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https://mathoverflow.net/questions/193378 | 4 | Is there a locally compact group $G$ such that the canonical map from $C^{\*}(G)$ to $C^{\*}\_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{\*}$ algebras are isomorphic $C^{\*}$ algebras via another morphism?
| https://mathoverflow.net/users/36688 | Can the full and reduced group $C^*$-algebras be "noncanonically" isomorphic? | If $C^\*(G)$ is isomorphic to $C^\*\_r(G)$, then $C^\*\_r(G)$ has a 1-dimensional representation, i.e. $G$ has a 1-dimensional representation $\chi$ weakly contained in the regular representation $\lambda\_G$. Then $1\_G=\chi\otimes\overline{\chi}$ is weakly contained in $\lambda\_G\otimes\overline{\lambda\_G}\simeq \i... | 20 | https://mathoverflow.net/users/14497 | 193383 | 94,479 |
https://mathoverflow.net/questions/192004 | 8 | Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is the same as over the complex numbers, as long... | https://mathoverflow.net/users/26635 | Closure order on nilpotent orbits in exceptional Lie algebras | I'm assuming your question involves just *good* prime characteristic $p$. Much of the literature focuses on *unipotent classes*, but Springer's $G$-equivariant isomorphism (for good $p$) between the unipotent variety in $G$ and the nilpotent variety in $\mathfrak{g}$ shows that the classes and orbits are in bijection a... | 5 | https://mathoverflow.net/users/4231 | 193390 | 94,480 |
https://mathoverflow.net/questions/192332 | 0 | Let $X$ be a $m$-dimensional Gaussian vector, and $Y$ a $q$-dimensional Gaussian vector, for some $m,q\geq 1$. Assume that the $X\_i$ and $Y\_j$ are centred and have unit variance. Assume that $E X\_i Y\_j \leq \mu$ for any $i,j$, for some $\mu< 1$.
Is there a constant $C\_{\mu,m,q}$ such that for every Borel $A,B$,
... | https://mathoverflow.net/users/16934 | reverse FKG type inequality for slightly correlated Gaussian vectors | In this formulation, it still isn't true.
Take $\{X\_i\}\_{i=1}^n$i.i.d standard Gaussian. Take $\{Y\_i\}\_{i=1}^{n-1}$ i.i.d. standard Gaussian. Take $Y\_n=\sum\_{i=1}^n X\_i/\sqrt{n}$. Then $|EX\_i Y\_j|\leq 1/\sqrt{n}$ for any $i,j$.
Fix $D\subset R$ measurable.
Take $A=\{\sum X\_i/\sqrt{n}\in D\}$ and $B=\{Y\_n\in ... | 2 | https://mathoverflow.net/users/35520 | 193398 | 94,482 |
https://mathoverflow.net/questions/193396 | 9 | I would like to have something like a linear order on classes, such that every instantiated predicate of classes has a minimal instance in that order. For my purposes, it is fine to assume V=L for sets.
| https://mathoverflow.net/users/38783 | What is known about global well ordering of classes in Gödel-Bernays? | In the context of models of second-order arithmetic, Mostowski showed that starting with the theory $Z\_2$ (the second-order arithmetic analogue of Kelley-Morse) together with the scheme of dependent choices, one can use forcing to add a well-ordering relation on classes so that the full second-order comprehension sche... | 10 | https://mathoverflow.net/users/5984 | 193399 | 94,483 |
https://mathoverflow.net/questions/193361 | 1 | This question is strictly related to [this one](https://mathoverflow.net/questions/190850/system-of-linear-first-order-pde-with-constant-coefficients). Let us consider the differential system with constant coefficients
$$\left(\begin{array}{ccc}
B\_{11} & B\_{12} & 0\\
-B\_{11} & 0 & 0\\
0 & -B\_{12} & 0
\end{arr... | https://mathoverflow.net/users/57571 | Cauchy problem for an overdetermined system of PDE | Here is a second attempt to answer the question:
There are three equations for three unknown functions. As Igor points out, if you add the three equations together, you get $0 = 0$. Therefore, the third equation is a consequence of the first two. It therefore suffices to solve the first two equations only. This is no... | 3 | https://mathoverflow.net/users/613 | 193401 | 94,484 |
https://mathoverflow.net/questions/193400 | 23 | By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{1}=1$, $\underline{x \cdot y}=\underline{x} \cdot \underline{y}$. Notice that the elements of this group have the form $... | https://mathoverflow.net/users/2841 | Non-abelian Grothendieck group | George Bergman refers to it as the **universal enveloping group** of the monoid. See Section 3.11, page 81, of his universal algebra notes/forthcoming book: <http://math.berkeley.edu/~gbergman/245/>
He also references P.M. Cohn's *Universal Algebra* book, and two papers of Mal'cev in which he establishes conditions f... | 19 | https://mathoverflow.net/users/3959 | 193403 | 94,486 |
https://mathoverflow.net/questions/193422 | 3 | I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products.
**Definition.** A "multi-monoidal category" consists of
* a category $\mathcal{C}$,
* for every $n \geq 0$ a functor $T\_n : \mathcal{C}^n \to \mathcal{C}$, denoted by $(A... | https://mathoverflow.net/users/2841 | Reference for "multi-monoidal categories" | Look at Section 3 of Leinster's *[Higher Operads, Higher Categories](http://arxiv.org/abs/math/0305049)*, where the term used is "unbiased monoidal category."
| 13 | https://mathoverflow.net/users/290 | 193423 | 94,491 |
https://mathoverflow.net/questions/192322 | 3 | Let $ f \in S\_k(\Gamma)$ be a weight k modular cusp form of level $\Gamma$, with modular curve $Y\_{\Gamma}$. Let $V^{k-2}$ be the homogenous polynomials in X and Y of degree k-2 with complex coefficients. Let $\Omega^1$ be the sheaf of differential 1 forms. We define the map:
$\phi : S\_k(\Gamma) \to H^0(Y\_{\Gamma... | https://mathoverflow.net/users/47195 | Proving the Eichler-Shimura Isomorphism defines a global section | What exactly is a global section of $\Omega^1 \otimes V^{k-2}$ on $Y\_\Gamma$? First consider $\Omega^1 \otimes V^{k-2}$ as a sheaf on $\mathbb{H}$. This sheaf is naturally endowed with a $\Gamma$ action through $\Gamma$'s action on $V^{k-2}$.
Cover $\mathbb{H}$ with open sets $\{U\_i\}$ such that $\gamma(U\_i)\cap U... | 3 | https://mathoverflow.net/users/13158 | 193428 | 94,492 |
https://mathoverflow.net/questions/180326 | 4 | Let $A = (\dotsc \twoheadrightarrow A\_2 \twoheadrightarrow A\_1 \twoheadrightarrow A\_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim\_i \,\mathsf{Mod}(A\_i)$: Objects are families of right $A\_i$-modules $M=(M\_i)$ together with isomorphisms $M\_{i+1}... | https://mathoverflow.net/users/2841 | Canonical presentation of pro-modules over pro-rings | Since this just got bumped ...
I think the answer <https://math.stackexchange.com/a/938076/88262>, to a special case of this question that Martin asked on Math.SE, settles this question by giving a counterexample.
In summary, there is a counterexample where $A\_i=k[x\_1,\dots,x\_i]$ modulo the ideal generated by po... | 2 | https://mathoverflow.net/users/22989 | 193438 | 94,496 |
https://mathoverflow.net/questions/193444 | 4 | Let $X \subseteq \mathbb{P}\_{\mathbb{C}}^n$ be an irreducible, projective, Cohen-Macaulay variety of dimension $k$. Let $L \subseteq \mathbb{P}\_{\mathbb{C}}^n$ be a linear space of dimension $n-k-1$ that does not intersect $X$. Then the linear projection $\pi: X \to \mathbb{P}\_{\mathbb{C}}^k$ from center $L$ is a fi... | https://mathoverflow.net/users/36563 | When do we get free modules from Noether normalization | $S$ is free if and only if it is a Cohen-Macaulay ring -- see e.g. Bourbaki, Algèbre commutative X, 4, no. 3, Corollaire (I am afraid this is not yet translated in english). In geometric terms, this means that $X$ is *projectively* (aka *arithmetically*) *Cohen-Macaulay* -- that is, $H^i(X,\mathcal{O}\_X(n))=0$ for all... | 7 | https://mathoverflow.net/users/40297 | 193452 | 94,500 |
https://mathoverflow.net/questions/193458 | 1 | Let $f:{\mathbb R}\rightarrow{\mathbb R}\_+$ be a [log-convex](http://en.wikipedia.org/wiki/Logarithmically_convex_function) function. Suppose that $f\_{\epsilon}$ is the smoothed version of $f$:
$$f\_{\epsilon}(x)=\int \varphi\_{\epsilon}(x-y)f(y)dy,$$
where $\varphi$ is a [Mollifier](http://en.wikipedia.org/wiki/... | https://mathoverflow.net/users/65764 | log-convexity of Mollified function? | Yes if you assume $\phi\_\epsilon\ge0$. Because $f\_\epsilon$ belongs to the convex cone spanned by the shifted functions $f(\cdot-h)$. Since every $f(\cdot-h)$ is logarithmically convex and the set of logarithmically convex functions is itself a convex cone, you are done.
| 3 | https://mathoverflow.net/users/8799 | 193462 | 94,502 |
https://mathoverflow.net/questions/191828 | 6 | Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$.
