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https://mathoverflow.net/questions/95217 | 2 | For simplicity, assume everythings occur on a smooth projective variaty $X$.
Dual bundle of the given line bundle $\mathcal L$ is determined by $\mathcal L$ and $c\_1(\mathcal L)$.
$\mathcal L^\*= \mathcal L (-2c\_1(\mathcal L)) $
My question is, is their any similar relation between a vector bundle of rank >1 an... | https://mathoverflow.net/users/21902 | dual of locally free sheaf | If $E$ is locally free of rank 2 then $E^\* \cong E(-c\_1(E))$. The isomorphism is induced by the nondegenerate pairing $E\otimes E \to \Lambda^2E \cong O(c\_1(E))$. For higher rank one can check that in general $E^\*$ is not a twist of $E$.
| 2 | https://mathoverflow.net/users/4428 | 95223 | 55,789 |
https://mathoverflow.net/questions/95191 | 6 | Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense of being stochastically dominated by an exponential and dominating a similar exponential. Formally:
**Conjecture** Give... | https://mathoverflow.net/users/19774 | If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense | $\newcommand{\eps}{\varepsilon}
\newcommand{\E}{\mathbb{E}}
\renewcommand{\P}{\mathbb{P}}$
Fix $\eps>0$ and let $a=\E[X]$ and $b=\E[X-\eps \mid X>\eps]$. Let $p=\P(X > \eps)$. Then
$$ (1-p)(b+\eps)\le a \le (1-p)(b+\eps) + p \eps \ .$$
Solving, we get
$$ 1-\frac{a+\eps}{b+\eps}+\frac{\eps}{b+\eps} < p < 1-\frac{a}{b}... | 2 | https://mathoverflow.net/users/1061 | 95230 | 55,792 |
https://mathoverflow.net/questions/95228 | 2 | I wonder if Morse Theory on pseudo-Hermitian manifold is developed. For example, I wonder if the following statement on pseudo-Hermitian manifold, which is corresponding to the Riemannian case, is true:
Suppose $f$ is a smooth real-valued non-degenerate function on a pseudo-Hermitian manifold $(M,\theta)$, $a < b$, ... | https://mathoverflow.net/users/14579 | Morse Theory on pseudo-Hermitian manifold | Checking the various definition of "pseudo-Hermitian manifolds" in on-line papers, it seems they are in particular Finsler manifolds (there is a norm on the $T\_x M$, continuously depending on $x$ ). In this case, one may apply the usual Lusternik-Schnirel'man theory (see e.g. *Lusternik-Schnirelman theory on Banach ma... | 3 | https://mathoverflow.net/users/6101 | 95232 | 55,793 |
https://mathoverflow.net/questions/95142 | 3 | $\newcommand{\bR}{\mathbb{R}}$ Suppose that $w:\bR\to \bR$ is a nonnegative, even smooth function decaying fast at $\infty$, $w\in\mathscr{S}(\bR)$.
Define
$$s\_m(w)= \int\_{\bR^m} w(|x|) dx,\;\; d\_m(w):=\int\_{\bR^m} x\_i^2 w(|x|) dx,\;\;\forall i $$
$$ h\_m(w) = \int\_{\bR^m} x\_i^2x\_j^2 w(|x|) dx,\;\;\forall... | https://mathoverflow.net/users/20302 | Inequalities involving moments | As explained by Jochen Wengenroth, the answer is no in general.
Indeed, as you notice, the inequality $(A)$ is equivalent to an inequality about the moments $B\_k=\int\_0^\infty t^k w(t) dt$, namely $B\_{m+1}^2 \geq C(m) B\_{m-1} B\_{m+3}$ for some $C(m)$ which is bounded above and below (my rapid computations give $... | 3 | https://mathoverflow.net/users/10265 | 95241 | 55,795 |
https://mathoverflow.net/questions/95248 | 7 | I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that:
i) they vanish if the bundle of unit spheres $S(E)$ of the vector bundle $E$ (viewed as a real vector bundle if $E$ is complex) is stably fiberwise-homotopically... | https://mathoverflow.net/users/23229 | Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles | An evident general construction is to take any multiplicative cohomology theory $E^\*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of... | 5 | https://mathoverflow.net/users/14447 | 95253 | 55,800 |
https://mathoverflow.net/questions/95252 | 10 | Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi\_{A}(x)$ be its characteristic polynomial:
$$\chi\_{A}(-x) = \det (A+xI) = x^n + a\_{1}x^{n-1}+\ldots+a\_{n-1}x + a\_{n}.$$
Let $g(A) = ( a\... | https://mathoverflow.net/users/17896 | Integration on the space of symmetric matrices | There is a Weyl integration formula that deals with this problem. $\newcommand{\bR}{\mathbb{R}}$ Denote by $G\_n$ the space of symmetric $n\times n$ real matrices. It states that for any $O(n)$-invariant $h:G\_n\to \bR$ we have
$$ (2\pi)^{-\frac{n(n+1)}{4}} \int\_{G\_n} h(A) e^{-\frac{{\rm tr} A^2}{4}} dA $$
$$
=\f... | 18 | https://mathoverflow.net/users/20302 | 95256 | 55,802 |
https://mathoverflow.net/questions/95259 | 5 | Conjugacy classes Of elements of $SL\_{2}(\mathbb{Z})$ can be characterized by its trace. If $\mid T(m)\mid=1 $, then it is an element of finite order, if $\mid T(m)\mid =2$, then m is conjugate to a matrix up to sign,$ (1 n) (0 1)$ ( row notation) where $ n $ a natural number. in other case, there is an expansion in w... | https://mathoverflow.net/users/23233 | Centralizers of elements in $Sl_{2}(\mathbb{Z})$ | This is a simple computation (write down your matrix, and see what it means for a two-by-two matrix to commute with it, or to normalize the subgroup). When doing the computation, it is useful to remember that two matrices commute if (and only if) they have the same eigenvectors.
**EDIT** In view of the fancy-pants ar... | 4 | https://mathoverflow.net/users/11142 | 95263 | 55,804 |
https://mathoverflow.net/questions/95258 | 5 | Dear all,
I came across the above group (for any fixed odd prime $p$) and would need to know if it's large or not. It seems like it shouldn't be...
Best wishes,
Elisabeth
| https://mathoverflow.net/users/23232 | Is $G=\left<b_1, b_2, b_3 | [b_i^p, b_j^p]=1, \forall i,j=1,2,3\right>$ large? | The group $G$ maps onto the free product $C\_p\*C\_p\*C\_p$ of three cyclic groups of order $p$
(just send each $b\_i^p$ to $1$). This free product is virtually free, as a free product of finite groups (by Kurosh theorem the kernel of the homomorphism onto $C\_p \times C\_p \times C\_p$ is free) and is not virtually cy... | 17 | https://mathoverflow.net/users/7644 | 95270 | 55,807 |
https://mathoverflow.net/questions/95237 | 2 | Let $X$ be a compact Hausdorff space. A P-point in $X$ is a point which does not lie in the boundary of the cozero set of a continuous real-valued function on $X$.
Question. Suppose that $X$ has no isolated points. Is the set of P-points a meagre subset of $X$? Failing this, is the set of non-P-points at least a Bair... | https://mathoverflow.net/users/22260 | non-P-points a Baire space | The set of non-P-points is a Baire space:
I´ll work with the usual definition of P-point: $p \in X$ is a *P-point* if the intersection of countably many neighborhoods of $p$ is again a neighborhood of $p$. This is equivalent to the definition in the question for compact $X$.
Fix a compact $X$, let $P \subseteq X$ b... | 5 | https://mathoverflow.net/users/17836 | 95271 | 55,808 |
https://mathoverflow.net/questions/95281 | 3 | Let $(X,d)$ be a compact metric space of finite diameter. Recall that we can always compute the Hausdorff distance between subsets $A, B \subset X$ via
$$ d\_H(A,B) = \max\left[\sup\_{a\in A}\inf\_{b\in B}d(a,b), \sup\_{b\in B}\inf\_{a\in A}d(a,b)\right]$$
Here is a simple question:
>
> Is there a nice (pseudo-)... | https://mathoverflow.net/users/18263 | Metric on the set of Polyhedral Decompositions of a Compact Metric Space | How about we look at the set $S$ of pairs of points $(x,y)$ that are in the same cell of $P\_1$ but in different cells in $P\_2$ or vice-versa. Then the volume of this set might work as a metric if you have a measure on $X\times X$.
**Edit**. On second thought, this might be more the kind of idea you're looking for: ... | 2 | https://mathoverflow.net/users/20186 | 95283 | 55,815 |
https://mathoverflow.net/questions/87154 | 11 | Does anyone have a clue where the "h" came from?
| https://mathoverflow.net/users/17847 | History question - why h in the definition of derivative? | I think that use of $h$ in the definition of derivative is linked to the relationship between Calculus of Finite Difference and Differential Calculus.
In the book *Leçons sur le Calcul des Fonctions*, Councier, **1806**, [Lagrange](http://archive.org/stream/leonssurlecalcu01lagrgoog#page/n12/mode/2up):
1. Assigns ... | 16 | https://mathoverflow.net/users/22714 | 95290 | 55,817 |
https://mathoverflow.net/questions/95289 | 7 | Let $G$ be a finite $p$-group, $k$ a field of characteristic $p$ and let $I(G)=(g-1\mid g \in G)$ be the augmentation ideal of the group ring $k[G]$. It's known that $I(G)$ is nilpotent, i.e. there is $n> 0$ such that $I(G)^n=0$. Call the least such $n$ the nilpotency degree of $I(G)$ and denote it by $\operatorname{ni... | https://mathoverflow.net/users/10194 | Nilpotency degree of the augmentation ideal | Let $G$ be a finite $p$-group, $k$ a field of characteristic $p$, and define a series $\Gamma\_i$ of
subgroups of $G$ by letting $\Gamma\_1 = G$ and
$$ \Gamma\_{i+1} = \langle [ \Gamma\_i,G ], \Gamma ^p \_{\lceil
(i+1)/p \rceil} \rangle .$$ Then $\Gamma\_i / \Gamma\_{i+1}$ is
elementary abelian, so we can fix elements ... | 5 | https://mathoverflow.net/users/6481 | 95296 | 55,818 |
https://mathoverflow.net/questions/95300 | 3 | When is it true that a local ring is embedded into its henselization? In the cases when it is true, it means that to check that a presheaf is separated in Nisnevich topology is the same as to check that it is separated in Zariski topology, am I right?
| https://mathoverflow.net/users/23251 | Henselization of a local ring | If $A$ is a local ring, its henselization $i:A\hookrightarrow A^h$ is always injective and even faithfully flat.
