parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/87265 | 5 | Let $(G, \mathcal D)$ be a densely defined operator on $C\_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ be a Markov process corresponding to it.
Assume that $\mathcal D\subset C\_K$ (continuous functions with compact su... | https://mathoverflow.net/users/21061 | Symmetric Feller processes and Dirichlet forms | I've decided to post an incomplete preliminary answer.
I ran into your problem when I was writing [1]. On page
258 you will see my resolution.
I should point out that in my case, the underlying space
$X$ was compact, and that $m$ was a finite measure with full
support. Thus, $C(X)$ embeds into $L^2(X;m)$ with a c... | 1 | https://mathoverflow.net/users/nan | 87470 | 51,869 |
https://mathoverflow.net/questions/87476 | 4 | Let $X$ be an algebraic variety, $S \subset X$ its singular locus, and $x \in S$. Say that $x$ is good if for any neighborhood $U$ of $x$, any top differential form $\omega$ on $U \setminus S$ and some resolution of singularities $U' \rightarrow U$, the pullback of $\omega$ to $U'$ is regular.
An obvious necessary co... | https://mathoverflow.net/users/4690 | Property of singularity | The condition you want is very close to having *rational singularities*.
**Definition:** A variety $X$ of characteristic zero has rational singularities if it is:
1. Cohen-Macaulay and,
2. for some resolution of singularities $\pi : Y \to X$ we have $\pi\_\* \omega\_Y = \omega\_X$.
Now, let me explain why condit... | 5 | https://mathoverflow.net/users/3521 | 87483 | 51,876 |
https://mathoverflow.net/questions/87445 | 4 | In his paper "[On the theory of indefnite quadratic forms](http://www.jstor.org/pss/1969191)", Siegel gives the formula (Thm. 1)
$$
\mu(S,T)=\prod\_p \alpha\_p(S,T),
$$
where
* $S$ is an $m\times m$ non singular integral symmetric matrix of signature $(r,m-r)$,
* $T$ is an $n\times n$ integral symmetric matrix,
* $\... | https://mathoverflow.net/users/20052 | Siegel's Mass Formula for ternary indefinite quadratic forms | Borovoi does not claim what you seem to think. Siegel's formalism discusses, for positive forms, the number of representations of an integer by an entire genus of positive ternary forms, each form weighted according to the number of its integral automorphs. For indefinite forms, the automorph groups are infinite and on... | 8 | https://mathoverflow.net/users/3324 | 87490 | 51,878 |
https://mathoverflow.net/questions/87478 | 2 | In continuation of my recent questions, here is the last one:
Is there a simple formula preferably existential that defines $\mathbb{Z}$ in $\mathbb{Z}^\ast/P\mathbb{Z}^\ast$ where $\mathbb{Z}^\ast$ is an elementary extension of $\mathbb{Z}$ and $P$ is an infinite prime?
| https://mathoverflow.net/users/nan | Defining $\mathbb{Z}$ in $\mathbb{Z}^\ast/P\mathbb{Z}^\ast$ where $P$ is an infinite prime | No. If $\phi(x,y)$ is any formula with two free variables $x$ and $y$, then it's true $\mathbb Z$ and therefore also in the elementary extension $\mathbb Z^\*$ that $(\forall y)$ [if $\phi(0,y)$ and if $(\forall x)\ (\phi(x,y)\implies \phi(x+1,y)\land\phi(x-1,y))$ then $(\forall x)\ \phi(x,y)$]. Now if $\psi(x)$, with ... | 3 | https://mathoverflow.net/users/6794 | 87494 | 51,881 |
https://mathoverflow.net/questions/87501 | 29 | Has someone constructed a programming language that can construct all the algorithms in P, and no others?
I'm interested in this restriction coming from the syntax naturally, as opposed to just being a normal Turing machine with a step-timer attached.
| https://mathoverflow.net/users/20886 | A programming language that can only create algorithms with polynomial runtime? | Yes, there is a whole research area devoted to this problem -- it's called "implicit complexity theory". The general idea is to use a lambda calculus based on linear logic. The linearity constraint on lambda-terms lets you control the complexity of cut-elimination (and hence of evaluation), giving natural programming l... | 30 | https://mathoverflow.net/users/1610 | 87507 | 51,889 |
https://mathoverflow.net/questions/87462 | 3 | Suppose $A\in\Re^{n\times n}\_{sym}$ is a symmetric matrix with eigenvalues $\lambda\_1,\dotsc,\lambda\_n$ in decreasing order. What I seek is a way to choose an eigenvector that is invariant to permutations of $A$.
Essentially, let $\phi:\Re^{n\times n}\mapsto\Re^{n}$ be a function that returns an eigenvector correspo... | https://mathoverflow.net/users/14393 | When can an eigenvector be chosen uniquely which is invariant to permutation? | You can always do it if there is any eigenvector not summing to 0, and some other times too. First define $Aut(A)$ to be the group of permutation matrices $R$ such that $RAR^T=A$. If $A\boldsymbol{x}=\lambda\boldsymbol{x}$, then $AR\boldsymbol{x}=RAR^TR\boldsymbol{x}=\lambda R\boldsymbol{x}$ so $R\boldsymbol{x}$ is als... | 2 | https://mathoverflow.net/users/9025 | 87508 | 51,890 |
https://mathoverflow.net/questions/87456 | 2 | Let $L=\mathbb{P}^l\subset\mathbb{P} ^ N \_ {\mathbb{C}}$ be a linear space and let $M=\mathbb{P}^{N-l-1}$ be a linear space skew to $L$, i.e. $L\cap M=\emptyset$.
Let $X\subseteq\mathbb{P}^N\_{\mathbb{C}}$ be a closed irreducible variety not contained in $L$ and
let $$ \pi\_L:X\dashrightarrow\mathbb{P}^{N-l-1}=M $$ ... | https://mathoverflow.net/users/15606 | A question on smoothness of varieties | The necessary condition is indeed the one given by Charles.
This is true if
$$\dim X=\dim\overline{\pi\_L(X)}+\dim\overline{\pi\_L^{-1}(\pi\_L(x))}\tag{$\star$}$$
1
-
Assume that $(\star)$ holds. The target is smooth, so in particular $\pi\_L(x)$ is a complete intersection. This implies that then (by the condit... | 2 | https://mathoverflow.net/users/10076 | 87510 | 51,891 |
https://mathoverflow.net/questions/87491 | 5 | Hi,
I am reading the lecture notes on Morse Homology written by M.Hutchings, in that notes definition of Hessian is defined in coordinate free way: given any connection $ H(f,p)= \nabla\_v(df)$ where $v$ is the tangent vector at critical point $p$, and $df$ is differential of $f$. I need to show this definition does no... | https://mathoverflow.net/users/8473 | definition of Hessian with respect to connection | (This is an elaboration on the comment of MG. I know I benefited a lot as an undergraduate from being shown this sort of argument once instead of having been told to check things in local coordinates, so I thought I'd do the same for you.)
The fact that $\nabla$ is a connection means that for every function $f$ $$\na... | 14 | https://mathoverflow.net/users/1306 | 87515 | 51,893 |
https://mathoverflow.net/questions/87480 | 13 | Hartshorne at the end of page 76 of his Algebraic Geometry book gives an example of a scheme which is not an affine scheme. The scheme is constructed by gluing two affine lines together along their maximal ideals obtained by removing a point P. There's also a figure accompanying the example:
\_\_*\_*\_\_*\_*\_\_*\_*... | https://mathoverflow.net/users/16649 | explanation on a scheme which is not affine scheme | Call $X $ your scheme over the field $k$, $P\_1$ and $P\_2$ the two special closed points, $A\_1$and $A\_2$ their respective open complements and $A\_{12}=A\_1\cap A\_2$, so that $A\_i\simeq \mathbb A^1\_k$ and $A\_{12}\simeq\mathbb G\_m$, all affine schemes.
Here are some (not independent) proofs that $X$ is not af... | 30 | https://mathoverflow.net/users/450 | 87516 | 51,894 |
https://mathoverflow.net/questions/87512 | 14 | The theory of group schemes seems to be well developed: there are many applications and examples, and the literature is vast.
On the other hand, a quick google search with some obvious keywords (ring, schemes, algebraic) does not yield any good reference on Ring Schemes, nor any interesting examples (other than Witt... | https://mathoverflow.net/users/4112 | On Ring Schemes | Some general results about ring schemes and algebraic rings can be found in the following papers by M. J. Greenberg: "Schemata over local rings" and "Algebraic rings", respectively. The concept of algebraic ring also appeared earlier in some work of Serre and Weil (see the references in the second paper above). There w... | 14 | https://mathoverflow.net/users/2381 | 87517 | 51,895 |
https://mathoverflow.net/questions/87143 | 4 | I have an unknown function $\psi(\xi\_1,\dots,\xi\_n)$, such that
$\psi$ satisfy an (unknown) polynomial equation with coefficients polynomials in the $\xi\_i$.
The function is homogeneous, that is, $\psi(t \xi\_1,\dots, t \xi\_n) = t^c \psi(\xi\_1,\dots,\xi\_n)$. I also know that $|\psi(e^{i \theta\_1},\dots,e^{i \t... | https://mathoverflow.net/users/1056 | Algebraic function with extra condition, what can it be? | So, here is my stab at a proof, which actually do not require algebraicness of $\psi$
Notice that we have $|\psi(\xi\_1, \dots \xi\_n)| = 1$ whenever $\xi\_1 \cdots \xi\_n = 1.$
Thus, using the homogeneity property, we may see that
$$\psi(t^{1-n}\xi\_1, t \xi\_2, t\xi\_3, \dots ,t\xi\_n) = \phi(\xi\_1,\dots,\xi\_n) e... | 1 | https://mathoverflow.net/users/1056 | 87519 | 51,897 |
https://mathoverflow.net/questions/87338 | 4 | Let $A$ be a noncommutative finitely generated algebra with a finitely generated set of relations. Moreover, assume that $A$ is finite dimensional as a vector space.
What I want to know is, can Mathematica (or any other package) be used to find the dimension of $A$ given only the generators of $A$ and the generators... | https://mathoverflow.net/users/3787 | Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer | As it was mentioned in my comment, you can use [GAP](http://www.gap-system.org/) and the noncommutative Gröbner bases package [gbnp](http://mathdox.org/products/gbnp/), written by Arjeh M. Cohen and Jan Willem Knopper.
