
https://mathoverflow.net/questions/227491 | 3 | Let $G=(V,E)$ be a simple, undirected graph.
We call a partition ${\cal P}$ of a non-empty subset of $V$ a *Hadwiger partition* if
1. every block (member of ${\cal P}$) is non-empty and connected, and
2. if $x, y \in {\cal P}$ are distinct blocks then there are $v\in x$ and $w \in y$ such that $\{v,w\} \in E$.
Th... | https://mathoverflow.net/users/8628 | Minimal number of vertices in a graph with special Hadwiger partitions | Octahedron$ .$
| 2 | https://mathoverflow.net/users/955 | 227746 | 106,257 |
https://mathoverflow.net/questions/227654 | 3 | Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha\_1,\dots,\alpha\_k)\vdash m$ with $\alpha\_1\ge\dots\ge \alpha\_k>0$ and nonnegative $ x\_1,\dots,x\_n$ define $x^\alpha :=\prod\_{i=1}^k x\_i^{\alpha\_i} $ and ${\tilde x^{\alpha}} :=\prod\_{i=1}^k x\_{i+1}^{\alpha\_i} $. All indices are cyclical... | https://mathoverflow.net/users/29783 | Majorization of cyclic products | This is an idea/a partial answer.
Let $x\_1,\dots,x\_n$ be positive nubmers such that $\prod x\_i=1$. Assume that $y\_i=\prod\_{k=0}^{n-1} x\_{i+k}^{\lambda\_k}$ for some exponents $\lambda\_k$, indices are modulo $n$. Note that we may replace $\lambda\_k$ to $\lambda\_k+c$ for any constant $c$. Now assume that $\sum... | 3 | https://mathoverflow.net/users/4312 | 227750 | 106,260 |
https://mathoverflow.net/questions/227672 | 5 | Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for
\begin{align\*}
\eta &:=
x\_1 \, dy\_1 - y\_1 \, dx\_1 + x\_2 \, dy\_2 - y\_2 \, dx\_2 \\
&=\tfrac{i}{2}(z \, d\bar{z}-\bar{z} \, dz +w \, d\bar{w} - \bar{w}\, dw).
\end{align\*}
The rest... | https://mathoverflow.net/users/69747 | Ascending surfaces in the 4-ball | To see why $T\_pF=\ker\alpha\_r$ at critical points $p$ of $\rho|\_F$, note that Morse-ness of $\rho|\_F$ means that $d(\rho|\_F)$ vanishes to precisely first order at $p$. On the other hand, we have
$$
(d\rho\wedge\eta)|\_F=fd\mathrm{vol}\_F
$$
for some $f\colon F\to\mathbb R\_{\ge0}$. Because $f$ is nowhere negative,... | 4 | https://mathoverflow.net/users/300 | 227758 | 106,262 |
https://mathoverflow.net/questions/227773 | 2 | Several [posts](https://mathoverflow.net/q/31545) (and [this](https://cs.stackexchange.com/q/41651/941)) on StackExchange sites like MO have some link-rot. For example, I've been looking into tree decomposition and keep coming across references to [treewidth.com](http://www.treewidth.com), but the link appears to be de... | https://mathoverflow.net/users/21157 | Does anyone have the correct link to treewidth.com? | In situations like this, <https://archive.org> helps.
In this case check:
[https://web.archive.org/web/\*/http://www.treewidth.com/](https://web.archive.org/web/*/http://www.treewidth.com/)
There are many snapshots over time, though the downloads fail for me.
Searching for "libtw.jar", showed the same site at:
... | 4 | https://mathoverflow.net/users/12481 | 227778 | 106,268 |
https://mathoverflow.net/questions/227757 | 4 | Which journals publish short papers/technical note in applied probability, for example stochastic approximation ?
| https://mathoverflow.net/users/77923 | Short papers in applied probability | You can try the [Journal of Applied Probability](http://journals.cambridge.org/action/displayJournal?jid=JPR).
| 1 | https://mathoverflow.net/users/10365 | 227796 | 106,272 |
https://mathoverflow.net/questions/227540 | 3 | For $q$ an odd prime power, one can construct $H:=O\_4^-(q)<U\_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U\_4(q)$ is isomorphic to $O\_6^-(q)$; I need to understand the geometric meaning of $H$ in $O\_6^-(q)$.
For $q \equiv 1\pmod {4}$, it appears that $H$ can be viewed as $O... | https://mathoverflow.net/users/11100 | embedding of $O_4^-(q)$ in $U_4(q)$ | As is pointed out in the question, our job is to display an (irreducible) copy of $H=O\_3(q^2)$ inside $G=O\_6^-(q)$.
As mentioned in comments, this can be done by taking $H$ to be a field-extension subgroup of $G$. An explicit construction for $q\equiv 3 \pmod 4$ is as follows.
We take $H$ to be the group of isome... | 2 | https://mathoverflow.net/users/801 | 227798 | 106,273 |
https://mathoverflow.net/questions/227799 | 11 | Being a novice with differential equations, I recently learned that if $\mu$ is an integrating factor for $\frac{dy}{dx}f(x,y)+ g(x,y)=0$, then the corresponding 1-form, $\mu fdy+\mu g dx$, is exact.
My question is, **is there some constructive proof that given a 1-form,
$fdy+gdx$, we can find a scalar function $\mu$... | https://mathoverflow.net/users/50796 | Existance of Integrating Factors, a Constructive Proof | You are basically asking for an «algorithm» deciding whether a first-order ODE (with, say, analytic coefficients $f, g$) has a (meromorphic,say) first-integral. This question (which Poincaré asked when $f, g$ are polynomials) is difficult and still open. I'll try to give here examples of undecidability from the sole kn... | 12 | https://mathoverflow.net/users/24309 | 227811 | 106,274 |
https://mathoverflow.net/questions/227752 | 6 | Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ can never be bounded if $\omega$ is a complete Kahler metric.
**Is the following claim true: the potential of a complet... | https://mathoverflow.net/users/4789 | Can the potential of a complete Kahler metric be bounded? | **The claim is false.** The potential $\psi$ of the complete Kahler metric on the disk given in Theorem 1.1 of <http://arxiv.org/pdf/math/0603530v7.pdf> is bounded.
Indeed, such potential $\psi$ is the pullback of the potential of $P := |z\_1|^2 + |z\_2|^2 + |z\_3|^2$ by the holomorphic map $X$. Since the image of $X$ ... | 3 | https://mathoverflow.net/users/43122 | 227824 | 106,281 |
https://mathoverflow.net/questions/227806 | 7 | While considering [this post](https://math.stackexchange.com/questions/1598329/can-we-find-x-1-x-2-x-n), it made me wonder about its generalization in another direction and from the perspective of Galois theory.
---
**Question:** Is it true that, given four constants ($\alpha,\beta,\gamma,\delta$), then the syst... | https://mathoverflow.net/users/12905 | Is this system always solvable in radicals by quartics, octics, $12$-ics, etc? | For $(a,b,c,d)=(1,2,3,20)$ and $(\alpha,\beta,\gamma,\delta)=(1,1,1,0)$ one computes that $x\_1$ is a root of the irreducible polynomial $4 x^{20} - 20 x^{19} + 40 x^{18} - 40 x^{17} + 195 x^{16} - 704 x^{15} + 1050 x^{14} - 700 x^{13} + 475 x^{12} - 900 x^{11} + 900 x^{10} - 300 x^{9} + 130 x^{8} - 260 x^{7} + 130 x^{... | 6 | https://mathoverflow.net/users/18739 | 227825 | 106,282 |
https://mathoverflow.net/questions/227784 | -2 | Given $p,q\in\mathbb Z^+$, and a vector $v=(x\_1,\dots,x\_{p+q})$ we consider the function $\chi(v)$:
$$\chi(v)=x\_1^2+\dots+x\_p^2-x\_{p+1}^2-\dots-x\_{p+q}^2$$
We wish to find solutions to $\chi(v)=0$ for all $p,q$ with the condition that $x\_i\in\mathbb Z^+$. This has no solutions if $p=0$ or $q=0$.