The rings $A$ and $A^h$ have the same dimension and share many properties :
$A$ is noetherian (resp. reduced, resp. a normal domain) $\iff$ $A^h$ is noetherian (resp. reduced, resp. a normal domain).
... | 14 | https://mathoverflow.net/users/450 | 95305 | 55,822 |
https://mathoverflow.net/questions/86102 | 11 | In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to:
If $T$ is a quasi-nilpotent (i.e., has only $0$ in its spectrum) trace class operator on a Hilbert space, then $T$ is the limit, in the trace class norm, of finite rank nilpot... | https://mathoverflow.net/users/2554 | Quasi-nilpotent trace class operators as limits of nilpotents | This question appears as the third exercise in the third chapter of Kenneth Davidson's textbook *Nest Algebras*. Based on the material presented in the third chapter, here is the intended proof (with the numbers referencing the text).
Let $Q$ be a quasinilpotent trace class operator. Since $Q$ is compact, there exist... | 7 | https://mathoverflow.net/users/23241 | 95313 | 55,826 |
https://mathoverflow.net/questions/95267 | 3 | Usually we would define a "densely defined, closed operator" on a Banach space $E$ to be a linear map $T:D(T)\rightarrow E$, where $D(T)$ is a dense subspace of $E$, and the graph of $T$, $G(T)=\{ (x,T(x)) : x\in D(T) \}$ is closed in $E\times E$. Then we can define an adjoint by setting
`\[ D(T^*) = \{ f\in E^* : \exi... | https://mathoverflow.net/users/406 | Closed operators and duality | See \S 36 of
G. K\"{o}the: Topological Vector Spaces, Vol. 2
| 2 | https://mathoverflow.net/users/23007 | 95327 | 55,838 |
https://mathoverflow.net/questions/95338 | 1 | Let $P(x,A)$ be a stochastic kernel on a measurable space $(E,\mathcal E)$ and $G = \sum\limits\_0^\infty P^n$ be its potential kernel. A $\sigma$-finite measure $\phi$ is called the irreducibility measure if $\phi(A)>0$ implies $PG(x,A)>0$ for all $x\in E$.
In [Numellin's book](http://ebooks.cambridge.org/ebook.jsf... | https://mathoverflow.net/users/11768 | One point on $\phi$-irreducibility | Well, $\phi P(A) > 0$ implies $\exists B\subset E: \phi(B)>0$ and $P(y,A)>0\ \forall y\in B$. This implies by definition of $\phi$ that $PG(x,B) > 0$ for all $x\in E$, whence $P^2G(x,A) = \int P(y,A) PG(x,dy) > 0$ for all $x\in E$. But $PG(x,A) \ge P^2G(x,A)$.
| 1 | https://mathoverflow.net/users/18032 | 95340 | 55,843 |
https://mathoverflow.net/questions/83350 | 8 | It is well known that for ordinary categories, if $C$ has finite limits and $D$ is cocomplete, and
$A:C \to D $ is left-exact (i.e. preserves finite limits) then the left-Kan extension of $F$ along the Yoneda embedding $y:C \hookrightarrow Set^{C^{op}}$ is left-exact. I'm pretty sure this is still true for $\left(\inf... | https://mathoverflow.net/users/4528 | Flatness for infinity functors | For reference, at least when $D$ is an infinity topos, which I believe is probably necessary, this is Proposition 6.1.5.2 in HTT.
| 2 | https://mathoverflow.net/users/4528 | 95343 | 55,845 |
https://mathoverflow.net/questions/95328 | 1 | (Tried asking this on math stackexchange, but no takers so far.)
I'm trying to prove something about matroids, which I have reduced to the following question:
Suppose I have a matrix $M$ which is a direct sum of submatrices $M\_1,M\_2,…,M\_k.$ When do the invariant factors of the $\{M\_i\}$ partition the set of inv... | https://mathoverflow.net/users/4078 | When do the invariant factors of a direct sum of matrices correspond to those of its summands? | By definition $M-XI\_n$ is equivalent, in $M\_n(k[X])$ to ${\rm diag}(d\_1,\ldots,d\_n)$. Likewise, $M\_j-XI\_{n\_j}$ is equivalent, in $M\_n(k[X])$ to ${\rm diag}(d\_{j,1},\ldots,d\_{j,n\_j})$. Therefore $M-X\_I$ is equivalent to the diagonal matrix with diagonal entries the polynomials $d\_{j,s}$ where $1\le j\le k$ ... | 1 | https://mathoverflow.net/users/8799 | 95347 | 55,847 |
https://mathoverflow.net/questions/95349 | 6 | Let $U$ and $V$ be two non-principal ultrafilters over **N**, and $R\_1$ and $R\_2$ the non-standard extensions of **R** given by $R\_1=R^N/U$ and $R\_2=R^N/V$. Are they always isomorphic (I think not, but could not prove it), and, if not, what axioms must be added to ZFC to ensure they are (or is it a consequence of A... | https://mathoverflow.net/users/17164 | Isomorphisms between non-standard reals. | If the continuum hypothesis holds, then both of these ultrapowers are saturated models of cardinality $\omega\_1$, and one can see that they are isomorphic by a back-and-forth argument.
When the CH fails, then they need not be isomorphic.
| 9 | https://mathoverflow.net/users/1946 | 95350 | 55,848 |
https://mathoverflow.net/questions/95314 | 6 | Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$.
Then, we get local systems $R^i \phi\_\* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$
In this case,there are tons of beautiful constructions even in such an elementary level:
infinitesimal VHS, Mixed Hodge str... | https://mathoverflow.net/users/21902 | motivating examples of family of Hodge structure | I guess the implied question is: what are good references containing explicit calculations of variations of Hodge structure etc.? I might suggest taking a look at Griffiths' early pioneering papers "On periods of certain rational integrals I, II" Annals 1969, and
"Periods of integrals on algebraic manifolds III" IHES 1... | 3 | https://mathoverflow.net/users/4144 | 95351 | 55,849 |
https://mathoverflow.net/questions/95341 | 2 | I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).
Is it possible to extend the proof also to polynomials
$P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$ ?
Thanks!
| https://mathoverflow.net/users/23261 | connectedness of the complement of the zero set of a polynomial $P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$ | $SL(n,\mathbb{C})^N$ is a smooth and irreducible variety, and thus a manifold. The zero set $Z$ of a nonzero polynomial is a subvariety with $\dim\_{\mathbb{C}}(Z)\leq Nn^2-1$ and so $\dim\_{\mathbb{R}}(Z)\leq 2Nn^2-2$. If $a$ and $b$ are points in $Z^c$ then we can choose a smooth path $\mathbb{R}\to SL(n,\mathbb{C})^... | 4 | https://mathoverflow.net/users/10366 | 95359 | 55,852 |
https://mathoverflow.net/questions/95360 | 1 | I want to know whether ot not Proposition 1.12 (2) in p. 120 of the book [KN] is typo.
[KN] : Foundations of differential geometry Volume II - Kobayashi and Nomizu
1969 Interscience Publishing
Let $\phi$ be a two form associated to Hermitian inner product $h$ such that
$\phi(X,Y) = h(X,JY)$ where $J$ is an alm... | https://mathoverflow.net/users/36572 | Algebraic preliminaries of complex manifold | Some authors define exterior products as quotient spaces of tensor products, letting $\alpha\wedge \beta$ be the coset represented by $\alpha\otimes\beta$. Others define exterior products as subspaces of tensor products, letting $\alpha\wedge \beta$ be $\alpha\otimes\beta-\beta\otimes\alpha$. Or one half of that. You c... | 5 | https://mathoverflow.net/users/6666 | 95364 | 55,853 |
https://mathoverflow.net/questions/95353 | 1 | So in [another question of mine](https://mathoverflow.net/questions/94657/weil-kostant-integrality-result-as-stated-by-brylisnki) there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^\* \ \underrightarrow{d\ log} \ \underline{A}^1\_{M, \mathbb{C}}$ ... | https://mathoverflow.net/users/19926 | Is this Sequences of Complexes of Sheaves Exact? | This is not related to sheafification. The sheaf $\mathbb{C}^{\*}$ of locally constant functions on $M$ is already a sheaf, so sheafification will not change it.
This sequence is not an exact sequence of complexes but it is an *exact triangle* of complexes. That is - it is an exact sequence of complexes, up to quasi... | 7 | https://mathoverflow.net/users/439 | 95365 | 55,854 |
https://mathoverflow.net/questions/95306 | 4 | I am reading a wonderful article of Arnaud Beauville, called *La théorie de Hodge et quelques applications*
<http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf>
There is one place on page 12 that I can not understand. Beauville seem to claim the following:
**Claim.** Denote by $K$ the field of meromorphic functions... | https://mathoverflow.net/users/13441 | Discussion of Luroth's problem in an article of Beauville | (As requested, I'm turning my comment into an answer.)
