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https://mathoverflow.net/questions/236879 | 6 | Suppose you have a pair of orthogonal factorization systems, $(E\_0, M\_0), (E\_1, M\_1)$ in a category $\cal C$ such that $M\_0\subseteq M\_1$; this [entails](https://ncatlab.org/nlab/show/ternary+factorization+system) that there is a ternary factorization
$$
X\xrightarrow{e\_1} A\xrightarrow{e\_0 m\_1} B\xrightarrow{... | https://mathoverflow.net/users/7952 | Factorization system "tilted" by $(L,R)$ | Yes.
First note that Galois connections are far more common than field theorists would have you believe: *any* binary relation gives rise to one and the two fixed sets a closed under any appropriate algebraic structure.
Write
* $P\perp Q$ for "every member of $P$ is orthogonal (in the sense of factorisation syste... | 8 | https://mathoverflow.net/users/2733 | 236918 | 109,574 |
https://mathoverflow.net/questions/236904 | 3 | This is a basic question (not research level) which has already been asked on SE by someone else but doesn't yet have an answer so I'd like to repost it on MO.
Let $R$ be a commutative ring with unity and $H$ be a normal subgroup of a finite group $G$. If $P$ is a finitely generated projective left $R[G]$-module the... | https://mathoverflow.net/users/90571 | An invariant submodule of a projective module | If $H$ is finite, then $R[G]^H$ is isomorphic to $R[G/H]$ as an $R[G/H]$-module (because both consist of the elements that are constant on cosets of $H$). However, if $H$ is infinite, then $R[G]^H = \{0\}$ (because elements of $R[G]$ have finite support, and therefore cannot be a nonzero constant on any coset of $H$), ... | 5 | https://mathoverflow.net/users/68305 | 236925 | 109,575 |
https://mathoverflow.net/questions/227318 | 4 | Let $\mathscr{M}\to\mathscr{N}$ be a map of (Deligne-Mumford) stacks. Recall that it is said to be representable by affine schemes if for all affine maps $\operatorname{Spec}R\to \mathscr{N}$, the pullback $\mathscr{M}\times\_\mathscr{N}\operatorname{Spec}R$ is equivalent to an affine scheme.
What is a criterion for ... | https://mathoverflow.net/users/nan | Representable map of Deligne-Mumford stacks | Of course, you mean representable *by affine schemes*, on usual schemes this is called an *affine* map. This is an important class of maps but quite restrictive, too. The question you ask is the main concern of the paper
>
> Powell, Geoffrey M. L. On affine morphisms of Hopf algebroids.
> Homology, Homotopy Appl. ... | 4 | https://mathoverflow.net/users/6348 | 236928 | 109,576 |
https://mathoverflow.net/questions/236706 | 22 | Let $\mathbb F\_q$ be a finite field, $C$ a curve over $\mathbb F\_q$ of genus $g\geq 2$, $\rho: \pi\_1(C) \to GL\_2(\overline{\mathbb Q}\_\ell)$ an irreducible local system. The geometric Langlands correspondence constructs a geometrically irreducible Hecke eigensheaf on $Bun\_2(C)$ associated to $\rho$, which is a pe... | https://mathoverflow.net/users/18060 | What do Hecke eigensheaves actually look like? | The characteristic cycle of Hecke eigensheaves (for irreducible local systems) is the zero fiber of the Hitchin fibration, counting with multiplicity. This is almost obvious for eigen-D-modules constructed from opers in characteristic zero (by Beilinson and Drinfeld), but for $GL\_2$ it should not be hard to check this... | 9 | https://mathoverflow.net/users/2653 | 236962 | 109,582 |
https://mathoverflow.net/questions/236913 | 10 | I'd be interested in doing some computations in quantum groups $ U\_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl\_5$), but are somewhat lengthy to do by hand. Is there software available that can do this? (I.e. take an element of the quantum group and reduce it to a PBW bas... | https://mathoverflow.net/users/2669 | Computing in quantum groups | There is the package QuaGroup by de Graaf for both GAP and Magma: see [QuaGroup](http://www.science.unitn.it/~degraaf/quagroup.html). I've used it in both systems and found it to be extremely helpful. Since there's a GAP package, you also have the option of using it inside Sage.
You will potentially want to be carefu... | 9 | https://mathoverflow.net/users/13215 | 236972 | 109,585 |
https://mathoverflow.net/questions/236978 | 3 | Let $(B,\pi)$ be an open book decomposition of a closed, connected, oriented 3-manifold $M$ with odd (even) binding number and with pages of Euler characteristic $\chi$. Is it possible to define another open book decomposition $(B',\pi')$ of $M$ with binding number one (two) and with pages of the same Euler characteris... | https://mathoverflow.net/users/88357 | Decreasing the binding number of an open book while increasing the genus of the pages | What you hope for is not in general possible.
For example, the overtwisted contact structure $\xi\_{-\frac{1}{2}}$ on $S^3$ with $d\_3 = -\frac{1}{2}$ (*ie.* the overtwisted contact structure in the same homotopy class as $\xi\_{\rm{std}}$) has a supporting open book which is a thrice-punctured sphere, by [[Etnyre–Oz... | 3 | https://mathoverflow.net/users/51178 | 236987 | 109,589 |
https://mathoverflow.net/questions/236654 | 5 | Suppose $X$ and $Y$ are two real-valued random variables with a specified joint probability distribution $P\_{X,Y}.$ I wish to determine if there is a $\sigma$-finite measure $\mu$ on the real line such that $P\_{Y|X=x} << \mu$ for $P\_X$-almost all $x\in\mathbb{R}$. Call this property $Q.$ Property $Q$ does not always... | https://mathoverflow.net/users/7576 | Conditions for existence of dominating $\sigma$-finite measure for all conditional distributions | In one direction it's clear: If $P\_{X,Y}\ll P\_X\otimes P\_Y$, then there is a jointly measurable density $f$, and (a version of) the conditional distribution $P\_{Y|X=x}$ is given by $f(x,y)P\_Y(dy)$, so the choice $\mu=P\_Y$ suffices.
Conversely, suppose that $Q$ holds. Then there is a jointly measurable function ... | 4 | https://mathoverflow.net/users/42851 | 236991 | 109,591 |
https://mathoverflow.net/questions/236568 | 9 | Let $(W,S)$ be a Coxeter group, $T=\bigcup\_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of [Dyer](http://www.sciencedirect.com/science/article/pii/002186939090149I) and [Deodhar](http://link.springer.com/article/10.1007%2FBF01199813), we know that the subgroup $W\_A$ generated by the elem... | https://mathoverflow.net/users/88858 | Bruhat order of reflection subgroups | The Bruhat graph of a reflection subgroup is an induced subgraph of the Bruhat graph of the larger group. This is proved by Dyer in "On the Bruhat graph of a Coxeter system." Thus if two elements that happen to be in the same reflection subgroup have a covering relation, then (the same) one also covers the other when y... | 5 | https://mathoverflow.net/users/62135 | 236995 | 109,593 |
https://mathoverflow.net/questions/236983 | 5 | Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda\_i$ and eigen-functions $e^i(x)$.
Let us define a sequence of "partial sum" operators $K\_N(x,y) = \sum\_{i=1}^N \lambda\_i e^i(x)e^i... | https://mathoverflow.net/users/89451 | Error estimate in the spectral theorem of compact operators on a Hilbert space | I'm following [Szego's book on orthogonal polynomials](https://books.google.co.il/books?hl=en&lr=&id=RemVAwAAQBAJ&oi=fnd&pg=PR9&dq=szego%20orthogonal%20polynomials&ots=l4bK485fTQ&sig=S6hN5M_fZqiHhr2ny7sOcV0oSpU&redir_esc=y#v=onepage&q=szego%20orthogonal%20polynomials&f=false).
In chapter III he considers $f \in L^2\l... | 2 | https://mathoverflow.net/users/42864 | 236999 | 109,596 |
https://mathoverflow.net/questions/236982 | 6 | For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S\_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least signifi... | https://mathoverflow.net/users/90645 | representations of $S_k \times S_j$ | I guess that $S\_n$ denotes the symmetric group. It is well known that all irreps of $S\_n$ over $\mathbb C$ are defined over $\mathbb Q$. Therefore $S(S\_k\times S\_{n-k},\mathbb{C})=S(S\_k\times S\_{n-k},\mathbb{Q})$. Also the fact that the sum of the squares of the dimensions is the group order and Cauchy-Schwarz im... | 8 | https://mathoverflow.net/users/89948 | 237011 | 109,600 |
https://mathoverflow.net/questions/236993 | 5 | A generic $k \times k$ block symmetric matrix $\Sigma$ is denoted as
\begin{align}
\Sigma = \begin{bmatrix}\Sigma\_{11} & \Sigma\_{12} & \ldots & \Sigma\_{1k} \\ \Sigma\_{21} & \Sigma\_{22} & \ldots & \Sigma\_{2k} \\ \ldots & \ldots & \ldots & \ldots \\ \Sigma\_{k1} & \Sigma\_{k2} & \ldots & \Sigma\_{kk}\end{bmatrix}.
... | https://mathoverflow.net/users/90066 | Recursively calculate the determinant | Perhaps you should write this more geometrically. Denote by $(-,-)\_i$ the canonical Euclidean inner product on $\newcommand{\bR}{\mathbb{R}}$ $\bR^{p\_i}$. Then we can identify $\Sigma\_{ii}$ with a symmetric positive operator. $\Sigma\_{ij}: \bR^{p\_j}\to\bR^{p\_i}$ is the operator that has the invariant description
... | 1 | https://mathoverflow.net/users/20302 | 237022 | 109,603 |
https://mathoverflow.net/questions/235070 | 1 | first some notation: $\langle x\rangle=\sqrt{1+x^2}$, $P\_{j}$ is the Littlewood Paley Projector and $P\_{\leq0}$ corresponds to the small frequencies.