Here you have an **example**:
Assume that you want to compute the dimension and a basis for the ... | 5 | https://mathoverflow.net/users/17845 | 87520 | 51,898 |
https://mathoverflow.net/questions/85005 | 0 | I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable.
Are there any known simple (e.g. exponential) upper bounds on the generalized Laguerre polynomials $L\_{n}^{(\alpha)}(x)$?
So far I've only found some asymptotic expansions, but I'd like an ... | https://mathoverflow.net/users/2586 | Upper bounds on generalized Laguerre polynomials | What kind of estimates exactly do you need? It is difficult to help if you are not more specific. At any rate, I believe there are many references you could check. You might find some useful inequalities in the papers:
1) "A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight" by Holly Car... | 1 | https://mathoverflow.net/users/7482 | 87521 | 51,899 |
https://mathoverflow.net/questions/87460 | 3 | I'm reading Deitmar's paper on [Schemes over $\mathbb{F}\_1$](http://arxiv.org/abs/math/0404185v7). Proposition 2.4. states that for a scheme $X$ over $\mathbb{F}\_1$ there is a bijection between $X(\mathbb{F}\_1)$ and the set of connected components of $X$. I don't understand the proof, which is quite sketchy. Here is... | https://mathoverflow.net/users/2841 | Connected components of schemes over $\mathbb{F}_1$ | I think this is a simple topological statement using the following facts:
1. Every affine set has a unique generic point.
2. Every open set contains an affine open subset.
Let $X$ be a scheme over $F\_1$.
If $U$ and $V$ are affine open subsets with $U\cap V\ne \emptyset$, then their generic points coincide, as both... | 5 | https://mathoverflow.net/users/nan | 87535 | 51,910 |
https://mathoverflow.net/questions/87514 | 4 | Let $K$ be a cubic extension of the rational numbers of discriminant $D$ and $\{ 1, \omega\_2, \omega\_3 \}$ be an integral basis for the ring of integers $\mathcal{O}\_K$ of $K$. Let $\alpha \in \mathcal{O}\_K$ be primitive so that no rational prime divides $\alpha$, let the norm of $\alpha$ be equal to $a^3$, with $a... | https://mathoverflow.net/users/17053 | Do there always exist integers $\beta_2$, $\beta_3$ such that $\{a, \beta_2 + \omega_2, \beta_3 + \omega_3 \}$ is an integral basis for the ideal $(a, \alpha )$ of a cubic field | If I understand this correctly, then the answer is no. If an ideal $I$ has a basis of the desired form, then clearly any element of the ring of integers is congruent to a rational integer modulo $I$. But this implies that $I$ is a product of ideals of inertia degree $1$.
Thus if $\alpha$ is the cube of an ideal of de... | 3 | https://mathoverflow.net/users/3503 | 87544 | 51,913 |
https://mathoverflow.net/questions/76908 | 12 | Cases where
$sup\_{\mu \in E(T)} h\_\mu(T)
\neq
\sup\_{\mu \in M(T)} h\_\mu(T)$.
Background
----------
For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of $T$-invariant probability measures
$M(T)$, and call the set of $T$-ergodic (probability) measures
$E(T)$.
It is ev... | https://mathoverflow.net/users/18191 | Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures. | You make a pretty common mistake here - your question has nothing to do with continuity, compactness etc. and belongs entirely to the measure category. What matters here is that you have a measure preserving transformation of a probability Lebesgue space (sometimes these spaces are also called standard probability spac... | 4 | https://mathoverflow.net/users/8588 | 87545 | 51,914 |
https://mathoverflow.net/questions/87538 | 1 | My problem is the following:
I have a finite surjective morphism $f: X\rightarrow Y$ of noetherian schemes and know that $Y$ is a regular scheme.
(Indeed, in my situation, the two schemes are topologically the same and the arrow is topologically the identity.)
I don't know if $f$ is étale or smooth. But I know that... | https://mathoverflow.net/users/18183 | When does regularity of the base scheme imply regularity of the top scheme? | I may be misunderstanding the question, but it seems rather straightforward to me.
a) As stated, without assuming, that $X$ is, say, reduced, it is certainly false:
Let $X=\mathrm{Spec}k[\varepsilon]=k[x]/(x^2)$, $Y=\mathrm{Spec} k$ and $f:X\to Y$ the structure map of $X$ as a $Y$-scheme. This is obviously a homeomo... | 3 | https://mathoverflow.net/users/10076 | 87550 | 51,917 |
https://mathoverflow.net/questions/87388 | 7 | Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL\_n\times GL\_{n-1}$ is defined over a number field $F$ , $H=GL\_{n-1}$ diagnoal embedded into $G$ as a subgroup and $P$ is some standard parabolic of $G$ .
The interesting point is that $H$ ... | https://mathoverflow.net/users/4245 | An interesting double coset in the theory of automorphic forms | First of all some remarks:
1. The pair that you discussed is spherical, so it is known that there is a finite number of such orbits (formally speaking it is implied only in char $0$ case, but it does not matter here)
2. A convenient way to think of the spherical space $G/H$ is as $GL\_n$ where the action of $G$ is gi... | 4 | https://mathoverflow.net/users/4690 | 87562 | 51,920 |
https://mathoverflow.net/questions/87555 | 6 | A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure cubic field? In particular, is there some theorem (or simple algorithm) that tells you when the class number will be o... | https://mathoverflow.net/users/21134 | Parity of class number of pure cubic fields | There is an algorithm for computing the 2-class number of pure cubic fields using $2$-Selmer groups of elliptic curves due to G. Frey and his diploma students Eisenbeis and Ommerborn: *Computation of the 2-rank of pure cubic fields*, Math. Comput. 32 (1978), 559-569.
This was generalized by U. Schneiders in *Estimati... | 8 | https://mathoverflow.net/users/3503 | 87564 | 51,921 |
https://mathoverflow.net/questions/87567 | 2 | I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172.
The lemma says that a special ribbon graph drawn on page 167 presents a cylinder.
I am sorry that I don't know how to show that ribbon graph here.
I especially don't understand ... | https://mathoverflow.net/users/21144 | A special ribbon graph presents a cylinder. | Turaev's book assumes familiarity with basic 3-dimensional geometric topology and especially Dehn surgery presentations of 3-manifolds. If you want to understand all the details in Tureav's book, then I strongly recommend first reading Rolfsen's ["Knots and Links"](http://books.google.com/books?id=s4eGEecSgHYC&dq=rolfs... | 5 | https://mathoverflow.net/users/284 | 87588 | 51,931 |
https://mathoverflow.net/questions/87573 | 4 | Let $h\_m$ is the class number of $\mathbb{Q}[\sqrt m]$ and let $p>2$ a prime number.
Is there a known connections between $h\_p$ and $h\_{-p}$? e.g. if $q^i$ divides $h\_p$ then it also divides $h\_{-p}$, or the other way around?
The only relevant result I've found is the following (ex' 10.6 in Washington's book "In... | https://mathoverflow.net/users/21148 | relations between class numbers of quadratic extensions | Let $m$ be a squarefree number, and let $d$ run through the discriminants of quadratic number fields coprime to $m$. Then the $2$-rank of the class group of ${\mathbb Q}(\sqrt{dm})$ is, up to a small term depending on the residue class of $m$ modulo $4$ and the sign of the fundamental
unit involved, essentially the num... | 2 | https://mathoverflow.net/users/3503 | 87590 | 51,933 |
https://mathoverflow.net/questions/86778 | 5 | Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 (3) of Chapter II.9 from the book *Metric spaces of non-positive curvature* by Bridson and Haefliger, there exist two points ... | https://mathoverflow.net/users/20913 | Flat sector in a proper cocompact CAT(0) space | Xie proved that this is true if $X$ is a 2 dimensional complex. I recommend reading his article: "The Tits Boundary of a CAT(0) 2-Complex", Trans. AMS., 357, no. 4, 1627-1661. I think that the answer to the more general question above is unknown. The techniques used in Xie's paper are further developed in an article by... | 2 | https://mathoverflow.net/users/7021 | 87592 | 51,934 |
https://mathoverflow.net/questions/87579 | 7 | *Given an embedded two-torus in three-dimensional Euclidean space, paint the inside of the torus red and the outside blue. Show that there is an oriented line in ${\mathbb R}^3$ that cuts the torus perpendicularly in (at least) two points at which it crosses from red to blue.*
This is true (I'll say why in a minute),... | https://mathoverflow.net/users/21123 | Oriented double normals | I suggest to look at my paper - P. E. Pushkar', “Generalization of the Chekanov Theorem. Diameters of Immersed Manifolds and Wave Fronts” Local and global problems of singularity theory, Collection of papers dedicated to the 60th anniversary of academician Vladimir
Igorevich Arnold, Tr. Mat. Inst. Steklova, 221, Nauka... | 4 | https://mathoverflow.net/users/2823 | 87593 | 51,935 |
https://mathoverflow.net/questions/87342 | 14 | Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H\_1(X)=0$, $H\_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi\_1(X)$ trivial?)