A **Frostman measure** on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r\_0>0$ with $\mu(B\_r(x)) \leq C r^t$ for all $x\in X$ and all $0<r\leq r\_0$. Let's call the supremum of such $t$ the **Frostman-Ex... | https://mathoverflow.net/users/3041 | Calculate Hausdorff measure with Frostman measures | I think I found an answer for $X\subseteq\mathbb{R}^n$ (which is good enough for me). In Mattila's book there is a theorem about densities stating that the upper density of the $t$-dimensional Hausdorff measure on a $\mathcal{H}^t$-finite set $A\subseteq\mathbb{R}^n$ is $\mathcal{H}^t$-almost everywhere bounded by 1, i... | 3 | https://mathoverflow.net/users/3041 | 193476 | 94,504 |
https://mathoverflow.net/questions/193477 | 2 | Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^\*$ be the set of infinite words. I was looking for an element $a = \lbrace a\_n \rbrace\_{n \in \mathbb{N}}$ of $A^\*$ so that, $\exists k\_0 \in \mathbb{N}$ so that $\forall k>k\_0$, any finite subsequence (of length $k \c... | https://mathoverflow.net/users/18974 | Totally aperiodic sequence | [Square free words](http://en.wikipedia.org/wiki/Square-free_word) exist over all alphabet sizes greater than 2, and cube free words exist over all alphabet sizes greater than or equal to 2 (same article).
| 6 | https://mathoverflow.net/users/1907 | 193479 | 94,505 |
https://mathoverflow.net/questions/193474 | 5 | I have the following problem, where $k$ is a fixed integer.
**Input:** Graph $G$.
**Output:** Minimum number of edges to remove from $G$ to obtain a graph such that every node has degree at most $k$.
Do you know the complexity of this problem?
| https://mathoverflow.net/users/59249 | Minimum number of edges to remove to have low degree | If you also insist that the bounded-degree subgraph is connected, then your problem is NP-Hard, since it includes the Longest Path problem when $k=2$.
On the other hand, without the connectivity constraint, your problem can be solved in polynomial-time using standard techniques from matching theory. See this [paper](... | 10 | https://mathoverflow.net/users/2233 | 193480 | 94,506 |
https://mathoverflow.net/questions/193482 | 1 | Find all natural solutions $m!=a^2n!$ .
It's clear that $m>=n$.
When $m=n$ we have solutions $(1,m,m)$.
When $m=n+1$ we have solutions $(a,a^2,a^2-1)$.
I think that when $m>n+1$ we have no solutions but I can't prove this.
| https://mathoverflow.net/users/65778 | Diophantine equation with factorials | [The product of consecutive integers is never a power.](https://projecteuclid.org/euclid.ijm/1256050816)
| 14 | https://mathoverflow.net/users/2384 | 193483 | 94,508 |
https://mathoverflow.net/questions/193352 | 4 | Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a *cover* of $H$ if $\bigcup C = V$.
A cover $M\subseteq E$ is said to be *strongly minimal* if for every cover $C$ of $H$ we have $$\text{card}(M\setminus C)\leq \text{card}(C\setminus M).$$
Is there ... | https://mathoverflow.net/users/8628 | Strongly minimal covers | Maybe I misunderstood something, but consider the following simple example.
Let $V=(0,\infty)$ and let the maximal edges be the (open) unit intervals, except $(0,1)$.
Any cover contains a sequence converging to $0$, so in fact there isn't any minimal cover at all.
**Update:** Noah asked whether there is an example wh... | 4 | https://mathoverflow.net/users/955 | 193494 | 94,512 |
https://mathoverflow.net/questions/193491 | 38 | Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a simplification of X if it is universal for maps from X to a simple space. Does Simp(X) exist for any connected space X? This wou... | https://mathoverflow.net/users/22 | Is there a "simplification" functor in algebraic topology? | The space $S^1\vee S^1$ does not have a simplification. Indeed, suppose $f:S^1\vee S^1\to X$ is a simplification and let $i:S^1\vee S^1\to S^1\times S^1$ be the standard inclusion. Then since $X$ is simple, the commutator of the generators of $\pi\_1(S^1\vee S^1)$ becomes nullhomotopic after composing with $f$, so $f$ ... | 40 | https://mathoverflow.net/users/75 | 193495 | 94,513 |
https://mathoverflow.net/questions/156715 | 2 | Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}\_n(k)$ *$p$-nilpotent* if $X^p=0$, and let $\mathcal{N}\_1=\mathcal{N}\_1(\mathfrak{gl}\_n(k))$ be the set of all $p$-nilpotent matrices, on which $\mathrm{GL}\_n(k)$ acts by conjugation. Each $\mathrm{GL}\_n(k)$-orbit of $... | https://mathoverflow.net/users/32261 | A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$ | This question is relatively old and not precisely enough formulated, so maybe it would help to clarify the questions actually being asked.
1) The header is misleading, since "$p$-nilpotent" isn't really at issue here. Instead, the prime $p$ is always taken to be large enough so that $p$-nilpotent is the same as nilp... | 4 | https://mathoverflow.net/users/4231 | 193497 | 94,515 |
https://mathoverflow.net/questions/193507 | 2 | As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ?
By fibration of groupoids I mean a fibration for the "natural" model structure where weak equivalences are categorical equivalences of groupoids, so by fibration I mean an isofi... | https://mathoverflow.net/users/nan | Does the nerve functor preserve fibrations? | Yes the nerve functor from groupoids to simplicial sets sends isofibrations to Kan fibrations. Being an isofibration means exactly that the nerve has the right lifting property against $\Lambda\_0^1\to\Delta^1$ and $\Lambda\_1^1\to \Delta^1$. The lifting property against higher dimensional horn inclusions is automatic.... | 8 | https://mathoverflow.net/users/10707 | 193523 | 94,521 |
https://mathoverflow.net/questions/193520 | 0 | In *Homology of $C\_{n+1}$-spaces, $n\geq 0$,* F.R. Cohen, *Lecture Notes in Mathematics, Vol. 533,* page 210 (the preface part before contents):
*Line 2:* ... is used to compute the precise algebra structure of $H^\*(F(\mathbb{R}^{n+1},p)/\Sigma\_p;\mathbb{Z}\_p)$;
Is there any theorem in later chapters stating th... | https://mathoverflow.net/users/41075 | cohomology algebra of unordered configuration space on Euclidean space | Yes, it is Theorem 5.2 on page 246. The preceding remarks explain the notation. The proof comes later (in Section 11).
| 3 | https://mathoverflow.net/users/8103 | 193526 | 94,522 |
https://mathoverflow.net/questions/192157 | 4 | For a semigroup $S,$ its power semigroup $P(S)$ is the semigroup of all non-empty subsets of $S$ with the operation given by $AB=\{ab\,|\,a\in A,b\in B\}.$ I would like to know about the cancellable elements of $P(S)$ given some knowledge of cancellability in $S.$
If $s\in S$ is left-cancellable in $S,$ then $\{s\}$ ... | https://mathoverflow.net/users/20803 | Cancellable elements of a power semigroup | Here is an example of a cancellable set with a non cancellable element. Take the semigroup with presentation $S=\langle a,b,c\mid ab=ac, ba=ca\rangle$
One checks that $ac\to ab$ and $ca\to ba$ is a complete rewriting system. The normal forms are elements with no $c$ next to an $a$. Notice that $b$ is cancellable sinc... | 3 | https://mathoverflow.net/users/15934 | 193529 | 94,523 |
https://mathoverflow.net/questions/193542 | 1 | I hope this is not too elementary.
Let $B: V\ \times V \to W$ be a skew-symmetric bilinear map where $V$, $W$ are
finite dimensional real vector spaces. Assume that $B (u, v)$ is never zero
for linearly independent $u$, $v\in V$.
>
> What can we say about the dimensions of $V$ and $W$?
>
>
>
An elementar... | https://mathoverflow.net/users/15155 | Image of skew-symmetric bilinear map which is never zero on linearly independent vectors | A better bound is $\dim W\geq 2\dim V-3$. This is obtained as follows. In the projective space $\mathbb{P}(\wedge^2V)$, the set of decomposable bivectors is the Grassmannian $\mathbb{G}(2,V)$, of dimension $2\dim V-4$. The bilinear map $B$ induces a linear map $b:\wedge^2V\rightarrow W$, and the hypothesis is $\mathbb{... | 3 | https://mathoverflow.net/users/40297 | 193550 | 94,532 |
https://mathoverflow.net/questions/193537 | 4 | I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology.
The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such that our sheaf is acyclic on the covering, i.e. Čech cohomology is $0$ on every finite intersection of elements in the cov... | https://mathoverflow.net/users/42995 | Leray's theorem up to some degree | Well, you're perfectly correct. In fact, for the proof, you only need that $H^1$ is vanishing on all the intersections of all the open sets in your cover. However,...you need that vanishing for **all** quasi-coherent sheaves (or whatever category you're working in) and the higher cohomology of any sheaf is the $H^1$ of... | 3 | https://mathoverflow.net/users/10076 | 193558 | 94,534 |
https://mathoverflow.net/questions/193560 | 19 | Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator in the setting of pure mathematics:
* Which underlying structure (metrics, bundles, etc.) a manifold have in order to be... | https://mathoverflow.net/users/62367 | Exact Definition of Dirac Operator | I would think that these [these notes](http://math.uchicago.edu/~amathew/dirac.pdf) by Akhil Mathew provide the "exact definition" you are asking for:
---
In response to the follow-up question "which is the first Clifford module used in the physics contex... | 11 | https://mathoverflow.net/users/11260 | 193565 | 94,536 |
https://mathoverflow.net/questions/192048 | 2 | Let $f(x,y)=0$ be irreducible elliptic curve over the rationals.
Are there $f$ for which:
Both $x,y$ are arbitrary large powers infinitely often,
i.e. infinitely many rational points $(u^k,v^m)$ with
both $k,m$ arbitrary large?
---
For $x,y$ squares (asked in previous revision) this is possible.
Take $f(x,y)=... | https://mathoverflow.net/users/12481 | Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often? | The method **joro** suggests works with little change for any $(k,m)$.
Fix an elliptic curve $E: P(X,Y)=1$ of positive rank.
Some simple examples are
$F(X,Y) = Y^2 - X^3 + 2X$ and $F(X,Y) = Y^2 - X^3 - 2$,
each with generator $(X,Y)=(-1,1)$.