I have a the following definition of the Besov norm:
$$||u||\_{B\_{p,q}^s}= ||P\_{\leq0}(f)||+(\sum^\infty\_{j=1}(2^{js}||P\_j(f)||\_{p})^{q})^{1/q}$$
and
$$||u||\... | https://mathoverflow.net/users/88808 | Different Besov-Norm Definitions | In this paper <http://arxiv.org/abs/1007.3418> one can find definitions for both inhomogenous and homogenous besov norms. The second definitions are the ones for the inhomogenous besov norms if you take away the ⟨⟩ brackets.
Furthermore it is obvious (since all norms on $\mathbb{R}^n$ are equivalent) that:
$$||P\_{\... | 1 | https://mathoverflow.net/users/88808 | 237024 | 109,604 |
https://mathoverflow.net/questions/237020 | 5 | Let $(z\_n)$ be a sequence of complex numbers satisfying $|z\_n|\to +\infty$ and such that $\{e^{z\_n}\mid n \in \mathbb{N}\}$ is infinite.
Is it always true that $\{(z\_n,e^{z\_n})\mid n \in\mathbb{N}\}$ is Zariski-dense in $\mathbb{C}^2$? In other words, if $p(x,y) \in \mathbb{C}[x,y]$ is a polynomial such that $p(... | https://mathoverflow.net/users/nan | A transcendence question involving the exponential function | It is false. Equation $\sin x=1/x$ has infinitely many real solutions $(x\_n)$, take $z\_n=ix\_n$, we get $z\_n(e^{2z\_n}-1)=-2e^{z\_n}$.
| 7 | https://mathoverflow.net/users/4312 | 237032 | 109,606 |
https://mathoverflow.net/questions/236763 | 6 | Let $\Gamma\_{(\lambda\_1, \dots, \lambda\_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda\_1, \dots, \lambda\_n)$ and let more specifically $X = \Gamma\_{(2\lambda, \dots, 0)}$ and $Y = \Gamma\_{(2\lambda\_1, \dots, 2\lambda\_n)}$, where at least one of the $\lambda\_j$ with $j>1$ is not equ... | https://mathoverflow.net/users/50047 | $U(n)$-submodules of ${\rm SO}(2n)$-modules | you can find branching rule ${\rm SO}(2n) \to U(n)$ in the book of Knapp, Lie groups beyond an introduction, and also in some paper of his, and in the book of Zelobenko, compact Lie groups and their rep's, also try the book of Tom Dieck.
best
jorge
| 3 | https://mathoverflow.net/users/67162 | 237037 | 109,608 |
https://mathoverflow.net/questions/236603 | 2 | I'm trying to better understand the consequences of representing a random set as a
1. Random element in the space of locally finite closed sets under the Borel sigma algebra generated by the Fell topology
2. Random element in the closed subset of locally finite integer-valued measures under the Borel sigma algebra g... | https://mathoverflow.net/users/90516 | Fell topology versus vague topology for representing random sets | Your intuition is correct at least in the case of locally compact second countable $S$. The map $\mu \to supp \, \mu$ is continuous from locally finite integer valued Borel measures on $S$ in the vague topology to closed subsets of $S$ in the Fell topology. If $\mu\_n \to \mu$ vaguely and $U$ is open in $S$ with $supp ... | 2 | https://mathoverflow.net/users/90719 | 237052 | 109,612 |
https://mathoverflow.net/questions/236979 | 3 | If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert\_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a discrete spectrum such that the eigenfunction of the $i^{th}$ largest eigenvalue is proportional to $e^{-\frac{(x-\mu)... | https://mathoverflow.net/users/89451 | About eigen-functions of the Gaussian kernel | for example, see [Positive Definite Kernels](http://www.math.iit.edu/~fass/PDKernels.pdf) by Gregory Fasshauer:
| 3 | https://mathoverflow.net/users/11260 | 237061 | 109,617 |
https://mathoverflow.net/questions/237064 | -1 | Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}\_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the center $\mathbf{Z}(\mbox{End}\_\mathbb{F}(\mathbf{V}))=\mathbb{F}$ implies that every maximal commutative subring $R\subse... | https://mathoverflow.net/users/89313 | Maximal commutative subrings of the endomorphism ring of a vector space | Even for a $2$-dimensional vector space (so we're looking at subrings of the ring $M\_2(\mathbb{F})$ of $2\times2$ matrices over $\mathbb{F}$) there are nonisomorphic maximal commutative subrings.
Both
$$\left\{\begin{pmatrix}x&0\\0&y\end{pmatrix}: x,y\in\mathbb{F}\right\}$$
and
$$\left\{\begin{pmatrix}x&y\\0&x\end{... | 4 | https://mathoverflow.net/users/22989 | 237066 | 109,619 |
https://mathoverflow.net/questions/237083 | 6 | Let's consider the Littlewood-Richardson coefficients $c^{\lambda}\_{\mu \nu}$ so that
\begin{equation}
V\_\mu \otimes V\_\nu = \bigoplus\_\lambda V\_\lambda^{\oplus c^{\lambda}\_{\mu \nu}}
\end{equation}
where $V\_\mu$ are representations of $GL\_n$.
Usually the basis elements of the (infinite dimensional) vector sp... | https://mathoverflow.net/users/13731 | "Diagonalizing" Littlewood-Richardson coefficients | If a diagonal basis existed, tensoring with a fixed representation would kill all but finitely many basis elements. This is not the case because e.g. tensoring with the $1$-dimensional trivial representation doesn't kill anything.
| 7 | https://mathoverflow.net/users/5263 | 237089 | 109,624 |
https://mathoverflow.net/questions/237028 | 1 | In the book [Algebras of Functions on Quantum Groups: Part I](http://bookstore.ams.org/surv-56), Remark 3.1.4, we have the following result.
Let $A$ be a Poisson Hopf algebra. That is, $A$ is both a Hopf algebra and a Poisson algebra and $\Delta: A \to A \otimes A$ is a Poisson algebra homomorphism. Then the antipod... | https://mathoverflow.net/users/11877 | Is the antipode anti-bracketed? | Yes. Suppose $A$ is a bracked Hopf algebra. Equip $A [\hbar]/\hbar^2$ with the multiplication $a \cdot\_\hbar b = ab + \hbar \{a,b\}$. That this is associative follows from the Leibniz rule --- you do not need Jacobi. Note that with undeformed $\Delta$, this is Hopf. (Indeed, it is a bialgebra by inspection. Now suppos... | 2 | https://mathoverflow.net/users/78 | 237110 | 109,631 |
https://mathoverflow.net/questions/237068 | 4 | Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, \psi\in\mathcal L\_\in\forall a:\omega \to V(\langle\phi\wedge \psi, a\rangle\in S\leftrightarrow \langle\phi, a\rangle\i... | https://mathoverflow.net/users/17968 | Can one satisfaction class code another? | In case of Peano Arithmetic the answer is yes (emphatically yes), if I understand your question correctly. This follows from Theorem 3.3 in Smith's "Nonstandard Definability" APAL 42 (1989) pp. 21-43 which says that for any recursively saturated countable model $M$ of PA and any set $A \subseteq M$ the set $A$ may be p... | 4 | https://mathoverflow.net/users/57888 | 237125 | 109,636 |
https://mathoverflow.net/questions/237120 | 4 | Let $C$, $Q \in \mathbb{R}[x\_0,\dots,x\_n]$ be homogeneous of degrees $3$ and $2$ respectively. Consider the scheme $V$ in $\mathbb{P}^n$ defined by
$$ V \; : \; C=Q=0$$.
Suppose
* $V$ is integral (over $\mathbb{C}$);
* $Q$ is indefinite of full rank $n+1$.
**Question:** Show that $V$ has a smooth real point.
**... | https://mathoverflow.net/users/4140 | Smooth real points on the intersection of a quadric and a cubic | I think that this is wrong for $n = 3$:
Let $C \subset {\mathbb P}^3\_{\mathbb R}$ be a
generic cubic surface not containing $(0:0:0:1)$ and take
$$Q \colon x^2 + y^2 + z^2 = \varepsilon w^2$$
with $\varepsilon > 0$ sufficiently small (the coordinates are $x,y,z,w$).
Then $C({\mathbb R}) \cap Q({\mathbb R})$ is empt... | 6 | https://mathoverflow.net/users/21146 | 237130 | 109,637 |
https://mathoverflow.net/questions/236919 | 4 | Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is *semistrict* iff the canonical map $\operatorname{Coker}(\ker (f))\rightarrow \operatorname{Ker}(\operatorname{coker}(f))$ is both a pseudomonomorphism and a pseudoepimorphism (a *pseudo... | https://mathoverflow.net/users/16639 | Are product / coproduct projections / inclusions 'semistrict'? | Here is, I claim, a counterexample. Consider the category $\mathrm{Cat}\_\*$ of pointed categories, i.e. the coslice category $1/\mathrm{Cat}$. Let $A$ be the walking involution, i.e. it has one object $a$ with one nonidentity morphism $e:a\to a$ such that $e e = 1\_a$. Let $B$ be the discrete category on two objects $... | 2 | https://mathoverflow.net/users/49 | 237136 | 109,640 |
https://mathoverflow.net/questions/236791 | 6 | Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S\_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-1}=(1,2,\ldots,n)\overbrace{(1,2)\cdots(1,2,\ldots,n)}^{2n-2}(1,2)=(1).$$ We want to know whether or not there exists a... | https://mathoverflow.net/users/75264 | A question about (unicity of certain cycles in a Cayley graph of a) symmetric group | The smallest $n$ for which there exist sequences as asked for is $n = 7$:
* $(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2)
\cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$
$(1,7,6,5,4,3,2) \cdot (1,2) \cdot (1,2,3,4,5,6,7) \cdot (1,2)
\cdot (1,7,6,5,4,3,2) \cdot (1,2) = ()$, and
* $(1,2,3,4,5,6,7) \cdot (1... | 8 | https://mathoverflow.net/users/28104 | 237138 | 109,641 |
https://mathoverflow.net/questions/237114 | 5 | Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C\_i, mi\_i)$ be a holomorphic current in $X$, where $C\_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m\_i$ is positive integer. Then we can define the ECH index of $\mathcal{C}$ as follow:
$$I(\mathcal{C})= \langle c\_1(TX), \mathc... | https://mathoverflow.net/users/89961 | How to understand Taubes' moduli space of holomorphic curves? | This is not Taubes' moduli space. Taubes has many more constraints on the currents, so that his "moduli space" is really a finite set of points (for generic $J$), and he requires special weightings on the multiply-covered curves (which are necessarily unbranched covers of tori due to his constraints) to get a well-defi... | 4 | https://mathoverflow.net/users/12310 | 237150 | 109,645 |
https://mathoverflow.net/questions/237155 | 3 | In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/D$, the space of all maximal integral submanifolds of $M$ with respect to $D$.