I asked this question among others [here before](https://mathoverflow.net/questions/87045/is-the-euler-chara... | https://mathoverflow.net/users/20995 | Are acyclic subcomplexes of finite contractible 2-complexes contractible? | According to Sergei Ivanov, *On balanced presentations of the trivial group*, Invent. Math. **165**, 525--549 (2006), for $n=m+1$, this is a particular case of Kervaire–Laudenbach conjecture which is also open, and according to one of Klyachko's results, the negative answer in this special case would imply the negative... | 10 | https://mathoverflow.net/users/20995 | 87596 | 51,938 |
https://mathoverflow.net/questions/87578 | 5 | Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$). In the sequel, two idempotents $e,f \in I(\mathbf{R})$ are said to be *orthogonal* if $e \cdot f = f \cdot e = 0$, an... | https://mathoverflow.net/users/8590 | Does every compact Hausdorff ring admit a decomposition into primitive idempotents? | This is proved more generally for pseudo-compact rings by Gabriel in Gabriel, Pierre
Des catégories abéliennes, Bull. Soc. Math. France 90 1962 323–448; see Page 393 Corollaries 1 and 2.
| 3 | https://mathoverflow.net/users/15934 | 87605 | 51,942 |
https://mathoverflow.net/questions/87486 | 12 | Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^\*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^\* = H^{-1}(\mathbb{R}^n)$.) We can then define the vector-valued Sobolev space $W^{1,2}([0,1]; V, V^\*)$ of functions $u \in L^2([0,1... | https://mathoverflow.net/users/4832 | Reference request: Simple facts about vector-valued Sobolev space | J. Wloka "Partial differential equations", § 25 (p. 390 on, in my 1992 CUP edition) has an account of the space $W(0,T)=W\_2^1(0,T)$ which is essentially the space $W^{1,2}([0,T];V,V^\*)$.
| 6 | https://mathoverflow.net/users/12643 | 87607 | 51,944 |
https://mathoverflow.net/questions/87613 | 7 | To my knowledge the notion of a t-category was first introduced Beilinson, Bernstein and Deligne's Faiseaux Pervers. But while they explain the name "perverse sheaf", they don't give any indication how they came up with the name t-category.
Does anyone know whether the "t" in "t-category"/"t-structure" stands for som... | https://mathoverflow.net/users/459 | What does the t in t-category stand for? | Though I don't have inside information, it's clear that the notes in Asterisque 100 (1982) by BBD gave the first formal definition of *t-category* and *t-structure* on a triangulated category (section 1.3). Since the nearest t-word involved is "triangulated" and no direct rationale is offered for the language introduce... | 9 | https://mathoverflow.net/users/4231 | 87623 | 51,956 |
https://mathoverflow.net/questions/87561 | 3 | Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional orbifold. If the orbifold fundamental group of Q is a hyperbolic group (in Gromov sense), then can we say that Q is a hype... | https://mathoverflow.net/users/19558 | Relationship between hyperbolicity in group theory and hyperbolicity in geometry | [See Peter Scott's Bulletin article for more information.] Typically, we say an orbifold $Q$ is hyperbolic if it comes to us as a quotient of hyperbolic space $H^n$ by the action of a discrete group $G$ of isometries. If the action $G$ is cocompact then $G$ will be a Gromov hyperbolic group. This is the "easy direction... | 8 | https://mathoverflow.net/users/1650 | 87624 | 51,957 |
https://mathoverflow.net/questions/87622 | 7 | I apologize if the following question ends up being too elementary for this website; I [asked it](https://math.stackexchange.com/questions/103160/do-real-vectors-attain-matrix-norms) on math.SE a week ago and it remains unanswered.
Let $A$ be an $n \times n$ matrix with real entries and let $p \geq 1$. I'm wondering... | https://mathoverflow.net/users/21162 | Do real vectors attain matrix norms? | Okay, so basically the answer can be found in here: <http://arxiv.org/pdf/math/0512608v1.pdf>
Here's how the argument works (a simplified version of what is done in the paper with finite dimensions and $p=q$):
*(Note: we define "$\Re$" of a vector by taking the real part componentswise)*
Lemma 3.4 says (applied t... | 7 | https://mathoverflow.net/users/17498 | 87625 | 51,958 |
https://mathoverflow.net/questions/87629 | 6 | I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H\_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H\_n$ in terms of the dominant weights of root systems of type $B\_n$ or $C\_n$. So far I haven't found an accessible reference that draws... | https://mathoverflow.net/users/4923 | schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups | The $O(n)$ version of Schur-Weyl duality involves Brauer algebras, the structure of which was not worked out completely until the 1980s by Hans Wenzl (Ann. of Math. (2) 128 (1988), no. 1, 173–193.) So perhaps you are looking for $H\_n$ as a subgroup of the Brauer algebras? Googling "Brauer algebra" and "Hyperoctahedral... | 6 | https://mathoverflow.net/users/6355 | 87630 | 51,961 |
https://mathoverflow.net/questions/87602 | 6 | After reviewing the (locally convex)
topological vector spaces that I know,
the only examples I could find where there is an isomorphism from the
space to its (anti)dual, are Hilbert spaces.
So my question is :
Are there topological vector spaces $V$ such that the topology does not come from a Hilbert structure, a... | https://mathoverflow.net/users/20756 | Topological vector spaces that are isomorphic to their duals | An interesting family of examples comes from number theory (or algebraic geometry, depending on who you ask): If you have a field $k$, the Laurent power series field $k((t))$ has an ultrametric topology where $\{ t^n k[[t]] \}\_{n \in \mathbb{Z}}$ form a neighborhood basis of zero. This space is isomorphic to its topol... | 7 | https://mathoverflow.net/users/121 | 87632 | 51,963 |
https://mathoverflow.net/questions/87599 | 5 | So my question is somewhat similar to [Restriction from $\mathfrak{gl}\_{2n}$ to $\mathfrak{sp}\_{2n}$](https://mathoverflow.net/questions/50729/restriction-from-mathfrakgl-2n-to-mathfraksp-2n); but I was having difficulty understanding the formula given in reference (Harris & Fulton) mentioned there.
Equation (25.3... | https://mathoverflow.net/users/2623 | Restriction of representation from GL(n) to O(n) | For the exterior powers (highest weight $(1,1,\dots,1)$), the restriction to the orthogonal group is still irreducible -- I think you are forgetting to take $\overline{\lambda} = 0$.
There are also issues with stable ranges in this formula (you have to assume that $\lambda$ doesn't have more than $n$ parts). If you ... | 5 | https://mathoverflow.net/users/321 | 87638 | 51,964 |
https://mathoverflow.net/questions/87584 | 3 | Let there be 2 players, p$\mathbb{N}$ and p$\mathbb{R}$. They are playing the Set Cardinality Game where p$\mathbb{N}$ has to present some number n that has not been used in the game so far by either of the players and it is in the set of natural numbers $\mathbb{N}$. Afterwards the player p$\mathbb{R}$ has to take som... | https://mathoverflow.net/users/21152 | Set Cardinality Game - Can a player with numbers in R win over a player with numbers in N as each of them in one turn has to present a new number? | 1. Yes.
2. Any proper superset of $N$ will do. Just exhaust $N$ first and only play a non-member of $N$ when $N$ is exhausted, winning in the next step.
| 1 | https://mathoverflow.net/users/21163 | 87645 | 51,966 |
https://mathoverflow.net/questions/87633 | 23 | Hi all,
As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?
Moreover, as we know all (elliptic) k3 surfaces are differential equivalent to each other, does this mean: topological... | https://mathoverflow.net/users/4874 | construct the elliptic fibration of elliptic k3 surface | Let $S$ be a smooth projective $K3$ surface, say over the complex numbers, and suppose that $S$ admits a (non-constant) fibration $\pi\colon S\to C$ over a curve $C$.
By the universal property of the normalization, we can suppose that $C$ is normal, hence smooth. Now, this curve $C$ must be $\mathbb P^1$, since othe... | 28 | https://mathoverflow.net/users/9871 | 87655 | 51,972 |
https://mathoverflow.net/questions/87656 | 3 | This is exercise 2.2.27 / 72 from Cohen Macaulay Rings, Bruns & Herzog
Let R be a Noetherian ring over which every finite module has a finite free resolution. Show R is a factorial domain.
| https://mathoverflow.net/users/13874 | R noetherian is factorial | This is well-known and references are presumably easy to find. Let me just give a few pointers.
It is enough to show that prime ideals of height one are principal. Let $I$ be such a prime ideal. Thanks to our hypotheses on $R$, the ideal $I$ has a finite free resolution. Hence, if $I$ is a projective module, then it ... | 3 | https://mathoverflow.net/users/2284 | 87664 | 51,974 |
https://mathoverflow.net/questions/87557 | 19 | Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the topological structure, so we are certainly looking for a complex which is contractible, but not collapsible. The only complexes I... | https://mathoverflow.net/users/21137 | A simplicial complex which is not collapsible, but whose barycentric subdivision is | [Lickorish and Martin](http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0238288-X/S0002-9947-1969-0238288-X.pdf) constructed, for each $r$, a triangulation of the $3$-ball whose $r$th barycentric subdivision collapses, but $(r-1)$th doesn't. The basic idea, going back to [Furch](http://infoshako.sk.tsukuba.... | 20 | https://mathoverflow.net/users/10819 | 87666 | 51,975 |
https://mathoverflow.net/questions/87661 | 2 | Recently I come up with an embarrassingly easy question. It should be known or elementary but I am still not able to find either a correct answer or references:
"Consider a random vector $x=(x\_1,...,x\_n)$ in the simplex $0\le x\_i, x\_1+..+x\_n=1$. It is easy to show that each $x\_i$ has beta distribution $B(1,n-1)... | https://mathoverflow.net/users/21174 | Random vector of fixed entry-sum | For the expected norm see equation (19) in [this paper](http://arxiv.org/abs/math/0505618) (or equation (12) in the published version of that paper), which, after some renormalizing, states that
$$
\mathbb{E} x\_1^{r\_1} \cdots x\_n^{r\_n} = \frac{(n-1)! r\_1! \cdots r\_n!}{(r+n-1)!},
$$
where $r\_i \ge 0$ and $r = r\_... | 5 | https://mathoverflow.net/users/1044 | 87671 | 51,978 |
https://mathoverflow.net/questions/87324 | 6 | Let $(A,H)$ be a von Neumann algebra in standard form (which means that $H=L^2A$), and
recall that the automorphism group $Aut(A)$ acts on both $A$ and $H$.
Let
$$N:=\{u\in U(H): uAu^\*=A\}$$
be the normalizer of $A$, equipped with the strong topoloy (the subspace topology from $U(H)$).
The group $N$ is canonically iso... | https://mathoverflow.net/users/5690 | Normalizer of a von Neumann algebra | Julien's comment to my previous answer leads to an even easier solution.