The issue seems to be that there are differing definitions of unirational. One is that there exists a dominant rational map $\mathbf{P}^n \dashrightarrow X$ for *some* $n$. (That corresponds to the inclusion of function fields $K \subset \mathbb C(y\_1,...,y\_n)$... | 6 | https://mathoverflow.net/users/nan | 95367 | 55,855 |
https://mathoverflow.net/questions/95371 | 27 | For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. To be more specific, here are some straight questions.
1. What can Dolbeault tell us that de Rham can't?
2. Does there ... | https://mathoverflow.net/users/12653 | de Rham vs Dolbeault Cohomology | Let $\Omega^{p,q}(M)$ be the $C^{\infty}$ $(p,q)$-forms. One always has a double complex with $\Omega^{p,q}(M)$ in position $(p,q)$. The cohomology in the $q$ direction is Dolbeault cohomology, the cohomology of the total complex is deRham cohomology. (In each case, essentially by definition.) Whenever you have a doubl... | 39 | https://mathoverflow.net/users/297 | 95377 | 55,860 |
https://mathoverflow.net/questions/95372 | 5 | If $M$ is a smooth paracompact manifold, then what is the usual topology of $C^\infty\_c(M) $, i.e., the smooth function with compact support?
| https://mathoverflow.net/users/16323 | What is the usual topology of $C^\infty_c(M) $ | Topologizing $C\_c^\infty(M)\subseteq C^\infty(M)$ with the subspace topology (where $C^\infty(M)$ has the Whitney topology, generated by the seminorms $\left|\sup\_K\frac\partial{\partial x^\alpha}f\right|$), makes it a dense subspace; in particular it is not itself complete. So I wouldn't really call this the "usual ... | 24 | https://mathoverflow.net/users/35353 | 95381 | 55,863 |
https://mathoverflow.net/questions/95384 | 6 | The following questions popped out while I was preparing a course on profinite groups.
Closed subgroups of free profinite groups are not necessarily profinite free (e.g. the p-sylow subgroups, or the kernel of the map on the maximum p quotient, and many more other examples) thus the Nielsen-Schreier theorem fails in... | https://mathoverflow.net/users/2042 | Open subgroups of free profinite groups | Luis Ribes and I gave a proof of Nielsen-Schreier for open subgroups of free profinite groups that avoids using completions and the discrete Nielsen-Schreier theorem (well, actually our proof does both at once). We use wreath products instead. But we do use the Schreier basis. The ArXiv version is <http://arxiv.org/pdf... | 7 | https://mathoverflow.net/users/15934 | 95391 | 55,866 |
https://mathoverflow.net/questions/95385 | 7 | Let $X\subset \mathbb{P}^n$ be a projective algebraic variety with coordinate ring $R$.
$X$ is said to be arithmetically Cohen-Macaulay if $R$ is a Cohen-Macauly ring. A equivalent definition is that the natural morphism
$$R\to \oplus\_{k\in\mathbb{Z}}\mathcal{O}\_X(k)$$ is bijictive and $H^i(X,\mathcal{O}\_X(k))=0$ ... | https://mathoverflow.net/users/2348 | Equivalent definitions of arithmetically Cohen-Macaulay varieties | Here's a proof. Let me assume that $X$ is not zero dimensional, but instead is equidimensional. I'll actually prove the whole thing (more than you want), but I'll prove the Cohen-Macaulay thing first.
Now, either condition implies that $X$ itself is Cohen-Macaulay, let me explain this.
Set $S$ to denote the section... | 9 | https://mathoverflow.net/users/3521 | 95402 | 55,872 |
https://mathoverflow.net/questions/95387 | 5 | The system has the form
$$
(n-2)f\_n^{(1)}=n(f\_{n-1}^{(1)}+1),
$$
$$
(n-2\cdot 2) f\_n^{(2)}=n(f\_{n-1}^{(2)}+f\_{n-1}^{(1)}),
$$
$$
\ldots
$$
$$
(n-2k)f\_n^{(k)}=n(f\_{n-1}^{(k)}+f\_{n-1}^{(k-1)}),
$$
for the unknown sequences $f\_n^{(1)},f\_n^{(2)},\ldots,f\_n^{(k)}$ with the initial conditions $f^{(i)}\_k=0,$ fo... | https://mathoverflow.net/users/23022 | How to solve the system of recurrence equations | This is not a full solution, but it reduces the system to another recurrence relation which involves only coefficients. Let
$$f\_n^{(k)}=a\_0^k n(n-1)...(n-k+1)+a\_1^k n(n-1)...(n-k)+...+a\_k^k n(n-1)...(n-2k+1).$$
Then the above system reduces to
$$a\_i^k={a\_i^{k-1}\over i-k}$$
for $i < k$ and $a\_k^k$ is determined ... | 4 | https://mathoverflow.net/users/12120 | 95405 | 55,874 |
https://mathoverflow.net/questions/95408 | 16 | So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.
I was reading Silverman's "Arithmetic of Elliptic Curves" and it seems that most of its treatment is over fields.
My question is, does it make sense to define an elliptic curve over a ri... | https://mathoverflow.net/users/22095 | Elliptic Curves over Rings? | A commutative ring, yes. This is treated to some extent in Silverman's second book; for the more general story of "abelian schemes," which is what you're really after, I might look at Milne's articles in the volume *Arithmetic Geometry* edited by Cornell and Silverman.
As for noncommutative rings, I'm afraid I have n... | 18 | https://mathoverflow.net/users/431 | 95409 | 55,875 |
https://mathoverflow.net/questions/95120 | 8 | For given integers $m$,$n$, what bounds are known on the number of positive integers $x$ such that $m-x^2$ and $n-x^2$ are both perfect squares? In particular, is the number of such $x$ bounded above by a constant independent of $m,n$?
A search for $m,n \le 5000$ gives three pairs for which there are 4 such x: (1370,... | https://mathoverflow.net/users/21519 | Number of x such that $m-x^2$ and $n-x^2$ are both squares | In comments Aaron asked about an example of Kevin's construction with $10$ points.
In Kevin's comment the rational points come from the group law on the elliptic curve and $D$ is the lcm of the denominators.
Here is magma online code and example with $10$ points -- as Kevin wrote this way you **get as many solution... | 5 | https://mathoverflow.net/users/12481 | 95419 | 55,881 |
https://mathoverflow.net/questions/95423 | 4 | If $f$ is an orientation-preserving diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that:
(1)$f=\tilde f$ on a neighborhood of $K$.
(2)There is a bounded set $V$ and $\tilde f=id$ outside $V$?
| https://mathoverflow.net/users/16323 | The orientation-preserving diffeomorphism of $\mathbb R^n$ | Yes. You may use the fact that ***f*** is isotopic to the identity to see it as the time-1 flow of a time-dependent vector field. Then you just have to modify the vector field so that it vanishes outside from a large ball.
| 19 | https://mathoverflow.net/users/23288 | 95425 | 55,882 |
https://mathoverflow.net/questions/95412 | 7 | I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.
I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter is the same sort of object as an extender, since whereas an ultrafilter on an uncountable $\kappa$ is in $V\_{\kappa + 2... | https://mathoverflow.net/users/23167 | An Extender is a Generalization of an Ultrafilter? | My preferred account of extenders is the following: an elementary
embedding $j:V\to M$ is an extender embedding if every element of
$M$ can be expressed in the form $j(f)(\alpha)$, where
$f:\kappa\to V$ and $\alpha\lt j(\kappa)$.
(With this extender concept, it is obvious that any ultrapower by
a normal measure on $\... | 11 | https://mathoverflow.net/users/1946 | 95429 | 55,884 |
https://mathoverflow.net/questions/95437 | 6 | Consider the algorithm that goes over all proofs in peano arithmetic. Allegedly, for a given multivariate polynomial equation we should find a proof or disproof of existence of an integer solution. Therefore, given the negaative solution of Hilbert's 10th problem, there should be a polynomial for which we could neither... | https://mathoverflow.net/users/23290 | Is there a polynomial equation whose solution over the integers is independent of ZFC | Hi Daniel.
The point with Hilbert's 10th problem is that diophantine equations are complex enough to encode Turing machines and other complicated stuff that has some kind of no-can-do-theorem. In particular: To each Turing machine there is a polynomial $p\in\mathbb{Z}[X\_1,\ldots,X\_n]$ such that $p$ has a solution i... | 7 | https://mathoverflow.net/users/3041 | 95442 | 55,892 |
https://mathoverflow.net/questions/95438 | 4 | I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references but no specific place where this is established.
Any help will be most appreciated.
Aakumadula
| https://mathoverflow.net/users/23291 | Connection between Deligne-Mostow monodromy and Gassner representation at roots of unity of the pure braid group | See
* Michael Kapovich, John J. Millson, *Quantization of bending deformations of polygons in Euclidean space, hypergeometric integrals and the Gassner representation*, Canadian Mathematical Bulletin, 44(1) (2001) 36-60, doi:[10.4153/CMB-2001-006-3](https://doi.org/10.4153/CMB-2001-006-3), arXiv:[math/0002222](https:... | 4 | https://mathoverflow.net/users/21684 | 95446 | 55,895 |
https://mathoverflow.net/questions/95406 | 6 | I know that sometime ago Vopenka proved this:
*Theorem*: Assume there is a strongly compact cardinal. Then for any set $A$, $V \neq L(A)$.
Can we get by with a consistency-wise strictly weaker assumption?
Namely, call an uncountable cardinal $\kappa$ *strong* if for any ordinal $\gamma$, there is an elementary em... | https://mathoverflow.net/users/23167 | Vopenka's Theorem on L(A) and Large Cardinals from Weaker Assumptions? | The way to think about it is this.
Whenever a property in $V$ is witnessed in an absolute manner by the existence of a single set with certain properties, then we will be able to take that set, $A$, and form the universe $L(A)$, which will have the witness and thus verify that the property is true.