Set-theoretically, given that through e... | https://mathoverflow.net/users/54780 | Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution | Assuming there is a smooth structure on $M/D$ such that $\pi$ is a submersion. Then a function $f\colon M/D\to\mathbb R$ is smooth if and only if $f\circ\pi\colon M\to\mathbb R$ is smooth. This means, a homeomorphism $\psi\colon U\to V\subset\mathbb R^n$ for $U\subset M/D$ open (in the quotient topology) is a smooth ch... | 9 | https://mathoverflow.net/users/70808 | 237157 | 109,646 |
https://mathoverflow.net/questions/237145 | 3 | Let $X$ be an algebraic variety over $\mathbb{F}\_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}\_{q^k}$ (Weil hypothesis). Is there an analogous statement for non-smooth varieties?
| https://mathoverflow.net/users/31356 | Number of rational points in a non-smooth variety | As per Daniel Loughran's comment, I found [this](https://terrytao.wordpress.com/2012/08/31/the-lang-weil-bound/) very nice exposition by Terence Tao which seems to answer the question in detail.
| 2 | https://mathoverflow.net/users/12218 | 237158 | 109,647 |
https://mathoverflow.net/questions/236812 | 1 | Does anyone know how to find explicit generators for the unit group of a number field on magma?
For example, in sage one could do
>
> K. = NumberField(x^3+x^2-2\*x-1)
>
>
> UnitGroup(K).gens()
>
>
>
and it gives you [-1, w^2 - 1, w + 1] which generate the unit group, but I can't figure out how to get this ... | https://mathoverflow.net/users/56362 | magma generators for unit group/ sage totally positive | In Magma, the unit group is returned as an abstract group together with a map from that group into the field. So you would do something like the following:
```
> Zx<x> := PolynomialRing(Integers());
> K<w> := NumberField(x^3 + x^2 - 2*x - 1);
> G,phi := UnitGroup(K);
> G;
Abelian Group isomorphic to Z/2 + Z + Z
Defi... | 2 | https://mathoverflow.net/users/90772 | 237176 | 109,655 |
https://mathoverflow.net/questions/235917 | 39 | I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff related to geometric representation theory ($D$-Modules, geometric quantization, Langlands & CFT).
It's fair to say I'... | https://mathoverflow.net/users/22810 | Roadmap to Geometric Representation Theory (leading to Langlands)? | Since this has many answers, I'll just put down what comes to mind.
It seems like you would be content to not worry about real semisimple or other fields and just grant yourself use of $\mathbb{C}$ and of $\mathbb{C}((t))$. That would be the safest option to specialize everything to this.
So you've already read som... | 14 | https://mathoverflow.net/users/69850 | 237180 | 109,658 |
https://mathoverflow.net/questions/164992 | 4 | Let $G=(V,E)$ be a finite simple graph, and let $\{X\_i\}\_{i \in V}$ be a collection of random variables associated with the vertices of $G$. The joint distributions of these r.v.s is a *Markov Random Field* if, for any three subsets $U,W,C \subset V$ such that $C$ separates $U$ from $W$ (i.e., any path from $U$ to $W... | https://mathoverflow.net/users/23661 | Local Markov implies global Markov | src: <http://web.engr.illinois.edu/~swoh/courses/IE598/handout/markov.pdf>
copied below
(3) being if $x\_A ⊥ x\_B|(x\_C, x\_D)$ and $x\_A ⊥ x\_C|x\_B ∪ x\_D,$ then $x\_A ⊥ (x\_B, x\_C)|x\_D$
which always holds for positive P.
Proof of (P)$⇒$(G) when (3) holds: [Pearl,Paz 1987]
by induction over s $\triangleq$ |... | 1 | https://mathoverflow.net/users/90778 | 237186 | 109,661 |
https://mathoverflow.net/questions/237017 | 5 | Consider $2n$ vertex balanced bipartite graph.
If total number of edges is $n^2$ then we have $n!$ perfect matchings.
Fix $c\in(0,\frac12)$ and consider collection of $2n$ vertex balanced bipartite graphs with at least $cn!$ perfect matchings. What fraction of graphs in this collection have at most $dn^2$ total num... | https://mathoverflow.net/users/10035 | On number of perfect matchings | Even if you have $0.999n^2$ edges, as $n \to \infty$ there can't be $0.0001n!$ matchings. For fixed $c, d \in (0,1)$ and large enough $n=n(c,d)$, among the bipartite graphs with $cn!$ perfect matchings, not only is there not a positive proportion of graphs with at most $dn^2$ edges, there are none at all.
If there a... | 6 | https://mathoverflow.net/users/2954 | 237189 | 109,662 |
https://mathoverflow.net/questions/237199 | 4 | For a given N-vertex similarity graph $ G=(V,A) $ the eigenvalues of the unrenormalized (graph) Laplacian may be denoted as
$$ 0= \mu\_0 \leq \mu\_1 \leq ... \leq \mu\_N $$
where the corresponding eigenvectors may be written as
$$ v\_0 , v\_1 , ... ,v\_N $$
In the case where $ 0= \mu\_0 = \mu\_1 < \mu\_2 $ I find c... | https://mathoverflow.net/users/90785 | exact definition of Fiedler vector | The concept of a Fiedler vector is defined for graphs that consist of one single connected component. Since the number of zero eigenvalues counts the number of connected components, the second largest eigenvalue $\nu\_1$ is then always nonzero. The multiplicity $m\_1$ of the second largest eigenvalue may be greater tha... | 7 | https://mathoverflow.net/users/11260 | 237203 | 109,668 |
https://mathoverflow.net/questions/237198 | 4 | There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, we know effective algorithms for construction irreducible polynomials, and for verifying that given polynomial is irred... | https://mathoverflow.net/users/31356 | Irreducible algebraic sets via irreducible polynomials | Bjorn Poonen and his coauthors have studied topics of this type in a series of papers on "Bertini Theorem over Finite Fields".
See for example the paper:
Poonen, Charles - Bertini irreducibility theorems over finite fields.
<http://www-math.mit.edu/~poonen/papers/bertini_irred.pdf>
Edit: I should clarify; this ... | 4 | https://mathoverflow.net/users/5101 | 237204 | 109,669 |
https://mathoverflow.net/questions/237209 | 4 | Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete lattice with respect to $\subseteq$.
Given $\tau\in\text{Top}(X)$ and $E\subseteq X$ we set $\tau\_E$ to be the topolo... | https://mathoverflow.net/users/8628 | "Discrete jumps" in the collection of all topologies on a set $X$ | The answer is negative.
For $X=\mathbb R$, let $\tau=\{(a,\infty)\subseteq\mathbb R\mid a\in[-\infty,\infty]\}$ and let $\tau'$ be the Euclidean topology. Choose any set $E\in\tau'\setminus\tau$. We can express $E$ as a countable disjoint union of open intervals $$E=\bigcup\_{n=1}^{\infty}(a\_n,b\_n),\qquad-\infty\le... | 2 | https://mathoverflow.net/users/16447 | 237227 | 109,677 |
https://mathoverflow.net/questions/237147 | 2 | Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\lambda$, i.e. $A(G) = \{\lambda\_{x,y} : x,y \in L^2(G)\}$ where, $\lambda\_{x,y}(g) = \langle \lambda(g)x,y\rangle$. From th... | https://mathoverflow.net/users/90755 | Does Fourier Algebra of locally compact group separate compact sets of the group? | For disjoint compact sets $A$ and $B$ in $G$ you can find an open set $U$ containing $A$ and an identity nbd $V$ such that both have compact closures and $UV^{−1}$ is disjoint from $B$. You can find $[0,1]$-valued continuous functions $\phi$ and $\psi$ supported on $U$ and $V$ correspondingly such that $\phi|\_A=1$ and... | 1 | https://mathoverflow.net/users/89334 | 237230 | 109,678 |
https://mathoverflow.net/questions/236417 | 6 | Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R\_{13} R\_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R\_{13} R\_{12}$$
It appears that these two equations are not equivalent, although I don't know a single example for an $R$ that s... | https://mathoverflow.net/users/13767 | When are the braid relations in a quasitriangular Hopf algebra equivalent? | The condition $R\_{21}\,R=I$ (to be triangular) implies the equivalence of the two equations.
For the second question, let $A$ be a finite abelian group, and $H=k^A$ the Hopf algebra of function on $A$. Functions $f:A\times A\to \mathbb{C}^\*$ (or equivalently maps from $A$ to $\operatorname{Maps}(A,\mathbb{C}^\*$)) ... | 3 | https://mathoverflow.net/users/6517 | 237244 | 109,683 |
https://mathoverflow.net/questions/237002 | 1 | Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y) = \{p\in P: x\leq p < y\}$, and $(x,y]$ is defined in an analogous manner. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. It is well-known that $\text{Top}(X)$ is a complete lattice with respect to $\subseteq$.