Consider the isomorphism map $\psi:Aut(A)\ltimes U(A^\prime)\rightarrow N$. The $u$-topology on $Aut(A)$ is the topology of pointwise convergence on $H$, which is the same as strong convergence when we consider the elements of $Aut(A)$ as unitar... | 4 | https://mathoverflow.net/users/2055 | 87676 | 51,980 |
https://mathoverflow.net/questions/87644 | 0 | I am working with a numerical application where some known techniques solve $\Delta u=\rho$ for $\rho(x,y)\geq0$ defined on a planar rectangle (given as the density of a large pointset). Since the application is related to transportation problems, I suspect that solving Monge-Ampere $\det(D^2 u)=\rho$ (with the same bo... | https://mathoverflow.net/users/17596 | Differences between the Poisson's and elliptic Monge-Ampere equations? | I agree with Deane Yang's comment and will try to expand on it in this answer.
First, the standard formulation of the Monge-Ampere equation is $ \det(D^2u)=\rho $.
Second, the Monge-Ampere equation and the Poisson equation are completely different, and the numerical techniques for the two problems are very differen... | 1 | https://mathoverflow.net/users/17113 | 87679 | 51,983 |
https://mathoverflow.net/questions/87658 | 11 | I'm trying to learn about and compute homotopy (co)limits. Specifically, if $\mathcal{C}$ is some Grothendieck site and $\mathcal{P}$ the simplicial model category of simplicial presheaves (say with the model structure in which weak equivalences and cofibrations are defined level-wise), then I would like to understand ... | https://mathoverflow.net/users/21028 | Computing homotopy (co)limits in a nice simplicial model category? | In practice, one tends to "compute" arbitrary homotopy colimits as bar constructions, especially when you have a simplicial model category.
If $X:J\to P$ is a simplicially enriched functor, where $J$ is small, then you get a "bar construction" $B=B(\*,J,X)$. This is a simplicial object in $P$, with
$$B\_0 = \coprod\... | 13 | https://mathoverflow.net/users/437 | 87684 | 51,984 |
https://mathoverflow.net/questions/87662 | 2 | Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is an intersection of 2 4-dimensional quadrics, and it is Fano.
If I recall correctly, all moduli spaces of bundles with ... | https://mathoverflow.net/users/4096 | fano moduli varieties of vector bundles | I assume you are asking about $SU(r,L),$ (semistable rank-$r$ bundles with determinant $L$) rather than $U(r,d)$ (semistable rank-$r$ bundles with determinant of degree d).
Drezet-Narasimhan showed that even when $SU(r,L)$ is not a fine moduli space, it is locally factorial with Gorenstein singularities, and that it... | 5 | https://mathoverflow.net/users/5496 | 87687 | 51,986 |
https://mathoverflow.net/questions/87688 | 0 | I'm writing an eigensolver and I'm trying to generate a guess for the next iteration in the solve that is orthogonal to to all known eigenvectors calculated thus far. This means that if I have only one eigenvector, that is say 2 million entries long, I need to generate a vector orthogonal to it. I don't think Gram Schm... | https://mathoverflow.net/users/21182 | Find an $N$-dimensional vector orthogonal to a given vector | Suppose the vector is $(x\_1,x\_2,\dots)$. The following algorithm should work:
1. If $x\_1=0$ then take $(1,0,0,\dots)$.
2. If $x\_1$ is non-zero and $x\_2=0$ then take $(0,1,0,0,\dots)$.
3. If $x\_1$ and $x\_2$ are non-zero then take $(-x\_2,x\_1,0,0,\dots)$
| 4 | https://mathoverflow.net/users/124862 | 87691 | 51,989 |
https://mathoverflow.net/questions/87669 | 3 | Basic question, but I found no reference.
Is the $\psi$ class the only one which is not a boundary class in the PIcard group of the Deligne-Mumford compactification of $\mathcal{M}\_{0,n}$? Or can it be expressed in terms of boundary divisors? If yes what is its expression?
| https://mathoverflow.net/users/4096 | $\psi$ class in $\overline{M}_{0,n}$ | Unless you mean something else when you write $\psi$ class, it is expressible in terms of boundary divisors.
That is, if $\psi\_i$ is the $i$-th cotangent bundle, then you can write it in terms of boundary divisors. One reference for this is the tome "Mirror Symmetry" by Hori, Katz, Klemm, et al. on p. 513, the compa... | 5 | https://mathoverflow.net/users/1703 | 87696 | 51,991 |
https://mathoverflow.net/questions/65892 | 12 | Hello!
Let $M$ be an almost complex manifold. Let $TM$ denote its tangent bundle. Then we have the decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ corresponding to the eigenvalues of the almost complex structure. This decomposition yields the decomposition:
$$
\Lambda^r(T^\star M\otimes\mathbb{C})=\Lambda... | https://mathoverflow.net/users/14806 | Differential forms on an almost complex manifold | Call $C^{\infty}\_{p,q}(M)$ the space of smooth complex sections of the bundle $\Lambda^{p,q}T^\*\_M$ and let $2n$ be the real dimension of $M$.
The fact that
$$
dC^{\infty}\_{p,q}(M)\subset C^{\infty}\_{p+2,q-1}(M)+C^{\infty}\_{p+1,q}(M)+C^{\infty}\_{p,q+1}(M)+C^{\infty}\_{p-1,q+2}(M)
$$
follows immediately from th... | 9 | https://mathoverflow.net/users/9871 | 87698 | 51,992 |
https://mathoverflow.net/questions/87699 | 3 | Assume we have a locally free sheaf $R$ of associative $O\_S$-algebras of rank $r^2$, possibly noncommutative, where $S=\mathbb{P}^2$ over $\mathbb{C}$. Furthermore given a locally free $O\_S$-module $E$ also of rank $r^2$, which is a simple $R$-module, i.e. $End\_R(E)=\mathbb{C}$.
We also know that all bundles come... | https://mathoverflow.net/users/3233 | Do cohomologically trivial line bundles affect morphisms? | If you are specifically interested in the case where $L=\mathcal{O}\_{X}(C\_{1}-C\_{2}),$ all that is needed for your determinant bundle argument to work is the vanishing $H^{0}(\mathcal{O}\_{X}(2C\_{1}-2C\_{2}))=0.$ But this follows from taking cohomology in the exact sequences $$0 \rightarrow \mathcal{O}\_{X}(-2C\_{2... | 4 | https://mathoverflow.net/users/5496 | 87701 | 51,993 |
https://mathoverflow.net/questions/87681 | 2 | Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
| https://mathoverflow.net/users/3922 | References for the categories: DIFF PL LIP TOP | I learnt of much of their properties when looking at the stuff on microbundles. A very clear source for lots of that is J. Lurie, Spring 2009, Topics in Geometric Topology (18.937) , notes for course 18.937. Look on Jacob's website and you will find the course notes. After that look back at the Haupvermutung book.
<h... | 3 | https://mathoverflow.net/users/3502 | 87706 | 51,996 |
https://mathoverflow.net/questions/87714 | 5 | Suppose $F$ is a field, and $F\_1, F\_2$ are two extension fields of $F$. Is it always the case that there is a field $L$, containing three subfields $F, K\_1, K\_2$ and two ring isomorphisms $\varphi\_{i}:F\_i\rightarrow K\_1$ fixing $F$?
Note 1: We lose no generality assuming $F$, rather than an isomorphic copy of... | https://mathoverflow.net/users/3199 | Does a "composite field" always exist? | The tensor product $F\_1 \otimes\_F F\_2$ is not 0, hence it has a quotient which is a field. This contains the images of both $F\_i$.
| 29 | https://mathoverflow.net/users/4790 | 87718 | 52,002 |
https://mathoverflow.net/questions/87719 | 15 | Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^\* E$ splits as a direct sum of line bundles (take e.g. the flag bundle of $E$). Is the analog true for holomorphic vector bundles (if we st... | https://mathoverflow.net/users/344 | Splitting principle for holomorphic vector bundles | The answer is positive. Let $P$ be the principal $\mathrm{GL}\_n$-bundle associated with $E$; then the space of flags is the quotient $P/B$, where $B$ is the Borel subgroup of $\mathrm{GL}\_n$ consisting of upper triangular matrices. Set $Z = P/T$, where $T$ is the maximal torus consisting of diagonal matrices. A point... | 20 | https://mathoverflow.net/users/4790 | 87724 | 52,005 |
https://mathoverflow.net/questions/87735 | 0 | Hi,
I've started studying brownian motion, and gathered some books on the subject but
something looks odd to me : All of the presentations I've seen this far consider the continuity of the brownian motion as an axiom.
Well, if you recast the brownian motion in the wider setting of Levy processes with stable indepen... | https://mathoverflow.net/users/20997 | Any reference on Brownian Motion continuity | This is a partial answer but shows the kind of subtlety that makes the continuity of Brownian motion non trivial. If you try and take the first three axioms of Brownian motion and try to prove that the process has continuous paths using a central limit theorem argument what you get is that on a probability space $(\Ome... | 4 | https://mathoverflow.net/users/11332 | 87737 | 52,012 |
https://mathoverflow.net/questions/87740 | 3 | There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.
Give some category $\mathcal{C}$ with the modifiers listed in the title of the question, and an object $A$ in that category, how does one define the "free monoid... | https://mathoverflow.net/users/11546 | Free Monoids in Closed Symmetric Monoidal Categories | Very simply, if a closed symmetric monoidal category has countable coproducts, and if monoid means monoid with respect to the monoidal product, then the free monoid on an object $A$ can be constructed as the "geometric series"
$$F(A) = \sum\_{n \geq 0} A^{\otimes n}.$$
The key fact needed to prove this is that $\... | 7 | https://mathoverflow.net/users/2926 | 87743 | 52,016 |
https://mathoverflow.net/questions/87739 | 1 | Given an unbiased estimator $\hat \theta\_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is consistent (i.e. that $\hat \theta\_n$ converges in probability to the constant $\theta$).
Is that actually true?
| https://mathoverflow.net/users/20639 | Is an unbiased estimator with arbitrarily small variance necessarily consistent? | Markov's inequality says that for a non-negative random variable $X$ with expected value $\mu$,
$$
\Pr(X>a) \le \frac \mu a.
$$
(E.g. no more than $1/15$ of the population can have more than $15$ times the average income (assuming all incomes are non-negative), etc.)