Let $u$ and $v$ be "random" rational functions on $E$
that generate the fu... | 9 | https://mathoverflow.net/users/14830 | 193588 | 94,546 |
https://mathoverflow.net/questions/193583 | 6 | Let $K$ be a number field, let $A$ be an abelian variety over $K$, and let $H$ be a torus over $K$. For a prime $l$, we have the natural map
$$\mathrm{Ext}^1(A, H) \otimes\_{\mathbb{Z}} \mathbb{Z}\_l \rightarrow \mathrm{Ext}^1\_{\mathrm{Gal}(\overline{K}/K)}(T\_l A, T\_l H),$$
where the first $\mathrm{Ext}$ is in the c... | https://mathoverflow.net/users/63877 | Extensions of an abelian variety by a torus vs. extensions of their $\ell$-adic Tate modules | It is affirmative (up to a compatibility check at the end of the argument given here). If $K'/K$ is a finite Galois splitting field of $H$ and $T' = H\_{K'}$ is the associated split $K'$-torus then we have an exact sequence of $K$-tori $$0 \rightarrow H \rightarrow {\rm{R}}\_{K'/K}(T') \rightarrow T \rightarrow 0.$$
Bu... | 6 | https://mathoverflow.net/users/61939 | 193593 | 94,549 |
https://mathoverflow.net/questions/168125 | 24 | It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods.
I've heard rumors that there was a proof which was purely set theoretic, and indeed much shorter than the original.
Does anyone know who wrote it, and whether or not it appears onlin... | https://mathoverflow.net/users/7206 | Short proof of $\frak p=t$ | One reference is given above by Inamdar that can be found here:
[$\frak p=t$, following Malliaris-Shelah and Steprans](https://www1.essex.ac.uk/maths/people/fremlin/n14528.pdf) ([Internet Archive](http://web.archive.org/web/20180605111554/https://www1.essex.ac.uk/maths/people/fremlin/n14528.pdf)).
Also there is ano... | 15 | https://mathoverflow.net/users/11115 | 193594 | 94,550 |
https://mathoverflow.net/questions/193601 | 1 | I went through several martingales concentration bounds, but none of them fit the settings I am interested in, which is the following. Suppose I have a sequence of nonnegative random variables $0=Y\_{0},\ldots,Y\_{n}$, where every $Y\_{i}$ is a function of some real random variables $X\_{1},\ldots,X\_{i}$ and I know th... | https://mathoverflow.net/users/27970 | Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases) | We can use the decomposition
$$Y\_n=\sum\_{i=0}^{n-1} Y\_{i+1}-Y\_i -\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ] + \sum\_{i=0}^{n-1}\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ],$$
hence by assumption, the bound
$$Y\_n\leqslant \sum\_{i=0}^{n-1} Y\_{i+1}-Y\_i -\mathbb E[Y\_{i+1}-Y\_i\mid X\_1,\dots, X\_i ] + \sum... | 1 | https://mathoverflow.net/users/17118 | 193605 | 94,552 |
https://mathoverflow.net/questions/193612 | 8 | It is my impression that the following question is open:
Does the existence of a basis for every vector space over the field K = the reals having a basis imply the axiom of choice?
I saw an answer from several years ago that indicated it was open. There was also a somewhat vague comment about the status of the ques... | https://mathoverflow.net/users/65844 | Axiom of choice and vector spaces over a given field | *Allow me to steal [my answer to a question from math.SE on the same topic](https://math.stackexchange.com/q/994632/622). With minor changes*
---
In their paper, Howard and Tachtsis discuss these sort of questions. The paper was published rather recently, and I suspect that there hasn't been any significant progr... | 10 | https://mathoverflow.net/users/7206 | 193615 | 94,554 |
https://mathoverflow.net/questions/193586 | 0 | Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^\* \in R$ it has a bounded smooth solution $u^\*$ in $R^n$ ($f$ is a nice smooth function). Under what conditoions one can guarantee that for $\alpha$ close to $\alpha^\*$ this equation has a solution $u$ close to $u^\*$?
... | https://mathoverflow.net/users/42326 | Implicit function theorem for elliptic partial differential equations | As your title indicates, you would do this using the implicit function theorem for Banach space applied to an appropriately defined functional. This in turn requires proving that there is a bounded right inverse to the linearized operator.
The second step requires solving a linear elliptic PDE and getting an a priori... | 4 | https://mathoverflow.net/users/613 | 193622 | 94,557 |
https://mathoverflow.net/questions/180122 | 16 | Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in
<http://arxiv.org/abs/q-alg/9609018>
In the more recent paper by Baez, Baratin, Freidel and Wise
<http://arxiv.org/abs/0812.4969>
a notion of infinite dimensional 2-vector space is di... | https://mathoverflow.net/users/2183 | Infinite dimensional 2-Hilbert spaces | An answer to the question in the second paragraph (what kind of objects should a quantum field theory assign to codimension 2 manifolds?) can be found in my paper <http://arxiv.org/abs/1304.7328v2>.
I will argue that this also provides an answer to the OP's main question (what is a possibly infinite dimensional 2-Hilbe... | 7 | https://mathoverflow.net/users/5690 | 193624 | 94,558 |
https://mathoverflow.net/questions/193631 | 2 | Let $X$ and $Y$ be locally compact, second countable spaces, and let $\varphi \colon X \to Y$ be a Borel map. Let $\mu$ be a sigma-finite measure on $X$. In general, the push-forward $\varphi\_\*\mu$ is not sigma-finite. Does there, however, exist a sigma-finite measure $\nu$ that is mutually absolutely continuous with... | https://mathoverflow.net/users/23661 | Is the push forward of a sigma-finite measure equivalent to a sigma-finite measure? | Yes, it is even equivalent to a finite measure.
Note that $\mu$ is equivalent to a finite measure $\nu$ on $X$. For example, if we can write $X = \bigcup\_n X\_n$ where $X\_n$ is Borel with $0 < \mu(X\_n) < \infty$, set $f = \sum\_{n=1}^\infty \frac{1}{2^n \mu(X\_n)} 1\_{X\_n}$ and take $d\nu = f\,d\mu$. Then $\nu$ i... | 3 | https://mathoverflow.net/users/4832 | 193634 | 94,561 |
https://mathoverflow.net/questions/191252 | 5 | Consider the 2-category $[S, H] $ of 2-functors $S\to H $ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z $
For simplicity, let's assume that every 2-functor $S\to H $ admits a pointwise (right) Kan extension along $ T $ (for instance, we may assume that ... | https://mathoverflow.net/users/18017 | Kan extension pseudonatural transformations | After thinking (and discussing with professor Steve Lack), I am sure that this extension of the Kan extension does not exist. Such an extension would give a 2-functor $ RAN\_T : [S,H]\_{PS}\to [Z,H]\_{PS} $. And, of course, this 2-functor would take pseudonatural isomorphic diagrams to pseudonatural isomorphic diagrams... | 3 | https://mathoverflow.net/users/18017 | 193635 | 94,562 |
https://mathoverflow.net/questions/193628 | 4 | Given a directed complete graph on $n$ vertices, is there an efficient algorithm for computing its automorphism group? Is there a nontrivial upper bound on the order of its automorphism group? How about the isomorphism of two such graphs?
| https://mathoverflow.net/users/56827 | Automorphism group of directed complete graph | I'm guessing that by "directed complete graph" you want each edge directed in exactly one of the two possible ways. If so, you have a *tournament*. Automorphism and isomorphism for tournaments is potentially easier than for general directed graphs but nobody has proved that. One thing to note is that the automorphism g... | 6 | https://mathoverflow.net/users/9025 | 193642 | 94,565 |
https://mathoverflow.net/questions/193616 | 14 | Let $p$ be a positive integer (which is not a power of $2$), and suppose we want to generate a number uniformly randomly in the set $\{ 0, 1, \dots , p-1 \}$ (to emulate a dice roll). We are given access to an unbiased coin (where successive coin flips are iid Bernoulli random variables with probability $\frac{1}{2}$ o... | https://mathoverflow.net/users/39521 | How to roll a $p$ | Here is an optimal method that I think is equivalent to your greedy method.
We start with a constant random variable, say $0$. Inductively, after $n$ steps we have a random element uniformly distributed on a set of $k$ elements where $k \equiv 2^n \mod p$ and $k \lt p$. At each step, we flip the coin and produce a ra... | 6 | https://mathoverflow.net/users/2954 | 193643 | 94,566 |
https://mathoverflow.net/questions/193576 | 1 | I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to express the degree of $\mathscr{L}$ in terms of the degrees of $\mathscr{L}|\_{X\_i}$, where the $X\_i$ are the irreduci... | https://mathoverflow.net/users/53197 | Moving a divisor on a (reducible, non-reduced) curve | First of all, given where you need this, let's start by passing to the algebraic closure of $K$. That does not effect the degree of divisors and makes our lives easier.
I started writing this up and realized that while I think what I was saying could be carried out, doing it precisely is a bit (i.e., a lot) of work. ... | 5 | https://mathoverflow.net/users/10076 | 193657 | 94,573 |
https://mathoverflow.net/questions/193659 | 4 | Let $M$ be a manifold. Then $F(M,k)/\Sigma\_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group action of $\Sigma\_k$, the symmetric group on $k$ letters.
Let $p$ be an odd prime. What is the relation between
$H^\*(... | https://mathoverflow.net/users/41075 | Relation between cohomology of ordered and unordered configuration spaces | The group $\Sigma\_k$ acts freely on $F(M,k)$, this implies that the orbits coincide with the homotopy orbits. In general, if a group $G$ acts on a topological space $X$ you have a spectral sequence
$$H\_\*(G,H\_\*(X,K))\implies H\_\*(X\_{hG},K)$$
where $K$ is any commutative field. The left hand side is the homolo... | 6 | https://mathoverflow.net/users/10707 | 193669 | 94,578 |
https://mathoverflow.net/questions/193672 | 5 | M.Newman raised several questions in his 1957 [paper](http://dx.doi.org/10.1112/plms/s3-7.1.334) on modular forms.