I have fou... | https://mathoverflow.net/users/70355 | Finding integer zeroes for a particular family of equations | Here is an easy argument, answering affirmatively both questions. I have interpreted the slightly ambiguous wording as meaning that for each choice of $p,q$, with both positive (which means *strictly* positive), there exists a solution with all entries (strictly) positive.
Lemma. Given $q \geq 1$, there exists
a tup... | 2 | https://mathoverflow.net/users/42278 | 227826 | 106,283 |
https://mathoverflow.net/questions/224124 | 9 | I've stumbled upon the statement that the morphism $\pi$ from a root stack of the form $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ (i.e. the "generic" version, not the one concentrated along a divisor) to its underlying stack $\mathscr{Y}$ is "étale".
Now, I know what "étale" means for representable morphisms (which the abov... | https://mathoverflow.net/users/4721 | Definition of étale (etc) for non-representable morphisms of algebraic stacks? | For non-representable morphisms of Artin (i.e. algebraic) stacks, different properties are defined in different ways, depending on their particular nature.
E.g. for properties which are smooth local on source and target, see [here](http://stacks.math.columbia.edu/tag/06FL) in the Stacks Project.
The property of bei... | 6 | https://mathoverflow.net/users/58057 | 227835 | 106,285 |
https://mathoverflow.net/questions/227803 | 4 | Denote by $P\_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by endowing it with the $\sigma$-algebra generated by open sets and Wiener-measure.
Let $F:P\_0(\mathbb{R}^d)\rightarrow ... | https://mathoverflow.net/users/84961 | Malliavin Calculus: directional derivatives of cylinder functions exist in what sense? | We can assume that all the first derivatives of $f$ are bounded by some polynomial function, let's say $|\nabla^j f(x)|\le C(1+|x|^p)$. Then we have $|F'(\omega + th)| \le n C(1 + (\|\omega\|\_\infty + t \|h\|\_\infty)^p)$ which can be bounded, independent of $t$, by something like $C\_1(1 + \|\omega\|\_\infty^p)$. But... | 2 | https://mathoverflow.net/users/4832 | 227836 | 106,286 |
https://mathoverflow.net/questions/227848 | 4 | In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where $\delta=1-\frac{1+\log \log 2}{\log 2}\approx 0.08607\ldots $ after defining $$H(x,y,z)\stackrel{\mathrm{def}}{=} \#\{n\leq x:\text{$n$ ... | https://mathoverflow.net/users/17773 | Improvement of a bound on divisor distributions from "Divisors" (Hall and Tenenbaum)? | [Kevin Ford](http://arxiv.org/pdf/math/0401223v5.pdf) has determined the order of $H(x,y,z)$ for all ranges of $x$, $y$, $z$. In particular, from his work (see Corollary 2) it follows that for $10 \le y\le \sqrt{x}$ one has
$$
H(x,y,2y) \asymp \frac{x}{(\log y)^{\delta} (\log \log y)^{3/2}}.
$$
In addition to the A... | 7 | https://mathoverflow.net/users/38624 | 227850 | 106,290 |
https://mathoverflow.net/questions/227457 | 11 | I have stumbled across [a related question](https://mathoverflow.net/q/87430/30186) asking which large cardinal properties can hold for $\aleph\_1$. [This question](https://mathoverflow.net/q/45448/30186) is probably also related, asking in what ways $\aleph\_0$ is a "large" cardinal.
To state my question:
>
> Fo... | https://mathoverflow.net/users/30186 | What sort of large cardinal can continuum be? | Let me add a few examples:
**(1)** If we start with a supercompact cardinal $\kappa$, and force with $Add(\omega, \kappa)$, then in the extension the cardinal $\kappa=2^\omega$ becomes generically supercompact. The same holds for many other large cardinals.
**(2)** If we start with a weakly compact cardinal, we can... | 7 | https://mathoverflow.net/users/11115 | 227856 | 106,293 |
https://mathoverflow.net/questions/227854 | 1 | Let $M$ and $N$ be $CW$-complexes.
**Definition. (different from the isotopy notion in geometry of submanifolds).** A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times [0,1]\longrightarrow N\times [0,1]
$$
such that $F$ maps the fibre $M\times t$ homeomorphically onto a subset of the fibre $N\times... | https://mathoverflow.net/users/41075 | isotopy equivalence (topological meaning) between $CW$-complexes | Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy condition implies that $fg$ is a homeomorphism onto the image, which coincides with the image of $f$ since $g$ is onto. T... | 4 | https://mathoverflow.net/users/12166 | 227857 | 106,294 |
https://mathoverflow.net/questions/227797 | 2 | Following Dan Willard (from his paper "Self-Verifying Axiom Systems, the Incompleteness Theorem, and Related Reflection Systems", found on his [homepage](http://www.cs.albany.edu/FacultyStaff/profiles/willard.htm), pdf [here](http://www.cs.albany.edu/%7Edew/m/jsl1.pdf)):
>
> "Define an axiom system $\alpha$ to be *... | https://mathoverflow.net/users/20597 | A question regarding the consistency of Nelson's Predicative Arithmetic | No, Nelson's system extends Robinson arithmetic, so it's subject to Gödel's incompleteness theorems. Willard's systems can't prove the totality of addition or multiplication (see the wiki article) and that lets them avoid being able to carry out the diagonalization needed to create unprovable sentences.
<https://en.w... | 6 | https://mathoverflow.net/users/84989 | 227859 | 106,295 |
https://mathoverflow.net/questions/227839 | 9 | Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any references?
| https://mathoverflow.net/users/41075 | embedding of quaternionic projective spaces | I. M. James, Lectures on algebraic and differential topology, pp. 134–174, Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972,
Theorems 1.2 and 1.3 show that $$N=13.$$
| 14 | https://mathoverflow.net/users/83633 | 227863 | 106,296 |
https://mathoverflow.net/questions/227860 | 8 | The symmetric group $S\_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the regular representation of $S\_n$, the basis of this vector space is indexed by the elements of $S\_n$. Then what are the ge... | https://mathoverflow.net/users/84990 | Ring of invariants for the regular representation | To the best of my knowledge this is an open problem. In fact, there is strong evidence that the problem is very hard indeed: Consider the action of $S\_n$ on the "two-sets," i.e., on the subsets of $\{1,\ldots,n\}$ of two elements. It is easy to see that this is a subrepresentation of the regular representation. So if ... | 11 | https://mathoverflow.net/users/82616 | 227872 | 106,297 |
https://mathoverflow.net/questions/227801 | 2 | A lot of recent literature in Bayesian approach to inverse problems involves Adaptive priors, i.e - priors that depend on noise level. A lot of articles deal with optimization of contraction rates using Adaptive priors.
My understanding is that in Bayesian statistics, the prior is constructed from information about the... | https://mathoverflow.net/users/65922 | Adaptive priors | The modification of the traditional Bayes method where the prior distribution is determined by the data goes by the name [empirical Bayes method.](https://en.wikipedia.org/wiki/Empirical_Bayes_method) The *motivation* for the empirical approach is a practical one: Not always is there enough information a priori availab... | 1 | https://mathoverflow.net/users/11260 | 227876 | 106,299 |
https://mathoverflow.net/questions/227656 | 1 | I asked this question two days ago om Math SE but didn't receive an answer: <https://math.stackexchange.com/questions/1597321/generalization-of-hkh-kk-cap-h>
Suppose we are given subgroups $H,K$ of a finite group $G$. Denote by $\langle H,K\rangle$ the subgroup generated by $H$ and $K$.