For example, i... | 6 | https://mathoverflow.net/users/1946 | 95451 | 55,897 |
https://mathoverflow.net/questions/95456 | 2 | Suppose we have a fiber square (pullback diagram) of schemes.
$$\begin{matrix}
\hspace{.7cm} X' \stackrel{v}{\longrightarrow} & X \cr
\downarrow^{g} & \downarrow^f \cr
\hspace{.7cm} Y' \stackrel{u}{\longrightarrow} & Y
\end{matrix}$$
It is well known that if $u$ is flat then $\mathbf{L}u^\ast \circ \mathbf{R}f\_\... | https://mathoverflow.net/users/23302 | pullback-pushforward isomorphism without derived functors | I don't think this is true in general.
For instance,
* Set $f : X \to Y = \mathbb{A}^2$ be the blowup of the origin.
* Set $Y' = \text{Spec} k$ and then fix $u : Y' \to Y$ to be the inclusion of the origin (note that $u$ is not flat).
It follows that $X'$ is the reduced exceptional divisor of $f$, in other word... | 3 | https://mathoverflow.net/users/3521 | 95464 | 55,902 |
https://mathoverflow.net/questions/95472 | 2 | I'm a non-mathematically inclined amateur whom is presently interested in Hamiltonian paths / Traveling Salesman problems.
I would like to request that someone be kind enough to tell me whether my understanding of the following sentence found in this research paper (<http://www.cs.technion.ac.il/~itai/publications/Al... | https://mathoverflow.net/users/22495 | Hamiltonian paths in grid graphs | 1. It means that there are some grid graphs for which there is some simple algorithm (or simply an existence/nonexistence proof), but this cannot be done for an arbitrary grid graph.
2.Traveling Salesman is not generally the same problem as hamiltonian path/circuit (usually there are costs involved).
1. The paper i... | 2 | https://mathoverflow.net/users/11142 | 95473 | 55,906 |
https://mathoverflow.net/questions/95490 | 5 | So I'm trying to understand a proof of Belyi's theorem from <http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf>
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t : X\rightarrow\mathbb{P}^1\_\mathbb{C}$ be a meromorphic function on $X$ thought of as a covering map of degree $n$... | https://mathoverflow.net/users/15242 | trying to understand the support of the sheaf of relative differentials | Ravi Vakil has a good explaination for the definition $\Delta^\*(I/I^2)$ in his notes. See his AG notes [here](http://math.stanford.edu/~vakil/0506-216/216class38.pdf) or [here](http://math.stanford.edu/~vakil/216blog/FOAGapr1312public.pdf) (chapter 23). In particular, I guess thinking about this locally makes it a lit... | 4 | https://mathoverflow.net/users/3996 | 95493 | 55,915 |
https://mathoverflow.net/questions/95495 | 4 | This question is related to [this](https://mathoverflow.net/questions/95384/open-subgroups-of-free-profinite-groups) mathoverflow question that I've asked recently.
The question rose while I prepared my lectures on Profinite Groups in an advance course in Tel Aviv University. Let $\mathcal{C}$ be a family of finite ... | https://mathoverflow.net/users/2042 | Open subgroups of free pro-C groups | No, it does not hold if $C$ is a class of finite nilpotent groups.
| 5 | https://mathoverflow.net/users/23311 | 95498 | 55,916 |
https://mathoverflow.net/questions/95475 | 1 | Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function space $Y^X$ (equipped with the product topology), where $Y$ is, say, some Banach space (if it helps simplify things, can... | https://mathoverflow.net/users/23306 | When LCS is isomorphic to subspace of some function space? | If I interpret the question correctly, Yaoliang would like to know which LCTVS are isomorphic to $\mathbb{C}^X$, where $X$ is a set (no topology), and $\mathbb{C}^X$ is given the product topology. If this is so, the answer is: very few. Actually, spaces of this form are fully determined by the cardinality of $X$.
Jus... | 3 | https://mathoverflow.net/users/23288 | 95504 | 55,917 |
https://mathoverflow.net/questions/95503 | 6 | Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$,
$$X\_m = {\frac{d}{d t}}\_{t=0} \Phi(m, t),$$
and by the symplectic form also with 1-forms
$$X \longleftrightarrow \; - \iota\_X \omega. $$
The following is well known (e. g. Abraham Marsden, Propos... | https://mathoverflow.net/users/17047 | Flow of a Hamiltonian vector field | The diffeomorphisms which are generated by (time-dependent) Hamiltonian vector fields are said to be Hamiltonian diffeomorphisms. Hamiltonian diffeomorphisms form a subgroup of the group of symplectic diffeomorphisms (actually, they are a subgroup of the connected component of the identity).
As you observe, locally ... | 14 | https://mathoverflow.net/users/23288 | 95507 | 55,918 |
https://mathoverflow.net/questions/95483 | 1 | I have a $5 \times 5$ parametric nonnegative matrix and want to show that it's stable (in the sense that all eigenvalues are positive). It is not symmetric, but I do know in advance that it has 5 real eigenvalues.
I have also shown that it's a $P$-matrix (i.e. all principal minors are positive). So now I can try to ... | https://mathoverflow.net/users/22051 | matrix stability criterion | Just a remark. Following your assumptions, your matrix is hyperbolic in the sense that you know that all eigenvalues are real-valued. I understand that it depends on some (real) parameters, then from a theorem due to Bronshtein, the characteristic roots are Lipschitz-continuous functions of the parameters.
Of course,... | 2 | https://mathoverflow.net/users/21907 | 95518 | 55,923 |
https://mathoverflow.net/questions/95519 | 10 | Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K\_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a reference in the literature for that? More generally:
**Question.** Is there any more concrete description of the Waldhausen'... | https://mathoverflow.net/users/2841 | K-Theory space of finite abelian groups | Everything is known. In fact as spectra we have canonically $K(\mathsf{finAb}) = \vee\_p K(\mathbb{F}\_p)$, and the spectra $K(\mathbb{F}\_p)$ are identified in the work of Quillen (see e.g. <http://www.math.uiuc.edu/K-theory/1006/>). In particular on $\pi\_0$ we find $K\_0(\mathsf{finAb}) = \oplus\_p \mathbb{Z}$, agre... | 11 | https://mathoverflow.net/users/3931 | 95520 | 55,924 |
https://mathoverflow.net/questions/95533 | 1 | give example of 2 languages A and B such that A and B are undecidable but there concatenation A.B is decidable.
| https://mathoverflow.net/users/23324 | example :concatenation of 2 undecidable language gives a decidable language | Let $A$ contain all even-length strings, plus an undecidable collection of odd-length strings. Let $B$ contain all odd-length strings, plus the empty string, plus an undecidable collection of even-length strings. So each is undecidable, but the concatenation $AB$ consists of all strings and hence is decidable.
| 4 | https://mathoverflow.net/users/1946 | 95534 | 55,931 |
https://mathoverflow.net/questions/95536 | 7 | Let $F$ be a field, and $E$ an extension field. Then two matrices in $GL\_n(F)$ are conjugate if and only if they are conjugate in $GL\_n(E)$. I'm curious whether the analogous fact holds for rings of integers.
Is the following true?
Two matrices in $GL\_n(\mathbb Z)$ are conjugate if and only if they are conjugat... | https://mathoverflow.net/users/15478 | Stable Conjugacy for Integer Matrices | Well, I think the answer is "no". Here's a construction: let R be the ring of integers of a real quadratic field K of class number > 1, let M be an invertible R-module of rank one which is not isomorphic to R, and let x be a fundamental unit in R. Then the action of x on M (viewed as a Z-module) determines a well-defin... | 11 | https://mathoverflow.net/users/3931 | 95539 | 55,933 |
https://mathoverflow.net/questions/95463 | 8 | Suppose that the ring homomorphism $R\rightarrow S$ is faithfully flat ($R$ and $S$ are Noetherian commutative rings). Let $A$ be an Artinian $R$-module. Do we have $0:\_S(A\bigotimes\_R S)=(0:\_RA)S$.
I know this the case when $A$ is Noehterian. Also by purity I know that $\big(0:\_S(A\bigotimes\_R S)\big)\bigcap R... | https://mathoverflow.net/users/23240 | Faithfully flat ring homomorphism and annihilator | No. Let $A=\mathbb{Z}[1/2]/\mathbb{Z}$. Let $R\to S$ be $\mathbb{Z}\to \mathbb{Z}\_2\times \mathbb{Z}\_3$, where $\mathbb{Z}\_n$ denotes $\mathbb{Z}[1/n]$.
Then $Ann\_{\mathbb{Z}}A=0$. But $(1,0)\in S$ kills $A\otimes\_{\mathbb{Z}}S$.
| 6 | https://mathoverflow.net/users/11218 | 95551 | 55,937 |
https://mathoverflow.net/questions/95514 | 17 | A useful tool in Algebraic Geometry is the incidence correspondence. Loosely speaking, it is a set of the form $$\{(p,X): p \text{ a fixed dimension subscheme of } Y \text{ and } X \text{ a specific type of subscheme} \}.$$ For example one could consider the incidence correspondence of lines in $\mathbb{P}^2$ with a po... | https://mathoverflow.net/users/12693 | Incidence Correspondence | There's several issues here.
(1) Is a given incidence correspondence actually a closed variety?
(2) What are explicit equations for the correspondence in the product of the relevant spaces?
(3) What are geometric properties of the incidence correspondence?
Most often, questions (1) and (3) are studied and littl... | 11 | https://mathoverflow.net/users/7399 | 95559 | 55,938 |
https://mathoverflow.net/questions/95549 | 0 | Hey all,
Not sure if this is a math problem or an algorithm problem - but hoping it has a math style answer.