>
> **Definition:** $H\_n$ is the subclass of all zero-free weakly modular forms of weight 0 on $\Gamma\_0(n)$, where $n$ is a **composite** number. Additionally, any $h\in H\_n$ is holomorphic at cusp... | https://mathoverflow.net/users/18286 | An old conjecture of M.Newman | First, I think your definition of $H\_{n}$ does not agree with Newman's definition. Newman says the following: "Let $H\_{n} \subset G\_{n}$ be the set of functions of $G\_{n}$ with non-negative valence at all parabolic points of $Q\_{n}$ other than $\tau = i\infty$." Here $G\_{n}$ is the set of modular functions that a... | 8 | https://mathoverflow.net/users/48142 | 193675 | 94,581 |
https://mathoverflow.net/questions/193653 | 10 | Is [Gabriel's theorem](https://en.wikipedia.org/wiki/Gabriel%27s_theorem) on the indecomposables of representations of quivers of finite type true over a commutative ring, i.e. not necessarily a field?
| https://mathoverflow.net/users/37002 | Gabriel's theorem over a commutative ring | Certainly it doesn't generalize in a really obvious way. One way to think about this is the following: Gabriel's theorem uses in really deep way that the category of quiver representations over a field is hereditary (in particular, $\mathrm{Ext}^2(M,N)=0$ for any representations). Over an arbitrary commutative ring, th... | 12 | https://mathoverflow.net/users/66 | 193681 | 94,582 |
https://mathoverflow.net/questions/193680 | -1 | We know that PA proves consistency of $I\Sigma\_{n}$ for any $n$. But does PA prove the sentence:
$\forall n (con(I\Sigma\_{n}))$?
| https://mathoverflow.net/users/65878 | Does PA prove a sentence asserting that all of I-sigma(n) theories are consistent? | No. Given any $\varphi\in \mathcal{L}\_{PA}$, if $PA\vdash \varphi$ then there exists $n$ such that $I\Sigma\_n\vdash \varphi$ (by finitarity of proofs). So $\forall n con(I\Sigma\_n)$ implies for any $n$ $I\Sigma\_n \not \vdash 0=1$ therefore $PA\not \vdash 0=1$, i.e. $con(PA).$
| 5 | https://mathoverflow.net/users/23835 | 193682 | 94,583 |
https://mathoverflow.net/questions/193678 | 9 | For any finite group $G$ and $n$ a divisor of $|G|$, consider the following subset of elements of "co-order" dividing $n$:
$$G(n) = \{ g \in G \mid g^{|G|/n} = 1 \}$$
* By a classical theorem of Frobenius, $\frac{|G|}{n} \mid |G(n)|$.
* A conjecture of Frobenius (which was proved via the classification of finite si... | https://mathoverflow.net/users/31469 | Condition for a certain subset being a subgroup | You have noted (accurately) that $G(p) =G$ unless $G$ has a cyclic Sylow $p$-subgroup. However, it is also clear that when $G$ has a cyclic Sylow $p$-subgroup and $G(p) \neq G,$
the group $G$ has a normal $p$-complement. For, otherwise, we have by Burnside's transfer theorem, there is a $p$-regular element $x \in N\_{G... | 5 | https://mathoverflow.net/users/14450 | 193683 | 94,584 |
https://mathoverflow.net/questions/193684 | 0 | Let $X$ be a topological space. Assume that $X$ admits a finite decomposition of the form $X=\bigsqcup\limits\_{i=1}^n V\_i$ where each $V\_i$ is homeomorphic (in the subspace topology of $X$) to an open cell of dimension $d\_i$. By convention, the open cell of dimension $0$ is a point. We are not assuming that the dis... | https://mathoverflow.net/users/11765 | Homology of a finite disjoint union of open cells | This can fail badly even if the closure of each cell only intersects lower-dimensional cells. For instance, take an open disk together with some subset of the boundary that has infinitely many connected components, and glue it to an open interval by identifying the boundary set with a homeomorphic subset of the interva... | 2 | https://mathoverflow.net/users/75 | 193687 | 94,587 |
https://mathoverflow.net/questions/193692 | 6 | I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts which demonstrate that early intuitionism (i.e. before the advent of modern intuitionistic set theories, which either d... | https://mathoverflow.net/users/37385 | Did Brouwer evade uncountability? | You are probably referring to Brouwer's considerations of the Creative Subject, which can be formulated mathematically as *Kripke's schema*. It implies that all subsets of $\mathbb{N}$ are countable, for example. I am having trouble finding good references, maybe these two will get you started:
* Göran Sundholm: ["Co... | 9 | https://mathoverflow.net/users/1176 | 193694 | 94,588 |
https://mathoverflow.net/questions/193691 | 6 | Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to (\Delta,\gamma)$$
and
$$\psi : (\mathbb{CP}^1 - \text{Int}\,\Delta,\gamma) \to (D^2,S^1)\,,$$
where $\text{Int}$ means int... | https://mathoverflow.net/users/39725 | Analytic diffeomorphisms of the circle from complex domains | This is the so-called *conformal welding problem*. One can ask the same question for any Jordan curve $\gamma$ (non necessarily analytic). With this domain of definition, your map $\Gamma$ is well-known to be neither injective nor surjective. There are even orientation-preserving homeomorphisms of the circle analytic e... | 9 | https://mathoverflow.net/users/1162 | 193695 | 94,589 |
https://mathoverflow.net/questions/193689 | 5 | Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?
| https://mathoverflow.net/users/3711 | Colimit density and monads | I guess I'll go out on a limb and assume arsmath means, when he says an object $c$ of $C$ is colimit-dense, that the full subcategory containing $c$ is a dense subcategory in the [usual sense](http://ncatlab.org/nlab/show/dense+functor).
There's some categorical lore which is helpful: a necessary condition for a cat... | 10 | https://mathoverflow.net/users/2926 | 193699 | 94,591 |
https://mathoverflow.net/questions/193664 | 14 | Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the [Peano existence theorem](http://en.wikipedia.org/wiki/Peano_existence_theorem). I give a counterexample [here](http://www.mathcounterexamples.net/co... | https://mathoverflow.net/users/41060 | ODE properties true in finite dimension but not in Banach spaces of infinite dimension | You can find some relevant information in Godunov, A. N. The Peano theorem in Banach spaces. Functional Anal. Appl. 9 (1975), no. 1, 53–55 and Pasika, E. E. An example of a first-order differential equation in a Hilbert space without continuous dependence of the solution on the initial condition. (Russian) Ukrain. Mat.... | 6 | https://mathoverflow.net/users/37822 | 193702 | 94,592 |
https://mathoverflow.net/questions/193701 | 4 | I would be grateful if you have an idea how to prove the following:
Let a finite $2$-group $P$ act on an elementary abelian group of odd order $N$ such that the centralizer $C\_P(N)=1$, and then form the semi-direct product $G:=NP$. An element $g\in G$ is said to be real if $g$ is $G$-conjugate to $g^{-1}$. Is it tru... | https://mathoverflow.net/users/64643 | A finite 2-group acts on an elementary abelian group of odd order | Note that your $P\_{2}$ is the Frattini subgroup $\Phi(P)$ since $P/P\_{2}$ is an elementary Abelian $2$-group while also $P\_{2} \leq \Phi(P)$ by the way you defined it.
Now $N = [N,P] \times C\_{N}(P)$ and $P$ acts faithfully on $[N,P]$, so we might as well suppose that $C\_{N}(P) =1$, and we do so. Now write $N = ... | 5 | https://mathoverflow.net/users/14450 | 193719 | 94,598 |
https://mathoverflow.net/questions/193510 | 6 | Let $A=\{a\_{ij}\}$ be a [normal](http://en.wikipedia.org/wiki/Normal_matrix) matrix such that $a\_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
\vdots & \ddots & a & b\\
b & \cdots & b & a\\
\end{pmatrix},\ where\ b>0.
$$
Does it follow that $A... | https://mathoverflow.net/users/29961 | Is a normal matrix satisfying $A^TA=...$ circulant? | **NO**.
Let $A$ satisfy the assumptions. If $P$ is a permutation matrix, then $B:=PA$ satisfies the assumptions too: on the one hand, we have $B^TB=A^TP^TPA=A^TA$. On the other hand (remark that the matrix in the question is permutation-invariant)
$$BB^T=PAA^TP^T=PA^TAP^T=A^TA=B^TB.$$
If the claim is true, we find ... | 5 | https://mathoverflow.net/users/8799 | 193723 | 94,599 |
https://mathoverflow.net/questions/193705 | 5 | Let $F$ be local field of characteristic zero and $\pi$ be a irreducible admissible representation of $GL\_n(F)$.
Let us consider its restriction to $GL\_{n-1}(F)$. Then I want to know whether $\pi|\_{GL\_{n-1}(F)}$ is completely reducible, namely, $\pi|\_{GL\_{n-1}(F)}=\oplus\_{i\in A}m\_i \cdot \tau\_i$ where $\tau... | https://mathoverflow.net/users/29422 | Is the restriction of a representation semisimple? | The anwser to your question is "no in general" since you already have a counter-example in the case $n=2$.
Take for $\pi$ an irreducible supercuspidal representation of ${\rm GL}(2,F)$. Then its restriction to ${\rm GL}(1,F)\simeq F^\times$ is known by the Kirillov model. This is the space $S(F^\times )$ of locally c... | 7 | https://mathoverflow.net/users/4767 | 193724 | 94,600 |
https://mathoverflow.net/questions/193725 | 6 | Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We know answer to this question when number of nodes is less than $12$. I did not see any research paper till now where auth... | https://mathoverflow.net/users/36977 | How many cospectral graphs available for a given number of nodes? | This appears an open problem according to a [paper](http://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n20/pdf).