If $H$ normalizes $K$, then... | https://mathoverflow.net/users/64639 | Generalization of $(HK:H)=(K:H\cap K)$ | I don't really know what you are looking for, but, for the record, if we know a full set $T$ of the $(H,K)$-double coset representatives in $G = \langle H,K \rangle$, ( so that $G$ is the disjoint union $\bigcup\_{t \in T}(HtK)$), then, for each $t$, we $|HtK| = |t^{-1}HtK| = \frac{|H||K|}{|t^{-1}Ht \cap K|}$. Hence $[... | 2 | https://mathoverflow.net/users/14450 | 227878 | 106,301 |
https://mathoverflow.net/questions/227817 | 1 | I encounter the following question.
$\textbf{Problem}$: For almost all Matrix $M\in\mathcal M\_{m\times n}(\mathbb R),$ all $y\in \mathbb R^m$ and any $N$, small $\epsilon>0$, there exists a constant $C$, depending on $\epsilon$ and $M$, and $p\in \mathbb Z^m$ , $q\in \mathbb Z^n$ satisfying $|q|<N$, such that the f... | https://mathoverflow.net/users/15214 | A quantitative Kronecker theorem | As far as I understand the question, we need to consider matrices $M=(m\_{ij})$ with entries in $[0,1]$ (the integer part does not matter), look at the vectors of $Mq$ modulo $1$ again with $\|q\|\_\infty\le N$ (the exact choice of the norm is not important because it merely changes the constant in the final answer) an... | 4 | https://mathoverflow.net/users/1131 | 227879 | 106,302 |
https://mathoverflow.net/questions/227708 | 2 | I have recently learned about the representation theorem for isotropic,
linear operators, which says the following:
**Defintion:**
Let $M\_n$ be the vector space of $n \times n$ real matrices. We say a linear operator $\alpha:M\_n \to M\_n$ is **isotropic** if:
$(\*) \, \, \alpha(S^TXS)=S^T\alpha(X)S \, , \, \for... | https://mathoverflow.net/users/46290 | Heuristics for counting degrees of freedom | Let $V$ be a finite-dimensional real inner product space. You want to know the dimension of $\text{End}\_G(V \otimes V^{\ast})$ where $G = O(V)$. Using the inner product we have an isomorphism $V \cong V^{\ast}$, so
$$V \otimes V^{\ast} \cong V \otimes V \cong S^2(V) + \wedge^2(V)$$
(where $+$ denotes direct sum).... | 3 | https://mathoverflow.net/users/290 | 227904 | 106,310 |
https://mathoverflow.net/questions/227892 | 10 | I posted this question first in Math.StackExchange one week ago [here](https://math.stackexchange.com/q/1596096/254733), but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function.
A sufficient condition... | https://mathoverflow.net/users/75968 | $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$ | That doesn't work because $H\_0^1$ functions are small near the boundary, so testing against them won't detect bad behavior of $u$ near $\partial\Omega$.
For a concrete example, take $\Omega$ as the unit ball and $u(x)=1/(1-|x|)\notin L^1$. Then
$$
\int |uv|\, dx \le \left( \int \frac{v^2\, dx}{(1-|x|)^{3/2}} \int \f... | 15 | https://mathoverflow.net/users/48839 | 227908 | 106,311 |
https://mathoverflow.net/questions/227889 | 20 | I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of Section 2 [here](http://www.maths.ed.ac.uk/~tom/belfastXXX.pdf)). As a theoretical physicist, I'm struggling to understand... | https://mathoverflow.net/users/22337 | Why are quantum groups so called? | Typically in math "quantum X" means a deformation of "X" which is in some sense "less commutative." So quantum groups should be deformations of groups which are "less commutative." Interpreting this is slightly tricky since groups are already non-commutative, but nonetheless they do have some "commutativity" built in w... | 24 | https://mathoverflow.net/users/22 | 227913 | 106,314 |
https://mathoverflow.net/questions/227909 | 1 | For a group $G$ and an inverse closed subset $S$ of $G\setminus \{1\}$, the Cayley graph $Cay(G,S)$, is the graph whose vertices are the elements of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy^{-1}\in S$. Let $a,a^{-1}$ are two elements of $G$ that are not in $S$. Also suppose $d$ be the diameter o... | https://mathoverflow.net/users/84430 | diameter of Cayley graphs | Suppose $G$ is a finite group and we choose $S'=G\setminus\{1,a,a^{-1}\}$. Then if we let $S=S'\cup \{a,a^{-1}\}$, the diameter of the cayley graph $Cay(G,S)$ is one, but the diameter of the Cayley graph $Cay(G,S')$ is not one (it seems it is two). In contrast, we have a lot of large Cayley graphs with diameter two, an... | 4 | https://mathoverflow.net/users/19885 | 227917 | 106,315 |
https://mathoverflow.net/questions/227905 | 3 | In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here).
Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is also fixed, and let $a\_i$ be independent variables.
We define $M$ by the formula: the $(i,j)$-th entry of $M$ is $a\_... | https://mathoverflow.net/users/80668 | On a determinantal equality | Let me prove a slightly more general claim.
>
> **Theorem 1.** Let $\mathbb{K}$ be a commutative ring. Let $R$ be a set. Let
> $n\in\mathbb{N}$ (where $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $). Set
> $\left[ n\right] =\left\{ 1,2,\ldots,n\right\} $. For every $i\in\left[
> n\right] $, $j\in\left[ n\right] $ and... | 5 | https://mathoverflow.net/users/2530 | 227922 | 106,316 |
https://mathoverflow.net/questions/227916 | 10 | In addition to the two reasonably well-known categories $\mathrm{SuperVect}\_{\mathbb R}$ and $\mathrm{SuperVect}\_{\mathbb C}$ of *real* and *complex* *super vector spaces*, each of which is monoidally equivalent to corresponding category of $\mathbb Z/2$-graded vector spaces but with the Koszul sign rules, there is a... | https://mathoverflow.net/users/78 | What is the "quaternionic" super Brauer group? | The Brauer-Picard 2-category of $SuperVect\_{\mathbb R}$ (let's call it $sBrPic\_\mathbb R$) is the homotopy fixed points of the Brauer-Picard 2-category of $SuperVect\_{\mathbb C}$ w.r.t. the involution given by complex conjugation (let's call that involution $C$).
It is plausible that the Brauer-Picard 2-category o... | 9 | https://mathoverflow.net/users/5690 | 227926 | 106,318 |
https://mathoverflow.net/questions/227918 | 7 | The matrices I am dealing with are $n\times n$ of the following type (with $n=7$):
$M\_7=\begin{pmatrix}1&0&0&0&0&0&1 \\ 1&1&0&0&0&0&0 \\ 0&1&1&0&0&0&0 \\ 0&0&1&1&0&0&0 \\ 0&0&0&1&1&0&0 \\ 0&0&0&0&1&1&0 \\ 0&0&0&0&0&1&1\end{pmatrix}$.
$M\_n$ only has $1$'s on the main diagonal, on the diagonal just below the main d... | https://mathoverflow.net/users/85022 | Covering the zeros of 0/1 matrix with submatrices | Divide-and-conquer works. It takes $O(\log n)$ submatrices to partition the $0$s. So, for large $n$, it takes fewer than $n$ submatrices.