Does there exist an inf... | https://mathoverflow.net/users/8628 | Lower neighbors in the lattice of topologies | The answer to the question ``Does there exist $\dots$'' is Yes.
Let $X = [0,\infty)$ be the nonnegative part of the real line.
Let $\tau$ be the topology on $X$ consisting of all sets $[0,r)$
for $r\in X$. Let $\sigma$ be the indiscrete topology on $X$ and let
$\rho$ be the restriction of the usual topology of the r... | 2 | https://mathoverflow.net/users/75735 | 237249 | 109,685 |
https://mathoverflow.net/questions/220629 | 2 | I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since $sn(u,k)$ is a real-valued function of $u\in\mathbb{R}$. For example, it holds
$$|sn(u,k)|\leq1, \quad \forall u\in\mathbb{R},... | https://mathoverflow.net/users/56553 | Challenging problems concerning Jacobian elliptic functions with complex modulus | The conjecture has been verified. For the proof and other interesting details, see <http://arxiv.org/abs/1512.06089>.
| 1 | https://mathoverflow.net/users/56553 | 237266 | 109,692 |
https://mathoverflow.net/questions/230874 | 26 | Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to decipher. I heard that this intellectual treasure has been stored even though Grothendieck did not want anyone to publish any... | https://mathoverflow.net/users/13625 | Have Grothendieck's notes in Montpellier already been investigated? | Here is a book (in French) where it is explained in detail how all of Grothendieck's private manuscripts were transported to Paris in November 2015, and saved there in a secure library:
>
> Philippe Douroux, **Alexandre Grothendieck- Sur les traces du dernier
> génie des mathématiques**.
>
>
>
| 8 | https://mathoverflow.net/users/34304 | 237267 | 109,693 |
https://mathoverflow.net/questions/237268 | 3 | The equation, $s\in\mathbb{C}$ with $0<\Re(s)<1$:
$$\frac{\zeta(2-s)}{\zeta(1+s)}=\frac{\Gamma(1+\frac{s}{2})}{\Gamma(1+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}$$
A general question: For which values the equation holds ? (Trivial case: $s=\frac{1}{2}$.)
Because I have not the technical possibility for a numerical evaluat... | https://mathoverflow.net/users/90369 | A condition for the Riemann Zeta-function by modification of its functional equation | This Mathematica plot indicates $s=1/2$ is the only solution for $|{\rm Im}|\,s<2$.
Plotted versus ${\rm Re}\,s\in(0,1)$ and ${\rm Im}\,s\in(-2,2)$ is the absolute value $|f(s)|$ of the function
$$f(s)=\frac{\zeta(2-s)}{\zeta(1+s)}-\frac{\Gamma(1+\frac{s}{2})}{\G... | 3 | https://mathoverflow.net/users/11260 | 237281 | 109,695 |
https://mathoverflow.net/questions/237277 | 2 | Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does *not* imply $g\_n \leq f\_n$ (here $g\_n = [x^n] g(x)$ denotes the $n$-th Taylor coefficient of $g$). However, does the inequality in the coefficients hold if we restrict to the clas... | https://mathoverflow.net/users/66274 | linear recurrence inequality | The answer is no. E.g., let
$\alpha = -1,\beta = 0,f\_0= 0,f\_1= 0,f\_2= -2,f\_3= 1$. Then $f\_{n+1} \leq \alpha f\_n + \beta f\_{n-1}$ for $n=1,2$, whereas
$$
f\_3 = 1 \not\leq 0= \left(\alpha ^2+\beta \right)f\_1 +\alpha \beta f\_0=[x^3] \dfrac{f\_0 + xf\_1 - \alpha x f\_0}{1-\alpha x - \beta x^2}.
$$
| 2 | https://mathoverflow.net/users/36721 | 237282 | 109,696 |
https://mathoverflow.net/questions/237050 | 1 | Some definitions: Let $(M,d)$, $(M',d')$ be metric spaces. For $f:M\to M'$, $x\in M$ and $r>0$, define $$D\_r(f)(x):= \sup\{r^{-1}d'(f(x),f(y)): y\in M,\,d(x,y)\leq r\}.$$ Define the *pointwise Lipschitz constant of $f$ at $x\in M$* as $\text{Lip}(f)(x):=\limsup\_{r\to 0} D\_r(f)(x)$, and say $f$ is *pointwise Lipschit... | https://mathoverflow.net/users/12248 | The pointwise Lipschitz-ness of a function on a dense set, implies its pointwise Lipschitz-ness everywhere? | Converting comment to an answer:
Let $M = [0,1]$ with its usual metric (which is a length space), $X= \mathbb{R}$ and let $f : [0,1] \to \mathbb{R}$ be the [Cantor function](https://en.wikipedia.org/wiki/Cantor_function). Let $S$ be the complement of the Cantor set, which is dense in $[0,1]$. Then we have $\operatorn... | 2 | https://mathoverflow.net/users/4832 | 237284 | 109,697 |
https://mathoverflow.net/questions/237283 | 1 | The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta\_n(x)$ spiked at the the origin, in the sense that:
$\lim\_{n \rightarrow \infty} \int\_{\mathbb{R}} f(x)\delta\_n(x-a)dx = f(a)$
I am aware that one can define a $p$-a... | https://mathoverflow.net/users/35431 | $p$-adic Dirac measure as a weak limit | The answer is yes. Moreover:
1) For every (first countable) locally compact group the $\delta$ measure at the identity is a weak limit of (a sequence of) continuous functions. You can find this in any abstarct harmonic analysis text, eg Folland, "A course in abstract harmonic analysis" Prop 2.42. Look up: *approximat... | 6 | https://mathoverflow.net/users/89334 | 237286 | 109,698 |
https://mathoverflow.net/questions/237288 | 1 | This is a follow up on my previous [linear recurrence inequality question](https://mathoverflow.net/questions/237277/linear-recurrence-inequality).
I have some matrices which satisfy a linear recurrence formula of the form
$$
A\_{n+1} = \alpha A\_{n} + \beta A\_{n-1},\qquad n\geq 1,
$$
with $A\_0 \in \mathbb{R}^{d\time... | https://mathoverflow.net/users/66274 | linear recurrence inequality of positive terms | If $\alpha$ and $\beta$ are nonnegative, then the conclusion is indeed true; the condition $f\_n\ge0$ is not needed.
First here, note that
$$[x^n] \dfrac{f\_0 + xf\_1 - \alpha x f\_0}{1-\alpha x - \beta x^2}=g\_n,$$
where $(g\_n)$ solves the linear recurrence
$$
g\_{n+1} = \alpha g\_n + \beta g\_{n-1},\qquad n\... | 1 | https://mathoverflow.net/users/36721 | 237289 | 109,700 |
https://mathoverflow.net/questions/237279 | 34 | The paper "[A proof of Liouville's theorem](http://www.ams.org/journals/proc/1961-012-06/S0002-9939-1961-0259149-4/)" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in $\mathbb{R}^n$ is a constant.
I presume there ... | https://mathoverflow.net/users/56624 | Nelson's proof of Liouville's theorem | This doesn't answer any of the three specific questions asked, but addresses an implicit question: "Why did the editor accept it?"
In 1961, the Proceedings of the AMS established a section called "Mathematical Pearls" devoted to, I quote:
>
> The purpose of this department is to publish very short papers of an un... | 26 | https://mathoverflow.net/users/3948 | 237294 | 109,701 |
https://mathoverflow.net/questions/237240 | 5 | For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum :
$$G\_q(n):=\sum\_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$
for $n$ a non-negative integer and $q$ a positive integer ($d'$ denotes the inverse of $d \pmod{q/d}$).
Does anyone recognize this sum?
Many thanks!
| https://mathoverflow.net/users/66686 | Does anyone recognize this exponential sum? | Yes, these sums occur as the arithmetic part of the Fourier expansion of
period kernels $\sum\_{ad-bc=1}(a\tau+b)^{-k}(c\tau+d)^{-k}$, the analytic part
being J-Bessel functions. The derivation is not difficult. I can send you
the paper where I compute them if you want (from 1980), private e-mail
please.
| 10 | https://mathoverflow.net/users/81776 | 237307 | 109,709 |
https://mathoverflow.net/questions/237302 | 3 | I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover their images are constrained in ways which are unimportant for this question.
If the space is first countable (points hav... | https://mathoverflow.net/users/35297 | Local "pathologies" in spaces arising naturally in algebraic topology | When a certain kind of homotopy theorist says "space," they don't mean a topological space, or even an object which in any sense has an underlying topological space. The simplest translation of what "space" means in this sense is "weak homotopy type," and it's not a meaningful question to ask whether a weak homotopy ty... | 3 | https://mathoverflow.net/users/290 | 237312 | 109,710 |
https://mathoverflow.net/questions/237311 | 24 | I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in his thesis or papers. Basically, the main difference is in the dimension of some singular sets (his works are mainly on... | https://mathoverflow.net/users/31676 | Adapting arguments and plagiarism | What you describe seems to me to be a normal mode of mathematical
progress, and I would urge you simply to carry on! Ride that train
as far as you can.
It often happens that someone's mathematical results can be
improved or generalized in various ways, and when this is
possible, it is mathematically desirable that th... | 97 | https://mathoverflow.net/users/1946 | 237315 | 109,711 |
https://mathoverflow.net/questions/237321 | 1 | I came to the following question, but I don't have quite a good idea how to approach.
Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position?
I believe that, since this is quite a simple question, this would already have an answer, but I could not find it.
| https://mathoverflow.net/users/68663 | Set of General Linear Position with Nonzero Measure | No, at least if $A$ is assumed to be measurable. Let $\pi \colon \mathbb{R}^n \to \mathbb{R}$ be the projection to the first coordinate. For every $t \in \mathbb{R}$, if $A$ is in general linear position, then $\pi^{-1}(t)$ is finite, and therefore has measure zero in $\mathbb{R}^{n-1}$. So Fubini's Theorem implies tha... | 4 | https://mathoverflow.net/users/68305 | 237323 | 109,716 |
https://mathoverflow.net/questions/237324 | 2 | Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf representative?