So
$$
\Pr(|\hat \theta\_n - \theta| > \varepsilon)... | 4 | https://mathoverflow.net/users/6316 | 87747 | 52,018 |
https://mathoverflow.net/questions/87752 | 4 | In my quest to understand spin representations, I am looking at the equivalent views of spin structures (on some oriented Riemannian $n$-manifold). Given such a manifold $M$, its tangent bundle $TM$ is described by gluing cocycles $g\_{ab}:U\_{ab}\rightarrow SO(n)$.
A *spin structure* would be some lift of $g\_{ab}$... | https://mathoverflow.net/users/12310 | Spin-c Structures viewed w.r.t. Cell Decomposition | A spin-c structure on a vector bundle $\pi : E \to B$ can be thought of as two things:
(1) A complex line bundle $c : C \to B$
together with
(2) A spin structure on $\pi \oplus c : E \oplus C \to B$. $\pi \oplus c$ is meant to be the fibrewise direct sum.
This appears in the Gompf reference given in the commen... | 5 | https://mathoverflow.net/users/1465 | 87754 | 52,021 |
https://mathoverflow.net/questions/87197 | 1 | I am trying to understand the Blattner-Kostant-Sternberg pairing as it applies to geometric quantization in real polarizations whose integral manifolds are, for simplicity, compact. I have been trying to follow Sniatycki's account (Geometric Quantization and Quantum Mechanics, pp 73-75). Thus, we have a symplectic mani... | https://mathoverflow.net/users/17913 | BKS pairing for distributional sections | I don't think you're missing anything. Yes, in the case when the Lagrangian foliation consists of tori, the $Q\_i$ will just be points. The `density' at one of these points will just be a number (see for example eqn (4.51) - now $k=n$, and so $\langle\sigma\_1,\sigma\_2\rangle\_{Q\_i}$ has no arguments), and the inner ... | 1 | https://mathoverflow.net/users/17945 | 87758 | 52,025 |
https://mathoverflow.net/questions/87751 | 6 | Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let
$$
O(F):=\{g\in GL\_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\}
$$
be the isotropy group of $F$.
Q: So how does one prove **in the simplest way possible** that if $O(F)=O(G)$ then
there exists $\lambda\in\mathbf{R}^{\times}$ s... | https://mathoverflow.net/users/11765 | On the determination of a quadratic form from its isotropy group | A relatively easy proof also follows from using the reflection identity: First, define the inner product associated to $F$, namely $v\ \cdot\_F\ w = {\frac12}\bigl(F(v{+}w)-F(v)-F(w)\bigr)$, and then, for any $v$ with $F(v)\not=0$, define the *reflection in $v$* by
$$
\rho^F\_v(w) = w - 2\ \frac{v\ \cdot\_F\ w}{v\ \cd... | 16 | https://mathoverflow.net/users/13972 | 87763 | 52,027 |
https://mathoverflow.net/questions/87759 | 0 | I want to find the a 3 term perturbation soln of
(i) $(1+x)^3 = ex$ where $e\ll1$
Direct substitution of the regular perturbation series $x = x\_0 + ex\_1 + e^2x\_2$
into (i) does not work
I think soln has the form: $x = x\_0 + e^{1/3}\*x\_1 + e^{2/3}\*x\_2$
Seems to work, but not sure it is correct
TIA,
Matt
| https://mathoverflow.net/users/21197 | Regular Perturbation Series soln to eqn | Since $e$ is small, the solution $x$ is close to $-1$. So write $x=-1+u$ and write your equation as $u(1-u)^{-1/3}=-e^{1/3}$. Then use the [Lagrange inversion formula](http://en.wikipedia.org/wiki/Formal_power_series#The_Lagrange_inversion_formula).
| 1 | https://mathoverflow.net/users/6101 | 87777 | 52,031 |
https://mathoverflow.net/questions/87780 | 9 | In the *"Morse-Bott theory and equivariant cohomology"* paper by D.M. Austin and P.J. Braam, the authors introduce the Morse-Bott-complex to calculate the de-Rham-cohomology of a compact manifold (using a Morse-Bott-function).
Where does this complex "come from", i.e. is it Austin and Braam's original work?
In the ... | https://mathoverflow.net/users/21205 | Who invented the Morse-Bott-complex? | I do not know a precise answer to your question (and perhaps someone else does), but I'm fairly certain the idea can be traced back to the paper
Bott, Raoul,
*The stable homotopy of the classical groups*.
Ann. of Math. (2) 70 1959 313–337.
This is the paper where Bott proves his famous periodicity theorem, and th... | 7 | https://mathoverflow.net/users/8103 | 87787 | 52,036 |
https://mathoverflow.net/questions/87672 | 4 | Recently i read that the space of completely holomorphic (also at the cusps) modular forms $M\_k(\Gamma(N))$ possesses a basis having Fourier coefficients in $\mathbb{Z}[\zeta\_N]$ where $\zeta\_N = e^{2 \pi i / N}$.
Can somebody point out a reference for this?
I already know the following things:
At least for $k \... | https://mathoverflow.net/users/20431 | Basis for $M_k(\Gamma(N))$ with Fourier coeffs in $\mathbb{Z}[\zeta_N]$? | The constant term of the Eisenstein series $G\_k^{0,v}$ in Diamond-Shurman is, up to a factor $N^k$, given by
$$\zeta(k,\frac{v}{N}) + (-1)^k \zeta(k,-\frac{v}{N})$$
where $\zeta(s,x) = \sum\_{\substack{n \in \mathbf{Q}\_{>0}, \\ n \equiv x \mod{1}}} \frac{1}{n^s}$ is the Hurwitz zeta function.
You can prove by h... | 7 | https://mathoverflow.net/users/6506 | 87789 | 52,037 |
https://mathoverflow.net/questions/87794 | 25 | The question was asked by a student, and I did not have a ready answer. I can think of the German word ``Einheit'', but since in German that is not how the identity element of a group is called, I doubt that is the origin. Any ideas?
| https://mathoverflow.net/users/3635 | Why is the identity element of a group denoted by $e$? | Heinrich Weber uses Einheit and e in his Lehrbuch der Algebra (1896).
| 25 | https://mathoverflow.net/users/2035 | 87797 | 52,040 |
https://mathoverflow.net/questions/87799 | 4 | Let $ \{ a \_ k \} \_{k\in\mathbb{N} \_ +} $ be a sequence of non-negative numbers, and let $MG(a\_1,\dots,a\_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum \_ {n\ge 1}MG(a\_1,\dots,a\_n) \le C\, \sum \_ {n\ge 1} a \_ n $$
holds, with $C=e$. This is quite elementary, although not obv... | https://mathoverflow.net/users/6101 | Bounding the series of the geometric means of the terms of a given positive series | You can’t have $C< e$. Fix $N$, and define
$$a\_n=\begin{cases}\tfrac1n&n\le N\\\\\\\\0&\text{otherwise.}\end{cases}$$
Then
$$\sum\_na\_n=H\_N=\log N+O(1),$$
and
\begin{multline}\sum\_n\mathrm{MG}(a\_1,\dots,a\_n)=\sum\_{n=1}^N\frac1{\sqrt[n]{n!}}=\sum\_{n=1}^N\frac en\left(1+O\left(\frac{\log n}n\right)\right)\\\\=eH\... | 2 | https://mathoverflow.net/users/12705 | 87807 | 52,045 |
https://mathoverflow.net/questions/87730 | 8 | It is known that the
[Cheeger constant](http://en.wikipedia.org/wiki/Cheeger_constant_%28graph_theory%29)
of a
[hypercube graph](http://en.wikipedia.org/wiki/Hypercube_graph) $Q\_n$
is exactly $1$, regardless of its dimension $n$. Is $1$ also an upper bound
on the Cheeger constant of nontrivial induced connected subgra... | https://mathoverflow.net/users/17806 | Is the Cheeger constant of an induced subgraph of a cube at most 1? | Your conjecture is true, every subgraph of the cube has expansion constant at most $1$.
*Proof:* Suppose we are given a subgraph $G\subset Q\_n$, $n>1$ and cut the cube into $2$ $(n-1)$-dimensional subcubes $A\_1,A\_2$. (So that $A\_1\cup A\_2=Q\_n$) The key is to notice that each vertex in $A\_1$ is connected to one... | 6 | https://mathoverflow.net/users/12176 | 87808 | 52,046 |
https://mathoverflow.net/questions/87768 | 1 | In the second paragraph on Page 71 of the book [Matrix Analysis by
Bhatia, 1997](http://books.google.com/books?id=F4hRy1F1M6QC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q=71&f=false), it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 fr... | https://mathoverflow.net/users/21199 | A question on gauge functions | Inequality III.12 is the famous Lidskii majorization for Hermitian matrices $A$ and $B$, which says that
$$\lambda^\downarrow(A) - \lambda^\downarrow(B) \prec \lambda(A-B) \prec \lambda^\downarrow(A) - \lambda^\uparrow(B).$$
Now, recall the following simple but crucial fact:
**Fact.** $x \prec y \implies$ $|x|\qu... | 1 | https://mathoverflow.net/users/8430 | 87812 | 52,049 |
https://mathoverflow.net/questions/87418 | 12 | **Background**
In [Berestycki and Lions](http://www.ams.org/mathscinet-getitem?mr=695535) it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$ S[u] = \int\_{\mathbb{R}^d} |\nabla u|^2 + G(u) dx $$
where $G... | https://mathoverflow.net/users/3948 | The ground state is signed and symmetric | A little bit more digging turned up [this paper of Mihai Mariş](http://www.ams.org/mathscinet-getitem?mr=2486598).