Let $M'\_n$ be the same as $M\_n$ except with a $0$ in position $(1,n)$. Let $f(n)$ be the minimum number of submatrices required to cover $M'\_n$. It takes at most $f(n)+2$ submat... | 10 | https://mathoverflow.net/users/2954 | 227927 | 106,319 |
https://mathoverflow.net/questions/227928 | 1 | Let a statistical model of a random variable $X$ with parameter $\theta \in R^m$ be represented by a density function $p(X=x|\theta)$. Assume that the prior, $q(\cdot)$, is on a lower dimensional function of $\theta$, $f(\theta): R^m\to R^n$ where $n<m$. Therefore we cannot translate the prior on $f(\theta)$ to $\theta... | https://mathoverflow.net/users/17276 | Bayesian estimation with lower dimensional prior | Here is one approach, [Optimal low-rank approximations of Bayesian linear inverse problems](http://arxiv.org/abs/1407.3463), see also reference 56. Typically, you construct a reduced basis for the parameter space.
| 1 | https://mathoverflow.net/users/11260 | 227946 | 106,324 |
https://mathoverflow.net/questions/227898 | 5 | Let $X$ be a compact oriented surface of genus at least two, equipped with a Riemannian metric $g$. By the uniformization theorem for Riemann surfaces, there is a conformal universal covering map $p:D\to X$, where $D$ is the unit disc. The standard hyperbolic metric on $D$ is thus $e^{2f\_0}p^\*(g)$ for some function $... | https://mathoverflow.net/users/10366 | Uniqueness of hyperbolic rescaling | Here is an answer which I learned from a set of notes titled "Conformal Geometry Seminar - The Poincare Uniformization Theorem" by Gilbert Weinstein (DOI: 10.13140/RG.2.1.4644.9767). Suppose we have found one metric $g$ in the original conformal class with curvature $K(g)=-1$. Suppose also that $K(e^{2f}g)=-1$; we need... | 3 | https://mathoverflow.net/users/10366 | 227949 | 106,325 |
https://mathoverflow.net/questions/227956 | 2 | I suspect that the following result should be true and more or less well known:
Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to \mathcal{M}$ and every $i \in I$, the natural map $holim\_I X \to X(i)$ is a weak equivalence.
Indeed this is a direct g... | https://mathoverflow.net/users/5181 | Homotopy limits of homotopically constant diagrams over contractible categories | It is true. You can reduce to the case of simplicial sets by using the fact that $\mathbf{R} \mathrm{Hom} (T, -)$ preserves homotopy limits and (allowing $T$ to vary) is jointly conservative. Finding where this is written down is another matter altogether...
I'll just sketch something more concrete for the case of a ... | 3 | https://mathoverflow.net/users/11640 | 227961 | 106,329 |
https://mathoverflow.net/questions/227880 | 4 | Let $\mathcal{O}\_\lambda$ be the set of hermitian $n+1 \times n+1$ matrices with Eigenvalues $\lambda = (\lambda\_1, \dots, \lambda\_{n+1})$.
and $\mathcal{O}^\mu$ the set of hermitian $n \times n$ matrices with Eigenvalues $\mu = (\mu\_1, \dots, \mu\_n)$.
Let $\pi \colon \mathcal{O}\_\lambda \to \mathbb{C}^{n\times... | https://mathoverflow.net/users/75382 | subset of hermitian matrices given by eigenvalues form a submanifold | The set $\pi^{-1}(\mathcal{O}^\mu)$ is always a smooth submanifold of $\mathcal{O}\_\lambda$ (though it may well be empty). When it is not empty, it is a single orbit of $\mathrm{U}(n)\subset \mathrm{U}(n{+}1)$, where $\mathrm{U}(n)$ is the subgroup of $\mathrm{U}(n{+}1)$ consisting of those unitary $(n{+}1)$-by-$(n{+}... | 3 | https://mathoverflow.net/users/13972 | 227964 | 106,330 |
https://mathoverflow.net/questions/227821 | 5 | In a Commentarii Mathematici Helvetici paper by Benno Eckman and Heinz Müller in 1980 (volume 50, pages 510-520) proved that poincaré Duality Groups of dimension 2 with positive first Betti number are surface groups.
Is there any development into proving that certain groups of (homological) or geometric dimension 2 ... | https://mathoverflow.net/users/21985 | homological 2 dimensional groups | See my paper with Bruce Kleiner "Geometry of quasiplanes" for general unital commutative rings, it also contains a reference to the paper by Eckmann and Linnell from 1983 where they prove the theorem for PD(2) groups over Z and to the one by Bowditch which works over Q.
| 4 | https://mathoverflow.net/users/21684 | 227966 | 106,332 |
https://mathoverflow.net/questions/207824 | 7 | I was trying to understand the paper "Forms of GL(2) from the analytic point of view", by Gelbart and Jacquet.
On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining operator $M(\eta\_{\nu})$ has codimension one.
However, I cannot immediately see this. Any ideas?
| https://mathoverflow.net/users/36035 | Analytic continuation of intertwining operator | This question is from a while ago, but for what it's worth, I agree with GH from MO: the fact in question is well-known but not particularly obvious. We didn't do any intertwining operators in the book (regrettably!) but you can see what you wanted fairly easily from the Bruhat pairing in corollary 8.10.11. The point i... | 4 | https://mathoverflow.net/users/21252 | 227974 | 106,334 |
https://mathoverflow.net/questions/227977 | 4 | Take $F$ a local field and $\chi\_1, \chi\_2$ two characters, write $M(\chi\_1, \chi\_2)$ for the standard intertwining integral
$$M(\chi\_1. \chi\_2).f(g) := \int\_{F} f\left( \begin{pmatrix} 0&-1\\ 1& 0 \end{pmatrix} \begin{pmatrix} 1& x \\ 0 & 1 \end{pmatrix}g \right) \, dx.$$ (Which varies meromorphically as a fun... | https://mathoverflow.net/users/21252 | Reference request: normalization of intertwining operators for GL(2, C) | This is on page 57-58 of M. Duflo, "Representations irreductibles des groupes semisimples complexes", in "Analyse harmonique sure les groupes de Lie", SLN 497, 1975, 26-88. V. Drinfeld also noted in email to me (2014) that Zelobenko wrote up SL(2,C) in 1963, in Izv. Akad. Nauk SSSR Ser. Mat. 27, 1343-1394, but I did no... | 4 | https://mathoverflow.net/users/15629 | 227980 | 106,336 |
https://mathoverflow.net/questions/227118 | 12 | First of all my apologies if this question is well known or obvious: this is not in my area of research.
Let $T(x)=\sum\_{n=0}^\infty t\_nx^n$, where $t\_n$ is the number of partitions $\lambda$ of $n$ into $m$ parts where, if $(m-i)\times (m-i)$ is the size of the Durfee square of $\lambda$, then the partition on th... | https://mathoverflow.net/users/45242 | Generating function for certain partitions (with a restriction on the Durfee square) | **Lemma.** Fix $n$ and $m$. Consider pairs of partitions $(\lambda,\mu)$ such that $\lambda$ has $m$ parts and $|\lambda|+|\mu|=n$. Let $A$ be the number of pairs for which $\max(\lambda)>\max(\mu)$ (where $\max(\emptyset)=-\infty$). Let $B$ be number of pairs for which $\lambda$ satisfies this condition with Durfee sq... | 6 | https://mathoverflow.net/users/4312 | 228021 | 106,348 |
https://mathoverflow.net/questions/227970 | 13 | By Friedman translation $HA$ and $PA$ prove the same $\Pi\_2$ formulas.
Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic?
Axioms of $Q$ are:
1. $\neg(Sx=0)$
2. $Sx=Sy\rightarrow x=y$
3. $y=0 \lor \exists x(Sx=y)$
4. $x+0=x$
5. $x+Sy=S(x+y)$
6... | https://mathoverflow.net/users/83598 | Intutionistic Robinson Arithmetic | Both are false. Consider the following Kripke model $M\vDash Q^e$ (in fact, it satisfies the intuitionistic version of $\mathrm{PA}^-$): it consists of two worlds $u,v$ such that $u$ sees $v$; the first-order structure at $v$ is the semiring $M\_v$ of polynomials $f\in\mathbb Z[x]$ with positive leading coefficient (an... | 7 | https://mathoverflow.net/users/12705 | 228027 | 106,352 |
https://mathoverflow.net/questions/228033 | -4 | Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}\_G(v) = \deg\_H(\varphi(v)).$$
Does this imply that $G\cong H$? In case the answer is positive, does it remain so if we consider infinite graphs... | https://mathoverflow.net/users/8628 | Reconstructing a graph from the multiset of degrees | No, consider the [Petersen graph](https://en.wikipedia.org/wiki/Petersen_graph) and another with no star in the middle but just a simple pentagon. The former has no 4-cycle, but the latter has one.
| 3 | https://mathoverflow.net/users/66559 | 228035 | 106,355 |
https://mathoverflow.net/questions/228040 | 9 | It is well-known that if the natural (partial) order on the class of cardinal numbers is a linear order, then it is in fact a well-order and the axiom of choice holds. I was, however, interested in how much choice we can recover given *some* linear ordering, or better — a well-ordering — of the class of cardinals.