Following arXiv:math/0111139, suppose $\mathcal{C}$ is semisimple and rigid with finitely many irredu... | https://mathoverflow.net/users/51107 | When are Morita classes represented by certain structured algebra objects? | No. In the category of supervector spaces with the Koszul signs, odd Clifford algebras are neither symmetric Frobenius nor Hopf.
Note that symmetric Frobenius structures transfer across Morita equivalences. Hopf structures do not but you can see that no algebra in the Morita class of an odd Clifford algebra can be Ho... | 4 | https://mathoverflow.net/users/78 | 237333 | 109,721 |
https://mathoverflow.net/questions/237245 | 1 | In a [different thread](https://mathoverflow.net/a/236999/42864), we stumbled upon the following question:
Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective orthogonal polynomials $\{p\_n \}\_{n=0}^{\infty}$. We follow Szego and define the ... | https://mathoverflow.net/users/42864 | Various limits of the Christoffel Darboux Kernel | It seems that both questions receive some answers in [this paper](http://www.jstor.org/stable/pdf/1990404.pdf?_=1461643206256).
In th $L^2$ norm, Section 9 seems to stipulate convergence and a limit $K$.
For the uniform and pointwise convergence, we have on p. 374, Theorem II, that since the orthogonal projections ... | 1 | https://mathoverflow.net/users/42864 | 237334 | 109,722 |
https://mathoverflow.net/questions/236224 | 6 | ***This question was asked on [math.stackexchange](https://math.stackexchange.com/questions/1647793/how-to-calculate-the-psd-of-a-stochastic-process) about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.***
Say we have a stochastic process described by a stochastic ... | https://mathoverflow.net/users/88535 | How to calculate the PSD of a stochastic process | You ask for the spectral analysis of a nonstationary stochastic process. Because the autocorrelation function $C(s,t)$ now depends on the two times $s$ and $t$ separately, and not only on their difference, the power spectral density $P(\omega,\omega')$ will depend on two frequencies, and not just on a single frequency.... | 7 | https://mathoverflow.net/users/11260 | 237335 | 109,723 |
https://mathoverflow.net/questions/237337 | -1 | In the setting of the [Hadwiger-Nelson problem](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem), two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and has uncountably many connected components.
If we tak... | https://mathoverflow.net/users/8628 | Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem | No. There is a finite $S$ with the same chromatic number. See [this](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)).
| 6 | https://mathoverflow.net/users/90863 | 237339 | 109,725 |
https://mathoverflow.net/questions/237344 | 5 | According to the paper [The emergence of open sets, closed sets, and limit points
in analysis and topology](http://dx.doi.org/10.1016/j.hm.2008.01.001) famous mathematician [Maurice Fréchet](https://en.wikipedia.org/wiki/Maurice_Ren%C3%A9_Fr%C3%A9chet) who introduced the concept of metric spaces has also introduced ano... | https://mathoverflow.net/users/54507 | Fréchet L-Spaces | I think that these spaces don't go under the name of $L$ spaces anymore. Actually, I am not sure if there is a consensus on how these structures are called today.
A good place to start is the fairly recent book
* [**Convergence Structures and Applications to Functional
Analysis**](https://books.google.com/books?id=... | 6 | https://mathoverflow.net/users/9652 | 237346 | 109,727 |
https://mathoverflow.net/questions/236610 | 3 | I'm trying to read the article [*"Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups"* by Bukhshtaber,V. M. Russian Mathematical Surveys(1990),45(3):213](http://iopscience.iop.org/article/10.1070/RM1990v045n03ABEH002361/pdf) Russian text is available at [MathNe... | https://mathoverflow.net/users/5712 | Functional equations associated with addition theorems for elliptic functions | There is a more simple proof in the book *V. M. Buchstaber, T. E. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, Amer. Math. Soc., 2015*, [arXiv: 1210.2368v3](http://arxiv.org/abs/1210.2368v3). This result is the theorem E5.4.
| 1 | https://mathoverflow.net/users/5712 | 237347 | 109,728 |
https://mathoverflow.net/questions/237340 | 1 | Is there a non trivial sequence $(T\_{n})$ of linear operators $T\_{n}$ on $M\_{n}(\mathbb{C})$ such that $$T\_{nm}(X\otimes Y)=T\_{n}(X)
\otimes T\_{m}(Y) $$ where $X$ and $Y$ are in $M\_{n}(\mathbb{C})$ and $M\_{m}(\mathbb{C})$, respectively?
By trivial sequence we mean $T\_{n}=Id$ for all $n$.
| https://mathoverflow.net/users/36688 | The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra | $T\_n(X) = n \times n$ matrix with all entries $trace(X)$.
This doesn't satisfy $T\_n(X\_1 X\_2) = T\_n(X\_1) T\_n(X\_2)$.
Edit: More examples:
1. $T\_n(X)$ has $trace(X)$ in upper left corner, 0 elsewhere.
2. $T\_n(X)$ has $trace(X)$ in lower right corner, 0 elsewhere.
3. $T\_n(X)$ has sum of anti-diagonal $(\S... | 4 | https://mathoverflow.net/users/59248 | 237358 | 109,731 |
https://mathoverflow.net/questions/237354 | 3 | Let $X$ be an algebraic variety over $\mathbb C$ and let $\mathcal F$ be a constructible sheaf on $X$. It is well-known that the Euler characteristic of the cohomology of $\mathcal F$ is equal to the Euler characteristic of cohomology with compact support. What is the right reference for this result?
In $l$-adic cohom... | https://mathoverflow.net/users/3891 | Euler characteristic - reference question | The paper you refer to by Gérard Laumon appears to be:
*Comparaison de caractéristiques d'Euler-Poincaré en cohomologie l-adique,*
C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 209–212.
According to the MathSciNet Review of this paper (reference MR0610321) by William E. Lang:
>
> "As a corollary, the... | 5 | https://mathoverflow.net/users/12218 | 237363 | 109,732 |
https://mathoverflow.net/questions/235688 | 6 | Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, which is $\displaystyle\frac{-1}{8}\tau(M)$ ($\tau$ is the signature). I don't know too much about the index theorem and i... | https://mathoverflow.net/users/47391 | Index of Modified Dirac Operator | For this particular case the two operators are conjugate (though note the conjugacy is not unitary unless $s$ is imaginary)
$$
D\_{f,s}=e^{-sf}D e^{sf}
$$
so that the dimension of both the kernel and cokernel are independent of $s$.
Note that Chris Gerig's comments on the other hand apply more generally. The **index... | 5 | https://mathoverflow.net/users/12605 | 237370 | 109,733 |
https://mathoverflow.net/questions/237213 | 16 | I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely).
The first big step which I find difficult to break to concise little pieces is how to get from general riemannian ... | https://mathoverflow.net/users/22810 | A careful roadtrip from locally symmetric spaces to algebra | Let me outline a different approach, namely the one É. Cartan himself took, one that doesn't depend on assuming completeness, doesn't rely on ideas about real-analyticity, and side-steps many of the issues that you would have to deal with in fleshing out the outline above.
So suppose that $(M^n,g)$ is a *locally* sym... | 18 | https://mathoverflow.net/users/13972 | 237391 | 109,740 |
https://mathoverflow.net/questions/237386 | 11 | It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry.
Note: Grothendieck view of Topos is as an "ultimate" generalization of space.
Also, Elementary topos has many good logical properties. I am intereste... | https://mathoverflow.net/users/83957 | "Spatial (geometrical)" realization of Elementary topos? | I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes.
My main point is that the geometrical aspect of Grothendieck toposes is not related to the fact that they are elementary toposes... | 19 | https://mathoverflow.net/users/22131 | 237392 | 109,741 |
https://mathoverflow.net/questions/234765 | 12 | While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, when studying the generalized transfer operator (or Ruelle operator) for the Gauss Map, Dieter Mayer points out that thi... | https://mathoverflow.net/users/36215 | Nuclear operators/spaces and transfer operators | I believe your question is: What is the importance of being nuclear of order zero for the transfer (=Perron-Frobenius) operator in Ergodic theory?
The importance of being nuclear is a theorem by Grothendieck: If $L$ is nuclear of order less or equal than $2/3,$ then $det(I-zL)$ is entire in $z$ and we have that $det(... | 4 | https://mathoverflow.net/users/39115 | 237397 | 109,744 |
https://mathoverflow.net/questions/232778 | 24 | Lemma D4.5.3 of Johnstone’s *Sketches of an Elephant* states:
>
> **Lemma.** For an object $A$
> of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent:
>
>
> 1. $A$ is internally projective [i.e. $\Pi\_A : \E/A \to \E$
> preserves epimorphisms];
> 2. $(−)^A : \E \to \E$ preserves epimorphism... | https://mathoverflow.net/users/2273 | Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”) | There is a counterexample, due to Todd Trimble, in [another nForum thread](https://nforum.ncatlab.org/discussion/4342/internally-projective-objects/?Focus=35649#Comment_35649); cf. the nLab entry on [internally projective objects](https://ncatlab.org/nlab/show/internally+projective+object). Theorem 2 of [loc. cit.](htt... | 9 | https://mathoverflow.net/users/15782 | 237406 | 109,748 |
https://mathoverflow.net/questions/237353 | 8 | I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not nee... | https://mathoverflow.net/users/90869 | Definitions of Hilbert Bundles | "Why not simply use Bochner spaces like $C(X;L^2(\Omega))$?" --- do you mean that this would be the space of continuous sections of the bundle with fiber $L^2(\Omega)$? Yes, that is correct if the bundle is *trivial*, i.e., a bundle of the form $X \times H$ where $H$ is the fiber Hilbert space. But of course not all bu... | 3 | https://mathoverflow.net/users/23141 | 237414 | 109,753 |
https://mathoverflow.net/questions/237377 | 6 | Whenever I read the [anecdote about Hardy, Ramanujan and the taxi number 1729](https://en.wikipedia.org/wiki/1729_(number)) I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two cubes in two different ways -- and that it is the smallest such number.