>
> In connection with the graph isomorphism problem, it is of interest what fraction of all graphs is uniquely determined by its spectrum. Haemers onjectures that the fraction of
> g... | 9 | https://mathoverflow.net/users/12481 | 193727 | 94,601 |
https://mathoverflow.net/questions/193720 | 3 | Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states that
$$
u\_1(x) = -\frac1{2\pi}\int\_{\mathbb R^2}\frac{x\_2-y\_2}{|x-y|^2}\omega(y) \, dy
$$
and
$$
u\_2(x) = \frac1{2... | https://mathoverflow.net/users/46298 | Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$ | Remember how the classical Biot-Savart law is obtained : first, you take the curl of the defining equation $\omega = \nabla \wedge u$, and then, you use the fact that you know the solution of the Poisson equation in terms of the source. So, your problem boils down to computing explicitly some kernel.
Let's go through... | 2 | https://mathoverflow.net/users/62629 | 193728 | 94,602 |
https://mathoverflow.net/questions/193528 | 1 | A hypergraph is a pair $H=(V,E)$ such that $V$ is a (possibly infinite) set and $E\subseteq \mathcal{P}(V)$. $C\subseteq E$ is said to be a *cover* if $\bigcup C = V$ and $C$is *minimal* if $C'\subseteq C$ and $C'\neq C$ imply $\bigcup C'\neq V$.
We call $H=(V,E)$ a *flag complex* if the following conditions are met... | https://mathoverflow.net/users/8628 | Minimality condition in a certain class of hypergraphs |
>
> **NOTATION:** $\ \mathbb Z\_+\ $ is the set of all non-negative integers $\ 0\ 1\ \ldots$.
>
>
>
The answer to the ***Question*** is YES, i.e.
**THEOREM** There exists a flag complex $\ H=(V,E)\ $ and a cover $\ M\subseteq Max(E)\ $ such that for every cover $\ K\subseteq M\ $ we have that K is not minim... | 2 | https://mathoverflow.net/users/8385 | 193729 | 94,603 |
https://mathoverflow.net/questions/193722 | 30 | *Context:* I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about $\mathcal{D}^{b}(\mathrm{Rep}\ kQ)$, i.e. derived categories for representations of quivers, and am going on later to talk about ... | https://mathoverflow.net/users/13215 | Which properties of a variety are detected by its derived category of coherent sheaves? | One can see some initial answers in Huybrechts' book "Fourier-Mukai Transforms in Algebraic Geometry"
1. ([Huybrechts], Prop 4.1) If two smooth projective varieties are derived equivalent, then they have the same dimension.
2. ([Huybrechts], Prop 3.10) If $X$ is Noetherian, then $X$ is connected if and only if $D^b(C... | 6 | https://mathoverflow.net/users/49611 | 193737 | 94,606 |
https://mathoverflow.net/questions/193463 | 1 | I want to read the topic "spaces of probability measures on a Polish space and the convergence". What is the best reference for that ?
| https://mathoverflow.net/users/nan | spaces of probability measures on a Polish space and the convergence | In addition to the "canonical" references mentioned above, the book ["A Basic Course in Probability Theory"](http://books.google.com/books?id=clQQVdUNBmcC&printsec=frontcover#v=onepage&q&f=false) by Bhattacharya and Waymire has a very nice treatment of this topic in Chapter 5.
| 2 | https://mathoverflow.net/users/7410 | 193743 | 94,608 |
https://mathoverflow.net/questions/193741 | 5 | I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is usually formulated as the existence of open book decompositions. Can this result be upgraded to knots? If not, I would be i... | https://mathoverflow.net/users/27433 | Are there spaces in which there are no fibered knots? | The answer for knots is still "no", because if you have an open book decomposition with disconnected binding then you can stabilize it (see section 2 of Etnyre's [lecture notes](http://arxiv.org/abs/math/0409402)) by attaching a handle to the page with feet on different binding components. This reduces the number of bi... | 8 | https://mathoverflow.net/users/428 | 193748 | 94,610 |
https://mathoverflow.net/questions/193750 | 13 | Let $HH\_\*(A,N)$ (or $HH^\*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$.
I was reading nlab's entry on Hochschild cohomology and I saw this term:
>
> Hochschild homology object of any bimodule over an monoid in a
> monoidal ($\infty ,... | https://mathoverflow.net/users/42768 | Relationship between Hochschild cohomology and Drinfeld centers | Classically, Hochschild cohomology is an invariant defined for associative algebras while the Drinfeld centre is an invariant defined for monoidal categories. The latter is a categorification of the classical notion of centre of a monoid. In particular the Drinfeld centre of the category with a single object and endomo... | 13 | https://mathoverflow.net/users/2503 | 193754 | 94,613 |
https://mathoverflow.net/questions/193711 | 2 | Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus.
Let $\lambda,\mu,\nu$ be dominant characters of $T$.
Let $V(\lambda)$ be the irreducible representation of highest weight $\lambda$.
If $\lambda=\mu+\nu$, then we know tha... | https://mathoverflow.net/users/27398 | highest weight representations inside tensor product | To expand my comment (and Robert Bryant's) a little further, the main point here is that tensor products of *finite dimensional* irreducible representations behave nicely. In the equivalent Lie algebra setting, the finite dimensional irreducibles are those whose highest weights are *dominant integral* in the dual of a ... | 5 | https://mathoverflow.net/users/4231 | 193760 | 94,614 |
https://mathoverflow.net/questions/131111 | 20 | I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in absolute value).
$\hspace{1in}$[$\Im\;$](http://mathworld.wolfram.com/ImaginaryPart.html)[$\psi^{(-2)}$](http://en.wiki... | https://mathoverflow.net/users/33664 | Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$ | The following identities hold for all real $x>0$. Your equality is the second identity at $x=1$.
$$
-x + \frac{\pi}{12}(6x^2-1) + x\log{x} - x \log(2\pi) + 2\Im{\psi^{(-2)}(ix)} + \frac{1}{2\pi}\text{Li}\_2(e^{-2\pi x}) = 0,\\
x + \frac{\pi}{12}(6x^2-1) - x\log{x} - x \log(2\pi) + 2\Im{\psi^{(-2)}(1+ix)} + \frac{1}{2\p... | 4 | https://mathoverflow.net/users/8410 | 193784 | 94,620 |
https://mathoverflow.net/questions/193790 | 2 | Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$.
My question:
How to get the fiber sequence
$G\simeq \Omega BG\to E\to B\to BG$?
| https://mathoverflow.net/users/41075 | fiber sequence of principal bundles | The bundle $E \rightarrow B$ is classified by a map $B \rightarrow BG$, which means that we have a pullback
$$ \require{AMScd}
\begin{CD}
E @>{}>> EG\\
@VVV @VVV \\
B @>{}>> BG
\end{CD}
$$
As $EG$ is contractible, this gives us a fiber sequence $E \rightarrow B \rightarrow BG$. As $E \rightarrow B$ is a $G$-principal b... | 3 | https://mathoverflow.net/users/14233 | 193797 | 94,627 |
https://mathoverflow.net/questions/12923 | 19 | Are there Néron models for Abelian varieties over higher dimensional ($> 1$) base schemes $S$, let's say $S$ smooth, separated and of finite type over a field?
If not, under what additional conditions?
| https://mathoverflow.net/users/nan | Are there Néron models over higher dimensional base schemes? | This is really a comment in response to JBorger and Qing Liu's questions about existence of Néron models after blowing up or altering the base, but is too long for the comment box.
In general, Néron models do not exist over bases of dimension greater than 1, even allowing alterations of the base. This non-existence ... | 10 | https://mathoverflow.net/users/4710 | 193802 | 94,628 |
https://mathoverflow.net/questions/193512 | 2 | Let $\mu\_n, \mu$ be a sequence of probability measures on a Polish space $S$ and $\mu\_n', \mu'$ be some kind of extension of $\mu\_n, \mu$ on $\bar{S}$ such that all the boundary points of $S$ gets a zero measure. Now if
$$\int fd\mu\_n' \to \int fd\mu'$$ for every $f \in C(\bar{S})$
then I think this claims tha... | https://mathoverflow.net/users/nan | Problem on convergence in space of probability measures | It seems that in the paper, $S$ is a Polish space, which is homeomorphic to a dense subset of a compact metric space denoted by $\overline S$. Without loss of generality, we shall work with this subspace instead of the original $S$.
By portmanteau theorem, it suffices to show the wanted convergence when $g$ is bounde... | 2 | https://mathoverflow.net/users/17118 | 193811 | 94,633 |
https://mathoverflow.net/questions/188886 | 9 | Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix.
My current approach is to use the Cayley-Hamilton theorem:
$$\text{adj}(A) = -(A^{n-1} + c\_1A^{n-2}+\ldots +c\_{n-2}A+c\_{n-1}\text{I})
$$
where $$\lambda^n + c\_1\lambda^{n-1} + \ldots + c\_{n-1}\l... | https://mathoverflow.net/users/62593 | Compute adjugate matrix over commutative ring | A reference dated 2001-2003 is <http://www4.ncsu.edu/~kaltofen/bibliography/01/KaVi01.pdf>
Over a commutative ring, the complexities of the calculation of $\det(A)$ and $adj(A)$ are the same. With the standard multiplication, the authors obtain a bit-complexity in $n^{10/3+\epsilon}N^{1+\epsilon}$ where $N$ is the le... | 4 | https://mathoverflow.net/users/9091 | 193818 | 94,635 |
https://mathoverflow.net/questions/193813 | 0 | I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and he seems to be making a claim that if you have a simply connected Lie group then the derived subgroup is always closed. I was just wondering if this statement was true or not. When I consulted with my supervisor he seemed to think that ... | https://mathoverflow.net/users/15482 | query about Jacques Tits' "Homorphismes `abstraits' de groupes de Lie" | This is true. The point is that the derived subgroup is the integral subgroup of the derived Lie algebra, therefore is a Lie subgroup (hence closed) if the group is simply connected. See Bourbaki, *Lie Groups and Lie Algebras*, ch. III, § 9, no. 2, Corollary of Prop. 4.
| 2 | https://mathoverflow.net/users/40297 | 193820 | 94,637 |
https://mathoverflow.net/questions/193816 | 5 | I'm trying to come up with the largest family of sets that obeys the following properties:
Consider $X = \{1,\dots,n\}$ and take $\mathcal{F} \subset 2^X$ such that for any three subsets $A,B,C \in \mathcal{F}$ we have that (at least) two of the numbers $|A \cap B|, |B \cap C|, |A \cap C|$ are the same size.