I ... | https://mathoverflow.net/users/30186 | How much choice does a linear or well-order on cardinals imply? | No.
In Cohen's first model, as constructed over $L$, you have a uniform linear ordering of the entire model. However the axiom of countable choice fails: there is a Dedekind-finite set of reals.
To see that the first claim holds, note that the model is $L(A)$ where $A$ is the of generic Cohen reals. Therefore there... | 5 | https://mathoverflow.net/users/7206 | 228042 | 106,357 |
https://mathoverflow.net/questions/228014 | 2 | Let $X$ and $Y$ be independent and identically distributed random variables.
Can $X+Y$ be a uniform distribution?
(Please prove.)
In other words, is a uniform distribution divisible?
The meaning of "divisible" is described at [Wikipedia](https://en.wikipedia.org/wiki/Indecomposable_distribution).
>
> If, furthe... | https://mathoverflow.net/users/85073 | divisibility of uniform distribution | Here is a direct argument.
Suppose independent $X\_1,X\_2 \sim X$, and $X\_1+X\_2$ is uniform on $[0,1]$.
$X$ is supported on $[0,1/2]$.
For any $0 \lt \alpha \lt 1/4$,
$\alpha = P\left(X\_1+X\_2 \in [0,\alpha]\right) \le P(X \in [0,\alpha])^2$ so $P(X \in [0,\alpha]) \ge \sqrt{\alpha}$. Similarly, $P(X \in [1/... | 3 | https://mathoverflow.net/users/2954 | 228044 | 106,358 |
https://mathoverflow.net/questions/228046 | 3 | I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly discrete would be great).
Now, if $G=\mathbb R^d$, then this is a consequence of the Paley-Wiener theorem, but $G$ i... | https://mathoverflow.net/users/11552 | Are the zeroes of the Fourier Transform of compactly supported functions isolated? | 1. Without the condition that $f$ is positive definite, the answer to the modified question is "no", even when $G=R$, the real line.
Suppose that the support of $f$ does not contain some neighborhood of $0$, say
$[-1,-1/2]\cup[1/2,1]$, and $f(-x)=\overline{f(x)}$, so that Fourier transform $F$ is real. Then $\sum F(t-t... | 5 | https://mathoverflow.net/users/25510 | 228055 | 106,364 |
https://mathoverflow.net/questions/227925 | 4 | Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the Lebesgue measure or the ideal of meager sets.
By theorem 17.41 of Kechris' book *Classical Descriptive Set Theory,* if $X$ ... | https://mathoverflow.net/users/41274 | Invariants of category in Polish spaces | Your suspicion is correct -- the answer is yes.
To see this, first observe that if $Y$ is a dense $G\_\delta$ subset of (a perfect Polish space) $X$, then $X$ and $Y$ must agree on the four cardinal invariants you mention. Second, I claim that that the Baire space $\omega^\omega$ is a dense $G\_\delta$ subspace of ev... | 3 | https://mathoverflow.net/users/70618 | 228063 | 106,366 |
https://mathoverflow.net/questions/185629 | 10 | Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
I know of the following:
$P$ is positive definite and elementwise nonnegative. Moreover, $p\_{jk}p\_{ii} \ge p\_{j... | https://mathoverflow.net/users/61028 | Product $PVPVP$ is elementwise nonnegative? | First, we repeat the arguments from this [stackexchange answer](https://math.stackexchange.com/a/989748). $P^{-1}$ is an $M$-matrix, and can thus be written as $s(I-A)$ for some positive $s$ and some $A$ with non-negative entries. As $P^{-1}$ is positive definite, the spectrum of $A$ lies to the left of $\{ z: \hbox{Re... | 11 | https://mathoverflow.net/users/766 | 228067 | 106,369 |
https://mathoverflow.net/questions/228038 | 4 | I asked this [question on stats.stackexchange](https://stats.stackexchange.com/questions/188371/negative-population-variable-importance) and even elsewhere, but it never received an answer.
I just state the probabilistic problem here. It is about the optimality of the conditional expectation.
Consider a random ve... | https://mathoverflow.net/users/21339 | Negative population variable importance | That is not possible: $I$ cannot be negative.
Indeed, let $X:=X\_1$, $X':=X'\_1$, $V:=(X\_2,X\_3)$. Let $E\_Z$ denote the conditional expectation given a random variable/random vector $Z$. Then $E\_{(X',V)}Y=E\_V Y$, since $X'$ is independent of $(Y,V)$ (see details below). So,
$$E(Y-f(X',V))^2\ge E(Y-E\_{(X',V)}Y)^... | 2 | https://mathoverflow.net/users/36721 | 228085 | 106,376 |
https://mathoverflow.net/questions/227869 | 18 | Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $\lambda=\sum \lambda\_i\omega\_i$ for $\lambda\_i\in\mathbb N$.
Is there a simple formula for the [Frobenius-S... | https://mathoverflow.net/users/5690 | Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group? | The Frobenius-Schur indicator (of a self-dual finite dimensional representation) is
$$
\chi\_\pi(\exp(2\pi i\rho^\vee))
$$
where $\chi\_\pi$ is the central character of $\pi$, $\rho^\vee$ is half the sum of the positive coroots, so $\exp(2\pi i\rho^\vee)$ is an element of order $2$ in the center of $G$. Equivalently i... | 12 | https://mathoverflow.net/users/6030 | 228088 | 106,378 |
https://mathoverflow.net/questions/228087 | 3 | (Cross-posted from [StackExchange](https://math.stackexchange.com/questions/1600159/euler-product-for-sum-of-multiplicative-function-times-log)). Let $g$ be a multiplicative function which satisfies $0 \le g(p) \ll 1/p$ and
$$ \sum\_{p\le x} g(p) = \log \log x + C + O((\log x)^{-10}). $$
[Iwaniec and Friedlander](h... | https://mathoverflow.net/users/70060 | Euler product for sum of multiplicative function times log | Generally speaking, Euler product type formulae involving logarithms can be derived (formally, at least) from Euler product formulae without logarithms via differentiation in the $s$ parameter.
In the specific situation, one can argue as follows.
For $s>0$ one has
$$ \sum\_n \frac{\mu(n) g(n)}{n^s} = \prod\_p (1 - ... | 11 | https://mathoverflow.net/users/766 | 228090 | 106,380 |
https://mathoverflow.net/questions/228082 | 4 | Let $\Sigma^b\_i$ and $\Pi^b\_i$ formulas be bounded formulas defined by Buss in language of $L\_b$. $PIND(\phi(x))$ is the formula:
$$\phi(0)\land \forall x(\phi(\left \lfloor \frac{x}{2} \right \rfloor)\rightarrow \phi(x))\rightarrow \forall x \phi(x)$$
* For $i\geq0, S^i\_2=BASIC+PIND\:\Sigma^b\_i$,
* For $i\geq... | https://mathoverflow.net/users/83598 | Weak Bounded Arithmetics | $\def\dotminus{\mathbin{\dot{-}}}$Actually, there are a number of papers on variants of $S^0\_2$, $T^0\_2$, and other theories axiomatized by $\Sigma^b\_0$ (sharply bounded) schemata, in particular:
[1] Gaisi Takeuti, *Sharply bounded arithmetic and the function $a\dotminus1$*, in: Logic and Computation (W. Sieg, ed.... | 5 | https://mathoverflow.net/users/12705 | 228102 | 106,385 |
https://mathoverflow.net/questions/228048 | 5 | [This was first posted on MSE but did not get any answer. I apologize if it is not suited for MO.]
Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext\_X^c(E,\omega\_X)$ be the Grothendieck d... | https://mathoverflow.net/users/30827 | Which curves have reflexive structure sheaf? | Indeed, this condition is equivalent to $C$ being CM (Regardless of $X$ as long as it is also CM). This "dual" is called the $\omega$-dual in [Kol13](https://goo.gl/1NwSOJ). You might be interested in reading section 2.5, or just the part directly dealing with this on pp.80-83. In particular, the statement you need is ... | 4 | https://mathoverflow.net/users/10076 | 228107 | 106,386 |
https://mathoverflow.net/questions/227975 | 7 | I have come across the following deceptively simple expression:
$$ H\_n^s=\sum\_{j=1}^n(-1)^{j-1}\left(\begin{array}{c}n\\j\end{array}\right)j^{-s} $$
We have (using eg mathematica, though probably not difficult to prove): $H\_n^0=1$, $H\_n^1=H\_n$ (the harmonic numbers), expressions involving hypergeometric series... | https://mathoverflow.net/users/nan | Alternating binomial Dirichlet series |
>
> 1. Does this reduce to values of a known special function for arbitrary real (or complex) $s$?
>
>
>
Answered by **Johannes Trost** in a comment: it's also known as a
"Roman harmonic number". But this and the associated references
do not yield answers to the next two questions, so I continue:
>
> 2. What... | 9 | https://mathoverflow.net/users/14830 | 228112 | 106,388 |
https://mathoverflow.net/questions/228114 | 4 | "Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have looked through e.g. Edgar's [Transseries for beginners](http://arxiv.org/abs/0801.4877). The transseries considered there... | https://mathoverflow.net/users/49 | Redundancy in transseries representation of functions? | How embarassing; I puzzled over this for several days, only to figure out the answer an hour after posting the question.
A transseries is only allowed to contain exponentials $e^T$ when $T$ is a "purely large" transseries: one all of whose terms "go to infinity as $x$ does". That rules out $1/x$ as an allowable expon... | 1 | https://mathoverflow.net/users/49 | 228116 | 106,389 |
https://mathoverflow.net/questions/228068 | 5 | In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't find any proof or explaination. It is related to rate of convergence of the algorithm.
Borwein brothers suggested two se... | https://mathoverflow.net/users/78387 | Rate of Convergence of Borwein Algorithm for computing Pi | Even for different starting values around $1$, both $x\_n$ and $y\_n$ rapidly approach $1$.
Define $\alpha\_n = x\_n-1$ and $\beta\_n = y\_n-1$. These will be very small quantities. $\beta\_5 \lt 10^{-40}$ and $\alpha\_5$ is smaller. We want to estimate $\alpha\_n/\beta\_n$.
$$\sqrt{x\_n} = \sqrt{1+\alpha\_n} = 1+... | 4 | https://mathoverflow.net/users/2954 | 228118 | 106,391 |
https://mathoverflow.net/questions/227944 | 0 | Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set
$$ K\_R:= Ext\_S^{s-d}(R,S), $$
where $d=\dim R$, $s=\dim S$.
If $K\_R$ is Cohen-Macaulay (i.e. $R$ is a canonical Cohen-Macaulay ring) and $R$ is unmixed (i.e. $dim \widehat{R}/p=d$ for all $p \in Ass \widehat{R}$), is there exi... | https://mathoverflow.net/users/42774 | Ring with Cohen-Macaulay canonical module | It seem to be we should consider the question when $R$ is Cohen-Macaulay and $R\_{\mathfrak{p}}$ is Gorenstein for all minimal primes $\mathfrak{p}$. In this case there exists canonical ideal $I \cong K\_R$ (see <http://arxiv.org/abs/1106.1301>, 2.7) and $\dim R/I = \dim R-1$. If $\dim R = 0$, there is nothing to do. I... | 1 | https://mathoverflow.net/users/17901 | 228120 | 106,392 |
https://mathoverflow.net/questions/228009 | 5 | **Motivation**
I encountered the following partial differential equation (PDE) in a [mathematical paper](http://www.sciencedirect.com/science/article/pii/S0362546X14000959)
$$\begin{array}{}
u\_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)
\\\qquad\quad-\Delta u\_{t}+\int\_{0}^{t}g(t-s)\Delta u(x,... | https://mathoverflow.net/users/81143 | Backgrounds of the p-Laplacian Operator | I'm writing about the scalar-valued equation, so the solution is $u \colon \Omega \to \mathbb{R}$ and $\Omega \subset \mathbb{R}^d$, $d \geq 1$. Maybe this is of some use.
---
Basic lecture notes on $p$-Laplace equation, mathematical aspects: <http://www.math.ntnu.no/~lqvist/p-laplace.pdf> . See also the bibliogr... | 8 | https://mathoverflow.net/users/1445 | 228122 | 106,393 |
https://mathoverflow.net/questions/121431 | 18 | I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they are defined). This topic seems pretty standard to me, but when I needed a basic reference on this (both to check my memori... | https://mathoverflow.net/users/9317 | Reference request for Plancherel measure | In my humble opinion the best reference is Dixmier $C^\*$-algebras. The first half of the book has a very complete explanation of what you need to know about $C^\*$-algebras. In chapter 8 he goes over what is the decomposition of a trace for $C^\*$-algebras. Then from Chapter 13 on he goes into the theory for a locally... | 3 | https://mathoverflow.net/users/40832 | 228123 | 106,394 |
https://mathoverflow.net/questions/228126 | -1 | Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties?
1. $a\in {\cal F} \implies |a|\geq 2$,
2. $a\neq b\in {\cal F} \implies |a\cap b| \leq 1$, and
3. there is no function $f: {\cal F} \to X$ such that
* $f(a... | https://mathoverflow.net/users/8628 | Selection problem in a collection of non-empty sets | Satisfying the new Condition 1, If $|X|\geqslant4$, let ${\cal F}$ consist of all subsets of $X$ with two elements.
Condition 1 can be weakened. For example, if $X=\{a\_{i,j}:1\leqslant i,j\leqslant 3\}$, then the following 12 subsets, each of size 3, satisfy the conditions.
$$
\{a\_{1,1},a\_{1,2},a\_{1,3}\},\{a\_{2,... | 3 | https://mathoverflow.net/users/34341 | 228148 | 106,403 |
https://mathoverflow.net/questions/228149 | 3 | The answer to whether this is possible for general fields is no. However, the [counterexamples](https://mathoverflow.net/questions/49913/factoring-a-field-extension-into-one-which-adds-no-roots-of-unity-followed-by-o) used two ingredients:
1) $\Bbb Q\_p$, whose extensions $K$ containing $\Bbb Q\_p(\sqrt[p^e]{u})$ mig... | https://mathoverflow.net/users/57771 | When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity? | No. Let $k = \mathbb{Q}(\sqrt{-17})$. The class group of $k$ is $\mathbb{Z}/4 \mathbb{Z}$, generated by $\langle 3, 1+\sqrt{-17} \rangle$. (I checked [this table](http://www.numbertheory.org/classnos/) for a field of class number $4$ and then checked that $\langle 3, 1+\sqrt{-17} \rangle$ and $\langle 3, 1+\sqrt{-17} \... | 7 | https://mathoverflow.net/users/297 | 228154 | 106,404 |
https://mathoverflow.net/questions/228101 | 3 | Is the following true?