... | https://mathoverflow.net/users/8628 | Generalizing Ramanujan's "1729 story" | There are many articles that study the quantity you call $r\_n(k)$ using sieve methods. Among them I mention the following, which give highly non-trivial bounds for the number of $k<X$ such that $r\_n(k)>1$. If you look at these, and also forward reference them using MathSciNet, you should be able to find the state of ... | 7 | https://mathoverflow.net/users/11926 | 237417 | 109,755 |
https://mathoverflow.net/questions/237384 | 23 | **Question:**
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $k$-sphere smoothly embedded in $\mathbb{R}^n$ (I suspect it is not always trivial)?
**What I know so far:**
For exampl... | https://mathoverflow.net/users/89166 | Is the normal bundle of a torus trivial? | You should be able to prove that the normal bundles in codimension $2$ are trivial as well. This is a little harder than codimension $1$; you need to know that such bundles are determined by their Euler class. The older literature (see papers cited below) tells you that the normal bundles of spheres are trivial once th... | 20 | https://mathoverflow.net/users/3460 | 237418 | 109,756 |
https://mathoverflow.net/questions/237425 | 1 | Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced morphisms from $A/\mathfrak{p} \to B/\mathfrak{q}$ are the same. Is it true that $f=g$? If not true in general, is there... | https://mathoverflow.net/users/58203 | Automorphisms of rings fixing all prime ideals | The condition just says that for each prime $\mathfrak{q}$ of $B$, both composite maps $A\rightrightarrows B\to B/\mathfrak{q}$ are equal. Hence for each $x\in A$, $f(x)-g(x)$ belongs to every prime of $B$, i.e. is nilpotent.
The converse is clear, so your condition is equivalent to the equality of both composites $... | 12 | https://mathoverflow.net/users/7666 | 237427 | 109,759 |
https://mathoverflow.net/questions/202642 | 2 | Among the Hausdorff compact spaces the closed interval is the simplest snake-like continuum. I'll present the definition after stating the problem.
The snake-like continua $\ S\ $ are *universal images for Hausdorff compact spaces* in the following sense:
**THEOREM** Let $\ S\ $ be an arbitrary snake-like continu... | https://mathoverflow.net/users/8385 | Snake-like continua and universal images | **EDIT.** As requested, I am extending the answer, including the relevant definitions. To do so, I have also re-arranged the answer somewhat. I am also now including the notions of surjective span and semi-span, which are directly relevant. As before, all continua in the following are assumed metrisable.
**Summary.**... | 1 | https://mathoverflow.net/users/3651 | 237466 | 109,772 |
https://mathoverflow.net/questions/237164 | 5 | There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}\_2$.
I mean there is polynomial reduction $F$ such that for every boolean circuit $g(x\_1,\ldots, x\_n)$ the corresponding system of polynomial equations $F(g) = \begin{cases} f\_1(x\_1, \ldots, x\_n)... | https://mathoverflow.net/users/31356 | an algebraic variety for a boolean circuit | Are you willing to allow adding additional coordinate variables $y\_1,\ldots,y\_m$ that are unused in $g$? (I mean that, for *any* $x\_1,\ldots,x\_n,y\_1,\ldots,y\_m \in \mathbb{F}\_2$, the variety will have the $\mathbb{F}\_2$-point $(x\_1,\ldots,x\_n,y\_1,\ldots,y\_m)$ *iff* $g(x\_1,\ldots,x\_n)=1$. Note that I am as... | 5 | https://mathoverflow.net/users/17064 | 237470 | 109,775 |
https://mathoverflow.net/questions/237460 | 1 | Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and restrict the $T$ to $E$. Say the matrix representing $T|\_{E}$ is $A\_{E}$. What is the range of singular values of $A\_{E}$ in ... | https://mathoverflow.net/users/69320 | range of singular values of sub-matrices | I assume that by singular values, you mean eigenvalues of $A^\*A$ (more precisely, their positive square roots). There are variational characterisations
\begin{align\*}
\lambda\_i&=\min\_{\begin{smallmatrix}U\subset V\\\dim U=i\end{smallmatrix}}\max\_{\begin{smallmatrix}v\in U\\|v|=1\end{smallmatrix}}|Av|\;,\\
&=\max\_... | 2 | https://mathoverflow.net/users/70808 | 237471 | 109,776 |
https://mathoverflow.net/questions/237462 | 3 | The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \psi=0$$
This follows from the Bochner-type formula $$D^2=\nabla^\*\nabla + \displaystyle\frac{R}{4}$$
by applying it to a... | https://mathoverflow.net/users/47391 | Converse to Lichnerowicz Vanishing Theorem? | I think the answer is No. You are essentially asking the following: If $0$ is not an eigenvalue of the Dirac operator $D$ on a compact Riemannian manifold, then does the underlying Riemannian metric have non-negative scalar curvature?
Well, pick some Riemannian manifold whose scalar curvature is very much not non-neg... | 7 | https://mathoverflow.net/users/2622 | 237484 | 109,782 |
https://mathoverflow.net/questions/163371 | 8 | I'm just trying to understand the following definition:
Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:
Let $O^\otimes$ be a unital $\infty$-operad and $C^\otimes \to O^\otimes$ a fibration of generalized $\infty$-operads. We let $Mod^O(C)^... | https://mathoverflow.net/users/19943 | understanding the definition of $\infty$-operad of module objects | I know this question is a little old but I just came across it.
Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg\_{/O}(C)$. However, you also get an *action* of $A$ on some object $M$ over $v$. The role of the semi-inert morphisms is as a "marking": a semi-inert morp... | 8 | https://mathoverflow.net/users/360 | 237491 | 109,785 |
https://mathoverflow.net/questions/237476 | 1 | Let $\Phi\_n(x)$ be the $n$ th cyclotomic polynomial. I've checked the values of $\Phi\_n(2)$ for some small $n\geq 2$ and noticed that there is always a divisor of $\Phi\_n(2)$ of the form $kn+1$ (regardless of $\Phi\_n(2)$ being a prime or not) with the only exception of $n=6$. Does anyone know any counterexample to ... | https://mathoverflow.net/users/73980 | Is it true that $\Phi_n(2)$ has a divisor of the form $kn+1$ for all $n\neq 6$? | Note that $\Phi\_{21}(2)$ is a multiple of 7. The actual divisibility criterion can be found in notes of G. J. O. Jameson mentioned here [Cyclotomic polynomials: $\Phi\_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?](https://mathoverflow.net/questions/221357) . In particular, non-Zsigmondy primes $q$ divide $\P... | 5 | https://mathoverflow.net/users/3402 | 237493 | 109,786 |
https://mathoverflow.net/questions/237474 | 4 | Let $\mathfrak t$ be a $t$-structure on a triangulated category $\cal T$. Let $\cal S$ be a thick (or even non-thick) triangulated subcategory, and ${\cal T}/\cal S$ the Verdier quotient.
Is there a canonical way to induce a $t$-structure on such a quotient, provided some assumptions on $\mathfrak t$ and $\cal S$?
| https://mathoverflow.net/users/7952 | t-structure induced on the Verdier quotient ${\cal T}/\cal S$ | Often, the answer is that there is a unique way to induce a $t$-structure on $T/S$ such that the quotient functor $T\rightarrow T/S$ is right $t$-exact. This is explained well in the sections on $t$-structures in Lurie's *Higher algebra*.
Here is one case where this works. Suppose that in fact $S\rightarrow T$ is a f... | 3 | https://mathoverflow.net/users/100 | 237502 | 109,790 |
https://mathoverflow.net/questions/237503 | 0 | I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an example about a p-subgroup that is radical and centric, but not essential. Anyone knows an example?
| https://mathoverflow.net/users/70864 | Radical and Centric not Essential P-group | Yes, take any prime $p$, and consider the fusion system for ${\rm GL}(4,p)$ ( ie objects the $p$-subgroups of ${\rm GL}(4,p)$ morphisms induced by conjugations).
Take a maximal parabolic $P$ such that $P/U \cong {\rm GL}(3,p)$, where $U$ is the unipotent radical of $P$. Then $U$ is centric and radical but is not essent... | 1 | https://mathoverflow.net/users/14450 | 237505 | 109,791 |
https://mathoverflow.net/questions/237371 | 6 | One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon M\to N$ with critical point $\kappa$.
I am looking for a reference, or at least a proof, that this is equivalent to ano... | https://mathoverflow.net/users/7206 | Embedding property of weakly compact cardinals | This is a selection on from my lecture notes text, Lectures on
Forcing and Large Cardinals, which I wrote long ago and which shows
the main equivalences, including the ones you mention.
**Theorem.** If $\kappa^{<\kappa}=\kappa$, then the following are
equivalent.
1. (weak compactness property) $\kappa$ is weakly co... | 14 | https://mathoverflow.net/users/1946 | 237508 | 109,793 |
https://mathoverflow.net/questions/237413 | 1 | I have the following theorem written on my whiteboard, but have misplaced the reference. I believe the probabilistic method may be involved in the proof. Any pointers appreciated.
**Theorem** Let $v\in\{0,1\}^n$ be nonzero. Let $k=v\_1+\ldots+v\_n$ be its Hamming weight. Then there exists $i\leq j$ such that
$$v\_i+v... | https://mathoverflow.net/users/17773 | Reference for a local density theorem for binary vectors | I've found the paper the result is from. It is Lemma 3.1 in
>
> "The unbounded error communication complexity of symmetric functions",
> by A. A. Shertsov, Combinatorica 31 (5) (2011) 583–614.