There ... | https://mathoverflow.net/users/61129 | Common sizes of intersections | Consider $\ \binom A2\ $ for an arbitrary finite set $\ A.\ $ Then you get a proper family with $\ \binom n2\ $ members for $\ n:=|A|.\ $ You get an improvement for every $\ n>3$.
**EDIT:** Let me coin the name *"Three Is a Crowd"* (or *TIsC* for short) for the families introduced in the ***Question*** by **James K... | 8 | https://mathoverflow.net/users/8385 | 193822 | 94,638 |
https://mathoverflow.net/questions/193817 | 1 | Let $M$ be a compact Riemannian manifold without a boundary. Let $p\_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that
$$|p\_t(x,y)| \leq C$$
for all $x,y \in M$ and $t > 1$.
Thanks for any help.
| https://mathoverflow.net/users/62229 | Heat kernel upper bound on compact Riemannian manifold | Let $C$ be the maximum of $p\_1(x,y)$ over both $x$ and $y$ (this exists because $p$ is smooth). Then from the semigroup property one obtains
$$p\_{1+t}(x,y) = \int p\_1(x,z)p\_t(z,y)\mathrm{d}z \le C\int p\_t(z,y)\mathrm{d}z = C,$$
so that you get the upper bound you where looking for.
This argument shows that the m... | 4 | https://mathoverflow.net/users/7631 | 193826 | 94,639 |
https://mathoverflow.net/questions/193459 | 14 | A plane partition is a subset of $\mathbb Z\_{\geqslant0}^3$ s.t. if it contains $(i+1,j,k)$ or $(i,j+1,k)$ or $(i,j,k+1)$ it also contains $(i,j,k)$.
>
> What is the generating function $R(q)$ of (volumes of) plane partitions *not containing the cell $(1,1,1)$*?
>
>
>
Since a plane partition containing $(1,1,... | https://mathoverflow.net/users/1556 | Plane partitions not containing (1,1,1) | It is indeed possible to get the result from your "Context 2".
Correcting typos, we have
$$R(q)=1+\sum\_{i,j,k=0}^\infty\frac{(q)\_{i+j}(q)\_{i+k}(q)\_{j+k}}{(q)\_i^2(q)\_j^2(q)\_k^2}\,q^{i+j+k+1} $$
(where $(q)\_k=\prod\_{j=1}^k(1-q^j)$). By the $q$-binomial theorem,
$$\frac{(q)\_{i+j}}{(q)\_i(q)\_j}=\sum\_{x=0}^i\... | 8 | https://mathoverflow.net/users/10846 | 193828 | 94,641 |
https://mathoverflow.net/questions/193825 | 4 | Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm looking for a proof that only assumes undergraduate real analysis of one variable.
Thank you.
| https://mathoverflow.net/users/27832 | Elementary Proof of the Uniqueness of Smooth Structures on R | You can assume that you have an atlas where you have charts on countably many open intervals. Then you need to check that you can replace two adjacent intervals with one interval. Iterating ths, you can a diffeomorphism between the whole thing and an open subset of $\mathbb R$. Using some standard diffeomorphisms, you ... | 7 | https://mathoverflow.net/users/18060 | 193832 | 94,645 |
https://mathoverflow.net/questions/193821 | 1 | Let $X$ be an algebraic variety and $W:X\rightarrow\mathbb{A}^1$ a regular map, then the triangulated category of matrix factorizations $D^b(X,W)$ is defined to be
$D^b(X,W)=\bigsqcup\_{t\in\mathbb{A}^1}D^b\big(W^{-1}(t)\big)/\mathrm{Perf}\big(W^{-1}(t)\big)$
Let $Y=\mathrm{Crit}(W)$ be the critical locus of $W$, t... | https://mathoverflow.net/users/43423 | A question on triangulated category of matrix factorizations | If W has a single critical value, then Y = Crit(W) is a finite collection of points, hence $D^{b}(Y) \cong \oplus\_{p \in Crit(W)} D^{b}(\mathbb{C}-vect)$. This will usually not be equivalent to the triangulated category of matrix factorizations of W.
For example, say that $W = x^{3} + y^{3} + z^{3}$ on $\mathbb{C}^{... | 2 | https://mathoverflow.net/users/65933 | 193841 | 94,650 |
https://mathoverflow.net/questions/192209 | 0 | I am reading *The Laplacian on A Riemannian Manifold*. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation and Fourier series to solve heat equations.
In addition, *A First Course in Harmonic Analysis* generalizes the concept... | https://mathoverflow.net/users/36119 | Heat Equation on LCA Group | I suspect there is a reference that covers what one can say about the heat equation on a locally compact abelian group. I do not know what this would say in general - for instance what is the heat equation on $\mathbb{Z}/n\mathbb{Z}$?
But things are known at least in the context of compact Lie groups. A reference fo... | 1 | https://mathoverflow.net/users/38379 | 193850 | 94,654 |
https://mathoverflow.net/questions/11644 | 10 | A C\*-algebra is *Rickart* if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\in S\}$$ and $R(x)\equiv R(\{x\})$.
In:
>
> Kazuyuki Saito and J. D. Maitland Wright. [$C^∗$-algebras which are Gr... | https://mathoverflow.net/users/25122 | Saito-Wright definition of Rickart C*-algebras | They are equivalent. See [*On Defining AW*\*-*algebras and Rickart C*\*-*algebras* by K. Saitô, and J.D.M. Wright](http://arxiv.org/abs/1501.02434) (arXiv:1501.02434)
| 10 | https://mathoverflow.net/users/65943 | 193854 | 94,655 |
https://mathoverflow.net/questions/192333 | 9 | A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a compact manifold. This is a special class of non-compact manifolds where elliptic theory of differential operators is well-... | https://mathoverflow.net/users/40090 | Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$ | After looking carefully for this type of manifolds I found that these manifolds are called manifolds with edges. Consider the manifold $M=\Omega\times[0,\infty)\times\mathbb{R}^{N}$ the boundary of this manifold is a fibration over $\mathbb{R}^{N}$ i.e. $\partial M=\Omega \times \left\{ 0 \right\}\times\mathbb{R}^{N}$.... | 5 | https://mathoverflow.net/users/40090 | 193858 | 94,657 |
https://mathoverflow.net/questions/193849 | 3 | I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int\_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge 0$.}
\tag{1}
$$
I have a such function $f$. It has a complicated form involving some special functions. But it is an ... | https://mathoverflow.net/users/36814 | A calculus question related to the nonnegative definite functions | see [On positivity of Fourier transforms](http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ74_01%2FS0004972700047511a.pdf&code=3b5549fe1a8edfaf28e851f47f408cbf); one sufficient condition is that $f''(x)>0$ for all $x>0$.
there are other sufficient conditions, see for example [On the positivity of Fourier t... | 3 | https://mathoverflow.net/users/11260 | 193866 | 94,659 |
https://mathoverflow.net/questions/193863 | 2 | This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F\_1$ case of $q$-binomial coefficients and identities? Here $F\_1$ is the field of one element.
| https://mathoverflow.net/users/38468 | Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients? | Yes, it is not only fair, but also often very useful. The analogy is that sets are like vector spaces over $\mathbb F\_1$, so for example the analog of $\binom{n}{k}$ counting $k$-element subsets of an $n$-element set, is that $\binom{n}{k}\_q$ counts the number of $k$-dimensional subspaces of $\mathbb F\_q^n$.
In la... | 4 | https://mathoverflow.net/users/2384 | 193868 | 94,660 |
https://mathoverflow.net/questions/192255 | 13 | We are familiar with the expansion of the j-function,
$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$
and maybe with the approximation,
$$e^{\pi\sqrt{652}} = (640320^3+744)^2-2\cdot196883.999999999918\dots$$
but can somebody give a *short*, non-specialist explanation (if it is even possib... | https://mathoverflow.net/users/12905 | On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections | I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional universe behave like thermodynamic objects. They have temperature, and entropy (reflecting some hidden microstates), and the en... | 5 | https://mathoverflow.net/users/121 | 193877 | 94,665 |
https://mathoverflow.net/questions/193862 | 1 | Consider a proper flat morphism of $k$-schemes ($k$ is an algebraically closed field) $ f:X\longrightarrow\mathbb P^1\_k$ such that every fiber $X\_p$ for $p\in\mathbb P^1\_{\mathbb C}$ is a reduced connected projective *stable* curve of genus $g$.
>
> $X$ is said an **isotrivial family of curves** if there is an ... | https://mathoverflow.net/users/47136 | Isotriviality: two definitions | Let $f:X\to B$ be a stable curve of genus $g$ over a finite type integral $\mathbb C$-scheme.
Lemma 1. The moduli map $B\to \bar {M\_g}$ is constant if and only if there exists a dense open $U$ of $B$ such that all fibers of $X$ over $U$ are isomorphic.