If I have a directed graph I can find all the closed loops - easy. (Actually not at all easy but I guess it is at least well studied). What I want to do is to find the arcs such that if one extra edge was a... | https://mathoverflow.net/users/13467 | finding missing edge in DAG which, when added, would create the longest cycle | Actually your context tells us you are interested in a much simpler problem. Rather than finding the longest directed simple path in the graph, you are only interested in the longest directed simple path which ends at a specified vertex (the person who is using the app at that moment). So if you just construct a new gr... | 4 | https://mathoverflow.net/users/3669 | 95561 | 55,940 |
https://mathoverflow.net/questions/95555 | 8 | To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not yet known.
Is there any example of an elliptic curve of rank 2 such that $p$-primary components of Ш are trivial for $p... | https://mathoverflow.net/users/20754 | Order of Ш (Sha) | No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically by performing $p^n$-descent for higher and higher $n$, until the... | 17 | https://mathoverflow.net/users/35416 | 95562 | 55,941 |
https://mathoverflow.net/questions/95570 | 3 | Let $X$ be a Banach space with continuous dual space $X'$ with norm topology. Let us regard the following property of $X$:
Property: Any linear subset $A \subset X'$ that satisfies $\bigcap\_{\alpha\in A} \ker\alpha = \{0\}$ is dense in $X'$.
Any reflexive space $X$ has this property. Can you classify the spaces th... | https://mathoverflow.net/users/2082 | If $A \subset X'$ annihilates only $0$, then $A$ is dense | Let $\Phi\in X''$ be non-zero, let $\alpha\_0\in X'$ be such that $\Phi(\alpha\_0)=1$, and let $A$ be the collection of $\alpha\in X'$ with $\Phi(\alpha)=0$. Then $A$ is a subspace of $X'$, and any $\beta\in X'$ is equal to
`\[ \beta = \Phi(\beta)\alpha_0 + (\beta - \Phi(\beta)\alpha_0) \in \mathbb K\alpha_0 \oplus A, ... | 3 | https://mathoverflow.net/users/406 | 95581 | 55,946 |
https://mathoverflow.net/questions/92603 | 1 | For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so that $X1\_m = Y1\_m = 1\_m$, where $1\_m$ is the m-length column vector with all entries equal to 1.
I know that if $... | https://mathoverflow.net/users/5066 | A matrix diagonalization problem | **UPDATE** Sorry, previous version was wrong. For $n=m=2$, this computation shows that there is such a matrix. **FURHTER UPDATE** However, for $m=n=3$, it shows there isn't.
Taking $n=m=2$, we see that $X$ and $Y$ are of the form $\begin{pmatrix} x & 1-x \\ 1-x & x \end{pmatrix}$ and $\begin{pmatrix} y & 1-y \\ 1-y ... | 3 | https://mathoverflow.net/users/297 | 95587 | 55,949 |
https://mathoverflow.net/questions/93828 | 42 | Friedman, in \_[Lectures notes on enormous integers](https://u.osu.edu/friedman.8/files/2014/01/EnormousInt112201-167h1l6.pdf) shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation denotes iteration). But actually, using the fast-g... | https://mathoverflow.net/users/17164 | How large is TREE(3)? | I believe I can state with some confidence that TREE(3) is larger than $f\_{\vartheta (\Omega^{\omega}, 0)} (n(4))$, given a natural definition of $f$ up to $\vartheta (\Omega^{\omega}, 0)$. I can state with certainty that TREE(3) is larger than $H\_{\vartheta (\Omega^{\omega}, 0)} (n(4))$, where H is a certain version... | 29 | https://mathoverflow.net/users/23338 | 95588 | 55,950 |
https://mathoverflow.net/questions/95316 | 21 | In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|\_{\partial\Omega} \equiv 0$,
Neumann
$D\_{\nu} u|\_{\partial\Omega}\equiv 0$
or Robin (for $\alpha \in \mathbb{R}$)
$(D\_{\nu} u + \alpha u)|\_{\partial \Omega} \e... | https://mathoverflow.net/users/1540 | Physical interpretation of Robin boundary conditions | Here is an example where $\Omega = \mathbb{R}^3$. One way to establish dispersion for the wave equation involves taking a temporal Fourier transform. In order to do this one has to multiply by a cutoff function supported in $t \in [0,\infty)$.
You then get the equation
$(\Delta+\omega^2)\psi = F$
where $\psi$ is t... | 8 | https://mathoverflow.net/users/4345 | 95589 | 55,951 |
https://mathoverflow.net/questions/95556 | 0 | I have some question about detail on Chapter 9 Riemannian submersion in the book [Be].
[Be] A. L. Besse, Einstein Manifolds, Springer-Verlag
In the above book, in the proof of 9.24 Proposition (See p. 240) it is described that
$[X,U]$ is vertical.
Let me explain details : On $(M = B\times\_f F, g=\hat{g} + f... | https://mathoverflow.net/users/36572 | Is [U,X] vertical on a warped product ? | Fix arbitrary $\phi:B\to \mathbb R$ and let $f=\phi\circ\pi$.
It is sufficient to show that $[X,U]f=0$.
Since $U$ is vertical,
$Uf=0$.
SInce $X$ is a horizontal lift,
$Xf$ is constant on each fiber and therefore $U(Xf)=0$.
Hence
$$[X,U]f=X(Uf)-U(Xf)=0.$$
(In case if $X$ is only horizontal, the function $Xf$ is arbi... | 2 | https://mathoverflow.net/users/1441 | 95592 | 55,952 |
https://mathoverflow.net/questions/95261 | 25 | Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-Waldspurger's classic "Spectral decomposition and Eisenstein series" , the couple tell us :
"...in particular the centre... | https://mathoverflow.net/users/4245 | Questions about the Bernstein center of a $p$-adic reductive group | *Abstract Definition* .
Let $Rep(G)$ be the abelian category of smooth complex representations of our $p$-adic group $G$. The Bernstein center is the endomorphism ring $\mathfrak Z(G)$ of the identity functor of $Rep(G)$. So it acts on any smooth representation, and this action commutes with any $G$-morphism.
*As a... | 48 | https://mathoverflow.net/users/21999 | 95594 | 55,953 |
https://mathoverflow.net/questions/94404 | 10 | This is in some sense a follow-up to my [question](https://mathoverflow.net/questions/93935/submersions-of-closed-manifolds) on submersions.
Let $f\colon\thinspace M\to N$ be a generic smooth map between closed manifolds of dimensions $m$ and $n$. Assume that the codimension $k = n-m$ is *negative*.
I am interested... | https://mathoverflow.net/users/8103 | Singular fibers of generic smooth maps of negative codimension | Fibers of a generic smooth map are polyhedra (so in particular CW-complexes) by the triangulation conjecture of Thom, proved by Andrei Verona [[*Stratified Mappings - Structure and Triangulability*, Springer LNM vol. 1102](http://www.calstatela.edu/faculty/averona/Triang.html)]. I'm not sure that the full strength of t... | 8 | https://mathoverflow.net/users/10819 | 95597 | 55,956 |
https://mathoverflow.net/questions/95572 | 27 | Let $C\_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p\_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently proved the following:
Theorem: Every tame knot (or link) $K$ has a (not necessarily minimal) stick presentation whic... | https://mathoverflow.net/users/12301 | Does this knot invariant distinguish trefoil chiralities? | I'm very curious where this came up. In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives.
Let $a\_0,\dots,a\_{N-1}$ be the roots of unity that are visited along the knot, in (cyclic) order. Suppose we have a minimal representative for some no... | 20 | https://mathoverflow.net/users/5010 | 95598 | 55,957 |
https://mathoverflow.net/questions/95579 | 17 | Hi,
I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve.
It is known that for a general presheaf on a Grothendieck Topology, we must in general apply the plus construction twice to obtain a she... | https://mathoverflow.net/users/7607 | Sheafification - Why does twice suffice? | If I understand you correctly you believe in the counter example and just want an intuitive reason. Here is how I think about it (I am open for remarks!):
Definition:
Let $\mathscr A$ be a (small) category with arbitrary (small) limits. A presheaf $F$ on a (small) site $\mathscr C$ with values in $\mathscr A$ is a sh... | 11 | https://mathoverflow.net/users/18744 | 95599 | 55,958 |
https://mathoverflow.net/questions/95510 | 16 | Let $X$ be a smooth $\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an adjunction
$$\mathcal{D}:Z^0(\mathbf{dgMod}\_{\mathrm{qcoh}}(\Omega^{\cdot}\_{X/\mathbf{C}}))\leftrightarrows \mathbf{Cplx}(\mat... | https://mathoverflow.net/users/23316 | $\mathcal{D}$-quasi-isomorphisms and coherent $\Omega$-modules | Let $\mathcal M$ be a left $\mathcal D$-module over a smooth curve $X$. Then the Koszul duality functor assigns to $\mathcal M$ the DG-module $\Omega\_X\otimes\_{\mathcal O\_X}\mathcal M$ over the de Rham complex $\Omega\_X$. Viewed as a complex of sheaves, $\Omega\_X\otimes\_{\mathcal O\_X}\mathcal M$ is a two-term co... | 15 | https://mathoverflow.net/users/2106 | 95610 | 55,966 |
https://mathoverflow.net/questions/95622 | 3 | If $L/K$ is a Galois extension, then any prime $\mathfrak{p}$ of $K$ splits into a product ${\mathfrak P}\_1^e\cdots {\mathfrak P}\_g^e$ of primes in $L$, and the exponents on the primes are equal since the Galois group acts transitively on the primes dividing $\mathfrak{p}$.
Question: Is the converse true? Namely, ... | https://mathoverflow.net/users/21857 | Proof of a Simple Converse in Algebraic Number Theory | If you throw in the residue degrees as well, then you get Zev's question. Otherwise, the converse is not true. As an example, consider a quadratic field that has an unramified everywhere $A\_5$ Galois extension (see e.g. [this MO thread](https://mathoverflow.net/questions/44801/a-5-extension-of-number-fields-unramified... | 3 | https://mathoverflow.net/users/35416 | 95624 | 55,974 |
https://mathoverflow.net/questions/95632 | 11 | Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in the literature that talks about properties of this kind of matrices?