**CONJECTURE:** $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ cannot be isometrically embedded in $\ C,\ $ for no $\ t>1.\ $ Then the center $c(C)$ of $C$ must belong to $B$, $\ c(C)... | https://mathoverflow.net/users/8385 | The center of a minimal convex superbody | Let $C$ be a $3\times 1$ rectangle centered at the origin, and $B$ a $1\times 1$ square "filling" the left side of $C$. Then $tB$ cannot be embedded in $C$ if $t>1$ but $B$ does not contain the origin, disproving the conjecture.
| 4 | https://mathoverflow.net/users/28128 | 228155 | 106,405 |
https://mathoverflow.net/questions/228152 | 3 | All of the famous examples for sequentially indistinguishable topologies on a set $X$ are provided on an uncountable set $X$ (an uncountable set $X$ with discrete and cocountable topology or the $l^1(\mathbb{N})$-space with norm and weak topology, see [this](https://mathoverflow.net/questions/36379/is-a-topology-determ... | https://mathoverflow.net/users/58682 | Sequentially indistinguishable topologies on a countable set | There are plenty of topologies on a countable set for which all convergent sequences are eventually constant.
The most constructive example I know is the Arens-Fort space, given as example 26 in Steen and Seebach's *Counterexamples in Topology*.
This topology is constructed on the set $\mathbb{N}^2 \cup \{\infty\}$... | 4 | https://mathoverflow.net/users/70618 | 228157 | 106,406 |
https://mathoverflow.net/questions/228136 | 7 | Unitary elements of a Banach space have been defined in [this](http://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwiUioeXvKHKAhUDC44KHQRiCeYQFggbMAA&url=http%3A%2F%2Fwww.siue.edu%2FMATH%2Fkj_papers%2FBan-Jar-Rao-unitaries.pdf&usg=AFQjCNGvRHJ9BIvY5DYrHgECjiPgby8ScA&sig2=baWxT_ImU3r6OUuoBVfaxw&bvm... | https://mathoverflow.net/users/33449 | Spectrum of unitary elements of a Banach algebra | The answer to your immediate question is no: there is no other possible solution for $\phi$. To see this, consider that
$$\phi(-1) + i\phi(1) = 2$$
implies
$${\rm Re}(\phi(-1)) + {\rm Re}(i\phi(1)) = 2.$$
If $\|\phi\|\_\infty = 1$ then both terms on the left are at most 1, and equal
1 if and only if $\phi(-1) = 1$ and ... | 8 | https://mathoverflow.net/users/23141 | 228163 | 106,408 |
https://mathoverflow.net/questions/228034 | 1 | Let $X$ be a solution to the boundary value problem
$$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X\_0, ~~X(t) = X\_1,$$
where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$, and $X\_0$, $X\_1$ are certain fixed matrices (suppose the problem has a unique solution for all $0<t \leq 1$.)
Does... | https://mathoverflow.net/users/16702 | ODE estimate for boundary value problem | If you mean $|X(s)|\leq C$ for $0\leq s\leq t$ and any $0<t\leq 1$,
then the answer is yes; if you want $|X(s)|\leq C$ for $0\leq s\leq 1$
and any $0<t\leq 1$, then the answer is no in general.
Indeed, let $Y\_j=Y\_j(s)$ for $j=0,1$ be the solutions of the differential equation with $Y\_0(0)=Y\_1'(0)=I$ and $Y\_0'(0)... | 1 | https://mathoverflow.net/users/69194 | 228173 | 106,409 |
https://mathoverflow.net/questions/228169 | 4 | I have the following exponential sum:
$\sum \_{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$
say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$.
My question is basically, how do I bound this?
I know of two methods used to bound exponential sums, the Weil way and the Van der Corput way. I think the ... | https://mathoverflow.net/users/110603 | An exponential sum over squares | Actually, $|\sum\_{M < n \leq N} e(x/n^2)| \sim c \sqrt{x}$, where $c \approx 0.016151690 + 0.0738060263i$.
To see this, write $$\sum\_{M < n \leq N} e(x/n^2) = \sum\_{M < n \leq \epsilon \sqrt{x}} e(x/n^2)+\sum\_{\epsilon \sqrt{x} < n \leq N} e(x/n^2),$$ where $\epsilon = \frac{1}{x^{1/6}}$ (say, actually $\epsilon ... | 9 | https://mathoverflow.net/users/50426 | 228182 | 106,413 |
https://mathoverflow.net/questions/220382 | 10 | I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too.
Let $k$ be a field and $R$ a $k$-algebra. The *stable module category* StMod(R) is the category of R-modules where we've modded out by all morphisms that factor through a ... | https://mathoverflow.net/users/11540 | Operads and the Stable Module Category | The answer to this question is now yes. I asked this question while working on the following paper with Donald Yau, about preservation of operad-algebra structure under right Bousfield localization:
<http://arxiv.org/abs/1512.07570>
After getting no answers here (as well as Dan Christensen sharing that he'd never hea... | 3 | https://mathoverflow.net/users/11540 | 228183 | 106,414 |
https://mathoverflow.net/questions/228179 | 7 | For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then with the new operation
$$
x\circ y := x^{1/2} y x^{1/2}
$$
we get a power-associative [loop](https://en.wikipedia.org/wi... | https://mathoverflow.net/users/37580 | For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group | As I mentioned in comments, I believe that H. Bender has used this operation in special situations(in particular for certain groups of odd order of nilpotence class 2) , and that his use of it appears in the book "Topics in Finite Groups" by T. Gagen ( published by LMS as I recall). I don't remember all details but her... | 8 | https://mathoverflow.net/users/14450 | 228196 | 106,419 |
https://mathoverflow.net/questions/228210 | 2 | I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\mathbb{R}$ known up to homeomorphism?
References would be greatly appreciated.
| https://mathoverflow.net/users/73530 | Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism? | Infinite, complete, separable linear order with at most countably many jumps and not both a greatest and least element.
See
<http://www.math.uni-hamburg.de/home/geschke/papers/SeparableLinearOrders2.pdf>
| 4 | https://mathoverflow.net/users/4600 | 228212 | 106,421 |
https://mathoverflow.net/questions/227937 | 14 | *In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does this construction already have a name?*
*(To me it feels in the same spirit as termspace forcing (<https://mihahabic.... | https://mathoverflow.net/users/8133 | Reverse-engineer forcing: am I reinventing the wheel? | Turns out I was indeed reinventing the wheel! The construction is due to Solovay (page 21 of <http://www.math.wisc.edu/~miller/old/m873-03/solovay.pdf>) and is generalized by Kanovei (see <http://arxiv.org/pdf/1403.5757.pdf>).
I can't find any work on niceness properties such as c.c.c.-ness being preserved (or not) b... | 4 | https://mathoverflow.net/users/8133 | 228213 | 106,422 |
https://mathoverflow.net/questions/228201 | 4 | It is shown by [Berest-Etingof-Ginzburg](http://arxiv.org/abs/math/0208138) that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H\_c(S\_n)$ of $A\_{n-1}$ type if and only if the deformation parameter $c$ takes the rational numbers of the form $c=m/n$.
Since the rational Cher... | https://mathoverflow.net/users/17644 | Finite-dimensional representations of DAHA | I believe that you should consult the book "Double affine Hecke algebras" by Cherednik, more specifically Section 3.7, more specifically Theorem 3.7.2.
| 2 | https://mathoverflow.net/users/1306 | 228214 | 106,423 |
https://mathoverflow.net/questions/228215 | 6 | This comes from a comment made by user bof in [this thread](https://mathoverflow.net/questions/228159/edge-chromatic-number-of-hypergraphs).
Let $X$ be a set, define ${\cal P}\_2(X) = \big\{\{a, b\}: a\neq b\in X\big\}$.