Proof. (Compare with jmc's comment.) If $B\to \bar{M\_g}$ is c... | 2 | https://mathoverflow.net/users/4333 | 193885 | 94,669 |
https://mathoverflow.net/questions/193892 | 5 | On page 338, *A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary,* Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group $\pi$ acts freely and properly, then there is a spectral sequence, homological type, with $$ E^2\_{p,q}=H\_p(\pi,H\_q(X... | https://mathoverflow.net/users/41075 | cohomology version of Cartan-Leray spectral sequence that deduces cup product | The Leray-Serre spectral sequence in cohomology is multiplicative, meaning that there is a multiplication $E\_{r}^{pq} \otimes E\_r^{p'q'} \to E\_r^{p+p',q+q'}$ for each $r$, and the multiplication on the $E\_{r+1}$ page is induced by that on the $E\_r$ page. Note however that this does not actually give you the cup pr... | 5 | https://mathoverflow.net/users/1310 | 193895 | 94,673 |
https://mathoverflow.net/questions/193851 | 3 | I am reading [these notes](http://www.massey.math.neu.edu/Massey/Massey_preprints/perverse_milnor.pdf) on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function $f:X\to \mathbb C$, assuming $X$ is closed in an open subset $U\subset \mathbb C^N$ and $f$ it is... | https://mathoverflow.net/users/30827 | Relation between Milnor fiber and its restriction via vanishing cycles | There is no general relationship between the cohomology of the Milnor fibers of $\tilde f$ and $f$, in this setting. At a point $p\in X\_0$, the Milnor fiber of $f$ is the intersection of $X$ and the Milnor fiber of $\tilde f$. Hence, the topologies of the two fibers may be very different.
If $X$ is a hyperplane or h... | 7 | https://mathoverflow.net/users/49345 | 193900 | 94,676 |
https://mathoverflow.net/questions/193903 | 3 | Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^\*$ considering that $\mathbf{A}$ is diagonally dominant? (see <http://en.wikipedia.org/wiki/Diagonally_dominant_matrix> for further details about diagonally dominant matrices).
... | https://mathoverflow.net/users/11825 | Positive solutions of linear systems with a diagonally dominant matrix | If you make no hypotheses on the positivity/negativity of the entries of $A$ and $b$, nothing can be proved in general: to see this, just pre- or post-multiply by diagonal matrices with $\pm 1$ on the diagonal. This won't affect diagonal dominance, but will change the signs.
The most common case is the one in which y... | 2 | https://mathoverflow.net/users/1898 | 193907 | 94,679 |
https://mathoverflow.net/questions/193787 | 6 | **I.** *Theorem*: "If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\qquad\tag3$$
with $(u,v,w)$ as the symmetric polynomials $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k + (b + u)^k + (c + u)^k + (d - u)^k + (e - u)^k + (f - u)^k = \\ ... | https://mathoverflow.net/users/12905 | The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$ | For the equation,
\begin{equation\*}
y^2=(-7x^3-21c\_1x+c\_2)(7c\_1x+c\_2)
\end{equation\*}
where $c\_1=(n^2+3)/8$ and $c\_2=(n^3-9n)/2$, and assuming $n$ is rational, we have a rational point
$(0,c\_2)$, so the quartic is birationally equivalent to an elliptic curve.
Using an ancient Ms-Dos version of Derive, it is ... | 9 | https://mathoverflow.net/users/56617 | 193911 | 94,681 |
https://mathoverflow.net/questions/193880 | 5 | I was hoping for an explicit reference to the description of the mod 2 cohomology of a cyclic group $C\_{2r}=\langle t \rangle$ of even order in terms of Stiefel-Whitney classes, i.e., that
$H^\*(BZ\_{2r};\mathbb{F}\_2)=\mathbb{F}\_2[x,y]/(x^2-ry)$, where $x=w\_1(\chi)$ is the 1st Stiefel-Whitney class of the represe... | https://mathoverflow.net/users/65956 | Reference for Mod 2 cohomology of $BZ_{2r}$ in terms of Stiefel-Whitney Classes | It is true that statements like this are completely classical and yet hard to find stated just like that in the literature! If noone has a reference, I would suggest to go as follows :
-- the cohomology groups are given in the book by Cartan and Eilenberg (oldest reference I can think of!)
-- the cohomology ring is... | 2 | https://mathoverflow.net/users/37021 | 193918 | 94,685 |
https://mathoverflow.net/questions/193905 | 5 | $A=diag\{\lambda\_1,...,\lambda\_n\}$ and $\lambda\_i>0$, $B$ is a positive definite symmetric matrix and $max\{B\_{ij} \}\ll min\{\lambda\_i\}$
Note that the perturbative calculation of square root of $I+B$ is very easy, where $B$ is a small matrix.
How to calculate the square root of $A+B$ perturbatively?
| https://mathoverflow.net/users/43941 | How to calculate the square root of matrix $A+B$ perturbatively? | EDIT. Finally user34669 is right. We assume that $A$ is fixed and $B$ tends to $0$. The following $3$ lines are not correct because, in general, $\sqrt{XY}\not=\sqrt{X}\sqrt{Y}$.
"$\Delta=\sqrt{A+B}-\sqrt{A}\approx (A^{-1/2}B)/2$. We may also write $\Delta\approx (BA^{-1/2})/2$ or in a symmetric form $\Delta\approx (... | 5 | https://mathoverflow.net/users/9091 | 193921 | 94,686 |
https://mathoverflow.net/questions/193867 | 5 | Let $G = SL\_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S\_{w\_1}$ and $S^-\_{w\_2}$ denote the $B$ and $B^-$ orbit corresponding to $w\_1, w\_2 \in W$ respectively.
So how much is known ... | https://mathoverflow.net/users/7780 | Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$ | Everything good happens: they are smooth, irreducible, affine, of the expected dimension $\ell(w\_1)-\ell(w\_2)$; the standard reference is Kleiman 1973. Even their closures, "Richardson varieties", are nice (normal, C-M, rational singularities), which one can blame on similar results for Schubert varieties: nearby any... | 6 | https://mathoverflow.net/users/391 | 193930 | 94,690 |
https://mathoverflow.net/questions/193341 | 5 | Let $\mathcal{A}$ be an essential arrangement of hyperplanes in $\mathbb{R}^n$. Zaslavsky's theorem says that the number of regions of $\mathcal{A}$ is given by $r(\mathcal{A})=(-1)^n\chi\_{\mathcal{A}}(-1)$ and the number of bounded regions is given by $b(\mathcal{A})=(-1)^n\chi\_{\mathcal{A}}(1)$. (See Theorem 2.5 of... | https://mathoverflow.net/users/25028 | Number of regions of a hyperplane arrangement avoiding a generic hyperplane | Sorry to answer my own question, but I found a reference. See Theorem 3.1 of Greene and Zaslavsky's "On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs", Trans. AMS 1983, available for free at <http://www.ams.org/journals/tran/1983-2... | 0 | https://mathoverflow.net/users/25028 | 193935 | 94,692 |
https://mathoverflow.net/questions/193933 | 36 | One of my daughters was having a small programming exercise.
Let's consider following algorithm:
* Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$.
* Remove every $2$nd number.
* From the resulting list, remove every $3$rd number.
* From the resulting list, remove every $4$th number.
* ... Follow on until t... | https://mathoverflow.net/users/41060 | Why does this sequence converges to $\pi$? | This problem was studied first by the founder of sieve theory, Brun himself, who proved this asymptotic. For a fairly recent paper on this subject look at [Andersson](http://matwbn.icm.edu.pl/ksiazki/aa/aa85/aa8542.pdf) who gives more precise estimates for $u\_n$. The MO question [Sequences with integral means](https:/... | 34 | https://mathoverflow.net/users/38624 | 193938 | 94,693 |
https://mathoverflow.net/questions/193894 | 4 | Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all homeomorphisms $\phi$ with $\phi^{n}=Id$ for some natural number $n$. Assume that $g\in X$ belongs to $\bar{Y}-Y$. Some questions about su... | https://mathoverflow.net/users/36688 | The closure of all periodic homeomorphisms of circle | The answer to both questions 1 and 2 is false, due to the following example.
Consider $(n+1)$ consecutive intervals $I\_0,...,I\_n$ of lengths $1,2,4,...,2^{n-1}, 2^n$. Let the map $T\_n$ cyclically permute them in an affine way. In fact, if you choose $I\_j=[2^j,2^{j+1}]$, then
$$
T\_n(x)=\begin{cases}
2x, & x \not\i... | 11 | https://mathoverflow.net/users/31371 | 193947 | 94,697 |
https://mathoverflow.net/questions/30619 | 9 | I would be grateful to anyone who could provide me with some reference concerning the behavior of the sum of Log Normal random variables (need not independent) with respect to a Log Normal random variable.
It is obvious that such a sum has no reason to be log normal but what is less obvious to me is "is it very far f... | https://mathoverflow.net/users/2642 | Sum of Log Normal random variables | you can see links for independent and correlated case:
<http://airccj.org/CSCP/vol4/csit43104.pdf>
<http://airccj.org/CSCP/vol4/csit43105.pdf>
Thanks
| 4 | https://mathoverflow.net/users/65994 | 193953 | 94,699 |
https://mathoverflow.net/questions/193914 | 6 | The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki's language). Here there are at most two possible lengths of roots, relative to a euclidean metric arising from the nond... | https://mathoverflow.net/users/4231 | Convention about "long" roots for simple Lie algebras of types ADE? | Let us define a coroot $\alpha^\vee \in \mathfrak g$ to be `short' if the corresponding group homomorphism $SU(2)\to G$ generates $\pi\_3(G)$ (the homomorphism has Dynkin index $1$).
With the above definition, in the ADE case, all coroots are short.