I have seen that there are some results for other... | https://mathoverflow.net/users/17246 | Maximum singular value of a random $\pm 1$ matrix | <http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf>
Theorem 5.39 (page 23) gives a non-asymptotic upper bound on the largest singular value
| 6 | https://mathoverflow.net/users/3669 | 95669 | 55,993 |
https://mathoverflow.net/questions/95670 | 0 | Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i\_{\*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times S^{2}$ and $Im i\_{\*}=0$, then
$$S^{1}\times S^{2}-i(T^{2})\cong K^{c}\cup (S^{1}\times D^{2})\sharp (S^{1}\times D^{2}... | https://mathoverflow.net/users/14500 | Embedding of $T^{2}$ on $S^{1}\times S^{2}$. | I remember this chestnut. Taken together, your two equations say that the torus $i(T)$ is contained in a 3-ball $B$ embedded in $S^2 \times S^1$: the first equation says that $i(T)$ is obtained up to isotopy by drilling a knotted hole through $B$ and taking the resulting boundary; the second equation says that $i(T)$ i... | 4 | https://mathoverflow.net/users/20787 | 95676 | 55,995 |
https://mathoverflow.net/questions/95403 | 2 | Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be the free groupoid on $C$ (i.e., fundamental groupoid of the geometric realization).
Is it possible to characterize (or express) connected... | https://mathoverflow.net/users/7494 | Connected covering spaces of a homotopy colimit | I have not quite thought this through but there are some facts which seem relevant:
1. For suitably nice $X$, the fundamental groupoid functor gives an equivalence of categories from covering maps over $X$, to covering groupoids over $\pi\_1X$; this is proved in my book "Topology and groupoids" (2006) and was in the ... | 1 | https://mathoverflow.net/users/19949 | 95679 | 55,997 |
https://mathoverflow.net/questions/95686 | 1 | Hi all.
I'm not even sure that this is a theorem, but a while ago I heard a topologist friend of mine (whom I haven't been able to reach) saying that given a continuous embedding $f:M^{n}\to N^{r}$ ($M$ and $N$ being topological manifolds), a sufficient condition for $f(M)$ to be (possibly) knotted in $N$ is that $r... | https://mathoverflow.net/users/10328 | Source on the proof that codimension 2 is sufficient for knottings? | This question is badly posed. If you have a not simply connected manifold (of any dimension $> 3$) then two connected closed curves are isotopic iff they are homotopic, hence you have as many isotopy classes of curves as the conjugacy classes in the fundamental group.
Also, following your comment, in codimension two (w... | 4 | https://mathoverflow.net/users/23193 | 95690 | 56,004 |
https://mathoverflow.net/questions/95692 | 12 | Hi,
For a group $G$, we say that $x\in G$ is a commutator if there exists $a,b \in G$ such that $x=a^{1}b^{-1}ab$, and we say that $x$ is a **non-commutator** if there is no $a,b \in G$ such that $x=a^{1}b^{-1}ab$.
Does there exist a non abelian simple group $G$ (finite or not) such that $G$ has at least one non-co... | https://mathoverflow.net/users/3958 | Non-commutator in simple group? | For infinite groups, you may find examples here:
<http://arxiv.org/abs/arXiv:0909.2294>
In fact, in the reference above you may find examples of infinite simple groups with infinite commutator width. In other words, examples of simple groups $G$ such that for every $n\in\mathbb{N}$ there exists an element $g\in G$ ... | 18 | https://mathoverflow.net/users/6206 | 95696 | 56,006 |
https://mathoverflow.net/questions/95688 | 8 | The following sentence is quoted from the paper *ON THE COHOMOLOGY OF SPLIT EXTENSIONS* by D. J. BENSON AND M. FESHBACH:
**In general, the differentials in the Lyndon-Hochschild-Serre spectral sequence
for a group extension are difficult to understand. In the case of a central extension,
the $E\_2$ page is easy to ca... | https://mathoverflow.net/users/23370 | Transgressions commute with the Steenrod operations on the base and fiber in a central group extension? | The statement in question refers to the Kudo-Serre transgression theorem: If $E\_r$ is, for example, the LHS spectral sequence of a group extension and $x \in E\_{2k+1}^{0,2k}$ is transgressive with $d\_{2k+1}(x)=y$, then $$d\_i(x^p)=0\;\;(i=2k+1,...,2kp)\quad \text{and }\;\; d\_{2kp+1}(x^p)=P^ky$$ where $P^k$ denotes ... | 8 | https://mathoverflow.net/users/10194 | 95700 | 56,008 |
https://mathoverflow.net/questions/95705 | 10 | Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff $s\supseteq t$, $A\subseteq B$ and $s\setminus t\subseteq B$.
Laver's forcing, on the other hand, consists of conditions $p... | https://mathoverflow.net/users/13059 | Difference between Laver's and Mathias's forcing | In the Mathias extension (extension by a single Mathias real) the collection $V\cap 2^\omega$ (the reals from the ground model) will have measure zero. This is not the case for the Laver extension.
Both facts can be found in the Bartoszynski-Judah book Set Theory: On the Structure of the Real Line. The former fact is... | 12 | https://mathoverflow.net/users/2436 | 95707 | 56,012 |
https://mathoverflow.net/questions/95699 | 5 | Let $A$ be something interesting like the C\*-algebra of compact operators on a separable infinite-dimensional Hilbert space and let $A^1$ be an ultrapower of $A$. Then $A^1$ is a primitive C\*-algebra strictly containing $A$ (Ge and Hadwin). Now let $A^2$ be an ultrapower of $A^1$. Then $A^2$ is a primitive C\*-algebr... | https://mathoverflow.net/users/22260 | iterating ultrapowers of C*-algebras | In the C\*-algebra ultrapower construction, instead of identifying two sequences if they agree on a set in the ultrafilter, you identify two sequences if their difference goes to zero along the ultrafilter. You also throw out those sequences whose norm goes to infinity along the ultrafilter.
I think the simple answer... | 2 | https://mathoverflow.net/users/23141 | 95713 | 56,014 |
https://mathoverflow.net/questions/95709 | 6 | Let $F$ be a fixed free group of finite rank. If $H$ is a finitely generated subgroup of $F$ and $A$ is a basis for $F$, then we can form the Stallings graph $\Gamma\_A(H)$ for $H$. It is the unique smallest ($|A|$-labelled) subgraph of the covering space of a bouquet of $A$-circles corresponding to $H$ that contains t... | https://mathoverflow.net/users/15934 | Are all free factors of finitely generated subgroups of free groups geometric? | If you take a free group on two generators to be $F=\langle a,b\rangle$, and take an index 2 subgroup to be $K$, then the subgroup $H$ generated by $[a,b]$ is not a geometric free factor in $K$ for any choice of generators for $F$ (but is a free factor). One may verify this for all three 2-generator subgroups of $F$ wi... | 11 | https://mathoverflow.net/users/1345 | 95716 | 56,016 |
https://mathoverflow.net/questions/95724 | 0 |
>
> **Possible Duplicate:**
>
> [Hyperbolic surfaces](https://mathoverflow.net/questions/95640/hyperbolic-surfaces)
>
>
>
Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance realizing geodesic in S?
(1) for S ... | https://mathoverflow.net/users/23358 | Uniqueness of distance realizing geodesic in hyperbolic surface. | For the pants, yes. In general, no. To prove this for the pants, classify *all* geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first one that came to my mind was taking a double cover.
EDIT - I see that this is a near-duplicate of a closed question. You... | 1 | https://mathoverflow.net/users/1650 | 95728 | 56,020 |
https://mathoverflow.net/questions/95537 | 31 | In their recent paper [The Quantum State Can Be Interpreted Statistically](http://arxiv.org/abs/1201.6554), Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for the foundations of quantum mechanics. In the hopes of getting some non-quantum math f... | https://mathoverflow.net/users/2575 | "psi-epistemic theories" in 3 or more dimensions | Since George Lowther seems to have a lot of late nights, I decided to express my gratitude to him by writing up his lovely answer myself and thereby saving him the trouble.
The answer to my (and Lewis et al.'s) question is that yes, maximally-nontrivial ψ-epistemic theories do exist for every finite dimension $d$.
... | 12 | https://mathoverflow.net/users/2575 | 95735 | 56,024 |
https://mathoverflow.net/questions/95747 | 5 | A very nice feature of W\*-algebras is the following:
once you have an element $a$ of a W\*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$.
It seems that it carries over to AW\*-algebras without pain. This is simply because (A)W\*-algebras have lots of projections, unlike general C\*-algebras.
... | https://mathoverflow.net/users/20746 | Polar decomposition in C*-algebras | There exist further generalizations but they do not go very far from AW\*-algebras. See
[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-137/issue-1/Three-quavers-on-unitary-elements-in-C-algebras/pjm/1102650543.full)
and
<http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1995_3... | 3 | https://mathoverflow.net/users/12205 | 95756 | 56,034 |
https://mathoverflow.net/questions/95748 | 6 | We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a compactification $Y$ that $Y-X$ has only $n$ elements, we say that $Y$ is the $n$ point compactification of $X$. I have so... | https://mathoverflow.net/users/23317 | A question about some special compactifications of $\mathbb{R}$ | Suppose a space $X$ has $k$ [ends](https://en.wikipedia.org/wiki/End_%28topology%29) and an $n$ point compactification. We can show that $k\geq n$. Indeed, there are disjoint neighborhoods $A\_1,\dots,A\_n$ of each of these points at infinity. Now let $Y$ be the complement of $\cup A\_i$. Then $Y$ is compact so under m... | 10 | https://mathoverflow.net/users/2384 | 95759 | 56,036 |
https://mathoverflow.net/questions/95751 | 9 | Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to second order logic, by which I mean, once we replace the induction schema with a single, second-order axiom. Are there n... | https://mathoverflow.net/users/7209 | Are there any complete, first-order and unstable theories which have non-categorical second-order formulations? | You had asked for theories that do not have a categorical second-order completion, but here is an answer to the dual question:
**Theorem**. Every consistent first order theory $T$ with an infinite model has a second-order completion that is not categorical.