>
>
>
Since there was an upvote for the question, I will give an outline of the proof below.
By symmetry, we can as... | 2 | https://mathoverflow.net/users/17773 | 237510 | 109,795 |
https://mathoverflow.net/questions/237506 | 7 | Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us say a finite poset $P$ on $\geq 2$ elements is *indecomposable* if $P=P\_1 \times P\_2$ implies that either $P\_1$ or $P... | https://mathoverflow.net/users/25028 | Unique factorization of posets | Basically repeating what I said in the comments:
* In full generality, the answer to your question is "No". For a counterexample, see: [Tadasi Nakayama and Junji Hashimoto, *On a problem of G. Birkhoff*, Proc. Amer. Math. Soc. 1 (1950), pp. 141--142](http://www.ams.org/journals/proc/1950-001-02/S0002-9939-1950-003527... | 8 | https://mathoverflow.net/users/2530 | 237514 | 109,797 |
https://mathoverflow.net/questions/237511 | 4 | I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon\_0$. Let $I=[t\_0,R]$ be a compact interval. Suppose that $S:I\to [0,\infty)$ is a continuous non-decreasing function such that $S(t\_0)=0$ and
$$S(T)\lesssim \epsilon\_0(S(T)+\epsilon)^4+\epsilon... | https://mathoverflow.net/users/90954 | A continuity/bootstrap argument | I tend to think about such continuity argument in this way:
Let $T^\*=\sup\{T\in I: S(t)\le\epsilon\}$ for all $t\le T$. Then it suffices to show $T^\*=R$.
Suppose not. Then by continuity we have $S(T^\*)=\epsilon$. Plugging this into the given inequality we get $\epsilon\ll \epsilon\_0(\epsilon^4+\epsilon^5)+\epsi... | 1 | https://mathoverflow.net/users/37103 | 237515 | 109,798 |
https://mathoverflow.net/questions/237526 | 3 | Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete lattice with respect to $\subseteq$.
Is there a set $X$ and $\tau \in \text{Top}(X)$ such that
* $\tau \neq \{\empty... | https://mathoverflow.net/users/8628 | Topology with no direct lower neighbor | An affirmative answer to the question appears in Claim 1 of
[this answer](https://mathoverflow.net/q/237249).
The claim that the topology generated by $\tau\cup\{\{x\}\}$ is a cover of $\tau$ in the lattice
of topologies seems not to be true. For example, if $\tau$ is the usual topology
on the real line and $x=0$, t... | 2 | https://mathoverflow.net/users/75735 | 237528 | 109,802 |
https://mathoverflow.net/questions/237440 | 7 | Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.
| https://mathoverflow.net/users/90928 | Character Values for Alternating Groups of degree $\geq 7$ | As Geoff thought, the answer is contained in James and Kerber (it's Theorem 2.5.13 in "The Representation Theory of the Symmetric Group", Encyclopedia of Mathematics and its Applications vol. 16, 1981).
Each row or column of the character table contains at most one pair of irrationalities.
If $\lambda$ is a self-co... | 13 | https://mathoverflow.net/users/22989 | 237532 | 109,804 |
https://mathoverflow.net/questions/237293 | 1 | Is there any "standard" reference for (rational) singularities on algebraic surfaces? I'm aware of Artin's papers and the one of Brieskorn (*Rationale Singularitäten komplexer Flächen*), but they seem pretty dense to me. I'm looking for a more extensive treatment using consistent notation, ideally (chapters of) a book.... | https://mathoverflow.net/users/80607 | General Reference for surface singularities | You could have a look at the nice textbook by S. Ishii *[Introduction to Singularities](http://www.springer.com/us/book/9784431550808)*. Chapter 7 is devoted to normal two-dimensional singularities, with an extensive treatment of rational ones.
You might be also interested in T. Okuma's book *[Plurigenera of Surface ... | 3 | https://mathoverflow.net/users/7460 | 237533 | 109,805 |
https://mathoverflow.net/questions/237521 | 8 | It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
**Question.** For any $K$, does there necessarily exist positive integers $a, b$ such that the sequence given by $x\_1 = a, x\_2 = b, x\_{n} ... | https://mathoverflow.net/users/90626 | Arbitrarily many primes in a Fibonacci-type sequence | I think the answer to this question is yes. The theorem of Green, Tao and Ziegler says that a collection of $K$ linear forms over the integers will all take prime values infinitely often provided that they are nondegenerate (which, in the homogeneous case, means not multiples of one another) and there are no local obst... | 12 | https://mathoverflow.net/users/5575 | 237534 | 109,806 |
https://mathoverflow.net/questions/237500 | 11 | Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
I've been looking for some references, but I can't find any dealing with this problem. I would appreciate if someone h... | https://mathoverflow.net/users/90952 | Triangulation with simplices of same volume | I don't know a reference, but I think that we can use a theorem of Moser to get what you want.
Start with any triangulation $\mathcal{T}$ of $M$, whose cardinal is denoted by $k$ and denote by $\omega$ the Riemannian volume form. There is a positive smooth function $f$ such that $\int\_M f\omega=\mathrm{vol}(M)$ (i.e... | 17 | https://mathoverflow.net/users/4961 | 237536 | 109,807 |
https://mathoverflow.net/questions/237540 | 1 | Suppose that $R$ is a ring, not necessarily commutative nor associative. Assume that for every non-zero $a \in R$, the left multiplication map
$$ \lambda\_a \colon R \to R \colon x \mapsto ax $$
is invertible. (We do *not* assume that its inverse is again a left multiplication map $\lambda\_b$ for some $b \in R$.)
Is... | https://mathoverflow.net/users/12858 | Invertibility of all left multiplication maps in non-unital rings | What if your ring is the $\mathbf{R}$-algebra $\mathbf{R}^2$ with the bilinear law $$(x,y)(z,t)=\begin{pmatrix}2x & -y \\ y & x\end{pmatrix}\begin{pmatrix}z \\ t\end{pmatrix}=(2xz-yt,yz+xt)\quad?$$ It's even commutative.
| 3 | https://mathoverflow.net/users/14094 | 237544 | 109,809 |
https://mathoverflow.net/questions/237539 | 1 | Some non-regular languages don't yield to the Pumping Lemma ($L\_1=a^nb^mc^m$ should work). But now consider the set of non-regular languages L only over the alphabet {a}. (Like $L\_2=a^{n^2}$ or whatnot). The Pumping Lemma applied to some L now can be expressed as a statement about arbitrarily long arithmetic sequence... | https://mathoverflow.net/users/11504 | Non-regular languages fulfilling the Pumping Lemma | **Initial observation.** Let me start by explaining that the answer to your final question is that no, the complement of $L\_2$ does not have the pumping property. To see this, suppose
that it had a pumping number $p$, so that any string of length at
least $p$ in the complement of $L\_2$ could be pumped. That is, if
$w... | 4 | https://mathoverflow.net/users/1946 | 237553 | 109,812 |
https://mathoverflow.net/questions/237543 | 3 | I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[0,1]$.
And I think the result of schauder fixed point theorem is stronger than that of Leray Schauder , Do I have any m... | https://mathoverflow.net/users/87922 | leray schauder fixed point and schauder fixed point | 1. Note that Leray-Schauder is usually proven by using the hypotheses to *construct* a mapping that satisfies the conditions of the Schauder fixed point theorem, and then appealing to the Schauder fixed point theorem. See, e.g. [these notes](https://cmouhot.files.wordpress.com/1900/10/lerayschauder.pdf) (Theorem 2.2 th... | 10 | https://mathoverflow.net/users/3948 | 237564 | 109,816 |
https://mathoverflow.net/questions/165105 | 18 | This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard large cardinal axiom?
I know the statement can be forced assuming a measurable cardinal, and it implies Chang's Conject... | https://mathoverflow.net/users/18128 | Namba forcing and semiproperness | Semiproperness of Namba forcing is indeed equivalent to SCC. Here by SCC I mean the version which appears in Chapter XII, Theorem 2.5 part (2) of Shelah's book: for all large $\theta$ and all wellorders $w$ on $H\_\theta$ and all countable $N \prec (H\_\theta,\in,w)$ and all $\alpha < \omega\_2$, there is an $N' \sqsup... | 14 | https://mathoverflow.net/users/26319 | 237569 | 109,817 |
https://mathoverflow.net/questions/230661 | 5 | Recently I have read Parry and Pollicott's book, [Zeta functions and the periodic orbit structure of hyperbolic dynamics](http://homepages.warwick.ac.uk/~masdbl/PP.pdf).
I have been interested in some technical properties of the Ruelle-Perron-Frobenius operator, RPF operator for short.
The RPF operator appears fist... | https://mathoverflow.net/users/nan | General properties of the Ruelle operator | I rather prefer the notation $S\_n w,$ instead of $w\_n.$
Proof of claim 1: $ \varphi\circ \sigma^n/\varphi=\alpha^n e^{iS\_nv},$ then
$$
\varphi(x) L\_u^n(g/\varphi)(x)=\sum\_{y\in\sigma^{-n}x}(\varphi\circ\sigma^n (y)/\varphi(y)) g(y) e^{S\_n u(y)}=\sum\_{y\in\sigma^{-n}x}\alpha^n e^{iS\_nv} g(y) e^{S\_n u(y)}\\=\... | 0 | https://mathoverflow.net/users/39115 | 237570 | 109,818 |
https://mathoverflow.net/questions/237519 | 13 | We are looking to run a working seminar about the Yang-Mills story. We hope that our seminars is of interest to analysts (working with curvatures and Ricci flows on Riemannian manifolds), the algebraists (working on Lie algebras and Lie groups), and, of course, math physicists. If we manage to understand something abou... | https://mathoverflow.net/users/41301 | References for Yang-Mills Theory | If your goal is to get some understanding of the Clay Problem, you can't really go wrong with first reading the [official problem statement](http://www.claymath.org/sites/default/files/yangmills.pdf) and then reading the papers referred to in the document.