Consider the statement
>
>
> >
> > ${\sf (S)}$ If $X$ is an infinite set, then there is ... | https://mathoverflow.net/users/8628 | Does "$|{\cal P}_2(X)| = |X|$ for $X$ infinite" imply ${\sf (AC)}$? | Yes, the usual proof that $|X|^2=|X|$ for all $X$ implies AC works for (S) as well. In detail, let $A$ be any infinite set, let $H$ be its Hartogs number (the least ordinal that does not inject into $A$), and let $X=A\sqcup H$. A bijection $\mathcal{P}\_2(X)\to X$ in particular restricts to an injection $i:A\times H\to... | 13 | https://mathoverflow.net/users/75 | 228217 | 106,424 |
https://mathoverflow.net/questions/228193 | 1 | Let $X \subset \mathbb{P}^3$ be a non-singular surface defined over $\mathbb{Q}$ of degree $d \geq 3$. It is a theorem of Colliot-Thelene (see the appendix to this paper: <http://www.jstor.org/stable/3062125?seq=1#page_scan_tab_contents>) that there are finitely many curves of degree $\delta \leq d-2$ contained in $X$.... | https://mathoverflow.net/users/10898 | A certain invariant of non-singular algebraic surfaces | I will explain what happens in the case of cubic surfaces. As is well known, any smooth cubic surface over an algebraically closed field contains $27$ lines. For the generic cubic surface over $\mathbb{Q}$, its configuration space of lines has Galois group $W(E\_6)$ (Weyl group of the $E\_6$ root system). Hence an appl... | 4 | https://mathoverflow.net/users/5101 | 228219 | 106,425 |
https://mathoverflow.net/questions/228218 | 4 | I stumbled upon the following elementary problem while trying to come up with a certain counterexample in category theory. (Basically, I am interested in the constant sheaf of $\mathbb F\_2$-vector spaces on the topological space $\mathbb Q$ of rational numbers; but understanding this motivation in not at all necessary... | https://mathoverflow.net/users/2106 | Compact, not local uniform convergence of sequences of functions on the rationals | No, such sequence of functions does not exist, even without the local constantness hypothesis.
For every $k$, choose some $n>k$ and $q\_k\in U\_{1/k}(q)$ such that $f\_n(q\_k)=1$. Then the set $K=\{q\_k\colon k\in\mathbb N\}\cup\{q\}$ is compact (since $q\_k\to q$), but $(f\_n)$ is not eventually vanishing on $K$.
| 4 | https://mathoverflow.net/users/17581 | 228223 | 106,427 |
https://mathoverflow.net/questions/228229 | 9 | Is there a theorem which classifies irreducible representations of semi-direct product of **finite** groups $G \rtimes A$, where $A$ is a finite abelian group and hence write down the character table for $G \rtimes A$?
In particular, I want to write down the character table for $M\_{12} \rtimes \mathbb{Z}\_2$ from the ... | https://mathoverflow.net/users/nan | Irreducible reps and characters of $G \rtimes A$ | I might as well turn my comment into an answer. I will just write $GA$ for the semidirect product ( with the normal subgroup being $G$). Clifford's theorem outlines a procedure for computing the character table, but it is usually not straightforward in practice.
Firstly, we need to compute a set of orbit representati... | 9 | https://mathoverflow.net/users/14450 | 228233 | 106,430 |
https://mathoverflow.net/questions/228226 | 0 | I have a question regrading the deck transformations. As we know, for the 2-torus $\mathbb T^2$, if we have a geodesic $\widetilde\gamma$ on the corresponding covering space $\mathbb R^2$, and $\alpha$ is a deck transformation not equal to the identity, then $\widetilde\gamma$ and $\alpha(\widetilde\gamma)$ are paralle... | https://mathoverflow.net/users/85168 | Geodesics under deck transformations on hyperbolic surface | The answer to your question depends on the behaviour of $\gamma$ in $M$. The following claims are easy to check.
1. There exists a deck transformation $\alpha$ such that $\alpha\tilde\gamma$ is a reparametrisation of $\tilde\gamma$ if and only if $\gamma$ is closed.
2. There exists $\alpha$ such that $\tilde\gamma$ i... | 3 | https://mathoverflow.net/users/70808 | 228240 | 106,431 |
https://mathoverflow.net/questions/226563 | 4 | Let $G$ denote a semisimple linear algebraic group over $\mathbb Q$ and let $r$ be its absolute rank.
For any prime $p$ let $v\_p$ be a vertex of the Bruhat-Tits building of $G({\mathbb Q}\_p)$ and let $D(p)$ denote the number of vertices neighboring $v\_p$. Can we say that there exists a constant $C>0$ such that
$$
D(... | https://mathoverflow.net/users/nan | Global bound for number of vertices in Bruhat-Tits building | This is not a complete answer, but only a reduction to computations with finite groups of Lie type.
To make things simpler assume that $G$ is simple so that its building is a genuine simplicial complex. Let $X$ be the Bruhat-Tits building of $G({\mathbb Q}\_p )$ and fix a vertex $v$ of $X$. Using Bruhat-Tits theory,... | 2 | https://mathoverflow.net/users/4767 | 228241 | 106,432 |
https://mathoverflow.net/questions/228235 | 3 | Suppose you have two rooted trees $T\_1$ and $T\_2$ with roots $r\_1$ and $r\_2$, respectively. Furthermore, for every $k\ge 0$, the number of walks of $T\_1$ starting at $r\_1$ of length $k$ is equal to the number of walks of $T\_2$ starting at $r\_2$ of length $k$. Is it true that $T\_1$ and $T\_2$ are isomorphic as ... | https://mathoverflow.net/users/41283 | A criterion for rooted trees to be isomorphic based on walks | Take a 9-cycle with vertices $\{0,\ldots,8\}$ and join new vertices of degree one to 0, 3, 6. Delete the vertex 8 from the cycle, producing a tree $T$ on 11 vertices. Then $T\backslash 2\cong T\backslash 5$, but no automorphism of $T$ maps 2 to 5.
Now there is a theorem that if $u$ and $v$ are vertices in graph $X$ ... | 8 | https://mathoverflow.net/users/1266 | 228245 | 106,435 |
https://mathoverflow.net/questions/228238 | 1 | Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by $\{h\_{\alpha}(t), \alpha \in \Phi, t \in K\}$. Here $\Phi$ is the set of roots. Then in page $44$ after lemma $28$ the author... | https://mathoverflow.net/users/69289 | Maximal split torus of universal chevalley group | By Lemma 28(b), $H$ is an abelian group generated by the $h\_i(t)$'s (where $h\_i = h\_{\alpha\_i}$), and since each $h\_i$ is multiplicative (by Lemma 28(a)), the existence follows.
To prove uniqueness, it suffices to show that if $\prod\_i h\_i(t\_i)=1$, then each $t\_i=1$. By Lemma 28(c) now, we have $\prod\_i t\_... | 3 | https://mathoverflow.net/users/12858 | 228247 | 106,436 |
https://mathoverflow.net/questions/228186 | 22 | Let $K$ be a finite extension of $\mathbb{Q\_p}$. The group $\ker H^1(G\_K, \mathbb{Q}\_p) \rightarrow H^1(G\_K, B\_{crys})$ is one-dimensional, which tells us that among all extensions of Galois modules
$$
0 \rightarrow \mathbb{Q}\_p \rightarrow E \rightarrow \mathbb{Q}\_p \rightarrow 0,
$$
which are classified ... | https://mathoverflow.net/users/84144 | Describing the crystalline extension of $\mathbb{Q}_p$ by $\mathbb{Q}_p$ | The representation $E$ in this case is not only crystalline, it is in fact unramified. This means we don't need much of the complicated machinery of $p$-adic Hodge theory to get a handle on the periods of $E$.
Whereas for general crystalline representations we need to use $\mathbf{B}\_\text{cris}$ to find periods, fo... | 13 | https://mathoverflow.net/users/362 | 228252 | 106,438 |
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