Proof. Consider all the models of $T$. By the Lowenheim-S... | 10 | https://mathoverflow.net/users/1946 | 95761 | 56,038 |
https://mathoverflow.net/questions/95743 | 46 | We know that
$$
\sum\_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c\_1 + O(1/x)
$$
where $c\_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum\_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x).
$$
Thus both these divergent series grow at the same rate. Mertens' theorem was proved withou... | https://mathoverflow.net/users/23388 | Why could Mertens not prove the prime number theorem? | Here is a heuristic that I plan to make into a blog post some day. Suppose that there were only finitely many primes with first digit $9$. Is your estimate good enough to see that?
To be more precise, suppose that there were no primes between $9 \times 10^k$ and $10^{k+1}$ for all sufficiently large $k$. And suppose ... | 82 | https://mathoverflow.net/users/297 | 95763 | 56,040 |
https://mathoverflow.net/questions/95742 | 4 | I came across this complex function in my work $f(z)=\frac{e^z-1}{z}$. Is there a reference to $f(z)$? What is its name in the literature? More importantly, is the function inversible? If so, what is $f^{-1}(z)$? Thanks.
| https://mathoverflow.net/users/22516 | What is the name of $\frac{e^z-1}{z}$ and how to invert it? | Let $y=(e^z-1)/z$ and $x=-1/y$. Then $xe^x=(x-z)e^{x-z}$. Hence
$$x-z=W(xe^x).$$
Here W is an appropriately chosen branch of the Lambert function (ProductLog[-1,.] in Mathematica).
| 12 | https://mathoverflow.net/users/12120 | 95775 | 56,048 |
https://mathoverflow.net/questions/95779 | 3 | Hello all,
Does anyone have an example in mind of a ring $R$ for which $R^n\cong R^m$ *as $R,R$ bimodules* for some positive integers $n\neq m$?
I would be a little surprised if someone showed no such thing could exist, but that would also be a welcome answer.
Thanks!
P.S.: Naturally such a ring could not have... | https://mathoverflow.net/users/19965 | Bimodule version of IBN | No. For a bimodule $M$ let $Z(M) = \{ m : rm = mr \forall r \in R \}$. Then $Z(R^n) \cong Z(R)^n$, so if $R^n \cong R^m$ as $(R, R)$-bimodules then $Z(R)^n \cong Z(R)^m$ as $Z(R)$-modules, and commutative rings satisfy IBN.
| 9 | https://mathoverflow.net/users/290 | 95780 | 56,049 |
https://mathoverflow.net/questions/95523 | 2 | Let $X$,$Y$ be two stochastic variables with probability distribution $\rho(X,Y)$. The
mutual information, $I(X;Y)$, represents the information shared by the two variables. This intuitive interpretation of $I(X;Y)$ suggests the following question. Given $\rho(X,Y)$, is there a process
$X\rightarrow Z\rightarrow Y$ so t... | https://mathoverflow.net/users/4274 | mutual information and minimal communication required for generating correlation | I have found this paper that is closely related with my question:
"The communication complexity of Correlation" by P. Harsha, R. Jain, D. McAllester, J. Radhakrishnan, IEEE Transactions on Information Theory 56, 438 (2010).
In this paper, it is shown that the communication cost required for generating
correlation c... | 2 | https://mathoverflow.net/users/4274 | 95793 | 56,058 |
https://mathoverflow.net/questions/95134 | 0 | Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the ***Voronoi region*** of the Leech lattice.
Best,
Farzad
| https://mathoverflow.net/users/15385 | Dither in Leech lattice quantization! | Hi all,
I refer readers to the paper, "Dithering in lattice quantization"by A. Kirac, in this paper he explains how to generate dither signals to have quantization error that is uniformly distributed on the voronoi cell.
Best,
Farzad
| 1 | https://mathoverflow.net/users/15385 | 95799 | 56,061 |
https://mathoverflow.net/questions/95790 | 4 | Let $f$ and $g$ be two cuspforms on $\Gamma \backslash \mathbb{H}$. They could be Maass cuspforms, or holomorphic modular forms. Let us say that they are holomorphic and also that $\Gamma = \operatorname{SL}\_2(\mathbb{Z})$ for simplicity. The product $f \overline g$ is not necessarily a cuspidal function on $\Gamma \b... | https://mathoverflow.net/users/16389 | Product of two cuspforms is not a cuspform | I don't think it's possible to find a "nice" (say, smooth) function $f \in L\_2(\Gamma \backslash \mathbb{H})$ such that $(1) \int\_0^{1} f(x+iy) dx = 0$ for all $y > 0$ and $\lim\_{y\rightarrow \infty} f(x+iy) \neq 0$. This may be total overkill, but consider the spectral decomposition of such an $f$, namely
$$(2) \qq... | 5 | https://mathoverflow.net/users/2627 | 95803 | 56,062 |
https://mathoverflow.net/questions/95795 | 3 | I'm trying to understand K\"ahler differentials for CDGAs (commutative differential graded algebras). A few minor things have been confusing me all afternoon. Pointers and references are welcome.
Let $A$ be a CDGA over a field of characteristic zero with differential of degree $1$ and with no cohomology in positive d... | https://mathoverflow.net/users/21028 | Sign convention for derivations in CDGAs | Yes, left modules for a graded-commutative ring can be routinely converted to right modules by defining $ma=(-1)^{|m||a|}am$, and then the "right" definition of degree zero derivation is
$\delta(ab)=a\delta(b)+\delta(a)b$.
More generally
$\delta(ab)=(-1)^{|\delta||a|}a\delta(b)+\delta(a)b$.
There really isn't a... | 4 | https://mathoverflow.net/users/6666 | 95807 | 56,065 |
https://mathoverflow.net/questions/95794 | 4 | Suppose we have a (possibly infinite) group given by generators and relations. One way to prove some inequality is to construct the representation of the group and show inequality in the representation. Is there some other methods?
| https://mathoverflow.net/users/8381 | Ways to prove an inequality in groups | The most general way is to use the complete presentations. See the [book by Sims, for example.](http://books.google.com/books?hl=en&lr=&id=k6joymrqQqMC&oi=fnd&pg=PP1&dq=complete+rewriting+system+in+finitely+presented+groups&ots=1_F416EoNn&sig=RRlKKyI9GGJIUWE20xyGGAsy_MY#v=onepage&q=complete%20rewriting%20system%20in%2... | 6 | https://mathoverflow.net/users/nan | 95816 | 56,070 |
https://mathoverflow.net/questions/95785 | 7 | We know that the space $\mathbb{R}$ has compactifications with one point remainder, and two point remainder. but there is no compactification of $\mathbb{R}$ with three point remainder and the same holds for every finite natural number $n$ greater than 3.
We know that the Stone-Čech compactification of $\mathbb{R}$ has... | https://mathoverflow.net/users/23317 | Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ remainder | I guess you are interested in Hausdorff compactifications. (It is easy to construct a non-Hausdorff compactification with 3-point remainder.)
Set $W=\beta\mathbb R\backslash \mathbb R$;
note that $W$ has two connected components.
If $Z$ is an other compactification of $\mathbb R$ then $V=Z\backslash \mathbb R$
is an ... | 12 | https://mathoverflow.net/users/1441 | 95823 | 56,074 |
https://mathoverflow.net/questions/95819 | 7 | Let $X$ be as smooth variety over a field $k$ of characteristic $0$.
Consider the following statements:
* The variety $X$ has no $k((t))$-rational points.
* No smooth compactification of $X$ has a $k$-rational point.
Are these equivalent? If not, what additional assumptions on $X$ would make them equivalent? I'm ... | https://mathoverflow.net/users/1107 | Rational points on smooth compactifications | Yes, this is true.
One implication is immediate: if $X$ has a $k((t))$ point then by the valuative criterion of properness there is a map $Spec(k[[t]])$ to any compactification of $X$, so the image of the closed point gives a $k$-point of the compactification.
For the converse, if a smooth compactification has a $k... | 12 | https://mathoverflow.net/users/519 | 95824 | 56,075 |
https://mathoverflow.net/questions/95846 | 0 | This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?
| https://mathoverflow.net/users/20300 | Does locally compact plus pseudocompact imply paracompact? | Any pseudocompact paracompact Hausdorff space is compact. So $\omega\_1$ with the order topology is a counterexample to your question since it is locally compact, pseudocompact (any real valued continuous function is eventually constant) and not compact.
| 2 | https://mathoverflow.net/users/17836 | 95854 | 56,087 |
https://mathoverflow.net/questions/95866 | 2 | In his paper [Gaussian maps and plethysm](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.12.771&rep=rep1&type=pdf), Manivel uses the term "total exterior product" of line bundles, appearing for the first time on page 3. Given two (projective) varieties $V\_1$ and $V\_2$ and sheaves of $\mathcal{O}\_{V\_i}$-mo... | https://mathoverflow.net/users/9947 | Total exterior Product | I believe what is meant by this is the usual *external* tensor product: if $p\_1$ and $p\_2$ are projections from $V\_1\times V\_2$ to $V\_1$ and $V\_2$ respectively, the external tensor product $\mathcal{F}\_1\boxtimes\mathcal{F}\_2$ is nothing but $p\_1^\*(\mathcal{F}\_1)\otimes p\_2^\*(\mathcal{F}\_2)$.
| 2 | https://mathoverflow.net/users/1306 | 95875 | 56,100 |
https://mathoverflow.net/questions/95857 | 0 | Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which is also paracompact must be strongly 0-dimensional (Engelking, 1989, p. 362). But the answer to a recent question post... | https://mathoverflow.net/users/20300 | Locally compact, 0-dimensional, pseudocompact space | An infinite collection $\mathcal{A}$ of infinite subsets of $\mathbb{N}$ is said to be almost disjoint (AD) if $A\cap B$ is finite whenever $A,B \in \mathcal{A}$ with $A \neq B$. If the family is maximal with respect to this property, then it is called a MAD family.