---
On the other hand, if your goal is not the quantum p... | 14 | https://mathoverflow.net/users/3948 | 237571 | 109,819 |
https://mathoverflow.net/questions/237567 | 18 | The Banach-Mazur distance $d(X, Y)$ between two normed spaces $X, Y$ of the same dimension is defined as $d(X, Y) = \log\inf \|T\| \cdot \|T^{-1}\|$, where the $T:X \to Y$ is a linear and invertible operator. The estimates between classical $\ell\_p^n$ spaces are known and in particular we have that $d\left(\ell\_1^n, ... | https://mathoverflow.net/users/69898 | Banach-Mazur distance between the cube and the octahedron | This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e\_i$ and $v\_j$ are the vertices of the unit balls in $\|\cdot\|\_1$ and $\|\cdot\|\_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e\_i\|\_\infty \le 1$, $\|T^{-1} v\_j\|\_1 \le s$ (or equival... | 15 | https://mathoverflow.net/users/13650 | 237580 | 109,823 |
https://mathoverflow.net/questions/237556 | 5 | Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H\_1,...,H\_r$, such that for any semi-simple $K$-subgroup $H\subset G$, $H$ is conjugate to one of the $H\_i$ by an element in $G(K)$?
I know the answer is... | https://mathoverflow.net/users/90978 | A finiteness property for semi-simple algebraic groups | For fields of characteristic zero one can argue as follows: Assume first that $K$ is algebraically closed. Since semisimple subgroups of $G$ correspond bijectively to semisimple subalgebras of $\mathfrak g={\rm Lie}\,G$ it suffices to consider the same problem in $\mathfrak g$. The advantage is that the subalgebras of ... | 12 | https://mathoverflow.net/users/89948 | 237585 | 109,825 |
https://mathoverflow.net/questions/237531 | 4 | Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an approximation from below.
Actually, I need something stronger:
Let $G$ be a locally compact group and let $C\_c^+(G)$ b... | https://mathoverflow.net/users/nan | Weil's Haar measure construction from below | The answer is Yes, provided that for every $\alpha$ and every $x\in G$, $f\_\alpha(x^{-1})=f\_\alpha(x)$.
Since we already completed the construction of the Haar measure we can actually use it in our proof. I suppose that one can show this also elementarily, but why bother?
Choose a left Haar measure on $G$.
Let me... | 1 | https://mathoverflow.net/users/89334 | 237592 | 109,827 |
https://mathoverflow.net/questions/237272 | 2 | Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A &B\\B^T &0\end{bmatrix}$$ We know that $C$ will have atleast $3$ zero eigenvalues. Removing those $3$ zeros, let $\lambda\_1\ge \lambd... | https://mathoverflow.net/users/70835 | Modified interlacing of eigenvalues | As I explained in the comments, the nonzero eigenvalues of $\left[\matrix{A&B\cr B^T&0}\right]$ and $\left[\matrix{A&2v\cr 2v^T&0}\right]$ are the same. So there's no reason you should expect interlacing between the eigenvalues of $C$ and the eigenvalues of a principal submatrix of $D$.
I went to an eigenvalue calcul... | 1 | https://mathoverflow.net/users/23141 | 237597 | 109,829 |
https://mathoverflow.net/questions/145489 | 3 | I came across the notion of information theoretic privacy in the paper of Yamamoto ("A source coding problem for sources with additional outputs to keep secret from the receiver or wiretappers "). The way this problem is formulated is as follows.
We have a random variable $X$ and a correlated random variable $Y$ with... | https://mathoverflow.net/users/41666 | Information theoretic privacy and distance of probability measures! | There is a large body of work on information-theoretic notions of secrecy, spread between the information theory and cryptography communities. For example, this Wikipedia page has lots of useful links:
<https://en.wikipedia.org/wiki/Information-theoretic_security>
I also recommend the recent paper of Bellare, Tessar... | 1 | https://mathoverflow.net/users/83481 | 237598 | 109,830 |
https://mathoverflow.net/questions/237596 | 8 | Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am requesting help here.
The question I face is:
>
> If we have a domain (like the unit disk) D in the complex plane and w... | https://mathoverflow.net/users/89375 | Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology? | For those who don't have the book (or have the wrong version), here is the proof that the topological vector space of holomorphic functions on the unit disk is not normable (i.e. whose topology is not defined by a norm).
**Definition**: A topological vector space (over $\mathbb R$ or $\mathbb C$) is *locally bounded*... | 14 | https://mathoverflow.net/users/37103 | 237606 | 109,833 |
https://mathoverflow.net/questions/237611 | 2 | Is there a subset of natural numbers that has the same additive and multiplicative structure as the set of ordered pairs of natural numbers under the classical operations of addition and multiplication i.e. is there a (primitive recursive) bijection $f: N \times N \rightarrow U \subset N$ with $f(x,y)+f(u,v)=f(x+u,y+v)... | https://mathoverflow.net/users/87821 | Is there a bijection $f: N \times N \rightarrow U \subset N$ with $f(x,y)+f(u,v)=f(x+u,y+v)$ and $f(x,y) \cdot f(u,v)=f(x \cdot u, y \cdot v)$? | There is no bijection satisfying the addition and multiplication
identities. In fact, even
without the assumption that $f$ is a bijection
there
are only three functions satisfying both identities: for all $x$ and $y$
either
$f\_1(x,y) = x$, $f\_2(x,y) = y$ or $f\_3(x,y) = 0$.
To see this, use the addition identity r... | 4 | https://mathoverflow.net/users/75735 | 237614 | 109,835 |
https://mathoverflow.net/questions/237602 | 14 | In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
>
> What is G'?
>
>
>
I know there is concrete description in terms of pairs (M,f) with M in G and f a function (see Lemma 2.14 of Huybrechts <http://arxiv.org/pdf/1111.1745v2.pdf>). But I... | https://mathoverflow.net/users/86649 | what is the universal cover of GL(2,R)? | For the sake of completeness here is an explicit proof. Let $m\in GL\_{2}(\mathbb{R})^{+}$, then $m\rightarrow \frac{m}{\det(m)}$ maps it to an element in $SL\_{2}(\mathbb{R})$. And we know $SL\_{2}(\mathbb{R})\cong \mathbb{R}^2\times \mathbb{S}^1$ via appropriate [parametrization](https://math.stackexchange.com/questi... | 9 | https://mathoverflow.net/users/18850 | 237616 | 109,836 |
https://mathoverflow.net/questions/237623 | 4 | It is said that the following proposition is true.
$\forall S \subset \mathbb{Z}, |S| = 2n-1.\ \exists A \subset S, |A| = n$ which satisfies
$$
n \ | \ \sum\_{a \in A}a.
$$
Could someone gives a proof or some infomation about this? Thx.
| https://mathoverflow.net/users/22954 | Find a subset such that its sum is divisible by $n$ | It is [Erdős–Ginzburg–Ziv theorem](https://en.wikipedia.org/wiki/Zero-sum_problem).
The standard proof is induction on number of prime divisors of $n$.
If $n$ is prime, let $0\leqslant x\_1\leqslant \dots \leqslant x\_{2n-1}\leqslant n-1$ be elements of $S$ (we may think so of course). If $x\_i=x\_{n-1+i}$ for some... | 9 | https://mathoverflow.net/users/4312 | 237635 | 109,841 |
https://mathoverflow.net/questions/237613 | 6 | Let $K =(K,| \cdot |)$ be a non-Archimedean valued field.
Let $E$ be a $K$-vector space. A **norm** on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that:
1. $||x||=0$ if and only if $x=0$,
2. $||\lambda x||=|\lambda|\,||x||$,
3. $||x+y||\leq\max\{||x||,||y||\}$,
for all $x,y\in E$.
A function $||\cdot||:E\to[0,\i... | https://mathoverflow.net/users/47542 | Norm vs A-norm in non-Archimedean Functional Analysis | I guess that putting an archimedean norm on a vector space over a nonarchimedean field gives just an uncorrelated product of something archimedean with something nonarchimedean. Number theorists sometimes look at *all* places of $\mathbb{Q}$ at once: all p-adic valuations and the archimedean valuation. I don't see a re... | 3 | https://mathoverflow.net/users/62434 | 237637 | 109,842 |
https://mathoverflow.net/questions/237559 | 3 | Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m\_1,...,m\_k$ at $p\_1,...,p\_k\in\mathbb{P}^n$ general points.
Let $S\subseteq |\mathcal{O}\_{\mathbb{P}^n}(d)|$ be the sublinear system to which $X$ belongs.
Is it possibile to find conditions on $d$ and the $m\_i$ ensuring tha... | https://mathoverflow.net/users/nan | Sections of a linear system splitting as a product of degree one polynomials | A union of $d$ hyperplanes has multiplicty at most $d$ at every point, and a hyperplane in $\mathbf P^n$ can be made to pass through at most $n$ general points. So the conditions
* $m\_i \leq d$ for each $i$
* $\sum\_{i=1}^k m\_i \leq dn$
are obviously necessary.
I claim they are also sufficient. (I think this i... | 1 | https://mathoverflow.net/users/75616 | 237639 | 109,844 |
https://mathoverflow.net/questions/237615 | 3 | Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth projective variety, central fiber $Y=f^{-1}(0)=\bigcup D\_i$, with reduced induced $Y\_{red}$ is normal crossing.
Question... | https://mathoverflow.net/users/63996 | Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber | On what group are you studying the monodromy? Begin with $\mathcal{Y} = \Delta \times \mathbb{P}^n$, $n\geq 2$, with its projection $\text{pr}\_1$ to $\Delta$. The central fiber is smooth of multiplicity $1$. Let $d\geq 2$ be an integer. Let $Z\subset \mathcal{Y}$ be a closed submanifold such that the restriction $\tex... | 2 | https://mathoverflow.net/users/13265 | 237643 | 109,846 |
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