Dually, it is then reasonable to call all roots `long' in the ADE ... | 8 | https://mathoverflow.net/users/5690 | 193954 | 94,700 |
https://mathoverflow.net/questions/193956 | 4 | I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what $a\_i$ in the asymptotic expansion of theorem 8.14.1 are. Any suggestions are welcomed.
| https://mathoverflow.net/users/18261 | Abstract stationary phase | the $a\_i$'s are complex coefficients in the asymptotic expansion $a(h)=\sum\_{i=0}^\infty a\_i h^i$; a more explicit expression is given on page 275-276 of [Guillemin and Sternberg:](http://www.math.harvard.edu/~shlomo/docs/Semi_Classical_Analysis_Start.pdf)
| 2 | https://mathoverflow.net/users/11260 | 193958 | 94,701 |
https://mathoverflow.net/questions/193955 | 6 | We know the [Shroeder-Bernstein](http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem) (SB) theorem can be proved in ZF, while the [Dual Schroeder-Bernstein](https://mathoverflow.net/questions/38771/dual-schroeder-bernstein-theorem) (DSB) can be proved in ZF+AC but not in ZF. Define as ISB the property ... | https://mathoverflow.net/users/2480 | Are two forms of the Dual Schroeder-Bernstein property equivalent? | What you call $\sf ISB$, is better known as $\sf WPP$ (Weak Partition Principle) and can be formulated as "*If there is a surjection from $X$ onto $Y$, then $X$ cannot have a strictly smaller cardinality than $Y$*" (alternatively, $|X|\leq^\*|Y|\leq|X|\rightarrow |X|=|Y|$ or $|X|\leq^\*|X|\rightarrow |X|\nless|Y|$).
... | 6 | https://mathoverflow.net/users/7206 | 193962 | 94,704 |
https://mathoverflow.net/questions/193716 | 13 | It is known that the smallest 4-chromatic graph of girth 4 is the [Grötzsch graph](http://en.wikipedia.org/wiki/Gr%C3%B6tzsch_graph) (11 vertices). What happens for girth 5?
The [Brinkmann graph](http://en.wikipedia.org/wiki/Brinkmann_graph) (21 vertices) has chromatic number 4, girth 5 and is 4-regular. Moreover it ... | https://mathoverflow.net/users/54628 | What is the smallest 4-chromatic graph of girth 5? | My computer tells me that there 195291625 graphs on 20 vertices with minimum degree at least 3 and maximum degree at most 6 and girth at least 5.
Sadly none of them have chromatic number 4, I just get 48 bipartite ones and the remainder being 3-chromatic.
Independent verification would be useful.
**Added** Here's... | 9 | https://mathoverflow.net/users/1492 | 193973 | 94,708 |
https://mathoverflow.net/questions/193943 | 11 | Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, [in this way](http://en.wikipedia.org/wiki/Turing_machine#Informal_description)) and then their definition is formalized (for example, [in this way](http://en.wikipedia.org/wiki/Turing_machine#Formal_definition)... | https://mathoverflow.net/users/nan | Define Turing machine with algebraic concepts/structures | Yes, there is now Pavlovic's characterization of Turing computability in terms of the monoidal computer, based on monoidal categories. <http://arxiv.org/abs/1208.5205>
| 8 | https://mathoverflow.net/users/4600 | 193983 | 94,712 |
https://mathoverflow.net/questions/193982 | 3 | In *Homology of $C\_{n+1}$-spaces, $n\geq 0$,* F.R. Cohen, *Lecture Notes in Mathematics, Vol. 533,* Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^\*(B(\mathbb{R}^{n+1},p),\mathbb{Z}\_p)$, for $p$ prime and $B(\mathbb{R}^{n+1},p)=F(\mathbb{R}^{n+1},p)/\Sigma\_p$, is obtained. A spectral sequence for fibratio... | https://mathoverflow.net/users/65800 | cohomology algebra of braid spaces, configuration spaces | For $p=2$ there are references from which you can extract the $\mathbb{Z}/2$ cohomology algebra of $B(M,2)$ for any closed manifold $M$. The answer in principle depends only on $H^\ast(M;\mathbb{Z}/2)$ as a module over the Steenrod algebra, together with the Stiefel-Whitney classes of $M$, but is usually not very pleas... | 3 | https://mathoverflow.net/users/8103 | 193987 | 94,714 |
https://mathoverflow.net/questions/193778 | 7 | I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:
>
> **Definition.** Let $H$ be an $\mathbb{R}$-Hilbert space and $A: D(A) \rightarrow H$ a diagonal linear operator on $H$ with point spectrum $\sigma\_P(A) \su... | https://mathoverflow.net/users/65910 | What is the idea behind interpolation spaces? | In the definition, "diagonal" does not make much sense, you probably mean "self-adjoint", although this can be relaxed to "m-sectorial" (powers of $A$ still make sense then). Also, it is irrelevant for the definition whether or not $A$ has some essential spectrum.
Anyway, the main reason why these interpolation spac... | 10 | https://mathoverflow.net/users/38566 | 193990 | 94,715 |
https://mathoverflow.net/questions/193993 | 3 | Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group?
Thanks in advance.
| https://mathoverflow.net/users/44172 | F.g group with infinite ends not Q.I to a free group | The free product $\mathbb{Z}^2 \ast \mathbb{Z}$ has infinitely-many ends, but is not quasi-isometric to a free group: furthermore, it is not hyperbolic since it has $\mathbb{Z}^2$ as a subgroup.
| 8 | https://mathoverflow.net/users/43559 | 194002 | 94,718 |
https://mathoverflow.net/questions/39676 | 6 | wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I got no key). (I guess the differential Bianchi identity is not due to Bianchi? So who did which?)
To clarify on the "v... | https://mathoverflow.net/users/9161 | Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa? | 1.)
The **algebraic Bianchi identity** must have been known to Riemann, as shown by his posthumously published prize essay to the Paris academy, "Commentatio mathematica ...". Cf. Spivak Vol. II which has an english translation and modern interpretation. (Interesting for German readers: Anmerkungen by Dedekind and Webe... | 1 | https://mathoverflow.net/users/9161 | 194004 | 94,719 |
https://mathoverflow.net/questions/194001 | 5 | This question follows up [a question I asked on math.SE](https://math.stackexchange.com/questions/1007928/embedding-g-in-a-zg-extension-of-operatornameautg). This is a refinement and a reference request.
>
> For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde G$) that c... | https://mathoverflow.net/users/12419 | Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$ | Yes, this situation has been studied. I think the first relevant reference is
S. Eilenberg and S. MacLane, Cohomology theory in abstract groups II, Ann. of Math. 48 (1947), 326-341.
The main result IIRC is that a group $H$ with $I={\rm Inn} G \le H \le A={\rm Aut} G$ determines an element $\sigma \in H^3(H/I,Z(G))... | 5 | https://mathoverflow.net/users/35840 | 194005 | 94,720 |
https://mathoverflow.net/questions/193999 | 1 | Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^\*$, such that we have $H = D^\*D$.
I will now consider the one-dimensional case on a compact set:
So assume that $H$ has a ground-state wavefunction $\psi\_0$ that does not have any nodes (this is possible for most boun... | https://mathoverflow.net/users/66012 | Witten index non-trivial in the context of Quantum Mechanics? | Consider the most classical example, $D=\frac{d}{dx}-x$, which is the creation operator. Note that $D$ is injective since, with $L^2$ norms and dot-products and say $u$ in the Schwartz class,
$$
\Vert{Du}\Vert^2\ge \langle[-x, \frac{d}{dx}] u, u\rangle=\Vert{u}\Vert^2.
$$
On the other hand, the range of $D$ has codimen... | 1 | https://mathoverflow.net/users/21907 | 194014 | 94,725 |
https://mathoverflow.net/questions/193994 | 4 | It is my understanding that that every $p$-adic representation of the absolute Galois group of a finite extension $K$ of $\mathbb{Q}\_p$ can be described in term of its associated $(\varphi,\Gamma)$-module over the Robba ring $\mathcal{R}\_K$.
This is due to Cherbonnier-Colmez and Kedlaya.
Do we have a similar pict... | https://mathoverflow.net/users/38010 | Robba ring and overconvergent (phi,Gamma)-modules | The theory of $(\varphi, \Gamma)$-modules works for $\mathbf{Z}\_p$-linear representations, but one has to use a slightly different coefficient ring $\mathbf{A}\_K$. This is explained very clearly in section 1.4 of the first Cherbonnier--Colmez paper ("Representations $p$-adiques surconvergentes") which you cite in you... | 5 | https://mathoverflow.net/users/2481 | 194017 | 94,726 |
https://mathoverflow.net/questions/194015 | 4 | Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it contain $0$. Nonetheless, it becomes an $R$-affine space in a natural way. Furthermore, for all natural numbers $n$ and $m$, t... | https://mathoverflow.net/users/26080 | Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where? | Let me work instead in the language of symmetric monoidal categories. There is a forgetful functor from categories to sets sending a category to its set of objects, and this functor has a left adjoint sending a set to the "discrete category" on that set, with no non-identity morphisms.
This left adjoint is symmetric... | 4 | https://mathoverflow.net/users/290 | 194020 | 94,727 |
https://mathoverflow.net/questions/194030 | 3 | Given structures $A\_i$ each of cardinality $<\kappa$ where $\kappa$ is a measurable cardinal, the cardinalities of the $A\_i$ are not uniformly bounded by a cardinal $\lambda <\kappa$, and $\mathcal{U}$ a $\kappa$-complete ultrafilter over $\kappa$, what is the cardinality of the ultraproduct $\prod A\_i/\mathcal{U}$?... | https://mathoverflow.net/users/38200 | Cardinality of an ultraproduct | It depends on the function $f : i \mapsto |A\_i|$. The ultraproduct $\prod A\_i/ U$ has the same cardinality as $\prod f(i)/U$. $f$ represents an ordinal $\alpha <j(\kappa)$ in the ultrapower $M$ of the universe $V$, where $j : V \to M$ is the canonical embedding. It's a standard fact that $2^\kappa < j(\kappa) < (2^\k... | 11 | https://mathoverflow.net/users/11145 | 194040 | 94,736 |
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