He shows that under two technical conditions:
* Minimizers are $C^1$ (which we have from elliptic regularity)
* If $u$ is an admissible function ($H^1$ in our case) and $v$ is a unit vector in $\mathb... | 3 | https://mathoverflow.net/users/3948 | 87816 | 52,051 |
https://mathoverflow.net/questions/87813 | 9 | Jacob's book titled "Categorical Logic and Type Theory" gives a nice description of Π and Σ types as adjunctions to substitution functors induced by display maps. Is there a similar categorical description of W-types (and maybe M-types while we are at it)?
| https://mathoverflow.net/users/4085 | Categorical semantics of W-types | The categorical semantics of W-types, as initial algebras, have been studied in the following paper of Moerdijk and Palmgren: "Wellfounded trees in categories", Annals of Pure and Applied Logic 104(2000), 189 - 218.
| 12 | https://mathoverflow.net/users/6485 | 87818 | 52,052 |
https://mathoverflow.net/questions/87819 | 4 | Dear MOs,
Let $\mathcal{D}(R):=C\_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak \*-topology induced by $\mathcal{D}(R)$, which makes it into a locally convex space (See Rudin Functional Analysis P.160... | https://mathoverflow.net/users/36814 | Is there dual space of the distributions $\mathcal{D}'(R)$? | Well, that depends on what topology you want to put on the space of distributions. The weak$^\*$ is probably not really the one you would like to take. Instead, the strong dual might be more useful. The seminorms of this topology are given by
$$p\_B(\varphi) = \sup\_{f \in B} |\varphi(f)|$$ where $B \subseteq \mathcal{... | 7 | https://mathoverflow.net/users/12482 | 87822 | 52,054 |
https://mathoverflow.net/questions/87814 | 5 | Given a category $\mathcal{C}$ with a notion of covering $\{ U\_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve
$$ \cdots \coprod\_{i}{U\_{ijk}} \substack{\rightarrow \\ \rightarrow \\ \rightarrow}\coprod\_{i}{U\_{ij}} \rightrightarrows \coprod\_{i} U\_{i... | https://mathoverflow.net/users/21028 | Cech nerve as homotopy colimit? | More generally, let $X\_\cdot$ be a simplicial presheaf. As such, we can consider it as a simplicial object in presheaves, which in particular may be thought of as a simplicial object in simplicial presheaves $X'\_\cdot.$ So we have:
$$X\_\cdot:\Delta^{op} \to Set^{C^{op}}$$ and $$X'\_\cdot =\left( \mspace{3mu} \cdot... | 3 | https://mathoverflow.net/users/4528 | 87824 | 52,055 |
https://mathoverflow.net/questions/87830 | 3 | Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C\_{\bullet}$ in $\mathcal{D}$, we have $C\_{\bullet} \cong H\_{\bullet}(C\_{\bullet})$, where the right hand side is the complex whose $i$-th term is $H\_i(C\_{\bul... | https://mathoverflow.net/users/297 | Earliest/most standard reference for derived categories of hereditary algebras | Dieter Happel's [book](http://www.ams.org/mathscinet-getitem?mr=935124) *Triangulated Categories in the representation theory of finite dimensional algebras* is a pretty canonical source, and it includes the result you mention.
(That particular result might be folkloric, though)
| 6 | https://mathoverflow.net/users/1409 | 87836 | 52,060 |
https://mathoverflow.net/questions/87827 | 8 | Let $B^{n} \_p= ${$ (x\_1, \dots, x\_n) : |x\_1|^p + \dots |x\_n|^p = 1 $} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm.
If $X\_1,\dots,X\_n$ are iid $\exp(1)$ -distributed random variables, then $(X\_1/D,\dots,X\_n/D)$, where $D =X\_1+ \dots + X\_n $ is uniformly distributed in $B^{n}\_1$.
If $X\_1,\dot... | https://mathoverflow.net/users/11541 | Sampling uniformly from a sphere | The result you want, I think, is in [Stationarity, Isotropy and Sphericity in $l\_p^\*$](http://www.springerlink.com/content/r2771gx9j2g40132/). It is behind a pay-wall, but the form of the distribution is stated in the abstract.
| 4 | https://mathoverflow.net/users/8719 | 87843 | 52,062 |
https://mathoverflow.net/questions/87842 | 3 | hallo,
i have the following question. let $M$ be a Riemann surface and $R \subset M$ be a compact totally real manifold (1 dimensional real). Furthermore assume there is a holomorphic 2-form $\alpha$ on $M$. Can one find a solution $\varphi$ of the equation $\partial \bar{\partial} \varphi = \alpha$ in a neighbourho... | https://mathoverflow.net/users/21202 | $\partial \bar{\partial}$ on a riemann surface | Because you are only interested in a neighborhood of $R$, you might assume that $M$ is (connected and) noncompact. A function $f$ satisfies $\partial \bar{\partial} f=0$ iff it is harmonic. Now pick an open cover $(U\_i)$ of $M$ and local solutions $\phi\_i$ of your problem.
The differences $\phi\_{ij}:=\phi\_i - \phi\... | 7 | https://mathoverflow.net/users/9928 | 87846 | 52,064 |
https://mathoverflow.net/questions/87849 | 2 | The question is in the title. The category $\bf pSet$ of partial functions has sets as objects and $\hom(X,Y)$ is the set of all triples $(X,Y,f)$ such that there exists $D\subseteq X$ and $f\colon D\to Y$. Composition of arrows is composition of [relations](http://en.wikipedia.org/wiki/Composition_of_relations).
| https://mathoverflow.net/users/7952 | Does $\bf pSet$ admit products? | If I'm reading your definition correctly, this category looks equivalent to the category ${\bf Set\_\*}$ of pointed sets and basepoint-preserving functions (the equivalence is by removing the basepoint from each pointed set: you're left with an ordinary set and possibly partial functions). So you should be able to take... | 5 | https://mathoverflow.net/users/1474 | 87851 | 52,065 |
https://mathoverflow.net/questions/87848 | 10 | I think the answer to the title question is "yes", but Gerald Edgar, in his comment on [Does antidifferentiability of continuous functions imply Dedekind completeness?](https://mathoverflow.net/questions/87640/does-antidifferentiability-of-continuous-functions-imply-dedekind-completeness) , points out an article (actua... | https://mathoverflow.net/users/3621 | Does Rolle's Theorem imply Dedekind completeness? | It turns out that Pelling and I mean different things by "Rolle's Theorem": his form of Rolle's Theorem treats only polynomials. I figured that out when I came across the following passage: "... Then $F$ is not real-closed since, e.g., $z^{1/2} \in L - F$, and it remains to show that Rolle's theorem is valid in $F$. By... | 7 | https://mathoverflow.net/users/3621 | 87859 | 52,071 |
https://mathoverflow.net/questions/87873 | 19 | Let $\zeta\_F$ denote the Dedekind zeta function of a number field $F$.
We have $\zeta\_F(s) = \frac{\lambda\_{-1}}{s-1} + \lambda\_0 + \dots$ for $s-1$ small.
Class number formula: We have $\lambda\_{-1} = vol( F^\times \backslash \mathbb{A}^1)$, where $\mathbb{A}^1$ denotes the group of ideles with norm $1$.
>... | https://mathoverflow.net/users/10400 | Dedekind Zeta function: behaviour at 1 | This is called the (generalised) Euler constant of the number field $K$, denoted $\gamma\_K$, as for $K = \mathbb{Q}$ we have $\gamma\_K = \gamma\_0$, the Euler--Mascheroni constant. There are many estimates known for $\gamma\_K$. For example, Theorem 7 of [this paper](https://dx.doi.org/10.1016/j.exmath.2006.08.001) h... | 23 | https://mathoverflow.net/users/3803 | 87876 | 52,076 |
https://mathoverflow.net/questions/87874 | 2 | Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale $k$-group scheme $G\_{s}:=G \times\_{S}$Spec$(k)$ is isomorphic to the constant group $\mathbb{Z} / p\mathbb{Z}$. Then is it ... | https://mathoverflow.net/users/19355 | Etale group schemes over a local ring | Take $p=3$, and take an étale double cover $X\to S=\mathrm{Spec}(R)$. Let $E$ be a copy of $S$, and $G=E\coprod X$. It is easy to see that there is a unique $S$-group scheme structure on $G$ with $E$ as unit section. Now there are plenty of examples where $X$ is nontrivial but is trivial over the closed point.
| 5 | https://mathoverflow.net/users/7666 | 87878 | 52,077 |
https://mathoverflow.net/questions/87877 | 28 | What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the rows indexed by $I$ and columns indexed by $J$. Then
$$ |\det A[I,J]| = | (\det A) \det A^{-1}[J^c,I^c]|,$$
where $I^... | https://mathoverflow.net/users/4923 | Jacobi's equality between complementary minors of inverse matrices | The key word under which you will find this result in modern books is "Schur complement". Here is a self-contained proof. Assume $I$ and $J$ are $(1,2,\dots,k)$ for some $k$ without loss of generality (you may reorder rows/columns). Let the matrix be
$$
M=\begin{bmatrix}A & B\\\\ C & D\end{bmatrix},
$$
where the block... | 23 | https://mathoverflow.net/users/1898 | 87881 | 52,079 |
https://mathoverflow.net/questions/74334 | 6 | For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^p(\Omega)^n=G^p(\Omega) \oplus D^p(\Omega)$,
where $G^p(\Omega)= \{ w\in L^p(\Omega)^n; w= \nabla p$ for some $p\in W^{1,p}... | https://mathoverflow.net/users/17544 | Helmholtz-Decomposition on compact Riemannian manifolds | Günter Schwarz, *Hodge Decomposition - A Method for Solving Boundary Value Problems*, Lecture Notes in Maths **1607** (1995)
| 3 | https://mathoverflow.net/users/9161 | 87887 | 52,082 |
https://mathoverflow.net/questions/87838 | 87 | This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, [Correspondence
between Borel algebras and topology](https://math.stackexchange.com/questions/88916/correspondences-between-borel-algebras-and-topological-spaces).
Since the question was not answered there after some time, I am... | https://mathoverflow.net/users/1946 | Is every sigma-algebra the Borel algebra of a topology? | Unfortunately, I can only provide a reference but no ideas since I don't have the paper.
In "On the problem of generating sigma-algebras by topologies", Statist. Decisions 2 (1984), 377-388, Albert Ascherl shows (at least according to the summary to be found on MathSciNet)
that there are $\sigma$-algebras which can't b... | 56 | https://mathoverflow.net/users/21051 | 87888 | 52,083 |
https://mathoverflow.net/questions/87857 | 5 | I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.
Let $M\_1$ and $M\_2$ be smooth $n$-manifolds (maybe closed, for simplicity), $N$ a closed $k$-manifold, $D$ a closed $(n-k)$-disk bundle over $N$ (so that $D$ is an $n$-dimensional manifold wh... | https://mathoverflow.net/users/17812 | Characteristic classes of a fibered sum | It depends what you mean by "formula", since you are talking about cohomology classes in different manifolds, so at the very least you need a way to relate them, which depends on context. So the "answer" is the Mayer-Vietoris sequence. Naturality of characteristic classes, together with the fact that the characteristic... | 5 | https://mathoverflow.net/users/3874 | 87895 | 52,087 |
https://mathoverflow.net/questions/87869 | 3 | Can i get the answer to the following problem. I am having a proof, i feel there is something wrong here..Can you please point out!
Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma: [0,1]\to D$, be a smooth embedding. Given a continuous one form $\phi$ along $\gamma$ and $\epsilon >0$, Does there ex... | https://mathoverflow.net/users/16031 | A corollary to Stone-Weierstrass theorem | In your case we can find a holomorphic function on the plane that uniformly approximates the
given continuous function .It is a consequence of the following .Suppose K is a compact measure zero subset of the plane whose complement in the plane is connected then every continuous function on K can be uniformly approximat... | 5 | https://mathoverflow.net/users/4696 | 87922 | 52,099 |
https://mathoverflow.net/questions/87919 | 15 | I have been informed that there is a reference out there which specifically details what goes wrong with the mod 2 Moore spectrum, i.e. why it is not $A\_\infty$ or something? I do not know the details, but I am interested in this from the point of view of trying to understand how to come up with the correct notion of ... | https://mathoverflow.net/users/11546 | Difficulties with the mod 2 Moore Spectrum | The actual statement is much stronger than you suggest, namely: the mod 2 Moore spectrum does not admit a *unital* multiplication (even if it is non-associative). I don't know a reference so I'll sketch the proof:
Let $R$ be a spectrum with unital product, with unit map $\eta\colon S^0\to R$ and product map $\mu: R\w... | 27 | https://mathoverflow.net/users/437 | 87924 | 52,100 |
https://mathoverflow.net/questions/87923 | 7 | First, a bit of notation. If we have an arithmetic progression $a, a+k, a+2k, \ldots, a+(n-1)k$ we will call $k$ the distance, and $n$ the length.
While trying to find an example for a paper I'm writing in ring theory, I was led to ask the question: Is there a sequence of 0's and 1's for which if there is an arithmet... | https://mathoverflow.net/users/3199 | Sequences without long arithmetic progressions | How about this? Fix your favourite irrational number $\phi$. I like the golden mean. Let $x\_s=[ s\phi ] - [(s-1)\phi]$ ($[t]$ means the integer part of $t$). These sequences are called Sturmian sequences.
Of course $x\_s$ is 1 if and only if $s\phi \bmod 1$ lies in $[0,\phi)$.
Now for any $a$ and $k$, you're aski... | 6 | https://mathoverflow.net/users/11054 | 87925 | 52,101 |
https://mathoverflow.net/questions/87905 | 5 | Let $(X\_i)$ be a sequence of compact metric spaces and $(f\_i)$ a sequence of transitive transformations $f\_i:X\_i \to X\_i$ with $0 < h\_{top}(f\_i) < \infty$.
The sequence of dynamical systems satifies:
* $X\_i \subset X\_{i+1}$, $h\_{top}(f\_i) < h\_{top}(f\_{i+1}) $;
* $X\_i$ converges to a compact metric spa... | https://mathoverflow.net/users/10518 | Limits of intrinsically ergodic systems | The answer is no. It's based on a (un?)published example of Crannell, Rudolph and Weiss.
The example is the following shift: $X$ is the subset of $\lbrace 0,\pm 1\rbrace ^{\mathbb Z}$ with the property that $x\_k\cdot x\_{k+2^n}$ is not allowed to be $-1$ for any values of $k$ and $n$.
What they prove is that there... | 6 | https://mathoverflow.net/users/11054 | 87927 | 52,103 |
https://mathoverflow.net/questions/87930 | 16 | There are two tools, generalizing a concept of a volume to the case of submanifolds in $\mathbb{R}^n$, namely the Hausdorff measure $H^k$ and the volume form. The question is how to show that if $M$ is an orientable $k$-submanifold in $\mathbb{R}^n$ with a volume form $dV$ then
$$
\int\limits\_{M} f(x) dV = \int\limit... | https://mathoverflow.net/users/17896 | Hausdorff measure and the volume form | I guess you know that it is true in $\mathbb R^k$.
Without loss of generality we can assume that $f\ge 0$.
Fix $\varepsilon>0$ and cover your manifold by $(1\mp\varepsilon)$-Lipschitz charts.
Break your integrals into pieces using subordinate partition of unity and put these pieces back together.
Since in $\mathbb{R}... | 17 | https://mathoverflow.net/users/1441 | 87933 | 52,107 |
https://mathoverflow.net/questions/35060 | 26 | David's question [Families of genus 2 curves with positive rank jacobians](https://mathoverflow.net/questions/35049/families-of-genus-2-curves-with-positive-rank-jacobians) reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of a hyperelliptic curv... | https://mathoverflow.net/users/2024 | When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve? | Over the complex numbers every product of two elliptic curves is isogenous to the jacobian of a genus 2 (hyperelliptic) curve. Indeed, the corresponding Siegel upper half-space $H\_2$ is an orbit of the real symplectic group $Sp(4,R)$. Since the subgroup $Sp(4,Q)$ of its rational points is everywhere dense in $Sp(4,R)$... | 13 | https://mathoverflow.net/users/9658 | 87936 | 52,109 |
https://mathoverflow.net/questions/87937 | 9 | Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can consider the operator $f(A)$ defined by functional calculous.
Is $f(A)$ again a pseudo-differential operator and if yes, ho... | https://mathoverflow.net/users/16702 | Functions of pseudodifferential operators | Here is a good reference for this
>
> Michael Taylor: Pseudodifferential Operators, Princeton University PRess, 1981
>
>
>
In Chapter 12 it explains how to construct $f(A)$ when $A$ is elliptic selfadjoint of order $1$, $A\geq 0$, and $f$ is a smooth symbol of order $m$, i.e., $f$ is smooth and
$$ f^{(k)}(\... | 10 | https://mathoverflow.net/users/20302 | 87938 | 52,110 |
https://mathoverflow.net/questions/87803 | 3 | Let $\bar{\rho} : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL\_2(\bar{\mathbb{F}}\_p)$ be an odd, irreducible Galois representation mod $p$ which is unramified outside $S$, where $S$ is a finite set of primes which contains $p$. Fix an integer $k \geq 2$ and a local Galois representation $\rho \_p ' : Gal(\bar{\m... | https://mathoverflow.net/users/10701 | Number of modular lifts with prescribed parameters | As Kevin said, I'm not sure if you can get a formula in any concrete sense, but there is a way to tell if the modular form that gives rise to $\bar\rho$ is the unique form (of a specific level). Say that $\bar\rho$ takes values in $GL\_2(k)$ and that $f \in S\_k(N,\mathcal{O})$ gives rise to $\bar\rho$ (so we have that... | 4 | https://mathoverflow.net/users/12750 | 87946 | 52,115 |
https://mathoverflow.net/questions/60596 | 14 | $\DeclareMathOperator\Cl{Cl}$**Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!).**
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:L\to k$ be a quadratic form, i. e., a map for which the map $L\times L\to k,\ \left(x,y\right)\mapsto g\left(x+y\r... | https://mathoverflow.net/users/2530 | Clifford PBW theorem for quadratic form | If someone could check the below I'd be very indebted.
Found the counterexample (to Question 1 and thus also to Question 2). It is inspired by the counterexample to ring-theoretical PBW in P. M. Cohn, *A remark on the Birkhoff-Witt theorem*. J. London Math. Soc. 38 1963 pp. 197-203, MR0148717 (though I still don't kn... | 5 | https://mathoverflow.net/users/2530 | 87958 | 52,122 |
https://mathoverflow.net/questions/87962 | 3 | I have a combinatorics problem motivated, of all things, by category theory.
Consider a two-dimensional grid of vertices and edges. Fix two points, $P$ and $Q$, such that $Q$ is $a$ steps right and $b$ steps down from $P$. I'm interested in counting pairs of shortest paths from $P$ to $Q$. That is two paths from $P$ ... | https://mathoverflow.net/users/18060 | How many double paths in a grid? | This number of such pairs will be
$$\frac{(a+b-1)!(a+b-2)!}{a!b!(a-1)!(b-1)!}.$$
This follows from the [Lindstrom-Gessel-Viennot lemma](http://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot_lemma) and the fact that the number of directed paths in an $m\times n$ rectangle is $\binom{n+m}{m}$.
| 8 | https://mathoverflow.net/users/2384 | 87963 | 52,124 |
https://mathoverflow.net/questions/87929 | 9 | *$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse $A(E)$ is contained in a disc of radius 10?*
Hopefully, this is really easy and it is only my ignorance in "reduction the... | https://mathoverflow.net/users/21123 | Ellipsoids and lattices: an enclosure problem. | Yes, such $A$ exists, and the radius can be much smaller than $10$; indeed radius $2^{1/2}$ suffices regardless of the area of $E$.
Write the ellipse $A(E)$ as $ax^2+bxy+cy^2 \leq 1$. By reduction theory of binary quadratic forms, we can choose the ${\rm SL}\_2({\bf Z})$ transformation $A$ to obtain coefficients sati... | 15 | https://mathoverflow.net/users/14830 | 87964 | 52,125 |
https://mathoverflow.net/questions/87967 | 10 | The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to define the reals is that it is (Dedekind- or Cauchy-) complete ordered field.
Consider the real numbers among other ord... | https://mathoverflow.net/users/nan | Is the reals the smallest connected ordered topological ring? | The following characterization of $\mathbb R$ and $\mathbb C$ among topological rings is due to Pontryagin and seems to be in the spirit of your question:
**Theorem**: If $F$ is a field with a Hausdorff ring topology which is locally compact and connected then $F$ is isomorphic as a topological field to either $\math... | 18 | https://mathoverflow.net/users/2384 | 87970 | 52,128 |
https://mathoverflow.net/questions/83582 | 5 | An equivalence relation $E$ is Borel-reducible to an equivalence relation $F$ if there is a $\Delta^1\_1$ function $f$ such that $xEy$ holds iff $f(x)Ff(y)$ holds. A set $A\subset \omega^{\omega}$ is Wadge reducible to a set $B\subset \omega^{\omega}$ if there is a continuous function $g$ such that $A=g^-1[B]$. How far... | https://mathoverflow.net/users/3859 | Borel reduction/Wadge hierarchy | Equivalence relations of the same complexity, when considered as sets, need not be mutually continuously reducible. A proof that the quasiorder of Borel equivalence relations up to continuous and Borel reducibility is ill-founded can be found in Louveau and Velickovic: 'A note on Borel equivalence relations' (1994).