Given an AD family $\mathcal{A}$, there is a well k... | 1 | https://mathoverflow.net/users/17836 | 95882 | 56,106 |
https://mathoverflow.net/questions/95833 | 4 | Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
| https://mathoverflow.net/users/3945 | density in SU(2,1) | Yes, the density you indicate certainly occurs. For $SU(n,n-1)$ and $SO(n,n-1)$ and many other classical groups, an analogous density holds, by a similar argument, although there are some minor complications (about compact real Lie groups...) for $SU(p,q)$ with $p>q+1$.
In the case at hand, first note that whatever t... | 5 | https://mathoverflow.net/users/15629 | 95886 | 56,108 |
https://mathoverflow.net/questions/95885 | 1 | Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space $$C=\{ u\in C^2([0,1]):u(0)=u(1)=0\}.$$
It's not hard to find the eigenvalues and eigenvectors(eigenfunctions) for $D$ by soloving the eigenvalue problem:
$$
-u''=\lambda u\qquad u(0)=u(1)=0.
$$
Here are my **questions**:
... | https://mathoverflow.net/users/nan | Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$? | This is easy. If the boundary condition for D is, say, $Lu=0$, then for $D^m$ you should pose the boundary conditions
$$Lu=LDu=LD^2u=...=LD^{m-1}u=0.$$
| 2 | https://mathoverflow.net/users/12120 | 95897 | 56,115 |
https://mathoverflow.net/questions/95902 | 31 | I suspect this is a bit basic for mathoverflow, seeing I'm still just an undergraduate
I've been playing around with quaternions as means to eliminate the gimbal lock. From what I understand, one place the gimbal lock occurs is when you rotate $\frac{\pi}{2}$ around the y-axis. If I create two rotation matrices, $R\... | https://mathoverflow.net/users/23427 | The gimbal lock shows up in my quaternions | There's no paradox here: you did the same calculation in two different ways and got the same answer, as you should. The issue is how to think about gimbal lock.
How should you represent a rotation in three dimensions? You can try using Euler angles to represent it using three rotation angles, but there's something fi... | 66 | https://mathoverflow.net/users/4720 | 95908 | 56,121 |
https://mathoverflow.net/questions/95889 | 2 | A [recent question](https://mathoverflow.net/questions/95537) here has convinced me that folks here have a warm heart for the foundations of quantum mechanics, so I decided to ask a question that has been bothering me for a while.
Quantum motivation
------------------
[Hardy](http://dx.doi.org/10.1016/j.shpsb.2003.... | https://mathoverflow.net/users/9211 | Cardinality of a certain set of distinct subsets of $\mathbb{N}$ | A simpler construction, which yields pairwise incomparable sets (but not almost disjoint sets):
For any subset $A\subseteq \mathbb N$, let $X\_A:= \{ 2n: n\in A\}$, and
let $Y\_A:= \{ 2n+1: n\notin A\}$, and let $Z\_A:= X\_A \cup Y\_A$.
Then the family of all $Z\_A$ has size continuum, and $Z\_A \subseteq Z\_B$
... | 9 | https://mathoverflow.net/users/14915 | 95926 | 56,130 |
https://mathoverflow.net/questions/95925 | 3 | Suppose $\Sigma$ is a signature in the sense of universal algebra and $\Sigma' \subseteq \Sigma$ a sub-signature. Every $\Sigma$-algebra is also a $\Sigma'$-algebra in a forgetful way. Suppose $A$ is a $\Sigma$-algebra and $B \subseteq A$ is a $\Sigma'$-subalgebra of $A$ viewed as a $\Sigma'$-algebra. Is there an accep... | https://mathoverflow.net/users/1176 | What to call substructures in universal algebra in which we restrict the signature? | If the domain of $B$ is the same as $A$, but you only forget the interpretation of the extra language elements, then $B$ is called a *reduct* of $A$ to signature $\Sigma'$. But you don't merely have a reduct, since you are taking a substructure in the smaller language. Thus, what you have is that $B$ is a substructure ... | 3 | https://mathoverflow.net/users/1946 | 95927 | 56,131 |
https://mathoverflow.net/questions/95939 | 50 | What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?
| https://mathoverflow.net/users/3621 | What is the difference between holonomy and monodromy? | Holonomy= monodromy iff the bundle is flat. In general, monodromy group is the quotient of holonomy group by the normal subgroup formed by parallel transports along homotopically trivial loops. One of the simplest examples when two groups are different is the holonomy of the tangent bundle of the standard Riemannian me... | 93 | https://mathoverflow.net/users/21684 | 95944 | 56,138 |
https://mathoverflow.net/questions/95949 | 4 | The usual binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!} $ can be generalized to real upper argument, lower argument still a nonnegative integer, by the definition
$\binom{\alpha}{k} = \frac{\alpha (\alpha-1)\dots (\alpha-k+1)}{k!}$.
In the same way we could generalize the multinomial coefficient
$ \binom{k... | https://mathoverflow.net/users/6494 | Generalized multinomial coefficient | We have
$$(1+y+z)^{\alpha} = \sum\_{t=0}^{\infty} \binom{\alpha}{t}(y+z)^t$$
The trinomial identity is just a rearrangement of the above by
$$\binom{\alpha}{r,s}=\binom{\alpha}{r+s}\binom{r+s}{s}$$
To have convergence, we assume $|y+z|<1$, $|y|<1$, and $|z|<1$.
The generalization follows similarly.
| 3 | https://mathoverflow.net/users/21090 | 95953 | 56,142 |
https://mathoverflow.net/questions/95954 | 13 | I want to find some condition to construct a continuous finitely additive measure on the natural numbers, i.e. $f:P(\mathbb{N})\rightarrow [0,1]$ such that $f(\{n\})=0$, and $f$ is an additive measure.
I know in ZFC we can use an ultrafilter $U$ and define $f$ by $f(A)=1\Leftrightarrow A\in U$, but this is too trival... | https://mathoverflow.net/users/22161 | How to construct a continuous finite additive measure on the natural numbers | In ZF you can't prove that there is a finitely additive measure on $\mathcal P(\mathbb N)$
(giving all singletons measure 0 and $\mathbb N$ measure 1).
The existence of such a measure gives you a subset of $\mathbb R$ without the Baire property,
and such a set cannot be constructed without some help of the axiom of cho... | 17 | https://mathoverflow.net/users/7743 | 95959 | 56,144 |
https://mathoverflow.net/questions/95923 | 8 | I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and cartesian closed. Can anyone comfirm or deny, with reference if possible?
| https://mathoverflow.net/users/17875 | Category of Uniform spaces | There are two possible meanings of "uniform space" in the literature. I'll follow Isbell's terminology where the definition of a *uniform* space includes the separation axiom, and one speaks of a *pre-uniform* space when it is not included.
The category of pre-uniform spaces and uniformly continuous maps has *concret... | 9 | https://mathoverflow.net/users/10819 | 95964 | 56,147 |
https://mathoverflow.net/questions/95930 | 1 | Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}\_u(\Omega,\mathbb{R})\subset\mathscr{C}(\Omega,\mathbb{R})$ the linear subspace of uniformly continuous real-valued functio... | https://mathoverflow.net/users/11211 | Uniformly continuous functions and Borel hierarchy in the compact-open topology | Let $(K\_n)\_{n\in\mathbb N}$ be a compact exhaustion of $\Omega$ (that is, every compact set is contained in some $K\_n$) and define
$$ A\_{n,m,k}=\lbrace f\in \mathscr C(\Omega,\mathbb R): \sup\lbrace |f(x)-f(y)|: x,y \in K\_n, d(x,y)<1/m\rbrace < 1/k\rbrace.$$
This set is open with respect to the semi-norm $\|f\|\_n... | 1 | https://mathoverflow.net/users/21051 | 95968 | 56,149 |
https://mathoverflow.net/questions/95962 | 2 | Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x\_1$, ... , $x\_n$ and complex coefficients.
$$ f = \sum\_I f\_I \mathbf x^I $$
This power series is said to be differentially finite (or D-finite) if the $\mathbb C(\mathbf x)$-vector-space generated by all the derivatives... | https://mathoverflow.net/users/19205 | Differential finiteness of power series with hypergeometric coefficients | Yes! In the multivariate setting, such a series is called proper hypergeometric by several authors. The main ingredient in the proof is the fact that multivariable D-finite functions are closed under Hadamard product (taking products of coefficients termwise), and is proved in
L. Lipshitz, D-finite power series, J. Alg... | 2 | https://mathoverflow.net/users/2384 | 95971 | 56,150 |
https://mathoverflow.net/questions/95965 | 10 | Joyal's [combinatorial species](http://en.wikipedia.org/wiki/Combinatorial_species), endofunctors in the category of finite sets with bijections $\mathbf B$ have found numerous applications. One generalisation is given by so-called "tensor species" (also "tensorial species", or, "linear species" - not to be confused wi... | https://mathoverflow.net/users/3032 | Has there been any application of tensor species? | In the theory of algebraic operads, the language of "tensor species" is often used,
see Chapter 5 of "Algebraic Operads, Jean-Louis Loday & Bruno Vallette, Grundlehren der mathematischen Wissenschaften, Volume 346, Springer-Verlag (2012).
For example one can define an operad very concise as a monoid in species under... | 3 | https://mathoverflow.net/users/2837 | 95973 | 56,151 |
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