... | 5 | https://mathoverflow.net/users/21255 | 87974 | 52,130 |
https://mathoverflow.net/questions/87976 | 5 | I asked this question first on math SE and was told that it would better fit here. So:
The following concept is due to Shelah and I have some issues with a claim using this notion:
Suppose that $\nu$ is a limit ordinal and that $P\_\nu$ is an iteration of forcing notions. We say that a $P\_\nu$ name $\dot{\alpha}$ o... | https://mathoverflow.net/users/4753 | Question about prompt names of ordinals | Here's proof for conclusion 1; I think 2 will then follow fairly easily. For brevity, I'll write things like $p\upharpoonright\eta$ when I really mean its extension by 1 to domain $\nu$. Suppose $\eta$ were a counterexample to 1. Since $p\upharpoonright\eta$ fails to force $\eta\leq\dot\alpha$, it must have an extensio... | 7 | https://mathoverflow.net/users/6794 | 87991 | 52,136 |
https://mathoverflow.net/questions/87939 | 3 | As part of the result of solving the [problem](https://mathoverflow.net/questions/87285/reference-request-steinbergs-1975-paper-on-a-paper-of-pittieretrieved) I am working on, my advisor and I translated the task of finding a basis for $R(T\_{sl\_{\mathbb{C}}(n)})$ in terms of $R(sl\_{\mathbb{C}}(n))$ into the followin... | https://mathoverflow.net/users/5175 | Reference Request: Basis in terms of ring of symmetric polynomials | Let $A$ be the ring $\mathbb{Z}[x\_1, \ldots, x\_n]$ and let $\Lambda$ be the subring of symmetric polynomials. Then $A$ is free as a $\Lambda$ module, and there are two fairly standard sets of bases.
The first is the monomials $x\_1^{a\_1} x\_2^{a\_2} \cdots x\_n^{a\_n}$ with $0 \leq a\_i \leq n-i$. (So $a\_n$ is a... | 5 | https://mathoverflow.net/users/297 | 87993 | 52,137 |
https://mathoverflow.net/questions/87998 | 10 | Is there a resolution of singularities for flat families?
More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in \mathbb{A} ^n$, the fiber map $Y\_p \rightarrow X\_p$ is a resolution of singularities? Can one require, moreover, that the map... | https://mathoverflow.net/users/4690 | Resolution of singularities for flat families. | I assume you want $Y \to X$ to be proper. The answer is a definite no, in general. For example, take a polynomial $f: \mathbb A^2 \to \mathbb A^1$; such a $Y$ would have to be finite over $\mathbb A^2$, and birational, so $Y = \mathbb A^2$. There are lots of counterexamples in higher dimension too: for example, it foll... | 11 | https://mathoverflow.net/users/4790 | 88003 | 52,140 |
https://mathoverflow.net/questions/88001 | 1 | It is known that any morphism is flat at an open set of points. I'd like to know if there is a relative version of this fact.
Let $f: X \rightarrow Y$ and $g:Y \rightarrow S$ be morphisms of varieties, and let $h=g \circ f$ be their composition. Suppose that $h$ and $g$ are flat. Is it true that the set
$\{x\in X| ... | https://mathoverflow.net/users/4690 | Relative generic flatness. | With these assumptions, the set in question is the set of points in $X$ where $f$ is flat; hence it is open. This is "flatness by fibers", EGA IV (11.3.10) (applied with $\mathcal{F}=\mathcal{O}\_X$).
| 2 | https://mathoverflow.net/users/7666 | 88007 | 52,143 |
https://mathoverflow.net/questions/87952 | 2 | Working with cellular automata I came across a system of equations for unknown integers $R\_{k}$ and $C\_{k}$ that looks like this.
$\binom{m}{k}=R\_{k}+C\_{k}+\sum\limits\_{j=1}^{k-1}R\_{j}C\_{k-j}.$
Where 0< k$\leq$ 2m
(for k>m we take $\binom{m}{k}=R\_{k}=C\_{k}=0$)
Given $R\_{1}$, the system has a unique s... | https://mathoverflow.net/users/18384 | A system of equations for integers | So piggybacking off the two two good answers so far, a system of equations $$a\_k=R\_k+C\_k+\sum\limits\_{j=1}^{k-1}R\_{j}C\_{k-j} \text{ for } 1 \le k \le M$$ is a system of $M$ equations in $2M$ variables which starts out
$$\begin{align}
R\_1 + C\_1 &= a\_1\\\
R\_2+C\_2&= a\_2-R\_1C\_1 \\\
R\_3+C\_3 &= a\_3- R\_... | 7 | https://mathoverflow.net/users/8008 | 88013 | 52,147 |
https://mathoverflow.net/questions/87985 | 4 | Hello,
Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I understand correctly (I don't understand much), I have a functor $Sh(X\_{et}) \to Sh(\overline{X}\_{et})^{\Gamma}$, from... | https://mathoverflow.net/users/2095 | Reference wanted - etale sheaves on $X$ versus on $\overline{X}$ | See SGA 7, XIII, Rappel 1.1.3; see also Geisser, Weil-étale cohomology over finite fields, Lemma 2.1 b).
| 2 | https://mathoverflow.net/users/nan | 88025 | 52,153 |
https://mathoverflow.net/questions/88012 | 3 | The lattice polytop $[0,n\_1]\times[0,n\_2]\times\dots\times[0,n\_{d-1}]\times[0,1]$ contains
$(n\_1+1)(n\_2+1)\cdots(n\_{d-1}+1)2$ integral points on the boundary and no integral points in
its interior. Its number of vertices, $2^d$, is however bounded by a function depending only
on its dimension $d$. Does there exis... | https://mathoverflow.net/users/4556 | Is the number of vertices of a convex $d-$dimensional lattice polytop without interior lattice points bounded? | Encouraged by Andre Henriques' comment (and not seeing a response which puts more constraints on the problem), I shall promote my comment to an answer.
Consider an arbitrary convex polygon P in R^2 which has n vertices for your favorite sufficiently large positive integer n. Then P x [0,1] (or an appropriate represnt... | 4 | https://mathoverflow.net/users/3568 | 88034 | 52,156 |
https://mathoverflow.net/questions/88029 | 1 | Suppose you are given a sequence of functions $f\_n \rightarrow F$ with a certain notion of convergence. Suppose that in your setting where this implies that $f\_n^{'} \rightarrow F^{'}$ with the same notion of convergence.
We will say that ``the rate of convergence of the sequence of derivatives is at least as fast ... | https://mathoverflow.net/users/21269 | Reference request: Rate of convergence of sequence of functions | Consider the special case of holomorphic functions. We can assume that $F=0$. Denote by $D\_R$ the disk $\lbrace |z|\leq R\rbrace$. Denote by $Z(n, R)$ the number of zeros of $f\_n$ in the interior of $D\_R$. Then
$$2\pi Z(n,R)= \left|\int\_{\partial D\_R} \frac{f\_n'}{f\_n} dz\right| \leq \int\_{\partial D\_R} \left... | 1 | https://mathoverflow.net/users/20302 | 88035 | 52,157 |
https://mathoverflow.net/questions/88026 | 22 | Recall that the space $P(M)$ of (smooth) pseudoisotopies of the compact manifold $M$ is defined as the space of all diffeomorphisms $M\times I\to M\times I$ that fix every point in $(M\times 0)\cup (\partial M \times I)$. I have worked on these things in high-dimensional cases, but I realize that there are big gaps in ... | https://mathoverflow.net/users/6666 | Pseudoisotopy in low dimensions | When $M$ is a compact $2$-manifold, with or without boundary, $P(M)$ is known. When $M$ is a 3-manifold there's bits and pieces known, especially once you get to more fine detail like pseudo-isotopy embedding spaces. But at the level of $P(M)$ I don't think there's a complete description for a single $3$-manifold. For ... | 18 | https://mathoverflow.net/users/1465 | 88040 | 52,158 |
https://mathoverflow.net/questions/88038 | 5 | For $X=\lbrace 0,\ldots,n-1\rbrace$, let $F\subseteq 2^X$ be a family of subsets of $X$ such that, for every $x\in X$, the singleton $\lbrace x\rbrace$ is the intersection of some elements of $F$. I am interested in the minimal families that have this property, in particular whether it is possible to have $|F|< n$. Can... | https://mathoverflow.net/users/21272 | Families of subsets containing every singleton as an intersection | You can achieve $\lvert F\rvert = 2\lceil\log\_2 n\rceil$ by using all subsets of the form $\{x\in X \vert i^{\text{th}}\text{ bit of }x\text{ is }j\}$ for $i \in \{0,1,\ldots,\lceil\log\_2 n\rceil-1\}$ and $j\in\{0,1\}$.
This rate is within a factor of two of best possible because there are at most $2^{\lvert F\rver... | 7 | https://mathoverflow.net/users/5963 | 88043 | 52,160 |
https://mathoverflow.net/questions/87788 | 15 | Let $\Gamma$ be a discrete group with a generating set $S$. Let $p\_c(\Gamma,S)$ be the critical probability for percolation of the Cayley graph of $\Gamma$. Is it known that if $\Gamma$ is non-amenable then there exists a generating set $S$ such that $p\_c(\Gamma,S)<\frac{1}{2}$ (or some other constant)?
The detail... | https://mathoverflow.net/users/8699 | The critical value of percolation on Cayley graphs. | Here's a more direct argument:
Let $S$ be a generating set and let $s\in S$. Pick some $x\_1,x\_2,\ldots , x\_k$ such that all of the $x\_i$'s and $x\_i^{-1}s$ are different from each other and are not in $S$. Do the same for all $s\in S$ in such a way that all of these elements are distinct. Let $S'$ be the resultin... | 6 | https://mathoverflow.net/users/1061 | 88050 | 52,164 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.